text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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title: Utilisateurs
---
## The User Entity
```typescript
import { BaseEntity, Entity, PrimaryGenerateColumn } from 'typeorm';
@Entity()
export class User extends BaseEntity {
@PrimaryGeneratedColumn()
id: number
}
```
The `User` entity is the core of the authentication and authorization system. It is a class that represents the `user` table in the database and each of its instances represents a row in this table.
The class definition is usually located in the file `src/app/entities/user.entity.ts`. Its attributes represent the columns of the table.
In FoalTS you can customize the `User` class to suit your needs. The framework makes no assumptions about the attributes required by the user objects. Maybe you'll need a `firstName` column, maybe not. Maybe the authentication will be processed with an email and a password or maybe you will use an authentication token. The choice is yours!
However, FoalTS provides abstract classes from which you can extend the `User` entity. Such classes, such as `UserWithPermissions`, have useful utilities for handling authentication and authorization, so that you do not have to reinvent the wheel.
## Creating Users ...
There are several ways to create users.
### ... Programmatically
```typescript
import { User } from './src/app/entities';
async function main() {
const user = new User();
user.foo = 1;
await user.save(); 1
});
}
```
### ... with a Shell Script (CLI)
You can use the `create-user` shell script (located in `src/scripts`) to create a new user through the command line.
```sh
npm run build
foal run create-user
```
## Example (email and password)
This section describes how to create users with an email and a password.
### The User Entity
Go to `src/app/entities/user.entity.ts` and add two new columns: an email and a password.
```typescript
import { hashPassword } from '@foal/core';
import { Column, Entity, PrimaryGeneratedColumn, BeforeInsert, BeforeUpdate } from 'typeorm';
@Entity()
export class User {
@PrimaryGeneratedColumn()
id: number;
@Column({ unique: true })
email: string;
@Column()
password: string;
@BeforeInsert()
@BeforeUpdate()
async hashPassword() {
// Hash the password before storing it in the database
this.password = await hashPassword(this.password);
}
}
```
> Note: The `BeforeInsert` and `BeforeUpdate` are TypeORM decorators for Entity Listeners that run before the entity is saved in the db. In this example they take care of hashing the password. More info about `Entity Listeners` in the [TypeORM docs](https://typeorm.io/#/listeners-and-subscribers)
### The create-user Shell Script
Running the `create-user` script will result in an error since we do not provide an email and a password as arguments.
Go to `src/scripts/create-user.ts` and replace its content with the following lines:
```typescript
// 3p
import { hashPassword } from '@foal/core';
// App
import { User } from '../app/entities';
import { dataSource } from '../db';
export const schema = {
additionalProperties: false,
properties: {
email: { type: 'string', format: 'email' },
password: { type: 'string' },
},
required: [ 'email', 'password' ],
type: 'object',
};
export async function main(args) {
await dataSource.initialize();
try {
const user = new User();
user.email = args.email;
user.password = await hashPassword(args.password);
console.log(await user.save());
} catch (error: any) {
console.error(error.message);
} finally {
await dataSource.destroy();
}
}
```
You can now create a new user with these commands:
```sh
npm run build
foal run create-user email=mary@foalts.org password=mary_password
```
## Using another ORM/ODM
In this document, we used TypeORM to define the `User` class and the `create-user` shell script. However, you can still use another ORM/ODM if you want to.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,584 |
<?php
declare(strict_types=1);
namespace Jh\Import\Filter;
use Jh\Import\Config;
use Jh\Import\Import\Record;
use Jh\Import\Import\RequiresPreparation;
use Jh\Import\Report\ReportItem;
class LoggingSkipNonExistingProducts implements RequiresPreparation
{
/**
* @var SkipNonExistingProducts
*/
private $skipNonExistingProducts;
/**
* @var string
*/
private $idField = '';
public function __construct(SkipNonExistingProducts $skipNonExistingProducts)
{
$this->skipNonExistingProducts = $skipNonExistingProducts;
}
public function prepare(Config $config): void
{
$this->skipNonExistingProducts->prepare($config);
$this->idField = $config->getIdField();
}
public function __invoke(Record $record, ReportItem $reportItem): bool
{
$found = $this->skipNonExistingProducts->__invoke($record);
if (!$found) {
//TODO: Make level configurable so we can cause import to fail
//warning will not cause a fail, whereas errror will
$reportItem->addWarning(sprintf('Product: "%s" does not exist.', $record->getColumnValue($this->idField)));
}
return $found;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,691 |
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Please answer all parts of Part (b) please only answer if you can do so correctly, and showing working out. Question 4 (a) A sound wave with a frequency of 250 Hz is transmitted through air at 25\u00b0 C (298.2K). The specific gas constant for air is 287 J\/kg K, and the specific heat ratio is y = 1.4... 5 answers ##### 16. If in = 6 A,i2 = 8 A R=6 9 , and 8 = 4, the current io (in amperes) is 1. -8 2. 8 3. -48 4. 48 5. 0 ic = Bix17. Ifvs =-6 V,R=20,and u = 4, the voltage Vo (in volts) is 1. -48 2. 48 3. -24 4. 24 Vc= Pvx 5. 018. Ifi =6 A,i2 =2A, Vs =14 V, and R = 2 0 , the voltage across the current sources (i1 and i2) is: 1. 4 2. 8 3. 6 4. 16. If in = 6 A,i2 = 8 A R=6 9 , and 8 = 4, the current io (in amperes) is 1. -8 2. 8 3. -48 4. 48 5. 0 ic = Bix 17. Ifvs =-6 V,R=20,and u = 4, the voltage Vo (in volts) is 1. -48 2. 48 3. -24 4. 24 Vc= Pvx 5. 0 18. 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Their work is below: Deterine if 14,A student was asked t0 find f'(x) if f(x) = (7 all errors: [3] work correct: If it is incorrect; then explain the student' 3x 6)= 3 (7 - Zlaxli.s}* Aa.s) f'l = 6x(7-x3U.syr @nCI.s_...","date":"2023-03-29 22:43:58","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\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4671623110771179, \"perplexity\": 6292.338378686289}, \"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-2023-14\/segments\/1679296949035.66\/warc\/CC-MAIN-20230329213541-20230330003541-00290.warc.gz\"}"} | null | null |
An Unexplainable UFO Landing in the San Francisco East Bay? Yup!
by Olav Phillips UFO's are one of those things. Many times an observer will report strange lights in the sky, and it is not uncommon for people in Contra Costa...
Spectral Students Haunt the Hallowed Halls of Pittsburg High…
by Olav Phillips Originally founded in 1924, Pittsburg High School has served the community of Pittsburg for over 90 years. A pillar of the community, it has produced many amazing...
The Happy Haunted Schoolhouse of Pleasant Hill…
By Olav Phillips The Old School House of Pleasant Hill is something of an institution. Having long since ceased to be a school, it was for many years home to...
Mr. Strange's Improbable Travel Destinations: Vasilyevka – The Mysterious Ghost Castle of the Ukraine
By Mr. Strange & Anastasiia Shkuro Recently my friend and I were discussing a very important topic, what makes people believe in ghosts and visit haunted places? Although both of...
Supernatural Conspiracy
by Eugenia Macer-Story Amazingly, a bit of casual map dowsing first made the solid connection for me between UFO "spirit light" phenomena and what appears to be an ancient form...
Mr. Strange's Improbable Travel Destinations: Tekal – The Mystery of Flowers Given to Lakshmi
By Mr. Strange The old religious chants praise this goddess as the bestower of wealth, knowledge, and family happiness. Those who meditate upon goddess Lakshmi claim to have felt a...
Unusual Nepal: How People Can Expierence Visions
by Anastasiia Shkuro Many people who arrive in Nepal are waiting for the happiness, love, and destiny changing for better. A lot of travelers with extra-sensual perception are assured that Nepal...
What Do You Say to a Man from Outer Space?
By Peter Guttilla Originally Published in SAGA UFO Report Summer 1974 In late January 1972, 16-year-old John Yeries, his brother James, and two small friends, Darrell Rich and Robbie Cross,...
Tulpamancers: How the Spiritual Practices of Buddhist Monks Became a Play of Children
by Anastasiia Shkuro It is widely believed this story has started from the social media. A couple of years ago the teenagers found themselves involved into creation of alternative multiple...
UFO's and Abominable Snowman – What Is Their Strange Connection?
By John Keel YOU are driving down a lonely country road late at night, lively music from your car radio helping to keep you alert. You haven't been drinking. You...
Karl Jung and Bollingen Tower: The Mystery of the Mandala
Karl Jung had a well-known standpoint. He believed in paranormal occasions and sometimes saw the harrowing creatures from the hell. According to some legend, he had an argument with Freud...
The Cursed Place: A Hospital with Ghosts
By PARANOIA Europe Staff, Old buildings are likely to gain a fame of ghosts-friendly houses. Let the scientists show us reasonable facts that explain the spirits with the concept of...
Paranoid Fun: Halloween – A Brief History of Halloween in Europe
As Halloween is just around the corner, it's only fair to put it in the limelight for a bit. The tradition has many names – Allhalloween, All Hallows' Eve, or...
4 Places in Romania That Will Give You the Creeps!
[caption id="attachment_1130363" align="alignright" width="300"] Corvin Castle, HunedoaraImage by Carcea Daniel - Own work, CC BY-SA 3.0 ro[/caption] Corvin Castle, Hunedoara Don't let the beauty of this magnificent castle fool you....
Ghostly stories from Ukraine: The Haunting of Chernobyl
You most likely know about the Chernobyl disaster. It is, to date, one of the most ravaging nuclear incidents the world has ever seen. But most of the time, an...
PARANOIA Explores: The Orang Bati
In the Indonesian island of Seram stories persist of a winged creature that is a hybrid of a monkey or orangutan and a bat. The fearsome creature is said to... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,315 |
It has been a remarkable March 2017 in Our World. As The Trump Administration passes the 50-Day Mark, a sense of uncertainty continues to be the order of the day. What has been amazing to us is the apparent sidelining of the Department of State in the realm of Foreign Policy as policymaking seems to have been concentrated in the White House with the profound changes in Europe and the Near East. Britain will be triggering Brexit Negotiations as elections are coming up in the Netherlands and France. This is as Turkey is to have a vote to expand the powers of President Erdogan. What is also of note is how Narenda Modi is dominating India as he gears up for the 2019 Elections. His Party, the BJP, won India's key state of Uttar Pradesh decisively.
Spring is also in the air and with it the advent of Nouroz. Nouroz has been recognized as by the UN and it is celebrated in a number of countries around the World--including Iran on March 21 of every year. The new year in Iran is also going to be a year of change as new Presidential Elections loom that positions the current incumbent against the conservatives within the ruling factions inside the Islamic Republic. The Israeli Prime Minister's recent Moscow trip was interesting in that it was trying to undermine Iranian influence in Syria as he deemed it, "..Unacceptable". It is interesting that he seems to be emboldened in the aftermath of President Trump's Election. It is also of interest, though, that Mahmoud Abbas has been invited to the White House by President Trump.
I'm sorry I'm not with you tonight. My absence is out of respect for the people of my country and those of other six nations whom have been disrespected by the inhuman law that bans entry of immigrants to the US. Dividing the world into the "us" and "our enemies" categories creates fear. A deceitful justification for aggression and war. These wars prevent democracy and human rights in countries which have themselves been victims of aggression. Filmmakers can turn their cameras to capture shared human qualities and break stereotypes of various nationalities and religions. They create empathy between us and others. An empathy which we need today more than ever."
What was quite remarkable was to be witness to the plight of one of the leading opposition figures inside Iran, Mehdi Khazali. Dr. Khazaili was detained after he called for a boycott to protest the internal repression inside the Country. What he noted after he was released was quite remarkable as he noted that a number of key Security Organs refused to accept the responsibility for his custody as he began a hunger strike to protest his detention.
The dialogue is in Farsi--However, the policies of the Islamic Republic was exposed in a major way. Although some of them are known--including for instance the work of the Quds Force in Iraq and Syria--the "soft war" is also remarkable. One of the elements of the Force is a brigade called the "Fatameyon"--Afghans who are approximately 18,000 who are supported. This is as an analyst appearing on this program compared this "Soft war" to Hollywood". One of the leading institutions, the Al Mustafa Organization, has apparently graduated some 40,000 graduates--in addition to having currently 30,000 students (including 10,000 religious students).
The "culture war" that the Islamic Republic has launched against the World is remarkable. One very interesting clip uncovered was what Khomeini said to undermine Iraq at the time. Iran has operations in over 80 Countries sponsoring cultural programs and training clerics to the tune of billions of dollars. Some of the support has included a 2 Billion Dollars to the Kumours Island, 2 Billion Dollars to Sri Lanka and a 1 Billon Dollar loan to Iraq. These are just very small examples that were exposed in this 47-Minute expose by this broadcast. The Islamic Revolutionary Guard Corp touted its' efforts in Lebanon rebuilding schools, fixing electricity for 73 villages and other economic development efforts. This is as after 30 years people in the South of the Country have hardly running water in the rural areas and as children are not able to attend school--one estimated that some 250 Villages have no running water at all. | {
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} | 7,204 |
The Malaysian Mermaid
What could be more suitable to round off the week, and the month, than some vintage Malaysian pop. Step forward please, Ms Uji Rashid.
Born in Brunei but moving to KL when still a child, Uji was a major pop and film star in Malaysia in the 1970s and 1980s. Or as the very wonky translation of her Malay Wikipedia entry puts it, "this fierce -eyed art man is endowed with a natural spoiled voice as well as synonymous with the title of 'spoiled voiced singer' by homeland art fans; especially the 70s and 80s generations".
As if being a star of stage and screen wasn't enough, she apparently has both a batik fabric pattern and a type of fish named after her. I could not find a photo of the batik pattern but this is the fish. I am not totally convinced that it is a compliment.
The real Uji is still with us and here she is doing what she does best. Both tracks come from a compilation album called "Kenangan Manis", which I am reasonably confident means "Sweet Memories" and which I probably picked up on a work trip to KL a few years back.
"Kemesraan" - Uji Rashid
"Berilah Kasih Sayang" - Uji Rashid
From a fish named after a singer to a singer named Fish. We don't just fling this together you know.
Mel & Him
Some super soul sounds from '72 for you today, courtesy of Mel & Tim. "Starting All Over Again" made the US Top 20 that year. Both tracks can be found on the album of the same name.
Before any of you ask, they are not a misspelt Mel & Kim, as I think their respective videos will prove.
"Starting All Over Again" - Mel & Tim
"Too Much Wheelin' And Dealin'" - Mel & Tim
Hiss Hitss
Over the weekend I was listening to "Quietly Blowing It", the new album by Hiss Golden Messenger. It's excellent. If you haven't bought it already you should go and do so now.
Hiss Messengerness is possibly my favourite artist to emerge over the last ten years or so - he's certainly up there. I first became aware of him through 2011's "Poor Moon" and first got to see him the following year in a pub basement supporting Michael Chapman. Since then I've been lucky enough to see him a few more times both solo and with a band, and I've picked up every album as well.
Here are a few golden oldies for you, in chronological order. The first is from "Country Hai East Cotton" (2009), the others from "Poor Moon" and "Haw" (2013).
"O Nathaniel" - Hiss Golden Messenger
"Blue Country Mystic" - Hiss Golden Messenger
"The Serpent Is Kind (Compared To Man)" - Hiss Golden Messenger
The second time I saw Hiss Golden Messenger it was just M.C. Taylor again, this time on a double bill with William Tyler. When he announced that for an encore he would be doing a song by Gary Stewart (one of my all-time heroes as long-serving readers might recall) and that Mr Tyler would be helping out I felt I had a take a sneaky video. With apologies for the quality, here it is.
And here he is the first time I saw him. I had forgotten that I had taken this video until I went to find the other one. The picture quality is a bit better is this one but the sound quality is worse, so apologies again.
Donny Reggae
Today's post goes out especially to our old friend George. He's a big Osmonds fan and doesn't mind who knows it. I considered featuring a couple of their singles but then realised he has probably got them all already, so here are a couple of Mandatory Reggae Versions instead.
"One Bad Apple" - Barry Biggs
"Love Me For A Reason" - The Fabulous Five
Even surfers of the zeitgeist and epicures of the exotic need to switch off sometimes. As much as I enjoy the rich stew of Patagonian funk and Uzbek prog that we serve up every week, sometimes I need to cleanse my palate with some good old-fashioned tunes I can tap my toes to.
Ladies and gentleman, please give a big jazz hand to Ms Liza Minnelli. Both tracks come from her 1973 LP "The Singer" on which she interprets in her own unique way the works of Stevie Wonder, James Taylor, Carly Simon, Donny Hathaway and... er... Hurricane Smith.
"Dancing In The Moonlight" - Liza Minnelli
"Baby Don't Get Hooked On Me" - Liza Minnelli
Do You Know The Way To São Tomé?
Something a bit special for you today, folks. Some vintage pop tunes from São Tomé and Principe.
Being the sophisticated bunch you are, you don't need me to tell you that the beautiful island nation of São Tomé and Principe can be found off the west coast of Africa (opposite Gabon to be precise) or that it was formerly a Portuguese colony.
What you may not know is that it is home to some fantastic music. To be fair, I did not know that either until I listened to "Léve-Léve", a brilliant compilation of music made in the islands during the 1970s and 1980s. The album was released last year by Bongo Joe records and is available on Bandcamp etc.
"Tólá Muandgi" - Sum Alvarinho
"Zimbabwe" - Africa Negra
"Cladênço Padê Cluço" - Tiny Das Neves
It has been ages since we last had a Single Song Sunday, and I feel the need to try to make it up to you. You may say it's too late. And you'd be right.
Carole King took her singer-songwriter standard to number one in the Billboard charts 50 years ago this month, which gives us all the excuse we need. Not that an excuse is really required to feature some of these cracking cover versions.
We kick off with Carole's original then swiftly follow it up with three great soul versions from 1972, including the Isley Brothers' ten minute psychodrama. Almost as long is the extended mix of the Quartz and Dina Carroll version which made the UK top ten thirty years ago. I should've got the single version. In this case, less is more.
At that point in proceedings Mr Ray Conniff and friends step in to liven things up, before introducing our international guests - one each from Finland, Germany and Cambodia. The event culminates, as always, in the Mandatory Reggae Version.
"It's Too Late" - Carole King
"It's Too Late" - Billy Paul
"It's Too Late" - Denise LaSalle
"It's Too Late" - The Isley Brothers
"It's Too Late" - Quartz (featuring Dina Carroll)
"It's Too Late" - Ray Conniff
"Jotain En Enää Saa" - Kisu
"Es Ist Spät, Zu Spät Für Uns" - Daliah Lavi
"Huos Haenuy Bong" - Poev Vannary
"It's Too Late" - The Chosen Few
Crazy Yama
Today's offering has been taken down from the weird but wonderful shelf and dusted off for your delectation and delight. Its "Le Monde Fabuleux Des Yamasuki", a 1971 French concept album featuring heavy drums, an actual Japanese judo master who shouts a lot and a female choir chanting real or pretend Japanese lyrics. That old chestnut.
The genius behind the album, Daniel Vangarde, has many other claims to fame. One of the tracks from the album, "Aie a Mwana" became a minor hit for Bananarama in 1981. His son is one half of Daft Punk. And he wrote and produced the two classics featured in the videos. He's one hell of a dude.
"Yama Yama" - The Yamasuki Singers
"Yamasuki" - The Yamasuki Singers
The Dog/Gun Dilemma
In today's episode of Trenchtown Eastenders, Dr Alimantado wants to acquire a gun but Nicodemus tries to talk him out of it.
"Gimme Mi Gun" - Dr. Alimantado
"Dog Better Than Gun" - Nicodemus
Loam On The Range
Today's selections are from "Lez Dance", the second album from Durham (North Carolina) based Loamlands. Released in 2019, it is available on Bandcamp, as is their first album "Sweet High Rise". Both are well worth checking out.
Loamlands describe themselves as "a distorted country music outfit based in North Carolina that places queer storytelling at the forefront", and go on to say that "their music intimately grapples with identity, pronouns and experiencing a queer existence in both modern-day and historical Southern culture".
That's not an experience that is particularly close to my own - although I did once inadvertently spend an evening in a lesbian bar in Raleigh (its a long story) - but I can tell a bloody good record when I hear one. This is one.
"Maureen" - Loamlands
"Wrong Days" - Loamlands
A Bagful Of Brinsleys
Ladies and Gentlemen, we give you Brinsley Schwarz. Founding fathers of public house rock music (to give it its full name). The primeval soup from which emerged Nick Lowe, Ian Gomm and The Rumour. But also so much more.
A few years ago I picked up their first five albums in a box set for about £10, but have only recently got round to listening to them all properly. I wouldn't say any of them are "must haves" to be honest, but there is an excellent compilation waiting to be made.
Because I'm a nice man I've got one track from each album for you, in chronological order. We start with their eponymous debut from 1970 and then follow it with "Despite It All" (1970), "Silver Pistol" (1972), "Nervous On The Road" (1972) and "Please Don't Ever Change" (1973).
"Hymn To Me" - Brinsley Schwarz
"The Slow One" - Brinsley Schwarz
"The Last Time I Was Fooled" - Brinsley Schwarz
"Surrender To The Rhythm" - Brinsley Schwarz
"Hooked On Love" - Brinsley Schwarz
Brinsley Schwarz the man and Brinsley Schwarz the band are two of my three favourite Brinsleys. The other? Brinsley Forde of Aswad of course.
Congo-A-Go-Go (Part 2)
We're back with Part 2 of our musical trip to the Democratic Republic of Congo. We have some more smashing sounds then.
I actually spent 24 hours in Kinshasa the best part of 30 years ago. It is a long and complicated story but in brief we had made the mistake of booking cheap flights to Johannesburg with the late lamented Air Zaire, which was not best known for its reliability
On the return journey the plane turned up a day late and the pilot then decided to terminate the flight in Kinshasa for some reason. Those of us lucky enough to find our luggage and nab a seat on the only coach that was available spent the night in a very dodgy hotel (better that than the floor of the transit lounge). Slightly to our surprise we were collected the next day and finally made it home 48 hours late.
All in all I don't think I saw the best of Kinshasa in my brief time there. Hopefully one day there will be a chance for me to go back and take in its many delights properly. Until then, we have the wonderful music.
"Magerita" - Jupiter & Okwess International
"Bye Bye Julie" - J.B. Mpiana
"Depression Nerveuse" - Alain & Bouro Mpela
There was great excitement at Leggies HQ last week when we had our first ever visitor from the Democratic Republic of Congo. Welcome to the gang!
It is becoming traditional here to patronise our new friends by playing them some of their own music. The DRC is a country that has produced mountainous amounts of magnificent music, much more than we could possibly do justice to in one post, so we are making this a two-parter. Even that is going to be woefully inadequate.
Today being a Monday, King Kester Emeneya seems an obvious place to start. It also has the merit of being a long song, as are today's other two selections, in a respectful nod to other people's traditions.
The middle video is even longer - 45 minutes of Franco and friends!
"Lundi" - King Kester Emeneya
"Ange Bokuma (Parts 1 and 2)" - Zaiko Langa Langa
"Ofelika" - Tony Dee
Well Gotten Gaines
Some vintage soul for you today, courtesy of Earl Gaines. Mr Gaines was born in Decatur, Alabama but moved to Nashville in his teens and was based there throughout his career. He had his biggest hit in 1955 as the uncredited lead vocalist on Louis Brooks & His Hi-Toppers' "It's Love Baby (24 Hours A Day)" - No. 2 in the R&B charts - but kept recording until the 1990s and performing right up until his death in 2009.
"Yearning And Burning (Deep Inside)" - Earl Gaines
"Don't Take My Kindness For Weakness" - Earl Gaines
Here's Earl performing his big hit, I would guess some time in the mid 1960s. The fashions sported by the band and the female dancers blend beautifully. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,843 |
\section{Introduction}
Rapid expansion of capacity in the automatic data generation and
acquisition has made a profound impact on statistics and machine learning, as it
brings data in unprecedented size and complexity. {These data are}
generally called as the ``massive data'' or ``big data''
\citep{Wu2014}. Massive data bring new opportunities of
discovering subtle population patterns and heterogeneities which are
believed to embody rich values and impossible to be found in
relatively small data sets. It, however, simultaneously leads to a
series of challenges such as the storage bottleneck, efficient
computation, etc.~\citep{Zhou2014}.
To attack the aforementioned challenges, a divide and conquer
strategy was suggested and widely used in statistics and machine
learning communities
\citep{li2013statistical,Mann2009,Zhang2013,Zhang2014,dwork2010differential,battey2015distributed,wang2015statistical}.
This approach firstly distributes a massive data set into $m$ data
blocks, then runs a specified learning algorithm on each data
block independently to get a {\it local estimate} $\hat{f}_{j},
j=1,\dots, m$ and at last transmits $m$ local estimates into one
machine to synthesize a {\it global estimate} $\bar{f}$, which is
expected to model the structure of original massive dataset. A
practical and exclusive synthesizing method is the {\it average
mixture} (AVM)
\citep{li2013statistical,Mann2009,Zhang2013,Zhang2014,dwork2010differential,battey2015distributed},
i.e., $\bar{f} =\frac{1}{m}\sum_{j=1}^m\hat{f}_{j}$.
In practice, the above divide and conquer strategy has many
applicable scenarios. We show the following three situations as
motivated examples. The first one is using limited primary memory to
handle a massive data set. In this situation, the divide and
conquer strategy is employed as a two-stage procedure. In the first
stage, it reads the whole data set sequentially {block by block},
each having a manageable sample size for the limited primary memory,
and derives a {local estimate} based on each block. In the
second stage, it averages {local estimates} to build up a global estimate~\citep{li2013statistical}. The second motivated
example is using distributed data management systems to tackle
massive data. In this situation, distributed data management systems
(e.g., Hadoop) are designed by the divide and conquer strategy.
They can load the whole data set into the systems and tackle
computational tasks separably and automatically. \citet{rhipe2012}
has developed an integrated programing environment of R and Hadoop
(called RHIPE) for expedient and efficient statistical computing. The third
motivated example is the massive data privacy. In this situation, it
divides a massive data set into several small pieces and combining
the estimates derived from these pieces for keeping the data
privacy~\citep{dwork2010differential}.
For nonparametric regression, the aforementioned
divide and conquer strategy has been shown to be efficient and
feasible for global modeling methods such as the kernel ridge
regression \citep{Zhang2014} and conditional maximum entropy model
\citep{Mann2009}. Compared with the global modeling methods, local
average regression (LAR)~\citep{gyorfi2006distribution,fan2000prospects,tsybakov2008introduction}, such as the Nadaraya-Watson kernel (NWK) and $k$ nearest neighbor (KNN)
estimates, benefits in computation and therefore, is also widely used in image processing~\citep{takeda2007kernel}, power
prediction~\citep{kramer2010power}, recommendation
system~\citep{biau2010statistical} and financial
engineering~\citep{Kato2012}. LAR is by definition a learning scheme
that averages outputs whose corresponding inputs satisfy certain
localization assumptions. To tackle massive data regression
problems, we combine the divide and conquer approach with LAR to
produce a new learning scheme, average mixture local average
regression (AVM-LAR), just as \cite{Zhang2014} did for kernel ridge
regression.
Our first purpose is to analyze the performance of AVM-LAR. After
providing the optimal learning rate of LAR, we show that AVM-LAR
can also achieve this rate, provided the number of data blocks, $m$,
is relatively small. We also prove that the restriction
concerning $m$ cannot be essentially improved. In a word, we provide
both the optimal learning rate and the essential restriction
concerning $m$ to guarantee the optimal rate of AVM-LAR. It should
be highlighted that this essential restriction is a bit strong and
makes AVM-LAR feasible only for small $m$. Therefore, compared with LAR, AVM-LAR does not bring the essential improvement, since we must pay much attention to
determine an appropriate $m$.
Our second purpose is to pursue other divide and conquer strategies to equip LAR efficiently. In particular, we provide two concrete variants of AVM-LAR in this
paper. The first variant is motivated by the distinction between KNN
and NWK. In our experiments, we note that the range of $m$ to
guarantee the optimal learning rate of AVM-KNN is much larger than
AVM-NWK. Recalling that the localization parameter of KNN depends
on data, while it doesn't hold for NWK, we propose a variant of
AVM-LAR such that localization parameters of each data block depend on data. We establish the optimal learning
rate of this variant with mild restriction to $m$ and verify its
feasibility by numerical simulations.
The second variant is based
on the definitions of AVM and LAR. It follows from the definition of
LAR that the predicted value of a new input depends on samples near
the input. If there are not such samples in a specified data block,
then this data block doesn't affect the prediction of LAR. However,
AVM averages {local estimates} directly, neglecting the concrete
value of a specified {local estimate}, which may lead to an
inaccurate prediction. Based on this observation, we propose another
variant of AVM-LAR by distinguishing whether a specified data block
affects the prediction. We provide the optimal learning rate of this
variant without any restriction to $m$ and also present the
experimental verifications.
To complete the above missions, the rest of paper is organized as
follows. In Section \ref{section2}, we present optimal learning
rates of LAR and AVM-LAR and analyze the pros and cons of
AVM-LAR. In Section \ref{section3}, we propose two new modified
AVM-LARs to improve the performance of AVM-LAR. A set of simulation
studies to support the correctness of our assertions are given in
Section \ref{section4}. In Section \ref{section5}, we
detailedly justify all the theorems. In Section \ref{section6}, we present the conclusion and some useful remarks.
\section{Divide and Conquer Local Average Regression}\label{section2}
In this section, after introducing some basic concepts of LAR, we
present a baseline of our analysis, where an optimal minimax
learning rate of LAR is derived. Then, we deduce the learning rate
of AVM-LAR and analyze its pros and cons.
\subsection{Local Average Regression}
Let $D=\{(X_i,Y_i)\}_{i=1}^N$ be the data set where $X_i\in\mathcal
X\subseteq\mathbb{R}^d$ is a covariant and $Y_i\in [-M,M]$ is the
real-valued response. We always assume $\mathcal X$ is a compact
set. Suppose that samples are drawn independently and
identically according to an unknown joint distribution $\rho$ over
$\mathcal X\times [-M,M]$. Then the main aim of nonparametric
regression is to construct a function $\hat{f}:\mathcal X\rightarrow
[-M,M]$ that can describe future responses based on new inputs. The
quality of the estimate $\hat{f}$ is measured in terms of the {\it
mean-squared prediction error} $\mathbf E\{\hat{f}(X)-Y\}^2$, which is
minimized by the so-called {\it regression function}
$f_{\rho}(x)=\mathbf E\left\{Y|X=x\right\}$.
LAR, as one of the most widely used
nonparametric regression approaches, constructs an estimate formed
as
\begin{equation}\label{local_estimation}
f_{D,h}(x)=\sum_{i=1}^NW_{h,X_i}(x)Y_i,
\end{equation}
where the localization weight $W_{h,X_i}$ satisfies $W_{h,X_i}(x)>0$ and
$\sum_{i=1}^NW_{h,X_i}(x)=1$. Here, $h>0$ is the so-called
localization parameter. Generally speaking, $W_{h,X_i}(x)$ is ``small'' if $X_i$
is ``far'' from $x$. Two widely used examples of LAR are
Nadaraya-Watson kernel (NWK) and $k$ nearest neighbor (KNN)
estimates.
\begin{example}(NWK estimate)
Let $K: \mathcal X\rightarrow\mathbb R_+$ be a kernel function~\citep{gyorfi2006distribution}, and $h>0$ be its bandwidth. The NWK estimate is defined by
\begin{equation}\label{Watson kernel estimate}
\hat{f}_h(x)=\frac{\sum_{i=1}^NK\left(\frac{x-X_i}{h}\right)Y_i}{\sum_{i=1}^NK\left(\frac{x-X_i}{h}\right)},
\end{equation}
and therefore,
$$
W_{h,X_i}(x)=\frac{
K\left(\frac{x-X_i}{h}\right)}{\sum_{i=1}^NK\left(\frac{x-X_i}{h}\right)}.
$$
It is worth noting that we use the convention $\frac{0}{0}=0$ in the following.
Two popular kernel functions are the naive kernel,
$K(x)={I}_{\{\|x\|\leq 1\}}$ and Gaussian kernel
$K(x)=\exp\left(-\|x\|^2\right)$, where $ I_{\{\|x\|\leq 1\}}$ is an indicator function with the feasible domain $\|x\|\leq 1$ and $\|\cdot\|$ denotes the Euclidean norm.
\end{example}
\begin{example}(KNN estimate)
For $x\in \mathcal{X}$, let $\{(X_{(i)}(x),Y_{(i)}(x))\}_{i=1}^N$ be
a permutation of $\{(X_i,Y_i)\}_{i=1}^N$ such that
$$
\|x-X_{(1)}(x)\|\leq\cdots\leq \|x-X_{(N)}(x)\|.
$$
Then the KNN estimate is defined by
\begin{equation}\label{KNNE estimate}
\hat{f}_k(x)=\frac1k\sum_{i=1}^k Y_{(i)}(x).
\end{equation}
According to (\ref{local_estimation}), we have
$$
W_{h,X_i}(x)=\left\{\begin{array}{cc} 1/k, &
\mbox{if}\ X_i\in\{X_{(1)},\dots,X_{(k)}\},\\
0, & \mbox{otherwise.}
\end{array}\right.
$$
Here we denote the weight of KNN as $W_{h,X_i}$ instead of
$W_{k,X_i}$ in order to unify the notation and $h$ in KNN depends
on the distribution of points and $k$.
\end{example}
\subsection{Optimal Learning Rate of LAR}
The weakly universal consistency and optimal learning rates of
some specified LARs have been justified by~\citet{stone1977consistent,stone1980optimal,stone1982optimal} and summarized in the book \citet{gyorfi2006distribution}. In particular, \citet[Theorem
4.1]{gyorfi2006distribution} presented a sufficient condition to
guarantee the weakly universal consistency of LAR. \citet[Theorem
5.2, Theorem 6.2]{gyorfi2006distribution} deduced optimal learning
rates of NWK and KNN.
The aim of this subsection is to present some sufficient conditions to guarantee optimal learning rates of general LAR.
For $r,c_0>0$, let $\mathcal
F^{c_0,r}=\{f|f:\mathcal{X}\rightarrow\mathcal{Y},|f(x)-f(x')|\leq
c_0\|x-x'\|^r,\forall x, x'\in\mathcal{X}\}$. We suppose in this
paper that $f_{\rho} \in \mathcal F^{c_0,r}$. This is a commonly
accepted prior assumption of regression function which is employed
in
~\citep{tsybakov2008introduction,gyorfi2006distribution,Zhang2014}.
The following Theorem \ref{THEOREM:SUFFICIENT CONDITION FOR LOCAL
ESTIMATE} is our first main result.
\begin{theorem}\label{THEOREM:SUFFICIENT CONDITION FOR LOCAL ESTIMATE}
Let $f_{D,h}$ be defined by (\ref{local_estimation}).
Assume that:
(A) there exists a positive number $c_1$ such that
$$
\mathbf E\left\{\sum_{i=1}^NW^2_{h,X_i}(X)\right\}\leq
\frac{c_1}{Nh^d};
$$
(B) there exists a positive number $c_2$ such that
$$
\mathbf
E\left\{ \sum_{i=1}^NW_{h,X_i}(X)I_{\{\|X-X_i\|>h\}}\right\}
\leq
\frac{c_2}{\sqrt{Nh^d}}.
$$
If $h\sim N^{-1/(2r+d)}$, then there exist constants $C_0$ and
$C_1$ depending only on $d$, $r$, $c_0$, $c_1$ and $c_2$ such that
\begin{equation}\label{theorem1}
C_0N^{-2r/(2r+d)}\leq \sup_{f_\rho\in\mathcal
F^{c_0,r}}\mathbf E\{\|f_{D,h}-f_\rho\|_\rho^2\}
\leq
C_1N^{-2r/(2r+d)}.
\end{equation}
\end{theorem}
Theorem \ref{THEOREM:SUFFICIENT CONDITION FOR LOCAL ESTIMATE}
presents sufficient conditions of the localization weights to ensure the
optimal learning rate of LAR. There are totally four constraints
of the weights $W_{h,X_i}(\cdot)$. The first one is the averaging
constraint $\sum_{i=1}^NW_{h,X_i}(x)=1, \ \mbox{for all}\
X_i,x\in\mathcal
X.$ It essentially reflects the word ``average'' in LAR.
The second one is the non-negative constraint. We regard it as a
mild constraint as it holds for all the widely used LAR such as
NWK and KNN. The third constraint is condition (A), which devotes
to controlling the scope of the weights. It aims at avoiding the
extreme case that there is a very large weight near 1 and others are
almost 0. The last constraint is condition (B), which implies the
localization property of LAR.
We should highlight that Theorem \ref{THEOREM:SUFFICIENT CONDITION
FOR LOCAL ESTIMATE} is significantly important for our analysis. On
the one hand, it is obvious that the AVM-LAR estimate (see Section \ref{AVM-LARsection}) is also a new LAR estimate. Thus, Theorem
\ref{THEOREM:SUFFICIENT CONDITION FOR LOCAL ESTIMATE} provides a
theoretical tool to derive optimal learning rates of AVM-LAR. On
the other hand, Theorem \ref{THEOREM:SUFFICIENT CONDITION FOR LOCAL
ESTIMATE} also provides a sanity-check that an efficient AVM-LAR
estimate should possess the similar learning rate as
(\ref{theorem1}).
\subsection{AVM-LAR}\label{AVM-LARsection}
The AVM-LAR estimate, which is a marriage of the classical AVM
strategy \citep{Mann2009,Zhang2013,Zhang2014} and LAR, can be formulated in the following Algorithm \ref{AVM-LAR}.
\begin{algorithm}\caption{AVM-LAR}\label{AVM-LAR}
\begin{algorithmic}[!h]
\STATE {{\bf Initialization}: Let $D=\{(X_i,Y_i)\}_{i=1}^N$ be $N$
samples, $m$ be the number of data blocks, $h$ be the bandwidth
parameter.}
\STATE{ {\bf Output}: The global estimate $\overline{f}_h$.}
\STATE{{\bf Division}: Randomly divide $D$ into $m$ data blocks
$D_1,D_2,\dots,D_m$ such that $D=\mathop{\bigcup}\limits_{j=1}^{m}
D_j, D_i\cap D_j=\varnothing, i\neq j$ and
$|D_1|=\dots=|D_m|=n=N/m$.}
\STATE{{\bf Local
processing}:
For $j=1,2,\dots,m$, implement LAR for the
data block $D_j$ to get the $j$th {\it local estimate}
$$
f_{j,h}(x)=\sum_{(X_i,Y_i)\in D_j}W_{X_i,h}(x)Y_i.
$$
}
\STATE{{\bf Synthesization}: Transmit $m$ {\it local estimates}
$f_{j,h}$ to a machine, getting a {\it global estimate} defined by
\begin{equation}\label{global_RERM}
\overline{f}_h=\frac{1}{m}\sum_{j=1}^mf_{j,h}.
\end{equation}}
\end{algorithmic}
\end{algorithm}
In Theorem \ref{THEOREM DISTRBUTED LAE POSITIVE}, we show that this
simple generalization of LAR achieves the optimal learning rate
with a rigorous condition concerning $m$. We also show that this condition is essential.
\begin{theorem}\label{THEOREM DISTRBUTED LAE POSITIVE}
Let $\overline{f}_h$ be defined by (\ref{global_RERM}) and $h_{D_j}$
be the mesh norm of the data block $D_j$ defined by $
h_{D_j}:=\max\limits_{X\in\mathcal X}\min\limits_{X_i\in
D_j}\|X-X_i\|.
$
Suppose that
(C) for all $D_1,\dots,D_m$, there exists a positive number $c_3$
such that
$$
\mathbf E\left\{\sum_{(X_i,Y_i)\in D_j}W^2_{h,X_i}(X)\right\}\leq
\frac{c_3}{nh^d};
$$
(D) for all $D_1,\dots,D_m$, there holds almost surely
$$
W_{X_i,h}I_{\{\|x-X_i\|>h\}}=0.
$$
If $h\sim N^{-1/(2r+d)}$, and the
event \{$ h_{D_j}\leq h$ for all $D_j$\} holds, then there exists a constant $C_2$ depending only on $d,r,M,c_0,c_3$ and $c_4$ such that
\begin{equation}\label{theorem1.11111}
C_0N^{-2r/(2r+d)}\leq \sup_{f_\rho\in\mathcal
F^{c_0,r}}\mathbf E\{\|\overline{f}_{h}-f_\rho\|_\rho^2\}
\leq
C_2N^{-2r/(2r+d)}.
\end{equation}
Otherwise, if the event \{$ h_{D_j}\leq h$ for all $D_j$\} dose not
hold, then for arbitrary $h\geq\frac12(n+2)^{-1/d}$, there exists a
distribution $\rho$ such that
\begin{equation}\label{theorem1.22222222222}
\sup_{f_\rho\in\mathcal
F^{c_0,r}}\mathbf E\{\|\overline{f}_{h}-f_\rho\|_\rho^2\}
\geq \frac{M^2\{(2h)^{-d}-2\}}{3n}.
\end{equation}
\end{theorem}
The assertions in Theorem \ref{THEOREM DISTRBUTED LAE
POSITIVE} can be divided into two parts. The first one is the
positive assertion, which means that if some conditions of the
weights and an extra constraint of the data blocks are imposed, then
the AVM-LAR estimate (\ref{global_RERM}) possesses the same learning
rate as that in (\ref{theorem1}) by taking the same localization
parameter $h$ (ignoring constants). This means that the proposed
divide and conquer operation in
Algorithm \ref{AVM-LAR} doesn't affect the learning rate under this circumstance. In fact,
we can relax the restriction (D) for the bound
(\ref{theorem1.11111}) to the following condition (D$^*$).
(D$^*$) For all $D_1,\dots,D_m$, there exists a positive number
$c_4$ such that
$$
\mathbf
E\left\{\sum_{(X_i,Y_i)\in D_j}W_{h,X_i}(X)I_{\{\|X-X_i\|>h\}}\right\}
\leq
\frac{c_4}{\sqrt{Nh^d}}.
$$
To guarantee the optimal minimax learning rate of AVM-LAR, condition
(C) is the same as condition (A) by noting that there are only $n$
samples in each $D_j$. Moreover, condition (D$^*$) is a bit
stronger than condition (B) as there are totally $n$ samples in
$D_j$ but the localization bound of it is $c_4/(\sqrt{Nh^d})$.
However, we should point out that such a restriction is also mild,
since in almost all widely used LAR, the localization bound either
is 0 (see NWK with naive kernel, and KNN) or decreases
exponentially (such as NWK with Gaussian kernel). All the above
methods satisfy conditions (C) and (D$^*$).
The negative assertion, however, shows that if the event \{there is
a $D_j$ such that $h_{D_j}>h$\} holds, then for any
$h\geq\frac12(n+2)^{-1/d}$, the learning rate of AVM-LAR isn't
faster than $\frac{1}{nh^d}$. It follows from Theorem
\ref{THEOREM:SUFFICIENT CONDITION FOR LOCAL ESTIMATE} that the best
localization parameter to guarantee the optimal learning rate
satisfies {$h\sim N^{-1/(2r+d)}$}. The condition
$h\geq\frac12(n+2)^{-1/d}$ implies that if the best parameter is
selected, then $m$ should satisfy $m\leq \mathcal O(N^{2r/(2r+d)})$.
Under this condition, from
(\ref{theorem1.22222222222}), we have
$$
\sup_{f_\rho\in\mathcal
F^{c_0,r}}\mathbf E\{\|\overline{f}_{h}-f_\rho\|_\rho^2\}
\geq \frac{C}{nh^d}.
$$
This means, if we select $h\sim N^{-1/(2r+d)}$ and $m\leq \mathcal
O(N^{2r/(2r+d)})$, then the learning rate of AVM-LAR is essentially
slower than that in (\ref{theorem1}). If we select a smaller $h$,
then the above inequality also yields the similar conclusion. If we
select a larger $h$, however, the
approximation error (see the proof of Theorem \ref{THEOREM:SUFFICIENT CONDITION FOR LOCAL ESTIMATE})
is $\mathcal O(h^{2r})$ which
can be larger than the learning rate in (\ref{theorem1}). In a word,
if the event \{$ h_{D_j}\leq h$ for all $D_j$\} does not hold, then
the AVM operation essentially degrades the optimal learning rate of
LAR.
At last, we should discuss the probability of the event \{$
h_{D_j}\leq h$ for all $D_j$\}. As $\mathbf P \{
h_{D_j}\leq h \ \mbox{for all}\ D_j \} =1-m\mathbf
P\{h_{D_1}>h\},$
and it can be found in \citep[P.93-94]{gyorfi2006distribution} that
$\mathbf P\{h_{D_1}>h\} \leq \frac{c}{nh^d}$, we have $\mathbf P \{
h_{D_j}\leq h \ \mbox{for all}\ D_j \} \geq
1-\frac{m}{nh^d}.$
When $h\sim (mn)^{-1/(2r+d)}$, we have
$$
\mathbf P \{
h_{D_j}\leq h \ \mbox{for all}\ D_j \} \geq
1-c' \frac{m^{(2r+2d)/(2r+d)}}{n^{2r/(2r+d)}}.
$$
The above quantity is small when $m$ is large, which means
that the event \{$ h_{D_j}\leq h$ for all $D_j$\} has a significant
chance to be broken down.
By
using the method in \citep[Problem 2.4]{gyorfi2006distribution}, we
can show that the above estimate for the confidence is essential
in the sense that for the uniform distribution, the equality holds
for some constant $c'$.
\section{Modified AVM-LAR}\label{section3}
As shown in Theorem \ref{THEOREM DISTRBUTED LAE POSITIVE}, if
$h_{D_j}\leq h$ does not hold for some $D_j$, then AVM-LAR cannot
reach the optimal learning rate. In this section, we propose two
variants of AVM-LAR such that they can achieve the optimal learning
rate under mild conditions.
\subsection{AVM-LAR with data-dependent parameters}
The event \{$ h_{D_j}\leq h$ for all $D_j$\} essentially implies
that for arbitrary $x$, there is at least one sample in the ball
$B_{h}(x):=\{x'\in \mathbb{R}^d: \|x-x'\|\leq h$\}. This condition holds for KNN as the parameter $h$ in KNN changes
with respect to samples. However, for NWK and other local average
methods (e.g., partition estimation~\citep{gyorfi2006distribution}), such a condition usually fails. Motivated by KNN, it is
natural to select a sample-dependent localization $h$ to ensure the
event \{$ h_{D_j}\leq h$ for all $D_j$\}. Therefore, we propose a
variant of AVM-LAR with data-dependent parameters in Algorithm
\ref{AVM-LAR-alg2}.
\begin{algorithm}\caption{AVM-LAR with data-dependent parameters}\label{AVM-LAR-alg2}
\begin{algorithmic}[!h]
\STATE {{\bf Initialization}: Let $D=\{(X_i,Y_i)\}_{i=1}^N$ be $N$
samples, $m$ be the number of data blocks.}
\STATE{ {\bf Output}: The global estimate $\tilde{f}_{\tilde{h}}$.}
\STATE{{\bf Division}: Randomly divide $D$ into $m$ data blocks
$D_1,D_2,\dots,D_m$ such that $D=\mathop{\bigcup}\limits_{j=1}^{m}
D_j, D_i\cap D_j=\varnothing, i\neq j$ and $|D_1|=\dots=|D_m|=n=N/m$.
Compute the mesh norms $h_{D_1},\dots, h_{D_m}$, and select
$\tilde{h}\geq h_{D_j}$, $j=1,2,\dots,m$.
}
\STATE{{\bf Local processing}:
For any $j=1,2,\dots,m$, implement LAR with bandwidth parameter $\tilde{h}$ for the
data block $D_j$ to get the $j$th {\it local estimate}
$$
f_{j,\tilde{h}}(x)=\sum_{(X_i,Y_i)\in D_j}W_{X_i,\tilde{h}}(x)Y_i.
$$
}
\STATE{{\bf Synthesization}: Transmit $m$ {\it local estimates}
$f_{j,\tilde{h}}$ to a machine, getting a {\it global estimate}
defined by
\begin{equation}\label{global_AVM_LAR1}
\tilde{f}_{\tilde{h}}=\frac{1}{m}\sum_{j=1}^mf_{j,\tilde{h}}.
\end{equation}}
\end{algorithmic}
\end{algorithm}
Comparing with AVM-LAR in Algorithm \ref{AVM-LAR}, the only
difference of Algorithm \ref{AVM-LAR-alg2} is the division step,
where we select the bandwidth parameter to be greater than all
$h_{D_j}, j =1,\dots,m$. The following Theorem \ref{THEOREM:
AVM-LAE} states the theoretical merit of AVM-LAR with
data-dependent parameters.
\begin{theorem}\label{THEOREM: AVM-LAE}
Let $r<d/2$, $\tilde{f}_{\tilde{h}}$ be defined by
(\ref{global_AVM_LAR1}). Assume (C) and (D$^*$) hold. Suppose
$$
\tilde{h}=\max\{m^{-1/(2r+d)}\max_j\{h_{D_j}^{d/(2r+d)}\},\max_j\{h_{D_j}\}\},$$
and
$m\leq\left(\frac{
c_0^2(2r+d)+8d(c_3+2c_4^2)M^2}{4r(c_0^2+2)}\right)^{d/(2r)}N^{2r/(2r+d)},
$ then there exists a constant $C_3$ depending only on $c_0,c_3,c_4,r,d$ and $M$ such that
\begin{equation}\label{theorem3}
C_0N^{-2r/(2r+d)}\leq \sup_{f_\rho\in\mathcal
F^{c_0,r}}\mathbf E\{\|\tilde{f}_{\tilde{h}}-f_\rho\|_\rho^2\}
\leq
C_3N^{-2r/(2r+d)}.
\end{equation}
\end{theorem}
Theorem \ref{THEOREM: AVM-LAE} shows that if the localization parameter
is selected elaborately, then AVM-LAR can achieve the optimal
learning rate under mild conditions concerning $m$. It should be
noted that there is an additional restriction to the smoothness
degree, $r<d/2$. We highlight that this condition cannot be removed.
In fact, without this condition, (\ref{theorem3}) may not hold for
some marginal distribution $\rho_X$. For example, let $d=1$, it can
be deduced from \citep[Problem 6.1]{gyorfi2006distribution} that
there exists a $\rho_{X}$ such that (\ref{theorem3}) doesn't hold.
However, if we don't aim at deriving a distribution free result, we
can fix this condition by using the technique in \citep[Problem
6.7]{gyorfi2006distribution}. Actually, for $d\leq 2r$, assume the
marginal distribution $\rho_X$ satisfies that there exist
$\varepsilon_0>0$, a nonnegative function $g$ such that for all
$x\in\mathcal X$, and $0<\varepsilon\leq \varepsilon_0$ satisfying
$\rho_X(B_\varepsilon(x))>g(x)\varepsilon^d,$ and $\int_{\mathcal
X}\frac{1}{g^{2/d}(x)}d\rho_X<\infty,$
then (\ref{theorem3}) holds for arbitrary $r$ and $d$. It is obvious
that the uniform distribution satisfies the above conditions.
Instead of imposing a restriction to $h_{D_j}$, Theorem
\ref{THEOREM: AVM-LAE} states that after using the data-dependent
parameter $\tilde{h}$, AVM-LAR doesn't degrade the learning rate
for a large range of $m$. We should illustrate that the derived
bound of $m$ cannot be improved further. Indeed, it can be found in
our proof that the bias of AVM-LAR can be bounded by $C\mathbf
E\{\tilde{h}^{2r}\}$. Under the conditions of Theorem \ref{THEOREM:
AVM-LAE}, if $m\sim N^{(2r+\varepsilon)/(2r+d)}$, then for arbitrary
$D_j$, there holds
$\mathbf E\{ {h_{D_j}}\}\leq Cn^{-1/d}
= C(N/m)^{-1/d}\leq CN^{(d-\varepsilon)/(2r+d)}.$
Thus, it is easy to check that $\mathbf E\{\tilde{h}^{2r}\}\leq
CN^{(-2r+\varepsilon)/(2r+d)}$, which implies a learning rate slower
than $N^{-2r/(2r+d)}$.
\subsection{Qualified AVM-LAR}
Algorithm \ref{AVM-LAR-alg2} provided an intuitive way to improve
the performance of AVM-LAR. However, Algorithm \ref{AVM-LAR-alg2}
increases the computational complexity of AVM-LAR, because we
have to compute the mesh norm $h_{D_j},j=1,\dots, m$. A natural
question is whether we can avoid this procedure while maintaining
the learning performance. The following Algorithm \ref{AVM-LAR-alg3}
provides a possible way to tackle this question.
\begin{algorithm}\caption{Qualified AVM-LAR}\label{AVM-LAR-alg3}
\begin{algorithmic}
\STATE {{\bf Initialization}: Let $D=\{(X_i,Y_i)\}_{i=1}^N$ be $N$
samples, $m$ be the number of data blocks, $h$ be the bandwidth parameter.}
\STATE{ {\bf Output}: The global estimate $\hat{f}_h$.}
\STATE {{\bf Division}: Randomly divide $D$ into $m$ data blocks,
i.e. $D=\cup_{j=1}^mD_j$ with $D_j\cap D_k=\varnothing$ for $k\neq
j$ and $|D_1|=\dots=|D_m|=n$.}
\STATE {{\bf Qualification}: For a test input $x$, if there exists
an $X_{0}^j\in D_j$ such that $|x-X_0^j|\leq h,$ then we qualify
$D_j$ as an active data block for the {\it local estimate}. Rewrite
all the active data blocks as $T_1,\dots,T_{m_0}$.}
\STATE {{\bf Local processing }: For arbitrary data block $T_j$,
$j=1,\dots,m_0$, define
$$
f_{j,h}(x)=\sum_{(X_i,Y_i)\in T_j}W_{X_i,h}(x)Y_i.
$$
}
\STATE{{\bf Synthesization}: Transmit $m_0$ {\it local estimates}
$f_{j,{h}}$ to a machine, getting a {\it global estimate} defined by
\begin{equation}\label{New AVM}
\hat{f}_h=\frac1{m_0}\sum_{j=1}^{m_0}f_{j,h}.
\end{equation}
}
\end{algorithmic}
\end{algorithm}
Comparing with Algorithms \ref{AVM-LAR} and \ref{AVM-LAR-alg2}, the
only difference of Algorithm \ref{AVM-LAR-alg3} is the qualification
step which essentially doesn't need extra computation. In fact,
the qualification and local processing steps can be implemented
simultaneously. It should be further mentioned that the
qualification step actually eliminates the data blocks which have
a chance to break down the event \{$h_{D_j}\leq h$ for all $D_j$\}.
Although, this strategy may loss a part of information, we show
that the qualified AVM-LAR can achieve the optimal learning rate
without any restriction to $m$.
\begin{theorem}\label{THEOREM: DLAE}
Let $\hat{f}_{h}$ be defined by (\ref{New AVM}). Assume
(C) holds and
(E) for all $D_1,\dots,D_m$, there exists a positive number
$c_5$ such that
$$
\mathbf
E\left\{ \sum_{i=1}^n|W_{h,X_i}(X)|I_{\{\|X-X_i\|>h\}}\right\}
\leq
\frac{c_{5}}{m\sqrt{nh^d}}.
$$
If $h\sim N^{-1/(2r+d)}$, then there exists a constant $C_4$
depending only on $c_0,c_1,c_3,c_5,r,d$ and $M$ such that
\begin{equation}\label{theorem dlae}
C_0N^{-2r/(2r+d)}\leq \sup_{f_\rho\in\mathcal
F^{c_0,r}}\mathbf E\{\|\hat{f}_{h}-f_\rho\|_\rho^2\}
\leq
C_4N^{-2r/(2r+d)}.
\end{equation}
\end{theorem}
In Theorem \ref{THEOREM: AVM-LAE}, we declare that AVM-LAR with
data-dependent parameter doesn't slow down the learning rate of
LAR. However, the bound of $m$ in Theorem \ref{THEOREM: AVM-LAE}
depends on the smoothness of the regression function, which is
usually unknown in the real world applications. This makes $m$ be a
potential parameter in AVM-LAR with data-dependent parameter, as we
do not know which $m$ definitely works. However, Theorem
\ref{THEOREM: DLAE} states that we can avoid this problem by
introducing a qualification step. The theoretical price of such an
improvement is only to use condition (E) to take place condition
(D$^*$). As shown above, all the widely used LARs such as the
partition estimate, NWK with naive kernel, NWK with Gaussian kernel
and KNN satisfy condition (E) (with a logarithmic term for NWK
with Gaussian kernel).
\section{Experiments}\label{section4}
In this section, we report experimental studies on synthetic data
sets to demonstrate the
performances of AVM-LAR and its variants. We employ three criteria
for the comparison purpose. The first criterion is the {\it global
error} (GE) which is the mean square error of testing set when $N$
samples are used as a training set. We use GE as a baseline that
does not change with respect to $m$. The second criterion is the
{\it local error} (LE) which is the mean square error of testing set
when we use only one data block ($n$ samples) as a training set. The
third criterion is the {\it average error} (AE) which is the mean
square error of AVM-LAR (including Algorithms \ref{AVM-LAR},
\ref{AVM-LAR-alg2} and \ref{AVM-LAR-alg3}).
\subsection{Simulation 1}
We use a fixed total number of samples
$N=10,000$, but assume that the number of data blocks $m$ (the data block size
$n=N/m$) and dimensionality $d$ are varied. The
simulation results are based on the average values of 20 trails.
We generate data from the following regression models $
y = g_j(x)+\varepsilon, \ j=1,2,
$ where $\varepsilon$ is the Gaussian noise $\mathcal{N}(0,0.1)$,
\begin{equation}\label{simulation1}
g_1(x)=\left\{
\begin{array}{cc}
(1-2x)_+^3(1+6x), & 0<x\leq 0.5 \\
0 & x>0.5%
\end{array}%
\right.,
\end{equation}
and
\begin{equation}\label{simulation2}
g_2(x)=\left\{
\begin{array}{cc}
(1-\|x\|)_+^5(1+5\|x\|) + \frac{1}{5}\|x\|^2, & 0<\|x\|\leq 1, x\in\mathbb{R}^5 \\
\frac{1}{5}\|x\|^2 & \|x\|>1%
\end{array}%
\right..
\end{equation}
\citet{Wendland2005} revealed that $g_1$ and $g_2$ are the so-called
Wendland functions with the property $g_1,g_2\in
\mathcal{F}^{c_0,1}$ and $g_1,g_2\notin \mathcal{F}^{c_0,2}$ for
some absolute constant $c_0$. The simulated $N$ samples are drawn
i.i.d. according to the uniform distribution on the (hyper-)cube
$[0,1]^d$. We also generate 1000 test samples $(X_i',Y_i')$ with
$X_i'$ drawn i.i.d. according to the uniform distribution and
$Y_i'=g_j(X_i'), j=1,2$.
On the basis of above setting, we illustrate two
simulation results. The first one is to compare the learning
performance between Algorithm \ref{AVM-LAR} and LAR. Both NWK and
KNN are considered. The second one is to show how Algorithms
\ref{AVM-LAR-alg2} and \ref{AVM-LAR-alg3} overcome Algorithm
\ref{AVM-LAR}'s weakness. Because Algorithms \ref{AVM-LAR},
\ref{AVM-LAR-alg2} and \ref{AVM-LAR-alg3} are the same for KNN, KNN is not considered in this part. The detailed implementation of
NWK and KNN is specified as follows.
\begin{itemize}
\item NWK: In Algorithm 1
and Algorithm \ref{AVM-LAR-alg3}, for each $m\in\{5,10,\dots,350\}$, the bandwidth
parameter satisfies $h\sim N^{-\frac{1}{2r+d}}$ according to
Theorem \ref{THEOREM DISTRBUTED LAE POSITIVE} and Theorem \ref{THEOREM: DLAE}.
In Algorithm \ref{AVM-LAR-alg2}, we set $\tilde{h}\sim
\max\{m^{-1/(2r+d)}\max_j\{h_{D_j}^{d/(2r+d)}\},\max_j\{h_{D_j}\}\}$
according to Theorem \ref{THEOREM: AVM-LAE}.
\item KNN: According to Theorem \ref{THEOREM DISTRBUTED LAE POSITIVE}, the parameter $k$ is set to $k\sim\frac{N^{\frac{2r}{2r+d}}}{m}$. However,
as $k\geq 1$, the range of $m$ should satisfy
$m\in\{1,2,\dots,N^{\frac{2r}{2r+d}}\}$.
\end{itemize}
To present proper constants for the localization parameters (e.g.,
$h=cN^{-\frac{1}{2r+d}}$), we use the 5-fold cross-validation method in simulations. Based on these strategies, we obtain the simulation results in the following Figures \ref{simulations1_NWK}, \ref{simulations1_KNN} and \ref{simulations2_NWK}.
As shown in Figure \ref{simulations1_NWK}, AEs are smaller than
LEs, which means AVM-NWK outperforms NWK with only one data block.
Furthermore, AEs of NWK are comparable with GEs when $m$ is not
too big and there exists an upper bound of the number of data
blocks, $m'$, lager than which the curve of AE increases
dramatically. Moreover, $m'$ decreases when $d$ increases.
\begin{figure*}[ht]
\begin{center}
\includegraphics[width=400pt]{SIM1.pdf}
\vspace{-15mm}
\caption{\emph{The first row shows AEs, LEs and GEs of NWK for
different $m$. The second row shows the number of inactive machines
which satisfy $h_{D_j}>h$. The vertical axis of the second row of
Figure \ref{simulations1_NWK} is the number of inactive data blocks
which break down the condition $h_{D_j}\leq h$.
}}\label{simulations1_NWK}
\end{center}
\end{figure*}
Let us explain these above phenomena. If only one data block is
utilized, then it follows from Theorem \ref{THEOREM:SUFFICIENT
CONDITION FOR LOCAL ESTIMATE} that $\min\limits_{j=1,\dots,m}\mathbf
E\{\|{f}_{j,h}-f_\rho\|_\rho^2\}=\mathcal{O}(n^{-\frac{2r}{2r+d}})$,
which is far larger than $\mathcal{O}(N^{-\frac{2r}{2r+d}})$ for
AVM-NWK due to Theorem \ref{THEOREM DISTRBUTED LAE POSITIVE}. Thus,
AEs are smaller than LEs. Moreover, Theorem \ref{THEOREM
DISTRBUTED LAE POSITIVE} implies that AEs are comparable with GE as
long as the event \{$ h_{D_j}\leq h$ for all $D_j, j=1,\dots, m$\}
holds. To verify this assertion, we record the number of data blocks
breaking down the condition $h_{D_j}\leq h$ for different $m$ in the
second row of Figure \ref{simulations1_NWK}. It can be observed that the
dramatically increasing time of the number of inactive data blocks
and AEs are almost same. This result is extremely consistent with
the negative part of Theorem \ref{THEOREM DISTRBUTED LAE POSITIVE}.
Because large $d$ and $m$ lead to a higher probability to break
down the event \{$ h_{D_j}\leq h$ for all $D_j, j=1,\dots, m$\}.
Thus $m'$ decreases when $d$ increases.
\begin{figure*}[ht]
\begin{center}
\includegraphics[width=430pt]{SIM1_KNN.pdf}
\vspace{-15mm}
\caption{\emph{ AEs, LEs and GEs of KNN for different
$m$.}}\label{simulations1_KNN}
\end{center}
\end{figure*}
Compared with AVM-NWK, AVM-KNN shows significant different results
in Figure \ref{simulations1_KNN}. In fact, there isn't a similar
$m'$ to guarantee comparable AEs and GE for AVM-KNN. The reason is
KNN selects a data-dependent bandwidth $h$ which makes the
event \{$ h_{D_j}\leq h$ for all $D_j, j=1,\dots, m$\} always holds.
This result is extremely consistent with the positive part of
Theorem \ref{THEOREM DISTRBUTED LAE POSITIVE}. However, we find that
AVM-KNN has a design deficiency. To be detailed, the range of $m$
must be in $\{1,2,\dots,N^{\frac{2r}{2r+d}}\}$ due to $k\geq 1$.
In Figure \ref{simulations2_NWK}, AEs of Algorithms \ref{AVM-LAR},
\ref{AVM-LAR-alg2} and \ref{AVM-LAR-alg3} which are denoted by
AE-A1, AE-A2 and AE-A3. We can find that AE-A1, AE-A2 and AE-A3 have
similar values which are comparable with GE when $m\leq m'$. The
reason is that the small number of data blocks can guarantee the
event \{$h_{D_j}\leq h$ for all $D_j, j=1,\dots,m$\}. Under this
circumstance, Theorems \ref{THEOREM DISTRBUTED LAE POSITIVE},
\ref{THEOREM: AVM-LAE} and \ref{THEOREM: DLAE} yield that all these
estimates can reach optimal learning rates. As $m$ increasing,
the event \{$h_{D_j}>h$ for some $j$\} inevitably happens, then
Algorithm \ref{AVM-LAR} fails according to the negative part of
Theorem \ref{THEOREM DISTRBUTED LAE POSITIVE}. At the same time,
AE-A1 begins to increase dramatically. As Algorithms
\ref{AVM-LAR-alg2} and \ref{AVM-LAR-alg3} are designed to avoid
the weakness of Algorithm \ref{AVM-LAR}, the curves of AE-A2 and
AE-A3 are always below that of AE-A1 when $m>m'$. Another
interesting fact is that AE-A3 is smaller than AE-A2,
although both of them all can achieve the same learning rate in theory.
\begin{figure*}[ht]
\begin{center}
\includegraphics[width=450pt]{S2.pdf}
\vspace{-25mm}
\caption{\emph{AE-A1, AE-A2, AE-A3 and GE of the simulation. The curves of AE-A2 and AE-A3 are always below AE-A1's to illustrate the improved capability of modified AVM-LARs.}}\label{simulations2_NWK}
\end{center}
\end{figure*}
\subsection{Simulation 2}
We make use of the same simulation study which is conducted by
\citet{Zhang2014} for comparing the learning performance of AVM-NWK
(including Algorithms \ref{AVM-LAR}, \ref{AVM-LAR-alg2} and
\ref{AVM-LAR-alg3}) and the divide and conquer kernel ridge regression
(DKRR for short).
We generate data from the regression model $y=g_3(x)+\epsilon$,
where $g_3(x)=\min(x,1-x)$, the noise variable $\epsilon$ is
normally distributed with mean 0 and variance $\sigma^2=1/5$, and
$X_i,i=1,\dots,N$ are simulated from a uniform distribution in
$[0,1]$ independently. In \citet{Zhang2014}'s simulation, DKRR used
the kernel function $K(x,x')=1+\min\{x,x'\}$, and regularization
parameter $\lambda = N^{-2/3}$ due to $g_3\in \mathcal{F}^{c_0,1}$
for some absolute constant $c_0$. We also use $N=10,000$ training
samples, and 1,000 test samples. The parameter setting of Algorithms
\ref{AVM-LAR}, \ref{AVM-LAR-alg2} and \ref{AVM-LAR-alg3} is the same
as that in Simulation 1.
\vspace{5mm}
\begin{figure*}[ht]
\begin{center}
\includegraphics[width=300pt]{s2comp.pdf}
\caption{\emph{AEs of Algorithm \ref{AVM-LAR}, \ref{AVM-LAR-alg2},
\ref{AVM-LAR-alg3} and DKRR.
}}\label{simulations2_com}
\end{center}
\end{figure*}
In Figure \ref{simulations2_com}, we plot AEs of Algorithms
\ref{AVM-LAR}, \ref{AVM-LAR-alg2}, \ref{AVM-LAR-alg3} and DKRR for
$m\in\{2^3,2^4,\dots, 2^{11}\}$. Figure \ref{simulations2_com} shows
AEs of Algorithm \ref{AVM-LAR}, \ref{AVM-LAR-alg2},
\ref{AVM-LAR-alg3} and DKRR are comparable when $m<256$. As long as
$m>256$, AEs of Algorithm \ref{AVM-LAR}, \ref{AVM-LAR-alg2} and DKRR
increase dramatically. However, AEs of Algorithm \ref{AVM-LAR-alg3}
are stable. The reason is that, to keep the optimal learning rates,
DKRR needs $m = \mathcal{O}(N^{1/3})$~\citep{Zhang2014}, and
Algorithm \ref{AVM-LAR-alg2} needs $m=\mathcal{O}(N^{2/3})$, while
Algorithm \ref{AVM-LAR-alg3} holds for all $m$.
\subsection{3D Road Network Data}
Building accurate 3D road networks is one of centrally important
problems for {\it Advanced Driver Assistant Systems} (ADAS).
Benefited from an accurate 3D road network, eco-routing, as an
application of ADAS, can yield fuel cost savings 8-12\% when
compared with standard routing~\citep{tavares2009optimisation}. For this reason, obtaining an accurate 3D road networks plays an important role for ADAS~\citep{kaul2013building}.
North Jutland (NJ), the northern part of Justland, Denmark, covers a
region of 185km$\times$130km. NJ contains a spatial road network
with a total length of $1.17\times 10^7m$, whose 3D ployline
representation is containing 414,363 points. Elevation values where
extracted from a publicly available massive Laser Scan Point Clod
for Denmark are added in the data set. Thus, the data set includes 4
attribute: {\it OSMID} which is the OpenStreetMap ID for each road
segment or edge in the graph; {\it longitude and latitude} with
Google format; {\it elevation} in meters. In practice, the acquired
data set may include missing values. In this subsection, we try to
use AVM-LAR based on Algorithms \ref{AVM-LAR}, \ref{AVM-LAR-alg2}
and \ref{AVM-LAR-alg3} for rebuilding the missing elevation
information on the points of 3D road networks via aerial laser scan
data.
To this end, we randomly select 1000 samples as a test set (record
time seed for the reproducible research). Using the other samples,
we run AVM-NWK based on Algorithm \ref{AVM-LAR}, \ref{AVM-LAR-alg2}
and \ref{AVM-LAR-alg3} to predict the missed elevation information
in the test set. Here, the bandwidth $h=0.13N^{-1/4}$ and
$N=413,363$. AE-A1, AE-A2, AE-A3 and GE for different
$m\in\{2,2^2,\dots,2^{10}\}$ are recorded in Figure
\ref{figure_road}.
\vspace{5mm}
\begin{figure*}[ht]
\begin{center}
\includegraphics[width=330pt]{3droad.pdf}
\caption{\emph{AE-A1, AE-A2, AE-A3 and GE of the 3D road network
data.}}\label{figure_road}
\end{center}
\end{figure*}
We can find in Figure \ref{figure_road} that AEs of Algorithm
\ref{AVM-LAR} are larger than GE, which implies the weakness of
AVM-NWK based on the negative part of Theorem \ref{THEOREM
DISTRBUTED LAE POSITIVE}. AE-A3 has almost same values with the GE
for all $m$, however, AE-A2 possesses similar property only when
$m\leq 32.$ Then, for the 3D road network data set, Algorithm
\ref{AVM-LAR-alg3} is applicable to fix the missed elevation
information for the data set.
\section{Proofs}\label{section5}
\subsection{Proof of Theorem 1}
Let $f_{\rho,h}(x)= \sum_{i=1}^NW_{h,X_i}(x)f_\rho(X_i).$
Then, it is obvious that $f_{\rho,h}(x)=\mathbf E^*\{f_{D,h}(x)\},$
where $\mathbf E^*\{\cdot\}=\mathbf E\{\cdot|X_1,X_2,\dots,X_n\}$.
Therefore, we can deduce
$$
\mathbf E^*\{(f_{D,h}(x)-f_\rho(x))\}=\mathbf E^*
\{(f_{D,h}(x)-f_{\rho,h}(x))^2\}+(f_{\rho,h}(x)-f_\rho(x))^2.
$$
That is,
$$
\mathbf E\{\|f_{D,h}-f_\rho\|_\rho^2\}
=
\int_{\mathcal X}\mathbf E\{\mathbf E^*
\{(f_{D,h}(X)-f_{\rho,h}(X))^2\}\}d\rho_X+\int_{\mathcal X}
\mathbf E\{(f_{\rho,h}(X)-f_\rho(X))^2\}d\rho_X.
$$
The first and second terms are referred to the {\it sample error} and {\it
approximation error}, respectively. To bound the sample error, noting
$\mathbf E^*\{Y_i\}=f_\rho(X_i)$, we have
\begin{eqnarray*}
&&\mathbf E^*
\{(f_{D,h}(x)-f_{\rho,h}(x))^2\}
=
\mathbf E^*
\left\{\left(\sum_{i=1}^NW_{h,X_i}(x)(Y_i-f_\rho(X_i))\right)^2
\right\}\\
&\leq&
\mathbf E^*\left\{\sum_{i=1}^N\left(W_{h,X_i}(x)(Y_i-f_\rho(X_i))\right)^2
\right\}
\leq
4M^2\sum_{i=1}^NW^2_{h,X_i}(x).
\end{eqnarray*}
Therefore we can use (A) to bound the sample error as
$$
\mathbf E\{(f_{D,h}(X)-f_{\rho,h}(X))^2\}\leq4M^2\mathbf E
\left\{\sum_{i=1}^NW^2_{h,X_i}(X)\right\}\leq
\frac{4c_1M^2}{Nh^d}.
$$
Now, we turn to bound the approximation error. Let $B_h(x)$ be the
$l^2$ ball with center $x$ and radius $h$, we have
\begin{eqnarray*}
&&\mathbf E\{(f_{\rho,h}(X)-f_\rho(X))^2\}
=
\mathbf
E\left\{\left(\sum_{i=1}^NW_{h,X_i}(X)f_\rho(X_i)-f_\rho(X)\right)^2\right\}\\
&=&
\mathbf
E\left\{\left(\sum_{i=1}^NW_{h,X_i}(X)(f_\rho(X_i)-f_\rho(X))\right)^2\right\}\\
&=&
\mathbf
E\left\{\left(\sum_{i=1}^NW_{h,X_i}(X)(f_\rho(X_i)-f_\rho(X))
\right)^2I_{\{B_h(X)\cap D=\varnothing\}}
\right\}\\
&+&
\mathbf E\left\{\left(\sum_{i=1}^NW_{h,X_i}(X)(f_\rho(X_i)
-f_\rho(X))\right)^2I_{\{B_h(X)\cap D\neq\varnothing\}}
\right\}.
\end{eqnarray*}
It follows from \citep[P.66]{gyorfi2006distribution} and
$\sum_{i=1}^NW_{h,X_i}(X)=1$ that
$$
\mathbf E\left\{\left(\sum_{i=1}^NW_{h,X_i}(X)(f_\rho(X_i)
-f_\rho(X))\right)^2I_{\{B_h(X)\cap D=\varnothing\}}
\right\}
\leq \frac{16M^2}{Nh^d}.
$$
Furthermore,
\begin{eqnarray*}
&&\mathbf
E\left\{\left(\sum_{i=1}^NW_{h,X_i}(X)(f_\rho(X_i)-f_\rho(X))
\right)^2I_{\{B_h(X)\cap D\neq\varnothing\}}
\right\}\\
&\leq&
\mathbf
E\left\{\left(\sum_{\|X_i-X\|\leq h}W_{h,X_i}(X)|f_\rho(X_i)-f_\rho(X)|
\right)^2I_{\{B_h(X)\cap D\neq\varnothing\}}
\right\}\\
&+&
\mathbf
E\left\{\left(\sum_{\|X_i-X\|> h}W_{h,X_i}(X)|f_\rho(X_i)-f_\rho(X)|
\right)^2
\right\}\\
&\leq&
c_0^2h^{2r}+\frac{4c^2_2M^2}{Nh^d},
\end{eqnarray*}
where the last inequality is deduced by $f_\rho\in\mathcal
F^{c_0,r}$, condition (B) and Jensen's inequality.
Under this circumstance, we get
$$
\mathbf E\{\|f_{D,h}-f_\rho\|_\rho^2\}
\leq
c_0^2h^{2r}+\frac{4(c_1+c_2^2+4)M^2}{Nh^d}.
$$
If we set $h=\left(\frac{4(c_1+c_2^2+4)M^2}{c_0^2N}\right)^{-1/(2r+d)},$
then
$$
\mathbf E\{\|f_{D,h}-f_\rho\|_\rho^2\}\leq
c_0^{2d/(2r+d)}(4(c_1+c_2^2+4)M^2)^{2r/(2r+d)}N^{-2r/(2r+d)}.
$$
This together with \citep[Theorem 3.2]{gyorfi2006distribution} finishes the proof
of Theorem \ref{THEOREM:SUFFICIENT CONDITION FOR LOCAL ESTIMATE}. $\square$
\subsection{Proof of Theorem 2}
Since $\mathbf E\left\{\|\overline{f}_h-f_\rho\|_\rho^2\right\}
=
\mathbf E\left\{\|\overline{f}_h-\mathbf
E\{\overline{f}_h\}+\mathbf
E\{\overline{f}_h\}-f_\rho\|^2_\rho\right\}$
and
$\mathbf E\{ \overline{f}_h \}=\mathbf
E\{ f_{j,h}\}, \quad j=1,\dots,m,$
we get
\begin{eqnarray}\label{distribution error decomposition}
\mathbf E\left\{\|\overline{f}_h-f_\rho\|^2_\rho\right\}
&=&
\frac1{m^2}\mathbf E\left\{\sum_{j=1}^m\left(\|f_{j,h }-\mathbf
E\{f_{j,h }\}\|_\rho^2+\|\mathbf
E\{f_{j,h }\}-f_\rho\|^2_\rho\right)\right. \nonumber \\
&+&
\left.2\sum_{j=1}^m\sum_{k\neq j}\langle f_{j,h}-\mathbf
E\{f_{j,h}\},f_{k,h}-\mathbf
E\{f_{k,h}\}\rangle_\rho\right\} \nonumber\\
&=&
\frac1m\mathbf E\{\| f_{1,h} -\mathbf
E\{f_{1,h}\}\|^2_\rho\}+ \|\mathbf
E\{f_{1,h}\}-f_\rho\|^2_\rho\\
&\leq&
\frac2m\mathbf E\{\| f_{1,h} -f_\rho\}\|^2_\rho\}+ 2\|\mathbf
E\{f_{1,h}\}-f_\rho\|^2_\rho. \nonumber
\end{eqnarray}
As $h\geq h_{D_j}$ for all $1\leq j\leq m$, we obtain that
$B_h(x)\cap D_j\neq\varnothing$ for all $x\in\mathcal X$ and $1\leq
j\leq m$. Then, using the same method as that in the proof of
Theorem \ref{THEOREM:SUFFICIENT CONDITION FOR LOCAL ESTIMATE}, (C)
and (D$^*$) yield that
$$
\mathbf E\{\| f_{1,h} -f_\rho\}\|^2_\rho\}
\leq c_0^2h^{2r}+\frac{4(c_3+c_4^2)M^2}{nh^d}.
$$
Due to the Jensen's inequality, we have
$$
\mathbf E\left\{\|\overline{f}_h-f_\rho\|_\rho^2\right\}
\leq
\frac{2c_0^2h^{2r}}{m}+\frac{8(c_3+c_4^2)M^2}{mnh^d}+
2\mathbf E\left\{\|\mathbf
E^*\{f_{1,h}\}-f_\rho\|^2_\rho\right\}.
$$
Noting $B_h(X)\cap D_j\neq\varnothing$ almost surely, the same
method as that in the proof of Theorem \ref{THEOREM:SUFFICIENT
CONDITION FOR LOCAL ESTIMATE} together with (D) yields that
$$
\mathbf E\left\{\|\mathbf
E^*\{f_{1,h}\}-f_\rho\|^2_\rho\right\}\leq
c_0^2h^{2r}+\frac{8c_4^2M^2}{mnh^d}.
$$
Thus,
$$
\mathbf E\left\{\|\overline{f}_h-f_\rho\|_\rho^2\right\}
\leq
\frac{16(c_3+c_4^2)M^2}{mnh^d}
+3c_0^2h^{2r}.
$$
This finishes the proof of (\ref{theorem1.11111}) by taking $
h=\left(\frac{16(c_3+c_4^2)M^2}{3c_0^2nm}\right)^{-1/(2r+d)}$ into
account.
Now, we turn to prove (\ref{theorem1.22222222222}). According to (\ref{distribution error
decomposition}), we have
$$
\mathbf E\left\{\|\overline{f}_h-f_\rho\|^2_\rho\right\}
\geq
\|\mathbf
E\{f_{1,h}\}-f_\rho\|^2_\rho
=\int_{\mathcal X}\left(\mathbf
E\left\{\sum_{i=1}^nW_{X_i,h}(X)f_\rho(X_i)-f_\rho(X)\right\}\right)^2
d\rho_X.
$$
Without loss of generality, we assume $h<h_{D_1}$. It then follows
from the definition of the mesh norm that there exits
$X\in\mathcal X$ which is not in $B_h(X_i)$, $X_i\in D_1$. Define the
separation radius of a set of points
$S=\{\zeta_i\}_{i=1}^n\subset\mathcal{X}$
via
$$
q_{_{S}}:=\frac12\min_{j\neq k}\|\zeta_j-\zeta_k\|.
$$
The mesh ratio $
\tau_{_{S}}:=\frac{h_{S}}{q_{_{S}}}\geq1
$ provides a measure of how uniformly points in $S$ are distributed
on $\mathcal X$. If $\tau_{_{S}}\leq 2$, we then call $S$ as the
quasi-uniform point set. Let $\Xi_l=\{\xi_1,\dots,\xi_l\}$ be
$l=\lfloor(2h)^{-d}\rfloor$ quasi-uniform points \citep{Wendland2005}
in $\mathcal X$. That is
$\tau_{_{\Xi_l}}=\frac{h_{_{\Xi_l}}}{q_{_{\Xi_l}}}\leq 2.$ Since
$h_{_{\Xi_l}}\geq l^{-1/d}$, we have $q_{_{\Xi_l}}\geq
\frac1{2l^{1/d}}\geq h.$ Then,
\begin{eqnarray}\nonumber
\mathbf
E\left\{\|\overline{f}_h-f_\rho\|^2_\rho\right\}
&=&
\|\mathbf E\{f_{1,h}\}-f_\rho\|_\rho^2\\\nonumber
&\geq&
\sum_{k=1}^l\int_{B_{q_{_{\Xi_l}}}(\xi_k)}\left(\mathbf
E\left\{\sum_{i=1}^nW_{X_i,h}(X)f_\rho(X_i)-f_\rho(X)\right\}\right)^2
d\rho_X.
\end{eqnarray}
{If $f_\rho(x)=M$, then
\begin{eqnarray*}
\mathbf E\left\{\|\overline{f}_h-f_\rho\|^2_\rho\right\}
& \geq&
M^2\sum_{k=1}^l\int_{B_{q_{_{\Xi_l}}}(\xi_k)}\left(\mathbf
E\left\{I_{\{D_1\cap B_{q_{_{\Xi_l}}}(\xi_k)=\varnothing\}}\right\}\right)^2
d\rho_X\\
&\geq&
M^2\sum_{k=1}^l\rho_X(B_{q_{_{\Xi_l}}}(\xi_k))\mathbf P\{D_1\cap
B_{q_{_{\Xi_l}}}(\xi_k)=\varnothing\}\\
&=&
M^2\sum_{k=1}^l\rho_X(B_{q_{_{\Xi_l}}}(\xi_k))(1-\rho_X(B_{q_{_{\Xi_l}}}(\xi_k)))^n.
\end{eqnarray*}
Since $h\geq\frac12(n+2)^{-1/d}$, we can let $\rho_X$ be the
marginal distribution satisfying
$$
\rho_X(B_{q_{_{\Xi_l}}}(\xi_k))=1/n, \ k=1,2,\dots, l-1.
$$
Then
$$
\mathbf E\left\{\|\overline{f}_h-f_\rho\|^2_\rho\right\}
\geq
M^2\sum_{k=1}^{l-1}\frac1n(1-1/n)^n
\geq\frac{M^2((2h)^{-d}-2)}{3n}.
$$
This finishes the proof of Theorem \ref{THEOREM DISTRBUTED LAE
POSITIVE}. $\square$
}
\subsection{Proof of Theorem 3}
Without loss of generality, we assume $h_{D_1}=\max_j\{h_{D_j}\}$.
It follows from (\ref{distribution error decomposition}) that
$$
\mathbf E\left\{\|\tilde{f}_{\tilde{h}}-f_\rho\|^2_\rho\right\}
\leq
\frac2m\mathbf E\{\| f_{1,\tilde{h}} -f_\rho\}\|^2_\rho\}+ 2\|\mathbf
E\{f_{1,\tilde{h}}\}-f_\rho\|^2_\rho.
$$
We first bound $\mathbf E\{\| f_{1,\tilde{h}} -f_\rho\}\|^2_\rho\}$.
As $\tilde{h}\geq h_{D_1}$, the same method as that in the proof of
Theorem \ref{THEOREM DISTRBUTED LAE POSITIVE} yields that
$$
\mathbf E\{\| f_{1,\tilde{h}} -f_\rho\}\|^2_\rho\}
\leq
c_0^2\mathbf E\{\tilde{h}^{2r}\}+\mathbf E\left\{
\frac{4M^2(c_3+c_4^2)}{n\tilde{h}^d}\right\}.
$$
To bound $\|\mathbf
E\{f_{1,\tilde{h}}\}-f_\rho\|^2_\rho$, we use the same
method as that in the proof of Theorem \ref{THEOREM DISTRBUTED LAE POSITIVE}
again. As $\tilde{h}\geq m^{-1/(2r+d)}h_{D_1}^{d/(2r+d)}$, it is
easy to deduce that
\begin{eqnarray*}
\|\mathbf
E\{f_{1,\tilde{h}}\}-f_\rho\|^2_\rho
&\leq&
\mathbf E\left\{\|\mathbf
E^*\{f_{1,\tilde{h}}\}-f_\rho\|^2_\rho\right\}
\leq
c_0^2\mathbf E\{\tilde{h}^{2r}\}+\mathbf
E\left\{\frac{8c_4^2M^2}{mn\tilde{h}^d}\right\}\\
&\leq&
c_0^2m^{-2r/(2r+d)}\mathbf E\{h_{D_1}^{2rd/(2r+d)}\}
+
c_0^2\mathbf E\{h_{D_1}^{2r}\}\\
&+&
8c_4^2M^2(mn)^{-1}\mathbf E\{m^{d/(2r+d)}h_{D_1}^{-d^2/(2r+d)}\}.
\end{eqnarray*}
Thus
\begin{eqnarray*}
\mathbf E\{\| \tilde{f}_{\tilde{h}} -f_\rho\}\|^2_\rho\}
&\leq&
c_0^2m^{-2r/(2r+d)}\mathbf E\{h_{D_1}^{2rd/(2r+d)}\}
+
(c_0^2+2)\mathbf E\{h_{D_1}^{2r}\}\\
&+&
8(c_3+2c_4^2)M^2(mn)^{-1}m^{d/(2r+d)}\mathbf E\{h_{D_1}^{-d^2/(2r+d)}\}.
\end{eqnarray*}
To bound $\mathbf E\{h_{D_1}^{2rd/(2r+d)}\}$, we note that for
arbitrary $\varepsilon>0$, there holds
$$
\mathbf P\{h_{D_1}>\varepsilon\}
=\mathbf P\{\max_{x\in\mathcal X}\min_{X_i\in D_1}\|x-X_i\|>\varepsilon\}
\leq \max_{x\in\mathcal X}
\mathbf E\{(1-\rho_X(B_\varepsilon(x)))^n\}.
$$
Let $t_1,\dots,t_l$ be the quasi-uniform points of $\mathcal X$.
Then it follows from \citep[P.93]{gyorfi2006distribution} that
$
\mathbf P\{h_{D_1}>\varepsilon\}\leq
\frac{1}{n\varepsilon^d}.
$
Then, we have
\begin{eqnarray*}
&&
\mathbf E\{h_{D_1}^{2rd/(2r+d)}\}
=
\int_0^\infty\mathbf
P\{h_{D_1}^{2rd/(2r+d)}>\varepsilon\}d\varepsilon
=
\int_0^\infty\mathbf
P\{h_{D_1}>\varepsilon^{(2r+d)/(2rd)}\}d\varepsilon\\
&\leq&
\int_0^{n^{-2r/(2r+d)}}1d\varepsilon+
\int_{n^{-2r/(2r+d)}}^\infty\mathbf
P\{h_{D_1}>\varepsilon^{(2r+d)/(2rd)}\}d\varepsilon\\
&\leq&
n^{-2r/(2r+d)}+\frac1n\int_{n^{-2r/(2r+d)}}^\infty\varepsilon^{-(2r+d)/(2r)}d\varepsilon
\leq
\frac{2r+d}dn^{-2r/(2r+d)}.
\end{eqnarray*}
To bound $\mathbf E\{h_{D_1}^{2r}\}$, we can use the above method
again and $r<d/2$ to derive
$
\mathbf E\{h_{D_1}^{2r}\}
\leq 4rd^{-1}n^{-2r/d}.
$
To bound $\mathbf E\{h_{D_1}^{-d^2/(2r+d)}\}$, we use the fact
$h_{D_1}\geq n^{-1/d}$ almost surely to obtain
$
\mathbf E\{h_{D_1}^{-d^2/(2r+d)}\}\leq n^{d/(2r+d)}.
$
Hence
$$
\mathbf E\{\| \tilde{f}_{\tilde{h}} -f_\rho\}\|^2_\rho\}
\leq
\left(\frac{c_0^2(2r+d)}d+8(c_3+2c_4^2)M^2\right)N^{-2r/(2r+d)}
+
\frac{4r(c_0^2+2)}{d}n^{-2r/d}.
$$
Since
$$
m\leq
\left(\frac{ c_0^2(2r+d)+8d(c_3+2c_4^2)M^2}{4r(c_0^2+2)}
\right)^{d/(2r)}N^{2r/(2r+d)},
$$
we have
$$
\mathbf E\{\| \tilde{f}_{\tilde{h}} -f_\rho\}\|^2_\rho\}
\leq
2\left(\frac{c_0^2(2r+d)}d+8(c_3+2c_4^2)M^2\right)N^{-2r/(2r+d)}
$$
which finishes the proof of (\ref{theorem3}). $\square$
\subsection{Proof of Theorem 4}
{\bf Proof.} From the definition, it follows that
$$
\hat{f_h}(x)=\sum_{j=1}^m\frac{I_{\{B_h(x)\cap D_j\neq
\varnothing\}}}{\sum_{j=1}^mI_{\{B_h(x)\cap D_j\neq
\varnothing\}}}\sum_{(X_i^j,Y_i^j)\in D_j}W_{h,X_{i}^j}(x)Y_{i}^j.
$$
We then use Theorem \ref{THEOREM:SUFFICIENT CONDITION FOR LOCAL
ESTIMATE} to consider a new local estimate with
$$
W^*_{h,X_{i}^j}(x)=\frac{I_{\{B_h(x)\cap D_j\neq
\varnothing\}}W_{h,X_{i}^j}(x)}{\sum_{j=1}^mI_{\{B_h(x)\cap D_j\neq
\varnothing\}}}.
$$
We first prove (A) holds. To this end, we have
\begin{eqnarray*}
\mathbf E\left\{\sum_{j=1}^m\sum_{(X_i^j,Y_i^j)\in
D_j}(W^*_{h,X_i^j}(X))^2\right\}
&\leq&
\mathbf E\left\{\sum_{j=1}^m\sum_{(X_i^j,Y_i^j)\in
D_j,X_{i}^j\in B_h(X)}(W^*_{h,X_i^j}(X))^2 \right\}\\
&+&
\mathbf E\left\{\sum_{j=1}^m\sum_{(X_i^j,Y_i^j)\in
D_j,X_{i}^j\notin B_h(X)}(W^*_{h,X_i^j}(X))^2\right\},
\end{eqnarray*}
where we define $\sum_{\varnothing}=0$. To bound the first term in
the right part of the above inequality, it is easy to see that if
$I_{\{X_{i}^j\in B_h(X)\}}=1$, then $I_{\{B_h(X)\cap D_j\neq
\varnothing\}}=1$, vice versa. Thus, it follows from (C)
that
\begin{eqnarray*}
\mathbf E\left\{\sum_{j=1}^m\sum_{(X_i^j,Y_i^j)\in
D_j,X_{i}^j\in B_h(X)}(W^*_{h,X_i^j}(X))^2 \right\}
&=&
\frac1{m^2}\mathbf E\left\{\sum_{j=1}^m\sum_{(X_i^j,Y_i^j)\in
D_j,X_{i}^j\in B_h(X)}(W_{h,X_i^j}(X))^2 \right\}\\
&\leq&
\frac1m\max_{1\leq j\leq m}\mathbf E\left\{\sum_{(X_i^j,Y_i^j)\in
D_j,X_{i}^j\in B_h(X)}(W_{h,X_i^j}(X))^2 \right\}\\
&\leq&
\frac1m\max_{1\leq j\leq m}\mathbf E\left\{\sum_{(X_i^j,Y_i^j)\in
D_j}(W_{h,X_i^j}(X))^2 \right\}\\
&\leq&
\frac{c_{3}}{Nh^d}
\end{eqnarray*}
To bound the second term, we have
\begin{eqnarray*}
&&\mathbf E\left\{\sum_{j=1}^m\sum_{(X_i^j,Y_i^j)\in
D_j,X_{i}^j\notin B_h(X)}(W^*_{h,X_i^j}(X))^2\right\}\\
&&=
\mathbf E\left\{\sum_{j=1}^m\sum_{(X_i^j,Y_i^j)\in
D_j,X_{i}^j\notin B_h(X)}\left(\frac{I_{\{B_h(X)\cap D_j\neq
\varnothing\}}W_{h,X_{i}^j}(X)}{\sum_{j=1}^mI_{\{B_h(X)\cap D_j\neq
\varnothing\}}}\right)^2\right\}
\end{eqnarray*}
At first, the same method as that in the proof of Theorem
\ref{THEOREM:SUFFICIENT CONDITION FOR LOCAL ESTIMATE} yields that
$
\mathbf E\{B_h(X)\cap D=\varnothing\}\leq \frac{4}{Nh^d}.
$
Therefore, we have
\begin{eqnarray*}
&&\mathbf E\left\{\sum_{j=1}^m\sum_{(X_i^j,Y_i^j)\in
D_j,X_{i}^j\notin B_h(X)}\left(\frac{I_{\{B_h(X)\cap D_j\neq
\varnothing\}}W_{h,X_{i}^j}(X)}{\sum_{j=1}^mI_{\{B_h(X)\cap D_j\neq
\varnothing\}}}\right)^2\right\}\\
&\leq&
\frac{4}{Nh^d}+m\max_{1\leq j\leq m}\mathbf E\left\{\sum_{(X_i^j,Y_i^j)\in
D_j}\left(W_{h,X_{i}^j}(X)I_{\|X-X_i^j\|>h}\right)\right\}\\
&\leq&
\frac{4+c_{3}+c_5}{Nh^d}.
\end{eqnarray*}
Now, we turn to prove (B) holds. This can be deduced directly by
using the similar method as the last inequality and the condition
(E). That is,
$$
\mathbf
E\left\{ \sum_{j=1}^m
\sum_{(X_i^j,Y_i^j)\in D_j}|W^*_{h,X^j_i}(X)|I_{\{\|X-X_i\|>h\}}\right\}
\leq
\frac{c_{5}}{\sqrt{Nh^d}}.
$$
Then Theorem \ref{THEOREM: DLAE} follows from Theorem
\ref{THEOREM:SUFFICIENT CONDITION FOR LOCAL ESTIMATE}. $\square$
\section{Conclusion}\label{section6}
In this paper, we combined the divide and conquer strategy with
local average regression to provide a new method called
average-mixture local average regression (AVM-LAR) to attack the
massive data regression problems. We found that the estimate
obtained by AVM-LAR can achieve the minimax learning rate under a strict
restriction concerning $m$. We then proposed two variants of AVM-LAR to either lessen the restriction or remove it. Theoretical analysis and
simulation studies confirmed our assertions.
We discuss here three interesting topics for future study.
Firstly, LAR cannot handle the high-dimensional data due to the
curse of
dimensionality~\citep{gyorfi2006distribution,fan2000prospects}. How
to design variants of AVM-LAR to overcome this hurdle can be accommodated as a desirable
research topic. Secondly, we have justified that applying the divide and conquer strategy on the LARs does not degenerate the order of learning rate under mild conditions. However, we did not show there is no loss in the constant factor. Discussing the constant factor of the optimal learning rate is an interesting project. Finally, equipping other nonparametric methods
(e.g., \citet{fan1994censored,gyorfi2006distribution,tsybakov2008introduction}) with the divide and conquer strategy can be taken into consideration for massive data analysis. For example, \citet{cheng2015computational} have discussed that how to appropriately apply the divide and conquer strategy to the smoothing spline method.
\bibliographystyle{asa}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,884 |
<?php
namespace Oneup\CssMin\Minifier\Plugin;
/**
* This {@link aCssMinifierPlugin} will convert hexadecimal color value with 6 chars to their 3 char hexadecimal
* notation (if possible).
*
* Example:
* <code>
* color: #aabbcc;
* </code>
*
* Will get converted to:
* <code>
* color:#abc;
* </code>
*/
class CssCompressColorValuesMinifierPlugin extends aCssMinifierPlugin
{
/**
* Regular expression matching 6 char hexadecimal color values.
*
* @var string
*/
private $reMatch = "/\#([0-9a-f]{6})/iS";
/**
* Implements {@link aCssMinifierPlugin::minify()}.
*
* @param aCssToken $token Token to process
* @return boolean Return TRUE to break the processing of this token; FALSE to continue
*/
public function apply(aCssToken &$token)
{
if (strpos($token->Value, "#") !== false && preg_match($this->reMatch, $token->Value, $m)) {
$value = strtolower($m[1]);
if ($value[0] == $value[1] && $value[2] == $value[3] && $value[4] == $value[5]) {
$token->Value = str_replace($m[0], "#" . $value[0] . $value[2] . $value[4], $token->Value);
}
}
return false;
}
/**
* Implements {@link aMinifierPlugin::getTriggerTokens()}
*
* @return array
*/
public function getTriggerTokens()
{
return array (
"CssAtFontFaceDeclarationToken",
"CssAtPageDeclarationToken",
"CssRulesetDeclarationToken"
);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,157 |
Q: sweetalert is not working if i unset the session I have a session named success and I am doing if the session is isset then show a sweetalert and it's working properly before i unset the session
<?php
if(isset($_SESSION['success'])){
// unset($_SESSION['success']);
?>
<script src="https://unpkg.com/sweetalert/dist/sweetalert.min.js"></script>
<script>
swal({
title: "Thank You!",
text: "Our Volunteer Will Contact You Soon!",
icon: "success",
button: "Close!",
});
</script>
<?php
}
?>
If i remove the comment on unset then alert is not coming
A: When you are redirecting to the page where you want to show the SweetAlert then use exit(0) after the header redirect.
header("Location: $self");
exit(0);
A: I found a way to make it work i hope it works on you too
Make a php file and this is the content
script.php content:
<script>
swal({
title: "Thank You!",
text: "Our Volunteer Will Contact You Soon!",
icon: "success",
button: "Close!",
});
</script>
Then do it like this
<?php
if(isset($_SESSION['success'])){
?>
<?php include('script.php'); ?>
<?php
unset($_SESSION['success']);
}
?>
This is my first answer on any question i hope it helps :)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,199 |
\section{Introduction}
Distributional representations of words, better known as word vectors, are a cornerstone of practical natural language processing (NLP). Examples of word vectors include Word2Vec \cite{Mikolov2013}, GloVe \cite{Pennington2014}, Eigenwords \cite{Dhillon2015}, and Fasttext \cite{Bojanowski2017}. These word vectors are usually referred to as \emph{distributional word vectors}, as their training methods rely on the \emph{distributional hypothesis} of semantics \cite{Firth1957}.
Recently, there has been interest in post-processing distributional word vectors to enrich their semantic content. The post-process procedures are usually performed in a lightweight fashion, i.e., without re-training word vectors on a text corpus. In one line of study, researchers used \emph{supervised methods} to enforce linguistic constraints (e.g., synonym relations) on word vectors \cite{Faruqui2015,Mrksic2016,Mrksic2017}, where the linguistic constraints are extracted from an external linguistic knowledge base such as WordNet \cite{Miller1995} and PPDB \cite{Pavlick2015}. In another line of study, researchers devised \emph{unsupervised methods} to post-process word vectors. Spectral-decomposition methods such as singular value decomposition (SVD) and principal component analysis (PCA) are usually used in this line of research \cite{Caron2001,Bullinaria2012,Turney2012,Levy2014,Levy2015,Mu2018}. The current paper is in line with the second, unsupervised, research direction.
Among different unsupervised word vector post-processing schemes, the \emph{all-but-the-top} approach \cite{Mu2018} is a prominent example. Empirically studying the latent features encoded by principal components (PCs) of distributional word vectors, \citet{Mu2018} found that the variances explained by the leading PCs ``encode the frequency of the word to a significant degree''. Since word frequencies are arguably unrelated to lexical semantics, they recommend removing such leading PCs from word vectors using a PCA reconstruction.
The current work advances the findings of \citet{Mu2018} and improves their post-processing scheme. Instead of discarding a fixed number of PCs, we \emph{softly} filter word vectors using \emph{matrix conceptors} \cite{Jaeger2014,Jaeger2017}, which characterize the linear space of those word vector features having high variances -- the features most contaminated by word frequencies according to \citet{Mu2018}. The proposed approach is mathematically simple and computationally efficient, as it is founded on elementary linear algebra. Besides these traits, it is also practically effective: using a standard set of lexical-level intrinsic evaluation tasks and a deep neural network-based dialogue state tracking task, we show that conceptor-based post-processing considerably enhances linguistic regularities captured by word vectors. A more detailed list of our contributions are:
\begin{enumerate}
\item We propose an unsupervised algorithm that leverages Boolean operations of conceptors to post-process word vectors. The resulting word vectors achieve up to 18.86\% and 28.34\% improvement on the SimLex-999 and SimVerb-3500 dataset relative to the original word representations.
\item A closer look at the proposed algorithm reveals commonalities across several existing post-processing techniques for neural-based word vectors and pointwise mutual information (PMI) matrix based word vectors. Unlike the existing alternatives, the proposed approach is flexible enough to remove lexically-unrelated noise, while general-purpose enough to handle word vectors induced by different learning algorithms.
\end{enumerate}
The rest of the paper is organized as follows. We first briefly review the principal component nulling approach for unsupervised word vector post-processing introduced in \cite{Mu2018}, upon which our work is based. We then introduce our proposed approach, \emph{Conceptor Negation} (CN). Analytically, we reveal the links and differences between the CN approach and the existing alternatives. Finally, we showcase the effectiveness of the CN method with numerical experiments\footnote{Our codes are available at \url{https://github.com/liutianlin0121/Conceptor-Negation-WV}}.
\subsection{Notation}
We assume a collection of words $w \in V$, where $V$ is a vocabulary set. Each word $w \in V$ is embedded as a $n$ dimensional real valued vector $v_w \in \mathbb{R}^n$. An identity matrix will be denoted by $\mathbf{I}$. For a vector $v$, we denote \text{diag}($v$) as the diagonal matrix with $v$ on its diagonal. We write $[n] = \{1, 2, \cdots, n\}$ for a positive integer $n$.
\section{Post-processing word vectors by PC removal}
This section is an overview of the all-but-the-top (ABTT) word vector post-processing approach introduced by \citet{Mu2018}. In brief, the ABTT approach is based on two key observations of distributional word vectors. First, using a PCA, \citet{Mu2018} revealed that word vectors are strongly influenced by a few leading principal components (PCs). Second, they provided an interpretation of such leading PCs: they empirically demonstrated a correlation between the variances explained by the leading PCs and word frequencies. Since word frequencies are arguably unrelated to lexical semantics, they recommend eliminating top PCs from word vectors via a PCA reconstruction. This method is described in Algorithm \ref{alg:abtt}.
\begin{algorithm}[htbp]
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{(i) $\{v_w \in \mathbb{R}^n : w \in V\}$: word vectors with a vocabulary $V$; (ii) $d$: the number of PCs to be removed.}
Center the word vectors: Let $\bar{v}_w \coloneqq v_w - \mu$ for all $w \in V$, where $\mu$ is the mean of the input word vectors. \\
Compute the first $d$ PCs $\{u_i \in \RR^n \}_{i \in [d]}$ of the column-wisely stacked centered word vectors $[\bar{v}_w]_{w \in V} \in \mathbb{R}^{n \times |V|}$ via a PCA. \\
Process the word vectors: $\tilde{v}_w^{\text{ABTT}} \coloneqq \bar{v}_w - \sum_{i = 1}^d u_i^\top u_i \bar{v}_w, \forall w \in V$. \\
\Output{$\{\tilde{v}_w^{\text{ABTT}}, w \in V\}$}
\caption{The all-but-the-top (ABTT) algorithm for word vector post-processing.}
\label{alg:abtt}
\end{algorithm}
In practice, \citet{Mu2018} found that the improvements yielded by ABTT are particularly impressive for word similarity tasks. Here, we provide a straightforward interpretation of the effects. Concretely, consider two arbitrary words $w_1$ and $w_2$ with word vectors $v_{w_1}$ and $v_{w_2}$. Without loss of generality, we assume $v_{w_1}$ and $v_{w_2}$ are normalized, i.e., $\| v_{w_1}\|_2 = \| v_{w_2}\|_2 = 1$. Given PCs $\{u_1, \cdots, u_n\}$ of the word vectors $\{v_w : w \in V\}$, we re-write $v_{w_1}$ and $v_{w_2}$ via linear combinations with respect to the basis $\{u_1, \cdots, u_n\}$: $v_{w_1} \coloneqq \sum_{i = 1}^n \beta_i u_i$ and $v_{w_2} \coloneqq \sum_{i = 1}^n \beta'_i u_i$, for some $\beta_i, \beta'_i \in \RR$ and for all $i \in [n]$. We see
\begin{eqnarray}
\text{cosine} (v_{w_1}, v_{w_2}) & \stackrel{( \ast)}{=} & v_{w_1}^\top v_{w_2}\label{eq:orig_cosine} \\
& = & \left (\sum_{i = 1}^n \beta_i u_i \right )^\top \left ( \sum_{i = 1}^n \beta'_i u_i \right ) \nonumber \\
& \stackrel{( \ast \ast)}{=} & \sum_{i = 1}^n \beta_i \beta'_i \label{eq:cosine}
\end{eqnarray}
where $(\ast)$ holds because the word vectors were assumed to be normalized and $(\ast \ast)$ holds because $\{u_1, \cdots, u_n\}$ is an orthonormal basis of $\RR^n$. Via Equation \ref{eq:cosine}, the similarity between word $w_1$ and $w_2$ can be seen as the overall ``compatibility'' of their measurements $\beta_i$ and $\beta'_i$ with respect to each latent feature $u_i$. If leading PCs encode the word frequencies, removing the leading PCs, in theory, help the word vectors capture semantic similarities, and consequently improve the experiment results of word similarity tasks.
\section{Post-processing word vectors via Conceptor Negation}
Removing the leading PCs of word vectors using the ABTT algorithm described above is effective in practice, as seen in the elaborate experiments conducted by \citet{Mu2018}. However, the method comes with a potential limitation: for each latent feature taking form as a PC of the word vectors, ABTT either completely removes the feature or keeps it intact. For this reason, \citet{Khodak2018} argued that ABTT is liable either to not remove enough noise or to cause too much information loss.
The objective of this paper is to address the limitations of ABTT. More concretely, we propose to use \emph{matrix conceptors} \cite{Jaeger2017} to gate away variances explained by the leading PCs of word vectors. As will be seen later, the proposed \emph{Conceptor Negation} method removes noise in a ``softer'' manner when compared to ABTT. We show that it shares the spirit of an eigenvalue weighting approach for PMI-based word vector post-processing. We proceed by providing the technical background of conceptors.
\subsection{Conceptors}
Conceptors are a family of regularized identity maps introduced by \citet{Jaeger2014}. We present a sketch of conceptors by heavily re-using \cite{Jaeger2014,He2018} sometimes verbatim. In brief, a \textit{matrix conceptor} $C$ for some vector-valued random variable $x$ taking values in $\mathbb{R}^N$ is defined as a linear transformation that minimizes the following loss function.
\begin{align}
\label{eq:matconobj}
\mathbb{E} \left [ \|x - Cx \|_2^2 \right ]+\alpha^{-2}\|C\|_{F}^2
\end{align}
where $\alpha$ is a control parameter called \textit{aperture}, $\|\cdot\|_{2}$ is the $\ell_2$ norm, and $\|\cdot\|_{F}$ is the Frobenius norm. This optimization problem has a closed-form solution
\begin{align}
\label{eq:conceptorsolution}
C = R(R+\alpha^{-2} \id )^{-1}
\end{align}
where $R=\mathbb{E}[xx^{\top}]$ and $\id$ are $N\times N$ matrices. If
$R = \Psi T \Psi^\top$ is the SVD of $R$, then the SVD of $C$ is given as $\Psi S \Psi^\top$, where the singular values $s_i$ of $C$ can be written in terms of the singular values $t_i$ of $R$: $s_i = t_i \slash (t_i + \alpha^{-2}) \in (0, 1)$ for $\alpha \in (0, \infty)$. In intuitive terms, $C$ is a soft projection matrix on the linear subspace where the samples of $x$ lie, such that for a vector $y$ in this subspace, $C$ acts like the identity: $Cy\approx y$, and when some $\epsilon$ orthogonal to the subspace is added to $y$, $C$ reconstructs $y$: $C(y+\epsilon)\approx y$.
Moreover, operations that satisfy most laws of Boolean logic such as NOT $\neg$, OR $\vee$, and AND $\wedge$, can be defined on matrix conceptors. These operations all have interpretation on the data level, i.e., on the distribution of the random variable $x$ (details in \cite[Section 3.9]{Jaeger2014}). Among these operations, the negation operation NOT $\neg$ is relevant for the current paper:
\begin{align}
\label{eq:bool}
\neg C \coloneqq \id -C.
\end{align}
Intuitively, the negated conceptor, $\neg C$, softly projects the data onto a linear subspace that can be roughly understood as the orthogonal complement of the subspace characterized by $C$.
\subsection{Post-processing word vectors with Conceptor Negation}
This subsection explains how conceptors can be used to post-process word vectors. The intuition behind our approach is simple. Consider a random variable $x$ taking values on word vectors $\{v_w \in \mathbb{R}^n : w \in V\}$. We can estimate a conceptor $C$ that describes the distribution of $x$ using Equation \ref{eq:conceptorsolution}. Recall that \cite{Mu2018} found that the directions with which $x$ has the highest variances encode word frequencies, which are unrelated to word semantics. To suppress such word-frequency related features, we can simply pass all word vectors through the negated conceptor $\neg C$, so that $\neg C$ dampens the directions with which $x$ has the highest variances. This simple method is summarized in Algorithm \ref{alg:cn}.
\begin{algorithm}[htbp]
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{(i) $\{v_w \in \mathbb{R}^n : w \in V\}$: word vectors of a vocabulary $V$; (ii) $\alpha \in \RR$: a hyper-parameter}
Compute the conceptor $C$ from word vectors: $C = R(R+\alpha^{-2} \id )^{-1}$, where $R$ is estimated by $\frac{1}{|V|} \sum_{w} v_w v_w^\top $ \\
Compute $\neg C \coloneqq \id - C$ \\
Process the word vectors: $\tilde{v}_w^{\text{CN}} \coloneqq \neg C v_w, \forall w \in V$\\
\Output{$\{\tilde{v}_w^{\text{CN}} : w \in V\}$}
\caption{The conceptor negation (CN) algorithm for word vector post-processing.}
\label{alg:cn}
\end{algorithm}
The hyper-parameter $\alpha$ of Algorithm \ref{alg:cn} governs the ``sharpness'' of the suppressing effects on word vectors employed by $\neg C$. Although in this work we are mostly interested in $\alpha \in (0, \infty)$, it is nonetheless illustrative to consider the extreme cases where $\alpha = 0$ or $\infty$: for $\alpha = 0$, $\neg C$ will be an identity matrix, meaning that word vectors will be kept intact; for $\alpha = \infty$, $\neg C$ will be a zero matrix, meaning that all word vectors will be nulled to zero vectors. The computational costs of the Algorithm \ref{alg:cn} are dominated by its step 1: one needs to calculate the matrix product $ R= \frac{1}{|V|} [v_{w}]_{w \in V} [v_{w}]_{w \in V}^{\top}$ for $[v_{w}]_{w \in V} \in \RR^{n \times |V|}$ being the matrix whose columns are word vectors. Since modern word vectors usually come with a vocabulary of some millions of words (e.g., Google News Word2Vec contains 3 million tokens), performing a matrix product on such large matrices $[v_{w}]_{w \in V}$ is computationally laborious. But considering that there are many uninteresting words in the vast vocabulary, we find it is empirically beneficial to only use a subset of the vocabulary, whose words are not too peculiar\footnote{This trick has also been used for ABTT by \citet{Mu2018} (personal communications).}. Specifically, borrowing the word list provided by \citet{Arora2017}\footnote{\url{https://github.com/PrincetonML/SIF/tree/master/auxiliary_data}}, we use the words that appear at least 200 times in a Wikipedia dump 2015 to estimate $R$. This greatly boosts the computation speed. Somewhat surprisingly, the trick also improves the performance of Algorithm \ref{alg:cn}. This might due to the higher quality of word vectors of common words compared with infrequent ones.
\section{Analytic comparison with other methods}
Since most of the existing unsupervised word vector post-processing methods are ultimately based on linear data transformations, we hypothesize that there should be commonalities between the methods. In this section, we show CN resembles ABTT in that both methods can be interpreted as ``spectral encode-decode processes''; when applied to word vectors induced by a pointwise mutual information (PMI) matrix, CN shares the spirit with the \emph{eigenvalue weighting} (EW) post-processing \cite{Caron2001,Levy2015}: they both assign weights on singular vectors of a PMI matrix. A key distinction of CN is that it does \emph{soft} noise removal (unlike ABTT) and that it is not restricted to post-processing PMI-matrix induced word vectors (unlike EW).
\subsection{Relation to ABTT}
In this subsection, we reveal the connection between CN and ABTT. To do this, we will re-write the last step of both algorithms into different formats. For the convenience of comparison, throughout this section, we will assume that the word vectors $\{v_w\}_{v \in V}$ in Algorithm \ref{alg:abtt} and Algorithm \ref{alg:cn} possess a zero mean, although this is not a necessary requirement in general.
We first re-write the equation in step 3 of Algorithm \ref{alg:abtt}. We let $U$ be the matrix whose columns are the PCs estimated from the word vectors. Let $U_{:,1:d}$ be the first $d$ columns of $U$. It is clear that step 2 of Algorithm \ref{alg:abtt}, under the assumption that word vectors possess zero mean, can be re-written as
\begin{align}
\tilde{v}_w^{\text{ABTT}} & \coloneqq \left ( I - U_{:,1:d} U_{:,1:d}^\top \right) v_w \nonumber \\
& = U \text{diag}([\underbrace{0, \cdots, 0}_{d~\text{copies~of~} 0}, 1, \cdots, 1]) U^\top v_w. \label{eq:abtt_rewrite}
\end{align}
Next, we re-write step 3 of the Conceptor Negation (CN) method of algorithm \ref{alg:cn}. Note that for word vectors with zero mean, the estimation for $R$ is a (sample) covariance matrix of a random variable taking values as word vectors, and therefore the singular vectors of $R$ are PCs of word vectors. Letting $R = U\Sigma U^\top$ be the SVD of $R$, the equation in step 3 of Algorithm \ref{alg:cn} can be re-written via elementary linear algebraic operations:
\begin{small}
\begin{align}
\tilde{v}_w^{\text{CN}} & \coloneqq \neg C \bar{v}_w \nonumber \\
&= \left ( \id - C \right ) v_w \nonumber \\
&= \left ( \id - R (R + \alpha^{-2} I )^{-1} \right ) v_w \nonumber \\
&= \left ( \id - U\Sigma U^\top (U\Sigma U^\top + \alpha^{-2} U U^\top )^{-1} \right ) v_w \nonumber \\
& = \left ( \id - U \text{diag}([\frac{\sigma_{1}}{\sigma_{1}+\alpha^{-2}}, \cdots, \frac{\sigma_{n}}{\sigma_{n}+\alpha^{-2}}]) U^\top \right) v_w \nonumber \\
& = U \text{diag}([\frac{\alpha^{-2}}{\sigma_{1}+\alpha^{-2}}, \cdots, \frac{\alpha^{-2}}{\sigma_{n}+\alpha^{-2}}]) U^\top v_w, \label{eq:cn_rewrite}
\end{align}
\end{small}
where $\sigma_{1}, \cdots, \sigma_{n}$ are diagonal entries of $\Sigma$.
Examining Equations \ref{eq:abtt_rewrite} and \ref{eq:cn_rewrite}, we see ABTT and CN share some similarities. In particular, they both can be unified into ``spectral encode-decode processes,'' which contain the following three steps:
\begin{enumerate}
\item \textbf{PC encoding}. Load word vectors on PCs by multiplying $U^\top$ with $v_w$.
\item \textbf{Variance gating}. Pass the PC-encoded data through the variance gating matrices $\text{diag}([0, \cdots, 0, 1, \cdots, 1])$ and $\text{diag}([\frac{\alpha^{-2}}{\sigma_{1}+\alpha^{-2}}, \cdots, \frac{\alpha^{-2}}{\sigma_{n}+\alpha^{-2}}])$ respectively for ABTT and CN.
\item \textbf{PC decoding}. Transform the data back to the usual coordinates using the matrix $U$.
\end{enumerate}
With the above encode-decode interpretation, we see CN differ from ABTT is its variance gating step. In particular, ABTT does a \emph{hard} gating, in the sense that the diagonal entries of the variance gating matrix (call them variance gating coefficients) take values in the set $\{0,1\}$. The CN approach, on the other hand, does a softer gating as the entries take values in $(0,1)$:
\[ 0 < \frac{\alpha^{-2}}{\sigma_{i}+\alpha^{-2}} \leq \frac{\alpha^{-2}}{\sigma_{j}+\alpha^{-2}} < 1,\] for all $1 \leq i < j \leq n$ and $\alpha \in (0, \infty)$. To illustrate the gating effects, we plot the variance gating coefficients for ABTT and CN for Word2Vec in Figure \ref{fig:gatingCoeff}.
\begin{figure}[htbp]
\centering
\includegraphics[width = 0.4\textwidth]{gatingCoeff.pdf}
\caption{The variance gating coefficients of ABTT and CN for Word2Vec. Hyper-parameters: $d = 3$ for ABTT and $\alpha = 2$ for CN.}
\label{fig:gatingCoeff}
\end{figure}
\subsection{Relation with eigenvalue weighting}
We relate the conceptor approach to the eigenvalue weighting approach for post-processing PMI-based word vectors. This effort is in line with the ongoing research in the NLP community that envisages a connection between ``neural word embedding'' and PMI-matrix factorization based word embedding \cite{Levy2014,Pennington2014,Levy2015}.
In the PMI approach for word association modeling, for each word $w$ and each context (i.e., sequences of words) $q$, the PMI matrix $M$ assigns a value for the pair $(w,q)$: $M(w, q) = \log \frac{\mathbb{P}( w, q)}{\mathbb{P}(w) \mathbb{P} (q)}$. In practical NLP tasks, the sets of words and contexts tend to be large, and therefore, directly working with $M$ is inconvenient. To lift the problem, one way is to perform a truncated SVD on $M$, factorizing $M$ into the product of three smaller matrices $M \approx \Theta_{:, 1:n} D_{1:n, 1:n} \Gamma_{:, 1:n}^\top$, where $\Theta_{:, 1:n}$ is the first $n$ left singular vectors of the matrix $M$, $D_{1:n, 1:n}$ is the diagonal matrix containing $n$ leading singular values of $M$, and $\Gamma_{:, 1:n}$ are the first $n$ right singular vectors of the matrix $M$. A generic way to induce word vectors from $M$ is to let \[ E \coloneqq \Theta_{:, 1:n} D_{1:n, 1:n} \in \RR^{|V| \times n}, \] which is a matrix containing word vectors as rows. Coined by \citet{Levy2015}, the term \emph{eigenvalue weighting}\footnote{It seems to us that a term ``singular value weighting'' is more appropriate because the weighting is based on singular values of a PMI matrix $M$ but not eigenvalues of $M$. The term ``eigenvalue'' is relevant here only because the singular values of $M$ are also the square roots of eigenvalues of $M^\top M$.} (EW) refers to a post-processing technique for PMI-matrix-induced word vectors. This technique has its root in Latent Semantic Analysis (LSA): \citet{Caron2001} first propose to define the post-processed version of $E$ as
\[ \tilde{E}^{\text{EW}} \coloneqq \Theta_{:, 1:n} D_{1:n, 1:n}^p,\]
where $p$ is the \emph{weighting exponent} determining the relative weights assigned to each singular vector of $\Theta_{:, 1:n}$. While an optimal $p$ depends on specific task demands, previous research suggests that $p < 1$ is generally preferred, i.e., the contributions of the initial singular vectors of $M$ should be suppressed. For instance, $p = 0, 0.25$, and $0.5$ are recommended in \cite{Caron2001,Bullinaria2012,Levy2015}. \citet{Bullinaria2012} argue that the initial singular vectors of $M$ tend to be contaminated most by aspects other than lexical semantics.
We now show that applying CN on the PMI-matrix-based word embedding $E \coloneqq \Theta_{:, 1:n} D_{1:n, 1:n}$ has a tantamount effect with ``suppressing initial singular vectors'' of EW. Acting the negated $\neg C$ on word vectors of $E$ (i.e., rows of $E$), we get the post-processed word vectors as rows of the $\tilde{E}^{\text{CN}}$:
\begin{small}
\begin{align*}
\tilde{E}^{\text{CN}} & \coloneqq (\neg C E^\top)^\top \\
& = E (I - R (R + \alpha^{-2} I)^{-1}) \\
& = E (I - \frac{1}{|V|} E^\top E (\frac{1}{|V|} E^\top E + \alpha^{-2} I)^{-1}) \\
& = \Theta_{:,1:n} D_{1:n, 1:n} (I - \frac{1}{|V|} D_{1:n, 1:n}^2 ( \frac{1}{|V|} D_{1:n, 1:n}^2 + \alpha^{-2} I)^{-1}) \\
& = \Theta_{:,1:n} D_{1:n, 1:n} \text{diag}([ \frac{|V| \cdot \alpha^{-2}}{ \lambda_1^2 + |V| \cdot \alpha^{-2}}, \cdots, \frac{|V| \cdot \alpha^{-2}}{ \lambda_n^2 + |V| \cdot \alpha^{-2}} ])\label{eq:cnPmi}
\end{align*}
\end{small}
Since
\[ 0 < \frac{|V| \cdot \alpha^{-2}}{ \lambda_i^2 + |V| \cdot \alpha^{-2}} \leq \frac{|V| \cdot \alpha^{-2}}{ \lambda_j^2 + |V| \cdot \alpha^{-2} } < 1,\] for all $1 \leq i < j \leq n$ and $\alpha \in (0, \infty)$, these weights suppress the contribution of the initial singular vectors, similar to what has been done in EW.
\section{Experiments}
We evaluate the post-processed word vectors on a variety of lexical-level intrinsic tasks and a down-stream deep learning task. We use the publicly available pre-trained Google News Word2Vec \cite{Mikolov2013}\footnote{\url{https://code.google.com/archive/p/word2vec/}} and Common Crawl GloVe\footnote{\url{https://nlp.stanford.edu/projects/glove/}} \cite{Pennington2014} to perform lexical-level experiments. For CN, we fix $\alpha = 2$ for Word2Vec and GloVe throughout the experiments\footnote{Analytical optimization methods for the aperture $\alpha$ are available from \cite{Jaeger2014}, remaining to be connected with the word vector post-processing scheme in the future.}. For ABTT, we set $d = 3$ for Word2Vec and $d = 2$ for GloVe, as what has been suggested by \citet{Mu2018}.
\paragraph{Word similarity} We test the performance of CN on seven benchmarks that have been widely used to measure word similarity: the RG65 \cite{Rubenstein1965}, the WordSim-353 (WS) \cite{Finkelstein2002}, the rare-words (RW) \cite{Luong2013}, the MEN dataset \cite{Bruni2014}, the MTurk \cite{Radinsky2011}, the SimLex-999 (SimLex) \cite{Hill2015}, and the SimVerb-3500 \cite{Gerz2016}. To evaluate the word similarity, we calculate the cosine distance between vectors of two words using Equation \ref{eq:orig_cosine}. We report the Spearman's rank correlation coefficient \cite{Myers1995} of the estimated rankings against the rankings by humans in Table \ref{tb:similarity}. We see that the proposed CN method consistently outperforms the original word embedding (orig.) and the post-processed word embedding by ABTT for most of the benchmarks.
\begin{table}[htbp]
\scalebox{0.85}{
\begin{tabular}{r c c c c c c}
\toprule
\multirow{2}{*}{} & \multicolumn{3}{c}{WORD2VEC} & \multicolumn{3}{c}{GLOVE} \\ \cmidrule(r){2-4} \cmidrule(r){5-7}
& orig. & ABTT & CN & orig. & ABTT & CN \\ \midrule
RG65 & 76.08 & 78.34 & \bf 78.92 & 76.96 & 74.36 & \bf 78.40 \\
WS & 68.29 & 69.05 &\bf 69.30 & 73.79 & 76.79 & \bf 79.08 \\
RW & 53.74 & 54.33 & \bf 58.04 & 46.41 & 52.04 & \bf 58.98 \\
MEN & 78.20 & \bf 79.08 & 78.67 & 80.49 & 81.78 & \bf 83.38 \\
MTurk & 68.23 & \bf 69.35 & 66.81 & 69.29 & 70.85 & \bf 71.07 \\
SimLex & 44.20 & 45.10 & \bf 46.82 & 40.83 & 44.97 & \bf 48.53 \\
SimVerb & 36.35 & 36.50 & \bf 38.30 & 28.33 & 32.23 & \bf 36.36 \\ \bottomrule
\end{tabular}}
\caption{Post-processing results (Spearman's rank correlation coefficient $\times$ 100) under seven word similarity benchmarks. The baseline results (orig. and ABTT) are collected from \cite{Mu2018}. }
\label{tb:similarity}
\end{table}
The improvement of results by CN are particularly impressive for two ``modern'' word similarity benchmarks SimLex and SimVerb -- these two benchmarks carefully distinguish \emph{genuine word similarity} from \emph{conceptual association} \cite{Hill2015}. For instance, \texttt{coffee} is associated with \texttt{cup} but by no means similar to \texttt{cup}, a confusion often made by earlier benchmarks. In particular, SimLex has been heavily used to evaluate word vectors yielded by \emph{supervised} word vector fine-tuning algorithms, which perform gradient descent on word vectors with respect to linguistic constraints such as synonym and antonym relationships of words extracted from WordNet and/or PPDB. When compared to a recent supervised approach of counter-fitting. Our results on SimLex are comparable to those reported by \citet{Mrksic2016}, as shown in Table \ref{tb:similarity_supervised}.
\begin{table}[htbp]
\scalebox{0.8}{
\begin{tabular}{l c c c } \toprule
&Post-processing method & WORD2VEC & GLOVE \\ \midrule
\multirow{3}{*}{supervised} & Counter-Fitting + syn. & 0.45 & 0.46 \\
& Counter-Fitting + ant. & 0.33 & 0.43 \\
& Counter-Fitting + syn. + ant. & \bf 0.47 & \bf 0.50 \\ \hdashline
unsupervised & CN & \bf 0.47 & 0.49 \\ \bottomrule
\end{tabular}}
\caption{Comparing the testing results (Spearman's rank correlation coefficient) on SimLex with those of Counter-Fitting approach (results collected from \cite[Table 2]{Mrksic2016} and \cite[Table 3]{Mrksic2017}). The linguistic constraints for Counter-Fitting are synonym (syn.) and/or antonym (ant.) relationships extracted from English PPDB.}
\label{tb:similarity_supervised}
\end{table}
\paragraph{Semantic Textual Similarity} In this subsection, we showcase the effectiveness of the proposed post-processing method using semantic textual similarity (STS) benchmarks, which are designed to test the semantic similarities of sentences. We use 2012-2015 SemEval STS tasks \cite{Agirre2012,Agirre2013,Agirre2014,Agirre2015} and the 2012 SemEval Semantic Related task (SICK) \cite{Marco2014}.
Concretely, for each pair of sentences, $s_1$ and $s_2$, we computed $v_{s_1}$ and $v_{s_2}$ by averaging their constituent word vectors. We then calculated the cosine distance between two sentence vectors $v_{s_1}$ and $v_{s_2}$. This naive method has been shown to be a strong baseline for STS tasks \cite{Wieting2016}. As in \citet{Agirre2012}, we used Pearson correlation of the estimated rankings of sentence similarity against the rankings by humans to assess model performance.
In Table \ref{tb:sts:detail}, we report the average result for the STS tasks each year (detailed results are in the supplemental material). Again, our CN method consistently outperforms the alternatives.
\begin{table}[htbp]
\scalebox{0.85}{
\centering
\begin{tabular}{ r c c c c c c}
\hline
\multirow{2}{*}{} & \multicolumn{3}{c}{WORD2VEC} & \multicolumn{3}{c}{GLOVE} \\ \cmidrule(r){2-4} \cmidrule(r){5-7}
& orig. & ABTT & CN & orig. & ABTT & CN \\ \midrule
STS 2012 & 57.22 & \bf 57.67 & 54.31 & 48.27 &54.06 &\bf 54.38 \\
STS 2013 &56.81 & 57.98 & \bf 59.17 &44.83 & 51.71 & \bf 55.51 \\
STS 2014 & 62.89. & 63.30 & \bf 66.22 & 51.11 & 59.23 & \bf 62.66 \\
STS 2015 &62.74 & 63.35 & \bf 67.15 &47.23 & 57.29 & \bf 63.74\\
SICK & 70.10 & {70.20} & \bf 72.71 & 65.14 & \bf{67.85} & 66.42 \\
\bottomrule
\end{tabular}
}
\caption{Post-processing results ($\times $100) on the semantic textual similarity tasks. The baseline results (orig. and ABTT) are collected from \cite{Mu2018}.}
\label{tb:sts:detail}
\end{table}
\paragraph{Concept Categorization} In the concept categorization task, we used $k$-means to cluster words into concept categories based on their vector representations (for example, ``bear'' and ``cat'' belong to the concept category of animals). We use three standard datasets: (i) a rather small dataset ESSLLI 2008 \cite{Baroni2008} that contains 44 concepts in 9 categories; (ii) the Almuhareb-Poesio (AP) \cite{Poesio2005}, which contains 402 concepts divided into 21 categories; and (iii) the BM dataset \cite{Battig1969} that 5321 concepts divided into 56 categories. Note that the datasets of ESSLLI, AP, and BM are increasingly challenging for clustering algorithms, due to the increasing numbers of words and categories.
Following \cite{Baroni2014,Schnabel2015,Mu2018}, we used ``purity'' of clusters \cite[Section 16.4]{Manning2008} as the evaluation criterion. That the results of $k$-means heavily depend on two hyper-parameters: (i) the number of clusters and (ii) the initial centroids of clusters. We follow previous research \cite{Baroni2014,Schnabel2015,Mu2018} to set $k$ as the ground-truth number of categories. The settings of the initial centroids of clusters, however, are less well-documented in previous work -- it is not clear how many initial centroids have been sampled, or if different centroids have been sampled at all. To avoid the influences of initial centroids in $k$-means (which are particularly undesirable for this case because word vectors live in $\RR^{300}$), in this work, we simply fixed the initial centroids as the average of original, ABTT-processed, and CN-processed word vectors respectively from ground-truth categories. This initialization is fair because all post-processing methods make use of the ground-truth information equally, similar to the usage of the ground-truth numbers of clusters. We report the experiment results in Table \ref{tb:categorization}.
\begin{table}[htbp]
\scalebox{0.85}{
\begin{tabular}{ r c c c c c c }
\toprule
\multirow{2}{*}{} & \multicolumn{3}{c}{WORD2VEC} & \multicolumn{3}{c}{GLOVE} \\ \cmidrule(r){2-4} \cmidrule(r){5-7}
& orig. & ABTT & CN. & orig. & ABTT & CN \\ \midrule
ESSLLI &100.0 & 100.0 & 100.0 & 100.0 & 100.0 & 100.0\\
AP & 87.28 & 88.3 & \bf 89.31 & 86.43 & 87.19 & \bf 90.95 \\
BM & 58.15 & 59.24 & \bf 60.19 & 65.34 & 67.35 & \bf 67.63 \\
\bottomrule
\end{tabular}
}
\caption{Purity ($\times$ 100) of the clusters in concept categorization task with fixed centroids.}
\label{tb:categorization}
\end{table}
The performance of the proposed methods and the baseline methods performed equally well for the smallest dataset ESSLLI. As the dataset got larger, the results differed and the proposed CN approach outperformed the baselines.
\paragraph{A Downstream NLP task: Neural Belief Tracker} The experiments we have reported so far are all intrinsic lexical evaluation benchmarks. Only evaluating the post-processed word vectors using these benchmarks, however, invites an obvious critique: the success of intrinsic evaluation tasks may not transfer to downstream NLP tasks, as suggested by previous research \cite{Schnabel2015}. Indeed, when supervised learning tasks are performed, the post-processing methods such as ABTT and CN can \emph{in principle} be absorbed into a classifier such as a neural network. Nevertheless, good initialization for classifiers is crucial. We hypothesize that the post-processed word vectors serve as a good initialization for those downstream NLP tasks that semantic knowledge contained in word vectors is needed.
To validate this hypothesis, we conducted an experiment using Neural Belief Tracker (NBT), a deep neural network based dialogue state tracking (DST) model \cite{Mrksic2017,Mrksic2018}. As a concrete example to illustrate the purpose of the task, consider a dialogue system designed to help users find restaurants. When a user wants to find a Sushi restaurant, the system is expected to know that Japanese restaurants have a higher probability to be a good recommendation than Italian restaurants or Thai restaurants. Word vectors are important for this task because NBT needs to absorb useful semantic knowledge from word vectors using a neural network.
In our experiment with NBT, we used the model specified in \cite{Mrksic2018} with default hyper-parameter settings\footnote{\url{https://github.com/nmrksic/neural-belief-tracker}}. We report the \emph{goal accuracy}, a default DST performance measure, defined as the proportion of dialogue turns where all the user's search goal constraints match with the model predictions. The test data was Wizard-of-Oz (WOZ) 2.0 \cite{Wen2017}, where the goal constraints of users were divided into three domains: \emph{food}, \emph{price range}, and \emph{area}. The experiment results are reported in Table \ref{tb:NBT}.
\begin{table}[h!]
\scalebox{0.85}{
\begin{tabular}{ r c c c c c c }
\toprule
\multirow{2}{*}{} & \multicolumn{3}{c}{WORD2VEC} & \multicolumn{3}{c}{GLOVE} \\ \cmidrule(r){2-4} \cmidrule(r){5-7}
& orig. & ABTT & CN. & orig. & ABTT & CN \\ \midrule
Food & 48.6 & \bf 84.7 & 78.5 & 86.4 & 83.7 &\bf 88.8\\
Price range & 90.2 & 88.1 & \bf 92.2 & 91.0 & 93.9 & \bf 94.7 \\
Area & 83.1 & 82.4 & \bf 86.1 & 93.5 & \bf 94.9 & 93.7 \\ \hdashline
Average & 74.0 & 85.1 & \bf 85.6 & 90.3 & 90.8 & \bf 92.4 \\
\bottomrule
\end{tabular}
}
\caption{The goal accuracy of food, price range, and area.}
\label{tb:NBT}
\end{table}
\paragraph{Further discussions}
Besides the NBT task, we have also tested ABTT and CN methods on other downstream NLP tasks such as text classification (not reported). We found that ABTT and CN yield equivalent results in such tasks. One explanation is that the ABTT and CN post-processed word vectors are different only up to a small perturbation. With a sufficient amount of training data and an appropriate regularization method, a neural network should generalize over such a perturbation. With a relatively small training data (e.g., the 600 dialogues for training NBT task), however, we found that word vectors as initializations matters, and in such cases, CN post-processed word vectors yield favorable results. Another interesting finding is that having tested ABTT and CN on Fasttext \cite{Bojanowski2017}, we found that neither post-processing method provides visible gain. We hypothesize that this might be because Fasttext includes subword (character-level) information in its word representation during training, which suppresses the word frequency features contained in word vectors. It remains for future work to validate this hypothesis.
\section{Conclusion}
We propose a simple yet effective method for post-processing word vectors via the negation operation of conceptors. With a battery of intrinsic evaluation tasks and a down-stream deep-learning empowered dialogue state tracking task, the proposed method enhances linguistic regularities captured by word vectors and consistently improves performance over existing alternatives.
There are several possibilities for future work. We envisage that the logical operations and abstract ordering admitted by conceptors can be used in other NLP tasks. As concrete examples, the AND $\wedge$ operation can be potentially applied to induce and fine-tune bi-lingual word vectors, by mapping word representations of individual languages into a shared linear space; the OR $\vee$ together with NOT $\neg$ operation can be used to study the vector representations of polysemous words, by joining and deleting sense-specific vector representations of words; the abstraction ordering $\leq$ is a natural tool to study graded lexical entailment of words.
\paragraph{Acknowledgement} We appreciate the anonymous reviewers for their constructive comments. We thank Xu He, Jordan Rodu, and Daphne Ippolito, and Chris Callison-Burch for helpful discussions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,698 |
{"url":"http:\/\/forums.xkcd.com\/viewtopic.php?f=7&t=61865&p=2213882","text":"## 0759: \"3x9\"\n\nThis forum is for the individual discussion thread that goes with each new comic.\n\nModerators: Moderators General, Prelates, Magistrates\n\nRabbitWho\nPosts: 284\nJoined: Sat May 01, 2010 10:16 pm UTC\nContact:\n\n### Re: \"3x9\" Discussion (#759)\n\nI'm delighted because this is the first time something mathy has been posted and I've understood it.\n\nvodka.cobra\nPosts: 371\nJoined: Thu Mar 27, 2008 6:50 pm UTC\nLocation: Florida\nContact:\n\n### Re: \"3x9\" Discussion (#759)\n\nPROTIP: I do not actually know how to do the gradient of a vector without using partial derivatives.\n\nYet, I aced the test on it. Similar concept to the comic, only it was calculus not arithmetic.\nIf the above comment has anything to do with hacking or cryptography, note that I work for a PHP security company and might know what I'm talking about.\n\nEebster the Great\nPosts: 3486\nJoined: Mon Nov 10, 2008 12:58 am UTC\nLocation: Cleveland, Ohio\n\n### Re: \"3x9\" Discussion (#759)\n\nrjk1994 wrote:\nShay Guy wrote:\nYou've heard of Graham's number, right? That absurdly huge number equivalent to 3\u2191\u2191\u2191...\u2191\u2191\u21913, with a much smaller but still absurdly huge number of up arrows?\n\nChange those 3s to 2s, and it reduces to 4.\n\nTwo and two is four, no matter how you do it.\n\nWhat? Are you serious? I really don't think it does.\n\n2^2 = 4\n2^2^2 = 2^4 = 16\n2^2^2^2 = 2^2^4 = 2^16 = 65536\n65536 != 4\nQ.E.D.\n\nAnyway, great comic. Wish I'd read this before Thursday's fail of an FP2 exam.\n\nYou are misunderstanding the Knuth up arrows. x\u2191y = xy. x\u2191\u2191y = x\u2191x\u2191x\u2191...\u2191x, where there are y x's. In general, x\u2191ny = x\u2191n-1x\u2191n-1x\u2191n-1x\u2191n-1...\u2191n-1x, where there are y x's.\n\nTherefore 2\u2191n2 = 2\u2191n-12 = 2\u2191n-22 = ... = 2\u21912 = 22 = 4.\n\nSimilarly, hyper(2, n, 2) = 4 for all n, and 2\u21922\u2192X = 4 for any Conyway chain X.\n\nTheoGB\nPosts: 70\nJoined: Fri Aug 22, 2008 1:38 pm UTC\n\n### Re: \"3x9\" Discussion (#759)\n\niGeek wrote:On a physics exam, a friend of mine couldn't remember whether a particular equation had \"squared\" or \"times two\" in it, so he decided to pick a number and try it both ways and see which one looked right. Unfortunately, he realized later, the number he picked was 2.\n\n2^2 = 2*2\n\nReminds me of when I've been trying to work out if I'm dealing with a plural in a sentence correctly and for some reason my brain always picks the word 'sheep' out, which is obviously the same as singular and plural and thus completely useless to me.\n\nmolnarm\nPosts: 1\nJoined: Mon Jun 28, 2010 7:23 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nThis trick works every time! Another useful one: if you don't understand something, write some grammatically incorrect or nonsense sentences containing words related to the problem. The examiner will decode them as the right answer.\n\nps: So this is the US notation for division?\n\nMr. Burke\nPosts: 99\nJoined: Wed Feb 04, 2009 4:56 pm UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nmolnarm wrote:ps: So this is the US notation for division?\n\nIt is. Apparently, they wanted to make it intentionally confusing.\n\nNow I'm off to C.L.A.S.S.\n\nsnowyowl\nPosts: 464\nJoined: Tue Jun 23, 2009 7:36 pm UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nI had a friend who did this all the time. I think he barely scraped a pass in Maths. I'll have to ask him.\n\nAs for me, I have a tendency to sneakily remove terms that shouldn't be there just when I'm cancelling everything else out (so it doesn't look too suspicious). I get caught out a lot though. I may have to be more humorous about it.\nThe preceding comment is an automated response.\n\nArakun\nPosts: 3\nJoined: Mon Apr 20, 2009 7:38 am UTC\nLocation: Sweden\n\n### Re: \"3x9\" Discussion (#759)\n\nWhat, no one posted this yet!?\nAbbott And Costello 13 \u00d7 7 is 28\n\nskeptical scientist\nclosed-minded spiritualist\nPosts: 6142\nJoined: Tue Nov 28, 2006 6:09 am UTC\nLocation: San Francisco\n\n### Re: \"3x9\" Discussion (#759)\n\nMax2009 wrote:\nairshowfan wrote:Well, I do usually manage to close up the seam. And when I don't, the gaping hole is pretty obvious (and I have to resist the temptation to write \"Then a miracle happens\" into the gap)\n\nI did that once.\nI actually wrote \"and then we conduct a voodoo ritual and transform x into y (or whatever step I was missing).\nI got 85% for that one and a big smiley. Proving that even though all mathematicians are crazy, some of them have a sense of humor.\n\nOr possibly the grader was just really happy that you didn't do what Randall was suggesting in the alt-text. If I were grading that, I would probably not be terribly amused by the joke, but I'd be much happier at you pointing out where the missing steps are rather than making me hunt for the error in the hopes that I wouldn't find it (or that I would give up and give you full credit out of laziness).\nI'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.\n\n\"With math, all things are possible.\" \u2014Rebecca Watson\n\nSwitch31\nPosts: 34\nJoined: Sun Jun 13, 2010 5:39 pm UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nI got to say, the alt text on this one was spot on. I have definitely done exactly that on an exam. I derived forward and backward and couldn't link them completely. I got a 9\/10 on the question and the professor pointed to the seam and just wrote \"how did you get here?\"\n\nThe day was once again saved by the TI-89's ability to solve complex integrals.\n\nskeptical scientist\nclosed-minded spiritualist\nPosts: 6142\nJoined: Tue Nov 28, 2006 6:09 am UTC\nLocation: San Francisco\n\n### Re: \"3x9\" Discussion (#759)\n\nmolnarm wrote:ps: So this is the US notation for division?\n\nAccording to the Wikipedia page on long division, this is standard notation throughout the English-speaking world, and also in Japan. It's not standard notation for division in general, but it's the standard way of writing things when you perform the long division algorithm.\nI'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.\n\n\"With math, all things are possible.\" \u2014Rebecca Watson\n\nshakeystacey\nPosts: 1\nJoined: Mon Jun 28, 2010 8:34 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nWe used to do this all the time in maths at college. It became a bit of a frustration for our teacher, especially when we started getting quite good at it, and she couldn't tell whether we actually knew the answer or not. Another trick she taught us was to write \"by inspection\" if there was a bit of an obvious gap in our workings out, implying we had made the leap in our heads and it was too obvious to write down. Don't know if it ever worked, but if it did we must have looked quite clever.\n\nIcalasari\nPosts: 107\nJoined: Wed May 26, 2010 5:11 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nDamnit, I was trying to figure out how he got 27 from that division. It took me about 30 minutes to figure it out\n\nAnd this is why reading XKCD at 3 AM is NOT a good idea...\n\nphlip\nRestorer of Worlds\nPosts: 7573\nJoined: Sat Sep 23, 2006 3:56 am UTC\nLocation: Australia\nContact:\n\n### Re: \"3x9\" Discussion (#759)\n\nmolnarm wrote:ps: So this is the US notation for division?\n\nIt isn't quite, but it's very similar...\n\nLong division looks like this:\\begin{align} 27 &\\\\ 3 \\overline{) 81} &\\\\ 6\\unicode{x2007}&\\\\ \\overline{21} &\\\\ 21 &\\\\ \\overline{\\unicode{x2007}0}& \\end{align}Or, possibly, with a straight vertical line instead of a curved one, between the divisor and dividend... especially when it's typed. Strangely, there doesn't seem to be a way of writing that symbol properly in LaTeX, so just imagine that the paren and the horizontal line are joined at a corner.\n\nSquare roots, on the other hand, look like this:$3\\sqrt{81}$\n\nThe comic is playing on the fact that the radical sign and the long division sign look similar - a number, then a mostly-vertical squiggle that ends as a horizontal line that goes over another number.\n\nNow, while searching Google Images, I did find one page that uses the radical sign for long division... but I think they're an anomaly... I don't think this is at all standard anywhere.\n\nCode: Select all\n\nenum \u0ca0_\u0ca0 {\u00b0\u25a1\u00b0\u2570=1, \u00b0\u0414\u00b0\u2570, \u0ca0\u76ca\u0ca0\u2570};void \u253b\u2501\u253b\ufe35\u200b\u2570(\u0ca0_\u0ca0 \u26a0) {exit((int)\u26a0);}\n[he\/him\/his]\n\nsendingsignal\nPosts: 21\nJoined: Mon Jun 28, 2010 9:14 am UTC\nLocation: Savannah, GA\nContact:\n\n### Re: \"3x9\" Discussion (#759)\n\nI did this all the time on math exams. Well, not exactly this, but I was terrible about showing my work.\n\nThe worst was when they wanted you to write it out in sentence form. Something about that seemed extremely counter productive to my early high school mind.\n\nbag\nPosts: 6\nJoined: Thu Dec 28, 2006 9:04 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nInteresting technique - i always ended up having to do similar thing back when i was in school, since they always wanted you to show workings out, even when it was easier to do in your head.\nSo i'd just answer them all, then write a load of math-shaped crap in unreadable scrawl next to them to pretend it was the working out -was faster than making something mathematically accurate. I'm pretty sure no one ever reads it as none of my stuff even remotely resembled making sense o.o\n\nit was literally;\n(2*8)\/4 = 4\nWith something along the lines of\n((7*4\/88 +11 -100) % 4 + 1000^15)*0+4 = 4\nWritten next to it - none of it having any relation to any actual workings out or method of gaining the answer, and more often that not, adding up to a totally different number than my answer too...\n\nDisjointing it from the answer so it wasn't obvious which question the \"workings out\" related to may have helped too\n\nsendingsignal\nPosts: 21\nJoined: Mon Jun 28, 2010 9:14 am UTC\nLocation: Savannah, GA\nContact:\n\n### Re: \"3x9\" Discussion (#759)\n\nThis also reminds me of a college math class where I would literally sleep through every class (I was somehow not allowed to test out, though it was basically algebra II...)\n\nThe teacher was really nice, but it was an 8am. I felt kind of bad, so at the end of the quarter I apologized to her. She said she totally understood, as it was pretty boring. With no trace of irony. Big smile.\n\nMan, I kind of feel sorry for her. She always had a huge cup of coffee - probably required to teach math at an art school.\n\nbear_tracks\nPosts: 1\nJoined: Mon Jun 28, 2010 10:10 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nIn a similar but alternative strategy I totally blanked on a physics exam and I used the units I knew the results were suppose to be in and the units of the things I had to figure it out. I got the problem completely right.\n\nLayrajha\nPosts: 6\nJoined: Tue Feb 03, 2009 9:49 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nIsn't the alt text's suggestion kind of close to what COQ does? And if so, does it validates this method as the state-of-the-art way to derive a proof and, consequently, to graduate?\n\nlittlelj\nPosts: 140\nJoined: Wed Feb 18, 2009 10:40 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nshakeystacey wrote:We used to do this all the time in maths at college. It became a bit of a frustration for our teacher, especially when we started getting quite good at it, and she couldn't tell whether we actually knew the answer or not. Another trick she taught us was to write \"by inspection\" if there was a bit of an obvious gap in our workings out, implying we had made the leap in our heads and it was too obvious to write down. Don't know if it ever worked, but if it did we must have looked quite clever.\n\nWish I'd had your teacher. I did the alt-text trick many a time (in fact, I'm pretty sure that's how I ended up with Maths A Further Maths A on my A-Levels) but nothing quite so neat as \"by inspection\".\n\nHaving said that, I have a frighteningly clever friend who can look at a big pile of hideous algebra and say \"oh that factorises by (x^3-x^2)\" without doing any working. Sometimes I hate him.\nDudes, I'm a woman.\n\nCiDhed\nPosts: 29\nJoined: Tue Jun 17, 2008 3:07 pm UTC\nLocation: Missouri\n\n### Re: \"3x9\" Discussion (#759)\n\nI always hated showing my work.\n\nBioTube\nPosts: 362\nJoined: Sat Apr 11, 2009 2:11 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nsendingsignal wrote:The worst was when they wanted you to write it out in sentence form. Something about that seemed extremely counter productive to my early high school mind.\nI remember having to do that once in middle school - it was part of a day devoted to writing as much crap as possible as \"preparation\" for the writing TAKS test(which is, for reasons only known to senseless bureaucrats, must be passed or your score for the entire English test is pegged one point below passing).\nFr\u00e9d\u00e9ric Bastiat wrote:Government is the great fiction through which everybody endeavors to live at the expense of everybody else.\n\ngraatz\nPosts: 87\nJoined: Thu Oct 29, 2009 7:24 pm UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nIt took me a few seconds to realize that the division ended up with the right answer.\n\nsween64\nPosts: 15\nJoined: Wed Jun 02, 2010 9:17 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\niGeek wrote:On a physics exam, a friend of mine couldn't remember whether a particular equation had \"squared\" or \"times two\" in it, so he decided to pick a number and try it both ways and see which one looked right. Unfortunately, he realized later, the number he picked was 2.\n\n2^2 = 2*2\n\nAnd 2+2=4. Crazy!\nLast edited by sween64 on Mon Jun 28, 2010 1:05 pm UTC, edited 1 time in total.\n\nLoyal Lurker\nPosts: 5\nJoined: Mon Jun 21, 2010 3:08 pm UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nAm I the only one who noticed that if you have to show work to do 3x9, you're doing it wrong?\n\nPosts: 208\nJoined: Fri Jul 27, 2007 7:07 am UTC\nLocation: Dutch-land\nContact:\n\n### Re: \"3x9\" Discussion (#759)\n\nBetter still, use QED at the end, according to my maths teacher, that would bring tears to the examiners eyes to see that, and forget whatever was in the \"proof\" and give you an A.\nOf course that's a load of rubbish, but so was the comic.\n:)\n\nYou are carrying:\n- a slightly paranoid Android\n- two left feet (not my own)\n- a still unfed and very hungry hippo\n- broadsword of +5 ridiculousness stained with the blood of the undead souls\n- a stetson Resistol, cuz stetson Resistols are cool.\n\nFreiberg\nPosts: 171\nJoined: Thu Oct 30, 2008 4:09 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nSwitch31 wrote:The day was once again saved by the TI-89's ability to solve complex integrals.\n\nThe ability of the TI-89 to get any answer is the only way I passed my AP Calc class. Sadly, it doesn't seem to work on the actual AP exam.\n\nLasher\nPosts: 11\nJoined: Wed Feb 25, 2009 2:25 pm UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nThis brings back some memories. This was called \"The confusion method\/rule\" among my peers quite useful to get partial credit.\nOn a side note, there's a urban legend that this caused a chronic migraine problem to a professor in the faculty.\n\nRplot\nPosts: 13\nJoined: Fri Feb 06, 2009 5:26 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nA long tradition. From another cartoonist.\n\nhttp:\/\/www.artistmarket.com\/artists\/har ... -%2001.htm\n\nslashme\nPosts: 22\nJoined: Mon Jul 28, 2008 9:08 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nReally, how did a bunch of self-professed geeks have so much discussion on the topic with no-one pointing out that \u00d7\u2260x?\n\nManial\nPosts: 212\nJoined: Thu Sep 27, 2007 11:52 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nWait, what?\n\nHow are you meant to show your work for 3 x 9? Draw 3 groups of 9 sheep or something?\n\nTophe\nPosts: 17\nJoined: Sat Dec 22, 2007 4:09 am UTC\nLocation: TN\n\n### Re: \"3x9\" Discussion (#759)\n\nhappysteve wrote:Kinda reminds me of this fun math tidbit...\n\nSolve for x:\n[imath]x^2 - x - 20 = 10[\/imath] ... hmm okay, factor the left hand side...\n[imath](x + 4)(x - 5) = 10[\/imath]\nah great, now it's just a matter of solving for (x + 4) = 10 and (x - 5) = 10\nx is either 6 or -5\nCheck with the original statement:\n[imath]6^2 - 6 - 20 = 10[\/imath] ... yup, that works\n[imath](-5)^2 - (-5) - 20 = 10[\/imath] ... yup, that works too.\n\nyay, problem solved.\n\nno, its not \"just a matter of solving for (x + 4) = 10 and (x - 5) = 10\"\nit's a matter of solving for (x + 4) = 10 AND (x - 5) = 1\nor (x + 4) = 1 AND (x - 5) = 10\nor (x + 4) = -10 AND (x - 5) = -1\nor (x + 4) = -1 AND (x - 5) = -10\n\n2 of those dependencies don't work\n\nthe better way to solve this is:\n\nx^2 - x - 20 = 10 ...subtract 10 from both sides\nx^2 - x - 30 = 0\n(x - 6)(x + 5) = 0\n\nx - 6 = 0 OR x + 5 = 0\n\nx = 6 or -5\nLast edited by Tophe on Mon Jun 28, 2010 2:11 pm UTC, edited 1 time in total.\n\nNeWtoz\nPosts: 6\nJoined: Mon Mar 16, 2009 6:08 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nwow, I saw the problem right away and easily would have marked it wrong. I was a TA for a semester for a very simple Algebra class, and I would never pass over this.\n\nVioletDaGrinder\nPosts: 2\nJoined: Wed Feb 28, 2007 5:10 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nI've noticed when dining with math geeks that they always take an unreasonably long time to figure out the tip. Whereas I, with my most advanced math being pre-calc, can do it in approximately 2 seconds. I've always suspected that something like this is going on in their heads.\n\nam3930\nPosts: 25\nJoined: Fri Jun 18, 2010 5:39 pm UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nsween64 wrote:\niGeek wrote:On a physics exam, a friend of mine couldn't remember whether a particular equation had \"squared\" or \"times two\" in it, so he decided to pick a number and try it both ways and see which one looked right. Unfortunately, he realized later, the number he picked was 2.\n\n2^2 = 2*2\n\nAnd 2+2=4. Crazy!\n\nThink back to elementary school definitions.\n\n2^2 is defined as 2*2 (until there are a total of two twos) and 2*2 is defined as 2+2 for the same reasons.\n\nCall it a property of two.\n\n-----------------\n\nI'm kind of reminded of something that happened on a linear algebra test. There was some strange matrix thingie that I can't remember what it's called or ever really saw the point in. The only thing I could remember on the test was how to find the inverse with it. I simply used a simpler method to find the inverse and work backwards and then stared awkwardly at the next question: Find the inverse.\nTake me out to the black.\nTell them I ain't comin' back.\nBurn the land and boil the sea.\nYou can't take the-\n\n****, they have.\n\nGhona\nPosts: 246\nJoined: Mon May 21, 2007 1:28 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nlittlelj wrote:Having said that, I have a frighteningly clever friend who can look at a big pile of hideous algebra and say \"oh that factorises by (x^3-x^2)\" without doing any working. Sometimes I hate him.\n\nWell, that's pretty straightforward, right? I mean, factoring by x^2 is obvious at a moment's inspection, and (x-1) isn't that bad.\n:p\nIf you're taking me too seriously, you probably are making a mistake.\n\nthinboy00\nPosts: 12\nJoined: Thu Oct 15, 2009 12:29 am UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nNot entirely related, but...\nWhen I was taking my AP CS exam (in Java; the whole thing was absurdly easy), one of the methods I had to write (on paper, no IDE) involved floating point math. So, just for kicks, I wrote \"should this whole class be strictfp?\" in a comment (to confuse the graders). strictfp is a really obscure Java keyword which might have made a small difference in the \uff5e10th decimal place or so... (unless you're unlucky and\/or using floats instead of doubles in which case it could be the \uff5e5th or so instead)\n\nalso: using weird tilde since the real one looks like this: ~\n\nApeiron\nPosts: 120\nJoined: Tue Feb 12, 2008 5:34 pm UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nGrumbleduke wrote:A quick bit of algebra shows that this will work for any two numbers where:\n$\\begin{eqnarray*}a \\times b & = & a \\times \\sqrt{b^2} \\\\ & = & a \\sqrt{b^2}\\\\ & = & \\frac{b^2}{a} \\\\ \\therefore a^2 & = & b \\end{eqnarray*}$\n\nIt's not all that exciting; try it with 2 and 4, or -3 and 9...\nHowever, the alt text is very good advice.\n\nEdit: Ah, assuming that a isn't 0 of course - although I think it still works algebraicly, if you use l'Hopital's rule on the fraction.\n\ni figured that it's:\n\nA * A2 = A3\n\nThe whole equation is an interestesting way of saying that multiplying a number times its square is the same as cubing it.\n\nnot baby Newt\nPosts: 110\nJoined: Wed Feb 03, 2010 11:30 pm UTC\n\n### Re: \"3x9\" Discussion (#759)\n\nP\nskeptical scientist wrote:\nmolnarm wrote:ps: So this is the US notation for division?\n\nAccording to the Wikipedia page on long division, this is standard notation throughout the English-speaking world, and also in Japan. It's not standard notation for division in general, but it's the standard way of writing things when you perform the long division algorithm.\n\nTo me it's fairly reasonable for actually doing division by hand, but nonsensical as part of an answer or equation.\n\nPosts: 350\nJoined: Wed Oct 08, 2008 1:58 am UTC\nLocation: Guarding the Super Missile\nContact:\n\n### Re: \"3x9\" Discussion (#759)\n\nrwald wrote:Initially this looks like another example of this, but it actually makes sense why this works; any time you multiply x by x^2, you'll get the same result as if you divide x^4 by x.\n\nWait, so x3 = x3?!\n\nWHOA NO WAY!!!\n\nnot baby Newt wrote:P\nskeptical scientist wrote:\nmolnarm wrote:ps: So this is the US notation for division?\n\nAccording to the Wikipedia page on long division, this is standard notation throughout the English-speaking world, and also in Japan. It's not standard notation for division in general, but it's the standard way of writing things when you perform the long division algorithm.\n\nTo me it's fairly reasonable for actually doing division by hand, but nonsensical as part of an answer or equation.\n\nAgreed. It completely breaks format with other notations; you could not put that in the middle of a one-line calculation without it being incredibly strange and confusing.\n\"It's easy to forget what a sin is in the middle of a battlefield.\" \"Opposite over hypotenuse, dipshit.\"","date":"2019-08-23 03:41:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7275359630584717, \"perplexity\": 2427.286335169378}, \"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\/1566027317817.76\/warc\/CC-MAIN-20190823020039-20190823042039-00514.warc.gz\"}"} | null | null |
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Weird, But Sweet (Complete Story, 1/18/23, Bonus Material added 1/25/23)
TQuintA posted a topic in Stories
Chapter 1 The bell above the door drew my attention as it clanged loudly. While the door swung open, I handed the customer at the counter in front of me their bread and change, but I was looking at Mason, who was walking through the door. I knew I loved Mason when he first walked through that same door two years ago. Same height as me (5' 10"), but all skinny and slight with an adorable hipster mustache and well-groomed scruff. The second I saw his floofy light brown hair and soulful brown eyes, I was in love, no matter how much I tried to deny it to myself. He dressed like a dork, but he didn't care. And because he didn't care, I didn't care. He had an easy smile, a way that made people feel comfortable, and a whimsical sense of humor. He has a youthfulness about him, and he's three years older than me. All about him is a quiet confidence. I was hiring an accountant to help save my bakery, but I found the man of my dreams. At the time, I didn't trust the instant attraction. For one, I was a solitary man. I wasn't entirely lonely, but I was alone. Sure, I dated, and I was friendly to all my customers. But I was also guarded. I had no close friends, and none of my boyfriends ever got to know the real me. There was physical intimacy, but no emotional intimacy. The only person I could be myself around was my Gramps. I had a secret to protect. More than that, before Mason, every man I'd ever dated had been bigger than me, and I'm a buff, lean, and muscular 190 pounds. I just liked guys who were bigger than me. Hell, I'd held myself back from getting as big as I truly wanted to be because I wanted a to be a big man in a relationship with a bigger man. Every guy I dated I'd met at the gym, and I go to a gay gym that has some bodybuilders—had for years. If you're picking up guys from a pool of bodybuilders, the chances that they're going to be bigger than you just go up. I think the fact that Mason was so unlike every other guy I'd dated was part of the appeal. We were the same, but different. Complementary. We both loved to joke around, we both were childish, but we both took the business seriously. If one of us got too silly, the other would bring him back to Earth, and if one of us got too serious, the other would send him into a fit of laughter. He dressed loudly and garishly, but had an unshakably professional demeanor. I dressed plainly and pragmatically, but would laugh and flirt with the customers. Even just how we styled our hair complemented—the hair on his head was free-flowing and wild with a devil-may-care flair, but his facial hair was strictly groomed with no follicle out of place and his mustache tightly curled at the ends. I kept my dark black hair in a youthful cut with the sides buzzed, kept in place with enough product so it didn't come loose as I worked. I tried to keep myself clean shaven (even though I love facial hair) because it made me look younger than my 30 years, but I often forget to shave, meaning I had lazy (and, if I may say so, sexy) scruff more often than not. In my eyes, Mason was perfect. Before we had even exchanged hellos, I knew I wanted him., on some deep, profound, mystical level, like my ancestors were calling out to me, saying, "That one!" I didn't even care that I technically had a boyfriend when I met Mason. I just knew I wanted Mason more than any man I'd ever wanted. Two years later, and the bakery and our relationship were thriving. A year ago, Mason had moved in with me, into the apartment above the bakery. All of the businesses on this street have residences above them. We live in a small city just outside New York, a tourist town really. It's big enough to have the amenities of a typical city, like a professional gym and a thriving arts scene, but small enough to have local charm, like a street full of people living in the apartments above their mom and pop businesses. Since Mason had moved in, we shared pretty much everything. Pretty much. Almost everything. I still had that one big secret I kept from him, but the secret was protecting him, not hurting him. Sometimes I felt guilty that I was holding back a big part of my life from him, but I'd never shared that part of my life with any of my boyfriends. Mason was different, though. I knew the day would come when I'd let him in on it. Just not yet. Mason had proposed to me the day before. We were too busy fucking each other's brains out the night prior to share any life-altering secrets. As soon as I had an opening, I'd tell him. If I was ready to marry the man, I was ready to share everything everything with him. This day, Mason was dressed unusually. I was used to him dressed like an overgrown toddler: t-shirts with cartoon characters, brightly colored socks, and blue jeans. It wasn't uncommon for him to wear crocs or suspenders. Sometimes both in the same outfit. This outfit, though… He was in a very sharp suit: a silvery trim fit number with an ice blue shirt (to match my eyes) and a bold purple tie. "Have a nice day," I told the customer, but my gaze never left Mason. My smile grew larger as he nervously came over to the counter, practically stuttering and stopping with each step. Odd again. I was used to Mason being smooth, charming, and confident—verging on cocky. Today, though, he was trying to hold back his anxiety. Thankfully, he relaxed a little when he saw my smile. "Someone's dressed up," I said, eyeing him up and down. "You're cutting the line," a customer complained to Mason. I realized I'd been neglecting my customers, turned to them, and said, "He's cutting nothing, ma' am. He works here. He's my boyfriend." "Fiancé," Mason corrected. "Fiancé. Right." It had been about twelve hours since Mason had popped the question; I was still adjusting. The only person I'd told was my Gramps. I smiled at my customer. "Just a moment, please." I turned back to Mason. "Where were we?" Mason leaned across the counter and kissed me. "You like?" he asked, already clearly knowing I loved it. I loved him in everything. Or nothing. The same customer, an irksome woman in a bright pink sweater, cleared her throat impatiently. "Kayla, get off prep and help the customers for a little bit," I said, taking the kerchief I used as a hairnet off my head. "I'm going to the office with Mason." "Got it, Roy," Kayla said as she swooped in to help the customers. I ducked under the opening in the counter (rather than opening it) and followed Mason to our office. Once the door was closed behind us, I turned around to undo my drawstring. "Again?" He asked, stopping my hands from untying my pants. "Three times last night wasn't enough?" He smirked with sexual tension as he walked over to the other side of the office, putting space between us. Our sexual dynamic was part of what I loved about him. He preferred to bottom, I preferred to top. I was usually the aggressor, and he usually gave the green light. But he said no just enough to keep me simmering like a teapot, and initiated just often enough to keep me on my toes. I couldn't help that I was blessed with a high libido and an 8-inch dick. Mason himself had a lovely 6 inches and had, in my experience, a higher than average sex drive. It just was a little lower than mine. Since sex was a no-go, I'd settle for some above the clothes fun. I slinked towards him, ready to practically leap on him—a dangerous prospect as I easily outweighed him by more than 50 pounds, all of it muscle—but he stopped me again. "You're covered in flour and god knows what else, and this is a new suit," he warned. Then, realizing he wanted his lips pressed against mine, he added, "Proceed with caution." I walked over to him slowly and kissed him tenderly. I wanted to throw my firm, large arms around him and cradle him against my proud chest, but I restrained myself. Mason moaned in delight. As the kiss neared its end, I slapped him firmly but playfully on the ass, and a cloud of particles erupted from the impact. I grinned widely, but he spun around trying to look at his own ass, his recent anxiety threatening to surface again. "Damn it, Roy," he said, laughing. "Did it leave a mark?" He looked at me the way one would look at a cute puppy that had peed on the rug. There was a white outline of my meaty hand on his tiny ass, and I said, "Looks perfect to me." "Really?" he asked. "I'm meeting your grandfather in an hour, and I want to look my best." I kissed him again quickly and then said, "Gramps will love you because I love you." Mason tensed a little. "I wish I were so certain, Roy. The man raised you all by himself. You call him every day. It's weird, but sweet." "It's not weird," I said, a little defensively. I knew it was weird, though. Everything about Gramps was weird. But, like me, he kept all of the true weirdness a deep secret. All of our weirdness. Mason implying that the nurturing, caring, intense bond between Gramps and me was weird… Well, if he knew Gramps as well as I did, those qualities wouldn't even make his list. "It's kinda weird," Mason said. "You blow him kisses goodbye over the phone." "He's my Gramps," I replied, unsure how that was weird. Reassuringly, he added, "It's more sweet than weird. If you weren't a little weird, I wouldn't be drawn to you. Weird but sweet is my favorite quality in a man." I blushed a little. Mason continued, "Your grandfather is your entire family. You've told me as much, and I listen." Mason began lecturing me to prove how much he listened, counting off important points on his fingers. "You have no brothers or sisters, no aunts or uncles. Your father was out of the picture before you were born, your mother died of kidney disease before your first birthday, and your grandmother died of a heart attack when you were three." He stopped counting and threw his hands in the air. "Honey, I'm not just meeting your grandfather. I'm meeting your entire family at once. I'm allowed to be a little nervous." With a kindness in my tone, I said, "I'm just not used to seeing this color on you." "He's never shown the slightest interest in meeting me, and we've invited him down eight or nine times this past year. I mean, you've visited my folks five times already, and I've never even spoken with your grandfather over the phone. Then, you tell him you're getting married, and he drives seven hours the very next day because, all of a sudden, he has to meet me? That bodes poorly. I should've met with him before I proposed to you." I threw a confused look at him. "To what? Get his permission? Barter for my dowry?" "Don't mock me." "I'm not. Gramps was excited when I told him we're getting married. He's about to take that two-month cruise for his 81st birthday." That was a lie. There was no cruise, but Gramps would be busy for two months, and it would take too much explaining to tell the truth. "That's the only reason he wants to meet you so fast. Relax." "You could take your own advice," Mason chided. "You've been an over-eager little boy since he said he was visiting." "Over-eager?" "I know every inch of your body." Mason prodded my chest. "Your pecs are pumped, mister. You went to the gym this morning. Think I wouldn't notice?" "I go to the gym five days a week." "Yeah," Mason said. "In the afternoons. Mornings are busy for bakers, you leave your evenings free for me, and you go to bed every night at 8. So, all you have left are afternoons. You wanted to look your biggest to impress him, and you wanted to have your whole afternoon free to play with him." I was guilty as charged. Mason added, "Even though he said he was only coming for lunch, you cleared your whole afternoon." "What's your point?" I asked. Matter-of-factly, as if delivering the verdict of a trial, he said, "This is a big deal to you too. Therefore, it has to go perfectly." "It will go perfectly because you're perfect. All's good with Gramps." Mason paced a little. "That's another thing. What do I call him?" "Whatever you want. Gramps is chill." "You've only ever called him Gramps. Is that what I should call him? Or do I call him sir? Or Mr. Whitaker?" "Why would you call him that?" I asked with a slight laugh. "My last name is Whitaker. Not his. His last name is Morrow." Mason stopped pacing. "Should I call him Mr. Morrow, then? What's his first name?" "Lucian," a grizzled voice came from the office door as it opened. "My name is Lucian Morrow."
muscle-growth
worship Black Cats
Ripped posted a topic in Stories
Black Cats Sequel to Black Cat. Chapter 1 | Chapter 2 | Chapter 3 | Chapter 4 Chapter 1 Early Tuesday morning I was awoken by the feeling of a pair of firm tits straddling my left thigh, a hard dick against my right foot and tongues licking my ironwood hard morning erection and pomegranate-sized balls. As I laid on my back with my eyes closed and soaked up the worship of the sister and brother I thought about all those fantastic things that happened to me in the past week. This whole adventure started last Monday when I found a little black cat in an alley downtown. When I picked up, cradled the purring furball in my arms and saw the beautiful electric blue eyes lovingly looking up at me I knew that I would do anything to keep her safe. I took the cat to a downtown veterinarian school for a check up and adopted Brenna, that's what the vet named her, as soon as her exam was finished. After my long term cheat'n bitch of a girlfriend walked out on me later in the week I found out that Brenna was actually a kind of therianthrope; a being that can shapeshift into any form she chose. I uttered a moan from deep in my chest and put my big hands behind my head as I basked in the sensations emanating from my monster cock. I opened my eyes and looked down. The first thing I saw were my magnificent pecs. I had to lift my head to look over my overpowered hyper-masculine chest to see my ten pack abs and shredded obliques. I flexed and squeezed every muscle and watched as my pecs turned into hemispherical boulders with a deep vertical crevasse and each segment of my abs rose to stretch the thin skin and thick veins covering them. The two bodies covering my legs were raised by my massive quads and hamstrings. "Oh, fuck Bruce," Brenna moaned. She kept her cat eyes and ears while taking the shape of an ultra-muscled sexy human female. "mmmm, I love it when you do that." I felt her pussy against my shin leak a copious amount of cunt juice and drank in the rush of lust she emanated. You see, I wasn't always the pinnacle of masculinity. Just a few days ago I was a 26 year old, short, obese, balding 9-to-5 office drone at an accounting firm in downtown Madison, right next to the capitol. The most exercise I ever got was doing mental gymnastics trying to figure out why a client thought it would be a good idea to try to claim their new Lamborghini as a business expense when they owned a small deli that barely broke even in West Towne Mall. That all changed Friday night. Brenna did something to me that allowed me to capture and store emotional energy directed at me. I can use this energy to change myself; increase height, gain muscle mass, grow stronger and slightly change my appearance. Hell, I can even make my balls large enough to produce gallons of cum and my cock into a telephone pole. To reward Brenna for the delicious pre-breakfast emotion drink I added an inch of length and girth to my cock. She responded appropriately by loudly purring, unhinging her jaw and greedily taking my glans deep down her throat. Not only had Bre changed me physically, she also tweaked my psyche, amplifying my Id. This change allowed me to become the dominant personality I always yearned to be. Then early Saturday morning, she sent me off to Madtown Fitness, a gym on the south side by the beltline, to test my new body. There I broke my limits, both physically and mentally. I exceeded any physical expectation I had as I crushed all house lift records. Took any throat, pussy or ass I wanted. Exerted my dominance over everything and everyone. With one very painful exception, it was a fucking amazing day. "Oh, Gods, Daaaddy. That's hot. That's sooo fucking BIG." Arthur, Brenna's brother, whimpered into my scrotum between licks with a long, wide, soft dog tongue. Bre's lips curled into a smile around my cock as I chugged his submission and rolled my eyes. I remembered how I met Art when I came home from the gym late Saturday night. Bre said he was an ex-boy friend that was in need of punishment. So I, now an uber dominant personality, made my physique explode. I must have grown well over seven feet tall and weighed over six hundred pounds, all muscle. I would have looked like a caricature of an over-the-top muscle bound freak if I didn't have an extremely dangerous looking, three foot long, ten inch thick, rock hard cock sticking straight up from a low hanging scrotum containing testicles the size of a fastpitch softball. I ravaged Art's throat and ass. I beat him to within an inch of death without realizing it. To save him I had to blow the whale sized load of cum. Consequently, Art says that now more than half of his DNA is from me, ergo, I'm his father. Yeah. That's what I said, too. 'Get our lazy ass up! Remember, I got a text last night from Keena saying we have a physical examination and lab work that has to be done before eight o'clock this morning or you can't start work tomorrow,' I heard in my head. That was another change. While meeting with Dr. Keena Panthera, the gym's owner, Saturday the resident muscle head, Terry "Zapp Brannigan" Limppernoodle, attacked her. After I subdued the bastard she told me that he had sexually assaulted her, raped her. Even though I just met Keena that day, I felt an overwhelming instinct to protect her. Learning that she had been violated threw me into an uncontrollable rage. That rage ended with me on a shower floor and in my own vomit. I had snapped both of Terry's femurs like twigs, ripped his arms off, crushed his head between my hands like it was a paper cup and ripped his genitals off. I tossed the bloody dick and balls at Keena's feet like a trophy. I don't know why I did that. Maybe I subconsciously wanted to indelibly stamp my dominance and superiority on her. My mom and uncle fought the last war. When my uncle returned he told me of fellow soldiers that were changed after their first kill, especially if it happened in hand-to-hand combat. I guess I changed that way, too. A part of me recoiled from the horror of what I did, splitting my mind into two equal and nearly opposite halves. The voice I just mentally heard calls himself Ego. He has an amplified intellect and is a fucking pain in the ass. He also can't keep his pronouns straight. 'I goddamn fucking asshole,' Ego bitched. 'I quit your job to work at the gym so he could fuck around all day long. Get my lazy ass out of bed!' I reminisced about Sunday and Monday as I try to get my legs out from under my worshipers. Saturday night my BFF with benefits, Deb, the maniacal Uber driver, aimed me at a bored wife of a supplements company executive. On Sunday I met the wife and the couple's maid in their mansion on Lake Mendota's north shore. I found that not only did the husband have impeccable taste in cars and women, he also was having an affair with, and beat, the petite French maid. The next day, after proving my superiority in every way, impregnating the wife and making him a submissive cuck, I ripped away from him the things he loved the most (in reverse order); his wife, his mistress, his company and his W Motors Lykan HyperSport. Let me say right now that the irony of me owning a car named after a human that can change into a wolf is not lost on me. Art lifted his head from my balls. I was greeted with a big, panting, corgi face. Arthur's ears wiggled as he said, "Are you gonna fuck me in the shower, Daddy? I've been a veeery bad doggy." Art's dog breath hit my nose like a slice of lemon wrapped around a large gold brick, "Geez, Art! Your breath stinks! What the hell, have you been licking your ass again?" Art tilted his head to the side the way dogs do when you try to explain the symmetrical beauty of double entry accounting. "No," he replied. "I've been licking yours." Brenna chuckled while I moaned and gruffly commanded, "Get off me and brush your fucking teeth!" He whimpered and gave me puppy-dog eyes, but obeyed. He climbed off me and made his way to the bathroom with his tail between his legs. Literally. Bre pulled my cock out of her throat. The head popped out of her mouth. "Oh, good. You're all mine now," Bre murmured just before taking my entire length, down to the root, in her throat. Her mighty neck muscles started to ripple against my unyielding god-rod. I wanted nothing more than her servicing me for the rest of the day, but, as Ego said, I had things to do. "YOU need to make coffee and breakfast while I take a shower," I declared. Her brow furrowed, eyes pleaded as she shook her head as best she could with a huge titanium rod stretching her esophagus and apple-sized glans in her stomach. I gripped the hair at the back of her head and started to pull her off me. "That wasn't a request, cat." At that she tightened her throat around my cock. Her neck muscles bulged, her traps thickened and rose up until they looked like steel I-beams and her back until it looked like an impenetrable mountain range as she fought against me. I smiled, took a drop of the emotional power I kept in reserve (stored somewhere off the Cayman Islands) and slowly pulled her off, easily overpowering her. When my cock head popped out of her mouth and smacked my chest she breathlessly whined, "Pleeeease, Master. I need you. I need your cock!" "Food now, play later," I said as I rotated my shoulder and brought the 350 pound female bodybuilder, who could easily overpower a middle-weight state bodybuilding champion, to her back like she was a rag doll. While I still held her hair I propped myself up on an elbow and captured her jaw in my free hand. "Here's a little taste for now," I said. Then released her jaw, moved my hand slowly over her magnificent, firm breasts. I tweaked a thick nipple between my index and middle fingers, then stroked her brick wall abs and, finally, burried a finger in her sex. Her eyes rolled back as she moaned and gasped for air when I flicked her clit. She was writhing and on the edge of a powerful orgasm. Then I stopped. She hissed and punched my shoulder with enough power to blast a hole in a foot thick solid steel wall. My deltoid only dented a little bit. "Now get going and I might buy you a toy mouse later," I teased and released her hair. She rolled off the bed and massaged the hand that punched me. With a look of lust, fury, resignation and pointing a finger at me she said, "That better be one hell of a mouse, buddy." She turned and picked up the button down shirt I wore yesterday. As she shrugged the shirt on I noticed that her back almost filled up the garment that could barely contain me. She lifted the front of the shirt and took a deep breath as she turned back to me. Her breasts pushed the shirt open, revealing her large, firm breasts, deep cleavage and ruggedly segmented abs. Her nipples grew hard, denting the fabric, her lower abs started to powerfully flex, her expression like a predator ready to strike its prey. "Fuck," she moaned. "I can still smell you on this." I growled and jabbed a finger at the bedroom door. She scowled and cat-walked, her glutes flexing and rolling against each other, out of my bedroom as her face assumed a more human visage. I laid for a few seconds more, then rolled out of bed. I yawned, raised my arms above my head and stretched to work out all the kinks in my sleepy mountains of muscle. I heard my back crack (twice), my sternum pop and heard Art still brushing his teeth. The urge to urinate was strong, so I willed my raging cock to settle down. It deflated as I moved towards the bathroom and immediately stubbed my foot on the corner of the bed. Cursing and promising for the 1,001st time to rearrange the furniture, I limped to the bathroom. Art was getting ready to spit into the sink when I moved up behind him. I studied the contrast of body types I saw in the mirror. Art morphed from a human/corgi mix into a complete human with the appearance and physique of an average height twenty year old blond, blue-eyed, olympic swimmer. I could see the well toned muscles roll and flex as he bent over the sink to spit as he shifted his weight between his strong legs. When he stood up the top of his head reached the bottom of my chin. His adequately developed chest had wisps of blond hair above a decent six pack. I applied a little emotional power to pack on tens of pounds of muscle and increase my height until his head was below my chest. I proportionally grew the rest of my body, except for my cock. Then I lengthed and thickened it too until I had a baseball bat swinging between my knees. After that I took a half step forward, pinning his narrow pelvis between me and the vanity. I thrust my chest forward, brought my arms up into a double bi pose and flexed. The bathroom vanity lights glinted off my hairless scalp and chest. A weekend's growth of thick whiskers gave my face a rugged appearance. My traps, delts and pecs grew to insane proportions. My biceps, already big, grew to dwarf Art's head. I gritted my teeth, growled and squeezed harder. The already apparent striations deepened into crevasses as the superficial veins that criss-crossed my body like a road map tried to burst through my skin. The two heads of each bicep grew and separated, causing a vein filled valley to form from my deltoids to my pronator tres and brachioradialis. Art's eyes grew wide, his mouth slack and he moaned as he leaned back into me. I think I even saw a little bit of saliva escape the corner of his mouth. He started leaking below too. Next to me, Schwarzenegger and Colman look like they don't even lift. I AM the Greatest Of All Time. "Oh, fuuuuck, Daddy," Art moaned as he flexed his ass against my cock. His strong glute muscles gave me more than enough friction stimulation to bring me to erection. "Do me right here. I don't care if you rip me in half. Please, just fuck me." I broke the pose and palmed Art's head, my hand nearly encompassed his skull, and slammed the side of his head against the mirror. A network of spider web cracks appeared under him in the mirror. Even though he whimpered from the pain, I could feel the overwhelming lust and pleasure pulsing off him. I soaked up every drop of that, sending it to my reserves as I caused my cock, which was now the size of his calf, to become harder and cleave Art's glutes, lifting him off his feet. I leaned in and gave him a predatory snarl, hot breath against his ear and I said, "If I did that, I'd have to blow an even bigger load than last time to bring you back. Then Bre would pound us both to pulp. But you'd like that, wouldn't you?" A weak, "uh-huh" escaped his lips as his hips started bucking against me and the vanity. He was about to cum from the mere suggestion of so much agony. I gave him a second or two of that mental image before I released his head, stood straight and stepped back. He barely caught himself before he fell to his knees. "Start my shower," I ordered. Then I moved to stand over the toilet and relieved myself. It took a moment for Art to realize I issued an order. When he pulled his head away from the mirror I saw that he left a few drops of blood and a sweaty impression of an ear and cheek on the fractured mirror. He gazed upon my infinitely superior physique with a mixture of anger, lust and anticipation and whined, "You're a fucking tease, you know that?" and punched my back as he passed behind me and into the shower. "Owww! Fuck!" he said and turned the handle. I heard the water cascade out of the shower head and Art yelling, "GODS DAMN this water is COLD!" As I relieved myself I happened to glance at my toothbrush in its holder beside the sink. "Art, did you use my toothbrush?" "Yeah, I hope you don't mind," he said. He stuck his head out of the shower and continued, "Dog saliva is very hypoallergenic," and licked the inside of one of his nostrils with a long dog tongue. I grumbled, flicked off the last drop of urine into the toilet and flushed. I leaned against the wall next to the shower and counted my fingers, waiting for the water to become scalding hot. As usual, right on the count of five Art screams, "JESUS FUCKING CHRIST!" and bolted out of the shower. I grabbed his neck before he got too far and threw him back into the shower. The house shuddered as he was driven into the shower wall. I ducked my head under the bar that holds the shower doors as I stepped into the stall. Then I turned the shower handle a quarter turn towards cold and slid the door closed behind me. Arthur looked at me with wide eyes and a slack jaw as my bulk almost filled the stall. I could feel the lust, pain and want cascading off of him. The delicious treat made me involuntarily flex my pecs against him, pushing him further into the wall. As the tile, drywall and wall frame cracked under him he said in a gasp, "Could you please not take up the entire shower?" I was disappointed he phrased it that way. If he had said, "Make yourself smaller," I would have given myself a hundred pounds of muscle, flexed everything and pushed him through the wall so hard he would have exploded into the next room. Instead, I compressed my height until the top of his head was just below my chin, just enough height that he'd have to crane his neck to look me in the eyes. I did that without losing any muscle; I looked even more fucking massive. Now that I could maneuver in the shower I grabbed Art by the neck, ripped him out of the wall and forcefully set his feet on the floor of the shower. "You will wash me. If you do a good enough job I might fuck you into next week," I growled and shoved the bar of soap from the shower caddy that hung from the showerhead into his hands. I then bent my left arm and flexed the bicep. Razor cut striations jutted across the twin peaks of the muscle. I straightened my arm and flexed again, this time including the forearm in the flex. The already insanely huge muscles grew larger and harder as even the smallest superficial vein doubled in size and pulsed with unbridled power. Art, with eyes wide, started to soap my upper left arm. "NO!" I barked. He flinched as if I had flicked a finger against his abs, expelling all air from his lungs and launching him across the room. He looked up timidly into my eyes. "Worship my might, worship the physique you dream about, worship the only true Alpha. Lick the sweat off me. Clean me with your tiny tongue. Then use the soap." He instantly dropped the soap and started tonguing, kissing and caressing my massive arm. I gulped down each delicious drop his submission and want. He emanated so much I started to get drunk on the power I was soaking up. When I sent most of it to my reserves my head cleared and I felt Art sucking on my anterior delt head. "Grrrrrr! That's it, little man. Show me how inferior you are," I said menacingly, then grabbed the back of his head with my right hand and dragged his mouth to the downward pointing left nipple. "You love how I can, how I HAVE, ripped you apart without even breaking a sweat, don't you?" I didn't need him to say anything; I already knew the answer. He fucking loved it. I brought my left arm down, put his head in the crook of the elbow and flexed the left side. His mouth yielded to my incredibly hard pec as my bicep and forearm's titanic brachioradialis and pronator teres bore into his skull. I kept increasing the pressure until I heard cracks and Art whimpering in pain. Then I released him. He fell on his hands and knees to the floor. The shower head sprayed water against the back of my head. What water didn't flow down my back was cascading off my pecs like Niagara Falls and pummeled his back and head. I clenched my hands into fists and slightly bent my arms while expanding my chest, flexing my arms, pecs and abs. I squeezed the muscles of my legs until they resembled the marble columns of a Greek temple. I looked down my nose at the pathetic excuse for a male at my feet. I am Colossus standing before a teenager. A teenager with an Olympic swimmer muscular body. When Art finally looked up his eyes became as big as saucers. The head of my semi-hard summer sausage sized cock was an inch from his face. When he reached up to my cock I grabbed his hands "Please," he pleaded. "I want...NEED your cock in me!" I could see his six inch dicklet was as hard as it could get over his inadequate balls. I put his trembling hands on my cinder block abs. "I will NOT repeat myself," I said in a low menacing voice. He almost cried in frustration as he slid his hands over my abs, down my adonis belt, over each bulging muscle group in my left quad's rectus femoris, vastus lateralis and vastus medialis, then onto the gastrocnemius and peroneus longus, the two halves of my diamond shaped left calf. From there he licked two days worth of sweat and grime off every inch of me. I flexed and squeezed the muscle under Art's mouth just to remind him how immensely powerful I am. After he tongue bathed both arms and armpits he placed his hands on my mountainous trapezius and stood on his toes to reach my neck. The sensation of his tiny body against my solid granite muscles started to arouse me. My rapidly hardening cock rising against his pintsized six pack made me groan. Thankfully Art finished his makeout session with my traps and neck quickly. I turned around, flared the muscles of my back; trapezius, latissimus dorsi, rhomboid major and teres, major and minor. Calmly as I could, commanded, "Now my back. Start at the bottom." I heard him drop to his knees and felt his hands on my hips as he buried his nose and tongue between unbreakable globes of my gluteus maximus. New sensations shot up my spine, ricocheted around my brain, back down my spine, bounced off my testicles, went up my cock and exploded in the glans. I involuntarily threw back my head and grunted as a literal pint of pre splashed against the shower wall. I saw two of the tiles crack after being hit with such pressure. Art must have noticed because he was taking his own sweet fucking time licking off any piece of shit I missed. 'Shit,' I thought. 'I'm going to have to make sure he brushes his teeth again.' 'Can't me think of any better puns than that?' Ego snidely interjected. 'Tell him to fucking hurry it up.' "Boy, if you keep that up there won't be any cum left for your ass. Move up." "Yes, Daddy," Art disappointedly replied. He licked, sucked and kissed up my wide, thick back until he swept the last speck of sweat off my trapezius at the back of my neck. When I felt his pathetic little wee-wee against me I spun around to face him, grabbed his neck and lifted him off his feet. He must have weighed 200 pounds, but it felt like I was lifting a paper marionette. "Are you ready to get FUCKED, boy?!" I growled from deep in my chest. He put his delicate hands on my pectoralis major, caressing them. "Yes, Daddy. Fuck your little boy." His hands slid downward, over my ten pack abs. "Show me how an Alpha takes what He wants…" He started spreading the pre flowing from the meatus over the shaft, hand over hand, with another eight inches of cock exposed. His hands couldn't even encircle half my girth. "...whenever He wants." He then slid his right hand up and thumbed the sensitive spot on the shaft. The sensation made my pelvic floor and prostate clench, forcing a heavy blast of pre to splash against his abs and chest. Arthur scooped the emissions off with his right hand while still stroking my rod from base to tip with his left. He brought his cupped hand, completely covered and overflowing with sticky precum, to his mouth and drank. As he drained his hand of the clear, thick liquid, I felt his throat contract four times as he swallowed the goo. "And now I know," he said with a quiet voice filled with lust and licentious inference, "what the mightiest Alpha on the planet and the last descendant tastes like." 'The WHAT?!' Ego said loudly. 'SHUT. THE FUCK. UP.' I internally growled as I flicked my wrist, tossing and twisting this average height, muscular boy in the air. I caught the back of his head when it was facing me and drove it into the tile on the shower wall. Blood, bone and broken tile were blasted out from the crater I created in the wall. Art's legs were instantly swept aside as my fully engorged cock rammed up into my totally helpless prey. With my cock supporting all his weight I released his head and put him into a Full Nelson hold, both of my hands were thrust under his arms, then I pressed against the back of his neck. As I pulled him away from the wall I saw blood splashing the destroyed tile; I must have crushed his nose. When I thrust up his weak hole he hooked his ankles around my dorsal calves and started to caress the rock-solid muscle. Now that I had him completely under my control I teased the fuck out of him by slowly and repeatedly only penetrating him with the glans. Still, he grunted from each breach of his rectum. "Fuck me...come on!", he pleaded. "Fuck...grunt..me..aieee!...HARD!" I drew a devilishly evil grin as I did exactly not that. For minutes I teased this boy cunt as he begged louder and louder until he was screaming at the top of his lungs. "FUUUCK MEE! HURT MEE! GODS DAMMIT PUNISH ME!!" Suddenly rage blasted off of him, he unhooked his feet from my legs and planted them on the shower wall. Huge claws exploded out of his toes and heals, fracturing the tile as they were buried deep in the wall. Each of his legs grew thicker than my waist with impossibly huge thighs. In an inhuman voice he bellowed, "I SAID HURT ME!" I could feel his rectum constrict around my cock tighter than any hole I've ever had, squeezing down and painfully compressing my shaft. "FUCK ME HARD!" Then he thrust back. I was slammed into the wall under the shower head. I felt the tile behind me fracture as his tight hole forced its way down my cock. I heard tearing sounds and saw his ass rupture. His copious blood provided a little bit of lubrication, but not enough; it felt like the skin on my shaft was being rubbed off. He kept screaming in a sound that shouldn't come from any living thing, "PUNISH ME!". When his ass finally impacted my root, faster than I could see, he pulled off and rammed back down. Between his battering assaults I saw my cock distend and almost rupture his abdomen several times. If he kept this up any longer he was going to fuck himself to death. When he was within reach I hooked my right elbow around his neck and clasped hands. "Arthur! You have to stop this!" I yelled. He bucked like a bronco, I almost lost my grip twice before I was able to squeeze and restrict the blood flow in his carotid artery. Just before he passed out he entire body tensed up, his rectum almost ripped my cock off. A blast of cum blew tile off the shower wall in front of us. Then he slowly went limp and I lowered him to the floor. When my cock popped out of him, blood geysered out of him and down the shower drain. I stood up and looked down at him. His face was a mass of shredded skin, broken bone and blood, yet I could see the devastation on his face already healing. My cock looked no better. It was red, swollen, covered with his blood and shit and hurt like hell. As I rinsed off I applied some power to heal myself. By the time I finished Arthur was whimpering in his sleep. 'What the hell just happened?' I asked Ego. 'You have no idea. I better get him in bed. Let him sleep,' he replied. I picked up Arthur off the shower floor and cradled him in my arms. As I took him to the bed he softly cried, "I'm sorry...I tried…not my fault...forgive..." He immediately went into a fetal position when I laid him down on the bed. He let out a sigh when I brought the comforter over him as, somehow, a large brown teddy bear appeared in his arms. I went back into the bathroom and found a fresh toothbrush under the sink. After unpacking the toothbrush I gazed at my stubble covered cheeks and jaw in the mirror. It had been three days since I last shaved. I rubbed a hand along my jawline; I kinda liked the way this protobeard made me look. I decided to keep it. As I brushed my teeth, Ego said, 'I don't like the way he seeks pain. There's something terribly wrong with him. Why does he feel he needs to be punished?' 'I don't know,' I mentally replied. 'Whatever it was, it couldn't be healthy for him to feel that way.' "Are you two almost done?" I heard Bre yell from the kitchen. "You had the hell better not let all this food go to waste or I'll barf a hairball in your shoes!" And I know she would. I exited the bathroom, expanded myself to the height I was earlier, packed on twice the muscle and walked towards the bedroom door leading to the hallway, looking at Arthur calmly lying on the mattress as I passed the bed. The comforter roiled and bulged in ways a human could never cause. I looked up just in time to bash my forehead on the top of the doorframe. 'Smooth move, dude,' I heard mockingly in my head. "Will you FUCK OFF!" I yelled aloud. "You had the HELL better not be talking to me, buster!" I heard from the kitchen. I grumbled, ducked my head and started down the hallway to the sun drenched kitchen. Rubbing my forehead I said, "I'm not. I just bashed my forehead on the door frame." When I entered the kitchen Brenna was right at my side, guiding me to a chair at the kitchen table. As I sat (and the chair creaked under my weight) my mouth started watering from sight and scent of the food overflowing the table; eight three-egg omelettes (four cheese, mushroom & bacon and four veggie), a stack of twenty pancakes smothered in real maple syrup topped with real butter, two pounds of bacon, three pounds of venison sausage (from my uncle), two loaves of whole grain and seed bread and, the pièce de résistance, a two full pound of cheese curds. I love those big chunks of cheese, especially how they squeak when you bite into them. Hey! I'm a Wisconsinite, don't judge me! Yes, I even have a Cheesehead Cheese Hat. My hungry eyes were torn away from the feast before me by a pair of ultra strong female hands on my cheeks. "Let me see," Bre said with concern and straddled my lap, the top of her head below my chin. The taste of that emotion was like taking a bite of an orange flavored strawberry. Interesting, but not as interesting as looking down her cleavage, deeply segmented abs and her little exposed clit. "Well, you're a little red up there," she motheringly said, "but I don't see any permanent damage." She brushed my forehead with her hand and pieces of wood and gypsum board fell into my line of sight. "You probably hurt the wall more than it hurt you." "Speaking of hurting, did you hear Arthur and me in the shower." She tilted her head down and her sad eyes slowly moved from my forehead to my abs. She put her hands lightly on my trapezius, feelings of worry and doubt made bile rise in my throat. "Something long ago…" she snapped her eyes to mine without lifting her chin "...happened to him. He won't talk about it, no matter how much I ask. I...I just try to emotionally support him...love him the best I can." Tears started to fall from her eyes. "I just don't know what to do." I wrapped my arms around and clutched her to my chest. She began to silently sob. 'Well, shit. There goes the mood,' Ego lamented. I silently agreed with him and slowly made myself decrease in size and mass. When Bre's chin touches my collar bone, her puffy, tear streaked eyes meet mine and she sniffed. "Wha...what are you doing?" "I thought that you wouldn't be in the mood since…" I said quietly, gently. "Oh, no you don't you FUCKER!" she interrupted me, jabbed a hard finger into my sternum and exclaimed with all the ferocity of an IRS conducting a multi-year, twenty million dollar audit. "You fucking TEASED ME by making me suck you off ALL FUCKING NIGHT! I had to listen as you got tongue bathed and kissed all over your perfect body! Do you know how hard it was NOT to plunge my fingers up my cunt when I heard that?! Oh, you're gonna fuck me, pal, and your gonna fuck me HARD! YOU WILL BE as big, as thick and as hard as you were when you walked in and YOU WILL pound me into submission! You're gonna prove to me that you're still my Master!" My cock rose against her ass in anticipation of feeling her hot, wet, tight hole and her admission of subservience. Her face then slowly went into the most evil grin I've ever seen and said, "But first you're eating all this food." I groaned and complained, "And I'm called a tease." "Yup!' Bre laughed as she rose off my lap to stand beside me. She stood up on her toes, leaned forward, patted the top of my head and said, "I'm the queen Molly of teasing, baby. Now EAT!" My stomach loudly growled as I picked up a fork and spoon and dug into the mountain of protein, carbs and fiber. I kept glancing at Brenna between shoving huge chunks of food in my mouth. She was still in my unbuttoned dress shirt, leaning her perfect, hard gluteus maximus against the kitchen counter and holding one of my biggest ceramic coffee mugs. It's the one that says, "Accountants take accounting of their accounts." Before you say ANYTHING, it was a UW graduation gift from my aunt. Every once and a while she would take a sip from the steaming liquid in the mug. A few seconds later she would sigh, tilt her head to the side and her eyelids would almost close. The liquid's scent wafting in my direction smelled like mint and chamomile. I fucking HATE herbal teas. When I was a teenager I had to help my uncle with bailing hay at the farm. After a whole day of breathing in dust and alfalfa flowers the crap running down my throat tastes exactly like herbal tea. Anyway, I finally shoveled in the last forkful of breakfast, leaned back in the chair and gave a hearty belch. "Now you have a choice, big boy." Bre was still leaning back against the kitchen counter only now she had a huge stainless steel travel mug in one hand and her tea in the other. "Coffee…" she raised the travel mug "…tea…" she raised the ceramic mug "…or me?" She shrugged her shoulders, the button down shirt fell down to her elbows and waistline revealing a body that a heavy weight bodybuilder dreams of; tight waist, tendinous inscriptions that were three inches thick, shredded serratus anterior, latissimus dorsi that imitated a cobra's hood, at least twenty-five inch upper arms, deltoids the size and hardness of bowling balls, trapezius that rose two-thirds of the way to her ears and pectorals that were hard and firm. If just those muscles were on a woman her size that woman would be considered pretty stacked. Brenna had those muscles AND incredible breasts with small brown areola and thick, inch long, nipples. Her torso supported by legs thick enough to squat a ton, yet shapely enough to give a guy whiplash as he turned his head for a second or fourth look. All this sculpted marble muscle on a five foot three inch frame made her look freaky...freaky sexy! I locked eyes with her and, deliberately, slowly stood up. As I rose from the chair I made myself grow in height until Bre's eyes were level with my navel. I flexed first my left bicep, then my right, then my cock, as I walked towards her. I made sure that each footstep made the house rattle. When my glans pressed into her impenetrable abs I raised my hand as if I was to stroke her cheek. She tilted her head in that direction expecting my touch. In the last instant, I grabbed the travel mug, downed the entire contents in two gulps, crushed the container and tossed the now useless scrap metal over my shoulder. If you remember from our first (one-sided) conversation, you know that I can't function in the mornings without a couple of cups of coffee in me. While the caffeine worked its magic on the A1 and A2A receptors I advanced on my prey. Bre held up a finger and said, "Waitaminute," and took a deep drink from her tea. Almost instantly her pupils dilated. She set the mug on the countertop behind her and reached up to put both of her muscled, veined, tiny hands on my pecs. Then she extended and dragged feline-like claws from the ends of her fingers across my taut skin. She took a step or two towards me, making my painfully hard cock rise up until it was nestled in her cleavage. "So, are you going to use this big thing to fuck me?" she seductively said. "Plunge it deep into my hot, wet cunt?" She started to slowly bounce her pectorals, making her breasts stroke my leaking cock. "I think you're just going to peter out, "she brought her elbows inward, using her upper arms to squeeze her breasts tighter, "just like the little boy you are." That did it. Now I was not just sexually aroused, but my dominance was thrown into question. Yes, I know she did that on purpose just to get me to fuck her hard. Well...it worked. I growled as I grabbed her gluteus maximus and lifted her 350 pound weight above my head like she was a feather. I easily squeezed and spread those unbreakable muscles in my hands to expose her ass and pussy. It took me a second or two to properly line up my cock with her leaking hole, then I explosively brought her down. As soon as I felt her hot, wet lips against my glans I flexed my rectus abdominis and thrust up into her. Whether by design or not, she was so virgin tight that I was only able to get the glans and an inch of shaft into her. She threw back her head and screamed so loud I'm sure the neighbors would call the cops. Her hands clenched my trapezius hard. Her claws pierced my skin, but couldn't penetrate the steel-hard muscle. I repeatedly pulled my cock out to the glans using my core muscles, then forced her to take another inch while I roared and she screamed at every fuck, "YOU THINK..." two inches in, pull out "...THAT THIS COCK...three inches in, pull out "...is a BOY'S DICK…" four inches in, pull out "THIS!" five in "IS" six in "A" seven in "MAN'S" eight in "COCK!" I rammed the remainder of my shaft into her as she started to scratch my back with those sharp claws. I still hadn't touched her cervix, she must have deepened her vagina just to tease me. I wasn't in the mood for any of that. In the next thrust I lengthen my cock until I felt the entrance to her inner core. The next time I withdrew I packed on five more inches, two more inches in girth and hardened the organ until I could punch a hole through a three inch steel plate. Then I shattered her core with a single devastating ram. She threw back her head, her eyes rolled to the back of her head and she used every last molecule of air in her lungs to scream again. At the same time I roared into her face with victory and dominance. I looked down and saw two or three inches of shaft were not buried in her. That would not do. I forced in the final inches until I'm balls deep in the most exquisite pussy on Earth. I removed my left hand from her ass, brought my arm up, bent the elbow and flexed my bicep so big it made Colman's arms look insignificant. I continued my devastation of her vigina and womb. I dropped my voice three octaves and, with each skewering thrust punctuating each word, said, "I...am...your...ALPHA...your...MASTER!" She grunted and cried out with every word. "Every...other...male...is...insignificant...next...to...ME!" She grabbed my incredibly hard, vein covered arm and threw her mouth onto my shoulder. She bit down as I felt her legs begin to spasm. "I...AM...A...YOUR…GOD!" With the last word I drop my voice into the infrasound level. That word couldn't be heard, yet windows and plates rattled, the house creaked. Brenna screamed around the muscle in her mouth and explosively squirted a gallon of girl cum on my abs and legs. Now that the natural order of things had been reestablished I lifted her off me and threw her against the kitchen counter in front of the sink and the window to the backyard. Her rectus abdominis shattered the marble countertop and bent the stainless steel sink. I waste no time in pinning her hands to the countertop at her side and ramming myself up her still quivering cunt. I took my time now, relishing each inch of her. While I slowly saw in and out of her I admired her shoulders and back. I ran a hand over her back. Her teres, major and minor, rhomboid major and infraspinatus almost rose above the larger trapezius and latissimus dorsi. Each muscle group was incredibly defined, stirated, thick and hard. She was the apex of muscular beauty. And my blood boiled with testosterone as I broke her. I cupped her chin and slowly pulled her head back. She resisted with all her considerable might, muscles flexed and pumped up with blood and superficial veins popped up under reddening skin. I could see her breasts swinging on her chest when her head met my upper abs. Her nipples were long, thick and diamond hard. I moved my hand down to her neck, slammed into her and demanded, "Who am I?" "AHHHHH!! You're Bruce Banderole!" She squeezed her eyes shut. I viciously ram up. "Wrong answer. LOOK AT ME!!" Brenna eyes fly open as she grunts from the impact, "You...you're my MMMaaasterrr!" Again, I smash through her cervix and punch the far wall of her uterus. "Try AGAIN!" "You're the...the LAST ONE!!" she screamed as another orgasm racked her body and she crushed the marble countertop under her hands. Not exactly what I was working towards. 'What the FUCK?!' Ego interjected. I ignored him and released Brenna's neck. She fell forward into the sink. Her back was covered with sweat and she was breathing like she just finished a marathon. I, however, was breathing like I was sitting in a chair and no moisture appeared on my brow. I looked out the window and saw Fred Metzer holding a hose in his hand in his backyard. His hand was at his side as he was totally engrossed with the carnal spectacle I was giving him. The hose looked like his limp two inch dick. His mischievous son had crimped the hose causing the water pressure to fall. I seized Bre's hair and lifted her head up so she could see Fred. "See that beta boy? See how limp his dick is?" She started breathing hard again. I released her hair and gave Fred the biggest fucking monster sized double bi flex he would ever see. His life changed forever; his eyes bugged out, he slouched and became the lesser male I always knew he was. Yeah, that's right, boy. You're a one, maybe two, on the male scale. I'm a one fucking hundred. "That's what every other male is next to me." I drank in the firehose of emotions Fred was directing at me; envy, submission and, wait a minute, need? That little fucker actually thinks I'd let him touch me. I gave him a little taste of what he would never have, never achieve; I bent my head down and licked a throbbing vein on a football-sized bicep. Brenna jerked and cried out when I took that power to thicken my already titanic cock, harden it until I could pulverize diamonds. Fred finally noticed that the water wasn't coming out of the hose and lifted it up to peer inside. His son chose that exact moment to release the pressure. The poor beta got a face full of water that blasted out of the hose. "And that happens every time those weakling BOYS see me fuck their women, claim their women, TAKE their women!" Her third orgasm hit her like a 10.0 Richter Scale earthquake. She pushed her hands forward, plowing through the two inch thick stone countertop as easily as you would push your hand through clay. When she comes down from the euphoria she's so out of it that her trembling legs aren't supporting her weight. The only thing keeping her upright is my cock. I take a step backwards, pulling her away from the countertop. I let her leg dangle in midair for a few seconds, then I lowered my still mind blowingly hard cock. She slid down my cock to lay like a puddle of muscle on the kitchen linoleum. I reached down, grabbed her hair, yanked her to her knees, bellowed, "I'm not done with you yet!" and rammed my cock down her throat. It took a few seconds to come to and realize I was fucking her throat, but when she does, oooh man, she took over. Her tongue lashed and whipped every inch of cock that wasn't in her throat, bobbed her head up and down my shaft so fast she became a blur. Under that assault it wasn't long before I reached and shattered my own orgasmic threshold. I arched my back, threw back my head, shoulders and arms and flexed every single muscle cell in my entire body in an effort to blast my very essence through the little slit at the tip of my cock and roared. Brenna grabbed my hips and hung on for dear life as I erupted for five minutes down her throat. She clawed at my legs as she slid down my satisfied cock. She purred when I looked down over my pecs at her. She eventually shifted herself to sit against the cabinet. She reached up and stroked my softening, yet still massive cock. "Feeling better now?" I said. Brenna, my cat, said, "Hmmmm, maybe a little," and smiled up at me. Bre then reached up and took the mug from the countertop. Somehow it had survived the destruction we wrought against the granite stone and sink. She took a sip and said, "Oooo yeah...that's better." She complained as I took the mug out of her hand, held it to my nose and sniffed. Yeah, it definitely smelled like a thirteen hour day of bailing hay, yet there was something else in it...something familiar. "What the hell is this?" I asked Bre. "It's catnip tea," Arthur said from the kitchen doorway looking only at his sister. Bre snatched the mug out of my hand and drained the remaining tea in one gulp. Art stood at five feet four inches tall and looked like a pale university freshman. His long brown hair hung over his hazel eyes and he was rail thin. The skinny jeans he wore were baggy, his dirty white shoes untied and his maroon wife beater shirt was three sizes too big. The shirt had a cartoon of Goldy Gopher, the University of Minnesota mascot, coming out of a hole in the ground giving a thumbs up. The caption, in a semicircle above the illustration, said, "GO MIGHTY GOPHERS!" "She's like you with coffee; needs it to start the day." Bre looked up at me, nodded her head and giggled. "We sometimes like to keep part of the animal when we are human. I, for instance, just like every dog, likes…" Bre perked up, raised herself on one arm while she pointed the mug at her brother. "Oh, no. DON'T YOU DARE say what I think you're going to say!" Undeterred Art continues "...to suck on a bone and lick some balls every once and awhile." I chuckled while Bre, who's probably heard every pun Art has uttered, put her head in her hand, pinched the bridge of her nose and said, "Oh, gods dammit. I was just about to have a good day." "Anyway, remember that thing that happened four days ago?" Bre immediately sobered up, turned serious, looked up to Art and put a trembling hand on my thigh. "Yeah," she said, trying to hide the concern in her voice. "I got a text. I have to do the thing. Don't wait up for me. I'll be gone for several days." I didn't like the sound of this. I turned to face him and returned myself to my default setting, six foot six inches tall, on-season championship bodybuilder with a thick, uncut seven inch flaccid penis and testicles to match. "Arthur, about what happened in the shower," I started to say. "I won't be able to contact you Bre..." he ignored and interrupted me. "...so don't worry. I'll be fine," he said, then turned towards the front door and walked away. "Whatever is troubling you..." Art stopped halfway to the door and stood still; his back tensed up. "...I know we can work it out together." Without acknowledging my offer he continued to the front door. He turned the doorknob and threw the door open. Even before the door slammed into the adjoining wall, he was gone. The door bounced off the wall, only closing half way. From the time Authur entered the kitchen to when he left, I only felt darkness from him. Bre walked up to my side. She slipped under my arm and wrapped her arm around my waist. Her free hand rested on my chest. Staring at the door I quietly said, "I'm worried about him." "I know." "He'll be beaten to a pulp if the football team sees him in that shirt." "I'm more worried about what he'd do to them." Bre then looked up, patted my chest and said, "Come on. You've got that human physical thing to do and it's getting late." She grabbed my hand, went up to the front door to close it then led me back to my bedroom where she proceeded to open my closet and rifled through my clothes. Three wardrobe changes later I'm wearing a tight red polo shirt, blue jeans and a pair of running shoes. Brenna had fastened all the buttons on the shirt. I immediately flexed my pectorals and trapezius, making the buttons burst off of me and ricochet off the walls. I looked down to see the canyon of my chest cleavage. She rolled her eyes and shoved me out the door. "Have fun, and don't worry about the cum stains everywhere. I'll lick them up!" she yelled after me and slammed the door. I turned around to get in the Lycan and saw old Ms. Chakancy, with her little white miniature poodle Killer on a leash, giving me an incredulous look. I just looked at her as I walked to the car and opened the door, gave her an award-winning smile and said, "Don't ya just love weekend long orgies?" and entered the car. "Well, I never!" she exclaimed. Ego snorted and said, 'No shit lady. You probably couldn't get laid giving blow jobs at a horny blind man's convention.' 'Now, now, let's be nice to the old bat,' I internally replied, started the car and burnt rubber rocketing out of the driveway and onto the street. 'Be nice! She calls the cops every time Debbie drives up with her stereo on!' Ahh, yes. Deb does love her Norwegian Death Metal music at maximum volume. I get the mental impression of Ego taking a deep breath and calming himself down. 'Alright. The doctor's office in Verona. Due to my morning extracurricular activities..." he emphasized the word 'activities' '…the morning rush is over so it should only take I seventeen minutes to get there. The appointment is forty-five minutes from now so me have plenty of time. And what the fuck is this 'You're the last descendent' and 'You're the one' bullshit?' I rolled my eyes and said aloud, "How the hell should I know and thanks, Siri." 'Oh, so I'm my digital assistant now, eh? In that case…' and I'm forced to listen to knock knock jokes until, three miles later, in the middle of South Midvale Boulevard, the car backfired and rapidly lost power.
alphamale
dom/sub
m/f/m
muscle growth with effort
muscle growth without effort
Author's note: if you don't care about backstory, skip to Chapter 2. Chapter 1 "Happy retirement!" Walter said, throwing confetti as I came through the living room door. I'd already torn down a hand-painted construction paper banner in the foyer saying the exact same thing. There was another brown paper banner in the hallway to the living room, which was lined in crepe paper and balloons. Here in the living room, which was festooned with multi-colored streamers and even more balloons, there was yet another banner saying the exact same thing, this one with characters from Winnie the Pooh on it, and the coffee table was scattered with a whole bunch of party favors he clearly bought at a Dollar Store. We're wealthy. Incredibly wealthy. Walter bought chintzy decorations just to get a laugh out of me. "You know I hate that word," I said through a snicker. As much as I wanted to get angry, the image of Tigger and Piglet on a retirement banner was too absurd. He knew today was going to be hard for me. I saw today more as a surrender or a bereavement than a retirement. My darling husband Walter was trying to wind me up and turn it into a joke. But I wasn't going to fall for it, instead dropping off my messenger bag in an arm chair, popping a balloon with my keys, and dropping my phone on the table, knocking some noisemakers to the floor. "I'm 30. That word makes me sound like I'm an old geezer." Walter came over and embraced me. "Use whatever word you want, Logan. This is a celebration. I'm just so happy you stopped banging your head against the wall. It was making you miserable." I held him tightly and breathed him in. He always smelled pleasantly of the sea. I don't know how, but it was true. Brine and wind and sunshine. Feeling him fill my lungs, I closed my eyes and felt myself relax for the first time all day, melting a little in his arms. Then I opened my eyes and saw Eeyore staring back at me. I pulled away from Walter a little. "If you're really sorry," I said, pointing to the banner, "you'll take that monstrosity down." "Right. On it," Walter said, running over to the sign and pulling it down with an overenthusiastic jump. I wasn't really retiring. Not really. I'd been trying to find success as an actor since my 18th birthday. I'd had some minor successes, one or two commercials, some paid community theater. Things like that. But, the closer I got to 30, the more I knew that if I didn't have a breakout success soon, I'd never have it. Now that my 30th birthday was well behind me, it felt futile to keep trying. I'd tried for twelve years with no career to show for it. My luck was giving out, so I was giving up and giving in. Walter, for his sake, had been entirely supportive of my career even though he hated how much it hurt me. He helped me get better representation, he helped me prepare for my auditions, and he paid for my acting classes. To his credit, he never just bought a production studio or bribed someone into casting me. He knew it was important to me to succeed or fail by my own merits, and while he gave me a helping hand, he never gave me a handout. And, most special of all, in my defeated moments, he reminded me that I could always just stop working, that I was choosing to endure this grind in pursuit of my dream. Reminding me it was my choice kept me grounded. A lot of acting was just being rejected over and over again. Too tall. Nose too big. Face too small. Too buff. Too blond. Not blond enough. His eyes don't match the set. Every flaw I had, real or imagined, was magnified into the whole of my existence. It was never about my talent. It was always about my looks. I spent hours staring in the mirror, grooming and preening, to make myself as presentable as possible. It was dehumanizing. But when I landed a gig, I felt seen and powerful and beautiful. When I was in front of an audience, I felt electric and vital. And not all the preening was a drag. I loved working out, and acting gave me an excuse to keep up my football physique from high school. The only downside to that was that I couldn't get too big. You get too big, you get no gig. So, acting was a head trip, but it was, as Walter lovingly reminded me, what I'd chosen. And it was my choice. When I said Walter could buy me a production studio, I meant it. Walter is Walter Jaffetz. That Walter Jaffetz. Tech billionaire Walter Jaffetz. You've probably seen me in some press coverage of him. I'm the handsome blond man on his arm in most of his publicity photos. Walter is enormously wealthy and happy to support me forever. I didn't need to work. I chose to. Most people—including every newspaper and magazine in print—think I'm some sort of trophy husband. They say it over and over again. I hate it. Most publications don't even bother asking my name. Before we got married, Walter wanted us to hyphenate our last names, but I decided to take his last name because "Logan Jaffetz" sounds like more of an actor's name than "Logan Smith." But that decision went nowhere. I'm not even "third-rate actor, Logan Jaffetz." Hell, they only refer to me if they're talking about both of us. And then, we're "Mr. and Mr. Jaffetz" or "Walter Jaffetz, and husband." Walter's tried to correct them, but still, I'm just "and husband." I swear I'm not a trophy husband. I try to take it as a compliment. As evidence that I'm movie-star-gorgeous. I am handsome, statuesque, and buff. I'm blond with green eyes. I work to keep my tan even. My stats are impressive: 6'1" and 200 pounds. I have a tight waist, broad shoulders, manicured chest hair, dazzlingly white smile, and strong chin. I really am that breathtaking. And I'm married to Walter. Physically, there's nothing outstanding about Walter—he's a plain-looking, regular Joe. He's got a very slight belly, is only 5'7", has curly reddish brown hair that he frequently leaves uncombed, and despite being 30 years old like me, he has grey at his temples, laugh lines, and the beginnings of crows feet, so he looks significantly older than he is. And, he's also a famous tech billionaire. Put all those facts together, and most people draw the wrong conclusion. I married Walter before he got rich. We were high school sweethearts. I was a popular jock; he was valedictorian. Both of us were well-known at our school. I was the openly gay star of the football team and the spring musicals all four years (even freshman year). Walter was the brilliant genius who was only doing school the normal way to make his parents happy (he probably could've graduated high school when he was 8). Yeah, like I said, he was average-looking and he forgot to comb his hair a lot, but he had the most dazzling brown eyes I'd even seen. Whenever he gave a speech (he was, of course, on student council), I would get lost in his words. I love words, and his words were beautiful. I was drawn to him even before I realized he was drawn to me. For the first two years of high school, it didn't even occur to me he might also be gay. Then, I caught him staring at me in the cafeteria. After that, I noticed he was coming to watch the football team practice even though I never saw him at even one game. He came to my practices to ogle me, and I knew it. And I wanted him to know it too. So, every time I saw him drooling over me, I'd wave. And whenever I waved at him, he'd turn bright red and hide. I found his mousiness and skittishness irresistible. And, I could tell—even through his highly unfashionable cargo pants—that he had a magnificently fuckable ass. Since he had trouble getting closer than 50 feet to me, I pursued him. I asked him out. Walter says if I hadn't made the first move, we never would've gotten together. On our first date, I took him to this little café I took all my men to (because it was right next door to a motel). I thought he would be a fun little tryst—I'd never bagged a brainiac before. But the affable little bugger is the most charismatic, animalistic, primal force I have ever encountered. He has an ineffable magnetic energy that that will not be denied. You wouldn't know it to look at him, but get in the same room as him, and its undeniable. According to Walter, he knew he only had one chance with me, so he turned the full force of his personality on me, a firehose of charm. I was in love with him before our waters were poured. I took his virginity soon afterwards (he made me wait a whole month and earn it), and, oh, was I right about his ass. Once he learned how to use his ass (once I taught him how to use his ass), he became a sex machine. He could ride me like a bucking bronco. He was a confirmed bottom—he tried topping me a few times but didn't like it. That was fine by me because I could think of nothing finer than blowing him or plowing him with my seven inches of steel. We started dating in the middle of junior year, and I proposed to him the day we graduated. I'd actually proposed to him the summer prior after a night of raucous love-making, but he said he wouldn't make up his mind until he graduated high school. As soon as we were declared graduates, I was down on one knee. This time, he said yes. It's not lost on me that I'm the only person Walter's had sex with. After he accepted my proposal but before our wedding, I gave him the opportunity to play the field a little bit. Have sex with some other men guilt and blame free—no tears and no recriminations—to make sure he wanted to commit to me. Something like a gay Rumspringa. Walter just shook his head and said, "Don't need to. You're my always." Our wedding was a modest affair in his parents' backyard. All of 18 people attended. Back then, we couldn't afford more. Walter didn't get fabulously wealthy until we'd been married for two years. He never went to college. He'd already been inventing things for years, so he didn't see the point. I never doubted him. I was convinced that if he could get over his shyness and show everyone else what he showed me on that first date, he could take over the world. Those first two years of marriage, though, were a financial struggle. We borrowed money from my parents more than once. They begrudgingly gave the financial support and hinted I should give up acting and/or find a more economically stable husband. Walter and I were in love, so we didn't care. But then, like lightning, one after the other, he invented a spate of medical devices he called "the money-makers." He got the patents in a snap, but no tech companies wanted to take a risk on a 20-year-old kid with no college degree. The first six companies he pitched his inventions to all said no. He was about to chuck it all in, acquiesce, and go to college just so people would take him seriously. Then, I gave him some advice. To me, it was just friendly advice, but according to Walter, they were words that changed his life forever. "Don't try to sell your inventions. Try to seduce the investors." Intrigued, he asked, "How would that even work?" "Just do it like you seduced me. Everyone—even me—thought I was way out of your league, but by the end of our first date, you had me wrapped around your little pinky. You picked the time and location of our next date. You even told me what to wear. And I would've done any dirty, kinky, nasty thing you could've dreamed up—in front of my grandmother even. But you wanted to wait and take things slow, so I went from fucking three different studs a week to waiting patiently for you. I was not used to guys calling the shots or saying no to me, and you kept me on simmer for a month. Bring that guy to the sales pitch." Then, I added, "And let me style your hair before you go." The very next company invested—gave twice as much as Walter was asking for—and the rest was history. I'm proud of him, naturally, but I don't fully understand his inventions. Walter's explained them to me more than once, but he's done it in excessive scientific and anatomic detail. I'm a bright guy, but I'm not Walter Jaffetz. He forgets that my understanding of biology ends at "the mitochondria is the powerhouse of the cell." What I do understand is that his machines make surgery easier and safer with less chance of infection. Hospitals all across the world use my Walter's creations. Once he became a billionaire, he even donated hospital equipment to places that couldn't afford it. Have I been saying "billionaire"? I suppose it's the correct term, but it understates the reality. Nowadays, "billionaire" doesn't do justice to his financial status. His patents churn out money faster than anyone could ever spend it. He repaid my parents every cent they lent us, with interest, and then bought them their dream home. He bought his parents a dream home. He bought the mansion we live in and both our luxury cars. He bought the café where we had our first date to make sure it never closes. I reassure him he doesn't have to buy my love, but if I have even a tiny whim, he buys it for me. Walter says he's glad I'm not materialistic because he would buy me anything I ask for. He could afford it. Hell, I could spend like the casts of all The Real Housewives for a decade, and Walter would still have money in the bank. So, when I say I chose to work, I meant it. "Since this isn't a retirement retirement," Walter said as he tore down the Winnie the Pooh banner, "What's next for Mr. Logan Jaffetz?" "A stiff drink and then sex with my husband," I answered, crashing onto the couch. "Both excellent ideas," Walter said, crashing next to me. He put his hand on my thigh and said, "But I mean life-wise. What's the next step? What's the plan? If I know my husband, you wouldn't have quit acting without a back-up plan." I honestly admitted, "I hadn't really thought about it." Walter genuinely looked shock. "You never cease to surprise me." I shrugged. "I just knew I had to quit acting." Walter tilted my head down so it rested on his shoulder, and held me in his arms. "Can I say that I'm glad you quit? Do I lose my Supportive Husband Card if I admit how thrilled I am that you quit?" I reveled in Walter's arms around me, breathing him in like an aromatherapy candle. "I knew you wanted me to quit." "Ah, but I helped you as best I could and never explicitly said anything." "True. You get credit for that. But now I have no idea what I'm doing with the rest of my life." Casually leaning over me, he asked, "What would help you most? Just sitting here in silence? Pointless shopping spree? Words of empathy and comfort? Crying and ice cream? Jumping right into problem solving mode?" I sat straight up, almost crashing into Walter. "Problem solving please," I said. "That sounds perfect." Walter nodded and said, "You got it." He turned his body so he was facing me. "Well, you're a witty fellow who's good with words. Way better than me. You could try to be a writer." I nodded noncommittally. Walter could tell I wasn't sold. "Oh! You're gorgeous and blond and a great public speaker. Politics!" "I'd need a degree to do it right." He waved his hands at me like a magician's assistant, saying, "Then, get a degree." I shook my head emphatically. "I don't think my heart's in that." "Heart, huh?" Walter thought for a second. "We could adopt a kid. Or a whole bunch. We'd be excellent fathers, and you could be a stay-at-home dad." There was definitely an appeal in that. But that was a stopgap, not a solution. "Adopting a kid because I have no plans wouldn't fix my problems, and it would be unfair to the kid." "Fair enough." Walter contemplated a few seconds before offering, "You could travel for a while to figure yourself out. See the world. You've commented how frequently I travel on business. You could join me. Keep me company en route and at the hotels, and then have your own adventures while I'm cooped up in meetings." That idea sat poorly with me. "Wouldn't that just be submitting to the whole trophy husband thing? Wouldn't that just make me look like a handsome idiot you keep around for sex and eye candy?" Walter stroked my cheek affectionately. "Logan, everyone these last ten years has already assumed you were my boy toy, and we both know they're wrong. They're the idiots. As soon as anyone gets to know us, they realize how amazing you are and that they were wrong too." I turned around so I could lean my head on Walter's shoulder again. "I could go out of my way to prove everybody wrong. Commit to becoming ugly," I suggested jokingly. "Lose 30 pounds of muscle." Walter could tell what game I was playing, so he played along. "Gain 100 pounds of fat." "Start dressing in stained t-shirts and sweatpants." "Shave a bald spot." "Pull out a few teeth." "Stop showering." By then I was laughing—full on belly-laughing. "Can you imagine?" "I can," Walter said, kissing the top of my head. "I'd still love you." I pulled him down closer to me to kiss his forehead. "Thanks." Then, as an afterthought, I added, "You know, I actually wish I had the guts to try all of that, just to see what the press would say." "They'd say, 'Walter Jaffetz's husband has really let himself go. How long until Walter files for divorce?'" That fell eerily accurate. "Yeah. You're right. Once a trophy husband, always a trophy husband. That's all they can see me as." "The comments aren't about you. They're a reflection of the people who write them and what they think their audience wants to read." "You're right," I repeated emphatically. "No matter what I do, they'll see me as the braindead hunk who tricked you into marrying me so I could spend your money." Just then, a brainstorm hit me, so I sat up and turned to face Walter. "I should go the other way." Walter raised an eyebrow, curious. "I like working out, and we both love my muscles. But I only ever worked out just enough to be viable as an actor. I could really lean into my workouts. Become a bloated mass of muscles straight out of a fetish video. Use your access to medical science to make it happen inhumanly fast." With a poker face, Walter asked, "Is that something you'd want?" I nodded enthusiastically. "Yes. I'd really like that." "Then let's go for it. I'll hire you a trainer in the morning." I tackled Walter and covered him in kisses. "You're the best!" Walter laughed. "Oh no!" he cried in sarcastic, faux panic. "My gorgeous, muscly husband wants to become even more muscly and gorgeous." Then, in his normal tone, he said, "It's no sacrifice on my part, Logan. You'd be making my wet dreams come true." "If we're going wet dream," I said, still lying on top of him, "we should make my cock bigger too. Pumping. Pills. Surgery. Whatever's most effective. We'll research." "Your cock is plenty big," he reassured me. I wriggled on top of him. "We're talking wet dreams, Jaffetz! Be bold!" Walter scooted out from under me, stood up, and held out his hand to help me up. "If we're talking dreams, let me show you your retirement present." "You got me a present?" "It's a present for both of us," Walter admitted, pulling me to my feet. "I didn't plan on giving it to you today. Today has taken some…" he paused for a second, looking for the right word, landing on, "interesting turns. I thought today was going to be crying and ice cream. The freezer's stocked. I didn't plan on giving this to you for a few weeks. But, you keep me on my toes. If you're in this frisky of a mood, we can go play." "It's a sex gift, then?" I leaned into Walter and thrust my crotch into his stomach. "Not specifically or uniquely, but I expect that's how we'll use it." "Cryptic," I said. "Into my workroom," Walter said, leading me by the hand. I followed demurely.
growth-without-effort
July 9th's "Black Cat"
Ripped posted a topic in 4th Annual Storiversary
Black Cat Chapter 1 | Chapter 2 | Chapter 3 Chapter 4 | Chapter 5 | Chapter 6 WARNING! Contains snuff. Chapter 7 | Chapter 8 | Chapter 9 Chapter 10 | Chapter 11 | Chapter 12 Epilogue Sequel: Black Cats Chapter 1 The wife lays on the bed on her side, passed out. Her long auburn hair covers her sweaty, cummy face, neck and tits. Cum leaks out of her pussy and ass onto the soaked hotel sheets. Sunlight from the window lands on her 36C breasts, her nipples still hard and extended from my manipulations. Those beautiful breasts jiggle, the bed creeks and the headboard hits the wall with each of my thrusts. "Oh, God! Fuuuuck!" the husband moans under me. I pin his head to the bed with my left hand. My right hand grips and lifts his pelvis keeping his ass inline with my cock. For the last twenty minutes I've been pulling all the way out, waiting for his sphincter to close, then, with a low-pitch growl, ramming in deep. "You like that, boy? hmmmggggggrrrrrr. Like getting fucked slow and hard?" The husband moans something that sounds like, "yes, Daddy", as his fists tightly clutch the sheets. I roll my head back and inhale deeply, relishing the musky scent of sex. I spike his ass with a quarter of my rod. "AAAAHHHHHHGGOOOOOODD!" the pitiful excuse for a male wails. "GRRRR! I don't give a FUCK what you like, cunt," I growl. Enough of treading lightly with this virgin ass. I grab his shoulders with both hands and use my abs for countless short hard strokes, smashing into his prostate with every fuck. The bed frame crashes against that wall with a loud BAM, BAM, BAM, BAM, BAMBAMBAMBAMBAM as I bore into him. I keep up the rapid-fire assault, grunting with each fuck, until he yells, "I'm cumming!" I wrap my left hand around his throat and lift him off the bed. I stand tall and press the back of his head into my chest. He utters a satisfying scream as he slides father down. A minuscule amount of cum bubbles through the tiny cock cage his little penis is trapped in. His legs swing and bump my shins as I walk to the floor-to-ceiling mirror. By the time I reach the mirror he's panting like a bitch in heat. I turn to face the mirror and look over the "man" impaled on my cock. Mid-twenties, crew-cut blond hair, handsome features. I guess you could say that he's got a middleweight bodybuilder physique. A decent chest above his six pack. An above average limp dick swings off of him. He could probably place in a regional bodybuilding competition if he didn't skip leg days. His eyes are squeezed tightly shut. "Open your eyes. Come on boy," I thump his head with my free hand, "open." When he doesn't obey I forcefully flex my abs, driving my ramrod deeper into his gut. "Aaaarrrg!" his eyes snap open and looks at me in the mirror with fear, lust and fear. He begins to hyperventilate. "Hey, calm down, little man." I slide my hand off of his throat putting him in a choke hold. I lightly squeeze his windpipe between my forearm and bicep. His hands immediately grab my arm and vainly tries to move it. I whisper into his ear, "Take slow deep breaths. That's it. Good boy." With his breathing slowing I find myself involuntarily slow fucking his tight ass and watching my hulking figure in the mirror. The husband's head, his mouth gaping open, is held between a forearm thicker than his upper arm and a bicep bigger than his head. Thick veins under my paper-thin skin look like a metropolitan subway map. Above my bowling ball delts thick traps rise like mountains to meet my corded neck. The husband whimpers when I flex my free arm into a Herculean ball of power. "You like that, puny boy? Grrrr, yeah, I think we both know who's superior." I feel a pair of tits against my wide lats and a sopping wet pussy grinding on my massive rippling leg. "Fuck him, baby." The wife wraps her tiny hands around my torso. One hand slowly strokes my ten-pack, fingers following the perimeter of each thick cobblestone segment. Her other hand travels up to my shelf-like pecs to try to squeeze the rock-hard muscle. Giving that up, she begins to pinch and tweak a downward pointing nipple. "Show him how to use that thing between his legs." She's kissing and licking my back. "He's never satisfied me. You…oh god…you made me cum more than I've ever had. Show him how a real man fucks." Never wanting to disappoint the ladies I break the bi pose and slap the husband's glute. He yelps and calls out for mercy to whatever deity he thinks is listening. I seize a leg and bring it to his chest as I start to pound his ass balls deep. I roar. The husband screams in terror and submission. The wife moans, "Make him your bitch like you made me your whore…Master" *** Good. Now I have your attention. With all the instant gratification, short news cycles and screaming 140 to 280 character dispatches from who-the fuck-cares you have to grab attention by the balls, squeeze and not let go. Even if the owner of said balls slaps you with an injunction. Which never happens to me…usually. Hi, I'm Bruce. Bruce Banderole. Ripped let me hijack his account to tell my story. I didn't used to be this way. The domineering alpha male, not the ball grabber. Well, maybe not that either. Anyway, I was just your typical, average office worker schmuck sitting in a nondescript cube surrounded by sappy inspirational posters from HR in the boring corporate world. To say I was the pinnacle of physical health would be laughable. Twenty-six years old at this time. Under the average height for a male, just under the definition of obese, nearsighted and balding. My idea of exercise was carrying a box of a dozen doughnuts to the office every Friday. The only thing in my life that brought a ray of sunshine into my gloom was my girlfriend, Val. Six weeks ago that would all start to change. On that Monday nothing could brighten my mood. I grabbed two different socks out of the sock drawer, my sandwich was moldy, the printer repeatedly jammed on duplication of a fifty page report and I had my review. See, there's me after work in line at the bus stop waiting for the 5:10 to my house. Hunched shoulders, thinking about the day and mumbling, "How the fuck did I get a 'adequate' on the Reynolds account? I busted my ass for that fucker!" I look up to the guy behind me, "Tell me why a guy that looks like an avocado had sex with an older more disgusting avocado complained?" The old lady behind him stepped back aghast. The guy looked at me and said, "Maybe it's your use of harsh language." When I realized I left my umbrella at home, it started to rain. I sighed defeat to the universe as the bus rolled to a stop. The doors opened and I heard something in the alley. I ignored it and shuffled forward. After a few steps I heard it again. This time I think I heard a cat. I had a cat once. My sister wanted to name it Dog. A few more shuffling steps and I definitely heard a cat. I reached the bus door, looked up at the driver and said "Wait for me, I'll be right back." I think I heard the driver mumble, "Yeah right buddy" as I turned into the ally. "Here, kitty kitty kitty." I heard a reply from the left ten feet down the alley. I called out again, followed the reply and found an average sized undernourished young adult black cat under a piece of cardboard. It looked up at me with pleading electric blue eyes. I knelt down to the cat, and slowly blinked, "Hey, there." I extended a finger in front of its nose. "You don't look too well." The cat sniffed my finger, blinked and replied with a meow that almost sounded like, "Help?" I extended the finger to scratch the cat's chin. I didn't see a collar when the cat lifted its head for more scratches. "Oh, so you're a stray huh? Well we can't have you walking the streets can we?" As I gently lifted the cat it began to purr. I turned around with the cat cradled in my arms just in time to see the back of the bus disappear down the street. I sigh and mutter, "Fuck." "Mew?" "It's just been one of those days, cat. There's a vet school a few blocks from here. How about we get you checked out and get something to eat?" "Purrrrmoowprrrr" "I'll take that as a 'yes'." Two hours later I ordered an Uber and headed for home with a $200 vet bill, a box of stuff and a very groggy cat. When the Uber pulled up I saw my favorite driver, Debbie, in her beat up Hyundai Accent. "What the fuck ya got there, Bruce?" she yelled at me over the blaring death metal as I put the cat carrier and box in the back seat. I closed the squeaky back door and yanked on the front passenger door until it opened. Most people don't like Debbie. It might be that she curses like a sailor or that she'd rather insult you than get a tip. I kinda liked her; the petite figure, the girl-next-door face, the long blond hair in pigtails just went so well with the spiked leather collar, ripped band t-shirt, short shorts and bowie knife strapped to her thigh. Think Harley Quinn without a psychopathic homicidal boyfriend with a pasty complexion. At least I hoped so. "I found a stray cat and decided to adopt her," I replied once I sat down in the seat and turned down the screaming Norwegians. "The vet named her Brenna. Apparently that's Gaelic for 'black hair'." Debbie lifted a single eyebrow when she glanced from the back seat to me, "You, a cat? What's that fucking bitch Val gonna say?" I buckled up and replied, "Yeah, a cat. I'm secure enough in my manhood that I don't need a hundred fifty pound slobbering dog that you have to get up at five o'clock in the morning to walk. Besides, everyone should have a little pussy." She blushed as she pulled the car out into that traffic and replied, "Uh-huh. You got food and a goddamn cat box yet?" I pointed my thumb at the box in the back, "Yeah. The vet gave me a box of stuff. Said it was a CCL Starter Kit." We almost get into an accident when Debbie snorted and started laughing. "You mind not killing us and tell me what's so funny?" She composed herself just enough to reply, "Shit! CCL stands for Crazy Cat Lady." My eyes involuntary rolled heavenwards, "Laugh it up, Deb. One cat doth not a crazy make." "Mrr, aarrr?" came from the back seat. "I didn't ask you." With a snicker she stated, "They say the fucking first step is talking to the them." "And yet you talk to your crappy car." "Hey, don't insult Reggy! He's very sensitive!" The car backfired in agreement. She started stroking the dashboard, winked at me and said, "There, there, Reg. Don't listen to that mean asshole. You still haven't said how your cocksucking bitch-friend's going to react." "A FUCKING CAT!" was the reply I got from Val to a text with a pic of Brenna on my lap. When we got home I set everything up for Brenna. She had some water and cat food from her bowls in the kitchen and used her box in the bathroom. With her belly full she constantly purred while I scratched and rubbed her chin, throat and belly. She held my arm with her paws to make sure I didn't stop. She especially wanted me to scratch under her new orange-red nylon collar. I tried to think why the love of my life would say that as I scrolled up the app and saw all the pics she sent of her Pekingese, Alcaeus. The many, many pictures of Alcaeus dressed in a toga, birthday hat, sunglasses. In a sombrero for Cinco de Mayo. In a Santa beard and stocking cap for Christmas. The pics of Alcaeus with a hair bow on its head, in a Che Guevara t-shirt and beret, dressed as a Minion, lapping up a strawberry smoothie (from MY glass, mind you), wearing a Mario cap and, the worst, with cat ears. I typed out my reply with the thumb the little spoiled shit bit last week when I tried to get him away from my smoothie, "I found her in an alley downtown. I'm not going to walk away from that. Come on over and meet her, babe." A minute later the reply chat bubble started bubbling. Two minutes later I see, "Ugh fine cu soon." "We're going to have company, Brenna," I told the purring ball of fur on my lap as I rubbed behind her ears. Then I noticed the pizza boxes on the coffee table and dirty socks on the sofa. I sighed and said, "Looks like I've got to clean this place up a bit…" "Myeah." "…so you're gonna have to…Wait a minute, did you just say 'yeah'?" She just blinked those beautiful electric blue eyes and purred. "Ooookay then. Let me just set," I lifted her up and settled her on the sofa, "you here while I straighten up the place." The garbage was tossed and a load of laundry started when I heard the familiar sounds of Cadaver's "Cannibalistic Dissection" outside, Val screaming, "You expect a TIP after THAT?!" and a creaky car door slam. I open my front door to Val in a tight blue mid-thigh dress, her auburn hair in a bun, oversized sunglasses and a floppy wide brimmed hat. Behind her I saw a Hyundai peeling rubber and the driver's arm sticking out the window with a middle finger extended. "Ugh! That cabbie is the WORST! I don't know why they don't just fire her ass!" "She's not a cabbie and they can't." I gave Val a kiss on the cheek as she blew past me. "You know she's an independent contractor and the best driver in the city." "WHATEVER. So," she scanned the room, "where's this cat?" When Val set her oversized purse down her "dog" squeaked and poked his head out. "She's on the sofa. Be gentle, she's not used to…" When Val saw Brenna she charged towards her. Loudly, she said, "Oh, aren't you just the cutest thing!" Brenna jumped to the back of the couch, yelled, "Moor? REEEOW!" and hissed. When Val started to reach for her she jumped down and hid under the couch. "Hey! That's not how you act around a new pet, Valerie!" Alcaeus jumped out of the purse and started yapping at the couch. "Pfft, what do you know. It's just a cat." Brenna came out and sat in front of Alcaeus. Alcaeus continued to yap away as if he was a fearsome beast. Brenna, just yawned and cleaned her paw. "Yeah, well, she MY cat and I won't have her becoming neurotic due to…" We both heard a cut off yip and looked down to see Brenna's paw pinning Alcaeus's head to the floor. Her tail swished and she gave a "humph" sound. "WHAT ARE YOU DOING TO MY DOG!" Val exclaimed. I suppressed a snicker and said pridefully, "I see there's no chance of that." Val reached down to pick up the dog Brenna sniffed her hand. "Get away!" She clutched the shivering dog to her breasts, "That cat is EVIL!" "No she's not. This is all new to her. It'll take time for her to adjust." "Yeah, well…" Val's phone sounded out a notification I haven't heard before. Her face went flush and her eyes widened when she looked at her phone. "Sorry, babe. Gotta go." "You just got here?" I said, my hands outspread. "I know, but somethings come up…at work," Val collected her purse, shoved the shivering mutt in it and headed toward the door. "Can I call you an Uber?" "With HER again. I don't think so. Besides, my ride is already outside. Bye!" And with that Val slammed the front door on her way out. I stood in the middle of the living room wondering what just happened when I felt Brenna winding herself around my legs. "Well, that could have gone better." I said looking down at her. She blinked twice. "She's really very nice, ya know." "Mnoow," was the only reply in the room. Did my cat just say, "No"?
muscle growth After Work Workout
PosingP posted a topic in Stories
Hi Everyone.... long time lurker, first time posting a story.... please go easy on me as I'm dyslexic! This one is probably a one off but I have some ideas for some other stories. _______________________________________________________________________________________________________________ After the first day at a new job, I walk into my flat and look around for my gym gear, catching my tired slender reflection in my bedroom mirror. I'm tempted to skip my workout but I had promised myself I will get in shape this time; my new job means I have more free time now. The new flat means I have access to a gym. I know I have no excuse for not following my dream of finally getting fit. Well, I dreamed of, and have wanked over, the idea of being a bodybuilder, but with my genetics, that's never going to happen, but if I could get some form of muscle and show a bit of definition, that would be nice. Unfortunately, my workout gear is all at the bottom of my washing basket from when I last managed to squeeze in a run. I fish around the back of my wardrobe for the random lycra outfit I had ordered online a few months ago. It was super cheap from a small website I'd never heard of before. The two-piece suit was two-tone blue and black, sadly the small size suit was baggy on my petite frame, and when I put it on, it had lots of wrinkles. It didn't matter if it didn't fit me though, nobody was going to see it. I was going to the gym in my apartment complex. It was small but surprisingly well stocked with weights. The few times I had walked past it whilst moving in it was empty. Nobody seemed to use it. This suited me as a small quiet gym was fine by me. I was determined to do a full-body workout; I had time to kill this evening, and given that I was just starting out, I thought I'd go through and practice my form on main compound exercises I'd learnt about; I may not have had time to train in the past, but I've spent far too many evenings enjoying youtube videos of muscle studs working out and giving training advice whilst all sweaty and shirtless. After a quick warmup, I look at my slender reflection in the mirror with the lycra suit hanging off me. I shake the idea of the fantasy of being a big beefy guy filling this suit and head to the squat rack. Putting on my headphones, I lift the unweighted bar, focusing on form, paying attention to each rep, trying to get 5-8 reps in a per set but concentrating on my body position and tensing and activating the right muscles. I add on some weight and start the next set, quickly getting in a rhythm and lose my mind to 'the zone', each set, I would up the weight ready for the next set then return to the focus on my form. I wasn't paying attention to how much weight I was adding to the bar and going again. I'm just focused, in a trance, squatting down and bringing the bar up. I don't notice the gradual reduction in wrinkles in the lycra suit. I finally feel my legs fatigued, only just managing to get the bar back on the rack. I unload the bar (not noticing how many plates I take off) and move onto the shoulder press. The wrinkles in my lycra suit are mostly gone, and around my legs, it's started to pull tight; I've not noticed though, I'm too busy lifting the bar and creating my slow controlled reps. All the time, my shoulders broadening and rounding, pushing against the lycra, not that I notice, I'm just doing this rep of this set… but which set is it? I've lost count… I found that manageable, so I better add some more weight and go again. Time passes; after moving onto the bench press, I pause to fill up my water bottle, ignoring the round pecs slowly expanding and pushing against my shirt, not noticing the crease forming from my overhanging balls of muscle. Nor have I noticed my arms starting to fill out the sleeves. No, it's time for deadlifts. Loading up the bar, then tensing my core getting ready to lift and thrust forward, concentrating on the squeeze, not noticing my back rounding and widening, traps and shoulders pulling the fabric of the lycra. As I stand up with each rep my core starts to push on the fabric, first as a small block pushing against it, and then, with each rep slowly starting to divide, first down the middle, then highlighting each individual ab. As I squeeze my butt at the top of each rep, it gets a little making it rounder, plumper, firmer; the lycra leggings start to conform to the dimples forming on each side, and yet I keep ongoing ignorant of the changes my body is going through. It's only when I have to pause mid-way through a set to adjust my feet position as my quads are pressing against each other that I realise, "Fuck my quads are pushing against each other! I gasp as l look down and see that the lycra tight over my body, contouring muscles that any fitness model would be happy with. My cock instantly hardens, growing down the side of my thick leg, pressing against my lycra suit as I take in my muscles. I glance over at the mirror, and I'm tempted to go and look: I've heard of a pump during a workout, but this is impossible! However, overpowering my sudden hornyenss and desire to examine my body is the desire to work out: I said I would do a full-body workout: I need to keep on lifting! I go back to lifting, but I'm now aware, I can feel my muscles swelling up and getting more pumped with every rep. My now ripped physique strains against my tight suit. I ignore the strain of my hard cock in my pants, pulsing bigger and harder, pushing forward as my balls and legs grow with each rep, forcing it against the tense fabric. I try to force the thought of how much I've grown out of my mind, my impossible new body, and focus on lifting… after all, if I carry on lifting, I'll grow more! I get on with my workout, each rep making me grow bigger, more defined. I try not to see my biceps swelling, my chest pushing against the lycra, the definition in my arms and legs showing through the straining lycra. Eventually, whilst doing a second round of bench presses, the suit can contain me no more. The lycra rips, pulled apart by my broad back, and as it does, I feel a wave of pleasure mixed with fatigue roll over my body, and I know the growth has stopped. I stop my exercise and walk over to the mirror, lycra tearing as it glides over my solid muscle. Reflected back in the mirror is a beast, as big as any picture of any bodybuilder I have ever jerked over. "FUCK YEAH" is all I can roar as I tense my chest; the lycra top explodes of my body. Seeing my pecs stand to attention as a shelf of glistening muscle, my deep abs and round shoulders cause my cock to go into overdrive, and it tears out of the ripped leggings. Now a foot and a half long and as thick as a beer can. I grab it with both hands and start stroking my thick pulsing meat, dripping with precum, flexing my body as I go. I tense my tree trunk legs pulsing with veins and bounce my pecs, watching the striations ripple. Grunting, I lick my sweaty hard bicep peak of my 22-inch arms, feeling the veins pulse under the paper-thin skin as I carry on jerking with my other hand. The feeling grows, both from the power I feel from my size and strength and all the new nerves from my massive cock. The sensation builds up to ecstasy, and I can't take it anymore rock-hard cock pulses; I moan with pleasure and cum. Squirting warm white liquid over my abs, chest, up into my face and with my mouth open to swallow the hot, warm liquid. As the orgasmic pulses subside, I collapse on the floor. The pump my muscles feel starts to subside, my body returning to where it was, almost, as I clear up and wrap a gym towel around me, I admire the little bit more definition of my body than earlier. I notice that my cock and balls press against my towel more than they usually would be. I saunter back to my room from the gym to put in a bulk order for those lycra suits. I'm going to buy the size XL and am already looking forward to tomorrow's workout.
m/m MaxandharryandmaX: A serial (?)
LJackson posted a topic in Stories
Okay, here's the start of something new. Let me know if you want me to carry on! What you need to understand is that Harry and I had been friends all our lives, or nearly. When we were in playgroup, I chewed on his toy car. When we went to school together, we taught one another to read. If a teacher asked me what my name was, I'd say: "Max and Harry." For years, right up till we went to university, people would call us Maxandharry or Harryandmax. Even at Uni, him studying Biological Sciences and me doing Business Studies, we texted nearly every day. For a while, after celebrating our Friendversary, we even changed our Faceboook names to MaxandharryandharryandmaX, till he pointed out it looked a bit gay. I felt a bit sad changing it, but my girlfriend of the time told me he was right. In the years since then, of course, we drifted apart a bit. We were both in London but we would only meet for a drink every couple of months. I guess he slowly started noticing a change in me about summer 2019 when I had made a special effort to get #BeachBodyReady. 'Shit, mate, you really do have a gym membership, don't you?' he said. 'When are Men's Health ringing you for a cover feature?' I was in a short sleeved shirt and I was feeling pretty good about myself back then, but still I knew he was talking shit. 'Bro, have you seen the guys on Love Island? All I've done so far is lose a bit weight really.' 'Impressive though,' he said, sinking the last of his pint. 'I don't think so,' I said. 'Not yet.' 'Well, I reckon Niamh would think so,' he said. I cuffed his shoulder. 'She's not superficial like other ho's,' I said. 'Girls like that sort of thing, though,' he said. He looked down at himself. 'They can't help themselves.' 'They like to see we're taking care of ourselves,' I told him. 'Anyway, you're far from overweight, dude.' 'Underweight, if anything,' he said. 'I do my heavy lifting with this.' He tapped his brow, and we both laughed. 'Mind you don't go impairing it with two many hipster beers, then,' I said, getting up to fetch another round. 'If you promise not to go crazy with this gym stuff,' he said. 'You're fine as you are.' But I knew he was wrong. By the time we met to toast Christmas, he couldn't resist squeezing my arm. 'Holy shit!' he said, eyes wide. 'I thought you just had a big jumper on under that coat. You're huge!' 'Not as big as I'd like,' I couldn't help saying, blushing. 'Still a way to go.' 'What do you weight though…?' 'About 70kg,' I said, automatically. 'Roughly 150lbs.' 'Whoa,' he said, 'I didn't expect you to actually know. You must be taking this seriously.' I shrugged. 'It's fun too.' 'Bet you're not drinking tonight.' 'Maybe just the one. My body fat's down to 7%.' I sipped my orange juice. 'It feels great, though. I'm so alive, so capable. And, bro, let me tell you, the girls are hanging round me like flies around shit.' God, it felt good.. Harry didn't seem to feel so, though. 'Mate, I didn't need to have another reason to feel inferior,' he sighed. His mouth was proper down-turned. 'Niamh,' I said, testingly. He nodded glumly. 'I wasn't enough for her, it seems.' 'You were too good for her, anyway,' I said, although I wasn't sure how true it was. There he sat, a weedy little guy with glasses, maybe half my size. He couldn't have lifted a weight above his head, let alone bench-pressed what I was doing. He looked pretty pathetic. 'Maybe you should join my gym. Just to cheer yourself up.' 'Can't think of anything worse,' he said. 'Come off it, mate,' I said. 'Build a bit of muscle and you'll feel amazing. Imagine if Niamh sees you on the beach next summer with your shirt off, biceps bulging, pecs rippling…' I tailed off, thinking how ridiculous this sounded. Even if I could persuade him to take some exercise, I thought, it probably wouldn't be a good idea for him to take his shirt off on camera. He was still as lean and hairless as when we both set off for University. I wondered, was I naturally superior to him? Niamh would certainly have picked me over him for a mate: law of the jungle. I've have shown her a better time, as well. I didn't like to think how embarrassing Harry must have been in bed. I shook my head to clear it. Harry had been talking to me and all the time I'd been thinking about banging his girlfriend, cucking him in his own bed. Crazy thinking. I interrupted him: 'I don't want to hear any shit, mate, you're coming with me to the gym tomorrow and that's that.' He sat back in surprise, held up his hands. 'Fine. Fine!' There was a long pause. Finally he laughed and got up to go to the bar. 'Better get a round in while I'm still allowed, then!' 'Skip the crisps this time, okay,' I laughed back. '2020's going to be your year.' And to be fair to him, he turned up. He obeyed me, as if I was in control of him. He did the exercises I told him to do. He changed his diet. It became quite fun, turning up to see him at the gym – maybe not as often as I would be there, but at least a couple of times a week. He didn't enjoy it but he worked hard. At the end of January, he had actually lost a bit of weight – but sadly, he hadn't put on a single shred of muscle. We stood looking at one another in the mirror. 'I can't fucking believe it,' he said. 'I'm actually smaller than before.' 'I've just grown bigger,' I said, 'that's all, mate.' 'Bullshit,' he said. 'Yes, you're bigger but look at me. I'm a shrimp. Nobody would guess we were the same age.' 'You're a week younger,' I told him, shrugging off my sweaty vest. 'Maybe you'll always be…' Again, I didn't want to say the truth. 'What?' he demanded. 'What were you going to say? Weaker? Smaller? Inferior?' 'But only to me,' I told him. 'You might put on some muscle by summer. You won't be strong but you'll be fit.' 'That's really what you think of me,' he said, eyes wide. Furious. He looked like he wanted to pick a fight with me, but something rational was stopping him. He'd be mullered in a second. Instead, he ran away. Just grabbed his bag and stormed out. I should have run after him, but I was meeting this girl for dinner. He had told me on Tinder that she wanted me to fuck her like an animal, and I was still wondering how best to do that. I was too distracted. That was half my trouble. MaxandharryandharryandmaX was in trouble for the first time in history. An unshakeable bond, broken by the gym. Or maybe more, I thought. Maybe by masculinity and testosterone. Maybe by alpha male power. It could never be equal again now that I had begun to dominate. February slipped past, and I thought about him every day. I dreamt about him. I dreamt he was watching while I effortlessly fucked Niamh. I was pumped and bigger than ever while I worked at her tight snatch, and he got smaller while he watched me. In March, I finally sent him a message. 'Miss lifting weights with you. Miss lifting pints even more.' He replied almost straight away. 'Maybe we should resume one of those activities. Guess which one.' 'Gym's open longer hours,' I replied, with a emoji to show I wasn't serious. 'Mate,' he texted back, 'I wouldn't go there right now.' 'Why?' I asked, expecting a joke. And so, Harry was the first person to tell me about coronavirus. I thought it was just him making an excuse about not exercising. I don't follow the news, don't really do social media; in fact, I was pretty buried in work and working out. I suppose, I also didn't want it to be true. I was in full denial. And so, inevitably, I caught it. I'm strong. I'm healthy. I don't even take roids, so no strain on my heart. So I guess I was well prepared for it. Even so, it knocked me out for longer than I expected. Those were some crazy weeks. Maybe the weeks that followed were even stranger. Apart from work, Harry was the main person I was in contact with during lockdown. At first we talked a bit about fitness: my gym had closed. I tried to buy gym equipment online, but there was none available. 'I must have got the last set,' Harry told me. He told me it was the one good thing in his life now. It was hard for him - he was still so weak - so I sent him tips over email, links to videos and online advice. He asked me questions about diets and supplements and steroids. I told him to do it all clean, which disappointed him, being the biology scientist – he wanted to make himself into his own experiment. But it was a good thing. It was like we were the same person after all, just slightly out of phase. Except I couldn't work out, not properly. I did some bodyweight stuff and started running. The muscle just melted away. I ended up looking as lean as a stick of celery. 'You're going to end up as my trainer,' he said, when we talked about it. I didn't like that. What if he caught me up? So we didn't talk about it. And the lockdown rolled by. The lockdown was raised. Life began to return to normal. It was September when we arranged to meet again. A few drinks in the same old pub we used to frequent – but first, we would work out together, side by side. It struck me, as I walked towards the bench press: we were equals at last. MaxandharryandharryandmaX had been brought closer together by the quarantine. We had both realised something: his innate power, my humility. At last, we would be best friends once again. The only problem was, he didn't appear to have showed up. The gym was pretty much empty, and the only person hanging around our agreed meeting spot was a big guy. Proper monster. Shaved head. Lats out to here, waist in here, a real triangle. He looked nearly a foot taller than me and I was worried for a second. In an empty gym, it's pretty bad etiquette to stand waiting for a piece of equipment. I didn't want him to think I was trying anything on; he could have ripped me apart. He turned around and the light glinted off his glasses. He smiled. 'Hey, Max, you made it!' 'Harry?!' I couldn't believe it. I thought it was a wind-up. He opened his arms inviting me for a hug. Each arm was as big as both of mine put together. The muscle was thick and pumped as if he had just completed a workout, not spent months in quarantine. 'Harry,' I said, looking up at him. 'This is insane. You're a fucking beast.' 'I used the lockdown well,' he said seriously. His voice had grown deeper. It was like talking to a different man with my friend's face, one who towered over me. 'All this came from working out?' 'Of course,' he said. 'I see you went the other way.' I looked down at myself, blushing. 'Yes, I'm pretty small these days.' 'How big were you before the lockdown?' he asked me. 'Let's not talk about it,' I said. 'Maybe we should –' 'Cut the crap,' he said. 'I asked you a question.' 'About 75kg,' I told him. '165lbs.' He shook his head in disbelief. 'You used to seem so big,' he said. He ran his hands over the huge globes of his chest and the hard, thick ripples of his abdominals, lightly brushing his fingers across the giant vascular boulders of his biceps and the vast shoulder-muscles that framed his firm chin. 'I'm about 270lbs. 125kg.' 'It's not possible,' I said. 'Feel it,' he said. 'Then you'll know how possible it is.' I laughed and held up my hands. 'That's fine, I'll believe you.' 'What is the problem with you?' he said. He didn't raise his voice but the authority was clear. 'I told you to feel it, you little bitch.' 'Are you okay?' I asked him. He shook his head for a second. 'Sorry, bro. It's the testosterone. Sometimes I lose control. I mean, it's true that you're little. And you're being a bitch.' He put a hand over his mouth and blushed. The redness spread down his huge neck where it got lost at the tops of his pecs. 'Jesus. Just bants, mate. Just bants!' 'Will it help you if I – if I feel your muscle?' I said, putting my hands on his physique. 'Be careful,' he said. 'I'm like a loaded gun nowadays.' 'More like a cannon,' I said, feeling his might. 'The Incredible Hulk has nothing on you. So you went for the steroids after all?' 'No way, you little cunt,' he said, breathing heavily. 'Ah, I'm sorry, Max. But no – not steroids, exactly. I did some research. With some friends online. Other biology academics into weightlifting. I was looking for short cuts, and they were looking for ways to build size and strength. I think between us we tapped into something that did both. Workouts combined with certain hormones and particular herbs. It's all perfectly safe, just a few side-effects.' 'Like the anger management issues,' I suggested. 'And all this fur.' I ran a hand through the pelt on his monstrous chest. 'You really are becoming a beast.' 'Fuck!' he gasped. 'You touched it! I should have warned you – but I thought it would sound gay.' 'Touched what?' I snatched my hand away but he caught it in a giant fist. 'You touched my nipple. Fucking sensitive nowadays.' 'So what – what does that mean?' 'It means,' he said, 'either I need to fuck a woman in the next ten minutes or…' 'Or what?' Harry looked toward the changing room door. 'Or, old friend, you'll have to milk the cum from my big swinging balls…' TO BE CONTINUED…
penis-growth
cum-flood
dumber
muscle growth Lucky Me
Author's note: Those familiar with my stories on this site know that I am prone to very long stories, dripping with dialogue, posted over the course of months. So, I gave myself a challenge: 5,000 words max, no direct dialogue, 24 hours to write the whole thing. It was a fun experiment. Let me know how you like the results. ============================================================ I only bought one fucking ticket. And I only did that because my boss was watching. The chances of me winning were astronomically small—no one ever won the big prizes. Ever. The ATL (Annual Transglobal Lottery) was supposed to be this big deal, a chance to win large sums of money, luxury items, designer vacations. You know, big lies to keep the humdrum workers humming with the thought that they might have a taste of the fabulous life. But in reality, in my 37 years of life, I'd never seen anyone win anything fancier than a bicycle. Sure, people always won the small prizes on the low tier—the gift cards, the restaurant vouchers. And who doesn't have an ATL pencil in their desk? The middle tier prizes—the weekend getaways, $5,000 cash—you occasionally heard of people winning those on the news, but I'd never met them personally. The top tier prizes—a two-story dream home, a sports car, cancellation of all debt—those prizes always went unclaimed. With over 10 billion adults playing the fucking game, you'd think someone would occasionally luck out. A number of cynics like me were convinced the top tier prizes were fundamentally unwinnable. Which is why I stopped buying the fucking tickets. But my boss doesn't like that. He sees people who aren't playing the ATL as people who've lost faith in the system. And people who've lost faith in the system were always the first fired. So, when the ATL salesgirl came around, I was about to fob her off, but my boss was right there watching. So, I bought a ticket. I then told her that I'd bought two dozen the day before and I couldn't afford more until my paycheck deposited. That was a lie she and my boss would believe. The tickets weren't cheap, and they were only available for the seven days leading up to the drawings. And yet, people still drove themselves into deep debt buying hundreds of tickets. In fact, of the people who still played the ATL, the average number of tickets purchased was 50. Fucking 50! Most people could pay a month's rent for that chunk of change. So, despite thinking the whole fucking thing was rigged—a fucking fairy tale to keep workers complacent—I fucking bought a fucking ticket. Then the fucking thing won. From the top fucking tier. Any other prize from the top tier would have changed my life in amazing ways. But with my luck, I won the one prize I couldn't have hated more. I won the fantasy body makeover. As far as I was concerned, I already had the fantasy body—I was a cute little 100-pound, 5'4" hairless cherub of a man. I wasn't even insecure about my 4-inch dick because how small I was really got my husband's engine revving. Albert, my husband, was a brute of a man—6'5" with 220 pounds of dense, hairy muscle with a 10-inch cock. And that's the way we liked it. He loved saying he was more than a foot taller than me, more than twice my weight, more than twice as hung. He worked hard to keep up his macho physique—his six pack was proof that the 220 pounds was all muscle. He went to the gym every fucking day. I also worked hard to maintain my petite measurements. Us short folk face the middle age spread too. When I won the makeover, I thought I could specify that I wanted to make myself even more diminutive for my man, but the makeover I won was preselected. It depended on the gender on my driver's license. Women who won this prize in the drawing won the deluxe women's makeover; men who won this prize the drawing won the deluxe men's makeover. Interestingly, if I'd had my gender as any of the other approved genders, I would've just won the cash equivalent. Why couldn't I have had one of those on my fucking ID? Albert and I could really have used the money. After an hour on the phone with the ATL reward people, I learned that the body mod industry had very specific ideas of what deluxe men wanted. At the end of the process, I would end up bigger than my husband—taller, more muscular, more hung. Likely, I'd also end up hairier and more ruggedly handsome. I wouldn't accept that lying down. I had too many anger issues to do that. So, I fucking fought back. But everything I tried just fell through. There's some fine print on these tickets that I never read. Some fun facts that I learned: 1. Once the drawing has begun, tickets are non-transferable. If you want to transfer your ticket before the drawing, you have to register the transfer with the ATL. This way, they said, no one could steal a winning ticket and declare it for themselves. With most of the prizes, you could just give it to someone after you won it, but there was no way of just giving Albert my growth. Upshot of this, I couldn't just give the prize to my husband even though we both would have fucking loved that. 2. Prizes cannot be refused unless you can declare a bona fide exemption (such as a devout Hindu winning a voucher to a steakhouse). If I wanted to declare that my religion disallows body modification, I had to have a religious affiliation declared on my file (I have none), and then report that to the ATL prior to the drawings. Upshot of this, I have to take the prize, whether the fuck I want it or not. 3. If you try to pretend that you didn't know your ticket won, you will be tracked down by the ATL so that they can inform you of your good luck. If you try to pretend that you threw away your ticket so you can't claim your prize, they show you your receipt of purchase, quite loudly, and in quite a celebratory way. This way, people who missed the drawings will still get their prizes. That was a fucking uncomfortable lunch break—ATL people everywhere, some with tambourines. Upshot of this, I can't just fucking ignore them. I tried to fight it further, but it just made things worse. I actually managed to get a decision-maker on the phone—a human decision-maker—and I insisted it should be my right as a winner to transfer my winnings or to refuse them outright. I even refrained from fucking swearing the whole call. He never considered someone actually refusing a top tier prize, and he was so upset that I was unhappy with my winnings that he decided to look into ways to compensate me appropriately. Two days later, an ATL computer called back. It apologized that I was unsatisfied with the rules of the ATL, so in their infinite generosity, they had reached a consensus that they would increase my winnings—also non-transferable, also irrefusable. Fighting it had actually gotten me upgraded to the ultra-deluxe men's makeover. This is how I found myself at the body modification center, rolling up my sleeve so they could inject me with the nanites that would, over the course of six months, reprogram my body to grow into the masculine ideal that global corporate interests had decided for me. And I just had to fucking take it. At first, my growing muscles and stature motivated Albert to push himself even harder at the gym. In that first month, as I grew three inches in height, added 1.5 inches to my dick, and put on 35 pounds (almost entirely muscle), Albert, with chemical assistance, managed to pack on another 10 pounds of muscle. He started wearing lifts in his shoes to keep us a foot apart, and he was still practically twice my size in dick. My increased boost in testosterone came with an increased sex drive, which was fun for us. That first month, things felt manageable when I was home. Work was another story. Most of the people on my floor were there for the day of a thousand tambourines. But even if they weren't, my win had made the global news. And even if they still somehow hadn't gotten the memo, my clothes gave me away. Nothing fit me anymore. All of my work clothes were fucking tight, and my shins showed off in all my pants. The world looked different at 5'7". I actually met some people's eyes and was taller than some of the employees. All the guys on my floor were jealous, the boss included, but I felt like a badly dressed clown. As part of the makeover, I'd get free hairstyling for a year and a new wardrobe once I reached my full size, but as I was going through my second puberty, I was left to my own devices. I guess the people who can actually afford the ultra-deluxe package can afford a new fucking wardrobe every month. The first month groaned into the second. By the end of the second month, I was now 5'10"—just on the taller side of average. And at 7 inches, my cock was just on the larger side of average. And at 170 muscular pounds, my weight in pounds was squarely average, but I was ripped. I had never been a big fellow, and it seemed that with the weight I was adding, most of the weight was going into my muscles. I suppose some of it had to be going into bone mass and healthily functioning organs, but I was getting stacked. I was still eating like I was 60 pounds ago—the nanites apparently pulled in the necessary building blocks from the air and sunshine—so my overall body fat was the same it had been, but it looked very different with all this muscle there. I was now an above-average male. Albert had put up a good fight, but even with chemical assistance, he was only able to put on 5 pounds that second month. He was still 65 pounds more than me, but the gap was closing every day. And with the increase in my height and cock, he was less and less turned on every day. It was only then that I realized he was never turned on by my body—not really. He was turned on by his body, and my body's smallness made him feel even more superior. So, as my sex drive was becoming sex overdrive, Albert began spending more time impressing the twinks at the local bars with his chemically-pumped muscles and daddy dick, while I stayed home altering his old clothes into something vaguely resembling work clothes. If we could afford a divorce, our marriage wouldn't have survived the second month. To work off all the fucking sexual energy, I began masturbating while staring at my body. My body was hot—I couldn't deny that. But I would stare at the mirror and cover my face with a sheet, pretending it was somebody else's body. I would watch his pecs hanging over his minuscule waist and taut six pack, the sweat glistening off the swollen mounds as his bicep flexed, his cock pistoned by his hand. This guy's cock was thick like a cock should be. And his muscles were made for fucking. Then I would climax and feel disgusted with myself. I didn't want to look like this; I wanted to be fucked by someone who looked like this. By now, work was ridiculous. My fucking boss, seething with envy, started treating me like shit. When he wasn't insulting me to my face, saying things like my brains had been sucked into my muscles, he was constantly writing me up for inappropriate work attire. On top of that, I started getting thankless grunt work. If he could've found a way to dock my pay, I'm sure he fucking would have. The third month brought more surprising changes. I had grown another three inches, so I was now over 6 feet—officially tall. If the world had looked different at 5'7", it looked like a different planet at my new height. Very few people were taller than me anymore, and they cleared a path for me when I moved through space. At 205 pounds of cut muscle, I had finally steadily increased the amount of food I ate. If I tried to eat the way I wanted to—the way I used to—I would get lightheaded and pass out. It just takes more food to fuel this behemoth I was becoming. While they built me even more massive, there was only so much the nanites could do without more food. My cock was also officially huge at 8.5 inches. It was getting harder to hide how fucking big it was getting at work, even though I was now wearing some clothes Albert no longer fit into. Albert always wore his pants a bit too tight in the crotch, so they all had faded bits that showed his dick print. And those dick prints just highlighted my growing bulge. By this point, Albert had essentially moved out of the house with a pretty young thing. A small pretty young thing. I won't pretend it didn't break my heart, because it did, but my hours were too full of monotony to really feel anything other than numb. I spent so much more time just eating and altering clothes and masturbating. Oh, was I masturbating. Although, I couldn't really call it that anymore. I was jacking off. The faceless stud in the mirror just swelled larger every day. The cleft between his pert pecs grew deeper as they rounded out and covered over with a fine dusting of hair. His arms looked flexed, even when they weren't. When they were at his sides, they sometimes even fought his chest for space. His legs were also becoming an obsession of mine. His thighs were thicker and thicker every day. They were getting as thick as my waist had been only three months ago. I would feel the striations as I would flex, and the stud in the mirror's thighs would blow out into large relief. I could also turn around and see the mirror-stud's ass. It was so round and thick. As the third month ended, I realized why men like Albert like to fuck them. Asses are just so fuckable. At the end of the fourth month, I was 6'4"—almost Albert's height. I had to shave every day—the hair came in fast and thick, hairier than Albert had ever dared dream. His clothes should have fit just fine, but I was more muscular than Albert had been at his biggest. His work clothes were painted onto me. My chest was bigger than his had ever been, so the top two buttons were always threatening to pop off, showing a peek of chest hair if I took a deep breath, and the shirt was starting to threaten at the sleeves and the shoulders. And my lats. People told me I looked even broader from the back. I had to take their word for it. With my growing traps, it was getting harder and harder to turn to look at the reflection of my back half. Even with all this new width, I had to tighten his belt one hole smaller. I used to think Albert's midsection looked powerful—my six pack looked angry. When I flexed them, I looked like I could stop a bullet. My stomach was always hard and ridged, even unflexed, and if I turned to the left or the right, all of my ridges were apparent through my shirt. Albert's pants were also a paradox. My legs and ass were thick enough to compensate for the missing mass in the middle, but the cock in my pants—the cock that should have been the same size as Albert's, was bigger. My soft cock was bigger than his soft cock. I could easily tell because I overfilled his dick print. I was thicker and longer flaccid. And I was definitely thicker when erect. It was getting harder to pretend the stud in the mirror wasn't me while jacking off. I could feel the mass of my cock as I stroked it. Everything about me felt heavy and took up space. My arm constantly bounded off my pecs if I jacked off too furiously. I had to hold my mighty quads further and further apart to give room to my swollen eggs. One night, I decided to end the pretense and uncover my face in the mirror. I'd seen my face every day over these last four months—especially with all the extra fucking shaving I was doing. But it wasn't until I looked at my face while jacking off that I realized how different I looked. How masculine. How severe. My thick brow furrowed in concentration, my face drawn tight with cheekbones and a jaw that exuded the effects of testosterone. The eyes were clearer and brighter, more alight with sexual fire. That was a stud's face staring back at me. I came so hard when I realized I didn't look like me anymore. But as soon as the blood was flowing properly to my brain, I realized how fucked up that was. I didn't even fucking look like me anymore. By the end of month five, work was untenable. My cubicle looked like it belonged to a child, and it was getting hard to type. Not only were my mitts too massive for the tiny keyboard, but my biceps and pecs were in a constant war these days, which added to the difficulty of typing. Even outside my cubicle, the office shrank around me. At 6'7", I was getting too huge for doorways. On top of all that, I was squeezed into Albert's old clothes—and they didn't even come close to fitting me. I was taller than him, so a row of my abs showed if I ever lifted my arms even slightly. And, I was 275 pounds—40 pounds more than Albert ever weighed. Even with my incredibly tight waist, the pants were getting too fucking tight around me. One day I sneezed, and the shirt split right down the middle, spraying buttons everywhere, and showing off my ponderous, hairy pecs to the whole office. The sight of those burgeoning glories started to stir my ever-present arousal. My soft cock was pornographically obscene—if I stiffened into my full 11.5 inches, I could destroy my pants too. My boss gave me vacation—well, unpaid leave—until I had reached a stable size. With all that time off, I just stared at every inch of my body naked in the mirror. I don't know who that man in the mirror was anymore, but he didn't even jack off anymore. He fucked his fist with his monster cock. His balls were so swollen with cum and manly juices that it took six orgasms to abate him, and even then, only for a few hours. I would gorge and fuck my fist and repeat the process. The sixth month was a haze of eating and fucking my fist. At the end of the sixth month, I was 6'10, 330 pounds of ripped furry muscle, and sported a 13-inch cock. My muscles and cock had tripled in size, and I was 18 inches taller than I started. My head was being swallowed by neck and trap muscles. My shoulders jutted out widely into infinity. I couldn't look down past the shelf of my pecs, my biceps and lats forced my arms out into a widespread stance, my thighs were officially thicker than my waist—even my calves looked like flexed biceps in the mirror. I was naked all the time now, but I was never cold thanks to the healthy carpet I'd sprouted. I could see all six of my abs through the forest of hair, and my intense cum gutters. I'd occasionally peel myself away from worshipping my body to stare at my face. The beard I now sported in addition to the increased effects of testosterone on my face had me looking stunning. I understood why Narcissus fell in love with himself. The man in the mirror didn't look like me, but he was a god. The sound of the front door closing took me out of my reverie. Albert will forever blame me for what happened, but it was his own fault for coming back to the house. He nearly fucking pissed himself when he saw me. He'd come back to officially move out. We still couldn't afford the divorce, but we could afford an amicable, no-fault separation, and he figured I'd be more agreeable now that I'd stopped growing. I thundered over to him, my footsteps shaking the floor as my thighs forced their way around each other. I smiled, looking down on him. I was five inches taller than him and 100 pounds more massive. When I saw how much bigger than him I was, I finally understood his obsession with comparing himself to my once minuscule size, and my cock sprang to full attention, three inches longer than his could ever strive to be. I agreed to the no-fault separation, if I got to fuck him first. He did not like that condition, but I came in his ass three times. He felt so fucking tight and small. This man who had dominated me with his size felt so small in my hands. Albert left, and I immediately felt a little disgusted with myself. Inside, I was still me, but with all the hormones swimming through my amplified magnificence, I wasn't sure who me was anymore. This new me was starting to act exactly like Albert had. I stayed in that state of naked introspection for two days, only breaking my shame spiral to occasionally eat or fuck my fist—animalistic actions that intensified the spiral. I also hated how much I liked this giant body I had. I wanted to return to my job, to some semblance of normalcy, but I couldn't until the ATL board came to finish the makeover, so I was in a bind. For two days, they didn't come. On day three, I called them. The six month timespan they'd quoted was only for the deluxe men's makeover. The ultra-deluxe took an additional three months. When I asked what to expect, they were pleased to tell me that I could expect more. More height, more muscle, denser muscle, increased masculinity, and an even bigger cock. I was just going to keep growing more hung, and more massive, and bigger for three more months. Lucky me.
forced growth
growth without effort An unexpected outcome
Makoto posted a topic in Stories
Before I start, I wanted to say that I've never written a story like this before. Having been inspired by Maxxxmuscle's comic Harvzilla. It is honestly like the hottest thing I think I have ever seen, and it's getting better every time he releases a page. So without further ado, here we go! Chapter 1: Coming out It started like it would with any love story. Josh was in love with his life long best friend, but had no clue how to express his affections. It was painfully obvious for all but Parker who was oblivious to his best friend's feelings. Parker had his own secret though... One which he had never shared with Josh. The two were madly in love with one another! Josh was actually quite a strapping young man. At 6'8 and 245lbs of pure muscle. He was what many teenage boys wished they could be. On top of that, Josh was every bit of a grower. Hidden behind his jeans, and boxer briefs lied a foot long by the likes nobody had ever seen. He often caught people ball gazing, but thought very little of it. Hell, he hardly cared if people drooled over him. Paired with the young Adonis' killer body, Josh's well known size was the source of rumor for many at Westwood University. Despite this fact, He had turned down everybody who made the attempt to try and date him. Josh had his eyes on one person, and he was mere minutes away from confessing to that person. Sitting at his Desk, Parker was in the middle of daydreaming again. He had just gotten through a painful Calculus lecture, and was eager to spend time with Josh later that night. The two had planned to see a movie at the local theater. But, what the two didn't know, was the others plan to confess feelings. While not as physically gifted as Josh. Parker was by no means a slacker. Standing at a smaller but still impressive 6'4 and weighing 215lbs. Parker packed a smaller, but still impressive anaconda of 10 inches. Like Josh, Parker was popular among the girls and the guys, but would reject everyone who bothered to ask. Like Josh, Parker was still a virgin, and it was a surprising fact. Few people expected someone of such an impressive size to still be one. "Hey, I'm on my way my dorm room. Wanna come hang out for a bit before the movie?" Josh texted Parker. Parker looked down at his phone and grinned, he loved hanging out with Josh. It wasn't even a question of if he wanted to or not. The answer was always yes, and Josh knew it. "Of course, I'll be there in a few minutes," Parker responded. Josh received the text as he unlocked the door to his room but not before an unknown package he failed to notice fell over. Upon inspecting the package he noticed it to be some kind of experimental protein powder. Wondering what it was, he closed the door to his dorm room and opened it up, Finding a red warning written in bold letters, he began to read the side effects of overdose. "Overuse of the experimental Hyperbulk powder can result in several side effects that include but aren't limited to, Extreme muscle growth, increased height, penis growth, an increased sex drive, and infectious cum" "Infectious cum"? Josh thought to himself. What do they mean by infectious? Before Josh had time to contemplate the message, he was overcome by a sense of hunger. "Well, I already have it. Why not try out some of this stuff while I wait on Parker?" Josh had been opening the bottle and gathering the other ingredients for a shake. The substance didn't include a scoop and he had never been one to use one. So he decided that half the smaller tin would be an appropriate amount. It was about the same amount he usually used when he made protein shakes anyway. Blending it to perfection, Josh was excited as he poured the shake into a cup for his consumption. He had made protein shakes a daily part of his routine, and sculpted his body over the years to near perfection. (Or so he thought) By the time he finished gulping the shake down, he heard a knock at his door. Parker was thirsty, as he had just ran 4 miles to get to Josh's dorm. It occurred to him that he hadn't really eaten anything either. He was always the book nerd compared to his best friend. Indeed, Josh often playfully teased Parker for being a math major while he was at college on a basketball scholarship. Despite the two being different in almost every single way, they had been friends since they were just four years old! Parker didn't really recall a time without his best bud. "Yo dude! Woah- you're drenched in sweat and panting," Josh noticed as he opened the door and greeted Parker. "Yeah, well I hate to keep you waiting. Besides I need the exercise," Parker replied Josh led Parker to his couch, and went to go make another shake. Using the remaining half of the tin without paying any particular attention to the serving size amounts. Josh blended the shake to perfection, and handed this one to Parker. "Try this stuff out," Josh said as he handed the drink to Parker Parker didn't have to be told twice, especially being as thirsty as he was now. He gulped the contents of the shake down, leaving no trace of it in the cup. "Thank you, that hit the spot," Parker let out a small burp of approval. Josh smiled and giggled a little bit. "No problem man, I actually just got that stuff delivered today. It was a sample of some new protein powder they're developing," Josh went on, as he began to felt a strange heat in his gut. "Well thanks man, but I wanted to tell you something.. Before we go out tonight. It's been weighing on my mind now for months," Parker began, as he too felt a weird warmth emanating from his gut. Josh sat on the couch next to Parker, and listened to what his friend had to say. Josh always listened to Parker, and cared a lot about what his friend had to say. "Well you see..." Parker began, "I like you a lot.. I have for some time. Everybody always drools over you and your body, and while I admit that's a bonus. I've always liked you for more than that Josh. You're my best friend, and know me better than anyone else. I'm more comfortable around you than I am anyone else." Parker had intended to continue, but not before being interrupted by a pair of firm yet soft lips on his own. "I've loved you ever since our Freshman year of Highschool. I feared telling you because I didn't want to lose our friendship. But are you sure I'm really what you want?" Josh blushed, as he exposed his more insecure side. "Of course you are...," Parker responded The two leaned in to kiss, both battling the others tongue in a display of hot fiery passion unlike anyone had ever seen. Their intense make out session would move to Josh's bed where the two gradually stripped clothing and continued making out. The two would begin to lose track of time as their bodies continued to rub against each other. However, the warmth in their guts began to spread throughout their bodies. Changes were beginning, that the two had yet to realize. Even Bigger changes were about to happen. Perhaps bigger than the two of them could begin to comprehend. End of chapter 1 Thanks to all those who have taken the time to read my first post. Hopefully you all enjoy this as I continue to develop it. I'll continue to work on improving my writing as the chapters go on. But please, let me know what you thought!
getting taller
muscle-sex
m/m THE NEW SUPERMEN: BIZARRO
Absman420 posted a topic in Stories
(AUTHOR'S NOTE: Well, here we are again -- fifteen years later and I'm kicking out some sequel. Weirdly, I've had a Parasite story brewing in my head for years, but this Bizarro story found its way out instead. Not a long one, I promise -- five parts. I just enjoy playing with these characters. (Previously, in THE NEW SUPERMEN: Superman has left Earth, seeking his destiny among the stars. To protect the planet -- and the American Way -- Superman has selected 100 of the nation's best police officers to take his place. Imbibing them with a serum to give them Kryptonian powers, plus a massive dose of solar radiation, these men transformed into hypermasculine superheroes. These "Super-Cops" were just adjusting to their new, super-powered bodies [and their inability to have sex with normal humans], when set upon by a plot to trap them through that very sexual weakness by Lex Luthor. Though nearly succumbing to Luthor's manipulations, the Super-Cops ultimately foiled his plan. Now, nearly a year later, there is a feeling of normalcy and routine, even if the New Supermen had to make some sexual compromises.) Pt1 REMOTE UTAH The helicopter had been flying for an hour without seeing even the slightest hint of civilization. These remote locations had their advantages in terms of regulations and governmental oversight, but they were a gigantic pain in the ass to travel to. Finally, just as the late afternoon sun was casting long shadows across the mountains, the destination came in sight -- the LexCorp symbol obvious on the helipad. That Lex Luthor himself was here only indicated how close they were to success. He and his bodyguard were greeted at the helipad by the Chief Scientist, Dr. Teng, the windrush of the slowing heli-blades blowing his white lab jacket up like Marilyn Monroe on a subway grating. "Greetings, Mr. Luthor," he shouted over the din. "We are honored to have you here -- we have shown much success in cloning…" "Yes, I know," Luthor said bluntly, walking past the doctor toward the facility. "Take me to him." The three entered the facility and immediately turned to a series of elevators. Had Teng not been the one with the ID card to summon the elevator, it would almost appear as if Luthor had no use for him at all. They plunged underground, deep enough that they felt their ears pop from the pressure change. Teng spent much of the time trying not to stare at Luthor's beautiful female bodyguard, Mercy -- she spent much of the time trying to resist the urge to kick him in the balls. Finally, they arrived in the Sub-Basement Labs, easily a mile below the surface, the elevator sliding open silently. Teng motioned, "This way," though Luthor already knew where he was going. There, in the center of the main lab, surrounded by cables and monitors and unidentifiable machines, was what could only be described as a huge glass tube, large enough to hold the overly-muscular male that floated inside it. "As you can see, Mr. Luthor, the clone is nearly ready." The clone wore a breathing apparatus, so its face wasn't clear, but the body more than gave away its genesis -- it was clearly a clone of Superman, the original Man of Steel -- regardless of the tight S-shield compression shirt and red speedo it wore. "And the triggers are in place?" Luthor asked, examining it. "The commands? Will it obey me?" Dr. Teng placed his hand almost affectionately on the tube. "Everything has gone according to plan, Mr. Luthor. The mental programming has been put in place. There is no reason to believe this clone won't be completely under your control -- your own personal Superman. Now… to be fair… we haven't run any tests since the… uh… virus was introduced…" Luthor looked at him quizzically. "Why would that make any difference?" Teng looked uncomfortable. "Well, it's an unexpected variable. As you can see, since it was introduced, the clone's testicles have swollen to at least five times their normal size. We had to create special trunks that would fit them…" Mercy rolled her eyes at Teng's awkwardness -- men put so much unnecessary importance on their genitals. Bigger balls meant a bigger target to kick as far as Mercy was concerned. "It's fine, Teng," Luthor said, glancing casually at the clone's big gonads. "It's all part of the virus." "This is the most success we've ever had cloning Kryptonian DNA, sir. It seems a little far-reaching to unnecessarily add…" "Are you questioning me, Teng?" Luthor asked calmly -- Mercy seemed to stand even more at alert. "No, Sir!" Teng said a little too quickly. He knew what happened to people who fell on the bad side of Lex Luthor. "I simply counsel caution. We're not even certain about the stability of the cloning process, much less…" "It'll be fine," Luthor said. "Even if he only kills one of them, it'll have been worth the money. Release him." "But, Sir, he's only at fifty-percent -- he hasn't been exposed to sufficient sunlight…" "RELEASE HIM!" Nervously, Dr. Teng went to the console. Before pressing the button, he shut his eyes and said a silent prayer -- even Luthor couldn't control his thoughts -- all he could do now was hope for the best. He pressed the release button. As they watched the fluid drain from the tube, Luthor mumbled, "FInally -- a superman of my own…" He knew it was only a matter of time before they could clone a super-body to transfer Luthor's MIND into -- then he'd personally settle the score. One step at a time. The tube raised, taking the breathing apparatus away with it, leaving the clone standing, semi-dazed as it came to consciousness, dressed only in the tiny, tight Superman shirt and stuffed red speedos. The clone looked pale, having never been exposed to sunlight -- it resembled an albino bodybuilder. It stood uncertainly on its feet, absorbing its surroundings, then looked down and noticed itself. It ran its hands over its muscular torso, flexing its biceps and smiling at their size. The process of self-discovery gave the clone an erection. "Horny," it mumbled… and began to clumsily masturbate. Luthor moved to take control, standing before the clone with his arm held out before him in a "stop" motion -- easily a foot shorter than the behemoth, Luthor barely came to its pecs. "I am Lex Luthor," he stated firmly. "Your creator. You will obey me!" The clone looked at Luthor with a semblance of recognition, smiling slightly, but making no move to stop playing with itself. "What's wrong with it?" Luthor asked Teng. "Why isn't it listening to me?" "It could be any number of things," Teng said hastily, carefully walking toward Luthor but making sure to face the clone. "It could be the Kryptonian gene-stock -- it could be the virus you introduced…" "Horny…" "There's so much we don't know about Superman…" "SUPERMAN!" the clone cried -- and shot a load that nearly hit Luthor and Teng. The clone's orgasm lasted nearly half-a-minute, going on and on. And the longer it lasted, the more the clone's moans turned to giggles -- it was as if his intelligence, not his ejaculate, that was cumming out of his oversized balls. "Oh my dear Lord!" whispered Teng, stepping back. "What is happening? All I did was mention Superman…" "SUPERMAN!" the cloned cried. "ME WANT SUPERMAN! ME WANT FUCK SUPERMAN!" "What…?" "Yes!" Luthor said, stepping toward the clone. "Superman. You fuck Superman!" "Mr. Luthor!" "Mercy, shut him up." Before Teng could speak another word of protest, Mercy kicked him in the stomach, doubling him over -- she would've preferred her target to be his balls, but whatever. Teng collapsed, moaning quietly -- scientists rarely put up a fight -- curled up on the floor. Luthor remained focused on the clone. "You find Superman," he said. "You fuck Superman." "ME FUCK SUPERMAN!" "Yes!" "SOOO HORNY…" "So fuck Superman... stupid." "YES! FUCK SUPERMAN STUPID!" Suddenly, the clone bucked his hips, orgasming again -- the front of his speedo soaking in his cum, dripping to his feet. His laughter was the giggle of a child. "FUCK SUPERMAN STUPID," he laughed. "YES… ME MAKE SUPERMAN HORNY AND STUPID!" "Okay," Luthor smiled. "Sure. Go and fuck Superman stupid." "ME GO! ME GO FUCK SUPERMAN STUPID!" They stood there staring at each other, the clone with an empty grin as he gently stroked himself. "Go NOW…" Luthor said, nearly exasperated. "YES!" the clone said, as if having the idea was brilliant. "ME GO NOW!" And with that, the clone looked around and, seeing only the elevator, bound toward it with a speed that belied his size. Before Luthor could react, the clone ripped the elevator doors open with his super-strength and then began ascending the shaft by lifting himself hand-over-hand up the cable. They stood there in the silence of the lab for nearly a minute, listening to the clone work his way up the mile-long shaft -- he clearly hadn't realized he could fly (or perhaps he hadn't been powered up enough) -- then, from very high above them, they heard the groan of creaking metal, a loud, childish laugh, and then the wreck of the elevator car crashed into the opening. "Looks like we're going to be here awhile," said Luthor, wryly. None of them laughed. NEXT PART
m/m The 12 Disciples
tereshky posted a topic in Stories
The Twelve Disciples (I wrote this story as a thank you to the first 12 guys who liked my work enough to follow my profile. This is my last story of the month so I just wanted give you guys one last thing to read before I went on vacation!) ---------------------------------------------------------------------------------------------------------------------- Shawn was your typical college athlete, always looking for any kind of an edge. He was one of thirteen people on his swim team and, while he was really good, he consistently came in third in their meets. His frustration at never coming in first was overwhelming. It didn't matter that he could have any girl he wanted or that his friends looked up to him like an idol. He wanted to be better than everyone. It was the end of August and a new swim season was about to begin. He was sharing his frustrations with an online friend late at night when they suggested an herbal supplement he'd never heard of before. Shawn went to the website and found it all pretty sketchy. The site was covered in obviously morphed photos and the company was operating out of some town in India that he'd never heard of before. But he was so desperate to win he'd try anything. When the package arrived he eagerly opened it up only to find that it contained one pill that said, "take with water." "Jesus Christ… I spent $50 bucks for this!?" He groaned. He was so pissed he almost threw it away but he figured he might as well swallow it. So he gulped down the pill and chased it with a glass of water and then hopped into the shower. As he was soaping his body up he felt warm all over. His muscles started to tingle with a strange sensation and then suddenly his entire groin felt like it was melting in pleasure. He felt his cum splashed against his chest as he fell against the side of the shower in the most intense orgasm of his life. His muscles were begging to swell. He gripped his pecs and felt the meat getting thicker in his hands. "Fuuuuuck." He closed his eyes as the sensations overwhelmed him. He shot load after load until his cum was spent and his cock was dry heaving. When he opened his eyes, his biceps looked like someone had added a baseball to their mass. His triceps flayed out more and his legs had crazy definition. "Shit that stuff really works!" He turned off the water and got out of the shower to check his body out in the mirror. He'd gained a good 20 lbs of muscle. He bounced his meatier pecs and smiled. "That shit's out of this world!" Then he immediately went online and ordered another package. He was going to do a lot better this season. The second package arrived in a few days, just before their first day of practice. He was so excited he took it immediately even though his friend was hanging out in his dorm. Rami was from India. He and Shawn had known each other since freshman year. "What's that stuff you're taking?" Rami asked him. "It's this great supplement I got from India. It's amazing! Watch this!" He said pulling off his shirt as he gulped down the pill with a glass of water and waited. Shawn and Rami stood in front of the bathroom mirror for 20 minutes with no obvious effect. "What are we looking for?" Rami said. Shawn's body looked amazing and he suspected this was some kind of joke his friend was using to show it off. "I'm going to start growing!" Shawn shouted, but nothing happened. "I'm bored with this, lets watch TV," Rami said. The two watched Netflix for about an hour and still there was no sign of any new growth on Shawn. Eventually it was time for the first swim practice of the season. "Ugh… so disappointing. It happened so quick last time!" He said gathering his things. Rami got up to go. "You don't have to leave man. I know you hate your room mate. You can just stay here while I'm out." He said, pulling on his shirt and walking out of the room. Rami sat down and continued watching "The Crown". Eventually he got up to go to the bathroom and nearly tripped over the packaging that the supplement came in. He picked up the packaging and looked at it. "Wow this is a really seedy company." He thought to himself. The instructions were all printed out in Hindi. Where there were English translations they were written by hand. "Odd… they translated this wrong. That doesn't say 'take WITH water' it says 'take IN water'." scratched his head wondering why on earth the instructions would say that. Then he read some of the untranslated section. "Allows the user to absorb water through skin… converts into muscle and genital growth… for safety do not expose yourself to more than fifty gallons at a time… holy shit!" Down at the pool Shawn's phone was vibrating like crazy as Rami tried to call him ten times but his phone was in his pants pocket and his pants were shoved to the back of a locker. Shawn was walking out to the pool deck in a tight speedo. His team mates were complementing him on the weight he'd gained since last season. Shawn was feeling confident, the second pill had been a dud but at least he had something to show for his efforts. His slightly larger ass was bulging out of the sides of his speedo as he dived into the pool and began swimming laps. He had gone five laps when Rami burst through the doors and started shouting at him to get out of the pool but Shawn was swimming too hard and couldn't hear. Suddenly Shawn's entire body began to tingle again. His cock poked out from the top of his tight speedo as it became erect. "Fuck… not now..." Shawn thought as he gripped the one of the lane dividers and started shooting white clouds into the water. The swim coach shouted at him, wondering what he was doing but Shawn couldn't respond, he was in too much pleasure as the water rushed into his skin and he began to bulk up. His shoulders looked like a rising volcano as his traps built up and his delts rounded out. His nipples were forced downward by the pounds of meat that were being added to his pecs. His cock continued to grow and thicken as clouds of white jizz erupted from it. His balls were swelling rapidly in his speedo until they popped out from either side, the size of tennis balls. His ass expanded until the speedo was jammed up in his crack like a thong before it snapped. By this time he'd captured the attention of his fellow teammates. To them his contorted face made it look like he was in incredible pain and they swam over into his cloud of cum filled water to see what was wrong. They touched his swelling body, unable to believe what they were seeing. "Bro… what the hell is going on!" One of them shouted as they crowded in to watch his biceps swell before their eyes and his back muscles develop a deep topography. He was getting taller too, his expanding feet were being pushed deeper into the pool, inch by inch. "Jesus man… should we get a… gnnnn" Another guy said. His skin was feeling tingly as the herbal infused cum began to soak into his skin. Shawn's balls were still swelling, now the size of melons and thick clouds of jizz were floating all over the entire team. They had been so concerned for Shawn that they hadn't seen what was happening in the cloudy water as their cocks all became rock hard. Suddenly they liked the sensation of their mostly naked bodies pressed against each other. They pressed in on each other more closely until their fit frames were rubbing the jizz into one another. Soon they were dry humping one another. "WHAT THE FUCK IS GOING ON!" The couch shouted, seeing his team all blissfully groping one another. Shawn continued to grow until he could touch the bottom of the deep end, his cock was as wide as a two liter bottle. His balls floated to the top of the water like shipping buoys as gallons of cum spewed into the pool. As he bounced his pecs in pleasure it created a wave pool. His muscular ass grew the size of a bean bag chair. Rami and the coach just stared in slack jawed awe. Shawns abs split in a twelve pack and the the team was climbing all over him to feel it, starting to swell themselves. He was finally number one on this 13 person team. He was their muscle god and they were his 12 disciples, and they worshiped their leader as all the water in the pool was slowly turned to cum.
height-growth
Cockfight - Final Part added February 19, 2019
anormorph posted a topic in Stories
For your reading pleasure. Thanks for being an awesome space. "You can do this. You can do this. You can do this." The thought runs through my head like a mantra as I climb the stairs. I find myself at the door to his loft before I realize I'm there. I clench my fingers into a fist in a final effort to steel myself and raise my hand to knock. The door rattles noisily as it slides open. His eyes widen in surprise at my hand in front of his face, still poised to rap on the door. "Whoa – that was quick. People don't usually make it up the four flights that fast." I look at him and shrug, doing my best to act non-chalant. "What's wrong, out of breath?" He grins toothily at me and gestures to come in. "Not yet." Well yes, but not from the stairs. Fucked if I was going to let on how he has me panting and terrified at the same time. Of course, he had answered the door in nothing but shorts. We were the same height at 5' 10", probably even the same foot size, but everything else was different. He was clearly the bulkier one. His profile listed him at 175 but he looked a little bigger in person. He was carrying a little extra bulk from the winter but that did nothing to hide the thickly muscled core underneath. The dark hair dusted across his chest trailed down his abdomen thickening above those shorts that did everything to accentuate the reason we were here today. The fabric outlined the meaty shape of his dick so clearly, he had to be going commando. "This isn't going to be much fun if I'm the only one who's getting naked." He slides the door shut behind me, audibly sliding the pin in to lock it. "I've got the heat cranked so we don't have to worry about any extra...shrinkage. Bathroom's to your left if you're shy to start." I ditch my shoes and head left, closing the bathroom door most of the way behind me. I manage socks and pants first, only getting tangled in my own clothes once. I turn to face the mirror and start unbuttoning my shirt, getting halfway down before my hands start to tremble. Cursing inwardly I stare at my hands, willing them to stop while I slow my breathing to help focus. My head snaps up at the sound of wood creaking behind me. There he was, right arm raised and bent to brace himself in the doorway leading to his bedroom. I didn't even notice the other door in my distraction. He looks me over slowly, feet to head, finally making eye contact in the mirror and before approaching from behind. "Hey." He turns me toward him and rests his forehead against mine. "Here.." He slowly grips my hands to steady them. Together, we finish undoing the buttons. He pulls the shirt off my arms, hanging it off a door handle. It was unexpectedly intimate and incredibly kind. "Do you forget how to undress yourself often?" His smile colour the words softer than they would have been. "Well, no, I just-" Fuck. The heat rising in my cheeks means I'm blushing, something I don't do often. I hate blushing. "I may be a little nervous. You?" "Terrified." "But you're not even-" "Home turf advantage" I groan. So corny. "And we both know I'm going to win." "Oh, yeah?" I step back from him then, raising my gaze defiantly. "You remember the rules, right?" He steps back and crosses his arms. I swear he's flexing to make sure I notice the taught balls of muscle that swell. "Oh I remember, all right. Looks like someone's been hitting the stair-master to draw out the inevitable as much as possible." I preen at the comment. Once we decided to get together and compare in-person instead of online, I had upped my cardio and tightened my diet. I had meant to drop the weight for a while and this real life meeting gave me the extra inspiration to finally do it. I was a lean 152 pounds now, down from the 160 my profile listed. A simple run over my lower body with clippers meant things everything was trim and neat. My upper body was bare until a triangular stripe of dark blonde flaring down below the navel. I also have the sense to wear briefs that flatter my average endowment in the front and really show off the work I put into my ass. " I just wanted to accentuate the difference as much as possible, you know?" I'm smiling now, finally finding the vibe that prompted us to agree to this in the first place. "It. Worked." At that he drops his shorts and steps out from them, fondling his sack gently after he does. About four and a half inches flaccid, his ample cock is forced forward to drape over his thick, heavy sack. The balls are a quite a bit tighter against him than I'm used to seeing them on cam; maybe he was telling the truth about being nervous. I raise an eyebrow at him and hook my thumbs in the band of my briefs. Slowly drawing them down, I watch his eyes widen as the material cleared my soft cock. His lips part slightly and his breath hitches as I finish sliding them over my feet and stand up for him to appraise. I barely clear three inches soft. It looks bigger because I took the time to trim, but a quick glance down at himself re-affirms that, while I'm almost textbook average, he, quite clearly, is more. Hips forward, he takes an involuntary step toward me, ready to compare the two together. Blood is already starting to redirect itself toward his thickening member. I clear my throat and he steps back again, imploring me with his eyes. "The Rules, remember?" "But we could just-" he starts to protest. "Trust me?" He sighs and nods before turning around to lead me through his bedroom and back into the open space of the loft. The furniture is all moved to the sides and a large mat covers the empty floor. He has the shades drawn down so the neighbours won't be getting a free show. Every light is on, lending the space a garish arena-like feel. I make special note of the island counter that separates kitchen area from the rest of the space – it could be of special use later, depending on exactly how this night plays out. "Wow. Do I even want to know why you happen to have a wrestling mat this size? Where would you even keep this?" That gets a chuckle but he stops just past the bedroom door and gestures with his chin to the other side of the room. I pad past him to turn to face him, legs bent and ready to spring. "That's quite the bed you just led me by. Pretty big, even for you, isn't it." I smirk to drive the barb home. "Custom-order." His voice is a little rougher now. We're both almost ready for the chase. "Guess you need something custom to hold the gigantic fucking weight of your sack. I mean, really, are you even strong enough to lift that thing up when you have to take a piss? I think you might have to start doing some extra curls to handle it." We're quickly heading full dark, no stars now; just our symbiotic kinks centred on his glorious endowment. He quickly rises to the bait. Literally. His cock rapidly approaches full mast, the skin stretching tight around the tip. It's so damn pretty. He glares and crouches down a bit to lunge at me. He gets a bit of a glazed look, clearly plotting what he's going to do if he fully wins. To win though, he has to catch and pin me first. His lip curl into a sneer and he bonafide growls at me,"I. Am going. To own. You." "You bet your sweet fucking ass you are." The words come out as a taunt, even though I'm already saluting his victory. It's on then. The temperature in the space rises noticeably as the chase begins. He isn't slow, but he doesn't stand a chance with all the cardio I've put in. Leg day is my favourite and, while I might not squat several hundred pounds, I have put in more than enough hours to add quite a bit of extra bounce to my step. I only have to outpace him for ten minutes for me to be the one to choose how I get to worship him tonight. I tease him through every sprint, every jump, every panting breath. He tries to banter back about my inadequacy but, really, we're here to 'talk' about him and we both know it. "Careful you don't throw out a hip with that thing!" "Awww, is your big thick dick slowing you down? Too much resistance keeping you from getting enough speed to catch me?" "Shut. Up. Pissant." "Muffin." Like that's going to happen. The rules specify he wants to know how big he is and for me to mock him about it. I'm not about to forfeit any time soon. The whole experience is like a football drill crossed with an intense zumba class all from somewhere in special-hell. We're both covered in a sheen of sweat and breathing hard from the constant exertion. He went flaccid right after the start – all the movement quickly solving any erection distraction problems that may have occurred. The flopping up, down, and side-to-side of his endowment could be comical if it wasn't so mesmerizing. We're closing on six minutes now – maybe I'm better at this than I thought? "Does any body else hear a helicopter? I swear I hear a helicopter...." Past eight minutes and approaching nine. The idea that I might call the shots is a little shocking. I had every expectation that I would lose. "What kind of host are you, leaving your mammoth cock out on the floor like that? Shouldn't that be draped on a couch before I trip on it or something?" And then 'or something' happens – kind of. I technically trip. There's a sensation of my foot connecting with something warm, soft, and heavy but when I look, there is nothing there. I tumble face-first to the mat, confused, and flip myself over onto my back. He's already there, looming over me with a face-splitting grin. He drops into full straddle and his splayed hand pushes hard on my chest, pinning me to the mat firmly. "Gotcha." I squirm uselessly. "You have any other tricks to try or are you ready to call it?" I relax and then try a sudden lunge forward but he already anticipated that. He slams me back to the mat, this time pinning my shoulders down and keeping my legs from bracing against the floor for any leverage. I feel somewhat like a pretzel and seriously turned on. "You win." He leaps to his feet instantly but with a hand extended to help me up as well. I gladly accept and the heady sensation of someone pulling me to my feet causes me to stumble against him. He catches me easily, holding my arms at my sides while keeping me flush against him. The heat of him is sublime, the aromas from our mutual exertion sending a lightning rod to my groin. We're the same height so it's awkward... for me. The weight of his larger equipment forces my struggling cock straight down. I try to wiggle out of his grip but he holds me still, mouth curled up in a slight smile. "Anything wrong?" "I can't, uh" "You can't what?" "I can't get it around...." "Oh? Does something have you pinned?" He's totally gloating. It makes me harder than I thought possible. "Hnnnhnn" "Maybe you need to do some curls." "Dick curls? That's not a thing." I was getting pretty desperate. "Bet you it is. What do you want to bet?" He sounds very confident, like he has a dick-curling-guideline page bookmarked on his laptop or something. "Or maybe we should make sure you stay the bigger man?" "I have no doubt about that. You ready for this?" "Fuck. Please?" I'm practically begging and he hears it. It's not like he's fully soft against me and he's starting to fill even more of the limited room down there. He finally lets me take the smallest step back, freeing my hard-on to spring up at it's full size, just brushing his skin. It's a pretty average five-and-a-half inches long and the same around. We both stare down as he expands freely between us now, his bigger rod gobbling up the new space. His bobbing cock knocks mine aside and under itself as he thickens and lengthens more, taking over the gap to accommodate its superiority. I can feel his slick tip straining against my skin - the sensation is sublime as his foreskin drags itself even wider around the head. "You might, mmmm, want to move back a bit. I'm not. Done. Yet." He finally lets go of my arms and poses with them behind his head. His cock stretches inexorably toward me as it continues to increase in size. He has to be getting close to his full seven-and-a-half inches. He's wider than me as well and my smaller cock is totally eclipsed; I can only see it if I glance from the side. The moist tip makes contact with my pubic bone again and I can't help letting out a moan. I rock ever so slightly against it, sure that he's done but I'm wrong. It pulses back and widens just a bit more. I am absurdly close to cumming. "You were right about the Chase. I don't think I have ever been this hard. Your smart mouth might even make me harder than seven-and-a-half. What do think about that?" "Fuck. Me." He smirks at that. Not like my response was unexpected. "If you're lucky. Guess I closed the gap...again. Fuck, just look at me." And I do, of course. There is no denying my absolute worship as I take in the flexed frame of his arms, his heaving chest, the spectacular trail down his abs, and, finally, the monolithic main event pushing against me between his meaty quads. It's physically digging into my pelvis now, ever harder and precious like diamond. He flexes his hips just slightly against me and the unbelievable happens. I stumble back and fall on my ass. The view is godlike, looking up at him like this and I moan again. His eyes go glassy and wide with unbridled lust – he just knocked a person to the ground with the size and strength of his fucking manhood. His first volley catches us both by surprise. It launches over my head almost entirely, a little of the trailing end catches me on the cheek. My mouth reflexively opens wide in shock and anticipation. He grips his cock for the second volley – he has excellent aim and now I've presented him with with an obvious target. It splashes against my lips and into my open mouth. That's more than enough for me to release my own shot up at him. I spray against his legs a few times as he lets loose at me with the rest of what he's got. The third makes contact against my chin spilling upward into my mouth again. Number four and five paint my chest and torso. He pushes out a sixth load and soaks my balls and cock. Seven is finally a dribble but even that makes its way onto my left ankle and heel. "Wow." He finally manages to pant out. "Yeah. Wow, alright." I have to lick my lips and wipe my face clear before I'm able to reply. I slowly get to my feet and cup my softening dick in my hands. The smell of him is everywhere - I have never been so thoroughly and utterly marked. "Um, you need to get over here." There's an edge of fear in his voice. "Hey, I'm right here." I finish clearing what I can from my face as I come closer. "Good, 'cause you, uh, you need to see this...." He's still holding his hard cock firmly in one hand and it isn't softening. The head of it is pulsing, the skin drawn taught and engorged with blood. The pulsing isn't just that though. His dick swells more, inching it's way ever closer to me with each second. It broadens in his hand, forcing the fingers apart. He starts making a low keening sound in the back of his throat and his eyes begin to flutter like he's cumming again. I brace him by the shoulders as his cock spools out inhumanly between us – when is it going to stop? His body gives a final tremble and his eyes widen as he glances down. He now has more than twice the cock I do – it might be obscene if I wasn't so hard from it. He touches the tips of our cocks together and the size disparity is comically overwhelming. His face breaks out into the largest smile. "I don't know what you did, but thanks? Like the upgrade? Wanna touch it?" I barely get a finger on the fat, meaty shaft before I'm cumming again against his tip. It's more of a dribble this close to my last one but he takes the obvious appreciation in stride and pulls me for a hug. "I'll take that as a yes." The hug is a little strange with his gigantic erection pressed up between us. It's only a momentary reprieve. We start to feel his dick pushing upward between us and rush to take a step apart. "Umm, you're not quite done yet?" And that's when I notice it. It isn't his cock getting bigger, it's him. His eyes have already risen so I'm looking just below them, then his nose comes into my line of sight, then my gaze falls to his mouth as he continues to grow, and then, at last the top of his chin fills my field of vision. I have to tilt my head back to see his eyes now. "Fuck. Yesssss." He closes his eyes in bliss as his body starts to gain mass next. The winter bulk melts inward so his already significant muscle mass stands out sharper relief beneath a thinner layer of flesh. His traps rise up a bit and the delts round out significantly to cap his frame. His already ample chest broadens further and the serratus start to pop into view down the sides. The biceps swell into higher, broader peaks in balance to the horseshoe expansion of the triceps. His lats flare out deliciously down to the taper of his waist while his abdomen sculpts itself into a gorgeous six pack. His quads are even more corded now and the calves have a diamond pattern below the skin. The growth ebbs to a halt leaving him a comic-book super-porn version of the already attractive man he was. I reach out in wonder to turn him around but he tackles me to the floor without warning. He straddles me again, this time with his knees pinning down my shoulders. His new-and-improved mammoth tool finally softening directly in front of my face. "Kiss it." Fuck did I want to. I close my eyes for what feels less than a second. Would that even be safe right now? What if he grew more, while on top of me? Would he just crush me into the floor? My panic must have been unmistakable because I hear his voice prompting me back to reality again, just like with the shirt before this began. "Hey, get out of your head. Open your eyes aaaaand, there you go. Not that tough, right? I don't think it'll bite. And really, you kissing my big dick is totally the kind of thing we've role-played online 500 times. Now there's just, more of it. So, here." He leans down a bit to make it as easy as possible in my pinned state and my lips meet that gorgeous, swollen glans. The skin in scorching and salty. I tease the tip of it with my tongue milking out some remaining semen. He takes a sharp breath in and I feel his cock stretch into my mouth. He pulls back quickly and leans back on his haunches. "Easy there, I don't think we're ready for what might happen if you do that yet. You remember what I said earlier?" Huh?" I'm entranced by his heavy, pendulous endowment swaying with each movement. "Eyes up here." He grips my jaw and moves my head so I look him in the face. "Who owns you?" "What?" This wasn't exactly my brightest moment. "I said, Who. Owns. You?" He's growling again. "You do. You won." It was a theme we often used. My final submission to his superiority – thank fuck he wasn't an asshole. It seemed even more fitting now that he was, well, this new version of himself. "That's right, you're mine." With that he leans back on his heels, grips my by the chest and stands up with me in his hands. "Hmm, that was easier than I thought." "Easier than you....?" I limp over to the island to steady myself. "Did you even look at yourself to see what you look like?" "Not...really? It's not like there's a mirror in here or anything." He's right of course, but shouldn't he have been able to see some of it? "Bathroom. Now." I start toward the door without waiting to see if he follows. "Mmhmm?" "Stop weighing your junk from hand to hand like a slinky and come look at yourself." "Yeah, but it's soooo heavy." He shoots me the dopiest grin. "So's the rest of you." "Is that a fat joke?" "Just get in here, you dork." "Fine." I hear his heavier footfalls approaching the door. He steps into the room and really looks at all of himself in the mirror. "Is that all me?" "Uh-huh." "Holy fuck! Really?" "Well it sure as shit isn't me, so, yeah." "That sass though. You sure that's such a great idea right now?" He flexes an arm into a curl in my direction, pitifully failing at a scowl. "Seriously? Get on your scale. We have to see this. Do you have a tape measure anywhere? Bedroom maybe?" He hustles to get on it, I have his full attention now with the numbers. "Right there, top drawer." Of course he kept in his bathroom. I fish it out while he steps gingerly onto the scale. "I can't. Fuck. I'm...I'm 260 pounds...." "Mmmm. Sweet daddy fuckballs!" "Did you just say sweet daddy fuckballs?" "Shut. It. Stand straight against the wall." He complies, silently for once. He's taller than I thought – this could be a bit of a challenge. "Need a stool?" He teases. "You'd like that, wouldn't you?" I look around the room for something to make a mark with. "...Maybe.... I mean, I'm already this big so what's a bit more?" Bullseye! Tweezers would do to scratch the paint the tiniest bit. "Okay step out from the wall." He hovers over me as I stretch the tape from the floor to my mark. I suck in an audible gasp. "Well, what is it?" I turn to look at him, shock and awe apparent. "Six feet, four inches." His giant dick is coming to life now. "Holy fuck! That's like six inches more I was!" Up and out his member comes. Fuck that thing gets huge. "And 260 lbs." Full on cock surge from him now. "That's like half a foot taller than you. And 110 pounds heavier." He eyes me up and down, calculating something while he gently strokes himself harder. I am positive it will be something devious. I also don't fail to notice the switch from inches to feet to describe how much bigger he is than me. "I'm measuring that next, aren't I." "Mmmhmm" He smirks and keeps stroking. I hold my breath as I do it. I don't think I would be able to stop fondling it if I were breathing normally. It was hot, thick, heavy, the skin blissfully silky to touch. Pulling the tape against it causes a final surge to its magnificent size. The flesh is unyielding as I wrap the tape around his giant fucking hard-on. It takes conscious effort to let out the breath I forget I was holding. I look up at him in full, mindless reverence. "You're 11 inches long and 7.8 inches around. I can't...I can't even...." He leans forward and grips my ass with his larger hand. Lifting me up onto my tip-toes so he doesn't have to lean down as much, he places his lips next to my ear to whisper. "I don't know what happened tonight but, you know what?" His voice is lower, throaty, indescribably sexy. I quiver at the sensation of the sound and breath against my ear. "Um, yes...no...I don't know?" Fuck. Words were not going well again. He was holding me so tight to himself that I felt the corners of his mouth raise into a smile. "I. Want. More." I was so fucked. Was beyond so fucked because I knew what was coming next. "You want to help me get more, right?"
plausible size difference
questionable consent
Priapus Pictures
Tattcub posted a topic in Stories
Hi All, This is a story that was inspired and given permission to play in by one of my fave authors Absman420 Anyway I hope you enjoy it Parts 1 to 5 Have fun TC I work for a guy named Saul Bennett. He's sort of a modern-day porn maven. Lots of money, loads of girls and guys. No morals. I work for him because I have no choice. (more on this later) I don't think anyone here does except maybe the guy who does all his hocus pocus for him. He's one scary bastard. Here's a story about one of the guys who got in his way. It wasn't long after Saul had fired Mick Masterson, he had been a real top dollar star for Bennett till like all things Mick's body had started to go south on him. Nothing major as the guy still had fans and his bod was still great just his age. That was Saul's excuse anyway. I heard him say it was because he'd fallen in love with some guy, pretty bodybuilder type and Saul didn't like split loyalties. Anyway, on with the story… James Fraser is or should I was an entertainment lawyer working out of west Hollywood. He was the guy who did contracts for studios big and small tying their actors up in red tape, so the studios got their monies worth. Except James was a rare type. He actually cared about his clients, so he had a specialisation that was more about helping the actors get out of their deals with the big sharks who own these studios. For example, his most recent case was against one Saul Bennett and his studio Priapus Pictures. This guy came to James about a contract he had signed some time back with Priapus and wanted out. His partner had just died and he just didn't have the heart to carry on in the industry any more. Bennett refused to cancel his contract even though the guy was happy to pay any penalty fee for doing so. So, the client came to James and asked for representation "James, your 2-o clock is here." Said his assistant Diane as James re-entered his office after his lunch with another client. "Thanks Di. "said the 33-year lawyer smiling as he took his jacket off and threw it over the chair in the outer room. He was a good-looking man standing about 5 ft 9 tall. He had dark hair cut short on the back and side and had twinkling green eyes that always seemed ready to smile. His taut athletic body was a good swimmers build from many hours in the pool at his apartment building and light gym three times a week. He was well liked by the entire company, always willing to help out and fight for any underdog cause that took his mind. He was a good man. He opened the door to his office and went in to greet his client Manuel Cortez. Manny to his friends. Manny stood and shook hands with him. "Hi James or is it Jim?" Said Manny smiling nervously as he stood to greet the lawyer. "Please, please sit and it's James." Said the lawyer smiling. "My dad was Jim or Big Jim as everyone called him." Said James as he made himself comfortable at his desk and reached for the clients file. Manny nodded in acknowledgement and said " So, any news on my case?" James looked at the last page on the file which contained a vitriolic letter from Bennett's team about what Manny and James could do with their suit. It was pretty much a good luck and see you in court letter. James explained this to Manny and the he just sat there and hung his head. When he raised it again to look at James he had tears in his eyes. "I can't do this James, not any more. The things that he gets us to do." Said the crying man. "It's not that I think they're disgusting, it's not that at all. I'm a gay man and gay sex and experimentation is fine. I'm even fine with gay does straight. It all pays the bills doesn't it and I like the sex. It's none of that. It's the fact we have no choice about what scenes we do. I'm pretty easy going and my partner used to just say go with the flow as we were both earning well out of it. Now he's gone I just don't feel it any more. "Manny wiped a tear from his eye as he talked. "When I said he gives us no choice I meant it." He paused for moment to catch his breath. "When we're on set something seems to come over the cast. No of us ever seem to argue with the directors on set. Ever. Something's going on and it's scaring the shit out of me. I don't want to be there. Since Rico died It's like veil was lifted from me. We were a great duo, did loads of pics together and we loved doing them. We never questioned Saul's methods because times were good and if I'm honest we were pretty high most of the time too. Saul provided all those sweeties too. Said he had his own alchemist as he called his dealer I think. I think there was more than dope and china in that mix because we all just toed the line you know ?" Finally stopping and taking a drink from his water glass. He looked over at James to see how he was reacting from his total honesty about this situation. James sat there for a moment, gathering his thoughts. "Are you saying that Bennett actually forced you all into sexual acts and drug taking ?" he said "No, it's not like that it's more that he gave this stuff away for free and we didn't even think to object to any of it." Said Manny. "It's only now that I haven't been on set in a couple of months that I am finally clear headed. There worst thing was I didn't touch any of that shit to start with. I was so nervous in the beginning I could only drink water on set." He finished. "Look, Manny this could be a criminal case too if Bennett can be proven to be a dealer or that he is somehow forcing you all to work against your wills." He Stood up from his desk and closed the file. "I think I should meet Mr Saul Bennett for myself and see what's going on at Priapus Pictures." Manny stood too. "Look man, please be careful. This man is dangerous and has a lot of power in this town. I wouldn't want anything to happen to you for getting involved." "Don't worry about me Manny." Said the lawyer reassuringly. "I've dealt with guys like this before. I have my own contacts in this town too." They shook hands and Manuel left the office. Turning and nodding his thanks before he closed the door. James picked up his mobile and dialled his wife "Hi Hun, really sorry I think I'm going to be late home tonight. Yeah…. A new case…..Don't worry I'll send out for something. Okay…bye…Love you too. So James had a plan, of sorts. He needed to rattle this guy, to get him out in the open and to make a mistake somewhere. A mistake that he, James could capture and exploit. He picked up his phone again. He had just the guy he could use for this job. "Hey Max." He said smiling. "I've got a job for you bro are you up for it?" He asked. "Ever heard of a guy called Saul Bennett? His company Priapus Pictures?" "Yeah, that's him." He said to his long-time friend Maxwell Calder. Max was a private detective and old friend of James' from way back in their college days. "I need you to do me a favour. Do you still do undercover shit ? I know you still love that James Bond stuff." Said James smiling as he remembered his friends first forays into the field as an eager much younger man. "Well, we should meet up and talk. Lunch tomorrow? At Deano's ? Yeah..Haha…" He laughed at his friends "Where else?" Comment . "Okay buddy. Yup, about 1.30 ?" He hung up. James looked back at the file on his desk. He had a few other calls to make. Insurance was always a great idea in this town. L.A. was not forgiving on the careless or the over eager. He had a long night ahead. He picked up his phone once more and dialled. It took a moment or two to answer. "Mz D." He said to the feminine voice that answered. "Good to speak to you." "Why James. "Said the throaty voice at the other end of the line. "It has been a while. It could be said that you have been remiss in your attentions." She chided gently. "Awww Mz D. You know you truly are the only woman for me." Said James smiling as he reacquainted himself with the accent and manners of the lady on the line. One who also had very, singular talents. One's that he, James felt were going to be needed sometime very soon. James sat back in his chair and turned to look out at the skyline onto L.A. and its sunset of promises, promises that all to often turned out to be smoke and mirrors. Well, maybe not this time. With any luck. "I need a favour sweet lady." Said the lawyer. MAX. Maxwell Calder was a a private detective in Lala land. He looked at his file on Priapus and Bennett and wasn't surprised by what he read there. He'd been through several of these joints in his years investigating Los Angeles seedier side. He was still young enough at 35 to still be open minded and old enough to not be surprised any more. He'd done his background checks with his various contacts and knew for damn sure that Bennett was dirty and had his fingers in many, many pies. Porn, mainstream and "other." Every form of fornication and filth was bread and butter to this guy and he'd made a fortune from it. James and by extension Manny James' client had warned him to be careful as Bennett was well known to be resourceful and unforgiving of betrayal. Max had decided to go in undercover on a low-level crew type security or roady type, etc. You know the drill, the polite escorting of over eager fans from sets etc or the occasional diverting of a loved one/ partner away from the stars dressing rooms when they were "resting" or otherwise engaged. Purgatory were big enough that they recruited regularly, and this industry was full of people trying to "break into the biz." He managed to get himself on a crew doing general security for a new movie that had the working title of New Fish. This was a prison scene in a mock up studio version of a prison block. It was going to be one of those gang bang movies. Max wasn't gay himself but the thought of seeing a room full of guys wasn't totally his thing. However he was professional and had no issues with gay guys at all so it wasn't a problem. He was due to start at the studio at 7am the next day in a studio lot in some warehouse district somewhere. Knowing what this Bennett was like meant Max wasn't going to go in wired or carrying a camera. He just had a mobile phone that took good pics and also recorded sound a lot longer than most of the other phones of it's type. He picked up the phone and called James. "Hey buddy, it's me." Said the detective. "Yeah, all sorted. Job starts tomorrow. Aha…Yeah low level security/go-fer sort of thing." "Ha ha…Very funny. I did it for you in college didn't I why wouldn't I run around for others for the job ?" he said smiling at his friends comments on the line. "Listen, if we start this we have to see it through. This guy has power and contacts. You know how much that's worth in L.A. don't you?" warned Max. "Yeah, I will. You too James. Don't expect to hear from me for a few days. Speak soon as I can." He hung up the phone. He looked at himself in the mirror. In looks he had that almost faded Cali surfer look going on. Slightly longer than normal naturally blond hair and clear blue eyes. Gave the impression of blankness which was handy in his business as many people underestimated him. He could turn on the dumb if he wanted to but in this instance, he decided to be a man of few words. He got his gear packed and decided to hit the shower before his job started tomorrow. James had waited about 4 days before picking up the phone and calling Max's number. As he had expected he got the answer phone. Max always used burner phones when he worked so his real one would be stashed elsewhere for now "Hey, buddy. Just checking in with you. I know you don't like to break your cover but give me a call when you can. Just looking for a prog rep." He put the phone back on the desk and started to work on the papers laid out in front of him. He just had a niggling feeling about this gig with Bennett, that it was going to be a lot bigger than he wanted. He leaned to the intercom. "Di could you bring in todays mail and a bottle of water please?" he asked his assistant. She came through the door moments later with a bundle in a tray and a bottle of cool water. "Here you go chief." She said smiling. James smiled at her. Laughing and shaking his head. "Let me guess, you're auditioning for a part later ?" he asked. "How did you guess? " She smiled sassily as she handed him the mail. The top of which was crowned with a brown padded envelope. He took the bundle and the water. "Haha…I dunno, calling me chief was the clue. Reporter maybe?" he said looking up at her with raised eyebrows. "Wow you should have been the P.I. I'll be gone for a couple of hours but will come back to finish off later if that's okay ?" she said as she turned to the door. "No, no don't worry about coming back. " He told her. "It's Friday you may as get your weekend on early." She smiled as she turned. "Thanks chief." And gave him a mock salute as she went back into the outer office. James looked down at the pile of mail. Looks like his weekend was not going to start early at all. His eyes were drawn to the envelope. It was hand written, badly by the look of it but it was familiar. It was also unstamped which meant it had been hand delivered too. He opened up the packet and found it contained two things. One was a cell phone and the other was a micro SD card. On the phone was stuck a note. PLAY US BOTH TOGETHER. He paused for a moment and looked out into the main office. Diane had gone so he couldn't ask who had dropped this off. He looked at the phone and switched it on. It didn't seem to have any service but could still be opened up. He put the SD card into his laptop and scanned it first, just in case. It came up clear. He clicked on the icon for the card when it came up and it came up with one file that read. PLAY ME. He did so and the screen blanked out for a moment and started up a slide show. PRESS PLAY ON THE PHONES SOUND RECORDER NOW. James did so. The voice that came through was disguised. "Mr Fraser, good afternoon. If my calculations are correct you should be receiving this package at sometime around 2 pm on Friday afternoon." There was a pause and the sound of someone making noise in the background. They sounded angry and sounded if they might be gagged or being stopped from talking. "I believe we have a friend of yours with us. He was found snooping about and before you stop this and call the police I can assure you that by the time the end of this recording has finished you will be aware that your friend Mr Calder is more than happy to be with us." "In fact you will be the first to witness our latest movie. It's a prison gang scene about a guy who's caught undercover by his fellow inmates. " The screen lit up on the laptop as the slideshow continued. On the screen was a man tied to a chair with his hands behind his back. His face was covered with a towel or something like it. His clothes were rumpled and the sleeve on his shirt was torn and bloody. Two very large muscular men stood either side. One of them was a very well presented man in a prison guards uniform. His hair was very short on the back and side and brushed back 1950's style. He was very athletic having the look of sportsman or coach in a uniform. He was good looking in a sexy daddy sort of way. Tached and salt and pepper colouring. The guy on the left-hand side was huge and dressed in an orange prison jumpsuit that hand the sleeves cut off. There was no way they would ever have been able to hold the arms that the man had. He looked like he had be hewn rather than born. Grown out of the stone floor he was standing on. His shaved head gleamed in the light of the room, accented with a heavy black goatee beard and pale grey eyes. His arms were folded around his chest and he seemed to unconsciously be flexing them. He was about 6ft and seemed to be staring like his guard companion blankly out of the picture and at James himself. "Mmmmmf….mmm…mmesss." Said the panicked voice of the person under the towel. "Yes, yes. " said the dark voice. "Warn him all you like." It said nonchalantly. "It won't make any difference." It paused for a moment. " Mr Fraser, please be aware if you attempt to contact the police or even stop this recording before we have shown you our work please know that they will find nothing and you will not be seeing your friend Max again." With that the next picture came up and it was of the guy in the chair and it confirmed James' suspicion as Max was revealed. His hair was a mess and he has a wild look in his eyes. He looked genuinely afraid. He had a bloodied nose and what seemed to be a black eye. His mouth was gagged with what looked like a rubber ball gag, a standard BDSM toy on a set for a porn film. Max looked out at the camera from the still photo as his voice was heard in the background. "We're just getting Mr Calder ready for his close up James." Said the sinister voice as the next picture was of the two large men ripping Max's shirt off. "First a little, preparation. Hold him." The next picture flashed up and was of the prison guard injecting Max with something just into his neck while the prisoner bull held him steady. "Mmmmmmm!....Ngghhh…!" James heard Max's muffled cries…Ove the next few drawn out minutes they got weaker and weaker until he heard a subdued groaning coming from his friends mouth. "Ahh that's better. Now to work. Max, Max can you hear me ?" Said the man. "Mfff." Said Max. The next picture flashed up and was of Max staring into the light of the camera with a hooded expression to his eyes. They seemed a little dull, even in the bright light. James rubbed his eyes. He couldn't believe what he was seeing. This bastard had his friend prisoner and was torturing him. "Now, James. Don't do anything rash." Said the voice again. "Take a sip of water and calm down. I'll give you some instructions after this is all over and we'll try and resolve this situation." "Now." Said the stranger. "Where were we? Oh yes. Take the gag off. He shouldn't be much trouble now." "Max,MAX…Can you hear me ?" "Huh ?..." said the slurred voice of his friend. "Look at me Max. That's good. You're a good boy well done." "Good booooy…" said his friends voice. It sounded deep and slow. Probably a result of the drug the bastards had given him thought James as he listened horrified and dreaded the next picture. The dark man's voice continued to talk to Max. "Now Max, in a moment we'll untie you from the chair and you will continue to sit there. Calm and still, calm and still. Do you understand?" "aarrrm and stiiilll…" came the reply as a new pic appeared. This was of Max. He had been untied and was sat in the chair and was looking up into the camera lens. James saw a little drool dribbling from his bottom lip. "That's good Max, very good. Now these two men are your friends and you want to help your friends don't you ?" "Aha..friends." Max's voice sounded thick and heavy, almost childish. "Stand up Max and let them help you undress. You'll be far more comfortable like that trust me." Max had obviously obeyed as the next picture came up. This time he was just standing there in the room, which did seem like a prison cell. He was stood there naked. The blank expression on his face, the drool and what seemed to be a raging hard on. "Well now, someone's happy to see his two friends isn't he?" said the monster on the phone. "They're happy to see you too. Can you see how happy they are Max ?" he said. James could feel the smirk in the bastards voice. He was enjoying this. James' mouth was dry. He took another sip of water from the bottle. He didn't want to continue but had no choice but to listen and watch as his friend was manipulated. "Max, in a moment you will start to feel a little warm and tingly. It'll feel a bit like when you go to the gym and have a good workout." "Mmmhhmm." Said his friend. "You understand ? Good." Said the man. "You're going to start to change and I want you to feel and see that it is all normal. This is a good change. You have nothing to fear." Said the man "Do you understand Max." he questioned. "No fear." Said Max's slow voice. It seemed more sure now but was also somehow deeper than James remembered. "Now I want you to kneel down and show your friends Adam and Rico your appreciation." Said the man. "You know what to do." He instructed. "I know…" The next picture flashed up and was off James' friend of many years. His college buddy was knelt there naked on the floor of this prison cell and had the cocks of the two big muscle men in his mouth. His eyes were wide and staring up at the two men. Slobber and drool from his efforts and the men's actions were dribbling off his chin. The screen blanked out and there were the sounds of someone giving and being given a blow job. This seemed to go on for a long time and the sounds of passion were punctuated with grunts, the sounds of animal passion. James tried to get the picture out of his head, he didn't know what to do but couldn't take his eyes off the screen and couldn't get the will to switch of the recording either. The sounds of the men's passions exploding brought James out of his reverie. "Ah…fuck…fuck…fuuuuuck…yeah…Give it to me." Said Max's voice. The two mens voices were just grunts. The sound of them cumming, hard was unmistakable. The screen lit up again with Max's face taking up most of the screen. His sweaty hair was plastered onto his face and his blank eyes stared at the camera in gratitude. On his face was what look like a gallon of cum. It looked like the poor man had been glazed. The screen went dark again. SWITCH OF THE LAPTOP. CONTINIUE TO LISTEN. Instructed the screen. "Do I truly have your attention James ?" Said the creepy man's smug voice. "Good. If you want to see your friend again and resolve this situation without further, action. Here are your instructions." James sat in horror and listened to the words. PART 3 "James?" said a faraway voice. "James, are you okay?" It asked. Louder this time. "JAMES!" shouted Diane his assistant. Bringing the lawyer out of his reverie. What had he been….? "Shit" he shouted loudly making Diane's concerned even more apparent as she touched his shoulder. "Are you okay boss?" She said worriedly, She'd come back to the office after her audition to grab some bits when she saw her boss just staring at the screen on his laptop. Which was weird in itself as the machine had gone into screen saver mode. James seem to be dazed. He ran his hands through his hair and shook his head. Trying to bring himself out of his daze. "Sorry Di. Dunno what happened to me there." He apologised to his worried assistant. "Must have dozed off or something." He said, not sounding entirely convinced by his own excuse. "Are you sure you're okay?" She said, making sure. "Yeah, yeah. " He assured her taking a swig from the now warm water bottle on his desk. "I'm fine. Must be tired or maybe coming down with something." He stood and straightened his tie. Looking at her with his beautiful green eyes. "Look, you head off home. I'm gonna head that way myself. "he said." Don't worry about me I'll be fine." He grabbed his jacket and popped his laptop into his bag then opened the office door and wave her out. Diane grabbed her coat and bag from her desk. "As long as you're sure you're okay. She said heading out the door. "See you Monday." He said smiling at her concern re-assuring her he was okay. "Monday." She said and was gone. James lent against the office door for a moment. His head was clearing. He remembered everything that had happened but had been unable to say a word of it to his friend and assistant. What had Bennett done to him ? How much power did the guy have that he could just get him with some pictures and a recorded voice ? The voice on the line had given him some instructions which he felt compelled to obey but they were vague, almost dreamlike in the exact memory of them. He knew he'd remember only when Bennetts voice wanted him to. He knew his first port of call was the gym where he went every day and often worked out with his old friend Max. Max… "Shit Max!" said James and ran out of the door. 25 minutes later he pulled his car up to the lot behind his gym. This wasn't an exclusive club or true iron gym it was a mid range place you could find in many towns and cities. It was busy enough not to be isolating to the customer and quiet enough that you could usually get on the machines you wanted to. It was about 4.30 and was beginning to fill with the pre-weekend crowd. The gym was called X-WORKS. He tapped his card as he went in and went into the changing rooms and got changed into his gear. Loose light blue vest and dark blue gym shorts just to above the knee and pale green trainers and sports socks. He wasn't worried about the whole gym fashion thing. He was looking over the gym floor and trying to decide where to start He went to the stretch mats carrying only his gym towel and a bottle of water. He started to warm up on the mat, slowly stretching his long swimmers muscles. He did a good fifteen minutes warm up and then got up ready to go and went to the cycles for some warm up cardio. As he did his eyes scanned the crowd of Friday nighters that were in the gym. After work dads and moms having their one free hour before the weekend with the kids. Older people walking quickly on the treadmills showing they still got. High schoolers trying to show how much they could lift. A whole mix of people including the pre-club workout crowd trying to get a pump before showing off their glistening bodies to anyone who wanted to gape at them. His eye was caught by one guy. Tall, blond type with blue eyes. This guy could have been a surfer if not for the muscle he carried. This guy was stacked. Must've weighed 265 if not at least 270 pounds. This was hard earned muscle, this guy was shredded to perfection, striations and veins cut across all of his major muscle groups his chest, shoulders and arms were covered in thick rope-like veins. His forearms were so thick that James realised he'd been staring at the guy. He looked away before the big man noticed. He went back to cycling and tried to concentrate on the tv screen ahead of him and started pedalling faster and faster. He was in the zone about 10 mins later when he heard a voice close to him over the gym's loud music. "Hey buddy." The voice said. "Any chance of helping me with a spot?" The deep voice asked James turned his and slowed his pedalling and there, next to him was the big guy he'd spotted earlier. Closer up he was a magnificent specimen of manehood. James wasn't gay but given his industry he was open and could still appreciate a male form. This guy was hotness personified. "Can I get a spot?" said the guy in a slow measured tone that seemed to be coming from somewhere deep below the ground. It was that type of voice that when heard managed to get most gay guys and straight woment right in the private parts. You know what I mean ? James stopped pedalling and took the guy in with his eyes. "Yeah, sure thing." He said. The big guy slapped him on the shoulder and James swore it went all the way through him the the floor. "Thanks buddy, the name's Mack." Said the behemoth. "Well Mack." Said James dismounting from the cycle. "It's good to meet you. " He said extended his hand to shake, The big guy stood there a moment and looked at James' hand as if confused and unsure what to do with it. Then it was as if he was receiving instructions from someone else he laughed. "Aha aha aha!" It was loud and forceful and lacking in any real intelligence. A true Jocks laugh. James groaned inwardly but he'd already agreed so followed the big man to a weight bench lined with free weights. "You can jump in too if you like." Said Mack "You've already done your warmups from what I could see." "Yeah, sure thing . " said James. What're we doing?" he asked the big bodybuilder. "Chest." Said the huge man. Getting his barbell ready with a warmup weight. As he settled on the bench and set it flat he lay down and looked up at James. "You okay with this weight to start?" "Yup. It's fine by me." Said the lawyer. It was a warm-up so he wasn't worried. The big guy pounded and James pounded out the warmup sets. Slightly increasing the weight each time. James or Mack counting out the others reps and helping at the end as required. Which wasn't often. James was surprised how much he was enjoying this. He hadn't worked out with anyone for a while expect with his pal Max…Max…He paused a moment.. Something started to niggle him. "Hey Jimmy you in there?" He felt a light tap on his head as the big guy got up from his last set and pretended to knock on his skull. James came back to himself."Oh…hi..sorry was miles away." He said. Where was he again? Oh yeah the gym. "Aha aha aha." That laugh again. James thought to himself. It was quite appealing rather than annoying. Quite sexy really. He sneaked a glance at the big guy as he turned to alter the weights again. His back was massive. He wore an old school, faded gold, World gym top that had seen a lot of use and what seemed at first glance to be compression shorts. On closer inspection they turned out to be jersey shorts stretched so tight James could see the veins on the man's thighs and his ass was spectacular. "Good view from back there buddy? " said Mack. "Oh err.."James would normally be flustered and rather than deny he had just been scoping out the mans ass he said. "Hell yeah." He blushed and felt awkward in the same way a teenager on their first date feels at kissing time. What was wrong with him? He could feel the small voice in his head saying that this wasn't him but he also wasn't really listening. "You're up Jimmy A bit more weight this time." Said the beast James meant to tell the guy it was James and not Jimmy but didn't want to spoil the moment by criticising the guy. He lay on the bench. Mack moved close the head of the bench where James' head was and lifted the barbell and lowered it to James' waiting hands. "Don't forget to breathe Jimmy." Said Mack smiling as he looked down and stepped closer to the bench and James head. James could just see the guys crotch at the top of his vision and his mouth went dry." "1-2-3.." counted the big man towering above him. James was enthralled by the play of his own muscles even as he watched the big guy stretch and lift above him. Time seemed to blur. James wasn't sure whether it had been a minute or an eternity when the set had finished. He felt dizzy for a moment and sat up slowly. "Take it easy buddy." Said the slow deep voice of his training partner. "Here, take this. You gotta be thirsty." Said Mack handing him a bottle of unopened water. James looked down to his and realised it was empty. When did he finish…? His chest felt like it was on fire as did his shoulders and arms. He'd only been working chest with this guy hadn't he ? "Well we're done for the day." Said Mack. "Huh…?" said Ji-j-James dully. He was stilled dazed from the workout obviously. "We've only just started.." James said. "Dude, we've been at it for nearly 3 hours. You said you weren't into heavy weights as a swimmer or sumthin' I knew you were a kidder." Said the behemoth who was now pumped to the Max… Max…Macks.. MAX! For a moment Jimmy looked at the big monster of a man in front of him and thought he knew him from somewhere else. "Max?" He question dully as they walked to the changing room. The big guy turned around and smiled at him. Big, toothy vacuous grin. "I was." Said Mack. The room was empty apart from the two of them. Mack was ripped and now pumped. He looked like a god stood there in the white light of the changing room. "Until you sent me to meet Saul Bennett." Said the big, muscular beast of a man. He took off the vest. As he did so all the muscle in his arms and shoulders bunched and flexed and he struggled for a moment to take the shirt off over his huge wing like lats. He laughed at that. That laugh again. For some reason the laugh made James/ Jimmy hard. He felt it in his groin the moment he heard that sound. He could feel his cock begin to stiffen in his shorts…? He looked down and realised he was wearing different clothes from when he'd entered the gym. His gym shorts were now tight under armour compression shorts and his t-shirt had gone replaced by a cut off tank. He couldn't make out the logo or words on the bright red cutoff for some reason. On his feet were a pair of Nike hi-tops. Bright red to match the shirt. He didn't notice this difference as he was to enthralled by the man in front of him who had just fished out the biggest cock Ja---Jimmy had ever seen. As he knelt in front of the big man with his mouth salivating he noticed a mirror to the side that showed the both of them. The big muscular man towering above him, Vast shoulder and lats, shoulders more like a range of mountains that held the must unfeasible set of arms Jimmy had ever seen. Massive ledgelike pecks and a thick waist with a slight roid belly look that supported all above. The legs, thigh and calves looked like they'd been hewn rather than grown. Topped off that all over the beautiful tanned body were veins that seemed to show every contour and contrast. Then there was the cock. Mack's cock was a beast to behold. Thick, almost two hander to hold, about 11 inches long. Mack had pulled back the foreskin and the thick, bloated, purple head of the monster was already leaking with precious juices. Jimmy caught sight of himself. Kneeling there in front of his god, salivating. In his gym gear and wearing a red baseball cap backwards on his head. His blank blue eyes staring at the mirror. Drool starting to drip from the corner of his mouth. His muscles were ridiculous. These were not the beautiful muscle of a cut body builder. Jimmy had the muscle of a laborious beast. He was more massive than the man above him. Crouched as he was he looked more Neanderthal than man. His dark eyes looked dim and his hair was shaggy where it peaked under the cap. His brow was thicker somehow and his eyebrow seemed to almost meet in the middle. His neck was so thick that it and his monstrous traps almost seemed to devour his skull. They were so high at the back. His shoulders and arms were almost grotesque in their massiveness. Veins didn't just cross his arms they ravaged them. His chest was beyond human and dusted with dark hair. They were almost pendulous in the way they hung there. The big brown nipples pointing to the ground because the pecks were so huge. He couldn't really see what his legs were like because he was kneeling but could feel the size and thickness of them. It was like they were made from steel. He turned back to the man in front of him and had only one thought in his mind. "JAMES!" said the voice. "JAMES ARE YOU OKAY." It said again. "FOR FUCKS SAKE JAMES. WAKE UP!" it said again. Jimmy no James looked up. "Diane?" He said confused…. Part 4 James shook his head as his vison cleared. "Phheww!" he sighed and shook his head again. "Are you okay," said Diane as she leaned over him, still sat in his desk chair. He was at the office. But he'd just been at the gym hadn't he? His thoughts were so muddled it was taking him a while just to get his mouth working again." "I'm fine Di." He said. Reassuring her in a way that he himself didn't feel. The attractive brunette got a bottle of water from the fridge in the outer office and ran a cloth under the tap in the kitchenette. "Here." She said in a matronly fashion as she handed him the water and put the cool, damp cloth on his neck where it sat on his hot skin soothingly. "I…I must've nodded off." He said look up at her blearily. His head felt thick, muddy and unclear. He just couldn't focus properly. "Do you want me to call the doctor?" She asked worried about her friend and boss. "No, no…I'll be alright. I must be coming down with something." He said. He did feel sore come to think of it. All over. It was a deep ache, almost gnawing ache. Like hunger but different, darker. He couldn't think of the words to describe it. "You sure?" she said. "I can cancel my plans and take you home, it's no biggie. Or I can call your wife to come get you?" "No, it's fine Di. I'm fine. Feeling much better now." He said and it was true. AS his senses returned to him he was feeling better, energised in fact. He stood up and removed the towel from his neck handing it back to her. "Thanks." He said as he started to gather his stuff and pack his bag. Again. Or at least that's what it felt like, déjà vu. "You go and get on with your weekend." He said as he all but shooed the woman out of the door smiling at her. "I'll be fine." He told her again. "As long as you're sure. " She said. She stopped at the door and turned around and put a light hand on his cheek. "You work too hard boss, please take it easy this weekend." Then she was off out the door and shouted as she went down the corridor. "Call me if you need anything." And the she was gone. James sat back in the chair heavily for a moment. He could still remember the gym. He could remember everything that happened in vivid detail. He remember Mack and his beautiful face and godlike body looming over him. He could remember how his gym gear pressed up against the thick, thight muscle of his hard steellike body. Veins running over the muscle just under the skin. Thick, rope like and full of pulsing life. Just like Mack's cock. Oh god the cock. James leaned back in the chair and closed his eyes. His breathing started to catch as he remembered the beast between Mack's legs. Thick and shining. Two hands to hold and dripping with pre-cum. The smell was beyond anything. It was like smelling joy to him. James wanted to run his tongue up and down the monster, he had to. He had to find Mack again. He opened his eyes and realised he was still in the office and he'd just closed his eyes for a second. That was enough for him to re-live that memory and then ejaculate. He'd cum in his trouser. Like a teenager,he thought, now horrified. What the hell was wrong with him ? He'd just had a fantasy about one of his work collegues. No, wait Mack wasn't a colleguege. Max was. Where was Max ? God he was so confused. He needed to go home and rest. He gathered his gear together, realising his gym bag was still there. He grabbed his laptop and stuffed it on top of the used and sweaty top and shorts. He hadn't actually been to the gym. Had he? Not trusting himself to drive he went down the stairs after closing the office and hailed a cab. He got in and gave the driver his address and heard a beep from his Wife. Apparently her sister had gone into labour early so she was heading off to the airport so she could be with her family and support her sister. He called her and wish her luck and told her to send his love to his sister in law. He also told he was feeling under the weather so it was just as well that she went as he would be no fun. She gave him her love and told him where he could get some good chicken soup near their home. He smiled and told her he loved her.He put the phone back in his pocket and lay back on the seat of the cab. He closed his eyes for a second just to still his senses. In his mind's eye all he could see were the massive tree trunk legs of Mack and that monster cock with it's pendulous balls. "Hey Buddy!" said the driver's voice. "Whha.." said James opening his eyes and looked around realising he was home. "Oh, right. Sorry." He apologised as he got out of the cab, he still felt sore and disorientated as he gave the driver the fare telling him to keep the change. He wasn't sure but he thought he heard the guy say "dumbass" as the cab pulled away at speed. He walked up to the house. There was a light burning in the front room and he heard music playing. His wife had probably left them on so he didn't feel so lonely coming home to an empty house. They did things like that. He smiled at the kind thought. He opened the door and dumped his bag on the floor just by the coat rack. There was a light classical music playing and room was lit by warm, mellow lights from the various lamps around it. The person standing in the room wasn't his wife. It was Saul Bennett. He was holding a glass half full of what appeared to be James' own 25 year scotch. As he walked into the room Bennett raised the glass in a salute and took a sip. "What the hell?" Said James. "Not bad." Said Bennett savouring the flavour of the alcohol and totally ignoring James indignation. "What are you doing in my house Bennett." Shouted James. "Get out now before I call the police." He said angrily. "Now, now James. Don't lose your pretty head." Said the monster with a smile. "I just wanted to take a few moments of your time to have a little chat, then I'll be gone." James crossed his arms and walked over to the bottle of his whisky and poured himself a good measure. He took a swig. "Then, talk." He said glaring at the man who had invaded his home. Bennett took another swig of his drink and smiled in much the same way as a crocodile would smile at a fox who'd just ask for a lift across the swamp on it's back. "Okay then.Talk." He demanded turning to keep the stare of this man with a confidence he really didn't feel. "Okay, then." Said Bennett. "We have a problem." He said. That smile again. "You are looking into things that really don't concern you and I'd like you to stop." Said the businessman. "If you do then I will pay you a fee, call it compensation for having to quit Manny's case." Offered Bennett. "No deal." Said James immediately. "Don't be so hasty James." Said Bennett sipping his drink again. "There will be a price to pay if you chose not to take this offer." He said looking over at the lawyer with something akin to pity. James next instinct was to curl his hands into fist with the intention of punching the sanctimonious pricks teeth down his throat. Fisted clenched at his sides in anger he took two steps forward towards the asshole. "STOP!" The words rang out, not just in his ears but inside his head. It was like a chorus of voices said it at once to him and he could only obey. There was no question of any other type of behaviour. So, he stood there. In silence while this man / monster just looked at him, looked at him as if weighing up what to do with a naughty puppy who has mess the kitchen up while his master's been at work. "One word James, that's all that was needed there." Said Bennett. "That's just pre-conditioning. You're in a highly suggestible state right now." "You have been for some time." He said. Pretty much since before lunch time when you ask Diane for a bottle of water. James' eyes widened as he thought back. Then' he looked to the whisky bottle in alarm. "Yes, that too." Bennett confirmed. "I have a certain flair with the dramatic don't you think?" he asked his audience of one. "I've actually had my eyes on you for a while." He sneered at James "You've been of little hindrance until recently, until Manny. "He said. "I really don't like people snooping into my affairs. There tends to be consequences for that." "Oh, you may speak, don't strain a blood vessel." Bennett gave his permission to James. "You fucking freak, how are you doing this ?What have you done with my wife and Diane ?" he shouted. "Do not, above all things, raise your voice to me James." Said Bennett, dangerously quietly. "I cannot abide rudeness". He said. "The ladies are fine and will remain that way. If you hear me out. I have some business to discuss with you first." "Talk, then." Demanded James still trying to move. "What did I say about rudeness Jim?" asked Bennett "kneel!" demanded the dark, smooth velvet of Bennetts voice. James Obeyed. Without a fight. He had no choice. He had to. Worse, he wanted to. It was a primordial urge, one that he could not even think of fighting. He knelt in front of his captor, head bowed in supplication almost. "What do you want from me?" He said, so quietly it was just a whisper. "That's easy Jim, Jimmy even." Said the face that loomed above him. "I want you." Bennett confirmed what James / Jimmy already knew "As I said we do have business to discuss." Bennett pressed some button on his phone. "Yes, you may come in now." James heard the front door opening and senses a large presence behind him. "I believe you've met Mack ?" said Saul Bennett. The figure that came in the door and into the light of the living room was huge. It was Jimmy's friend Max, although not the Max he remembered, apart from in his Laptop fantasy (That cock). This was Max 5.0 This was Mack the monster. Every bit as shredded and packed with muscle as he pictured him from his fevered dream back in the office. "Max ?" he said, not quite believing what he was seeing. The behemoth was dressed in gym gear. A raggedy Golds gym top in faded blue. Straps stretched so thin over the monstrous traps, shoulders and pecks that it just seemed unfeasible a person would be able to move if at all with all that bulk. The pecks themselves were so large that the thick nipples pointed down to the floor because of the sheer mass of the slabs of beef they were attached too. All of this bulk seemed to cinch itself inwards with the shape of the monstrous lats at the back and the ridiculous arms, that Jimmy was sure wouldn't serve any practical purpose other than to lift weight. Biceps, thick hanging triceps and forearms that were from the fevered dreams of the most dedicated muscle fetishists. All of this growth and mass was on top of a pair of legs, that were themselves encased in white full leg compression leggings. The legs so powerful and filled with strength Jimmy could see the veins almost pulsing under the tight white fabric. The monsters' huge feet encased in a pair of white Hi-top adidas trainers. Right at Jimmy's eye levels was a bulge that was doing very little to hide itself. "Jimmy" said a voice that Jim heard as if it was drawn from the bottom of a well. Although he was sure he felt it start in his balls. It was that deep and that hot. Jimmy looked up in to the face of the beast and it was Max, not the nice gentle man he had known for years. This was a creature of stone, hewn, rather than grown. His brow was markedly thicker. His beautiful blue eyes were still intense and held an almost blank animal cunning a lust even. The thick black hair on his head was tousled and rough and longer than he remembered. Mack put a rough calloused hand on Jimmy's shoulder briefly and he nearly shot his load where he knelt. "What the fuck?" He breathed. Realising that it wasn't a question it was more a statement of awe. "Well" said the ringmaster, still holding the remains of his whisky glass. "To business." He sat on the arm of the chair and leaned forward to Jimmy in a friendly manner. "You have a choice here Jimmy." He offered. "You can walk out of here now, tomorrow you will pack up your business and within a month you and your lovely wife will have a new set up in a part of the country you choose. You will forget about any of this and all will be well." Jimmy couldn't keep his eyes off the man beast in front of him. He was that close he could feel the heat radiating from him. "Or?" Jimmy asked weakly. "Or…." Said Bennett drawing out the word. "Or, you will end up working for me as part of my stable. A new and exciting partner for Mack here. Our latest attractions if you will while I'm waiting for another to arrive from another part of the country." "Fuck you." Spat Jimmy (no James) suddenly finding the strength from somewhere to resist the devil in a suit before him. He tried to stand and almost did before a ton of stone seemed to crash onto his shoulders in the form of Mack's monstrous hands. James turned to Bennett and spat at him, caught him clean in the left eye. The air seemed to crackle and turn heavy for a moment as if lightening was about to strike. Then it cleared as Saul Bennett started laughing as he grabbed a pocket square and wiped he face. He shook his head as he looked down on his prey. "Well, I must say I didn't see that one coming. I certainly didn't think you'd have had the strength to even turn your head." He smiled and dropped the square on the coffee table. "So you've made your choice then ? No Job? Can't see us working well together ?" He taunted "Oh well, I did try. I gave you a choice, an out if you will." Said Bennett. "Mack, he's all yours." Said Bennett as he got up and brushed himself down. He headed to the door. "James it's been interesting. Jimmy, I will be seeing you very soon." Then he was gone. James felt the strong hands of Mack again. His attention was again drawn to the huge bulge in front of him. Mack put his hands into the front of the leggings and brought out the monstrous cock that was oh so vivid in Jimmy's mind (JAMES not Jimmy please…) James could smell the odour of sweat and pheromones. Male sweat was emanation from the beast in Mack's hands. He caught the smell of cock and he knew that he had started to drool. He couldn't help himself. He dove on the cock, now released from the hold that Bennett had placed on him. He was now entranced by another master. He gave himself to it completely, he couldn't help it. He took the beast in his mouth as if born to it. Mack put both his thick muscular hands on either side of Jimmy's head (Definitely Jimmy now) and wouldn't let him pull away. He could feel the thick piece of meat grow in his mouth but he didn't care. He didn't care if it choked him he felt he would die happy. Jimmy worked his mouth up and down the huge member, slathered it in his own spit and worked it with both of his hands. Mack started to fuck his face, groaning slightly, in and out. In and out. Slowly at first. Jimmy was moaning in lust. He couldn't help it. He couldn't get enough of it. In and out. In and out. The huge man started to build up speed and his breath started to come in more urgent, ragged breaths. For about 15 minutes they were like this. Jimmy working the monstrous tool while the big bodybuilder alternated between slow and deep and out and out face fucking. Deep throating almost every stroke. Jimmy took it all and loved it. What Jimmy didn't notice was that during this process he appeared to be wearing the gym gear that had been in his bag earlier. It still smelled of the gym. His body had started to slowly grow and change too. Mack reached behind himself as he was getting towards his final strokes and took a bright red snapback out of nowhere and placed if on Jimmy's head. Jimmy had started to wank his own splendid cock now, but was really still riveted on the one that was still invading his face. All thoughts of his wife, Diane, Max and even Saul Bennett were about to be washed away. Mack drew himself up to his full height as he knew he was close. His think veined muscle seemed to jostle on his massive frame as he started to tremble. He clenched all of his power into a huge pose. Most muscular for the gods and with a roar he shot his wad into the mouth of the waiting cock slut beneath him. It felt like hours as he continued to pump the man's mouth until he'd been sucked dry by the dumbass jock on his knees in front of him. Jimmy looked up at his lover with nothing less than vacuous wonder. He drew his hand across his mouth and wiped away some of the drool. He felt clear if not empty headed. His own thoughts now were limited to the needs of the beast in front of him and his master Saul Bennett. Part V The lady sat under a large sun umbrella and sipped at a cup of tea that had been served in a delicate powder blue china cup and saucer. It had the delicate bergamot perfume of Earl Grey and was served, quite properly with a slice of lemon. She was dressed in pale blue herself. Knitted pale blue suit and hat to match. She had a small set of pinz nez on a silver chain around her neck and would occasionally lift them to her eyes and gaze at the world or the people around her. Not that she needed them when she looked at you it pretty much felt like she already knew everything about you and what you were going to say. Her amethyst eyes were laser-like in their luminosity. She sat very properly, as ladies should, with her knees close together and her feet crossed at the ankles. I asked her about the incident with James Fraser and Maxwell Calder. She took a slow, delicate sip of her tea and looked into me for a moment. Then she dabbed her lips gently with her napkin and set it and her cup on the table beside her. When she spoke it was the voice of everyone's Grandma. Gentle, warm and easy. It had a southern drawl to it that said this lady was a Georgia woman somewhere down the line. "Well, my dear you found me to talk to me about it. I wass wondering when we would get to it." She said. "How did you first get involved? I didn't think this would be something you would have got involved in." I asked. "Okay, this was unusual." She said. " I have a lot of friends and contacts around the world. Especially in my field I am a very valued expert. I have a magic touch if you will." "Usually I rely on wish fulfilment or revenge events but every now and then someone will call me and ask for specific help. If it, or they are worthy I help. On this occasion they really needed my help. " She took another sip of her tea. "A week or so ago I had a call from a friend of mine called James, he works in L.A. (A cesspool of scum and iniquity but in the end money is money to some.) He had come across a situation that wasn't really in his purview and asked for my help. "We'd worked together previously, and I'd taught him the basics in how to recognise manipulation, alchemy and magic. "She said and look at me intensely. "Anyway, the call went like this… Oh and yes dear I did say magic… "It does exist as you will find out if you bother to read all of the story rather than trying to skip through to the horny bits that you always do (Yes dear. You. )" she seemed to say to no-one in particular." "I taped the call." she said as she drew out a small recording device and placed it on the table. She looked at it testily when it wouldn't start and then just glared at it for a second. I swear I think the thing started out of embarrassment… Anyway. The call went like this : Ms D "Hello James." James: "Ms D.Good to speak to you." Ms D "Why James it has been a while. It could be said that you have been remiss in your attentions." James "Aww Ms D. You know you truly are the only woman for me." Said James Ms D "What can this old southern gal do for a big city lawyer like yourself?" James "I think I have one of those problems that only you can deal with." Ms D "Intriguing, it has been a while since we crossed paths and I know that I taught you well enough to recognise meddling when it's around." James "That's why I called. I think there's someone who is definitely at it here in L.A." He pauses for a moment on the phone. James "It's a bit out of my league and to be honest I'm a little worried about it. It might be good to have a little back-up." Ms D "I sense that there's something more about this one James. What's the issue my dear?" James "This guy changes people. Actually transforms them. I know it's possible to hypnotise and entrance people but this ? It's high level and way out of my experience. I'm worried. Max Calder is going in tomorrow and I want to make sure he's protected. Look I know you're busy but if you can help in any way…" Ms D "I'll help dear boy, of course I will. What's the name of the reprobate we're dealing with so I can have a little look see?" James "His name is Saul Bennett." Ms D "Saul B.E.N.N.E.T.T ?" James "Yup that's him. He's a big time porn producer with a rep for really owning his guys and girls if you know what I mean?" Ms D "I do indded, go on." James "Well I have heard stories over the last 6 months or so about guys either disappearing or changing enough that their own families and friends hardly recognise them. It's scary. I've met one of the guys whose partner has changed. It's scary." Ms D Pauses a moment Ms D "I can only imagine what the poor souls are going through. James, does this man have two different coloured eyes ?" James "Why yes, he does. How'd you know?" Ms D "Years of experience and a quick search of my database while we were chatting. I may be an old maid dear but I'm far from decrepit yet." James (Laughing) "No-one who's ever met you would call you that dear, lady." Ms D "I've got a couple of good likenesses here now, give me a few hours and then call me back. Also contact Max and tell him to call me before he goes in to. Tell him not to make any contact there unless he's spoken to me first." James "I will, thanks for this. I really do appreciate it. I'll have to pay you back one day." Ms D (Gentle laughter) "You can do so by visiting a little more than you have and maybe taking an old lady out to dinner." James "You'll outlive me I'm certain of that. Dinner's a date. Speak to you soon." Ms D "You shall indeed. Be safe James. This person is dangerous I can sense it. This isn't a stage magician you're dealing with. Bye for now." The Line goes dead The Lady picks up her device and pops it back into her bag and closes it with a snap. She takes a final sip of her tea and looks at me over the rim of the cup. She finishes it and pops it back on it's saucer and on the table. "Well?" she asks "Did that answer all of your questions?" I pause for a minute to gather my thoughts and then soldier on under this woman's intense stare. "To be honest for each one it answer I think I have about 20 more." I admit honestly. I cannot be anything but honest in this lady's presence. It would feel wrong. I look at her as she sits primly and properly in her wicker chair and cannot understand for the life of me where she seems to get this aura of calm, authority from. It's as if someone took every grandmotherly emotion and condensed it into some sort of protective cloak about her. I know that I would do anything for her. It's that sort of feeling. "Why, what a lovely compliment young man." She says brightly and fans herself with her hand. "I find myself quite, quite flattered." She smiles. "I-I, err.. " I stammer slightly. "How did I know? " she asked smiling still. "Come on dear boy." She chided gently "You heard the recording and you've seen the two boys." She said. "It's true?" I asked already knowing the answer I had already, in truth known all along. "All of it, every word." She affirmed. "Wow." I said. "The big question here Michael my dear." She said pausing and leaning forward out of her chair and lightly gripping my chin so she raises my eyes to her blue lasers. "Is what are we going to do about Saul Bennett?"
mind-control
m/m The Jock Hunter
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{"url":"https:\/\/math.stackexchange.com\/questions\/4236281\/radius-of-convergence-of-fz-10-sum-n-1-infty-fracpnnzn","text":"Radius of convergence of, $f(z)=10+\\sum_{n=1}^\\infty \\frac{p(n)}{n!}z^{n!}$\n\nProblem statement\n\nIf $$p$$ is a polynomial with positive integer coefficients, what is the radius of convergence of, $$\\displaystyle f(z)=10+\\sum_{n=1}^\\infty \\frac{p(n)}{n!}z^{n!}$$\n\nMy Attempt\n\nThe value of the constant 10 is irrelevant. I considered the term $$\\frac{p(n)}{n!}$$.\n\nSince any arbitrary order positive polynomial will always increase slower than $$n!$$, I made the assumption that the denominator increases asymptotically faster than the numerator and thus always converges for all values. And the radius of convergence must be infinity when applying the root test.\n\nThis was a problem from my complex analysis assignment, and the correct answer is apparently a radius of 1 not infinity.\n\nHow do I go about solving this problem correctly?\n\nThanks for any and all help!\n\n\u2022 Well, for example $2^{n!}$ grows a lot faster than $n!$ Aug 30, 2021 at 3:28\n\u2022 A common mistake in dealing with this type of power series is to treat $p(n)\/n!$ as the coefficient of $z^n$, instead of $z^{n!}$. Aug 30, 2021 at 4:36\n\nWe consider first the case where $$p(z)=z^m$$, then $$p(n?)^{1\/n}=(n?)^{m\/n}\\leq (n^{1\/n})^m$$. Moreover $$1=1^{m\/n}\\leq n?^{m\/n}=p(n?)^{1\/n}$$. Where $$n?$$ denotes the inverse factorial (or inverse of $$\\Gamma(n+1)$$). What we have shown is that the sequence $$p(n?)^{1\/n}$$ tends to $$1$$ as $$n$$ tends to $$\\infty$$.\nWe then get $$\\lim_{n\\to\\infty} \\frac{p(n?)^{1\/n}}{n^{1\/n}}=\\frac{1}{1}=1.$$\nIt now also follows that if $$a_n$$ is the sequence $$p(n?)\/n$$ if $$n?$$ is an integer and $$0$$ otherwise, then\n$$\\limsup_{n\\to\\infty} a_n^{1\/n}=1$$\nwhich is exactly what we need. Hence the radius of convergence of $$\\sum_{n=1}^{\\infty} a_nz^n=\\sum_{n=0}^{\\infty} \\frac{p(n)}{n!}z^{n!}$$ is $$1$$. Now all we have to do to finish the proof for any polynomial, is to realize that for large $$|z|$$ we have $$|p(z)|=|a_mz^m+a_{m-1}z^{m-1}+...+a_1z+a_0|\\leq |a_mz^m|+|a_{m-1}z^{m-1}|+...+|a_1z|+|a_0|\\leq$$$$|a_mz^m|+|a_{m-1}z^m|+...+|a_1z^m|+|a_0z^m|=(|a_m|+...+|a_0|)|z|^m.$$\nThe estimate now follows similarly with $$C^{1\/n}\\to 1$$ for any nonzero constant.","date":"2023-04-02 05:12:12","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\": 25, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9856646656990051, \"perplexity\": 73.14401078623546}, \"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-2023-14\/segments\/1679296950383.8\/warc\/CC-MAIN-20230402043600-20230402073600-00748.warc.gz\"}"} | null | null |
- [Configuration](#configuration)
- [Handling Errors](#handling-errors)
- [HTTP Exceptions](#http-exceptions)
- [Handling 404 Errors](#handling-404-errors)
- [Logging](#logging)
<a name="configuration"></a>
## Configuration
The logging handler for your application is registered in the `app/start/global.php` [start file](/docs/lifecycle#start-files). By default, the logger is configured to use a single log file; however, you may customize this behavior as needed. Since Laravel uses the popular [Monolog](https://github.com/Seldaek/monolog) logging library, you can take advantage of the variety of handlers that Monolog offers.
For example, if you wish to use daily log files instead of a single, large file, you can make the following change to your start file:
$logFile = 'laravel.log';
Log::useDailyFiles(storage_path().'/logs/'.$logFile);
### Error Detail
By default, error detail is enabled for your application. This means that when an error occurs you will be shown an error page with a detailed stack trace and error message. You may turn off error details by setting the `debug` option in your `app/config/app.php` file to `false`.
> **Note:** It is strongly recommended that you turn off error detail in a production environment.
<a name="handling-errors"></a>
## Handling Errors
By default, the `app/start/global.php` file contains an error handler for all exceptions:
App::error(function(Exception $exception)
{
Log::error($exception);
});
This is the most basic error handler. However, you may specify more handlers if needed. Handlers are called based on the type-hint of the Exception they handle. For example, you may create a handler that only handles `RuntimeException` instances:
App::error(function(RuntimeException $exception)
{
// Handle the exception...
});
If an exception handler returns a response, that response will be sent to the browser and no other error handlers will be called:
App::error(function(InvalidUserException $exception)
{
Log::error($exception);
return 'Sorry! Something is wrong with this account!';
});
To listen for PHP fatal errors, you may use the `App::fatal` method:
App::fatal(function($exception)
{
//
});
If you have several exception handlers, they should be defined from most generic to most specific. So, for example, a handler that handles all exceptions of type `Exception` should be defined before a custom exception type such as `Illuminate\Encryption\DecryptException`.
### Where To Place Error Handlers
There is no default "home" for error handler registrations. Laravel offers you freedom in this area. One option is to define the handlers in your `start/global.php` file. In general, this is a convenient location to place any "bootstrapping" code. If that file is getting crowded, you could create an `app/errors.php` file, and `require` that file from your `start/global.php` script. A third option is to create a [service provider](/docs/ioc#service-providers) that registers the handlers. Again, there is no single "correct" answer. Choose a location that you are comfortable with.
<a name="http-exceptions"></a>
## HTTP Exceptions
Some exceptions describe HTTP error codes from the server. For example, this may be a "page not found" error (404), an "unauthorized error" (401) or even a developer generated 500 error. In order to return such a response, use the following:
App::abort(404);
Optionally, you may provide a response:
App::abort(403, 'Unauthorized action.');
This method may be used at any time during the request's lifecycle.
<a name="handling-404-errors"></a>
## Handling 404 Errors
You may register an error handler that handles all "404 Not Found" errors in your application, allowing you to easily return custom 404 error pages:
App::missing(function($exception)
{
return Response::view('errors.missing', array(), 404);
});
<a name="logging"></a>
## Logging
The Laravel logging facilities provide a simple layer on top of the powerful [Monolog](http://github.com/seldaek/monolog) library. By default, Laravel is configured to create a single log file for your application, and this file is stored in `app/storage/logs/laravel.log`. You may write information to the log like so:
Log::info('This is some useful information.');
Log::warning('Something could be going wrong.');
Log::error('Something is really going wrong.');
The logger provides the seven logging levels defined in [RFC 5424](http://tools.ietf.org/html/rfc5424): **debug**, **info**, **notice**, **warning**, **error**, **critical**, and **alert**.
An array of contextual data may also be passed to the log methods:
Log::info('Log message', array('context' => 'Other helpful information'));
Monolog has a variety of additional handlers you may use for logging. If needed, you may access the underlying Monolog instance being used by Laravel:
$monolog = Log::getMonolog();
You may also register an event to catch all messages passed to the log:
#### Registering A Log Listener
Log::listen(function($level, $message, $context)
{
//
});
| {
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The Spanish Cultural Heritage Institute (IPCE) is the institution of the Ministry of Culture and Sports dedicated to the research, conservation and documentation of our heritage.
In order to develop this work, the IPCE counts with specialists in various disciplines: architects, archaeologists, ethnographers, conservators, scientists, archivists, computer specialists and curators, among others. These professionals face their work from the necessary multidisciplinary perspective, demanded by the complex task of conserving and valorizing cultural property.
Functions
The Institute carries out the following functions:
Development and implementation of conservation plans of movable and immovable Spanish Cultural Property.
Setting up of main lines of research in heritage conservation criteria, methodology and techniques.
Filing and systematizing the work carried out by the Institute.
Training of specialists who attend the functions of the Institute.
Conservation, research, training and diffusion are the basis of the IPCE structure. It is based on the solid track record of its predecessor, the Spanish Historical Heritage Institute, responding to a dynamic collaboration with local and autonomous Administrations. All this is upgraded by the constant contact between international professionals and institutions.
IPCE Functional Areas
Conservation
One of the main functions of the Spanish Cultural Heritage Institute (IPCE) is the conservation of our heritage legacy.
The Institute develops many projects and activities on all types of heritage: monumental architecture, works of art, archaeological objects, textiles, bibliographic and documentary heritage, etc.
The IPCE also coordinates the generation and implementation of the National Cultural Heritage Plans, joint management tools for different types of cultural goods. Their drafting involves professionals of the Central Government, Autonomous Communities and independent experts.
A National Plan sets the criteria and develops methods to protect, preserve, investigate, document and transmit cultural heritage.
Currently there are thirteen National Cultural Heritage Plans for the following matters: Cathedrals, abbeys, monasteries and convents; defensive architecture; industrial heritage; cultural landscapes; research in heritage conservation; preventive conservation; intangible heritage; twentieth century heritage; education and heritage; traditional architecture; conservation of photographic heritage; and emergencies in heritage.
Research
Research should be the leitmotif of any conservation intervention and valorisation of cultural heritage. On this premise, the Institute specialists develop research in different areas: heritage applied sciences, conservation, architecture, archaeology, art history, ethnology and documentation.
Training
The quality of the interventions developed by the managerial and technical stall, responsible for heritage protection, depends on their complete and updated training. The IPCE is aware of the need to develop a training policy,keeping with national and international criteria in the heritage field. We, therefore, run various training activities, such as courses, conferences, technical seminars, etc.
Documentation
The ample documentary, bibliographic and photographic archives of the IPCE require constant conservation and valorisation. Thus the information stored in the Archive, the Library and Photo Library of the Institute contributes to increase constantly the knowledge of Spanish Cultural Heritage.
Diffusion
The functions carried out by the Institute, mainly those related to research, conservation and diffusion of cultural goods, demand an active communications policy addressed to government, public and private institutions. Among other initiatives, this diffusion policy brings about a yearly publication plan that includes scientific monographs and magazines - "Patrimonio Cultural de España" and "Informes y Trabajos"-.
IPCE Internal Organization
Conservation Intervention Area
The main functions of this area are the design, management and implementation of conservation interventions in cultural property, the draft and management of National Heritage Plans and the promotion of architectural, archaeological, historical and ethnographic research.
The area develops the following works:
Conservation of cultural property.
Advice on conservation projects of Autonomous Communities and other public or private institutions.
Draft and management of National Heritage Plans.
Technical reports about projects applying for finance through 1% Cultural (1% of the budget for civil works is spent on art projects).
Ethnographic research and implementation of projects.
Management of the Finance Program for the International Archaeological Projects, developed by the Ministry of Culture and Sports.
Drafting of documents on intervention criteria, methodology and action protocol in cultural property.
Reports on loan agreements and condition reports for exhibitions promoted by public institutions.
Coordination of European Heritage Days.
Research and Training Area
The general functions are the development, application and research of experimental sciences in conservation of cultural heritage.
The lines of works are:
Advice on the selection of methods and materials preparation used in cultural property treatments.
Selection of methodology and analysis techniques. Standardization and equipment start up.
Sampling and analyzing.
Study of original and added materials in cultural property.
Design of analytical methods that contribute to the diagnosis of the cultural property condition.
Setting environmental conditions of storage, display and transportation of cultural property.
Study of environmental and biological deterioration agents.
IPCE Training Program design and management.
Documentation and Diffusion Area
The main duties are:
Conservation, management and valorisation of cultural heritage documents kept in our archives.
Management of our specialized Cultural Heritage Library.
Research, preservation and diffusion of our Heritage Photo Library.
Development, coordination and centralization of information produced by the Institute activities.
Attending information requests of the Ministry of Culture and Sports and other institutions, as well as the diffusion of the work generated by the Institute activities.
Publication management.
Exhibition organization.
Guided tours of the Institute and monuments during their conservation processes.
Coordination of web space and IPCE profile in social networks.
Headquarters
The headquarters of the Cultural Heritage Institute of Spain (Spanish: Instituto del Patrimonio Cultural de España) are located in Madrid, Spain. They were declared Bien de Interés Cultural in 2001.
Building was projected in 1967 by architects Fernando Higueras and Antonio Miró Valverde, and built from 1967 to 1970.
References
External links
Presentation of the Spanish Cultural Heritage Institute
Official website
Buildings and structures in Ciudad Universitaria neighborhood, Madrid
Bien de Interés Cultural landmarks in Madrid
Buildings and structures completed in 1970 | {
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Q: Using map function in python If you have the following code, how exactly is it following the documentation: map(function, iterable,...)?
x = sorted(map(int, dat[0].split()))
Is int a function and if so, why isn't it expressed as such?
A: int is a constructor so it's callable so you can use it with map
A: In your case dat[0] is as string, and split() generates a list of strings, by splitting the input string at whitespaces.
Eg
"1 11".split()
returns
["1", "11"]
The map function has two input arguments:
*
*The first argument is something which can be called (in Python you say it is a callable), eg a function. int is not realy a function but such a thing (Python slang: it is an object).
Eg int("3") return 3. So int when applied to a string tries to convert this string to an integer, and gives the integer value back.
*The second argument is something you can iterate over, in your case it is a list.
If you then call the map function, the first argument is applied to all elements from the second argument.
So
map(int, ["1", "11"])
returns
[1, 11]
If you combine what I explained you understand that
map(int, "1 11".split())
returns
[1, 11]
A: When you ask "why isn't it expressed as such" I suppose you mean, why doesn't it have brackets like a function? The answer is that if you put the brackets in then you get what the function does instead of the function itself. Compare what happens when you enter int() versus int.
A: Think of it like this
def map( function, iterable ):
return ( function(x) for x in iterable )
In x = sorted(map(int, dat[0].split())) the function, int, is being named, not evaluated. This code provides a function object to the map function. The map function will evaluate the given function.
A: The syntax of map in the simplest form is:
map(func, sequence)
It will apply the function "func" on each element of the sequence and return the sequence. Just in case you don't know, int() is a function.
>>> int(2.34)
2
>>> int("2")
2
>>> a = ["1", "101", "111"]
>>> map(int, a)
[1, 101, 111]
Now, I will give you my implementation of map().
>>> for index in range(len(a)):
... a[index] = int(a[index])
...
>>> a
[1, 101, 111]
If you have understood the concept, let's take a step further. We converted the strings to int using base 10 (which is the default base of int() ). What if you want to convert it using base 2?
>>> a = ["1", "101", "111"]
>>> for index in range(len(a)):
... a[index] = int(a[index], 2) # We have an additional parameter to int()
...
>>> a
[1, 5, 7]
To get the same result using map(), we will use a lambda function
>>> a = ["1", "101", "111"]
>>> map(lambda x: int(x, 2), a)
[1, 5, 7]
| {
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\section{Introduction}
RS Ophiuchi (RS Oph) is a well-observed recurrent nova (recurrence period $\sim$ 20 years) and is one of the
ten confirmed recurrent novae that belong to our galaxy (Kato $\&$ Hachisu 2012, Schaefer 2010).
The RS Oph system is composed of a massive ($\sim$ 1.35 M$_{\odot}$, Kato, Hachisu $\&$ Luna 2008) white dwarf (WD) primary accompanied by a red giant secondary of estimated spectral class around M2 III (Worters et al. 2007 and references therein). Brandi et al. (2009) estimated the orbital period to be 453.6 days, the red giant mass, M$_{g}$ = 0.68 - 0.80 M$_{\odot}$ and the orbital inclination, i = 49$^{\circ}$ - 52$^{\circ}$ for the system.
The outburst takes place due to a thermonuclear runaway (TNR) on the WD surface
that accretes matter from the secondary red giant companion. The outburst causes ejection of mass $\sim$ 10$^{-6}$ - 10$^{-8}$ M$_{\odot}$
at a high speed of $\sim$ 4000 km s$^{-1}$ (e.g. Buil 2006). Previous studies of outbursts indicate that
the WD mass of RS Oph is possibly increasing due to the accumulation of a percentage of the accreted matter on its surface.
Consequently, the mass of the WD in RS Oph may gradually reach the Chandrasekhar limit and explode as a Type Ia supernova -
this has made RS Oph an object of great significance to the astrophysicists. However, there have been considerable debates about
this hypothesis (Starrfield et al. 2004, Wood-Vasey $\&$ Sokoloski 2006). \\
RS Oph was detected in outburst previously in 1898, 1933, 1958, 1967, 1985; its latest outburst was discovered on 2006
February 12.83 UT (Hirosawa 2006). The reason for the much shorter recurrence period in the RS Oph system,
in comparison to classical novae (CNe),
is due to combined effect of the high WD mass and a high accretion rate (Starrfield et al. 1985, Yaron et al. 2005).
The two recent outbursts of RS Oph in 1985 and 2006 have been observed intensively over a wide range of wavelengths,
from X-rays to the radio regions (Bode 1987; Evans, Bode, O'Brien $\&$ Darnley 2008). Detailed studies
display very similar characteristics of the outbursts. In the early phase of the outburst, the spectra show broad, low-ionization emission features
of H, He, N, O and Fe; the nova enters quickly (about a month after outburst) to the nebular phase with the
emergence of strong coronal (e. g., [Fe XIV] 0.5303 $\mu$m, [Ar X] 0.5535 $\mu$m, [Fe X] 0.6374 $\mu$m, [Si VI] 1.9641 $\mu$m,
[Al IX] 2.0444 $\mu$m, [Mn XIV] 2.0894 $\mu$m) and nebular lines (e.g., [O III] (0.4363, 0.4959 and 0.5007 $\mu$m) and
[N II] 0.5755 $\mu$m) (Iijima 2009, Banerjee et al. 2009). The nova light curves also behave similarly; they decline fast with $t_{2}$ $\sim$ 6
and $t_{3}$ $\sim$ 17 days (Rosino 1987, Munari et al. 2007). The key result of the 1985 and 2006 observations was the detection of
a shock that is generated while the ejecta interacts with the surrounding wind of the red giant secondary (Bode $\&$ Kahn
1985, Das et al. 2006 and references therein) and a non-spherical bi-polar shape of the nova ejecta (e.g., Taylor
et al. 1989, Chesneau et al. 2007, Bode et al. 2007).
Further investigations also helped to determine a few important parameters viz. determinations of the distance,
d = 1.6 $\pm$ 0.3 kpc (Hjellming et al. 1986), the interstellar hydrogen column density, N $\sim$ 2.4 $\times$ 10$^{21}$ cm$^{-2}$ (Hjellming et al. 1986), and an interstellar reddening of E(B - V) = 0.73 (Snijders 1987).\\
However, despite plenty of observations of RS Oph, the abundance analysis of the nova ejecta has not been done adequately. A Few values
have been calculated, for example, from optical studies of the 1985 outburst, Anupama $\&$ Prabhu (1985)
derived a helium abundance of $n(He)/n(H)$ = 0.16; Evans et al. (2007) estimated the O/Ne ratio
(by number) to be $\gtrsim$ 0.6 from IR
studies of the 2006 outburst. A complete knowledge of elemental abundances in the
ejecta is of crucial importance for several reasons, for example, to understand the TNR process that leads to the nova explosion,
the composition of material of the WD\textbf{,} as there is a possibility of mixing of WD material with the ejecta,
the contribution of novae to the chemical evolution of galaxy etc. In this paper, we report the results of an elemental abundance
analysis of the ejecta of RS Oph by modeling its available optical and near-infrared (NIR) spectra observed during the 2006
eruption. We have used the photoionization code CLOUDY (version 13.02; Ferland et al., 2013) to generate spectra, by varying
the parameter values. Model generated spectra are then compared with the observed emission line spectra, the best fit model is
chosen by calculating the corresponding $\chi ^{2}$ values. The procedure of modeling is described in section 3; results obtained from the analysis
is described in section 4.
\section{Photoionization model analysis}
We use the CLOUDY photoionization code, C13.02 (Ferland et al., 2013) for the abundance analysis in RS Oph.
The benefit of using photoioinization models is that in addition to elemental abundances, they also provide
estimate of several other parameters, e.g., density, source luminosity, source temperature etc. Previously, this method
was used to determine the elemental analysis and physical characteristics of a few novae by modeling the observed spectra,
for example, LMC 1991 (Schwarz 2001), QU Vul (Schwarz et al. 2002), V1974 Cyg (Vanlandingham et al. 2005),
V838 Her $\&$ V4160 Sgr (Schwarz et al. 2007a), V1186 Sco (Schwraz et al. 2007b), V1065 Cen (Helton et al. 2010).
The photoionization code CLOUDY uses a set of parameters that specify the initial physical conditions of the source and the ejected shell.
The source is described by the spectral energy distribution of the continuum source, its temperature and luminosity. The physical condition
of the shell is described by the density, inner and outer radii, geometry, covering factor (fraction of 4$\pi$ sr enclosed by the model shell),
filling factor (ratio of the contribution of the dense shell to the diffuse shell) and elemental abundances (relative to solar).
The density of the shell is set by a hydrogen density parameter and the elemental abundances, relative to hydrogen, are set by the abundance
parameters. The hydrogen density, n(r), and filling factor, f(r), may vary with the radius as given by the following relations,
\begin{equation}
n(r) = n(r_{0}) (r/r_{0})^{\alpha} cm^{-3} \,\,\,\,\, \,\,\&\,\,\,\,\,\, f(r) = f(r_{0}) (r/r_{0})^{\beta}
\end{equation}
where, r$_{o}$ is the inner radius, $\alpha$ and $\beta$ are exponents of power laws.
We choose $\alpha$ = -3, the filling factor = 0.1 and the filling factor power-law exponent, ($\beta$) = 0, which
are the typical values used in similar kind of studies (e.g., Schwarz 2002, Vanlandingham et al. 2005, Helton et al. 2010).\\
CLOUDY solves the equations of thermal and statistical equilibrium using the above mentioned set of input parameters to
generate output spectra from the non-LTE ejecta illuminated by the central source. Its calculations incorporate effects of
important ionization processes, e.g., photo, Auger, collisional $\&$ charge transfer and recombination
process viz. radiative, dielectronic, three-body recombination, and
charge transfer. We assume the continuum shape to be a blackbody of a high temperature $T_{BB}$ $\ge$ 10$^{5}$ K,
as done in the previous investigations, to ensure that it supplies the correct amount of photons for photoionization.
The output predicts the flux of emission lines, which is compared to the measured line
fluxes in the observed spectra.\\
\begin{table*}
\small
\centering
\caption{A log of the used spectroscopic data of RS Oph (2006).}
\smallskip
\begin{threeparttable}
\centering
\begin{tabular}{l c c c c c c c}
\hline\
Epochs & Wavelength & Date of & Days after & Telescopes / & Resolution & Refer- \\
& band & observation & outburst & Instruments & & ences \\
\hline
D31 & Optical & March 15 & 31 & Observatorio Astr\'{o}nomico & $\sim$ 1000 - & 1 \\
& & & & Nacional en San Pedro & 3500 & \\
& & & & M\'{a}rtir/ Boller \& Chivens & \\
& NIR & March 16 & 32 & Mt. Abu Telescope/ NICMOS & $\sim$ 1000 & 2 \\
\hline
D49 & Optical & April 4 & 50 & Astrophysical Observatory of & $\sim$ 1000 & 3 \\
& & & & Asiago/ Boller \& Chivens & \\
& NIR & April 2 & 49 & Mt. Abu Telescope/ NICMOS & $\sim$ 1000 & 2 \\
\hline
\end{tabular}
\begin{tablenotes}
\item[] 1 = Riberio et al. 2009; 2 = Das et al. 2006 $\&$ Banerjee et al. 2009; 3 = Iijima 2009
\end{tablenotes}
\end{threeparttable}
\end{table*}
\section{Modeling procedure}
For the present analysis, we use observed optical and NIR $JHK$ spectra of the 2006 outburst of RS Oph. Moedeling of both optical and NIR
data enables to sample over a broader range of ionization and excitation levels in the emission lines and thus helps to constrain
the results more accurately. We choose two epochs of observations taken at different times of the nova evolution, that had nearly simultaneous
optical and NIR spectra and form two data sets represented by D31 and D49.
D31 consists of optical and NIR spectra that is observed, respectively, on 2006 March 15 $\&$ March 16 i.e. approximately 31 days after outburst;
whereas, D49 consists of optical and NIR spectra observed, respectively, on 2006 April 4 $\&$ April 2, i.e. approximately 49 days after outburst.
Here, for simplicity, we assume that the physical condition and corresponding parameters, in the ejecta remain unchanged over 1-2 days.
Details about the used spectra are presented in Table 1.
A detailed modeling using more data sets extended over a larger time period and including other wavelength regions
is in progress and will be published later.\\
We assume a spherically symmetric expanding shell geometry of the ejecta that is illuminated by the central source.
Several spectra are generated by varying the free parameters, one after one, viz. hydrogen density, underlying luminosity,
effective blackbody temperature and abundances of only those elements
which showed observed lines. The abundances of other elements, which do not show any emission line, were
fixed at solar values. Since, novae ejecta are not homogenous in density, we assume that the ejecta is composed of at least two different density
regions - one for the higher density to fit the lower ionization lines and the other for the lower density to fit the higher ionization lines.
To reduce the number of free parameters in the final model, each component is subjected to the same parameters except
the hydrogen densities at the inner radius and the covering factors assuming that the sum of the two covering factors be less than or equal to 1. Thus,
the overall number of free parameters increases by 2 due to the second component's initial density and covering factor.
The final model line ratios were calculated by adding the line ratios of each component after multiplying by its covering factor.
Thus, this method is only a first-order approximation to incorporate density gradients into the photoionization
analysis; there is no coupling or interaction between the components, as expected in reality.
However, this procedure has been used successfully in the case of other novae, e. g., V838 Her (Schwarz et al. 2002), V1974 Cyg (Vanlandingham et al. 2005), V838 Her $\&$ V4160 Sgr (Schwarz et al. 2007a), V1186 Sco (Schwraz et al. 2007b) and V1065 Cen (Helton 2010).\\
Since CLOUDY uses a large number of parameters and many of the parameters are interdependent, it is
difficult to determine the uniqueness of any solution by checking the generated spectra visually. Hence, we choose the the best fit model
by calculating $\chi^{2}$ and reduced $\chi^{2}$ ($\chi^{2}_{red}$) of the model given by,
\begin{equation}
\chi ^{2} = \sum\limits_{i=1}^n (M_{i} - O_{i})^{2}/ \sigma_{i}^{2}, \,\,\,\,\&\,\,\,\, \chi^{2}_{red} = \chi^{2}/\nu
\end{equation}
where, $n$ = number of observed lines, $n_{p}$ = number of free parameters, $\nu$ = degree of
freedom = n - n$_{p}$, M$_{i}$ = the modeled ratio of line flux to hydrogen line flux, O$_{i}$ = measured flux ratio,
and $\sigma_{i}$ = error in the observed
flux ratio. We estimate an error in the range of 10 - 30$\%$, depending upon the strength of a
spectral line relative to the continuum, possibility of blending with other lines,
and formal error in the measurement of line flux. A model is considered good if the value of $\chi^{2}$ $\sim$ $\nu$ such that the
($\chi^{2}_{red}$) value is low (typically in the range of 1 - 2).\\
To minimize the number of free parameters, the inner ($R_{in}$) and outer ($R_{out}$) radii of the ejected shell are held constant during the
iterative process of fitting the spectra. For simplicity,
we assume a spherical geometry of the expanding shell with ($R_{in}$) and ($R_{out}$) defined by the minimum ($V_{min}$) and maximum ($V_{max}$) expansion velocities, respectively. We adopt $V_{min}$ = 3500 km s$^{-1}$ and $V_{max}$ = 4500 km s$^{-1}$ during explosion based on values calculated from the optical and NIR emission lines (e.g., Das et al. 2006, Skopal et al. 2008). The velocities remain constant during the free expansion phase that lasted for first $\sim$ 3 days (taking an average of 2 and 4 days, derived by Sokoloski et al. 2006 and Das et al. 2006, respectively) and then decrease gradually with time (t) as $t^{-0.6}$ (Das et al. 2006) due to interaction with surrounding matter from the red giant companion. Following this process, we calculate and find $V_{min}$ $\sim$ 1950 km s$^{-1}$ and $V_{max}$ $\sim$ 2500 km s$^{-1}$ on day 5.5. These values are consistent with the interferometric results (Chesneau et al., 2007) that show the evidence of two different radial velocity fields inside the ejecta, with $V_{min}$ $\le$ 1800 km s$^{-1}$ and $V_{max}$ $\sim$ 2500 - 3000 km s$^{-1}$, at day 5.5. The agreement between the two results gives us confidence about the model we adopted here. Proceeding further, we calculate the velocities and expansion of the shells on each date, then add them up to calculate the final values of $R_{in}$ and $R_{out}$ on D31 and D49.\\
\begin{table*}
\tiny
\centering
\caption{Observed and Best-Fit Cloudy Model Line Fluxes $^{a}$.}
\smallskip
\begin{threeparttable}
\centering
\begin{tabular}{l c c c c c c c}
\hline
& & D31 & & & & D49 & \\
Line & $\lambda$ ($\mu$ m)& Observed & Modeled & $\chi^{2}$ & Observed & Modeled & $\chi^{2}$\\
\hline
& & & Optical & & & & \\
\hline
$[$Fe VII] & 0.3760 & ... & ... & ... & 0.21 & 0.12 & 0.80 \\
$[$Ne III] & 0.3868 & ... & ... & ... & 0.32 & 0.32 & 0.00 \\
H I, He I & 0.3889 & ... & ... & ... & 0.43 & 0.36 & 0.49 \\
H $\epsilon$, [N III] & 0.3970 & ... & ... & ... & 0.24 & 0.45 & 4.40 \\
He I, He II & 0.4029 & 0.01 & 0.05 & 0.12 & 0.16 & 0.05 & 1.21\\
H $\delta$& 0.4103 & 0.14 & 0.20 & 0.34 & 0.51 & 0.31 & 3.99 \\
Fe II, [Ni XII] & 0.4233 & 0.02 & 0.17 & 2.02 & 0.14 & 0.07 & 0.42 \\
H $\gamma$& 0.4341 & 0.38 & 0.40 & 0.04 & 0.50 & 0.55 & 0.25\\
$[$Fe II] & 0.4415 & 0.02 & 0.10 & 0.65 & 0.07 & 0.07 & 0.00 \\
He I & 0.4472 & 0.05 & 0.12 & 0.56 & 0.11 & 0.10 & 0.01 \\
Fe II & 0.4523 & 0.04 & 0.10 & 0.46 & 0.08 & 0.06 & 0.04\\
Fe II & 0.4549 & 0.04 & 0.11 & 0.47 & 0.08 & 0.06 & 0.04\\
Fe II & 0.4590 & 0.04 & 0.11 & 0.45 & 0.08 & 0.04 & 0.16\\
He II & 0.4666 & 0.35 & 0.57 & 4.60 & 0.73 & 0.79 & 0.36 \\
H $\beta$ & 0.4863 & 1.00 & 1.00 & 0.00 & 1.00 & 1.00 & 0.00 \\
He II, Fe II & 0.4923 & 0.07 & 0.11 & 0.18 & 0.15 & 0.07 & 0.64 \\
$[$O III], He I, Fe II& 0.5017 & 0.23 & 0.46 & 5.29 & 0.43 & 0.29 & 1.96 \\
$[$Fe II], [Fe VII] & 0.5158 & 0.08 & 0.18 & 0.92 & 0.16 & 0.28 & 1.44 \\
Fe II & 0.5235 & 0.03 & 0.04 & 0.01 & ... & ... & ... \\
Fe II & 0.5276 & 0.04 & 0.15 & 1.12 & ... & ... & ... \\
Fe II & 0.5317 & 0.01 & 0.05 & 0.14 & ... & ... & ... \\
He II & 0.5411 & 0.01 & 0.10 & 0.25 & 0.08 & 0.10 & 0.03\\
$[$Ar X] & 0.5535 & 0.05 & 0.08 & 0.11 & 0.14 & 0.6 & 0.64\\
$[$N II] & 0.5755 & ... & ... & ... & 0.12 & 0.16 & 0.16\\
He I & 0.5876 & ... & ... & ... & 0.42 & 0.43 & 0.01\\
$[$Fe X] & 0.6374 & ... & ... & ... & 0.33 & 0.14 & 3.61\\
He I & 0.6678 & ... & ... & ... & 0.23 & 0.13 & 1.00\\
$[$Ar XI] & 0.6919 & ... & ... & ... & 0.10 & 0.04 & 0.36\\
He I & 0.7065 & ... & ... & ... & 0.22 & 0.40 & 3.20\\
He I & 0.7281 & ... & ... & ... & 0.13 & 0.09 & 0.16\\
\hline
&&& J-band & & & & \\
\hline
He I & 1.0830 & 4.39 & 4.70 & 2.40 & 11.80 & 11.45 & 3.06 \\
Pa $\gamma$ & 1.0938 & 1.02 & 0.78 & 1.44 & 1.79 & 0.85 & 9.82 \\
He II & 1.1626 & 0.11 & 0.10 & 0.02 & 0.40 & 0.26 & 1.96 \\
Pa $\beta$ & 1.2818 & 1.00 & 1.00 & 0.00 & 1.00 & 1.00 & 0.00 \\
\hline
&&& H-band & & & & \\
\hline
Br 20 & 1.5184 & 0.57 & 0.50 & 0.49 & 0.33 & 0.29 & 0.16\\
Br 19 & 1.5256 & 0.31 & 0.24 & 0.49 & 0.38 & 0.25 & 1.69\\
Br 18 & 1.5341 & 0.45 & 0.37 & 0.64 & 0.47 & 0.38 & 0.81\\
Br 17 & 1.5439 & 0.45 & 0.31 & 1.96 & 0.60 & 0.38 & 4.84\\
Br 16 & 1.5570 & 0.71 & 0.63 & 0.64 & 0.91 & 0.73 & 3.24\\
Br 15 & 1.5685 & 0.66 & 0.53 & 1.69 & 0.60 & 0.60 & 0.00\\
Br 14 & 1.5881 & 0.76 & 0.69 & 0.49 & 0.77 & 0.70 & 0.49\\
Br 13 & 1.6109 & 0.82 & 0.55 & 1.82 & 0.69 & 0.59 & 1.00\\
Br 12 & 1.6407 & 1.00 & 1.00 & 0.00 & 1.00 & 1.00 & 0.00\\
Br 11 & 1.6806 & 1.81 & 1.82 & 0.01 & 1.34 & 1.20 & 1.96\\
Fe II & 1.6872 & 1.01 & 0.91 & 0.99 & 0.91 & 0.78 & 1.69\\
He I & 1.7002 & 0.69 & 0.79 & 1.00 & 0.74 & 0.90 & 2.56\\
Br 10 & 1.7362 & 2.18 & 2.26 & 0.64 & 2.10 & 2.26 & 2.56\\
Fe II & 1.7406 & 0.34 & 0.28 & 0.36 & 0.69 & 0.58 & 1.21\\
\hline
&&& K-band & & & & \\
\hline
Br 8 & 1.9446 & ... & ... & ... & 0.8 & 0.86 & 0.36 \\
$[$Si VI] & 1.9621 & 0.46 & 0.29 & 2.28 & 1.3 & 1.80 & 0.64 \\
$[$Al IX] & 2.0444 & 0.04 & 0.08 & 0.13 & 0.13 & 0.10 & 0.09 \\
He I & 2.0581 & 0.71 & 0.44 & 1.80 & 0.87 & 0.40 & 5.50 \\
He I & 2.1120 & 0.14 & 0.08 & 0.36 & 0.13 & 0.12 & 0.04 \\
Br $\gamma$& 2.1655 & 1.00 & 1.00 & 0.00 & 1.00 & 1.00 & 0.00 \\
\hline
Total & & & & 37.99 & & & 69.06 \\
\hline
\end{tabular}
\begin{tablenotes}
\item[a] Relative to H $\beta$ in optical, Pa $\beta$ in $J$ band, Br 12 in $H$ band and Br $\gamma$ in $K$ band.
\end{tablenotes}
\end{threeparttable}
\end{table*}
\begin{figure}
\centering
\includegraphics[bb= 265 80 1280 460, width=6.0 in, height = 2.6 in, clip]{figure1.eps}
\caption{Best Cloudy model (dotted line) fit to the observed optical and NIR $JHK$ spectra (solid lines) of RS Oph observed during the epochs D31 and D49 during the 2006 outburst. The spectra were normalized to H $\beta$ in optical, Pa $\beta$ in $J$ band, Br 12 in $H$ band and Br $\gamma$ in $K$ band. Also few of the strong features have been marked (see text for details).}
\end{figure}
\section{Results}
The results of our analysis are presented in Table 2 $\&$ 3. Table 2 shows the relative fluxes of the best-fit model predicted and observed lines
along with corresponding $\chi^{2}$ values. Here, we have
considered only the lines which are present both in the model\textbf{-}generated and observed spectra. We assume a
distance of 1.6 kpc for RS oph to match the predicted luminosities with the
reddening-corrected observed flux. The measured line fluxes were dereddened using E(B - V ) = 0.73 (Snijders 1987)
and compared to the output of each CLOUDY model to calculate $\chi ^{2}$ for the fit.
The line intensities are determined from direct integration of the line; for features with multiple components, the profiles were decomposed
with multiple Gaussians using IRAF tasks. To minimize the errors associated with flux calibration
between different epochs and wavelength regions, we calculate the modeled and observed flux ratios relative to prominent hydrogen lines, which are not blended with any other features, within a given wavelength region, e. g., relative to H$\beta$ in the optical region, relative to Paschen $\beta$ in the $J$-band, relative to Brackett 12 in the $H$-band and relative to Brackett $\gamma$ in the $K$-band.\\
\begin{table*}
\centering
\caption{Best-fit CLOUDY Model parameters}
\smallskip
\begin{threeparttable}
\centering
\begin{tabular}{l c c c c c c c}
\hline
Parameters & D31 & D49 & Predicted Abundances \\
\hline\\
$T_{BB}$ ($\times$10$^{5}$ K) & 5.8 & 5.5 & ... \\
Source luminosity ($\times$10$^{36}$ erg s$^{-1}$) & 6.3 & 8.0 & ... \\
Clump hydrogen density ($\times$10$^{8}$ cm$^{-3})$ & 10.0 & 6.3 & ...\\
Diffuse hydrogen density ($\times$10$^{8}$ cm$^{-3})$ & 1.6 & 1.0 & ...\\
$\alpha$\tnote{a} & -3 & -3 & ...\\
Inner radius ($\times$10$^{14}$ cm ) & 2.1 & 2.8 & ...\\
Outer radius ($\times$10$^{14}$ cm) & 4.8 & 6.8 & ...\\
Clump to diffuse covering factor & 90/10 & 85/15 & ... \\
Filling factor & 0.1 & 0.1 & ... \\
$\beta$\tnote{b} & 0.0 & 0.0 & ... \\
He/He$_\odot$\tnote{c} & 1.8 (11) & 1.9 (16) & 1.8 $\pm$ 0.1\\
N/N$_\odot$ & ... & 12.0 (2) & 12.0 $\pm$ 1.0 \\
O/O$_\odot$ & 1.0 (1) & 1.0 (1) & 1.0 $\pm$ 0.4 \\
Ne/Ne$_\odot$ & ... & 1.5 (1) & 1.5 $\pm$ 0.1 \\
Si/Si$_\odot$ & 0.3 (1) & 0.5 (1) & 0.4 $\pm$ 0.1\\
Fe/Fe$_\odot$ & 3.0 (12) & 3.5 (11) & 3.2 $\pm$ 0.2 \\
Ar/Ar$_\odot$ & 4.9 (1) & 5.2 (2) & 5.1 $\pm$ 0.1\\
Al/Al$_\odot$ & 0.9 (1) & 1.1 (1) & 1.0 $\pm$ 0.1\\
Ejected Mass ($\times$10$^{-6}$ M$_\odot$) & 3.4 & 4.9 & ...\\
Number of observed lines (n) & 42 & 51 & ... \\
Number of free parameters (n$_{p}$) & 11 & 13 & ... \\
Degrees of freedom ($\nu$) & 31 & 38 & ... \\
Total $\chi^{2}$ & 38.0 & 69.1 & ... \\
$\chi^{2}_{red}$ & 1.2 & 1.8 & ... \\
\\
\hline
\end{tabular}
\begin{tablenotes}
\item[a] Radial dependence of the density $r^{\alpha}$
\item[b] Radial dependence of filling factor $r^{\beta}$
\item[c] Abundances are given in logarithmic scale, relative to hydrogen. All other elements which are not listed
in the table were set to their solar values. The number in the parentheses represents number of lines used in
determining each abundance.
\end{tablenotes}
\end{threeparttable}
\end{table*}
The best fit modeled spectra (dotted line) together with the observed optical and NIR
spectra (solid lines) on D31 and D49, are shown in Figure 1. We have marked the strong spectral lines on the figure, a
detailed observed line list is presented in Banerjee et al. (2009) and (Iijima 2009).
The spectra on D31 are dominated by prominent features of low ionization lines of H$\epsilon$ (0.3970 $\mu$m), H$\delta$ (0.4103 $\mu$m), H$\gamma$ (0.4341 $\mu$m), He II (0.4666 $\mu$m), H $\beta$ (0.4863 $\mu$m) in the optical region; He I (1.0830 $\mu$m), Pa $\gamma$ (1.0938 $\mu$m), O I (1.1287 $\mu$m), Pa $\beta$ (1.2818 $\mu$m), Brackett series lines, Fe II (1.6872 $\mu$m), He I (1.7002 $\mu$m) in the $H$-band; and
He I (2.0581 $\mu$m), Br $\gamma$ (2.1655 $\mu$m) in the $K$-band. There are also higher ionization lines, viz. [Fe VII] (0.3760 $\mu$m), a blended feature of [O III], He I $\&$ Fe II (0.5017 $\mu$m), \textbf{a} blended feature of [Fe II] $\&$ [Fe VII] (0.5158 $\mu$m), [Ar X] (0.5535 $\mu$m), [Fe X] (0.6374 $\mu$m)
in the optical; and [Si VI] (1.9641 $\mu$m), [Al IX] (2.0444 $\mu$m), [Mn XIV] (2.0894 $\mu$m) in the NIR region.
As mentioned in the earlier section, a single shell of ejecta could not generate all of these lines.
For example, if we consider a clumpy shell of high density, e.g. 10$^{9}$ cm$^{-3}$ only, the modeled spectra fit the majority of the
lines in the observed spectra, but systematically under represents the higher ionization lines, for example, [Al IX] (2.0444 $\mu$m), Fe II (1.7406$\mu$m), [Fe X] (0.6374 $\mu$m) and [Fe VII] (0.3760 $\mu$m). This one component shell also increases the strength of the K-band He I line enormously. On the other hand, only a diffuse shell of lower density can produce the higher-ionization lines successfully but can not produce the He I and Fe II lines sufficiently to match the observed lines. Hence, for a better fitting of the lines, it was required to consider a two-component model consisting of a dense shell (hydrogen densities, by number: 10 x 10$^{8}$ cm$^{-3}$ and 6.3 x 10$^{8}$ cm$^{-3}$ on D31 and D49, respectively) and a diffuse shell
(1.6 x 10$^{8}$ cm$^{-3}$ and $\&$ 1.0 x 10$^{8}$ cm$^{-3}$ on D31 and D49, respectively). The ratio of the
clump to diffuse components are 90:10 and 85:15, respectively, on D31 and D49. This
indicates that the ejecta volume was dominated more by the dense gas.
The best-fit models use a blackbody temperature (T$_{BB}$) of 5.8 $\times$ 10$^{5}$ K and 5.5 $\times$ 10$^{5}$ K and
a luminosity of 6.3 $\times$ 10$^{38}$ ergs s$^{-1}$ and 8.0 $\times$ 10$^{38}$ ergs s$^{-1}$, respectively, on D31 and D49.
This is in agreement with the WD temperature of around 8 $\times$ 10$^{5}$ K derived from x-ray studies by Nelson et al. (2008).
The low $\chi^{2}_{red}$ values of 1.2 and 1.8 for the two dates, respectively, indicate that the fits are satisfactory.
The best-fit model parameters for each epoch are described in Table 3.\\
Despite the low values of $\chi^{2} _{red}$, the model still has some problems.
An inspection of Table 2 shows that a few lines make the highest contributions to the total $\chi^{2}$, for
example, He II (0.4666 $\mu$m), the blended feature of [O III], He I $\&$ Fe II (0.5017 $\mu$m)
on D31 and H $\epsilon$ $\&$ [N III] (0.3970 $\mu$m), H $\delta$, [Fe X] (0.6374 $\mu$m),
He I (0.7065, 1.0830 and 2.0581 $\mu$m), Pa $\gamma$, Br 16 and 17 on D49. The possible reason for these misfits
is that a two-component model is not sufficient to adequately reproduce the complex density structure of the nova ejecta;
consequently, the exact conditions inside the ejecta in which these lines are formed, have not been not reproduced correctly.
In addition, CLOUDY was unable to adequately reproduce a few of the observed lines, e.g., Si II (0.5041, 0.5056 $\mu$m),
Fe II (0.5235, 0.5276, 0.5317 $\mu$m), [Fe XIV] (0.5303 $\mu$m), N III (0.4641 $\mu$m), Raman emission band (0.6830 $\mu$m),
He I (1.1969, 1.2527 $\mu$m), N I (1.2074, 1.2096, 1.2470 $\mu$m), O I (1.1287, 1.3164 $\mu$m), [Mn XIV] (2.0894 $\mu$m) and He I (2.1120 $\mu$m).
So, we have excluded these lines from our analysis. The problem of reproducing the NIR O I emission
lines (1.1287 $\mu$m and 1.3164 $\mu$m) and He I (2.0581 $\mu$m) has also been reported by
Helton et al. (2010). They attributed these difficulties to related excitation mechanisms which are not properly
included in the present CLOUDY code.\\
Overall abundance values for RS Oph, calculated from the mean of these modeled results, are presented in Table 3. Abundances are
given as logarithm of the numbers relative to hydrogen and relative to solar.
The derived values show that the helium, nitrogen, neon, iron and argon abundances are all enhanced relative to solar, while
the oxygen and aluminium abundances are solar with respect to hydrogen, and silicon abundances are subsolar with respect to hydrogen.
We note that there is no prominent line of oxygen in the observed spectra. Our determination of the oxygen abundance is based on the feature
at 0.5017 $\mu$m that is blended with He I and Fe II. Therefore the calculated abundance value of oxygen may incorporate a significant error. Similarly, the calculation of abundances of silicon, argon and aluminium is based on one or two observed lines only. Modeling of multi-wavelength
spectra, observed over a longer time scale may improve the level of accuracy.
\subsection{Estimation of \textbf{the} ejecta mass}
Using the parameter values determined from the present analysis, we can estimate the hydrogen ejected mass predicted by the two component
models using the following relation (e.g., Schwarz 2001, 2002),
\begin{equation}
M_{shell} = n(r_{0}) f(r_{0}) \int \limits_{R_{in}}^{R_{out}} (r/r_{0})^{\alpha + \beta} 4\pi r^{2} dr
\end{equation}
The resulting mass is then multiplied by corresponding covering factors to obtain mass of the dense and diffuse shells;
the final mass is calculated by adding the masses of these two shells. Using this method, we find an ejected mass
of 3.4 $\times$ 10$^{-6}$ M$_{\odot}$ and 4.9 $\times$ 10$^{-6}$ M$_{\odot}$
for D31 and D49, respectively. Similar values of 1 - 3 $\times$ 10$^{-6}$ M$_{\odot}$ and 3 $\times$ 10$^{-6}$ M$_{\odot}$ were estimated
from other studies (e.g., Das et al. 2006; Kato, Hachisu $\&$ Luna 2008) of the 2006 outburst of RS Oph.
From the light curve analysis, Hachisu $\&$ Kato (2001) estimated an ejecta mass of $\sim$ 2 $\times$ 10$^{-6}$ M$_{\odot}$ for a WD of mass
of 1.35 M$_{\odot}$. Therefore the present result also favors the possibility of a high mass of $\sim$ 1.35 M$_{\odot}$ in the RS Oph system.
\section{Summary}
We have used the photoionization code CLOUDY to model the observed optical and NIR emission-line spectra of RS Oph observed on two epochs, 31 and 49 days after the outburst, during 2006. We generated a set of spectra by varying several parameters and assuming a spherical geometry of the ejecta that consists of two different shells of different densities. The model\textbf{-}generated spectra were then compared with observed spectra, the best fit parameters were chosen using the $\chi^{2}$ technique. The best-fit model parameters are in agreement with a hot WD with temperature of $\sim$ 5.5 - 5.8 $\times$ 10$^{5}$ K and luminosity of 6 - 8 $\times$ 10$^{36}$ ergs s$^{-1}$. The abundance analysis shows that the ejecta are significantly enhanced, relative to solar, in helium, nitrogen, neon, iron and argon as well as silicon was found to be subsolar. We estimate an ejected mass in the range of 3.4 - 4.9 $\times$ 10$^{-6}$ M$_{\odot}$ which fits a high mass $\sim$ 1.35 $M_{\odot}$ of the associated WD in the RS Oph system. In continuation of the present work, a complete and detailed analysis of abundances and related parameters, covering a larger wavelength region and evolution period, will be the subject of a future work.
\section*{Acknowledgments}
The research work at S N Bose National Centre for Basic Sciences is funded by the Department
of Science and Technology, Government of India.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,115 |
Q: json_encode with mysql content and umlauts in utf-8 i feel my beard growing while trying to find out the Problem here.
Basic the Problem is, that Umlauts/Special Signs äöß ... don't work. I guess everyone is sick and tired of that questions but all the solutions found online don't seem to work.
Im having utf-8 content in a utf-8 Mysql Database. I feel the Problem ist somewhere in the Database connection but i just can't figure out.
character_set_client utf8
character_set_connection utf8
character_set_database utf8
character_set_filesystem binary
character_set_results utf8
character_set_server latin1
character_set_system utf8
Im not sure if the problem is the latin1 for character_set_server because im not into that mysql stuff. I also dont know how to change cause i can't access the mysql server's config files.
Whatever is confusing me, that if i get my results from the Database and echo it, print_r gives the right result.
ini_set('default_charset','utf-8');
header('Content-Type: text/plain; > charset=utf-8');
Firefox says char encode is utf-8 but if when i output:
print_r($listnew);
echo json_encode($listnew[5]);
print_r results everything right but json_encode does wrong.
print_r:
[5] => Array (
[id] => 5
[data] => U-Bahnhof Theresienstraße
[size] => 17
)
json_encode:
{"id":5,"data":"U-Bahnhof Theresienstra\u00dfe","size":17}
i know json_encode needs a utf-8 string to work properly there and i feel im having a encode trouble here but i just can't firgure out where it is.
Any help would be appreciated,
thanks in advance.
i3
A: Ummm... I think that is the correct way actually. \u00df is a correct unicode representation of ß. When you json_decode() it back, it will become ß again.
Where is this making problems for you? Is the receiving end not decoding it properly? It should if you use standard json_* functions.
All the examples in the manual show the same thing - characters beyond the ASCII range are turned into numeric sequences.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,030 |
.dtpicker-cont{top:25%}.dtpicker-overlay{-ms-filter:"progid:DXImageTransform.Microsoft.gradient(startColorstr=#20000000, endColorstr=#20000000)";filter:progid:DXImageTransform.Microsoft.gradient(startColorstr=#20000000, endColorstr=#20000000);zoom:1!important} | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,879 |
Want a quick and dirty way to Truncate all the tables in a MySQL DB? You can use the following one line script.
What this does is uses mysqldump to output all the tables in the database "YourDBName" into a dump file. This dump file contains drop statements, and create table statements for each table in the database. What this effectively does is drops every table, and recreates it WITHOUT any data inside. Be very careful, since this is a very powerful script, it has the ability to completely wipe out your database with no hope of recovery unless you have a current backup.
For additional ways to execute specific commands against all tables in a DB, look into using mk-find in Maatkit, it is an excellent tool for advanced MySQL scripting.
MySQL Delimiters – Or why I hate stored procedures and Error Code : 1064 You have an error in your SQL syntax.
MySQL – DateTime vs. TimeStamp – When to use?
2 Weird Lotus Elise Brake Pad Facts..
What database does Facebook use? | {
"redpajama_set_name": "RedPajamaC4"
} | 5,173 |
{"url":"http:\/\/blogs.fau.de\/hager\/archives\/1073","text":"# OpenMP, ccNUMA and C++\n\n## Sprungmarken\n\nPosted on by\n\nOpenMP, ccNUMA and C++\nIf you are interested in programming with C++ and OpenMP, the just-finished diploma thesis of Holger Stengel might be interesting for you (in German \u2013 available on request). It studies ccNUMA effects in C++ and ways to circumvent them. To fuel your appetite, there is a nice English poster with most of the results: poster_cppnuma.pdf\n\nThis whole work was kicked off by some of the problems I had encountered during my PhD thesis where I had parallelized a C++ code from condensed matter physics. At that time, nobody had even thought about what would happen if standard C++ elements (arrays of objects, std::vector<> etc.) were used on a ccNUMA machine with OpenMP. Another inspiration came from Matt Austern\u2018s article about Segmented Iterators and Hierarchical Algorithms. The segmented iterator described in this paper could by useful for many purposes, of which NUMA placement is only one. In the thesis we implemented a version in which you could exactly control data placement by configurable padding.\n\nI would be glad to continue on this topic with another diploma\/bachelor\/masters student. If you are hooked, feel free to contact me.\n\nThis entry was posted in HPC and tagged , , by Georg Hager. Bookmark the permalink.\n\n### One comment on \u201cOpenMP, ccNUMA and C++\u201d\n\n1. Gary Howell says:\n\nIt would be great if average engineers could write code to use NUMA machines fairly efficiently. You guys are making that more feasible .. so good work. I particularly appreciate that your efforts also apply to Fortran .. a bit simpler to write in ..","date":"2014-12-18 20:55:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.4827919602394104, \"perplexity\": 3192.9926158160733}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-52\/segments\/1418802767873.65\/warc\/CC-MAIN-20141217075247-00153-ip-10-231-17-201.ec2.internal.warc.gz\"}"} | null | null |
Als Rekombinationsleuchten werden Leuchterscheinungen von Gasatomen und Molekülen bezeichnet, die nach der Ionisation durch starke Strahlungsquellen ihre zusätzliche Energie in Form von Photonen wieder abgeben. Diese strahlende Rekombination besteht in der Vereinigung elektrisch positiver und negativer Ladungsträger (Ionen und Elektronen) zu elektrisch neutralen Teilchen, stellt also den Umkehrprozess zur Ionisation dar und ist für Astronomie und Plasmaphysik bedeutsam.
Erdatmosphäre
In der Erdatmosphäre werden die Luftmoleküle – insbesondere Sauerstoff, Stickstoff und Natrium – tagsüber vor allem vom UV-Anteil des Sonnenlichts ionisiert. Nach Sonnenuntergang bewirkt ihr Rekombinationsleuchten eine leichte, natürliche Aufhellung des Nachthimmels. Dazu kommt allerdings noch die Streuung von Sternenlicht und von terrestrischen Lichtquellen, die vor allem in den unteren Kilometern der Atmosphäre (der Troposphäre) erfolgt, sowie an schwebenden Staubteilchen (Aerosolen).
Rekombination ist auch für das Aufleuchten von Sternschnuppen und Feuerkugeln in der Hochatmosphäre verantwortlich. Die kleineren dieser aus dem interplanetaren Raum kommenden Körper verdampfen schon in etwa 100 km Höhe durch Reibung an den Luftmolekülen, wobei sie diese ionisieren. Die Leuchtspuren kommen jedoch nicht vom Verglühen selbst, sondern aus der Rekombination kurz danach.
Astronomie
In der Astrophysik ist das Rekombinationsleuchten vor allem in Gasnebeln von Bedeutung, wenn sie durch heiße Sterne in ihrer Umgebung angeregt werden. In Gebieten von Sternentstehung bilden sich die jungen, stark im UV-Licht strahlenden Sterne meist im Innern solcher Molekülwolken, wodurch die Leuchterscheinungen besonders stark ausfallen. In den besonders hellen Emissionsnebeln wie dem Lagunennebel (M8) im Schützen oder dem Orionnebel (M42) ist es vor allem die Rekombination ionisierten Wasserstoffs und Heliums. Teilweise erfolgt sie in richtiggehenden Kaskaden zwischen den Elektronenschalen n → n−1 → n−2 usw., also beim Wasserstoff in mehreren Spektrallinien violett bis rot der Balmer-Serie. Auch in H-II-Gebieten – dem expandierenden Nachfolgestadium der Sternbildung – und in Planetarischen Nebeln um ehemalige Riesensterne herrscht das Rekombinationsleuchten vor.
Literatur
Arnold Hanslmeier: Einführung in Astronomie und Astrophysik. 3. Auflage. Springer-Spektrum-Verlag, Berlin/Heidelberg 2014.
S. & P. Friedrich: Handbuch Astronomie. Oculum-Verlag, Erlangen 2015.
Walter Baier et al.: Elektronik-Lexikon. Franck-Verlagsbuchhandlung, Stuttgart 1974.
Plasmaphysik
Astronomie | {
"redpajama_set_name": "RedPajamaWikipedia"
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Q: .ebextensions with Docker on elasticbeanstalk I am attempting to configure nginx on an elastic beanstalk Docker instance to allow the use of websockets. Using the information found here I added a .ebextensions/01_files.config to the zip I am deploying.
The zip contains .ebextensions/01_files.config, a Dockerfile, and a tgz of my application.
The content of 01_files.config is the following:
files:
"/etc/nginx/conf.d/websocketupgrade.conf" :
mode: "000755"
owner: root
group: root
content: |
proxy_set_header Upgrade $http_upgrade;
proxy_set_header Connection "upgrade";
Elastic beanstalk seemingly ignores my .ebextensions. The file /etc/nginx/conf.d/websocketupgrade.conf is never written.
What is the correct way to use .ebextensions with a Docker container on elastic beanstalk?
A: I'm pretty sure that ElasticBeanstalk overwrites /etc/nginx/conf.d/ as part of the set up an application, so anything you put there with ebextensions will be gone by the time the application has started up.
To get around this, you need to make your changes to the staging area, rather than the final destination. I am not sure what the staging area is for elasticbeanstalk nginx deployments, but if you look in
/var/log/eb-activity.log
You should be able to figure it out.
You should use the:
container_command:
directive to make changes to the staging area.
A: Elastic Beanstalk now supports docker with a lot less hacking, you should be using Dockerfile and Dockerrun.aws.json.
Just base a new container off the offical nginx image and add the conf file you have posted above. Then pull in your nginx image using the dockerrun file (See example below).
{
"AWSEBDockerrunVersion": "1",
"Authentication": {
"Bucket": "my-bucket",
"Key": "mydockercfg"
},
"Image": {
"Name": "me/my-nginx-image",
"Update": "true"
},
"Ports": [
{
"ContainerPort": "80"
}
],
"Volumes": [
{
"HostDirectory": "/var/",
"ContainerDirectory": "/etc/mysql"
}
],
"Logging": "/var/log/nginx"
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,294 |
Pilip Siamienawicz Piestrak (biał. Піліп Сямёнавіч Пестрак, ur. 27 listopada 1903 w miejscowości Sokołuce w obecnym obwodzie brzeskim, zm. 22 sierpnia 1978) – białoruski pisarz i działacz polityczny.
Życiorys
Urodził się w rodzinie chłopskiej jako syn pracownika budowy trasy kolejowej i pracownicy gospodarstwa wiejskiego. Po wybuchu I wojny światowej rodzina ewakuowała się do wsi Maksimowka k. Buzułuka w guberni samarskiej, gdzie Pilip Piestrak uczył się w miejscowej szkole ludowej do 1915, po czym został pomocnikiem rachmistrza w towarzystwie kredytowym. W latach 1917-1918 uczył się w gimnazjum w Buzułuku, 1918-1920 był pomocnikiem księgowego w gminnym komitecie wykonawczym, w 1920 uczył się na rabfaku (fakultecie robotniczym), po ukończeniu którego podjął studia na Uniwersytecie Samarskim. W 1921 wraz z rodzicami wrócił do zachodniej Białorusi, gdzie pracował jako leśnik, a od 1924 do 1926 odbywał służbę w WP. W 1926 wstąpił do KPZB i został członkiem Komitetu Powiatowego Białoruskiej Robotniczho-Chłopskiej Gromady w Kosowie. W 1927 został sekretarzem podziemnego Komitetu Rejonowego KPZB i członkiem Komitetu Okręgowego KPZB w Słonimiu, a 1933-1934 był instruktorem KC KPZB. Za działalność komunistyczną był niejednokrotnie aresztowany i więziony, łącznie spędził w polskich więzieniach 11 lat. We wrześniu 1939 został uwolniony z więzienia przez Armię Czerwoną, po czym został sekretarzem Komitetu Miejskiego KP(b)B w Grodnie, a w 1940 przewodniczącym Związku Pisarzy Białorusi w Białymstoku. Po ataku Niemiec na ZSRR w 1941 brał udział w radzieckim ruchu partyzanckim. W 1943 został szefem Zarządu ds. Sztuk przy Radzie Komisarzy Ludowych Białoruskiej SRR, od 1946 pracował w Instytucie Języka, Literatury i Historii Sztuki, a od 1948 był konsultantem literackim Związku Pisarzy Białorusi. W 1940 opublikował pierwszy zbiór wierszy, Na straże. Pisał po białorusku, rosyjsku i polsku. Tworzył liryki, opowiadania i powieści o tematyce społeczno-obyczajowej z życia wsi. W 1946 wydał zbiór wierszy Za swoju otcziznu na temat walki partyzantów z niemieckimi okupantami Białorusi. W 1948 i 1952 w dwóch tomach wydał powieść historyczno-rewolucyjną Sustrenemsja na barykadach (samodzielne wydanie w 1954). W 1963 opublikował powieść Sriediborje, za którą otrzymał nagrodę literacką Białoruskiej SRR im. Jakuba Kołasa. Tłumaczył również polską literaturę. Był odznaczony pięcioma orderami, a także medalami. W 1968 otrzymał tytuł zasłużonego działacza kultury Białoruskiej SRR.
Przypisy
Białoruscy pisarze
Członkowie Komunistycznej Partii Zachodniej Białorusi
Urodzeni w 1903
Zmarli w 1978 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,381 |
\section{Introduction}
The thermalization of strongly-interacting systems causes information about the initial configuration to become ``scrambled'' at late times, wherein two initial states (with the same conserved quantities) become indistinguishable without measuring a macroscopic number of observables \cite{deutsch1991quantum,srednicki1994chaos,tasaki1998quantum,rigol2008thermalization}.
Recent studies on the dynamics of such information scrambling have sharpened our understanding of chaos in quantum many-body systems \cite{Hayden07, Sekino08, Lashkari13, Kitaev:2014t1, Maldacena:2016ac, Shenker:2013pqa, Traversable2017, Roberts:2014isa, Shenker:2013yza, Roberts:2014ifa, Hosur:2015ylk, Roberts:2017aa, Blake:2016wvh, Blake:2016aa, Kitaev:2014t2, Nahum:2017aa, Keyserlingk:2017, Cotler:2017aa, Khemani:2017aa, Davison:2017aa, Gu:2017aa, Gao:2016aa}, and have led to new insights on a variety of questions ranging from the black hole information paradox \cite{Page93,Hayden07, Hosur:2015ylk, Traversable2017, Yoshida:2017aa} to transport phenomena in non-Fermi liquids \cite{banerjee2017solvable,patel2017quantum}.
While a precise definition of quantum scrambling remains elusive, a powerful proxy for characterizing its behavior is provided by out-of-time order correlation (OTOC) functions, which take the general form: $\langle V(0)W(t)V(0)W(t)\rangle$, where $V, W$ are operators that act on sufficiently small subsystems \cite{larkin1969quasiclassical,Kitaev:2014t1,Shenker:2013pqa,Roberts:2014isa}.
The intuition behind this correlator is an attempt to measure the influence of one observable at earlier times on another observable at later times --- in essence, a quantum version of the so-called butterfly effect.
To do this however, requires the precise reversal of time evolution and thus, poses a daunting challenge for any experiment.
\begin{figure}
\centering\includegraphics[width=1.\linewidth]{fig1_schem_v3}
\caption{a) In the Hayden-Preskill variant of the black-hole decoding problem \cite{Hayden07}, one asks whether Bob can decode the state of Alice's quantum spin using only Hawking radiation and an entangled partner of the black-hole. Any successful ``decoding'' serves as affirmation for the existence of scrambling dynamics. b) For an arbitrary unitary $U$, one can utilize a teleportation-based decoding protocol to probe the scrambling behavior of the circuit \cite{Yoshida:2017aa}. Crucially, even in the presence of arbitrary noise and imperfections, the teleportation fidelity acts as a metric for quantum scrambling and enables the bounding of the mutual information between Alice and Bob's reference quantum registers.
}
\label{fig-setup}
\end{figure}
Despite this challenge, a tremendous amount of interest has been devoted to the development of protocols \cite{swingle2016measuring,yao2016interferometric,zhu2016measurement} and platforms \cite{garttner2017measuring,li2017measuring} for the direct measurement of OTOCs.
The crucial difficulty in interpreting such measurements can be summarized as follows:
For a generic interacting system without symmetries, the scrambling of quantum information will cause out-of-time order correlation functions to decay to zero.
However, both decoherence and imperfect experimental controls (e.g.~time reversal) will \emph{also} cause OTOCs to decay to zero.
At present, the only way to distinguish between these two contributions --- namely, true chaotic scrambling versus noise and decoherence --- is to perform full quantum tomography on the many-body system, requiring exponentially many measurements in the number of qubits \cite{hradil1997quantum,dodonov1997positive,haffner2005scalable,gross2010quantum}.
To this end, the ability to distinguish between genuine quantum information scrambling and extrinsic decoherence remains an essential open question.
In this paper, we analyze a quantum teleportation protocol that explicitly enables such differentiation.
We present three main results.
First, we demonstrate that within our protocol, one can extract a ``noise'' parameter, which quantifies the \emph{non-scrambling} induced decay of OTOCs. Here, we focus on two illustrative examples: i) depolarization (i.e.~a non-unitary error) and ii) imperfect ``backwards'' time evolution (i.e.~a unitary error).
Second, using this noise parameter, we provide a \emph{bound} on the true scrambling-induced decay of the OTOC.
Again, we analyze two cases, one which applies specifically to the situation of unitary errors and another which applies to arbitrary errors.
Finally, we describe two simple realizations of our protocol amenable to near-term, intermediate scale qubit and qutrit systems as well as their generalizations to include Grover search \cite{grover1996fast}.
The essence of our approach is based upon a recent decoding algorithm for the Hayden-Preskill variant of the black hole information problem \cite{Hayden07, Yoshida:2017aa}.
The connection between this decoding algorithm and information scrambling can be understood as follows: If the dynamics of a black hole are unitary, then one should in principle, be able to retrieve a quantum state that is thrown in from the Hawking radiation that comes out (Fig.~1a). Crucially, it turns out such a successful ``decoding'' of the original quantum state serves as smoking-gun evidence for the existence of true scrambling dynamics.
Our manuscript is organized as follows. In Sec.~II, we begin by reviewing the information theoretic interpretation of scrambling and OTOCs. Then, using the example of a depolarizing quantum channel, we illustrate the fact that decoherence can result in the decay of OTOCs even in the absence of scrambling dynamics. This allows us to propose a sharp measure which quantifies the ratio of scrambling-induced versus decoherence-induced OTOC decay. Moreover, it reveals that the genuine metric for scrambling should be taken as the mutual information between subsystems and not simply the measured OTOC.
The groundwork being laid, in Sec.~III, we introduce the teleportation-based decoding protocol and clarify its operation in the ideal case without noise and decoherence. Then in Sec.~IV, we turn to an analysis of the protocol in the presence of \emph{arbitrary} noise and decoherence. Here, we demonstrate that the protocol provides a quantitative estimate for the amount of dissipation in the system. In Sec.~V, motivated by recent experiments, we restrict ourselves to a sub-class of noise and imperfections, with a focus on \emph{coherent} errors. Under this restriction, we show that one can explicitly bound the ideal value of the OTOC (i.e. in the absence of errors), using the experimentally \emph{measured} value of the OTOC.
In Sec.~VI, we generalize such a bound to the case of arbitrary errors and prove that one can utilize the teleportation fidelity to bound the mutual information between subsystems (and hence the amount of scrambling).
Finally, in Sec.~VII, we propose and analyze two experimental implementations of our protocol in near-term intermediate scale quantum simulators. We focus on a class of Clifford scramblers that
saturate the lower bound for OTOCs.
In Sec.~VIII, we offer some concluding remarks and intriguing directions to be pursued.
\section{Characterizing scrambling and decoherence}
\subsection{Definition of scrambling in terms of OTOCs}
Let us begin by providing a definition for quantum scrambling in terms of the behavior of out-of-time order correlation functions~\cite{Roberts:2017aa, Yoshida:2017aa}:
\begin{align}
\langle O_{X}O_{Y}(t)O_{Z}O_{W}(t) \rangle \approx \langle O_{X}O_{Z}\rangle\langle O_{Y} \rangle\langle O_{W} \rangle + \nonumber \\ \langle O_{X}\rangle \langle O_{Z}\rangle\langle O_{Y} O_{W} \rangle - \langle O_{X}\rangle \langle O_{Y}\rangle\langle O_{Z} \rangle\langle O_{W} \rangle \label{eq:def}
\end{align}
where $O_{X},O_{Z}$ are operators that act on sub-system $A$ (at time zero) and $O_{Y},O_{W}$ are operators that act on sub-system $D$ (at time $t$), as depicted in Fig.~\ref{fig-setup}.
This equation becomes exact in the thermodynamic limit for chaotic systems at late times and can also be derived from the eigenstate thermalization hypothesis \cite{deutsch1991quantum,srednicki1994chaos,tasaki1998quantum,rigol2008thermalization,Huang:2017aa}.
While we will focus on infinite temperature systems with $\rho=\frac{1}{d}\mathbb{I}$, we note that this definition naturally generalizes to finite temperatures.
Our above definition of scrambling is required to hold for \emph{all} local operators, but a slightly a more coarse-grained characterization of scrambling (and one which is easier to probe experimentally) can be achieved via the averaged OTOC~\cite{Hosur:2015ylk}:
\begin{align}
\overline{\langle \OTOC \rangle} \equiv \iint d O_{A} d O_{D} \langle O_{A} O_{D}(t)O_{A}^{\dagger} O_{D}^{\dagger}(t)\rangle
\end{align}
where $\int d O_{R}$ is the Haar-average over all unitary operators on sub-system $R$. This Haar integral can be replaced by an average over Pauli operators:
\begin{align}
\iint d O_{A} d O_{D} \langle O_{A} O_{D}(t)O_{A}^{\dagger} O_{D}^{\dagger}(t)\rangle = \nonumber \\ \frac{1}{d_{A}^2d_{D}^2}\sum_{P_{A},P_{D}} \langle P_{A} P_{D}(t)P_{A}^{\dagger} P_{D}^{\dagger}(t)\rangle \label{eq:1-design},
\end{align}
where $P_{A(D)}$ are Pauli operators and $d_{A(D)}$ is the dimension of the sub-system \footnote{Eq.~\eqref{eq:1-design} holds since the Pauli operators form a unitary $1$-design \cite{renes2004symmetric,dankert2009exact}. Note that there are $d_{A}^2,d_{D}^2$ Pauli operators (including the identity operator) on regions $A,D$, respectively.}.
Working at infinite temperature and using Eqn.~\eqref{eq:def} then yields the scrambled value of the averaged OTOC as \footnote{We note that there are unitary operators which satisfy Eq.~\eqref{eq:ave-def}, but not Eq.~\eqref{eq:def}. For example, a random Clifford operator is scrambling for Eq.~\eqref{eq:ave-def}, since the Clifford operators form a unitary $2$-design. However, OTOCs for a Clifford unitary are always $\pm1$ if $O_{X}=O_{Z}$ and $O_{Y}=O_{W}$ are Pauli operators, and thus do not satisfy Eq.~\eqref{eq:def}. In this sense, a random unitary from a $2$-design is not enough to achieve full scrambling. Rather, to achieve full scrambling, it suffices to pick a random operator $U$ from a unitary $4$-design.}:
\begin{align}
\overline{\langle \OTOC \rangle}_{\text{S}} \approx \frac{1}{d_{A}^2} + \frac{1}{d_{D}^2} - \frac{1}{d_{A}^2d_{D}^2} \label{eq:ave-def}.
\end{align}
This scrambled value, $\overline{\langle \OTOC \rangle}_{\text{S}}$, is achieved for a Haar random unitary as $d\rightarrow \infty$ \cite{Hosur:2015ylk}.
On the other hand, for arbitrary unitary time evolution, $\overline{\langle \OTOC \rangle}$ is bounded from above by unity and from below by $\max(\frac{1}{d_{A}^2}, \frac{1}{d_{D}^2} )$; the fact that it never fully decays to zero is because it contains contributions from cases where $P_{A}=\mathbb{I}$ or $P_{D}=\mathbb{I}$.
We note that the minimal scrambled value is only asymptotically achieved for large systems with $d \gg d_{A}\gg d_{D}$ or $d \gg d_{D}\gg d_{A}$ \cite{suppinfo}.
\subsection{Decoding as a route to scrambling}
In order to characterize the effect of decoherence on the averaged OTOC, it will be useful to first recall the information theoretic interpretation of $\overline{\langle \OTOC \rangle}$ in terms of the mutual information between sub-systems.
To do so, we will utilize the so-called state representation of the time-evolution operator, $U$ \cite{Hayden07, Hosur:2015ylk}. This representation allows us to view a unitary operator $U$, acting on an $n$-qubit Hilbert space $\mathcal{H}_{AB}$, as a pure quantum state, supported on a $2n$-qubit Hilbert space $\mathcal{H}_{AB}\otimes \mathcal{H}_{RB'} (\simeq \mathcal{H}_{RCDB'} )$:
\begin{align}
\hspace{-10mm} |\Psi\rangle \equiv (I_{R}\otimes U_{AB}\otimes I_{B'}) |\text{EPR}\rangle_{RA}\otimes |\text{EPR}\rangle_{BB'} = \nonumber \hspace{-10mm} \\
\figbox{1.0}{fig-world-state}\qquad \label{eq:world-state}
\end{align}
where time runs upward and the horizontal lines in the diagram represent EPR pairs ($|\text{EPR}\rangle \equiv\frac{1}{\sqrt{d}}\sum_{j=1}^{d} |j\rangle \otimes |j\rangle$), while the dots capture the $\frac{1}{\sqrt{d}}$ normalization factor in the EPR pair.
Crucially, this representation allows us to characterize the scrambling behavior of the time evolution, $U$, via the entanglement properties of the pure state, $|\Psi\rangle$!
Three remarks are in order.
First, for non-interacting time evolutions, including free-fermion dynamics or SWAP operators, $|\Psi\rangle$ contains mostly bipartite entanglement among subsystems.
On the other hand, for strongly-interacting time evolutions that lead to scrambling, $|\Psi\rangle$ consists of \emph{multipartite} entanglement delocalized over the full Hilbert space $RCDB'$.
Second, we note that $|\Psi\rangle$ is precisely the state of interest in the Hayden-Preskill thought experiment \cite{Hayden07}.
In particular, the Hilbert spaces $A,B,C,D$ [Eqn.~\eqref{eq:world-state}] support, respectively, Alice's input states, the initial black hole, the remaining black hole and the Hawking radiation. Meanwhile, $R$ serves as a reference for Alice's input state, while $B'$ is the entangled partner of the black hole.
The Hayden-Preskill decoding problem can then be stated as follows: when can Bob decode Alice's quantum state using only the Hawking radiation $D$ and the entangled black-hole partner $B'$.
The answer, somewhat naturally, is when the (von Neumann) mutual information between $R$ and $B'D$ is maximal.
More precisely, when this is the case, there exists a unitary operator acting on $B'D$ which distills an EPR pair between $R$ and $B'D$ with high fidelity, thereby faithful recovering Alice's input state~\cite{Hayden07, Hayden:2008aa}.
\begin{figure}
\centering\includegraphics[width=0.55\linewidth]{Fig2_setup_otoc}
\caption{Schematic image of the setup associated with an out-of-time ordered correlation function, $\langle O_{A}(0)O_{D}(t)O_{A}(0)O_{D}(t) \rangle$. Time runs upward and evolution is generated by the unitary $U$. The Hilbert space decomposes as $\mathcal{H}=\mathcal{H}_{A}\otimes \mathcal{H}_{B}=\mathcal{H}_{C}\otimes \mathcal{H}_{D}$. Ideally, operators $O_{A}$ and $O_{D}$ act on sufficiently small subsystems of the full system.
}
\label{fig-setup}
\end{figure}
Third, we note that for maximally mixed states the R\'{e}nyi-$2$ mutual information, $I^{(2)}(R,B'D)$, lower bounds the von Neumann mutual information, $I(R,B'D)$ \footnote{It is possible to generalize our results to finite temperature (factorable ensembles) using the R\'{e}nyi divergence \cite{suppinfo}.}.~This is particularly useful since $I^{(2)}(R,B'D)$ is in fact, directly related to our previously defined averaged OTOC~\cite{Hosur:2015ylk, Roberts:2017aa}:
\begin{align}
\overline{\langle \OTOC \rangle} = 2^{ - I^{(2)}(R,B'D) }, \label{eq:OTOC-unitary}
\end{align}
where $I^{(2)}(R,B'D) \equiv S_{R}^{(2)} + S_{B'D}^{(2)} - S_{RB'D}^{(2)}$ and the R\'{e}nyi-$2$ entropies, $S^{(2)}$, are evaluated with respect to the state $|\Psi\rangle$. To this end, in an \emph{ideal} (noise-less) system, the smallness of $\overline{\langle \OTOC \rangle}$, which characterizes the amount of scrambling in the system, is \emph{also} sufficient to diagnose Bob's faithful recovery of Alice's state
\footnote{While the R\'{e}nyi-$2$ mutual information is a measurable quantity as the average of OTOCs, the standard mutual information ($\alpha=1$) is often more convenient as it satisfies useful monotonicity inequalities. For the case of maximally mixed ensembles $\rho=\frac{1}{d}I$, one can derive $I(A,B'D)\geq I^{(2)}(A,B'D)$ using the monotonicity of R\'{e}nyi entropy. This analysis can be generalized to cases where the input and output ensembles factorize \cite{suppinfo}; $\rho_{AB}=\rho_{A}\otimes \rho_{B}$ and $\rho_{CD}=\rho_{C}\otimes \rho_{D}$ where the R\'{e}nyi-$2$ mutual information is replaced with a certain expression involving the R\'{e}nyi-$2$ divergence from which the standard mutual information can be lower bounded. See appendix for details.
}.
However, the essential point is that in a system with noise and imperfections, the smallness of $\overline{\langle \OTOC \rangle}$ can either result from decoherence or from true scrambling behavior. Crucially, only the latter will contribute to Bob's ability to decode Alice's quantum state!
In the following subsections, we will first focus on identifying the effects of decoherence on the averaged OTOC. With this in hand, we will then provide a precise metric to distinguish between decoherence and scrambling.
\subsection{Effects of decoherence on the OTOC}
To understand the effects of decoherence, let us consider the following quantum channel $\mathcal{Q}$:
\begin{align}
\rho \rightarrow \mathcal{Q}(\rho) = (1-p) U \rho U^{\dagger} + p \frac{1}{d} \Tr(\rho) \label{eq:noise}
\end{align}
which suffers from depolarization with probability $p$ \cite{nielsen2010quantum}.
For traceless operators, one finds that the out-of-time order correlators behave as:
\begin{align}
\langle O_{X}\widetilde{O_{Y}}(t)O_{Z}\widetilde{O_{W}}(t) \rangle = (1-p)^2 \langle O_{X}O_{Y}(t)O_{Z}O_{W}(t) \rangle
\label{eq:OTOCdefn}
\end{align}
where we use the tilde to indicate observables time-evolved under the quantum channel $\mathcal{Q}$, while time-evolved operators without a tilde are evolved under the unitary portion of the channel, $U$ \footnote{We note that in the presence of arbitrary forms of noise and decoherence, the experimentally ``measured'' value of the OTOC may depend on the specific measurement protocol. For example, the OTOC measured via interferometric protocols \cite{swingle2016measuring,yao2016interferometric,zhu2016measurement,garttner2017measuring,li2017measuring} will generically differ from the OTOC measured via our teleportation protocol. However, for the important case of a purely depolarizing channel as per Eqn.~\eqref{eq:noise}, all such protocols will measure the same OTOC given by Eqn.~\eqref{eq:OTOCdefn}. Moreover, while the quantitative values of noisy OTOCs may differ between protocols, their qualitative decay in the presence of decoherence is generic. To this end, a key difference between our decoding protocol and previously proposed interferometric protocols is the initial preparation of EPR pairs; this preparation is not present in the case of interferometric protocols and underlies the reason why our teleportation-based method can verify the existence of scrambling dynamics while prior methods cannot. }.
Thus, even in the absence of information scrambling (i.e.~in the actual behavior of $\langle O_{X}O_{Y}(t)O_{Z}O_{W}(t) \rangle$), the \emph{measured} OTOCs for the channel $\mathcal{Q}$, can become small owing to decoherence; in particular, undergoing depolarization with a finite probability per unit time induces an exponential decay of the measured values of OTOCs.
The difference between scrambling and decoherence can be further sharpened and made precise by considering the late-time asymptotics of OTOCs, which serve as our operational definition of quantum scrambling in Eqn.~\eqref{eq:def}. Specifically, under a completely depolarizing channel (e.g.~$p=1$), the out-of-time order correlators decompose as follows:
\begin{align}
\langle O_{X}O_{Y}(t)O_{Z}O_{W}(t) \rangle = \langle O_{X}O_{Z}\rangle\langle O_{Y} \rangle\langle O_{W} \rangle,
\end{align}
which contains only the first term in Eqn.~\eqref{eq:def}.
As before, one can also examine the averaged OTOCs associated with the channel $\mathcal{Q}$:
\begin{align}
\langle \widetilde{\OTOC} \rangle \equiv \iint d O_{A} dO_{D} \langle O_{A}\widetilde{O_{D}}(t)O_{A}^{\dagger}\widetilde{O_{D}^{\dagger}}(t) \rangle.
\label{eq:OTOCtildechannel}
\end{align}
Note that for an arbitrary quantum channel, the value of $\langle \widetilde{\OTOC} \rangle$ is now lower bounded by $\min(\frac{1}{d_{A}^2},\frac{1}{d_{D}^2})$, whereas in the absence of imperfections, $\langle \overline{\OTOC} \rangle$ was previously lower bounded by $\max(\frac{1}{d_{A}^2},\frac{1}{d_{D}^2})$.
In the above discussion, we have implicitly assumed that both $O_{Y}$ and $O_{W}$ are evolved with the same (possibly imperfect) quantum channel $\mathcal{Q}$.
However, it is certainly of interest to consider the situation where they evolve under two different quantum channels, which is precisely the experimental scenario if one performs backwards time evolution imperfectly.
We will address this case in detail a bit later.
\subsection{Distinguishing decoherence from scrambling}
As we have shown, for an arbitrary quantum channel, the decay of OTOCs is not sufficient to experimentally diagnose the scrambling behavior of the system.
To this end, we now provide a formal metric for distinguishing between scrambling and decoherence in noisy quantum systems.
Let us consider the state representation of the channel $\mathcal{Q}$ defined as follows:
\begin{align}
\hspace{-10mm} \rho \equiv \mathcal{Q}(|\text{EPR}\rangle \langle\text{EPR}|_{RA} \otimes |\text{EPR}\rangle \langle \text{EPR}|_{BB'}) = \nonumber \hspace{-10mm} \\ \figbox{0.9}{fig-world-channel}\ ,
\end{align}
where $\rho$ is the system's density matrix.
To gain some intuition, let us consider the two limiting cases: $p=0$ (no decoherence) and $p=1$ (full depolarization). In the first case, $\mathcal{Q}$ is purely unitary and can be decomposed into two separate boxes corresponding to $U$, $U^{\dagger}$, wherein $\rho = |\Psi \rangle \langle \Psi|$ is a pure state with $|\Psi \rangle$ as defined in Eqn.~\eqref{eq:world-state}.
In the second case, $\mathcal{Q}$ induces complete depolarization and the corresponding quantum state, $\rho$, is a maximally mixed state on $RCDB'$ with graphical representation:
\begin{align}
\rho = \frac{1}{d^2} \mathbb{I}_{R}\otimes \mathbb{I}_{C}\otimes \mathbb{I}_{D}\otimes \mathbb{I}_{B'}= \ \figbox{0.8}{fig-world-depolarization}\ .
\end{align}
As one can see from this example, for non-unitary time evolution, $\rho$ is \emph{not} a pure state.
However, one nevertheless finds that the averaged OTOC can be re-expressed in terms of R\'{e}nyi-$2$ entropies evaluated with respect to $\rho$ \footnote{
We do not know the terminology for the expression $S_{BD}^{(2)} + S_{D}^{(2)} - S_{B}^{(2)}$, but it is worth noting that $S_{BD} + S_{D} - S_{B} \geq 0$ corresponds to the celebrated Araki-Lieb inequality.
}:
\begin{align}
\langle \widetilde{\OTOC} \rangle= 2^{- (S_{B'D}^{(2)} + S_{D}^{(2)} - S_{B'}^{(2)}) }.
\label{eq:OTOCtilde_entropy}
\end{align}
The astute reader may wonder why this looks quite similar to the aforementioned result in the ideal, noiseless case [Eqn.~\eqref{eq:OTOC-unitary}]?
Since $S_{R}^{(2)}+S_{B'}^{(2)}=S_{C}^{(2)}+S_{D}^{(2)}=n$ (where $n$ is the total number of qubits in $RB'$), if $\rho$ was in fact a pure state, then one would have $S_{RB'D}^{(2)} = S_{C}^{(2)}$ and hence:
\begin{align}
S_{B'D}^{(2)} + S_{D}^{(2)} - S_{B'}^{(2)}= I^{(2)}(R,B'D).
\end{align}
Thus, when $\mathcal{Q}$ is unitary (e.g.~when the depolarizing probability $p=0$), the averaged OTOC indeed reduces to our previous result for the ideal system [Eqn.~\eqref{eq:OTOC-unitary}].
Crucially, for a generic noisy quantum channel, the state $\rho$ is \emph{not} pure and $S_{B'D}^{(2)} + S_{D}^{(2)} - S_{B'}^{(2)} \neq I^{(2)}(R,B'D)$!
Herein lies the essence of our result: The genuine metric for scrambling, the mutual information, is \emph{not} directly measured via the OTOC, which instead only measures the entropy, $S_{B'D}^{(2)} + S_{D}^{(2)} - S_{B'}^{(2)}$.
The deviation between these two quantities serves as a natural metric or ``noise parameter'' capturing the decoherence present in the channel $\mathcal{Q}$:
\begin{align}
\delta \equiv \frac{2^{I^{(2)}(R,B'D)}}{2^{S_{B'D}^{(2)} + S_{D}^{(2)} - S_{B'}^{(2)}}} = 2^{S_{C}^{(2)} -S_{RB'D}^{(2)} },
\label{eq:deltanoiseparam}
\end{align}
where $\delta=1$ for unitary time evolution while $\delta = 1/d_{D}^2$ for a completely depolarizing channel.
Note that for any $\delta < 1$, one knows that decoherence is at least partially responsible for the observed decay in the averaged OTOC.
More succinctly, there are two physical mechanisms that cause $\langle \widetilde{\OTOC}\rangle$ to decay. First, entangling $B'D$ with $R$ (as per unitary scrambling) and second, entangling $B'D$ with the environment (as in a depolarizing channel); $\delta$ captures the ratio between these two contributions.
In the following sections, we will turn to the experimental measurement and characterization of $\delta$, via a quantum teleportation decoding-protocol \cite{Hayden07, Yoshida:2017aa}. In Sec.~III, we will begin by setting up the framework of the protocol in the ideal case (decoherence and noise free), while in Sec.~IV, we will shift our attention to investigate a variety of imperfections (i.e.~both unitary and non-unitary errors).
\section{Teleportation-based Decoding Protocol (ideal case)}
\subsection{Representing the OTOC as a thermofield double state}
To begin, let us consider the diagrammatic representation of the OTOC in the case of unitary time-evolution $U$:
\begin{align}
\langle O_{A}O_{D}(t)O_{A}^{\dagger}O_{D}^{\dagger}(t) \rangle =\ \figbox{0.7}{fig-OTOC-unfold} \label{eq:OTOC-unfold}
\end{align}
where again, time runs upward and the expectation value is taken at infinite temperature; in our diagrammatic representation, connecting the legs of the input and output corresponds to taking a trace with respect to a maximally mixed state.
While the OTOC, $\langle O_{A}O_{D}(t)O_{A}^{\dagger}O_{D}^{\dagger}(t) \rangle$, is defined on the Hilbert space $\mathcal{H}_{AB}$, it can be recast as the expectation value of local operators on the doubled Hilbert space $\mathcal{H}_{AB}\otimes \mathcal{H}_{B'A'}$. In particular, consider the following state, $|\Phi_{O_{A}}\rangle \equiv (U_{AB}\otimes U_{B'A'}^{*}) (O_{A}\otimes I_{BB'A'}) | \text{EPR} \rangle_{ABB'A'}$, which lives in $\mathcal{H}_{AB}\otimes \mathcal{H}_{B'A'}$.
This is the so-called thermofield double state (at infinite temperature) perturbed by local operator $O_{A}$ and then time-evolved by $U\otimes U^{*}$. Taking the expectation value of $I_{C}\otimes O_{D} \otimes O_{D}^{*}\otimes I_{C'}$ in this state results in:
\begin{align}
\hspace{-10mm} \langle \Phi_{O_{A}} | I_{C}\otimes O_{D} \otimes O_{D}^{*}\otimes I_{C'}| \Phi_{O_{A}} \rangle = \nonumber \hspace{-10mm} \\ \figbox{0.9}{fig-OTOC} \label{eq:OTOC}
\end{align}
which is exactly equivalent to the OTOC defined in Eqn.~\eqref{eq:OTOC-unfold}. This equivalence is most easily seen by ``unfolding'' the diagram of Eqn.~\eqref{eq:OTOC-unfold} while noting that $(U\otimes I)|\text{EPR}\rangle=(I\otimes U^{T})|\text{EPR}\rangle$, or in diagrammatic form:
\begin{align}
\figbox{1.0}{fig-U} = \figbox{1.0}{fig-UT}\ .
\end{align}
Since $\langle O_{A}O_{D}(t)O_{A}^{\dagger}O_{D}^{\dagger}(t) \rangle = \langle \Phi_{O_{A}} | I_{C}\otimes O_{D} \otimes O_{D}^{*}\otimes I_{C'}| \Phi_{O_{A}} \rangle$, one can directly measure OTOCs as an expectation value of $O_{D} \otimes O_{D}^{*}$ in the doubled Hilbert space $\mathcal{H}_{AB}\otimes \mathcal{H}_{B'A'}$; then to compute $\overline{\langle\OTOC \rangle}$, one can simply average over the various operators: $O_{A}, O_{D}$.
As aforementioned, a more elegant and efficient method for measuring $\overline{\langle\OTOC \rangle}$ has recently emerged in the form of a probabilistic decoding protocol (via postselected teleportation) for the Hayden-Preskill thought experiment \cite{Yoshida:2017aa}.
\subsection{Decoding protocol in the ideal case}
In the decoding protocol, in addition to Alice's reference state, Bob also prepares an additional EPR pair $|\text{EPR}\rangle_{A'R'}$ before applying $U^{*}$ to both the entangled black-hole partner $B'$ and the $A'$-part of his EPR pair.
In order to decode Alice's state, Bob must create an EPR pair between Alice's reference state $R$ and his remaining register qubit, $R'$.
After time evolution, the system is in the state:
\begin{equation}
\begin{split}
|\Psi_{\text{in}}\rangle &=(I_{R}\otimes U_{AB} \otimes U^{*}_{B'A'}\otimes I_{R'}) \\ &\hspace{5mm}|\text{EPR}\rangle_{RA}\otimes|\text{EPR}\rangle_{BB'}\otimes|\text{EPR}\rangle_{A'R'} \\
&= \figbox{1.0}{fig-in-state}\ .
\end{split}
\end{equation}
Next, Bob collects pairs of qubits on $DD'$ and performs a projective measurement onto $|\text{EPR}\rangle_{DD'}$, resulting in the state:
\begin{align}
|\Psi_{\text{out}}\rangle = \frac{1}{\sqrt{P_{\text{EPR}}}} I_{RC}\otimes \Pi_{DD'}\otimes I_{C'R'} |\Psi_{\text{in}}\rangle =\nonumber \\ \frac{1}{\sqrt{P_{\text{EPR}}}}\ \figbox{1.0}{fig-out-state} \label{psiout}
\end{align}
where $P_{\text{EPR}}$ represents the probability of measuring $|\text{EPR}\rangle_{DD'}$. Noting that $\langle \Psi_{\text{out}}|\Psi_{\text{out}}\rangle = \frac{1}{P_{\text{EPR}}} \langle \Psi_{\text{in}}| I_{RC}\otimes \Pi_{DD'}\otimes I_{C'R'} | \Psi_{\text{in}}\rangle = 1$, yields the diagram for $P_{\text{EPR}}$:
\begin{equation}
\begin{split}
P_{\text{EPR}} \equiv \langle \Psi_{\text{in}}| I_{RC}\otimes \Pi_{DD'}\otimes I_{C'R'} | \Psi_{\text{in}}\rangle
= \\ \figbox{0.9}{fig-P} \label{eq-P}\ ,
\end{split}
\end{equation}
where $\Pi_{DD'}=|\text{EPR}\rangle\langle\text{EPR}|_{DD'}$ represents the projective measurement.
The fidelity of Bob's decoding (of Alice's state) can then be computed via the EPR projection fidelity on $RR'$:
\begin{align}
F_{\text{EPR}} \equiv \langle\Psi_{\text{out}}|I_{CDD'C'} \otimes \Pi_{RR'} |\Psi_{\text{out}}\rangle.
\end{align}
It has been shown \cite{Yoshida:2017aa} that if the time-evolution $U$ is scrambling, an EPR pair $|\text{EPR}\rangle_{RR'}$ can be distilled with high fidelity by post-selecting the measurement result on $|\text{EPR}\rangle_{DD'}$.
Thus, the projection, $\Pi_{DD'}$, not only serves to decouple Bob's register $R'$ from the remaining black holes, $CC'$, but also teleports Alice's quantum state $|\psi\rangle$ to Bob's register (Fig.~1b).
In the ideal, noiseless case, this probabilistic decoding protocol enables one to measure the averaged OTOC associated with $U$ in two \emph{different} ways, using the values of $P_{\text{EPR}}$ and $F_{\text{EPR}}$, respectively.
First, noting that $\int dO_{D}\ O_{D}\otimes O_{D}^{*} = \Pi_{DD'}$, one finds via a simple graphical derivation that~\cite{Roberts:2017aa},
\begin{align}
P_{\text{EPR}} = \overline{\langle \OTOC \rangle}.
\end{align}
Thus, by keeping track of the probability associated with the projective measurement, $\Pi_{DD'}$, one directly measures the averaged OTOC.
In the case of $F_{\text{EPR}}$, one can use the following equation:
\begin{equation}
\begin{split}
P_{\text{EPR}}F_{\text{EPR}} &=\langle \Psi_{\text{in}}| \Pi_{RR'}\Pi_{DD'}\otimes I_{CC'} | \Psi_{\text{in}}\rangle \\
&= \frac{1}{d_{A}^2}\ \figbox{0.9}{fig-PF-U} \\ &= \frac{ 1}{d_{A}^2}
\end{split} \label{pf-nonoise}
\end{equation}
to derive
\begin{align}
F_{\text{EPR}} = \frac{1}{d_{A}^2 \overline{\langle \OTOC \rangle}}.
\end{align}
To this end, the teleportation fidelity of Alice's state into Bob's register also directly encodes the averaged OTOC!
While both $P_{\text{EPR}}$ and $F_{\text{EPR}}$ measure $\langle \overline{\OTOC} \rangle$, there is an important (but subtle) distinction from the perspective of experiments; in particular, for a scrambling unitary, the former becomes small while the latter becomes large.
Thus, when using $P_{\text{EPR}}$, an experiment cannot distinguish between a decay in signal arising from scrambling or decoherence.
On the other hand, when using $F_{\text{EPR}}$, since decoherence can never enhance the fidelity, a successful decoding always serves as a definite signature of quantum scrambling.
This difference will become more apparent in Sec.~IV when we explicitly consider the effects of noise and decoherence.
\subsection{Teleportation of a quantum state}
In the previous subsection, we have formulated the decoding protocol in terms of the distillation of EPR pairs on $RR'$.
This formulation implicitly assumes an average over Alice's input state $|\psi\rangle$.
However, in the context of experiments, one necessarily perform the teleportation protocol for individual quantum states.
Moreover, for dynamics that are not fully scrambling, the dependence of the decoding fidelity on the initial state can be used to discern certain properties of the unitary.
An example of this is provided by a system evolving under classical random dynamics, where teleportation only occurs for computational basis states.
To this end, we now consider the decoding protocol for a specific input wavefunction, where Alice prepares $|\psi\rangle$ on $A$, and Bob checks to see if he obtains $|\psi\rangle$ on $R'$ (Fig.~1b):
\begin{align}
\figbox{1.0}{fig-no-ref}
\end{align}
Interestingly, such a setup for decoding specific states can probe more fine-grained properties of OTOCs. The probability of measuring an EPR pair on $DD'$ is given by
\begin{align}
P_{\psi} =\ \figbox{0.9}{fig-P-no-ref}\ .
\end{align}
This probability can be re-expressed in terms of OTOCs as follows:
\begin{align}
P_{\psi} = \iint d O_{D} d\phi \ \langle O_{A} O_{D}(t)O_{A}^{\dagger}O_{D}^{\dagger}(t)\rangle,
\end{align}
where $O_{A}= |\psi\rangle \langle \phi |$ and the average over $O_{A}$ is performed by integrating over $|\phi\rangle$.
It suffices to take an average over any set of orthogonal states (i.e.~$\{|0\rangle,|1\rangle,|2\rangle,\ldots\}$), since the above Haar-integral involves only the first moment of $|\phi\rangle$. By inserting an EPR projection onto $CC'$, one arrives at the following lower bound,
\begin{align}
P_{\psi}\geq \ \figbox{0.9}{fig-P-no-ref-bound}\ = \frac{1}{d_{A}^2}.
\end{align}
To recover $P_{\text{EPR}}$, one simply averages over (orthogonal) states, $P_{\text{EPR}} = \int d\psi \ P_{\psi}$.
Since the minimal value of $P_{\text{EPR}}$ is also $\frac{1}{d_{A}^2}$, this minimum is achieved when $P_{\psi} = \frac{1}{d_{A}^2}$ for all states.
Letting $F_{\psi}$ be the decoding fidelity after postselection, one finds
\begin{align}
P_{\psi}F_{\psi} = \frac{1}{d_{A}}\ \figbox{1.0}{fig-PF-no-ref}\ .
\end{align}
leading to the bound \footnote{To derive this lower bound, we again insert an EPR projector on $CC'$ into the diagram for $P_{\psi}F_{\psi}$.},
\begin{align}
P_{\psi}F_{\psi}\geq \frac{1}{d_{A}^2}\quad \Rightarrow \quad F_{\psi} \geq \frac{1}{d_{A}^2P_{\psi}}.
\label{eq:bound-state}
\end{align}
Thus, in the ideal case, a small value of $P_{\psi}$ guarantees the faithful post-selected teleportation of $|\psi\rangle$ from Alice to Bob.
In contrast to the previous subsection, we note that the value of $P_{\psi}F_{\psi}$ depends on the initial state $|\psi\rangle$.
One can also recast $P_{\psi}F_{\psi}$ as an OTOC,
\begin{align}
P_{\psi}F_{\psi} = \int d O_{D} \langle O_{A} O_{D}(t)O_{A}^{\dagger}O_{D}^{\dagger}(t)\rangle,
\label{eq:pfotocstates}
\end{align}
where $O_{A}= |\psi\rangle \langle \psi |$. Then, by averaging over input states \footnote{This integral over $|\psi \protect \rangle$ can be replaced with an average over a set of states that form a $2$-design. One example is the set of eigenstates of the Pauli operators.}, one obtains
\begin{align}
\int d\psi P_{\psi}F_{\psi} = \frac{1}{d_{A}+1}\Big( P_{\text{EPR}} +\frac{1}{d_{A}}\Big)
\end{align}
for the ideal, noise-free case.
\subsection{Physical interpretation of EPR projection}
\BY{}
Interestingly, Eqn.~\eqref{eq:pfotocstates} suggests that all of the accessible information about OTOCs probed in a state decoding experiment are averaged over operators $O_{D}$ on subsystem $D$.
The physical intuition, as well as the operational interpretation of taking this average is as follows.
In classical physics, chaos refers to the sensitive dependence of the system's dynamics on the initial conditions.
In particular, one can imagine preparing two identical objects, adding a small perturbation to one of them, and then letting them evolve under the same Hamiltonian.
If the system is chaotic, the outcomes will be drastically different, since a small initial perturbation has an exponentially growing effect.
In quantum systems, chaos can be probed by preparing a pair of objects with macroscopic entanglement, i.e.~in an EPR pair (or the thermofield double state at finite temperature).
Once again, one can imagine adding a small perturbation to one of the objects, and then letting them evolve under two Hamiltonians, $H$ and $H^{*}$, forward and backward in time, respectively. Without the perturbation, the system remains in an EPR pair since,
\begin{align}
e^{-iHt} \otimes e^{iH^{*}t} |\text{EPR}\rangle = |\text{EPR}\rangle.
\end{align}
However, with a perturbation, the resulting dynamics can cause the system to deviate.
In the above, our perturbation is $O_{A}$ and we are interested in studying how the initial $|\text{EPR}\rangle$ entanglement changes in time.
Operationally, we use the EPR projector, $\Pi_{DD'}$, to check if the entanglement (as seen from $DD'$) has been disturbed by the perturbation or not.
This now provides a natural bridge to explain the averaging over operators $O_{D}$ seen in Eqn.~\eqref{eq:pfotocstates}, since $\int d O_{D}\ O_{D}\otimes O_{D}^{*} = \Pi_{DD'}$.
In addition to verifying entanglement, the projector $\Pi_{DD'}$ also has the intriguing effect of ``undoing'' the chaotic dynamics.
To see this, let us assume that the time-evolution operator $U$ is strongly scrambling so that $\langle \overline{\OTOC} \rangle$ is close to its theoretical minimum $\sim \frac{1}{d_{A}^2}$. By postselecting on $|\text{EPR}\rangle_{DD'}$, one obtains an output state $|\Psi_{\text{out}}\rangle$ [Eqn.~\eqref{psiout}]. Since $\langle \Psi_{\text{in}} |\Pi_{RR'} \Pi_{CC'}\Pi_{DD'} | \Psi_{\text{in}} \rangle = \frac{1}{d_{A}^2}$, one has
\begin{align}
\langle \Psi_{\text{out}} |\Pi_{RR'} \Pi_{CC'}\Pi_{DD'} | \Psi_{\text{out}} \rangle = \frac{1}{d_{A}^2 \langle \overline{\OTOC} \rangle } \approx 1.
\end{align}
Thus, the projector $\Pi_{DD'}$ not only distills an EPR pair on $RR'$, but also undoes the chaotic time-evolution associated with $U$, returning the entire system to a set of EPR pairs!
In particular, if one prepares a quantum state $|\psi\rangle$ on $A$, then the output state will be close to $|\text{EPR}\rangle_{CC'}|\text{EPR}\rangle_{DD'}|\psi\rangle_{R'}$.
The fact that the projector $\Pi_{DD'}$ can halt the chaotic dynamics of $U$ is consistent with the traversable wormhole interpretation of the Hayden-Preskill thought experiment \cite{Hayden07,Traversable2017,Yoshida:2017aa}. Indeed, it has been found that the growth of the wormhole interior can be stopped or slowed down by applying certain interactions, and here, $\Pi_{DD'}$ plays the role of resetting the growth of the wormhole.
Most importantly, this observation provides an additional verification method for our teleportation-based decoding protocol. Once one measures an EPR pair on $DD'$, it is very likely that one will measure EPR pairs on other pairs of qubits if the experimental procedures are perfect and there is no decoherence.
\section{Teleportation-based Decoding Protocol: Arbitrary noise and decoherence}
In the previous section, we saw that in the \emph{absence} of decoherence, both $P_{\text{EPR}}$ and $F_{\text{EPR}}$ provide the same information, namely, the value of the averaged OTOC, which in the ideal case, precisely captures the scrambling behavior of the unitary.
We now turn to our piece de resistance, an analysis of the decoding protocol in the presence of arbitrary noise and imperfections, as characterized via a generic
quantum channel $\mathcal{Q}$.
The intuition behind the protocol's ability to distinguish between scrambling and decoherence is the redundancy provided by the pair of measurements, $P_{\text{EPR}}$ and $F_{\text{EPR}}$, in inferring the scrambling behavior of the unitary.
The protocol proceeds in exactly the same fashion as in the previous section, except that $\mathcal{Q}$ and $\mathcal{Q}^{*}$ are now applied (rather than $U$ and $U^{*}$).
A straightforward graphical calculation then yields the probability, $P_{\text{EPR}}$, associated with $\Pi_{DD'}$ as:
\begin{equation}
\begin{split}
&P_{\text{EPR}} = \langle \widetilde{\OTOC} \rangle \\
&= \figbox{0.85}{fig-P-channel} \label{eq-P-channel}\ ,
\end{split}
\end{equation}
As one might recall [Eqn.~\eqref{eq:OTOCtilde_entropy}], $\langle \widetilde{\OTOC} \rangle$ is directly related to the values of the R\'{e}nyi-$2$ entropies, $S_{B'D}^{(2)} + S_{D}^{(2)} - S_{B'}^{(2)}$, meaning that it contains effects from both decoherence and scrambling.
To measure the mutual information, $I^{(2)}(R,B'D)$, which encodes the true scrambling behavior of the channel $\mathcal{Q}$,
we return to our previous equation for $P_{\text{EPR}}F_{\text{EPR}}$, wherein one finds:
\begin{equation}
\begin{split}
P_{\text{EPR}}F_{\text{EPR}} &=\langle \Psi_{\text{in}}| \Pi_{RR'}\Pi_{DD'}\otimes I_{CC'} | \Psi_{\text{in}}\rangle \\
&= \frac{1}{d_{A}^2}\ \figbox{0.9}{fig-PF} \\ &= \frac{d_{C} \Tr(\rho_{RB'D}^2) }{d_{A}^2} = \frac{ \delta }{d_{A}^2},
\end{split}
\label{eq:pfdelta}
\end{equation}
where $\delta \equiv \frac{2^{I^{(2)}(R,B'D)}}{2^{S_{B'D}^{(2)} + S_{D}^{(2)} - S_{B'}^{(2)}}}= 2^{I^{(2)}(R,B'D)}\times P_{\text{EPR}}$ is precisely our previously defined noise parameter [Eqn.~\eqref{eq:deltanoiseparam}]!
We emphasize that equations \eqref{eq-P-channel} and \eqref{eq:pfdelta} are precisely the ``noisy-quantum-channel'' analogs of equations \eqref{eq-P} and \eqref{pf-nonoise} for the noise-free case.
The decoding fidelity after post-selection is then given by:
\begin{align}
F_{\text{EPR}} = \frac{ 2^{I^{(2)}(R,B'D) } }{ d_{A}^2}.
\end{align}
Thus, the success of teleportation implies true scrambling (i.e.~large $I^{(2)}(R,B'D)$) for a generic quantum channel, $\mathcal{Q}$.
Moreover, by measuring both $P_{\text{EPR}}$ and $F_{\text{EPR}}$, one can directly compute $\delta$, thereby characterizing the amount of noise in the quantum channel.
To see this in action, let us now return to the case where $\mathcal{Q}$ reflects a depolarizing channel [Eqn.~\eqref{eq:noise}].
In this situation, the measurement of $\delta$ via $P_{\text{EPR}}$ and $F_{\text{EPR}}$ immediately provides insight into the amount of dissipation in the system (given by probability $p$), since
\begin{align}
\delta =
\Big[ (1-p)^2 + (2p -p^2)\frac{1}{d_{D}^2} \Big].
\end{align}
While experimental decoherence cannot always be recast simply as depolarization, this expression serves as an operational (and quantitative) measure of extrinsic experimental noise.
For the case of state decoding, an analogous calculation reveals that the error parameter $\delta$ is given by:
\begin{align}
\int d\psi\ P_{\psi}F_{\psi} = \frac{1}{d_{A}+1}\Big( P_{\text{EPR}} +\frac{\delta}{d_{A}}\Big).
\end{align}
Interestingly, we note that as an alternative strategy, one can also study the effect of decoherence for a specific input state by observing possible violations of the bound in Eqn.~\eqref{eq:bound-state}.
\section{Teleportation-based Decoding Protocol: Coherent Errors}
\subsection{Distinguishing scrambling from coherent errors}
In the previous section, we focused on the case of a generic noisy quantum channel and more specifically, on the effects of depolarization.
In this subsection, motivated by recent experiments \cite{garttner2017measuring,li2017measuring}, we will consider the case of coherent unitary errors (i.e.~systematic over or under-rotations), which lead to imperfect ``backwards'' time-evolution (but no non-unitary decoherence).
In particular, we will investigate the situation where the time-evolution operator is given by $U\otimes V^{*}$ (rather than $U\otimes U^{*}$, which we assume to be the desired ideal case). For simplicity, let us assume that all other operations, including the initial preparation of EPR pairs and the final readout measurements are error-free \footnote{Note that coherent errors in the initial EPR preparation can also be absorbed into the definition of $V$.}.
In this scenario, the probability of measuring $|\text{EPR}\rangle_{DD'}$ is given by:
\begin{align}
P_{\text{EPR}} = \iint d O_{A}dO_{D} \langle O_{A}O_{D}(t) O_{A}^{\dagger}O^{\dagger}_{D_{V}}(t)\rangle = \nonumber \\ \figbox{0.9}{fig-P-cohe}
\label{pcoherenterrors}
\end{align}
where $O_{D}(t)=UO_{D}U^{\dagger}$ and $O_{D_V}(t)=VO_{D}V^{\dagger}$ are time-evolved by different unitaries, $U$ and $V$, respectively.
A simple graphical calculation yields the product, $P_{\text{EPR}}F_{\text{EPR}}$, as
\begin{align}
P_{\text{EPR}}F_{\text{EPR}} = \frac{1}{d_{A}^2}\ \figbox{1.0}{fig-PF-U-cohe} \ \leq \frac{1}{d_{A}^2},
\label{pfcoherenterrors}
\end{align}
which is strictly smaller than the ideal case (i.e. when $U=V$), where $P_{\text{EPR}}F_{\text{EPR}}=\frac{1}{d_{A}^2}$. Again, we emphasize that equations \eqref{pcoherenterrors} and \eqref{pfcoherenterrors} are precisely the ``coherent-error'' analogs of equations \eqref{eq-P} and \eqref{pf-nonoise} in the ideal case.
By analogy to Eqn.~\eqref{eq:pfdelta}, this suggests that one can define a noise parameter, $\eta$, for \emph{coherent} errors as follows
\begin{align}
P_{\text{EPR}}F_{\text{EPR}} = \frac{\eta}{d_{A}^2}.
\end{align}
While $\eta$ and $\delta$ effectively measure the same diagram, $\eta$ cannot be interpreted in terms of entropy since $U\otimes U^{*}$ is performed incorrectly.
Moreover, the physical interpretation of $\eta$ is quite different from that of $\delta$, which characterizes the strength of decoherence.
In particular, we note that a natural measure of the amount of coherent error is provided by the composite unitary operator, $E = U^\dagger V$. In the error-free, ideal case, $E$ simply corresponds to the identity operation. The noise parameter, $\eta$, is related to $E$ as follows
\begin{align}
\eta &= \ \figbox{1.0}{fig-delta-hat} \nonumber \\ &=\text{Tr}\left(\mathbb{I}\otimes \Pi_{DD'}E\Big( \frac{\mathbb{I}}{d_{C}}\otimes \Pi_{DD'} \Big) E^{\dagger} \right).
\label{eq:etaerror}
\end{align}
The right hand side of Eqn.~\eqref{eq:etaerror} is the $2$-norm overlap between $E|_{D}$ and $\mathbb{I}_{D}$; here, $E|_{D}$ is the quantum channel defined on $D$ by tracing out the degrees of freedom on $C$. If one takes $D$ to be the whole system, then $\eta$ is simply the $2$-norm overlap between $E$ and $\mathbb{I}$, precisely capturing the amount of deviation between the composite unitary, $U^{\dagger}V$, and the identity.
Finally, we note that $0\leq \eta \leq 1$, where the lower bound is saturated when $E= \mathbb{I}\otimes O_{D} $ for any traceless operator $O_{D}$. This contrasts with the decoherence noise parameter, $\delta$, which is lower bounded by $\min(\frac{1}{d_{A}^2},\frac{1}{d_{D}^2})$.
\subsection{Bounding the OTOC via $\eta$ in the case of coherent errors}
Intriguingly, under certain physical assumptions, one can utilize the measured value of $\eta$ to upper bound the true value of the OTOC that would have been measured without coherent errors (i.e.~if $U=V$).
In particular, we would like to compare the following two quantities:
\begin{align}
\langle O_{A}O_{D}(t)O_{A}^{\dagger}O_{D}^{\dagger}(t)\rangle \qquad
\langle O_{A}O_{D}(t)O_{A}^{\dagger}O_{D_V}^{\dagger}(t)\rangle,
\label{eq:twootocseqn}
\end{align}
and use the second, which is measured via $P_{\text{EPR}}$, to bound the first. For simplicity, let us assume that $O_{A},O_{D}$ are Pauli operators.
We will also make and justify a second assumption. In particular, consider an OTOC of the form,
$\langle O_{A}(0) O_{D}(t) O_{A}^{\dagger} O_{D}'^{\dagger}(t) \rangle$
where the two operators, $O_{D}$ and $O_{D}'$, are both time-evolved by $U$.
We will assume that
\begin{align}
\langle O_{A}(0) O_{D}(t) O_{A}^{\dagger} O_{D}'^{\dagger}(t) \rangle \approx 0 \label{eq:assumption-OTOC}
\end{align}
so long as $\Tr( O_{D} O_{D}'^{\dagger}) =0$.~The intuition behind this assumption is as follows.~At $t=0$, if regions $A$ and $D$ do not overlap, then $\langle O_{A}(0) O_{D}(t) O_{A}^{\dagger} O_{D}'^{\dagger}(t) \rangle=\Tr( O_{D} O_{D}'^{\dagger})= 0$.
Then, since OTOCs generically decay under ergodic time-evolution, one expect the above expectation value to remain small \emph{throughout} the time-evolution.
To proceed, it will be useful to define a new un-evolved (e.g.~time $t=0$) operator $O_E = E O_{D}E^{\dagger}$, which corresponds to the conjugation of $O_{D}$ by the composite unitary $E$. The subsequent time evolution of this operator via the unitary $U$ is given by: $UEO_{D}E^{\dagger}U^{\dagger} = VO_{D}V^{\dagger} = O_{D_V}(t)$. Then, we have
\begin{align}
\langle O_{A}O_{D}(t)O_{A}^{\dagger}O_{D_V}^{\dagger}(t)\rangle = \langle O_{A}O_{D}(t)O_{A}^{\dagger}O_{E}^{\dagger}(t)\rangle \label{eq:OTOC-exp}
\end{align}
where $O_{E}(t)=UO_{E}U^{\dagger}$.
Let us now expand the composite unitary, $E$, in terms of Pauli operators, $P$ and $Q$:
\begin{align}
E = \sum_{P,Q} \alpha_{P,Q} P \otimes Q,
\end{align}
where $P$, $Q$ act on subsystems $C$, $D$, respectively and $\sum_{P,Q}|\alpha_{P,Q}|^2 = 1$ \footnote{Note that this normalization condition is implied by the unitarity of $E$.}. Plugging this into our expression for $\eta$, one obtains
\begin{align}
\eta = \sum_{P} |\alpha_{P,\mathbb{I}}|^2.
\end{align}
Let us also expand $O_{E}$ in terms of Pauli operators,
\begin{align}
O_{E}= \sum_{P,Q} \beta_{P,Q} P \otimes Q,\label{eq:OD-hat}
\end{align}
where again $\sum_{P,Q}|\beta_{P,Q}|^2 = 1$. Plugging this expression back into Eqn.~\eqref{eq:OTOC-exp} yields,
\begin{align}
\langle O_{A}O_{D}(t)O_{A}^{\dagger}&O_{E}^{\dagger}(t)\rangle =
\sum_{P,Q} \beta_{P,Q} \langle O_{A}O_{D}(t)O_{A}^{\dagger}(P\otimes Q)(t)\rangle \nonumber \\
&\approx \beta_{\mathbb{I},O_{D}}\langle O_{A}O_{D}(t)O_{A}^{\dagger}O_{D}^{\dagger}(t)\rangle,
\label{betadecompotoc}
\end{align}
where we have used our assumption [Eqn.~\eqref{eq:assumption-OTOC}] to drop all terms with $Q \neq O_D$ in going from the first to second line.
Noting that $\beta_{\mathbb{I},O_{D}} = \frac{1}{d}\Tr ( O_{D} O_{E}^{\dagger} )$, allows us to bound it as follows:
\begin{align}
&\beta_{\mathbb{I}, O_{D} }
= \frac{1}{d} \Tr\left[ (\mathbb{I} \otimes O_{D} ) E(\mathbb{I} \otimes O_{D}^{\dagger} )E^{\dagger} \right] \nonumber \\
&= \frac{1}{d} \Tr \left[ \sum_{P,Q} |\alpha_{P,Q}|^2 ( \mathbb{I} \otimes O_{D} ) (P \otimes Q )( \mathbb{I} \otimes O_{D}^{\dagger} ) (P \otimes Q ) \right] \nonumber \\
&= \sum_{P} |\alpha_{P,\mathbb{I}}|^{2} + \sum_{P}\sum_{Q\not=\mathbb{I}} \pm|\alpha_{P,Q}|^{2} \geq 2\eta -1. \label{eq:bound-beta}
\end{align}
Here, the $\pm$-signs in the final line correspond to the case where $O_{D}$ and $Q$ commute/anti-commute, respectively.
Thus, the lower bound corresponds to the case where all non-zero $\alpha_{P,Q}$ come with a negative sign.
Finally, combining Eqns.~\eqref{eq:OTOC-exp}, \eqref{betadecompotoc}, and \eqref{eq:bound-beta} yields the following bound:
\begin{align}
\langle O_{A}O_{D}(t)O_{A}^{\dagger}&O_{D_V}^\dagger(t)\rangle =\langle O_{A}O_{D}(t)O_{A}^{\dagger}O_{E}^\dagger(t)\rangle \nonumber \\
&= \beta_{\mathbb{I},O_{D}}\langle O_{A}O_{D}(t)O_{A}^{\dagger}O_{D}^{\dagger}(t)\rangle \nonumber \\
&\geq ( 2\eta - 1) \langle O_{A}O_{D}(t)O_{A}^{\dagger}O_{D}^{\dagger}(t)\rangle.
\end{align}
Thus, in the case of coherent errors corresponding to imperfect backwards time evolution, the experimentally \emph{measured} value of the averaged OTOC (via for example $P_\text{EPR}$) explicitly bounds the actual ideal OTOC:
\begin{align}
\langle \overline{\text{OTOC}} \rangle \le \frac{P_{\text{EPR}}}{ 2\eta -1 }.
\end{align}
We note that this bound is only valid for $\eta > 0.5$.
Two additional remarks. First, it is worth pointing out that the value of $\beta_{\mathbb{I}, O_{D} }$ can be directly measured via $\langle O_{A}O_{D}(t)O_{A}^{\dagger}O_{D_V}^{\dagger}(t)\rangle$ with $O_{A}=\mathbb{I}$, since $\beta_{\mathbb{I}, O_{D} } = \langle O_{D}O_{E}^{\dagger}(t)\rangle=\langle O_{D}(t) O_{D_V}^{\dagger}(t)\rangle$.
Second, in a generic chaotic system, one expects the $\pm$-signs in Eqn.~\eqref{eq:bound-beta} to appear randomly. Under this assumption, one can make the following approximation:
\begin{align}
\beta_{\mathbb{I}, O_{D} } \approx \sum_{P} |\alpha_{P,\mathbb{I}}|^2 = \eta,
\end{align}
which enables us to obtain an estimate for the actual \emph{value} of the OTOC and not simply a bound,
\begin{align}
\langle O_{A}O_{D}(t)O_{A}^{\dagger}O_{D}^{\dagger}(t)\rangle \approx \frac{1}{\eta}\langle O_{A}O_{D}(t)&O_{A}^{\dagger}O_{D_V}^\dagger(t)\rangle.
\end{align}
\section{Bounding the Mutual Information via the decoding fidelity}
\subsection{Mutual Information Bound}
In the previous section, we have shown that in the case of coherent errors, one can utilize $\eta$ as extracted from $P_{\text{EPR}}$ and $F_{\text{EPR}}$ to formally bound the true (i.e.~error-free) value of the averaged OTOC. However, this proof explicitly hinges on the unitarity of the composite channel $E$ and is thus inapplicable to the generic situation with decoherence. Moreover, in the presence of decoherence, it becomes ambiguous to define what precisely the \emph{value} of the OTOC is \footnote{The intuition behind this ambiguity is that for a generic quantum channel, one can decompose its action using Kraus operators but this decomposition is not unique \cite{nielsen2010quantum}.}; rather, as we have previously seen, a better characterization for quantum scrambling is provided by the mutual information.
To this end, in this section, we demonstrate that for \emph{arbitrary} quantum channels, one can derive a bound
on the mutual information, $I^{(2)}(R,B'D)$, using only the decoding fidelity, $F_{\text{EPR}}$ \footnote{The key point here is that $F_{\text{EPR}}$ always provides a lower bound on the \emph{mutual information} between $R$ and $B'D$, regardless of the nature of experimental errors. Stated differently, in the context of the black hole information problem, the fact that one can retrieve a quantum state from the Hawking radiation (i.e.~the teleportation is successful) implies that the system has scrambled, regardless of how one performs the decoding.}. When applied to the case of purely unitary errors (i.e.~the previous section), this leads to a somewhat weaker bound on $\langle \overline{\text{OTOC}} \rangle$.
To treat experimental imperfections on a fully general footing, we consider time-evolution via the quantum channel $\mathcal{Q}$ and an arbitrary decoding operation $\Phi$, acting non-trivially only on $B'D$ (Fig.~\ref{fig-general}). As previously discussed, the goal of this decoding operation is to distill an EPR pair on $R\bar{R}$, where $\bar{R}$ represents a subset of the qubits in $B'D$ with the same dimension as $R$ (e.g. $|R|=|\bar{R}|$).
Let us assume that $\Phi$, an arbitrary completely-positive trace-preserving map, outputs a normalized state supported on $R\bar{R}$:
\begin{align}
\Phi: \rho_{RB'D} \rightarrow \sigma_{R\bar{R}}.
\end{align}
Since the decoding operation acts locally on $B'D$, it \emph{cannot} increase entanglement between $R$ and $B'D$, i.e.~the mutual information satisfies $I(R,B'D)\geq I(R,\bar{R})$ \footnote{Note that we are using the von Neumann mutual information here.}.
Since $I(R,\bar{R})$ can be lower bounded via $F_{\text{EPR}}$, any non-trivial decoding fidelity always signifies quantum scrambling even in the presence of arbitrary imperfections.
\begin{figure}
\centering\includegraphics[width=1.5in]{fig3_v2}
\caption{Schematic representation of the decoding protocol as the distillation of an EPR pair on $R\bar{R}$. The initial state is $|\text{EPR}\rangle_{RA}|\text{EPR}\rangle_{BB'}$. The quantum channel $\mathcal{Q}$ describes the time-evolution of the system and $\Phi$ represents an arbitrary decoding operation acting only on $B'D$.
}
\label{fig-general}
\end{figure}
Deriving a rigorous lower bound on either the von Neumann or R\'{e}nyi-$2$ mutual informations in terms of $F_{\text{EPR}}$ is tremendously challenging, owing in part, to the existence of fine-tuned adversarial examples that tend to make the bounds loose in physically relevant situations.
This is ameliorated by making the assumption that the decoding fidelity is independent of the input state $|\psi\rangle$.
One expects this assumption to be approximately valid for strongly interacting systems without conserved quantities after the system locally thermalizes. This assumption also excludes the trivial decoding protocol which returns a fixed state regardless of input state.
To begin, we note that $\rho_{R}$ is a maximally mixed state [Eqn.~(11)], implying that $\sigma_{R}$ is also maximally mixed, since $\Phi$ acts only locally on $B'D$.
Moreover, our assumption that the decoding fidelity does not depend on the input state implies that $\sigma_{\bar{R}}$ is a maximally mixed state as well.
While generally true, this statement is particularly easy to see in the case where $R$ and $\bar{R}$ consist of only single qubits.
In particular, we can use our previous trick and decompose $\sigma_{R\bar{R}}$ in terms of Pauli operators: $\sigma_{R\bar{R}} = \sum_{P,Q} \gamma_{P,Q} P\otimes Q$, for $P,Q \in \{\mathbb{I}, X,Y,Z\}$.
Since $\sigma_{R}$ is maximally mixed, one has that $\gamma_{P, \mathbb{I}}=0$ for all $P\not=\mathbb{I}$.
For an input state $|\psi\rangle$, the quantum state on $\bar{R}$ is given by
\begin{align}
d_{R}(|\psi\rangle\langle \psi| \otimes \mathbb{I}_{\bar{R}}) \sigma_{R\bar{R}}(|\psi\rangle\langle \psi| \otimes \mathbb{I}_{\bar{R}})
\end{align}
where $d_{R}$ is a normalization constant. The decoding fidelity can then be written as
\begin{align}
F_{\psi}= d_{R}\Tr ( |\psi\rangle\langle \psi| \otimes |\psi^{*}\rangle\langle \psi^{*}| \sigma_{R\bar{R}} ).
\end{align}
Noting that $|0\rangle\langle 0 | = \frac{I+Z}{2}$ and $|1\rangle\langle 1 | = \frac{I-Z}{2}$, implies $\gamma_{Z, \mathbb{I}}+\gamma_{\mathbb{I},Z}=0$; since $\gamma_{Z,\mathbb{I}}=0$, one also has that
$\gamma_{\mathbb{I}, Z}$=0. The same analysis leads to $\gamma_{\mathbb{I},P}=0$ for all non-identity Pauli operators $P$. Thus, $\sigma_{\bar{R}}$ is also a maximally mixed state.
Having shown that $\sigma_{\bar{R}}$ is a maximally mixed state, we are now ready to lower bound the mutual information. The EPR projector and the decoding fidelity of the distilled quantum state, $\sigma_{R\bar{R}}$, are given by: $\Pi_{R\bar{R}}= |\text{EPR}\rangle\langle \text{EPR}|_{R\bar{R}}$ and $F_{\text{EPR}} = \Tr( \Pi_{R\bar{R}} \rho_{R\bar{R}} )$, respectively.
Then using the Cauchy-Schwartz inequality, one immediately arrives at the following bound:
\begin{align}
&S^{(2)}_{R\bar{R}}= - \log_{2} \Tr( \rho_{R\bar{R}}^2 ) \leq \nonumber \\ &- \log_{2} \Tr( \Pi_{R\bar{R}}\rho_{R\bar{R}}) \Tr( \Pi_{R\bar{R}}\rho_{R\bar{R}}) = - 2\log_{2}F_{\text{EPR}},
\end{align}
implying that the mutual information satisfies:
\begin{align}
I^{(2)}(R,\bar{R}) = S_{R} + S_{\bar{R}} - S_{R\bar{R}} \ge 2 \log_{2} d_{R} + 2 \log_{2} F_{\text{EPR}}.
\end{align}
In order to utilize the monotonicity of mutual information \cite{nielsen2010quantum}, we will make the additional technical assumption that the R\'{e}nyi-$2$ and von Neumann entropies are close to one another \footnote{Recall that R\'{e}nyi-$2$ mutual information is not monotonically decreasing in general \cite{suppinfo}.}.
This then leads to our final result, lower bounding the mutual information in terms of the decoding fidelity:
\begin{align}
I(R,B'D)\approx I^{(2)}(R,B'D) \ge 2 \log_{2} d_{R} + 2\log_{2} F_{\text{EPR}}.
\end{align}
\subsection{OTOC Bound for Coherent Errors}
While the previous subsection focused on the case of \emph{arbitrary} quantum channels, one can also apply the derived bound to the situation where only coherent errors are present.
To this end, let us return to scenario described in Sec.~IVb, where the time-evolution is given by $U\otimes V^{*}$.
As we have already seen, the measurement of $P_{\text{EPR}}$ corresponds to
\begin{align}
P_{\text{EPR}} = \int dO_{A} d O_{D} \langle O_{A}O_{D}(t)O_{A}^{\dagger}O_{D_V}^{\dagger}(t) \rangle,
\end{align}
which includes the effect of unitary errors associated with $E = U^\dagger V \neq \mathbb{I}$. In analogy to Sec.~IVc, the true OTOC, which would have been measured if the experiment did not contain such unitary errors is given by:
\begin{align}
\overline{\langle \OTOC \rangle} = \int dO_{A} d O_{D} \langle O_{A}O_{D}(t)O_{A}^{\dagger}O_{D}^{\dagger}(t) \rangle.
\end{align}
Since $ \overline{\langle \OTOC \rangle}= 2^{-I^{(2)}(A,BD)}$, our above bound on the mutual information also immediately bounds $ \overline{\langle \OTOC \rangle}$ in the case of purely coherent errors:
\begin{align}
\overline{\langle \OTOC \rangle} \leq \frac{1}{d_{R}^2 F_{\text{EPR}}^2}.
\end{align}
\section{Experimental Implementation}
Having detailed a teleportation protocol that explicitly enables experiments to distinguish between decoherence and quantum information scrambling \footnote{Notable examples of fast quantum information scramblers include: the SYK model~\cite{Kitaev:2014t2}, $k$-local random spin models~\cite{Erdos:2014aa} and random quantum circuits~\cite{Dankert09}. }, we now propose two specific examples of scrambling Clifford circuits \cite{suppinfo} amenable to near-term experiments in small-scale quantum simulators \cite{zhang2017observation,bernien2017probing}.
\subsection{Qubit Clifford Scrambler}
Let us consider the following $3$-qubit unitary operator:
\begin{align}
U = \figbox{0.9}{fig-qubit-unitary}
\end{align}
where $H$ represents a Hadamard gate, while two-qubit, control-$Z$ gates ($|i,j\rangle \rightarrow (-1)^{ij}|i,j\rangle$) are depicted as horizontal lines (ending in dots).
This unitary is maximally scrambling since all one-body Pauli operators are delocalized into three-body Pauli operators under $U$ \cite{Pastawski15b}. From the perspective of decoding, this delocalization implies that Bob can collect any pair of qubits (from among the three possible pairs in Figure 3) and perform a projective measurement in order to decode Alice's state.
To be concrete, the full decoding protocol is illustrated in Figure 3.
\begin{figure}
\centering\includegraphics[width=2.3in]{fig2_expt}
\caption{Decoding circuit based upon a 3-qubit Clifford scrambler. Alice's quantum state $|\psi\rangle$ is supported on qubit-$1$, while Bob's quantum register corresponds to qubit-$7$.
The left (beige) and right (gray) Hilbert spaces have the following correspondences $1 \leftrightarrow 6$, $2 \leftrightarrow 5$ and $3 \leftrightarrow 4$ (with respect to $U\otimes U^{*}$).
By performing an EPR projection on qubits $3$ and $4$, Bob teleports Alice's quantum state to his register qubit. In the case of this Clifford scrambler, Bob could also have achieved teleportation by performing EPR projections on either qubits $\{1,6\}$ or $\{2,5\}$. This distinguishes the Clifford scrambler from other more trivial (non-scrambling) unitaries (i.e.~a SWAP gate), where teleportation only occurs for EPR projection on a specific pair of qubits.
}
\label{fig-qubitexpt}
\end{figure}
Two comments are in order.
In particular, for a Haar random unitary, one expects $\langle \overline{\OTOC}\rangle_{\text{S}} = \frac{7}{16}$, whereas our circuit exhibits: $\langle \overline{\OTOC}\rangle_{\text{S}} = \frac{1}{4}$.
This discrepancy arises from finite size effects, since one expects a Haar random unitary to saturate the lower bound of $1/4$ only in the limit of large systems, i.e.~$d,d_{D}\rightarrow \infty$ while fixing $d_{R}=2$.
On the other hand, our Clifford circuit saturates this lower bound by construction but has certain non-generic features \cite{suppinfo}.
Second, as we briefly alluded to in Sec.~IIIc, it is also possible to explore circuits that scramble only classical information:
\begin{align}
U = \figbox{1.0}{fig-qubit-unitary-classical}.
\end{align}
In this case, teleportation occurs only for computational basis states.
\subsection{Qutrit Clifford Scrambler}
While we presented the minimum case of interest for qubits in the previous subsection, by increasing the on-site Hilbert space, one can realize slightly more complex circuits in even smaller systems. To this end, motivated by the advent of physical qutrit implementations ranging from solid-state spin defects and superconducting circuits to orbital angular momentum states of photons, we describe a simple qutrit Clifford scrambler.
To begin, we denote a qutrit as a three-state quantum spin with basis: $|0\rangle, |1\rangle, |2\rangle$. An elementary entangling gate between two qutrits can be achieved via the following controlled-NOT gate:
\begin{align}
\text{CNOT}_{1\rightarrow 2}|i,j\rangle = |i,i+j\rangle \qquad \text{modulo $3$}
\end{align}
where the subscript $1\rightarrow 2$ indicates that the control is qutrit-$1$ and the target is qutrit-$2$. Switching the control and target realizes an analogous operation: $\text{CNOT}_{2\rightarrow 1}|i,j\rangle = |i+j,i\rangle
\hspace{1mm}\text{mod $3$}$.
Let us now consider the following qutrit unitary:
\begin{align}
U = \text{CNOT}_{2\rightarrow 1}\text{CNOT}_{1\rightarrow 2},
\end{align}
which can be explicitly decomposed as $U|i,j\rangle = |2i+j,i+j\rangle$ or graphically re-expressed as:
\begin{align}
U = \figbox{1.0}{fig-qutrit-unitary}.
\label{eq:qutritunitarycnots}
\end{align}
To understand the scrambling properties of this unitary, we will explore how the qutrit Pauli operators are transformed under the circuit; in particular, let us consider the following qutrit Pauli's: $X = \sum_{j=0}^{2} |j+1\rangle\langle j|$ and $Z = \sum_{j=0}^2 \omega^j |j\rangle\langle j|$ where $\omega = e^{i\frac{2\pi}{3}}$.
One finds that these operators are transformed as follows:
\begin{align}
&U(Z\otimes I )U^{\dagger} = Z\otimes Z^2 \nonumber \\ &U(I\otimes Z )U^{\dagger} = Z^2\otimes Z^2 \nonumber \\
&U(X\otimes I )U^{\dagger} = X^2\otimes X \nonumber \\ &U(I\otimes X )U^{\dagger} = X\otimes X.
\end{align}
Thus, as in the qubit case, we observe that the unitary transforms any non-identity one-body Pauli operator into a two-body operator.
This property is essential for the delocalization of quantum information and enables the construction of a similar decoding protocol:
\begin{align}
\figbox{1.0}{fig-qutrit-unitary-whole}.
\end{align}
By performing an EPR projection on either qutrits $\{2,3\}$ or $\{1,4\}$, Bob successfully teleports Alice's quantum state from qutrit-$1$ to qutrit-$5$.
\subsection{Distinction from conventional quantum teleportation}
The importance of being able to perform teleportation by projecting \emph{either} pair of qutrits (or in the previous case, any of the three qubit pairs) is most easily seen by considering the effect of a SWAP gate, $\text{SWAP}|i,j\rangle = |j,i\rangle$, or graphically:
\begin{align}
\textrm{SWAP} = \figbox{1.0}{fig-SWAP}.
\end{align}
From the perspective of scrambling, a SWAP gate is totally trivial since it does not generate any entanglement; thus, its decoding behavior must be markedly different from that of the maximally scrambling $U$ in Eqn.~\eqref{eq:qutritunitarycnots}.
Replacing $U$ with the SWAP gate in the decoding protocol leads to the following:
\begin{align}
\figbox{1.0}{fig-SWAP-whole},
\end{align}
which is simply ordinary quantum teleportation \cite{nielsen2010quantum, bouwmeester1997experimental,furusawa1998unconditional}. Crucially, this teleportation \emph{only} works when Bob projects on qutrits $\{2, 3\}$ and fails if he attempts to project on qutrits $\{1, 4\}$. Herein lies the essential feature of a maximally scrambling unitary: Successful decoding and teleportation occur regardless of which pair of qutrits (or qubits) one chooses to collect and project, precisely indicating the full delocalization of quantum information across the circuit.
\section{Conclusion}
We have demonstrated that one can distinguish between scrambling and decoherence in strongly interacting quantum systems by utilizing a teleportation-based decoding protocol. Our protocol enables the explicit extraction of a ``noise parameter'', which can be used to bound the value of out-of-time-ordered correlation functions in the case of coherent errors. More generally, even for arbitrary imperfections, the teleportation fidelity acts as a metric
for quantum scrambling and enables the bounding of the mutual information between subsystems. Thus, our protocol represents the first example of an experimental method, which can unambiguously characterize the
delocalization of quantum information within a system's own degrees of freedom and differentiate this from entanglement with an extrinsic environment.
Our work opens the door to a number of intriguing future directions.
First, by systematically exploring the state-dependent decoding fidelity in the presence of different forms of decoherence, one may be able to study the transition from classical to quantum chaos.
Second, in this work, we have mainly focused on decoherence as an adversary to quantum scrambling.
However, the pronounced sensitivity of scrambling dynamics to the presence of decoherence suggests that one may be able to utilize our protocol as a particularly efficient ``noise'' spectroscopy tool.
Finally, an interesting question that has received much recent attention, and which goes under the moniker of quantum supremacy, is whether quantum devices without error correction can perform computational tasks beyond the capabilities of classical computers \cite{preskill2018quantum}.
It has been suggested that the simulation of random quantum circuits may be an ideal platform for this purpose~\cite{boixo2016characterizing}.
%
Since OTOCs are natural probes of pseudorandomness, it may be possible to generalize our protocol to explore such questions.
%
\emph{Acknowledgements---}We gratefully acknowledge the insights of and discussions with P. Hess, A. Kyprianidis, G. Pagano, J. Zhang, V. Ramasesh, M. Blok, I. Siddiqi, C. Monroe, and Q. Zhuang.
We are particularly indebted to T. Schuster for a careful reading of the manuscript and many helpful discussions.
This work was supported by the DOE under contract PH-COMPHEP-KA24 and the Office of Advanced Scientific Computing Research, Quantum Algorithm Teams Program.
\mciteSetMidEndSepPunct{}{\ifmciteBstWouldAddEndPunct.\else\fi}{\relax}
\section{Clifford scramblers versus Haar random unitaries}
The aforementioned $2$-qutrit and $3$-qubit unitary operators are Clifford operators which transform Pauli operators into Pauli operators. This is a rather restrictive class of unitary operators. For instance, a Haar random unitary $U$ will transform a Pauli operator $P$ into $UPU^{\dagger}=\sum_{Q \in \text{Pauli}}C_{Q}Q$ where $Q$ are Pauli operators, and $C_{Q}$ are coefficients that are almost uniformly distributed over non-identity Pauli operators. Because of the special property of Clifford operators which preserve the Pauli group, the decoding task can be performed even without post-selection. To see this, let us explicitly consider the case with the $3$-qubit unitary operator. Bob performs measurements in the Bell basis:
\begin{align}
\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) \quad \frac{1}{\sqrt{2}}(|10\rangle + |01\rangle) \quad \frac{1}{\sqrt{2}}(|10\rangle - |01\rangle) \quad \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)
\end{align}
which can be also written as
\begin{align}
(I\otimes I)|\text{EPR}\rangle \quad (X\otimes I)|\text{EPR}\rangle \quad (Y\otimes I)|\text{EPR}\rangle \quad (Z\otimes I)|\text{EPR}\rangle.
\end{align}
The original probabilistic decoding protocol succeeded only when Bon measures $|\text{EPR}\rangle$ on $DD'$. Suppose that Bob had instead measured $(X\otimes I)|\text{EPR}\rangle$ instead of $|\text{EPR}\rangle$. The outcome can be represented graphically as follows
\begin{align}
\figbox{1.0}{fig-Clifford}.
\end{align}
where a Pauli $X$ operator is inserted on a horizontal line connecting $DD'$. Since $U$ is a scrambling operator, there exists some operator $V$ supported on $C'R'$ which satisfies:
\begin{align}
\figbox{1.0}{fig-Clifford-map1}\approx \figbox{1.0}{fig-Clifford-map2}.
\end{align}
This statement follows from the fact that $I(A,BD)=I(D,AC)$ is nearly maximal. In general, the operator $V$ cannot be written as a tensor product of two operators acting on $C'$ and $R'$. In the case of $U$ being a Clifford operator, $V$ can be written as
\begin{align}
V = P\otimes Q
\end{align}
where $P$ and $Q$ are some Pauli operators acting on $C'$ and $R'$ respectively. This implies that, by applying $Q$ on $R'$, one can reconstruct the original quantum state even if $(X\otimes I)|\text{EPR}\rangle$ was measured. It should be emphasized that this phenomena crucially relies on the fact that the time-evolution operator was a Clifford operator, and does not occur for generic scrambling unitary operators, such as a Haar random unitary.
Another subtle difference between Haar random unitary operators and Clifford operators is the value of averaged OTOCs. Let us consider the case where $d_{A}\leq d_{D}$. If $U$ is drawn uniformly at random, the late-time asymptotic value is given by
\begin{align}
\langle \overline{\OTOC} \rangle \approx \frac{1}{d_{A}^2} + \frac{1}{d_{D}^2} - \frac{1}{d_{A}^2d_{D}^2}.
\end{align}
Namely, the values of $\langle \overline{\OTOC} \rangle$ for different random unitary operators will not differ much as the variance is suppressed by the system size. On the other hand, if $U$ is drawn from Clifford operators, we have the same ensemble average
\begin{align}
\int_{U \in \text{Clifford}} dU\ \langle \overline{\OTOC} \rangle \approx \frac{1}{d_{A}^2} + \frac{1}{d_{D}^2} - \frac{1}{d_{A}^2d_{D}^2}.
\end{align}
However, the statistical variance of $\langle \overline{\OTOC} \rangle$ is not suppressed by the system size. The reason why this variance remains unsuppressed is simple: the values of OTOCs taken with respect to Pauli operators, are either $\pm1$, and become small only after taking an average over $O_{A}$ and $O_{D}$. Thus, according to our fine-grained definition of scrambling, random Clifford operators are not scrambling as four-point OTOCs do not decompose as in Eqn.~(1). On the other hand, such random Clifford operators do satisfy our coarse-grained definition of average scrambling.
Finally, Clifford operations are typically assumed to be (relatively) easy to implement while non-Clifford gates are significantly more challenging. Let us consider a scenario where we would like to check if a given unitary operator $U$ is a Clifford operator or not. One approach, which is motivated by OTOCs, is to measure the commutator
\begin{align}
\langle P(t)Q(0)P(t)Q(0) \rangle = \frac{1}{d}\text{Tr} ( UPU^{\dagger}Q UPU^{\dagger}Q)
\end{align}
for randomly chosen Pauli operators $P,Q$. If $U$ is a Clifford operator, then the above quantity should be either $+1$ or $-1$. The merit of this method is that one can tell if $U$ is Clifford or not after only a few trials with reasonable confidence via relatively simple operations.
\section{Towards a finite temperature generalization}
We have treated the cases where the quantum state is maximally mixed in OTOCs; $\langle O_{X}O_{Y}(t)O_{Z}O_{W}(t)\rangle = \Tr( O_{X}O_{Y}(t)O_{Z}O_{W}(t) \frac{1}{d} \mathbb{I})$. The relation between the mutual information $I(A,BD)$ and OTOCs can be generalized to cases where the input and output ensembles factorize; $\rho_{AB}=\rho_{A}\otimes \rho_{B}$ and $\rho_{CD}=\rho_{C}\otimes \rho_{D}$. This generalization was originally discussed in~\cite{Kitaev-Haar,Yoshida:2017aa}. Note that previously, at infinite temperature, $\rho_{A}=\frac{1}{d_{A}}\mathbb{I}_{A}$, $\rho_{B}=\frac{1}{d_{B}}\mathbb{I}_{B}$, $\rho_{C}=\frac{1}{d_{C}}\mathbb{I}_{D}$, $\rho_{D}=\frac{1}{d_{D}}\mathbb{I}_{D}$.
In order to consider the state representation of a unitary operator $U$ with the initial quantum state $\rho_{AB}$, we simply replace each dot with $\rho_{R}^{1/2}$ on the corresponding Hilbert space $R$ in Eqn.~(21):
\begin{align}
\figbox{1.0}{fig-state} = \figbox{1.0}{fig-U-in}= \figbox{1.0}{fig-U-out}.
\end{align}
The goal is to lower bound the mutual information $I(A,BD)$ of the above pure state $|\Psi\rangle$ from OTOCs. To do so, we think of preparing the thermofield double states for $\rho_{A}$ on $RA$ and $A'R'$, and the thermofield double state for $\rho_{B}$ on $BB'$. Note that the thermofield double state can be created by replacing a dot in the EPR pair with $\rho_{R}^{1/2}$ on the corresponding Hilbert space $R$. We then performs a projective measurement onto the thermofield double state for $\rho_{D}$ on $DD'$. Defining the density matrix $\rho=|\Psi\rangle\langle \Psi|$, the amplitude for this projective measurement is given by
\begin{align}
P = \Tr \Big[ \rho_{BD} (\rho_{B}\otimes \rho_{D}^{-1})^{-1/2} \rho_{BD} (\rho_{B}\otimes \rho_{D}^{-1})^{-1/2} \Big].
\end{align}
Using the sandwiched R\'{e}nyi-$\alpha$ divergence, defined as $D_{\alpha}(f||g)=\frac{1}{\alpha - 1} \log \left( \frac{1}{\text{Tr} (f)} \text{Tr} \Big[\big( g^{\frac{1-\alpha}{2\alpha}}f g^{\frac{1-\alpha}{2\alpha}}\big)^{\alpha} \Big] \right)$, the amplitude can be written as
\begin{align}
\log_{2} P = D_{2}(\rho_{BD} | \rho_{B}\otimes \rho_{D}^{-1} ) \label{eq:P-divergence}
\end{align}
Using the monotonicity of R\'{e}nyi-$\alpha$ divergence~\cite{beigi2013sandwiched}, it can be bounded as follows:
\begin{align}
\log_{2}(P) \geq D_{1}(\rho_{BD} | \rho_{B}\otimes \rho_{D}^{-1} ) = -S_{BD} + S_{B} - S_{D}.
\end{align}
The righthand side is equal to $- I(A,BD)$ due to unitarity of $U$, so we have
\begin{align}
I(A,BD) \geq - \log_{2} P.
\end{align}
The rest is to relate the amplitude $P$ to OTOCs. One may consider the following two types of OTOCs:
\begin{equation}
\begin{split}
\langle O_{X}O_{Y}(t)O_{Z}O_{W}(t)\rangle_{1} =\Tr( O_{X}O_{Y}(t)O_{Z}O_{W}(t) \rho_{AB})\\
\langle O_{X}O_{Y}(t)O_{Z}O_{W}(t)\rangle_{2} =\Tr( O_{X}O_{Y}(t) \sqrt{\rho_{AB}} O_{Z}O_{W}(t) \sqrt{\rho_{AB}})
\end{split}
\end{equation}
where subscripts correspond to ``one-sided'' or ``two-sided'' geometries of a black hole. It is not difficult to see that the amplitude $P$ can be expressed as a certain weighted average of the two-sided OTOCs $\langle O_{X}O_{Y}(t)O_{Z}O_{W}(t)\rangle_{2}$. One may consider a similar weighted average of the one-sided OTOCs $\langle O_{X}O_{Y}(t)O_{Z}O_{W}(t)\rangle_{1}$ which upper bounds the average of the two-sided OTOCs $\langle O_{X}O_{Y}(t)O_{Z}O_{W}(t)\rangle_{2}$ via the Cauchy-Schwarz inequality. Therefore, smallness of certain averaged OTOCs, either one-sided or two-sided, is sufficient to lower bound $I(A,BD)$.
\section{Deterministic decoder with Grover search}
In the main part of the paper, we have discussed a probabilistic decoding protocol which works with probability $\approx \frac{1}{d_{A}^2}$ where $d_{A}$ is the size of the Hilbert space for Alice's message. In this appendix, we briefly describe a deterministic decoding protocol which incorporates a procedure similar to the Grover search algorithm~\cite{Grover:1996}. The circuit complexity of the deterministic decoding protocol is proportional to $d_{A}$ for large $d_{A}$, and the whole process is related to higher-point OTOCs. For a qubit input ($d_{A}=2$), the protocol requires only one iteration, and the whole process is related to six-point and eight-point OTOCs.
The initial state of the protocol is
\begin{align}
|\Psi_{\text{in}}\rangle
= \: \figbox{1.0}{fig-in-state}\:, \label{eq-in-state}
\end{align}
where Bob has already applied $U^{*}$ to his share of qubits. Define the following unitary operators
\begin{align}
W_{D} = 1 - 2 (I_{RC}\otimes P_{D}) \qquad W_{A} = 2 (I_{RC}\otimes \widetilde{P}_{A})-1
\end{align}
where $P_{D}$ is a projector onto EPR pairs on $DD'$, $P_{A}$ is a projector onto EPR pairs on $A'R'$, and $\widetilde{P}_{A}=(I_{D}\otimes U^{*}\otimes I_{R'})P_{A}(I_{D}\otimes U^{T}\otimes I_{R'})$, or graphically:
\begin{align}
P_{A} = \figbox{1.0}{fig-PA}, \qquad\quad \widetilde{P_{A}} = \figbox{1.0}{fig-tilde-PA}\,\:.
\end{align}
Bob's decoding strategy is to implement a unitary operator $V\equiv V_{A}V_{D}$ multiple ($\approx \frac{\pi d_{A} }{4}$) times to obtain a good approximation of $|\Psi_{\text{out}}\rangle$. The protocol is summarized in Fig.~\ref{Figure-Grover}(a).
\begin{figure}
\centering\includegraphics[width=0.7\linewidth]{suppfig}
\caption{Schematic depiction of the the deterministic decoder (a) and the Grover rotation (b).}
\label{Figure-Grover}
\end{figure}
To illustrate how the protocol works, we will use some equations that hold in the ideal case with $I^{(2)}(A,BD)=\log_{2}d_{A}^2$:
\begin{equation}
\begin{split}
I_{RC}\otimes P_{D}|\Psi_{\text{in}}\rangle = \frac{1}{d_{A}}|\Psi_{\text{out}}\rangle \qquad I_{RC}\otimes P_{D}|\Psi_{\text{out}}\rangle= |\Psi_{\text{out}}\rangle \\
I_{RC}\otimes \widetilde{P}_{A}|\Psi_{\text{in}}\rangle = |\Psi_{\text{in}}\rangle \qquad I_{RC}\otimes \widetilde{P}_{A} |\Psi_{\text{out}}\rangle = \frac{1}{d_{A}}|\Psi_{\text{in}}\rangle. \label{eq:PAPD}
\end{split}
\end{equation}
The rest is the standard analysis of the Grover search algorithm. Consider a two-dimensional plane spanned by $|\Psi_{\text{in}}\rangle$ and $|\Psi_{\text{out}}\rangle$ with real coefficients. Notice that applications of $W_{A},W_{D}$ keep wavefunctions on the two-dimensional plane. Let $|\Psi_{\perp}\rangle $ be a wavefunction which lies on this plane and is orthogonal to $|\Psi_{\text{out}}\rangle$; $\langle\Psi_{\perp}|\Psi_{\text{out}}\rangle=0$. Such a wavefunction can be constructed by observing $|\Psi_{\perp}\rangle\propto(1-P_{D})|\Psi_{\text{out}}\rangle$. Notice that $V_{D}$ is a reflection across $|\Psi_{\perp}\rangle$, so this induces a rotation by angle $\theta$ with $\sin \frac{\theta}{2} = \frac{1}{d_{A}}$ when applied to $|\Psi_{\text{in}}\rangle$. Similarly, $W_{A}$ is a reflection across $|\Psi_{\text{in}}\rangle$ (see Fig.~\ref{Figure-Grover}(b)). Therefore, by applying $W=W_{A}W_{D}$, one can rotate $|\Psi_{\text{in}}\rangle$ on the two-dimensional plane by angle $\theta$. After $m$ steps, we have
\begin{align}
|\Psi(m)\rangle = \sin\Big(\big(m+\frac{1}{2}\big)\theta \Big)|
\Psi_{\text{out}}\rangle +
\cos\Big(\big(m+\frac{1}{2}\big)\theta \Big)|
\Psi_{\perp}\rangle.
\end{align}
So, the probability of obtaining $|\Psi_{\text{out}}\rangle$ is $\sin^{2} \Big( \big( m+ \frac{1}{2} \big)\theta \Big)$.
When $I^{(2)}(A,BD)\approx 2\log_{2}d_{A}$ and $d_{A}=2$ (the input is a qubit), an almost perfect decoding is possible by applying $U^{*}$ and $U^{T}$. This is because $\theta=\pi/3$. Namely, the following iteration implements the deterministic decoding:
\begin{align}
\figbox{1.0}{fig-no-ref-dA}
\end{align}
If one further applies $U^{*}$ on the right hand side and $W_{D}$ on $DD'$, we will have an EPR pair on $DD'$:
\begin{align}
\figbox{1.0}{fig-no-ref-dA-check}
\end{align}
One may postselect the experiment by using an EPR pair on $DD'$.
\mciteSetMidEndSepPunct{}{\ifmciteBstWouldAddEndPunct.\else\fi}{\relax}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,696 |
rm -f generated-runtime-deps-list.txt
rm -f generated-runtime-deps-tree.txt
mvn dependency:list -DincludeScope=runtime -DoutputFile=generated-runtime-deps-list.txt -Dsort=true
mvn dependency:tree -Dscope=runtime -DoutputFile=generated-runtime-deps-tree.txt
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,094 |
Game and Freezzer play-time: Unturned - 2,444 Hours / I've been playing on Freezzer for around 1 year now i quit Unturned some months ago and came back.
Recent/current weekly + daily play-time: This week 42 Hours because i've got exams, which are done now / 5 hours today atleast / I can probably be online 10 Hours a day because Exams Finish.
About yourself: My name is Kevin Aabo. I like to play Unturned around 20 hours a day on many diffrent servers. I'm 15 years old and i live in (Alanya , Antalya , Turkey), but i'm Danish. My timezone is UTC +3 and i do not have a work in real life Yet. I've got a girlfriend, she is on holiday at the moment.
Why you would be well suited to become a Support-Staff: I think i am suited to be a Support-Staff because i've got a lot of experience i've been working with unturned servers for a long time and been staff on many servers and currently still are on some. I can put a lot of time into the servers now because my Exams are over. I'm very dedicated into helping out servers it's my habbit. I've owned some popular servers in the past and still own some. I know how to develope unturned servers and i'm descend at it i would say. I've been staff on many big communities also.
My VAC Ban on my account is because i used a glitch in CS:GO and it's about 1 year ago, i don't know if that will effect the application. I can bring a lot of new fresh members in the discord , steam group and Server, i've got 242 people in my group in 1 day around 1 year ago. I have something so i can take screenshot of things, i can record videos on the server and take videos of hackers or glitchers for evidence etc... I agree to help everyone in the server no matter what unless they are asking for something that would make me abuse then i would not. If somebody was going around being racist, i would ask him why, and Warn him if he keep's doing it it's going to be a kick and if that is not enough i would punish him for real with a temp-ban. I would gladly send some proof of the servers i'm staff on and was staff on same with Communities. I've never really hacked in Unturned before only banned a few.
The post was edited 2 times, last by Puppy ( Jun 16th 2018 ).
Firstly, thank you for your interest in joining the team. However, don't you think your active roles that you already have will affect your activity as a Freezzer team member? | {
"redpajama_set_name": "RedPajamaC4"
} | 8,108 |
Your Contact at DiversityWorking.com
My name is Steven Garcia, and I am the V.P. of Sales and Operations here at DiversityWorking.com, The Largest Diversity Job Board Online. I am here to help you find a satisfying career and to help Employers find the right candidate. In the coming months you will be viewing video feeds with answers to your questions and information from Employers looking to hire someone just like you. I bring over 20 years experience specifically in the diversity recruitment area.
Northrop Grumman Launches Education Initiative and Other Diversity News
One of DiversityWorking.com's prestigious diversity company partners, Northrop Grumman - a leading global security company providing innovative systems, products and solutions – is in the news lately and here are some of its big stories.
A press release by Business Wire announced the company's Northrop Grumman Foundation , in collaboration with the National Science Teachers Association (NSTA), just launched the Northrop Grumman Foundation Teachers Academy, a comprehensive education initiative for middle school STEM teachers (grades 5-8) that aims to promote student achievement in science, technology, engineering and math (STEM), by aiding in the enhancement of teacher confidence and classroom excellence in science, engineering and technology, at the same time, increasing the understanding of teachers regarding the skills needed for a scientifically literate workforce. According to the press release published on MarketWatch, the program will start to support 25 teachers from school districts in select Northrop Grumman communities. - Read more at:
The promotion of and commitment to diversity and inclusion are part of Northrop Grumman's Corporate Responsibility. It believes that valuing diversity and fostering inclusion is essential to promoting innovation and increasing productivity and profitability.
Likewise, Northrop Grumman is an Equal Opportunity Employer, and is committed to employ and advance in employment qualified minorities, women, individuals with disabilities, and covered veterans.
The company has long been recognized for this, and attesting to this are the numerous diversity awards it has received from various organizations and communities that value diversity. One of the latest recognitions, according to a Globe Newswire press release, is the recent awarding of two Northrop Grumman's employees, Kaiann Fu and Padmapriya Vijayasarathi, at the 20th Ammual Women of Color STEM Conference for their achievements in their particlular fields. - See more here:
One of the company's latest achievements is seen to provide job opportunities, according to the Los Angeles Times. Aerospace giant Northrop Grumman Corp. won an intense four-year competition Tuesday to build the nation's new fleet of stealth bombers, a project likely to create thousands of jobs in Southern California. - Read more here:
DiversityWorking.com, the largest diversity job board online, is a career opportunity resource and job search engine for the cultural diversity marketplace. Through DiversityWorking.com, you can post jobs for veterans who can readily fill your diversity and equal employment opportunity requirements. Visit DiversityWorking.com now.
Posted by Your Diversity Career Consultant at 7:39:00 PM 0 comments
Is This Diversity Working: Asian Americans and the "Model Minority" Myth
Is this diversity working? The 2nd installment of this series focuses on Asian Americans and the 'model minority' myth, a label deemed offensive to many.
Recently, the column "The Asian Advantage" by Nicholas Kristof of the New York Times, stirred uproar among the Asian American community, and the 'model minority' resurfaced once again in conversations, according to The Washington Post.
While many Asian American commenters said they appreciated Kristof's attempt to clarify his points, the post likely befuddled others. What could be objectionable, after all, about a column representing as fact the achievements of Asian immigrants in America? But to many Asian Americans, the column's opening gambit isn't just awkward. It's offensive — and dangerous. - Read more at:
The report above quotes Kristof as saying, "My column last weekend on Asian-Americans sparked lots of conversation and criticism," he wrote, addressing at length the various objections to and interpretations of the column. "Thanks for joining the conversation, whether you were patting me on the back or whacking me over the head."
In his column, in which he discusses factors that contribute to the "Asian Advantage," Kristof wrote as an opening, and to which many Asian Americans took offense: THIS is an awkward question, but here goes: Why are Asian-Americans so successful in America? Later in his column, he continues:
"So the Asian advantage, Nisbett argues, isn't intellectual firepower as such, but how it is harnessed. Some disagree, but I'm pretty sure that one factor is East Asia's long Confucian emphasis on education. […] There's also evidence that Americans believe that A's go to smart kids, while Asians are more likely to think that they go to hard workers...." - Read more at:
Many responded to Kristof's column, and among these is an article by Sahra Vang Nguyen, who wrote: There is no "Asian Advantage" -- there are only skewed stats to purport the model minority myth and a divide within the racial justice movement.[...] First off, when people say "Asian American," please remember that this describes a massive conglomerate of 48 countries, with distinct cultural differences and political histories in the United States (from exploited railroad labor, to the brain drain, to war refugees). - Read more at:
In another article, Asian American students explain how the 'model minority' tag does not accurately picture their particular situations.
"I'm Chinese American, studying economics at an Ivy League school, good at piano, class valedictorian from high school. I guess you could say I fit the model minority myth," she says. "My aspirations are not culturally programmed. I have struggled with anxiety, fought with my parents about my future and faced microaggressions growing up in rural America," Zhang says.
Katie Zdunek, a journalism major at Western Kentucky University, agrees. "These stereotypes negate individual needs, talents, and experiences. It's demeaning!" Zdunek says. - Read more:
Asian Americans have raised their voices against being labeled as a 'model minority' time and time again, yet the tag remains. Aside from the fact that Asian Americans come from diverse ethnic and cultural groups, and therefore, the phrase overlooks the struggles within each of these diverse subgroups, the phrase also "justify oppressive racial structures," one article notes.
The "model minority" stereotype distorts the causality of differential academic and career success among various minority groups. A failure to succeed is attributed to inherent flaws within the character of an individual or the minority group as a whole. However, the historical facts indicate that the structuring of U.S. policy (mostly by white males) – be it through the Immigration Act, NAFTA or slavery – heavily determined the success of any given minority group. Likewise, structural discrimination against African Americans and Latinos in prison sentencing, employment and police brutality today is ignored in favor of blame-the-victim explanations promoted by the "model minority" ideology. - Read more at:
Thus, if the phrase carries negative implications and perpetuates unjust structures, policies, and discrimination, is it not indeed time for such myth to be dismantled? Is this diversity working? You may share your thoughts with us.
Middle School Election Results Held Up For Winners Not Diverse, Principal Says
Diversity or democracy?
This has been the question raised partly in protest of a middle school principal's delaying the recently held student election, saying the winners are not diverse enough.
It is also a question that has opened discussions on the influence of race/race issues in politics - in schools and outside.
When San Francisco middle school principal Lena Van Haren saw which kids on her campus had been elected to the student council, she was disturbed at the lack of diversity among the winners. There were no Latino or black candidates chosen for the top four spots. - Read more at:
The incident happened at Everett Middle School in San Francisco's Mission District. The voting was held Oct. 10, but the principal sent an email to parents on Oct. 14 saying the results would not be released because the candidates that were elected as a whole do not represents the diversity that exists at the school…. - Read more at:
According to the above report by The Washington Post, school principal Haren said that her school consists of 80 percent students of color and 20 percent white students, yet the results of the election did not show enough representation of the student population. "That is concerning to me because as principal I want to make sure all voices are heard from all backgrounds," Van Haren said…., the WP continues.
Reports also say this action angered many students and parents, and there are those who did not agree with that particular move by the principal, who according to reports eventually announce the election results on Monday after pressure from parents and the district administration.
Opinion writers also disagree with the prinicpal's action, including the writer of the WP article, Eugene Volokh: And exactly what "learning experience" would the children get this way, whether about racial tolerance or democracy?
I'm inclined to say, by the way, that attempting to "add positions in an effort to be more equal" by (presumably) filling them with children of a particular race would likely violate the Equal Protection Clause and federal civil rights law.
From a Fox News article: "The problem, it appears, is that the principal didn't like who got the most votes. In other words – too many white kids got elected.[...]"This is complex, but as a parent and a principal, I truly believe it behooves us to be thoughtful about our next steps here so we have a diverse student council that is truly representative of all voices at Everett," she wrote in a message to moms and dads.
So the principal seems to think that boys and girls should vote for representatives based on the color of their skin and not the content of their character?"
The same article above also mentions a student's reaction: "I feel like it is disrespectful to all the people who were running," student Sebastian Kaplan told television station KRON. "The whole school voted for those people, so it is not like people rigged the game, but in a way – now it is kind of being rigged."
From one parent, as reported by the SFGate article above, reacted: "The thing that's so frustrating to me, as a parent and an engaged citizen, is you release the results and then you form committees," David said. "How can you say, 'In the name of social justice, we're going to withhold election results'?"
Another parent says that school officials should have decided, prior to the election and students' campaigns, how they wanted to ensure the election results would be more diverse. [...]"So if we can't teach them the right way of how it's supposed to work, then we are really letting our kids down," Gutierrez said. "My heart goes out to the kids because they're confused still about what's the issue." - Read more here:
It seems like from the different news curated above, the principal's action -- which she resorted to if only to highlight the value of diversity -- is deemed bad for democracy, bad for the students, bad for school politics.
Here are some questions to ponder. We invite our readers to share their thoughts.
Now, is this diversity working at all?
In the business world, and other sectors of society, diversity is already being realized as a good value to pursue, because the pursuit of diversity promotes inclusion, thus help in the eradication of discrimination of any form.
Why do you think the principal's intention to have an elected student council as diverse as the school's student population stirred negative reactions?
What makes people feel strongly against diversity?
What are some ways to show diversity's working in a democracy?
This article is the 1st of a series on the topic, "Is Diversity Working?"
Posted by Your Diversity Career Consultant at 12:45:00 AM 0 comments
Compass Group North America, Diversity-Committed, To Be Awarded with Diversity Leadership Award for Corporations
Compass Group North America, the leading foodservice management and support services company, and a recent addition to DiversityWorking's prestigious list of diversity company partners, is indeed diversity-committed, staying true to one of its guiding principles: Developing our People and Valuing Diversity.
As such, it has been recognized for its diversity and inclusion efforts, and is to be awarded with the Diversity Leadership Award for Corporations by the Society for Diversity, an organization that acknowledges corporations, non-profits, educational institutions and individuals who have exceptional practices and define the leadership standard for diversity and inclusion. […] - See more at:
The company's winning attitude towards diversity and inclusion is reflected in dealing with its people who find themselves openly accepted and valued for their contribution. They all work as a team to create a work environment that is challenging and provides the opportunities and support for everyone to develop, learn and succeed.
These are the values the company and its people live by helping each and everyone establish Compass Group as a reputable name in the industry:
Openness, Trust and Integrity;
Passion for Quality;
Win Through Teamwork;
Responsibility ;
These values and the other Compass Groups' guiding principles, which also include Safety, Health and Environment First; Delivering for Clients and Consumers, among others, remind each one in the company of their mission: "to consistently delivering superior service in the most efficient way, for the shared benefit of our customers, shareholders and associates."
True enough, Compass Group North America has consistently grown though all the years since it began in 1994, with more than 220,000 associates in all 50 states, ten provinces and two territories. And it is still on its way to greater success.
IHG Execs Calls for Cooperation As Company Continues Expansion
The InterContinental Hotels Group (IHG), one of the world's established names in the hotel industry continues its expansion, as it adds more hotels and locations to its portfolio. To ensure greater success as it meets the demands of the time, its corporate leaders have called for cooperation among its franchisees.
The executive leadership of InterContinental Hotels Group preached the need for cooperation with franchisees during the first day of the 2015 IHG Americas Owners Conference Wednesday, according to Hotel News Now. The report also mentions four areas wherein its excutives have seen the need for cooperation: technology, loyalty, quality assurance, and revenue management. It likewise quotes IHG's CEO Richard Solomons: "When we think and act as one, we're unstoppable," Solomons said. "We need one another. Not just as business owners and hotel owners, but as real people." - See more at:
HospitalityNet reports on one of IHG's expansions: InterContinental Hotels Group (IHG), one of the world's leading hotel companies, has announced the signing of four Holiday Inn Express® hotels in Germany in four key cities, reinforcing the company's strong position in the German market.
According to this report, IHG has at present the largest share in Germany's branded hotel market with ove 4,400 rooms, based on figures gathered on June 30, 2015. - Read more at:
Another IHG expansion announced through a press release involves the joint venture of Noble Investment Group ("Noble") and IHG to develop a dual-branded EVEN® Hotels and Staybridge Suites® hotel in downtown Seattle. The nine-story building, expected to open in mid-2017, will be located at the intersection of Mercer Street and Fairview Avenue North adjacent to the Amazon urban campus in the rapidly growing South Lake Union area. This new dual-branded hotel will be the first EVEN Hotels property on the West coast and the third Staybridge Suites property in the Seattle market. - Read more:
The West Coast's first hotel under InterContinental Hotels Group's new Even brand is scheduled to open in Seattle in 2017. A 123-room Even hotel and a 112-room Staybridge Suites will be part of a development in Seattle's South Lake Union district. - See more at:
The InterContinental Hotels Group (IHG) is one of DiversityWorking.com's long-time prestigious diversity company partners.
Microsoft Launching of Surface Book Brings PC Industry Out from Slumps
Microsoft Corporation is stirring things up in the world of personal computers with the recent launching early this week of Surface Book, the company's first laptop, together with its other products that includes the smartphones Lumia 950 and Lumia 950XL. At the same time, it is set to open its new NYC superstore later this month.
According to an article by Nick Wingfield of the New York Times, Microsoft' s mission to rev up the PC industry seems to be getting positive results and bringing optimism at a time when people are mostly keeping on with their old PCs and turning more to mobile devices for their computing needs.
"Initiatives like Surface and Surface Book have helped the industry wake up and say, 'We've got to make the industry cool and sexy again,'" said Frank Azor, executive director and general manager of Dell's XPS line of PCs.
The stated reason that Microsoft got into the PC hardware business three years ago, with the original Surface, was not to put PC companies out of business. The company said the goal was to better illustrate the capabilities of its software, providing devices that would inspire PC makers to be more innovative. - Read more at:
In its own article, International Business Times also tells of how Microsoft is giving Apple and other PC makers strong, healthy competition. Microsoft on Tuesday left little doubt it plans to remake itself in the mold of archrival Apple, introducing a number of sleek, artfully crafted devices that rival and, in some cases, surpass the best Tim Cook and company have produced this year, including the iPhone 6S and iPad Pro. It's the clearest sign yet that Microsoft CEO Satya Nadella sees hardware as integral to his company's future as software is, and he is bent on creating an ecoystem that extends from smartphones and tablets to apps and the cloud. [...]"We're making great progress toward the aspiration we shared with you in January," said Nadella, in his customary black shirt and jeans, at an event in midtown Manhattan. "Moving people from needing Windows, to choosing Windows, to loving Windows." - Read more at:
Microsoft's new flagship store along New York City's famed Fifth Avenue is certainly massive, though significantly less jawdropping than Apple's iconic location just five blocks away.
The store is set to open on Oct. 26, the same day Microsoft will begin selling a slew of devices the company announced at a press event on Tuesday. - Read more at:
Microsoft is one of DiversityWorking.com's long-time prestigious diversity company partners.
DiversityWorking.com Announces Silicon Valley Bank (SVB) As New Diversity Company Member
DiversityWorking.com a career opportunity resource and job search engine for the cultural diversity marketplace, is pleased to announce the inclusion of Silicon Valley Bank, to its prestigious list of diversity company members.
Named by Forbes one of America's best banks (2015) and one of America's best-managed companies (2014), as well as one of Fortune's 100 Best Companies to Work For, Silicon Valley Bank provides targeted financial services and expertise through its offices in innovation centers around the world. It was established in 1983, and for over 30 years, has been a strong leader in the industry, helping innovative companies and their investors move bold ideas forward, fast.
A great part of its success lies in its unified team with diverse talents, passionate about ideas and driven to make a difference who work in a welcoming and collaborative environment, a safe and healthy workplace, organizational transparency, and attractive compensation. Thus, SVB is recognized a great place to work.
In its continual search for engaged, able and qualified employees, Silicon Valley Bank has partnered with DiversityWorking.com, whose mission is to help companies promote their jobs, build their brand and send targeted and qualified diversity candidates directly to the best jobs possible.
To know more about Silicon Valley Bank and its job opportunities, please click here.
Posted by Your Diversity Career Consultant at 8:16:00 AM 0 comments
DiversityWorking.com Announces Compass Group North America as New Diversity Company Member
DiversityWorking.com, a career opportunity resource and job search engine for the cultural diversity marketplace, is pleased to announce Compass Group North America as one of its newest diversity company members. Compass Group USA is the leading foodservice management and support services company which began operations in the USA in 1994, and with headquarters in Charlotte, North Carolina.
Through the years, it has built its good reputation and established a strong foothold not only in the US, but also in Canada as well, serving eight million meals a day, and with more than 220,000 associates in all 50 states, ten provinces and two territories.
Working dedicatedly towards its vision to be a world-class provider of contract foodservice and support services, renowned for our great people, our great service, and our great results, Compass Group North America has indeed been recognized as such. Among the many honors it has received is being named by the National Business Group on Health as One of Best Employers for Healthy Lifestyles in 2010, 2011, 2012 and 2013.
It has also been recognized for its commitment to Diversity and Inclusion. Compass Group is a diversity growth-oriented organization, committed to providing equal growth opportunities for its associates. In its search for the best, dedicated diverse talents, it has partnered with DiversityWorking.com, whose mission is to help companies promote their jobs, build their brand and send targeted and qualified diversity candidates directly to the best jobs possible.
To know more about Compass Group North America and its job opportunities, please click here.
Harmony of Diversity Marked Pope's US Visit
In the aftermath of Pope Francis' six-day visit to the U.S., one of the striking imprints he must have left in the hearts of many is the 'harmony of diversity' brought to the fore in many of the papal events and in his speeches.
One of these events was the interfaith religious ceremony held at the ground zero site at the 9/11 Museum in lower New York. New York City is nothing if not bold and we, its residents, are nothing if not in your face about our beliefs. This service was a New York City in-your-face gracious presentation that showed the world not only what we are, but also who we are. The news commentators on MSNBC kept pointing out the diversity of New York City, and so, of course did the pope. This event was organized precisely to show the harmony in diversity, which actually does work in New York, writes Jo-Ann Mort in her article on Haaretz. - Read more at:
Other events that highlighted Pope Francis' message of inclusive love which made a big impact on those who witnessed this included his visits to member of minority groups, such as people with disabilities.
Janice L. Benton helped plan participation by people with disabilities in the pope's events in Washington and Philadelphia.[...]"The church was vibrant with diversity" at the Serra Mass, and "inclusive of all gifts and abilities. The entire 'family' was at the table together," said Mary O'Meara, executive director of the archdiocesan Department of Special Needs Ministries. "Persons with disabilities were welcomed, valued and celebrated," she said later in a statement.- Read more at:
In one of his homilies, delivered during the Mass at Madison Square Garden in New York, the Pope spoke of and praised the richness of diversity of large cities.
During his homily, he praised large cities like New York for their diversity but also said their size can make people feel alienated unless they allow God to bring them together.
"Living in a big city is not always easy. Yet big cities are a reminder of the hidden riches present in our world, in the diversity of its cultures, traditions and historical experiences. In the variety of its languages, costumes and cuisine," he said. - Read more at:
The Pope also expressed his strong support for immigrants in the U.S.
"Do not be discouraged by whatever challenges and hardships you face," the pope told the many Hispanics and other recent immigrants in the crowd on Saturday, adding that he felt "particular affection" toward them.
During his first visit to the country, the leader of the world's 1.2 billion Roman Catholics on Thursday had urged Americans in a historic speech to Congress to reject "a mindset of hostility" toward immigrants. - Read more at:
Posted by Your Diversity Career Consultant at 11:57:00 PM 0 comments
Your Diversity Career Consultant
NY, NY, United States
As Vice President of Sales and Operations at DiversityWorking.com, Mr. Garcia is responsible for sales operations, product development, and customer satisfaction. Mr. Garcia brings over 20 years' experience specifically in the diversity recruitment area. Before joining DiversityWorking.com, Mr. Garcia was the National Sales Manager at a popular online diversity job board. He held the responsibility of team building, sales operations, and overall growth of the organization. Mr. Garcia was instrumental in developing the business from a startup organization to a multi-million dollar business. Mr. Garcia graduated from the University of California at Santa Barbara with a BA in Psychology.
Northrop Grumman Launches Education Initiative and...
Is This Diversity Working: Asian Americans and the...
Middle School Election Results Held Up For Winners...
Compass Group North America, Diversity-Committed, ...
IHG Execs Calls for Cooperation As Company Continu...
Microsoft Launching of Surface Book Brings PC Indu...
DiversityWorking.com Announces Silicon Valley Bank...
DiversityWorking.com Announces Compass Group North...
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Your Contact at DiversityWorking.com. Some rights reserved. © 2007 DiversityWorking.com | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,043 |
Minnah Karlsson er et coveralbum af Minnah Karlsson. På albummet findes sange, som hun har optrådt med i den svenske Idol 2010.
Albummet blev udgivet den 19. december 2010.
Adam Lambert har rost Minnah Karlsson for hendes kopiudgave af hans sang "Whataya Want From Me", der også er at finde på albummet.
Sangene på albummet
Albummets sange er alle coverversioner. Den oprindelige kunstner er anført i parentes.
"Whataya Want From Me" (Adam Lambert)
"Twist And Shout " (The Beatles)
"Just the Way You Are" (Bruno Mars)
"Always On My Mind " (Elvis Presley)
"Calleth You, Cometh I" (The Ark)
"Total Eclipse of the Heart" (Bonnie Tyler)
"When Love Takes Over" (David Guetta og Kelly Rowland)
"(Everything I Do) I Do It for You" (Bryan Adams)
"Alone" (Heart)
"Piece Of My Heart" (Beverley Knight)
"Not ready to make Nice" (Dixie Chicks)
Album fra 2010 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,047 |
package main
import (
"bytes"
"encoding/json"
"fmt"
"io"
"os"
"path/filepath"
"strings"
"github.com/ugjka/tag"
)
//
//Process audio files, extract info/artwork
//
func getSongs(searchdir string, flac bool) (songs songs) {
cache := make(map[string]song)
cachef, err := os.OpenFile("cache.json", os.O_RDWR|os.O_CREATE|os.O_APPEND, 0644)
if err != nil {
panic(err)
}
defer cachef.Close()
err = json.NewDecoder(cachef).Decode(&cache)
if err != nil {
srvlog.Warn("could not decode cache json", "error", err)
}
filepath.Walk(searchdir, func(path string, info os.FileInfo, err error) error {
if info.IsDir() {
return nil
}
if !(strings.HasSuffix(path, ".mp3") || (strings.HasSuffix(path, ".flac") && flac)) {
//srvlog.Info("skipping invalid file", "file", info.Name())
return nil
}
if v, ok := cache[path]; ok {
v.path = path
songs = append(songs, v)
return nil
}
f, err := os.Open(path)
if err != nil {
srvlog.Warn("could not read file", "path", path)
return nil
}
defer f.Close()
m, err := tag.ReadFrom(f)
if err != nil {
srvlog.Warn("could not read tags", "file", info.Name())
return nil
}
artist := strings.TrimSpace(m.Artist())
title := strings.TrimSpace(m.Title())
album := strings.TrimSpace(m.Album())
track, _ := m.Track()
hash, err := tag.Sum(f)
if err != nil {
srvlog.Warn("could not hash the file", "file", info.Name())
return nil
}
if m.Picture() != nil {
art, err := os.OpenFile(fmt.Sprintf("./artcache/%s", hash), os.O_RDWR|os.O_CREATE|os.O_APPEND|os.O_TRUNC, 0644)
if err != nil {
panic(err)
}
io.Copy(art, bytes.NewReader(m.Picture().Data))
art.Close()
}
result := song{Artist: artist, Title: title, Album: album, Track: track, ID: hash, path: path}
songs = append(songs, result)
cache[path] = result
return nil
})
err = cachef.Truncate(0)
if err != nil {
srvlog.Warn("could not truncate cache file", "error", err)
return
}
enc := json.NewEncoder(cachef)
enc.SetIndent("", "\t")
err = enc.Encode(cache)
if err != nil {
srvlog.Warn("could not encode cache json", "error", err)
}
return
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,147 |
Les eleccions prefecturals de Tòquio de 1951 (1951年東京都議会議員選挙, 1951-nen Tōkyō-to Gikai giin senkyo) es van celebrar el 30 d'abril de 1951, dins de les eleccions locals unificades de 1951, per tal d'elegir els 120 membres que conformarien el parlament de Tòquio, l'Assemblea Metropolitana de Tòquio.
La campanya electoral per a les eleccions començà el 3 d'abril. El resultat de les eleccions va dur un triomf de la dreta, amb majoria absoluta del Partit Liberal (PL) i el Partit Democràtic Popular (PDP) com a quarta força. Per la seua banda, l'esquerra va patir una sensible davallada, amb el Partit Socialista del Japó (PSJ) com a tercera força per darrere dels candidats independents i el Partit Comunista del Japó (PCJ) que va millorar lleugerament els seus anteriors resultats; el Partit Socialdemòcrata i el Partit dels Treballadors i els Camperols, els dos escissions del PSJ, no van obtindre representació. Yuriko Motojima, del PSJ, qui havia estat la primera dona membre de l'Assemblea Metropolitana, tornà a ser reelegida. Per altra banda, entre les files del PL també van resultar electes dues dones, donant un total de tres membres femenins en l'assemblea.
Resultats
Generals
Per circumscripció
Referències
1951
Eleccions del 1951 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,733 |
Q: How to highlight "ini files" in Eclipse How do I make Eclipse to highlight .ini files?
In the menu Window/Preferences, General/Content Types, Test/Configutation files, in the box "File associations:" I can see:
*.cfg (locked)
*.ini (locked)
but I don't know either if this has something to do with the matter, or what can I do see the ini files highlighted instead of flat-black.
Does eclipse support ini higlighting natively? How do I activate it? Or maybe do I have to install a plugin? Which one?
A: I've just found this plugin that seems to do the trick. It is a little overzealous and assigns itself to all known filetypes but editing Preferences > General > Editors > File Associations can solve this.
http://colorer.sourceforge.net/eclipsecolorer/index.html
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,482 |
«Разрисованная вуаль» () — чёрно-белая драма Metro-Goldwyn-Mayer (1934). Экранизация романа «Узорный покров» британского писателя У. С. Моэма (1925), режиссёром которого выступил Ричард Болеславский. Арт-директор Седрик Гиббонс, дизайнер по костюмам . Главные роли в киноленте исполнили Грета Гарбо и Герберт Маршалл.
В 2006 Джон Кёрран также экранизировал роман Моэма с Наоми Уоттс и Эдвардом Нортоном.
Сюжет
После того, как младшая дочь семьи Кёрбер Ольга выходит замуж, старшая сестра, Кэтрин, также желает покинуть отчий дом и мечтает о самостоятельной жизни вне Австрии. И когда британский бактериолог Уолтер Фейн просит её руки, девушка не раздумывая соглашается, хотя они разные люди и она его совсем не любит.
Супруги переезжают в Гонконг. Львиную часть дня Уолтер посвящает своей работе, в то время как у Кэтрин завязываются романтические отношения с помощником губернатора колонии женатым Джеком Таунсендом. Таунсенд увлечён прелестной женой бактериолога, он показывает ей экзотические достопримечательности города, одновременно обольщая новобрачную. Миссис Фейн не одобряет поведение Таунсенда и прерывает отношения, но вновь мирится с ним на буддистском фестивале танца. В этом буйстве красок и загадочной дикой атмосфере Джек объясняется в своей любви Кэтрин, заставляя её признаться в нелюбви к мужу.
В то же время у Кэтрин накаляются отношения с Уолтером: его хронические задержки на работе и усталость раздражают её. Дабы загладить свою вину перед любимой супругой, на следующий день Фейн возвращается домой пораньше, но обнаруживает закрытую дверь в спальню Кэтрин и оставленную внизу шляпу Джека. Вечером того же дня Уолтер высказывает Кэтрин свои подозрения, и ей ничего не остаётся, как признаться мужу в любви к другому. Обезумевший от горя Уолтер говорит жене, что даст ей развод только в случае письменного обещания Джека развестись со своей женой и жениться на Кэтрин. Когда Кэтрин осведомляет любовника об условиях мужа, Таунсенд говорит ей, что развод исключён, ведь он разрушит и его карьеру, и его репутацию, и его семью.
Убитая горем Кэтрин против своей воли сопровождает Уолтера во внутренний Китай (Мэй-дань-фу), где бушует эпидемия холеры. В то время как Уолтер изо всех сил пытается остановить повальную болезнь, Кэтрин становится всё более подавленной и одинокой. Уолтер, видя нескончаемую смерть и горе вокруг, начинает относиться к разногласиям с женой как к незначащим. Он говорит ей, что всё ещё любит её, и просит отправиться назад в Гонконг, в то время как сам уезжает в более отдалённую деревню, в которой только началась эпидемия. Кэтрин, не до конца разобравшаяся в своих отношениях к Джеку, всё же остаётся с Уолтером. Теперь она понимает, как оскорбила мужа, и стыдится прошлого.
Кэтрин помогает в местном приюте. Пока отсутствует Фейн, в Мэй-дань-фу приезжает осознавший свою истинную любовь Таунсенд. Уолтер возвращается из деревни, чтобы сжечь всю инфекцию и не дать ей распространиться дальше. Во время беспорядков бактериологу наносят тяжёлое ножевое ранение. Пока Кэтрин ждёт разрешения войти к изувеченному мужу, она встречает Джека, но теперь с лёгкостью противостоит ему, говоря, что любит только Уолтера и понимает наконец жертвы, которые он принёс во имя медицины. После прощания с Джеком Кэтрин уверяет чуть живого Уолтера, что любит его.
Актёрский состав
Ссылки
Фильмы-драмы США
Фильмы-мелодрамы США
Чёрно-белые фильмы США
Экранизации произведений Сомерсета Моэма
Фильмы MGM
Фильмы о Китае
Экранизации романов Великобритании
Фильмы о собаках
Экранизации литературных произведений
Фильмы про свадьбу
Фильмы Ричарда Болеславского | {
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Kanye West donates $2 million, pays college tuition for George Floyd's daughter
Kanye West has donated $2 million to support the families of George Floyd, Ahmaud Arbery and Breonna Taylor, a representative for West told CNN.
This donation will include funding to go towards legal fees for Arbery and Taylor's families, along with funding for black-owned businesses in crisis in his native Chicago and other cities.
According to CNN, West has reportedly established a 529 education plan to fully cover college tuition for Gianna Floyd, the 6-year-old daughter of George Floyd.
Public demonstrations have taken place all around the world in response to the deaths of George Floyd, Ahmaud Arbery and Breonna Taylor, which has led to a number of folks in the entertainment industry joining the cries for an end to systematic racism, police brutality and action from political leadership.
CNN has also stated that "Three of the Minneapolis Police officers arrested in Floyd's death make their first court appearance Thursday afternoon, as the Floyd family holds the first of several planned memorial services."
Meanwhile, a touching video of Floyd's daughter Gianna proclaiming that her "Daddy changed the world" has made waves on social media, and with this proposed funding coming from Kanye West, it appears Gianna is going to have all the tools to change the world herself. | {
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Oberamt (, plural ) was the designation of an administrative unit in the German state of Württemberg, introduced in 1758 instead of Amt. Literally translated, the term means Upper, Senior, Higher or Superior Office. It was in use until 1934, after the nazi seizure of power, when the were renamed Kreise with the Kreisordnung of Württemberg and their number was considerably reduced by mergers in 1938.
History
Duchy
The subdivision of the Duchy of Württemberg (until 1495 county) into public administration called reflected in its diversity the gradual growth of the territory. In addition to the secular offices, which made up the largest part of the state, there were monastic, rentier and chamber offices. Usually, a secular office consisted of the eponymous town and the surrounding villages as or , but the districts differed considerably in area and population, and complicated borderlines with many exclaves marked the map. Some larger offices, such as the Amt Urach, were divided into several ("sub-offices"). For clarification, the offices themselves were called Oberamt from 1758 on, without structural reforms being connected with this renaming. The ducal civil servant, who was traditionally called Vogt (about equal to "bailiff") and who managed the administrative affairs at the official level, held the title from 1759. From that day on, "all and every secondary title with the bailiff′s word was to cease immediately and only the Oberamtmann′s name was to be valid." He was responsible for the implementation of government measures in his bailiwick, for example by publishing new laws, receiving complaints from subjects and forwarding them to the appropriate higher authorities. He also warned persons who only slightly violated laws. In the , representatives of the official town and places of office discussed common matters. For example, it was decided here how the road construction in the district was to be financed. The official assembly also elected its representatives for the so-called "Landschaft".
Kingdom
After the areas that had been assigned to the House of Württemberg as a result of the upheavals of the Napoleonic era since 1803 were initially administered separately as "Neuwürttemberg", the organisational edict of 1806 - Württemberg had in the meantime risen to become the Kingdom of Württemberg - initiated the creation of uniform structures. In the following years the declaration of intent
"An expedient division and merger of the senior and staff offices will be made gradually.
and the whole country, regardless of historical and denominational circumstances, will be newly divided into approximately equal senior offices, the number of which was reduced to 64 by 1810 and to 63 by 1819 with the abolition of the Albeck senior office. A special role was played by the Residence City Stuttgart, where the fulfilled the corresponding tasks.
The higher offices were subordinate to the Ministry of the Interior and were responsible for all essential areas of state administration, only the financial system was in the hands of the Kameralämter since 1806. Since 1814 every senior office received a public health officer under the title (senior physician). According to the understanding of the state at that time, administration and jurisdiction were not separate, rather the senior civil servant presided over the High Court in personal union. Municipal self-governance and the right to a say of the estates, which had already been temporarily restricted under Duke Carl Eugen, was suspended by King Friedrich.
King Wilhelm I. took over the government in 1816 and immediately began comprehensive reforms, which led to the constitution of 1819 and thus changed Württemberg from an absolute to a constitutional Monarchy. The edicts issued on 31 December 1818 regulated various aspects of the restored local self-government:
The sheriff's offices became Selbstverwaltungskörper.
The municipalities of an together formed the , a territorial authority with its own parliament () and its own assets ().
This resulted in a double function of the senior civil servant, who was not only a civil servant as before, but also functioned as organ of the official authority.
Administration and justice were separated from each other at the level.
Chapter V of the Constitution contained detailed information on the administrative structure and rights of municipalities and official bodies. In particular, § 64 provided that limits could only be changed by law, i.e. with the consent of parliament. This possibility was only used very sparingly; only in 1842 were major changes made, affecting around thirty municipalities. A bill introduced by the government in 1911 to simplify administration in the sense of cost savings provided for only 42 , but was rejected by the Chamber of Deputies.
People's state
In 1919, renewed consideration was given to reducing the number of and restoring the uniformity lost due to the different population trends. After the Landtag had agreed to the abolition of the Cannstatt on 1 October 1923, the government attempted to dissolve the of Blaubeuren, Brackenheim, Neresheim, Spaichingen, Sulz, Weinsberg and Welzheim on 1 April 1924 by emergency decree, covered by an enabling act. The protests caused by this led to the resignation of the government, the emergency decree was withdrawn and subsequently only the Weinsberg was abolished (on 1 April 1926).
Nazi dictatorship
In 1933, the organs of local self-government were dissolved. After the Oberamtmann had already been titled Landrat since 1928, following the Prussian style, the Kreisordnung of 1934 replaced the names Oberamt by Kreis and Amtskörperschaft by Kreisverband, but did not yet include a change of boundaries. Only with the administrative district reform of 1938 were 27 of the remaining 61 districts abolished.
Descriptions of the local authorities
From 1824 to 1886, all the were statistically processed and their history, communities, population figures and the characteristics of their inhabitants were elaborately described in print.
The mainly catholic "new Württemberg" areas, e.g. in Oberschwaben, as described from the point of view of the Württemberg bureaucracy in evangelically influenced Stuttgart, often bear certain characteristics. Quote from the description of Oberamt Ravensburg, p. 29: "The character of the inhabitants is generally praised more than in other neighbouring districts, it is described as simple and trusting".
The Oberamt descriptions have become sought-after and expensively paid collector's items; therefore all volumes were reprinted in the 1970s. Most of these reprints meanwhile are out of print again. All of them are now available in digital form, see Wikisource.
Today's traces of the boundaries
In the former Württemberg region of contemporary Baden-Württemberg, often the courts are located in the former Oberamt cities.
The ecclesiastical administrative structures of the Evangelical-Lutheran Church in Württemberg also largely reflect the former higher offices. In most of the former cities of the Oberamt cities there is still the seat of a deanery, whose area of responsibility is the same as the former Oberamt. Deviations from this mainly occur in the predominantly Catholic areas and wherever new deaneries were established due to an increase in church members, such as in Ditzingen or Bernhausen.
List of the Württemberg (1811 to 1934)
References
Literature
Walter Grube: Vogteien, Ämter, Landkreise in Baden-Württemberg. Stuttgart 1975,
Historischer Atlas von Baden-Württemberg, Karten VII,4 und VII,5 mit Beiwort. Stuttgart 1976
Weblinks
Württemberg | {
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Subsequently RAI submitted a representation to Hon'ble Joint Secretary, Ministry of Agriculture and Farmer Welfare to exempt retailers from this mandatory requirement. The new amendment poses certain impediments to the retail industry which can be addressed without increasing the risk exposure. Most retail outlets are engaged in the sale of "household insecticides" which pose considerably less risk to life as compared to other industrial or commercial insecticides which indeed need to be regulated more strictly.
Please find the representation here. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,527 |
TT Unified Teachers Association (TTUTA) Tobago officer Orlando Kerr is calling on the Tobago House of Assembly (THA) Division of Education, Innovation and Energy to address indiscipline, bullying and violence at schools across the island.
Kerr was speaking of Wednesday's melee at the Signal Hill Secondary School, after which 17-year-old Jaden Pereira had to be flown to the Port of Spain General Hospital with serious head injuries. Though sources are still tight-lipped, Newsday Tobago understands that on Wednesday, Pereira, a form five student, was hit unconscious with a concrete block by another student. The incident was said to have taken place while four boys were gambling at the school.
Pereira was taken to the Scarborough General Hospital by ambulance and later airlifted to Trinidad. According to a television report, Stacy Pereira, mother of the injured Jaden, said she cannot bear to see her son in his current condition.
Police are investigating. A video of the incident has gone viral on social media.
Kerr, speaking with Newsday Tobago by telephone, said he wants the division to be proactive in dealing with fighting in schools, rather than being reactive.
Kerr recalled a recent incident at the Mason Hall Secondary School, and asked, "Are we going to wait until a student is killed?"
As recently as Monday, he said, "I was at the Mason Hall Secondary School, where they had an incident of a scratch bomb. As everybody knows now, scratch bombs are dangerous. A teacher was in a classroom and a student threw a scratch bomb into the classroom.
"So these are things that we are faced with and we don't want to wait until someone is seriously injured or even dead before we start dealing with this issue and that is all we are saying.
Kerr said officials at the division need to understand how to treat with students who are showing insubordination.
The THA Chief Secretary, who holds the portfolio of Secretary of Education, Innovation and Energy, Kelvin Charles and Assistant Secretary Marisha Osmond could not be reached for comment, as calls to their cellphones went unanswered.
However, minutes later, a press release from the Office of the Chief Secretary said the division strongly condemned the incident.
It said schools supervisor Sherry-Anne Hackett confirmed Pereira had surgery at the PoSGH on Wednesday night and is under observation.
It quoted Charles as saying incidents such as these will not be tolerated at the island's schools and will be taken very seriously.
"Our schools are supposed to be safe havens for learning. Incidents such as these are totally unacceptable, and all necessary actions will be taken to ensure that there is fair justice. It is extremely sad that parents now have to worry about their children's safety, when they have sent them to gain an education," the release read.
Reply to "Signal Hill student knocked unconscious with brick" | {
"redpajama_set_name": "RedPajamaC4"
} | 5,081 |
PRAISE FOR NOTHING LEFT OVER
"Toinette Lippe is a woman whose wisdom is as deep as it is unassuming. The ideas in Nothing Left Over are seeds bursting with vitality and her book is a primer in grateful living. As you come to know her in a delightful intimacy, you come to know yourself from unsuspected perspectives."
—Brother David Steindl-Rast, author of Gratefulness
"A magnificent piece of writing. Toinette Lippe's lucid memoir puts into practice what we all intuitively know makes sense but somehow never quite get round to doing."
—Stephen Batchelor, author of Buddhism Without Beliefs
"I can think of no greater praise than to say this is an honest book; it helps us understand what the values of 'simple living' are really meant to impart in a complicated and unsimple world, not just theoretically, but in the details of our everyday existence."
—Jacob Needleman, author of An Unknown World
"I find myself thrown into a state of wonder that a life story so authentic in its telling and so fascinating to read could be lived amid the turmoil of a modern metropolis."
—Thomas Berry, author of The Great Work
"Toinette Lippe has used her marvelous gift of word imagery to make her point that a lifetime of simply complete moments emerges as a very full life indeed."
—Sylvia Boorstein, author of It's Easier Than You Think
"Read Nothing Left Over slowly in a quiet place. Let it lead you to do some interior housekeeping. Live with those changes for a while, and then return to the book for another gentle suggestion, and then another, about what it really means to be contained and satisfied."
—Spirituality & Health
"This is the elegance of the book: the idea that any moment contains all we really need to know in order to live simply and plainly; every instance presents an opportunity to refine our awareness so that we can be present and of service to others."
—Turning Wheel
"Nothing Left Over reveals an independently won spiritual maturity. Readers will appreciate her practical advice on attitudes toward everyday life."
—Shambhala Sun
"The great pleasure of Nothing Left Over is exactly in its honesty and attention to detail, which flow from Toinette Lippe's life to her writing and back again. I encourage you to listen to her plain and heartfelt words as I did, and to consider the mysterious and ordinary lessons within: find balance and travel light."
—Alan Senauke, Inquiring Mind
"An important book for anyone who has ever wondered what really matters."
—Rachel Naomi Remen, M.D., author of Kitchen Table Wisdom
Nothing Left Over: A Plain and Simple Life
Copyright © 2002, 2014 by Toinette Lippe
All rights reserved. No part of this book may be used or reproduced in any manner without written permission from the publisher except in critical articles and reviews. Contact the publisher for information.
Originally published in hardcover by Jeremy P. Tarcher/Penguin, New York, 2002 and in trade paperback, 2004. Monkfish Book Publishing trade paperback edition, 2014. First ebook edition Monkfish Book Publishing, 2014.
Grateful acknowledgment is made to Alfred A. Knopf, a division of Random House, Inc., for permission to use chapter 11 from Tao Te Ching, by Lao Tsu, translated by Gia-fu Feng and Jane English with Toinette Lippe. Copyright © 1972 by Gia-fu Feng and Jane English.
Cover illustration: Wild Orchids © 2008 by Toinette Lippe
Frontispiece: Hummingbird 10 © 2011 by Toinette Lippe
Library of Congress Cataloging-in-Publication Data
Lippe, Toinette, date.
Nothing left over: a plain and simple life / Toinette Lippe.
p. cm.
eISBN 9781939681140
I. Simplicity. 2. Conduct of life. 3. Lippe,Toinette, date. I. Title.
BJI496.L57 2002 2001052582
I79'.9—dc2I
Monkfish Book Publishing ISBNs
Paperback: 978-1-939681-25-6
Ebook: 978-1-939681-14-0
Printed in the United States of America
Book and cover design by Sherry Williams and Danielle Ferrara
Monkfish Book Publishing Company
22 E. Market Street, Suite 304
Rhinebeck, New York 12572
www.monkfishpublishing.com
For Joel Fotinos,
who caught me on the cusp,
inviting me to reflect on the lessons I have learned so far
and watch what happens as I enter the unknown future.
Without this gift I would still be living an unexamined life.
Now I find myself perched on the fulcrum of the present.
I offer him my love and thanks.
With deep gratitude I acknowledge my debt to numberless friends, acquaintances, and even enemies—without whose teaching I would not have been able to write this book.
Table of Contents
PRAISE FOR NOTHING LEFT OVER
Title Page
Copyright Page
Dedication
ONE - Teach Me What Life Is For
TWO - Taking Stock
THREE - How Much Is Enough?
FOUR - At Play
FIVE - Nothing Unnecessary
SIX - Do It Now
SEVEN - The Company We Keep
EIGHT - Giving Attention
NINE - Offer No Resistance
TEN - The Habit of Truth
ELEVEN - What Nourishes Us
TWELVE - A Balancing Act
THIRTEEN - Law and Order
FOURTEEN - Traveling Light
FIFTEEN - In Conclusion
Afterword
About the Author
PRAISE FOR CAUGHT IN THE ACT: REFLECTIONS ON BEING, KNOWING, AND DOING
Experience is the fuel;
I would live my life burning it up as I go along,
so that at the end nothing is left unused,
so that every piece of it has been consumed in the work.
May Sarton, Plant Dreaming Deep
ONE
Teach Me What Life Is For
It wasn't until I was seventeen that I began to wonder what life was for. I had gone to boarding school at four for a year (because my English parents felt I would be safer there than staying with them on an RAF base during World War II), and after that I was cared for at home by a strict Victorian nanny, the daughter of a Royal Marine, who had been hired to look after my little brother. When I was ten, I went away to boarding school once more, and for seven years I worked hard and tried to do whatever was expected of me. Eventually, I passed all my exams with flying colors; but instead of being accepted at a university where I had dreamed of exploring the universe and finding the friends I had never had at high school, I found myself with a state scholarship but no place at any college. It was hard in those days for a girl to gain admission. I believe I could have got in if I had said that I wanted to teach. But I thought that the whole point of going to college was to discover what you might want to do from the vast array of possibilities. Why would you make up your mind before you even knew what these might be? I felt that I couldn't make such a commitment at the outset, and so when I was asked whether I wanted to be a teacher, I answered, "I don't know."
Since I wasn't going to a university like everyone else, there I was with the whole of my life before me, and I didn't know what direction to take. I remember asking my parents what life was all about. What was everyone supposed to be doing? How could I decide how to spend my life if no one had ever explained the point of it? My education had filled my head with information that might or might not turn out to be useful, but there had never been any discussion of why we were learning all this. Now I was launched, so to speak, yet without any guiding principles.
My parents were alarmed at my questions. They never seemed to have had any doubt about what their function was. They met while working with the poor in the East End of London and went on to become mayor of Harrow (each of them in turn). Their lives were devoted to public service. I didn't see this as necessarily the only path, although I respected their approach. When I started demanding answers, they summoned the family doctor, the genial Dr. Waller (house calls were still possible in those days), to whom I tearfully posed my questions. He listened for a few minutes and then told me that people just didn't ask questions like those. They weren't appropriate. He reported to my father and mother that I was suffering from a nervous breakdown and prescribed a program of rest and exercise (I was not naturally athletic but soon found myself enrolled in fencing, skating, swimming, and "keep-fit" classes). I had been about to set off for Switzerland to become the secretary to the director of the Swiss Everest Foundation, but Dr. Waller decided that I had worked too hard at my exams and should not be even more stressed by earning my living in another language in a foreign country. So my questions were squelched, and I was made to feel as though I had done something unforgivable by even raising them.
I bring all this up because I suspect it may be a far from uncommon experience, even if the details of the dilemma are different for each of us. Our parents and teachers try to do their best for us, but either there never seems to be a suitable moment to address questions like these or they themselves have not found the answers and so hesitate to offer advice for fear of being more of a hindrance than a help. It is a little like the way sex education was then: no one spoke about it, and it was assumed that each person would discover the facts for him- or herself. There is, of course, one other possibility, and that is that a great many people simply never allow themselves to voice these questions. The aim of modern education seems to be to equip us to be suitably employed so that we can support ourselves and our families. And once we are employed, the idea is to get through our working lives so that we can enjoy ourselves in our retirement. How on earth did society lose sight of its reason for existence? And how can we know what to do now, in this moment, if we don't know where we are going?
After three months of exercise and no intellectual stimulation, my questions faded into the background, and I set about earning my living like everyone else. Then, quite by chance it seemed, I started attending evening classes at the School of Economic Science (not to be confused with the London School of Economics). It came about this way: I was dating a rather diffident young legal student who mentioned that he planned to sign up for philosophy classes and asked me if I would like to accompany him. In the London Underground there were these plain black-and-white posters with the word Philosophy printed in large navy blue letters. (In those days I was unaware that this word meant "love of wisdom.") While I waited for the train, I would study the few paragraphs of course description and was always puzzled by the fact that afterward I couldn't articulate what I had read. Even now I cannot tell you what was on the poster, but I was so intrigued by this phenomenon that I decided to enroll together with Anthony and find out what the course had to offer.
I have to say that I found the first two lectures rather boring and simplistic. The course was taught through the Socratic method: raising questions. Answers were not often forthcoming, but at last many of the questions that had been troubling me were being asked. We had a rather benevolent middle-aged tutor who listened carefully to the response of each student and then smiled and said, "Yes." I noticed that he had responded this way to two diametrically opposite answers, and so I challenged him. "When I say 'Yes," he replied, "I mean that I've heard you, not that I agree with what you say."
Halfway through the first term, we were invited to sign up for the following one. At the coffee break, small groups of students were discussing their plans, and I hovered at the edge of one of these. There was one man in our class I particularly respected. He was tall, good-looking, intelligent, and solid as a rock. He appeared to know his own mind, and I always appreciated what he had to say. He remarked that although he himself had not yet found much of value in the classes, he had a close friend, an older man, who had been attending them for years. If his friend had found something worthwhile here and he had not, perhaps the fault was his. He said that he was going to sign up for the next term to see if he could discover what his old friend had found. And I signed up also, just because this logic was so persuasive. At least I had found a community where it was acceptable to ask important questions.
I remained in this organization for a considerable number of years, studying the philosophy of many of the world's great traditions, and what I heard and put into practice there laid the groundwork for the rest of my life. At that time I believed that "the truth" was available only through this one conduit. Now I see that it is available anywhere and everywhere. But I probably would not have realized this without my early schooling in this particular discipline.
It was my friend and original publisher, Joel Fotinos, who dreamed up the idea for this book. The thought would never have occurred to me on my own. We were having dinner one night, I was telling him of my plans to leave full-time employment, and he instantly suggested that I write a book for him with this title. Once I had got over the shock, the notion seemed irresistible. When I stopped to consider it, I realized that everything I do is governed by the principle of not having anything left over. Still, I'm baffled that he was able to size me up so accurately.
I see now that living economically and wanting to be of service to other people and share with them whatever has come my way have always been themes for me. When I first came to the United States, I met with the head of the fledgling New York philosophy school, which went under a different name than the School of Economic Science, and she asked me, "What do you want from us?" I was taken aback by this question because both at home and in the school's parent organization in London, I could not recall anyone ever asking me what I wanted. As far back as I remembered, I was simply told what to do, and usually I did it. I don't wish to imply that I was a yes-person. I certainly challenged authority a great deal, but this took a certain amount of courage because challenge was not considered an option by Those-in-Charge. (This was England fifty years ago. In the United States there are always so many options...)
For a moment there was silence as I tried to collect my thoughts, and then I said, "I just want to be useful." I understand now that this desire has characterized my whole life. I like things to be put to good use. For me, economy is all. I never buy or cook any more than is necessary. I am always going through my closets to see what I can pass on to someone else. I feel guilty if I am not using whatever I own—books, sweaters, shoes, you name it. And when I went through my files to see if I had ever written anything on the subject of economy, I found a quote from May Sarton (the epigraph of this book) that I had squirreled away five years before. Come to think of it, none of this should have surprised me, since I began my training at a place called the School of Economic Science! (The school had begun by teaching the economics of the American Henry George, author of Progress and Poverty, and had gone on to add philosophy to its curriculum. I never found the economic aspect of what was taught there very appealing—or so I thought until now. I'd always been under the impression that I had gone there for the philosophy.)
As I thought about it, I realized that the result of gathering about you only what you need and relinquishing everything else is self-sufficiency—a lack of emotional neediness. This is another way of saying that it is wise to be satisfied with what you have. Lately, I have been mulling over the word content. I find it wonderful that it means both "that which is contained" and also "being satisfied." Both meanings come from the past participle of the Latin verb continere. Contentment is a peaceful and unruffled state, but nowadays it is all too rare.
So everything you read in these pages is an exploration of how to live so that supply does not exceed demand or consumption; how to share whatever you may have with everyone else, not holding anything back in a miserly way; and how to trust that the universe will respond to you in the same way that you respond to it.
I visited the island of Bali in 1986, and my most enduring impression of it is its fecundity. Bali is located seven degrees from the equator, but since it is a small island, it never seems to get too hot. It does, however, rain every day for a little while. The result is that there doesn't seem to be a square inch of land where things aren't growing in profusion. And, in many ways, that is how it is with the world. Everything is available in profusion. It is just not always obvious when you are not in Bali.
I had always thought that I lived a plain and simple life until I began to write about it, and then I became amazed and almost overcome by the wealth and quality of my experiences. I now feel very blessed.
I have tried to put down my observations on how the world works so that you, too, can share in this abundance. In order to get back to the essentials, we must first identify what they are. Only then are we in a position to do something about them. What follows is not so much about what needs to take place at the physical level—practical instructions on what to do or not to do—as about what goes on in the mind. Don't try to change things on the outside—in the "real" world. The work has to be done on the inside. It is a kind of interior housekeeping. Each time you drop an old attitude or habit, it is like spring cleaning: more space becomes available. There is room to move about and examine the situation from a new perspective, and everything feels freer and lighter. If you clarify the mind and relinquish some of the curious ideas that have become lodged there (most of the time we aren't even aware of them), then you will be free to enjoy all the glorious things that are in the physical realm. So here are a few principles for clearing the mind of clutter.
TWO
Taking Stock
It is August and just over a week since I quit corporate America and launched out on my own. For the last eleven years I have juggled two almost full-time jobs at Random House with only half an assistant—selling reprint rights for Alfred A. Knopf (something that became so familiar to me after thirty-two years that I felt as though I could do it in my sleep) and publishing nine books a year at Bell Tower, a spiritual imprint I set up as part of Harmony Books in 1989. Earlier this year I admitted to myself that I was unenjoying a great deal of each day, and I decided to stop living my work and start living my life. I have taken my pension and made an arrangement to continue editing my Bell Tower books as an independent contractor.
I planned to set things up in a corner of my living room so that my home office would be almost as invisible as my small, elderly television set. Perhaps that will eventually happen but, for the time being, it is just not possible. The office is there, but it is certainly not invisible. My son, Adam, graduated from college in May and announced that he didn't want to live at home. I supported his decision (after having had the place more or less to myself for the last four years, I have grown to like living alone) but pointed out that first he needed a job and a checking account. I suggested he wait to get an apartment until he had settled into a job.
Unfortunately, he was miserable in his first job (not surprising since he was trying to sell telephone service door-to-door), quit after just a few weeks, and is now a little at sea as to what to do next. So it looks as though it will be some months before he and all his accumulated possessions from college will leave the apartment. Currently, every closet is jammed with his goods and chattels. And then I purchased a computer, printer, and scanner for my new life and acquired with them three huge boxes I dare not abandon for a year in case anything goes wrong. After which I brought back from the office nine boxes of my own stuff that I felt I needed to set up shop. So every available surface—windowsills, tables, and floor space—is stacked with files and books and manuscripts. This is hard for someone like me who thrives on a great deal of empty space. But this is the situation, and there is no choice but to relax into it and not fret.
This new phase of my life has forced me to reexamine and question the smallest details: Why do I do this rather than that? Is there a real reason for it? Or is it just the encrustation of habit? This happened once before in 1964 when I came to live in the United States. I had thought I was coming here to discover what America was like. Instead, it was myself I discovered. For a short while, silhouetted against the backdrop of a new country and culture, I was able to see who I thought I was more clearly than ever before.
Human beings are creatures of habit. What normally happens is that we develop bad habits, but it is not so hard to develop good habits instead. Once these are set in motion, they operate on their own. In some ways this is a blessing because the moving mind (the part of the brain that learns how to perform physical movements) takes over physical and mental activities, and we come to rely on it. This is what I hope will take place with all this new equipment with which I have encumbered myself. There is a jungle of wires and tefillin-like black plugs seething beneath my desk—evidence of a terrifying amount of electricity being consumed in the name of simplicity. At the moment I am still in the learning stage, trying to figure out exactly what is in the computer Dell shipped me and how to make good use of it. Presumably, at some point this will all recede into my moving memory and become second nature. However, after sixty-one years of living, far too many things have become automatic for me. Half the time I do not even realize that I am doing them, and I welcome the opportunity to take a fresh look at everything.
We get very attached to our habits, and it can be amusing to try to dislodge some of them occasionally. As we grow older, we tend to close down our options. We think of it as refining our taste. We prefer broccoli to cabbage and cauliflower, so we gradually stop buying cabbage and cauliflower. There is nothing wrong with eating broccoli (my favorite vegetable), but a variety is always healthier. It is not a very good idea to eat broccoli every night of your life.
We place our slippers in a particular way under a chair. We can no longer even imagine placing them any other way. It is okay to eliminate unnecessary things, but we need to keep an eye open to make sure that we don't go overboard. I noticed that I always hold the telephone in my right hand. The other day I decided to pick it up with my left hand. It felt very strange. The telephone was heavy in my hand. I pressed it too hard against my ear. I didn't seem able to relax with it the way I normally do. Also, I wasn't used to listening with the other ear, and there seemed to be a great deal of echo either in the receiver or in my head. When I thought about this for a moment, I realized that all my life I have been overtaxing one ear and causing pressure on it. Also, I have not been using 50 percent of my listening (and holding) capacity. This really woke me up. We need to be flexible—willing to cross our legs the opposite way for a change, open the refrigerator door with the other hand. We have to guard against not becoming rigid. Flexibility offers a slender tree the freedom to sway in the wind, and so it is with us.
Since I am someone who has lived a very orderly life up to now, I had nursed the idea that in the future I would not be so rigid. My days would proceed naturally, and I would not set aside special times for this and that. I would cope with e-mail (something I had not permitted in the house before), the phone and fax, work on manuscripts as the occasion demanded, and an opportunity would spontaneously arise each day for me to write a portion of this book. I would never feel pressured again. There would be time for walks and expeditions, meals with friends, joining a community garden in Riverside Park that I had long admired, tai chi chuan, a second period of meditation in the late afternoon, and so on.
I had been invited to write this book at the beginning of April, and now it was August. It had taken all these months to organize things so that I would have the time to write the book. I had to give up my job at Knopf before I could take on another project. So I thought that on the first day of my newfound freedom, there would come a moment when I would just start writing the first chapter.
But this did not happen. Each day I was busy from seven in the morning until Adam returned at the end of the day. It rained a great deal, so I did not go out much. There was always more Bell Tower work to do; it was right there in front of me, and there never seemed to be a natural break when I could turn to what I saw as my own work. After four days of this, I realized that I had trained myself for forty-three years to put my employer's interests before my own, and this was going to be a tough habit to break. The first lesson in my new life was perhaps how to establish boundaries.
I thought back to the days when I had begun to meditate and remembered how at first it was so difficult to fit two half-hours into an already crowded day. Then it became clear that if you waited for a space to appear in your schedule, it never did. But if you meditated before the day began and again at the beginning of the evening, there was never any problem fitting everything else in. So I determined that in the weeks ahead I simply had to sit down at the computer early in the day and not peek at my e-mail until the time for writing was over—not that I needed to set a specific time to start and finish but that I had to take care of this one thing, my writing, before all the demands of the day claimed my attention.
That was August, and I was full of good intentions. Now it is February. It has been just over six months since I abandoned a full-time job for the freedom of a life of my own, and it is hard. For all these months I have been coasting along on the energy accumulated over a lifetime of push, push, push. I had had a full (too full), active, and successful professional life, and now that I have broken loose from the old mold, I am in limbo. I am no longer being driven by outside circumstances, and inertia has set in to some extent. This is truly a bleak, dry, and uneasy time. In my previous life I was, of course, usually on automatic, reacting to outside stimuli.
Whoever I talk to or whatever I read points out that the way to get going again is to make myself a schedule and stick to it. As the months trickled by I noticed that, although I planned to start writing at 8 a.m. and round off the day by practicing tai chi chuan at 5 p.m., neither of these things ever happened. Yes, it is possible to schedule every moment of the day and not allow yourself any wiggle room. That way I would certainly get things done, but would I be any more present while I was doing them than I had been in my old job? I do not want to exchange one kind of servitude for another. Not that you can't choose to be present at any moment, but being present does tend not to happen if you get pulled or pushed into the next activity through sheer force of habit. I know only too well that I can have a bath, get dressed, do my mini-stint of yoga, eat breakfast, and listen to the news on National Public Radio, without really knowing what I am doing.
Early on in my time at the philosophy school, we were instructed to pause between actions and remember who we were. One of the reasons for this was to prevent the energy from one task from being carried forward into the next. If you make a clean break when you finish something and come to a full stop, then you can start fresh with whatever is necessary for the next action. This made an inordinate amount of sense to me, and I resolved to introduce the practice into my life. That was over thirty years ago, and I haven't managed it yet because I have always been moving so fast that I have been unable to come to a stop. This is a terrible admission for someone who believes she has a lot of willpower, but it is true. I have failed absolutely, so it is not surprising that I am having such a difficult time with the two things I have recently tried to add to my schedule.
It occurs to me that in my old life I was like one of those little Volkswagens, beetling along the highway en route to events other people had organized. In this new life I am determined to leave the day open and experiment with formlessness. I know this is harder than making myself a new timetable, but I have a long history of doing things the hard way, if only to prove that it is possible. I believe that operating without a regular schedule is the only way I will discover the space around and between things and events. In the past I led a tethered life, and now perhaps I can learn to glide on the available air currents.
So: I no longer feel as though I am running my life. I am not running after it or ahead of it. It is just running on its own, and if I stay in the present from time to time, I catch a glimpse of what is approaching and am ready to welcome it when it arrives. You could say that I am here expecting the unexpected.
To the extent to which I am able I am trying to live without props. This doesn't mean that I don't plan anything (this is, after all, New York in the twenty-first century). At my age I suspect that it isn't possible to change the time I awake naturally in the morning (5:30 to 5:45 a.m.), or the time my eyelids close in the evening (10-ish). Anyway, I don't think that I need to alter those things. There is plenty of time during the day between waking and sleeping, and if I can relax my grip on all those hours, perhaps I will also find a way to make a shift between the sheets.
What it means is that I am not dragooning myself to do things. This way, I don't have to deal with the regret that will definitely arise when I don't manage to do them. For instance, I have just started a Japanese brush-painting class, which is something I have been meaning to do for longer than I care to admit. On my shelf I have a slew of books on the subject. Over the years different friends have given me not only books but also rice paper, brushes, an inkstone, and sticks of red and black ink. I never asked for any of this stuff, but my friends were convinced that I would enjoy brush painting. I even have a manuscript a would-be author sent me from California in 1985, complete with brush-writing equipment (this was for Chinese brush writing and painting). The manuscript arrived unsolicited, with a letter that encouraged me to follow the instructions in the teaching manual. "There is no SASE," it continued. "If you like the experience, think about publishing the book. If the answer is still no, just let me know." The author and I wrote back and forth for some time and eventually he said, "I am moving you from my book publisher file to my correspondence course file." I never actually started grinding the ink and using the brushes, however, and neither did I publish his book.
Then, last fall, I went to China in a group led by Kazuaki Tanahashi, a master Japanese brush painter now living in California. We were following in the footsteps of Dogen, the thirteenth-century founder of Soto Zen Buddhism, but Kaz initiated us into painting the enso, or circle, for several mornings in Shaoxing (the Orchid Pavilion there is renowned for its calligraphers, particularly Wang Xizhi, the inventor of Chinese running script, which is like clouds floating in the sky), Hangzhou, and elsewhere. Kaz showed us how to grind the ink (hold the ink stick vertically and make sure its whole surface is in contact with the inkstone as you dilute the ink with water) and then draw a circle on the flimsy paper. He demonstrated how to do it but didn't give us any verbal instructions. Later I questioned one of my fellow travelers who had once taken a course with him, and he said that Kaz had told him that the brush should be vertical and the mouth horizontal. (I think I have that right. This meant that the mouth should be in a half smile, which certainly helps to keep you relaxed.)
While we were in Hangzhou we paid a visit to the Shi-lin Seal Engravers Society on an island in the graceful West Lake, where many of the group had individual chops made (these are the traditional seals with which painters and calligraphers sign their work). I couldn't imagine that my work would ever be good enough to sign, so while I waited for the others, I made a momentous purchase of my own: a set of three beautiful brushes in a green brocade box.
On my return to New York early in November, I put them on the shelf with my growing collection and eyed them ruefully from time to time. Then, in January, I was in SoHo at lunchtime, and I walked past a little storefront with a sign that said KOHO SCHOOL OF SUMI-E. Outside was a bunch of flyers, so I took one home and read it from beginning to end. It sounded very forbidding, but it did offer the possibility of watching a class before signing up for a ten-week course. A few weeks later I was again in the vicinity and the lights were on in the store, so I walked in. The gracious Japanese-American teacher and her cat who greeted me were both very welcoming, and so I decided to come a few days later to observe the Sunday-morning class. While waiting for the other students to arrive, I learned that Koho Yamamoto was almost eighty years old and had established her school on the corner of Houston and Macdougal Streets twenty years before—around the time I had started gathering brushes, ink, and books. I think that this fact brought home to me that I couldn't afford to wait any longer. I had already lost the possibility of learning from her for the last twenty years. Who knew how much longer she would be teaching? I made a commitment to join the class the following weekend and now, after three lessons, am happily experimenting with bamboo leaves and stalks. Plum blossom and pine trees to follow.
It is fascinating that I have chosen to study this particular discipline without really knowing why. Just yesterday, I came across this quote by Motoi Oi: "The aim of the Sumi-e artist is not the reproduction of the subject matter but the elimination of the inessential." The book in which I found the quote, The Book of Bamboo, by David Farrelly, goes on to say that the aim is "not to record every rock on the mountain or each leaf in the grove, but to capture—with a spare economy of strokes—the moment-by-moment urgency of life itself."
Once you join a class there is an expectation of practice, regular practice. (What other kind is there?) So the next question was: When would I fit this in so that it actually got done? I decided that the evenings would be the best time since then I would not be tempted by e-mail. But evenings came and went, and I usually felt too tired for the unfamiliar discipline of holding the brush wedged between my middle and ring fingers, keeping my elbow down, pushing or pulling the brush with the whole arm, not just the fingers or wrist, and keeping my attention where the ink was flowing onto the newsprint (which is what is used to begin with). Eventually, I realized that the only way to practice is to do so when it occurs to me, immediately—not planning it for some time in the future, because that time will probably never arrive.
I know that that was rather a long story, but I wanted to trace the maneuverings of my mind over the last twenty years and illustrate the lengths to which a seemingly intelligent person will go to avoid doing the obvious. Now that I have actually started the Sumi-e painting, I feel as though I have a responsibility toward it. I practice whenever the impulse arises, and this might be in the middle of the morning, sometime in the afternoon, or in the evening. It doesn't matter, as long as it gets done. And I don't kid myself that it is going to happen absolutely every day. It gets done as often as it gets done—probably three or four times a week in addition to the two-hour lesson on the weekend. It is a tremendous relief to me that I have resolved this amicably with myself.
This approach is working with other things, too. Writing and answering e-mail is compulsive, but I have now given myself the freedom to get up from the computer and walk around the house when I feel like it, do a little cleaning (rather than launch a major blitz once a week), correct galleys while snuggled under a blanket on the sofa, run a few errands on Broadway, or go for a walk in the park with a friend. I have even made a date recently to go to a foreign film at lunchtime. This last sortie seemed almost sinful to me after so many years attached to my desk during the workweek, but I am learning (slowly) to relinquish the attachment to my desk.
You may be wondering what all this has to do with living a simple life. What I am endeavoring to do is abandon as many outmoded ideas that I have been harboring as I can, so that my head and heart are clearer and my step lighter. I recently attended a two-day seminar on the variety of meditation practices available from different traditions. It was given by the elderly, inimitable, Sufi teacher, Pir Vilayat Khan. I took copious notes about different states of consciousness and what needed to be done to reach them, but the deepest impression that remains with me is his instruction that we should become like gossamer. This is a wonderful image and illustrates precisely how I am trying to live these days.
Not only am I doing my best to eliminate the inessential, but I am also trying to do what is essential. Like the S-shaped duo of yin and yang, these are two aspects of the same principle. You cannot have one without the other.
THREE
How Much Is Enough?
Most people believe that they expend just the right amount of energy for whatever it is they are doing. If you are like me, however, when the phone rings not only do you try to pick it up immediately but you also grip it far harder than you need to. The verb that is generally used regarding phones is "cradle," but that is rarely what people do with them. The person at the other end is not going to give up after just a couple of rings, so you don't have to act as though the call is an emergency. When you pick up the receiver, notice how tightly you are holding it and see if you can relax. Whatever tension there is in your hand will communicate itself to the rest of your body and vice versa, because the two parts are physically connected. If you can relax the muscles in your hand, that release will be felt throughout your body. What a relief. Enlightenment begins with relaxation. The hint is right there in the word itself: en-lighten-ment.
Hardly anyone handles a tool with just the right amount of energy. When you want to sweep the floor, all you need is the minimum amount of energy to stop the broom from falling out of your hand. The broom does not need to be pushed or pulled across the surface of the floor. There is a natural wide and gentle movement from the shoulder, in which just the tips of the bristles whisper across the wood catching any dust and transporting it into your little pile. This is a very different activity from the one we normally engage in.
Observe what you do when you open a door. We tend to put about 100 percent more force into every movement than we need. Only a very small amount of energy is needed to turn the handle of the door and push it open. That is the way the door handle and the door hinges are designed. Experiment with actions like these to see how little force is really needed. It will be a revelation. Each time we relinquish this effort, we will be able to save strength and energy for doing something else. One of the reasons we all get so tired and run out of energy is that we usually expend far more than we need at any given moment.
Much of this added tension comes about because we are not content to simply perform actions. We add into them layers of feeling and desire that are counterproductive, and we often start off with the wrong aim, so it is a good idea to look for your intention and see if you can recognize it at the same time as you check on the level of effort you are putting into everything.
Recently I was home visiting my eighty-eight-year-old mother and discovered that everything I touched in the kitchen was grimy and greasy. As I scrubbed away I was seething over the fact that the person who was being paid to do this job on a regular basis was either lazy, unskilled, or in dire need of glasses. At times I became so concentrated on my resentment that I lost track of the cleaning.
I was already an adult when I started to learn how to play the piano, and I found it inordinately difficult. At the end of my lesson, I would be wiped out for the rest of the evening. The man I worked for at the time found this very puzzling because to him playing the piano was a real joy (and this is certainly what I had hoped for, too). I explained to him that it was draining because I made so many mistakes and I saw this as failure. I wanted so much to get it right. He pointed out to me that if I could play perfectly, I would have no more need of instruction. My teacher did not expect me to get everything right, so why should I? It was then that I realized that I had been putting my effort into the trying—trying to get the notes right. But it was misguided nonetheless. I had two desires operating. One was to perform perfectly and the other to make music. The former was stronger than the latter. I had put the cart before the horse.
If you can catch a glimpse of your desires and let go of them, then you can devote yourself completely to whatever it is you want to do so that your effort is untrammeled—which brings to mind the word impedimenta, the Latin for "baggage." We need to surrender any unnecessary baggage so that we can travel lightly.
This may be a rather revolutionary concept to many people in the United States, but it is worth experimenting with. When you sit down to a meal, help yourself to no more than you are certain you can eat. You can always have more; once something is on your plate, however, it tends to get thrown out if you don't finish it. I am not sure why so many of us put so much food on our plates. It can't be that we fear starvation—not in this generation and in this country. Even after almost forty years in New York City, I am uncomfortable with the amount of food restaurants serve. Meal after meal and day after day, they discard our leftovers. Has it never occurred to them that we might prefer to eat less, pay less, and weigh less?
My parents were scrupulously honest and upright, but I was always troubled by one of my mother's quirks. When she ate at restaurants that sported ashtrays with their names emblazoned on them, she would often bring one home with her. Over and over again I would say, "Darling, that's stealing."
"No, it's not," she would retort. "It's ashtrays. Restaurants expect you to take them home. This is their way of advertising." On reflection, the explanation seemed a reasonable one, but it left me with a certain degree of uneasiness.
I think there were two aspects of it that did not sit well with me. One was the principle of not taking something that doesn't belong to you and is not freely offered (the maître d' hadn't pressed an ashtray into her hands as she left). The other was that people tend to help themselves to whatever is available, even if it is something they don't need. If I remember rightly, my mother had given up smoking years earlier and so had absolutely no use for all those ashtrays. She just found the concept of free stuff irresistible.
A few weeks ago I was passing one of the new MetroCard-dispensing machines at a subway station. Out of the corner of my eye I watched a man take his card and walk away, forgetting to pick up his change. The machine kept blinking its message about the change, and the coins lay in the slot beneath. I experienced a momentary tug toward the money, but then a little voice said, "That money isn't yours. And even if the man doesn't come back to claim it, there are other people who could really use it. You don't actually need it." I admitted this was true and walked on.
Many people treat life like a smorgasbord. They pile their plates high with a huge assortment of delicacies, simply because the food or whatever is available. Their selection and the amount of it they take bear no relation to their need at the moment. I am reminded of the ad that the Archer Daniels Midland Company ran on PBS. It said that there is no shortage of food in the world: The problem is not the food; it is the politics. So the reason that there often doesn't seem to be enough of anything to go around is often a question of inappropriate appropriation or hogging it.
Some years ago I visited an old high-school friend in Washington, D.C., and she threw a party for me. Since her husband worked for the World Bank and she was connected to American University, almost all the guests moved in these circles, both of which were unfamiliar to me. Everyone I encountered seemed to be studying at night, in addition to working during the day. Eventually someone asked me what courses I was taking. I explained that I worked in publishing and so there was not any need to take courses. I found the question very puzzling.
"But if you learned something new, you could get promoted or move on to a better job," the woman said.
"You can't learn publishing in a class," I said. "You just have to plunge in and learn as you go. I have been doing this for many years, and I know how to do it. I love what I do and I don't want to do anything else or work anywhere else, so what would be the point of taking classes?" Neither of us could comprehend the other's point of view. We were like two ships passing in the night. All these people were on a restless search for something more or something else.
A slogan familiar to many people is "Less is more," and once I saw the beauty of it in operation, I espoused it completely. But lately I have begun to detect a basic fallacy in it. In this consumer society, we have been conditioned to believe that more is always better. (I once thought of getting in touch with Macy's to offer them what I thought would be the perfect slogan: "Macy's has more.") Now I realize that I don't want more. I simply want enough. Although "Less is more" sounds as though it is a way of cutting back, of returning to simplicity, it contains the subtle message that if you have less, you will receive more. It is still a promise that more is better.
I went to Havana in 1997 and saw the conditions under which the Cubans lived because of the U.S. embargo. This was the first time I understood that "Less is more" doesn't apply to every situation. The people I met had hardly anything, but they were cheerful and not resentful. Less isn't more when you don't have very much; however, the Cubans did have just enough to keep going, and so it was enough. It is the same way with the rest of us. Less doesn't necessarily have to be more. We simply need to have whatever is sufficient to deal with the situation we find ourselves in. So I propose that we change the adage to "Less is enough." What do you think?
The whole question of why our lives seem so unsatisfying needs close examination. Why is it that our experiences or possessions never seem to bring us lasting happiness or a sense of completion? We always want something more, and it is always eluding us. Not only do we want to hold on to what we already have but we also want to acquire as much more as we can. I think of possessions as possessing me rather than vice versa. If you own something, then you are responsible for taking care of it and are continually worrying that it might get harmed or you might lose it.
We try to fill the vacuum that we believe to be inside us, but we need to remember that we didn't come into this life to shop, to chalk up experiences, to amass objects we can't take with us when we go, or even to make a lot of money.
In truth, it is not the number and diversity of our possessions that are the problem but our attachment to them. When the attachment grows thin and the filament breaks, then we discover that we do not really want so much anymore. What we need to relinquish, therefore, is our attachment to possessions and experiences, not the things themselves. The freedom we are all seeking is freedom from the fear of losing what we believe we own.
Among the notes I have kept over the years is a small scrap of paper on which I typed out a passage from a book by Robert Pilpel, entitled Between Eternities. It speaks with extraordinary clarity on this whole matter:
You wonder about the next life because this life's not enough for you. And this life's not enough for you because you're not living it but thinking about it.
I thought that there had to be more to life than being alive and I resolved never to be satisfied with my existence until that something more, whatever it was, had been savored to the full. I felt, moreover, that once my great goal had been achieved I would be prepared to die....
Why are we afraid of death? Surely it is not because the process of dying is painful—because the process of living is infinitely more so. And we don't fear living—at least, not as much as we fear dying. We are afraid to die because we are not ready. Does death stand for our final failure to achieve the unattainable? And if it does, what then does the unattainable stand for? Would I want it so much if I knew what it was?
When I think back to what I believed would be the most memorable moments in my life—confirmation, the first time I made love, my wedding ceremony, the birth of my son—I remember that each time I had expected to feel different in some way. I anticipated that something in me would be transformed forever. But nothing like that ever happened, and the next day it was always recognizably the same me who woke up in the morning.
Why is it that we yearn to be more or other than we are? It so rarely occurs to us that what we are looking for may be—indeed, always is—already within us, simply undiscovered.
FOUR
At Play
If you want to know someone well, watch how she walks, talks, stands, sits, and speaks. This will reveal exactly how her mind operates. Just observing fellow travelers in the subway tells you a great deal about them without your ever being introduced. If you want to know even more about the person, watch what she does in her leisure time. Knowing that will help you understand what makes another person tick.
Take me, for example. What have I ever done in my spare time but work? Yes, occasionally, I have taken time off, but I have had to force myself to do it. I was brought up (how on earth did this begin?) to finish my work before I began to play. Of course, there is always more work and so I rarely, if ever, got around to playing.
I remember an occasion many years ago when an author and I had been working very hard for months on his book. One afternoon everything was finally done, and I said, "The work is over. Now we can play." And then we realized that neither of us even knew where or how to begin. Most of the time I still feel that way, but I am determined to make up for all the time I have lost.
I don't know whether my son is typical of the current generation, but we have always had a difference of opinion about work. I believe that you do your chores first and, if there is time left over, then you are free to play. He believes the opposite, probably because he has observed me slaving away for the last twenty-three years. All he has ever seen me do is work; perhaps he has come to the conclusion that this is no way to spend a life, so (as far as I can tell) he gets in as much play as possible first and manages to squeeze in a little work at the end.
When I consider the implications of how I have spent my life so far, I am appalled. Somehow, I have completely overlooked the importance of playing and what its significance is. Many of the sacred teachings I have studied (particularly Vedanta) go out of their way to point out that creation is one great play and that whatever or whoever produced this play did so for the sake of enjoyment—his/hers/its and ours. We came into this world, the current production, as players not workers. We have roles that may shift from moment to moment, depending on the director, but we don't have to write the script, paint the scenery, or sell the tickets. We just have to play our parts. Why is this something that I fail to remember over and over again? Most of the time I, and probably you too, labor under the delusion that I am responsible for all aspects of the play when the truth is that I just have to be there for the performance. And if I'm not, neither I nor the audience will enjoy it.
Perhaps this can be most clearly seen on stage. Once you have experienced it there, it is easier to put it into practice in your daily life. The most memorable performances I have witnessed were those of Mikhail Baryshnikov dancing at the New York City Ballet when he first came to the West, and Douglas Perry who sang the part of Gandhi in Philip Glass's Sanskrit opera Satyagraha, which I saw at the Brooklyn Academy of Music. In each case, the performer was having such a good time, he positively glowed. The light was flooding out of him irrepressibly. He was enjoying himself in every sense of the word. And so we, the audience, shared his enjoyment. This is the way to go through life. And if you are wondering what all this has to do with not living to excess, here's the answer: There was an article in Reader's Digest umpteen years ago in which I learned that far fewer muscles are required in smiling than looking sad. Less tension is involved.
So, whether it is someone else you want to know about or yourself, have a look at what this person does when not working and also see if he or she is enjoying it. Are you spending all your time watching television (whatever the excuse)? If so, what does this say about you? Do you seem to do nothing but cook? Does he devote himself to music?
I wonder how I got to this ripe age feeling so driven by my work? Although it is not uncommon in the United States, remember that I grew up in England where to talk of how you earn a living (or religion, politics, or sex) is just not done. In this country, almost the first thing you say to someone when you meet is "What do you do?" This is not a question that would ever be asked in the United Kingdom. Here in New York I love to talk about my work with and for authors. But when I go home, no one wants to know about it. No one asks, and so I don't tell.
My first job was in a public library. The salary was so small that by the time I got my paycheck at the end of the month, it wasn't even enough to cover my expenses although I was living at home. I cycled to work and took a sandwich for my lunch, but even these economies didn't seem to help. My time was spent stamping books and putting the volumes that had been returned back on the shelves. The high point of my day would be when someone asked me, "Can you recommend me a good book, miss?" (I would hastily scan the first and last pages of whatever I was holding at the time so that I could describe the kind of book it was) or once, "Do you have books on learning foreign languages? I'm going to Switzerland for my holidays, and I want to learn Swiss." After three months of this, I had managed to reimburse my parents for the money they had had to lend me during my time at the library. Then I quit, having discovered that the only way for advancement as a librarian was to study for the equivalent of a university degree in my spare time. This would then qualify me to choose purchases for the library from lists of books that circulated once a week and also, presumably, rule the roost over the other bluestockings who worked at the branch. The prospects were not thrilling. Surely there was more to life than that?
So I enrolled in a secretarial college for young ladies in Mayfair. In those days that meant Pitman's shorthand (I took rapid dictation in minuscule glyphs), typing at a speed of at least sixty words a minute on a manual typewriter, learning how to lay out a letter or document so that it was good enough to be framed, and commercial French, Spanish, and German. I managed to cram the year's course into six months. Then I applied for a job at the Automobile Association, where I spent six months translating the descriptions of members' cars into French for their carnets (passports for cars to travel in Europe). There turned out to be no future in that job, either, so I moved to the foreign franchise department of Schweppes, working for six roving managers, and from there to Pirelli (the footwear, not the tire division), and then on to a chemical company, and eventually a small engineering firm in Chelsea.
Each time that I found myself in a new situation, it took me about three months to understand the business and how it functioned, and then I would revamp all the office systems to make them work more efficiently. That would take me another three months, but then there would be no further challenge. All the executives were always men, and all the secretaries young women. Even if I thought I could do the job better than most of the men, no one would ever have considered asking me to. I didn't have a degree, and I wasn't a man.
So I consulted someone about what other possibilities there might be for an intelligent woman, and he told me that there were two avenues I could pursue without a degree: advertising and publishing. In both these spheres women had already made their way, and there was no reason why I might not do the same. I had no interest in advertising and I had started out with books in the first place, so I applied for a job at André Deutsch, a small independent publishing house (they were almost all independent in those days, but they weren't all small).
Now that I think about it, I am amazed I took the job. I was interviewed and hired as André Deutsch's assistant by a young American editor who herself had just started work there. My salary was to be £12 a week rather than the £15 I was then earning (it took me three years to get back to £15), and I would not meet André himself until the following week.
I suppose that it was there that I learned to work so hard and think of almost nothing else. I shared an office with another young woman. Perhaps office is too dignified a word for our space. We sat side by side at the end of a corridor. Beside us on the floor was a huge pile of unopened manuscripts that we attacked once a week, carefully saving the envelopes in which they had arrived so that we could reuse them later. On the other side was another pile of manuscripts waiting to be rejected. Shirley worked for the production manager and two editors (each of these jobs was a full-time occupation). I worked for André. And between us we coped with the manuscripts that ebbed and flowed around us, confiding in each other that we thought we might be able to get everything done if it were not for the authors. But, of course, without the authors there would be nothing to publish.
André was a short, charming, distinguished-looking man from Budapest. It was common knowledge that the three most important things in his life were his work. And it was also rumored that you could trust André as far as you could throw George Weidenfeld (in those days, this physically large and successful publisher had not yet been knighted). This was both unfair and unwarranted. I worked for André for more than three years, and I never caught him doing anything illegal or even unethical. He made sure that he always received whatever he was entitled to in full measure but that was as far as it went. When it was clear that paying a bill could no longer be avoided, he would instruct the business manager to send a check to the American publisher "by fast sea mail" (perhaps he thought that the post office used ships that traveled at two different speeds?).
He would appear at our end of the corridor just as we were about to leave for the day, with his shirtsleeves rolled up and a letter that had to be dictated immediately. And he would stand just behind one of us, almost touching but not quite, until the letter was finished. He had enormous animal magnetism, and he used it. He knew that there was no way we were strong enough to refuse to stay late if he flashed his smile. By the time each of us had taken the train home, it would be late and we would be so exhausted that we would often cry over our supper.
I wondered how to extricate myself from this situation, and then the young woman who handled the foreign rights decided to leave, so I asked if I could take on her job. André was willing for me to do so but insisted that I continue as his secretary as well. I asked to be paid more if I was going to take on the additional work, and he said no: If I wanted the job, I could have it. If not, that was fine too. In retrospect, I see that this was blackmail. He would have had to pay someone else to handle foreign rights if I had not shouldered the responsibility. But I suppose he reckoned that it was a good gamble, and he won. I agreed to do the extra work without remuneration so that I could learn a new skill and move to a better job elsewhere.
André was a hard taskmaster, but I learned a lot from him. He knew how to sell anything. Anything. It just so happened that he had ended up as a publisher, but he would probably have been equally successful in another business. I noticed that he hardly ever named a price himself. If he was selling the translation rights to a book, he invited the other publisher to make an offer and then insisted that the figure was too low, no matter how much money was involved. This was a very neat trick. I suppose that occasionally he must have been put in a situation where he needed to name a sum himself but, if I remember rightly, he would then ask for something preposterous so that the other publisher would be forced to offer more than he had originally anticipated.
A year or so after I had taken on the second job I went to the United States for a month's vacation. It wasn't that this was somewhere I had yearned to visit, but a Pakistani friend from high school (the one who threw a party for me many years later in Washington, D.C.) was now living in upstate New York and had sent me a Christmas card asking when I was coming to visit. My mother said, "Why not?" and so that spring I went.
The following year I spotted on André's calendar that Bob Gottlieb, then managing editor at Simon and Schuster, was coming to see him. About ten minutes into the appointment, André's phone rang, he picked it up, and then stayed on the phone for an unconscionably long time. I had grown used to this behavior, and I knew how much New York editors fretted at being kept waiting when they had half a dozen more appointments to get to later in the day. I stopped what I was doing, slipped into the room, sat down next to the visitor, and said, "You must be Bob Gottlieb. My name is Toinette Rees. We met last year when I went to New York. How is Jean?"
Jean Jollett was Bob's assistant, and I had had an introduction to her from an editor at Deutsch who had recently returned from an eighteen-month stint working for Bob. I had really enjoyed the time I had spent with Jean, and so I asked Bob whether there was a chance of her coming to London on some kind of working vacation.
"Oh, no," he said. "Actually, my editorial assistant has just left, and I'm promoting Jean to this position. If you want to see Jean, you'll have to come to New York."
"In publishing it takes a lifetime to save enough to go to America," I replied. "I did save and I went, but now I would have to save for another lifetime. I couldn't come unless I had a job." I was just chatting idly in the hope that André would be courteous enough to get off the phone soon and let me get back to what I had been doing.
"Well," said Bob. "Why don't you come and work for me? I need a new assistant."
I was dumbfounded. "But you don't even know me," I said. I'd spoken to him for about three minutes one day after I had had lunch with Jean in New York. I don't think I would have recognized him in the street. He looked like a tall Woody Allen, but at that time I had never heard of Woody Allen. This was, after all, September 1964.
"No, but Jean knows you and likes you and that's good enough for me," he responded. "Can you type?"
"Of course, I can type," I said. "Why do you ask?"
"Because I want to make sure that you are a good typist."
"If I come, may I come just for a year?" I asked. I had recently explored the possibilities of getting a job selling rights in other London publishing houses. There were only three houses that I really wanted to work for, and none of them had a vacancy at that time. It occurred to me that if I went away for a year and got more experience, I could come back and perhaps command a better position. After all, the chief editor at Deutsch had got her job after working as Bob's assistant. Bob, who had achieved fame as the editor of Joseph Heller's Catch-22, had the reputation of being the best young editor in the English-speaking world.
"I want someone forever," he replied.
"But you might not live that long," I pointed out.
"No, I might crash on the way home on Sunday and then you wouldn't even get the job." It transpired that he was terrified of flying and always crossed the Atlantic by boat. This would be the first time he had boarded a plane.
"Well, if you don't crash and I do come, may I come just for a year? And how much would you pay?"
"I believe we pay new assistants ninety dollars a week, but I would need to check with my colleagues," he said.
"I'm not a beginner. I have been working for seven years and have been working for André for more than three. When I was in New York, I was told that a British secretary could earn one hundred and twenty dollars a week."
"Not in publishing," Bob replied.
"You begin to sound like André already," I retorted. "Why don't you go back and find out how much you could pay me, then write me a letter offering me a job in your editorial department reading French and German novels. I promise you I won't take you up on that, since I don't like French and German novels even though I speak both languages. It wouldn't be worth your while to ask me to read them because I would always write negative reports. But there's currently an embargo on British secretaries in New York City (where on earth could I have picked up this snippet of information?), so if I am going to get a visa, I'll need evidence that I have a job offer for something else. It takes six weeks to get an American visa (again, I wonder how I knew this) and five days to cross the Atlantic in a boat. If I am bringing my belongings for a year, I won't be able to fly. So I could be with you about seven weeks after I got your offer."
On reflection, this was a very saucy way of responding to a prospective employer, but at the time I believed that I was just doing my best to distract a visiting American editor. I wasn't taking the conversation seriously in any way. But suddenly everything shifted.
Before Bob could respond to my proposition, André finally hung up the phone. What I haven't yet mentioned is why he had remained on this call for so long. He was inordinately proud of the fact that he had published The Magic Christian and Flash and Filigree by the American novelist Terry Southern before they had appeared in the United States. Then Terry had written Candy, which had been published pseudonymously as Lollipop in Paris by Maurice Girodias at Olympia Press. André had just heard that the book was being published in America by Putnam's, and he wanted to publish it in the U.K. But André was a Hungarian Jew who had come to England at the beginning of World War II, and he didn't want to risk going to prison. Much as he hated to give up any potential profit, he was trying to persuade two other editors to copublish the book with him. He believed that, although the director of public prosecutions might send one publisher to prison for obscenity, it was unlikely that he would send three. André had already got the relatively new young publisher, Anthony Blond, to come in with him, and he had spent the last twenty minutes trying to persuade the much older and very distinguished Fred Warburg of Secker & Warburg to do the same. Fred had been arguing that he didn't believe that Candy was a satire on pornography. As far as he was concerned, it was simply pornography. And he didn't think it was particularly well written. He was not willing to publish the book at all. André never achieved his goal. In the end it was published in London many years later by another publisher.
"Can this girl type?" Bob asked.
"Of course she can type. Why do you ask?" André replied.
"Because she is going to come and be my secretary," Bob announced. This was news to me, but I didn't say anything. I thought I had just been distracting a nice American editor while my employer was being rude.
"What about me?" asked André.
"You are welcome to come and be my secretary if you like, but you probably wouldn't want to for a hundred dollars a week," said Bob.
"So what would I get in return?"
"What would you like?" responded Bob. (I began to feel like merchandise being haggled over. For the moment, both men seemed to have forgotten that I was still in the room.)
"Well, I will want a lot of titles," said André, always having the profit motive in mind. What he meant was that he expected Bob to give him an exclusive look at a large number of Simon and Schuster books that he might purchase for publication in the U.K.
"You mean after all these years all you want in exchange for Toinette is some books?" Bob and I looked directly at each other (only at that moment did all this suddenly become real for me) and then at André.
André saw that he had made a grave error. "Oh, I didn't mean that," he said, but, of course, he did.
The next thing I remember was boarding the SS France for an exceedingly tempestuous voyage across the Atlantic Ocean. I was twenty-five years old when I arrived in New York on November 10, 1964, and I started work at Simon and Schuster the next day.
I was so accustomed to having far more to do than was physically possible that each day I soon finished everything that Bob gave me. I don't think he was used to someone who worked with such intensity. He found my empty desk a reproach, and when I asked for more to do, his response was, "We don't pay you to work here. We pay you to be here." It is interesting that he put it like this. I have told this story many times in my life, but only now do I grasp the full significance of what he was saying. Being present (and knowing it) is one of the most important things that one can do in life, while work can be achieved without complete presence. (Perhaps the work itself will not be perfect but most people won't spot the difference.)
Shortly after I arrived in New York, I went to live in an apartment on the Upper West Side with two young women who also worked in publishing and one and a half Siamese cats (one was only half-Siamese). I had thought that I would be better off sharing with others in a strange city where I had no friends, but the cats were always trying to climb the net curtains and they were very vocal. I soon realized that what I really wanted was to be on my own.
Then, after three months, a dear little apartment fell into my lap, but not in a way that made me happy. I had come to New York to work for a brilliant editor and to improve my chances of getting the job I wanted on my return to London, but I had also come because I wanted to spend time with Jean, whom I had really enjoyed meeting on my earlier visit. When I arrived back in New York, I found that Jean was in the throes of a romance. She had little time for me and soon decided to get married and leave Simon and Schuster. She bequeathed to me her sixth-floor walkup apartment on East Seventh Street between Second and Third Avenues, a few steps from Cooper Union. For $300 key money I inherited the contents of the apartment. (It included everything I could possibly need, plus a large library of paperback books. All she took with her was her coffee pot.) And the rent was only $37.24 a month—a steal even in those days.
From my kitchen window I had a view of the green-blue onion domes of the Ukrainian Orthodox church next door (alas, this lovely little church was pulled down to make way for the large new brick edifice for which the congregation was saving in the early sixties). This was a time of flux in the East Village. For years it had been the Ukrainian section of the city, but during the three years I lived there the face of the neighborhood changed almost unrecognizably. There had always been old bums drifting over from the Bowery in a drunken stupor, but now young panhandlers appeared asking for money to support their drug habits. I felt sorry for the old-timers, even though I didn't give them any donations for liquor. I would smile and shake my head and they would generally say, "God bless you anyway, lady." But I saw absolutely no reason why the nineteen-year-olds should expect the world to help them out.
When I moved in, friends advised me to take a taxi home if I was ever out alone after dark, so that no one would follow me into the tenement building. But time and again I would huff and puff my way up the six flights of stairs and collapse onto the bed, only to realize that I had once more forgotten to take a cab.
As the year wore on, people would ask me whether I really planned to return to London in November. I would look at Bob, he would look at me, and neither of us would say anything. The more I considered it, the more ridiculous it seemed for me to go back to England at the end of my year simply because I had said I would. I had no job offer, no place to live, and no pressing relationship to return to. I stayed. Bob never said anything, but several years passed and I thoroughly enjoyed my work with him. There was always something new to learn—manuscripts to read, authors to meet, books to edit, jacket copy to write. We all used to joke that working at S&S was like having an extra family you didn't need. If you were out sick one day, three or four people would call to find out what had happened and what they could do to help. I had never been part of a community like that before.
We all have different experiences at work, whether we are employed in a large organization or a small company or whether we work on our own. But the one thing that we share is the belief that the results of the work belong to us. It goes further than taking pride in our work. We put a claim on it. We are convinced that what we have produced bears our mark, that we deserve recognition, remuneration, advancement, whatever. But here is the truth about the situation:
All that lives is full of the Lord. Claim nothing; enjoy. Do not covet His property. Then hope for a hundred years of life doing your duty. No other way can prevent deeds from clinging, proud as you are of your human life.
For me, this quotation from the beginning of the three-thousand-year-old Isha Upanishad is one of the most powerful teachings I have ever encountered. It goes so much further than Jesus' teaching "Render unto Caesar the things that are Caesar's and unto God the things that are God's." It tells us that everything, no matter what it is, belongs to God, that it is all His property, that we cannot claim any of it. It also instructs us to enjoy the creation. This is a commandment, one that we fail to keep most of the time. The text doesn't say: "Try to enjoy things." It just says, "Enjoy." This view of work is similar to that taken in the Bhagavad Gita, where a great deal is said about actions, who does them, and who is entitled to the results. There Krishna declares to Arjuna that we have "only the right to work but none to the fruit thereof," and he urges him not to let the fruit be the motive for work, as it is for so many of us.
Some people may find this view difficult to accept. Surely, if we work hard and do a good job, we should be entitled to something? As Eartha Kitt used to sing, "If I can't take it with me when I go, I just ain't gonna go." But even she had no choice. Better to admit that we don't really own anything, that none of it—possessions, cash, reputation, know-how—is going with us on our journey into the beyond. We can have fun doing things, but the experience is fleeting. We can take delight in the results, but we can't say that we own them in any way. Once we admit this, a huge burden is lifted from our shoulders.
FIVE
Nothing Unnecessary
Some years ago I was invited to attend a conference on inner science at which His Holiness the Fourteenth Dalai Lama spoke. I listened to him elucidating Buddhist dialectics for three days and was for the most part unable to understand the content or direction of his argument; however, I soon became aware that his actual teaching—at least for me—was going on at another level. I noticed that whatever he did or said, he did with his whole being—whether it was laughing, talking, or just resting. Part of him was not doing something else. He was completely concentrated in the moment, and the power of his unsplintered attention was electrifying. Not only was all his attention given to whatever he chose, but mine was also. Since he was not distracted, neither was I. I left the conference in some amazement, never before (or since) having met anyone who appeared able to focus in this way. This teaching was a tremendous gift.
This morning I heard from an old colleague musing in a wistful tone about when he could retire from his job and just do one thing at a time. He felt completely overwhelmed and torn apart by the multitude of tasks in front of him. The truth is, you don't have to wait for retirement to do only one thing at a time. There really isn't any other way to do things. People who believe that they can do more than one thing at a time are just fooling themselves. If you split your attention among, say, three jobs, all you are doing is giving your attention to one of them, then leaving that one for a minute and moving to the next, and so on. I know it looks and feels as though you are doing everything at the same time, but look again.
Most of us are brought up to believe that it is advantageous to do as many things as possible at the same time; however, if we observe carefully, we will discover that this is not only undesirable but also counter-productive. Unfortunately, we not only believe that doing more than one thing at a time is good, but we have also drawn a veil over our activities so that for the most part we are oblivious to what is actually happening.
Take ironing, for example. You may remain aware of what you are doing while you are laying the garment on the ironing board. As soon as your hand begins to steer the iron over the garment, however, your thoughts are off and away. Ironing is one of the dreamiest activities. It is a useful exercise to give full attention to the ironing and see if you can spot the mind's tendency to wander off. Each time it does, gently bring it back to the task at hand. The ironing will be accomplished better and in less time if it is done without the mind doing something else. In addition, you will discover that ironing—or anything else, for that matter—is not a boring activity. Usually, what makes something appear boring is that we are not giving it any attention. When we give it our undivided attention, many details become clear—things that we would ordinarily miss—and the result may be intriguing.
Obviously, we can walk along the street and look into store windows as we pass them. However, we can give our attention to only one of these activities. If the walking is going along fine by itself, it is easy to look in a window. Yet, if we stumble, knock into someone, or hear a screech of brakes, our attention immediately leaves the store window and is drawn to whatever requires it. It goes there instantly, and the window is forgotten. Most of us walk down the street with our attention neither on the activity of walking nor on the windows of the stores we are passing; not on the architecture of the buildings or on the behavior of the drivers a few feet from us; not on the clouds or planes in the sky, and probably not often on the people walking alongside us or approaching us. We are simply lost in thought, rehearsing something that happened and we wish it hadn't, or something that we would like to happen, going through a mental checklist of things that need doing or people we hope to see—anything but giving attention to where we are and what is taking place. These mental conversations that we have with ourselves are generally not very fruitful because we cannot rewrite history, and if we are scripting a future conversation, the chances are that other people will not be aware of their cues when the time comes. And while this energy-consuming activity is taking place in our heads, the world is turning and we are missing so much that is taking place. It seems sad not to embrace the fullness of the moment in which anything may be revealed.
I once attended a press conference and watched the reporters scribbling furiously (it was in the days before people used tape recorders to do their work for them). Then I noticed one man sitting very still and not taking notes. He just listened. Afterward I asked him why he had not written anything down, and he replied that if he took notes, he would be doing two things at a time and would therefore undoubtedly miss something. He did not wish to split his attention.
I was thinking about this recently, in particular with regard to studying and learning, because our training has been to take notes on everything we are taught in school so that we can refer back to them later, and I wondered why we all do this. When we read an item in a book or newspaper, hear about an event, or discover an idea by ourselves, we have no difficulty whatever remembering it and reporting it afterward. This is because it is something that interests us, something that caught and held our attention. The trouble comes only when we are not interested in something: for instance, if we are taking a course and need to study something that is required and we wish it weren't. What happens then is that we tend to focus on our resistance to the matter at hand rather than whatever it is we are supposed to be learning. So the secret is to be interested in whatever we are learning or doing. Then the full power of our consciousness flows to it, it goes directly into the mind, and we have no difficulty retrieving it later. It does not have to be "learned." The learning happens naturally.
Until a short time ago, while I was still working in a big office, I would be sitting at my desk typing away when the phone would ring and I would answer it immediately. Before I could get back to what I was doing, someone would walk into my office to discuss something he or she felt was much more important than whatever I might be doing. Then I would check my e-mail to make sure that I wasn't missing anything. After I'd fired off three e-mails and answered the phone again and made a note to myself of something I had planned to do first thing in the day, I would remember that I had been in the middle of typing a letter about half an hour earlier. This is the way our lives tend to run. Even now that I work at home, it is no different. Yes, it would be wonderful if there were no interruptions, but even if you are alone with no phone and no e-mail, there are still your thoughts to distract you.
There is away to approach your work that takes all this into account. Usually while this maelstrom of distractions is taking place, we are dragged from one demand to another, trailing our attention behind us. By which I mean that we don't relinquish whatever we are doing when we are interrupted. You need to make a clean break. Just drop whatever it is. That way you are doing only one thing at a time, even if it is in small drafts. Carrying your regrets about not finishing the first task into the second and then the third is counterproductive. We have enough to do these days without bearing this extra weight. The important thing is to come to each job fresh, as though it were the only thing in the world. When the right moment comes, you will get back to the first task. Good luck.
Postscript: A few days after I wrote the last paragraph, I was cheerfully answering e-mail, dreaming up fact sheets on next fall's books for the sales force, receiving a fax, and trying to write some of my own book. I thought I had all my balls in the air, and then the computer crashed, the screen froze, and my options dwindled down to rebooting the computer or rebooting the computer. Alack and alas, the copy I had concocted for one of the fall books dematerialized because I hadn't saved it. The lesson to be learned from this, I suppose, is that it is okay to shift your attention from one job to another, but you must make sure that everything is in good order before you do so, whether this means putting the saw out of harm's way so that no one will trip over it or turning off the flame under the pan of milk before you leave the kitchen. One moment's inattention can cause a lot of heartache. Now to reconstruct that brilliant description that went the way of all flesh...
It is always important to be aware of what is going on in our minds because whatever it is, it is absorbing our energy and attention. We are giving ourselves to it. This constant activity, of which most of us are completely unaware, can be exhausting and wasteful of our resources. Whatever we give our attention to grows, so we should know what that is. It may be fear, loneliness, anxiety, or any number of things, but it behooves us to take a look. If what we had planned to do was take a walk, why not just walk? It could be fun, but if all we are doing is continuing an inner conversation that has been going on all day, the walk will not prove very refreshing.
Nowadays people tend to look for distraction, any distraction that will take their mind off whatever they don't want to think about. This tends to take the form of "entertainment." People will do almost anything in order to avoid being where they are, doing whatever it is that needs to be done. Sometimes they are seduced (by themselves) into thinking that whatever anyone else is doing is bound to be more interesting than what they are doing. This assumes many guises, but all of them make us restless and discontent, unable to settle down to what we really need to do, and usually unable to enjoy it.
There is one small rule that can be of enormous benefit to us, not only when we are engaged in work but also throughout our lives, and it is this: Do and say nothing unnecessary. In order to observe this rule, we need to remain in the present or we will not be able to tell if something is necessary, so right from the start it can be seen that this is a useful thing to do. This maxim also implies that we will do whatever is necessary to accomplish the task at hand, while giving up what is not germane.
One of the places where it is important to recognize what is necessary and what isn't is in our own homes. I remember reading a proposal for a book about this many, many years ago. It suggested that you sit down and make a list of the things that you actually do at home most of the time and then plan your furniture and its arrangement around them. It pointed out that many people buy a three-piece suite simply because they believe it is the thing to do. But how many people actually sit on a sofa at any given time? Usually one. People feel a little crowded if they have to share a sofa with someone else. So in my living room I have two small love seats rather than one big sofa. Love seats are more intimate and more practical. I like to put my legs up when I read, so I have love seats with firm, straight backs to support me, and when I have guests, I encourage them to take their shoes off and put their legs up just the way I do.
I am lucky enough to have an eat-in kitchen with a round table and two rush-seated chairs. Because I generally avoid giving dinner parties and prefer to invite only one person at a time for a meal, I rarely use the dining room table, so I've relegated this to the far end of my living room. Luckily it is one of those tables with fold-down leaves, so it takes up very little room when it is not being used.
Every morning I do half an hour's yoga in ten minutes (this is New York, the world capital of fast living) and twenty minutes of meditation. I therefore keep my huge carpet free of furniture so that there is plenty of space for these two activities. I don't want to feel closed in or cramped for either one. In the bedroom I have very little except a double bed low on the floor, a small bedside table with lamp, clock, coaster for my nighttime glass of water, cough lozenges (just in case), pad of paper, and pencil. Over by the window there is a ficus tree and a collection of prayer plants nestling next to a couple of African baskets filled with sweaters I don't often wear. A plain rosewood chest of drawers, a standing lamp, a love seat for draping my clothes on when I undress, and a thick, cream carpet to welcome my toes when I get out of bed in the morning. On the wall above the chest is a rack for earrings so that I can see everything at a glance, and beside it an almost floor-length mirror. All the hanging clothes are hidden away in the wall closet (shirts on one side, pants in the middle, dresses and skirts on the other side). I have to admit that there is also a spinning wheel that I haven't used in twenty-five years, plus two baskets of fleece, spindles, and carders. I can't bear the thought of surrendering these non-necessities that I plan to incorporate back into my life one day. It's the exception that proves the rule.
What each person needs in a particular room of the house will differ, but the trick is not to take into that room anything that isn't really necessary. "Necessary" will, of course, include beautiful things and not just useful ones, but the fewer of (each of) these there are, the better. Avoid clutter of all kinds.
About clothes: Do not be tempted to decide what you are going to wear tomorrow today. For years I would take out of my closet whatever I thought would be appropriate for the next day before I went to bed each night, under the illusion that it would save me time in the morning. To begin with, you have no idea what the weather will be like tomorrow. How reliable is the weather forecast? Yes, it may rain tomorrow, but when and where are always a mystery. More important is the fact that emotional weather also changes. There are some days when I dress in pastel shades. On others I gravitate toward browns and rusts, and so on. For some years I would arrive in the office each day and find to my astonishment that one of the production editors was attired in exactly the same colors, even though we were not close friends. Something in us was resonating. On the rare occasion when I was wearing red and she was wearing blue, we would joke about it. So the thing to do is go to the closet first thing in the morning with no fixed idea in mind and see which garments call to you. That way you will be in tune with the day and you won't have to put away a whole other set of clothes that you took out the night before but just don't seem right now.
When I was still working at Knopf, I used to receive a flood of free books, some of which we published and some of which were gifts. Still, I tried to take home only those books that I truly believed I would read. The others I returned to their donors or found other willing recipients for. And, once I have read a book, if I think that I will never reread it or refer to it again, I find someone else who would enjoy it.
I endeavor to do the same thing with my clothes. I go through my closet periodically and try to be honest with myself. If I haven't worn something in the last year, the chances are I won't wear it in the next year either. So I either find a friend who would like it or give it to a thrift store or emergency relief. I suspect that dwelling in Manhattan makes it easier to live this way because of the paucity of storage space. Most city dwellers have a hard enough time finding a home for the things they use every day. Still, it is remarkable the objects that creep into your apartment. I currently have an old phone I am kidding myself I might need if both of my current phones stopped functioning at the same time. I also have a microwave oven my son hardly used at college. I have never used a microwave and hope to go to my grave without doing so, but I have not yet managed to bring myself to take it across the street to the Salvation Army. Perhaps someone I know would welcome it?
Not introducing unwanted possessions into your home and/or surrendering those that are not being used creates not just physical space but also psychological space. People often comment on how empty my apartment is, and I think this is because it feels free and clear. There is very little in it (apart from the phone and the microwave) that is just gathering dust.
One major thing I don't bring into the house is the newspaper. Six weeks after I started work at Knopf, there was a big piece in The New York Times about the circumstances of Bob Gottlieb's hiring. I was shocked not by what was included in the article but what was not. The major facts had been left out; therefore, the whole report was skewed and, it seemed, malicious. This was an event that I knew about. Thereafter, I hesitated to read accounts in the newspaper of things of which I had no prior knowledge because I wouldn't know what to add in to make the picture accurate. I stopped reading The Times, and people continue to be amazed by this. As someone said to me the other day, "Not buying The New York Times represents a level of simple living far beyond anything I've considered." It's not that I am completely uninformed; I do listen to the news on the radio in the morning and watch it on television at night. And there are always friends who press upon me clippings that they have saved. But can you imagine the time, money, and energy I have saved over the last thirty-two years?
When I leave town for any reason, I have got into the habit of surrendering my apartment to a friend or an acquaintance. I don't go out and look for anyone. What happens is that a week or so before I am leaving, I hear from someone who is hoping to find accommodation for a short period. Sometimes it is a person I know well, but often it is someone I know only by reputation. I don't charge rent. I am just happy that the space is being occupied and enjoyed. And I like the idea of someone being there in my absence to deal with any emergencies that might arise (but don't). In a way, it is odd to have guests I can't entertain personally, but I believe that they appreciate what my little home has to offer, and they will return to share it with me sometime in the future. I am also always touched by the gifts I find on my return: a bowl of tall white narcissus, a loaf of crusty bread, a new brush for the toilet—whatever each visitor perceives is my need.
As I was walking home along Broadway today, I saw a sign outside the corner store announcing that it was closing and that most of the merchandise was being sold at 30 percent discount. I was on my way to "my" Korean store to buy butter, but it occurred to me that the reason the store was closing was perhaps because it was in financial difficulty and I might be able to make a small contribution by purchasing anything I needed there rather than where I usually go. So I bought the butter (which wasn't reduced), plus some dried cranberries, and a jar of artichoke hearts (okay, I'll admit I also bought two bars of Toblerone because I've been experiencing a recurring desire for chocolate). I looked around at the array of other foodstuffs, but the truth is: I didn't really need anything else. For a moment I was torn between wanting to save money and realizing that there wasn't anything else in the store that I was likely to use in the next year or so, so I wouldn't really be saving any money if I bought other things. The things I purchased are things I will definitely eat over the next few weeks. I emerged unscathed, happy that I had saved a little money of my own and contributed to their cash flow while avoiding the trap of tying up my money in unnecessary goods.
All this talk of what is unnecessary raises the question: What is necessary? A very big question. I once gave a talk to the twelfth grade of my son's school, and the teacher who introduced me asked how I would describe the kind of books I published. "Necessary books," I replied. "I have to believe that the world really needs a book before I will take it on." This caused him to ask me to write an essay for the school magazine on "What Constitutes a Necessary Book?"
At the time (1985), I consulted my dictionary and discovered that necessary was defined as "needed for the continuing existence or functioning of something; essential; indispensable," and to those definitions I added "useful; a tool." On reflection, I realized that what I try to do is supply readers with something vital to body, mind, or heart, and what I see as my function is to supply this something in its most appropriate form to all those who really need it. When someone picks up a book, the title and subtitle, the color and design of the jacket, the copy on the flaps, the choice of typeface and interior design, the paper, and the binding should all inform the prospective reader of the writer's intent. I believe that the physical aspect of each book should be a reflection of everything that is within it. Because I am writing this in 2001, I must add one more thing: In publishing during the 1980s all you had to do was produce the book and float it out into the bookstores where it would find a ready audience. Now that so many more books are published each year by companies large and small and even by individuals, it is no longer enough to manufacture books; you also have to find a way of bringing them to the attention of customers, which is a much more difficult proposition. But if you cannot do this, your books will have no chance in the marketplace, and all your effort will have been for naught.
The principle behind all this is not to waste precious energy where it is not needed. We do tend to fritter it away if we are not aware of what we are doing. Sometimes this takes the form of a nervous tapping of the foot or drumming of the fingers. I notice that television cameras often focus on people's hands and what they are doing while they are making speeches or testifying, so I am not the only person who has spotted this. I catch myself walking down the street with my hands clasped in front of me. Usually I am walking in order to get some exercise, and the best way to achieve this is not to restrict the body in any way. Let the arms swing. I don't know how I started this unfortunate habit, which creates tension rather than relaxation and has the opposite effect from the one I am hoping for, but perhaps by keeping an eye on it now, I will be able to let go enough to just walk and not clutch.
There is one thing I must record here because it was the impetus for this book: When Joel suggested that I write this book, I asked him what would be in it, and he said, "Well, when you go to make yourself a cup of tea, you just boil one cup of water." "Doesn't everyone?" I responded. "Of course not," he said. "Well, then, they are wasting not only water but time and energy too." He smiled and nodded. Since then I have spoken to quite a few people about this. It turns out that I am not the only person who fills the kettle with exactly the amount of water I need, but there are enough people who have never even given it a thought for me to tuck this advice into the book. Small note: It's not that I measure the water in the kettle exactly, but from years of experience I know how heavy the kettle feels when it has enough water in it for one person.
The other day I went to visit a friend with two small children. After an hour, I had had enough and was about to leave when I realized that this had been a very small respite in her day. It was possible for me to offer something I had completely overlooked. She was exhausted but hadn't mentioned it, so I asked if she would like to take a nap while I entertained the small boy (the baby was already asleep). I had been so caught up with my own agenda (did you know that this Latin word means "doing"?) that I hadn't spotted how depleted she was.
Asking yourself what is truly necessary can make an enormous difference in your life. Ask it in all kinds of circumstances—when you are tempted to criticize or gossip but also when whoever you are with is silently crying out for something and you are not noticing it because you are filled with your own thoughts.
SIX
Do It Now
In the late spring of 1999 I attended the World Sacred Music Festival in Fes. On our first full day we set off on a tour of the city (which, according to the guide, was "the third crowdiest city in Morocco") and then entered the medina—the old walled city of Fes el Bali, bustling with markets and life. It is reminiscent of the souks in the Old City of Jerusalem but on a scale that is breathtaking. There are fourteen gates to this medina, and the narrow, labyrinthine passageways (there are said to be a thousand blind alleys) afford no access to cars or bicycles, so people go about on foot, and merchandise is transported on the backs of donkeys and mules that make their way hither and yon at an alarming pace. On their backs or in panniers on either side are huge loads. The men or boys who lead them or are perched on top of the packages yell out to warn everyone to get out of the way, and people flatten themselves sideways or get trampled underfoot. This is no place to become drowsy. Visitors are advised not to wander around on their own because even Fasis (inhabitants of Fes) get lost sometimes. I don't know how large the medina is, but I was told that it would take a whole hour to walk from one side to the other. Inside the medina are not just booths but workshops of every craft, medersas (koranic schools), mosques (there are said to be 785 in the whole of Fes, and 360 of them are in the medina), dozens of palaces, fountains, and many-chambered restaurants. And, of course, thousands of residents.
We were taken to a carpet shop that had once been a koranic school. It had very high ceilings and intricately carved white pillars, with carpets hanging on all the walls. The building was so high that there was room for three carpets, one above the other. First we gathered in the central area and were offered the proverbial mint tea. We were treated to a display of rugs from different parts of the country, made by various tribes. Two men would unroll each rug with a flourish as the third man described what we were looking at. The weavers use natural dyes; saffron for yellow, bougainvillea for peach, indigo for blue, poppy for red, and mint for green. Afterward we were led away to individual nooks where more tea was served and we were pressured by a salesman into buying whatever our hearts desired.
I certainly had no intention of buying a carpet, but within twenty minutes I found myself settling on a magnificent thick cream Berber rug with a traditional design. I picked the colors of the decoration and before long was signing away my money. A lot of haggling went on, both about whether the insurance was included in the price of shipping (as had been quoted to me) and whether they were going to charge me an extra 6 percent for using a MasterCard (I refused). The carpet would either arrive in "four to five months," or I would learn the hard way not to trust carpet salesmen. The guide who took us there (and who gets an enormous cut of the profits) said that he guaranteed I would receive it, and I certainly had his name and address if something should go wrong, but he lived a long way away and I could hardly go around to his house to complain. Five months later the carpet did arrive at my apartment on the Upper West Side of Manhattan. It was rolled up and sewn beautifully into a protective sheath. Not only was it a miracle that it was finally there, but it also turned out to be bigger (the difference between meters and inches had been beyond my ability to calculate) and thicker than I had anticipated.
That was almost a year ago, and I have received untold pleasure from the carpet just by gazing at it and squishing my toes into it when I get out of bed in the morning—until two days ago, when I walked into the room and detected some movement out of the corner of my eye. I walked over and saw that there was a cluster of small brown moths fluttering just above it. All my good Buddhist intentions about not taking life fled. I began to pounce on the moths and speed them onto the next life by grinding them between my thumb and forefinger. They didn't put up much resistance. When I could see no more activity, I retired to the kitchen to ponder what I should do. Where there was one moth, there was always the possibility of future moths. Later in the day, I found several more on another corner of the carpet and dispatched those to kingdom come also.
I went out to the hardware store searching for a solution. There were "old-fashioned" mothballs (I bought a box of these) and a can of lethal spray (which I decided might kill me as well as the moths, so I left it in the store). I took the mothballs home and read the directions. It seemed that these too were hazardous to your health, and I worried about sleeping a few inches away from them. So I deposited the unopened box on top of the place where I had found the first colony of moths and hoped that the fierce smell emanating from it would be sufficient to deter any more eggs from hatching. Yes, I know that this wasn't a very reasonable attitude, but the instructions on the box said to place the "garment" with the mothballs inside a sealed container. Since my "garment" was larger than any container I owned, I wasn't sure how to proceed. Also, my friends were all urging me to call a carpet cleaner instantly and have the problem solved professionally. My experience with carpet cleaners (I have employed two in the past) has not been good, and I couldn't bring myself to heed this advice. You see: My luscious rug has a one-and-a-half-inch pile. It is really fluffy and shaggy, and I was afraid of what else the carpet cleaners might do to it when they deinfested it. It might never be the same rug again (as had happened when I had called in the cleaners before. Since then I bought a carpet-cleaning machine and do the work myself).
When I got up this morning I decided to take things into my own hands and see if I could effect a cure. First I vacuumed the carpet as thoroughly as I could. Then I took a stiff-bristled brush and, with the aid of my fingers, raked the pile as deeply as possible, skimming off handfuls of loose fluff. I vacuumed again. Then something made me turn over a corner of the carpet and inspect the other side. There, to my horror, was a host of tiny white eggs clinging to the back of the carpet. More brushing and vacuuming ensued. I repaired to the other site where I had found errant moths, and there were more eggs. Eventually I rolled back the entire rug and brushed until I could detect no more small white specks. Actually, I didn't find any more, but I decided to brush and beat the rug as much as I could in case some of them had been laid a little farther from the original site.
As I was clearing up, a vivid memory popped into my mind: The day I had unfurled the carpet and laid it lovingly on the floor, it had occurred to me that perhaps I should brush and clean the underside before I put it down, but then I had ignored that little warning voice. And I had paid for my "ignorance," perhaps not in spades but certainly in moths.
There have been far too many times in my life when I have ignored that voice, and I don't know why I keep doing it. There are also plenty of times when I have obeyed it, so why don't I do this all the time? It is obvious that it is better to get something right at the beginning than undo a mistake further along in the process. We know this deep down inside, but then there is our rational mind that tells us that the information being supplied by our sixth sense is "unreasonable." Believe me, it is not. If it occurs to you to do something now, just do it. Once it is done, then the weight of it, the responsibility, is released. When you see that something needs to be done, the energy to do it arises at the same moment. The longer you put off doing it, the more that energy is dissipated and the larger the task appears to be. Eventually, a task can loom so large you are unable to contemplate it. The thing to spot in all this is that it is all the extraneous thoughts that crowd into the mind unnoticed that obscure what needs to be done. These trains of thought naturally carry us away. Trains of thought, like any other trains, are designed to carry us to destinations that are anywhere but here.
Which brings me to a mini-law that has governed my life for many years. In Sanskrit these mini-laws are known as sutras, short, pithy sentences that embody the essence of a universal law. Here is how this one was presented to me: "Step by step, at every step, there is that which is conducive to each step," but you could also translate it as "Whatever you need is right here." This applies not only to the energy necessary to complete a job but also, often, to the wherewithal to do it.
So much of our time is taken up in worrying about how we are going to cope with things that have not happened yet and may indeed never happen. At every moment things shift, so even if you could work out the "right" action to take at some point in the future, the situation would have changed in many ways by the time that moment arrived. The only thing we need to concern ourselves with is the present moment.
This has become exceedingly clear to me from the minute I sat down to write this book. To begin with, I almost panicked at how blank my mind was. How would I find enough things to write about for a whole book? But one day as I was walking down the street on an errand, it occurred to me that I didn't have to know the answer to that question then. I couldn't write a whole book immediately. Just like everyone else, I could write only one chapter... No: one page... No: one paragraph... No: one sentence... No: one word at a time. The thing to do was get started and see what appeared on each page. So this is what I am doing. I have stopped worrying about all the other chapters I may or may not write. I am just typing this sentence, and right here at this moment there is everything I need to be able to complete it.
This is really the way it is with everything else, too.
I had arrived in New York City at the ripe old age of twenty-five with my virginity still intact. I discovered to my dismay that other people found this amusing. "What were you saving it for?" they asked. In England I had defended myself against all comers (pun not intended), certain in the belief that my future husband, whoever he might be, would prefer it that way. Now I wondered how I could divest myself of this embarrassment as quickly as possible. There were certainly plenty of prospects. Young men invited me out all the time, but they seemed to expect that if they paid for the dinner I would sleep with them afterward, and I never liked any of them enough to reward them in this way.
Eventually a nice young man did turn up. I cannot recall the details of the dinner that must have preceded it. What I remember is that the bed broke under us. It wasn't exactly the way Hemingway described the earth moving under his fictional lovers. The bed, apparently, had a habit of falling apart. For me the experience (of losing my maidenhead) was quite painful and not very enjoyable, but at least and at last the deed was done. Either the young man in question was turned off by my inexperience, or he met someone else the next week. History doesn't relate. But in a curious turn of events, he became my divorce lawyer many years later when I bumped into him and found out that he lived across the street from me. I don't have much faith in lawyers, but I felt that because I had once entrusted him with my body, he was probably safe with my divorce proceedings.
A friend called me the other day. She had been for two job interviews and was worrying about what she should say if one place offered her a job and she hadn't heard from the other but would prefer to work there. This seemed like a nonissue to me. I pointed out that no decision had to be made yet since neither organization had made an offer. When the decision needed to be made she would know exactly what to do, but at the time we were talking any decision would be irrelevant. Many weeks passed, and neither job ever materialized. People waste a lot of time agonizing over something that may never happen.
At one time I was working on a book by Lanza del Vasto, then the leader of the nonviolent movement in France. The translation arrived from him many weeks late, and I had very little time before I had to hand it into production. So I worked all weekend to edit the manuscript. On Monday I wrote to him with various questions. He wrote back very upset. He felt that because I had done all the editing in two days, I could not have been very thorough. "Sometimes," he said, "I agonize all night over just one word." I was nettled. I had had to make up for his tardiness in my spare time, and here he was complaining that I could not have done a proper job so quickly. I responded by saying that I never agonized about just the right word because I didn't believe that agony was helpful. If the right word doesn't come to mind immediately, I simply put a penciled query beside it and come back to it later. By then I will probably be able to come up with what is needed. This method has always worked for me. I don't like agony.
I tend to size up a situation as quickly as possible and then make a move. One of my favorite Michael Crichton dictums from his little-known book, Travels, is, "Don't expect other people to read your mind. Read your own mind."
After I had worked at Simon and Schuster for a little while, I asked for an electric typewriter to replace the old manual one I had inherited. Even at a small, impoverished British publishing house like André Deutsch, I had had an electric typewriter. The office manager said that one would be forthcoming, but the months passed and nothing happened. Eventually, I simply went on strike. She marched up to my desk and informed me that I couldn't go on strike because I didn't belong to a union. "But I've already struck," I said. "I have plenty of work to do that isn't typing: filing, reading, etc. When the new typewriter arrives, I'll start to type again."
"But it will take time to get a new typewriter," she responded.
"It's all right," I said. "I'll wait."
An electric typewriter appeared miraculously the next day.
And then there was the time when there was trouble over the dress code. In the mid-sixties people had started wearing miniskirts. This was in the days before tights, and my desk was at the end of a very long corridor. The way the desk had been designed left my legs and much else in full view of the world. This was a very unfortunate situation to be in, and I decided to remedy it. I went out and bought a pantsuit. It was a navy blue wool suit with brass buttons, and I wore it with a long-sleeved, high-collared white ruffled shirt. I looked demure but efficient.
Immediately the publisher sent for me. There had been a complaint from the president's office (from his middle-aged narrow-minded female assistant, actually).
"We don't want you to wear that outfit in the office again," he said. "It's not appropriate."
"It's far more appropriate than what all the other women are wearing," I said. "At least I'm not revealing parts of myself that are best kept covered in public."
"If you were working in a bank, you wouldn't be allowed to wear pants," he retorted.
"But I don't work in a bank, Peter. I work in a publishing house. Surely the only person who might have some cause to object is Bob, and he doesn't mind at all."
"You are just being a prima donna," he said. "Are you planning to wear this suit every day?"
"Of course not," I replied. "I never wear the same thing two days running."
"Will you promise me that you won't wear it to the office in the future?"
"I promise you that on the days when I wear this I will not come into the office," I said and turned on my heel and walked out of the room.
Sometimes when you do things quickly, other people get alarmed. After I had been in New York for many years, a handsome young man asked me to marry him. I had had offers before (all in England) but had never felt tempted. It seemed to me that you had to be very certain of something like this before you embarked on it. But this time I was not in any doubt. There was something about Neil Lippe that I recognized. It was as though we had known each other for eternity. It came about like this.
When I first saw Neil I was immediately attracted to him, but something inside me warned me that if I went in that direction, I would get burned, and so I looked away. He told me later that he had looked at me and felt the same strong pull but decided to wait before he did anything about it. He was in his early twenties and not yet ready for any kind of commitment. This was a playboy of the Western world, experimenting with everything that came his way, and I was a rather straitlaced Englishwoman of almost thirty.
Circumstances brought us together again about seven years later, and after just one date, Neil proposed, I was very taken aback and asked if I could have a few days to think about it. But the more I thought about it, the more I realized that there was really no decision to make. The memory of our connection was ancient, and one evening when he was an hour late arriving at my apartment and I began to imagine him under a bus, I accepted, telling him that I had discovered that I couldn't live without him.
Bob Gottlieb did his best to stop me.
"Why are you rushing into this?" he asked. "Maria and I lived together for five years before we got married."
"But that was you and Maria, and this is Neil and me," I responded.
"Marriage is hard," he said. "Take some time to think about this."
"I know that marriage is hard. But life itself is hard. Why not get married and share the difficulties? I know the difference between a wedding and a marriage. I understand that the wedding is just the beginning."
Almost everyone else also did their best to dissuade me. Neil had made no secret of the fact that he had had many gay relationships, but I knew that he had also had two long-term affairs with women and that none of his liaisons with men had lasted as long as his three-year relationships with women. He wanted to put all that behind him and settle down with me. I thought that this took tremendous courage and that with his talents, which were many and varied, and my strength, we could go far together. I had no hesitation in saying yes, and the more people tried to dissuade me, the more we wanted to go ahead.
A few years later Neil came home one evening and said, "In my first year of business I have made three thousand five hundred dollars, so now you can have your baby." (This was in 1977 but, even then, this wasn't a large sum.)
"Three thousand five hundred gross or net?" I asked.
"Gross," he said.
"Well, that's not a great deal, but if everybody waited until they could afford a baby, none would ever be born. Still, it can't be 'my' baby. I need help getting it started, and I'll need even more help once it arrives. Also, if you don't want a baby, we shouldn't even contemplate this. It has to be 'our' baby, not 'my' baby."
"I want a baby, too," he said.
And so that night I conceived a child. Again, there were no fireworks or other epiphanies (unfortunately). You may wonder how I can be so sure that the seed was sown that night. What I heard was that this was the moment to get pregnant, so immediately I became open to the opportunity. Also, I had had an infection for the previous ten days, and so there had been no possibility of conception until that moment. Neil said later that he prayed that night that we would have a child who would love God. That may be true, but he didn't mention it at the time.
Exactly two weeks later I started retching. To begin with, I thought it was a recurrence of the hepatitis that I had contracted earlier from Neil. He hadn't realized that he was a carrier or even that he had had hepatitis at all, but it turned out to be in his blood. I was sick from the day of our wedding for many months, and I believed at the time that if our marriage could survive that illness, it could survive anything. On that score I turned out to be wrong.
When the nausea didn't go away, I went to a friend's doctor, and he performed every test under the sun and could find nothing wrong. After two visits and several large bills, I called from my sickbed to say that I still couldn't stop throwing up. I thought that I might be pregnant.
"Why didn't you mention it before?" he asked tartly.
"Doctor," I said, "if an unknown woman comes to you and she hasn't kept any food down for days on end, surely pregnancy is always a possibility? Even if I were not married, there is still such a thing as immaculate conception. It's happened before and it could happen again."
"When did you last see your gynecologist?" he asked.
I told him that it was several weeks earlier but that it had been before I thought I might have conceived.
"How are your breaths?" he asked—or I thought that was what he said. It turned out that he had said "breasts." What on earth did he mean? Apparently, he wanted to know if they were tender or different in any way, but they were not. So he instructed me to come in the next day and bring him a urine sample.
When I arrived in his office, he looked in disdain at what I had brought him and said that there was not enough liquid to test. I told him that he was lucky there was something in the bottle, since I had not managed to keep anything down for days. He gave me an internal exam and said that I had no visible signs of pregnancy. Then he drew some blood and told me to call back a few days later.
When I called, he told me that the result was positive, but he sounded doubtful.
"So I'm pregnant," I announced.
"I'm from Missouri, the 'Show-Me' state," he said. He recommended that I wait a couple of weeks and then go and see my gynecologist again.
"Doctor," I said, "you have known women who have tested negative and then turned out to be pregnant, but not the reverse. The only thing that might cause you to think that I was pregnant is a change in my hormone levels, and you've tested me for that and everything else and you could find nothing."
So I telephoned Neil and said, "Either you have very strong squiggles or you have good aim or both."
"What do you mean?" he asked.
I announced that we were going to have a baby. I think we were both very shocked. It is one thing to decide to get pregnant, and another to have your wish granted a few hours later. I was thirty-eight years old and had never been pregnant before. I had thought that we might have to try for a couple of years before anything happened, and I wouldn't have been surprised if it had turned out to be already too late for me.
In the first couple of months I went from 112 pounds to 100 pounds. I felt queasy for the whole pregnancy, but once my obstetrician threatened to put me on an IV I stopped vomiting (he said that this threat always worked with his patients). And then I ballooned. By my seventh month I was up to 152 pounds, and all of it was in the front. I lumbered along the street with a watermelon in front of me. I found it impossible to sleep at night because it was then that the baby became active. But it was possible to sleep for an hour at four o'clock in the afternoon. And because I preferred to be horizontal at the moment I fell asleep, I left work each day at 3 so that I would be lying down when 4 p.m. came. Not being able to sleep properly for so many months did teach me one thing: You don't actually have to sleep deeply at night (although it is much nicer if you do), but you do need to rest your body. So if you lie there and rest, a certain amount of refreshment takes place. The main thing is not to fret about the fact that you can't sleep. It is the fretting that makes you feel so terrible. This discovery has stood me in good stead on the very few nights since that time when I have not been able to sleep.
I had no prior experience of taking care of babies, but once Adam arrived, I did not give my inexperience a second thought. I just proceeded from one stage to the next, and somehow the solutions presented themselves naturally. I felt as if I had always known how to hold a baby and care for him. Perhaps I was lucky enough to have an easy child, because I don't remember any disastrous occasions. However, I often found myself puzzled by the attitudes of other parents I met, such as the young couple whose two-year-old insisted on trying to stick his fingers in every electrical socket in our apartment. I kept darting forward to prevent catastrophe, and they just stood by and watched as I told the little boy in no uncertain terms to cease and desist.
"When do you start discipline?" the mother asked. "Is it time to begin?"
"You start from the moment a child is born," I answered. "And you never stop. It is a little late already, but I suggest you start now."
It is usually not a good idea to wait around wondering when to begin something. If an idea comes to you, it is probably because it is the appropriate moment for it. The thought has presented itself because it is needed. See if it can be implemented now rather than letting it molder or fester in the back of your brain. I have found that if I let things languish, they either start to weigh me down or they never come to fruition at all.
SEVEN
The Company We Keep
Human beings are gregarious by nature. We like to spend time alone, but most of us prefer to spend more time with other people than by ourselves. It is our nature to share—a meal, a hug, a conversation, a time and place. I know that just having another person in the room or in the house keeps my consciousness more alert. I can sit quietly all evening with no need to talk or exchange anything out loud, as long as there is someone else there. Now that I am usually alone in the evenings, I find it difficult to maintain the attention necessary to do all I would like to do and often fall asleep over a book or manuscript.
When I visited China, I noticed that I was often unwilling to do certain things unless I had company. By which I mean that I was willing to enter the subway in Shanghai without having any idea how it worked, where it went, and how to pay (and, of course, I do not speak Chinese) simply because I was exploring with another woman. I don't think I would have had the nerve to do this alone. We often draw courage from someone else and can be terrified on our own.
A few years ago I was staying in a wooden cabin in New Mexico, and in the middle of the night I was awoken by the worst thunderstorm I had ever experienced—two whole hours of forked lightning and thunder directly overhead, all reverberating around the wooden walls and visible through all fifteen windows. I was completely terrified. There was nowhere to hide. There was no phone, no lightning conductor, and my hosts in a trailer a few hundred yards away were too far away to hear me scream (so I didn't). I wondered if they would find me in the morning—fried. Of course, I did survive to tell the tale, but I know that my fear would have been less if I had not been alone. What is it that we draw from other people that is so comforting? It is not that we think that they are braver or more capable than we are. Why is there so much strength in numbers?
Not only does the company of others give us the courage to do things we would not do on our own, but we also do all kinds of things for them that we would not be willing (or able) to do for ourselves. I am not just thinking of stories of mothers who have lifted cars off children trapped beneath them.
Recently I joined a small "sitting" group that meets once a week. We sit for three-quarters of an hour. I used to meditate twice a day for half an hour, but for the last few years this has crept back to twenty minutes in the morning and nothing in the evening. Nowadays, after twenty minutes my body has programmed itself to get up from the cushion. So I wasn't sure that I would be able to manage three-quarters of an hour comfortably. I was curious to see what would happen.
The first evening I was fidgety. The following week I settled down immediately, and when the bell sounded at the end of the session, I had a hard time surfacing again. It has continued thus ever since, and I am grateful. Perhaps the reason I am able to sit there longer has to do with not losing face. I don't want to disturb anyone else by getting up after twenty minutes. But I think there are two other factors at work: If someone says "three-quarters of an hour," then I don't question it. The second harks back to what I said earlier about there being strength in numbers. If the other people in the room can do it, well then, so can I. It is a question of meeting the challenge.
One interesting facet of what happens when we are in the company of others is the shifting roles we play. My brother Chris and his middle son recently went on a safari in Africa. The group consisted of six teenage boys, a father, and a guide. Everyone ate and slept outdoors and pitched in with all the chores. I was privileged to read the communal diary that was kept, and the entry that struck me the most was by one of the boys who, after a couple of days of following behind the guide, was asked to be the leader. Instead of relying on a professional to be on the alert and not step on a snake or disturb wild animals, the young man found himself responsible for the safety, indeed the lives, of all his companions. Suddenly all his antennae were out, his senses alert. He went from being rather bored and caught up in his mosquito bites to this new and wondrous state because of his different point of view. He was traveling along the same track in the high grass, but it was now a matter of life and death. He found this enormously empowering. His whole mind and body rose to the challenge. He didn't make a single movement without being aware of what he was doing. He wrote that taking the lead forced him to remain in the present without letting his mind wander off into dreams.
Living alone tends to make you a little selfish. In some ways, you close down rather than open up. Because you do not have to accommodate anyone else, you arrange everything just so (particularly if you happen to be me). This was brought home some years back when I invited someone to live with me. The first evening after he moved in, I returned from the office and he was already sitting in the living room watching television. My hackles rose. For longer than I could remember I had come through the door and decompressed gently. I didn't want to talk to anyone for a little while after I had got off the subway. I just wanted to be quiet, to put my things down, open my mail, make a cup of tea, take my shoes off, and collapse on the sofa. Only then did I feel up to communicating with other people.
That first evening I watched how I felt, and I didn't say anything. After all, I was the one who had invited him into the house. He came, I realized ruefully, with his own habits, and I would need to welcome those too. But the second night it upset me even more, and so I told him how I felt and asked him if I could have just five minutes' quiet. This was all I needed. After that, he was nice enough to turn the television off as soon as he heard my key in the door and ask me later if it was okay to turn it back on. It is always the little things that get on our nerves.
As I said earlier, Neil and I went out on only one date before he asked me to marry him ten days later. I suspect that it was something he said that first evening that convinced me to say yes. We were eating dinner at his favorite French restaurant, and I was regaling him with my customary patter. For some time he watched me without saying anything. Indeed, I don't think he was really listening. He was just observing me. And then he leaned forward and said, "Why do you always hold yourself back?"
I stopped in my tracks. No one had ever noticed this before. Indeed, I had never even acknowledged to myself that I used my ability to tell stories as a kind of articulate armor that would protect me from all slings and arrows. I didn't know how to respond, but I was grateful that someone had at last spotted this. I felt both uncovered and discovered—disarmed but acknowledged. This was a level of honesty I had rarely encountered. I suspect that I hoped that this would be just the first of many revelations. Here was a man who would act as my mirror, reflecting back to me the things about myself that I needed to know.
Unfortunately, this was the only occasion on which he did this. Still, it provided enough of a shock to show me that my defenses needed to come down fast if I wanted a true meeting with another human being. I became aware that it is not possible to meet anyone if you are not available. Many of us try to protect ourselves in this way, but we need to remember that walls keep out not only our enemies but also our friends. Since that time I have done my best to be available to anyone or anything that needs me. I don't always manage it, but I try.
This word available is an interesting word. It turns out that avail, meaning "to be of use, assist, or help" comes from the Latin valere, "to be strong, to be worthy." This is remarkable, in that when I was younger, I wanted to be useful, and now I try to be available. I had thought that they were two different concepts but now that I have consulted a dictionary, I see how they are related. Also, I understand why being available to others carries with it such a sense of strength and value. It is right there in the root of the word.
When I began working at Knopf, it was as Bob Gottlieb's assistant, but gradually I took on other roles as well. For one thing, I started to sell reprint rights, which are the licensing rights to reproduce a company's books in paperback and other formats. It was a given at Knopf in the early 1970s that anyone could edit books, provided they didn't stint on the job for which they were being paid. So, although I had no degree, I soon found myself acquiring books that interested me, particularly books of spiritual teaching. As my small list of books and authors grew, I saw them as providing "good company" for the people who read them. Plotinus expressed it well:
For the absolute good is the cause and source of all beauty, just as the sun is the source of all daylight, and it cannot therefore be spoken or written; yet we speak and write of it, in order to start and escort ourselves on the way, and arouse our minds to the vision: like as when one showeth a pilgrim on his way to some shrine that he would visit: for the teaching is only of whither and how to go, the vision itself is the work of him who hath willed to see.
After I had edited fifty books, I asked if I could start an imprint at Random House devoted exclusively to spiritual books, books that nourish the soul, illuminate the mind, and speak directly to the heart, and in 1989 I was given the go-ahead to create an imprint as part of Harmony which already published books of a similar nature. When the question of a name came up, I suggested "In Good Company," my idea being that each title would be "A Good Companion" that would "escort pilgrims on the way." The powers-that-be vetoed this idea, declaring that it sounded like a bevy of old ladies in a retirement home, so reluctantly I chose another name: Bell Tower.
I worked for some months to distill the essence of what I had learned in my life so far, teachings I wanted to share with the world at large. When Bell Tower's first three books were published in the spring of 1991, this message appeared in calligraphy on a small bookmark in each copy:
The pure sound of the bell summons us into the present moment.
The timeless ring of truth is expressed in many different voices,
each one magnifying and illuminating the sacred.
The clarity of its song resonates within us
and calls us away from those things which often distract us
—that which was, that which might be—
to That Which Is.
At the end of 1996 I was on a retreat with meditation teacher Toni Packer at the Santa Sabina Center in San Rafael, California. One day, as I was silently pacing around the cloister, I spotted something familiar on one of the tables. There was a big poster of Thomas Merton and before it a candle and a tall, thin piece of calligraphy. When I stopped to read it, my heart stopped too. It was a blow-up of the bookmark. Later (when we were allowed to speak again), I discovered that the nuns liked the statement so much they used it at the beginning of many of their workshops. Finding what I had written five years earlier on display three thousand miles away was a very heartwarming experience. It showed me that there was indeed a universal sangha, a company of men and women on the path of truth who appreciated the signposts I had erected.
I spoke earlier about not taking what is not yours unless it is freely offered. The other side of the coin is making sure to share with others what you believe is yours. By which I mean everything that is currently in your possession. I mentioned earlier that all the stuff that we are currently responsible for—goods, money, a house, a car—is just ours for the time being. Like us, it is just passing through creation. Sometimes when I spend more money on something than I feel comfortable with, I tell myself that I am helping the economy. What goes around comes around. If you continually offer to others what you have right now, the chances are that whatever you need later will find its way to you from one source or another. I visited Russia not long after the breakup of the Soviet Union. None of the people I met seemed to have a ruble to their name but I discovered that whenever a friend, neighbor, or relative needed money for an emergency, everyone pitched in. Wherever this loan money came from (and I never found out the answer to that question), it seemed to circulate in the atmosphere, always becoming available to whoever needed it. If we don't put a strong claim on our possessions, then we don't feel a lack when we pass them on to others.
Although I enjoy being alone for periods of time, I am happiest in the company of others. This is because in another person it is always possible to see our own reflection, to see who we really are. You need to be still for this to happen, or there is a distortion in the mirror. You also need to be quiet. It doesn't happen often, but when it does... Perhaps this is why we are all so drawn to babies. We want to gaze directly into their eyes. We want to see and be seen. The same thing happens when we are in love. In ordinary life so much time is spent in talking, in avoiding other people's eyes. But when (at last) you are comfortable with someone else alone and in silence, just looking, there is a true meeting, and words drop away. They no longer have a place. They are a distraction, a limitation, a hindrance to the blazing fullness of it all.
EIGHT
Giving Attention
Years later, when my ex-husband, Neil, was dying, I remember asking Ram Dass what I could do for him. Somehow, until that moment, my loved ones had always died offstage, and I had no experience in such a situation. He told me just to give Neil my full attention, if possible coordinating my breathing with his. Giving others our complete attention is such a simple thing, and yet we rarely remember to do it. Whether the person we are with is dying or living, this is always one of the greatest treasures we can give them. When I thought about it, I realized that when I myself was in the hospital with hepatitis, my friend Katharine just came and sat there, waiting on me, as it were. Everyone else seemed under the impression that they had to entertain or engage me in some way. It was such a relief when she came. After each visit I felt as though she had actually brought me energy while the others had drained it away.
From time to time I will be in the middle of a conversation with a friend and am suddenly aware that he is no longer listening. If that happens, I stop speaking and wait a minute. The silence is often enough to bring him back. Perhaps he was off on his own train of thought, but wherever he was, he was no longer hearing me. Of course, I have often been guilty of this myself, although no one I know has tried the same trick on me. I generally return from my "vacation" and desperately try to figure out what my companion has just said.
Not only do most of us not listen very carefully to what other people are saying, we also rarely listen to ourselves. And if we are not willing to listen to our own voices, then why should anyone else bother? You will find that if you start listening to what you say as you say it (and I am not talking about rehearsing it in your head beforehand), you will discover that you speak with more clarity and it is much easier for the other person to hear you. There is a great power in attention.
Attention is at its most miraculous when you start to use it. To begin with, it is only when you bring all your senses to bear on whatever you plan to do next that you will know how to proceed. We tend to approach most things with preconceived ideas. We carry forward whatever we noticed on earlier occasions in the belief that it will be very useful this time. In fact, there is an unspoken assumption that we do not really need to pay that much attention this time because "we've done it before." But no two situations are exactly the same, and such a habit can be very costly. Centuries ago Heraclitus pointed out that you cannot step twice into the same river, and hardly any of us have taken his observation to heart. I am not a very technical person, but I have discovered that I can solve many household problems just by bringing myself into the present, contemplating what is in front of me, and waiting for inspiration to arrive. One day I came home from the office and went to the closet to get something. When I glanced down, I saw a note and a few screws on the floor right next to the polisher. It was from my Brazilian cleaning lady, and it read: "This machine it dusnt work." I hauled the polisher out and sat down on the floor beside it. I started to take it apart, one piece at a time, hoping to catch sight of where the missing pieces might fit. And, indeed, about twenty minutes later I had reassembled everything and got it to work (don't ask me how).
The main thing to remember when you are faced with a physical task is that you need to put your full attention at the place where the work is being done. For instance, if you are hammering a nail into the wall, you may think that the place to focus on is the head of the nail. However, the work is actually taking place where the nail is going into the wall, so put your mind at that precise point while your eyes watch the hammer hit the nail. This will enable the nail to go straight into the wall. If you are trying to unscrew a recalcitrant jar lid, let your attention rest in that space between the metal and the glass as you twist. You may think that there isn't any space there, but obviously there must be or the two would be welded together. I once applied this principle to pulling poison ivy roots out of the undergrowth. I let my mind travel the length of the root, allowing it to come to rest at the point where the root went into the ground, and I left my attention there in that space between the root and the earth as I gently pulled the plant toward me. And the long, white root came out of the earth without any argument.
The other place where there is a space and yet we overlook it is between ourselves and the situation. The Tibetan Buddhist teacher Chögyam Trungpa said, "Once one is aware of the space between the situation and oneself, then anything can happen in that space. Whatever occurs does so in the midst of space. Nothing takes place 'here' or 'there' in terms of relationship or battle."
Over the years I have noticed a remarkable thing, and that is that I can trust each situation. What I am talking about is that I like to get things taken care of the moment I spot them, if not before. If there is dust on the sideboard, I want to clean it now and not wait until Sunday morning, which is when I would normally do it. However, from time to time something needs to be done (such as calling the painter to come in and repaint my son's room now that he has left home. This is something I have been meaning to do since he left eight days ago), and I just don't do it. When I first observed this seeming reluctance, I decided that it was sloth, but gradually I came to the conclusion that it was because it was simply not the right moment. If I am not taking care of something that is obvious, it is because I need to wait for the universe to get into the right mode. You certainly need to discriminate about this because for some people it may indeed be laziness. However, knowing myself the way I do, I find it amusing to watch how the days go by and I am still not moved to take action. So I just let whatever it is sit quietly in my mind for the time being. Then, suddenly, I make my move. I guess this is rather like a lizard catching its lunch. (Sorry: this is not the happiest of metaphors, but it is what sprang into my mind. I see this lizard immobile on a rock, and then its tongue streaks out and catches a fly or whatever.)
It is the same when you cannot remember someone's name. It is no good searching for it desperately. If I do this I usually draw a blank. However, if I let everything go, a few minutes later the name will drift into my mind without more ado (well, more often than not). Things happen when they happen, in their own good time.
It is the nature of thoughts to circulate. If they have floated into your mind once, they will undoubtedly float in again, so there is no need to grasp them. I find that they operate rather like a Lazy Susan. When I sit down to meditate (or lie in the bath. I have trained myself to stay in the bath for a little while after I am clean—just lying there in the water with my mind at rest. I have discovered that all kinds of things come into my mind that I am normally too busy to entertain), there is a great temptation to open my eyes and make a note of something that I need to do once I get up from my cushion. But I have learned that this isn't really necessary. Whatever it was that came up surfaced because I was no longer suppressing it or overwhelming it with a host of other urgent things. If I just allow it to be there along with all the other things that arise, it will still be there later. The same thing happens as I go to sleep. Whatever drifts in as I am about to go to sleep, will be right there first thing in the morning. There is no need to turn the light on and search for a pencil and pad and make your mind active again. A little trust is called for.
In 1972 I edited a translation of the Tao Te Ching by Gia-fu Feng and Jane English. Eventually the time came when I had to write jacket copy for the book. This is the moment that every editor dreads because you have to find a way to express the essence of the book in a few sentences so that a reader will grasp immediately what the book has to offer. I was daunted at the prospect of reducing this sixth-century-B.C. classic to a single paragraph. Weeks went by, and finally I had only twenty minutes left. I sat down before the typewriter, and my mind went blank (not deliberately). I waited a little while and then began to type. What appeared on the page arrived whole. Now, thirty years later, I would not alter a word of the copy on the outside of the book:
Accept what is in front of you without wanting the situation to be other than it is. Study the natural order of things and work with it rather than against it, for to try to change what is only sets up resistance. Nature provides everything without requiring payment or thanks, and also provides for all without discrimination—therefore let us present the same face to everyone and treat all people as equals, however they may behave.... We serve whatever or whoever stands before us, without any thought for ourselves. Te—which may be translated as "virtue" or "strength"—lies always in Tao, or "natural law." In other words: Simply be.
After this I realized, somewhat ruefully, that it was not writing the copy that took time but the not-writing of it. The not-writing was a tremendous burden that I had carried around unnecessarily for weeks.
I have never quite understood what people mean when they say that making decisions is hard. My experience is that if I have to do something, I look at it as clearly as possible and just move forward. There seems to be only one possibility, and I pursue it. If I really don't know what to do, then I do nothing. Eventually, as I said before, the universe shifts, and whatever needs doing becomes obvious. Until it is the right moment, there is nothing that can be done.
Think of all those days when you have gone shopping for a particular item and you are thwarted at every turn. We all have those days when nothing seems to go right, and the hard part is accepting that this is not one of those times when your plans are going to come to fruition. Yesterday, for instance, the painter was going to come to give me an estimate for painting the room, but he canceled. This left the whole day free for writing and other things, but everything I had wanted to do didn't get done. My sister-in-law, Valery, had written me a long e-mail regarding a paper she had just given at a conference in Florence on the Neoplatonist Marsilio Ficino and his views on philosophy and politics. The paper contained the word myrobalan. I was mystified, and then I remembered that I had the Oxford English Dictionary on CD-ROM and here was the perfect moment to refer to it. I had used it only a handful of times since I had received it as a present a couple of months back. That's when the first disaster struck: The computer refused to read the data disc, and even though I uninstalled the program and reinstalled it, the whole thing kept freezing. I consulted the booklet and saw to my horror that the warranty lasted only sixty days and had therefore expired the day before. I kept hoping that a miracle would occur. I dusted the CD-ROM. I treated it gently. But however often I loaded it in, no miracle happened. I won't bore you with all the other small things that went awry (but I will tell you that a myrobalan is a very bitter medieval plum). I just wanted to give you the flavor of how a myriad of details can appear to thwart the progress of a happy day. Of course, I did get some things done but not the ones that satisfied my heart. And therein lies the secret. We are so set on accomplishing the things we want to do—fulfilling our desires—that we don't want to see what needs to be done right in front of us.
Yes, I walked to the farmers' market (twenty blocks there and twenty blocks back; got my body energized). And I bought long, slim, mauve eggplants, crisp green beans, a bunch of young leeks, small sweet potatoes, and shining vermilion and yellow peppers, and then made three pans of roasted vegetables drizzled with oil and sprinkled with herbs, so that I have the makings of meals for myself for several days. I also cleaned some silver, re-covered a disintegrating cushion, and so on, but I saw all those things as hindrances to what I really wanted to do. Yet all the chores I took care of were necessary, and it would have been much more enjoyable if I hadn't had this internal argument going on about the fact that I should have been writing, that I was "wasting" time. What we are doing right now is what we are doing right now. Wanting a different scenario is useless. This is the movie we have rented, so why not watch it? The other movies aren't available or haven't been released yet.
One of the most practical teachings I have received in my life is "Go through the door that's open." We get all kinds of ideas in our heads as to what we think we would like to do, and most of them bear no relation to what is likely or possible. This is not like trying to unlock a door to which you have no key; it is more like trying to walk through a wall. Have a clear look at what is in front of you, and you will recognize the way to go. But it does have to be a very clear look.
I have only just seen that this is the answer to one of my besetting sins. Those that know me well are aware that I have an annoying habit: I complain a lot—even though, as someone pointed out to me last year, there is really no point in complaining unless it is to someone who can remedy the situation. But (at least in my case) reason doesn't always kick in when things seem unfair: I start to whine more often than not. This is something I know I need to work on, and I apologize now to all those friends to whom I should have apologized long ago. When we (read "I") complain, it is because we believe that things should be other than they are. Who was it who said, "The rain falls equally on the just and upon the unjust, but the unjust has the just's umbrella"? Why didn't I take to heart the first sentence of that Tao Te Ching copy I wrote so long ago?
If the computer has a conniption and you are put on hold by the techies for twenty minutes, that's the way it is. Complaining about it upsets you, and it also upsets those who have to listen to your complaints, so what have you achieved? It does not solve anything. In the end it all boils down to our attitude. If you eat nothing for two weeks, you could call it a fast or you could see it as starvation. If you are confined to one room for several years, you could view it as a punishment or as an opportunity. One of my favorite authors, Charles Morgan, was taken prisoner as a British naval officer during World War I and held captive in Holland for four years. He used the time away from the responsibility of earning a living to write a novel. He considered this period in his life a blessing and went on to write many more novels and win several prestigious literary prizes. Next time I am on hold, I need to remember this and put the time to good use.
I have a friend who lives in Arizona who tells me he likes to come to New York City, where he is constantly caught up in traffic. It gives him the chance to sit and meditate for a little while. This isn't possible where he lives because in the desert there are long stretches of road with no traffic lights and no chance to do anything but keep driving. He is the only person I know who seems to enjoy stopping at the lights.
A key issue we often overlook is our own attitude. Someone once said to me: "Everything is fine as it is. Your view of it may or may not be." In his book Travels, Michael Crichton pointed out that the most valuable thing we can possess is a perspective, a new way of looking at things. "The purpose of education is to provide perspectives.... Any new perspective alters consciousness."
What I have discovered is that not only does a different view of things change the outcome but so does a different form. When I make bread, I always use the same recipe (whole-wheat flour, dry yeast, salt, water), but I don't necessarily make a round loaf each time. Sometimes I make a long loaf, and sometimes I make rolls. When I take the bread out of the oven, the texture and the taste are different, depending on the shape and size. This never fails to surprise me. Early in the 1970s I shared a brownstone with some friends. Since we all enjoyed fresh bread, we would take turns doing the baking each day, and we signed up for however much we needed: half a loaf, two loaves, whatever. We were all using the same ingredients and the same recipe, but the bread always turned out differently, depending on who made it. In fact, you could tell who had baked bread that day by looking at it and picking it up. The taste and appearance of the bread usually reflected the character of the baker. Some people baked high, fluffy loaves with lots of holes in them but little taste. Others produced indigestible, stone-like offerings. What did I bake? Very compact, tasty loaves, of course. I could never make my loaves rise the way I wanted them to, but they always tasted delicious.
I encountered a couple of tai chi chuan teachers recently who focus on this principle. They have understood that if you free the body from mechanical movements, you also free the mind from mechanical thoughts. As one put it, "The moment you change your physical position, it changes the way you think."
The opposite is also true: If you change the flow of energy, you change the structure. When a moment of clarity comes, we move naturally from a contorted position and vice versa. Straighten your back and become balanced, and your head will also clear.
A year or so after Neil left, I was at a party and someone came over and asked about my husband and new baby. I told him that my baby was fine but that my husband had left when Adam was a year and a quarter. He looked at me for a moment and then said, "So you must be almost over it by now?" The question hung in the air as he held my gaze. This was not the usual cocktail chitchat. I took a deep breath and replied, "I don't think you ever get over it, but perhaps with time things recede into perspective." And then I burst into tears. Perspective is indeed all-important.
I was just on the phone with my friend David, describing how events in my life seem to arise and make their way into this book within a few days. It is not exactly recycling; it is more like cycling. "Re" makes it sound as though something is being repeated, but that is not the way it feels. He suggested that I am calling these situations out of the universe (I am not sure how he expressed it, but that was what I heard). Come to think of it, one of my criteria for selecting a book for publication is whether what the author has written in the manuscript changes my own spiritual practice. If it can do that for me, I believe it can do the same for others. So much of the teaching I receive these days comes from the manuscripts that people send to me. I know that these writers think that they are looking for a publisher, but perhaps their manuscripts keep arriving here because they contain whatever it is I need to learn next. What a revolutionary idea! And yet the sages have always known this. "When the student is ready, the teacher will appear" (and vice versa). The teacher doesn't have to be a flesh-and-blood person. Anything or anyone can be a teacher. In fact, whatever is right in front of you is your teacher.
The idea that whatever is in front of you has something to teach you is extraordinary. A few years back, Adam, who knows more about movies than I ever will, mentioned that he was going to see Pocahontas. I thought it strange that a college kid would go to see a Disney movie, but he doesn't do things lightly, and so I inquired why he was going. I should mention that he was majoring in film and video. "I can always learn something from a movie," he said, "even if it's how not to make one." It is sometimes hard to glean a scrap of wisdom from a trip to the supermarket or wherever, but it is there if you can spot it.
Whatever it is, you can learn from it. No need to waste any experience, even (particularly) the telephone conversation I just had with David. There I was bemoaning to him (I had already bemoaned to myself) that the morning had disappeared and I still hadn't got down to any writing. I felt guilty. This happens day after day. I think it is because almost everyone has been telling me that good writing happens in the morning. Armed with that idea, I have been failing morning after morning for the last month or so. But the thing is: I find that the writing does happen spontaneously (providing I start typing. Without that, it certainly doesn't happen at all), which is the way I had hoped it would when I decided to write this book. Earlier today, I didn't have this material to write about. Now that David has called, I do. So we are back to just accepting things as they are and not wanting them to be some other way—or perhaps this is another way of seeing how we have no alternative but to go through the door that is open.
Sometimes I am not at all sure what circumstances are teaching me. As I said, I prefer to do everything immediately, just as soon as I think about it. This can be awkward because thoughts can distract you at any moment of the day. In an ideal world you keep your mind on what you are doing, and no stray thought would dare enter your mind, but the ideal often seems very unattainable. If you drop everything the moment a new thought occurs to you, you will just leave a trail of unfinished jobs behind you. The best solution I have come up with is to make a note of whatever it is and go firmly back to what I was doing before. That way, at least I haven't completely lost track of it for the day. The thought will, of course, return at some later moment, but if you want to take care of whatever it was next, it is safer to make a note now (unless you are meditating or about to go to sleep). The main thing is not to panic when all kinds of new ideas come flooding into your head one after the other. Think of them as a plus rather than a minus: The juices are flowing. Hurrah! Oh, and make sure that when you make your note, it is legible. Far too often I have made a note to myself that I have not been able to read later. This is because I have very bad handwriting. I am ashamed of how bad it is. I think of myself as a very orderly person, and yet why is it that after all these years my handwriting is still so terrible?
One of the lessons I learned quite early in my life is that whatever you seem worst at is perhaps where an undiscovered talent lies. It is as though we deliberately ignore our own talents and go out of our way to deny them. I discovered this when I was attending classes at the philosophy school and I was the one chosen to go back to the London school for a week, learn the rudiments of a particular calligraphy discipline, and return to New York to teach it to the other students. At first I thought the person who had asked me to do this was out of her mind. I pointed out that I had the most dreadful handwriting in the whole group. But apparently that was one of the reasons I had been chosen. And so I flew to London, devoted an entire week to calligraphy, and went on to be in charge of calligraphy in New York for many years, delighting in the forms of the letters and the spaces they described. (It did wonders for my calligraphy but unfortunately nothing for my handwriting.) I observed the same principle in action with another student who was very feisty and always causing problems. Eventually this man was asked to teach the class on a day when the tutor was absent, and he took it over as if to the manner born. He never caused trouble in the class after that.
In order to see what is going on at any given moment, we need to have our attention directed out rather than in. Instead of being carried along by the flow of events or withdrawing from it, we just stay right here. There is only this moment. This is the only method of appreciating the divine.
We are always anticipating that something better (or worse) is about to happen, but we would do well to keep in mind that wonderful Gahan Wilson cartoon in which two men are sitting on cushions in the zendo and have obviously been there for some time. The monk leans over and whispers into the ear of the novice: "Nothing happens next. This is it."
I don't quite know how we got it into our heads that whatever is happening right now is probably not that important and can be dispensed with, but it is a very pervasive view even though in 1242 Zen Master Dogen observed:
Each day is valuable.... Do not compare it with a dragon's bright pearl. A dragon's pearl may be found. But this one day out of a hundred years cannot be retrieved once it is lost.
I just paused before I typed the last sentence. I was mulling over what Dogen had said and marveling that more than seven centuries had elapsed and that most people have still not grasped its significance. As I sat here, I suddenly heard the sparrows chirping outside my window, the trundling of the road-working equipment, the tires swishing on the wet street, as well as the underlying hum of electricity in my office. Up until that moment I had been deaf to all this, having narrowed my attention to the screen in front of me. What is even stranger is that as I started to acknowledge all these sounds, I realized that I had lost track of how my body felt or whether I could smell or taste anything. We have five senses, and we tend to use only one at a time. Such a wealth of impressions is available to us at any given moment, and yet we crowd it out by paying attention only to whatever is going on in our heads.
Our attention is either in or out. This is very clear in meditation but not always apparent at other times. If we are focusing on some idea or problem, we miss everything going on outside. If we welcome impressions from the outside, then we do not get seduced by our own thoughts. It is obvious when you think about it: You can't be in two places at once. So, stay here. Don't go away.
All of us can rise to an emergency. When such a thing happens, it does not occur to us to opt out. We just move forward to do whatever needs to be done, drawing on reserves of knowledge and strength we didn't know we possessed. There is undoubtedly more than one factor involved. An emergency calls us into the present, and immediately the adrenaline necessary for the task floods our system. It is in many ways an impersonal thing; it has to do with the way nature preserves the species. What I find interesting about these situations is that once you are "there," completely present, you feel as though you could do anything. This can happen even in very small ways. For instance, from time to time I have been standing at the kitchen sink, wiping glasses with a dish towel, and suddenly a glass has slipped from my grasp. I have very quick reflexes, and almost without exception I will catch the glass before it smashes to the ground. I may have been in a dream when the glass escaped, but immediately there is a need all my senses are working and I am ready to tackle anything. Most of the time we are off somewhere in our heads and don't experience this sense of urgency, which is why accidents happen in the first place.
NINE
Offer No Resistance
When I lived in England, I often answered "No" to people and events, or, at least, that is how I remember it. Once I had said "Yes" to Bob Gottlieb and his offer of a job in New York—which I did almost on a whim—I continued to respond that way, and it was a profound transformation. I'm not sure how this happened, but I am glad that it did.
Saying no takes so much more effort than saying yes. This doesn't mean that you meekly accept other people's outrageous behavior. What I am talking about is looking at what is before you and welcoming it if it seems appropriate. An opportunity that is presented to you is a present (a gift) and it calls you into the present (now). What most of us tend to do with the maelstrom of opportunities that besiege us is ignore them. Ignoring implies that you know something is there but you choose not to pay attention to it. This can take up a lot of psychic energy. It is a very deliberate act. I usually accept what comes along unless there is a very good reason not to. For instance, if someone calls to invite me to dinner or an event when I have nothing planned for that particular time and it is not something I am sure I would hate, I am reluctant to say no because refusing is like turning my back on what the world is offering. You just never know what each invitation or opportunity holds or where it might lead.
I was discussing this point with my friend Deb one day while circling the reservoir in Central Park, and she disputed my reasoning. A mutual friend had given her my name and number and encouraged her to call me at least a year or so before we met, but she kept not doing anything about it. Eventually we encountered each other through a different connection, and she was quite taken aback that I had materialized in her life anyway. She told me that it had been easier not to call me than to call me, but I don't agree with her. Not doing something may feel easier, but it really isn't. All those months when she didn't call me I was hovering in her unconscious and taking up unnecessary room.
After I had been working at Simon and Schuster for more than three years, Bob was hired to run Alfred A. Knopf, and he said that I was welcome to go with him if I wanted to but that I should think carefully before making a decision. There might be many jobs opening up at S&S (two other senior executives were moving to Knopf in addition to Bob). It would not have occurred to me to leave Bob because I had become very attached to him, so I accompanied him to Knopf. Nineteen years later, when he came to tell me that he was quitting Knopf for The New Yorker; he didn't ask me to go with him, and so it didn't occur to me to raise the possibility.
I also embraced the situation as it presented itself when I was in my early thirties and a number of students at the philosophy school were looking for apartments and someone suggested that instead of us all paying rent, we should buy a house. I had never owned property. Once the idea arose, however, it seemed the obvious thing to do. There were ten of us needing somewhere to live and five who could put up money for a down payment. A brownstone on the Upper West Side of Manhattan materialized almost immediately, and the only problem seemed to be that no bank was willing to give a mortgage to five professional men and women who had no debts and no financial responsibilities. This was my first experience of the peculiar way the American economy works. It transpired that a bank would lend you money only if you already had debt and had therefore established yourself as creditworthy. Never mind that if you had incurred one debt and had not finished paying it off you were probably not in a position to take on another one. The banks told us that if we formed a corporation, they would give us a mortgage, but we didn't want to do that because then we would have been unable to claim individual tax deductions. Luckily the building's owner was willing to give us a purchase money mortgage at an extraordinarily reasonable rate, and so the deed was done.
We were the happy owners of a five-story brownstone that had been turned into ten apartments, although much of the renovation still remained to be done. So the next thing we did was start renovating. I learned a great deal about sanding, painting, electricity, and plumbing—all on-the-job training. Our spare time was given over to working on the house, and when I left to get married four years later, the renovation was still in progress.
Those of us who lived in the building all had different responsibilities. One person was in charge of the finances, another the boiler, a third the garbage, a fourth the garden, and so on. My responsibility was to care for everyone who lived in the house in a general way—someone had to be nominally in charge—and to keep the peace. This led to some interesting situations, including the following one.
Two of the people who moved in were a high-school teacher and her sixteen-year-old ward, Lorraine. As it turned out, there was some friction between them, and shortly after their arrival the guardianship came up for renewal, and it was decided not to continue it. I offered to become responsible for Lorraine, and we went to family court to make this official. This was a very long-drawn-out procedure requiring several appearances and much investigation. Lorraine had been assigned a guardian by the Bureau of Child Welfare when her mother had taken her to court three years earlier on a PINS petition (Person in Need of Supervision). The bureau suspected that we were running a commune and tried to block the transfer of guardianship. I pointed out to our legal aid lawyer that if it were a commune, we would be sharing expenses, which we were not—unfortunately for me. Then they said that the building was unsuitable because there were single men living in it. I countered by saying that each of the two single men occupied his own apartment on the garden floor while Lorraine was living on the fifth floor but that also, if there were an apartment building in the city in which young men did not live, there was probably something peculiar about it. In the end we got the bureau disallowed in the courtroom because of its prejudice. When I eventually came before the black woman judge, she asked me only one question: "What do you see as your responsibility toward this child?"
I took a deep breath and replied, "To know where she is at all times—physically, emotionally, spiritually..." I was about to continue, but the judge cut in:
"Precisely. The child is placed on probation, and the condition of the probation is that she live with Miss Rees. Case dismissed."
The proceedings were over almost before they had begun. We all filed out, and I asked the psychiatric social worker what the ruling meant. She explained that because the Bureau of Child Welfare had not been allowed in the courtroom, Lorraine had been reassigned from the bureau to the Probation Office. Unfortunately, the Probation Office had no funds and so, although we had achieved our aim, there was not going to be money from any of the agencies to support Lorraine. She offered to take me to the Welfare Department and work with me to obtain some funding. She was as good as her word. She went with me over and over again, but I had to wait until a new mayor was elected before I got any money. I was grateful to John Lindsay when he overhauled the system and we began to get an allowance and food stamps. In the interim, various friends helped us out.
My mother was dismayed when she heard about Lorraine. I pointed out that she had been after me for years about a grandchild. Now I had provided her with one, and she shouldn't complain. Apparently she had not expected a sixteen-year-old grandchild. She had been thinking of a husband (first) and then a baby. She asked me why I had done this, and I explained that I could tell her how I had done it but not why. Lorraine had needed a home, and I happened to be available. I quoted to her the words of George Mallory when he was asked why he had climbed Everest: "Because it's there."
Lorraine was in my care for three years. They were very thorny years because she was a tomboy and a real handful, but it never occurred to me not to do my best for her. I had made a commitment to the judge and to Lorraine, and I wasn't about to renege on either commitment.
Eventually, things got really out of control. Lorraine would disappear overnight, and I wouldn't know where she was. I told her that if she was not going to use her room, we would offer it to someone else. Space is always at a premium in New York City. I said that the next time she stayed out all night without permission she could look for somewhere else to live. A short time later she vanished for a whole weekend. I called the kind psychiatric social worker to find out what the legal situation was and discovered that if no term is set for probation, then it is just for one year. No one had bothered to let me know this. Apparently, after the first year the responsibility should have been her parents'. So Lorraine left our house on West Ninety-fifth Street and set off for "fresh woods and pastures new." A couple of times she has been back to visit, and she still calls me from California every few years to chat and to ask my advice but not take it.
If I had said no to taking on Lorraine in the beginning, I would have missed out on so much. I really had no experience with young people, and everything I learned during our time together stood me in good stead when my own offspring finally arrived. Even if you start out with little or no affection for someone, once you care for her over a long period, this "caring for" turns into "caring about," and then into love. People often asked me whether I would have picked her to be my child if I had had a choice. We were so different from each other. It always seemed a ridiculous question to me. Even if your child emerges from your own body, you don't get a choice about the kind of person he or she is. Babies just arrive, willy-nilly, and you start loving them right away.
The other trick we often try (on ourselves) in order to avoid dealing with what is in front of us is changing the circumstances. This usually does not solve anything. It just postpones the moment when we have to face whatever it is. Some people are always changing jobs, and others go from one "life" partner to another. For instance, the few men I have fallen in love with have always been unavailable. Often they live thousands of miles away. Sometimes they are married to other people, and the man I did marry was gay. Perhaps if these men had been available, I would have overlooked them. One day I need to see who is right in front of me instead of looking somewhere else and repeating the same mistake over and over.
Both Neil and I tried to make our marriage work (I think), but one evening when Adam was a year and a quarter, Neil said that he needed three months by himself. He didn't know whether he would return. I was shocked and bewildered. In many respects I am very old-fashioned. If I make a vow, I keep it, no matter what the cost. Divorce was not in my vocabulary. However difficult the marriage had been, it would never have occurred to me to leave it. I asked him why he was doing this.
"Our marriage has problems," he said.
"Every marriage has problems," I replied. "But once you identify what they are, you try to fix them. You don't just walk out."
I offered to go with him to a marriage counselor if he thought that would help, even though I myself had never contemplated any kind of therapy. But he said that he had all the tools he needed from the philosophy school (where we had met and which we were both still attending). I pointed out that having the tools and using them were two different things, but the next morning he left and, in fact, never returned.
When three months had passed, I asked Neil if there was any chance of his coming back to live with us. I told him that I was considering renting out half of the apartment, both to bring in extra income and also to help one night a week with the baby-sitting. I didn't want to embark on this if there was a possibility he might return. But he said that there was not, and so I moved my bed into a corner of the living room and offered the spare bedroom and bathroom to the universe, wondering who would come and occupy it.
I never advertised, but for some years there was a steady stream of people who came and went—a young Swiss woman who enjoyed mountaineering and liked to walk up and down the six flights of stairs to keep herself in shape, an Episcopalian priest who worked in the city during the week and went back to his home on weekends, an organist and choirmaster who eventually departed without paying the last month's rent, someone who wanted to take a sabbatical from her marriage, a couple from New Jersey who needed a pied-à-terre, and several others. I rented the space furnished and asked each tenant to pay half the rent and baby-sit one evening a week. The latter was always a problem. Even though there were seven nights in each week, no one wanted to make such a commitment. Some offered alternatives: a week's groceries from the supermarket her husband owned (since I never knew when this would kick in, it was possible to ask only for paper goods and Kitty Litter, things that didn't spoil), a massage (this from the woman taking a sabbatical from her marriage, and who was training to be a masseuse. This would have been a great trade but didn't often happen because she came home from school so late, and I didn't like to remind her of our bargain), and so on.
I really enjoyed the company, and it brought me into contact with people I might never have met, but eventually I began to feel as though I was living in Grand Central Terminal. This being New York, almost no one stayed more than six months. After that, a major shift would occur in their lives and they would move on. So at a certain point I went to Bob Gottlieb and said that I didn't want to do this anymore. I wanted my home to myself. He said that he thought this was a great idea, but I pointed out that unless he was willing to give me a raise equal to the amount I would be losing, I couldn't do it. He gave it to me without hesitation, and I was torn between gratitude and relief—and anger that he had not offered this chunk of money years earlier.
This year when I went home to take care of my mother, I fought a silent battle with her every evening. Books no longer hold her attention, and she likes to watch television. She turns the volume up very loud because she doesn't like to wear her hearing aid (or cannot find it). I sat beside her because I know how much she treasures my company, but I have no interest in the antics being displayed on the screen and I was trying to work on my writing. I could have gone to sit in my bedroom, but then she would have felt snubbed. Still, it was hard for me to sit there with all that noise. I have often told her that I like silence, but she doesn't believe me. From time to time she asked if I would like the television turned off, and I said that it was fine as long as she watched it and allowed me to work. But every minute or two she made a comment and demanded a response from me: "Look at that woman's hair! See the beautiful flowers. Isn't this an old film?" and when I kept silent, she asked whether I had heard her. I would explain that I was fine with her watching and me working. It's just that she wanted to involve me in her activity, and I found this infuriating.
I was able to ignore the flickering and chattering of the television, but I could not ignore her voice. She would promise to be quiet and not interrupt me again, but she suffers increasingly from short-term memory loss. Looked at in one way, this is a blessing: She has reached a state that many of us are still aiming for in that she is more often than not in the present moment and does not refer back to the past. Still, for those around her, it does test the extent of their patience. As I often remark to my brother, my mother is more ecologically conscious than the rest of us, and she recycles her conversation every minute or so. Before I could get to the end of another sentence, she would be asking me innocently to look at the television again. Why did I find it so hard to give her this time? There is probably little enough of it left. I was trying to deny her my attention—the one thing that it is in my power to give but which I was selfishly withholding from her. Much of what she says and does these days is no longer under her control. I was ashamed of my lack of generosity and resolved not to tussle with her that way again. What on earth did I think I was achieving by punishing her in this way?
Gradually I became aware that it was simpler (and kinder, of course) to say yes to what was happening and let it be, to "suffer" it, in the real sense of the word. After that it was not nearly so hard to sit in the living room with her hour after hour, joining her in her activity rather than attempting to flee in my mind. After all, I had traveled three thousand miles to be with her. What was the point of wishing myself somewhere else?
Last night I was invited to accompany someone to a concert of art songs. I didn't quite grasp that all the songs had been commissioned especially for this performance, and indeed all the composers were present and took their bows. Much of the music was atonal (I think that is the right word). There were no tunes you could hum, i.e., not the kind of music I enjoy listening to. It all took a huge effort on the part of the performers and the audience. This was not a demand I wanted to respond to, particularly since the soprano soloist was large, enthusiastic, and didn't really need a microphone (there were moments when I had to put my hands over my ears to protect them from the inexorable, piercing sound). My first impulse was to escape, but we were right at the front and my companion would have been upset if I had walked out, so I decided that I had better stay put. I didn't try to blank out what was taking place on the stage. Instead, I looked for features of the performance that I could appreciate. It is a rare event that has absolutely nothing to recommend it. The two music directors, who were also the pianists and hosts for the evening, gave a superb performance in each of their roles. The singers were very accomplished. Possibly the composers were gifted also, even if their work didn't appeal to me. So it was not a total loss. I would certainly go to a concert organized by this group in the future—once I was sure that the music was at least a hundred years old. I was just unlucky in my choice of program.
Far too often in my life I have spent time imagining that other people are doing all kinds of wonderful things while I am home alone. This is another form of resistance—a refusal to be wherever you happen to be, doing whatever you happen to be doing. In fact, it is a double denial: You are not out there with the phantom revelers, and neither are you back here on your own.
I heard an interview the other day on National Public Radio with a tai chi instructor. He was describing how much more energy people feel they have once they start practicing tai chi. He always points out to them that what has occurred is that they are not using up so much of their energy tensing their bodies. Once you stop contracting your muscles unnecessarily, a great deal of energy is no longer being wasted.
The other side of offering no resistance is setting things up so that resistance doesn't arise in the first place. Once I had learned the rudiments of playing the piano, a young Dutch woman invited me to join her in a Mozart piece for four hands. Knowing that she was a skilled pianist, I panicked. First I explained that I was an absolute beginner and then that I couldn't sight-read, but she was not to be deterred. What I didn't know about her was that she was also an extraordinary teacher. We sat side by side at the keyboard, and she slowed her pace to my painful picking out of the notes. Each time I made a mistake, she would say, "Try stretching your finger a little farther" or "Yes, now play those two notes together." There was no hint of criticism in her demeanor. She supported me at every turn. Eventually we got to the end of a couple of pages of music, and she allowed me to escape. Both of us were all smiles. I don't remember the piece we played, but to this day I recall how she made me feel. She had only said "Yes." She never said "No." She taught by encouragement alone, and this allowed me to gradually drop all my fear and hesitation and enter into the spirit of the playing itself.
TEN
The Habit of Truth
I do not remember anyone instilling in me the importance of telling the truth, although I do recall how I explained it to Adam when he was small. I told him that not only was it the right thing to do but it was also the simplest. I could never understand why people lied. It seemed to me such a complicated way of operating. Once you had told someone something, you had to remember whom you had said what to and hope that this person would never interact with anyone to whom you had told a different tale. Those who lie appear to be setting themselves up for disaster, but perhaps the people who do it are unable to stop themselves.
The first time truthfulness was called into question for me was at boarding school when I was just four years old. After the customary afternoon nap, I had put on clean clothes and gone outside to play. I was watching several of the other girls drawing patterns in the dust with their white-socked feet when suddenly the headmistress appeared, and everyone scuttled to put their sandals back on. She asked the head girl, Georgina, who the culprits were, and I was identified as one of them. To this day I don't know whether this was because Georgina didn't like me, she wasn't sure who had been involved, or she thought that I had participated. Several of us were singled out, and while the rest of the group were sent off on that beautiful sunny afternoon to hunt for hens' eggs in the fields, we were taken to the gymnasium. There we were informed that because we liked taking our clothes off, we were now to take all of them off. (This was an ingenious form of making the punishment fit the crime.) Once this was done, we would be free to get dressed again and join the others outside. I remember that I was wearing a navy blue dress with a white Peter Pan collar and rows of little white elephants marching down to the hem. I stood there all afternoon, long after the other little girls had stripped, put their clothes back on, and gone off to play, because I adamantly refused to admit to a wrongdoing I hadn't committed. Even at that age truthfulness was of vital importance to me.
And in all these years I haven't shifted my position. My friends and acquaintances are often amused by how much of a stickler for the truth I am, but behind their smiles lurk both respect and relief. In business dealings they can rely on the fact that I will not fudge, that I will go out of my way to provide accurate details and correct any view that may lead to misunderstanding. They can also be sure that I will reveal things that may be to my disadvantage. I want them to have the full picture, believing that they are quite capable of drawing their own conclusions from what I tell them. And this is not something they can be sure of with most people. When I have discussed this with colleagues who also sell subsidiary rights, they give me the impression that they don't feel it is safe to give out this much information. They believe it will result in a less lucrative deal. I find this puzzling because as far as I can tell, I never got less money for something by doing this. I suspect that sometimes I got even more money than they might have because people find this way of doing business so refreshing.
In her poem "Ladder," Jane Hirshfield says, "Rarely are what is spoken and what is meant the same." I was astonished when I came across that line but on reflection realized that, indeed, this is unfortunately the case. Over the last decade there has been a degradation in the value of words. People no longer seem to have faith in the words of others, particularly anyone in public office or the press, and so they don't give what these people say any credence. And I have come to the reluctant conclusion that many people don't even understand what they themselves are saying. There is an imprecision that has crept into the language that is a little terrifying.
I try to be meticulous in my speech, in that I say exactly what I mean in the fewest possible words. From time to time I see that other people are taken aback by this forthrightness, because they are not used to it. There are certainly times when it is better to say nothing at all, but if you are going to speak, why not say what you mean? Get right to the point. Avoid the preamble. This doesn't mean that you have to be brutal. You can cut to the chase and still be gentle, but it is a kindness to others not to hold back if there is really something you want to communicate. When I talk, there is no hidden agenda because nothing is hidden.
It is no secret that many English people find it hard to express a preference and are inclined to defer to the other person, not wishing to upset him or her in any way: "What would you like to do?" "Oh, I don't mind. What would you like to do?" "It doesn't matter to me, really. Would you like to go to a movie?" "Would you?" This exchange can go on for some time and even when it is over, neither English person is quite sure whether the final choice is mutually satisfying. Better to be clear in the beginning and get on to the next thing. Someone called today to ask me if I would like to go and see Gladiator, since it had just won an Oscar. I was pleased that I had been invited, but my answer was brief and to the point: "No, thank you. Go with someone else. I saw Quo Vadis when I was a teenager, and that is enough Roman stuff for me." If you say something this straight, other people know exactly where you stand.
I recall the broadcast that Saddam Hussein made to the American people during the Gulf War. It went on for a very long time, and almost no one could grasp his message because he never seemed to say whatever it was he wanted us all to know. All I took away from listening to it was the phrase "the mother of all battles" and the impression that Arabic was an even more indirect language than I had thought. I couldn't tell whether his style was more flowery and oblique than other Iraqis', but I suspect that it was. And this was sad because I think that many of us were curious to understand his point of view.
Telling the truth is completely uncomplicated; you don't have to work out any strategy. You just tell it like it is. Also, there is no residue. You don't find yourself wondering if you should or shouldn't have said something. This saves an inordinate amount of time and energy (two things at a premium these days).
Straight talk is vital when it comes to children. I learned this lesson when I was taking care of Lorraine. I had warned her that if she misbehaved in a certain way again (I no longer recall the circumstances), I would put her across my knee and spank her. When I said this, it was just a threat, and I thought it would be enough to deter her, particularly since she was at that time about seventeen. But then she disobeyed me once more, and I was honor bound to deliver. I applied a hairbrush ten times to her rump. When it was over there were tears in her eyes and I was taken aback, but I thought that perhaps the indignity had upset her. I hadn't struck her hard at all. About a year later she told me that it was not the punishment itself that had hurt so much but the quality of my attention. She said that anyone else would have brought the brush down in a different spot each time but because I was so careful, I hit her repeatedly in the same place and almost broke the skin. The moral of this story is not that I recommend corporal punishment. I don't. What is important here is that you shouldn't tell a child that you will do something (or not do something) unless you mean it, because, if you don't follow through and it is an empty promise, the child will no longer believe you or have any respect for you. (The same is true for adults, of course.) I know a woman who is always lashing out at her children verbally, but they no longer pay much attention to what she says because they know that she is speaking from anger and that she has no intention of fulfilling her threats.
Intimately connected to this is the practice of asking a child to do something when you yourself act differently. Children learn through mimicry and osmosis. They copy the way you behave. If you are always anxious, the chances are that they will learn to be anxious too. It's in the air. If you remain calm, then they will not panic. You are the one providing the cues.
It follows that using reverse psychology also sends children a hollow message, and the repercussions later in their lives may be irreversible. And, last, I don't subscribe to the "If you eat one more mouthful, then you may..." school. Bargaining with children is an unfortunate practice. I admit that I was tempted to use both these methods when Adam was little, but I didn't succumb. I found that if I spoke with enough authority and really meant what I said, he heard me and would fall into line.
At one time in my life I was put in charge of the advertising for the philosophy school. I was instructed that all I needed to do was present what we were offering accurately and concisely, never promising that any particular result would come from trying it. After all, how could one know what the result might be for other people? The point of advertising is to offer customers goods and services they might need. It is not about tempting them to buy something they might want. There is a vast difference between necessity and desire. I don't think that anyone remembers this anymore. Copywriters nowadays imagine that their job is to arouse desire. They do this by pinpointing a perceived need, but it is rarely a true need.
The first person we have to tell the truth to is ourselves, If we are not going to be honest with ourselves, there is little possibility that we will treat anyone else differently. This is a thorny issue. I sometimes sweep everything under the rug by saying "Perhaps I am naive..." when I don't want to admit that I have been foolish enough to hoodwink myself. I pretend that someone else has seduced me into seeing it her or his way, but if I went along with it, then the responsibility was mine just as much as the other person's. Time and again the evidence is right in front of me, and yet I choose to ignore it. I may call it "giving someone the benefit of the doubt," but we all know that this is often a reluctance to face the facts. We prefer to believe the illusion we have laid over the situation. We need to look at our relationships with other people as well as our relationship with ourselves. Both are generally a little murky.
Many people felt that my marriage was a mistake (and some were unkind enough to tell me so. Honesty is fine when it is called for, but kindness should temper full disclosure). When our marriage fell apart, these people were quick to blame my husband—which was unfortunate. I don't believe that he was any more at fault than I was. We both chose not to see certain things in ourselves and in each other. Like many couples, we put on blindfolds when we fell in love. Years later when he was in therapy, he said to me, "I am just coming out of my cave. When we were married, I didn't even know I was in a cave." I had never met anyone who lived in a psychological cave. He certainly seemed more private than anyone I had been close to, but I had put it down to shyness. I thought that once we were living together, that would fall away. But people are the way they are, and they don't change just because they move in together—at least, not in my experience. What you see is what you get. The other person may not be seeing you very clearly, so you need to be certain that you see who he or she is.
While I was typing the last paragraph, I received a call from a man in Florida who had courted me shortly after the breakup of my marriage. He had called me out of the blue two days ago, saying that he had been sorting through some of his papers and had come across my name. It transpired that his own marriage had now fallen apart, and when he saw my name he recalled the feelings he had had for me and felt impelled to get in touch. We talked for a little while, and I was as free and easy with him as I had been all those years ago. When he called just now he told me that he had been thinking of me very warmly for the last two days and he was wondering... You can imagine the rest of the conversation without my filling you in. I told him that I was happy for us to be friends but that it could not go any further than that. I had refused him over twenty years ago (I had told him gently that I could not marry anyone without a sense of humor), and the situation hadn't changed. He was sad. He had been building up this whole picture in the interval between Monday and Wednesday. All I could do was read him what I had written in the last paragraph and point out that it applied to me, to Neil, but also to him. We have to face the truth.
There are certain sentences and phrases that really hit home when you first hear them, and then continue to reverberate throughout your life. For me, one of the most powerful is the statement in Isaiah 43: "Ye are my witnesses, saith the Lord." Witness is a strong word, and nowadays we tend to think of a witness as someone who observes dispassionately. The original meaning of the word is much deeper. The root is the Anglo-Saxon witan, "to know," from which we get the expression "to have your wits about you." So witnessing has to do with transparency, seeing through the scrim that generally covers things and looking deeply into them so that we know and understand what they truly are. When God says, "Ye are my witnesses," it means that the role of human beings is to know divinity and to know it now. The sentence is in the present tense. We are here to see and experience God in all things. Or, as Kabir Helminski describes it in his book A Knowing Heart: A Sufi Path of Transformation: "The true human being embodies the divine presence."
So often we see only what we want to see. Our desires get in the way of our observation, and we look at part of something rather than all of it or we distort what is right in front of us in other ways. Everyone wears different glasses—metaphorically speaking—and has a different prescription. Some people have an astigmatism (I do). Some like to wear tinted glasses. And then there are those who like to wear dark glasses or shades. Presumably they like to look on the gloomy side.
When I was growing up, history was taught as though the account we were given was the only view. Many years later, when I started working at Knopf, which had a fine American history list, I gradually came to the realization that there could be as many views of what took place at a battle as the number of soldiers who fought in it, not to mention all those who reported from the front or wrote about it later. One of the most honest titles for a memoir I have come across is Nicolas Bentley's A Version of the Truth. Nick was a cartoonist and also an editor at André Deutsch. He understood that no perspective is complete. So much of the time we see things a certain way because of our background and experience, and we fail to grasp that what we are aware of is only one of a multitude of possibilities. In addition, all of us interpret (or misinterpret) circumstances differently. Each of us focuses on different facets and glosses over what doesn't interest us or those things we would prefer to avoid (whether this is something we are willing to admit or not). Consequently, everyone formulates an individual picture of what is happening. No two people inhabit precisely the same spot on the planet.
This became even clearer to me when I was attending an art class and the week's assignment was to draw the same object each day from a different angle. I was astonished to look at the variety that appeared in my sketchbook. The following week we were asked to focus just on the spaces described by the object we had chosen and record them on the paper. I was again fascinated at how different a chair seemed when I drew the gaps between the decorative woodwork slats on the back, then the trapezoid shapes caused by the legs, and finally the edge of the enormous space surrounding the whole chair. The third week we were told to choose a photograph of a familiar object and turn it upside down before we drew it. It is extraordinary how this transforms your view.
These three exercises revolutionized the way I saw the world. As I write this I am sitting outside my friend Lynn's house in New Jersey, observing the slivers of the next-door garden visible between the trunks of the trees at the edge of her property. Normally, I would focus on the adjacent garden and ignore the trees in the foreground. My mind would fill in what was hidden from view with what could reasonably be expected to be there. The trees themselves would vanish from existence, because subliminally I considered them to be an obstruction. Now that I remember that art class, however, I see the blocks of space between the trunks and branches which all slope in different directions—a geometrical profusion of possibilities. I become aware of the distance between me and the trees, and also of the grass and bushes and wildflowers, the play of sunlight and shadow, the unseen area behind my back, the invisible birds calling to one another in their liquid language, the rustling leaves, and the fading drone of an airplane in the sky. My perspective shifts with each of these acknowledgments, and my appreciation of the garden expands. But, of course, the ant scurrying past my foot is probably unaware of most of this, yet cognizant of a whole universe I shall never know.
It is good to bear all this in mind when you read or listen to other people's stories. Recognize that while they may believe that what they are telling you are the facts, their "truth" undoubtedly comes from a limited viewpoint.
This is probably as good a place as any to make a disclaimer about the stories I tell in this book: I have not knowingly distorted anything, but as I grow older my memory is increasingly unreliable. If those who appear in my stories remember the circumstances differently, I hope they will forgive me. What I describe here is the way I perceived things at the time they happened, but, of course, I was looking only from my own point of view, and, in addition, much time may have elapsed in the interim.
Here is the way that Jungian analyst Helen Luke describes the process in her book Old Age. She reimagines Hermes speaking to Odysseus on his final journey of "a spirit of discriminating wisdom, separating moment by moment the wheat of life from the chaff, so that you may know in both wheat and chaff their meaning and their value in the pattern of the universe."
Nowadays many people no longer understand what it means to discriminate. They think that it is choosing one thing over another, but, in fact, discrimination is as Helen Luke describes it: recognizing exactly what's what; not being confused about the nature of things. Adultery happens not just between consenting adults. To adulterate is to mix things so that they are no longer pure, whether the substance is metal, food, or the present moment. In fact, if you remain in the present and avoid impurity now by not importing the past or the future, then the chances are that nothing will spill over and contaminate other moments.
ELEVEN
What Nourishes Us
Whatever we put into our bodies, minds, and hearts is what feeds them; but I am not sure that we appreciate the implications of this, and we are often very careless about what we consume. We think, "Oh, it's just this once," but these "onces" become habitual and can soon add up to a lifetime of neglect.
Our life depends on the air we breathe, yet we draw air into our lungs without really considering where it is coming from and what it will do to us. We know the difference it makes when the air is crystal clear and we take deep drafts of it as we stride across the fields. We feel invigorated, and everything around us sparkles. Contrast this with how we feel in a stuffy train or office where the air is recycled and who knows when there will be any new oxygen available. The mind becomes dull and the body sluggish. More and more places are suffering from air pollution because neither the authorities nor the population take the consequences of their actions seriously. I was recently in Shanghai, a city of thirteen million with a floating population of another three million itinerant construction workers, and I learned that all the children there are already affected by the polluted quality of the air, caused by heavy traffic.
When I grew up in England, everyone kept their windows open a little, no matter how cold or wet it was. We believed that the way to remain healthy was to keep the air circulating and allow fresh air to enter the house at all times. I still believe this and shut my windows only on those rare days when fierce winds cause grit to accumulate on the windowsills. I can't honestly say whether it is because I have a strong constitution, I eat a healthy diet, or because the air in my home is always fresh, but I rarely suffer from colds and flu the way other people seem to. In addition to breathing fresh air in our homes, we need to get out every day and walk vigorously, so that our blood doesn't start to curdle. This will do wonders for our constitution and doesn't cost nearly as much as membership in a health club.
We also need to be careful what we drink. I don't subscribe to the school of thought that says we should imbibe eight glasses of water a day. Each person needs a different amount of liquid, depending on her or his build and activities. For many years all I drank was three cups of strong coffee a day and perhaps a few mouthfuls of water at lunch, but on reflection I don't think that this was a very wise habit and I am lucky that my body didn't rebel. I do know that drinking a great deal of anything during a meal is harmful to the digestion because it dilutes the digestive juices both in our mouths and in our stomachs. I prefer to drink either before or after a meal, and I have found that my system prefers it too. I also drink only when I am thirsty. Whatever your favorite beverage is, take a moment to consider how much of it you drink each day (be honest) and what the cumulative effect of this consumption may be.
A simple and delicious way to eat is to choose food that is as close to its natural state as you can find and then enjoy it without adding or taking anything away from it, that is, without cooking or seasoning. Lanza del Vasto described it as "putting as little space and time as possible between the earth and your mouth." Choose fruits and vegetables that are fresh and in season and that have not traveled too long or too far before you buy them. If you can discover fruit that has ripened on the tree and was not sprayed in the process, so much the better. Then eat it at its best, with the taste of the sun still there. Resist the temptation to garnish everything, and you will find that an avocado, if it is a good avocado, has a flavor all its own. In fact, each avocado (or apple or apricot) seems to taste completely different from any other you have ever eaten.
From time to time I stock up my shelves with dried herbs; beans in all colors, shapes, and sizes; and a variety of grains, cookies, and crackers. They form a wonderful display in their glass jars, but months go by and I forget to use any of them. What I actually consume are the things in my short-term memory—whatever I have bought in the last few days. I go to the fruit and vegetable market and buy whatever is fresh and firm (beware of produce that is bruised or flabby). I am, after all, the granddaughter of a Covent Garden wholesale fruit and vegetable merchant. I do not buy more than I can use within two or three days because then it will no longer be fresh. This is hard to stick to when I visit a farmers' market because there is so much wonderful stuff, but you have to be stern with yourself and buy with your head and not your belly. I also find it difficult to restrain myself when it comes to quantities of wondrous-looking fruit and vegetables. When I am putting them into plastic bags, I have to remind myself to buy enough for only one dish. This is particularly hard when I am cooking just for myself. But I believe that it is a crime to take home more than I can use.
When the moment comes to prepare a meal, I look in the refrigerator and see what is there and what combination of foods seems right for the day and hour. This is just the way I choose what to wear in the morning. The resulting meals (and outfits) can be quite stunning if you don't have preconceived ideas of what goes together and what doesn't. It is not that I never use cookbooks, but I tend to use them for inspiration rather than information. If I have someone coming to dinner, I occasionally consult a cookbook. I leaf all the way through and always come up with a recipe that includes ingredients that are not in season. This is because we are always attracted by something unavailable (or, at least, I am). Then I try to figure out how I can adapt the recipe I have chosen for ingredients that I can actually find.
Even if you have a large family, try to estimate quantities accurately. Almost everyone would prefer a new dish rather than leftovers day after day. Yes, I know that nowadays there are such things as freezers, and people are always encouraging me to make enough for several meals and tuck portions away in the deep freeze. But psychologically this does not work for me. I just can't believe that eating food that is canned, bottled, dried, or frozen is as good as eating produce that is only a few hours old. Of course it is possible to survive by eating foods that have had all these things done to them, but over the long haul I suspect that they take their toll. You can also be nourished by foods already prepared and available in stores; however, you will be far more nourished by food you have prepared yourself, and it will also be cheaper and simpler.
I well remember going home to England shortly after I had become a strict vegetarian. My mother had saved the first fruit and vegetables of the season so that she could offer them to me when I visited her, but she had frozen each and every one. I adamantly refused to eat any of them, even though she claimed that they were "freshly frozen." I have relaxed quite a bit since those days. I am no longer a complete vegetarian, and when I am in someone else's house, I do my best to eat what is put in front of me. It probably won't kill me, I may never go to heaven (being a vegetarian is probably not a good enough reason for ending up there), and it is rude and unkind to make a fuss if you are a guest unless the food is going to make you ill.
At one period in my life, I ate everything raw except for bread and cheese, and I also ate only one thing at a time. I shared an apartment in an old Federal house in the Village with two other women, and we would bake our own bread and put an extraordinary selection of fruit, vegetables, cheese, nuts, and honey on the table every day. Whatever we had in the house was offered at every meal. This simple diet is not as boring as it may sound because the variety is constantly changing. Visitors were wide-eyed at the cornucopia.
Each of us would choose what to eat by consulting our stomachs. Generally we carry around some idea about what we would like to eat, but it is often a desire we have been nurturing for some time. If you sit at the table and actually ask your body what it wants, the answer may surprise you. So you might start with some cherries. You would put on your plate only as many cherries as you were sure you could eat. When you had finished them, you could certainly take more or you could move on to something else, but you couldn't combine foods except with bread, i.e. bread and cheese or bread and honey. This measured way of eating gives your body just what it needs at any given meal, and there are never any leftovers.
Perhaps you are wondering how this works with children? In my house it worked very well. I provided a large selection at every meal, and Adam could pick anything he wanted from what was there. Because everything was already on the table, he was not tempted to ask for something else because he knew we didn't have it. Children like a choice, just as we all do, and they should be given the freedom to make it. Having in the house only foods I believed would be good for him meant that he exercised his choice from what was available and I never had to say "No."
Eating like this simplifies life enormously. The food is quick and easy to prepare (most of it just needs washing or arranging on a plate), there are no pots and pans to cope with afterward, and when you rise from the table, you never feel weighed down.
I no longer eat this way (except occasionally when I am lunching at home alone) although I really like it, but if you have never tried it, do experiment with it a little. It really fine-tunes your ability to recognize and respond to what your body actually needs. The way most of us tend to eat is by putting a lot on our plates and confusing our taste buds by putting two or more things in our mouths at the same time. Keep in mind that if fruit and vegetables are not in a good enough condition to eat raw—if they have no taste or are not yet ripe—adorning them with a sauce or adding flavoring other than salt is not going to benefit you or your stomach. Nowadays I cook my vegetables by steaming, stir-frying, or roasting them in the oven with a little olive oil, and I keep the cooking time to a minimum.
I have focused on fruit, vegetables, and grains because they are the foundation of what I eat, but much of what I have said also applies to meat and fish. The principles are what are important here.
One thing that may change your attitude toward food is saying grace. In 1992, Bell Tower published a little book by Marcia and Jack Kelly entitled One Hundred Graces. In the introduction I wrote about the function and nature of mealtime blessings:
Saying grace is an ancient and vital tradition the world over. To begin with, it provides a space, a moment of stillness, in which to relinquish the activities of the day, and allow the mind to settle. Then, as we acknowledge the source of our nourishment, we are filled with astonishment at the generosity of the Creator, with gratitude, and with praise. In bringing the body, mind, and heart together, we come to ourselves, and remember who we are and why we are here. For some families, a meal is the only time everyone is present and so the opportunity to enjoy one another and really celebrate the occasion is not to be lost. For many, a meal is also the only time that there is any memory of the Divine. Saying grace establishes an immediate connection with that memory. In such a moment, when our minds are clear and the truth is reinforced by being sounded aloud, we can dedicate the meal and the strength we receive from it to the service of whoever or whatever is before us.
Once you have the food in front of you (cooked or raw), the next thing is to remember to taste every mouthful; otherwise, it is such a waste. How many times have you wolfed something down because it was your favorite food and realized when your plate was empty that you did not actually taste any of it? So smell it, taste it, chew it, and swallow it only when you are sure you have experienced it.
I am not going to talk about smoking or taking drugs because I have little experience of the former and absolutely none of the latter. I have never found either habit appealing, and I cannot believe that introducing these substances into our bodies can be beneficial in the long run.
During my pregnancy I amassed a few books on child care that I trusted would stand me in good stead for the first few years of my baby's life. I was ill for most of the nine months and did not feel like focusing on how to cope once the pregnancy was over. Once Adam arrived, there never seemed any time to read, so the books remained almost untouched on my shelf until he grew up. I did consult Dr. Spock once but wasn't thrilled to discover that he offered three solutions to whatever the problem was. I had been expecting just one foolproof answer, and I never tried Spock again.
However, I got the very best advice one day from a huge black grandmother sitting next to me on the Broadway bus. I don't know how we got into conversation, but to this day I can hear her telling me that the most important thing you can do for a boy is give him his independence as early as possible. What she said went right in and stayed there.
I wasn't one of those mothers who fret about their children when they leave them at home with someone else. I stayed home for the first seven weeks, but Neil was out of a job, and, to be honest, I itched to get back to the office. So I started work again quickly. I worked at home three days a week and on the other two days went in to the office from 10 a.m. to 3 p.m. I found a friend with a small child. She was willing to earn some extra money by coming to my house to look after both her child and my own two days a week, and that left me free for a few hours. It never occurred to me that disaster could strike in my absence, and Adam tended not to notice that I was leaving. I kept up this schedule for a year, but after the first six months, he was so active that it was hard for me to get any work done at home. He wanted to be out in the park running around, and I felt tied to the telephone during office hours.
So then I found a wonderful West Indian woman who had a raft of children of her own. She came to my house every day, enabling me to go back to the office full-time, and Adam would often go home with her for the weekend and sleep in a big bed with several other kids if I had to be away for some reason. People regarded me with some horror, I think, because when I left town, I didn't call every day to find out how Adam was and speak to him. I reckoned that he was probably fine. If he wasn't fine, she would let me know. After all, I had left him with a woman I trusted completely. It seemed to me that calling him all the time would remind him of my absence, and he was better off without this thought.
Once I returned to work, the most difficult moment was coming home in the evening. As I walked through the door, both Adam and Miss Grace, our blue-gray Persian cat, would fling themselves at me, wanting both food and affection. There was no chance to sit down for a while and catch my breath. I talked to Ram Dass about this, and he pointed out that I was teaching Adam how to come home and deal with whatever the situation was without missing a beat or losing my temper. This was an important lesson he might not learn during the "quality" time I never seemed to have an opportunity to spend with him.
Just how quickly independence took over became very clear on Adam's first day at preschool. When I returned from work, I was anxious to know how everything had gone. (You guessed right: I was not one of those mothers who hovered in the wings for the first day. I took him to the front door and simply handed him over with a hug.) So I asked him to tell me what he had done.
"Ate cheese sandwich," was the only response I got. Because I myself had made this cheese sandwich, his eating it was not a great surprise. I was curious to know about everything else, but he either couldn't or wouldn't tell me. Perhaps the cheese sandwich was the high point of his day. It was then that I realized that never again would I know what he was up to all day. But he was fine. He was enjoying everything, and now he had lots of company to play with and learn from.
Although motherhood was the most exhausting experience of my life, it was also the most rewarding. The strength of the bond that I felt with Adam was something I had never anticipated. And then there were all those moments that are etched into my memory forever.
I remember him asking me one afternoon when he was a year or so older, "Does God have a birthday every day?"
I wasn't sure where he had picked up this idea, but I liked it.
"Yes," I responded. "I think you could say that he does. He creates everything new every day."
"And does Mrs. God celebrate his birthday with him?" Adam asked next.
God, it seemed, was often on his mind. There was another occasion the same year, when he was six. This time he wanted to know:
"Why didn't God interfere when they were trying to kill his son?"
"God doesn't interfere. He just makes the rules and watches to see what happens," was the best I could manage. (Why didn't Dr. Spock have some ready answers for this kind of question?)
"I would have interfered," Adam announced.
"So would I, if someone tried to kill you," I said.
"It's a shame Jesus didn't become an angel before he died," was his next remark. This was a little bewildering. I wondered where he was going with this one.
"Why do you think he should have become an angel?"
"Because he was such a nice man," Adam replied. I had never heard Jesus described this way, but it was probably true. Still, I was curious:
"What makes you say that?" I asked.
"He always wore a sun on his head," came the answer.
Finally, there was a conversation about death. I don't think that any of us are ever ready for this, but we do our best. We were walking down Broadway one day when he asked me what happens when we die.
"No one really knows," I responded, "but many people believe that first you have a rest with God for a little while and then you come back here and live another life."
"I'm not coming back," he announced with some conviction.
"Well, I don't really think you have a choice," I offered, no longer certain of my ground.
"You mean I may have to come back, whether I want to or not?" said Adam. And that is where we left it.
It became increasingly clear to me that not only does a baby in the womb use all a mother's resources—literally feeding on her body—but this process also continues throughout childhood. A mother provides nourishment for her child at every level, whether she is aware of it or not. Some of this food may be good and some not so good, but everything that she thinks, feels, says, and does affects her offspring as well as her. There is no holding any of it back. It is an open system.
When he was about nine years old, he told me that he wanted to come home from school on his own. He felt too old to be met by a baby-sitter every day. I would walk him to school in the morning before I went to work. I would take him as far as Seventy-eighth Street and West End Avenue and watch as he crossed the street and entered the gate of Collegiate School. In my heart I would say good-bye to him, knowing that there is never any guarantee that we will see our loved ones again, although we always count on doing so.
I was comfortable (well, almost) with the idea of his walking home on his own. I had taught him not just to check the traffic lights before he crossed the street but to watch the eyes of the driver of each car. I explained that some drivers weren't very awake and you could not always count on the fact that they would obey the lights. But if you observed them carefully, you could easily tell what they were planning to do.
Neil freaked out when I told him that Adam was now making his own way home, but I explained firmly that this was New York and the sooner Adam learned how to be self-reliant the safer he would be.
As it turned out, only one incident occurred on the way home, and it was about a year later. I was very proud of the way Adam handled it. He was walking up Broadway where there were always lots of people, and several boys he did not know jumped him and ran off with his backpack. He chased them down the block and eventually, I think, they abandoned his backpack because it was so heavy. Then he went to the nearest phone and called the police to report what had happened, before walking the rest of the way home.
How we choose to feed our minds is just as important as what we put into our bodies. At every moment of the day we are receiving impressions through sight, sound, smell, taste, and touch. If we are deprived of these sensations, we start to feel disembodied. So we need to take a fresh look at everything we are inviting into our minds.
It is no secret these days that older people who keep learning new things enjoy themselves and live longer than those who don't. In addition to stretching the mind with some form of study (it doesn't have to be something new, but it is good to keep the mind engaged in intellectual pursuit), have a look at the books you read, the art and sculpture you admire, the music you listen to, and so on. Spectator sports such as football and the Internet are fine occasionally, but we need to watch that they do not become a steady diet so that we forget to eat anything else.
On subways and buses when I am forced to share some of the music other people are listening to because their headphones are leaking, I worry about what such an insistent beat and screech is doing to their brains. Some music appeals to our sense of rhythm, some to our emotions, and some to our intelligence, and we need to make sure that none of these three is being starved. There are times when I realize that I have spent too long in front of the computer screen and I am all in my head. This is the moment for me to put on some Cuban music and dance myself loose for a little while, but I wouldn't dream of listening to this music all the time. There needs to be a balance here as in all things. There are times when I yearn for Mozart or Hildegard of Bingen, and there are times when I don't.
Again, I am concerned when people I know spend a lot of time watching horror or action movies. I go to the movies to be moved (isn't that why they are called "movies"?), either to tears or laughter. I don't want to be shocked or scared for entertainment. There is enough of that in real life.
Then there is conversation. How rare it is these days to participate in a real conversation. So much of what we share with our friends is gossip, complaints, conjecture, or reminiscence, and none of this is nourishing in the least. For three years I hosted a monthly meeting for friends and acquaintances for whom this kind of talk was no longer enough. We gathered together for a couple of hours, beginning with a short meditation and then proceeding to an open discussion of "what really matters." There was no set theme, no one was in charge, and once someone had begun to speak we all listened carefully without interrupting. The one rule we had was that during this short and precious time we wouldn't talk of anything else. Just once a month we attempted to devote at least two hours to speaking the truth.
We all need good company, friends we can depend on for spiritual sustenance. It is not always possible to meet on a regular basis the way we did, but everyone needs friends and relations who will always be there for them and vice versa.
Each day it is good to set aside some time for nourishing the body with exercise, the mind with reading and contemplating scripture or wisdom teachings, and the spirit with some form of devotion such as meditation or prayer. These things are as vital to us as a whole person as those foods with which we choose to feed our bodies. For each person, the timetable for doing this will be different, but I have found that starting the day with yoga before breakfast, and meditation and reading afterward, is a good preparation for everything that lies ahead. Then, again, in the evening, it is good to preface it with more exercise (in my case, tai chi chuan) and meditation so that there is a clear separation from the workday. The quality of what we feed ourselves on every level will manifest itself in our lives and in our relationships with other people.
TWELVE
A Balancing Act
Nowadays most of us feel as though we are being dragged along at breakneck speed in directions we would really rather not go. Somehow we need to find a way of slowing down, stepping off this path, and reestablishing a balance in our lives. It might be helpful to think of it as reeducation in that it is not as though we need to introduce something new. We already have what it takes. It has just become a little buried, and we need to lead it out to pasture again. The Latin root of educate does indeed mean "leading out." We need to slough off the accumulation of so many of the cultural ideas we have acquired over the years.
Balance has to do with remaining in the moment, neither looking back wistfully nor looking too far forward. The not-looking-back is a lot easier for some people than others. I happen to be one of those lucky people who don't miss other people and objects once I am no longer in their presence. This may be because I went away to boarding school at a very early age and felt compelled to learn self-sufficiency. I don't know. What I do know is that this sometimes causes distress to my mother. Every time I fly back to New York, she says, "I'll miss you. Will you miss me?" And I tell her that I wish she would not ask that, because she knows that I rarely miss people. My answer can only upset her. If I am with people, fine, but if not, so be it. I don't like to be torn between two realities.
The same goes for looking too far forward. I try not to think about what may happen at some event that I hope to go to. You never know these days. Maybe you'll get there but maybe you won't. There is an infinite number of possibilities in any given moment, but we endeavor to control our destiny most of the time. The older I get, the less set in stone things seem to be. If I don't have expectations that are too high, I will not be disappointed.
For instance, when I knew that a friend from northern California was going to be in New York, I purchased tickets for a film festival. Later I discovered that she had changed her dates and would no longer be in town that day. I was disconcerted, but I invited another friend to go with me. Then, a few days before the event, I had to fly to London to be with my mother, who had suffered a stroke. I mailed both tickets to my New York friend, and she asked someone else to accompany her. At the last minute she herself fell ill and had to give both tickets to her friend. Luckily, neither she nor I had anything but mild regret over what had happened.
Living in the present doesn't mean that you can't or shouldn't plan ahead. You need to have an intention in mind when you embark on something, but the thing to do is make plans and then wait to see what happens. There are no guarantees. For instance, last fall I gathered seeds from both the red and the pink hollyhocks in our community garden, and I sowed them all along the edge of the path. I had this vision of an army of old-fashioned blossoms guarding the path like sentinels. Now that it is early summer I am watching to see if any or all of them will come up. Life is just like this. You can sow all kinds of seeds but you never know which ones will germinate or what will befall any of them once they sprout.
Another way of allowing more freedom into the mind is not holding fixed ideas about people and things. This is a tough one. Over the years we observe how other people behave in certain situations, and we come to the conclusion that they will always react in the same way. But in our heart of hearts, we know this is not true. For instance, I have built up a picture of one of the doormen in my building as being an old curmudgeon, someone who never does the right thing. It is true that ever since he had a heart attack his behavior has been erratic, but he means well most of the time, and the fact that I carry around this negative idea most probably produces the result I expect.
I once published a book by Jean Liedloff called The Continuum Concept, which described how children almost without exception do what you expect them to do. If you are afraid that they will fall into the fishpond, they will. Just keep the idea in your mind. You don't have to say anything out loud. Children have an uncanny ability to connect with what you are thinking and feeling rather than what you may be saying. Your mind is the environment in which they dwell. In the years when I was bringing up my son, I expected him to go to bed when I asked him to, and he always did. I myself had always gone to bed when I was told to when I was small. It did not occur to me that Adam would not go willingly to bed and then to sleep, and so it did not occur to him either. I had a theory that the reason so many American children seemed to have problems at bedtime was that the parents were not sure whether the children would go to bed, and so the children were not sure either: i.e. the parents created the problem in the first place.
What I am suggesting here is that we hold neither positive nor negative ideas, that we try to stay as present as possible and just see what happens. I know this is hard, particularly if you have a history with a friend who seems to cancel or postpone every appointment you make. But if you can bring yourself to do this, both you and the friend will taste a wonderful freedom in your relationship with each other.
Lately I have become very aware of the way we introduce stress into our speech, particularly in New York City. It is almost as though we are beating time as we speak. We come down stronger on some parts of a sentence than others. We probably think of it as "emphasis," but it is a very different way of speaking than is customary, say, in England. If you listen to an English person say "the White House" or "the weekend," you will notice that all the syllables have the same weight. They are evenly balanced. Yet in this country we put the stress on the first syllable, and the second one is almost insignificant. I suspect that we have built the stress we are all experiencing into the very way we speak.
Those people whose voices are most restful are those who do not add this push. You can hold people's attention more easily if you neither drop nor raise your voice at the end of each sentence. Your audience is left hanging on your words. It is a fascinating phenomenon. Your voice floats on the air, and people's minds have less of a tendency to get carried off into their own trains of thought. One person who seems to understand this is Toni Packer from the Springwater Center for Meditative Inquiry near Rochester, New York. When I was on a retreat with her, once a day she gave a short talk (the retreat was otherwise silent). Her words and phrases came and went, the sound of her voice never falling at the end of each sentence, so that we all felt gentled, then released, and borne aloft like milkweed parachutes drifting up and away.
The way to walk the middle path in relation to money is not so very different. You need to find a way to stay balanced.
I have never made a concerted effort to make money but I was brought up during World War II, and it comes naturally to me to be frugal. So anything I don't really need, I don't acquire (well, of course, sometimes I slip from this, but I have to make a real effort!). When any money accumulates in my checking account, I either give it to a good cause or salt it away somewhere. When I salt it away, I make sure that it doesn't just sit there. I treat it like one of the plants on my windowsill. I water it and watch it carefully, waiting for it to grow. If nothing seems to be happening after a reasonable amount of time, I repot it. This is the method I use for investing, and it has served me well. Although I was a single parent for most of the period during which Adam was growing up, I managed to pay all his school bills without taking out a loan and emerged at the other end with a tidy sum still in the account.
I remember one year in the 1970s when I earned $12,500, and the man I was working for earned $100,000 (I knew this because we were trying to get his tax return in on time). I had saved money out of my pittance and he hadn't, but of course he had two more wives to support than I did. What I am trying to say is that I don't save money in order to save it. It gets saved because I don't spend it as often as other people seem to.
It is really a question of how you view money. I don't make a budget, but perhaps on some level there's a budget in the back of my mind. I don't allocate a specific sum for this or that. If I need something, I always look for the best quality as well as the best value. If I don't really need something, I simply don't buy it.
Although the best quality is often the most expensive, in the long run this usually works out cheaper. The writer Laurie Colwin once took me down to Canal Street, where she frequented secondhand-clothes shops on a regular basis. She was astonished to find that I had never been to one. I trotted around after her all Saturday morning and by the time we sat down to lunch in a café, she was in despair, but I was elated and amused. She felt as though the whole expedition had been a failure. I had watched her buy a dun-colored cashmere cardigan for twenty-five dollars because she admired the buttons. In my opinion, the color was so awful, I couldn't imagine ever choosing to put it on. If this was the case, why spend twenty-five dollars for the buttons? At one point she took me to a store where they had a sale on men's alpaca overcoats. The coats were luscious, and I tried one on. Not surprisingly, it hung on me, making me look like a dwarf. She bunched handfuls of the silky cloth behind my back to show me how it would look if I had it altered, but I pointed out to her that the whole coat would have to be remade to fit me and so in the end it would cost far more than if I bought a coat that was the right size. In addition, I didn't need a new winter coat.
I told her that I had really enjoyed accompanying her because I thought that perhaps I had been missing something by not visiting thrift shops, and now I knew that I had not. Also, it made my heart feel glad to have seen all those clothes and not bought any. I felt as though I had saved a fortune. I was not at all disappointed. I explained to her that I wouldn't be surprised if I spent less money on clothes than she did, even though I usually bought good-quality garments in classic styles. I considered clothes to be a kind of investment. I bought the best, only if and when there was a need, in styles that I knew suited me, and then expected them to last for twenty years. She, on the other hand, sallied forth every week and bought something cheap that might or might not go with what was already in her wardrobe. I would have been curious to tally up what each of us spent over the year, but I didn't want to push her too far. Also, of course, she was using shopping as a form of entertainment, and I was not.
I have to admit that there is one aspect of spending where I often get hooked, and that is with mail-order catalogs. These, of course, arrive in avalanches, but I have learned to distinguish at a glance which ones have clothes in natural—not man-made—fabrics that suit me (nothing tight; there must be plenty of room for movement). Mail order is less exhausting than traipsing around stores, and if you can discover a couple of catalogs that offer merchandise you are consistently happy with, buying this way is a real boon.
I tend to go through my favorite catalogs almost as soon as they arrive in the mail (if you wait too long, the things you want are often no longer available in your size). I turn down the corners of the pages of my selections, and I put the catalog aside for a little while. Catalogs are very seductive, and the only way to deal with them is not to pick up the phone immediately. Exercise restraint. Look at the catalog again about a week later and be honest with yourself: Will you actually wear what looks so splendid on the page or will it lurk in your closet for the next few years, haunting you each time you peer in? When exactly would you wear it? Once I have examined my conscience, I am ready to make my purchases. Sometimes I decide against everything I thought I wanted. Sometimes I go for broke. Sometimes it is half and half.
This is all about not being pulled too far in any direction. We are all subject to impulses, but if we are aware that we are being carried away, we can redress the balance and return to the fulcrum. From there we can move out when the need arises. We are level-headed—not up in the clouds or down in the doldrums. The point of balance is all-important.
One of the characteristics of the mind is that it is always expecting something, chasing after something it wants, or retreating from or resisting something it doesn't want. It has an extraordinary tendency to slip away from the present moment. And yet it is only in the present that anything happens. All spiritual work takes place in the present at those times when we can disengage from the march of progress or whatever it is that is swiftly carrying us away from where we are. Perhaps the present could be likened to being in neutral, ready to move into another gear but not yet having done so. For a brief period we are "free, free at last." No one can stay in neutral, but it is good to acknowledge that it is from this place that the next move comes and to this place that it will eventually return. This is the common ground. If we succeed in remaining still with a situation, a decision, or a problem, instead of scurrying after various distractions, both the mind and heart will open, and unlooked-for opportunities will emerge.
The other night I was caught unawares. I had two guests to supper, and shortly after they arrived, my phone rang. It was Adam's girlfriend asking if he was with me. She had just returned home from work, it was 7:45 p.m., and there was no message from him. This was an unusual situation. Adam is very responsible, and I don't remember any occasion when such a thing had happened before. If he is going to be late, he always calls ahead. I suggested that he might be stuck in the subway, but she uses the same line and had not had any difficulty getting home herself. She had already tried his office, and he wasn't there. Not knowing what else to suggest, I simply asked that he call me when he did arrive home. As soon as I hung up, I started to panic. Earlier in the day he had called to say that he was not feeling at all well, and I counseled him to phone the doctor immediately. So now I envisioned him lying unconscious somewhere.
I tried to bring myself back to the dining room table and the conversation going on around it. I struggled to be a good hostess, but in vain. I was grateful for the demand on my attention, for the company, and for the support it provided, but I felt myself so far from what was taking place in the room that I could no longer connect to it. I was distraught, undone, frozen. My mind went wild, scrabbling around like a frightened animal, and paralysis set in.
Why is it that it is so difficult to accept the condition of not knowing? The times when all there is to do is wait are so hard. Energy arises to meet the need—any need—and the system is flooded, overwhelmed. So it simply shuts down.
Over the years there have been many occasions when I have stood silently at the window watching what was going on in the street below. Sometimes, years ago, I was waiting for my ex-husband and our toddler son to return from the beach. As each light changed and their taxi didn't appear, I would try to calm myself and wait for the next light, the next cab to slow down across from our building. Sometimes I would wait there for an hour and they wouldn't arrive, and I would try not to think of what disasters might have overtaken them. Sometimes I stood there waiting in vain for my lover to return. The truth is that I have stood there over and over again in different situations, but always the people I was expecting have eventually arrived and no harm has befallen them. Still, this time it might be different... Fear is always lurking around the corner.
On this occasion, Adam turned up after an hour. He had thought that his girlfriend was arriving home from work later. He had gone to a video store and lost track of the time. He seemed taken aback that we had been worrying about him. As soon as I knew that he was safe and that nothing was wrong, I lay down on the sofa and closed my eyes, allowing all the anxiety to seep away. I became aware that one of the women had her arms gently around me and that the other one was clearing the dishes. I just let go.
Somehow I, we, need to learn to let go at the beginning and not just at the end. We have to find a way to remain centered. This is a big lesson that most of us put off learning all our lives.
There is something in the third chapter of Genesis that addresses the question of fear and presence. Remember when God calls to Adam and says, "Where art thou?" and Adam responds, "I heard thy voice in the garden, and I was afraid, because I was naked; and I hid myself." This is the first time in the Bible that God asks man where he is. Until then, presumably, this had not been necessary because Adam and Eve were simply present. But once they had eaten of the fruit of the tree of knowledge of good and evil, they wandered and their minds wandered, and both God and they no longer knew where they were. Once fear enters, we are no longer present. Fear is fear of the unknown. In the present, fear doesn't exist.
There is actually a reference in the next chapter that I have always found both fascinating and relevant. It says, "And Cain went out from the presence of the Lord and dwelt in the land of Nod." When I was studying Hebrew I discovered that the word nod means "restless" or "wandering." I don't know why the word was not translated in the King James Version. Giving it a capital letter like that makes us think that it is a country in its own right. I believe that what is meant here is that after killing his brother, Cain no longer knew how to be still and in the present moment, and that he became a wanderer, or nomad.
I always have the sense that traveling in a plane is very similar to being out of gear. The moment a plane lifts off the runway into the air is a moment of sheer delight. My heart experiences a release—of attachment to the ground, perhaps—and I am filled with a deep smile. Once you are up above the clouds with their fleeces turned toward the sun, you are in a hiatus, having left one place and not yet reached your destination. This in itself can be a very liberating experience. On one such journey to London about twenty years ago, I made the following observation:
I have just seen that it is attachment and identification that prevent true work. When one is content and there is no desire, no preoccupation claiming the light and strength of consciousness, then the mind naturally dwells in the present. Desire exists only when tied to a future, not-yet-arrived moment. The subtle attachment to an illusion is what produces tension—a net drawn tightly across the surface of the mind preventing entry.
We have all had the experience of trying to do something while our attention was still on something else, and thoughts seemed to bounce off our minds. It is only later that we realize that we were not open to whatever needed to be done. Recently someone brought me a publicity release to check through. I had been in an all-day meeting and was on the phone to an author. I read through the release and made a few grammatical corrections, but it was not until the following day at home that I came to and remembered that the reason for publishing the book in question had been completely omitted from the release. I had been so caught up in what I had been doing, in addition to everything that still had to be done, that I had forgotten to come into the present. The lines of tension surrounded me and prevented me from appreciating the situation clearly.
Perhaps the reason so many of us feel driven by exterior circumstances, be it household chores or the volume and intensity of our workload, is that we often have a fixed idea of what we still have to do (or are avoiding thinking about it) and when we believe we should get it all done. None of us can measure up to this "tyranny of the shoulds." For some strange reason, we have the impression that everyone else is living a perfectly ordered life, even though it is obvious that they are not. We all have things that never quite get done, but the truth is that however long you live and however hard you work, you cannot finish everything. Susan Strasser's history of American housework expresses this perfectly. It is called Never Done.
Some of the pressure we feel comes from the fact that we focus on what we haven't accomplished rather than what we have. My friend Sarah Jane once pointed out that after God had worked for six days, he saw that everything he had made was very good and he ceased (shavat) from his labor, but that is not what we do. Almost all of us don't see or don't admit that what we have done is good (let alone very good), and we rarely stop working because we believe that whatever we have done is not quite good enough. The concept of a Sabbath, first described in the Book of Genesis, was and still is a revolutionary one. It is important to stop at a certain point, no matter what is happening.
Observing the Sabbath is in many ways like practicing meditation. When I began to meditate, it seemed as though I could not possibly fit it into my already busy life, but I found that if I put it first, then a measure of rest entered my life and from that place I was able to move on, refreshed and restored. It was as though the day expanded to accommodate these two half-hours.
It is the same with reserving one day a week for respite, one day when nothing is scheduled and we are free to delight in whatever arises—a day to acknowledge the divine and be glad. On this day we focus not on doing but on being.
It is our addiction to "doing" that causes much of the trouble and all the frenzy. We cram more and more into our days, and none of it ultimately satisfies us because we ourselves are the driving force behind it, seeking to achieve this and that, and coming up empty-handed every time. All this activity doesn't bring us the serenity and contentment we seek. It just exhausts us. If we stop to think about it, whatever it is will either get done or it won't. If we were to die today, either someone else would take care of it or not.
It is the claim that we put on this doing that is the problem. Somewhere deep inside us we believe that we are what we do. We identify with our actions. We invest ourselves in every action, under the illusion that if we are not doing something, then perhaps we don't exist. Invest means "clothe in." It is a habit, something we don. Somehow, we persuade ourselves that it is our responsibility to do every job. We become identified with both the work and the results. But the truth is that it is not our work: it is the work. If we can find a way to relax our grip on our actions and what comes out of them, there is great freedom. Just watching the activity rather than becoming completely identified with it is restful rather than exhausting. Nowadays people will claim almost any work they think they have to do. I hear people say that they have to do awash when what they mean is that they have to carry laundry to the washing machine and press a button. However, because they believe that they are doing the washing, they may be using up as much energy as they would if they had to go down to the river and beat the sheets on a rock.
The space in the mind that I refer to from time to time has many blessed qualities. In a way, it is this space that is the fabric of the universe. Everything we do happens in this space.
Thirty spokes share the wheel's hub;
It is the center hole that makes it useful.
Shape clay into a vessel;
It is the space within that makes it useful
Cut doors and windows for a room;
It is the holes which make it useful.
Therefore benefit comes from what is there;
Usefulness from what is not there.
—Tao Te Ching, by Lao Tsu, chapter 11
In the Hindu tradition there are five elements: earth, water, fire, air, and ether, or space. Space is often the element that is overlooked, and yet it is the one we yearn for—the one where we feel at home. Without space, there would be no place for the other elements to manifest themselves. Or, to put it another way, without a "here" there can be no "now." Once we begin to catch sight of our desires and then let go of them, we can take an extra step: We can actually rest. It has been said that true rest takes place only between one desire and the next, between the moment when you have relinquished one thought and before you have been hooked by the next. When worries awake me in the middle of the night, it is obvious that I have not been resting. I was not able to surrender my problems before I went to sleep. I am not resting either in my bed or in the infinity of space.
THIRTEEN
Law and Order
There is a strong tradition of cleaning in my family. Most likely it started with our Victorian nanny, Miss Rizpah Smith (known to one and all as Moth, because one of the incipient maharajahs she had looked after in India felt that her white hair made her look like a moth). It continued at boarding school, became honed at work parties at the philosophy school, and is now second nature. I have always suspected it is true that cleanliness is next to godliness.
As you have probably gathered, things in my house are neat and orderly. Moth used to come to stay with us when we had grown up, and she always said that looking into my closets made her uncomfortable because they were so tidy. My rejoinder was that she had only herself to blame because I would never have learned this skill without her.
Some people probably find my passion for having everything in its place a little unreasonable, but I simply cannot rest if my apartment or desk is messy. Once my surroundings are in order, my thoughts are sorted out too. Leaving everything ready for its next use means that you do not have to worry about what you have left undone. If you come to visit me, you will never catch me unprepared. I would not dream of going out in the morning without having made my bed. I do not go to sleep at night without washing the supper dishes. If a garment gets stained or torn, I clean or mend it the same day. Readiness is all. I understand that it may be hard for other people to be this organized, and I certainly don't expect it of them, but I can report that it does wonders for your mind—and perhaps even for your heart.
Neil once told me about a woman he knew who couldn't face ironing. She used to stash all the wrinkled clothes in the closet, and at one point when she hadn't ironed for three months, she had to take a running jump at the door to get it closed. This tale sounded apocryphal to me because I simply couldn't imagine such a situation.
When Adam was a teenager and from time to time I got him to tidy up and clean his room, he would admit (without prompting) that he felt a great deal better afterward and he didn't even know the principle behind what I had asked him to do. I was never able to understand how he managed to do any homework in his room the way it usually looked. As I get older, it's so easy to lose track of things that if I do not put them in the right place to begin with, I may never find them again. And even when I do put them in the right place, they seem to walk off sometimes.
In my experience what goes on inside us is reflected on the outside, and vice versa. Perhaps they are two aspects of the same thing. You can either clean up your act in what we think of as the real world and this will clarify what is within, or you can work from the other direction.
Whenever I see a layer of dust on my dining room table or sideboard, I have to take care of it or I will have the distinct impression that the dust will settle into my mind. Still, I do not see this as a chore. I actually feel that I am burnishing my mind each time I clean. When I go to visit my mother and her home is not as pristine as mine, I am concerned not just for her exterior health but for her inner health too. I want her to have as much clarity as possible in her last years. I don't explain why I am cleaning, but I was really touched on my last visit when my sister-in-law, Valery, came to see us and commented on the lightness she found that hadn't been there before. We all know how grungy we feel when we enter a room that hasn't been cleaned for a long time.
Sometimes the way to clear the cobwebs from your mind is to take a brisk walk outside. It has been snowing on and off for the last few days and my body hasn't felt like volunteering for exercise, but today the sun was bright and the sky blue and I practically danced down the street as I went in search of ink cartridges and dark-chocolate cookies. Now here I am back in front of the screen, and I have the energy to write once more. Even if you can't go outside, when you find yourself falling asleep over what you are doing or reaching an impasse, get up and do some physical movement. I find that cleaning always does the trick. If I get stuck on this page for too long, I shall go to the hall closet and do a little spring-cleaning. As I hung up my jacket just now, I noticed that fluff was accumulating on the floor between the shoes.
In some ways I see my editorial function as one of cleaning and polishing. I am not one of those editors who dream up ideas for books. I take the Michelangelo approach: If you offer me a block of Carrara marble (and it does have to be the best quality), then I can see the David within and help the author carve his or her vision out of the block of stone. I had one author who used to say that I edited his mind rather than his manuscript. Obviously, if you can do your editorial work there, there is less to do with a pencil later.
When I was growing up my mother would often say, "Don't throw away dirty water until you have clean," and I believed her. I don't know where this maxim came from, but recently I started thinking about it and realized that it doesn't make any sense. As long as your basin is full of dirty water, there is nowhere to put clean water, and there is also little chance that anyone will offer you any. While you are holding on to one object, your hands are not free to accept something else. While you have one idea firmly in place, it is impossible to entertain another.
Perhaps it is the inclination to have everything ready and available that enables me to be prompt for appointments. There is nothing dragging me back or holding me down. I am not running behind as so many people seem to be. In fact, I have an annoying habit of arriving early no matter how late I start off. I really do try to be late but I have rarely succeeded, and on the few occasions when circumstances like the traffic conspire against me, I do not worry because I know in my heart that if I am going to be late, other people will be even later.
I used to get upset when someone I was meeting for lunch was late at the restaurant, particularly if the wait lasted more than fifteen minutes. But eventually I realized that this hiatus was a gift. I was already in place, and there was nothing else to do but rest. For once, I didn't have to go anywhere or do anything. When the other person finally turns up, you can be much more gracious about it, whatever his or her excuse, if you see the time alone as a plus rather than a minus.
One of the concepts in Judaism that continues to fascinate me is that of tikkun olam. This expression from the Hebrew root TKN, "to set straight or put in order," is generally translated as "repair, restoration, or healing of the world," the idea being that we are here on this earth in order to look after whatever has gone awry or needs mending. But I think of it more in terms of our being caretakers. The word caretaker is used to describe someone who is responsible either for a sick or elderly person or for a building. In reality, the word has a much richer meaning. A caretaker is someone who takes care, who is careful, i.e. full of care. This puts a different complexion on the concept. It implies that our function is to care for one another and the whole of creation however we can. It means that our attention naturally goes out to other people and things wherever it is needed. There is a quality of devotion about it that doesn't normally come to mind when we say "caretaker."
Lately, I have been rereading the first five books of the Bible, studying both the English and the Hebrew texts, which is very rewarding. Each time, one comes across something new. What I discovered this time was that the Hebrew verb shamar, meaning "watch over, care for, protect, give attention to, observe, and revere," is used in (at least) four significant places. The first time is in Genesis 2:15 when God tells Adam to "till and care for" the Garden of Eden. In Genesis 4:9 comes Cain's famous response, "Am I my brother's keeper?" It turns out that "keeper," as it is usually translated, is not very accurate. The word really means "caretaker" or "someone who watches out for you." Then, in Genesis 28:15, in the passage about Jacob's dream, God says, "Remember, I am with you: I will protect you wherever you go..." And I had not realized that in the second rendering of the Ten Commandments, in Deuteronomy 5:12, the instruction regarding the Sabbath day is to "observe" it rather than "remember" it, as it says in Exodus. So it turns out that caring is what we are here in this world to do for one another. God cares for, protects, and watches over us, and, in the same way, we do this for others in whatever way we can.
On my bedroom wall is a calligraphed and illuminated quote by the Tibetan Buddhist teacher Chögyam Trungpa, which says:
Generosity is giving whatever you have. It is not for you to make judgments; it is for the recipients to make the gesture of receiving. If the recipients are not ready for your generosity, they will not receive it.
This ties in closely with the Jewish practice of tzedakah, which is often translated as "charity," but, in fact, the two concepts are different. Charity, as described in the New Testament, is choosing to give to someone when you perceive his or her need. It has an optional quality about it. Tzedakah, on the other hand, comes from the root meaning "wise" or "just." In Judaism, tzedakah is a requirement. (I believe that taking care of the poor is also one of the Five Pillars of Islam.) What this means is that if you are on the subway and a scruffy man asks you for a handout, you don't size him up and decide whether or not he deserves it. You simply give. There is no way you can know for certain what his situation is, and the rabbis taught that it is better to err on the side of generosity. God forbid you should refuse the one person in dire need.
This is not easy to put into practice. When I first learned about tzedakah, I was very moved and for the next week or so, I was able to just give, whoever asked me. But gradually this goodwill dissipated, and I started to judge people again. It was particularly hard when I went to India where women clutching babies drag on your arm, beseeching you for money, and when you relent and give them some, they ask for more. They figure that if you have already given something, you undoubtedly have more available and the next person may not. Also, in their culture, there is more merit in giving than receiving, so the way they see it, they are doing you a favor by allowing you to give again. This is hard for Westerners to grasp, particularly since no thanks are ever forthcoming. I didn't mind giving, but I wasn't able to resist feeling entitled to some kind of acknowledgment!
Of course, this is related to the idea that none of us really owns anything. Everything currently in my possession is just passing through my hands. I have been entrusted with it, or, to put it another way, I am its caretaker. Through the luck of the draw I was born into more favorable circumstances than millions of other people. I have never gone hungry or been without a roof over my head. Trungpa said, "Generosity is giving whatever you have." No equivocation there.
Here is one of my favorite stories about the Buddha. He was teaching one day, when someone started to heckle him. The Buddha stopped and asked the man: "If someone offers you a wonderful gift and you refuse it, to whom does the gift belong?" The man answered that it would remain the property of the giver. "Even so," said the Buddha, "I do not accept your abuse." It is said that the man's attitude was so changed that he became one of the Buddha's disciples.
This morning I admitted to myself that I was unlikely to ever again wear the pinafore dress I wore last Thursday. For some reason, although I was delighted with it when I bought it, it no longer suits me, and I know that it will lurk in my closet forevermore unless I Take Steps. So I extracted it and looked along the rack to see what else might join it on a trip to what my tax accountant calls "the Slavation Army" (What does he know that I don't?). I managed to put together a good shopping bagful of clothes that others might like. It is amazing how often you can do this and still find garments you are no longer attached to. But the miraculous effect of this surrendering and clearing out is the space that it creates. It is the space in and around things that is so wondrous. We all know that when our closets and drawers are crammed full, we cannot tell what is in them. When there is sufficient space to see what is there, we have a real choice about what to wear because we are able to get a clear view.
It is also healthy to relinquish larger items. Once I had both a child and a job, I started using the piano as a receptacle for manuscripts rather than for making music, and it sat there month after month like a reproach. One day a colleague came to dinner and admired it. I asked her if she played, and she said that she would if she had a piano. So I offered it to her. I explained that having something that large in the corner of the living room not being put to the use for which it was intended was a minus for me rather than a plus. She would be doing me a big favor by taking it away and putting it to good use. I also gave her the piano stool and my music.
What I sense is happening during this period of my life is a gradual draining or spring-cleaning of my mind, as I let go of one thing after another, whether it is attachment to particular garments or long-held ideas about people or things. In their stead, I find that from time to time an idea or a small revelation will drift in. It arrives on its own and doesn't come confused with a multitude of other stuff, so I am able to appreciate it fully and decide how to make the best use of it.
The important lesson here is to see what you can leave out of your life. If you consider music, for example, it is obvious that the space between the notes makes all the difference—both how wide the interval is and how long the silence lasts. It isn't any different with your clothes and, indeed, all your possessions. There is an inordinate amount of freedom to be found in giving things up.
FOURTEEN
Traveling Light
Once we are born, only two things are certain. One is that we will die and the other that we will experience change. In The Mahabharata, Prince Arjuna is asked, "What is the greatest wonder in the world?" and his response is, "Each day death strikes, and yet we live as if we were immortal." Nothing in the universe stands still, so why is it that we are so often surprised by change and shocked by death? The whole of this book is about ways in which we can live more freely in the universe, without most of the stuff we generally carry around. This is, in some measure, a preparation for our final journey. But this particular chapter is about preparing for all the other journeys we take in this life.
Many of the changes in our lives appear to be thrust upon us, but there are also those times when we deliberately seek out change. The most obvious occasion is when we travel. Nowadays many of us find ourselves on the road or in the air at frequent intervals. Travel has become a "fixture" in our lives, and we do not always acknowledge its implications.
Whether we are going away on business, for pleasure, out of obligation, or as recreation, leaving the comfort of our own home causes an upheaval, even if it offers some measure of respite. For a time you are separated and released from customary chores and routines, but being in an unfamiliar environment entails other forms of stress.
The way I deal with this is to create a small comfort cocoon, in much the same way that astronauts do as they rove the skies. If you take a survival kit along with you, it provides an insurance you can always fall back on. The makeup of this kit will differ from person to person, from one location to another, and also according to the season. In the 1960s many girls would not leave home without their eyelash curlers. The main thing is to take with you whatever will enable you to settle into and enjoy your new surroundings without fretting over something you feel is indispensable but lacking. I once lived with a man unencumbered enough to carry whatever he needed in a waist pack, but most of us are under the impression that we cannot manage with just a comb, a toothbrush, and our ID.
If we are traveling because of our work or in response to a family crisis, we are relatively clear as to the purpose of our trip, but this isn't always the case when we are "just getting away from it all."
What are we looking for when we go on vacation? Isn't it a break from our life in general? We want a change of scene, different circumstances, another view of the world, perhaps a new passion of some kind.
I find it interesting that Americans go on vacation while the British go on holiday. The original meaning of each of these words is quite different. Vacation comes from the Latin and has to do with emptiness and freedom, while the Old English holiday is a day or time dedicated to the sacred, wholeness, and health. Maybe the reason I prefer to go on holiday rather than on vacation is that I am English, but I suspect it is because the promise of what I might receive on holiday is far greater than what I might get on vacation.
It is important to be clear about the aim of your journey. Some people like to go back to the same place every year. I assume what they are looking for is comforting and familiar surroundings where they can relax and regroup. Others, like me, have something very different in mind. We want to stretch all our muscles—physical, mental, emotional, and spiritual. This often means that the beds and the food are not as wonderful as we would like, but it offers so many other possibilities that it is always worth the trade-off, and sometimes the accommodations do turn out to be good. Many people dream of going to a particular place, but they don't think it through any further than that. This is rather like planning your wedding but not having any idea of what marriage itself is. It makes sense, therefore, to prepare for the trip, however long it may be, so that you can take full advantage of your time away once you are there.
I am drawn to places where I think I will be nourished on every level. I don't go to great lengths to seek out these places. I use the same method as I do as an editor. With books, I keep an open and welcoming mind, and they arrive out of the blue. My vacation decisions come about in much the same way. Two weeks ago I opened a magazine and saw an advertisement for a trip to Turkey "On the Path of Rumi" and knew immediately that I would go on it. For many years I had wanted to go to Turkey, but it had somehow never worked out. When opportunities like this present themselves, don't hesitate.
Over the years I have visited the Golden Ring churches and monasteries in the countryside around Moscow, spent the turn of the millennium at St. Peter's in Rome with the pope (and ten thousand other pilgrims), explored arts and crafts on the enchanted island of Bali, traced the River Ganges to its source in the Himalayas, traveled with a group of Buddhists to practice centers in the southwestern United States, gone on a nine-day silent retreat in California, and made a short trip to a small town in Germany to be in the presence of Mother Meera, a young Indian woman considered by many to be an incarnation of the Divine Mother. In each case I was seeking to reconnect my spirit with the ground of my being, so that I could return home refreshed and invigorated, and in some measure this always happened.
When we are in new surroundings it is easier for us to see and hear more clearly. We don't labor under the impression that we have "seen it all before." When I got to Bali, I wished I had not boned up ahead of time by reading the Insight Guide. The full-color photos in the guide were so good that when I saw the actual places, I was rather disappointed because I felt I already knew what they looked like. It is always a toss-up about how much to read before you go. On my recent trip to China, I was so overwhelmed at the size of the country and the fact that I had no knowledge of the language and very little of its history that I threw up my hands and did nothing. I landed in Beijing in a state of tabula rasa, and everything I drank in was wondrously new. Generally, I riffle through a guidebook and read one or two other books to give me some background before I set off. This provides a context for whatever happens once I arrive.
I always hope for the kind of experience that happened to me in an industrial town called Ningbo on the eastern coast of China. An old Chinese friend in Connecticut had asked me to see if I could buy him a Chinese edition of the classic Zen collection of koans, The Blue Cliff Record. I marched into a three-story bookstore near my hotel and tried to find someone who spoke English. People looked at me in complete incomprehension but kept pointing upward (where it turned out the foreign-language books were) so I moved from escalator to escalator, always hoping that there would be someone with whom I could communicate. After fifteen minutes of nonsuccess, I came across two girls about age fourteen, one of whom said, "I speak English," and they tried to help me. As I discovered much later, the name of the book in Chinese is nothing like The Blue Cliff Record, and so no one recognized the ideographs written on my little piece of paper. But just before I left the store, one of the girls asked me, "What are you doing here? Why are you here?" I was rather taken aback and didn't know how to answer. She was asking what I was doing in Ningbo of all places, but her question aroused a deeper question within me: What was I doing here (or anywhere, for that matter) and why? I find that being away from home often puts me in wake-up situations like this, and I am grateful.
Whether I am going away for twenty-four hours, a weekend, or much longer, once I have left home, I am on my way. This may sound obvious, but what I mean is that once I have locked the door to my apartment, everything that has gone before is in the past. I don't carry my ordinary life with me in my head. I do indeed vacate (in some respects I have become an American). I can travel an hour's distance to stay with a friend and not think twice all the time I am away of the things that normally occupy me, so that sometimes twenty-four hours is all the break I need. When I go to a foreign country, it is sometimes hard for me to remember my job and my family back home. People will ask what books I am working on, and I cannot dredge up a single title.
I always carry with me a mini-notebook that fits into the palm of my hand, and when I am on vacation I make brief notes as I travel. I have bad camera karma, and so I have not bothered with photos for many years. My father once lent me his Nikon and I dutifully took thirty-six shots of Norway, only to find when I had them developed that the roller had stuck on the first frame. I felt as though I had missed Norway while looking through the lens, and I determined that I would never waste my holiday in this way again.
So nowadays I just scribble down a phrase here and there to remind me of a scene or something fascinating that I hear. This kind of shorthand enables me to write up a twenty-page journal of the trip on my return. Nowadays I have a small army of people who travel with me vicariously. They say that they are glad that I am making these trips and not them. They can enjoy traveling without leaving their armchairs. My Chinese professor friend and his wife have even put in a request for me to go to the Galapagos Islands because they feel that they are now too old to journey there themselves but they would still like to go (I declined. That part of the world doesn't speak to me—yet).
I have tried to keep this book free of instructions on how to do this or that on the physical level, because there are already many books available that provide that kind of advice. However, friends and traveling companions always marvel at how lightly I travel. So, by popular demand, I am including my checklist of travel essentials that I keep handy for those stressful days just before you set off on a journey. If you have a list like this, you don't have to wonder what you might be forgetting. Of course, your list will look slightly different from mine (particularly if you are a man), so just use this one as a model and make up a list of your own.
As you will see, I plan for every eventuality. You never know these days how unpredictable the weather will be or what pickle you will find yourself in, such as an extra eight hours on the runway, so it is best to be prepared.
Don't travel with more than you can lift (or trundle). There might not be anyone to help you at the other end. Also, it is far safer to take the suitcase into the plane with you than entrust it to airline personnel. Airlines lose track of your luggage at the most inconvenient times. Well, I suppose that there really are no convenient times for losing luggage.
Some years ago I flew to India on Gulf Air, and we changed planes in Bahrain. It had been an interminable journey, and we arrived in Delhi at 5 a.m. on a Sunday morning only to discover that all our luggage (there were twenty-five in the group) had remained in Bahrain and wouldn't arrive for another forty-eight hours. One official with a blunt pencil was detailed to fill in the forms describing each piece of luggage. This took two hours. While we were awaiting our turn, the tour guide brought us cups of hot chai (the aromatic Indian tea) in thick, chipped cups. I had been warned never to drink anything out of a cup that was not my own unless I was in a hotel or restaurant, because there was no way to tell whether the cup had been washed properly. My own metal cup was, of course, in Bahrain. For about half an hour I hesitated, but when I saw all my fellow travelers drinking and realized that it would be two days before I would meet up with my cup again, I nervously accepted the hot tea. By the time we arrived at the hotel, I had begun to feel very ill. I spent the next two days lying on my bed, shaking from head to foot. I couldn't eat anything and sipped just a little water provided by the hotel when it was absolutely necessary. I hadn't caught any dire disease. It was simply fear. But it prevented me from running around town like the others who were enjoying themselves hugely, buying clothes and other things that they needed to tide them over. The moral of this story is that you should not lose sight of your own luggage. This means packing very economically.
Always keep your toilet bag in a state of readiness, so that you don't have to check it out and make last-minute purchases for necessities you thought you already had. The time to check everything is before you put the bag away at the end of your vacation. Make certain that each little bottle or tube is full enough to last two weeks or however long you usually go away for.
Leave in your suitcase the things that you use only when you travel so that you don't have to collect them from wherever each time. For instance: shoe bags, extra hangers, tea bags, airplane socks, and toilet bag.
And, lastly, leave room in your suitcase to bring back goodies from your trip. If you have to sit on the suitcase to get it closed before you depart, you are going to be in trouble when you find something irresistible in a market somewhere.
Here is my list:
TRAVELING ESSENTIALS
(This includes what you are wearing)
Permanent resident card (I've never become an American), passport, money, and ticket (including copies of the card and passport to be kept in a separate place. If your passport is lost or stolen, the agony will be less if you at least have all the details with you)
Map + list of addresses and telephone numbers of people to visit in duplicate (keep each list in a different place so that when you mislay one, you can still find the other)
Food for the journey (Why would you want to wait until the airline decides to serve its unnourishing meals?)
Ziploc bags (including one for taking small bottles and tubes of paint through security)
Dried fruit and nuts (in case the hotel food leaves something to be desired or an excursion lasts four hours longer than anticipated)
Tea bags (English breakfast)
Ground coffee and gold filter (it's unlikely that the coffee will be as good as your own)
Hard candy (for when there's no water)
Strap in case suitcase breaks (mine never has, but it just might)
Toiletries (including tissues. Take half a roll of toilet paper in a plastic bag and keep it in your purse or pocket. You don't want to be without in an emergency)
Small and large shoulder bags + money belt (keep your passport and extra cash in your money belt. It's worth the trouble, and you will no longer feel the belt around your waist after a few hours)
Swimsuit(s) (you never know which hotels will have good pools or when you might come across a mountain spring)
Umbrella and raincoat
Laundry bag
3 hangers
2 books
Notebook plus extra pen and pencil
Painting equipment
Sunglasses and reading glasses
Sun hat
Binoculars and camera
Alarm clock (check the batteries before you leave)
Flashlight small enough to carry in your purse (ditto)
Washcloth (these are unknown in many foreign climes)
Pull-on slippers for the plane
Warm jacket or coat, gloves, and Polartec headband (in winter)
2 pairs of shoes
2 pairs of socks
Nightgown
Pashmina (no nice Jewish Buddhist would travel without her pashmina shawl)
Jewelry
4 sets of separates that can be mixed and matched (don't include anything that needs ironing)
1 outfit for out-of-season weather (the world climate has already gone berserk, so be prepared)
1 warm overblouse
1 sweater or vest
4 pairs of tights
3 undershirts
5 pairs of underpants
When you pack, you will find you can get more in if you fold things as little as possible. Align both legs of your pants and fold them over just once. Fold sweaters and blouses horizontally, and then fold the sleeves on top. This small step can make a huge difference, not only in the amount of space you save but also in the number of wrinkles the clothes don't have.
The first time I met my editor, he was very intrigued to see that I was going off to stay with a friend with nothing but a large shoulder bag. He wanted to know how that was possible and asked exactly what was in my purse. This is what I told him:
If I am going for just one night (as I was on that occasion), I figure I can wear the same thing the day I arrive as the day I leave, although normally I wouldn't do this. So all I need are clean underclothes and tights or socks, plus a sweater or shawl of some kind because it is usually ten degrees colder in the country than in New York City. My cosmetic case always contains a traveling toothbrush, a tiny tube of toothpaste, lipstick, rouge, face cream, mirror, comb, safety pins, toothpick, aspirin, fold-up scissors, cough lozenges, Band-Aids, a few sheets of toilet paper (and the cosmetic case is small, believe me), so I don't need anything more for one night. I like to sleep in my nothings, so no nightgown is necessary (unless I am staying in a hotel and think I might need to flee from a fire in the night). What else? One book. A pad and paper to write on. An umbrella: I often carry a fold-up umbrella, so that wouldn't really be extra for the weekend, and I also carry a Japanese fan for unair-conditioned subway cars, plus a small flashlight for descending emergency staircases in tall buildings in the dark (I had to do this from the twenty-second floor at my office once and don't want to be caught that way again). I make sure to wear shoes that will work for almost every occasion and are good for walking or loafing in. Oh, and a small gift for my host.
That's it. I don't know why people make such a fuss about packing. I don't like to haul a great deal around with me, and I rarely miss all the things that others seem to bring along, particularly since I have noticed that most people do not actually use much of what they take away with them.
FIFTEEN
In Conclusion
Here is a passage by Abraham Joshua Heschel from A Passion for Truth:
All worlds are in need of exaltation, and everyone is charged to lift what is low, to unite what lies apart, to advance what is left behind.... All facts are parables; their object is God. All things are tales the Teacher relates in order to render intelligible issues too difficult to comprehend literally, directly. Through things seen, God accommodates Himself to our level of understanding. What a shame it is that people do not comprehend the greatness of things on earth. They act as if life were trivial, not realizing that every trifle is filled with Divinity. No one makes a move that does not stir the highest Heaven.
When a young woman who was my assistant for a brief period came to tell me that she was leaving to go to work for a new magazine, she explained, "You need someone good at detail, which I'm not."
I explained to her gently that there are only details. There isn't anything else. If you step onto a bridge and don't look where you are going, you may lose your footing and plunge into the water. I had earlier tried to get her to understand that it is better to pay attention and get something right to begin with than to make a small mistake and expend a great deal of time and energy correcting it. Mistakes proliferate at an alarming rate. In that last conversation I encouraged her to understand the nature of details not for my sake but for her own. Not paying attention to a detail might one day cost her her life.
It made me a little sad that during the five months she worked for me, she had not learned one of the most important things I had to teach her—scrupulous attention to everything. I have this down to such a fine art that it sometimes backfires! Publishing books is all about detail, and those who work with me sometimes assume that I have checked everything carefully, and therefore (I suspect) they don't double-check what I hand over to them the way they do for other people. They just assume that I have probably caught any mistakes. I am as fallible as everyone else, which is why I feel the need to be so careful. I am never sure that I have caught everything.
Because even the smallest detail is vital, I find it important to keep in mind the four laws of ecology that Barry Commoner shared with the world in his book The Closing Circle. They are as useful to me now as they were when I first encountered them in 1971:
**1.** Everything is connected to everything else.
**2.** Everything must go somewhere.
**3.** Nature knows best.
**4.** There's no such thing as a free lunch.
One of the remarkable aspects of these principles is that they are all saying the same thing, and so it doesn't really matter at which end you start.
Everything that you think or do or say is connected not only to everything else in your life, but also to everything in everybody else's life. You cannot do something in isolation, something that does not have an effect on the world at large.
Not only is everything connected to everything else, but each person is also connected to everyone else. We are all joined at the hip, so to speak. Looked at in this way, it becomes evident that there is only one of us. Buddhists explain this by saying that each of us is like a different part of the same body and we tend to strut about, believing that we have a life of our own, although, in truth, we may be just a cell or an eyelash.
The corollary to all this is that not only does what you or I do affect the whole world, but what everyone else in the world does affects us also. This is why it is impossible (at least for us) to trace karma back to its roots. There are strands of it coming from every conceivable direction.
It follows that if someone takes you out for a meal, sends you an armful of tulips, or pays you a compliment, it comes with a price. The price is not necessarily visible on a tag, but it is definitely there. The donor may believe that he or she is giving you something freely, but that may not be the way you receive the gift—or vice versa. There are subtle strings joining everything in creation. I don't mean that you will necessarily "pay back" the same person. The response you have to what you received may take any number of forms. For instance, "random acts of kindness" are not nearly as random as people believe. I heard a story about someone paying the toll for the car behind without knowing who was driving that car. The person who paid the toll did so because of something that had happened to her. And the driver who was on the receiving end undoubtedly passed on the goodwill to someone else in one form or another. As I said, it doesn't have to be a direct quid pro quo.
This is all part of the law of karma. Many people believe that karma is linear: If you do some terrible (or wonderful) thing, you will receive your just desserts. However, the way it seems to be set up is not as personal as that. You have a thought or take some action, and there is a result. The consequences will be experienced, but not necessarily by you, and nobody knows when.
Around the same time that I discovered Barry Commoner, I started delving into the work of Alan Watts. The first book I read was The Book on the Taboo Against Knowing Who You Are, and there is one story he tells in it that is relevant here. He describes a high fence with just a chink in it through which you can peer in order to see what is going on on the other side. If you put your eye close to the chink, you sometimes get a glimpse of whiskers, then a nose, two ears, a body with two legs, two more legs, and finally a tail. Sometime later you may see this sequence in reverse order, and you may very well deduce from your observations that whiskers are always followed by a tail. However, if you looked over the fence, you would see at once that it was all part of the same cat.
It becomes apparent that what may look like a thought, a word, or an action, and its eventual consequences is, in reality, all part of the same cat.
This started out as a book on how to live a simple life. What I discovered as I examined the nature of simplicity is that it all depends on integrity and impeccability. If you focus on these two qualities, your life simplifies itself.
The scraps of wisdom I offer here—my gleanings—have been gathered over many years. Each grain lingered dormant in a crevice of my mind until now. Perhaps you will connect with one of these seeds and store it away for another decade before you, too, actually water it with your attention and it starts to sprout.
The book is a sharing of all that I have received of value. I have tried to record faithfully whatever I thought would be useful to others. I didn't want to have anything left over! I cannot claim that any of it is original, but I hope that some of it may be new to you. As I was nearing the end and scouring my mind for any overlooked seeds, a friend in London sent me the following poem that was written by an American over a hundred years ago. He must have been a very frugal fellow because he set down on one sheet of paper what it has taken me a whole book to say:
To live content with small means,
to seek elegance rather than luxury
and refinement rather than fashion,
to be worthy, not respectable,
and wealthy, not rich,
to study hard, think quietly,
talk gently, act frankly,
to listen to stars and birds,
babes and sages, with open heart,
to hear all cheerfully,
do all bravely,
await occasions,
hurry never—
in a word, to let the spiritual,
unbidden and unconscious,
grow up through the common.
This is to be my symphony.
William Henry Channing (I8I0-I884)
Afterword
In my "retirement" I have made many new and wonderful friends, both in person and on the page. Authors no longer send me manuscripts offering the very teaching I need at this moment, but books I never had the time to read when I was in publishing provide mini-revelations. I think of the authors of these books as "companions on the path," and what they have to say may be just what you need too. So here is a little of their wisdom:
"I had learnt so much from the people I'd met along my journeys: people for whom.... landscape was intricately involved with self-perception, and for whom certain places or weathers brought yields of grace," writes Robert Macfarlane in The Old Ways, his luminous book on walking his way through the world. "Yields of grace," indeed. That's what I experience every step of the way—as long as I keep my eyes and ears open.
And this from Jonah Lehrer in Imagine: How Creativity Works. "The German philosopher Heidegger believed that the reality of things is naturally obscured by the clutter of the world, by all those ideas and sensations that distract the mind. The only way to see through this clutter is to rely on the knife of conscious attention, which can cut away the excess and reveal 'the things themselves.'" "The knife of conscious attention"—I wish I had thought of this phrase myself.
The poet Jack Gilbert was once interviewed in Paris Review. The whole interview is wonderful, but this passage struck me deeply: "When I read the poems that matter to me, it stuns me how much the presence of the heart—in all its forms—is endlessly available there.... I really want to say something to someone that they will feel significantly inside themselves, and if I'm not doing that then I'm wasting our time....Being alive is so extraordinary.... What is most important to writing poetry [is] presence, feeling, passion—not passion, but love.... I want to experience or discover ways of feeling that are fresh. I love it when I have perceived something fresh about being human and being happy."
In Pilgrim at Tinker Creek Annie Dillard offers a wealth of extraordinary perceptions. Here are two such: "Experiencing the present purely is being emptied and hollow; you catch grace as a man fills his cup under a waterfall." And "You wait in all naturalness without expectation or hope, emptied, translucent, and that which comes rocks and topples you; it will shear, loose, launch, winnow, grind."
After reading biologist David George Haskell's marvelous The Forest Unseen: A Year's Watch in Nature, I went to hear him give a talk and came away with this gem: "Wonder increases as speed decreases." Let us all take this to heart as we rush into the future.
In Tricycle magazine I found meditation teacher Ken McLeod's account of a three-week retreat where he was instructed to do nothing. In "Something from Nothing," he asks, "What is it like to do nothing? I mean, really do nothing, nothing at all—no recalling what has happened, no imagining what might happen, no reflecting on what is happening, no analyzing or explaining or controlling what you experience. Nothing! I had to receive whatever arose, experience it, and not do anything with it. Be willing to receive, not control, whatever arises—that is, not only allow but embrace every sensation, feeling, and thought, everything we experience." This is what I have been endeavoring to practice for many years now, but have not seen it described so succinctly anywhere else.
In 1961, when I was initiated into TM, I was asked what my aim in life was, and I replied: "To be able to love." Loving someone seemed so remote then, so unattainable. But thirteen years later, when I was almost thirty-four, I fell in love for the first time. Yes, I had swooned over a few men before, but they had never really acknowledged my feelings or showed the slightest hint of reciprocating them, so I had simply worshipped each of them from afar. This time it was different. One evening I found myself saying to one of my authors, whom I had just met after a long correspondence, "Do you know that I love you?" No one could have been more astonished at what came out of my mouth than I was, but he smiled and said that he did know. I have little memory of the meal we shared, but afterward he took me in his arms and kissed me. I looked into his eyes and they were alive with liquid fire—dancing, merry, eternal. I told him that I had never seen such a thing before and he said that he had never experienced anything like this either. That relationship did not blossom in the way I'd hoped it would, but from then on I knew that loving someone was possible because it had happened and, if something had happened once, there was always the chance that it could happen again. And it did when shortly thereafter I met and married my husband Neil. Many years have gone by since then and I had (almost) stopped hoping that love would arrive again and this time it would come to stay. Then, one day, I found this line on a blog by a woman whose name I'm embarrassed to say I didn't make a note of: "You know, I realize, after all my searching, that the love of my life is—my life."
This describes my life too. I am no longer looking for one person to love. I love so many people now—and so much else besides. What I love is my life and the abundance the universe delivers to my door time and time again. Everything unfolds and I love life itself.
New York City, December 2013.
About the Author
Toinette Lippe was born in London, where she began her publishing career. In 1964 she came to New York City "for a year," worked at Simon and Schuster for three years, and then at Alfred A. Knopf as reprint rights director and editor for thirty-two years. In 1989, while still at Knopf, she founded Bell Tower, an imprint publishing books that nourish the soul, illuminate the mind, and speak directly to the heart, and she remained its editorial director for twenty years. Her own first book, Nothing Left Over: A Plain and Simple Life was first published in 2002 and her second, Caught in the Act: Reflections on Being, Knowing, and Doing, in 2004. In 2011 Vintage published a new edition of the Gia-fu Feng/Jane English translation of the Tao Te Ching, which Toinette edited at Knopf/Vintage in 1972 and which has sold more than a million copies. She and Jane English refreshed and revised the original translation, adding over a hundred new photographs by Jane and a foreword by Toinette, who is now listed as one of the translators.
After fifty years in publishing Toinette abandoned editorial work and devotes herself to East Asian brush painting, which she taught at the Educational Alliance for four years and now teaches on the Upper West Side of New York City. You can view and purchase her paintings and cards at www.toinettelippe.com.
PRAISE FOR CAUGHT IN THE ACT: REFLECTIONS ON BEING, KNOWING, AND DOING
"Caught in the Act is about surrender on several levels: surrendering what's unnecessary in life, surrendering attitudes that keep you from pure enjoyment, surrendering yourself to the moment—letting it take you where it wants to go rather than where you wish to be taken. The book is unconventional in form and manner, and is told in a brisk, conversational style that's immediate and engaging. It has a mystery and beauty all its own."
—Robert Leiter, Jewish Exponent
"If we're fortunate, there are a couple of junctures in our lives when we have the leisure and impetus to reflect on the past and consider our future. Caught in the Act witnesses a mature woman contemplating how what you do becomes what you are. She has a curious mind, an indefatigable eye for detail and a serious intellect. To see what a mind like that does with semi-retirement, read Caught in the Act."
—Shambhala Sun
"This is no instructional manual of advice, thankfully; instead it is a book about learning to surrender. An informed and well-grounded wisdom shines forth on every page.
While the stories from her life give the reader a sense of connection to her, somehow the book magically becomes about you. Her questions become your questions, too. She writes unpretentiously, as one who finds it unnecessary to state the obvious. It is tempting to credit her decades as a book editor for the clarity of her writing, but the ability to turn a rigorous, analytical mind back on itself requires a degree of personal honesty that only comes with years of spiritual practice and contemplation. This is what makes her insight so recognizably human and relevant. It takes both humility and courage to first see, and then reveal oneself so forthrightly.
Never mind that she calls herself an "almost-Buddhist," her grasp of the issues centered around "aimless aim" is right up there with Zen in the Art of Archery. If we don't have any goals or intentions with whatever activity we are doing, we may go nowhere. Yet if we are too focused on results, we burden our actions with heavy expectations. This book is about finding that balance in your daily life."
—Gloria Lee, Nonduality Highlights
"All those on a spiritual path will identify with Lippe's struggles to simply be.... Control is what so many of us aim for and yet surrender to experience is what all the sages in every mystical tradition tell us to do. Lippe counsels herself to be welcoming to everything and to relinquish the attachments that stop her from being present."
—Frederic and Mary Ann Brussat, Spirituality & Health
"Lippe's anecdotes arise from her travels, her friendships, her reading, and her meditation practice. Her style is clear and revealing. Her book will appeal to those searching for what is authentic in life, as well as those dealing with transitions, including retirement. In becoming intimate with her journey, readers have the opportunity to tune in to their own."
—Tova Green, Turning Wheel
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\section{Introduction}\label{s:1}
The standard generators $T(x)=x+1$, $S(x)=-1/x$ of the modular group $SL(2,\Z)$ were used classically to define piecewise continuous maps acting on the extended real line $\bar \mathbb R=\mathbb R\cup \{\infty\}$ that led to well-known continued fraction algorithms.
In this paper we present a general method of constructing such maps suggested by Don Zagier, and study their dynamical properties and associated generalized continued fraction transformations.
Let $\mathcal P$ be the two-dimensional parameter set
\[
\mathcal P=\{(a,b)\in \mathbb R^2\, |\, a\leq 0\leq b,\,b-a\geq 1,\,-ab\leq 1\}
\]
and consider the map $f_{a,b}:\bar\mathbb R\rightarrow\bar\mathbb R$ defined as
\begin{equation}\label{fab}
f_{a,b}(x)=\begin{cases}
x+1 &\text{ if } x< a\\
-\displaystyle\frac{1}{x} &\text{ if } a\le x<b\\
x-1 &\text{ if } x\ge b\,.
\end{cases}
\end{equation}
Using the first return map of $f_{a,b}$ to the interval $[a,b)$, denoted by $\hat f_{a,b}$, we introduce a two-dimensional family of
continued fraction algorithms and study their properties. We mention here three classical examples: the case $a=-1/2$, $b=1/2$ gives the ``nearest-integer" continued fractions considered first by Hurwitz in \cite{Hurwitz1}, the case $a=-1$, $b=0$ described in \cite{Z, K3} gives the ``minus" (backward) continued fractions, while the situation $a=-1$, $b=1$ was presented in \cite{S1, KU2} in connection with a method of symbolically coding the geodesic flow on the modular surface following Artin's pioneering work \cite{Artin}. Also, in the case $b-a=1$, the class of one-parameter maps $f_{b-1,b}$ with $b\in [0,1]$ is conceptually similar to the ``$\alpha$-transformations" introduced by Nakada in \cite{N1} and studied subsequently in \cite{LM, MCM, NN1,NN2, Sw}.
The main object of our study is a two-dimensional realization of the natural extension map of $f_{a,b}$, $F_{a,b}:\bar\mathbb R^2\setminus\Delta\rightarrow \bar\mathbb R^2\setminus\Delta$, $\Delta=\{(x,y)\in \bar\mathbb R^2| x=y\}$,
defined by
\begin{equation}\label{Fab}
F_{a,b}(x,y)=\begin{cases}
(x+1,y+1) &\text{ if } y<a\\
\left(-\displaystyle\frac{1}{x},-\displaystyle\frac{1}{y}\right) &\text{ if } a\le y<b\\
(x-1,y-1) &\text{ if } y\ge b\,.
\end{cases}
\end{equation}
The map $F_{a,b}$ is also called the \emph{reduction map}.
Numerical experiments led Don Zagier to conjecture that such a map $F_{a,b}$ has several interesting properties for all parameter pairs $(a,b)\in \mathcal P$ that we list under the
{\bf Reduction theory conjecture.}
\begin{enumerate}
\item The map $F_{a,b}$ possesses a global attractor set $D_{a,b}=\cap_{n=0}^\infty F^n(\bar\mathbb R^2\setminus\Delta)$
on which $F_{a,b}$ is essentially bijective.
\item The set $D_{a,b}$ consists of two (or one, in degenerate cases) connected components each having {\em finite rectangular structure}, i.e. bounded by non-decreasing step-functions with a finite
number of steps.
\item Every point $(x,y)$ of the plane ($x\ne y$) is mapped to $D_{a,b}$ after finitely many iterations of $F_{a,b}$.
\end{enumerate}
\begin{figure}[htb]
\psfrag{a}[l]{\small $a$}
\psfrag{b}[l]{\small $b$}
\includegraphics[scale=1.1]{donexample.eps}
\caption{Attracting domain for Zagier's example: $a=-\frac{4}{5},\,b=\frac{2}{5}$}
\label{don-a}
\end{figure}
Figure \ref {don-a} shows the computer picture of such a the set $D_{a,b}$ with $a=-4/5$, $b=2/5$.
It is worth mentioning that the complexity of the domain $D_{a,b}$
increases as $(a,b)$ approach the line segment $b-a=1$ in $\mathcal P$, a situation fully analyzed in what follows.
The main result of this paper is the following theorem.
\begin{result} There exists an explicit one-dimensional Lebesgue measure zero, uncountable set $\mathcal E$ that lies on the diagonal boundary $b= a+ 1$ of $\mathcal P$ such that:
\begin{itemize}
\item[(a)] for all $(a,b)\in\mathcal P\setminus\mathcal E$ the map $F_{a,b}$ has an attractor $D_{a,b}$ satisfying properties (1) and (2) above;
\item[(b)] for an open and dense set in $\mathcal P\setminus\mathcal E$ property (3), and hence the Reduction theory conjecture, holds. For the rest of $\mathcal P\setminus\mathcal E$ property (3) holds for almost every point of $\bar\mathbb R^2\setminus \Delta$.
\end{itemize}
\end{result}
We point out that this approach gives explicit conditions for the set $D_{a,b}$ to have finite rectangular structure that are satisfied, in particular, for all pairs $(a,b)$ in the interior of the maximal parameter set $\mathcal P$.
At the same time, it provides an effective algorithm for finding $D_{a,b}$, independent of the complexity of its boundary (i.e., number of horizontal segments). The simultaneous properties satisfied by $D_{a,b}$, attracting set and bijectivity domain for $F_{a,b}$, is an essential feature that has not been exploited in earlier works. This approach makes the notions of reduced geodesic and dual expansion natural and transparent, with a potential for generalization to other Fuchsian groups. We remark that
for ``$\alpha$-transformations" \cite{N1,LM}, explicit descriptions of the domain of the natural extension maps have been obtained only for a subset of the parameter interval $[0,1]$ (where the boundary has low complexity).
The paper is organized as follows. In Section \ref{s:2} we develop the theory of $(a,b)$-continued fractions associated to the map $f_{a,b}$. In Section \ref{s:3} we prove that the natural extension map $F_{a,b}$ possesses a {\em trapping region}; it will be used in Section \ref{s:6} to study the attractor set for $F_{a,b}$.
In Section \ref{s:4} we further
study the map $f_{a,b}$. Although it
is discontinuous at $x=a$, $b$, one can look at two orbits of each of the discontinuity points. For generic $(a,b)$, these orbits meet after finitely many steps, forming a {\em cycle} that can be {\em strong} or {\em weak}, depending on whether or not the product over the cycle is equal to the identity transformation.
The values appearing in these cycles play a crucial role in the theory. Theorems \ref{b-cycle} and \ref{a-cycle} give necessary and sufficient conditions for $b$ and $a$ to have the {\em cycle property}. In Section \ref{s:5} we introduce the {\em finiteness condition} using the notion of {\em truncated orbits} and prove that under this condition the map $F_{a,b}$ has a bijectivity domain $A_{a,b}$ with a finite rectangular structure that can be ``computed" from the data $(a,b)$ (Theorem \ref{thm:recstructure}). In Section \ref{s:6} we define the attractor for the map $F_{a,b}$ by iterating the trapping region, and
identify it with the earlier constructed set $A_{a,b}$ assuming the finiteness condition (Theorem \ref{attractor}). In Section \ref{s:7}
we prove that the Reduction theory conjecture holds under the assumption that both $a$ and $b$ have the strong cycle property, and that under the finiteness condition property, (3) holds for almost every point of $\bar\mathbb R^2\setminus\Delta$.
In Section \ref{s:8} we prove that the finiteness condition holds for all $(a,b)\in\P$ except for
an uncountable set of one-dimensional Lebesgue measure zero that lies on the boundary $b=a+1$ of $\P$, and we present a complete description of this exceptional set. We conclude by showing that the set of $(a,b)\in\mathcal P$ where $a$ and $b$ have the strong cycle property is open and dense in $\mathcal P$.
And, finally, in Section \ref{s:9} we show how these results can be applied to the study of invariant measures and ergodic properties of the associated Gauss-like maps.
\section{Theory of $(a,b)$-continued fractions}\label{s:2}
Consider $(a,b)\in \P$. The map $f_{a,b}$ defines what we call \emph{$(a, b)$-continued fractions} using
a generalized integral part function $\lfloor x\rceil_{a,b}$ :
for any real $x$, let
\begin{equation}
\lfloor x\rceil _{a,b}=\begin{cases}
\lfloor x-a \rfloor &\text{if } x< a\\
0 & \text{if } a\le x<b\\
\lceil x-b \rceil & \text{if } x\ge b\,,
\end{cases}
\end{equation}
where $\lfloor x\rfloor$ denotes the integer part of $x$ and $\lceil x\rceil=\lfloor x\rfloor+1$.
Let us remark that the first return map of $f_{a,b}$ to the interval $[a,b)$, $\hat f_{a,b}$, is given by the function
\[
\hat f_{a,b}(x)=-\frac{1}{x}-\left\lfloor-\frac{1}{x}\right\rceil_{a,b}=T^{-\lfloor-1/x\rceil_{a,b}}S(x) \text{ if } x\ne 0, f(0)=0.
\]
We prove that any irrational number $x$ can be expressed in a unique way as an infinite $(a,b)$-continued fraction
\[
x=n_0-\cfrac{1}{n_1 -\cfrac{1}{n_2-\cfrac{1}{\ddots}}}
\]
which we will denote by $\lfloor n_0,n_1,\dots \rceil_{a,b}$ for short.
The
``digits" $n_i$, $i\ge 1$, are non-zero integers determined recursively by
\begin{equation}\label{a-b}
n_0=\lfloor x\rceil_{a,b},\,x_1=-\frac1{x-n_0},\text{ and }
n_{i}=\lfloor x_i \rceil_{a,b},\,x_{i+1}=- \frac1{x_i-n_i}.
\end{equation}
In what follows, the notation $(\alpha_0,\alpha_1,\dots,\alpha_k)$ is used to write formally a ``minus" continued fraction expression, where $\alpha_i$ are real numbers
\begin{thm}\label{convergence}
Let $x$ be an irrational number, $\{n_i\}$ the associated sequence of integers defined by (\ref{a-b})
and
\[
r_k=( n_0,n_1,\dots, n_k)\,
\]
Then the sequence $r_k$ converges to $x$.
\end{thm}
\begin{proof}\footnote{The authors proved initially the convergence statement assuming $-1\le a \le 0 \le b\le 1$, and two Penn State REU students, Tra Ho and Jesse Barbour, worked on the proof for $a,b$ outside of this compact triangular region. The unified proof presented here uses some of their ideas.} We start by proving that none of the pairs of type $(p,1)$, $(-p,-1)$, with $p\ge 1$
are allowed to appear as consecutive entries of the sequence $\{n_i\}$. Indeed, if $n_{i+1}=1$, then
$$b\le x_{i+1}=-\frac{1}{x_i-n_i}<b+1\,,$$
therefore $-\displaystyle\frac{1}{b}\le x_i-n_i<-\frac{1}{b+1}\le (b-1)$, and $n_i<0$. If $n_{i+1}=-1$, then
$$a-1\le x_{i+1}=-\frac{1}{x_i-n_i}<a\,,$$
so $-\displaystyle\frac{1}{a-1}\le x_i-n_i<-\frac{1}{a}$. But $a+1\le -\frac{1}{a-1}$, thus $n_i>0$.
With these two restrictions, the argument follows the lines of the proof for the classical case
of minus (backward) continued fractions \cite{K3}, where $n_i\ge 2$, for all
$i\ge 1$. We define inductively two sequences of integers
$\{p_k\}$ and $\{q_k\}$ for $k\ge -2$:
\begin{equation}\label{pkqk}
\begin{split}
&p_{-2}=0\;,\;p_{-1}=1\;;\; p_{k}=n_{k}p_{k-1}-p_{k-2}\; \text{ for } k\ge 0\\
&q_{-2}=-1\;,\;q_{-1}=0\;;\;q_{k}=n_{k}q_{k-1}-q_{k-2}\; \text{ for } k\ge 0\;.
\end{split}
\end{equation}
We have the following properties:
\begin{itemize}
\item[(i)] there exists $l\ge 1$ so that $|q_l|<|q_{l+1}|<\dots<|q_k|<\dots$;
\item[(ii)] $(n_0,n_1,\dots,n_k,\alpha)=\displaystyle\frac{\alpha p_k-p_{k-1}}{\alpha q_k-q_{k-1}}$, for any real number $\alpha$
\item [(iii)] $p_kq_{k+1}-p_{k+1}q_k=1$;
\end{itemize}
Let us prove property (i). Obviously $1=q_0\le |q_1|=|n_1|$, $q_2=n_2q_1-q_0=n_2n_1-1$. Notice that $|q_2|>|q_1|$ unless
$n_1=1, n_2=2$ or $n_1=-1$, $n_2=-2$. We analyze the situation $n_1=1$, $n_2=2$. This implies that $q_3=n_3(n_2n_1-1)-n_1=n_3-n_1$, so $|q_3|>|q_2|$, unless $n_3=2$. Notice that it is impossible to have $n_i=2$ for all $i\ge 2$, because $x$ is irrational and the minus continued fraction expression consisting only of two's, $(2,2,\dots)$, has numerical value $1$. Therefore, there exists $l\ge 1$ so that
$n_{l+1}\ne 1,2$. This implies that $|q_{l+1}|>|q_l|$. We continue to proceed by induction. Assume that property (i) is satisfied up to $k$-th term, $k>l$. If $|n_{k+1}|\ge 2$, then
$$
|q_{k+1}|\ge |n_{k+1}|\cdot |q_k|-|q_{k-1}|\ge 2|q_k|-|q_{k-1}|>|q_{k}|\,.
$$
If $n_{k+1}=1$, then
$q_{k+1}=q_k-q_{k-1}$. Since $q_k=n_{k}q_{k-1}-q_{k-2}$ with $n_k<0$, one gets
$$q_{k-1}=\frac{q_k+q_{k-2}}{n_{k}}\,.$$
We analyze the two possible situations
\begin{itemize}
\item If $q_k>0$ then $|q_{k-2}|<q_k$, so $q_k+q_{k-2}>0$ and $q_{k-1}<0$. This implies that
$q_{k+1}=q_k-q_{k-1}\ge q_{k}>0\,.$
\item If $q_k<0$, then $|q_{k-2}|<-q_k$, so $q_k+q_{k-2}<0$ and $q_{k-1}>0$. This implies that
$q_{k+1}=q_k-q_{k-1}<q_k<0\,.$
\end{itemize}
Thus $|q_{k}|<|q_{k+1}|$. A similar argument shows that the inequality remains true if $n_{k+1}=-1$.
Properties (i)--(iii) show that $r_k=p_k/q_k$ for $k\ge 0$.
Moreover, the sequence $r_k$ is a Cauchy sequence because
$$|r_{k+1}-r_k|=\frac{1}{|q_kq_{k+1}|}\le \frac{1}{(k-l)^2}\, \text{ for } k>l.$$
Hence $r_k$ is convergent.
In order to prove that $r_k$ converges to $x$, we write $x=(n_0,n_1,\dots,n_k,x_{k+1})$, and
look only at those terms $(n_0,n_1, \dots,n_k,x_{k+1})$ with $|x_{k+1}|\ge 1$.
There are infinitely many such terms: indeed, if $-1\le a<b\le 1$, then $|x_{k+1}|\ge 1$ for all $k\ge 1$; if $a<-1$, and $|x_{k+1}|<1$, then $b\le x_{k+1}<1$, so $x_{k+2}=-1/(x_{k+1}-1)\ge 1$; if $b>1$, and $|x_{k+1}|<1$, then $-1<x_{k+1}<a$, so $x_{k+2}=-1/(x_{k+1}+1)\ge 1$.
Therefore, the corresponding subsequence $r_k=p_k/q_k$ satisfies
\begin{equation*}
\begin{split}\left|\frac{p_k}{q_k}-x\right|=&\left|\frac{p_k}{q_k}-\frac{p_k x_{k+1}-p_{k-1}}{q_k x_{k+1}-q_{k-1}}\right|
=\frac{1}{|q_k(q_k x_{k+1}-q_{k-1})|}\\ \le & \frac{1}{|q_k|(|q_k||x_{k+1}|-|q_{k-1}|)}\le \frac{1}{|q_k|}\rightarrow 0.
\end{split}
\end{equation*}
We showed that the convergent sequence $r_k=p_k/q_k$ has a subsequence convergent to $x$, therefore the whole sequence converges to $x$.
\end{proof}
\begin{rem}
One can construct $(a,b)$-continued fraction expansions for rational numbers, too. However, such expansions will terminate after finitely many steps if $b\ne 0$. If $b=0$, the expansions of rational numbers will end with a tail of $2$'s, since $0=(1,2,2,\dots)$
\end{rem}
\begin{rem} \label{quadratic}It is easy to see that if the $(a,b)$-continued fraction expansion of a real number is eventually periodic, then the number is a quadratic irrationality.
\end{rem}
It is not our intention to present in this paper some of the typical number theoretical results that can be derived for the class of $(a,b)$-continued fractions. However, we state and prove a simple version about $(a,b)$-continued fractions with ``bounded digits". For the regular continued fractions, this is a classical result due to Borel and Bernstein (see \cite[Theorem 196]{HW} for an elementary treatment). We are only concerned with
$(a,b)$-expansions that are written with two consecutive digits, a result explicitly needed in Sections \ref{s:7} and \ref{s:8}.
\begin{prop}\label{bdigits1}
The set $\Gamma^{(m)}_{a,b}=\{x=\lfloor 0, n_1,n_2,\dots\rceil_{a,b} \:|\: n_k\in\{m,m+1\}\}$
has zero Lebesgue measure for every $m\ge 1$.
\end{prop}
\begin{proof}
First, notice that if $m=1$, then the set $\Gamma^{(1)}_{a,b}$ has obviously zero measure, since the pairs $(2,1)$ and $(-2,-1)$ are not allowed in the $(a,b)$-expansions.
Assume $m\ge 2$. Notice that $\Gamma^{(m)}_{a,b}\subset \Gamma^{(m)}_{0,-1}$ since a formal continued fraction $x=(0, n_1,n_2,\dots)$ with $n_k\in\{m,m+1\}$ coincides with its ``minus" (backward) continued fraction expansion ($a=-1, b=0$), $x=\lfloor 0, n_1,n_2,\dots\rceil_{-1,0}$. The reason is that any sequence of digits $n_i\ge 2$ gives a valid ``minus" continued fraction expansion.
In what follows, we study the set $\Gamma^{(m)}_{0,-1}$. For practical reasons we will drop the subscript $(0,-1)$.
It is worth noticing that the result for $\Gamma^{(m)}_{0,-1}$ does not follow automatically from the result about regular continued fractions, since there are numbers for which the $(0,-1)$-expansion has only digits $2$ and $3$, while the regular continued fractions expansion has unbounded digits.
We follow the approach of \cite[Theorem 196]{HW} and estimate the size of the set $\Gamma^{(m)}_{n_1,n_2,\dots,n_k}\subset \Gamma^{(m)}$ with the digits $n_1,n_2,\dots,n_k\in \{m,m+1\}$ being fixed. In this particular case, the recursive relation \eqref{pkqk} implies that $1=q_1<q_2<\dots<q_k$.
If $x\in \Gamma^{(m)}_{n_1,n_2,\dots,n_k}$, then
$$(0,n_1,n_2,\dots ,n_k-1)\le x<(0, n_1,n_2,\dots, n_k)\,.$$
Using property (iii), the endpoints of such an interval $I^{(m)}_{n_1,\dots,n_k}$ are given by
$$
\frac{(n_k-1)p_{k-1}-p_{k-2}}{(n_k-1)q_{k-1}-q_{k-2}}\text{ , }
\frac{n_kp_{k-1}-p_{k-2}}{n_kq_{k-1}-q_{k-2}}
$$
and the length of this interval is
\[
l(I^{(m)}_{n_1,\dots,n_k})=\frac{1}{(n_kq_{k-1}-q_{k-2})((n_k-1)q_{k-1}-q_{k-2})}=\frac{1}{q_{k}(q_k-q_{k-1})}
\]
by using that $p_{k-2}q_{k-1}-p_{k-1}q_{k-2}=1$ and $q_k=n_kq_{k-1}-q_{k-2}$.
Denote by $\Gamma^{(m)}_k$ the set of numbers in $[-1,0)$ with $(-1,0)$-continued fraction digits $n_1, n_2, \dots$, $n_k\in\{m,m+1\}$. The set $\Gamma^{(m)}_k$ is part of the set
$$I^{(m)}_k=\bigcup_{n_1,\dots, n_k\in\{m,m+1\}}I^{(m)}_{n_1,\dots,n_k}\,.$$ We have the following relation:
$$
I^{(m)}_{k+1}=\bigcup_{n_1,\dots, n_k\in\{m,m+1\}}I^{(m)}_{n_1,\dots,n_k,m}\cup I^{(m)}_{n_1,\dots,n_k,m+1}
$$
If $x$ lies in $I^{(m)}_{n_1,\dots,n_k,m}\cup I^{(m)}_{n_1,\dots,n_k,m+1}$, then
$$(0,n_1,n_2,\dots ,n_k,m-1)\le x<(0, n_1,n_2,\dots, n_k,m+1)\,.$$
The length of this interval is
$$l(I^{(m)}_{n_1,\dots,n_k,m}\cup I^{(m)}_{n_1,\dots,n_k,m+1})=\frac{2}{((m+1)q_{k}-q_{k-1})((m-1)q_{k}-q_{k-1})}
$$
Now we estimate the ratio
\begin{equation*}
\begin{split}
\frac{l(I^{(m)}_{n_1,\dots,n_k,m}\cup I^{(m)}_{n_1,\dots,n_k,m+1})} {l(I^{(m)}_{n_1,n_2,\dots,n_k})} &= \frac{2q_{k}(q_k-q_{k-1})}{((m+1)q_{k}-q_{k-1})((m-1)q_{k}-q_{k-1})}\\
&\le \frac{2q_{k}}{(m+1)q_{k}-q_{k-1}}\\
&\le \frac{2q_{k}}{3q_{k}-q_{k-1}}=\frac{2}{3-q_{k-1}/q_k}\\
&\le\frac{2k}{2k+1}
\end{split}
\end{equation*}
since $\displaystyle\frac{q_{k-1}}{q_{k}}\le \frac{k-1}{k}$. Indeed, if $n_1=\dots=n_k=2$, then $q_{k-1}/q_k=(k-1)/k$; if some $n_j>2$, then $q_{k-1}/q_k\le 1/2$ from \eqref{pkqk}. This proves that for every $k\ge 1$
$$
I^{(m)}_{k+1}\le \frac{2k}{2k+1}I^{(m)}_k
$$
so
$$l(I^{(m)}_{k})\le \frac{2\cdot 4 \cdots (2k-2)}{3\cdot 5\cdots(2k-1)} \cdot l(I^{(m)}_1)\longrightarrow 0 \text { as } k\rightarrow \infty.$$
Therefore, in all cases, $l(I^{(m)}_{k})\rightarrow 0$ as $k\rightarrow \infty$. Since $\Gamma^{(m)}\subset I^{(m)}_{k}$ for every $k\ge 1$, the proposition follows.
\end{proof}
\begin{rem}\label{bdigits2}
By a similar argument, the set $\Gamma^{(-m)}_{a,b}=\{x=\lfloor 0, n_1,n_2,\dots\rceil_{a,b}\:|\: n_k\in\{-m,-m-1\}\}$
has zero Lebesgue measure for every $m\ge 1$.
\end{rem}
\section{Attractor set for $F_{a,b}$}\label{s:3}
The reduction map $F_{a,b}$ defined by \eqref{Fab} has a trapping domain, i.e. a closed
set $\Theta_{a,b}\subset \bar\mathbb R^2\setminus \Delta$ with the following properties:
\begin{itemize}
\item[(i)] for every pair $(x,y)\in \bar\mathbb R^2\setminus \Delta$, there exists a positive integer $N$
such that $F_{a,b}^N(x,y)\in \Theta_{a,b}$;
\item[(ii)] $F_{a,b}(\Theta_{a,b})\subset \Theta_{a,b}$.
\end{itemize}
\begin{thm}\label{Delta-trapping} The region $\Theta_{a,b}$ consisting of two connected components (or one if $a=0$ or $b=0$)
defined as
\begin{equation*}
\Theta^u_{a,b}=
\begin{cases}
[-\infty,-1]\times[b-1,\infty]\cup [-1,0]\times[-\frac{1}{a},\infty] & \text{ if } b\ge 1, a\ne 0\\
\emptyset & \text{ if } a=0\\
\begin{split}
[-\infty,-1]\times[b-1,\infty]&\cup [-1,0]\times[\min (-\frac{b}{b-1}, -\frac1 a),\infty]\\
&\cup[0,1]\times[-\frac{1}{b-1},\infty]
\end{split} & \text{ if } 0<b<1
\end{cases}
\end{equation*}
\begin{equation*}
\Theta^l_{a,b}=
\begin{cases}
[0,1]\times[-\infty,-\frac{1}{b}]\cup [1,\infty]\times[-\infty,a+1] & \text{ if } a\le -1, b\ne 0\\
\emptyset & \text{ if } b=0\\
\begin{split}
[-1,0]\times[-\infty,-\frac{1}{a+1}]&\cup [0,1]\times[-\infty,\max (\frac{a}{a+1}, -\frac{1}{b})]\\
&\cup[1,\infty]\times[-\infty,a+1]
\end{split} &\text{ if } a>-1
\end{cases}
\end{equation*}
is the trapping region for the reduction map $F_{a,b}$.
\end{thm}
\begin{figure}[htb]
\includegraphics[scale=0.6]{trap1.eps} $\quad$ \includegraphics[scale=0.6]{trap2.eps}
\caption{Typical trapping regions: case $a<-1, 0<b<1$ (left); case $-1<a<0<b<1$ (right)}
\label{trap}
\end{figure}
\begin{proof}
The fact that the region $\Theta_{a,b}$ is $F_{a,b}$-invariant is verified by a direct calculation.
We focus our attention on the attracting property of $\Theta_{a,b}$. Let $(x,y)\in \mathbb R^2\setminus\Delta$, write
$y=\lfloor n_0,n_1,\dots\rceil_{a,b}$, and construct the following sequence of
real pairs $\{(x_k,y_k)\}$ ($k\ge 0$) defined by $x_0=x$, $y_0=y$
and:
$$y_{k+1}=ST^{-n_k}\dots ST^{-n_1}ST^{-n_0}y\,,\quad x_{k+1}=ST^{-n_k}\dots ST^{-n_1}ST^{-n_0}x\,.$$
If $y$ is rational and its $(a,b)$-expansion terminates $y=\lfloor n_0,n_1,\dots,n_l\rceil_{a,b}$, then $y_{l+1}=\pm \infty$,
so $(x,y)$ lands in $\Theta_{a,b}$ after finitely many iterations. If $y$ has an infinite $(a,b)$-expansion, then
$y_{k+1}=\lfloor n_{k+1},n_{k+2},\dots\rceil_{a,b}$, and $y_{k+1}\ge -1/a$ or $y_{k+1}\le -1/b$ for $k\geq 0$. Also,
\begin{equation*}
\begin{split}
y&=T^{n_0}ST^{n_1}S\dots T^{n_k}S(y_{k+1})=\frac{p_{k}y_{k+1}-p_{k-1}}{q_{k}y_{k+1}-q_{k-1}}\\
x&=T^{n_0}ST^{n_1}S\dots T^{n_k}S(x_{k+1})=\frac{p_{k}x_{k+1}-p_{k-1}}{q_{k}x_{k+1}-q_{k-1}}\,,
\end{split}
\end{equation*}
hence
\begin{equation*}\label{eq:uk}
x_{k+1}=\frac{q_{k-1}x-p_{k-1}}{q_kx-p_k}=\frac{q_{k-1}}{q_k}+\frac{1}{q_k^2(p_k/q_k-x)}=\frac{q_{k-1}}{q_k}+\varepsilon_{k}
\end{equation*}
where $\varepsilon_{k}\rightarrow 0$. This shows that for $k$ large enough $x_{k+1}\in [-1,1]$. We proved that there exists $N>0$, such that
$$F_{a,b}^{N}(x,y)=ST^{-n_k}\dots ST^{-n_1}ST^{-n_0}(x,y)\in [-1,1]\times ([-1/a,\infty]\cup [-\infty,-1/b])\,.$$
The point $F^N_{a,b}(x,y)=:(\tilde x,\tilde y)$ belongs to $\Theta_{a,b}$, unless $b<1$ and $(\tilde x,\tilde y)\in [0,1]\times [-1/a,-1/(b-1)]$ or
$a>-1$ and $(\tilde x,\tilde y)\in [-1,0]\times [-1/b,-1/(a+1)]$.
Let us study the next iterates of $(\tilde x,\tilde y)\in [0,1]\times [-1/a,-1/(b-1)]$. If $\tilde y\ge b+1$ then
$$F^2_{a,b}(\tilde x,\tilde y)=(\tilde x-2,\tilde y-2)\in [-1,1]\times [b-1,\infty]\,,$$ so $F^2_{a,b}(\tilde x,\tilde y)\in \Theta_{a,b}$. If it so happens that $-1/a\le \tilde y<b+1$, then
$$F_{a,b}(\tilde x,\tilde y)=(\tilde x-1,\tilde y-1)\in [-1,0]\times [0,b]$$ and
$$F^2_{a,b}(\tilde x,\tilde y)=ST^{-1}(\tilde x,\tilde y)\in [0,\infty]\times [-1/b,\infty] \subset \Theta_{a,b}\,.$$
Similarly, if $(x,y)\in [-1,0]\times [-1/b,-1/(a+1)]$, then $F^2_{a,b}(x,y)\in \Theta_{a,b}$.
Notice that if $a=0$, then $y_{k+1}\le -1/b$ for all $k\ge 0$ (so $\Theta_{a,b}^u=\emptyset$) and if $b=0$, then $y_{k+1}\ge -1/a$ for al $k\ge 0$ (so $\Theta_{a,b}^l=\emptyset$)
\end{proof}
Using the trapping region described in Theorem \ref{Delta-trapping} we define the associated \emph{attractor set}
\begin{equation}\label{def-atrac}
D_{a,b}=\bigcap_{n=0}^\infty D_n,
\end{equation}
where $D_n=\bigcap_{i=0}^n F_{a,b}^i(\Theta_{a,b})$.
\begin{rem}
In the particular cases when $a=0$ and $b\ge 1$, or $b=0$ and $a\le -1$ or $(a,b)=(-1,1)$ the trapping regions
\begin{eqnarray*}
\Theta_{0,b}&=&[-1,0]\times [-\infty,-1]\cup [0,1]\times[-\infty,0]\cup[1,\infty]\times [-\infty,1]\\
\Theta_{a,0}&=&[-\infty,-1]\times[-1,\infty]\cup [-1,0]\times [0,\infty]\cup [0,1]\times [1,\infty]\\\
\Theta_{-1,1}&=&
[-\infty,-1]\times[-1,\infty]\cup [-1,0]\times [1,\infty]\\
& &\!\!\!\cup \,\,[0,1]\times [-\infty,-1]\cup [1,\infty]\times [-\infty,0]
\end{eqnarray*}
are also bijectivity domains for the corresponding maps $F_{a,b}$. Therefore, in these cases the attractor $D_{a,b}$ coincides with the trapping region $\Theta_{a,b}$, so the properties mentioned in the introduction are obviously satisfied. In what follows, all our considerations will exclude these degenerate cases.
\end{rem}
\section{Cycle property}\label{s:4}
In what follows, we simplify the notations for $f_{a,b}$, $\lfloor,\cdot\rceil_{a,b}$, $\hat f_{a,b}$ and $F_{a,b}$ to $f$, $\lfloor,\cdot\,\rceil$, $\hat f$ and $F$, respectively, assuming implicitly their dependence on parameters $a,b$. We will use the notation $f^n$ (or $\hat f^n$) for the $n$-times composition operation of $f$ (or $\hat f$). Also,
for a given point $x\in (a,b)$ the notation $\hat f^{(k)}$ means the transformation of type $T^iS$ ($i$ is an integer) such that
\[
\hat f^k(x)=\hat f^{(k)}\hat f^{(k-1)}\cdots \hat f^{(2)}\hat f^{(1)}(x),
\]
where $\hat f^{(1)}(x)=\hat f(x)$.
The map $f$ is discontinuous at $x=a$, $b$, however, we can associate to each $a$ and $b$ two forward orbits: to $a$ we associate the {\em upper orbit} ${\mathcal O}_u(a)=\{f^{n}(Sa)\}$, and the {\em lower orbit} ${\mathcal O}_\ell(a)=\{f^n(Ta)\}$, and to $b$ --- the {\em lower orbit} ${\mathcal O}_\ell(b)=\{f^n(Sb)\}$ and the {\em upper orbit} ${\mathcal O}_u(b)=\{f^n(T^{-1}b)\}$. We use the convention
that if an orbit hits one of the discontinuity points $a$ or $b$, then the next iterate is computed according to the lower or upper location: for example,
if the lower orbit of $b$ hits $a$, then the next iterate is $Ta$,
if the upper orbit of $b$ hits $a$ then the next iterate is $Sa$.
Now we explore the patterns in the above orbits. The following property plays an essential role in studying the map $f$.
\begin{defn}\label{def:cycles}
We say that the point $a$ has the {\em cycle property} if for some non-negative integers $m_1,k_1$
\[
f^{m_1}(Sa)=f^{k_1}(Ta)=c_a.
\]
We will refer to
the set
\[
\{Ta, fTa,\dots ,f^{k_1-1}Ta\}
\]
as the {\em lower side of the $a$-cycle}, to the set
\[
\{Sa, fSa,\dots ,f^{m_1-1}Sa\}
\]
as the {\em upper side of the $a$-cycle}, and to $c_a$ as the {\em end of the $a$-cycle}.
If the product over the $a$-cycle equals the identity transformation, i.e.
\[
T^{-1}f^{-k_1}f^{m_1}S=\rm Id,
\]
we say that $a$ has {\em strong cycle property}, otherwise, we say that $a$ has {\em weak cycle property}.
Similarly, we say that $b$ has {\em cycle property} if for some non-negative integers $m_2,k_2$
\[
f^{k_2}(Sb)=f^{m_2}(T^{-1}b)=c_b.
\]
We will refer to
the set
\[
\{Sb, fSb,\dots ,f^{k_2-1}Sb\}
\]
as the {\em lower side of the $b$-cycle}, to the set
\[
\{T^{-1}b, fT^{-1}b,\dots ,f^{m_2-1}T^{-1}b\}
\]
as the {\em upper side of the $b$-cycle}, and to $c_b$ as the {\em end of the $b$-cycle}.
If the product over the $b$-cycle equals the identity transformation, i.e.
\[
Tf^{-m_2}f^{k_2}S=\rm Id,
\]
we say that $b$ has {\em strong cycle property}, and otherwise we say that $b$ has {\em weak cycle property}.
\end{defn}
It turns out that the cycle property is the prevalent pattern. It can be analyzed and described explicitly by partitioning the parameter set $\mathcal P$ based on the first digits of $Sb$, $STa$, and $Sa$, $ST^{-1}b$, respectively. Figure \ref{fig-P} shows a part of the countable partitions, with $B_{-1},B_{-2},\dots$ denoting the regions where $Sb$ has the first digit $-1, -2,\dots$, and $A_1,A_2,\dots$, denoting the regions where $Sa$ has the first digit $1,2,\dots$.
For most of the parameter region, the cycles are short: everywhere except for the very narrow triangular regions shown in Figure \ref{fig-P} the cycles for both $a$ and $b$ end after the first return to $[a,b)$.
However, there are Cantor-like recursive sets where the lengths of the cycles can be arbitrarily long. Part of this more complex structure, studied in details in Section \ref{s:8}, can be seen as very narrow triangular regions close to the boundary segment $b-a=1$.
\begin{figure}[htb]
\psfrag{A}[l]{\small $a$}
\psfrag{B}[l]{\small $b$}
\psfrag{a}[c]{\small $B_{-1}$}
\psfrag{b}[c]{\small $B_{-2}$}
\psfrag{c}[l]{\small $B_3$}
\psfrag{d}[l]{\small $A_1$}
\psfrag{e}[c]{\small $A_2$}
\psfrag{f}[c]{\small $A_3$}
\psfrag{u}[l]{\small $\vdots$}
\psfrag{v}[l]{\small $\dots$}
\includegraphics[scale=1]{domain.eps}
\caption{The parameter set $\P$ and its partition}
\label{fig-P}
\end{figure}
By symmetry of the parameter set $\P$ with respect to the line $b=-a$, $(a,b)\mapsto (-b,-a)$, we may assume that $b\leq -a$ and concentrate our attention to this subset of $\P$.
The structure of the set where the cycle property holds for $b$ is described next for the part of the parameter region with $0<b\le -a<1$. We make use extensively of the first return map $\hat f$.
\begin{thm}\label{b-cycle
Let $(a,b)\in\P$, $0<b\leq -a<1$ and $m\ge 1$ such that $a\le T^mSb<a+1$.
\begin{itemize}
\item[\bf{(I)}] Suppose that there exists $n\geq 0$ such that
\[
\quad\quad\quad \hat f^{k}T^mSb\in \Big(\frac{b}{b+1},a+1\Big) \text{ for } k<n, \text{ and } \hat f^{n}T^mSb\in \Big[a,\frac{b}{b+1}\Big].
\]
\begin{itemize}
\item[(i)] If $\hat f^{n}T^mSb\in (a,\frac{b}{b+1})$, then $b$
has the cycle property; the cycle property is strong if and only if $\hat f^{n}T^mSb\neq 0$.
\item[(ii)] If $\hat f^{n}T^mSb=a$, then $b$ has the cycle property if and only if $a$ has the cycle property.
\item[(iii)] $\hat f^{n}T^mSb=b/(b+1)$, then $b$ does not have the cycle property, but the orbits of $Sb$ and $T^{-1}b$ are periodic.
\end{itemize}
\item[\bf{(II)}] If $\hat f^{k}T^mSb\in (\frac{b}{b+1},a+1)$ for all $k\geq 0$, then $b$ does not have the cycle property.
\end{itemize}
\end{thm}
\begin{proof} {\bf{(I)}}
In the case $m=1$, and assuming $a<TSb<a+1$ we have
\begin{equation}\label{m=1}
a<1-\frac1{b}< \frac{b}{b+1},
\end{equation}
and the cycle relation for $b$ can be explicitly described as
\begin{equation}\label{m1}
\begin{diagram}
& & \boxed{b-1} &\rTo{S} & \boxed{-\frac 1{b-1}}\\
&\ruTo{T^{-1}}& & & &\rdTo{T^{-1}} \\
\boxed{b}& & & & & &\quad\boxed{c_b=\frac{b}{1-b}}\\
&\rdTo{S} & & & &\ruTo{S} \\
& &\boxed{-\frac 1{b}}&\rTo{T}& \boxed{\frac{b-1}{b}}
\end{diagram}
\end{equation}
In the particular situation that $TSb=a$,
the lower orbit of $b$ hits $a$ and continues to $a+1$, while the upper orbit hits $\frac{b}{1-b}=-1/a$. This means that the iterates will follow the lower and upper orbits of $a$, respectively, thus statement (ii) holds.
Since the second inequality (\ref{m=1}) is strict, the case (iii) cannot occur.
\medskip
For the case $m=2$ (and assuming $T^2Sb\ne a$) we analyze the following situations: if $b<\frac12$, then
$
2-\frac1{b}<0,
$
and the cycle relation is
\begin{equation}\label{m=2}
\begin{diagram}
& & \boxed{b-1} &\rTo{S} & \boxed{-\frac 1{b-1}} &\rTo{ST^{-2}}& \boxed{1+\frac{b}{1-2b}}\\
&\ruTo{T^{-1}}& & & & & &\rdTo{\;\;T^{-1}}\\
\boxed{b}& & & & & & & & \boxed{c_b=\frac{b}{1-2b}}\\
&\rdTo{S}& & & & & \ruTo(3,2){S} & & &\\
& &\boxed{-\frac 1{b}}&\rTo{T^{2}}& \boxed{-\frac{1-2b}{b}}
\end{diagram}
\end{equation}
If $b>\frac12$ we have
\[
0<2-\frac1{b}\leq \frac{b}{b+1},
\]
since we must also have $2-\frac1{b}<a+1$, i.e. $b\leq \frac1{1-a}$,
and
the cycle relation is
\begin{equation}
\begin{diagram}
& & \boxed{b-1} &\rTo{S} & \boxed{-\frac 1{b-1}} & & & &\\
&\ruTo{T^{-1}}& & & & \rdTo(4,2){ST^{-2}} & \\
\boxed{b}& & & & & & & & \boxed{c_b=1+\frac{b}{1-2b}}\\
&\rdTo{S}& & & & & & \ruTo_{T} \\
& &\boxed{-\frac 1{b}}&\rTo{T^{2}}& \boxed{-\frac{1-2b}{b}} & \rTo^S & \boxed{\frac{b}{1-2b}}&
\end{diagram}
\end{equation}
The above cycles are strong.
If $b=\frac12$ the cycle relation is
\begin{equation}
\begin{diagram}
& & \boxed{b-1} &\rTo{S} & \boxed{-\frac 1{b-1}}\\
&\ruTo{T^{-1}}& & & &\rdTo(2,2){T^{-2}} \\
\boxed{b}& & & & & &\boxed{c_b=-\frac{1-2b}{b}=0}\\
&\rdTo{S}& & & \ruTo(3,2){T^2} &\\
& &\boxed{-\frac 1{b}}& & &
\end{diagram}
\end{equation}
It is easy to check that this cycle is weak. In the particular situation when
$T^2Sb=a$, the lower orbit of $b$ hits $a$, and continues with $a+1$, while the upper orbit still hits $\frac{b}{1-2b}=-1/a$. This means that the iterates will follow the lower and upper orbits of $a$, respectively, and statement (ii) holds.
The relation $2-\frac1{b}=\frac{b}{b+1}$ implies $b=\frac{-1+\sqrt{5}}2$ that does not have the cycle property and the orbits
of $Sb$ and $T^{-1}b$ are periodic; this is the only possibility for (iii) to hold.
\medskip
The situation for $m\geq 3$ is more intricate. First we will need the following lemmas.
\begin{lem} \label{xy} Suppose $STSx=y$. The following are true:
\begin{itemize}
\item[(a)] if $TSb\leq x<a$, then $b-1\leq y<\frac{a}{1-a}$;
\item[(b)] if $a\le x<\frac{b}{b+1}$, then $\frac{a}{1-a}\le y< b$
\item[(c)] if $\frac{b}{b+1}\le x< a+1$, then $b\le y<\frac{a}{1-a}+1$;
\item[(d)] if $x=0$, then $y=0$.
\end{itemize}
\end{lem}
\begin{proof} Applying $STS$ to the corresponding inequalities
we obtain
\smallskip
(a)
\(\qquad
b-1=STSTSb\leq y<STSa=\displaystyle\frac{a}{1-a}.
\)
\smallskip
(b)
\(\qquad
\displaystyle \frac{a}{1-a}=STSa\le y<STSTSTb=b
\)
\smallskip
(c)
\(\qquad
b=STSTSTb\le y<STSTa=T^{-1}Sa\le \displaystyle\frac1{1-a}=\frac{a}{1-a}+1,
\)
\smallskip
\noindent where the last inequality is valid for $a\le \frac{1-\sqrt{5}}{2}$,
which is true in the considered region $b\le\frac1{2-a}$.
Relation (d) is obvious.
\end{proof}
\begin{lem}\label{main-argument}
Suppose that for all $k<n$
\begin{equation}\label{continuation}
\frac{b}{b+1}<\hat f^{k}T^mSb<a+1.
\end{equation}
Then
\begin{enumerate}
\item for $0\leq k\leq n$, in the lower orbit of $b$, $\hat f^{(k)}=T^mS$ or $T^{m+1}S$; in the upper orbit of $b$, $\hat f^{(k)}=T^{-i}S$ with $i=2$ or $3$;
\item
there exists $p>1$ such that
\begin{equation}\label{STS}
(STS)\hat f^{n}T^mS=(T^{-2}S)\hat f^{p}T^{-1}.
\end{equation}
\end{enumerate}
\end{lem}
\begin{proof}
(1) Applying $T^mS$ to the inequality (\ref{continuation}), we obtain
\[
a-1\leq T^{m-1}Sb=T^mSTSTb<T^mS\hat f^k T^mSb\leq T^mSTa\leq T^mSb< a+1,
\]
therefore $\hat f^{(k+1)}=T^mS$ or $T^{m+1}S$. Since $\hat f^{(0)}=T^mS$, we conclude that $\hat f^{(k)}=T^mS$ or $T^{m+1}S$ for $0\leq k\leq n$.
(2) In order to determine the upper side of the $b$-cycle,
we will use the following relation in the group $SL(2,\Z)$ obtained by concatenation of the ``standard" relations (from right to left)
\begin{equation}\label{stand}
(STS)T^iS=(T^{-2}S)^{i-1}T^{-1}\quad (i\geq 1),
\end{equation}
and Lemma \ref{xy} repeatedly.
The proof is by induction on $n$. For the base case $n=1$ we have
\[
\frac{b}{b+1}< T^mSb< a+1.
\]
Then for $1\leq i\leq m-1$ $T^iSb$ satisfies condition (a) of Lemma \ref{xy}, hence
\[
b-1<(T^{-2}S)^{i-2}T^{-1}b<\frac{a}{1-a},
\]
which means that on the upper side of the $b$-cycle $\hat f^{(1)}=T^{-1}$ and $\hat f^{(i)}=T^{-2}S$ for $1<i\leq m-1$.
Using (\ref{stand}) for $i=m$ we obtain
\[
(STS)T^mS=(T^{-2}S)^{m-1}T^{-1}=(T^{-2}S)\hat f^{m-2}T^{-1},
\]
i.e. (\ref{STS}) holds with $p=m-2$.
Now suppose the statement holds for $n=n_0$, and for all $k<n_0+1$ we have
\[
\frac{b}{b+1}<\hat f^{k}T^mSb< a+1.
\]
By the induction hypothesis,
there exists $p_0>1$ such that
\begin{equation}\label{cont}
(STS)\hat f^{n_0}T^mS=(T^{-2}S)\hat f^{p_0}T^{-1}.
\end{equation}
But since
\[
\frac{b}{b+1}<\hat f^{n_0}T^mSb< a+1,
\]
condition (c) of Lemma \ref{xy} is satisfied, and hence
\[
b<(T^{-2}S)\hat f^{p_0}T^{-1}b<\frac{a}{1-a}+1,
\]
which is equivalent to
\[
b-1<(T^{-3}S)\hat f^{p_0}T^{-1}b<\frac{a}{1-a},
\]
i.e. $\hat f^{p_0+1}=T^{-3}S$. Using the relation
\(
(STS)T^2S=T^{-1}(STS),
\)
we can rewrite (\ref{cont}) as
\begin{equation}\label{ind}
(STS)T^2S\hat f^{n_0}T^mS=(T^{-3}S)\hat f^{p_0}T^{-1}=\hat f^{p_0+1}T^{-1}.
\end{equation}
Let $\hat f^{(p_0+1)}=T^qS$. We have proved in (1) that $q=m$ or $m+1$, hence $q\geq 3$.
Let
\[
b_0=T^2S\hat f^{n_0}T^mSb \text{ and } c_0=(T^{-3}S)\hat f^{p_0}T^{-1}b.
\]
Then by (\ref{ind})
$(STS)b_0=c_0$.
Using the relation $(STS)T=T^{-2}S(STS)$, we obtain
\[
(STS)T^i=(T^{-2}S)^i(STS),
\]
and therefore,
\begin{equation}\label{STSi}
(STS)T^ib_0=(T^{-2}S)^i(STS)b_0=(T^{-2}S)^ic_0
\end{equation}
Since for $0\leq i<q-2$ $T^ib_0$ satisfies condition (a) of Lemma \ref{xy},
we conclude that
\[b-1< (T^{-2}S)^ic_0<\frac{a}{1-a}.
\]
Therefore $\hat f^{(i)}=T^{-2}S$ for $p_0+1<i\leq p_0+q$,
and (\ref{STSi}) for $i=q-2$ gives us the desired relation
\[
(STS)\hat f^{n_0+1}T^mS=(T^{-2}S)\hat f^{p_0+q}T^{-1}
\]
with $p=p_0+q$.
\end{proof}
Now we complete the proof of the theorem. In what follows we introduce the notations
\[
I_\ell=\Bigl( a,\frac{b}{b+1}\Bigr)\,,\; I_u=\Bigl(\frac{a}{1-a},b\Bigr)
\]
and write $\overline I_\ell$, $\overline I_u$ for the corresponding closed intervals.
(I) If
\(
\hat f^nT^mSb\in I_\ell
\), then
condition (b) of Lemma \ref{xy} is satisfied, and
\[
(T^{-2}S)\hat f^{p}T^{-1}b\in I_u.
\]
It follows that $\hat f^{(p+1)}=T^{-2}S$,
therefore (\ref{STS}) can be rewritten as
\[
(STS)\hat f^{n}T^mS=\hat f^{p+1}T^{-1},
\]
which means that we reached the end of the cycle. More precisely,
(i) if $\hat f^{n}T^mSb\in (0,\frac{b}{b+1})$, then
\[
TS\hat f^{n}T^mSb=S\hat f^{p}T^{-1}b=c_b;
\]
$b-1<\hat f^{j}T^{-1}b<\frac{a}{1-a}$ for $j<p$, and $\hat f^{p}T^{-1}b\in (0,b)$. In this case $c_b<Sb$.
If $\hat f^{n}T^mSb\in (a,0)$, then
\[
S\hat f^{n}T^mSb=T^{-1}S\hat f^{p}T^{-1}b=c_b;
\]
$b-1<\hat f^{j}T^{-1}b<\frac{a}{1-a}$ for $j<p$, and $\hat f^{p}T^{-1}b\in (\frac{a}{1-a},0)$.
In this case $c_b>Sa$.
Since the cycle relation in both cases is equivalent to the identity (\ref{STS}), the cycle property is strong, and (i) is proved.
If $\hat f^{n}T^mSb=0$, then
\[\hat f^{n}T^mSb=\hat f^{p}T^{-1}b=0
\]
is the end of the cycle; for $j<p$, $b-1<\hat f^{j}T^{-1}b<\frac{a}{1-a}$. In this case the cycle ends ``before" the identity (\ref{STS}) is complete, therefore the product over the cycle is not equal to identity, and the cycle is weak.
\medskip
(ii) If $\hat f^{n}T^mSb=a$, then following the argument in (i) and using relation \eqref{STS} we obtain that the upper orbit of $b$ hits $T^{-1}S\hat f^{p}T^{-1}b=S\hat f^{n}T^mSb=Sa=-1/a$, while the lower orbit hits the value $a+1$, hence $b$ satisfies the cycle property if and only if $a$ does.
\medskip
(iii) If $\hat f^{n}T^mSb=\frac{b}{b+1}$, then following the argument in (i) we obtain
\[
(T^{-2}S)\hat f^{p}T^{-1}b=b.
\]
However, one needs to apply one more $T^{-1}$ to follow the definition of the map $f$, hence $\hat f^{(p+1)}=T^{-3}S$, not $T^{-2}S$, and the cycle will not close.
One also observes that in this case the $(a,b)$-expansions of $Sb$ and $T^{-1}b$ will be periodic, and therefore the cycle will never close.
(II) If
\[
\hat f^{k}T^mSb\notin \overline {I_\ell}
\]
for all $k\geq 0$, by the argument in the part (I) of the proof, on the
lower orbit of $b$
each $\hat f^{(k)}=T^q S$, where $q=m$ or $m+1$, and on the
upper orbit of $b$
each $\hat f^{(p)}=T^{-r}S$, where $r=2$ or $3$, and
for all $p\geq 1$
\[
\hat f^pT^{-1}b\notin\overline{I_u}.
\]
This means that for all images under the original map $f$ on the
lower orbit of $b$
we have
\[
f^kSb\in\left(-1-\frac1{b},a\right)\cup \left(\frac{b}{b+1},a+1\right)
\]
while for the images on the
upper orbit of $b$
\[
f^kT^{-1}b\in\left(b-1,\frac{a}{1-a}\right)\cup\left(b,1-\frac1{a}\right).
\]
Since these ranges do not overlap, the cycle cannot close, and $b$ has no cycle property.
\end{proof}
A similar result holds for the $a$-cycles. First, if $Sa$ has the first digit $1$, i.e. $b\le Sa<b+1$, then one can easily write the $a$-cycle, similarly to \eqref{m=1}. For the rest of the parameter region we have
\begin{thm}\label{a-cycle}
Let $(a,b)\in\P$, $0<b\leq -a<1$ with $Sa\ge b+1$ and $m\ge 1$ such that $a\le T^mSTa<a+1$.
\begin{itemize}
\item[\bf{(I)}] Suppose that there exists $n\geq 0$ such that
\[
\quad\quad\quad \hat f^{k}T^mSTa\in \Big(\frac{b}{b+1},a+1\Big) \text{ for } k<n, \text{ and } \hat f^{n}T^mSTa\in \Big[a,\frac{b}{b+1}\Big].
\]
\begin{itemize}
\item[(i)] If $\hat f^{n}T^mSTa\in (a,\frac{b}{b+1})$, then $a$
has the cycle property; the cycle property is strong if and only if $\hat f^{n}T^mSTa\neq 0$.
\item[(ii)] If $\hat f^{n}T^mSTa=a$, then $a$ does not have the cycle property, but the $(a,b)$-expansions of $Sa$ and $Ta$ are eventually periodic.
\item[(iii)] $\hat f^{n}T^mSTa=b/(b+1)$, then $a$ has the cycle property if and only if $b$ has the cycle property.
\end{itemize}
\item[\bf{(II)}] If $\hat f^{k}T^mSTa\in (\frac{b}{b+1},a+1)$ for all $k\geq 0$, then $a$ does not have the cycle property.
\end{itemize}
\end{thm}
\begin{proof} The proof follows the proof of Theorem \ref{b-cycle} with minimal modifications. In particular, the relation
(\ref{STS}) should be replaced by relation
\begin{equation}\label{STS*}
(STS)\hat f^nT^mST=(T^{-2}S)\hat f^p.
\end{equation}
For (iii), since $\hat f^nT^mSTa=\frac{b}{b+1}$, on the lower side we have $TS f^nT^mSTa=Sb$, and on the upper side, using (\ref{STS*}), $(T^{-2}S)\hat f^pb=b$. As in the proof of Theorem \ref{b-cycle}, $\hat f^{p+1}=T^{-3}S$, so $(T^{-3}S)\hat f^pb=T^{-1}b$. Therefore $a$ has (strong or weak) cycle property if and only if $b$ does.
\end{proof}
Let us now
describe the situation when $a\le -1$.
\begin{thm}\label{ab-cycle
Let $(a,b)\in\P$ with $0<b\leq -a$ and $a\le -1$. Then $a$ and $b$ satisfy the cycle property.
\end{thm}
\begin{proof}
It is easy to see that $a=-1$ has the degenerate weak cycle:
\begin{equation}\label{a=-1}
\begin{diagram}
& & \boxed{1} \\
&\ruTo{S}& &\rdTo{T^{-1}} \\
\boxed{a=-1} & & \rTo{T} & & \boxed{0} \\
\end{diagram}
\end{equation}
while $a<-1$ satisfies the following strong cycle relation:
\begin{equation}\label{acycle}
\begin{diagram}
& & \boxed{-\frac{1}{a}} &\rTo{T^{-1}} & \boxed{-\frac{1}{a} - 1} & \rTo{S} &\boxed{\frac{a}{a+1}}\\
&\ruTo{S} & & & & & &\rdTo{T^{-1}} \\
\boxed{a}& & & & & & & & \boxed{c_a=-\frac{1}{a+1}} \\
&\rdTo{T} & & & & & & \ruTo(6,2){S} \\
& &\boxed{a+1}
\end{diagram}
\end{equation}
In order to study the orbits of $b$, let $m\ge 0$ such that $a\le T^mSb<a+1$. If $m=0$, then $Sb=a$ (since $Sb\le a$), and the cycle of $b$ is identical to the one described by \eqref{a=-1}. If $m\ge 1$, then one can use relation \eqref{stand} to construct the $b$-cycle. More precisely, if $a<T^mSb<a+1$, then we have:
\begin{equation}
\begin{diagram}
& & \boxed{b-1} &\rTo{S} & \boxed{-\frac 1{b-1}} &\rTo{(ST^{-2})^{m-1}}& \boxed{1+\frac{b}{1-mb}}\\
&\ruTo{T^{-1}}& & & & & &\rdTo{\;\;T^{-1}}\\
\boxed{b}& & & & & & & & \boxed{c_b=\frac{b}{1-mb}}\\
&\rdTo{S}& & & & & \ruTo(3,2){S} & & &\\
& &\boxed{-\frac 1{b}}&\rTo{T^{m}}& \boxed{-\frac{1-mb}{b}}
\end{diagram}
\end{equation}
If $T^mSb=a$, then it happens again that the lower orbit of $b$ hits a, and then $Ta$, while the upper orbit hits $Sa$. Following now the cycle of $a$ described by \eqref{acycle}, we conclude that $b$ satisfies the strong cycle property.
If $T^mSb=0$, i.e. $b=1/m$, then a minor modification of the above b-cycle gives us the following weak cycle relation:
\begin{equation}
\begin{diagram}
& & \boxed{b-1} &\rTo{S\;\;} & \boxed{-\frac 1{b-1}} &\rTo{T^{-1}(ST^{-2})^{m-2}}& \boxed{\frac{b}{1-mb+b}=1}\\
& \ruTo_{T^{-1}} & & & & & &\rdTo_{\;\;T^{-1}}\\
\boxed{b}& & & & & & & & \boxed{c_b=0}\\
&\rdTo{S\;\;}& & & & & \ruTo(3,2){T} & & &\\
& &\boxed{-\frac 1{b}=-m}&\rTo{T^{m-1}}& \boxed{-1}
\end{diagram}
\end{equation}
\end{proof}
The following corollaries are immediate from the proof of Theorems \ref{b-cycle}, \ref{a-cycle}, \ref{ab-cycle}.
\begin{cor} \label{b-norepeats}If $b$ has the cycle property, then the upper side of the $b$-cycle
\[
\{T^{-1}b, fT^{-1}b,\dots ,f^{m_2-1}T^{-1}b\}
\]
and the lower side of the $b$-cycle
\[
\{Sb, fSb,\dots ,f^{k_2-1}Sb\}
\]
do not have repeating values.
\end{cor}
\begin{cor} \label{a-norepeats}If $a$ has the cycle property, then the upper side of the $a$-cycle
\[
\{Sa, fSa,\dots ,f^{m_1-1}Sa\}
\]
and the lower side of the $a$-cycle
\[
\{Ta, fTa,\dots ,f^{k_1-1}Ta\}
\]
do not have repeating values.
\end{cor}
\section{Finiteness condition implies finite rectangular structure}\label{s:5}
In order to state the condition under which the natural extension map $F_{a,b}$ has an attractor with finite rectangular structure mentioned in the Introduction, we follow the split orbits of $a$ and $b$
\[
\mathcal{L}_a=\begin{cases} {\mathcal O}_\ell(Ta)&\text{ if $a$ has no cycle property}\\
\text{lower part of $a$-cycle}&\text{ if $a$ has strong cycle property}\\
\text{lower part of $a$-cycle $\cup\{0\}$}&\text{ if $a$ has weak cycle property},
\end{cases}
\]
\[
\mathcal{U}_a=\begin{cases} {\mathcal O}_u(Sa)&\text{ if $a$ has no cycle property}\\
\text{upper part of $a$-cycle}&\text{ if $a$ has strong cycle property}\\
\text{lower part of $a$-cycle $\cup\{0\}$}&\text{ if $a$ has weak cycle property},
\end{cases}
\]
and, similarly, $\mathcal{L}_b$ and $\mathcal{U}_b$ by
\[
\mathcal{L}_b=\begin{cases} {\mathcal O}_\ell(Sb)&\text{ if $b$ has no cycle property}\\
\text{lower part of $b$-cycle}&\text{ if $b$ has strong cycle property}\\
\text{lower part of $b$-cycle $\cup\{0\}$}&\text{ if $b$ has weak cycle property},
\end{cases}
\]
\[
\mathcal{U}_b=\begin{cases} {\mathcal O}_u(T^{-1}b)&\text{ if $b$ has no cycle property}\\
\text{upper part of $b$-cycle}&\text{ if $b$ has strong cycle property}\\
\text{lower part of $b$-cycle $\cup\{0\}$}&\text{ if $b$ has weak cycle property},
\end{cases}
\]
We find it useful to introduce the map $\rho_{a,b}:\bar\mathbb R\to\{T,S,T^{-1}\}$
\begin{equation}
\rho_{a,b}(x)=\begin{cases}
T &\text{ if } x < a\\
S &\text{ if } a\le x<b\\
T^{-1} &\text{ if } x\ge b
\end{cases}
\end{equation}
in order to write $f_{a,b}(x)=\rho_{a,b}(x)x$ and $F_{a,b}(x,y)=(\rho(y)x,\rho(y)y)$.
\begin{rem}It follows from the above definitions that $\rho(y)=S$ or $T$ if $y\in\mathcal{L}_a\cup \mathcal{L}_b$, and $\rho(y)=S$ or $T^{-1}$ if $y\in\mathcal{U}_a \cup\mathcal{U}_b$
\end{rem}
\begin{defn} We say that the map $f_{a,b}$ satisfies the {\em finiteness condition} if the sets of values in all four truncated orbits $\mathcal{L}_a,\mathcal{L}_b,\,\,\mathcal{U}_a,\,\,\mathcal{U}_b$ are finite.
\end{defn}
\begin{prop} \label{shift} Suppose that the set $\mathcal{L}_b$ is finite. Then
\begin{enumerate}
\item either
$b$ has the cycle property or
the upper and lower orbits of $b$ are eventually periodic.
\item The finiteness of $\mathcal{L}_b$ implies the finiteness of $\mathcal{U}_b$.
\end{enumerate}
Similar statements hold for the sets $\mathcal{L}_a$, $\mathcal{U}_a$ and $\mathcal{U}_b$ as well
\end{prop}
\begin{proof} The two properties follow from Theorem \ref{b-cycle} and its proof. If $b$ does not have the cycle property, but its lower orbit is eventually periodic, then one uses Lemma \ref{main-argument} to conclude that the upper orbit of $b$ has to be eventually periodic.
\end{proof}
\begin{rem}
If $b$ has the strong cycle property, then the set $\mathcal{L}_b$ coincides with the lower side of the $b$-cycle and $\mathcal{U}_b$ coincides with the upper side of the $b$-cycle. If $b$ does not have the cycle property, but the lower and upper orbits of $b$ are eventually periodic then $\mathcal{L}_b$ and $U_b$ are identified with these orbits accordingly, until the first repeat.
\end{rem}
\begin{thm} \label{thm:recstructure} Let $(a,b)\in \mathcal P$, $a\ne 0$, $b\ne 0$, and assume that the map $f_{a,b}$
satisfies the finiteness condition. Then there exists a set $A_{a,b}\underset{\neq}\subset\bar\mathbb R^2$ with the following properties:
\begin{enumerate}
\item[(A1)] The set $A_{a,b}$ consists of two connected components each having {\em finite rectangular structure},
i.e. bounded by non-decreasing step-functions with a finite number of steps.
\item[(A2)] $F_{a,b}: A_{a,b}\to A_{a,b}$ is a bijection except for some images of the boundary of $A_{a,b}$.
\end{enumerate}
\end{thm}
\begin{proof}
(A1)
We will construct a set $A_{a,b}$ whose upper connected component is bounded by a step-function with values in the set $\mathcal{U}_{a,b}=\mathcal{U}_a\cup\mathcal{U}_b$ that we refer to as {\em upper levels}), and whose lower connected component is bounded by a step-function with values in the set $\mathcal{L}_{a,b}=\mathcal{L}_a\cup\mathcal{L}_b$ that we refer to as {\em lower levels}. Notice that each level in $\mathcal{U}_a$ and $\mathcal{U}_b$ appears exactly once, but if the same level appears in both sets, we have to count it twice in $\mathcal{U}_{a,b}$. The same remark applies to the lower levels.
Now let $y_\ell\in\mathcal{L}_{a,b}$ be the closest $y$-level to $Sb$ with $y_\ell\geq Sb$, and $y_u\in\mathcal{U}_{a,b}$ be the closest $y$-level to $Sa$ with $y_u\leq Sa$. Since each level in $\mathcal{U}_a$ and in $\mathcal{L}_b$ appears only once,
if $y_u=Sa$, $y_u$ can only belong to $\mathcal{U}_b$, and if $y_\ell= Sb$, $y_\ell$ can only belong to $\mathcal{L}_a$.
We consider the rays $[-\infty,x_b]\times \{b\}$ and $[x_a,\infty]\times \{a\}$, where $x_a$ and $x_b$ are unknown, and
``transport" them (using the special form of the natural extension map $F_{a,b}$) along the
sets $\mathcal{L}_b,\,\,\mathcal{U}_b,\,\,\mathcal{L}_a$ and $\mathcal{U}_a$ respectively until we reach the levels $y_u$ and $y_\ell$ (see Figure \ref{fig-levels}).
Now we set-up a system of two fractional linear equations by equating the right end of the segment at the level $Sb$ with the left end of the segment at the level $y_\ell$, and, similarly, the left end of the segment at the level $Sa$ and the right end of the level $y_u$.
\begin{figure}[htb]
\psfrag{8}[c]{\tiny $STa$}
\psfrag{7}[l]{\tiny $ST^{-1}b$}
\psfrag{5}[c]{\tiny $Sa$}
\psfrag{6}[c]{\tiny $Sb$}
\psfrag{3}[c]{\tiny $y_u$}
\psfrag{4}[c]{\tiny $y_\ell$}
\psfrag{1}[c]{\tiny $x_b$}
\psfrag{2}[c]{\tiny $x_a$}
\psfrag{a}[c]{\small $a$}
\psfrag{b}[c]{\small $b$}
\includegraphics[width=2.7in, height=3.1in]{levels.eps}
\caption{Construction of the domain $A_{a,b}$}
\label{fig-levels}
\end{figure}
\begin{lem}\label{2-sys} The system of two equations at the consecutive levels $y_u$ and $Sa$, and $y_\ell$ and $Sb$,
has a unique solution with $x_a\geq 1$ and $x_b\leq -1$.
\end{lem}
\begin{proof}
In what follows, we present the proof assuming that $0<b\le -a<1$. The situation $a\le -1$ is less complex due to the explicit cycle expressions described in Theorem \ref{ab-cycle} and will be discussed at the end.
Let $m_a,m_b$ be positive integers such that $a\le T^{m_a}STa<a+1$ and $a\le T^{m_b}Sb<a+1$. For the general argument we assume that $m_a,m_b\geq 3$, the cases $m_a$ or $m_b\in \{1,2\}$ being considered separately.
The level $y_u$ may belong to $\mathcal{U}_a$ or $\mathcal{U}_b$, and the level $y_\ell$ may belong to $\mathcal{L}_a$ or $\mathcal{L}_b$, therefore we need to consider $4$ possibilities.
\noindent{\bf Case 1: $y_u\in\mathcal{U}_a,\,\,y_\ell\in\mathcal{L}_a$.} Then we have
\[
Sx_a= T^{-1}S\hat f_-^{n_1}(\infty)\,,\quad
Sx_b= TS \hat f_+^{n_2}Tx_a,
\]
where $\hat f_-^{n_1}$ is a product of factors $T^{-i}S$ (that appear on the upper orbit of $a$) with $i=2$ or $3$, and $\hat f_+^{n_2}$ is a product of factors $T^{i}S$ (that appear on the lower orbit of $a$) with $i=m$ or $m+1$. Using (\ref{STS*}) we rewrite the first equation as
\[
x_a=ST^{-1}S\hat f_-^{n_1}(\infty)=ST^{-1}SST^2STS\hat f_+^{k_1}T^mST(\infty)
=T^{-1}\hat f_+^{k_1}T^mST(\infty)\,.
\]
Since $\hat f_+^{k_1}$ is a product of factors $T^{i}S$ with $i=m$ or $m+1$, $m\geq 3$, we conclude that $Tx_a$ has a finite formal continued fraction expansion starting with $m'\geq 3$, i.e. $Tx_a>2$, and $x_a>1$.
Furthermore, from the second equation
\[
x_b= STS \hat f_+^{n_2}Tx_a,
\]
hence $ \hat f_+^{n_2}Tx_a$ has a finite formal continued fraction expansion starting with $m'\geq 3$, i.e. $ \hat f_+^{n_2}Tx_a>2$, and $x_b<-2$.
\noindent{\bf Case 2: $y_u\in\mathcal{U}_a,\,\,y_\ell\in\mathcal{L}_b$.} Then
\[
Sx_a= T^{-1}S\hat f_-^{n_1}(\infty)\,,\quad
Sx_b= TS \hat f_+^{n_2}(-\infty)\,.
\]
Like in Case 1 we see that $x_a>1$, and
\[
x_b= STS \hat f_+^{n_2}(-\infty)<-2,
\]
since $ \hat f_+^{n_2}(-\infty)$ has a formal continued fraction expansion starting with $m'\geq 3$, and therefore is $>2$.
\noindent{\bf Case 3: $y_u\in\mathcal{U}_b,\,\,y_\ell\in\mathcal{L}_a$.} Then
\[
Sx_a= T^{-1}S\hat f_-^{n_1}T^{-1}x_b\,,\quad
Sx_b= TS \hat f_+^{n_2}Tx_a\,.
\]
Using (\ref{STS})
\[
x_a= ST^{-1}S\hat f_-^{n_1}T^{-1}x_b=ST^{-1}SST^2STS\hat f^{k_2}T^mSx_b,
\]
and using the second equation and simplifying, we obtain
\[
Tx_a=\hat f^{k_2}T^mSSTS\hat f_+^{n_2}(Tx_a)=\hat f^{k_2}T^{m+1}S\hat f_+^{n_2}(Tx_a).
\]
Since all its factors are of the form $T^iS$ with $i\geq 3$, the matrix $\hat f^{k_2}T^{m+1}S\hat f_+^{n_2}$ is hyperbolic and its attracting fixed point $Tx_a$ has periodic formal continued fraction expansion starting with $m'\geq 3$ (see Theorem 3.1 of \cite{KU1}), hence $x_a>1$. Finally, as in Case 1,
\[
x_b=STS\hat f_+^{n_2}Tx_a<-2
\]
since $\hat f_+^{n_2}Tx_a$ has formal continued fraction expansion with $m'\geq 3$, hence $>2$.
\noindent{\bf Case 4: $y_u\in\mathcal{U}_b,\,\,y_\ell\in\mathcal{L}_b$.} Then
\[
Sx_a= T^{-1}S\hat f_-^{n_1}T^{-1}x_b\,,\quad
Sx_b= TS \hat f_+^{n_2}S(-\infty)\,.
\]
From the second equation we obtain
\[
x_b=STS\hat f_+^{n_2}S(-\infty)<-2
\]
since $\hat f_+^{n_2}S(-\infty)$ has formal continued fraction expansion with $m'\geq 3$, hence $>2$.
Finally,
\[
x_a= ST^{-1}S\hat f_-^{n_1}T^{-1}x_b=T^{-1}\hat f^{k_2}T^{m+1}S\hat f_+^{n_2}S(-\infty),
\]
hence
\[
Tx_a=\hat f^{k_2}T^{m+1}S\hat f_+^{n_2}S(-\infty)>2
\]
since it has formal continued fraction expansion with $m'\geq 3$, therefore $x_a>1$.
Now we analyze the particular situations when $m_a$ or $m_b\in\{1,2\}$, using the explicit cycle descriptions that exist for these situations as described by Theorems \ref{b-cycle} and \ref{a-cycle}.
(i) If $m_a=m_b=1$, then relation \eqref{m1} for the $b$-cycle and a similar one for the $a$-cycle shows that $y_\ell=-\frac1{b}+1$ and $y_u=-\frac1{a}-1$, therefore $x_a=1$ and $x_b=-1$
\medskip
(ii) If $m_a=1$, $m_b=2$, following the explicit cycles given by \eqref{m=2} we obtain $y_\ell=-1/b +1$, and $y_u=-1/(b-1)-1$, therefore $x_a=2$, $x_b=-1$.
\medskip
(iii) If $m_a=1$, $m_b\ge 3$, using the cycle structure in Theorem \ref{b-cycle} we obtain $y_\ell=1/b+1$ and $y_u=T^{-1}(ST^{-2})^{m_b-2}ST^{-1}b$, therefore, $x_a=m_b$, and $x_b=-1$.
\medskip
(iv) If $m_a=2$, $m_b=2$, using the cycle structure in Theorems \ref{b-cycle} and \ref{a-cycle} we obtain $y_\ell=-\frac1{a+1}+1$ and $y_u=-\frac1{b-1}-1$, and a calculation in this particular case, like in Lemma \ref{2-sys}, Case 3 implies that $x_a>1$ and $x_b<-1$.
\medskip
(v) if $m_a=2$, $m_b>2$, an analysis of the four cases above for this particular situation (with an explicit cycle relation for $a$) yields $x_a\ge 1$ and $x_b\le -1$. Indeed, in Case 1, we have $y_u=-1/a-1$, hence $x_a=1$ and $x_b=-2$. In Case 2, we get $x_a=1$ and $x_b<-2$. Cases 3 and 4 are treated similarly.
\end{proof}
Now, since $x_a$ and $x_b$ are uniquely determined, by ``transporting" the rays $[-\infty, x_b]\times\{b\}$ and $[x_a,\infty]\times\{a\}$ along the sets $\mathcal{L}_b,\,\,\mathcal{U}_b,\,\,\mathcal{L}_a$ and $\mathcal{U}_a$ we obtain the $x$-coordinates of the right and left end of the segments on each level.
\begin{defn} We say that two consecutive levels $y_1\leq y_2$ of $\mathcal{L}_{a,b}$, respectively, $\mathcal{U}_{a,b}$, are called {\em connected by a vertical segment} (we will refer to this as {\em connected}) if the $x$-coordinate of the right end point of the horizontal segment on the level $y_1$ is equal to the the $x$-coordinate of the left end point of the horizontal segment on the level $y_2$.
\end{defn}
We will prove that all levels of $\mathcal{L}_{a,b}$ and all levels of $\mathcal{U}_{a,b}$ are connected
We first look at the levels in $\mathcal{L}_{a,b}$. By Lemma \ref{2-sys} the levels $y_u$ and $Sa$, and the levels $Sb$ and $y_\ell$ are connected.
\begin{lem} \label{next} The levels $Sb\in\mathcal{L}_b$ and $STa\in\mathcal{L}_a$ are two consecutive levels of $\mathcal{L}_{a,b}$ connected by a vertical segment at $x=0$.
The levels $Sa\in\mathcal{U}_a$ and $ST^{-1}b\in\mathcal{U}_b$ are two consecutive levels of $\mathcal{U}_{a,b}$ connected by a vertical segment at $x=0$.
\end{lem}
\begin{proof} Suppose there is $y\in \mathcal{L}_{a,b}$ such that $STa\leq y\leq Sb$. Then $y\in \mathcal{L}_a$ or $\mathcal{L}_b$. In either case,
since by Lemmas \ref{a-norepeats} and \ref{b-norepeats} the truncated orbits $\mathcal{L}_a,\mathcal{L}_b$ do not have repeated values, neither $STa=y$ nor $y=Sb$ is possible. Thus the only case we need to consider is
\[
STa<y<Sb.
\]
Then, either $y=Sy'$ for some $y'\in\mathcal{L}_{a,b}$ ($0<y'\le a+1$) or $y=Ty''$ for some $y''\in\mathcal{L}_{a,b}$. These would imply that either $y'>Ta$, which is impossible, or $Ty''<Sb$, i.e. $y''<T^{-1}Sb$, which is also impossible (if $y''<T^{-1}Sb$ then $y=Ty''$ must be the end of the $a$-cycle, by Theorem \ref{a-cycle}). The $x$-coordinate of the right end point of the segment at the level $STa$ and of the left end point of the segment at the level $Sb$ is equal to $0$
The second part of the proof is similar
\end{proof}
The following proposition will be used later in the proof.
\begin{prop} \label{tech} Suppose that the set $\mathcal{L}_{a,b}$ is finite and $y\in \mathcal{L}_{a,b}$ with $y>STa$.
\begin{enumerate}
\item If $y\in\mathcal{L}_a$, then
there exists $n_0>0$ such that
$\rho(f^{n}y)=\rho(f^n STa)$
for all $0<n<n_0$ and $\rho(f^{n_0}y)\neq \rho(f^{n_0}STa)$, or $f^{n_0}y=0$;
\item If $y\in \mathcal{L}_b$, then $y>Sb$, and there exists $n_0>0$ such that
$\rho(f^{n}y)=\rho(f^n Sb)$
for all $n<n_0$ and $\rho(f^{n_0}y)\neq \rho(f^{n_0}Sb)$, or $f^{n_0}y=0$.
\end{enumerate}
\end{prop}
\begin{proof}
Suppose that $y\in\mathcal{L}_a$ and $a$ satisfies the cycle property. It follows that such an $n_0$ exists or $f^{n_0}y$
is the end of the $a$-cycle. We will show that the latter is possible only if $f^{n_0}y=0$, i.e. it is the end of a weak cycle.
Suppose $f^{n_0}y$ is the end of the $a$-cycle.
Then if
\[
\rho(f^{n_0-1}y)=\rho(f^{n_0-1}STa)=S,
\]
we must have
$f^{n_0-1}y<0$ since otherwise the cycle would not stop at $S$, but $f^{n_0-1}(STa)>0$ since for $STa$ we have not reached the end of the cycle. This contradicts the monotonicity of $f^{n_0-1}$ and the original assumption $y>STa$, thus is impossible. The other possibility is
\[
\rho(f^{n_0-1}y)=\rho(f^{n_0-1}STa)=T.
\]
But this either implies that $f^{n_0-1}y<T^{-1}Sb$, and by monotonicity of $f^{n_0-1}$, $f^{n_0-1}(STa)<f^{n_0-1}y<T^{-1}Sb$, which implies that we have reached the end of the cycle of $STa$ as well, a contradiction, or,
$f^{n_0}y=0$, i.e. it is the end of a weak cycle.
Now suppose $y\in\mathcal{L}_b$. Then by Lemma \ref{next} $y\geq Sb$, but
since each level in $\mathcal{L}_b$ appears only once, we must have but $y>Sb$. Now the argument that $f^{n_0}y$ cannot be the end of the $b$-cycle is exactly the same as for the $a$-cycle.
In the periodic case, let us assume that no such $n_0$ exists. Then, in case (1) the $(a,b)$-expansions of $STa$ and $y$, which is the lower part of the former, are the same, i.e. $(a,b)$-expansions of $STa$ is invariant by a left shift.
In case (2), we have seen already that we must have $y>Sb$
Then the $(a,b)$-expansions of $Sb$ and $y$, which is the lower part of the former, are the same, i.e. $(a,b)$-expansions of $Sb$ is invariant by a left shift.
The proof that this is impossible
is based on the following simple observation: if $\sigma= (a_1,a_2,\dots , a_k, \overline{a_{k+1},a_{k+2},\dots a_{k+n}})$ is an eventually periodic symbolic sequence with the minimal period $n$ and invariant under a left shift by $m$, then $\sigma$ is purely periodic and $m$ is a multiple of $n$.
By the uniqueness property of $(a,b)$-expansions, this would imply that
$y=STa$ or $y=Sb$, a contradiction.
\end{proof}
Let $y_b^-,y_b^+\in\mathcal{U}_{a,b}$ be two consecutive levels with $y_b^-\leq b< y_b^+$, and $y_a^-,y_a^+\in\mathcal{L}_{a,b}$ be two consecutive levels with $y_a^-< a\leq y_a^+$
\begin{lem}\label{a-connected}
There is always one level connected with level $a+1$, and
the levels $y_a^-$ and $y_a^+$ are connected by the vertical segment at $x_a$.
\end{lem}
\begin{proof
By Lemmas \ref{2-sys} and \ref{next}, we know that three consecutive levels $STa\leq Sb\leq y_\ell$ are connected. Moreover, their images remain connected under the same transformations in $SL(2,\Z)$.
Since each level in $\mathcal{U}_a$ and in $\mathcal{L}_a$ appears only once, at least one of the two inequalities must be strict, i.e. if $STa= Sb$, then $STa=Sb< y_\ell$, and if $Sb= y_\ell$, then $STa< Sb= y_\ell$.
First we prove that
$y_\ell<TSb$. Suppose $y_\ell\geq TSb$. Its pre-image must be $y'_\ell=T^{-1}y_\ell$ since for any $y,\,0<y<Ta$,
$Sy<STa\leq Sb<TSb$, and
we would have
$Sb\leq y'_\ell<y_\ell$ that contradicts the assumption that $y_\ell$ is {the next level above} $Sb$. Therefore, if the first digit in the $(a,b)$-expansion of $Sb$ is $-m$,
then the first digit of $y_\ell$ is $-(m-1)$ or $-m$. In the first case, the three level
\[
T^{m-1}Sb< a\le T^{m-1}y_{\ell}
\]
are connected and satisfy $T^{m-1}Sb=y^-_a, T^{m-1}y_{\ell}=y^+_a$.
Therefore, the levels $T^{m}Sb$ and $a+1$ are connected.
For the second case, we know that $Sb\leq y_\ell$ and
\[
a\le T^{m}Sb\leq T^{m}y_{\ell}< a+1.
\]
If $Sb=y_\ell$, then $y_\ell\in \mathcal{L}_a$, and $STa<y_\ell$.
If $Sb<y_\ell$, then $y_\ell\in \mathcal{L}_b$, or $y_\ell\in \mathcal{L}_a$ and $STa<y_\ell$.
Let us assume that $y_\ell$ belongs to $\mathcal{L}_a$.
Since $STa<y_\ell$, by Proposition \ref{tech}, there are two possibilities:
\begin{enumerate}
\item $f^{n_0}y_\ell$ is the end of a weak cycle.
\item There exists
$n_0$ such that
$\rho(f^{n}y_\ell)=\rho(f^n STa)$
for all $n<n_0$, and $\rho(f^{n_0}y_\ell)\neq \rho(f^{n_0}STa)$.
\end{enumerate}
In the first case, we have $f^{n_0}STa=y_a^-$ and $f^{n_0}Sb=y_a^+$, or $f^{n_0}Sb=y_a^-$ and $ f^{n_0}y_{\ell}=y_a^+$. Therefore, either
$f^{n_0+1}STa$ or $f^{n_0+1}Sb$ is connected with level $a+1$.
In the second case,
we notice that
\[
\rho(f^{n_0-1}y_\ell)=\rho(f^{n_0-1}STa)=T
\]
otherwise, $\rho(f^{n_0-1}y_\ell)=\rho(f^{n_0-1}STa)=S$
would imply
\[
\rho(f^{n_0}y_\ell)=\rho(f^{n_0}STa)=T
\]
in contradiction with the choice of $n_0$.
Further,
there are two possibilities:
\[
\text{(i) } \rho(f^{n_0}STa)=S,\,\,\rho(f^{n_0}y_\ell)=T\,,\quad \text{(ii) } \rho(f^{n_0}STa)=T,\,\,\rho(f^{n_0}y_\ell)=S.
\]
In case (i) we obtain
\[
f^{n_0}y_\ell<a\le f^{n_0}STa
\]
which contradicts the monotonicity of $f$ and the original assumption $y_\ell>STa$. Thus the only possibility is
\[
f^{n_0}y_\ell \ge a>f^{n_0}STa.
\]
By using the monotonicity of $f^{n_0}$ we have
\[
f^{n_0}y_\ell>f^{n_0}Sb>f^{n_0}STa
\]
and conclude that $f^{n_0}STa=y_a^-$ and $f^{n_0}Sb=y_a^+$, or $f^{n_0}Sb=y_a^-$ and $ f^{n_0}y_{\ell}=y_a^+$. Therefore, either
$f^{n_0+1}STa$ or $f^{n_0+1}Sb$ is connected with level $a+1$.
The case when $y_\ell$ belongs to $\mathcal{L}_b$ is very similar, and in this case $f^{n_0}Sb=y_a^-$, $ f^{n_0}y_{\ell}=y_a^+$, and $f^{n_0+1}Sb$ is connected with $a+1$. By construction, in both cases the common $x$-coordinate of the end points is equal to $x_a$.
\end{proof}
After an application of $S$ the level connected with $a+1$ will be connected with $STa$, and now, instead of $3$ connected levels $STa\leq Sb\leq y_\ell$ (with at least one strict inequality) we have at least $4$ connected levels $y'\leq STa\leq Sb\leq y_\ell$ (with no more than two equalities in a row).
The process continues with a growing number of connected levels, the highest being $a+1$. Since on each step we cannot have more than two equalities in a row, the number of distinct levels in this sequence will also increase. Therefore, we obtain a sequence of connected levels
\begin{equation}\label{lowersnake}
a+1\geq y_1\geq \dots\geq y_{s}>\frac{b}{b+1}\geq y_{s+1}.
\end{equation
It is evident from the construction that
there are no unaccounted levels $y\in \mathcal{L}_{a,b}$, $a+1\geq y\geq y_{s+1}$
Now we prove a similar result for $\mathcal{U}_{a,b}$.
\begin{lem}\label{b-connected}
There is always one level connected with level $b-1$, and
the levels $y_b^-$ and $y_b^+$ are connected by a vertical segment at $x_b$.
\end{lem}
\begin{proof}
By Lemmas \ref{2-sys} and \ref{next}
we know that the three consecutive levels $y_u\leq Sa\leq ST^{-1}b$ are connected.
It is easy to see that the first digit in $(a,b)$-expansion of $ST^{-1}b$ is $2$,
and the first digit in $(a,b)$-expansion of $Sa$ is either $1$ or $2$. Therefore, the first digit in $(a,b)$-expansion of $y_u$ is either $1$ or $2$.
In the first case either
\[
T^{-1}Sa< b\leq T^{-1}ST^{-1}b
\]
or
\[
T^{-1}y_u<b\leq T^{-1}Sa
\]
are the connected levels. Therefore either $T^{-1}Sa=y_b^-$ and $T^{-1}ST^{-1}b=y_b^+$, or $T^{-1}y_u=y_b^-$ and $T^{-1}Sa=y_b^+$ are connected. So either $T^{-2}ST^{-1}b$ or $T^{-2}Sa$ is connected with level $b-1$.
In the second case, we know that $y_u\leq Sa$ and
\[
b-1\leq T^{-2}y_u\leq T^{-2}Sa<b.
\]
If $y_u=Sa$, $y_u$ must belong to $\mathcal{U}_b$, in which case $y_u<ST^{-1}b$. If $y_u<Sa$, then $y_u\in \mathcal{U}_a$, or $y_u\in \mathcal{U}_b$ and $y_u<ST^{-1}b$.
Let us assume that $y_u$ belongs to $\mathcal{U}_b$. Since $y_u<ST^{-1}b$, by
Proposition \ref{tech} there are two possibilities:
\begin{enumerate}
\item $f^{n_0}y_u$ is the end of a weak cycle,
\item there
exists $n_0$ such that $\rho(f^{n}y_u)=\rho(f^nST^{-1}b)$
for all $n<n_0$, and $\rho(f^{n_0}y_u)\neq \rho(f^{n_0}ST^{-1}b)$.
\end{enumerate}
In the first case, either $f^{n_0}ST^{-1}b=y_b^+$ and $f^{n_0}Sa=y_b^-$, or $f^{n_0}Sa=y_b^+$ and $ f^{n_0}y_u=y_b^-$, so either
$f^{n_0+1}ST^{-1}b$ or $f^{n_0+1}Sa$ is connected with level $b-1$.
In the second case, we first notice that
\[
\rho(f^{n_0-1}y_u)=\rho(f^{n_0-1}ST^{-1}b)=T^{-1}
\]
since if we had $\rho(f^{n_0-1}y_u)=\rho(f^{n_0-1}ST^{-1}b)=S$, then
we would have
\[
\rho(f^{n_0}y_u)=\rho(f^{n_0}ST^{-1}b)=T^{-1}
\]
in contradiction with the choice of $n_0$.
Further,
there are two possibilities:
\[
\text{(i) } \rho(f^{n_0}ST^{-1}b)=S,\,\,\rho(f^{n_0}y_u)=T^{-1},\;
\text{(ii) } \rho(f^{n_0}ST^{-1}b)=T^{-1},\,\,\rho(f^{n_0}y_u)=S.
\]
In the first case we obtain
\[
f^{n_0}y_u>b>f^{n_0}ST^{-1}b
\]
which contradicts the monotonicity of $f^{n_0}$ and the original assumption $y_u<ST^{-1}b$. Thus the only possibility is
\[
f^{n_0}y_u<b<f^{n_0}ST^{-1}b.
\]
By monotonicity of $f^{n_0}$ we have
\[
f^{n_0}y_u<f^{n_0}Sa<f^{n_0}ST^{-1}b.
\]
Therefore either $f^{n_0}y_u=y_b^-$ and $f^{n_0}Sa=y_b^+$,
or $f^{n_0}Sa=y_b^-$ and $f^{n_0}ST^{-1}b=y_b^+$ are connected. So either $T^{-1}f^{n_0}ST^{-1}b$ or $T^{-1}f^{n_0}Sa$ is connected with level $b-1$. The case when $y_u$ belongs to the $a$-cycle is very similar, and in this case $f^{n_0}y_u=y_b^-$ and $f^{n_0}Sa=y_b^+$ and $T^{-1}f^{n_0}Sa$ is connected with level $b-1$. By construction, in both cases the common $x$-coordinate of the end points of the segments at the levels $y_b^-$ and $y_b^+$ is $x_b$
\end{proof}
After an application of $S$ the levels (2) will be connected with $ST^{-1}b$, and now, instead of $3$ connected levels $y_u\leq Sa\leq ST^{-1}b$ we have at least $4$ connected levels $y_u\leq Sa\leq ST^{-1}b\leq y''$.
The process continues with a growing number of connected levels, the lowest being $b-1$. Also the number of distinct levels will increase, and we obtain a sequence of connected levels
\begin{equation}\label{uppersnake}
b-1\leq \bar y_1\leq \dots\leq \bar y_t<\frac{a}{1-a}\leq \bar y_{t+1}.
\end{equation}
It is evident from the construction that
there are no unaccounted levels $y\in \mathcal{U}_{a,b}$, $b-1\leq y\leq \bar y_{t+1}$
Now we complete the proof that all levels of $\mathcal{L}_{a,b}$ are connected. For that it is sufficient to find a sequence of connected levels with the distance between the highest and the lowest level $\geq 1$ and the lowest level $\geq T^{-1}Sb$. This is because the set of levels in $y\in\mathcal{L}_{a,b}$ satisfying $T^{-1}Sb\leq y\leq a+1$ is periodic with period $1$, and each $y\in \mathcal{L}_{a,b}$ uniquely determines a horizontal segment on level $y$, as was explained just before Lemma \ref{next}.
If $y_{s+1} \leq a$, then
all levels in $\mathcal{L}_{a,b}$ are connected.
Suppose now that $y_{s+1}> a$. If
$y_{s+1}=y_a^+$, then, since $y_a^+$ is already connected with $y_a^-$, all levels of $\mathcal{L}_{a,b}$ are connected.
Now assume that $y_{s+1}>y_a^+$. Then either
\[ y_{s+1}=\frac{b}{b+1} \quad \text{ or } \quad y_{s+1}<\frac{b}{b+1}.
\]
In the first case either $TSy_{s+1}=y_\ell=Sb$ (this can only happen if $y_{s+1}\in\mathcal{L}_a$), or $TSy_{s}>Sb$ is the next level above $Sb$, and hence $TSy_{s}=y_\ell$. In either case
$Sy_{s+1}\leq Sy_{s}\leq \cdots \leq STa\leq Sb=TSy_{s+1}$ are the connected levels with the distance between the lowest and the highest equal to $1$,
thus we conclude that all levels of $\mathcal{L}_{a,b}$ are connected.
In the second case,
the two levels $y_a^+<y_{s+1}$ will produce the ends of the cycles (one of them can be weak if one of $y_a^+$ or $y_{s+1}$ is equal to $0$).
By the cycle property (Proposition \ref{main-argument}(ii)), there exists a level $z\in\mathcal{U}_{a,b}$, $\frac{a}{1-a}< z<b$ such that $z=(STS)y_{s+1}$. We claim that $z=y_b^-$. Suppose not, and $z<y_b^-$. Then $y_b^-$ gives rise to the second cycle, and again by the cycle property, there exists $y\in\mathcal{L}_{a,b}$, $y<\frac{b}{b+1}$, such that $y_b^-=STSy$. Since $STS(z)=-\frac{z}{z-1}$
is monotone increasing for $z<1$, we conclude that
$y>y_{s+1}$ in contradiction with (\ref{lowersnake}). Thus $y_b^-=(STS)y_{s+1}$.
Then $TSy_{s+1}=Sy_b^-$ which implies that the right end of the segment at the level $Sy_b^-$, which is equal to the right end of the segment at the level $Sb$, is equal to the right end of the segment at the level $TSy_{s+1}$ (notice that this level may belong to $\mathcal{L}_{a,b}$, $\mathcal{U}_{a,b}$ or be at infinity if $y_{s+1}=0$). Since $y_{s}$ and $y_{s+1}$ were connected, the left end of the segment at the level $TSy_{s}$ is equal to the right end of the segment at the level $TSy_{s+1}$ even though they may belong to the boundaries of different connected components. Since $TSy_{s}\in \mathcal{L}_{a,b}$, we conclude that the segment at the level $TSy_{s}$ is adjacent to the segment at the level $Sb$, i.e. $TSy_{s}=y_\ell$. Thus $Sy_{s}\leq Sy_{s-1}\leq \cdots \leq STa\leq Sb\leq TSy_{s}$ are the connected levels with the distance between the lowest and the highest equal to $1$, and therefore
all levels in $\mathcal{L}_{a,b}$ are also connected. The proof for $\mathcal{U}_{a,b}$ follows exactly the same lines.
\medskip
(A2) In order to prove the bijectivity of the map $F$ on $A_{a,b}$ we write it as a union of the upper and lower connected components, $A_{a,b}=A_{a,b}^u\cup A_{a,b}^\ell$, and subdivide each component into $3$ pieces: $A_{a,b}^u=\cup_{i=1}^3U_i$, and $A_{a,b}^\ell=\cup_{i=1}^3L_i$, where
\[\begin{aligned}
U_1=&\{(x,y)\in A_{a,b}^u\,:\, y\ge b\}\\
U_2=&\{(x,y)\in A_{a,b}^u\,:\, b-1\le y\le 0\}\\
U_3=&\{(x,y)\in A_{a,b}^u\,:\, 0\le y\le b\}\\
L_1=&\{(x,y)\in A_{a,b}^\ell\,:\, y\le a\}\\
L_2=&\{(x,y)\in A_{a,b}^\ell\,:\, 0\le y\le a+1\}\\
L_3=&\{(x,y)\in A_{a,b}^\ell\,:\, a\le y\le 0\},
\end{aligned}
\]
Now let
\[
U'_1=T^{-1}(U_1),\,\,
U'_2=S(U_2),\,\,
U'_3=S(U_3),\,\,
L'_1=T(L_1),\,\,
L'_2=S(L_2),\,\,
L'_3=S(L_3)
\]
be their images under the transformation $F$ (see Figure \ref{fig-bij}).
\begin{figure}[htb]
\psfrag{a}[l]{\tiny $a$}
\psfrag{b}[l]{\tiny $b$}
\psfrag{x}[c]{\tiny $x_b$}
\psfrag{y}[c]{\tiny $x_a$}
\psfrag{M}[c]{\tiny $F_{a,b}$}
\psfrag{A}[c]{\tiny $\mathbf{U_1}$}
\psfrag{C}[c]{\tiny $\mathbf{U_2}$}
\psfrag{E}[c]{\tiny $\mathbf{U_3}$}
\psfrag{B}[l]{\tiny $\mathbf{U'_1}$}
\psfrag{D}[c]{\tiny $\mathbf{U'_2}$}
\psfrag{F}[c]{\tiny $\mathbf{U'_3}$}
\psfrag{G}[c]{\tiny $\mathbf{L_2}$}
\psfrag{I}[c]{\tiny $\mathbf{L_3}$}
\psfrag{K}[c]{\tiny $\mathbf{L_3}$}
\psfrag{H}[c]{\tiny $\mathbf{L'_2}$}
\psfrag{J}[c]{\tiny $\mathbf{L'_3}$}
\psfrag{N}[c]{\tiny $\mathbf{L'_1}$}
\psfrag{L}[c]{\tiny $\mathbf{L_1}$}
\includegraphics[scale=0.5]{bijectionv2.eps}
\caption{Bijectivity of the map $F_{a,b}$}
\label{fig-bij}
\end{figure}
Since the set $A_{a,b}$ is bounded by step-functions with finitely many steps, each of the pieces $U_i,L_i$ have the same property, and so do their images under $F$.
By the construction of the set $A_{a,b}$ we know that the levels corresponding to the ends of the cycles $c_a$ and $c_b$, if the cycles are strong, do not appear as horizontal boundary levels; the corresponding horizontal segments, let us call them the {\em locking segments} lie in the interior of
the set $A_{a,b}$. Furthermore, the images of all levels except for the levels next to the ends of the cycles, $f^{k_1-1}Ta$, $f^{m_1-1}Sa$, $f^{m_2-1}Sb$, and $f^{k_2-1}T^{-1}b$, also belong to $\mathcal{U}_{a,b}\cup\mathcal{L}_{a,b}$.
The exceptional levels are exactly those between $0$ and $b$ and above $TSa$ in $\mathcal{U}_{a,b}$, and between $a$ and $0$ and below $T^{-1}Sb$ in $\mathcal{L}_{a,b}$. The images of the horizontal segments belonging to these levels are the locking segments. Notice that the exceptional levels between $0$ and $b$ and between $a$ and $0$ constitute the horizontal boundary of the regions $U_3$ and $L_3$.
Transporting the rays $[-\infty,x_b]$ and $[x_a,\infty]$ (with $x_a$ and $x_b$ uniquely determined by Lemma \ref{2-sys}),
along the corresponding cycles, and using the strong cycle property, we see that the ``locking segment" in the horizontal boundary of $U_1'$ coincides with the locking segment of the horizontal boundary of $L_3'$, and the locking segment in the horizontal boundary of $L_1'$ coincides with the locking segment of the horizontal boundary of $U_3'$. It can happen that both ``locking segments" belong to $A_{a,b}^u$ or $A_{a,b}^\ell$. If only one of the numbers $a$ or $b$ has the strong cycle property, then there will be only one locking segment.
If the cycle property is weak or the $(a,b)$-continued fraction expansion of one or both $a$ and $b$ is periodic, then
all levels of $\mathcal{L}_a,\,\,\mathcal{L}_b,\,\,\mathcal{U}_a$ and $\mathcal{U}_b$ will belong to the boundary of $A_{a,b}$,
and there will be no locking segments. In these cases $L_3=[x_1,\infty]\times[a,0]$, and $L'_3=[-1/x_1,0]\times[-1/a,\infty]$, where $x_1=x_a$. Let $x_2$ be the $x$-coordinate of the right vertical boundary segment of $U_2$. Then the $x$-coordinate of the right vertical boundary segment of $U_1$ is $-1/x_2$. Let us denote the highest level in $\mathcal{U}_{a,b}$ by $y_2$.
Since $y_2\leq -1/a+1$, $y_2-1\leq -1/a$ is the next level after $-1/a$ in $\mathcal{U}_{a,b}$. This is since if we had $y\in\mathcal{U}_{a,b}$ such that $y_2-1<y<-1/a$, its preimage $y'=Ty$ would satisfy $y_2<y'<-1/a+1$, a contradiction. By construction of the region $A_{a,b}$ the segments at the levels $y_2-1$ and $-1/a$ are connected, therefore $Sx_1=T^{-1}Sx_2$.
This calculation shows that $L'_3$ and $U'_1$ do not overlap and fit together by this vertical ray.
Thus in all cases
the images $U'_i,\,L'_i$ do not overlap, and
$A_{a,b}=(\cup_{i=1}^3U'_i)\cup(\cup_{i=1}^3L'_i)$. This proves the bijectivity of the map $F$ on $A_{a,b}$ except for some images of its boundary. This completes the proof in the case $0<b\le -a<1$.
\medskip
Now we return to the case $a\le -1$ dropped from consideration before Lemma \ref{2-sys}.
The explicit cycle relations for this case have been described in Theorem \ref{ab-cycle}. Notice that
all lower levels are connected, and $T^mSb$ is connected with $a+1$. Therefore $y_\ell=TSb$, and this implies that $x_a=m$.
The upper levels in the positive part are
$$ST^{-1}b<ST^{-2}ST^{-1}b< ... <(ST^{-2})^{m-1}ST^{-1}b<a/(a+1)$$
and $y_u=T^{-1}(ST^{-2})^{m-2}ST^{-1}b$.
Lemma \ref{2-sys} in this case holds with $x_a=m$ and $x_b=-1$ since
the equation for adjacency of the levels $y_u$ and $Sa$ is
$$T^{-1}(ST^{-2})^{m-2}ST^{-1}x_b=ST^{m-1}Sx_b=-1/m\, ,$$ which implies $x_b=-1$. Lemma \ref{a-connected} also holds with $y_a^-=ST^{m-1}b$ and $y_a^+=ST^{m}b$. Lemma \ref{b-connected} holds with $y_b^-=T^{-1}Sa$ and $y_b^+=T^{-1}ST^{-1}b$ and all upper level will be connected by an argument similar to one described obove. To prove the bijectivity of $F$ on $A_{a,b}$ one proceeds the same way as above, the only modification being that level $L_2$ does not exist, and
$L_3=\{(x,y)\in A_{a,b}^\ell, a\le y\le a+1\}$.
\end{proof}
The following corollary is evident from the proof of part (ii) of the above theorem.
\begin{cor} \label{cor:bry} If both $a$ and $b$ have the strong cycle property, then for any boundary component
$h$ of $A_{a,b}$ (vertical or horizontal) there exists $N>0$ such that $F^N(h)$ is in the interior of $A_{a,b}$
\end{cor}
\section{Finite rectangular structure of the attracting set}\label{s:6}
Recall that the attracting set $D_{a,b}$ was defined by \eqref{def-atrac}: starting with the trapping region $\Theta_{a,b}$ described in Theorem \ref{Delta-trapping}, one has
\[ D_{a,b}=\bigcap_{n=0}^\infty D_n, \text{ with } D_n=\bigcap_{i=0}^n F^i(\Theta_{a,b})\,.
\]
\begin{lem} \label{Dn} Suppose that the map $f$ satisfies the finiteness condition. Then,
for each $n\geq 0$, $D_n$ is a region consisting of two connected components, the upper one, $D_n^u$, and the lower one, $D_n^\ell$, bounded by non-decreasing step-functions.
\end{lem}
\begin{proof}
The proof is by induction on $n$. The base of induction holds by the definition of the trapping region $\Theta_{a,b}$. For the induction step, let us assume that the region $D_n$ consists of two connected components, the upper one
$D_n^u$ and the lower one $D_n^\ell$, bounded by non-decreasing step-functions.
We will show that
the region $D_{n+1}$ consists of two connected components,
$D_{n+1}^u$ and $D_{n+1}^\ell$, bounded by non-decreasing step-functions.
In what follows, we present the proof assuming that $0<b\le -a<1$. The situation $a\le -1$ is less complex due to the explicit cycle expressions described in Theorem \ref{ab-cycle} and can be treated similarly with some minor modifications
We decompose the regions $D_n^u$ and $D_n^\ell$ as follows
\[
\begin{split}
U_n^{11}&=\{(x,y)\in D_n^u\,:\, y\ge TSa\}\\
U_n^{12}&=\{(x,y)\in D_n^u\,:\, b\le y\le TSa\}\\
U_n^3&=\{(x,y)\in D_n^u\,:\, 0\le y\le b\}\\
U_n^{21}&=\{(x,y)\in D_n^u\,:\, \frac{a}{1-a}\le y\le 0\}\\
U_n^{22}&=\{(x,y)\in D_n^u\,:\, b-1\le y\le \frac{a}{1-a}\}\\
L_n^{11}&=\{(x,y)\in D_n^\ell\,:\, y\le T^{-1}Sb\}\\
L_n^{12}&=\{(x,y)\in D_n^\ell\,:\, T^{-1}Sb\le y\le a\}\\
L_n^3&=\{(x,y)\in D_n^\ell\,:\, a\le y\le 0\}\\
L_n^{21}&=\{(x,y)\in D_n^\ell\,:\, 0\le y\le \frac{b}{b+1}\}\\
L_n^{22}&=\{(x,y)\in D_n^\ell\,:\, \frac{b}{b+1}\le y\le a+1\}.\\
\end{split}
\]
By induction hypothesis, the regions $U_{12},\,U_n^3,\,U_n^{21}$ and $U_n^{22}$ are bounded below and above, and $U_n^{11}$ only below, by a ray and on the right by a non-decreasing step-function. Similarly, the regions $L_n^{12},\,L_n^3,\,L_n^{21}$ and $L_n^{22}$ bounded above and below, and $L_n^{11}$ only above, by a ray and on the left by a non-decreasing step-function.
If $B\subset D_n^u$
is one of the upper subregions, let $\partial B$ be the union of the boundary components of $B$ that belong to the boundary of $D_n^u$, and, similarly, if
$B\subset D_n^\ell$
is one of the lower subregions, let $\partial B$ be the union of the boundary components of $B$ that belong to the boundary of $D_n^\ell$.
Since $\Theta_{a,b}$ is a trapping region, $F(\Theta_{a,b})\subset \Theta_{a,b}$, $D_{n+1}=F(D_n)\subset D_n$, and hence $D_{n+1}^u\subset D_n^u$ and $D_{n+1}^\ell\subset D_n^\ell$.
The natural extension map $F$ is piecewise fractional-linear, hence it maps regions bounded by non-decreasing step-functions to regions bounded by non-decreasing step-functions. More precisely, we have
\[
\begin{split}
U_{n+1}^u&=S(U_n^{22}\cup U_n^{21})\cup T^{-1}(U_n^{11}\cup U_n^{12})\cup S(L_n^3)\\
U_{n+1}^\ell&=S(L_n^{22}\cup L_n^{21})\cup T(L_n^{11}\cup L_n^{12})\cup S(U_n^3)\,.
\end{split}
\]
In order to show that the region $D^u_{n+1}$, is connected, we notice
that the region $T^{-1}(U_n^{11}\cup U_n^{12})$ is inside the ``quadrant" $[-\infty,0]\times[b-1,\infty]$
while $S(U_n^{22}\cup U_n^{21})$ is inside the strip $[0,1]\times[ST^{-1}b,\infty]$. Therefore, they either intersect by a ray of the $y$-axis, or are disjoint. In the first case, either $T^{-1}ST^{-1}b<Sa$, which implies that $S(L_n^3)$ is inside the connected region
$S(U_n^{22}\cup U_n^{21})\cup T^{-1}(U_n^{11}\cup U_n^{12})$, or $Sa\leq T^{-1}ST^{-1}b$ which implies that the level $Sa$ belongs to the boundary of the trapping region, and again $S(L_n^3)$ is inside the connected region
$S(U_n^{22}\cup U_n^{21})\cup T^{-1}(U_n^{11}\cup U_n^{12})$.
Now suppose that the regions $T^{-1}(U_n^{11}\cup U_n^{12})$ and $S(U_n^{22}\cup U_n^{21})$ are disconnected. Notice that the right vertical boundary of the region $S(L_n^3)$ is a ray of the $y$-axis, thus $S(L_n^3)\cup S(U_n^{22}\cup U_n^{21})$ is a connected region bounded by a non-decreasing step-function. Since $T^{-1}(U_n^{12})\cap S(L_n^3)=\emptyset$, the non-connectedness situation may only appear from the intersection of $T^{-1}(U_n^{11})$ and $S(L_n^3)$,
i.e. inside the strip $[-1,0]\times[-1/a,\infty]$.
Since $f$ satisfies the finiteness condition,
Theorem \ref{thm:recstructure} is applicable, and the set $A_{a,b}$ constructed there belongs to each $D_n$. This is because $A_{a,b}\subset \Theta_{a,b}$, and if $A_{a,b}\subset D_n$, we have $A_{a,b}=F(A_{a,b})\subset F(D_n)=D_{n+1}$. The set
$A_{a,b}$ has finite rectangular structure
and contains the strip $[-1,0]\times[-1/a,\infty]$. Thus the connectedness
of the region $D^u_{n+1}$ is proved. Moreover, this argument shows that $\partial T^{-1}(U_n^{11})$ is inside $D^u_{n+1}$ and therefore does not contribute to its boundary, and
\[
\partial U_{n+1}^u=\partial (T^{-1}( U_n^{12}))\cup\partial (S(U_n^{22}\cup U_n^{21})\cup S(L_n^3)).
\]
Since $\partial (T^{-1}( U_n^{12})$ and $\partial (S(U_n^{22}\cup U_n^{21})\cup S(L_n^3))$ are given by non-decreasing step-functions, one $<Sa$, and the other $\ge Sa$, it follows that $\partial U_{n+1}^u$ is also given by a non-decreasing step-function. A similar argument proves that $D_{n+1}^\ell$ is connected and bounded by a non-decreasing step-function.
\end{proof}
\begin{lem}\label{l:all-levels} Suppose that, for each $n$, $D_n$ consists of two connected components as in Lemma \ref{Dn}.
Then
\begin{enumerate}
\item all horizontal levels of the boundary of $D_n^u$ belong to $\mathcal{U}_{a,b}$ (resp., $D_n^\ell$ belong to $\mathcal{L}_{a,b}$) and remain as horizontal levels of $D_{n+1}^u$ (resp., $D_{n+1}^\ell$);
\item all levels of $\mathcal{U}_{a,b}$ appear in the boundary of some $D_n^u$, and all levels of $\mathcal{L}_{a,b}$ appear in the boundary of some $D_n^\ell$;
\item
the attractor $D_{a,b}$ consists of two connected components bounded by non-decreasing step-functions; the upper boundary function takes all values from the set $\mathcal{U}_{a,b}$, and the lower boundary function takes all values from the set $\mathcal{L}_{a,b}$.
\item The map $F: D_{a,b}\to D_{a,b}$ is surjective.
\end{enumerate}
\end{lem}
\begin{proof} (1) We prove this by induction. For the base case, $D_0^u$ contains the horizontal levels $T^{-1}b$, $ST^{-1}b$ and $\min(T^{-1}ST^{-1}b,Sa)$. The levels $T^{-1}b$, $ST^{-1}b$ belong to the boundary of $D_1^u$. If $Sa<T^{-1}ST^{-1}b$, then $ST^{-1}b>TSa$ and therefore is the end of the cycle and does not belong to $\mathcal{U}_{a,b}$. If
$Sa>T^{-1}ST^{-1}b$, then $T^{-1}ST^{-1}b$ appears as a boundary segment of $D_1^u$.
A similar argument applies to $D_0^\ell$ that contains the horizontal levels $Ta$, $STa$, and either $TSTa$ or $Sb$.
For the induction step we assume that (1) holds for $k=n-1$, and prove that it holds for $k=n$.
Let $y\in\partial D_n$ be a horizontal segment of the boundary, $y\geq ST^{-1}b$, and $y\in \mathcal{U}_{a,b}$. Then $y=Sy'$, where $y'\in\partial D_{n-1},\,b-1\leq y'<0$.
By inductive hypothesis, $y'\in\partial D_n$, hence $y=Sy'\in\partial D_{n+1}$. Now let $y\in\partial D_n$ be a horizontal segment of the boundary, $b-1\leq y<Sa$. Then $y=T^{-1}y'$, where $y'\in\partial D_{n-1},\,0< y'<TSa$. By inductive hypothesis, $y'\in\partial D_n$, hence $y=Sy'\in\partial D_{n+1}$.
The level $y=Sa$ appears as a boundary segment of $D_n^u$ since $T^{-1}(\partial(U_{n-1}^{11})\cup \partial(U_{n-1}^{12}))$ and $S(\partial(L_{n-1}^3))$ do not overlap. Then $y=Sy'$, where $y'=a$ is the $y$-coordinate of the horizontal lower boundary of $L_{n-1}^3$. Since $L_{n}^3\subset L_{n-1}^3$ and $U_{n}^{11}\cup U_{n}^{12}\subset U_{n-1}^{11}\cup U_{n-1}^{12}$, we get that $T^{-1}(\partial(U_{n}^{11})\cup \partial(U_{n}^{12}))$ and $S(\partial(L_{n}^3))$ do not overlap, and $y=Sa$ will appear as a boundary segment of $D_{n+1}^u$.
On the other hand, assume $y\in\partial D_{n+1}$ was not a horizontal level of $\partial D_n$. Then $y=Sy'$ for some $y'\in\partial (U_n^{22}\cup U_n^{21})$, $y=T^{-1}y'$ for some $y'\in\partial (U_n^{12})$, or $y=Sa$. In all cases $y\in \mathcal{U}_{a,b}$
by the structure of the sets $\mathcal{U}_{a}$ and $\mathcal{U}_b$ established in Theorems \ref{a-cycle} and \ref{b-cycle}.
(2)
We start with level $-\frac1{b-1}$ which belongs to the boundary of the trapping region $\Theta_{a,b}$ by definition. We have seen that if $T^{-1}ST^{-1}b\in \mathcal{U}_b$,
then the level appears in the boundary of $D_1^u$. Now, if $b-1<T^{-k}ST^{-1}b<\frac{a}{1-a}$ (for the smallest $k=2$ or $3$), then the expansion continues, each $T^{-i}ST^{-1}b,\,i\leq k$ appears for the first time in the boundary of $D_i^u$ for $i\leq k$, and
the next element in the cycle, $ST^{-k}ST^{-1}b$, appears in the boundary of $D_{k+1}^u$. Using the structure of the set $\mathcal{U}_b$
established in Theorem \ref{b-cycle} we see that all levels of the set $\mathcal{U}_b$ appear as boundary levels of some $D_n^u$. We use the same argument for level $-\frac1{a}$ which appears for the first time in the boundary of some $D_{n_0}^u$, to see that all elements of the set $\mathcal{U}_a$ appear as boundary levels of all successive sets $D_n^u$. The same argument works for the lower boundary.
(3) Thus starting with some $n$, all sets $D_n$ have two connected components bounded by non-decreasing step-functions whose $y$ levels coincide with the sets $\mathcal{U}_{a,b}$ and $\mathcal{L}_{a,b}$.
Therefore, the attractor $D_{a,b}=\cap_{n=0}^\infty D_n$ has the same property.
(4) The surjectivity of the map $F$ on $D_{a,b}$ follows from the nesting property of the sets $D_n$.
\end{proof}
A priori the map $F$ on $D_{a,b}$ does not have to be injective, but in our case it will be since we will identify $D_{a,b}$ with an earlier constructed set $A_{a,b}$.
\begin{cor} \label{FRS}If the map $f$ satisfies the finiteness condition, then the attractor $D_{a,b}$ has finite rectangular structure, i.e. bounded by non-decreasing step-functions with a finite number of steps.
\end{cor}
\begin{thm} \label{attractor} If the map $f$ satisfies the finiteness condition, then the set $A_{a,b}$ constructed in Theorem \ref{thm:recstructure} is the attractor for the map $F$.
\end{thm}
\begin{proof} We proved in Theorem \ref{thm:recstructure} that the set $A_{a,b}$ constructed there is uniquely determined by the prescribed set of $y$-levels $\mathcal{U}_{a,b}\cup\mathcal{L}_{a,b}$. By Corollary \ref{FRS}, the set $D_{a,b}$ has finite rectangular structure with the same set of $y$-levels. Now we look at the $x$-levels of the jumps of its boundary step-functions.
Take the vertex $(x, b-1)$ of $D_{a,b}$. From the surjectivity of $F$ on $D_{a,b}$, there is a point $z\in D_{a,b}$ s.t. $F(z)=(x, b-1)$.
Then $z$ must be the intersection of the ray at the level $b$ with the boundary of $D_{a,b}$, i.e. $z=(\tilde x_b, b)$, hence $x=\tilde x_b-1$. Continue the same argument: look at the vertex at the level $-1/(b-1)$. It must be $F(\tilde x_b-1, b-1)$, etc. Since each $y$-level of the boundary has a unique ``predecessor" in its orbit, all $x$-levels of the jumps obtained by ``transporting" the rays $[-\infty,\tilde x_b]$ and $[\tilde x_a,\infty]$ over the corresponding cycles, satisfy the same
equations that defined the boundary of the set $A_{a,b}$ of Theorem \ref{thm:recstructure}. Therefore $\tilde x_a=x_a,\,\,\tilde x_b=x_b$,
the step-functions that define the boundaries are the same, and $D_{a,b}=A_{a,b}$.
\end{proof}
\section{Reduction theory conjecture}\label{s:naturalextension}\label{s:7}
Don Zagier conjectured that the Reduction Theory properties, stated in the Introduction, hold for every $(a,b)\in\mathcal P$.
He was motivated by the classical cases and computer experimentations with random parameter values $(a,b)\in\mathcal P$ (see Figures \ref{don-a} and \ref{fig:A} for attractors obtained by iterating
random points using Mathematica program).
The following theorem gives a sufficient condition for the Reduction Theory conjecture to hold:
\begin{thm} \label{RTC}If both $a$ and $b$ have the strong cycle property, then for every point $(x,y)\in \bar\mathbb R^2\setminus \Delta$ there exists $N>0$ such that $F^N(x,y)\in D_{a,b}$.
\end{thm}
\begin{proof} Every point $(x,y)\in\bar\mathbb R^2\setminus \Delta$ is mapped to the trapping region by some iterate $F^{N_1}$. Since the sets $D_n$ are nested and contain $D_{a,b}$, for large $N$, $F^N(x,y)$ will be close to the boundary of $D_{a,b}$. By Corollary \ref{cor:bry}, for any boundary component $h$ of $D_{a,b}$ there exists $N_2>0$ such that $F^{N_2}(h)$ is inside $D_{a,b}$. Therefore, there exists a large enough $N>0$ such that $F^N(x,y)$ will be in the interior of $D_{a,b}$.
\end{proof}
The strong cycle property is not necessary for the Reduction theory conjecture to hold. For example, it holds for the two classical expansions $(-1,0)$ and $(-1,1)$ that satisfy only a weak cycle property. In the third classical expansion $(-1/2,1/2)$ that also satisfies a weak cycle property, property (3) does not hold for some points $(x,y)$ with $y$ equivalent to $r=(3-\sqrt{5})/2$.
\noindent\begin{figure}[thb]
\begin{minipage}[b]{.3\textwidth}
\begin{center}
\includegraphics[height=1.5in]{gafullv3.eps}
\center{$A_{-1,0}$}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{.3\textwidth}\mbox{ }
\begin{center}
\includegraphics[height=1.5in]{afullv3.eps}
\center{$A_{-1,1}$}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[b]{.3\textwidth}\mbox{ }
\begin{center}
\includegraphics[height=1.5in]{hfullv3.eps}
\center{$A_{\tiny -1/2,1/2}$}
\end{center}
\end{minipage}
\caption{Attractors for the classical cases}\label{fig:A}
\end{figure}
The next result shows that, under the finiteness condition, almost every point $(x,y)\in \bar\mathbb R^2\setminus \Delta$ lands in the attractor $D_{a,b}$ after finitely many iterations.
\begin{prop}
If the map $f_{a,b}$ satisfies the finiteness condition, then for almost every point $(x,y)\in \bar\mathbb R^2\setminus \Delta$, there exists $N>0$ such that
$F_{a,b}^N(x,y)\in D_{a,b}$.
\end{prop}
\begin{proof}
Let $(x,y)\in \mathbb R^2$ with $y$ irrational and $y=\lfloor n_0,n_1,n_2,\dots\rceil_{a,b}$. In the proof of Theorem \ref{Delta-trapping}, we showed that there exists $k>0$ such that
$$(x_{j+1},y_{j+1})=ST^{-n_j}\dots ST^{-n_1}ST^{-n_0}(x,y)\in [-1,1]\times ([-1/a,\infty]\cup [-\infty,-1/b])$$
for all $j\ge k$. The point $F^{N}_{a,b}(x,y)=(x_{k+1}, y_{k+1})$ is in $A_{a,b}$, if $(x_{k+1},y_{k+1})\in [-1,0]\times [-1/a,\infty]$ or
$(x_{k+1},y_{k+1})\in [0,1]\times [-\infty,-1/b]$. Also, $F^{N+1}(x,y)=F(x_{k+1},y_{k+1})$ is in $A_{a,b}$ if $(x_{k+1},y_{k+1})\in [0,1]\times [-1/a+1,\infty]$
or $(x_{k+1},y_{k+1})\in [-1,0]\times [-\infty,-1/b-1]$. Thus we are left with analyzing the situation when the sequence of iterates
$$(x_{j+1},y_{j+1})=ST^{-n_j}\dots ST^{-n_1}ST^{-n_0}(x,y)$$ belongs to $[0,1]\times [-1/a,-1/a+1]$ for all $j\ge k$ (or $[-1,0]\times [-1/b,-1/b-1]$ for all $j\ge k$). Assume that we are in the first situation: $y_{j+1}\in [-1/a,-1/a+1]$ for all $j\ge k$. This implies that all digits $n_{j+1}$, $j\ge k$ are either $\lfloor -1/a\rceil$ or $\lfloor -1/a\rceil+1$. In the second situation, the digits $n_{j+1}$, $j\ge k$ are either $\lfloor -1/b\rceil$ or $\lfloor -1/b\rceil-1$. Therefore the continued fraction expansion of $y$ is written with only two consecutive digits (starting from a certain position). By using Proposition \ref{bdigits1} and Remark \ref{bdigits2} we obtain that the set of all such points has zero Lebesgue measure. This proves our result.
\end{proof}
\begin{rem}
In the next section we show that there is a non-empty Cantor-like set ${\mathcal E}\subset \Delta$ belonging to the boundary segment $b=a+1$ of $\mathcal P$ such that for $(a,b)\in{\mathcal E}$ the set $\mathcal{U}_{a,b}\cup\mathcal{L}_{a,b}$ is infinite. Therefore, for $(a,b)\in{\mathcal E}$ either the set $D_n^u$ or $D_n^\ell$ is disconnected for some $n>0$, or,
by Lemma \ref{l:all-levels}(3), the attractor $D_{a,b}$ consists of two connected components whose boundary functions are not step-functions with finitely many steps.
\end{rem}
\section{Set of exceptions to the finiteness condition}\label{s:8}
In this section we study the structure of the set ${\mathcal E}\subset\mathcal P$ of exceptions to the finiteness condition.
We write ${\mathcal E}={\mathcal E}_{\mathfrak{b}}\cup{\mathcal E}_{\mathfrak{a}}$
where ${\mathcal E}_{\mathfrak{b}}$ (resp., ${\mathcal E}_{\mathfrak{a}}$) consists of all points $(a,b)\in \P$ for which $b$ (resp., $a$) does not satisfy the finiteness condition, i.e. either the truncated orbit $\mathcal{U}_b$ or $\mathcal{L}_b$ is infinite (resp., $\mathcal{U}_a$ or $\mathcal{L}_a$).
We analyze the set ${\mathcal E}_{\mathfrak{b}}$. Recall that, by Proposition \ref{shift}(2), the set $\mathcal{U}_b$ is infinite if and only if $\mathcal{L}_b$ is infinite, therefore it is sufficient to analyze the condition that the orbit $\mathcal{U}_b$ is not eventually periodic and its values belong to the interval $(\frac{b}{b+1},a+1)$. As before, we restrict our analysis (due to the
symmetry considerations) to the parameter subset of $\P$ given by $b\le -a$ and write
${\mathcal E}_{\mathfrak{b}}=\cup_{m=3}^\infty{\mathcal E}^m_{\mathfrak{b}}$ where $b\in{\mathcal E}^m_{\mathfrak{b}}$ if $b\in {\mathcal E}_{\mathfrak{b}}$ and $T^mSb\in (\frac{b}{b+1},a+1)$.
By Theorem \ref{b-cycle} and its proof, it follows that if $b\in {\mathcal E}^m_{\mathfrak{b}}$, then the first digit of the $(a,b)$-continued fraction expansion of $Sb$ is $-m$ and all the other digits are either $-m$ or $-(m+1)$.
We describe a recursive construction of the exceptional set ${\mathcal E}^m_{\mathfrak{b}}$. One starts with the `triangular' set
$${\mathcal T}^m_{\mathfrak{b}}=\{(a,b)\in \P : \frac{b}{b+1}\le T^mSb\le a+1\}.$$
The range of possible values of $b$ in ${\mathcal T}^m_{\mathfrak{b}}$ is given by the interval $[\underline b, \bar b]$ where $T^mS\bar b=\bar b$ and $T^mS\underline b=\underline b/(\underline b+1)$. Since
\[
\frac{b}{b+1}\leq b \text{ for all } b\ge 0,
\]
and the function $T^mSb$ is monotone increasing, we obtain that $\underline b<\bar b$, and $\underline b$ is the horizontal boundary of ${\mathcal T}^m_{\mathfrak{b}}$, while $\bar b$ is the $b$-coordinate of its `vertex'.
At the next stage we obtain the following regions:
$${\mathcal T}^{m,m}_{\mathfrak{b}}=\{(a,b)\in {\mathcal T}^m_{\mathfrak{b}}: \frac{b}{b+1}\le T^{m}ST^mSb\le a+1\}$$
$${\mathcal T}^{m,m+1}_{\mathfrak{b}}=\{(a,b)\in {\mathcal T}^m_{\mathfrak{b}}: \frac{b}{b+1}\le T^{m+1}ST^mSb\le a+1\}\,.$$
By the same argument as above each region is `triangular', i.e. the $b$-coordinate of its lower (horizontal) boundary is less than the $b$-coordinate of its vertex. We show that its intersection with the triangular region obtained on the previous step is either empty or has `triangular' shape. The horizontal boundary of ${\mathcal T}^{m,m}_{\mathfrak{b}}$ has the $b$-coordinate given by the relation $T^mST^mSb=b/(b+1)$ (call it $\tilde b$).
We have
\[
T^mST^mS\underline b=T^mS\left(\frac{\underline b}{\underline b+1}\right)=T^mS\underline b-1=-\frac1{\underline b+1}<\frac{\underline b}{\underline b+1},
\]
so $\underline b<\tilde b$. On the other hand,
\[
T^mST^mS\bar b=T^mS\bar b=\bar b,
\]
which shows that the hyperbola $T^mST^mSb=b$ intersects the diagonal side $b=a+1$ at the point with $b$-coordinate $\bar b$. It follows that the region ${\mathcal T}^{m,m}_{\mathfrak{b}}$ is triangular and non-empty with $\underline b<\tilde b<\bar b$.
The upper boundary of ${\mathcal T}^{m,m+1}_{\mathfrak{b}}$ is given by the hyperbola $T^{m+1}ST^mSb=a+1$. Notice that, if $\underline a+1=T^mS\underline b$, then the point $(\underline a,\underline b)$ lies on the curves $T^mSb=a+1$ (obviously) and $T^{m+1}ST^mSb=a+1$ because
$$T^{m+1}ST^mS\underline b=T^{m+1}S(\underline b/(\underline b+1))=T^mS\underline b=\underline a+1\,.$$
This shows that the entire horizontal boundary of ${\mathcal T}^m_{\mathfrak{b}}$ belongs to that of ${\mathcal T}^{m,m+1}_{\mathfrak{b}}$.
Moreover, the hyperbola $T^{m+1}ST^mSb=a+1$ intersects the diagonal side $b-a=1$ at the point $\hat b$ satisfying $T^{m+1}ST^mS\hat b=\hat b$. Therefore, $T^{m}ST^mS\hat b=\hat b-1<\frac{\hat b}{\hat b+1}$, i.e. $\hat b<\tilde b$.
In this case we have
$\underline b<\hat b<\tilde b<\bar b$, and the two triangular regions ${\mathcal T}^{m,m}_{\mathfrak{b}}$ and ${\mathcal T}^{m,m+1}_{\mathfrak{b}}$ are disjoint and non-empty.
The situation becomes more complicated as we proceed recursively. Let ${\mathcal T}^{n_1,n_2,\dots,n_k}_{\mathfrak{b}}$ be one of the regions obtained after $k$ steps of this construction, with $n_1=m$ and $n_i\in\{m,m+1\}$ for $2\le i\le k$. At the next step we get two new sets (possible empty) (see Figure \ref{fig-exc}):
$${\mathcal T}^{n_1,n_2,\dots,n_k,m}_{\mathfrak{b}}=\{(a,b)\in {\mathcal T}^{n_1,n_2,\dots,n_k}_{\mathfrak{b}}: \frac{b}{b+1}\le T^{m}ST^{n_k}S\dots T^{n_1}Sb\le a+1\}$$
$${\mathcal T}^{n_1,n_2,\dots,n_k,m+1}_{\mathfrak{b}}=\{(a,b)\in {\mathcal T}^{n_1,n_2,\dots,n_k}_{\mathfrak{b}}: \frac{b}{b+1}\le T^{m+1}ST^{n_k}S\dots T^{n_1}Sb\le a+1\}\,.$$
\begin{figure}[ht]
\psfrag{B}[r]{${\mathcal T}^{n_1,n_2,\dots,n_k,m}_{\mathfrak{b}}$}
\psfrag{A}[l]{${\mathcal T}^{n_1,n_2,\dots,n_k,m+1}_{\mathfrak{b}}$}
\psfrag{c}[c]{$\underline b$}
\psfrag{d}[c]{$\hat b$}
\psfrag{e}[c]{$\tilde b$}
\psfrag{f}[c]{$\bar b$}
\psfrag{u}[c]{$\underline a$}
\psfrag{v}[c]{$\underline a'$}
\centerline{\includegraphics[scale=0.8]{fig-except.eps}}
\caption{Set ${\mathcal T}^{n_1,n_2,\dots,n_k}_{\mathfrak{b}}$ and its two subregions}
\label{fig-exc}
\end{figure}
As in the base case, the inequality $T^{m}ST^{n_k}S\dots T^{n_1}Sb\le a+1$ of ${\mathcal T}^{n_1,n_2,\dots,n_k,m}_{\mathfrak{b}}$ is satisfied by all points of ${\mathcal T}^{n_1,n_2,\dots,n_k}_{\mathfrak{b}}$ because of the monotone increasing property of $T,S$ and the fact that $T^{n_k}S\dots T^{n_1}Sb\le a+1$ implies
$$T^{m}ST^{n_k}S\dots T^{n_1}Sb\le T^mS(a+1)\le T^mS(b)\le a+1\,.$$
Thus the upper boundary of the region ${\mathcal T}^{n_1,n_2,\dots,n_k,m}_{\mathfrak{b}}$ (if nonempty) is part of the upper boundary of ${\mathcal T}^{n_1,n_2,\dots,n_k}_{\mathfrak{b}}$; it is the lower (horizontal) boundary that changes. Similarly, the defining inequality $\frac{b}{b+1}\le T^{m+1}ST^{n_k}S\dots T^{n_1}Sb$ of ${\mathcal T}^{n_1,n_2,\dots,n_k,m+1}_{\mathfrak{b}}$ is satisfied by al points of ${\mathcal T}^{n_1,n_2,\dots,n_k}_{\mathfrak{b}}$ because
$$T^{m+1}ST^{n_k}S\dots T^{n_1}Sb\ge T^{m+1}S\frac{b}{b+1}=m-\frac{1}{b}=T^mSb\ge\frac{b}{b+1}\,.$$
Thus the lower boundary of ${\mathcal T}^{n_1,n_2,\dots,n_k,m+1}_{\mathfrak{b}}$ (if nonempty) is part of the lower boundary of ${\mathcal T}^{n_1,n_2,\dots,n_k}_{\mathfrak{b}}$.
Therefore, we can describe the above sets as
\begin{eqnarray}
& &{\mathcal T}^{n_1,n_2,\dots,n_k,m}_{\mathfrak{b}} = \{(a,b)\in {\mathcal T}^{n_1,n_2,\dots,n_k}_{\mathfrak{b}}: \frac{b}{b+1}\le T^{m}ST^{n_k}S\dots T^{n_1}Sb\}\\
& &{\mathcal T}^{n_1,n_2,\dots,n_k,m+1}_{\mathfrak{b}} = \{(a,b)\in {\mathcal T}^{n_1,n_2,\dots,n_k}_{\mathfrak{b}}: T^{m+1}ST^{n_k}S\dots T^{n_1}Sb\le a+1\}\,.
\end{eqnarray}
By the same reason as in the base case, the two regions ${\mathcal T}^{n_1,\dots,n_k,m}_{\mathfrak{b}}$ and ${\mathcal T}^{n_1,\dots,n_k,m+1}_{\mathfrak{b}}$ do not overlap.
The set ${\mathcal E}^m_{\mathfrak{b}}$ is now obtained as the union of all sets of type
\begin{equation}\label{eni}
{\mathcal E}^{(n_i)}_{\mathfrak{b}}=\bigcap_{k=1}^\infty{\mathcal T}^{n_1,n_2,\dots,n_k}_{\mathfrak{b}}
\end{equation}
where
$n_1=m$, $n_i\in\{m,m+1\}$ if $i\ge 2$, and the sequence $(n_i)$ is not eventually periodic. If such a set ${\mathcal E}^{(n_i)}_{\mathfrak{b}}$ is non-empty and $(a,b)$ belongs to it, then $b$ is uniquely determined from the $(a,b)$-expansion of $Sb=\lfloor -n_1,-n_2,\dots\rceil$.
First we need some additional lemmas:
\begin{lem}\label{lem-cf} $ $
\begin{itemize}
\item[(i)] A point $b\in [0,1]$ satisfying $T^{n_k}S\dots T^{n_1}Sb=b$ with $|n_i|\ge 2$ can be written formally using a periodic ``$-$" continued fraction expansion
\begin{equation}\label{bbar}
b=-1/(\overline{-n_1,-n_2,\dots,-n_k})=(0,\overline{-n_1,-n_2,\dots,-n_k})\,.
\end{equation}
If $b$ is in ${\mathcal T}^{n_1,n_2,\dots,n_k}_b$, then $Sb$ has the $(a,b)$-continued fraction expansion
$$\lfloor Sb\rceil_{a,b}=\lfloor\overline{-n_1,-n_2,\dots,-n_k}\,\rceil\,.$$
\item[(ii)] A point $b$ in $[0,1]$ satisfying $T^{n_k}S\dots T^{n_1}Sb=b/(b+1)$ can be written formally using the periodic ``$-$'' continued fraction expansion
\begin{equation}\label{underb}
b=(0,-n_1,\overline{-n_2,\dots,-n_k,-(m+1)})\,.
\end{equation}
If the point $b\in T^{n_1,n_2,\dots,n_k}_b$, then $\lfloor Sb\rceil_{a,b}=\lfloor -n_1,\overline{-n_2,\dots,-n_k,-(m+1)}\rceil$.
\end{itemize}
\end{lem}
\begin{proof}
One can verify directly that the point $b$ given by \eqref{bbar} is the fixed point of the hyperbolic transformation $T^{n_k}S\dots T^{n_1}S$ and $b\in [0,1]$ (see also \cite[Proposition 1.3]{KU1}).
The equation in part (ii) can be written as $STST^{n_k}S\dots T^{n_1}Sb=b$ and one verifies directly that the value $b$ given by \eqref{underb} is the fixed point of that hyperbolic transformation and $b\in [0,1]$.
\end{proof}
Notice that the relation $(0,-n_1,-n_2,\dots)=-(0,n_1,n_2,\dots)$ is satisfied, assuming that the formal ``$-$" continued fraction expansions are convergent (from the proof of Theorem \ref{convergence}, the convergence property holds if $|n_i|\ge 2$ for all $i\ge 1$).
\begin{defn}
We say that two sequences (finite or infinite) $\sigma_1=(n_i)$ and $\sigma_2=(p_j)$ of positive integers are in lexicographic order, $\sigma_1\prec \sigma_2$, if on the first position $k$ where the two sequences differ one has $n_k<p_k$ ,or if the finite sequence $(n_i)$ is a starting subsequence of $(p_j)$.
\end{defn}
The following property follows from the monotonicity of $T,S$.
\begin{lem}\label{order}
Given two infinite sequences $\sigma_1=(n_i)$ and $\sigma_2=(p_j)$ of integers $n_i\ge 2$ and $p_j\ge 2$ such that $\sigma_1\prec \sigma_2$ then
$$(0,n_1,n_2,\dots) < (0,p_1,p_2,\dots)\,.$$
\end{lem}
The next lemma provides necessary conditions for a set ${\mathcal E}^{(n_i)}_{\mathfrak{b}}$ to be non-empty. Denote by $\mathfrak{l}_m$ the length of the initial block of $m$'s and by $\mathfrak{l}_{m+1}$ the length of the first block of $(m+1)$'s in $(n_i)$.
\begin{lem}\label{lemE} $ $
\begin{itemize}
\item[(i)] If a set ${\mathcal E}^{(n_i)}_{\mathfrak{b}}$ in the upper region ${\mathcal T}^{m,m}_{\mathfrak{b}}$ is non-empty then the sequence $(n_i)$ contains no consecutive $(m+1)$'s and the length of any block of $m$'s is equal to $\frak{l}_m$ or $\frak{l}_m-1$.
\item[(ii)] If a set ${\mathcal E}^{(n_i)}_{\mathfrak{b}}$ in the lower region ${\mathcal T}^{m,m+1}_{\mathfrak{b}}$ is non-empty then the sequence $(n_i)$ contains no consecutive $m$'s and the length of any block of $(m +1)$'s is
equal to $\frak{l}_{m+1}$ or $\frak{l}_{m+1}+1$.
\end{itemize}
\end{lem}
\begin{proof}
(i) Assume that the sequence $(n_i)$ contains two consecutive $(m+1)$'s.
Then some ${\mathcal T}^{n_1,n_2,\dots,n_k,m+1,m+1}_{\mathfrak{b}}$ (with $n_1=n_2=n_k=m$) is non-empty. The upper vertex of such a triangular set satisfies the inequality
\[
\begin{split}
\bar b&\le -(0,\overline{n_1,n_2,\dots,n_k,m+1,m+1})\\
&=-(0,m,m,\dots,m,m+1,\boxed{m+1},\dots)
\end{split}
\]
while the lower (horizontal) boundary satisfies
\[
\begin{split}
\underline{b}&\ge -(0,n_1,\overline{n_2,\dots,n_k,m+1})\\
&=-(0,m,m,\dots,m,m+1,\boxed{m},\dots)\qquad
\end{split}
\]
This implies that $\underline{b}>\bar b$ because the entries of the corresponding continued fractions
with positive entries are in lexicographic order (they coincide on the first $k+1$ places, and on the $(k+2)^{th}$
position the first continued fraction has digit $m+1$ while the second one has digit $m$), i.e. the set ${\mathcal T}^{n_1,n_2,\dots,n_k,m+1,m+1}_{\mathfrak{b}}$ is empty.
Now assume that there exists a non-empty set ${\mathcal T}^{n_1,n_2\dots,n_k, m, m, \dots, m}$ ($n_k=m+1$) with the final block of $m$'s of length greater than $\frak{l}_m$. The upper vertex of this set is given by
\[
\begin{split}
\bar b&\le -(0,\overline{n_1,n_2,\dots,n_k})=-(0,\overline{\underbrace{m,m,\dots,m}_{\frak{l}_m},m+1,\dots,n_k})\\
&=-(0,\underbrace{m,m,\dots,m}_{\frak{l}_m},m+1,\dots,n_k,\underbrace{m,m,\dots,m}_{\frak{l}_m},m+1,\dots)
\end{split}
\]
while the lower horizontal segment is given by
\[
\underline{b}\ge -(0,n_1,\overline{n_2,\dots,n_k,\underbrace{m,m\dots, m}_q,m+1})\,.
\]
If $\frak{l}_m<q$ then the two continued fractions coincide on the first $k+p$ entries. Looking at the $k+p+1$ entry, we get that $\bar b<\underline{b}$, hence the set ${\mathcal T}^{n_1,n_2\dots,n_k, m, m, \dots, m}_{\mathfrak{b}}$ would be empty.
Assume now that there exists a non-empty set of type ${\mathcal T}^{n_1,n_2\dots,n_k, m, m, \dots, m,m+1}_{\mathfrak{b}}$ ($n_k=m+1$) with the last block of $m$'s of length $q$ strictly less than $\frak{l}_m-1$. Because $n_k=m+1$, $n_{k-1}=m$, and ${\mathcal T}^{n_1,n_2\dots,n_k, m, m, \dots, m,m+1}_{\mathfrak{b}}\subset {\mathcal T}^{n_1,n_2\dots,n_k}_{\mathfrak{b}}$ we have that the lower limit of the set ${\mathcal T}^{n_1,n_2\dots,n_k, m, m, \dots, m,m+1}_{\mathfrak{b}}$ satisfies the relation
\[
\begin{split}
\underline b&\ge -(0,n_1\overline{n_2,\dots,n_{k-1},m+1})=-(0,n_1\overline{n_2,\dots,n_{k-1},n_k})\\
&=-(0,\underbrace{m,m,\dots,m}_{\frak{l}_m},m+1,\dots,n_k,\underbrace{m,\dots,m}_{\frak{l}_m-1},m+1,\dots)
\end{split}
\]
while the upper limit of the same set satisfies the relation
\[
\bar b\le -(0,\overline{n_1, n_2,\dots,n_k,\underbrace{m,m\dots, m}_q,m+1}]\,.
\]
This implies that $\bar b<\underline b$ because the two continued fractions coincide on their first $k+q$ entries, and the $k+q+1$ entries are $m$, and $m+1$ respectively. Therefore the set ${\mathcal T}^{n_1,n_2\dots,n_k, m, m, \dots, m,m+1}_{\mathfrak{b}}$ is empty.
\medskip
(ii) Assume that a set ${\mathcal T}^{n_1,n_2,\dots,n_k,m,m}_{\mathfrak{b}}$ (with $n_1=m$, $n_2=m+1$ and $n_k=m+1$) is non-empty. The upper vertex of such a set satisfies the inequality
\[
\bar b\le -(0,\overline{n_1,n_2,\dots,n_k})
=-(0,m,m+1,\dots,n_k,m,\boxed{m+1},\dots)
\]
while the lower horizontal segment satisfies the relation
\[
\underline{b}\ge -(0,n_1,\overline{n_2,\dots,n_k,m,m,m+1})
=-(0,m,m+1,\dots,n_k,m,\boxed{m},m+1,\dots).
\]
Then $\underline{b}>\bar b$ because the sequences of the corresponding continued fractions with positive entries are in lexicographic order, i.e. the set ${\mathcal T}^{n_1,n_2,\dots,n_k,m,m}_{\mathfrak{b}}$ is empty.
Now assume that there exists a non-empty set ${\mathcal T}^{n_1,n_2\dots,n_k, m+1, m+1, \dots, m+1}_{\mathfrak{b}}$ ($n_k=m$) with the final block of $(m+1)$'s of length $q$ greater than $\frak{l}_{m+1}+1$. The upper vertex of this set satisfies
\[
\bar b\le -(0,\overline{m,\underbrace{m+1,\dots,m+1}_{\frak{l}_{m+1}},m,\dots,n_k,\underbrace{m+1,\dots,m+1}_q})
\]
while the lower horizontal segment satisfies the relation
\[
\begin{split}
\underline{b}&\ge -(0,n_1,\overline{n_2,\dots,n_k,m+1})\\
&=-(0,m,{\underbrace{m+1,\dots,m+1}_{\frak{l}_{m+1}},m,\dots, n_k,\underbrace{m+1,\dots,m+1}_{\frak{l}_{m+1}+1},m,\dots}).
\end{split}
\]
Since the two continued fraction expansions with positive entries coincide on the first $k+\frak{l}_{m+1}+1$ entries and their $k+\frak{l}_{m+1}+2$ entries are $m+1$ and $m$, respectively, we obtain $\bar b<\underline{b}$, i.e. the set ${\mathcal T}^{n_1,n_2\dots,n_k, m+1, m+1, \dots, m+1}_{\mathfrak{b}}$.
Finally, suppose that there exists a non-empty set ${\mathcal T}^{n_1,n_2\dots,n_k, m+1, m+1, \dots, m+1,m}_{\mathfrak{b}}$ ($n_k=m$) with the final block of $(m+1)$'s of length $q$ less than $\frak{l}_{m+1}$. The upper vertex of this set satisfies
\[
\bar b\le -(0,\overline{m,\underbrace{m+1,\dots,m+1}_{\frak{l}_{m+1}},m,\dots,n_k,\underbrace{m+1,\dots,m+1}_{\frak{l}_{m+1}}})
\]
while the lower horizontal segment satisfies the relation
\[
\begin{split}
\underline{b}&\ge -(0,n_1,\overline{n_2,\dots,n_k,\underbrace{m+1,\dots,m+1}_{q},m,m+1})\\
&=-(0,m,{\underbrace{m+1,\dots,m+1}_{\frak{l}_{m+1}},m,\dots, n_k,\underbrace{m+1,\dots,m+1}_{q},m,\dots}).
\end{split}
\]
Since the two continued fraction expansions with positive entries coincide on the first $k+\frak{l}_{m+1}$ entries and their $(k+\frak{l}_{m+1}+1)^{th}$ entries are $(m+1)$ and $m$, respectively, we obtain $\bar b<\underline{b}$, i.e. the set ${\mathcal T}^{n_1,n_2\dots,n_k, m+1, m+1, \dots, m+1,m}_{\mathfrak{b}}$ is empty.
\end{proof}
In what follows, we describe in an explicit manner the symbolic properties of a sequence $(n_i)$ for which ${\mathcal E}^{(n_i)}_{\mathfrak{b}}\ne \emptyset$. Notice that in both cases of Lemma \ref{lemE} there are two admissible blocks that can be used to express the admissible sequence $(n_i)$:
\medskip
case (i): $A^{(1)}=(\underbrace{m,\dots,m}_{\frak{l}_{m}},m+1)$ and $B^{(1)}=(\underbrace{m,\dots,m}_{\frak{l}_{m}-1},m+1)$;
\medskip
case (ii): $A^{(1)}=(m,\underbrace{m+1,\dots,m+1}_{\frak{l}_{m+1}})$ and $B^{(1)}=(m,\underbrace{m+1,\dots,m+1}_{\frak{l}_{m+1}+1})$.
\medskip
\noindent with $\frak{l}_{m}\ge 2$, $\frak{l}_{m+1}\ge 1$. In both situations $A^{(1)}\prec B^{(1)}$.
One could think of $A^{(1)}$ as being the new `$m$' and $B^{(1)}$ the new `$m+1$', and treat the original sequence of $m$'s and $m+1$'s as a sequence of $A^{(1)}$'s and $B^{(1)}$'s. Furthermore, the next lemma shows that such a substitution process can be continued recursively to construct blocks $A^{(n)}$ and $B^{(n)}$ (for any $n\ge 1$), so that the original sequence $(n_i)$ may be considered to be a sequence of $A^{(n)}$'s and $B^{(n)}$'s. Moreover, only particular blocks of $A^{(n)}$'s and $B^{(n)}$'s warrant non-empty triangular regions of the next generation
Let us also introduce the notations $A^{(0)}=m$ and $B^{(0)}=m+1$. Assume that ${\mathcal E}^{(n_i)}_{\mathfrak{b}}$ is a nonempty set. We have:
\begin{lem}\label{lem-rec}
For every $n\ge 0$, there exist integers $\frak{l}_{A^{(n)}}\ge 2$, $\frak{l}_{B^{(n)}}\ge 1$ such that the sequence $(n_i)$ can be written as a concatenation of blocks
\begin{equation}\label{case1}
A^{(n+1)}=(\underbrace{A^{(n)},\dots,A^{(n)}}_{\frak{l}_{A^{(n)}}},B^{(n)}) \,,\quad B^{(n+1)}=(\underbrace{A^{(n)},\dots,A^{(n)}}_{\frak{l}_{A^{(n)}}-1},B^{(n)})
\end{equation}
or
\begin{equation}\label{case2}
A^{(n+1)}=(A^{(n)},\underbrace{B^{(n)},\dots,B^{(n)}}_{\frak{l}_{B^{(n)}}}) \,,\quad B^{(n+1)}=(A^{(n)},\underbrace{B^{(n)},\dots,B^{(n)}}_{\frak{l}_{B^{(n)}}+1})\,.
\end{equation}
\end{lem}
\begin{proof}
Notice that Lemma \ref{lemE} proves the above result for $n=0$ with $\frak{l}_{A^{(0)}}=\frak{l}_{m}$, $\frak{l}_{B^{(0)}}=\frak{l}_{m+1}$.
We show inductively that
\begin{equation}\label{eqorder}
A^{(n)}\prec B^{(n)}
\end{equation}
and if a finite sequence $\sigma$ starts with an $A^{(n)}$ block and ends with a $B^{(n)}$ block, $\sigma=(A^{(n)},\tau,B^{(n)})$, then the lower boundary $\underline b(\sigma)$ of ${\mathcal T}^\sigma_{\mathfrak{b}}$ (if nonempty) satisfies
\begin{equation}\label{bng}
\underline b(\sigma)\ge-(0,A^{(n)},\overline{\tau,B^{(n)}})\,.
\end{equation}
Relation \eqref{eqorder} is obviously true for $n=0$; \eqref{bng} is also satisfied if $n=0$, since one applies Lemma \ref{lem-cf} part (ii) to the sequence $\tilde\sigma=(A^{(0)},\tau)$ where ${\mathcal T}^{\tilde\sigma}_{\mathfrak{b}}\supset {\mathcal T}^\sigma_{\mathfrak{b}}$.
We point out that by applying Lemma \ref{lem-cf} part (i) to the region ${\mathcal T}^{\sigma}$ we have
\begin{equation}\label{ubng}
\bar b(\sigma)\le-(0,\overline{\sigma})=-(0,\overline{A^{(n)},\tau,B^{(n)}})\,.
\end{equation}
To prove the inductive step, suppose that for some $n\ge 1$, we can rewrite the sequence $(n_i)$ using blocks $A^{(n+1)}$ and $B^{(n+1)}$ as in case \eqref{case1} or \eqref{case2}.
\medskip
\noindent\textbf{Case 1.} Assume $A^{(n+1)}$ and $B^{(n+1)}$ are given by \eqref{case1}.
It follows immediately that $A^{(n+1)}\prec B^{(n+1)}$ since $A^{(n)}\prec B^{(n)}$. Also, if a sequence $\sigma$ starts with an $A^{(n+1)}$ block and ends with a $B^{(n+1)}$ block (thus, implicitly, $\sigma$ starts with an $A^{(n)}$ block and ends with a $B^{(n)}$ block),
$$\sigma=(A^{(n+1)},\tau,B^{(n+1)})=(\underbrace{A^{(n)},\dots,A^{(n)}}_{\frak{l}_{A^{(n)}}},B^{(n)},\tau,\underbrace{A^{(n)},\dots,A^{(n)}}_{\frak{l}_{A^{(n)}}-1},B^{(n)})\,$$ then, by applying \eqref{bng} to $\tilde \sigma=(\underbrace{A^{(n)},\dots,A^{(n)}}_{\frak{l}_{A^{(n)}}},B^{(n)},\tau)=(A^{(n)},B^{(n+1)},\tau)$ (which starts with $A^{(n)}$ and ends with $B^{(n)}$) we get
$$\underline b(\sigma)\ge b(\tilde\sigma)\ge -(0,A^{(n)},\overline{B^{(n+1)},\tau})=-(0,A^{(n)},B^{(n+1)},\overline{\tau, B^{(n+1)}})\,.$
Therefore, \eqref{bng} holds for $n+1$, since $(A^{(n)},B^{(n+1)})=A^{(n+1)}$.
Now assume that $(n_i)$ starts with a block of $A^{(n+1)}$'s of length $\frak{l}_{A^{(n+1)}}>1$. We prove that the sequence $(n_i)$ cannot have two consecutive $B^{(n+1)}$'s and any sequence of consecutive blocks $A^{(n+1)}$ has length $\frak{l}_{A^{(n+1)}}$ or $\frak{l}_{A^{(n+1)}-1}$. Suppose the sequence $(n_i)$ contains two consecutive blocks of type $B^{(n+1)}$:
\[
(n_i)=(A^{(n+1)},A^{(n+1)},\dots,A^{(n+1)},B^{(n+1)},B^{(n+1)},\dots).
\]
We look at the set
$${\mathcal T}^{A^{(n+1)}A^{(n+1)}\dots A^{(n+1)}B^{(n+1)}B^{(n+1)}}$$
and remark that the upper boundary satisfies (from \eqref{ubng})
\begin{equation}\label{tb}
\bar b\le -(0,\overline{A^{(n+1)},A^{(n+1)},\dots,A^{(n+1)},B^{(n+1)},B^{(n+1)}})
\end{equation}
and the lower boundary satisfies (from \eqref{bng})
\begin{equation}\label{lb}
\underline b\ge -(0,A^{(n+1)},\overline{A^{(n+1)},\dots,A^{(n+1)},B^{(n+1)}})\,.
\end{equation}
But \eqref{tb} and \eqref{lb} imply that $\underline b>\bar b$, because the two corresponding continued fractions
with positive entries are
in lexicographic order. Thus, there cannot be two consecutive $B^{(n+1)}$ blocks in the sequence $(n_i)$.
Now, let us check that the sequence $(n_i)$ cannot have a block of $A^{(n+1)}$'s of length $q>\frak{l}_{A^{(n+1)}}$. Assume the contrary
$$(n_i)=(\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{\frak{l}_{A^{(n+1)}}},B^{(n+1)},\tau,B^{(n+1)}, \underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{q},B^{(n+1)},\dots)\,.$$
Then the set ${\mathcal T}^{(n_i)}_{\mathfrak{b}}$ has the upper bound $\bar b$ satisfying
$$\bar b\le -(0,\overline{\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{\frak{l}_{A^{(n+1)}}},B^{(n+1)},\tau,B^{(n+1)}})$$
while the lower bound $\underline b$ satisfies by \eqref{bng}
$$
\underline b\ge -(0,A^{(n+1)},\overline{\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{\frak{l}_{A^{(n+1)}}-1},B^{(n+1)},\tau,B^{(n+1)},\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{q},B^{(n+1)}}).
$$
Comparing the two continued fractions, we get that $\bar b<\underline b$ (since $A^{(n+1)}\prec B^{(n+1)}$ and $q>\frak{l}_{A^{(n+1)}}$).
Now assume that $(n_i)$ starts with $A^{(n+1)}$ and then continues with a block of $B^{(n+1)}$'s of length $\frak{l}_{B^{(n+1)}}\ge 1$. We prove that the sequence $(n_i)$ cannot have two consecutive $A^{(n+1)}$'s and any sequence of consecutive blocks $B^{(n+1)}$ has length $\frak{l}_{B^{(n+1)}}$ or $\frak{l}_{B^{(n+1)}}+1$. Suppose the sequence $(n_i)$ contains two (or more) consecutive blocks of type $A^{(n+1)}$:
$$(n_i)=(A^{(n+1)},B^{(n+1)},\tau,B^{(n+1)},\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{q\ge 2},B^{(n+1)},\dots)\,.$$
We study the region ${\mathcal T}^{A^{(n+1)},B^{(n+1)},\tau,B^{(n+1)},A^{(n+1)},\dots,A^{(n+1)},B^{(n+1)}}$
and remark that its upper boundary satisfies (from \eqref{ubng})
\begin{equation}\label{ttb}
\bar b\le -(0,\overline{A^{(n+1)},B^{(n+1)},\tau,B^{(n+1)}})
\end{equation}
and the lower boundary satisfies (from \eqref{bng})
\begin{equation}\label{llb}
\underline b\ge -(0,A^{(n+1)},\overline{B^{(n+1)},\tau,B^{(n+1)},\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{q\ge 2},B^{(n+1)}}).
\end{equation}
But \eqref{ttb} and \eqref{llb} implie that $\underline b>\bar b$ because the two corresponding continued fractions
with positive entries are
in lexicographic order. Thus, there cannot be two consecutive $A^{(n+1)}$ blocks in the sequence $(n_i)$.
Now, let us check that the sequence $(n_i)$ cannot have a block of $B^{(n+1)}$'s of length $q>\frak{l}_{B^{(n+1)}}+1$. Assume the contrary
$$(n_i)=(A^{(n+1)},\underbrace{B^{(n+1)}\dots,B^{(n+1)}}_{\frak{l}_{B^{(n+1)}}},A^{(n+1)},\tau,A^{(n+1)}, \underbrace{B^{(n+1)},\dots,B^{(n+1)}}_{q},A^{(n+1)},\dots)\,.$$
Then the set ${\mathcal T}^{(n_i)}$ has the upper bound $\bar b$ satisfying
$$\bar b\le -(0,\overline{A^{(n+1)},\underbrace{B^{(n+1)}\dots,B^{(n+1)}}_{\frak{l}_{B^{(n+1)}}},A^{(n+1)},\tau,A^{(n+1)}, \underbrace{B^{(n+1)},\dots,B^{(n+1)}}_{q},A^{(n+1)}})
$$
while the lower bound $\underline b$ satisfies by \eqref{bng}
$$
\underline b\ge -(0,A^{(n+1)},\overline{\underbrace{B^{(n+1)}\dots,B^{(n+1)}}_{\frak{l}_{B^{(n+1)}}},A^{(n+1)},\tau,A^{(n+1)}, B^{(n+1)}}).
$$
Comparing the two continued fractions, we get that $\bar b<\underline b$.
\medskip
\noindent\textbf{Case 2.} Assume $A^{(n+1)}$ and $B^{(n+1)}$ are given by \eqref{case2}. It follows that $A^{(n+1)}\prec B^{(n+1)}$ since $A^{(n+1)}$ is the beginning block of $B^{(n+1)}$.
Also, if a sequence $\sigma$ starts with an $A^{(n+1)}$ block and ends with a $B^{(n+1)}$ block (thus, implicitly, $\sigma$ starts with an $A^{(n)}$ block and ends with a $B^{(n)}$ block),
$$\sigma=(A^{(n+1)},\tau,B^{(n+1)})=(A^{(n)},\underbrace{B^{(n)},\dots,B^{(n)}}_{\frak{l}_{B^{(n)}}},\tau,A^{(n)},\underbrace{B^{(n)},\dots,B^{(n)}}_{\frak{l}_{B^{(n)}}+1})\,$$
then by applying \eqref{bng} to $\tilde \sigma=(A^{(n)},\underbrace{B^{(n)},\dots,B^{(n)}}_{\frak{l}_{B^{(n)}}},\tau,A^{(n)},B^{(n)})$, which starts with $A^{(n)}$ and ends with $B^{(n)}$, we get
\begin{equation*}
\begin{split}
\underline b(\sigma)\ge \underline b(\tilde\sigma)\ge -(0,A^{(n)},\overline{\underbrace{B^{(n)},\dots,B^{(n)}}_{\frak{l}_{B^{(n)}}},\tau,A^{(n)},B^{(n)}})\\=-(0,A^{(n+1)},\overline{\tau, A^{(n)},\underbrace{B^{(n)},\dots,B^{(n)}}_{\frak{l}_{B^{(n)}+1}}})
\end{split}
\end{equation*}
so \eqref{bng} holds for $n+1$.
Assume that $(n_i)$ starts with a sequence of $A^{(n+1)}$'s of length $\frak{l}_{A^{(n+1)}}>1$. Similar to the analysis of the first case, one proves that the sequence $(n_i)$ cannot have two consecutive $B^{(n+1)}$'s and any sequence of consecutive blocks $A^{(n+1)}$ has length $\frak{l}_{A^{(n+1)}}$ or $\frak{l}_{A^{(n+1)}}-1$.
If the sequence $(n_i)$ starts with $A^{(n+1)}$ and then continues with a sequence of $B^{(n+1)}$'s of length $\frak{l}_{B^{(n+1)}}\ge 1$, one can prove that the sequence $(n_i)$ cannot have two consecutive $A^{(n+1)}$'s and any sequence of consecutive blocks $B^{(n+1)}$ has length $\frak{l}_{B^{(n+1)}}$ or $\frak{l}_{B^{(n+1)}}+1$.
\end{proof}
Additionally, we prove
\begin{lem}\label{lem-order}
If the block $\tau_1=(n_i,\dots, n_l)$ is a tail of $A^{(n)}$ and $\tau_2=(p_j,\dots,p_h)$ is a tail of $B^{(n)}$, then
$A^{(n)}\prec \tau_1$ and $B^{(n)}\prec \tau_2$.
\end{lem}
\begin{proof}
The statement is obviously true if $n=1$. Assume it is true for some $n$ both for $A^{(n)}$ and $B^{(n)}$. We analyze the case of $A^{(n+1)}$ being given by \eqref{case1}, $A^{(n+1)}=(\underbrace{A^{(n)},\dots, A^{(n)}}_{\frak{l}_{A^{(n)}}},B^{(n)})$. Consider an arbitrary tail $\tau$ of $A^{(n+1)}$; $\tau$ could start with a block $A^{(n)}$ or a tail of $A^{(n)}$ or $\tau$ coincides with $B^{(n)}$ or a tail of $B^{(n)}$. In all situations, the inductive hypothesis and the fact that $A^{(n)}\prec B^{(n)}$ prove that $A^{(n+1)}\prec \tau$. The case of $A^{(n+1)}$ given by \eqref{case2} is treated similarly.
\end{proof}
\begin{rem}\label{rem-ineq}
Using the relations \eqref{bng} and \eqref{ubng}, notice that a set ${\mathcal T}^{A^{(n+1)}}_{\mathfrak{b}}$ (if nonempty) has the upper vertex satisfying
\begin{equation}\label{bn}
\bar b_{n+1}\le -(0,\overline{A^{(n+1)}})
\end{equation}
and a lower horizontal boundary that satisfies
\begin{equation}\label{ubn1}
\underline b_{n+1}\ge -(0,A^{(n+1)},\overline{B^{(n+1)}})
\end{equation}
if $A^{(n+1)}$ is given by the substitution rule \eqref{case1}, and
\begin{equation}\label{ubn2}
\underline b_{n+1}\ge -(0,A^{(n)},\overline{B^{(n)}})
\end{equation}
if $A^{(n+1)}$ is given by \eqref{case2}.
\end{rem}
We will prove that the above inequalities are actually equality relations. For that we construct a starting subsequence of $A^{(n+1)}$ defined inductively as:
$$
\sigma^{(1)}=
\begin{cases}
(\underbrace{m,\dots,m}_{{\frak l}_m}) & \text{if } A^{(1)}=(\underbrace{m,\dots,m}_{{\frak l}_m},m+1)\\
({m}) & \text{if } A^{(1)}=(m,\underbrace{m+1,\dots,m+1}_{{\frak l}_{m+1}})\\
\end{cases}
$$
\noindent{Case 1.} If $A^{(n)}$ is given by a relation of type \eqref{case1}, i.e. $A^{(n)}=(A^{(n-1)}, \dots,A^{(n-1)}, B^{(n-1)})$, then
\begin{equation}\label{snc1}
\sigma^{(n+1)}=
\begin{cases}
(\underbrace{A^{(n)},\dots,A^{(n)}}_{{\frak l}_A^{(n)}-1},\sigma^{(n)}) & \text{if } A^{(n+1)}=(\underbrace{A^{(n)},\dots,A^{(n)}}_{{\frak l}_{A^{(n)}}},B^{(n)})\\
\sigma^{(n)} & \text{if } A^{(n+1)}=(A^{(n)},\underbrace{B^{(n)},\dots,B^{(n)}}_{{\frak l}_{B^{(n)}}})
\end{cases}
\end{equation}
\noindent{Case 2.} If $A^{(n)}$ is given by a relation of type \eqref{case2}, i.e. $A^{(n)}=(A^{(n-1)}, B^{(n-1)}, \dots, B^{(n-1)})$, then
\begin{equation}\label{snc2}
\sigma^{(n+1)}=
\begin{cases}
(\underbrace{A^{(n)},\dots,A^{(n)}}_{{\frak l}_A^{(n)}},\sigma^{(n)}) & \text{if } A^{(n+1)}=(\underbrace{A^{(n)},\dots,A^{(n)}}_{{\frak l}_{A^{(n)}}},B^{(n)})\\
(A^{(n)},\sigma^{(n)}) & \text{if } A^{(n+1)}=(A^{(n)},\underbrace{B^{(n)},\dots,B^{(n)}}_{{\frak l}_{B^{(n)}}})
\end{cases}
\end{equation}
We introduce the notation $f^{\sigma}$ to denote the transformation $T^{n_k}S\dots T^{n_1}S$ if $\sigma=(n_1,\dots,n_k)$.
\begin{lem}\label{relbn}
Let $\sigma^{(n+1)}$ be the starting block of $A^{(n+1)}$ defined as above. Then the equation
$$f^{\sigma^{(n+1)}}b=\frac{b}{b+1}$$ has a unique solution $b\in [0,1]$ given by
\begin{equation}\label{eqbn}
b_{n+1}=\begin{cases}
-(0,A^{(n+1)},\overline{B^{(n+1)}}) & \text{if } A^{(n+1)} \text{ given by } \eqref{case1}\\
-(0,A^{(n)},\overline{B^{(n)}}) & \text{if } A^{(n+1)} \text{ given by } \eqref{case2}
\end{cases}
\end{equation}
\end{lem}
\begin{proof}
We proceed with an inductive proof, and as part of it we also show that
\begin{equation}\label{sigma}
(\sigma^{(n+1)},m+1,\tilde A^{(n)})=
\begin{cases}
A^{(n+1)} & \text{if } A^{(n+1)}=(\underbrace{A^{(n)},\dots,A^{(n)}}_{{\frak l}_{A^{(n)}}},B^{(n)})\\
(A^{(n)},B^{(n)}) & \text{if } A^{(n+1)}=(A^{(n)},\underbrace{B^{(n)},\dots,B^{(n)}}_{{\frak l}_{B^{(n)}}})
\end{cases}
\end{equation}
where $A^{(n)}=(m,\tilde A^{(n)})$.
The relation \eqref{eqbn} is true for $n=0$ due to Lemma \ref{lem-cf}(ii). Also, \eqref{sigma} follows immediately.
Suppose now that the inductive relations hold for some $n$. We analyze the solution of $f^{\sigma^{(n+2)}}b=\frac{b}{b+1}$.
Assume that $A^{(n+1)}=(\underbrace{A^{(n)},\dots,A^{(n)}}_{{\frak l}_{A^{(n)}}}, B^{(n)})$. We look at the two possible cases:
\noindent (i) If $A^{(n+2)}=(\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}}, B^{(n+1)})$,
$\sigma^{(n+2)}=(\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}-1},\sigma^{(n+1)})$.
Using Lemma \ref{lem-cf}(ii), we have that the solution to $f^{\sigma^{(n+2)}}b=\frac{b}{b+1}$ is given by
\begin{equation*}
\begin{split}
b_{n+2}&=-(0,m,\overline{\tilde A^{(n+1)},\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}-2},\sigma^{(n+1)},m+1})\\
&= -(0,m,\tilde A^{(n+1)},\overline{\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}-2},\sigma^{(n+1)},m+1,\tilde A^{(n+1)}})\\
&=-(0,m,\tilde A^{(n+1)},\overline{\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}-2},\sigma^{(n+1)},m+1,\tilde A^{(n)},B^{(n+1)}})\\
&=-(0,A^{(n+1)},\overline{\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}-2},A^{(n+1)},B^{(n+1)}}\\
&=-(0,A^{(n+1)},\overline{B^{(n+2)}}=-(0,A^{(n+2)},\overline{B^{(n+2)}}).
\end{split}
\end{equation*}
Also,
\begin{equation*}
\begin{split}
(\sigma^{(n+2)},m+1,\tilde A^{(n+1)})&=(\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}-1},\sigma^{(n+1)},m+1,\tilde A^{(n)}, B^{(n+1)})\\
&=(\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}-1},A^{(n+1)},B^{(n+1)})=A^{(n+2)}.
\end{split}
\end{equation*}
\noindent (ii) If $A^{(n+2)}=(A^{(n+1)},\underbrace{B^{(n+1)},\dots,B^{(n+1)}}_{{\frak l}_{B^{(n+1)}}})$, then
$
\sigma^{(n+2)}=\sigma^{(n+1)}\,,
$
and the induction step gives us the solution of $f^{\sigma(n+2)}b=\frac{b}{b+1}$ as
$
b_{n+2}=-(0,A^{(n+1)},\overline{B^{(n+1)}})
$. Also,
\begin{equation*}
(\sigma^{(n+2)},m+1,\tilde A^{(n+1)})=(\sigma^{(n+1)},m+1,\tilde A^{(n)},B^{(n+1)})=(A^{(n+1)},B^{(n+1)}).
\end{equation*}
\noindent Now assume that $A^{(n+1)}=(A^{n},\underbrace{B^{(n)},\dots,B^{(n)}}_{{\frak l}_{B^{(n)}}})$.
We look again at the two possible cases:
\noindent (i) If $A^{(n+2)}=(\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}}, B^{(n+1)})$,
$
\sigma^{(n+2)}=(\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}},\sigma^{(n+1)})
$.
Using Lemma \ref{lem-cf}(ii), we have that the solution to $f^{\sigma(n+2)}b=\frac{b}{b+1}$ is given by
\begin{equation*}
\begin{split}
b_{n+2}&=-(0,m,\overline{\tilde A^{(n+1)},\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}-1},\sigma^{(n+1)},m+1})\\
&= -(0,m,\tilde A^{(n+1)},\overline{\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}-1},\sigma^{(n+1)},m+1,\tilde A^{(n+1)}})\\
&=-(0,m,\tilde A^{(n+1)},\overline{\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}-1},\sigma^{(n+1)},m+1,\tilde A^{(n)},\underbrace{B^{(n)},\dots,B^{(n)}}_{{\frak l}_{B^{(n)}}})}\\
&=-(0,m,\tilde A^{(n+1)},\overline{\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}-1},A^{(n)},B^{(n)},\underbrace{B^{(n)},\dots,B^{(n)}}_{{\frak l}_{B^{(n)}}})}\\
&=-(0,A^{(n+1)},\overline{\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}-1},B^{(n+1)}})=-(0,A^{(n+2)},\overline{B^{(n+2)}}).
\end{split}
\end{equation*}
A similar approach gives us that $(\sigma^{(n+2)},m+1,\tilde A^{(n+1)})=A^{(n+2)}$.
\medskip
\noindent (ii) If $A^{(n+2)}=(A^{(n+1)},\underbrace{B^{(n+1)},\dots,B^{(n+1)}}_{{\frak l}_{B^{(n+1)}}})$, then
$
\sigma^{(n+2)}=(A^{(n+1)},\sigma^{(n+1)}).
$
Using Lemma \ref{lem-cf}(ii),
we have that the solution to $f^{\sigma(n+2)}b=\frac{b}{b+1}$ is given by
\begin{equation*}
\begin{split}
b_{n+2}&=-(0,m,\overline{\tilde A^{(n+1)},\sigma^{(n+1)},m+1})\\
&= -(0,m,\tilde A^{(n+1)},\overline{\sigma^{(n+1)},m+1,\tilde A^{(n+1)}})\\
&=-(0,m,\tilde A^{(n+1)},\overline{\sigma^{(n+1)},m+1,\tilde A^{(n)},\underbrace{B^{(n)},\dots,B^{(n)}}_{{\frak l}_{B^{(n)}}}})\\
&=-(0,m,\tilde A^{(n+1)},\overline{A^{(n)},B^{(n)},\underbrace{B^{(n)},\dots,B^{(n)}}_{{\frak l}_{B^{(n)}}}})\\
&=-(0,A^{(n+1)},\overline{B^{(n+1)}}).
\end{split}
\end{equation*}
Also,
\begin{equation*}
\begin{split}
(\sigma^{(n+2)},m+1,\tilde A^{(n+1)})&=(A^{(n+1)},\sigma^{(n+1)},m+1,\tilde A^{(n)},\underbrace{B^{(n)},\dots,B^{(n)}}_{{\frak l}_{B^{(n)}}})\\
&=(A^{(n+1)},A^{n},B^{(n)},\underbrace{B^{(n)},\dots,B^{(n)}}_{{\frak l}_{B^{(n)}}})=(A^{(n+1)},B^{(n+1)}).
\end{split}
\end{equation*}
\end{proof}
\begin{thm}\label{thm-exact}
Any sequence $(n_i)$ constructed recursively using relations \eqref{case1} and \eqref{case2} provides a non-empty set $\mathcal E^{(n_i)}_b$.
\end{thm}
\begin{proof}
We prove inductively that any set ${\mathcal T}^{A^{(n+1)}}_{\mathfrak{b}}$ is nonempty and the relations \eqref{bn} and \eqref{ubn1} or \eqref{ubn2}
are actual equalities, i.e.
\begin{equation}\label{bne}
\bar b_{n+1}= -(0,\overline{A^{(n+1)}})
\end{equation}
and a lower horizontal boundary that satisfies
\begin{equation}\label{ube1}
\underline b_{n+1}= -(0,A^{(n+1)},\overline{B^{(n+1)}})
\end{equation}
if $A^{(n+1)}$ is given by the substitution rule \eqref{case1} or
\begin{equation}\label{ube2}
\underline b_{n+1}= -(0,A^{(n)},\overline{B^{(n)}})
\end{equation}
if $A^{(n+1)}$ is given by \eqref{case2}. As part of the inductive proof, we also show that any tail block $\tau$ of $A^{(n+1)}$, $\tau\ne \tau^{(n+1)}$ satisfies $\tau\prec \tau^{(n+1)}$, where $\tau^{(n+1)}$ denotes the tail block of $A^{(n+1)}$ obtained by eliminating the starting block $\sigma^{(n+1)}$ defined by \eqref{snc1} or \eqref{snc2}.
Indeed for $n=0$, one can check directly that the sets ${\mathcal T}^{m,m,\dots,m,m+1}_{\mathfrak{b}}$ and ${\mathcal T}^{m,m+1,\dots,m+1}_{\mathfrak{b}}$ satisfy the above equalities using the fact that an ``$m$" digit does not change the position of the upper vertex, while an ``$m+1$'' digit does not change the position of the horizontal segment of such a triangular set. Also, for any tail $\tau\ne \tau^{(1)}$ of $A^{(1)}$, $\tau\prec \tau^{(1)}$.
Now, let us assume that ${\mathcal T}_{\mathfrak{b}}^{A^{(n+1)}}$ obtained from $A^{(n+1)}=(\underbrace{A^{(n)},\dots,A^{(n)}}_{\frak{l}^{(n)}},B^{(n)})$ is nonempty and satisfies \eqref{bne} and \eqref{ube1}.
For ${\mathcal T}^{A^{(n+2)}}_{\mathfrak{b}}$ we look at the two possible cases:
\noindent (i) $A^{(n+2)}=(\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}}, B^{(n+1)})$. By Remark \ref{rem-ineq},
\[
\bar b_{n+2}\le -(0,\overline{A^{(n+2)}})=-(0,\overline{\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{\frak{l}_{A^{(n+1)}}},B^{(n+1)}})=:\hat b
\]
and
\[
\underline b_{n+2}\ge -(0,A^{(n+2)},\overline{B^{(n+2)}})=-(0,A^{(n+1)},\overline{\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{\frak{l}_{A^{(n+1)}-1}},B^{(n+1)}})=:\tilde b
\]
where $\tilde b$ was obtained by applying Lemma \ref{lem-cf} part (ii) to the starting block
$$
\sigma^{(n+2)}=(\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_{{\frak l}_{A^{(n+1)}}-1},\sigma^{(n+1)})
$$
of $A^{(n+2)}$.
We prove first the other inductive step: any tail block $\tau$ of $A^{(n+2)}$, $\tau\ne\tau^{(n+2)}$ satisfies $\tau\prec \tau^{(n+2)}$. Notice that $\tau^{(n+2)}=(\tau^{(n+1)}, B^{(n+1)})$. There exists $\tau'$ a tail block of $A^{(n+1)}$ with the property that
$$\tau=(\tau',\underbrace{A^{(n+1)},\dots,A^{(n+1)}}_l,B^{n+1}), \quad 0\le l\le \frak{l}_{A^{(n+1)}-1}$$
or $\tau=\tau'$.
The latter case holds when $\tau$ is just a tail of $B^{(n+1)}$ (which itself is a tail of $A^{(n+1)}$). It is possible that $\tau'=\emptyset$, but in this case $\tau\prec\tau^{(n+2)}$
because $A^{(n+1)}\prec \tau^{(n+1)}$ by Lemma \ref{lem-order}. If $\tau'\ne\emptyset$, we also get
that $\tau\prec \tau^{(n+2)}$ by using the inductive hypothesis relation $\tau'\prec \tau^{(n+1)}$.
Now we show that the points $(\tilde b-1,\tilde b)$ and $(\hat b-1,\hat b)$ belong to the set ${\mathcal T}^{A^{(n+2)}}_{\mathfrak{b}}$. The point $(\hat b-1,\hat b)$ belongs to ${\mathcal T}^{A^{(n+1)}}_{\mathfrak{b}}$ so $f^{A^{(n+1)}}\hat b\le \hat b$. If $\sigma$ is an intermediate block between $A^{(n+1)}$ and $A^{(n+2)}$, $A^{(n+1)}\subset\sigma\subset A^{(n+2)}$, then
$$f^\sigma(\hat b)=-(0,\tau,\overline{A^{(n+2)}})\le -(0,\overline{A^{(n+2)}})=\hat b$$
The inequality is due to the fact that $\tau$ is a tail block of $A^{(n+2)}$ obtained by eliminating $\sigma$, so
$A^{(n+2)}\prec \tau$.
Now we show that $f^\sigma(\tilde b)\ge \tilde b/(\tilde b+1)$ for any intermediate block $\sigma$ between $A^{(n+1)}$ and $A^{(n+2)}$. We have that $f^{\sigma^{(n+2)}}(\tilde b)=\tilde b/(\tilde b+1)$ by Lemma \ref{relbn}, and
\[
f^{\sigma^{(n+2)}}(\tilde b)=-(0,\tau^{(n+2)},\overline{B^{(n+2)}})\, ,
\]
where $\tau^{(n+2)}=(\tau^{(n+1)},B^{(n+1)})$. Also $f^\sigma(\tilde b)=-(0,\tau,\overline{B^{(n+2)}})$ with $\tau$ being the tail block of $A^{(n+2)}$ obtained by eliminating $\sigma$. But $\tau\prec\tau^{(n+2)}$ as we have just proved, hence $f^\sigma(\tilde b)\ge f^{\sigma(n+2)}(\tilde b)$.
In conclusion, any intermediate block
$\sigma$ between $A^{(n+1)}$ and $A^{(n+2)}$ satisfies
\[
\tilde b/(\tilde b+1)\le f^\sigma(\tilde b)\le f^\sigma(\hat b)\le \hat b\,,
\]
therefore the points $(\tilde b-1,\tilde b)$ and $(\hat b-1,\hat b)$ belong to the intermediate set ${\mathcal T}^\sigma_{\mathfrak{b}}$. This proves the induction step for ${\mathcal T}^{A^{(n+2)}}_{\mathfrak{b}}$.
\medskip
\noindent (ii) $A^{(n+2)}=(A^{(n+1)},\underbrace{B^{(n+1)},\dots,B^{(n+1)}}_{{\frak l}_{B^{(n+1)}}})$. By Remark \ref{rem-ineq}, we have that
\[
\bar b_{n+2}\le -(0,\overline{A^{(n+2)}})=-(0,\overline{A^{(n+1)},\underbrace{B^{(n+1)},\dots,B^{(n+1)}}_{\frak{l}_{B^{(n+1)}}}})=:\hat b
\]
and
\[
\underline b_{n+2}\ge -(0,A^{(n+1)},\overline{B^{(n+1)}})=:\tilde b
\]
where $\tilde b$ was obtained by applying Lemma \ref{lem-cf} part (ii) to the starting block
$
\sigma^{(n+2)}=\sigma^{(n+1)}
$
of $A^{(n+2)}$.
We prove first the other inductive step: any tail block $\tau$ of $A^{(n+2)}$, $\tau\ne\tau^{(n+2)}$, satisfies $\tau\prec \tau^{(n+2)}$. There exists $\tau'$ a tail block of $A^{(n+1)}$ with the property that
$$\tau=(\tau',\underbrace{B^{(n+1)},\dots,B^{(n+1)}}_l), \quad 0\le l\le \frak{l}_{B^{(n+1)}}$$
(again, using the fact that $B^{(n+1)}$ is a tail block of $A^{(n+1)}$).
Since
$$\tau^{(n+2)}=(\tau^{(n+1)}, \underbrace{B^{(n+1)},\dots,B^{(n+1)}}_{{\frak l}_{B^{(n+1)}}}))\,,$$
we get that $\tau\prec \tau^{(n+2)}$ by using the inductive hypothesis $\tau'\prec \tau^{(n+1)}$.
Now we show that the points $(\tilde b-1,\tilde b)$ and $(\hat b-1,\hat b)$ belong to the set ${\mathcal T}^{A^{(n+2)}}_{\mathfrak{b}}$. The point $(\hat b-1,\hat b)$ belongs to ${\mathcal T}^{A^{(n+1)}}_{\mathfrak{b}}$ so $f^{A^{(n+1)}}\hat b\le \hat b$. If $\sigma$ is an intermediate block between between $A^{(n+1)}$ and $A^{(n+2)}$ then
$$f^\sigma(\hat b)=-(0,\tau,\overline{A^{(n+2)}})\le -(0,\overline{A^{(n+2)}})=\hat b$$
because $\tau$ is a tail block of $A^{(n+2)}$ obtained by eliminating $\sigma$, so
$A^{(n+2)}\prec \tau$.
Now we show that $f^\sigma(\tilde b)\ge \tilde b/(\tilde b+1)$. We have that $f^{\sigma^{(n+2)}}(\tilde b)=\tilde b/(\tilde b+1)$ by Lemma \ref{relbn}, and
\[
f^{\sigma^{(n+2)}}(\tilde b)=-(0,\tau^{(n+1)},\overline{B^{(n+1)}})\, \quad f^\sigma(\tilde b)=-(0,\tau,\overline{B^{(n+1)}})
\]
with $\tau$ being the end block of $A^{(n+2)}$ obtained by eliminating $\sigma$. But $\tau\prec\tau^{(n+2)}$ as we have just proved, hence $f^\sigma(\tilde b)\ge f^{\sigma(n+2)}(\tilde b)$. In conclusion, any intermediate sequence $\sigma$ between $A^{(n+1)}$ and $A^{(n+2)}$ satisfies
\[
\tilde b/(\tilde b+1)\le f^\sigma(\tilde b)\le f^\sigma(\hat b)\le \hat b\,,
\]
therefore the points $(\tilde b-1,\tilde b)$ and $(\hat b-1,\hat b)$ belong to the intermediate set ${\mathcal T}^\sigma_{\mathfrak{b}}$.
We proved the induction step for ${\mathcal T}^{A^{(n+2)}}_{\mathfrak{b}}$, when $A^{(n+1)}$ is given by \eqref{case1}. A similar argument can be provided for the case when $A^{(n+1)}$ is given by \eqref{case2}, so the conclusion of the theorem is true.
\end{proof}
We prove now that each set nonempty set ${\mathcal E}^{(n_i)}$ with $(n_i)$ not eventually aperiodic sequence is actually a singleton.
\begin{thm}
Assume that $(n_i)$ is a not eventually periodic sequence such that the set ${\mathcal E}^{(n_i)}_b$ is nonempty. Then the set ${\mathcal E}^{(n_i)}_{\mathfrak{b}}$ is a point on the line segment $b-a=1$.
\end{thm}
\begin{proof}
The sequence $(n_i)$ satisfies the recursive relations \eqref{case1} or \eqref{case2}. We look at the set ${\mathcal T}^{A^{(n+1)}}_{\mathfrak{b}}$ and estimate the length of its lower base. In case \eqref{case1}
its upper vertex is given by \eqref{bne} and its lower base satisfies \eqref{ube1}. The lower base is a segment whose right end coordinate is
$$\underline a^r_{n+1}= -(0,A^{(n+1)},\overline{B^{(n+1)}})-1$$ and left end coordinate is
\[
\underline a^l_{n+1}=f^{A^{(n +1)}}(- (0,A^{(n+1)},\overline{B^{(n+1)}})) -1=-(0,\overline{B^{(n+1)}}) -1\,.
\]
Hence the length of the lower base is given by
$$
L_{n+1}=\underline a^r_{n}-\underline a^l_{n+1}=(0,\overline{B^{(n+1)}})-(0,A^{(n+1)},\overline{B^{(n+1)}})\,.
$$
In case \eqref{case2}, the lower base is a segment whose right end coordinate is
$$
\underline a^r_{n+1}=-(0,A^{(n)},\overline{B^{(n)}})-1
$$ and the left end coordinate is given by
\[
\underline a^l_{n+1}=f^{A^{(n +1)}}(- (0,A^{(n)},\overline{B^{(n)}})) -1=-(0,\overline{B^{(n)}}) -1\,.
\]
Hence the length of the lower base is given by
$$
L_{n+1}=\underline a^r_{n+1}-\underline a^l_{n+1}=(0,\overline{B^{(n)}})-(0,A^{(n)},\overline{B^{(n)}})\,.
$$
Notice that in the first case the two continued fraction expansions have in common at least the block $A^{(n)}$, while in the second case they have in common at least the block $A^{(n-1)}$. This implies that in both cases $L_{n+1}\rightarrow 0$ as $n\rightarrow \infty$.
Moreover, the bases of the sets ${\mathcal T}^{n_1,\dots n_k}_{\mathfrak{b}}$ have non-increasing length and we have found a subsequence of these bases whose lengths converge to zero. Therefore the set ${\mathcal E}^{(n_i)}_{\mathfrak{b}}$ consists of only one point $(b-1,b)$, where
$b=-(0,n_1,n_2,\dots)$.
\end{proof}
The above result gives us a complete description of the set of exceptions ${\mathcal E}_{\mathfrak{b}}$ to the finiteness condition. It is a subset of the boundary segment $b=a+1$ of $\P$. Moreover, each set ${\mathcal E}^m_{\mathfrak{b}}$ is uncountable because the recursive construction of a nonempty set ${\mathcal E}^{(n_i)}_{\mathfrak{b}}$ allows for an arbitrary number of successive blocks $A^{(k)}$ at step $(k+1)$. Formally, one constructs a surjective map $\mathfrak j: {\mathcal E}^m_{\mathfrak{b}}\rightarrow \mathbb N^{\mathbb N}$ by associating to a singleton set ${\mathcal E}^{(n_i)}_b$ a sequence of positive integers defined a
\medskip
\centerline{$\mathfrak j({\mathcal E}^{(n_i)}_b)(k)=\#$ of consecutive $A^{(k)}$-blocks at the beginning of $(n_i)$.}
\medskip
The set ${\mathcal E}_{\mathfrak{b}}$ has one-dimensional Lebesgue measure $0$. The reason is that all associated formal continued fractions expansions of $b=-(0,n_1,n_2,\dots)$ have only two consecutive digits; such formal expansions $(0,n_1,n_2,\dots)$ are valid (-1,0)-continued fractions. Hence the set of such $b$'s has measure zero by Proposition \ref{bdigits1}.
Analogous conclusions hold for ${\mathcal E}_{\mathfrak{a}}$. Thus we have
\begin{thm} \label{ex}
For any $(a,b)\in\P$, $b\neq a+1$, the finiteness condition holds. The set of exceptions ${\mathcal E}$ to the finiteness condition
is an uncountable set of one-dimensional Lebesgue measure $0$ that lies on the boundary $b=a+1$ of $\P$.
\end{thm}
Now we are able to provide the last ingredient in the proof of part (b) of the Main Result:
\begin{prop}\label{open-dense}
The strong cycle property is an open and dense condition.
\end{prop}\begin{proof}
It follows from Theorems \ref{b-cycle} and \ref{a-cycle} that the condition is open. Theorem \ref{ex} asserts that for all $(a,b)\in\P$, $b\neq a+1$ the finiteness condition holds, i.e. all we need to show is that if $b$ has the week cycle property or the $(a,b)$-expansions of $Sb$ and $T^{-1}b$ are eventually periodic, then in any neighborhood of it there is a $b$ with the strong cycle property. For, if $b$ has the weak cycle property, it is a rational number obtained from the equation $\hat{f}^nT^mSb=0$, and any small perturbation of it will have the strong cycle property. Similarly, if the $(a,b)$-expansions of $Sb$ and $T^{-1}b$ are eventually periodic, then $b$ is a quadratic irrationality (see Remark \ref{quadratic}), and for any neighborhood of $b$ will contain values
satisfying the strong cycle property. A similar argument holds for $Sa$ and $Ta$.
\end{proof}
\section{Invariant measures and ergodic properties}\label{s:9}
Based on the finite rectangular geometric structure of the domain $D_{a,b}$ one can study the measure-theoretic properties of the Gauss-type map $\hat f_{a,b}:[a,b)\rightarrow [a,b)$,
\begin{equation}\label{1dGauss}
\hat f_{a,b}(x)=-\frac{1}{x}-\left\lfloor -\frac{1}{x}\right\rceil_{a,b}\,,\quad \hat f_{a,b}(0)=0
\end{equation}
and its associated natural extension map $\hat F_{a,b}:\hat D_{a,b}\rightarrow \hat D_{a,b}$
\begin{equation}\label{2dGauss}
\hat F _{a,b}=\left(\hat f_{a,b}(x),-\frac{1}{y-\lfloor -1/x\rceil_{a,b}}\right).
\end{equation}
We remark that $\hat F_{a,b}$ is obtained from the map $F_{a,b}$ induced on the set $D_{a,b}\cap \{(x,y)| a\le y <b\}$ by a change of coordinates $x'=y$, $y'=-1/x$. Therefore the domain
$\hat D_{a,b}$ is easily identified knowing $D_{a,b}$ and may be considered its ``compactification".
We present the simple case when $\displaystyle 1\le -\frac{1}{a}\le b+1$ and $a-1\le -\frac{1}{b}\le -1$. The general
theory is the subject our paper in preparation \cite{KU6}.
The truncated orbits of $a$ and $b$ are
\begin{eqnarray*}
& &\mathcal{L}_a =\left\{a+1,-\frac{1}{a+1}\right\}, \quad \mathcal{U}_a=\left\{-\frac{1}{a},-\frac{a+1}{a}\right\} \\
& &\mathcal{L}_b =\left\{-\frac{1}{b},\frac{b-1}{b}\right\}, \quad \mathcal{U}_b=\left\{b-1,-\frac{1}{b-1}\right\}
\end{eqnarray*}
and the end points of the cycles are $c_a=\frac{a}{a+1}$, $c_b=\frac{b}{1-b}$.
\begin{thm} If $1\le -\frac{1}{a}\le b+1$ and $a-1\le -\frac{1}{b}\le -1$, then the domain $\hat D_{a,b}$ of $\hat F_{a,b}$ is given by
\[
\begin{split}
\hat D_{a,b}&=[a,-\frac{1}{b}+1]\times[-1,0]\cup [-\frac{1}{b}+1,a+1]\times[-1/2,0]\\
&\quad \cup [b-1,-\frac{1}{a}-1]\times [0,1/2]\cup [-\frac{1}{a}-1,b]\times [0,1]
\end{split}
\]
and $\hat F_{a,b}$ preserves the Lebesgue equivalent probability measure
\begin{equation}\label{dnu}
d\nu_{a,b}=\frac{1}{\log[(1+b)(1-a)]}\frac{dxdy}{(1+xy)^2}\,.
\end{equation}
\end{thm}
\begin{proof}
The description of $\hat D_{a,b}$ follows directly from the cycle relations and the finite rectangular structure. It is a standard computation that the measure $\frac{dxdy}{(1+xy)^2}$ is preserved by $\hat F_{a,b}$, by using the fact any M\"obius transformation, hence $F_{a,b}$, preserves the measure $\frac{du\,dw}{(w-u)^2}$, and $\hat F_{a,b}$ is obtained from $F_{a,b}$ by coordinate changes $x=w, y=-1/u$.
Moreover, the density $\frac{1}{(1+xy)^2}$ is bounded away from zero on $\hat D_{a,b}$ and
$$
\int_{\hat D_{a,b}}\frac{dxdy}{(1+xy)^2}=\log[(b+1)(1-a)]<\infty
$$
hence the last part of the theorem is true.
\end{proof}
\begin{figure}[htb]
\includegraphics[scale=0.8]{case1ab.eps}
\caption{Typical domain $\hat D_{a,b}$ for the case studied}
\end{figure}
The Gauss-type map $\hat f_{a,b}$ is a factor of $\hat F_{a,b}$
(projecting on the $x$-coordinate), so one can obtain its smooth
invariant measure $d\mu_{a,b}$ by integrating $d\nu_{a,b}$ over $\hat
D_{a,b}$ with respect to the $y$-coordinate as explained in
\cite{AF3}. Thus, if we know the exact shape of the set $D_{a,b}$, we
can calculate the invariant measure precisely.
The measure $d\mu_{a,b}$ is ergodic and the measure-theoretic entropy
of $\hat f_{a,b}$ can be computed explicitly using Rokhlin's formula.
\begin{thm} $ $
The map $\hat f_{a,b}:[a,b)\rightarrow [a,b)$ is ergodic with respect to Lebesgue equivalent invariant probability measure
\begin{equation}
d\mu_{a,b}=\frac{1}{C_{a,b}}\left(\frac{\chi_{(a,-\frac{1}{b}+1)}}{1-x}+\frac{\chi_{(-\frac{1}{b}+1,a+1)}}{2-x}+\frac{\chi_{(b-1,-\frac{1}{a}-1)}}{x+2}+
\frac{\chi_{(-\frac{1}{a}-1,b)}}{x+1}\right)dx
\end{equation}
where $C_{a,b}=\log[(1+b)(1-a)]$. The measure-theoretic entropy of $\hat f_{a,b}$ is given by
\begin{equation}\label{entropy}
h_{\mu_{a,b}}(\hat f_{a,b})=\frac{\pi^2}{3\log[(1-a)(1+b)]}\,.
\end{equation}
\end{thm}
\begin{proof}
The measure $d\mu_{a,b}$ is obtained by integrating $d\nu_{a,b}$ over $\hat D_{a,b}$. Ergodicity follows from a more general result concerning one-dimensional expanding maps (see \cite{AF3,Zw}). To compute the entropy, we use Rokhlin's formula
\[
\begin{split}
h_{\mu_{a,b}}(\hat f_{a,b})&=\int_a^b\log|\hat f'_{a,b}|d\mu_{a,b}=-2\int_a^b\log|x|d\mu_{a,b}\\
&=\frac{-2}{C_{a,b}}\left(\int_a^{-\frac{1}{b}+1}\frac{\log|x|}{1-x}dx+\int_{-\frac{1}{b}+1}^{a+1}\frac{\log|x|}{2-x}dx\right.\\
&\qquad+\left.\int_{b-1}^{-\frac{1}{a}-1}\frac{\log|x|}{x+2}dx+\int_{-\frac{1}{a}-1}^b\frac{\log|x|}{x+1}dx \right)
\end{split}
\]
Let $I(a,b)$ denote the sum of the four integrals. The function depends smoothly on $a,b$, hence we can compute the partial derivatives $\partial I/\partial a$ and $\partial I/\partial b$. We get that both partial derivatives are zero, hence $I(a,b)$ is constant. Using $a=-1, b=1$, we get
$$I(a,b)=I(-1,1)=2\int_0^1\frac{\log|x|}{1+x}dx=-\pi^2/6\,,$$
and the entropy formula \eqref{entropy}.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 410 |
Townes Van Zandt
Waylon Arnold Jennings (June 15, 1937-February 13, 2002) was an American country music singer, songwriter, and musician. He rose to prominence as a bassist for Buddy Holly following the break-up of The Crickets. Jennings escaped death in the February 3, 1959, plane crash that took the lives of Buddy Holly, Ritchie Valens, and J.P. "The Big Bopper" Richardson, when he gave up his seat to Richardson who had been sick with the flu. Urban legend and Hollywood folklore have it that Jennings and The Big Bopper flipped a coin for the last seat on the plane, with Jennings losing. It was, in fact, Tommy Allsup who flipped the coin for the fated plane trip, losing his seat to Valens.
By the 1970s, Jennings had become associated with so-called "outlaws," an informal group of musicians who worked outside of the Nashville corporate scene. A series of duet albums with Willie Nelson in the late '70s culminated in the 1978 crossover hit, "Mammas Don't Let Your Babies Grow Up to Be Cowboys." In 1979, he recorded the theme song for the hit television show The Dukes of Hazzard, and also served as the narrator ("The Balladeer") for all seven seasons of the show.
He continued to be active in the recording industry, forming the group The Highwaymen with Nelson, Johnny Cash, and Kris Kristofferson. Jennings released his last solo studio album in 1998. In 2001, he was inducted into the Country Music Hall of Fame.
Biography from Wikipedia | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,104 |
Who's (Still) Above the Social Security Payroll Tax Cap?
The Social Security payroll tax cap is the earnings level above which no further Social Security taxes are collected. The cap is currently at $110,100, though legislation has been introduced in Congress to apply the Social Security payroll tax to earnings above $250,000 (but not between the current cap and this level). This issue brief updates earlier work, finding that 5.8 percent of workers would be affected if the Social Security cap were eliminated entirely and 1.4 percent would be affected if the current tax were applied to earnings over $250,000. It breaks down these numbers further by gender, race or ethnicity, age, and state of residence.
Title: Who's (Still) Above the Social Security Payroll Tax Cap? | {
"redpajama_set_name": "RedPajamaC4"
} | 782 |
Yes, you read it right—because knowing what not to do can be a surprisingly positive approach to successful Customer Relationship Management.
CRM weaves a vital connecting grid integrating all aspects of your business. From marketing and sales to service and finance, it serves as the central repository for information. Identifying and avoiding common weak areas is an essential part of building a strong process.
1. Never keep contacts in email address books or spreadsheets.
Clients prefer to do business in a place 'where everybody knows your name.' But your CRM is much more than a listing of customer names and phone numbers. It provides a history and analysis of interactions with all past, present and prospective customers and vendors. By tracking personalities, preferences and most-valued attributes, relationship-building and sales opportunity information is readily available in real time for all functions across your business.
Such data not only needs to be accessible to all, but also controlled in how it's used.
Only a carefully created and monitored singular CRM can be your company's reliable 'source of truth.' No spreadsheet can offer that.
2. Avoid entries in multiple locations.
This is one of those more-is-not-better scenarios. Every employee across your company should have confidence that the client information and history they're accessing is correct and up to date. If there's more than one source for data entry and retrieval, there's obvious room for error and inconsistency. With a properly developed and maintained CRM, there's no need for wasted time and effort on any additional client data sources.
3. Don't overlook a very important process for VIPs.
As intuitive as it might sound, remember to have a process for noting VIP status in company contact data. All customer contact is important, but let's face it, if your client's CEO calls in, you want to make sure everyone from your receptionist to service team is aware and can respond at the appropriate level.
4. Do not forget to link CRM and operations.
It only makes sense for CRM data to effectively feed into all areas of business operations including administrative tasks, contracts, billing, reporting, finance and accounting. In addition to the most current contact info for such tasks as invoicing, CRM data is valuable in quoting and reviewing sales opportunities.
As the single source for company and contact data, your CRM is a critical part of all company functions. As such, a team approach works best in keeping it 'healthy.' Every colleague should take personal responsibility for updating, maintaining, and using the CRM data. Make data entry easy and use keywords or statuses to identify information gaps for follow-up. Make sure information is verified and updated regularly.
Your CRM plays an overarching role in all aspects of your business. Taking time to mistake-proof the process will help ensure it's a productive partner in your business.
See a Powerful CRM in Action!
Find out even more ways a CRM (if used effectively) can help you increase efficiency and throughput in your business. Sign up for a ConnectWise Manage CRM demo! | {
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} | 2,691 |
Q: Firefox Extension Content Script Invocation/Lifecycle I created a simple firefox add-on with the add-on SDK as below, but it only ever works once. I am using version 1.8 of the Add-On SDK.
$ mkdir test_ext
$ cd test_ext
$ cfx init
$ cat >package.json
{
"name": "no2",
"license": "MPL 2.0",
"author": "",
"version": "0.1",
"fullName": "test_ext",
"id": "my-id",
"description": "a basic add-on"
}
$ cat >lib/main.js
<!-- language: lang-js -->
require("page-mod").PageMod({
include: ["*.org"],
contentScript: 'document.body.innerHTML = ' +
'"<h1>this page has been eaten</h1>";'
});
$ cfx run
Now in the browser if I go to w3.org it works. If I reload or go to a different .org domain like wwf.org nothing happens. How can I make it test and apply any new page?
A: The way I would write this is:
require("page-mod").PageMod({
include: ["*.co.uk"],
contentScript: 'document.body.innerHTML = ' +
'"<h1>this page has been eaten</h1>";'
});
I think the problem with your code is the use of the 'new' keyword?
| {
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} | 8,574 |
{"url":"https:\/\/math.stackexchange.com\/questions\/2556618\/show-that-if-xn-is-a-bounded-sequence-then-xn-converges-if-and-only-if-lim-sup","text":"Show that if xn is a bounded sequence, then xn converges if and only if lim sup(xn) = lim inf (xn).\n\nI know that limit inferior describes the smallest number, v, that a subsequence can converge to, and I know that limit superior describes the largest number, u, that a subsequence can converge to.\n\nSo if lim inf = lim sup, then all subsequences must converge to the same number, and if all convergent subsequences converge to a limit, L, then the sequence itself converges to a limit L. Is this correct thinking?\n\nFor the converse, would I say that if a sequence converges to a limit, L, then any subsequence converges to L, and by proxy, if all subsequences converge to L, lim sup = lim inf?\n\nLooks good. For the other direction, you might bring into play the fact that there are subsequences of $\\{x_n\\}$ that converge to $\\limsup x_n$ and $\\liminf x_n$ respectively. You also don't need the boundedness hypothesis if you define convergence of a sequence $\\{x_n\\}$ to $\\pm\\infty$ in the natural way.","date":"2021-10-20 22:18:38","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\": 0.9929077625274658, \"perplexity\": 81.4775326414222}, \"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-2021-43\/segments\/1634323585353.52\/warc\/CC-MAIN-20211020214358-20211021004358-00469.warc.gz\"}"} | null | null |
Reading: All About Jason Kidd's Son: Trey Jason Kidd
All About Jason Kidd's Son: Trey Jason Kidd
Trey American pop culture icon Jason Kidd is a young man. Trey Jason Kidd is well-known for being the son of Jason Kidd, a famous former American basketball player, and coach. Jason is one of the nation's favorite American basketball players. The Dallas Mavericks, the most well-known team in America, are coached by Trey's dad, Jason.
Some Quick Fcts about KiddTrey Jason Kidd's CareerWho is Jason Kidd, Trey's Father?Jason Kidd's Childhood and CareerEarly Years:Professional Experience:Joumana, Trey Jason Kidd's MotherRelationships and Affairs of TreyFinancial Status and Net Worth
Trey is from a respectable American household. Trey Jason Kidd is not a moderate extrovert who thrives on fame and the spotlight. Trey avoids attention and maintains a low profile while providing excessive personal information online. In his early years, Trey Jason Kidd benefited from the genuine love and support of his father's fans. Trey is a podcast host and a student at the moment.
Some Quick Fcts about Kidd
Full Name: Trey Jason Kidd
Known as: TJ Kidd
Birthplace: United States
Residence: Los Angeles, California, United States
Mother's name: Joumana Marie Samaha
Father's name: Jason Kidd
Siblings: Jazelle Kidd and Miah Kidd
Education: Bergen Catholic High School (New Jersey, United States)
Height: 1.87 m (6 feet 2 inches)
Profession: Basketball coach, Student, Blogger, Podcast host
Popular as: Jason Kidd's son
Ethnicity: Mixed (African American)
Trey Jason Kidd's Career
Trey recently graduated from high school, so he is still a young man. As far as we can tell from his most recent interviews, he is currently a university student in Los Angeles, California. Surprisingly, TJ is active in various extracurricular activities, which is unusual for someone his age. As soon as he entered college, he began mentoring high school students.
At Beverly Hills High School in California, he had previously served as an assistant coach for the varsity squad. After that, he was named the same organization's high school basketball team's head coach. He did, however, stop coaching after working there for around two years. He also engages in various other activities, such as blogging & producing podcasts. You can see him discuss NBA happenings on the 360 Degree Podcast YouTube account.
Who is Jason Kidd, Trey's Father?
Jason Kidd is a former basketball player & the current head coach of the Dallas Mavericks of the National Basketball Association (NBA). Kidd has been selected to the NBA All-Star team 10 times, five times made the All-NBA First Team, and nine times made the NBA All-Defensive Team. Jason Kidd is one of the league's top point guards and passers.
In 2011, while playing for the Dallas Mavericks, he captured the NBA title. Along with the American national team, he also caught two Olympic gold medals in 2000 and 2008. The Naismith Memorial Basketball Hall of Fame accepted him as a player. One of the greatest players in league history, Senior Kidd was chosen for the NBA 75th Anniversary Team in October 2021.
Jason Kidd's Childhood and Career
Early Years:
Kidd was raised in upper-middle-class Oakland after being born in San Francisco. His mother, Anne, is Irish-American, and Steve is an African-American. As a youngster, Kidd received numerous all-star and MVP awards while highly recruited for AAU teams and competitions. He played on Oakland's municipal courts, frequently facing Gary Payton, a future NBA Hall of Famer, and attended the East Oakland Youth Development Center.
Professional Experience:
The Dallas Mavericks chose Kidd as their second overall pick, ahead of Grant Hill of Duke and just behind Purdue's Glenn Robinson. He shared the 1995 NBA First of the Year award with Hill of the Detroit Pistons, who also averaged 11.7 points, 5.4 rebounds, and 7.7 assists in his rookie season, leading the NBA in triple-doubles. The Mavericks had the worst record in the NBA when the season ended the year before Kidd was selected, going 13-69. After Kidd's first season with the franchise, the Mavericks' record improved to 36-46, the most significant improvement in the NBA by 23 games.
Joumana, Trey Jason Kidd's Mother
Joumana, TJ's mother, was born in Foster City, California, on September 28, 1972. She attended high school in Foster City before relocating to San Francisco to pursue her post-secondary education. She eventually earned a Bachelor of Communications from San Francisco State University and started working as a writer. However, she married Jason Kidd, TJ's father, shortly after.
After having children, she stopped working but returned to the workforce, this time as a television host for the program Extra. That was the last time she looked back. She did a great job presiding over the Grammy and Rock n Roll Hall of Fame ceremonies. In 2010, Joumana participated in the VH1 reality series Let's Talk About Pep. She did not get married after her split from Kidd, and there is no information on her current mate. She is probably not married.
Relationships and Affairs of Trey
Trey American citizen Jason Kidd is currently single. Trey When it comes to his romantic situation, Jason Kidd is reticent. Due to his disconnected personality, the basketball coach and podcaster have kept his love life a secret. It isn't easy to show proof of Jason Kidd's present relationship because he has never been identified as dating anywhere.
Jason Kidd might be seeing someone else and waiting to tell anyone until the time is right. Jason Kidd was still preoccupied with his career. Therefore he couldn't possibly have the time for such pursuits. Despite the lack of knowledge, Trey Jason Kidd might need to be coupled.
Financial Status and Net Worth
Trey Blogger and podcaster Jason Kidd is American. Because he is the son of the most well-known basketball coach, Jason Kidd, Trey is currently a student who is interested in basketball. At Beverly Hills High School in California, Trey previously served as an assistant trainer for the varsity team. Trey Jason then took over as the squad's head coach for the high academy basketball team at the same institution.
But Trey's current net worth needs to be clarified. His famous father, Jason Kidd, has a net worth of about $10 million. He earns a sizable income in his dual roles as an active basketball coach and former American basketball player. It is safe to say that Jason Kidd has led a high-quality existence.
Also read: Delonte West's Wife Caressa Suzzette Madden: Know All About Her
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Legends League Cricket 2023: Gayle, Pathan & Panesar Confirm Participation in LLC 2023
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Laura Harrier, Klay Thompson's Girlfriend, Affairs and Wealth
Otto Porter: Girlfriend, Net Worth, & NBA
Bio of Emily Vakos: College, Husband, Net Worth, and Wedding | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,877 |
Michigan Travel Site Exploring Breweries, Waterfalls, Lighthouses, and Family Friendly Vacations
Search Travel the Mitten
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Browse: Home / Arts and Entertainment / Michigan A to Z: Travel the Mitten's Summer Guide
Michigan A to Z: Travel the Mitten's Summer Guide
May 27, 2016 · by jonkatje · in Arts and Entertainment, Family Trips, Sports
Michigan is all set for a hot summer packed full of festivals, events, sports and more. Fairs, festivals, races, and can't-miss vacation spots are all highlighted in our summer guide. From Detroit and Grand Rapids to the far reaches of the Upper Peninsula, there's something for everyone on this list and there's a good chance at least one is close to where you live. If you're looking for an idea for vacation or an add-on event to plans already on the calendar, check out the list below for our A to Z look at June, July, and August events in Michigan.
http://www.gaylordalpenfest.com/
Alpenfest – From July 12 – 16, downtown Gaylord is transformed into the Alpenstrasse, home to more than 60 vendors with crafts, food an more. Other events include a carnival, a parade, the Queen's Pageant, live music, and much more. Find out all the details at: http://www.gaylordalpenfest.com/, where you'll also find a full schedule of events.
tallshipcelebration.com
Bay City Tall Ship Celebration – From July 14-17, vessels including the Appledore IV (Gaff Schooner), Appledore V (Gaff Schooner), S/V Denis Sullivan (3-masted Schooner), Draken Harald Harfagre (Viking Longship), El Galeon Andalucia (Spanish Galleon), Pride of Baltimore II (Topsail Schooner) and more will make port in Bay City for the Tall Ship Celebration. There are events for the entire family, including ship tours and excursions, concerts, a street party, "Ballads and Brews," art, crafts for kids, games, and more. Find a full schedule of events and more details on the Tall Ships Challenge at http://www.tallshipcelebration.com/#intro.
Coast Guard Festival, Grand Haven – West Michigan's premier summer event is Grand Haven's Coast Guard Festival, which is a weeklong celebration that features parades, fireworks, concerts, food, children's activities, a car show, a carnival, ship tours, and more. This year's festival runs from July 29 to August 7 and will have three US Coast Guard vessels in port as Coast Guard City USA honors those who have served and those who continue to serve. Check out an events schedule and all the other necessary information: http://www.coastguardfest.org/.
Detroit Lions Summer Camps and Training Camp – Beginning in early June the Detroit Lions will be hosting youth summer football camps in cities across Michigan. You can find a full schedule of events at http://www.detroitlions.com/youth-programs/summer-camps-schedule.html. Academies focus on specific positions and are more advanced, while other events teach basic skills in a relaxed and fun environment. In August, select training camp sessions are open to the public and offer a chance to meet players and get autographs. The Lions' training facility is located in Allen Park at 333 Republic Drive. Family Day is always a popular event with plenty of events and mascot Roary on hand for photos with kids' club members. Keep checking www.detroitlions.com for details as they are posted.
Engineers Day at the Soo Locks – One day each year, visitors to the Soo Locks are allowed up-close access to large freighters and the chance to walk across the lock walls. This year's event takes place on June 24, and the Cloverland Hydroelectric Power Plant will be open for tours as well. The International Bridge Walk takes place the following day, making for a great weekend full of events in Sault Ste. Marie. More information is available at http://www.saultstemarie.com/soo-locks-engineers-day-june-24-2016/.
Fayette Historic State Park – One of the hidden gems of Michigan's state park system is Fayette Historic State Park in the Upper Peninsula, a restored "living museum" showing what life was like in an industrial community in the late 1800s. Two special events this summer offer extra fun for visitors: Christmas in June takes place on June 18th and will feature outdoor games, children's projects, food and more while Fayette Heritage Day takes place on August 13th and features period displays, food, music, and a vintage baseball game.
Gilmore Car Museum – One of Michigan's best car museums can be found in Hickory Corners, where more than 300 vehicles are displayed in vintage buildings. Summer events this year include a vintsage motorcycle show (June 12), an antique tractor, engine and machinery show (June 24, 25, 26), Deutsche Marques: A German Auto Event (July 9), Mad Dogs & Englishmen's British Auto Faire XXVI (July 10), and the Relix Riot Show & Swap Meet (August 20). Find out more about this awesome automobile attraction just north of Kalamazoo: http://www.gilmorecarmuseum.org/.
http://www.hastingssummerfest.com/
Hastings Summerfest – This annual festival in Barry County features something for everyone in the family, and draws thousands of visitors to downtown Hastings each year. The 2016 Summerfest takes place August 26-28, and "the event features arts & crafts vendors on the courthouse lawn, concessions, free trolley rides, children's activities, a softball tournament, 3-on-3 basketball, 10K/5K run, 3-on-3 soccer, roller hockey tournament, weight lifting contest, parade, car show, soapbox derby, live entertainment and a refreshment tent." http://www.hastingssummerfest.com/
insanesinflatable5k.com
Insane Inflatable 5K – This unique and exciting event gets away from the timed competition of a normal 5K and instead focuses on fun. Inflatable obstacles like stairs, slides, wrecking balls, bounces and more are mixed in to the race course. This summer will see events take place in Kalamazoo (July 9 at Gilmore Car Museum), Lansing (July 30 at Hope Sports Complex), Detroit (July 23 at David Shepherd Park), and Grand Rapids (August 20 at Millennium Park). Find out more at http://insaneinflatable5k.com/locations/.
http://www.experiencejackson.com/
Jackson Events – The city of Jackson is where it's at for summer festivals and fun. Check out the Bluesfest (June 2-4) http://jacksonbluesfest.com/, ART BEER WINE 2016 at the Ella Sharp Museum (June 4) http://ellasharpmuseum.org/?page_id=886, the Jackson County Rose Festival Parade & Picnic in the Park (June 5) http://jacksonrosefestival.org/, the Michigan Shakespeare Festival (July 9-24) http://www.michiganshakespearefestival.com/, the Hot Air Jubille (July 15-17) http://www.hotairjubilee.com/, and How Bazaar Jackson (July 30-31) http://www.howbazaarjackson.com/.
Kid's Fishing Tournaments at MI State Parks – June 11-12 is a Free Fishing weekend, and several state parks will be hosting kids' fishing tournaments. Check out events at Young State Park, the Cheboygan Field Office, Bald Mountain Recreation Area, Bewabic State Park, Bay City State Recreation Area to take advantage of this great opportunity to introduce a younger generation to Michigan's amazing fishing opportunities.
Lakenenland – You won't find another roadside attraction in Michigan quite like the Lakenenland Junkyard Art Sculpture Park. Easy to find on M-28 near Marquette, this free attraction features more than 50 scrap iron sculptures made by Tom Lakenen. Check out our Photo Gallery Friday: Lakenenland post for more pictures, and visit http://lakenenland.com/ for directions and more information.
Meteors & S'mores at Michigan State Parks – Michigan will be treated to a Perseid meteor shower in early August, and several state parks are hosting "Meteors & S'mores" events. Head out to Bald Mountain Recreation Area (August 13), Clear Lake State Park (August 12), Fort Wilkins Historic State Park (August 10-11), Holland State Park (August 13), Leelanau State Park (August 10) for a night full of campfires and stargazing.
National Trails Day – June 4th is National Trails Day for 2016, and Michigan offers many options for anyone looking to get out and celebrate. The Chief Noonday Chapter, Grand Traverse Hiking Club and the Spirit of the Woods chapters of the North Country Trail Association are all hosting events, or you can check out the Petoskey Trail Town Celebration or the Grasslands Birds Hike hosted by Oakland County Parks. You can find other ideas by checking out our Trails tab, or check out https://northcountrytrail.org/ or http://www.michigantrails.org/ for more ideas and maps.
Ontonagon Historical Museum and Lighthouse Tours – The Ontonagon Historical Museum features a collection of memorabilia detailing the area's history of logging, mining, farming and social events. The museum also coordinates tours to the historic Ontonagon lighthouse three times a day (Monday-Saturday) from May to September. This is the only way to gain access to the lighthouse grounds and climb the tower for stunning views of Lake Superior and the city and harbor. The lighthouse was built in 1866 and is a brick building in the schoolhouse style. Find out more about the museum and tours at: http://www.ontonagonmuseum.org/lighthouse1.htm. This makes for a great stop if you're in the area to visit the Porcupine Mountains or if you're headed north to the Keweenaw Peninsula.
Petoskey Rocks! – Friday nights in one of Northern Michigan's favorite destination towns will be extra fun this summer thanks to the addition of live music, movies, and more. You can find a full listing of music and movies at: http://www.northernmichiganeventscalendar.com. While in the area be sure to check out some of our favorite spots: Petoskey Brewing, Beards Brewery, and Mackinaw Trail Winery and Brewing Company. You can also take a walking tour of places from author Ernest Hemingway's time in the area.
www.visitludington.com
Quilt Barns of Mason County – There are eleven barns in the Ludington area that feature 8′ x 8′ squares colorfully painted to look like quilt patterns. Much like the barns we profiled on the Old Mission Peninsula, many of these barns are more than 100 years old and an important part of the area's farming and cultural heritage. Find a full list and addresses: www.pureludington.com. Also consider checking out Mason County's Agricultural Trail, Lumber Heritage Trail, and Sculpture Trail.
www.saultstemarie.com
Rendezvous in the Sault – If you've ever wondered what life was like in the late 1600s when Michigan's oldest city was founded, head to Sault Ste. Marie July 30-31 to experience Rendezvous in the Sault. A $10 admission gets families in for both days, where storytellers, musicians, period authentic food, colonial games, historic re-enactors, guided tours of the area's historic homes, and more will keep everyone entertained. Visit www.historicwaterstreet.org for more information.
Silver Lake State Park – If you're looking for sand dunes this summer, consider heading to Silver Lake State Park, where there's more than 2,000 acres of dunes to explore and even ride on/over. There is a campground with 200 modern sites, and an ORV area that is the only place east of Utah where a private vehicle can be driven in a dune area. If you don't want to drive it yourself, another section of the park is rented to Mac Woods Dune Rides. A southern section of the park features a day use beach and the historic Little Sable Point Lighthouse, which can be toured during summer months.
Torch Lake – You may not want to head there for the party on July 4th, but consider a summer trip to Michigan's longest and second largest inland lake. Well known for its turquoise colored water and massive sandbar, Torch Lake is popular for swimming, kayaking, boating, and fishing. It's also just a short drive from Traverse City, and pretty close to Short's Brewing Company.
www.upstatefair.org
Upper Peninsula State Fair – Michigan's only state fair (since 2010 when the downstate Michigan State Fair was discontinued) takes place in Escanaba, August 15-21. Visitors can check out exhibits, experience Skerbeck Entertainment Group rides, munch on fair food, midway entertainment, concerts and more. Head over to http://www.upstatefair.org/ for more information, and be sure to take advantage of special events for Family Day and Ladies' Day.
www.frankenmuth.org
Volkslaufe – Consider heading to Frankenmuth on July 4th weekend to take part in "the people's race" – the annual Volkslaufe. Featuring 5K, 10K, 20K and 2K (fun run) races, this family friendly event begins and ends at heritage park and features many nods to the area's rich German heritage. The race route runs through historic Frankenmuth and crosses the Holz Brucke covered bridge. Head over to http://volkslaufe.runfrankenmuth.org/ for more information, and also consider checking out Bronner's CHRISTmas Wonderland (World's Largest Christmas store), Zehnder's restaurant and Splash Village, and the famous Bavarian Inn.
West Michigan Whitecaps Tiger Fridays – The Detroit Tigers' Class A affiliate in Grand Rapids will bring in several former Tigers players this season as part of the ever-popular Tiger Fridays. Appearances include: Jack Billingham on June 10 (Michigan and Trumbull beach towel giveaway), Joel Zumaya on July 8 (bobblehead giveaway), Frank Tanana on July 22 (oven mitt giveaway), Duke Sims on August 5 (cooking apron giveaway), Mickey Lolich on August 12 (bobblehead giveaway), Vince Coleman on August 19 (mini base giveaway), and Lloyd Moseby on September 2 (bobblehead giveaway). Tickets are more information at: www.whitecapsbaseball.com.
www.thextremeexperience.com
Xtreme Xperience at M1 Concourse – The Detroit area's newest motorsports facility features a 1.5-mile track and more, and in August it will host the Xtreme Xperience. You can experience the 10 turns and 2,000 foot straightaway of the course while driving dream cars like the Ferrari 458 Italia, Ferrari F430, Lamborghini Huracan LP610-4, Nissan GT-R, Porsche 911 GT3, McLaren 570S, or the Lamborghini LP560-4. TO find out more about this awesome experience with "Real Supercars. Real Racetracks. Real Instruction.," head to http://www.thextremexperience.com/tracks/m1-concourse/ and keep an eye on social media for potential special offers and discounts.
Yoopers Ride Free Days – Upper Peninsula residents can experience the Pictured Rocks Cruises for free on June 11 or 12 by showing a valid photo ID with their UP address – limited to the first 1,500 riders each day. The Alger County Chamber of Commerce will also be hosting Pictured Rocks Days that weekend, with vendors, food, free music, kids activities and more at Bayshore Park in Munising. Find out more at the Alger County Chamber of Commerce website.
Zoo celebrates 125 years – Grand Rapids' John Ball Zoo will continue their celebration of 125 years of operation into the summer, with a handful of special events planned. Be sure to check out the tigers, lions and bears as well as Sting Ray Lagoon. RendeZoo (a casual, outdoor zoo party and fundraiser) takes place on June 10 and the public celebration week is August 15 to August 19. Find out more at: http://www.jbzoo.org/125/events.
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Photo Gallery Friday: Van Riper State Park in Marquette County
Things To Do In Michigan Summer 2018 – A to Z Guide
12 Things to See and Do at Belle Isle Park in Detroit, Michigan
Tags: A to Z, Bay City Tall Ships, Beach, Festival, Hiking, John Ball Zoo, Lighthouse, Michigan Summer, Michigan Vacation, State Park, UP State Fair
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\section{Introduction}
In this paper, we focus on the integral of the following form:
\begin{equation}\label{MixedIntegral}
f_{\Gamma}(z)=\int_\Gamma e^{h_{0}(x)}h_{1}(x)^{-\gamma_1}\cdots h_{k}(x)^{-\gamma_k}x^{c}\omega,
\end{equation}
where $h_l(x;z)=h_{l,z^{(l)}}(x)=\sum_{j=1}^{N_l}z_j^{(l)}x^{{\bf a}^{(l)}(j)}$ ($l=0,\dots,k$) are Laurent polynomials in torus variables $x=(x_1,\dots,x_n)$, $\gamma_l\in\C$ and $c={}^t(c_1,\dots,c_n)\in\C^{n\times 1}$ are parameters, $x^c=x_1^{c_1}\dots x_n^{c_n}$, $\Gamma$ is a suitable integration cycle, and $\omega$ is an algebraic relative $n$-form which has poles along $D=\{ x\in\Gm^n\mid x_1\dots x_n h_1(x)\dots h_k(x)=0\}$. As a function of the independent variable $z=(z_j^{(l)})_{j,l}$, the integral (\ref{MixedIntegral}) defines a hypergeometric function. We call the integral (\ref{MixedIntegral}) the Euler-Laplace integral representation.
We can naturally define the twisted de Rham cohomology group associated to the Euler-Laplace integral (\ref{MixedIntegral}). We set $N=N_0+\dots+N_k$ $\Gm^n={\rm Spec}\;\C[x_1^\pm,\dots,x_n^\pm]$, and $\A^N={\rm Spec}\;\C[z_j^{(l)}]$. For any $z\in\A^N$, we can define an integrable connection $\nabla_z=d_x+d_xh_0\wedge-\sum_{l=1}^k\gamma_l\frac{d_xh_l}{h_l}\wedge+\sum_{i=1}^nc_i\frac{dx_i}{x_i}\wedge:\mathcal{O}_{\Gm^n}(*D)\rightarrow\Omega^1_{\Gm^n}(*D)$. Setting $U=\Gm^n\setminus D$, the algebraic de Rham cohomology group $\Homo_{\rm dR}^*\left( U;(\mathcal{O}_U,\nabla_z)\right)$ is defined as the hypercohomology group $\mathbb{H}^*\left( \Gm^n;(\cdots\overset{\nabla_z}{\rightarrow}\Omega_{\Gm^n}^\bullet(*D)\overset{\nabla_z}{\rightarrow}\cdots)\right)$. Under a genericity assumption on the parameters $\gamma_l$ and $c$, we have the vanishing result $\Homo_{\rm dR}^m\left( U;(\mathcal{O}_U,\nabla_z)\right)=0$ $(m\neq n)$. Moreover, we can define a perfect pairing $\langle\bullet,\bullet\rangle_{ch}:\Homo_{\rm dR}^n\left( U;(\mathcal{O}_U,\nabla_z)\right)\times \Homo_{\rm dR}^{n}\left( U;(\mathcal{O}_U^\vee,\nabla_z^\vee)\right)\rightarrow\C$ which is called the cohomology intersection form (see (\ref{RDc.i.n.}) and the proof of \cref{thm:QuadraticRelation}). The main result of this paper (\cref{thm:QuadraticRelation}) is on the explicit formula of the cohomology intersection number.
In order to extract the information of the cohomology intersection number from (\ref{MixedIntegral}), it is important to observe that (\ref{MixedIntegral}) satisfies a special holonomic system called GKZ system introduced by I.M.Gelfand, M.I.Graev, M.M.Kapranov, and A.V.Zelevinsky (\cite{GelfandGraevZelevinsky}, \cite{GKZToral}). Let us recall the definition of GKZ system. The system is determined by two inputs: an $d\times n$ ($d<n$) integer matrix $A=({\bf a}(1)|\cdots|{\bf a}(n))$ and a parameter vector $\delta\in\C^{d}$. GKZ system $M_A(\delta)$ is defined by
\begin{subnumcases}{M_A(\delta):}
E_i\cdot f(z)=0 &($i=1,\dots, d$)\label{EulerEq}\\
\Box_u\cdot f(z)\hspace{-0.8mm}=0& $\left(u={}^t(u_1,\dots,u_{n})\in L_A=\Ker(A\times:\Z^{n\times 1}\rightarrow\Z^{d\times 1})\right)$,\label{ultrahyperbolic}
\end{subnumcases}
where $E_i$ and $\Box_u$ are differential operators defined by
\begin{equation}\label{HGOperators}
E_i=\sum_{j=1}^{n}a_{ij}z_j\frac{\partial}{\partial z_j}+\delta_i,\;\;\;
\Box_u=\prod_{u_j>0}\left(\frac{\partial}{\partial z_j}\right)^{u_j}-\prod_{u_j<0}\left(\frac{\partial}{\partial z_j}\right)^{-u_j}.
\end{equation}
\noindent
Throughout this paper, we assume an additional condition $\Z A\overset{def}{=}\Z{\bf a}(1)+\dots+\Z{\bf a}(n)=\Z^{d\times 1}$. Denoting by $\DD_{\A^n}$ the Weyl algebra on $\A^n$ and by $H_A(\delta)$ the left ideal of $\DD_{\A^n}$ generated by operators (\ref{HGOperators}), we also call the left $\DD_{\A^n}$-module $M_A(\delta)=\DD_{\A^n}/H_A(\delta)$ GKZ system. The fundamental property of GKZ system $M_A(\delta)$ is that it is always holonomic (\cite{Adolphson}), which implies that the stalk of the sheaf of holomorphic solutions at a generic point is finite dimensional.
In our case, we should set $d=k+n$, $n=N$, $\delta=(\gamma_1,\dots,\gamma_k,c)$, and
\begin{equation}\label{IntroCayley}
A
=
\left(
\begin{array}{ccc|ccc|ccc|c|ccc}
0&\cdots&0&1&\cdots&1&0&\cdots&0&\cdots&0&\cdots&0\\
\hline
0&\cdots&0&0&\cdots&0&1&\cdots&1&\cdots&0&\cdots&0\\
\hline
&\vdots& & &\vdots& & &\vdots& &\ddots& &\vdots& \\
\hline
0&\cdots&0&0&\cdots&0&0&\cdots&0&\cdots&1&\cdots&1\\
\hline
&A_0& & &A_1& & &A_2& &\cdots & &A_k&
\end{array}
\right),
\end{equation}
\noindent
where $A_l=\left({\bf a}^{(l)}(1)|\dots|{\bf a}^{(l)}(N_l)\right)$. The matrix (\ref{IntroCayley}) is a variant of the Cayley configuration (\cite{GKZEuler}). It is proved in \cref{thm:mainDresult} that (\ref{MixedIntegral}) is a solution of $M_A(\delta)$. Therefore, we expect that the theory of GKZ system tells us some information of the integral (\ref{MixedIntegral}).
An important combinatorial structure of GKZ system discovered by Gelfand, Kapranov and Zelevinsky is the secondary fan (\cite[Chapter 7]{GKZbook}, \cite{DeLoeraRambauSantos}). If we denote by $L_A^\vee$ the dual lattice of $L_A$, the secondary fan ${\rm Fan}(A)$ is a special (possibly incomplete) fan in $L_A^\vee\otimes_{\Z} \R$. Moreover, each cone of the secondary fan has a combinatorial interpretation as a convex polyhedral subdivision of $\R_{>0}A$, the positive cone spanned by the column vectors of the matrix $A$. Any triangulation $T$ of $\R_{>0}A$ arising in this way is called a regular triangulation. It was an important discovery of Gelfand-Kapranov-Zelevinsky which was later sophisticated by M.-C. Fern\'andez-Fern\'andez that $T$ can be interpreted as a set of independent solutions of the GKZ system $M_A(\delta)$. Namely, for each simplex $\s\in T$, we can associate a finite Abelian group $G_{\s}$ and hypergeoemtric series $\{ \varphi_{\s,g}(z;\delta)\}_{g\in G_{\s}}$ which are solutions of $M_A(\delta)$. Though these series $\varphi_{\s,g}(z;\delta)$ may diverge, there is at least one regular triangulation $T$ such that $\varphi_{\s,g}(z;\delta)$ is convergent for any simplex $\s\in T$ and $g\in G_{\s}$. We say such a regular triangulation $T$ is convergent in this paper. Then, it is known that the set $\Phi_T=\bigcup_{\s\in T}\{\varphi_{\s,g}(z;\delta)\}_{g\in G_{\s}}$ is a basis of solutions of $M_A(\delta)$ (\cite{FernandezFernandez}). Geometrically speaking, the secondary fan ${\rm Fan}(A)$ defines a toric variety $X({\rm Fan}(A))$ which contains the quotient torus $(\C^*)^N/j_A(\C^*)^{n+k}$ where $j_A$ is defined by $j_A(t)=t^A$ for any $t\in(\C^*)^{n+k}$. Any series $\varphi_{\s,g}(z;\delta)$ can be interpreted as a local solution near the torus fixed point $z_T^\infty$ of the toric variety $X({\rm Fan}(A))$ corresponding to $T$.
In the main statement of this paper, we focus on a particular class of the integral (\ref{MixedIntegral}) that the corresponding GKZ system $M_A(\delta)$ admits a unimodular triangulation. A regular triangulation $T$ is said to be unimodular if for any simplex $\s\in T$, the corresponding Abelian group $G_{\s}$ is trivial. The study of the matrix $A$ admitting a (special) unimodular triangulation is an active area of research in combinatorics (\cite{SantosZiegler}, \cite{HibiOhsugiRootConfiguration}, \cite{HibiOhsugiCentrallySymmetric}). In relation to special functions, the most famous example is Aomoto-Gelfand system (\cite{AomotoKita}, \cite{GelfandGeneralTheory}). In our notation, this corresponds to the case that $h_0\equiv 0$ and other Laurent polynomials $h_l$ are all linear polynomials. Another important class is the configuration matrix $A$ associated to a reflexive polytope (\cite{Batyrev}, \cite{StienstraMirror}, \cite{NarumiyaShiga}). This class of GKZ system has been studied in the context of toric mirror symmetry. These classes define regular holonomic GKZ systems (\cite{Hotta}), and have Euler integral representations, i.e., we can take $h_0(x)\equiv 0$. However, they have other integral representations with non-zero exponential factor $h_0(x)$. Moreover, there are various examples of (\ref{MixedIntegral}) admitting a unimodular triangulation and satisfying an irregular GKZ system. An important class is a ``confluence'' of Aomoto-Gelfand system (\cite{KimuraHaraokaTakano}, \cite{MarkovTarasovVarchenko}) which is discussed in \S\ref{E(21...1)}. The main result of this paer is an explicit formula of the cohomology intersection number in terms
of series solutions associated to regular triangulations.
\begin{thm}\label{thm:IntroTheorem}
Suppose that four vectors ${\bf a},{\bf a}^\prime\in\Z^{n\times 1},{\bf b},{\bf b}^\prime\in\Z^{k\times 1}$ and a convergent unimodular regular triangulation T are given. Under the assumption on the parameters $\gamma_l$ and $c$ of \cref{thm:QuadraticRelation}, one has an identity
\begin{align}
&(-1)^{|{\bf b}|+|{\bf b}^\prime|}\gamma_1\cdots\gamma_k(\gamma-{\bf b})_{\bf b}(-\gamma-{\bf b}^\prime)_{{\bf b}^\prime}
\sum_{\s\in T}\frac{\pi^{n+k}}{\sin\pi A_\s^{-1}\delta}\varphi_{\s,0}\left(z;
\begin{pmatrix}
\gamma-{\bf b}\\
c+{\bf a}
\end{pmatrix}
\right)\varphi_{\s,0}^\vee\left(z;
\begin{pmatrix}
\gamma+{\bf b}^\prime\\
c-{\bf a}^\prime
\end{pmatrix}
\right)\nonumber\\
=&\frac{\langle x^{\bf a}h^{\bf b}\frac{dx}{x},x^{{\bf a}^\prime}h^{{\bf b}^\prime}\frac{dx}{x}\rangle_{ch}}{(2\pi\ii)^n}\label{IntroFormula}
\end{align}
when $z$ is in the non-empty open set $U_T$.
\end{thm}
\noindent
In the theorem above, we put $(\gamma-{\bf b})_{\bf b}=\prod_{l=1}^k(\gamma_l-{b}_l)_{ b_l}=\prod_{l=1}^k\frac{\Gamma(\gamma_l)}{\Gamma(\gamma_l-b_l)}$ and $x^{\bf a}h^{\bf b}=x_1^{a_1}\dots x_n^{a_n}h_1^{b_1}\dots h_k^{b_k}$ if ${\bf a}={}^t(a_1,\dots,a_n)$ and ${\bf b}={}^t(b_1,\dots,b_k)$. The symbol $\sin\pi A_\s^{-1}\delta$ denotes the product of values of sine functions at each entry of the vector $\pi A_\s^{-1}\delta$. $U_T$ is an open neighbourhood of the point $z_T^\infty$. The formula (\ref{IntroFormula}) gives a convergent Laurent series expansion of the cohomology intersection number $\langle x^{\bf a}h^{\bf b}\frac{dx}{x},x^{{\bf a}^\prime}h^{{\bf b}^\prime}\frac{dx}{x}\rangle_{ch}$. Note that cohomology classes of the form $[x^{\bf a}h^{\bf b}\frac{dx}{x}]$ generate the algebraic de Rham cohomology groups $\Homo_{\rm dR}^*\left( U;(\mathcal{O}_U,\nabla_z)\right)$ and its dual $\Homo_{\rm dR}^*\left( U;(\mathcal{O}_U^\vee,\nabla_z^\vee)\right)$ (\cite{HibiTakayamaNishiyama}).
The derivation of the formula (\ref{IntroFormula}) is based on the twisted analogue of Riemann-Hodge bilinear relation, commonly referred to as the twisted period relation. The twisted period relation expresses the cohomology intersection number in terms of the homology intersection number and period integrals. The essential part of this paper \S\ref{SectionEuler} and \S\ref{IntersectionNumbers} provides with a concrete method of constructing a good basis of the rapid decay homology group associated to the Euler-Laplace integral (\ref{MixedIntegral}). In particular, we have closed formulas of the homology intersection matrix and the period integrals. Behind the construction, we fully make use of the combinatorial structure of GKZ system. In the rest of the introduction, we explain the background of our study.
\subsection{The role of twisted (co)homology intersection numbers}
It seems that the interest on computing (co)homology intersection numbers is confined in a small community of specialists. Therefore, it is reasonable to recall the importance of computing the exact formula of (co)homology intersection numbers.
It was discovered in \cite{ChoMatsumoto} that a family of functional identities of hypergeometric functions called quadratic relations can be derived in a systematic way from the twisted version of Riemann-Hodge bilinear relation. This relation is a compatibility among cohomology intersection numbers, homology intersection numbers, and twisted period pairings. Therefore, the computation of cohomology intersection numbers can be used to derive quadratic relations. K.Cho and K.Matsumoto developed a method of evaluating cohomology intersection numbers for $1$-dimensional integrals, which was generalized to generic hyperplane arrangement case by Matsumoto in \cite{MatsumotoIntersection}. Another application of cohomology intersection number is the derivation of Pfaffian system that Euler-Laplace integral (\ref{MixedIntegral}) satisfies. In \cite{GotoKanekoMatsumoto}, \cite{GotoMatsumotoContiguity}, \cite{FrellesvigMastroliaMizera}, \cite{MastroliaMizera}, \cite{MizeraIntersection}, the authors deal with various Pfaffian systems from this point of view.
On the other hand, the nature of homology intersection number is more topological. Let us recall that the Euler-Laplace integral (\ref{MixedIntegral}) can be regarded as a result of the period pairing between the algebraic de Rham cohomology group $\Homo_{\rm dR}^n\left( U;(\mathcal{O}_U,\nabla_z)\right)$ and the rapid decay homology group $\Homo_n^{\rm r.d.}\left( U;(\mathcal{O}_{U^{an}}^\vee,\nabla_z^{{\rm an}\vee})\right)$ of M.Hien (\cite{HienRDHomology}). Therefore, the homology intersection pairing $\langle\bullet,\bullet\rangle_h$ can naturally be defined as a perfect pairing between two rapid decay homology groups $\Homo_n^{\rm r.d.}\left( U;(\mathcal{O}_{U^{an}}^\vee,\nabla_z^{{\rm an}\vee})\right)$ and $\Homo_n^{\rm r.d.}\left( U;(\mathcal{O}_{U^{an}},\nabla_z^{\rm an})\right)$, which is compatible with the cohomology intersection pairing $\langle\bullet,\bullet\rangle_{ch}$ through period pairings. For the precise definition of $\langle\bullet,\bullet\rangle_{ch}$, see \S\ref{RDIntersection}. It is customary in the theory of hypergeometric integrals to call the rapid decay homology groups $\Homo_n^{\rm r.d.}\left( U;(\mathcal{O}_{U^{an}}^\vee,\nabla_z^{{\rm an}\vee})\right)$ and $\Homo_n^{\rm r.d.}\left( U;(\mathcal{O}_{U^{an}},\nabla_z^{\rm an})\right)$ the twisted homology groups.
The most typical application of homology intersection number is probably the monodromy invariant hermitian matrix in the study of period maps (\cite{DeligneMostow}, \cite{MSY}, \cite{YoshidaMyLove}). It plays an essential role when one determines the image of the period map. Therefore, it is crucial that the homology intersection numbers can be evaluated exactly for a given basis of the twisted homology group. In these studies, the integral (\ref{MixedIntegral}) is reduced to Aomoto-Gelfand hypergeoemtric integral (\cite{AomotoKita}). An explicit formula of the homology intersection numbers can be found in \cite{KitaYoshida}, \cite{KitaYoshida2}.
Another application of the homology intersection number is the global study of the hypergeometric function that the integral (\ref{MixedIntegral}) represents. This is based on the standard fact that the rapid decay homology group $\Homo_n^{\rm r.d.}\left( U;(\mathcal{O}_{U^{an}}^\vee,\nabla_z^{{\rm an}\vee})\right)$ is isomorphic to the solution space of the Gau\ss-Manin connection associated to the integral (\ref{MixedIntegral}). Once this isomorphism is established, one wants to understand the parallel transport of a cycle $[\Gamma(z)]\in\Homo_n^{\rm r.d.}\left( U;(\mathcal{O}_{U^{an}}^\vee,\nabla_z^{{\rm an}\vee})\right)$ from a singular point $z=z_1$ to another one $z=z_2$. This can be achieved once we find a good basis $\{ \check{\Gamma}_i(z)\}_{i}$ of $\Homo_n^{\rm r.d.}\left( U;(\mathcal{O}_{U^{an}},\nabla_z^{{\rm an}})\right)$ near $z=z_2$ with which one can compute the homology intersection numbers $\langle \Gamma(z), \check{\Gamma}_i(z)\rangle_h$. The interested readers may refer to \cite{GotoPhDThesis}, \cite{MatsumotoFD}, \cite{MatsumotoYoshida} and references therein.
In these studies, the most important part is the choice of a good basis of twisted homology groups with which one can compute the exact value of homology intersection numbers. Let us review the method of constructing a basis of the twisted homology group.
\subsection{Constructing a good basis of cycles I: The method of stationary phase}
Let us recall the construction of a good basis of the twisted homology group by means of the method of stationary phase. For the detail, see \cite{AomotoLesEquations}, \cite{AomotoKita}, \cite{ArnoldStationaryPhase}, \cite{Fedoryuk}, or \cite{PhamMethodeDeCol}.
Introducing a real parameter $\tau$ and positive rational numbers $\eta_1,\dots,\eta_{k+n}\in\Q_{ >0}$, we consider an integral
\begin{equation}\label{OscillatoryIntegral}
f_{\Gamma}^{\pm}(z)=\int_{\Gamma}e^{\tau\varphi}h_{1,z^{(1)}}(x)^{\mp\gamma_1}\cdots h_{k,z^{(k)}}(x)^{\mp\gamma_k}x^{\pm c}\omega
\end{equation}
where $\varphi=\varphi(x;z,\eta)=h_{0}(x;z)+\sum_{l=1}^k\eta_{n+l}\log h_{l}(x;z)+\sum_{i=1}^n\eta_i\log x_i$. Note that the integral (\ref{OscillatoryIntegral}) is essentially same as the integral (\ref{MixedIntegral}). The recipe of the method of stationary phase is as follows: We first detect all the critical points of $\varphi(x;z,\eta)$ where $z$ and $\eta$ are regarded as fixed constants. The set of critical points is denoted by ${\rm Crit}$. Assume that any critical point $p$ is Morse. For each critical point $p\in{\rm Crit}$, we associate the contracting (resp. expanding) manifold $L_p^+$ (resp. $L_p^-$) as a set of all trajectories that have the point $p$ as the limit point at $t=+\infty$ (resp. $t=-\infty$) of the gradient vector field of $\Re\varphi$. Assuming the completeness of the gradient flow, we obtain $n$-dimensional cycle $L_p^+$ (resp. $L_p^-$) which is called a positive (resp. negative) Lefschetz thimble (to be more precise, one must replace the Euclidian metric by a complete K\"ahler metric as in \cite[Chapter 4]{AomotoKita}). Under the generic condition that $L_p^\pm$ does not flow into another critical point $q\in {\rm Crit}$, the method of stationary phase tells us that the asymptotic expansion of the integral $f_{L_p^\pm}^{\pm}(z)$ as $\tau\rightarrow \pm\infty$ has the form $e^{\tau\varphi(p;z,\eta)}a_0^\pm\tau^{-\frac{n}{2}}(1+o(\tau))$ where $a_0^\pm$ is a non-zero constant which can be computed from the Hessian of $\varphi(x;z,\eta)$. Moreover, by construction, we have the natural orthogonality relation $\langle L_p^+,L_q^-\rangle_h=0$ if $p\neq q$ and $\langle L_p^+,L_p^-\rangle=1$ (Smale's transversality). In this fashion, it is expected that we obtain a good basis $\{ L_p^+\}_{p\in{\rm Crit}}\subset\Homo_n^{\rm r.d.}\left( U;(\mathcal{O}_{U^{an}}^\vee,\nabla_z^{{\rm an}\vee})\right)$ and $\{ L_p^-\}_{p\in{\rm Crit}}\subset\Homo_n^{\rm r.d.}\left( U;(\mathcal{O}_{U^{an}},\nabla_z^{\rm an})\right)$ of the twisted homology groups.
\subsection{Constructing a good basis of cycles I\hspace{-.1em}I: Regularization of chambres}
When the integral (\ref{MixedIntegral}) is associated to a hyperplane arrangement, positive Lefschetz thimbles $\{ L_p^+\}_{p\in{\rm Crit}}$ can be understood in a combinatorial way. For simplicity, let us consider the case when $h_l$ are polynomials of degree $1$, $h_0\equiv 0$, the variable $z$ is real and parameters $\gamma_l$ and $c$ are generic. In this case, the real part $U^{\rm an}\cap\R^N$ is decomposed into finitely many connected components which are called chambers. In particular, a relatively compact chamber is called a bounded chamber. Then, each positive Lefschetz thimble is represented by exactly one bounded chamber $\Delta$ with the property $p\in\Delta$. Let us describe it more precisely. We denote by $\mathcal{L}^+$ the local system on $U^{\rm an}$ of flat sections of $\nabla_z^{{\rm an}\vee}$. The dual local system of $\mathcal{L}^+$ is denoted by $\mathcal{L}^-$. For each bounded chamber $\Delta$, we denote by $\Delta^+$ (resp. $\Delta^-$) an element of the locally finite homology group $\Homo^{\rm lf}_n(U,\mathcal{L}^+)$ (resp. $\Homo^{\rm lf}_n(U^{\rm an},\mathcal{L}^-)$) represented by $\Delta$. Then, the set $\{ \Delta^\pm\}_{\Delta:\text{bounded chambres}}$ is a basis of $\Homo^{\rm lf}_n(U^{\rm an},\mathcal{L}^\pm)$ (\cite{KohnoHomology}, \cite[Proposition 6.4.1]{OrlikTerao}) and $\{ \Delta^+\}_{\Delta:\text{bounded chambres}}$ coincides with the basis $\{ L_p^+\}_{p\in{\rm Crit}}$.
It is important that the intersection numbers with respect to these bases $\{ \Delta^\pm\}_{\Delta:\text{bounded chambres}}$ can be computed explicitly. Let ${\rm reg}:\Homo^{\rm lf}_n(U^{\rm an},\mathcal{L}^\pm)\tilde{\rightarrow}\Homo_n(U^{\rm an},\mathcal{L}^\pm)$ be the inverse of the natural isomorphism $\Homo_n(U^{\rm an},\mathcal{L}^\pm)\tilde{\rightarrow}\Homo_n^{\rm lf}(U^{\rm an},\mathcal{L}^\pm)$. The isomorphism ${\rm reg}$ is called the regularization map. For a bounded chamber $\Delta$, there is an explicit description of the cycle ${\rm reg}(\Delta^+)$. For simplicity, we suppose that the hyperplane arrangement $D$ is generic, i.e., $D$ is a normal crossing divisor. For a given element $[\Delta^+]\in\Homo_n^{\rm lf}(U^{\rm an},\mathcal{L}^+)$ represented by a bounded chamber $\Delta$, we cut off a small neighbourhoods of the faces of $\Delta$ and consider a small chamber $\s$. We can regard $\s$ as a finite chain whose orientation is induced from that of $\Delta^+$. At this stage, $\s$ is not yet a cycle. Then, to each face of codimension $p$, we put $p$ ``pipes'' encircling it with a suitable coefficient in the local system $\mathcal{L}^+$. Repeating this process, we obtain a cycle ${\rm reg }(\Delta^+)$ whose homology class in $\Homo_n^{\rm lf}(U^{\rm an},\mathcal{L}^+)$ is $[\Delta^+]$ (\cite{KitaWronskian}). From this construction, we see that the regularized cycle ${\rm reg}(\Delta^+)$ is, up to a constant multiplication, equal to the multidimensional Pochhammer cycle (see e.g. \cite{Beukers}). As for the description of the regularization map for a particular non-generic arrangement, see \cite[Chapter 5]{TsuchiyaKanie}. With this construction, we have an explicit formula of the homology intersection number $\langle {\rm reg}(\Delta^+_1),\Delta_2^-\rangle_h$, which turns out to be a periodic function of the parameters (\cite{KitaYoshida} and \cite{KitaYoshida2}).
\begin{figure}[h]
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{tikzpicture
\draw[-] (-1,0)--(4,0);
\draw[-] (0,-1)--(0,3);
\draw[-] (-1,3)--(3,-1);
\draw[-] (-2,-1)--(4,0.1);
\filldraw[fill=lightgray] (0,0) -- (2,0) -- (0,2);
\node at (0.6,0.6) {$\Delta^+$};
\end{tikzpicture}
\end{center}
\end{minipage}
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{tikzpicture
\draw[-] (-1,0)--(4,0);
\draw[-] (0,-1)--(0,3);
\draw[-] (-1,3)--(3,-1);
\draw[-] (-2,-1)--(4,0.1);
\filldraw[fill=lightgray] (0.2,0.2)--(1.5,0.2)--(0.2,1.5)--cycle;
\draw[-] (0.5,1.8) -- (1.8,0.5);
\draw[-] (0.2,-0.2) -- (1.5,-0.2);
\draw[-] (-0.2,0.2) -- (-0.2,1.5);
\draw[-] (0.2,1.5) to [out=100,in=130] (0.5,1.8);
\draw[-] (1.5,0.2) to [out=100,in=130] (1.8,0.5);
\draw[-] (0.2,1.5) to [out=20,in=-80] (0.5,1.8);
\draw[dashed] (1.5,0.2) to [out=0,in=-80] (1.8,0.5);
\draw[-] (1.5,0.2) to [out=-50,in=30] (1.5,-0.2);
\draw[dashed] (1.5,0.2) to [out=-100,in=100] (1.5,-0.2);
\draw[-] (0.2,0.2) to [out=-50,in=30] (0.2,-0.2);
\draw[-] (0.2,0.2) to [out=-120,in=120] (0.2,-0.2);
\draw[-] (0.2,0.2) to [out=110,in=70] (-0.2,0.2);
\draw[-] (0.2,0.2) to [out=-110,in=-70] (-0.2,0.2);
\draw[-] (0.2,1.5) to [out=110,in=70] (-0.2,1.5);
\draw[dashed] (0.2,1.5) to [out=-110,in=-70] (-0.2,1.5);
\node at (2,2){reg$(\Delta^+)$};
\node at (0.6,0.6) {$\sigma$};
\end{tikzpicture}
\end{center}
\end{minipage}
\caption{A bounded chamber and its regularization}
\end{figure}
\subsection{Our method: The viewpoint of GKZ system}
In the previous subsection, we explained that there is a combinatorial method of constructing a basis of the homology group whose homology intersection matrix has an explicit formula when the integrand of (\ref{MixedIntegral}) defines a hyperplane arrangement. Now it is natural to ask to what extent the construction can be generalized. There are many papers discussing constructions of bases of the rapid decay homology groups in various contexts (e.g., \cite{AomotoConfigurations}, \cite{AomotoHypersphereSchlafli}, \cite{EsterovTakeuchi}, \cite{GotoPhDThesis}, \cite{MarkovTarasovVarchenko}). To our knowledge, however, there is no systematic method of constructing a basis of the rapid decay homology group whose intersection matrix has a closed formula.
We propose a systematic method of constructing a basis of the rapid decay homology group without any restriction on the Laurent polynomials $h_l(x)$. We no longer expect that the ``visible cycles'' such as chambers or their regularizations are sufficient to construct a basis. Instead, we make use of a convergent regular triangulation $T$ of the cone $\R_{>0}A$. Let us illustrate our construction by a simple example.
Suppose that $n=1$ and the integrand of (\ref{MixedIntegral}) is $e^{z_1x+z_2x^{-1}}(z_3+z_4x)^{-\gamma}x^c$. For simplicity, we assume that $z_1$ and $z_2$ are positive. We consider the real oriented blow-up (\cite[\S 8.2]{SabbahIntroductionToStokes}) $(\C\setminus\{ 0,\zeta=-\frac{z_3}{z_4}\})\sqcup S^1\infty\sqcup S^10$, where $S^1\infty$ (resp. $S^10$) is the circle at $\infty$ (resp. at the origin). Under the standard identification $S^1=\R/2\pi\Z$, the twisted homology group $\Homo_1^{\rm r.d.}\left( U;(\mathcal{O}_{U^{an}}^\vee,\nabla_z^{{\rm an}\vee})\right)$ is simply given by a relative homology group $\Homo_1\left( (\C\setminus\{ 0,\zeta\})\sqcup (\pi,\frac{3\pi}{2})\infty\sqcup (\pi,\frac{3\pi}{2}) 0,(\pi,\frac{3\pi}{2})\infty\sqcup (\pi,\frac{3\pi}{2}) 0;\mathcal{L}\right)$ where $\mathcal{L}$ is the local system of flat sections of the connection $\nabla^{{\rm an}\vee}_z$. As a convergent regular triangulation, we take $T=\{ 14,34,23\}$. For the simplex $14\in T$, we associate a limit $z_2,z_3\rightarrow 0$ with $z_1,z_4$ fixed. At the limit, we observe that the exponential factor $e^{z_1x+z_2x^{-1}}$ becomes $e^{z_1x}$ while the point $\zeta$ goes to the origin. The key observation is that, at the limit, we may pretend as if the rapid decay homology group is associated to the integrand $e^{z_1x}x^{c-\gamma}$. Since the rank of this rapid decay homology group is $1$, we can take a generator $[\Gamma_{14}(z)]$. Though this cycle is defined when $z_2$ and $z_3$ are small, it can be defined for any generic $z$ after parallel transportation (see the argument in \S\ref{SectionEuler}). Applying this process to other simplices $34,23$, we obtain a basis $\left\{ [\Gamma_{14}(z)],[\Gamma_{34}(z)],[\Gamma_{23}(z)]\right\}$ of the twisted homology group.
After a suitable modification, we can generalize the construction above to the general case: We take a convergent regular triangulation $T$ of $\R_{>0}A$. To each simplex $\s\in T$, we associate a limit $z_j\rightarrow 0$ ($j\notin\s$). At the limit, we can pretend as if the rank of the rapid decay homology group is the cardinality $|G_{\s}|$ of the finite Abelian group $G_{\s}$. We can take out a set of independent cycles $\{\Gamma_{\s,h}\}_{h\in\widehat{G_{\s}}}$ labeled by the dual group $\widehat{G_{\s}}$. After performing the parallel transportation, the union $\Gamma_T=\bigcup_{\s\in T}\{\Gamma_{\s,h}\}_{h\in\widehat{G_{\s}}}$ is a basis of the rapid decay homology group $\Homo_n^{\rm r.d.}\left( U;(\mathcal{O}_{U^{an}}^\vee,\nabla_z^{{\rm an}\vee})\right)$. The same construction works for the dual group $\Homo_n^{\rm r.d.}\left( U;(\mathcal{O}_{U^{an}},\nabla_z^{{\rm an}})\right)$ and we denote this basis by $\check{\Gamma}_T=\bigcup_{\s\in T}\{\check{\Gamma}_{\s,h}\}_{h\in\widehat{G_{\s}}}$.
We denote by $z_\s^{\infty}$ the point near $z_T^\infty$ the $j$-th coordinate of which is $0$ unless $j\in \s$. In a sense, $z_\s^\infty$ plays the role of a critical point in the method of stationary phase. Namely, the cycles $\Gamma_{\s,h}$ are characterized by the behaviour of the integrand of (\ref{MixedIntegral}) near the special point $z_\s^{\infty}$. The general construction above can be carried out in quite an explicit manner. Indeed, after a suitable sequence of coordinate transformations, each cycle $\Gamma_{\s,h}$ can be constructed as a product of Pohhammer cycles and the Hankel contour. Thanks to this explicit construction, we can also expand the integral (\ref{MixedIntegral}) along the cycle $\Gamma_{\s,h}$ into a hypergeometric series converging near the point $z_T^\infty$. In \S\ref{SectionEuler}, we give an explicit transformation matrix between two basis of solutions $\Phi_T$ and $\Gamma_T$ in terms of the character matrices of the finite Abelian groups $G_{\s}$ (\cref{thm:fundamentalthm3}). The homology intersection matrix $\left(\langle \Gamma_{\s_1,h_1},\check{\Gamma}_{\s_2,h_2}\rangle_h\right)$ is naturally block-diagonalized in the sense that $\langle \Gamma_{\s_1,h_1},\check{\Gamma}_{\s_2,h_2}\rangle_h=0$ unless $\s_1=\s_2$. Finally, we can evaluate the diagonal term $\langle \Gamma_{\s,{\bf 0}},\check{\Gamma}_{\s,{\bf 0}}\rangle_h$ as a periodic function in parameters when the group $G_{\s}$ is the trivial Abelian group ${\bf 0}$. A table comparing our construction with the method of stationary phase is the following.
\begin{figure}[t]
\begin{center}
\scalebox{0.5}[0.5]{
\begin{tikzpicture
\draw[-,domain=92:268] plot ({3*cos(\x)},{3*sin(\x)});
\draw[-,domain=92:268] plot ({cos(\x)},{sin(\x)});
\draw[dashed,domain=-88:88] plot ({3*cos(\x)},{3*sin(\x)});
\draw[dashed,domain=-88:88] plot ({cos(\x)},{sin(\x)});
\node at (0,3){$\circ$};
\node at (0,-3){$\circ$};
\node at (0,1){$\circ$};
\node at (0,-1){$\circ$};
\node at (2,0){$\circ$};
\node at (2,-0.5){$\zeta$};
\node at (0,0){\scalebox{1.8}{$S^10$}};
\node at (1,3.6){\scalebox{1.8}{$S^1\infty$}};
\draw[-,domain=92:268] plot ({8+2*cos(\x)},{6+2*sin(\x)});
\draw[dashed,domain=-88:88] plot ({8+2*cos(\x)},{6+2*sin(\x)});
\node at (8,8){$\circ$};
\node at (8,4){$\circ$};
\node at (8,6){$\circ$};
\node at (8.3,6){$\circ$};
\node at (7.7,5.7){\bf O};
\node at (8.5,5.7){$\zeta$};
\node at (9.2,6.6){$\Gamma_{14}$};
\draw[->-=.5,domain=0:2.2] plot ({8+2*cos(200)+(\x)},{6+2*sin(200)});
\draw[-<-=.5,domain=0:2.2] plot ({8+2*cos(160)+(\x)},{6+2*sin(160)});
\coordinate (A) at ({10.2+2*cos(160)},{6+2*sin(160)});
\coordinate (B) at ({10.2+2*cos(200)},{6+2*sin(200)});
\draw[-<-=.5] (A) to [out=0,in=0] (B);
\draw[dashed,domain=0:360] plot ({8+2*cos(\x)},{2*sin(\x)});
\node at (8,0){$\circ$};
\node at (9,0){$\circ$};
\node at (7.2,0){\bf O};
\node at (9.8,0){$\zeta$};
\node at (8.5,-1){$\Gamma_{34}$};
\draw[-,domain=-140:140] plot ({9+0.4*cos(\x)}, {0.2+0.4*sin(\x)});
\draw[-,domain=40:320] plot ({8+0.4*cos(\x)}, {0.2+0.4*sin(\x)});
\draw[-,domain=-140:140] plot ({9+0.4*cos(\x)}, {-0.2+0.4*sin(\x)});
\draw[-,domain=40:320] plot ({8+0.4*cos(\x)}, {-0.2+0.4*sin(\x)});
\coordinate (A1) at ({8+0.4*cos(40)}, {0.2+0.4*sin(40)});
\coordinate (A2) at ({9+0.4*cos(-140)}, {-0.2+0.4*sin(-140)});
\coordinate (B1) at ({9+0.4*cos(140)}, {-0.2+0.4*sin(140)});
\coordinate (B2) at ({8+0.4*cos(40)}, {-0.2+0.4*sin(40)});
\coordinate (C1) at ({8+0.4*cos(320)}, {-0.2+0.4*sin(320)});
\coordinate (C2) at ({9+0.4*cos(140)}, {0.2+0.4*sin(140)});
\coordinate (D1) at ({9+0.4*cos(-140)}, {0.2+0.4*sin(-140)});
\coordinate (D2) at ({8+0.4*cos(320)}, {0.2+0.4*sin(320)});
\draw[-] (A1) -- (A2);
\draw[-] (B1) -- (B2);
\draw[-] (C1) -- (C2);
\draw[-] (D1) -- (D2);
\draw[dashed,domain=0:360] plot ({8+2*cos(\x)},{-6+2*sin(\x)});
\draw[-,domain=96:264] plot ({8+0.5*cos(\x)},{-6+0.5*sin(\x)});
\draw[dashed,domain=-84:84] plot ({8+0.5*cos(\x)},{-6+0.5*sin(\x)});
\node at (8,-5.5){$\circ$};
\node at (8,-6.5){$\circ$};
\node at (9.5,-6){$\circ$};
\node at (9.5,-6.5){$\zeta$};
\node at (8,-7.5){$\Gamma_{23}$};
\draw[-<-=.5,domain=-90:90] plot ({8+cos(\x)},{-6+sin(\x)});
\coordinate (AA) at ({8+cos(90)},{-6+sin(90)});
\coordinate (BA) at ({8+cos(-90)},{-6+sin(-90)});
\coordinate (AB) at ({8+0.5*cos(150)},{-6+0.5*sin(150)});
\coordinate (BB) at ({8+0.5*cos(210)},{-6+0.5*sin(210)});
\draw[-] (AB) to [out=150,in=180] (AA);
\draw[-] (BB) to [out=210,in=180] (BA);
\node at (4,3){$\rotatebox{45}{\Huge $\Rightarrow$}$};
\node at (5,0){\Huge $\Rightarrow$};
\node at (4,-3){$\rotatebox{-45}{\Huge $\Rightarrow$}$};
\node at (20,6){\Huge $e^{z_1x+z_2x^{-1}}(z_3+z_4x)^{-\gamma}x^c\approx e^{z_1x}x^{c-\gamma}$};
\node at (22,0){\Huge $e^{z_1x+z_2x^{-1}}(z_3+z_4x)^{-\gamma}x^c\approx (z_3+z_4x)^{-\gamma}x^c$};
\node at (20,-6){\Huge $e^{z_1x+z_2x^{-1}}(z_3+z_4x)^{-\gamma}x^c\approx e^{z_2x^{-1}}x^c$};
\end{tikzpicture}
}\label{TheFirstFigure}
\caption{Our construction of cycles}
\end{center}
\end{figure}
\begin{table}[ht]
\begin{tabular}{|c|c|c|}
\hline
&Method of stationary phase & Our method \\
\hline
Special point &Critical point &$z_\s^\infty$ ($\s\in T$) \\
\hline
Integration cycle &Lefschetz thimble& (Pochhammer cycle)$\times$(Hankel contour)\\
\hline
Intersection matrix & Identity matrix& Periodic function in parameters\\
\hline
Expansion of the integral &Asymptotic series in $\tau\approx\infty$& Hypergeometric series in $z\approx z_T^\infty$ \\
\hline
\end{tabular}
\end{table}
\subsection{Quadratic relations for Aomoto-Gelfand hypergeoemtric functions and its confluence}
In the last two sections, we apply \cref{thm:IntroTheorem} to Aomoto-Gelfand system (\S \ref{QuadraticRelationsForGrassman}) and its confluence (\S \ref{E(21...1)}). The special feature of these classes is that there is a special convergent unimodular regular triangulation $T$ called the staircase triangulation (\cite{GelfandGraevRetakh}). Since there is a one-to-one correspondence between simplices of $T$ and spanning trees of a complete bipartite graph, we can express the homology intersection numbers in terms of these graphs. It is noteworthy that our basis of cycles is different from that consisting of regularizations of bounded chambers $\{ {\rm reg}(\Delta^\pm)\}_{\Delta:\text{bounded chambers}}$. Namely, our cycles may go around several divisors $\{ h_{l}(x)=0\}$ simultaneously so that they are linked in a more complicated way. In this sense, our cycles can be referred to as ``linked cycles''.
Besides the precise description of the twisted homology group, there is a basis of the algebraic de Rham cohomology group whose cohomology intersection matrix has a closed formula (\cite{MatsumotoIntersection} and \cref{prop:E(21...1)c.i.n.}). Therefore, \cref{thm:IntroTheorem} gives rise to a general quadratic relation for Aomoto-Gelfand hypergeometric functions (\cref{thm:QuadraticRelationsForAomotoGelfand}) and its confluence (\cref{thm:QuadraticRelationsForAomotoGelfand2}). The simplest example of such an identity is the following relation:
\begin{align}
&(1-\gamma+\alpha)(1-\gamma+\beta){}_2F_1\left(\substack{\alpha,\beta\\ \gamma};z\right){}_2F_1\left(\substack{-\alpha,-\beta\\ 2-\gamma};z\right)-\alpha\beta{}_2F_1\left(\substack{\gamma-\alpha-1,\gamma-\beta-1\\ \gamma};z\right)
{}_2F_1\left(\substack{1-\gamma+\alpha,1-\gamma+\beta \\ 2-\gamma};z\right)\nonumber\\
=&(1-\gamma+\alpha+\beta)(1-\gamma).\label{QRGauss}
\end{align}
\noindent
Here, ${}_2F_1\left(\substack{\alpha,\beta\\ \gamma} ;z\right)$ is the usual Gau\ss' hypergeometric series
\begin{equation}\label{series}
{}_2F_1\left(\substack{\alpha,\beta\\ \gamma} ;z\right)=\displaystyle\sum_{n=0}^\infty\frac{(\alpha)_n(\beta)_n}{(\gamma)_n(1)_n}z^n
\end{equation}
with complex parameters $\alpha,\beta\in\C$ and $\gamma\in\C\setminus\Z_{\leq 0}$.
\subsection{The structure of the paper}
We give a plan of this paper. The first three sections \S\ref{SectionDModules}, \S\ref{SectionRapidDecay} and \S\ref{RDIntersection} are devoted to the foundations of the study of the integral (\ref{MixedIntegral}). In \S\ref{SectionDModules}, we establish an isomorphism between the Gau\ss-Manin connection associated to the integral (\ref{MixedIntegral}) and GKZ $\DD$-module $M_A(\delta)$ under the non-resonance assumption of $\delta$ (\cref{thm:mainDresult}). The technique of the proof is an adaptation of the arguments of \cite[\S 3]{AdolphsonSperber} and \cite[\S 9]{DworkLoeser} in the framework of $\DD$-modules. Based on \S\ref{SectionDModules}, we establish an isomorphism between the rapid decay homology group $\Homo_n^{\rm r.d.}\left( U;(\mathcal{O}_{U^{an}}^\vee,\nabla_z^{{\rm an}\vee})\right)$ and the stalk of the solutions of $M_A(\delta)$ at a generic point $z$ in \S\ref{SectionRapidDecay}. The essential part of the construction is a compactification of $U$ in the spirit of \cite{KhovanskiNewton} and \cite{EsterovTakeuchi}. In \S\ref{RDIntersection}, we give a foundation of the (co)homology intersection pairings. The definitions of the pairing is a natural generalization of that of \cite{MajimaMatsumotoTakayama}.
From \S\ref{SectionSeries}, we combine the general theory above with the combinatorial structure of GKZ system. In \S\ref{SectionSeries}, we summarize the construction of hypergeoemtric series in \cite{FernandezFernandez}. The readers familiar with this topic can skip this section. In \S\ref{SectionEuler} we construct a basis $\Gamma_T$ of the rapid decay homology group for a given convergent regular triangulation $T$. We give an explicit transformation formula between the basis of series solutions $\Phi_T$ and the basis $\Gamma_T$ (\cref{thm:fundamentalthm3}). In \S\ref{IntersectionNumbers}, we establish a closed formula of intersection matrix with respect to the bases $\Gamma_T$ and $\check{\Gamma}_T$ when the convergent regular triangulation $T$ is unimodular. In \S\ref{QuadraticRelationsForGammaSeries}, we give a proof of \cref{thm:IntroTheorem}.
In the last two sections \S\ref{QuadraticRelationsForGrassman} and \S\ref{E(21...1)}, we discuss the application of \cref{thm:IntroTheorem} to Aomoto-Gelfand system and its confluence. After recalling/establishing closed formulas of the cohomology intersection numbers (\cite{MatsumotoIntersection}, \cref{prop:E(21...1)c.i.n.}) we give closed formulas of quadratic relations associated to the staircase triangulation in terms of bipartite graphs (\cref{thm:QuadraticRelationsForAomotoGelfand} and \cref{thm:QuadraticRelationsForAomotoGelfand2}).
\section{General framework of Euler-Laplace integral representation}\label{SectionDModules}
We begin with revising some basic notation and results of algebraic $\DD$-modules. For their proofs, see \cite{BorelEtAl} or \cite{HTT}. Let $X$ and $Y$ be smooth algebraic varieties over the complex numbers $\C$ and let $f:X\rightarrow Y$ be a morphism. Throughout this paper, we write $X$ as $X_x$ when we emphasize that $X$ is equipped with the coordinate $x$. We denote $\DD_X$ the sheaf of linear partial differential operators on $X$ and denote $D^b_{q.c.}(\DD_X)$ (resp. $D^b_{coh}(\DD_X)$, resp. $D^b_{h}(\DD_X)$) the derived category of bounded complexes of left $\DD_X$-modules whose cohomologies are quasi-coherent (resp. coherent, resp. holonomic). We denote by $D^b_{*}(\DD_X)$, one of two categories $D^b_{q.c.}(\DD_X)$ or $D^b_{h}(\DD_X)$. For any coherent $\DD_X$-module $M$, we denote ${\rm Char}(M)$ its characteristic variety in $T^*X$. In general, for any object $M\in D^b_{coh}(\DD_X)$, we define its characteristic variety by ${\rm Char}(M)=\displaystyle\cup_{n\in\Z}{\rm Char}\left(\Homo^n(M)\right)$. We denote ${\rm Sing }(M)$ the image of ${\rm Char}(M)$ by the canonical projection $T^*X\rightarrow X$. For any object $N\in D^b_*(\DD_Y)$, we define its inverse image $\LL f^*N\in D^b_*(\DD_X)$ (resp. its shifted inverse image $f^\dagger N\in D^b_*(\DD_X)$) with respect to $f$ by the formula
\begin{equation}
\LL f^*N=\DD_{X\rightarrow Y}\overset{\LL}{\underset{f^{-1}\DD_Y}{\otimes}}f^{-1}N \;\;\;(\text{resp. } f^{\dagger}N=\LL f^*N[\dim X-\dim Y]),
\end{equation}
where $\DD_{X\rightarrow Y}$ is the transfer module $\mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_Y}f^{-1}\DD_Y.$ For any object $M\in D^b_*(\DD_X)$, we define its holonomic dual $\HD_X M\in D^b_*(\DD_X)^{op}$ by
\begin{equation}
\HD_X M=\R\mathcal{H}om_{\DD_X}(M,\DD_X)\otimes_{\mathcal{O}_X}\Omega_X^{\otimes -1}.
\end{equation}
Note that $\HD_X$ is involutive, i.e., we have $\HD_X\circ\HD_X\simeq {\rm id}_X$. Next, for any object $M\in D^b_*(\DD_X)$, we define its direct image $\int_fM\in D^b_*(\DD_Y)$ (resp. its proper direct image $\int_{f!}M\in D^b_*(\DD_Y)$) by
\begin{equation}
\int_fM=\R f_*(\DD_{Y\leftarrow X}\overset{\LL}{\underset{\DD_X}{\otimes}}M),\;\;\;(\text{resp. } \int_{f!}=\HD_Y\circ\int_f\circ\HD_X M),
\end{equation}
where $\DD_{Y\leftarrow X}$ is the transfer module $\Omega_X\otimes_{\mathcal{O}_X}\DD_{X\rightarrow Y}\otimes_{f^{-1}\mathcal{O}_Y}f^{-1}\Omega_Y$. If $X$ is a product variety $X=Y\times Z$ and $f:Y\times Z\rightarrow Y$ is the natural projection, the direct image can be computated in terms of (algebraic) relative de Rham complex
\begin{equation}\label{RelativeDeRhamFormula}
\int_fM\simeq \R f_*(\DR_{X/Y}(M)).
\end{equation}
In particular, if $Y=\{*\}$ (one point), and $M$ is a connection $M=(E,\nabla)$ on $Z$, then for any integer $p$, we have a canonical isomorphism
\begin{equation}
\Homo^p\left( \int_fM\right)\simeq\mathbb{H}_{dR}^{p+\dim Z}(Z,(E,\nabla)),
\end{equation}
where $\mathbb{H}_{dR}$ denotes the algebraic de-Rham cohomology group. If a cartesian diagram
\begin{equation}
\xymatrix{
X^\prime \ar[r]^{f^\prime } \ar[d]_{g^\prime}&Y^\prime\ar[d]^{g}\\
X \ar[r]^{f} &Y
}
\end{equation}
is given, for any object $M\in D^b_*(\DD_X)$, we have the base change formula
\begin{equation}\label{BaseChange}
g^\dagger\int_fM\simeq\int_{f^\prime}g^{\prime\dagger}M.
\end{equation}
For objects $M,M^\prime\in D^b_*(\DD_X)$ and $N\in D^b_*(\DD_Y)$, the tensor product $M\overset{\mathbb{D}}{\otimes}M^\prime\in D^b_*(\DD_X)$ and external tensor product $M\boxtimes N\in D^b_*(\DD_{X\times Y})$ are defined by
\begin{equation}
M\overset{\mathbb{D}}{\otimes}M^\prime=M\overset{\mathbb{L}}{\underset{\mathcal{O}_X}{\otimes}}M^\prime,\;\; M\boxtimes N=M\underset{\C}{\otimes}N.
\end{equation}
Note that for any objects $N,N^\prime\in D^b_{*}(\DD_Y)$, we have a canonical isomorphism
\begin{equation}
\mathbb{L}f^*(N\overset{\D}{\otimes}N^\prime)\simeq (\mathbb{L}f^*N\overset{\D}{\otimes}\mathbb{L}f^*N^\prime).
\end{equation}
For any objects $M\in D_*^b(\DD_X)$ and $N\in D^b_*(\DD_Y)$, we have the projection formula
\begin{equation}
\int_f\left( M\overset{\D}{\otimes}\mathbb{L}f^*N\right)\simeq\left(\int_fM\right)\overset{\D}{\otimes}N.
\end{equation}
\noindent
Let $Z$ be a smooth closed subvariety of $X$ and let $i:Z\hookrightarrow X$ and $j:X\setminus Z\hookrightarrow X$ be natural inclusions. Then, for any object $M\in D_*^b(\DD_X)$, there is a standard distinguished triangle
\begin{equation}\label{sdt}
\int_ii^\dagger M\rightarrow M\rightarrow\int_j j^\dagger M\overset{+1}{\rightarrow}.
\end{equation}
If we denote by $\Gamma_{[Z]}$ the algebraic local cohomology functor supported on $Z$, it is also standard that there are canonical isomorphisms
\begin{equation}
\R\Gamma_{[Z]}(\mathcal{O}_X)\overset{\D}{\otimes}M\simeq\R\Gamma_{[Z]}M\simeq\int_ii^\dagger M.
\end{equation}
\noindent
For any (possibly multivalued) function $\varphi$ on $X$ such that $\varphi$ is nowhere-vanishing and that $\frac{d\varphi}{\varphi}$ belongs to $\Omega^1_X(X)$, we define a $\DD_X$-module $\mathcal{O}_X\varphi$ by twisting its action as
\begin{equation}
\theta\cdot h=\Big\{\theta+\Big(\frac{\theta\varphi}{\varphi}\Big)\Big\}h\;\;\;(h\in\mathcal{O}_X,\;\theta\in\Theta_X).
\end{equation}
For any $\DD_X$-module $M,$ we define $M\varphi$ by $M\varphi=M\underset{\mathcal{O}_X}{\otimes}\mathcal{O}_X\varphi.$ We denote $\underline{\C}\varphi$ the local system of flat sections of $\left(\mathcal{O}_X\varphi^{-1}\right)^{an}$ on $X^{an}.$
\noindent
Lastly, for any closed smooth subvariety $Z\subset X$, we denote $\mathcal{I}_{Z^{an}}$ the defining ideal of $Z^{an}$ and denote $\iota:Z^{an}\hookrightarrow X^{an}$ the canonical inclusion. We set $\mathcal{O}_{X^{an}\hat{|}Z^{an}}=\underset{\underset{k}{\leftarrow}}{\lim}\mathcal{O}_{X^{an}}/\mathcal{I}_{Z^{an}}^k$. Then, for any object $M\in D^b_{*}(\DD_X),$ we have a canonical isomorphism
\begin{equation}\label{FormalRestriction}
\R\Hom_{\DD_{Z^{an}}}\left( \mathbb{L}\iota^*M^{an},\mathcal{O}_{Z^{an}}\right)\simeq\R\Hom_{\iota^{-1}\DD_{X^{an}}}\left( \iota^{-1}M^{an},\mathcal{O}_{X^{an}\hat{|}Z^{an}}\right).
\end{equation}
Now, we are going to prove the isomorphism between Laplace-Gauss-Manin connections associted to Euler-Laplace and Laplace integral. We first prove the following identity which is ``obvious'' from the definition of $\Gamma$ function.
\begin{prop}\label{prop:GammaInt}
Let $h:X\rightarrow\A^1$ be a non-zero regular function such that $h^{-1}(0)$ is smooth, $\pi:X\times (\Gm)_y\rightarrow X$ be the canonical projection, $j:X\setminus h^{-1}(0)\hookrightarrow X$ and $i:h^{-1}(0)\hookrightarrow X$ be inclusions, and let $\gamma\in\C\setminus\Z$ be a parameter. In this setting, for any $M\in D^b_{q.c.}(\DD_X)$, one has a canonical isomorphism
\begin{equation}\label{TheFirstIsomorphism}
\int_{\pi}(\mathbb{L}\pi^*M)y^\gamma e^{yh}\simeq\int_j (j^\dagger M)h^{-\gamma}.
\end{equation}
and a vanishing result
\begin{equation}\label{TheVanishingResult}
\int_{\pi}\R\Gamma_{[h^{-1}(0)\times(\Gm)_y]}(\mathbb{L}\pi^*M)y^\gamma e^{yh}=0.
\end{equation}
\end{prop}
\noindent
For the proof, we insert the following elementary
\begin{lem}\label{lem:gamma}
Let $pt:(\Gm)_y\rightarrow\{ *\}$ be the trivial morphism. If $\gamma\in\C\setminus \Z$ and $h\in\C,$ one has
\begin{equation}
\int_{pt}\mathcal{O}_{(\Gm)_y}y^\gamma e^{hy}=
\begin{cases}
0&(h=0)\\
\C&(h\neq 0).
\end{cases}
\end{equation}
\end{lem}
\begin{proof}
By the formula (\ref{RelativeDeRhamFormula}), we have equalities
\begin{equation}
\int_{pt}\mathcal{O}_{(\Gm)_y}y^\gamma e^{hy}=\Big(\Omega^{\bullet+1}((\Gm)_y), \nabla\Big)=\Big(0\rightarrow \overset{\overset{-1}{\smile}}{\C[y^{\pm}]}\overset{\nabla}{\rightarrow}\overset{\overset{0}{\smile}}{\C[y^\pm]}\rightarrow 0\Big),
\end{equation}
where $\nabla=\frac{\partial}{\partial y}+\frac{\gamma}{y}+h.$ In view of this formula, the lemma is a consequence of an elementary computation.
\begin{comment}
Take any element $\displaystyle\sum_{n\in \Z}a_ny^n\in\C[y^\pm]$. By the definition of $\nabla$, we have
\begin{equation}
\nabla\Big(\sum_{n\in\Z}a_ny^n\Big)=\sum_{n\in\Z}\lef (n+\gamma)a_n+ha_{n-1}\righ y^{n-1}.
\end{equation}
If $h=0$, it is clear that $\nabla$ is an isomorphism of $\C$ vector spaces since $\gamma\notin\Z$. Thus, (1) is true. In the following, we assume $h\neq 0$. If $\displaystyle\sum_{n\in\Z}a_ny^n\in\Ker\nabla,$ we have $a_{n-1}=-\frac{(n+\gamma)}{h}a_n$ for any $n$, hence $a_n=0$ for all $n$. Now, let us show that $1\notin\im\nabla.$
Suppose the converse. Then, $\nabla\Big(\displaystyle\sum_{n\in\Z}a_ny^n\Big)=1$ implies
\begin{equation}
(1+\gamma)a_1+ha_0=1,\;\; (n+\gamma)a_n+ha_{n-1}=0 \; (\forall n\neq 1).
\end{equation}
Suppose that $a_1\neq 0.$ Then, for any $n\in\Z_{\geq 2},$
\begin{equation}
a_n=-\frac{h}{n+\gamma}a_{n-1}=\dots=\frac{(-1)^{n-1}h^{n-1}}{(n+\gamma)\dots (2+\gamma)}a_1\neq 0.
\end{equation}
This contradicts the fact that $\displaystyle\sum_{n\in\Z}a_ny^n\in\C[y^\pm].$ Similarly, if we assume that $a_0\neq 0,$ the formula $a_{n-1}=-\frac{(n+\gamma)}{h}a_n$ implies a contradiction.
On the other hand, since
\begin{equation}
\nabla(y^n)=ny^{n-1}+\gamma y^{n-1}+hy^n,
\end{equation}
we can conclude that
\begin{equation}
\C\ni 1\mapsto [1]\in \C[y^\pm]/\im\nabla\neq 0
\end{equation}
is surjective. Hence this is an isomorphism.
\end{comment}
\end{proof}
\noindent
(Proof of proposition)
\noindent
By projection formula, we have isomorphisms
\begin{equation}
\int_{\pi}(\mathbb{L}\pi^*M)y^\gamma e^{yh}\simeq M\overset{\D}{\otimes}\int_\pi\mathcal{O}_{X\times (\Gm)_y}y^\gamma e^{yh}
\end{equation}
and
\begin{equation}
\int_j (j^\dagger M)h^{-\gamma}\simeq M\overset{\D}{\otimes }\int_j\mathcal{O}_{X\setminus h^{-1}(0)}h^{-\gamma}.
\end{equation}
\noindent
Therefore, the first isomorphism of the proposition is reduced to the case when $M=\mathcal{O}_X$. Consider the following cartesian diagram:
\begin{equation}
\xymatrix{
h^{-1}(0)\times(\Gm)_y \ar[r]^{\tilde{i}} \ar[d]_{\tilde{\pi}}&X\times(\Gm)_y \ar[d]^{\pi}\\
h^{-1}(0) \ar[r]^{i} &X.}
\end{equation}
\noindent
By base change formula and \cref{lem:gamma}, we have
\begin{equation}
i^\dagger\int_\pi\mathcal{O}_{X\times (\Gm)_y}y^\gamma e^{yh}=\int_{\tilde{\pi}}\tilde{i}^\dagger\mathcal{O}_{X\times (\Gm)_y}y^\gamma e^{yh}=\int_{\tilde{\pi}}\mathcal{O}_{h^{-1}(0)\times (\Gm)_y}y^\gamma [-1]=0.
\end{equation}
Therefore, by the standard distinguished triangle (\ref{sdt}), we have a canonical isomorphism
\begin{equation}
\int_\pi\mathcal{O}_{X\times (\Gm)_y}y^\gamma e^{yh}\simeq\int_jj^\dagger\int_\pi\mathcal{O}_{X\times (\Gm)_y}y^\gamma e^{yh}.
\end{equation}
We are going to compute the latter complex. We consider the following cartesian square:
\begin{equation}
\xymatrix{
\Big(X\setminus h^{-1}(0)\Big)\times(\Gm)_y \ar[r]^{\;\;\;\;\;\;\tilde{j}} \ar[d]_{\tilde{\pi}^\prime}&X\times(\Gm)_y \ar[d]^{\pi}\\
X\setminus h^{-1}(0) \ar[r]^{ \;\;\;\;\;\; j} &X.}
\end{equation}
Again by projection formula, we have
\begin{equation}
j^\dagger\int_\pi\mathcal{O}_{X\times (\Gm)_y}y^\gamma e^{yh}\simeq\int_{\tilde{\pi}^\prime}\tilde{j}^\dagger\mathcal{O}_{X\times (\Gm)_y}y^\gamma e^{yh}.
\end{equation}
\noindent
We consider an isomorphism $\varphi: \Big( X\setminus h^{-1}(0)\Big)\times(\Gm)_y\tilde{\rightarrow}\Big( X\setminus h^{-1}(0)\Big)\times(\Gm)_y$ defined by $\varphi(x,y)=(x,h(x)y).$ Since $\tilde{\pi}^\prime=\tilde{\pi}^\prime\circ\varphi,$ we have
\begin{equation}
\int_{\tilde{\pi}^\prime}\tilde{j}^\dagger\mathcal{O}_{X\times (\Gm)_y}y^\gamma e^{yh}\simeq\int_{\tilde{\pi}^\prime}\int_\varphi\mathcal{O}_{\Big(X\setminus h^{-1}(0)\Big)\times (\Gm)_y}y^\gamma e^{yh}\simeq\int_{\tilde{\pi}^\prime}\mathcal{O}_{X\setminus h^{-1}(0)}h^{-\gamma}\boxtimes \mathcal{O}_{(\Gm)_y}y^\gamma e^{y}\simeq \mathcal{O}_{X\setminus h^{-1}(0)}h^{-\gamma}.
\end{equation}
Thus, the first isomorphism (\ref{TheFirstIsomorphism}) follows. As for the vanishing result (\ref{TheVanishingResult}), we have a sequence of isomorphisms
\begin{align}
\int_\pi\R\Gamma_{[h^{-1}(0)\times (\Gm)_y]}\left( (\mathbb{L}\pi^*M)y^\gamma e^{yh}\right)&\simeq \int_\pi \int_{\tilde{i}}\tilde{i}^\dagger\left( (\mathbb{L}\pi^*M)y^\gamma e^{yh}\right)\\
&\simeq \int_{\pi\circ\tilde{i}}\left( \mathbb{L}(\pi\circ\tilde{i})^*My^\gamma\right)[-1]\\
&\simeq M\overset{\D}{\otimes}\int_{\pi\circ\tilde{i}}\mathcal{O}_{h^{-1}(0)\times (\Gm)_y}y^\gamma[-1]\\
&\simeq M\overset{\D}{\otimes}\int_{i\circ\tilde{\pi}}\mathcal{O}_{h^{-1}(0)\times (\Gm)_y}y^\gamma[-1]\\
&\simeq 0.
\end{align}
\qed
\begin{rem}
In the proof above, we have used the following simple fact: Let $X$ be a smooth algebraic variety, and $f:X\rightarrow X$ be an isomorphism. Then, we have an identity
\begin{equation}\label{Cart}
\int_f\simeq (f^{-1})^{\dagger}=\mathbb{L}(f^{-1})^*.
\end{equation}
Indeed, base change formula applied to the following cartesian diagram gives the identity (\ref{Cart}):
\begin{equation}
\xymatrix{
X \ar[r]^{{\rm id}_X } \ar[d]_{f^{-1}}&X\ar[d]^{{\rm id}_X}\\
X \ar[r]^{f} &X.
}
\end{equation}
\end{rem}
Repeated applications of the \cref{prop:GammaInt} give the following
\begin{cor}\label{cor:gamma}
Let $X$ be a smooth algebraic variety over $\C$, $h_l:X\rightarrow\A^1$ $(l=1,\cdots, k)$ be non-zero regular functions such that $h_l^{-1}(0)$ are smooth, $\pi:X\times (\Gm)_y^k\rightarrow X$ be the canonical projection, $j:X\setminus \lef h_1\dots h_k=0\righ\hookrightarrow X$ be the inclusion, and let $\gamma_l\in\C\setminus\Z$ be parameters. In this setting, for any object $M\in D^b_{q.c.}(\DD_X)$, one has a natural isomorphism
\begin{equation}
\int_{\pi}(\mathbb{L}\pi^*M)y_1^{\gamma_1}\dots y_k^{\gamma_k} e^{y_1h_1+\dots+y_kh_k}\simeq\int_j(j^\dagger M)h_1^{-\gamma_1}\cdots h_k^{-\gamma_k}.
\end{equation}
\end{cor}
The following theorem proves the equivalence of Laplace integral representation and Euler-Laplace integral representation.
\begin{thm}[Cayley trick for Euler-Laplace integrals]\label{thm:CayleyTrick}
Let $h_{l,z^{(l)}}(x)=\displaystyle\sum_{j=1}^{N_l}z_j^{(l)}x^{{\bf a}^{(l)}(j)}$ $(l=0,1,\dots,k)$ be Laurent polynomials on $(\Gm)^n_x$. We put $N=N_0+\cdots+N_k$, $z=(z^{(0)},\dots,z^{(k)})$, $X_0=\A^N_z\times (\Gm)_x^n\setminus\lef (z,x)\in\A^N\times(\Gm)^n\mid h_{1,z^{(1)}}(x)\cdots h_{k,z^{(k)}}(x)=0\righ$, and $X_k=\A^N_z\times(\Gm)_y^k\times(\Gm)_x^n$. Let $\pi:X_0\rightarrow\A^N_z$ and $\varpi: X_k\rightarrow\A^N_z$ be projections and $\gamma_l\in\C\setminus\Z$ be parameters. Then, one has an isomorphism
\begin{equation}\label{Desired}
\int_\pi\mathcal{O}_{X_0 }e^{h_{0,z^{(0)}}(x)}h_{1,z^{(1)}}(x)^{-\gamma_1}\cdots h_{k,z^{(k)}}(x)^{-\gamma_k}x^c \simeq\int_\varpi\mathcal{O}_{X_k}y^\gamma x^c e^{h_z(y,x)},
\end{equation}
where $h_z(y,x)=h_{0,z^{(0)}}(x)+\displaystyle\sum_{l=1}^ky_lh_{l,z^{(l)}}(x).$
\end{thm}
\begin{proof}
Note first that hypersurfaces $\{ (z,x)\in\A^N\times (\Gm)^n\mid h_{l,z^{(l)}}(x)=0\}\subset\A^N_z\times(\Gm)^n_x$ $(l=1,\dots,k)$ are smooth. Now, consider the following commutative diagram:
\begin{equation}
\xymatrix{
X_0 \ar[d]_{\pi } \ar[dr]_{j}& \\
\A^N_z & \A^N_z\times(\Gm)_x^n \ar[l]_{\tilde{\pi}}\\
X_k \ar[ur]^{p} \ar[u]_{\varpi} & .
}
\end{equation}
\noindent
By projection formula,
\begin{align}
&\int_j\mathcal{O}_{X_0 }h_{1,z^{(1)}}(x)^{-\gamma_1}\cdots h_{k,z^{(k)}}(x)^{-\gamma_k}x^c e^{h_{0,z^{(0)}}(x)}\nonumber\\
\simeq&\int_j\Big( \mathcal{O}_{X_0 }h_{1,z^{(1)}}(x)^{-\gamma_1}\cdots h_{k,z^{(k)}}(x)^{-\gamma_k}\Big)\overset{\mathbb{D}}{\otimes}\mathcal{O}_{\A^N_z\times(\Gm)^n_x}x^ce^{h_{0,z^{(0)}}(x)}.\label{CIsom1}
\end{align}
\noindent
By \cref{cor:gamma}, we have
\begin{equation}
\int_j\Big( \mathcal{O}_{X_0 }h_{1,z^{(1)}}(x)^{-\gamma_1}\cdots h_{k,z^{(k)}}(x)^{-\gamma_k}\Big)\simeq\int_p\mathcal{O}_{X_k}y^\gamma e^{y_1h_{1,z^{(1)}}(x)+\cdots+y_kh_{k,z^{(k)}}(x)}.
\end{equation}
Again by projection formula, we have
\begin{equation}\Big(\int_p\mathcal{O}_{X_k}y^\gamma e^{y_1h_{1,z^{(1)}}(x)+\cdots+y_kh_{k,z^{(k)}}(x)}\Big)\overset{\mathbb{D}}{\otimes}\mathcal{O}_{\A^N_z\times(\Gm)^n_x}x^ce^{h_{0,z^{(0)}}(x)}\simeq\int_p\mathcal{O}_{X_k}y^\gamma x^c e^{h_z(y,x)}\label{CIsom2}
\end{equation}
\noindent
Since one has canonical isomorphisms
\begin{equation}
\int_\pi\simeq\int_{\tilde{\pi}}\circ\int_j\;\;\;\;\;\;\;\;\int_\varpi\simeq\int_{\tilde{\pi}}\circ\int_p\;\;\;,
\end{equation}
applying the functor $\int_{\varpi}$ to the left hand side of (\ref{CIsom1}) and to the right hand side of (\ref{CIsom2}) gives the desired formula (\ref{Desired}).
\end{proof}
\begin{cor}\label{cor:Cor2.5}
Under the assumption of \cref{thm:CayleyTrick}, one has a canonical isomorphism
\begin{equation}
\int_{\pi!}\mathcal{O}_{X_0 }e^{h_{0,z^{(0)}}(x)}h_{1,z^{(1)}}(x)^{-\gamma_1}\cdots h_{k,z^{(k)}}(x)^{-\gamma_k}x^c \simeq\int_{\varpi!}\mathcal{O}_{X_k}y^\gamma x^c e^{h_z(y,x)}
\end{equation}
\end{cor}
\begin{proof}
Let $\iota:X_k\rightarrow X_k$ be an involution defined by $\iota(z,y,x)=(z,-y,x)$. Then, we see that $\varpi\circ\iota=\varpi$. This identity implies an equality $\int_\varpi=\int_\varpi\circ\int_\iota$, from which we obtain an identity
\begin{equation}
\int_{\varpi}\mathcal{O}_{X_k}y^\gamma x^c e^{h_z(y,x)}=\int_{\varpi}\mathcal{O}_{X_k}y^\gamma x^c e^{h_z(-y,x)}.
\end{equation}
In view of this identity and two equalities $\HD_{\A^N_z}\circ\int_\pi=\int_{\pi!}\circ\HD_{X_0}$ and $\HD_{\A^N_z}\circ\int_\varpi=\int_{\varpi!}\circ\HD_{X_k}$, we obtain the desired isomorphism by applying $\HD_{\A^N_z}$ to
(\ref{Desired}) and replace $-\gamma$, $-c$ and $-z^{(0)}$ by $\gamma$, $c$ and $z^{(0)}$.
\end{proof}
Let us refer to the result of M.Schulze and U.Walther (\cite[Corollary 3.8]{SchulzeWaltherLGM}, see also \cite{SchulzeWaltherIrreducibility}) which relates $M_A(c)$ for non-resonant parameters to Laplace-Gauss-Manin connection. It is stated in the following form.
\begin{thm}[\cite{SchulzeWaltherLGM}]
Let $\phi:(\Gm)^n_x\rightarrow\A^N$ be a morphism defined by $\phi(x)=(x^{{\bf a}(1)},\dots,x^{{\bf a}(N)})$. If $c$ is non-resonant, one has a canonical isomorphism
\begin{equation}\label{SWisom}
M_A(c)\simeq \FL\circ\int_\phi\mathcal{O}_{(\Gm)^n}x^c,
\end{equation}
where $\FL$ stands for Fourier-Laplace transform.
\end{thm}
\noindent
Recall that the parameter $c$ is non-resonant (with respect to $A$) if for any face $\Gamma<\Delta_A$ such that $0\in \Gamma$, one has $c\notin\Z^{n\times 1}+\spanning_{\C}\Gamma$.
For readers' convenience, we include a proof of an isomorphism which rewrites the right-hand side of (\ref{SWisom}) as a direct image of an integrable connection. The following result is essentially obtained in \cite{EsterovTakeuchi}.
\begin{prop}\label{prop:EsterovTakeuchi}
Let $f_j\in\mathcal{O}(X)\setminus\C$ $(j=1,\dots,p)$ be non-constant regular functions. Put $f=(f_1,\dots,f_p):X\rightarrow\A^p_\zeta$. Define the Fourier-Laplace transform $\FL:D^b_{q.c.}(\mathcal{D}_{\A^p_\zeta})\rightarrow D^b_{q.c.}(\mathcal{D}_{\A^p_z})$ by the formula
\begin{equation}
\FL(N)=\int_{\pi_z}(\mathbb{L}\pi^*_\zeta N)\overset{\mathbb{D}}{\otimes}\mathcal{O}_{\A^p_\zeta\times\A^p_z}e^{z\cdot\zeta},
\end{equation}
where $\pi_z:\A^p_z\times\A^p_\zeta\rightarrow\A^p_z$ and $\pi_{\zeta}:\A^p_z\times\A^p_\zeta\rightarrow\A^p_\zeta$ are canonical projections. Let $\pi:X\times\A^p_z\rightarrow\A^p_z$ be the canonical projection. Under these settings, for any object $M\in D^b_{q.c.}(\DD_X),$ one has an isomorphism
\begin{equation}
\FL\Big(\int_f M\Big)\simeq\int_\pi\left\{ (M\boxtimes\mathcal{O}_{\A^p_z})\overset{\mathbb{D}}{\otimes}(\mathcal{O}_{X\times\A^p_z}e^{\sum_{j=1}^pz_jf_j})\right\}.
\end{equation}
\end{prop}
\begin{proof}
Consider the following commutative diagram
\begin{equation}
\xymatrix{
X\times\A^p_z \ar[d]_{\pi } \ar[r]^{f\times \id}& \A^p_\zeta\times\A^p_z \ar[dl]_{\pi_z}\\
\A^p_z &
}.
\end{equation}
\noindent
By the projection formula, we have canonical isomorphisms
\begin{align}
\FL\Big(\int_fM\Big)&\simeq\int_{\pi_z}\left\{ \Big(\left(\int_fM\right)\boxtimes\mathcal{O}_{\A^p_z}\Big)\overset{\mathbb{D}}{\otimes}\mathcal{O}_{\A^p_z\times\A^p_\zeta}e^{z\cdot\zeta }\right\}\\
&\simeq\int_{\pi_z}\left\{ \Big(\int_{f\times\id_z}M\boxtimes\mathcal{O}_{\A^p_z}\Big)\overset{\mathbb{D}}{\otimes}\mathcal{O}_{\A^p_z\times\A^p_\zeta}e^{z\cdot\zeta }\right\}\\
&\simeq\int_\pi\left\{ \left(M\boxtimes\mathcal{O}_{\A^p_z}\right)\overset{\mathbb{D}}{\otimes}(\mathcal{O}_{X\times\A^p_z}e^{\sum_{j=1}^pz_jf_j})\right\}.
\end{align}
\end{proof}
\noindent
If we take $X$ to be $(\Gm)^n_x,$ $M$ to be $\mathcal{O}_{(\Gm)^n_x}x^c,$ and $f$ to be $f=(x^{{\bf a}(1)},\dots,x^{{\bf a}(N)}),$ we have
\begin{equation}
\FL\Big(\int_f\mathcal{O}_{(\Gm)^n_x}x^c\Big)\simeq\int_\pi\mathcal{O}_{(\Gm)^n_x\times\A^N_z}x^ce^{h_{z}(x)},
\end{equation}
where $h_z(x)=\displaystyle\sum_{j=1}^Nz_jx^{{\bf a}(j)}.$ Therefore, we obtain a
\begin{cor}\label{cor:Cor2.9}
If $c$ is non-resonant, one has a canonical isomorphism
\begin{equation}
M_A(c)\simeq\int_\pi\mathcal{O}_{(\Gm)^n\times\A^N}x^ce^{h_z(x)}.
\end{equation}
\end{cor}
We have a similar result for the Fourier transform of the proper direct image. For the proof, we need a simple
\begin{lem}\label{lem:Duality}
For any objects $M,N\in D^b_{coh}(\DD_X)$, if the inclusion ${\rm Ch}(M)\cap{\rm Ch}(N)\subset T^*_XX$ holds, one has a canonical quasi-isomorphism $\D_X(M\overset{\D}{\otimes}N)\simeq \D_X M\overset{\D}{\otimes}\D_X N.$
\end{lem}
\noindent
The proof of this lemma will be given in the appendix.
\begin{prop}\label{prop:Prop2.11}
Under the setting of \cref{prop:EsterovTakeuchi}, for any $M\in D^b_{coh}(\DD_X)$, one has
\begin{equation}
\FL\Big(\int_{f!}M\Big)\simeq\int_{\pi!}\left\{ (M\boxtimes\mathcal{O}_{\A^p_z})\overset{\mathbb{D}}{\otimes}(\mathcal{O}_{X\times\A^p_z}e^{\sum_{j=1}^pz_jf_j})\right\}.
\end{equation}
\end{prop}
\begin{proof}
By \cite[PROPOSITION2.2.3.2.]{Daia}, for any $N\in D^b_{coh}(\DD_{\A^N_\zeta})$, we have a canonical isomorphism ${\rm FL}(N)\simeq\int_{\pi_z!}(\mathbb{L}\pi_\zeta^*N)\overset{\HD}{\otimes}\mathcal{O}_{\A^p_z\times\A_\zeta^p}e^{z\cdot\zeta}.$ We remark that the convention of inverse image functor in \cite{Daia} is different from ours. By \cite[Theorem 2.7.1.]{HTT}, we see that functors $\mathbb{L}\pi^*_\zeta$ and $\HD$ commute. Therefore, by \cref{lem:Duality}, we have
\begin{align}
\FL\Big(\int_{f!}M\Big)&\simeq\int_{\pi_z!}\left(\mathbb{L}\pi_\zeta^*\left( \int_{f!}M\right)\right)\overset{\HD}{\otimes}\mathcal{O}_{\A^p_z\times\A_\zeta^p}e^{z\cdot\zeta}\\
&\simeq \HD\circ\int_{\pi_z}\left(\mathbb{L}\pi_\zeta^*\left( \int_{f}\HD M\right)\right)\overset{\HD}{\otimes}\mathcal{O}_{\A^p_z\times\A_\zeta^p}e^{-z\cdot\zeta}\\
&\overset{\cref{lem:Duality}}{\simeq} \HD\circ\int_{\pi_z}\left(\D \mathbb{L}\pi_\zeta^*\left( \int_{f!} M\right)\right)\overset{\HD}{\otimes}\mathcal{O}_{\A^p_z\times\A_\zeta^p}e^{-z\cdot\zeta}\\
&\simeq\HD\circ\int_{\pi_z}\left\{ ((\HD M)\boxtimes\mathcal{O}_{\A^p_z})\overset{\mathbb{D}}{\otimes}(\mathcal{O}_{X\times\A^p_z}e^{-\sum_{j=1}^pz_jf_j})\right\}\\
&\simeq \HD\circ\int_{\pi_z}\left\{ \HD (M\boxtimes\mathcal{O}_{\A^p_z})\overset{\mathbb{D}}{\otimes}(\mathcal{O}_{X\times\A^p_z}e^{-\sum_{j=1}^pz_jf_j})\right\}\\
&\overset{\cref{lem:Duality}}{\simeq} \int_{\pi_z!}\left\{ (M\boxtimes\mathcal{O}_{\A^p_z})\overset{\mathbb{D}}{\otimes}(\mathcal{O}_{X\times\A^p_z}e^{\sum_{j=1}^pz_jf_j})\right\}.
\end{align}
\end{proof}
\noindent
Now, we use the same notation as \cref{thm:CayleyTrick}. We put
\begin{equation}
\Phi=\Phi(z,x)=e^{h_{0,z^{(0)}}(x)}h_{1,z^{(1)}}(x)^{-\gamma_1}\cdots h_{k,z^{(k)}}(x)^{-\gamma_k}x^c,\;\; \Phi_k=y^\gamma x^c e^{h_z(y,x)}
\end{equation}
to simplify the notation. Let us formulate and prove the main theorem of this section. We put $N=N_0+N_1+\cdots+N_k$, define an $n\times N_l$ matrix $A_l$ by $A_l=({\bf a}^{(l)}(1)\mid\cdots\mid {\bf a}^{(l)}(N_l))$. Then, we define the Cayley configuration $A$ as an $(n+k)\times N$ matrix by
\begin{equation}\label{CayleyConfig}
A
=
\left(
\begin{array}{ccc|ccc|ccc|c|ccc}
0&\cdots&0&1&\cdots&1&0&\cdots&0&\cdots&0&\cdots&0\\
\hline
0&\cdots&0&0&\cdots&0&1&\cdots&1&\cdots&0&\cdots&0\\
\hline
&\vdots& & &\vdots& & &\vdots& &\ddots& &\vdots& \\
\hline
0&\cdots&0&0&\cdots&0&0&\cdots&0&\cdots&1&\cdots&1\\
\hline
&A_0& & &A_1& & &A_2& &\cdots & &A_k&
\end{array}
\right).
\end{equation}
We define a morphism $j_A:(\Gm)^k_y\times (\Gm)^n_x\rightarrow\A^N_z$ by $j_A(y,x)=(y,x)^A$. In view of the proof of \cite[LEMMA 4.2]{EsterovTakeuchi}, one has a canonical isomorphism $\int_{j_A!}\mathcal{O}_{(\Gm)^k_y\times (\Gm)^n_x}y^\gamma x^c\overset{\sim}{\rightarrow}\int_{j_A}\mathcal{O}_{(\Gm)^k_y\times (\Gm)^n_x}y^\gamma x^c$. Combining \cref{thm:CayleyTrick}, \cref{cor:Cor2.5}, \cref{cor:Cor2.9}, and \cref{prop:Prop2.11}, we have the following main result of this section.
\begin{thm}\label{thm:mainDresult}
Suppose that the parameter
$\delta=\begin{pmatrix}
\gamma_1\\
\vdots\\
\gamma_k\\
c
\end{pmatrix}
$ is non-resonant and $\gamma_l\notin\Z$ for $l=1,\dots,k$. Then, one has a sequence of canonical isomorphisms of $\DD_{\A^{N}_z}$-modules
\begin{equation}\label{EulerLaplaceGaussManin}
M_A(\delta)\simeq\int_\varpi\mathcal{O}_{X_k}\Phi_k\simeq\int_\pi\mathcal{O}_{X_0 }\Phi.
\end{equation}
\noindent
Moreover, the regularization conditions
\begin{equation}
\int_\varpi\mathcal{O}_{X_k}\Phi_k\simeq\int_{\varpi!}\mathcal{O}_{X_k}\Phi_k
\;\;\;\text{ and }\;\;\;
\int_\pi\mathcal{O}_{X_0}\Phi\simeq\int_{\pi!}\mathcal{O}_{X_0 }\Phi
\end{equation}
hold.
\end{thm}
Recall that a point $z\in\A^{N}$ is said to be Newton non-degenerate if for any face $\Gamma$ of $\Delta_A$ which does not contain the origin, the set $\{ (y,x)\in(\Gm)^k\times (\Gm)^n\mid d_{(y,x)}h^\Gamma_z(y,x)=0\}$ is empty. Here, we denote by $h^\Gamma_z(y,x)$ the Laurent polynomial associated to the face $\Gamma$ (before LEMMA 3.3 of \cite{Adolphson}). We set $d_x+\sum_{i=1}^nc_i\frac{dx_i}{x_i}\wedge-\sum_{l=1}^k\gamma_l\frac{d_xh_{l,z^{(l)}}(x)}{h_{l,z^{(l)}}(x)}\wedge+d_xh_{0,z^{(0)}}(x)\wedge$. The following result is also obtained in \cite{AdolphsonSperber}
\begin{cor}
Under the same assumption of \cref{thm:mainDresult}, if $z$ is a Newton non-degenerate point, the algebraic de Rham cohomology group $\Homo^*_{dR}\left( \pi^{-1}(z);(\mathcal{O}_{\pi^{-1}(z)},\nabla_z)\right)$ is purely $n$-codimensional.
\end{cor}
\begin{proof}
We denote by $\iota_z:\{ z\}\hookrightarrow\A^N$ the canonical inclusion. Since $\iota_z$ is non-characteristic with respect to the $\DD$-module $M_A(\delta)$ (\cite[LEMMA 3.3]{Adolphson}), the complex $\mathbb{L}\iota_z^*M_A(\delta)$ is concentrated in degree $0$. On the other hand, we have $\mathbb{L}\iota_z^*\int_\pi\mathcal{O}_{X_0 }\Phi=DR_{\pi^{-1}(z)/\{ z\}}(\mathcal{O}_{\pi^{-1}(z)},\nabla_z)$ by the projection formula. Therefore, the result follows from the isomorphism (\ref{EulerLaplaceGaussManin}).
\end{proof}
\section{Description of the rapid decay homology groups of Euler-Laplace integrals}\label{SectionRapidDecay}
We inherit the notation of \S\ref{SectionDModules}. We begin with proving an explicit version of \cref{thm:mainDresult}. Let $Y$ be a smooth product variety $Y=X\times Z$, $X$ be Affine and let $M=(E,\nabla)$ be a (meromorphic) integrable connection on $Y$. We denote $\pi_Z:Y\rightarrow Z$ the canonical projection. We revise the explicit $\DD_Z$-module structure of $\int_{\pi_Z}M$. We can assume that $Z$ is Affine since the argument is local. From the product structure of $Y$, we can naturally define a decomposition $\Omega_{Y}^1(E)=\Omega^1_{Y/X}(E)\oplus\Omega^1_{Y/Z}(E)$. Here, $\Omega^1_{Y/X}(E)$ and $\Omega^1_{Y/Z}(E)$ are the sheaves of relative differential forms with values in $E$. By taking a local frame of $E$, we see that $\nabla$ can locally be expressed as $\nabla=d+\Omega\wedge$ where $\Omega\in\Omega^1(\End (E))$. We see that $\Omega$ can be decomposed into $\Omega=\Omega_x+\Omega_z$ with $\Omega_x\in \Omega^1_{Y/Z}(\End (E))$ and $\Omega_z\in \Omega^1_{Y/X}(\End (E))$. Then, $\nabla_{Y/Z}=d_x+\Omega_x\wedge$ and $\nabla_{Y/X}=d_z+\Omega_z\wedge$ are both globally well-defined and we have $\nabla=\nabla_{Y/X}+\nabla_{Y/Z}$. Here, $\nabla_{Y/X}:\mathcal{O}_{Y}(E)\rightarrow\Omega^1_{Y/X}(E)$ and $\nabla_{Y/Z}:\mathcal{O}_{Y}(E)\rightarrow\Omega^1_{Y/Z}(E)$. Note that the integrability condition $\nabla^2=0$ is equivalent to three conditions $\nabla_{Y/X}^2=0, \nabla_{Y/Z}^2=0,$ and $\nabla_{Y/X}\circ\nabla_{Y/Z}+\nabla_{Y/Z}\circ\nabla_{Y/X}=0$. For any (local algebraic) vector field $\theta$ on $Z$ and any form $\omega\in\Omega_{Y/Z}^*(E)$, we define the action $\theta\cdot \omega$ by $\theta\cdot \omega=\iota_\theta(\nabla_{Y/X}\omega)$, where $\iota_\theta$ is the interior derivative. In this way, ${\rm DR}_{Y/Z}(E,\nabla)=(\Omega^{\dim X+*}_{Y/Z}(E),\nabla_{Y/Z})$ is a complex of $\DD_Z$-modules. It can be shown that ${\rm DR}_{Y/Z}(E,\nabla)$ represents $\int_{\pi_Z}M$ (\cite[pp.45-46]{HTT}).
For any non-constant regular function $h$ on $Y$ and a parameter $\gamma\in\C\setminus\Z$, we are going to give an explicit version of the isomorphism
\begin{equation}
\int^0_{\pi_Z\circ\pi}(\mathbb{L}\pi^*M)y^\gamma e^{yh(x,z)}\simeq \int^0_{\pi_Z}\int_j(j^\dagger M)h^{-\gamma},
\end{equation}
where $\pi:Y\times(\Gm)_y\rightarrow Y$ is the canonical projection, $j:Y\setminus h^{-1}(0)\rightarrow Y$ is the canonical inclusion, and $\int^0$ is the $0$-th cohomology group. We denote $(E_1, \nabla_1)$ the integrable connection $(\mathbb{L}\pi^*M)y^\gamma e^{yh}$. We set $D=h^{-1}(0)\times (\Gm)_y$ and consider a short exact sequence of complexes of $\DD_Z$-modules
\begin{equation}
0\rightarrow{\rm DR}_{Y\times(\Gm)_y/Z}(E_1, \nabla_1)\rightarrow
{\rm DR}_{Y\times(\Gm)_y/Z}((E_1, \nabla_1)(*D))
\rightarrow
\frac{
{\rm DR}_{Y\times(\Gm)_y/Z}((E_1, \nabla_1)(*D))
}
{
{\rm DR}_{Y\times(\Gm)_y/Z}(E_1, \nabla_1)
}
\rightarrow 0.
\end{equation}
Here, the first and the second morphism are natural inclusion and projection respectively. Since the third complex is quasi-isomorphic to $\int_{\pi_Z\circ\pi}\R\Gamma_{[D]}((\mathbb{L}\pi^*M)y^\gamma e^{yh})$, this is quasi-isomorphic to $0$ by (\ref{TheVanishingResult}).
Now, we consider an isomorphism $\varphi:(Y\setminus h^{-1}(0))\times(\Gm)_y\rightarrow (Y\setminus h^{-1}(0))\times(\Gm)_y$ defined by $\varphi(z,x,y)=(z,x,\frac{y}{h(z,x)})$. For any $\omega\in \Omega^p_{Y\times(\Gm)_y/Z}(E_1)(*D)$, we define $\varphi_z^*\omega$ to be the pull-back of $\omega$ by $\varphi$ freezing the variable $z$. More precisely, we consider the decomposition $\Omega_{Y\times(\Gm)_y}^p=\Omega_{Y\times(\Gm)_y/Z}^p\oplus\Omega^1_Z\wedge\Omega^{p-1}_{Y\times(\Gm)_y}$. Then, $\varphi_z^*\omega$ is defined to be the projection of $\varphi^*\omega$ to the component $\Omega_{Y\times(\Gm)_y/Z}^p$. We put $(E_2,\nabla_2)$ to be the meromorphic integrable connection $\left(\int_j (j^\dagger M)h^{-\gamma}\right)\boxtimes \mathcal{O}_{(\Gm)_y}y^\gamma e^y.$ By a direct computation, we can verify that $\varphi_z^*$ induces a $\C$-linear isomorphism of complexes $\varphi_z^*:{\rm DR}_{Y\times(\Gm)_y/Z}((E_1, \nabla_1)(*D))\rightarrow {\rm DR}_{Y\times(\Gm)_y/Z}(E_2, \nabla_2)$. However, this is not a morphism of $\DD_Z$-modules. None the less, we can prove the following
\begin{prop}\label{prop:Prop312}
\begin{equation}
\Homo^{0}(\varphi_z^*):\Homo^{0}({\rm DR}_{Y\times(\Gm)_y/Z}((E_1, \nabla_1)(*D)))\rightarrow \Homo^{0}({\rm DR}_{Y\times(\Gm)_y/Z}((E_2, \nabla_2))
\end{equation}
is an isomorphism of $\DD_Z$-modules.
\end{prop}
\begin{proof}
Remember that the connection $(E,\nabla)$ can locally be expressed as $\nabla=d+\Omega\wedge=d_x+\Omega_x\wedge+d_z+\Omega_z\wedge$. Therefore, we locally have $\nabla_1=\nabla+\gamma\frac{dy}{y}\wedge +d(yh)\wedge=(d_{x,y}+\Omega_x\wedge+\gamma\frac{dy}{y}\wedge+h{dy}\wedge +yd_xh\wedge)+(d_z+\Omega_z\wedge+yd_zh\wedge)$ and
$\nabla_2=\nabla-\gamma\frac{dh}{h}\wedge+\gamma\frac{dy}{y}\wedge+dy\wedge=(d_{x,y}+\Omega_x\wedge-\gamma\frac{d_xh}{h}\wedge+\gamma\frac{dy}{y}\wedge+dy\wedge)+(d_z+\Omega_z-\gamma\frac{d_zh}{h}\wedge)$.
Let us take any element $\xi\in {\rm DR}^{\dim X+1}_{Y\times(\Gm)_y/Z}((E_1, \nabla_1)(*D))$. By definition, $\xi$ can be written in the form $\xi=a(z,x,y)\frac{dy}{y}\wedge\omega(x)$ where $\omega(x)\in\Omega^{\dim X}_{Y/Z}(E)$ and $a(z,x,y)$ is a regular function on $Y\times(\Gm)_y$ having poles along $h^{-1}(0)$. In the following we fix a vector field $\theta$ on $Z$ and compute its actions to $\xi$ and $\varphi^*_z\xi$. In order to emphasize that the actions are different, we write the resulting elements as $\theta\overset{(1)}{\bullet}\xi$ and $\theta\overset{(2)}{\bullet}(\varphi^*_z\xi)$. Firstly, we have an equality
\begin{equation}\label{EquationAbove}
\theta\overset{(1)}{\bullet}\xi=(\theta a)(z,x,y)\frac{dy}{y}\wedge\omega(x)+\Omega_z(\theta)\xi+y(\theta h)(z,x)\xi.
\end{equation}
Applying $\varphi_z^*$ to (\ref{EquationAbove}), we have
\begin{align}
\varphi_z^*(\theta\overset{(1)}{\bullet}\xi)=&(\theta a)(z,x,\frac{y}{h(z,x)})\frac{dy}{y}\wedge\omega(x)+\Omega_z(\theta)a(z,x,\frac{y}{h(z,x)})\frac{dy}{y}\wedge\omega(x)\nonumber\\
&+\frac{y}{h(z,x)}(\theta h)(z,x)a(z,x,\frac{y}{h(z,x)})\frac{dy}{y}\wedge\omega(x).
\end{align}
\noindent
Secondly, by a direct computation, we have an equality
\begin{equation}
\theta\overset{(2)}{\bullet}(\varphi^*_z\xi)=(\theta a)(z,x,\frac{y}{h(z,x)})\omega(x)\wedge\frac{dy}{y}-\frac{y (\theta h)(z,x)}{h(z,x)^2}a_y(z,x,\frac{y}{h(z,x)})\omega(x)\wedge\frac{dy}{y}+\Omega_z(\theta)\varphi_z^*\xi-\gamma\frac{(\theta h)(z,x)}{h(z,x)}\varphi_z^*\xi.
\end{equation}
\noindent
Finally, we also have an equality
\begin{align}
& (\nabla_2)_{Y\times(\Gm)_y/Z}\left(\frac{(\theta h)(z,x)}{h(z,x)}a(z,x,\frac{y}{h(z,x)})\omega(x)\right)\nonumber\\
=&\frac{y(\theta h)(z,x)}{h(z,x)^2}a_y(z,x,\frac{y}{h(z,x)})\frac{dy}{y}\wedge\omega(x)+\gamma\frac{(\theta h)(z,x)}{h(z,x)}a(z,x,\frac{y}{h(z,x)})\frac{dy}{y}\wedge\omega(x)\nonumber\\
&+\frac{y (\theta h)(z,x)}{h(z,x)}a(z,x,\frac{y}{h(z,x)})\frac{dy}{y}\wedge\omega(x),
\end{align}
from which we obtain a relation
\begin{equation}
\varphi_z^*(\theta\overset{(1)}{\bullet}\xi)+(\nabla_2)_{Y\times(\Gm)_y/Z}\left(\frac{(\theta h)(z,x)}{h(z,x)}a(z,x,\frac{y}{h(z,x)})\omega(x)\right)=\theta\overset{(2)}{\bullet}(\varphi^*_z\xi).
\end{equation}
Taking the cohomology groups, we can conclude that $\varphi^*_z$ is a morphism of $\DD_Z$-modules.
\end{proof}
We denote $(E_3,\nabla_3)$ the meromorphic connection $\int_j(j^\dagger M)h^{-\gamma}$. The relative de Rham complex for $\int_{\pi_Z}(E_3,\nabla_3)$ is explicitly given by the formula ${\rm DR}_{Y/Z}(E_3,\nabla_3)=\left( \Omega^{\dim X+*}_{Y/Z}(*h^{-1}(0)),\nabla-\gamma\frac{d h}{h}\wedge\right).$
\begin{prop}\label{prop:TrivialProp}
Wedge product induces an isomorphism of complexes of $\DD_Z$-modules
\begin{equation}
{\rm DR}_{(\Gm)_y/{\rm pt}}\left(\mathcal{O}_{(\Gm)_y},d_y+\gamma\frac{dy}{y}\wedge+dy\wedge\right)\boxtimes {\rm DR}_{Y/Z}(E_3,\nabla_3) \overset{\sim}{\rightarrow} {\rm DR}_{Y\times(\Gm)_y/Z}(E_2, \nabla_2).
\end{equation}
\end{prop}
\noindent
The proof of the \cref{prop:TrivialProp} is straightforward. Therefore, in view of \cref{lem:gamma}, we have a quasi-isomorphism of complexes of $\DD_Z$-modules ${\rm DR}_{Y/Z}(E_3,\nabla_3) \overset{\sim}{\rightarrow} {\rm DR}_{Y\times(\Gm)_y/Z}(E_2, \nabla_2)$ which sends any relative $p$-form $\xi\in\Omega^p_{Y/Z}(E)(*h^{-1}(0))$ to $\frac{dy}{y}\wedge\xi$.
Now, we apply the construction above to Euler-Laplace integral representation. For given Laurent polynomials $h_{l,z^{(l)}}(x)$ ($l=0,1,\dots,k$), we put $D_l=\{ h_{l,z^{(l)}}(x)=0\}\subset X_0$. Then, $\int_{\pi}\mathcal{O}_{X_{0} }\Phi$ is isomorphic to the complex
\begin{equation}
{\rm DR}_{\A^N_z\times (\Gm)^n_x/\A^N_z}\left( \mathcal{O}_{\A^N_z\times (\Gm)^n_x}\left(*\left(\sum_{l=1}^kD_l\right)\right),d+\sum_{i=1}^nc_i\frac{dx_i}{x_i}\wedge-\sum_{l=1}^k\gamma_l\frac{dh_{l,z^{(l)}}(x)}{h_{l,z^{(l)}}(x)}\wedge+dh_{0,z^{(0)}}(x)\wedge\right).
\end{equation}
\noindent
In the same way, $\int_{\varpi}\mathcal{O}_{X_{k} }\Phi_k$ is isomorphic to the complex
\begin{equation}
{\rm DR}_{\A^N_z\times (\Gm)^n_x\times (\Gm)^k_y/\A^N_z}\left( \mathcal{O}_{X_k},d+\sum_{i=1}^nc_i\frac{dx_i}{x_i}\wedge+\sum_{l=1}^k\gamma_l\frac{dy_l}{y_l}\wedge+dh_{z}(y,x)\wedge\right).
\end{equation}
\noindent
We set $\frac{dx}{x}=\frac{dx_1\wedge\dots\wedge dx_n}{x_1\dots x_n}$ for brevity. Applying \cref{prop:Prop312} and \cref{prop:TrivialProp} repeatedly, we obtain a
\begin{thm}\label{thm:CyclicGenerator}
There is an isomorphism
\begin{equation}
\int_{\pi}^0\mathcal{O}_{X_{0} }\Phi
\rightarrow
\int_{\varpi}^0\mathcal{O}_{X_{k} }\Phi_k,
\end{equation}
of $\DD_{\A^N_z}$-modules which sends $[\frac{dx}{x}]$ to $[\frac{dy}{y}\wedge\frac{dx}{x}]$.
\end{thm}
\begin{cor}\label{cor:Corollary216}
If the parameter $d$ is non-resonant and $\gamma_l\notin\Z$ for any $l=1,\dots,k$, $M_A(\delta)\ni[1]\mapsto [\frac{dx}{x}]\in \int_{\pi}^0\mathcal{O}_{X_{0} }\Phi$ defines an isomorphism of $\DD_{\A_z^N}$-modules.
\end{cor}
\begin{proof}
In \cite[Lemma 4.7]{EsterovTakeuchi}, it was proved that $[\frac{dy}{y}\wedge\frac{dx}{x}]$ is a cyclic generator (Gauss-Manin vector) of $\int_{\pi_{k}}^0\mathcal{O}_{X_{k} }\Phi_k$. Therefore, by \cref{thm:CyclicGenerator}, $[\frac{dx}{x}]$ is a cyclic generator of $\int_{\pi}^0\mathcal{O}_{X_{0} }\Phi$. On the other hand, it can easily be proved that $M_A(\delta)\ni[1]\mapsto [\frac{dx}{x}]\in \int_{\pi}^0\mathcal{O}_{X_{0} }\Phi$ defines a morphism of $\DD_{\A_z^N}$-modules. When the parameter $d$ is non-resonant, this is an isomorphim since $M_A(\delta)$ is irreducible by \cite{SchulzeWaltherIrreducibility}.
\end{proof}
\begin{comment}
The differentiation can be computed in the following way. For any $u\in\Z^{N\times 1}_{\geq 0}$, and a decomposition
$u=
\begin{pmatrix}
u^{(0)}\\
u^{(1)}\\
\vdots\\
u^{(k)}
\end{pmatrix}
$ we have
$\partial_z^u\bullet[\frac{dx}{x}]=h^{\gamma}e^{-h_0}\partial_z^uh^{-\gamma}e^{h_0}[\frac{dx}{x}]=\prod_{l=1}^k(-1)^{|u^{(l)}|}(\gamma_l)_{|u^{(l)}|}\prod_{l=1}^kh_{l,z^{(l)}}(x)^{-|u^{(l)}|}x^{Au}[\frac{dx}{x}]$. Thus, for any positive integer vectors $u,v\in\Z^{N\times 1}_{\geq 0}$ such that $Au=Av$, we have $\partial_z^u\bullet[\frac{dx}{x}]=\partial_z^v\bullet[\frac{dx}{x}].$
Moreover, we see that $\partial_{z_j^{(0)}}[\frac{dx}{x}]=x^{{\bf a}^{(0)}(j)}[\frac{dx}{x}]$ and $\partial_{z_j^{(l)}}[\frac{dx}{x}]=-\gamma_l\frac{x^{{\bf a}^{(l)}(j)}}{h_{l,z^{(l)}}(x)}[\frac{dx}{x}]$. Therefore, we have $\sum_{j=1}^{N_l}\theta_{z_j^{(l)}}[\frac{dx}{x}]=-\gamma_l[\frac{dx}{x}]$ ($l=1,\dots,k$). Moreover, we also have
\begin{align*}
\nabla_x\left( (-1)^{i-1}\frac{dx_{\hat{i}}}{x_{\hat{i}}}\right)&=c_i\frac{dx}{x}-\sum_{l=1}^k\gamma_l\frac{x_i\frac{\partial h_l}{\partial x_i}}{h_l}\frac{dx}{x}+x_i\frac{\partial h_0}{\partial x_i}\frac{dx}{x}\\
&=c_i\frac{dx}{x}-\sum_{l=1}^k\gamma_l
\sum_{j=1}^{N_l}
\frac{
a_{ij}^{(l)}z_j^{(l)}x^{{\bf a}^{(l)}(j)}
}
{
h_l
}\frac{dx}{x}+\sum_{j=1}^{N_0}a_{ij}^{(0)}z_j^{(0)}x^{{\bf a}^{(0)}(j)}\frac{dx}{x}\\
&=c_i\frac{dx}{x}+\sum_{l=1}^k\sum_{j=1}^{N_l}a_{ij}^{(l)}\theta_{z_j^{(l)}}\frac{dx}{x}+\sum_{j=1}^{N_0}a_{ij}^{(0)}\theta_{z_j^{(0)}}\frac{dx}{x}.
\end{align*}
\end{comment}
Now we discuss the solutions of Laplace-Gauss-Manin connection $\int_{\pi}\mathcal{O}_{X_{0} }\Phi$. For the convenience of the reader we repeat the relevant material from \cite{EsterovTakeuchi} and \cite{HienRDHomology} without proofs, thus making our exposition self-contained. Let $U$ be a smooth complex Affine variety, let $f:U\rightarrow \A^1$ be a non-constant morphism, and let $M=(E,\nabla)$ be a regular integrable connection on $U$. We consider an embedding of $U$ into a smooth projective variety $X$ with a meromorphic prolongation $f:X\rightarrow\mathbb{P}^1$. We assume that $D=X\setminus U$ is a normal crossing divisor. We decompose $D$ as $D=f^{-1}(\infty)\cup D_{irr}$. Then, we denote $\widetilde{X^D}=\widetilde{X}$ the real oriented blow-up of $X$ along $D$ and denote $\varpi:\widetilde{X}\rightarrow X$ the associated morphism (\cite[\S 8.2]{SabbahIntroductionToStokes}). We denote $\widetilde{\PP^1}$ the real oriented blow-up of $\PP^1$ at infinity and $\varpi_\infty:\widetilde{\PP^1}\rightarrow\PP^1$ the associated morphism. Note that the closure of the ray $[0,\infty)e^{\ii\theta}$ in $\widetilde{\PP^1}$ and $\widetilde{\PP^1}\setminus\C$ has a unique intersection point which we will denote by $e^{\ii\theta}\infty$. Now, a morphism $\tilde{f}:\widetilde{X}\rightarrow \widetilde{\PP^1}$ is naturally induced so that it fits into a commutative diagram
\begin{equation}
\xymatrix{
\widetilde{X} \ar[r]^{\tilde{f} } \ar[d]_{\varpi}&\widetilde{\PP^1}\ar[d]^{\varpi_\infty}\\
X \ar[r]^{f} &\PP^1.
}
\end{equation}
\noindent
We set $\widetilde{D^{r.d.}}=\tilde{f}^{-1}\left( \{e^{\ii\theta}\infty\mid \theta\in(\frac{\pi}{2},\frac{3\pi}{2})\}\right)\setminus\varpi^{-1}({D_{irr}})\subset\widetilde{X}$.
\noindent
We put $\mathcal{L}=\left(\Ker \left(\nabla^{an}:\mathcal{O}_{X^{an}}(E^{an})\rightarrow\Omega_{X^{an}}^1(E^{an})\right)\right)^\vee$, where ${}^\vee$ stands for the dual local system. We consider the natural inclusion $U^{an}\overset{j}{\hookrightarrow} U^{an}\cup \widetilde{D^{r.d.}}$. Then, the rapid decay homology group of M.Hien $\Homo^{r.d.}_*\left(U^{an},(E^\vee,\nabla^\vee)\right)$ is defined in this setting by
\begin{equation}
\Homo^{r.d.}_*\left(U^{an},(Me^f)^\vee\right)=\Homo_*\left(U^{an}\cup \widetilde{D^{r.d.}},\widetilde{D^{r.d.}};j_*\mathcal{L}\right)
\end{equation}
(\cite{HienRDHomology}, see also \cite{EsterovTakeuchi} and \cite{MatsubaraRapidDecay}). Note that $U^{an}\cup \widetilde{D^{r.d.}}$ is a topological manifold with boundary and that $j_*\mathcal{L}$ is a local system on $U^{an}\cup \widetilde{D^{r.d.}}$. We set $\Homo_{\rm dR}^{*+\dim U}(U,Me^f)=\Homo^*({\rm DR}_{U/{\rm pt}}(Me^f))$. The main result of \cite{HienRDHomology} states that the period pairing $\Homo^{r.d.}_*\left(U^{an},(Me^f)^\vee\right)\times \Homo_{\rm dR}^{*}(U,Me^f)\rightarrow \C$ is perfect.
\begin{rem}\label{rem:RemarkRDHomology}
We put $\widetilde{D^{r.d.}_0}=\tilde{f}^{-1}\left( \{ e^{\ii\theta}\infty\mid\theta\in (\frac{\pi}{2},\frac{3\pi}{2})\}\right)$ and denote by $\bar{j}$ the natural inclusion $U^{an}\hookrightarrow U^{an}\cup \widetilde{D^{r.d.}_0}$. It can easily be seen that the inclusion $\left( U^{an}\cup \widetilde{D^{r.d.}}, \widetilde{D^{r.d.}}\right)\hookrightarrow \left( U^{an}\cup\widetilde{D_0^{r.d.}},\widetilde{D_0^{r.d.}}\right)$ is a homotopy equivalence (\cite[Lemma 2.3]{MatsubaraRapidDecay}). Therefore, the rapid decay homology group can be computed by the formula $\Homo^{r.d.}_*\left(U^{an},(E^\vee,\nabla^\vee)\right)=\Homo_*\left(U^{an}\cup \widetilde{D_0^{r.d.}},\widetilde{D_0^{r.d.}};\bar{j}_*\mathcal{L}\right).$ Note that this realization is compatible with the period pairing.
\end{rem}
\begin{rem}
The formulation of \cite{HienRoucairol} is not suitable in our setting. In their formulation, $\widetilde{X}$ is taken to be the fiber product $X\underset{\PP^1}{\times}\widetilde{\PP^1}$. However, the corresponding embedding $j:U^{an}\hookrightarrow U^{an}\cup \widetilde{D^{r.d.}}$ may have higher cohomology groups $R^pj_*\underline{\C}_{U^{an}}$. None the less, under a suitable genericity condition of eigenvalues of monodromies of $\mathcal{L}$, we can recover the vanishing of higher direct images $R^pj_*\mathcal{L}$. We do not discuss this aspect in this paper.
\end{rem}
We construct a family of good compactifications $X$ associated to the Laplace-Gauss-Manin connection $\int_{\pi}^0\mathcal{O}_{X_{0} }\Phi.$ First, we put $\Delta_0=\text{\rm convex hull}\{ 0,{\bf a}^{(0)}(1),\dots,{\bf a}^{(0)}(N_0)\}$ and $\Delta_l=\text{\rm convex hull}\{ {\bf a}^{(l)}(1),\dots,{\bf a}^{(l)}(N_l)\}$ ($l=1,\dots,k$). For any covector $\xi\in (\R^n)^*$, we set $\Delta^\xi_l=\{ v\in \Delta_l\mid \langle \xi,v\rangle=\underset{w\in\Delta_l}{\rm min}\langle \xi,w\rangle\}$ and $h^\xi_{l,{z}^{(l)}}(x)=\displaystyle\sum_{{\bf a}(j)\in\Delta^\xi_l}z_jx^{{\bf a}^{(l)}(j)}$. Now, we consider the dual fan $\Sigma$ of the Minkowski sum $\Delta_0+\Delta_1+\dots+\Delta_k$. By taking a refinement if necessary, we may assume that $\Sigma$ is a smooth fan. Then, the associated toric variety $X=X(\Sigma)$ is sufficiently full for any $\Delta_l$ in the sense of \cite{KhovanskiNewton}. We denote $\{ D_j\}_{j\in J}$ the set of torus invariant divisors of $X$.
\begin{defn}\label{defn:Nonsingularity}
We say that a point $z=(z^{(0)},z^{(1)},\dots,z^{(k)})\in\C^N$ is nonsingular if the following two conditions are both satisfied:
\begin{enumerate}
\item For any $1\leq l_1<\dots <l_s\leq k$, the Laurent polynomials $h_{l_1,z^{(l_1)}}(x),\dots,h_{l_s,z^{(l_s)}}(x)$ are nonsingular in the sense of \cite{KhovanskiNewton}, i.e., for any covector $\xi\in (\R^n)^*$, the $s$-form $d_xh^\xi_{l_1,z^{(l_1)}}(x)\wedge\dots\wedge d_xh^\xi_{l_s,z^{(l_s)}}(x)$ never vanishes on the set $\{ x\in(\C^\times)^n\mid h^\xi_{l_1,z^{(l_1)}}(x)=\dots=h^\xi_{l_s,z^{(l_s)}}(x)=0\}$.
\item For any covector $\xi\in (\R^n)^*$ such that $0\notin\Delta_0^\xi$ and for any $1\leq l_1<\dots <l_s\leq k$ ($s$ can be $0$), the $s+1$-form $dh^\xi_{0,z^{(0)}}(x)\wedge dh^\xi_{l_1,z^{(l_1)}}(x)\wedge\dots\wedge dh^\xi_{l_s,z^{(l_s)}}(x)$ never vanishes on the set $\{ x\in(\C^\times)^n\mid h^\xi_{l_1,z^{(l_1)}}(x)=\dots=h^\xi_{l_s,z^{(l_s)}}(x)=0\}$.
\end{enumerate}
\end{defn}
\begin{prop}
The set of nonsingular points is Zariski open and dense.
\end{prop}
\begin{proof}
We say $z\in\A^N$ is singular if it is not nonsingular. We prove that the set $Z\overset{def}{=}\{ z\in\A^N\mid z\text{ is singular}\}\subset\A^N$ is Zariski closed. For this purpose, it is enough to prove that there is a Zariski closed subset $\widetilde{Z}\subset\A^N\times X$ such that $\pi_{\A^N}(\widetilde{Z})=Z$, where $\pi_{\A^N}:\A^N\times X\rightarrow\A^N$ is the canonical projection. Indeed, since $\Sigma$ is a complete fan, $X\rightarrow pt$ is a proper morphism, its base change $\pi_{\A^N}$ is also a closed morphism. We consider the case when the condition 1 of \cref{defn:Nonsingularity} fails. We take a maximal cone $\tau\in\Sigma$. Since $\Sigma$ is taken to be smooth, there are exactly $n$ primitive vectors $\kappa_1,\dots,\kappa_n\in\Z^{n\times 1}\setminus \{0\}$ such that $\tau\cap\Z^{n\times 1}=\Z_{\geq 0}\kappa_1+\dots+\Z_{\geq 0}\kappa_n$. We set $m_i^{(l)}=\underset{a\in\Delta_l}{\rm min}\langle\kappa_i,a\rangle$ for $l=0,\dots,k$, $i=1,\dots,n$. We put $m^{(l)}=(m_1^{(l)},\dots,m_n^{(l)})$. We also choose a coordinate $\xi=(\xi_1,\dots,\xi_n)$ so that the equality $\C[\tau^\vee\cap\Z^{n\times 1}]=\C[\xi]$ holds. Then, $\tilde{h}_{l,z^{(l)}}(\xi)=\xi^{-m^{(l)}}h_{l,z^{(l)}}(\xi)$ ($l=1,\dots,k$) is a polynomial with non-zero constant term. For any subset $I\subset\{ 1,\dots,n\}$, we set $\tilde{h}^I_{l,z^{(l)}}(\xi_{\bar{I}})=\tilde{h}_{l,z^{(l)}}(\xi)\restriction_{\cap_{i\in I}\{ \xi_i=0\}}$. Then, the condition 1 of \cref{defn:Nonsingularity} fails if and only if $d_{\xi_{\bar{I}}}\tilde{h}^I_{l_1,z^{(l_1)}}(\xi_{\bar{I}})\wedge\dots\wedge d_{\xi_{\bar{I}}}\tilde{h}^I_{l_s,z^{(l_s)}}(\xi_{\bar{I}})=0$ for some $\xi_{\bar{I}}\in\left\{ \xi_{\bar{I}}\in\C^{\bar{I}}\mid \tilde{h}^I_{l_1,z^{(l_1)}}(\xi_{\bar{I}})=\dots=\tilde{h}^I_{l_s,z^{(l_s)}}(\xi_{\bar{I}})=0\right\}$. This condition is clearly a Zariski closed condition.
As for condition 2 of \cref{defn:Nonsingularity}, we rearrange the index $\{ 1,\dots,n\}=\{ 1,\dots,i_0,i_0+1,\dots, n\}$ so that $m_i^{(0)}<0$ for $i=1,\dots,i_0$ and $m_i^{(0)}=0$ for $i=i_0+1,\dots,n$. For any subset $I\subset\{ 1,\dots,n\}$ such that $I\cap\{1,\dots,i_0\}\neq\varnothing$, we set $\tilde{h}^I_{0,z^{(0)}}(\xi_{\bar{I}})=\displaystyle\prod_{i\in I}\xi_i^{-m_i^{(0)}} h_{0,z^{(0)}}(\xi)\restriction_{\cap_{i\in I}\{ \xi_i=0\}}$. Then, condition 2 of \cref{defn:Nonsingularity} fails if and only if $\tilde{h}^I_{0,z^{(0)}}(\xi_{\bar{I}})d_{\xi_{\bar{I}}}\tilde{h}^I_{l_1,z^{(l_1)}}(\xi_{\bar{I}})\wedge\dots\wedge d_{\xi_{\bar{I}}}\tilde{h}^I_{l_s,z^{(l_s)}}(\xi_{\bar{I}})=0$ and
$d_{\xi_{\bar{I}}}\tilde{h}^I_{0,z^{(0)}}(\xi_{\bar{I}})\wedge d_{\xi_{\bar{I}}}\tilde{h}^I_{l_1,z^{(l_1)}}(\xi_{\bar{I}})\wedge\dots\wedge d_{\xi_{\bar{I}}}\tilde{h}^I_{l_s,z^{(l_s)}}(\xi_{\bar{I}})=0$ for some $\xi_{\bar{I}}\in\left\{ \xi_{\bar{I}}\in\C^{\bar{I}}\mid \tilde{h}^I_{l_1,z^{(l_1)}}(\xi_{\bar{I}})=\dots=\tilde{h}^I_{l_s,z^{(l_s)}}(\xi_{\bar{I}})=0\right\}$. This is also a Zariski closed condition. Finally, the non-emptiness of nonsingular points follows immediately from the description above and Bertini-Sard's lemma.
\end{proof}
\begin{rem}
If $k=0$, the nonsingularity condition is equivalent to the non-degenerate condition of \cite[p274]{Adolphson}. In general, nonsingularity condition is stronger than non-degenerate condition. Never the less, it is still a Zariski open dense condition as we saw above.
\end{rem}
In the following, we fix a nonsingular $z$ and a small positive real number $\varepsilon.$ We denote $\Delta(z;\varepsilon)$ the disk with center at $z$ and with radius $\varepsilon$. By abuse of notation, we denote $D_j$ the product $\Delta(z;\varepsilon)\times D_j$. By the condition 1 of \cref{defn:Nonsingularity}, for any subset $I\subset\{ 1,\dots,k\}$, the closure $Z_I=\overline{\displaystyle\bigcap_{l\in I}\{ (z^\prime,x)\in\Delta(z;\varepsilon)\times(\C^\times)^n_x\mid h_{l,z^{\prime(l)}}(x)=0\}}$ $\subset \Delta(z;\varepsilon)\times X$ intersects transversally with $D_{J^\prime}=\displaystyle\bigcap_{j\in J^\prime}D_j$ for any $J^\prime\subset J$. Let us rename the divisors $D_j$ so that $D_j$ with $j\in J_1$ is a part of the pole divisor of $h_{0,z^{\prime(0)}}(x)$ on $X$ and that any $D_j$ with $j\in J_2$ is not. Then by the condition 2 of \cref{defn:Nonsingularity}, the closure $Z_0=\overline{\{ (z^\prime, x)\in\Delta(z;\varepsilon)\times(\C^\times)^n_x\mid h_{0,z^{\prime(0)}}(x)=0\}}\subset \Delta(z;\varepsilon)\times X$ intersects transversally with $Z_I\cap D_{J^\prime}$ such that $J^\prime\cap J_1\neq \varnothing$.
Now we take a small positive real number $\varepsilon $ and consider the canonical projection $p:\Delta(z;\varepsilon)\times X\rightarrow\Delta(z;\varepsilon)$. We remember the blowing up process of \cite{EsterovTakeuchi} (see also \cite{MatsuiTakeuchi}). We consider a sequence of blow-ups along codimension $2$ divisors $Z_0\cap D_j$ ($j\in J_1$). If the pole order of $h_{0,z^{\prime(0)}}(x)$ along $D_j$ is $m_j\in\Z_{>0}$, one needs at most $m_j$ blow-ups along $Z_0\cap D_j$. Repeating this process finitely many times, we obtain a non-singular complex manifold $\bar{X}$. We denote $\bar{p}:\bar{X}\rightarrow \Delta(z;\varepsilon)$ the composition of the natural morphism $\bar{X}\rightarrow \Delta(z;\varepsilon)\times X$ with the canonical projection $\Delta(z;\varepsilon)\times X\rightarrow \Delta(z;\varepsilon)$. We also denote $\bar{Z}_l$ and $\bar{D}_j$ the proper transforms of $Z_l$ and $D_j$. We equip $\bar{X}$ with the Whitney stratification coming from the normal crossing divisors $\bar{D}=\{ \bar{Z}_l\}_{l=1}^k\cup\{ \bar{D}_j\}_{j\in J}\cup\{\text{exceptional divisors of blow-ups}\}$. We have a diagram $\Delta(z;\varepsilon)\overset{\bar{p}}{\leftarrow} \bar{X}\overset{\bar{h}_{0,z^{\prime(0)}}}{\rightarrow}\mathbb{P}^1$. By construction, we see that $\bar{h}_{0,z^{\prime(0)}}^{-1}(\infty)$ intersects transversally with any stratum of $\bar{p}^{-1}(z^\prime)$. Let us consider a real oriented blow-up $\widetilde{X}=\widetilde{\bar{X}^{\bar{D}}}$ of $\bar{X}$ along $\bar{D}$. We naturally have the following commutative diagram
\begin{equation}
\xymatrix{
\widetilde{X} \ar[r]^{\tilde{h}_{0,z^{\prime(0)}} } \ar[d]_{\varpi}&\widetilde{\PP^1}\ar[d]^{\varpi_\infty}\\
\bar{X} \ar[r]^{\bar{h}_{0,z^{\prime(0)}}} &\PP^1.
}
\end{equation}
We also equip $\widetilde{X}$ with the Whitney stratification coming from the pull-back of the normal crossing divisor $\bar{D}$. We set $\tilde{p}=\bar{p}\circ\varpi$. Then, $\tilde{p}^{-1}(z^\prime)$ for any $z^\prime\in\Delta(z;\varepsilon)$ is naturally equipped with an induced Whitney stratification. By construction, $\tilde{h}_{0,z^{\prime(0)}}^{-1}(e^{\ii\theta}\infty)$ intersects transversally with any stratum of $\tilde{p}^{-1}(z^\prime)$ for any $\theta$. Now it is routine to take a ruguous vector field $\Theta$ on $\widetilde{X}$ with an additional condition
\begin{equation}\label{AdditionalCondition}
\Theta (\tilde{h}_{0,z^{\prime(0)}}(x))=0
\end{equation}
near $\tilde{h}_{0,z^{\prime(0)}}^{-1}(S^1\infty )$ (\cite{Verdier}, see also \cite[\S3.3.]{HienRoucairol}). Taking the flow of $\Theta$, we have a stratified trivialization of $\widetilde{X}$ with respect to the morphism $\tilde{p}:\widetilde{X}\rightarrow\Delta(z;\varepsilon)$. We define $\widetilde{D^{r.d.}}\subset\widetilde{X}$ by the formula $\widetilde{D^{r.d.}}=\tilde{h}_{0,z^{\prime(0)}}^{-1}\left( (\frac{\pi}{2},\frac{3\pi}{2})\infty\right)$ and put $\widetilde{D^{r.d.}_{z}}=\widetilde{D^{r.d.}}\cap \tilde{p}^{-1}(z)$. With the aid of the additional condition (\ref{AdditionalCondition}), we have a local trivialization
\begin{equation}\label{Trivialization}
\xymatrix{
\left(\pi^{-1}(z)\cup \widetilde{D^{r.d.}_{z}}\right)\times \Delta(z;\varepsilon) \ar[d]_{ } \ar[r]^{\Lambda}& \pi^{-1}(\Delta(z;\varepsilon))\cup \widetilde{D^{r.d.}} \ar[dl]_{\tilde{\pi}}\\
\Delta(z;\varepsilon) &
}
\end{equation}
with an additional condition $\Lambda\left(\widetilde{D^{r.d.}_{z}}\times \Delta(z;\varepsilon)\right)\subset \widetilde{D^{r.d.}}$. Here, the first vertical arrow is the canonical projection. It is clear that $\tilde{p}^{-1}(z^\prime)$ is a good compactification for any $z^\prime\in\Delta(z;\varepsilon)$. For any $z^\prime\in\A^N$, we denote $\Phi_{z^\prime}$ the multivalued function on $\pi^{-1}(z^\prime)$ defined by $\pi^{-1}(z^\prime)\ni x\mapsto \Phi(z^\prime,x)$. Denoting $j_{z}:\pi^{-1}(z)^{an}\hookrightarrow \pi^{-1}(z)^{an}\cup \widetilde{D^{r.d.}_{z}}$ the natural inclusion, we set
\begin{equation}
\Homo^{r.d.}_{*,z}=\Homo_*\left( \pi^{-1}(z)^{an}\cup \widetilde{D^{r.d.}_{z}},\widetilde{D^{r.d.}_{z}};j_{z*}\left(\underline{\C} \Phi_{z}\right)\right).
\end{equation}
\begin{thm}\label{thm:SolutionDescription}
For any nonsingular $z\in\C^N$, the map
\begin{equation}
\int:\Homo^{r.d.}_{n,z}\ni[\Gamma]\mapsto\left( [\omega]\mapsto\int_\Gamma \Phi \omega \right)\in\Hom_{\DD_{\C^N}}\left( \left(\int_{\pi}^0\mathcal{O}_{X_{0} }\Phi\right)^{an},\mathcal{O}_{\C^N}\right)_{z}\label{Morphism}
\end{equation}
is well-defined and injective.
\end{thm}
\begin{proof}
Note first that, for any $[\omega]\in \int_{\pi}^0\mathcal{O}_{X_{0} }\Phi$, the integral
\begin{equation}
f(z)=\int_\Gamma \Phi\omega
\end{equation}
is well-defined for any $z^\prime$ sufficiently close to $z$. Indeed, with the aid of the trivialization (\ref{Trivialization}), one can construct a continuous family $\{\Gamma_{z^\prime}\}_{z^\prime\in\Delta(z;\varepsilon)}$ of rapid decay cycles such that $\Gamma_{z}=\Gamma$. For any $z^\prime$ close to $z$, $\Gamma_{z^\prime}$ is homotopic to $\Gamma$. Moreover, if $f(z)=0$ for any $[\omega]$, by the duality theorem of \cite{HienRDHomology}, we have $[\Gamma]=0$.
\end{proof}
\begin{rem}
The assumption that $z$ is nonsingular is important. As a simple example, we consider a Laplace-Gauss-Manin connection $\int_\pi \mathcal{O}_{\A^2_z\times \Gm}e^{z_1x+z_2x^2}x^c$ with $c\notin\Z$. In this case, we can easily see that $z$ is nonsingular (non-degenerate) if $z_2\neq 0$. Let us fix a point $z=(1,0).$ Then, the Hankel contour $\Gamma$ which begins from $-\infty$ turns around the origin and goes back to $-\infty$ belongs to $\Homo^{r.d.}_{1,z}.$ However, as soon as ${\rm Re}(z_2)> 0$, the integral $\int_\Gamma e^{x+z_2x^2}x^c\frac{dx}{x}$ diverges.
\end{rem}
As an application of \cref{thm:SolutionDescription}, we have the following
\begin{thm}\label{thm:Theorem221}
Suppose the parameter vector $\delta$ is non-resonant and $\gamma_l\notin\Z$ for any $l=1,\dots,k$. Suppose that $z\in\C^N$ is nonsingular. Then the morphism (\ref{Morphism}) is an isomorphism.
\end{thm}
\begin{proof}
In view of (\ref{FormalRestriction}), \cref{thm:mainDresult} and projection formula, we have isomorphisms
\begin{align}
\R\Hom_{\C}\left( \int_{\pi}\mathcal{O}_{\pi^{-1}(z)}\Phi_{z},\C\right)&\simeq\R\Hom_{\DD_{\C^{N}}}\left( \int_{\pi}\mathcal{O}_{X_0}\Phi,\hat{\mathcal{O}}_{z}\right)\\
&\simeq\R\Hom_{\DD_{\C^{N}}}\left( M_A(\delta),\hat{\mathcal{O}}_{z}\right),
\end{align}
where $\hat{\mathcal{O}}_{z}$ is the ring of formal power series with center at $z$. Taking the $0$-th cohomology groups of both sides, we obtain an equality
\begin{equation}
\dim_{\C}\Homo^{r.d.}_{n,z}=\dim_{\C}\Hom_{\DD_{\C^{N}}}\left( M_A(\delta),\hat{\mathcal{O}}_{z}\right).
\end{equation}
By \cref{thm:SolutionDescription} and the inequality
\begin{equation}\label{TheTrivialInequality}
\dim_{\C}\Hom_{\DD_{\C^{N}}}\left( M_A(\delta),\mathcal{O}_{z}\right)\leq \dim_{\C}\Hom_{\DD_{\C^{N}}}\left( M_A(\delta),\hat{\mathcal{O}}_{z}\right),
\end{equation}
we obtain the theorem.
\end{proof}
By \cref{cor:Corollary216}, an isomorphism
\begin{equation}
\Hom_{\DD_{\C_z^N}}(\int_{\pi}^0\mathcal{O}_{X_{0} }\Phi,\mathcal{O}_{\C^N})\rightarrow\Hom_{\DD_{\C^N_z}}(M_A(\delta),\mathcal{O}_{\C^N})
\end{equation}
is induced. In view of \cref{thm:Theorem221}, we obtain the second main result of this section.
\begin{thm}\label{thm:EulerLaplaceRepresentationTheorem}
Suppose the parameter vector $d$ is non-resonant and $\gamma_l\notin\Z$ for any $l=1,\dots,k$. Suppose that $z\in\C^N$ is nonsingular. Then the morphism
\begin{equation}\label{Integration}
\Homo^{r.d.}_{n,z}\overset{\int}{\rightarrow}\Hom_{\DD_{\C^N_z}}(M_A(\delta),\mathcal{O}_{\C^N})_{z}
\end{equation}
given by
\begin{equation}
[\Gamma]\mapsto \int_\Gamma \Phi \frac{dx}{x}
\end{equation}
is an isomorphism of $\C$-vector spaces.
\end{thm}
\begin{rem}
We denote $\Omega$ the Zariski open dense subset of $\A^N$ consisting of nonsingular points. It is straightforward to construct a local system $\mathcal{H}^{r.d.}_{n}=\displaystyle\bigcup_{z\in\Omega^{an}}\Homo^{r.d.}_{n,z}\rightarrow\Omega^{an}$ and an isomorphism $\mathcal{H}^{r.d.}_{n}\overset{\int}{\rightarrow}\Hom_{\DD_{\C^N_z}}(M_A(\delta),\mathcal{O}_{\C^N})\restriction_{\Omega^{an}}$ whose stalks are identical with (\ref{Integration}). See the proofs of \cite[Proposition 3.4. and Theorem 3.5.]{HienRoucairol}.
\end{rem}
\section{Rapid decay intersection pairing and its localization}\label{RDIntersection}
In this section, we develop an intersection theory of rapid decay homology groups along the line of the preceding studies \cite{ChoMatsumoto}, \cite{GotoCyclesForFC}, \cite{IwasakiWittenLaplacian}, \cite{KitaYoshida2}, \cite{MajimaMatsumotoTakayama}, \cite{MimachiYoshida}, and \cite{OharaSugikiTakayama}. We use the same notation as \S\ref{SectionRapidDecay}. Namely, we consider a smooth complex Affine variety $U$ and a regular singular connection $(E,\nabla)$ on $U$. In order to simplify the discussion and the notation, we assume that $E$ is a trivial bundle and $\nabla$ is given by $\nabla=d+\sum_{i=1}^k\alpha_i\frac{df_i}{f_i}\wedge$ for some regular functions $f_i\in\mathcal{O}(U)\setminus\C$ and complex numbers $\alpha_i$. For another regular function $f\in\mathcal{O}(U)$, we set $\nabla_f=\nabla+df\wedge$. We take a standard orientation of $\C^n$ so that for any holomorphic coordinate $(z_1,\dots,z_n)$, the real form $\left(\frac{\ii}{2}\right)^ndz_1\wedge\dots\wedge dz_n\wedge d\bar{z}_1\wedge\dots\wedge d\bar{z}_n$ is positive. Note that this choice of orientation is not compatible with the product orientation. For example, our orientation of $\C^2$ is different from the product orientation of $\C\times\C$.
Let us recall several sheaves on the real oriented blow-up $\widetilde{X}$. We denote by $\mathcal{P}^{<D}_{\widetilde{X}}$ (resp. $\mathcal{P}^{{\rm mod}D}_{\widetilde{X}}$) the sheaf of $C^\infty$ functions on $U^{an}$ which are flat (resp. moderate growth) along $D$. We denote by $\Omega^{(p,q)}_{X^{an}}$ the sheaf of $(p,q)$-forms on $X^{an}$. We set $\mathcal{A}^{?D}_{\widetilde{X}}=\Ker\left( \bar{\partial}:\mathcal{P}^{?D}_{\widetilde{X}}\rightarrow \mathcal{P}^{?D}_{\widetilde{X}}\otimes_{\varpi^{-1}\mathcal{O}_{X^{an}}}\varpi^{-1}\Omega^{(0,1)}_{X^{an}}\right)$ for $?=<,{\rm mod}$. As in \cite{HienRDHomology}, we set
\begin{equation}\label{RapidDecayComplex}
DR^{?D}_{\widetilde{X}}(\nabla_f):=\mathcal{A}^{?D}_{\widetilde{X}}\otimes_{\varpi^{-1}\mathcal{O}_{X^{an}}}\varpi^{-1}DR_{X^{an}}(\nabla_f).
\end{equation}
With this notation, we set ${}^\vee\mathcal{S}^{<D}:=\mathcal{H}^0\left( DR^{<D}_{\widetilde{X}}(\nabla^\vee_f)\right)$ and $\mathcal{S}^{{\rm mod}D}:=\mathcal{H}^0\left( DR^{{\rm mod}D}_{\widetilde{X}}(\nabla_f)\right)$.
The main result of \cite{HienRDHomology} is an explicit description of the perfect pairing between $\mathbb{H}^{2n-*}(\widetilde{X},{}^\vee\mathcal{S}^{<D})$ and $\mathbb{H}^*(\widetilde{X},\mathcal{S}^{{\rm mod}D})$. Namely, $\mathbb{H}^{2n-*}(\widetilde{X},{}^\vee\mathcal{S}^{<D})$ is described by the rapid decay homology group, $\mathbb{H}^*(\widetilde{X},\mathcal{S}^{{\rm mod}D})$ is the algebraic de Rham cohomology group, and the pairing is given by the exponential period pairing. For later use, we need other realizations of these cohomology groups and the period pairing.
We first remark that $DR_{\widetilde{X}}^{<D}(\nabla_f^\vee)$ and $DR_{\widetilde{X}}^{{\rm mod}D}(\nabla_f)$ are resolutions of ${}^\vee\mathcal{S}^{<D}$ and $\mathcal{S}^{{\rm mod}D}$ respectively (\cite[Proposition 1]{HienRDHomology}). Combining this result with the quasi-isomorphism
\begin{equation}
\mathcal{A}^{?D}_{\widetilde{X}}\otimes_{\varpi^{-1}\mathcal{O}_{X^{an}}}\varpi^{-1}\Omega^{r}_{X^{an}}\tilde{\rightarrow}\left(\mathcal{P}^{?D}_{\widetilde{X}}\otimes_{\varpi^{-1}\mathcal{O}_{X^{an}}}\varpi^{-1}\Omega^{(r,\bullet)}_{X^{an}},\bar{\partial}\right),
\end{equation}
we see that ${}^\vee\mathcal{S}^{<D}$ and $\mathcal{S}^{{\rm mod}D}$ are quasi-isomorphic to
$
\mathcal{P}DR^{<D}_{\widetilde{X}}(\nabla^\vee_f)\overset{def}{=}\left(\mathcal{P}^{<D}_{\widetilde{X}}\otimes_{\varpi^{-1}\mathcal{O}_{X}}\varpi^{-1}\Omega^{(\bullet,\bullet)}_{X},\nabla^\vee_f,\bar{\partial}\right)$
and
$\mathcal{P}DR^{{\rm mod}D}_{\widetilde{X}}(\nabla_f)\overset{def}{=}\left(\mathcal{P}^{{\rm mod}D}_{\widetilde{X}}\otimes_{\varpi^{-1}\mathcal{O}_{X}}\varpi^{-1}\Omega^{(\bullet,\bullet)}_{X},\nabla_f,\bar{\partial}\right)
$
respectively. We set
\begin{equation}
\Homo_{r.d.}^*\left( U,\nabla_f^\vee\right)\overset{def}{=}\mathbb{H}^{*}\left( \widetilde{X}; {}^\vee\mathcal{S}^{<D}\right).
\end{equation}
\noindent
Since $\mathcal{P}DR^{<D}_{\widetilde{X}}(\nabla^\vee_f)$ is a soft resolution of ${}^\vee\mathcal{S}^{<D}$, $\Homo_{r.d.}^*\left( U,(\mathcal{O}_U,\nabla_f^\vee)\right)$ can be computed by taking global sections of the complex $\mathcal{P}DR^{<D}_{\widetilde{X}}(\nabla^\vee_f)$. We also remark that the factorization $\Gamma_{\widetilde{X}}=\Gamma_X\circ\R\varpi$ and the relation $\R\varpi_*\left( \mathcal{A}^{{\rm mod}D}_{\widetilde{X}}\right)=\mathcal{O}_{X^{an}}(*D)$ (\cite[CHAPITRE I\vspace{-.1em}I, Corollaire 1.1.8]{SabbahDim2}) implies $\mathbb{H}^*(\widetilde{X},\mathcal{S}^{{\rm mod}D})=\Homo^*\left(\R\Gamma_X\R\varpi\left( \mathcal{P}DR^{{\rm mod}D}_{\widetilde{X}}(\nabla_f)\right)\right)=\mathbb{H}^*\left(X^{an};\mathcal{O}_{X^{an}}\otimes_{\varpi^{-1}\mathcal{O}_{X^{an}}}\varpi^{-1}DR_{X^{an}}(\nabla_f)\right)=\Homo_{\rm dR}^*\left( U,\nabla_f\right)$. Here, the last equality is a consequence of GAGA (\cite{SerreGAGA}).
Hence, if we denote by $\tilde{j}:U^{an}\rightarrow \widetilde{X}$ the natural inclusion, the canonical duality pairing $DR^{<D}_{\widetilde{X}}(\nabla^\vee_f)\otimes DR^{{\rm mod}D}_{\widetilde{X}}(\nabla_f)\rightarrow\tilde{j}_!\C_{U^{an}}$ (\cite{HienRDHomology}[Theorem 3]) yields a perfect pairing
\begin{equation}\label{RDc.i.n.}
\begin{array}{cccc}
\langle\bullet,\bullet\rangle_{ch} &\Homo_{\rm dR}^{*}\left( U; \nabla_f\right)\times\Homo_{r.d.}^{2n-*}\left( U,\nabla_f^\vee\right) &\rightarrow&\C\\
&\rotatebox{90}{$\in$}& &\rotatebox{90}{$\in$}\\
&(\omega,\eta) &\mapsto&\int\omega\wedge\eta.
\end{array}
\end{equation}
\noindent
We also give a realization of $\mathbb{H}^*(\widetilde{X},\mathcal{S}^{{\rm mod}D})$ in terms of a certain relative homology group. We set $\widetilde{D^{r.g.}}=\left( \widetilde{D_{irr}}\setminus\tilde{f}^{-1}(S^1\infty)\right)\cup\tilde{f}^{-1}\left( \left\{ e^{\theta\ii}\infty\mid -\frac{\pi}{2}<\theta<\frac{\pi}{2}\right\}\right)$. We consider a sequence of natural inclusions $U^{an}\overset{l}{\rightarrow} U^{an}\cup \widetilde{D^{r.g.}}\overset{k}{\rightarrow}\widetilde{X}$. By the local description of $\mathcal{S}^{{\rm mod}D}$ (\cite{HienRDHomology}[p12]), we can easily confirm that the equality $\mathcal{S}^{{\rm mod}D}=k_!l_*\mathcal{L}^\vee$ holds. Moreover, if we set
\begin{equation}
\mathcal{C}^{-p}_{U^{an}\cup \widetilde{D^{r.g.}},\widetilde{D^{r.g.}}}(\mathcal{L}^\vee)=\left( V\mapsto S_p\left( U^{an}\cup \widetilde{D^{r.g.}},\left(U^{an}\cup \widetilde{D^{r.g.}}\setminus V\right)\cup\widetilde{D^{r.g.}};l_*\mathcal{L}^\vee\right)\right)^\dagger,
\end{equation}
we see that $\mathcal{S}^{{\rm mod}D}_{\widetilde{X}}[2n]\simeq k_!\mathcal{C}^{-*}_{U^{an}\cup \widetilde{D^{r.g.}},\widetilde{D^{r.g.}}}(\mathcal{L}^\vee)$ as in the arguments after Proposition 2.1 of \cite{MatsubaraRapidDecay}. Therefore, we have a realization $\mathbb{H}^*\left( \widetilde{X};\mathcal{S}^{{\rm mod}D}\right)=\Homo^{r.g.}_{2n-*}\left( U,\nabla_f\right)\overset{def}{=}\Homo_{2n-*}\left( U^{an}\cup \widetilde{D^{r.g.}},\widetilde{D^{r.g.}};l_*\mathcal{L}^\vee\right)$. Moreover, the same argument as \S 5 of \cite{HienRDHomology} proves the perfectness of the pairing
\begin{equation}
\begin{array}{ccc}
\Homo_{r.d.}^{*}\left( U,\nabla_f^\vee\right)\times \Homo^{r.g.}_{2n-*}\left( U,\nabla_f\right) &\rightarrow&\C\\
\rotatebox{90}{$\in$}& &\rotatebox{90}{$\in$}\\
(\eta,\delta^\vee) &\mapsto&\int_{\delta^\vee}e^{-f}\prod_{i=1}^kf_i^{-\alpha_i}\eta.
\end{array}
\end{equation}
With these setups, we can naturally define the Poincar\'e duality isomorphism $\Phi:\Homo^{r.d.}_{*}\left( U^{an},\nabla_f^\vee\right)\overset{\sim}{\rightarrow}\Homo_{r.d.}^{2n-*}\left( U,\nabla_f^\vee\right)$ and $\Phi^\vee:\Homo^{r.g.}_{*}\left( U,\nabla_f\right)\overset{\sim}{\rightarrow}\Homo^{2n-*}_{dR}\left(U,\nabla_f\right)$. Namely, for any element $[\gamma]\in\Homo^{r.d.}_{*}\left( U^{an},\nabla_f^\vee\right)$, $\Phi(\gamma)\in\Homo_{r.d.}^{2n-*}\left( U,\nabla_f^\vee\right)$ is the unique element such that the equality $\int_{\gamma}e^f\prod_{i=1}^kf_i^{\alpha_i}\omega=\int \Phi(\gamma)\wedge\omega$ holds for any $\omega\in\Homo^{*}_{dR}\left(U,\nabla_f\right)$. In the same way, for any element $[\delta^\vee]\in\Homo^{r.g.}_{*}\left( U,\nabla_f\right)$, $\Phi^\vee(\delta^\vee)\in\Homo^{2n-*}_{dR}\left(U,\nabla_f\right)$ is the unique element such that the equality $\int_{\delta^\vee}e^{-f}\prod_{i=1}^kf_i^{-\alpha_i}\eta=\int \Phi^\vee(\delta^\vee)\wedge\eta$ holds for any $\eta\in \Homo_{r.d.}^{*}\left( U,\nabla_f^\vee\right)$. We define the homology intersection pairing $\langle\bullet,\bullet\rangle_h$ by
\begin{equation}
\begin{array}{cccc}
\langle\bullet,\bullet\rangle_h:&\Homo_*^{r.d.}\left( U,\nabla^\vee_f\right)\times\Homo^{r.g.}_{2n-*}\left( U,\nabla_f\right)&\rightarrow&\C\\
&\rotatebox{90}{$\in$}& &\rotatebox{90}{$\in$}\\
&(\gamma,\delta^\vee) &\mapsto&\int\Phi(\gamma)\wedge\Phi^\vee(\delta^\vee).
\end{array}
\end{equation}
We are ready to state the twisted period relation (cf. \cite[Theorem 2]{ChoMatsumoto}) for rapid decay homology groups. Let us fix four bases $\{\omega_i\}_{i=1}^r\subset\Homo_{dR}^{*}\left( U; \nabla_f\right)$, $\{\gamma_i\}_{i=1}^r\subset\Homo_*^{r.d.}\left( U;\nabla_f^\vee\right)$, $\{\eta_i\}_{i=1}^r\subset\Homo_{r.d.}^{2n-*}\left( U;\nabla_f^\vee\right)$, and $\{\delta_i^\vee\}_{i=1}^r\subset\Homo^{r.g.}_{2n-*}\left( U;\nabla_f\right)$. We set $I_{ch}=(\langle \omega_i, \eta_j\rangle_{ch})_{i,j}$, $I_h=(\langle \gamma_i,\delta_j^\vee\rangle_h)_{i,j}$, $P=\left( \int_{\gamma_j}e^{f}\prod_{l=1}^kf_l^{\alpha_l}\omega_i\right)_{i,j}$, and $P^\vee=\left( \int_{\delta_j^\vee}e^{-f}\prod_{l=1}^kf_l^{-\alpha_l}\eta_i\right)_{i,j}$. With these setups, we can state a basic
\begin{prop}
The following identity is true:
\begin{equation}\label{GeneralQuadraticRelation}
I_{ch}=P{}^tI_h^{-1}{}^tP^\vee.
\end{equation}
\end{prop}
\noindent
The proof is exactly same as that of the twisted period relation \cite[Theorem 2]{ChoMatsumoto}.
Now, we need to prove an important technique to compute homology intersection numbers. For any open subset $\widetilde{V}$ of $\widetilde{X}$, we set $V=\widetilde{V}\cap U$, $DR^{?D}_{\widetilde{V}}(\nabla_f)=DR^{?D}_{\widetilde{X}}(\nabla_f)\restriction_{\widetilde{V}}$ and $\mathcal{S}^{?D}_{V}=\mathcal{S}^{?D}_{\widetilde{X}}\restriction_{\widetilde{V}}$. Let $a_{\widetilde{V}}:\widetilde{V}\rightarrow pt$ denote the morphism to a point. We set $\widetilde{D_V^{r.d.}}=\widetilde{D^{r.d.}}\cap\widetilde{V}$, $\widetilde{D_V^{r.g.}}=\widetilde{D^{r.g.}}\cap\widetilde{V}$, $\Homo^{r.g.}_*\left( V; \nabla_f\right)=\Homo_{*}^{lf}\left( V\cup \widetilde{D_V^{r.g.}},\widetilde{D_V^{r.g.}};l_*\mathcal{L}^\vee\right)$, $\Homo^*_{r.g.}\left(V; \nabla_f\right)=\mathbb{H}^{*}\left( \widetilde{V};DR^{{\rm mod}D}_{\widetilde{V}}(\nabla_f)\right)$,
$\Homo^{r.d.}_*\left(V;\nabla_f^\vee\right)=\Homo_{*}\left( V\cup \widetilde{D_V^{r.d.}},\widetilde{D_V^{r.d.}};\mathcal{L}\right)$, and $\Homo_{r.d.}^*\left(V;\nabla_f^\vee\right)=\mathbb{H}_c^{*}\left( \widetilde{V};DR^{<D}_{\widetilde{V}}(\nabla_f^\vee)\right)$. Here, $\Homo^{lf}$ stands for the locally finite (or Borel-Moore) homology group. We denote by $\tilde{j}_V:V\rightarrow\widetilde{V}$ the natural inclusion. By the perfectness of the duality pairing $DR_{\widetilde{V}}^{<D}(\nabla_f^\vee)\otimes DR^{\rm mod D}_{\widetilde{V}}(\nabla_f)\rightarrow \tilde{j}_{V!}\C$ and the identity $a_{\widetilde{V}}^!\C\simeq\tilde{j}_{V!}\C[2n]$ we get a sequence of isomorphisms
\begin{align}
\R\Gamma_{\widetilde{V}}\left( DR^{\rm mod D}_{\widetilde{V}}(\nabla_f)[2n]\right)&\simeq\R\Gamma_{\widetilde{V}}\R\mathcal{H}om\left( DR_{\widetilde{V}}^{<D}(\nabla_f^\vee),\tilde{j}_{V!}\C_V[2n]\right)\\
&\simeq\R\mathcal{H}om\left( a_{\widetilde{V}!}DR_{\widetilde{V}}^{<D}(\nabla_f^\vee),\C\right).
\end{align}
\noindent
Here, the last isomorphism is a result of Poincar\'e-Verdier duality. Since we have identities $\mathbb{H}^*\left( \widetilde{V};\mathcal{S}^{{\rm mod}D}_{V}\right)=\Homo^{r.g.}_{2n-*}\left( V; \nabla_f\right)$ and $\mathbb{H}^*\left( \widetilde{V};\mathcal{S}^{<D}_{V}\right)=\Homo^{r.d.}_{2n-*}\left(V;\nabla_f^\vee\right)$, this isomorphism gives rise to perfect pairings
\begin{equation}
\begin{array}{ccc}
\Homo^{r.d.}_*\left( V;\nabla_f^\vee\right)\times \Homo^{*}_{r.g.}\left( V;\nabla_f\right)&\rightarrow&\C\\
\rotatebox{90}{$\in$}& &\rotatebox{90}{$\in$}\\
(\gamma,\omega)&\mapsto&\int_{\gamma}e^{f}\prod_{i=1}^kf_i^{\alpha_i}\omega,
\end{array}
\end{equation}
\begin{equation}
\begin{array}{ccc}
\Homo_{r.d.}^*\left( V;\nabla_f^\vee\right)\times \Homo_{*}^{r.g.}\left(V;\nabla_f\right)&\rightarrow&\C\\
\rotatebox{90}{$\in$}& &\rotatebox{90}{$\in$}\\
(\eta,\delta^\vee)&\mapsto&\int_{\delta^\vee}e^{-f}\prod_{i=1}^kf_i^{-\alpha_i}\eta,
\end{array}
\end{equation}
and
\begin{equation}
\begin{array}{cccc}
\langle\bullet,\bullet\rangle_{ch}:&\Homo^{*}_{r.g.}\left(V;\nabla_f\right)\times \Homo_{r.d.}^{2n-*}\left( V;\nabla_f^\vee\right)&\rightarrow&\C\\
&\rotatebox{90}{$\in$}& &\rotatebox{90}{$\in$}\\
&(\omega,\eta)&\mapsto&\int\omega\wedge\eta.
\end{array}
\end{equation}
\noindent
Then, the Poincar\'e duality morphisms $\Phi_{V}:\Homo^{r.d.}_*\left( V;\nabla_f^\vee\right)\tilde{\rightarrow}\Homo_{r.d.}^{2n-*}\left( V;\nabla_f^\vee\right)$ and its dual counterpart $\Phi_{V}^\vee:\Homo^{r.g.}_*\left( V;\nabla_f\right)\tilde{\rightarrow}\Homo_{r.g.}^{2n-*}\left( V;\nabla_f\right)$ are naturally defined with the aid of the perfect pairings above. Namely, the definitions of $\Phi_V$ and $\Phi^\vee_V$ are same as those of $\Phi$ and $\Phi^\vee.$ Therefore, we define the homology intersection pairing by
\begin{equation}
\begin{array}{cccc}
\langle\bullet,\bullet\rangle_h:&\Homo^{r.d.}_*\left( V;\nabla_f^\vee\right)\times\Homo^{r.g.}_{2n-*}\left( V;\nabla_f\right)&\rightarrow&\C\\
&\rotatebox{90}{$\in$}& &\rotatebox{90}{$\in$}\\
&(\gamma,\delta^\vee) &\mapsto&\int\Phi(\gamma)\wedge\Phi^\vee(\delta^\vee).
\end{array}
\end{equation}
With these set-ups, we can discuss localization of intersection pairings. If we denote by $\iota_{\widetilde{X}\widetilde{V}}:\widetilde{V}\rightarrow\tilde{X}$ the natural inclusion, the natural transform
$
id_{\widetilde{X}}\rightarrow\iota_{\widetilde{X}\widetilde{V}*}\iota_{\widetilde{X}\widetilde{V}}^{-1}
$
induces a commutative diagram
\begin{equation}
\xymatrix{
\Homo_{*}^{r.g.}\left( U;\nabla_f\right) \ar[r]^{rest } \ar[d]_{\Phi^\vee}&\Homo_{*}^{r.g.}\left( V;\nabla_f\right)\ar[d]^{\Phi^\vee}\\
\Homo_{\rm dR}^{2n-*}\left( U;\nabla_f\right) \ar[r]^{rest} &\Homo^{2n-*}_{r.g.}\left( V;\nabla_f\right).
}
\end{equation}
Here, the first horizontal morphism is nothing but the usual restriction morphism of locally finite homology groups and the second horizontal morphism is induced by taking pull-backs of differential forms.
On the other hand, the natural transform $\iota_{\widetilde{X}\widetilde{V}!}\iota_{\widetilde{X}\widetilde{V}}^{-1}\rightarrow id_{\widetilde{X}}$ induces a commutative diagram
\begin{equation}
\xymatrix{
\Homo_{*}^{r.d.}\left( U;\nabla^\vee_f\right) \ar[d]_{\Phi}&\Homo_{*}^{r.d.}\left( V;\nabla_f^\vee\right)\ar[l]^{\iota_{\widetilde{X}\widetilde{V}!} } \ar[d]^{\Phi}\\
\Homo^{2n-*}_{r.d.}\left( U;\nabla_f^\vee\right) &\Homo^{2n-*}_{r.d.}\left( V;\nabla_f^\vee\right)\ar[l]^{\iota_{\widetilde{X}\widetilde{V}!}} .
}
\end{equation}
\noindent
Both horizontal morphisms are given by extension by zero. By definition, we see that the morphisms $rest$ and $\iota_{\widetilde{X}\widetilde{V}!} $ satisfy an adjoint relation
\begin{equation}\label{Adjoint}
\langle \iota_{\widetilde{X}\widetilde{V}!}(\gamma),\delta^\vee\rangle_h=\langle \gamma,rest(\delta^\vee)\rangle_h\;\;\;\;\left(\gamma\in\Homo_{*}^{r.d.}\left( V;\nabla_f^\vee\right),\delta^\vee\in\Homo_{2n-*}^{r.g.}\left( U;\nabla_f\right)\right).
\end{equation}
\noindent
We consider a commutative diagram
\begin{equation}\label{ThisCD}
\xymatrix{
\mathcal{S}^{<D}_{\widetilde{X}}\ar[r]^{can}&\mathcal{S}^{{\rm mod}D}_{\widetilde{X}}\ar[dr]&\\
\iota_{\widetilde{X}\widetilde{V}!}\mathcal{S}^{<D}_{\widetilde{V}}\ar[u]\ar[r]&\iota_{\widetilde{X}\widetilde{V}*}\mathcal{S}^{<D}_{\widetilde{V}}\ar[r]&\iota_{\widetilde{X}\widetilde{V}*}\mathcal{S}^{{\rm mod}D}_{\widetilde{V}},
}
\end{equation}
where the morphism $can:\mathcal{S}^{<D}_{\widetilde{X}}\rightarrow\mathcal{S}^{{\rm mod}D}_{\widetilde{X}}$ is induced by the canonical morphism $\mathcal{A}^{<D}_{\widetilde{X}}\rightarrow\mathcal{A}^{{\rm mod}D}_{\widetilde{X}}$ and the vertical arrows are induced by natural transforms
$
id_{\widetilde{X}}\rightarrow\iota_{\widetilde{X}\widetilde{V}*}\iota_{\widetilde{X}\widetilde{V}}^{-1}
$ and
$\iota_{\widetilde{X}\widetilde{V}!}\iota_{\widetilde{X}\widetilde{V}}^{-1}\rightarrow id_{\widetilde{X}}$.
\noindent
By taking hypercohomologies, the diagram (\ref{ThisCD}) induces another commutative diagram
\begin{equation}\label{RDCommutativity}
\xymatrix{
\Homo_*^{r.d.}\left( U;\nabla_f\right)\ar[r]^{can_U}&\Homo_{*}^{r.g.}\left( U;\nabla_f\right)\ar[d]^{rest}\\
\Homo_{*}^{r.d.}\left( V;\nabla_f\right)\ar[u]^{\iota_{\widetilde{X}\widetilde{V}!}}\ar[r]^{can_V}&\Homo_{*}^{r.g.}\left( V;\nabla_f\right).
}
\end{equation}
\noindent
Combining (\ref{Adjoint}) and (\ref{RDCommutativity}), we have the following localization formula
\begin{prop}\label{prop:TheLocalizationFormula}
For any $\gamma\in\Homo_{*}^{r.d.}\left( V;\nabla_f^\vee\right)$ and $\gamma^\vee\in\Homo_{2n-*}^{r.d.}\left( V;\nabla_f\right)$, the identity
\begin{equation}
\langle\iota_{\widetilde{X}\widetilde{V}!}(\gamma),can_U\circ\iota_{\widetilde{X}\widetilde{V}!}(\gamma^\vee)\rangle_h=\langle\gamma,can_V(\gamma^\vee)\rangle_h
\end{equation}
holds.
\end{prop}
Finally, we discuss cross products of chains. Let us consider another smooth complex Affine variety $W$ and a regular connection $\nabla^\prime=d+\sum_{l=1}^m\beta_l\frac{dg_l}{g_l}\wedge:\mathcal{O}_W\rightarrow\Omega^1_W$. We compactify $W$ into a smooth projective variety $Y$ so that $D^\prime=Y\setminus W$ is a normal crossing divisor. Then, we see that $X\times Y$ is a good compactification of $(\mathcal{O}_{U\times W},\nabla_f+\nabla^\prime)$. We set $D_{U\times W}=X\times Y\setminus U\times W.$ By considering K\"unneth isomorphism $\Homo_{\rm dR}^*\left(U\times W,(\mathcal{O}_{U\times W},\nabla_f+\nabla^\prime)\right)\simeq\bigoplus_{p+q=*}\Homo_{\rm dR}^p\left(U,(\mathcal{O}_{U},\nabla_f)\right)\boxtimes\Homo_{dR}^q\left(W,(\mathcal{O}_{ W},\nabla^\prime)\right)$ induced by the wedge product, we see that there is an isomorphism
\begin{equation}
\mathbb{H}^{*}\left( \widetilde{X\times Y};DR^{<D_{U\times W}}_{\widetilde{X\times Y}}(\nabla_f+\nabla^\prime)\right) \simeq \bigoplus_{p+q=*}\mathbb{H}^{p}\left( \widetilde{X};DR^{<D}_{\widetilde{X}}(\nabla_f)\right)\boxtimes\mathbb{H}^{q}\left( \widetilde{Y};DR^{<D_{W}}_{\widetilde{Y}}(\nabla^\prime)\right)
\end{equation}
induced again by the wedge product. Therefore, if we set $\mathcal{L}_W^\vee=\Ker\left( \nabla^\prime:\mathcal{O}_{W^{an}}\rightarrow\Omega^1_{W^{an}}\right)$, for any $\delta^\vee\in\Homo^{r.g.}_p\left( U, (\mathcal{O}_U,\nabla_f)\right)$ and $\delta_W^\vee\in\Homo^{r.g.}_q\left( W, (\mathcal{O}_W,\nabla^\prime)\right)$, we can define the cross product $\delta^\vee\times\delta^\vee_W$ so that the formula
\begin{equation}
\int_{\delta^\vee\times\delta^\vee_W}e^{-f}\prod_{l=1}^kf_l^{-\alpha_l}\prod_{l=1}^mg_l^{-\beta_l}\eta\wedge\eta_W=\left(\int_{\delta^\vee}e^{-f}\prod_{l=1}^kf_l^{-\alpha_l}\eta\right)\left(\int_{\delta_W^\vee}\prod_{l=1}^mg_l^{-\beta_l}\eta_W\right)
\end{equation}
\noindent
is true for any $\eta\in\mathbb{H}^{p}\left( \widetilde{X};DR^{<D}_{\widetilde{X}}(\nabla_f)\right)$ and $\eta_W\in\mathbb{H}^{q}\left( \widetilde{Y};DR^{<D^\prime}_{\widetilde{Y}}(\nabla^\prime)\right)$. Likewise, we can also define the cross product $\gamma\times \gamma_W$ for any $\gamma\in\Homo^{r.d.}_p\left( U,\nabla_f\right)$ and $\gamma_W\in\Homo^{r.d.}_p\left( W,\nabla^\prime\right)$ (\cite[Lemma 2.4]{MatsubaraMellinBarnesKyushu}). If we denote by $n^\prime $ the complex dimension of $W$, we have a
\begin{prop}\label{prop:TheProductFormula}
For $\gamma\in\Homo^{r.d.}_p\left( U,(\mathcal{O}_U,\nabla_f)\right)$, $\gamma_W\in\Homo^{r.d.}_q\left( W,(\mathcal{O}_W,\nabla^\prime)\right)$, $\delta^\vee\in\Homo^{r.g.}_{2n-p}\left( U, (\mathcal{O}_U,\nabla_f)\right)$ and $\delta_W^\vee\in\Homo^{r.g.}_{2n^\prime-q}\left( W, (\mathcal{O}_W,\nabla^\prime)\right)$, one has an identity
\begin{equation}
\langle \gamma\times\gamma_W,\delta^\vee\times\delta^\vee_W\rangle_h=(-1)^{nn^\prime+pq}\langle \gamma,\delta^\vee\rangle_h\langle \gamma_W,\delta^\vee_W\rangle_h.
\end{equation}
In particular, if $p=n$ and $q=n^\prime$, one has $\langle \gamma\times\gamma_W,\delta^\vee\times\delta^\vee_W\rangle_h=\langle \gamma,\delta^\vee\rangle_h\langle \gamma_W,\delta^\vee_W\rangle_h$.
\end{prop}
\noindent
The readers should be aware of our choice of the orientation of $\C^n.$
\section{Review on the combinatorial structure of series solutions}\label{SectionSeries}
In this section, we briefly recall the construction of a basis of solutions of GKZ system in terms of $\Gamma$-series following the exposition of M.-C. Fern\'andez-Fern\'andez (\cite{FernandezFernandez}). For any commutative ring $R$ and for any pair of finite sets $I$ and $J$, we denote by $R^{I\times J}$ the set of matrices with entries in $R$ whose rows (resp. columns) are indexed by $I$ (resp. $J$). For any univariate function $F$ and for any vector $w={}^t(w_1,\dots,w_n)\in\C^{n\times 1}$, we define $F(w)$ by $F(w)=F(w_1)\cdots F(w_n)$. In this section, $A$ is a $d\times n$ $(d<n)$ integer matrix whose column vectors generate the lattice $\Z^{d\times 1}$. Under this notation, for any vector $v\in\C^{n\times 1}$ such that $Av=-\delta,$ we put
\begin{equation}\label{GammaSeries1}
\varphi_v(z)=\displaystyle\sum_{u\in L_A}\frac{z^{u+v}}{\Gamma(1+u+v)}.
\end{equation}
It can readily be seen that $\varphi_{v}(z)$ is a formal solution of $M_A(\delta)$ (\cite{GKZToral}). We call (\ref{GammaSeries1}) a $\Gamma$-series solution of $M_A(\delta)$.
For any subset $\tau\subset\{1,\dots,n\}$, we denote $A_\tau$ the matrix given by the columns of $A$ indexed by $\tau.$ In the following, we take $\sigma\subset\{1,\dots,n\}$ such that the cardinality $|\sigma|$ is equal to $d$ and $\det A_\sigma\neq 0.$
Taking a vector ${\bf k}\in\Z^{\bar{\sigma}\times 1},$ we put
\begin{equation}
v_\sigma^{\bf k}=
\begin{pmatrix}
-A_{\sigma}^{-1}(\delta+A_{\bar{\sigma}}{\bf k})\\
{\bf k}
\end{pmatrix},
\end{equation}
where $\bs$ denotes the complement $\{ 1,\dots,n\}\setminus \s$. Then, by a direct computation, we have
\begin{equation}\label{seriesphi}
\varphi_{\s,{\bf k}}(z;\delta)\overset{\rm def}{=}\varphi_{v_\sigma^{\bf k}}(z)=
z_\sigma^{-A_\sigma^{-1}\delta}
\sum_{{\bf k+m}\in\Lambda_{\bf k}}\frac{(z_\sigma^{-A_\sigma^{-1}A_{\bar{\sigma}}}z_{\bar{\sigma}})^{\bf k+m}}{\Gamma({\bf 1}_\sigma-A_\sigma^{-1}(\delta+A_{\bar{\sigma}}({\bf k+m}))){\bf (k+m)!}},
\end{equation}
where $\Lambda_{\bf k}$ is given by
\begin{equation}\label{lambdak}
\Lambda_{\bf k}=\Big\{{\bf k+m}\in\Z^{\bar{\sigma}\times 1}_{\geq 0}\mid A_{\bar{\sigma}}{\bf m}\in\Z A_\sigma\Big\}.
\end{equation}
The following lemmata can be confirmed immediately from the definitions (\cite[Lemma 3.1,3.2, Remark 3.4.]{FernandezFernandez}).
\begin{lem}
For any ${\bf k},{\bf k^\prime}\in\Z^{\bar{\sigma}\times 1}$, the following statements are equivalent
\begin{enumerate}
\item $v^{\bf k}-v^{\bf k^\prime}\in\Z^{n\times 1}$
\item $[A_{\barsigma} {\bf k}]=[A_{\barsigma} {\bf k^\prime}]$ in $\Z^{d\times 1}/\Z A_\sigma$
\item $\Lambda_{\bf k}=\Lambda_{\bf k^\prime}.$
\end{enumerate}
\end{lem}
\begin{lem}
Take a complete set of representatives $\{ [A_{\barsigma}{\bf k}(i)]\}_{i=1}^{r_\s}$ of the finite Abelian group $\Z^{d\times 1}/\Z A_\sigma.$ Then, one has a decomposition
\begin{equation}
\Z^{\barsigma\times 1}_{\geq 0}=\bigsqcup_{j=1}^{r_\s}\Lambda_{{\bf k}(j)}.
\end{equation}
\end{lem}
\noindent
Thanks to these lemmata, we can observe that $\{\varphi_{\sigma,{\bf k}(i)}(z;\delta)\}_{i=1}^{r_\s}$ is a set of $r_\s$ linearly independent formal solutions of $M_A(\delta)$ unless $\varphi_{\sigma,{\bf k}(i)}(z;\delta)=0$ for some $i$. In order to ensure that $\varphi_{\s,{\bf k}(i)}(z;\delta)$ does not vanish, we say that a parameter vector $\delta$ is very generic with respect to $\sigma$ if $A_\sigma^{-1}(\delta+A_{\bar{\sigma}}{\bf m})$ does not contain any integer entry for any ${\bf m}\in\mathbb{Z}_{\geq 0}^{\bar{\sigma}\times 1}.$ Using this terminology, we can rephrase the observation above as follows:
\begin{prop}\label{prop:independence}
If $\delta\in\C^{d\times 1}$ is very generic with respect to $\sigma$, $\Big\{\varphi_{\s,{\bf k}(i)}(z;\delta)\Big\}_{i=1}^{r_\s}$ is a linearly independent set of formal solutions of $M_A(\delta)$.
\end{prop}
\noindent
As is well-known in the literature, under a genericity condition, we can construct a basis of holomorphic solutions of GKZ system $M_A(\delta)$ consisting of $\Gamma$-series with the aid of regular triangulation. Let us recall the definition of a regular triangulation. In general, for any subset $\sigma$ of $\{1,\dots,n\},$ we denote $\cone(\sigma)$ the positive span of the column vectors of $A$ $\{{\bf a}(1),\dots,{\bf a}(n)\}$ i.e., $\cone(\sigma)=\displaystyle\sum_{i\in\sigma}\R_{\geq 0}{\bf a}(i).$ We often identify a subset $\sigma\subset\{1,\dots,n\}$ with the corresponding set of vectors $\{{\bf a}(i)\}_{i\in\sigma}$ or with the set $\cone(\s)$. A collection $T$ of subsets of $\{1,\dots,n\}$ is called a triangulation if $\{\cone(\sigma)\mid \sigma\in T\}$ is the set of cones in a simplicial fan whose support equals $\cone(A)$. We regard $\Z^{1\times n}$ as the dual lattice of $\Z^{n\times 1}$ via the standard dot product. We denote $\pi_A:\Z^{1\times n}\rightarrow L_A^\vee$ the dual of the natural inclusion $L_A\hookrightarrow \Z^{n\times 1}$ where $L_A^\vee$ is the dual lattice $\Hom_{\Z}(L_A,\Z)$. By abuse of notation, we still denote $\pi_A:\R^{1\times n}\rightarrow L_A^\vee\underset{\Z}{\otimes}\R$ the linear map $\pi_A\underset{\Z}{\otimes}{\rm id}_{\R}$ where ${\rm id}_{\R}:\R\rightarrow\R$ is the identity map. Then, for any generic choice of a vector $\omega\in\pi_A^{-1}\left(\pi_A(\R^{1\times n}_{\geq 0})\right),$ we can define a triangulation $T(\omega)$ as follows: A subset $\sigma\subset\{1,\dots,n\}$ belongs to $T(\omega)$ if there exists a vector ${\bf n}\in\R^{1\times d}$ such that
\begin{align}
{\bf n}\cdot{\bf a}(i)=\omega_i &\text{ if } i\in\sigma\\
{\bf n}\cdot{\bf a}(j)<\omega_j &\text{ if } j\in\barsigma.
\end{align}
A triangulation $T$ is called a regular triangulation if $T=T(\omega)$ for some $\omega\in\R^{1\times n}.$ For a fixed regular triangulation $T$, we say that the parameter vector $\delta$ is very generic if it is very generic with respect to any $\sigma\in T$. Now suppose $\delta$ is very generic. Then, it was shown in \cite{FernandezFernandez} that we have $\rank M_A(\delta)=\vol_\Z(\Delta_A).$ Let us put $H_\sigma=\{ j\in\{ 1,\dots, n \}\mid |A_\sigma^{-1}{\bf a}(j)|=1\}$. Here, $|A_\sigma^{-1}{\bf a}(j)|$ denotes the sum of all entries of the vector $A_\sigma^{-1}{\bf a}(j)$. We set
\begin{equation}
U_\sigma=\left\{z\in(\C^*)^n\mid {\rm abs}\left(z_\sigma^{-A_\sigma^{-1}{\bf a}(j)}z_{j}\right)<R, \text{for all } a(j)\in H_\sigma\setminus\sigma\right\},
\end{equation}
where $R>0$ is a small positive real number and abs stands for the absolute value.
\begin{defn}
A regular triangulation $T$ is said to be convergent if for any $n$-simplex $\s\in T$ and for any $j\in \bs$, one has the inequality $|A_\sigma^{-1}{\bf a}(j)|\leq 1$.
\end{defn}
\begin{rem}
By \cite[Remark 2.1]{FernandezFernandezLocalMonodromy}, there exists at least one convergent regular triangulation.
\end{rem}
With this terminology, the following result is a special case of \cite[Theorem 6.7.]{FernandezFernandez}.
\begin{prop}
Fix a convergent regular triangulation $T$. Assume $\delta$ is very generic. Then, the set
$\displaystyle\bigcup_{\sigma\in T}\left\{ \varphi_{\sigma,{\bf k}(i)}(z;\delta)\right\}_{i=1}^{r_\s}$
is a basis of holomorphic solutions of $M_A(\delta)$ on $U_{T}\overset{def}{=}\displaystyle\bigcap_{\sigma\in T}U_\sigma\neq\varnothing$ where $r_\s=\vol_\Z(\sigma)=|\Z^{d\times 1}/\Z A_\sigma|.$
\end{prop}
\begin{rem}
We define an $N\times\bs$ matrix $B_\sigma$ by
\begin{equation}
B_\sigma=
\begin{pmatrix}
-A_\sigma^{-1}A_{\barsigma}\\
{\bf I}_{\barsigma}
\end{pmatrix}
\end{equation}
and a cone $C_\sigma$ by
\begin{equation}
C_\sigma=\Big\{ \omega\in\R^{1\times n}\mid \omega\cdot B_\sigma>0\Big\}.
\end{equation}
Here, ${\bf I}_{\barsigma}$ is the identity matrix. Then, $T$ is a regular triangulation if and only if $C_{T}\overset{def}{=}\displaystyle\bigcap_{\sigma\in T}C_\sigma$ is a non-empty open cone. In this case, the cone $C_T$ is characterized by the formula
\begin{equation}
C_T=\Big\{ \omega\in\R^{1\times n}\mid T(\omega)=T\Big\}.
\end{equation}
From the definition of $U_\sigma$, we can confirm that $z$ belongs to $U_T$ if $(-\log|z_1|,\dots,-\log|z_n|)$ belongs to a sufficiently far translation of $C_T$ inside itself, which implies $U_T\neq\varnothing.$
\end{rem}
We conclude this section by quoting a result of Gelfand, Kapranov, and Zelevinsky (\cite[Chapter 7, Proposition 1.5.]{GKZbook},\cite[Theorem 5.2.11.]{DeLoeraRambauSantos}).
\begin{thm}[\cite{GKZbook},\cite{DeLoeraRambauSantos}]
There exists a polyhedral fan ${\rm Fan}(A)$ in $\R^{1\times n}$ whose support is \newline $\pi_A^{-1}\left(\pi_A(\R^{1\times n}_{\geq 0})\right)$ and whose maximal cones are exactly $\{ C_T\}_{T: \text{regular triangulation}}$. The fan ${\rm Fan}(A)$ is called the secondary fan.
\end{thm}
\begin{rem}
Let $F$ be a fan obtained by applying the projection $\pi_A$ to each cone of ${\rm Fan}(A)$. By definition, each cone of ${\rm Fan}(A)$ is a pull-back of a cone of $F$ through the projection $\pi_A$. Therefore, the fan $F$ is also called the secondary fan.
\end{rem}
\section{Combinatorial construction of integration contours via regular triangulations}\label{SectionEuler}
In this section, we construct integration contours associated to Euler-Laplace integral representation
\begin{equation}\label{EulerInt2}
f_{\Gamma}(z)=\frac{1}{(2\pi\ii)^{n+k}}\int_\Gamma e^{h_{0,z^{(0)}}(x)} h_{1,z^{(1)}}(x)^{-\gamma_1}\cdots h_{k,z^{(k)}}(x)^{-\gamma_k}x^{c}\frac{dx}{x}.
\end{equation}
with the aid of a convergent regular triangulation. Without loss of generality, we may assume $N_l\geq 2$ for any $l=1,\dots,k$. This is because $N_l=1$ implies that the corresponding Laurent polynomial $h_{l,z^{(l)}}$ is a monomial hence (\ref{EulerInt2}) is reduced to the integral with $k-1$ powers of Laurent polynomials.
\noindent
Suppose we are given Euler-Laplace integral (\ref{EulerInt2}). We denote by ${\bf e}_l\;(l=1,\dots,k)$ the standard basis of $\Z^{k\times 1}$, and put ${\bf e}_0=0\in \Z^{k\times 1}$. We set $I_l=\{ N_0+\dots+N_{l-1}+1,\dots,N_0+\dots+N_l\}$ or equivalently, $I_l=\left\{
\begin{pmatrix}
{\bf e}_l\\
\hline
{\bf a}^{(l)}(j)
\end{pmatrix}
\right\}_{j=1}^{N_l}$ ($l=0,\dots,k$). This induces a partition of indices
\begin{equation}\label{partition}
\{1,\dots,N\}=I_0\cup\dots\cup I_k.
\end{equation}
\noindent
In the following we fix an $(n+k)$-simplex $\s\subset\{1,\dots,N\}$, i.e., a subset with cardinality $n+k$ and $\det A_\s\neq 0$. We also assume an additional condition $|A_\s^{-1}{\bf a}(j)|\leq 1$ for any $j\in\bs$. According to the partition (\ref{partition}), we have an induced partition $\s=\s^{(0)}\cup\dots\cup\s^{(k)}$, where $\s^{(l)}=\s\cap I_l$. By $\bs^{(l)}$, we denote the complement $I_l\setminus\s^{(l)}$. Since $\det A_\s\neq 0$, we have $\s^{(l)}\neq \varnothing$ for any $l=1,\dots,k$. For any finite set $S$, we denote by $|S|$ the cardinality of $S$.
Let us consider an $n$-dimensional projective space $\mathbb{P}^n$ with a homogeneous coordinate $\tau=[\tau_0:\cdots:\tau_n]$. Let $\alpha_0,\dots,\alpha_{n+1}\in\C$ be parameters such that $\alpha_0+\dots+\alpha_{n+1}=1$ and $\omega(\tau)$ be the section of $\Omega_{\mathbb{P}^n}^n(n+1)$ defined by $\omega(\tau)=\sum_{i=0}^n(-1)^i\tau_i d\tau_0\wedge\dots\wedge\widehat{d\tau_i}\wedge\dots\wedge d\tau_n$. We consider an affine open set $U=\{\tau_0\neq 0\}$. We define the coordinate $t=(t_1,\dots,t_n)$ of $U$ by $\frac{\tau_i}{\tau_0}=e^{\pi\ii}t_i$ and $t_{n+1}$ by $t_{n+1}=1-t_1\dots- t_n$. Let $P_{\tau}$ denote the $n$-dimensional Pochhammer cycle in $U$ as in \cite[\S 6]{Beukers} with respect to these coordinates (see also the appendix of this paper). Then we have the following
\begin{lem}\label{lemma:ProjBeukers}(\cite[Proposition 6.1]{Beukers})
For any complex numbers $\alpha_0,\dots,\alpha_{n+1}\in\C$ such that $\alpha_0+\cdots+\alpha_{n+1}=1,$ one has
\begin{align}
\int_{P_\tau}\tau_0^{\alpha_0-1}\cdots\tau_n^{\alpha_n-1}(\tau_0+\dots+\tau_n)^{\alpha_{n+1}-1}\omega(\tau)
&=\frac{(2\pi\ii)^{n+1}e^{-\pi\ii\alpha_{n+1}}}{\Gamma(1-\alpha_0)\cdots\Gamma(1-\alpha_{n+1})}.
\end{align}
\end{lem}
\noindent
We note that the equality
\begin{align}
&\int_{P_\tau}(e^{\pi\ii} t_1)^{\alpha_1-1}\dots (e^{\pi\ii}t_n)^{\alpha_n-1}t_{n+1}^{\alpha_{n+1}-1}d(e^{\pi\ii}t_1)\dots d(e^{\pi\ii}t_n)\nonumber\\
=&\int_{P_\tau}\tau_0^{\alpha_0-1}\cdots\tau_n^{\alpha_n-1}(\tau_0+\dots+\tau_n)^{\alpha_{n+1}-1}\omega(\tau)
\end{align}
implies the original formula \cite[Proposition 6.1]{Beukers}
\begin{equation}
\int_{P_\tau} t_1^{\alpha_1-1}\dots t_n^{\alpha_n-1}t_{n+1}^{\alpha_{n+1}-1}dt_1\dots dt_n=\frac{(2\pi\ii)^{n+1}e^{-\pi\ii(\alpha_1+\dots+\alpha_{n+1})}}{\Gamma(1-\alpha_1)\dots\Gamma(1-\alpha_{n+1})\Gamma(\alpha_1+\dots+\alpha_{n+1})}.
\end{equation}
Now we consider projective spaces $\PP^{|\s^{(l)}|-1}$. Writing $\s^{(l)}=\{i^{(l)}_0,\dots,i^{(l)}_{|\s^{(l)}|-1}\}$ so that $i^{(l)}_0<\dots<i^{(l)}_{|\s^{(l)}|-1}$, we equip $\PP^{|\s^{(l)}|-1}$ with a homogeneous coordinate $[\tau_{\s^{(l)}}]=\left[\tau_{i^{(l)}_0}:\dots:\tau_{i^{(l)}_{|\s^{(l)}|-1}}\right]$. Here, we use the convention $\PP^0=\{ *\}$ (one point). We define the covering map
\begin{equation}
p:(\C^\times)^n_x\rightarrow(\C^\times)^{\s^{(0)}}_{\xi_{\s^{(0)}}}\times\displaystyle\prod_{l=1}^k\left(\PP^{|\s^{(l)}|-1}_{\tau_{\s^{(l)}}}\setminus\bigcup_{i\in\s^{(l)}}\{ \tau_i=0\}\right)
\end{equation}
by $p(x)=\left( z_{\s^{(0)}}({}^t{\bf 1}_k,x)^{A_{\s^{(0)}}}, \left([z_{\s^{(l)}}\cdot({}^t{\bf 1}_k,x)^{A_{\s^{(l)}}}]\right)_{l=1}^k\right)$, where ${\bf 1}_k={\bf e}_1+\dots+{\bf e}_k$ and $z_{\s^{(l)}}({}^t{\bf 1}_k,x)^{A_{\s^{(l)}}}=\left( z_{i}({}^t{\bf 1}_k,x)^{{\bf a}^{(l)}(i)}\right)_{i\in\s^{(l)}}$ for $l=0,\dots,k$. We define $\omega(\tau_{\s^{(l)}})$ by $\omega(\tau_{\s^{(l)}})=\sum_{j=0}^{|\s^{(l)}|-1}(-1)^j\tau_{i_j} d\tau_{i_0}\wedge\dots\wedge\widehat{d\tau_{i_j}}\wedge\dots\wedge d\tau_{i_{|\s^{(l)}|-1}}.$ We denote the product $\prod_{j=0}^{|\s^{(l)}|-1}\tau_{i_j}$ by $\tau_{\s^{(l)}}$. We set $\tau_\s=\prod_{l=1}^k\tau_{\s^{(l)}}.$ By a direct computation employing Laplace expansion, we have the identity
\begin{equation}
p^*\left(\frac{d\xi_{\s^{(0)}}}{\xi_{\s^{(0)}}}\wedge\frac{\omega(\tau_\s)}{\tau_\s}\right)=p^*\left(\frac{d\xi_{\s^{(0)}}}{\xi_{\s^{(0)}}}\wedge\frac{\omega(\tau_{\s^{(1)}})}{\tau_{\s^{(1)}}}\wedge\cdots\wedge\frac{\omega(\tau_{\s^{(k)}})}{\tau_{\s^{(k)}}}\right)=\text{sgn} (A,\s)(\det A_\s)\frac{dx}{x},
\end{equation}
where we have put $\text{sgn} (A,\s)=(-1)^{k|\s^{(0)}|+(k-1)|\s^{(1)}|+\dots+|\s^{(k-1)}|+\frac{k(k-1)}{2}}$.
Now we use the plane wave expansion coordinate. Let us introduce a coordinate transform of $\xi_{\s^{(0)}}$ by $\xi_i=\rho u_i\;\;(i\in\sigma^{(0)}),$
where $\rho$ and $u_i$ are coordinates of $\C^\times$ and $\{u_{\s^{(0)}}=(u_i)_{i\in\s^{(0)}}\in(\C^\times)^{\s^{(0)}}\mid\displaystyle\sum_{i\in\sigma^{(0)}}u_i=1\}$ respectively. Then, it is standard that we have an equality of volume forms $d\xi_{\s^{(0)}}=\rho^{|\s^{(0)}|-1}d\rho du_{\s^{(0)}},$
where $du_{\s^{(0)}}=\sum_{j=1}^{|\s^{(0)}|}(-1)^{j-1}u_{i_j}du_{\widehat{i_j}}$ with $du_{\widehat{i_j}}=du_{i_1}\wedge\cdots\wedge \widehat{du_{i_j}}\wedge\cdots\wedge du_{i_{|\s^{(0)}|}}$ and $\s^{(0)}=\{ i_1,\dots,i_{|\s^{(0)}|}\}$ ($i_1<\dots<i_{|\s^{(0)}|}$).
Using formulae above, we obtain
\begin{align}
f_{\Gamma}(z)
=&\frac{sgn(A,\s)}{\det A_\s}\frac{z_\s^{-A_\s^{-1}\delta}}{(2\pi\ii)^{n+k}}\displaystyle\int_{p_*\Gamma}\prod_{l=1}^k\left(\sum_{i\in\s^{(l)}}\tau_i+\sum_{j\in\bar{\s}^{(l)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j(\xi_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}\right)^{-\gamma_l}\times\nonumber\\
&\exp\left\{ \sum_{i\in\s^{(0)}}\xi_i+\sum_{j\in\bs^{(0)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j(\xi_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}\right\}
(\xi_{\s^{(0)}},\tau_\s)^{A_\s^{-1}\delta}\frac{d\xi_{\s^{(0)}}\omega(\tau_\s)}{\xi_{\s^{(0)}}\tau_\s}\\
=&\frac{sgn(A,\s)}{\det A_\s}\frac{z_\s^{-A_\s^{-1}\delta}}{(2\pi\ii)^{n+k}}\displaystyle\int_{p_*\Gamma}\prod_{l=1}^k\left(\sum_{i\in\s^{(l)}}\tau_i+\sum_{j\in\bar{\s}^{(l)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}\right)^{-\gamma_l}\times\nonumber\\
&\exp\left\{
\rho+\sum_{j\in\bs^{(0)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j
\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}
(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}
\right\}
\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}\delta}
(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}\delta}\nonumber\\
&
\frac{d\rho d u_{\s^{(0)}} \omega(\tau_\s) }{\rho u_{\s^{(0)}} \tau_\s},\label{PlaneWaveIntegral}
\end{align}
where $\Gamma$ is an integration contour to be clarified below. We have also used the convention that $\tau_i$ for $i\in\s^{(l)}$ with $|\s^{(l)}|=1$ is equal to $z_i({\bf 1}_k,x)^{{\bf a}(i)}$.
Let us construct the cycle $\Gamma$. For this purpose, we consider a degeneration of the the integrand $\Phi$. Namely, we consider the following limit: variables $z_j\neq 0$ with $j\in\bs $ are very small while variables $z_j\neq 0$ with $j\in\s$ are frozen. Symbolically, we write this limit as $z\approx z_{\infty}^\s$. The corresponding degeneration of the integrand is
\begin{equation}
\Phi\approx e^{\sum_{i\in\s^{(0)}}z_ix^{{\bf a}^{(0)}(i)}}\left(\displaystyle\sum_{i\in\s^{(1)}}z_ix^{{\bf a}^{(1)}(i)}\right)^{-\gamma_1}\dots \left(\displaystyle\sum_{i\in\s^{(k)}}z_ix^{{\bf a}^{(k)}(i)}\right)^{-\gamma_k}.
\end{equation} We first set $\rho=1$ and construct a cycle in $u_{\s^{(0)}}$ and $\tau_\s$ directions. We take a cycle $\Gamma_{0}$ in $\{\rho=1\}\times\displaystyle\prod_{l=1}^k\left(\PP^{|\s^{(l)}|-1}_{\tau_{\s^{(l)}}}\setminus\bigcup_{i\in\s^{(l)}}\{ \tau_i=0\}\cup\left\{ \sum_{i\in\s^{(l)}}\tau_i=0\right\}\right)$ as a product cycle $\Gamma_0=P_{u_\s^{(0)}}\times\displaystyle\prod_{l=1}^kP_{\tau_{\s^{(l)}}}$. We take a $(n-1)$-dimensional twisted cycle $\tilde{\Gamma}_{\s,0}$ in $\{ \rho=1\}\subset(\C)^n_x$ so that $p_*\tilde{\Gamma}_{\s,0}=\Gamma_0.$ For the construction of such a cycle, see Appendix 3. Note that we determine the branch of multivalued functions $h_{l,z^{(l)}}(x)^{-\gamma_l}$ so that the expansion
\begin{align}
h_{l,z^{(l)}}^{-\gamma_l}(x)=&\left(\sum_{i\in\s^{(l)}}z_ix^{{\bf a}^{(l)}(i)}+\sum_{j\in\bar{\s}^{(l)}}z_jx^{{\bf a}^{(l)}(j)}\right)^{-\gamma_l}\\
=&
\displaystyle
\sum_{{\bf m}_l\in\Z^{\bar{\s}^{(l)}}_{\geq 0}}
\frac{(-1)^{|{\bf m}_l|}(\gamma_l)_{|{\bf m}_l|}}{{\bf m}_l!}
\left(\sum_{i\in\s^{(l)}}z_ix^{{\bf a}^{(l)}(i)}\right)^{-\gamma_l-|{\bf m}_l|}
z_{\bs^{(l)}}^{{\bf m}_l}
({\bf 1}_k,x)^{A_{\bs}{\bf m}_l}
\end{align}
is valid. Thus, the branch of $h_{l,z^{(l)}}(x)^{-\gamma_l}$ is determined by that of $\left(\displaystyle\sum_{i\in\s^{(l)}}z_ix^{{\bf a}^{(l)}(i)}\right)^{-\gamma_l}$, which is determined by the choice of $\Gamma_{\s,0}$. Note that the expansion above in $(\rho,u_{\s^{(0)}},\tau_{\s^{(l)}})$ coordinate is
\begin{align}
&\left(\sum_{i\in\s^{(l)}}\tau_i+\sum_{j\in\bar{\s}^{(l)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}\right)^{-\gamma_l}\\
=&
\displaystyle
\sum_{{\bf m}_l\in\Z^{\bar{\s}^{(l)}}_{\geq 0}}
\frac{(-1)^{|{\bf m}_l|}(\gamma_l)_{|{\bf m}_l|}}{{\bf m}_l!}
\left(\sum_{i\in\s^{(l)}}\tau_i\right)^{-\gamma_l-|{\bf m}_l|}
\left(z_\s^{-A_\s^{-1}A_{\bs^{(l)}}}z_{\bs^{(l)}}\right)^{{\bf m}_l}
\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}A_{\bs}{\bf m}_l}(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}A_{\bs}{\bf m}_l},\label{TheExpansion}.
\end{align}
In $\rho$ direction, we take the so-called Hankel contour $C_0$. $C_0$ is given by the formula
$C_0=(-\infty,-\delta]e^{-\pi\sqrt{-1}}+l_{(0+)}-(-\infty,-\delta]e^{\pi\sqrt{-1}},$
where $e^{\pm\pi\sqrt{-1}}$ stands for the argument of the variable and $l_{(0+)}$ is a small loop which encircles the origin in the counter-clockwise direction starting from and ending at the point $-\delta$ for some small positive $\delta$. Using this notation, we have
\begin{lem}\label{lemma:lemma}
Suppose $\alpha\in\mathbb{C}.$ One has an identity
\begin{equation}
\int_{C_{0}}\xi^{\alpha-1}e^\xi d\xi=\frac{2\pi\sqrt{-1}}{\Gamma(1-\alpha)}.
\end{equation}
\end{lem}
\begin{figure}[t]
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{tikzpicture}
\draw[->-=.5,domain=-175:175] plot ({-1+cos(\x)}, {sin(\x)});
\draw[-<-=.5] ({-1+cos(-175)},{sin(-175)}) -- (-6, {sin(-175)});
\draw[->-=.5] ({-1+cos(175)},{sin(175)}) -- (-6, {sin(175)});
\node at (-1,0){$\cdot$};
\draw (-1,0) node[below right]{O};
\end{tikzpicture}
\caption{Hankel contour}
\end{center}
\end{minipage}
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{tikzpicture}
\node at (0,0){$\cdot$};
\node at (4,0){$\cdot$};
\draw (0,0) node[below]{$t_1=0$};
\draw (4,0) node[below]{$t_1=1$};
\draw[-<-=.5,domain=-140:140] plot ({4+cos(\x)}, {0.4+sin(\x)});
\draw[-<-=.5,domain=40:320] plot ({cos(\x)}, {0.4+sin(\x)});
\draw[->-=.5,domain=-140:140] plot ({4+cos(\x)}, {-0.4+sin(\x)});
\draw[->-=.5,domain=40:320] plot ({cos(\x)}, {-0.4+sin(\x)});
\coordinate (A1) at ({cos(40)}, {0.4+sin(40)});
\coordinate (A2) at ({4+cos(-140)}, {-0.4+sin(-140)});
\coordinate (B1) at ({4+cos(140)}, {-0.4+sin(140)});
\coordinate (B2) at ({cos(40)}, {-0.4+sin(40)});
\coordinate (C1) at ({cos(320)}, {-0.4+sin(320)});
\coordinate (C2) at ({4+cos(140)}, {0.4+sin(140)});
\coordinate (D1) at ({4+cos(-140)}, {0.4+sin(-140)});
\coordinate (D2) at ({cos(320)}, {0.4+sin(320)});
\draw[->-=.75] (A1) -- (A2);
\draw[->-=.5] (B1) -- (B2);
\draw[->-=.75] (C1) -- (C2);
\draw[->-=.5] (D1) -- (D2);
\end{tikzpicture}
\caption{Pochhammer cycle $P_1$}
\end{center}
\end{minipage}
\end{figure}
\noindent
We wish to integrate the integrand along the product contour $C_0\times \Gamma_0$. To do this, we need a simple
\begin{lem}\label{lem:sum2}
For any $l=1,\cdots,k$ and for any $j\in\barsigma^{(l)}$, one has
\begin{equation}\label{Formula4.17}
\sum_{i\in\sigma^{(m)}}{}^t{\bf e}_iA_\sigma^{-1}{\bf a}(j)=
\begin{cases}
1\; (m=l)\\
0\; (m\neq 0,l).
\end{cases}
\end{equation}
Moreover, if $j\in\barsigma^{(0)}$, one has
\begin{equation}\label{WeirdEq}
\sum_{i\in\sigma^{(m)}}{}^t{\bf e}_iA_\sigma^{-1}{\bf a}(j)=
0\;\; (m=1,\dots,k).
\end{equation}
\end{lem}
\begin{proof}
Observe first that, if we write $A$ as $A=({\bf a}(1)|\cdots|{\bf a}(N)),$ then for any $j\in\bar{\sigma}^{(l)}$ ($l=1,\dots,k$) and $m=1,\dots,k$, we have
\begin{equation}
\transp{
\begin{pmatrix}
{\bf e}_m\\
\hline
O
\end{pmatrix}
}
{\bf a}(j)=
\begin{cases}
1\; (m=l)\\
0\; (m\neq l)
\end{cases}
\end{equation}
\noindent
This can be written as
\begin{equation}
\left(
\begin{array}{c|c}
I_k& \\
\hline
&\mathbb{O}_n
\end{array}
\right)
{\bf a}(j)=
\begin{pmatrix}
{\bf e}_l\\
\hline
O
\end{pmatrix}.
\end{equation}
\noindent
We thus have
\begin{align}
\begin{pmatrix}
{\bf e}_l\\
\hline
O
\end{pmatrix}
&=
\left(
\begin{array}{c|c}
I_k& \\
\hline
&\mathbb{O}_n
\end{array}
\right)
{\bf a}(j)\\
&=
\left(
\begin{array}{c|c}
I_k& \\
\hline
&\mathbb{O}_n
\end{array}
\right)
A_\sigma
A_\sigma^{-1}
{\bf a}(j)\\
&=
\left(
\begin{array}{ccc|ccc|ccc|c|ccc}
0&\cdots&0&1&\cdots&1&0&\cdots&0&\cdots&0&\cdots&0\\
\hline
0&\cdots&0&0&\cdots&0&1&\cdots&1&\cdots&0&\cdots&0\\
\hline
&\vdots& & &\vdots& & &\vdots& &\ddots& &\vdots& \\
\hline
0&\cdots&0&0&\cdots&0&0&\cdots&0&\cdots&1&\cdots&1\\
\end{array}
\right)
A_\sigma^{-1}
{\bf a}(j).
\end{align}
\noindent
The formula above clearly shows (\ref{Formula4.17}). On the other hand, for any $j\in\bar{\sigma}^{(0)}$ we have
\begin{equation}
\transp{
\begin{pmatrix}
{\bf e}_m\\
\hline
O
\end{pmatrix}
}
{\bf a}(j)=
0\;\;(m=1,\dots,k).
\end{equation}
Thus, the same argument as above shows (\ref{WeirdEq}).
\end{proof}
\noindent
From \cref{lem:sum2} and the equality
\begin{equation}
\displaystyle\sum_{m=0}^k\sum_{i\in\sigma^{(m)}}{}^t{\bf e}_iA_\sigma^{-1}{\bf a}(j)=|A_\sigma^{-1}{\bf a}(j)|,
\end{equation}
we obtain two inequalities on the degree of divergence
\begin{equation}\label{inequality1}
\displaystyle\sum_{i\in\sigma^{(0)}}{}^t{\bf e}_iA_\sigma^{-1}{\bf a}(j)\leq 0\;\;(j\in\barsigma^{(l)},\; l=1,\dots,k)
\end{equation}
and
\begin{equation}\label{inequality2}
\displaystyle\sum_{i\in\sigma^{(0)}}{}^t{\bf e}_iA_\sigma^{-1}{\bf a}(j)\leq 1\;\;(j\in\barsigma^{(0)}).
\end{equation}
From these inequalities we can verify that the expansion (\ref{TheExpansion}) is valid uniformly along $C_0\times \Gamma_0$ and the integral (\ref{PlaneWaveIntegral}) is convergent if $z\approx z_\infty^\s$.
In order to define the lift of the product cycle $C_0\times\Gamma_0$ to $x$ coordinate, we need a
\begin{lem}\label{lem:ThomMather}
Let $z_j\neq 0$ ($j=1,\dots,N$) be complex numbers and let $\varphi(x)=\sum_{j=1}^Nz_jx^{{\bf a}(j)}$ be a Laurent polynomial in $x=(x_1,\dots,x_n)$. If there is a vector $w=(w_1,\dots,w_n)\in\Z^{1\times n}$ and an integer $m\in\Z\setminus\{ 0\}$ such that for any $j$, one has $w\cdot {\bf a}(j)=m$, then the smooth map $\varphi:\varphi^{-1}(\C^\times)\rightarrow\C^\times$ is a fiber bundle.
\end{lem}
\begin{proof}
Define an action of a torus $\C^\times_\tau$ on $(\C^\times)^n_x$ (resp. on $\C^\times_t$) by $\tau\cdot x=(\tau^{w_1}x_1,\dots,\tau^{w_n}x_n)$ (resp. by $\tau\cdot t=\tau^mt$). Then, it can readily be seen that for any $\tau\in\C^\times$ and $t\in\C^\times$, we have $\tau\cdot\varphi^{-1}(t)=\varphi^{-1}(\tau\cdot t)$. Therefore, if $\varphi$ is a trivial fiber bundle on an open set $U\subset\C^\times_t$, it is again trivial on the open subset $\tau\cdot U$. By Thom-Mather's 1st isotopy lemma (\cite[(4.14) Th\'eor\`eme]{Verdier}), $\varphi$ defines a locally trivial fiber bundle on a non-empty Zariski open subset of $\C^\times_t$. Thus, we can conclude that $\varphi$ is locally trivial on $\C^\times_t$.
\end{proof}
\noindent
In view of \cref{lem:ThomMather}, let us define the twisted cycle $\Gamma_{\s,0}$ as the prolongation of $\tilde{\Gamma}_{\s,0}$ along the Hankel contour $C_0$ with respect to the map $\rho=\sum_{i\in\s^{(0)}}z_j({\bf 1}_k,x)^{{\bf a}(i)}:(\C^\times)^n_x\rightarrow\C$. Computing the integral on this contour, we obtain
\begin{align}
&f_{\s,0}(z;\delta)\nonumber\\
\overset{\rm def}{=}&f_{\Gamma_{\s,0}}(z)\\
=&\frac{sgn(A,\s)}{\det A_\s}\frac{z_\s^{-A_\s^{-1}\delta}}{(2\pi\ii)^{n+k}}\displaystyle\int_{C_0\times\Gamma_0}\prod_{l=1}^k\left(\sum_{i\in\s^{(l)}}\tau_i+\sum_{j\in\bar{\s}^{(l)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}\right)^{-\gamma_l}\times\nonumber\\
&\exp\left\{
\rho+\sum_{j\in\bs^{(0)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j
\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}
(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}
\right\}
\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}\delta}
(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}\delta}
\frac{d\rho d u_{\s^{(0)}} \omega(\tau_\s) }{\rho u_{\s^{(0)}} \tau_\s}\\
=&\frac{sgn(A,\s)}{\det A_\s}\frac{z_\s^{-A_\s^{-1}\delta}}{(2\pi\ii)^{n+k}}\displaystyle\sum_{{\bf m}\in\Z_{\geq 0}^{\bs}}\frac{\prod_{l=1}^k(-1)^{|{\bf m}_l|}(\gamma_l)_{|{\bf m}_l|}}{{\bf m}!}(z_\s^{-A_\s A_{\bs}}z_{\bs})^{{\bf m}}\nonumber\\
&\int_{C_0\times \Gamma_0}\prod_{l=1}^k\left(\sum_{i\in\s^{(l)}}\tau_i\right)^{-\gamma_l-|{\bf m}_l|}e^{\rho}
\rho^{
\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}(\delta+A_{\bs}{\bf m})
}
(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}(\delta+A_{\bs}{\bf m})}
\frac{d\rho du_{\s^{(0)}}\omega(\tau_\s)}{\rho u_{\s^{(0)}}\tau_\s}.
\end{align}
\noindent
We put
$
\tilde{\bf e}_l=
\begin{pmatrix}
{\bf e}_l\\
{\bf O}
\end{pmatrix}
\in\Z^{(k+n)\times 1}
$. Since
$
{}^t\tilde{\bf e}_l={}^t\tilde{\bf e}_lA_\s A_\s^{-1}=\displaystyle\sum_{i\in\s^{(l)}}{}^t{\bf e}_iA_\s^{-1},
$
we have
\begin{equation}
\sum_{i\in\s^{(l)}}{}^t{\bf e}_iA_\s^{-1}(\delta+A_{\bs}{\bf m})={}^t{\bf e}_l(\delta+A_{\bs}{\bf m})=\gamma_l+|{\bf m}_l|.
\end{equation}
Therefore the assumption on the parameters in \cref{lemma:ProjBeukers} is satisfied. Moreover, in view of \cref{lem:sum2}, for any $l\geq 1$ such that $|\s^{(l)}|=1$, we also have that if $\{ i\}=\s^{(l)}$ then ${}^t{\bf e}_iA_\s^{-1}={}^t{\bf e}_l$ and $\Gamma(1-{}^t{\bf e}_iA_\s^{-1}(d+A_{\bs}{\bf m}))=\Gamma(1-\gamma_l-|{\bf m}_l|)$. Let $\{ A_{\bs}{\bf k}(i)\}_{i=1}^{r_\s}$ be a complete system of representatives of $\Z^{(n+k)\times 1}/\Z A_\s$. Using \cref{lemma:ProjBeukers} and employing the formula
\begin{equation}
(\gamma_l)_{|{\bf m}_l|}=\frac{2\pi\ii e^{-\pi\ii\gamma_l}(-1)^{|{\bf m}_l|}}{\Gamma(\gamma_l)\Gamma(1-\gamma_l-|{\bf m}_l|)(1-e^{-2\pi\ii\gamma_l})},
\end{equation}
we obtain the basic formula
\begin{equation}
f_{\s,0}(z;\delta)=\frac{{\rm sgn}(A,\s)\displaystyle\prod_{l:|\s^{(l)}|>1}e^{-\pi\ii(1-\gamma_l)}\displaystyle\prod_{l:|\s^{(l)}|=1}e^{-\pi\ii\gamma_l}}{\det A_\s\Gamma(\gamma_1)\dots\Gamma(\gamma_k)\displaystyle\prod_{l:|\s^{(l)}|=1}(1-e^{-2\pi\ii\gamma_l})}\sum_{i=1}^{r_{\s}}\varepsilon_\s(\delta,{\bf k}(i))\varphi_{\s,{\bf k}(i)}(z).
\end{equation}
\noindent
Here, we have put
\begin{equation}
\varepsilon_\s(\delta,{\bf k})=
\begin{cases}
1&(|\s^{(0)}|\leq 1)\\
1-\exp\left\{ -2\pi\ii\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}(\delta+A_{\bs}{\bf k})\right\}&(|\s^{(0)}|\geq 2).
\end{cases}
\end{equation}
To any integer vector $\tilde{\bf k}\in\Z^{\s\times 1}$, we associate a deck transform $\Gamma_{\s,\tilde{\bf k}}$ of $\Gamma_{\s,0}$ along the loop $(\xi_{\s^{(0)}},[\tau_\s])\mapsto e^{2\pi\ii{}^t\tilde{\bf k}}(\xi_{\s^{(0)}},[\tau_\s])$. By a direct computation, we have
\begin{align}
f_{\s,\tilde{\bf k}}(z;\delta)\overset{def}{=}&f_{\Gamma_{\s,\tilde{\bf k}}}(z)\nonumber\\
=&e^{2\pi\ii{}^t\tilde{\bf k}A_\s^{-1}d}\frac{{\rm sgn}(A,\s)\displaystyle\prod_{l:|\s^{(l)}|>1}e^{-\pi\ii(1-\gamma_l)}\displaystyle\prod_{l:|\s^{(l)}|=1}e^{-\pi\ii\gamma_l}}{\det A_\s\Gamma(\gamma_1)\dots\Gamma(\gamma_k)\displaystyle\prod_{l:|\s^{(l)}|=1}(1-e^{-2\pi\ii\gamma_l})}\times\nonumber\\
&\sum_{i=1}^{r_\s}e^{2\pi\ii{}^t\tilde{\bf k}A_\s^{-1}A_{\bs}{\bf k}(i)}
\varepsilon_\s(d,{\bf k}(i))
\varphi_{\s,{\bf k}(i)}(z;\delta).
\end{align}
\noindent
We take a complete system of representatives $\{ \tilde{\bf k}(i)\}_{i=1}^{r_\s}$. Since it can readily be seen that the pairing $\Z^{\s\times 1}/\Z {}^tA_\s\times\Z^{(n+k)\times 1}/\Z A_\s\ni ([\tilde{\bf k}],[{\bf k}])\mapsto {}^t\tilde{\bf k}A_\s{\bf k}\in\Q/\Z$ is perfect in the sense of Abelian groups, we can easily see that the matrix
$\Big(
\exp\left\{
2\pi\ii\transp{
\tilde{\bf k}(i)
}
A_{\s}^{-1}A_{\bs}{\bf k}(j)
\right\}
\Big)_{i,j=1}^{r_\s}$ is the character matrix of the finite Abelian group $\Z^{(n+k)\times 1}/\Z A_\s$, hence it is invertible.
Let us take a convergent regular triangulation $T$. With the aid of the trivialization (\ref{Trivialization}), we can take a parallel transport of $\Gamma_{\s,\tilde{\bf k}(j)}$ constructed near $z^\s_\infty$ to a point $z_\infty\in U_T$. The resulting cycle is also denoted by $\Gamma_{\s,\tilde{\bf k}(j)}$.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture
\draw[-] (0,0)--(4,-1);
\draw[-] (0,0)--(4,3);
\draw[-] (0,0)--(3,4);
\draw[-] (0,0)--(-1,4);
\node at (3,3){$\cdot$};
\node at (3.3,3){$z_\infty$};
\node at (3,0){$\cdot$};
\node at (0,3){$\cdot$};
\node at (3,-0.5){$z_\infty^\s$};
\node at (0,2.5){$z_\infty^{\s^\prime}$};
\node at (1,0){$U_\s$};
\node at (0.3,1.3){$U_{\s^\prime}$};
\node at (1.6,1.6){$U_T$};
\draw[->-=.5] (3,0) to [out=120,in=200] (3,3);
\draw[->-=.5] (0,3) to [out=40,in=100] (3,3);
\draw[<->,domain=-15:50] plot ({0.5*cos(\x)}, {0.5*sin(\x)});
\draw[<->,domain=40:100] plot ({cos(\x)}, {sin(\x)});
\draw[<->,domain=40:50] plot ({1.5*cos(\x)}, {1.5*sin(\x)});
\end{tikzpicture}
\caption{Parallel transport}
\end{center}
\end{figure}
It is worth pointing out that the cycles $\Gamma_{\s,\tilde{\bf k}(j)}$ constructed above are locally finite cycles rather than finite ones. It is routine to regard $\Gamma_{\s,\tilde{\bf k}(j)}$ as a rapid decay cycle: We use the notation of \S\ref{SectionRapidDecay}. For simplicity, let us assume that $z\approx z^\s_\infty$ is nonsingular. Then, we regard $\Gamma_{\s,\tilde{\bf k}(j)}$ as a subset of $\tilde{\pi}^{-1}(z)$ and take its closure $\overline{\Gamma_{\s,\tilde{\bf k}(j)}}\subset\tilde{\pi}^{-1}(z)$. By construction, $\overline{\Gamma_{\s,\tilde{\bf k}(j)}}\subset \pi^{-1}(z)\cup\widetilde{D^{r.d.}_0}$. This is a (closure of) semi-analytic set. By \cite[THEOREM 2.]{LojasiewiczTriangulations}, we can obtain a semi-analytic triangulation of $\overline{\Gamma_{\s,\tilde{\bf k}(j)}}$ which makes it an element of $\Homo_{n,z}^{r.d.}$ in view of \cref{rem:RemarkRDHomology}.
\begin{comment}
To be more precise, we need to cut off the cycle $\Gamma_{\s,\tilde{\bf k}(j)}$ into finite pieces $\Gamma_{\s,\tilde{\bf k}(j)}=\cup_j\Gamma_j$ so that on each $\Gamma_j$, the integrand is single valued. Then we apply the argument above to each piece $\Gamma_j.$
\end{comment}
Summing up all the arguments above and taking into account \cref{thm:EulerLaplaceRepresentationTheorem}, we obtain the main
\begin{thm}\label{thm:fundamentalthm3}
Take a convergent regular triangulation $T$. Assume that the parameter vector $d$ is very generic and that for any $l=1,\dots,k$, $\gamma_l\notin\Z_{\leq 0}$. Then, if one puts
\begin{equation}
f_{\s,\tilde{\bf k}(j)}(z;\delta)=\frac{1}{(2\pi\ii)^{n+k}}\int_{\Gamma_{\s,\tilde{\bf k}(j)}} e^{h_{0,z^{(0)}}(x)}h_{1,z^{(1)}}(x)^{-\gamma_1}\cdots h_{k,z^{(k)}}(x)^{-\gamma_k}x^{c}\frac{dx}{x},
\end{equation}
$\displaystyle\bigcup_{\s\in T}\{ f_{\sigma,\tilde{\bf k}(j)}(z)\}_{j=1}^{r_\s}$ is a basis of solutions of $M_A(\delta)$ on the non-empty open set $U_T$, where $\{\tilde{\bf k}(j)\}_{j=1}^{r_\s}$ is a complete system of representatives of $\Z^{\s\times 1}/\Z{}^tA_\s$. Moreover, for each $\sigma\in T,$ one has a transformation formula
\begin{equation}
\begin{pmatrix}
f_{\sigma,\tilde{\bf k}(1)}(z;\delta)\\
\vdots\\
f_{\sigma,\tilde{\bf k}(r_\s)}(z;\delta)
\end{pmatrix}
=
T_\sigma
\begin{pmatrix}
\varphi_{\sigma,{\bf k}(1)}(z;\delta)\\
\vdots\\
\varphi_{\sigma,{\bf k}(r_\s)}(z;\delta)
\end{pmatrix}.
\end{equation}
Here, $T_\sigma$ is an $r_\s\times r_\s$ matrix given by
\begin{align}
T_\sigma=&\frac{{\rm sgn}(A,\s)\displaystyle\prod_{l:|\s^{(l)}|>1}e^{-\pi\ii(1-\gamma_l)}\displaystyle\prod_{l:|\s^{(l)}|=1}e^{-\pi\ii\gamma_l}}{\det A_\s\Gamma(\gamma_1)\cdots\Gamma(\gamma_k)\displaystyle\prod_{l:|\s^{(l)}|=1}(1-e^{-2\pi\ii\gamma_l})}
\diag\Big( \exp\left\{
2\pi\ii\transp{
\tilde{\bf k}(i)
}
A_{\s}^{-1}\delta
\right\}\Big)_{i=1}^{r_\s}\times\nonumber\\
&\Big(
\exp\left\{
2\pi\ii\transp{
\tilde{\bf k}(i)
}
A_{\s}^{-1}A_{\bs}{\bf k}(j)
\right\}
\Big)_{i,j=1}^{r_\s}
\diag \left( \varepsilon_\s(\delta,{\bf k}(j))\right)_{j=1}^{r_\s}.
\end{align}
In particular, if $z$ is nonsingular, $\gamma_l\notin\Z$ for any $l=1,\dots,k$, and $d$ is non-resonant, $\displaystyle\bigcup_{\s\in T}\left\{ \Gamma_{\s,\tilde{\bf k}(j)}\right\}_{j=1}^{r_\s}$ is a basis of the rapid decay homology group $\Homo_{n,z}^{r.d.}.$
\end{thm}
For later use, we also give a formula for dual period integral. Consider an integral of the form
\begin{equation}\label{DualEulerInt}
f_{\check{\Gamma}}^\vee(z)=\frac{1}{(2\pi\ii)^{n+k}}\int_{\check{\Gamma}} e^{-h_{0,z^{(0)}}(x)} h_{1,z^{(1)}}(x)^{\gamma_1}\cdots h_{k,z^{(k)}}(x)^{\gamma_k}x^{-c}\frac{dx}{x}.
\end{equation}
\noindent
Using plane wave coordinate as before, we get
\begin{align}
f_{\check{\Gamma}}^\vee(z)
=&\frac{sgn(A,\s)}{\det A_\s}\frac{z_\s^{A_\s^{-1}\delta}}{(2\pi\ii)^{n+k}}\displaystyle\int_{p_*\check{\Gamma}}\prod_{l=1}^k\left(\sum_{i\in\s^{(l)}}\tau_i+\sum_{j\in\bar{\s}^{(l)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j(\xi_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}\right)^{\gamma_l}\times\nonumber\\
&\exp\left\{ -\sum_{i\in\s^{(0)}}\xi_i-\sum_{j\in\bs^{(0)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j(\xi_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}\right\}
(\xi_{\s^{(0)}},\tau_\s)^{-A_\s^{-1}\delta}\frac{d\xi_{\s^{(0)}}\omega(\tau_\s)}{\xi_{\s^{(0)}}\tau_\s}\\
=&\frac{sgn(A,\s)}{\det A_\s}\frac{z_\s^{A_\s^{-1}\delta}}{(2\pi\ii)^{n+k}}\displaystyle\int_{p_*\check{\Gamma}}\prod_{l=1}^k\left(\sum_{i\in\s^{(l)}}\tau_i+\sum_{j\in\bar{\s}^{(l)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}\right)^{\gamma_l}\times\nonumber\\
&\exp\left\{
-\rho-\sum_{j\in\bs^{(0)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j
\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}
(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}
\right\}
\rho^{-\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}\delta}
(u_{\s^{(0)}},\tau_\s)^{-A_\s^{-1}\delta}\nonumber\\
&
\frac{d\rho d u_{\s^{(0)}} \omega(\tau_\s) }{\rho u_{\s^{(0)}} \tau_\s},\label{DualPlaneWaveIntegral}
\end{align}
\noindent
The cycle $\check{\Gamma}$ in $(u_{\s^{(0)}},\tau_\s)$-direction is the product of Pochhammer cycles $\check{\Gamma}_0=\check{P}_{u_\s^{(0)}}\times\displaystyle\prod_{l=1}^k\check{P}_{\tau_{\s^{(l)}}}$. In $\rho$ direction, we take the dual Hankel contour $\check{C}_0$. $\check{C}_0$ is given by the formula
$\check{C}_0=-[\delta,\infty)+l_{(0+)}+[\delta,\infty)e^{2\pi\sqrt{-1}},$
where $e^{2\pi\sqrt{-1}}$ stands for the argument of the variable and $l_{(0+)}$ is a small loop which encircles the origin in the counter-clockwise direction starting from and ending at the point $\delta$ for some small positive $\delta$. Therefore, we take the cycle $\check{\Gamma}_{\s,0}$ so that $p_*\check{\Gamma}_{\s,0}=\check{C}_0\times \check{\Gamma}_0$. Note that the change of coordinate $\tilde{\rho}=e^{-\pi\ii}\rho$ transforms $\check{C}_0$ to $C_0$. Thus a simple computation gives the formula
\begin{align}
f^\vee_{\s,0}(z;\delta)\overset{def}{=}&f^\vee_{\check{\Gamma}_{\s,0}}(z)\nonumber\\
=&e^{-\pi\ii\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}\delta}\times\nonumber\\
&f_{\s,0}\left( z_{\s^{(0)}},\left( -e^{\pi\ii\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}z_j\right)_{j\in\bs^{(0)}},(z_{\s^{(l)}})_{l=1}^k,\left\{ \left( e^{\pi\ii\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}z_j\right)_{j\in\bs^{(l)}}\right\}_{l=1}^k;-\delta\right)
\end{align}
\noindent
We set
\begin{equation}\label{DualSeriesPhiVee}
\varphi^\vee_{\s,{\bf k}}(z;\delta)=z_\sigma^{A_\sigma^{-1}\delta}
\sum_{{\bf k+m}\in\Lambda_{\bf k}}\frac{(-1)^{\bf k_0+m_0}e^{\pi\ii\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}A_{\bs}({\bf k+m})}(z_\sigma^{-A_\sigma^{-1}A_{\bar{\sigma}}}z_{\bar{\sigma}})^{\bf k+m}}{\Gamma({\bf 1}_\sigma+A_\sigma^{-1}(\delta-A_{\bar{\sigma}}({\bf k+m}))){\bf (k+m)!}}.
\end{equation}
\noindent
Then, it is easy to see the formula
\begin{align}
f_{\s,0}^\vee(z;\delta)=&e^{-\pi\ii\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}\delta}\frac{{\rm sgn}(A,\s)\displaystyle\prod_{l:|\s^{(l)}|>1}e^{-\pi\ii(1+\gamma_l)}\displaystyle\prod_{l:|\s^{(l)}|=1}e^{\pi\ii\gamma_l}}{\det A_\s\Gamma(-\gamma_1)\dots\Gamma(-\gamma_k)\displaystyle\prod_{l:|\s^{(l)}|=1}(1-e^{2\pi\ii\gamma_l})}\times\nonumber\\
&\sum_{j=1}^{r_{\s}}\varepsilon_\s(-\delta,{\bf k}(j))\varphi_{\s,{\bf k}(j)}^\vee(z;\delta)
\end{align}
holds. As before, to any integer vector $\tilde{\bf k}\in\Z^{\s\times 1}$, we associate a deck transform $\check{\Gamma}_{\s,\tilde{\bf k}}$ of $\check{\Gamma}_{\s,0}$ along the loop $(\xi_{\s^{(0)}},[\tau_\s])\mapsto e^{2\pi\ii{}^t\tilde{\bf k}}(\xi_{\s^{(0)}},[\tau_\s])$. We have the dual statement of Theorem \ref{thm:fundamentalthm3}.
\begin{thm}\label{thm:dualfundamentalthm3}
Take a convergent regular triangulation $T$. Assume that the parameter vector $\delta$ is very generic and that for any $l=1,\dots,k$, $\gamma_l\notin\Z_{\geq 0}$. Then, if one puts
\begin{equation}
f_{\s,\tilde{\bf k}}^\vee(z;\delta)=\frac{1}{(2\pi\ii)^{n+k}}\int_{\check{\Gamma}_{\s,\tilde{\bf k}(j)}} e^{-h_{0,z^{(0)}}(x)}h_{1,z^{(1)}}(x)^{\gamma_1}\cdots h_{k,z^{(k)}}(x)^{\gamma_k}x^{-c}\frac{dx}{x},
\end{equation}
for each $\sigma\in T,$ one has a transformation formula
\begin{equation}
\begin{pmatrix}
f^\vee_{\sigma,\tilde{\bf k}(1)}(z;\delta)\\
\vdots\\
f^\vee_{\sigma,\tilde{\bf k}(r_\s)}(z;\delta)
\end{pmatrix}
=
T_\sigma^\vee
\begin{pmatrix}
\varphi_{\sigma,{\bf k}(1)}^\vee(z;\delta)\\
\vdots\\
\varphi_{\sigma,{\bf k}(r_\s)}^\vee(z;\delta)
\end{pmatrix}.
\end{equation}
Here, $T_\sigma^\vee$ is an $r_\s\times r_\s$ matrix given by
\begin{align}
T_\sigma^\vee=&e^{-\pi\ii\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}\delta}\frac{{\rm sgn}(A,\s)\displaystyle\prod_{l:|\s^{(l)}|>1}e^{-\pi\ii(1+\gamma_l)}\displaystyle\prod_{l:|\s^{(l)}|=1}e^{\pi\ii\gamma_l}}{\det A_\s\Gamma(-\gamma_1)\dots\Gamma(-\gamma_k)\displaystyle\prod_{l:|\s^{(l)}|=1}(1-e^{2\pi\ii\gamma_l})}
\times\nonumber\\
&
\diag\Big( \exp\left\{
-2\pi\ii\transp{
\tilde{\bf k}(i)
}
A_{\s}^{-1}\delta
\right\}\Big)_{i=1}^{r_\s}
\Big(
\exp\left\{
2\pi\ii\transp{
\tilde{\bf k}(i)
}
A_{\s}^{-1}A_{\bs}{\bf k}(j)
\right\}
\Big)_{i,j=1}^{r_\s}
\diag \Big( \varepsilon_\s\left(-\delta,{\bf k}(j)\right)\Big)_{j=1}^{r_\s}.
\end{align}
In particular, if $z$ is nonsingular, $\gamma_l\notin\Z$ for any $l=1,\dots,k$, and $d$ is non-resonant, $\displaystyle\bigcup_{\s\in T}\left\{ \check{\Gamma}_{\s,\tilde{\bf k}(j)}\right\}_{j=1}^{r_\s}$ is a basis of the rapid decay homology group $\check{\Homo}_{n,z}^{r.d.}\overset{def}{=}\Homo_n^{r.d.}\left( \pi^{-1}(z);\nabla_z\right).$
\end{thm}
\begin{comment}
\begin{align}
f_{\s,\tilde{\bf k}}^\vee(z;d)\overset{def}{=}&f_{\Gamma_{\s,\tilde{\bf k}}^\vee}^\vee(z)\nonumber\\
=&e^{-\pi\ii\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}d}\frac{{\rm sgn}(A,\s)\displaystyle\prod_{l:|\s^{(l)}|>1}e^{-\pi\ii(1+\gamma_l)}\displaystyle\prod_{l:|\s^{(l)}|=1}e^{\pi\ii\gamma_l}}{\det A_\s\Gamma(-\gamma_1)\dots\Gamma(-\gamma_k)\displaystyle\prod_{l:|\s^{(l)}|=1}(1-e^{2\pi\ii\gamma_l})}e^{-2\pi\ii{}^t\tilde{\bf k}A_\s^{-1}d}\times\nonumber\\
&\sum_{j=1}^{r_{\s}}e^{2\pi\ii{}^t\tilde{\bf k}A_\s^{-1}A_{\bs}{\bf k}(j)}\varepsilon_\s(-d,{\bf k}(j))\varphi_{\s,{\bf k}(j)}^\vee(z;d)
\end{align}
\end{comment}
\begin{exa}
We consider a $3\times 5$ matrix
$
A=
\left(
\begin{array}{ccc|cc}
1&1&1&0&0\\
\hline
0&0&0&1&1\\
\hline
0&1&-1&0&1
\end{array}
\right)
$
and a $5\times 2$ matrix
$
B
=
\begin{pmatrix}
-1&-1\\
1&0\\
0&1\\
1&-1\\
-1&1
\end{pmatrix}
$ so that $L_A=\Z B$ holds. For a parameter vector $\delta=
\begin{pmatrix}
\gamma_1\\
\gamma_2\\
c
\end{pmatrix}
$, the GKZ system $M_A(\delta)$ is related to the Horn's $G_1$ function (\cite{DworkLoeser}). By considering an exact sequence $0\rightarrow \R^{1\times 3}\overset{\times A}{\rightarrow}\R^{1\times 5}\overset{\times B}{\rightarrow}\R^{1\times 2}\rightarrow 0$, we can draw a projected image of the secondary fan ${\rm Fan}(A)$ in $\R^{1\times 2}$ as in Figure \ref{SecondaryFanOfG1}.
The Euler integral representation we consider is $f_\Gamma(z)=\frac{1}{(2\pi\ii)^3}\int_\Gamma (z_1+z_2x+\frac{z_3}{x})^{-\gamma_1}(z_4+z_5x)^{-\gamma_2}x^c\frac{dx}{x}.$ Let us describe the basis of solutions associated to the regular triangulation $T_4$. We first consider the simplex $345\in T_4$. This choice of simplex corresponds to the degeneration $z_1,z_2\rightarrow 0$. This induces a degeneration of the configuration of branch points of the integrand. We denote by $\zeta_\pm$ the zeros of the equation $z_1+z_2x+\frac{z_3}{x}=0$ in $x$. The induced degeneration is $\zeta_\pm\rightarrow\infty$. If we put $\zeta=-\frac{z_4}{z_5}$, the cycle $\Gamma_{345,0}$ is just a Pochhammer cycle connecting $\zeta$ and the origin as in Figure \ref{Cycle345G1}. Since $\sharp(\Z^{\{345\}\times 1}/\Z {}^tA_{345})=1$, we are done for this simplex.
\begin{figure}[H]
\begin{center}
\begin{tikzpicture
\draw (0,0) node[left]{O};
\draw[thick, ->] (0,0)--(1,0);
\draw[thick, ->] (0,0)--(0,1);
\draw[thick, ->] (0,0)--(-1,1);
\draw[thick, ->] (0,0)--(-1,-1);
\draw[thick, ->] (0,0)--(1,-1);
\draw[-] (0,0)--(4,0);
\draw[-] (0,0)--(0,4);
\draw[-] (0,0)--(-4,4);
\draw[-] (0,0)--(-3,-3);
\draw[-] (0,0)--(3,-3);
\node at (2,2){$T_1=\{ 125,134,145\}$};
\node at (-2,4){$T_2=\{ 124,134,245\}$};
\node at (-3,0){$T_3=\{ 234,245\}$};
\node at (0,-2){$T_4=\{ 235,345\}$};
\node at (3,-1){$T_5=\{ 125,135,345\}$};
\end{tikzpicture}
\caption{The secondary fan of Horn's $G_1$ in $\R^{1\times 2}$}\label{SecondaryFanOfG1}
\end{center}
\end{figure}
\begin{figure}[H]
\begin{center}
\begin{tikzpicture}
\node at (0,0){$\cdot$};
\node at (4,0){$\cdot$};
\draw (0,0) node[below]{O};
\draw (4,0) node[below]{$\zeta$};
\draw[-<-=.5,domain=-140:140] plot ({4+cos(\x)}, {0.4+sin(\x)});
\draw[-<-=.5,domain=40:320] plot ({cos(\x)}, {0.4+sin(\x)});
\draw[->-=.5,domain=-140:140] plot ({4+cos(\x)}, {-0.4+sin(\x)});
\draw[->-=.5,domain=40:320] plot ({cos(\x)}, {-0.4+sin(\x)});
\coordinate (A1) at ({cos(40)}, {0.4+sin(40)});
\coordinate (A2) at ({4+cos(-140)}, {-0.4+sin(-140)});
\coordinate (B1) at ({4+cos(140)}, {-0.4+sin(140)});
\coordinate (B2) at ({cos(40)}, {-0.4+sin(40)});
\coordinate (C1) at ({cos(320)}, {-0.4+sin(320)});
\coordinate (C2) at ({4+cos(140)}, {0.4+sin(140)});
\coordinate (D1) at ({4+cos(-140)}, {0.4+sin(-140)});
\coordinate (D2) at ({cos(320)}, {0.4+sin(320)});
\draw[->-=.75] (A1) -- (A2);
\draw[->-=.5] (B1) -- (B2);
\draw[->-=.75] (C1) -- (C2);
\draw[->-=.5] (D1) -- (D2);
\node at (6.3,0) {$\cdot$};
\draw (6.3,0) node[below]{$\zeta_+$};
\node at (-2,0) {$\cdot$};
\draw (-2,0) node[below]{$\zeta_-$};
\draw[->] (6.4,0.1) -- (9.7,0.9);
\draw[->] (-2.1,0.1) -- (-4.8,0.9);
\node at (10,1) {$\infty$};
\node at (-5,1) {$\infty$};
\node at (2,-1) {$\Gamma_{345,0}$};
\end{tikzpicture}
\caption{Degeneration of an arrangement associated to a simplex $345$}
\label{Cycle345G1}
\end{center}
\end{figure}
\begin{figure}[H]
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{tikzpicture}
\node at (0,0){$\cdot$};
\node at (2,0){$\cdot$};
\draw (0,0) node[below]{O};
\draw (2,0) node[below]{$\zeta_+$};
\draw[-<-=.5,domain=-140:140] plot ({2+(0.5)*(cos(\x))}, {(0.5)*(sin(\x))});
\draw[->-=.5] ({2+(0.5)*(cos(-140))},{(0.5)*(sin(-140))}) -- ({(0.5)*(cos(-40))},{(0.5)*(sin(-40))});
\draw[->-=.5,domain=-40:-200] plot ({(0.5)*cos(\x)},{(0.5)*sin(\x)});
\draw[->-=.5] ({(0.5)*cos(-200)},{(0.5)*sin(-200)})--({-2+(0.5)*cos(20)},{(0.5)*sin(20)});
\draw[->-=.5,domain=20:340] plot ({-2+(0.5)*cos(\x)},{(0.5)*sin(\x)});
\draw[->-=.5] ({-2+(0.5)*cos(340)},{(0.5)*sin(340)})--({(0.7)*cos(-165)},{(0.7)*sin(-165)});
\draw[->-=.5,domain=-165:25] plot ({(0.7)*cos(\x)},{(0.7)*sin(\x)});
\draw[->-=.5] ({(0.7)*cos(25)},{(0.7)*sin(25)}) -- ({2+(0.5)*cos(140)},{(0.5)*sin(140)});
\node at ({(0.7)*cos(25)},{(0.7)*sin(25)}) {$\bullet$};
\draw ({(0.7)*cos(25)},{(0.7)*sin(25)}) node[above]{$\arg x=0$};
\node at (2,0) {$\cdot$};
\draw (2,0) node[below]{$\zeta_+$};
\node at (-2,0) {$\cdot$};
\draw (-2,0) node[below]{$\zeta_-$};
\node at (0.3,0) {$\cdot$};
\draw (0.3,0) node[right]{$\zeta$};
\draw[->] (0.2,0) -- (0.1,0);
\end{tikzpicture}
\caption{The cycle $\Gamma_{235,0}$}
\label{Cycle2350G1}
\end{center}
\end{minipage}
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{tikzpicture}
\node at (0,0){$\cdot$};
\node at (2,0){$\cdot$};
\draw (0,0) node[below]{O};
\draw (2,0) node[below]{$\zeta_+$};
\draw[-<-=.5,domain=-140:140] plot ({-2-(0.5)*(cos(\x))},{(-0.5)*(sin(\x))});
\draw[->-=.5] ({-2-(0.5)*(cos(-140))},{(-0.5)*(sin(-140))}) -- ({(-0.5)*(cos(-40))},{(-0.5)*(sin(-40))});
\draw[->-=.5,domain=-40:-200] plot ({(-0.5)*cos(\x)},{(-0.5)*sin(\x)});
\draw[->-=.5] ({(-0.5)*cos(-200)},{(-0.5)*sin(-200)})--({2+(-0.5)*cos(20)},{(-0.5)*sin(20)});
\draw[->-=.5,domain=20:340] plot ({2+(-0.5)*cos(\x)},{(-0.5)*sin(\x)});
\draw[->-=.5] ({2-(0.5)*cos(340)},{(-0.5)*sin(340)})--({(-0.7)*cos(-165)},{(-0.7)*sin(-165)});
\draw[->-=.5,domain=-165:25] plot ({(-0.7)*cos(\x)},{(-0.7)*sin(\x)});
\draw[->-=.5] ({(-0.7)*cos(25)},{(-0.7)*sin(25)}) -- ({-2-(0.5)*cos(140)},{(-0.5)*sin(140)});
\node at ({(-0.7)*cos(25)},{(-0.7)*sin(25)}) {$\bullet$};
\draw ({(-0.7)*cos(25)},{(-0.7)*sin(25)}) node[below]{$\arg x=\pi$};
\node at (-2,0) {$\cdot$};
\draw (-2,0) node[below]{$\zeta_-$};
\node at (0.3,0) {$\cdot$};
\draw (0.3,0) node[below]{$\zeta$};
\draw[->] (0.2,0) -- (0.1,0);
\end{tikzpicture}
\caption{The cycle $\Gamma_{235,1}.$}
\label{Cycle2351G1}
\end{center}
\end{minipage}
\end{figure}
\begin{comment}
The relation between series solution and the integral is explicitly given by
\begin{equation}
f_{\Gamma_{345,0}}(z)=\frac{e^{-\pi\ii(1-\gamma_2)}e^{-\pi\ii\gamma_1}}{\Gamma(\gamma_1)\Gamma(\gamma_2)(1-e^{-2\pi\ii\gamma_1})}\varphi_{345,0}(z).
\end{equation}
We can explicitly write down $\varphi_{345,0}$:
\begin{align}
\varphi_{345,0}(z)=&z_3^{-\gamma_1}z_4^{\gamma_1-\gamma_2+c}z_3^{-\gamma_1-c}\times \nonumber\\
&\sum_{m_1,m_2\geq 0}\frac{(z_3^{-1}z_4z_5^{-1}z_1)^{m_1}(z_3^{-1}z_4^2z_5^{-2}z_2)^{m_2}}{\Gamma(1-\gamma_1-m_1-m_2)\Gamma(1+\gamma_1-\gamma_2+c+m_1+2m_2)\Gamma(1-\gamma_1-c-m_1-2m_2)m_1!m_2!}.
\end{align}
\end{comment}
On the other hand, the simplex $235$ induces a different degeneration. This choice of simplex corresponds to the limit $z_1,z_4\rightarrow 0$. Therefore, the corresponding degeneration of branch points of the integrand is $\zeta\rightarrow 0$ and $\zeta_\pm\rightarrow \pm\sqrt{-\frac{z_3}{z_2}}$. Since $\Z^{\{235\}\times 1}/\Z{}^t A_{235}\simeq\Z/2\Z$, we have two independent cycles as in Figure \ref{Cycle2350G1} and \ref{Cycle2351G1}.
\end{exa}
\begin{exa}
We consider a $2\times 4$ matrix
$
A=
\left(
\begin{array}{cc|cc}
0&0&1&1\\
\hline
1&-1&0&1
\end{array}
\right)
$
and a $4\times 2$ matrix
$
B
=
\begin{pmatrix}
1&1\\
1&0\\
0&1\\
0&-1
\end{pmatrix}
$ so that $L_A=\Z B$ holds. For a parameter vector $\delta=
\begin{pmatrix}
\gamma\\
c
\end{pmatrix}
$, the GKZ system $M_A(\delta)$ is related to Horn's $\Gamma_2$ function (\cite{DworkLoeser}). The Euler-Laplace integral representation is of the form $f_\Gamma(z)=\frac{1}{(2\pi\ii)^2}\int_\Gamma e^{z_1x+z_2x^{-1}}(z_3+z_4x)^{-\gamma}x^c\frac{dx}{x}$. We take $T_2$ as our regular triangulation. All the simplexes have normalized volume 1. Let us consider $\s=14$. We set $\zeta=-\frac{z_3}{z_4}$. Then, the simplex $\s=14$ corresponds to the limit $z_2,z_3\rightarrow 0$ which induces a degeneration of the integrand $ e^{z_1x+z_2x^{-1}}(z_3+z_4x)^{-\gamma}x^c\rightarrow e^{z_1x}x^{c-\gamma}.$ Therefore, the resulting integration contour $\Gamma_{14,0}$ is as in the upper right one in Figure \ref{TheFirstFigure}. We can construct the contour $\Gamma_{23,0}$ in the same way as in the lower right picture of Figure \ref{TheFirstFigure}. Finally, the cycle $\Gamma_{34,0}$ is nothing but the Pochhammer cycle connecting $0$ and $\zeta$, hence bounded.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture
\draw (0,0) node[left]{O};
\draw[thick, ->] (0,0)--(1,0);
\draw[thick, ->] (0,0)--(0,1);
\draw[thick, ->] (0,0)--(1,1);
\draw[thick, ->] (0,0)--(0,-1);
\draw[-] (0,0)--(4,0);
\draw[-] (0,0)--(0,4);
\draw[-] (0,0)--(4,4);
\draw[-] (0,0)--(0,-3);
\node at (2,4){$T_1=\{ 14,24\}$};
\node at (4,1){$T_2=\{ 14,23,34\}$};
\node at (2,-1.5){$T_3=\{ 13,23\}$};
\end{tikzpicture}
\caption{The secondary fan of Horn's $\Gamma_2$ in $\R^{1\times 2}$}
\end{center}
\end{figure}
\begin{comment}
\begin{figure}[h]
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{tikzpicture}
\draw[->-=.5,domain=-175:175] plot ({-1+cos(\x)}, {sin(\x)});
\draw[-<-=.5] ({-1+cos(-175)},{sin(-175)}) -- (-6, {sin(-175)});
\draw[->-=.5] ({-1+cos(175)},{sin(175)}) -- (-6, {sin(175)});
\node at (-1,0){$\cdot$};
\draw (-1,0) node[below right]{O};
\draw (-1.6,0) node{$\cdot$};
\draw (-1.6,-0.4) node{$\zeta$};
\draw[->] (-1.4,0) -- (-1.1,0);
\end{tikzpicture}
\caption{$\Gamma_{14,0}$}
\end{center}
\end{minipage}
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{tikzpicture}
\draw[->-=.5] (0,0) to [out=150,in=180] (-0.5,1.5);
\draw[->-=.5] (-0.5,1.5) to [out=0,in=90] (1,0);
\draw[-<-=.5] (0,0) to [out=-150,in=-180] (-0.5,-1.5);
\draw[-<-=.5] (-0.5,-1.5) to [out=0,in=-90] (1,0);
\node at (2,0){$\cdot$};
\draw (0,0) node[below right]{O};
\draw[->] (2,0) -- (3,0);
\node at (3.5,0){$\infty$};
\node at (2,-0.5){$\zeta$};
\end{tikzpicture}
\caption{$\Gamma_{23,0}$}
\end{center}
\end{minipage}
\end{figure}
\end{comment}
\end{exa}
\begin{exa}
\begin{figure}[h]
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{tikzpicture
\draw (0,0) node[left]{O};
\draw[thick, ->] (0,0)--(1,0);
\draw[thick, ->] (0,0)--(0,1);
\draw[thick, ->] (0,0)--(-1,1);
\draw[thick, ->] (0,0)--(0,-1);
\draw[-] (0,0)--(5,0);
\draw[-] (0,0)--(0,4);
\draw[-] (0,0)--(-4,4);
\draw[-] (0,0)--(0,-3);
\node at (2.5,2){$T_1=\{ 125,145,235,345\}$};
\node at (-2,4){$T_2=\{ 125,135,235\}$};
\node at (-2,-1){$T_3=\{ 123\}$};
\node at (2,-2){$T_4=\{124,234\}$};
\end{tikzpicture}
\caption{Projected image of the secondary fan of Horn's ${\bf H}_4$ in $\R^{1\times 2}$}
\end{center}
\end{minipage}
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{tikzpicture}[domain=0:3, samples=100]
\draw (0,0) node[anchor= north west]{O};
\draw[->] (-4,0)--(4,0);
\draw[->] (0,-4)--(0,4);
\draw[-] (-1.8,-0.2)--(-0.2,-1.8);
\node at (4.4,0){$z_1x$};
\node at (0,4.2){$z_2y$};
\draw[-] (-0.2,-1.8) to [out=20,in=100] (0.2,-2.2);
\draw[dashed] (-0.2,-1.8) to [out=-90,in=-120] (0.2,-2.2);
\draw[-] (-1.8,-0.2) to [out=90,in=80] (-2.2,0.2);
\draw[dashed] (-1.8,-0.2) to [out=180,in=270] (-2.2,0.2);
\draw[->-=.5,dashed] (-3,-3.2) to (0.4,0.2);
\draw[-<-=.5] (-3,-2.8) to (0.2,0.4);
\draw[->-=.5,dashed] (0.4,0.2) to [out=45, in=45] (0.2,0.4);
\draw[-,domain=0.02:3] plot ( \x,0.06/\x );
\draw[-,domain=-3:-0.02] plot ( \x,0.06/\x );
\end{tikzpicture}
\caption{cycle $\Gamma_{125,0}$}
\label{Cycle1250H4}
\end{center}
\end{minipage}
\end{figure}
\begin{figure}[H]
\begin{center}
\begin{tikzpicture
\draw[-] (-8,0)--(-3.5,0);
\draw[-] (-6,-1)--(-6,3);
\node at (-6,4) {$\C^2$};
\draw[-,domain=-1:2] plot (\x-5.5,1.5-\x);
\node at (-2.7,0) {$\{\eta=0\}$};
\node at (-6,-1.5) {$\{\xi=0\}$};
\node at (-3.5,-1) {$\{1-\xi-\eta=0\}$};
\draw[-] (-5.8,0.2) -- (-4.6,0.2) -- (-5.8,1.4) -- cycle ;
\draw[->-=.5] (-5.2,0.2) to [out=-50,in=60] (-5.2,-0.4);
\draw[-<-=.5, dashed] (-5.2,0.2) to [out=-110,in=120] (-5.2,-0.4);
\draw[->-=.5] (-5.2,0.8) to [out=70,in=130] (-4.7,1.3);
\draw[-<-=.5,dashed] (-5.2,0.8) to [out=20,in=-50] (-4.7,1.3);
\draw[->-=.5] (-5.8,0.8) to [out=110,in=50] (-6.2,0.8);
\draw[-<-=.5,dashed] (-5.8,0.8) to [out=-110,in=-50] (-6.2,0.8);
\draw[-] (1,0) --(6,0);
\draw[-] (3,-1.7) -- (3,3);
\node at (3,4) {$Bl_{(0,0)}(\C^2)$};
\draw[-,domain=-1:2] plot (\x+3,2+\x/2);
\draw[-] (5,-1) to [out=110, in=-90] (4,3);
\node at (6.7,0) {$\{y=0\}$};
\node at (3,-2) {$\{x=0\}$};
\node at (5,-1.4) {$\{ z_3+z_4x+z_5xy=0\}$};
\node at (7.5,3) {proper transform of $\{\xi=0\}$};
\draw[-] (3.2,0.3) -- (4,0.3) -- (4,2) -- cycle;
\draw[->-=.5] (4,1.15) to [out=60, in=110] (4.5,1.3);
\draw[-<-=.5, dashed] (4,1.15) to [out=-60, in=-70] (4.5,1.3);
\draw[->-=.5] (3.6,0.3) to [out=-70,in=60] (3.6,-0.3);
\draw[-<-=.5, dashed] (3.6,0.3) to [out=-110,in=120] (3.6,-0.3);
\draw[->-=.5] (3.6,1.15) to [out=80,in=0] (3.2,2.3);
\draw[-<-=.5, dashed] (3.6,1.15) to [out=120,in=220] (3.2,2.3);
\draw[->] (0,0) -- (-2,0);
\node at (-1,0.5) {$p$};
\end{tikzpicture}
\caption{cycle $\Gamma_{345,0}$}
\label{Cycle3450H4}
\end{center}
\end{figure}
We consider a $3\times 5$ matrix
$
A=
\left(
\begin{array}{cc|ccc}
0&0&1&1&1\\
\hline
1&0&0&1&1\\
0&1&0&0&1
\end{array}
\right)
$
and a $5\times 2$ matrix
$
B
=
\begin{pmatrix}
1&0\\
0&1\\
1&0\\
-1&1\\
0&-1
\end{pmatrix}
$ so that $L_A=\Z B$ holds. For a parameter vector $\delta=
\begin{pmatrix}
\gamma\\
c_1\\
c_2
\end{pmatrix}
$, the GKZ system $M_A(\delta)$ is related to Horn's ${\bf H}_4$ function (\cite{DworkLoeser}). The Euler-Laplace integral representation is of the form $f_\Gamma(z)=\frac{1}{(2\pi\ii)^2}\int_\Gamma e^{z_1x+z_2y}(z_3+z_4x+z_5xy)^{-\gamma}x^{c_1}y^{c_2}\frac{dx\wedge dy}{xy}$. We take $T_1$ as our convergent regular triangulation. All the simplexes have volume 1. Let us consider $\s=125$. The simplex $\s=125$ corresponds to the limit $z_3,z_4\rightarrow 0$ which induces a degeneration of the integrand $ e^{z_1x+z_2y}(z_3+z_4x+z_5xy)^{-\gamma}x^{c_1}y^{c_2}\rightarrow z_5^{-\gamma}e^{z_1x+z_2y}x^{c_1-\gamma}y^{c_2-\gamma}.$ Therefore, the resulting integration contour $\Gamma_{125,0}$ is as in Figure \ref{Cycle1250H4}. The construction is as follows: we consider a change of coordinate $(z_1x,z_2y)=(\rho u,\rho v)$ with $u+v=1$. Then the cycle $\Gamma_{125,0}$ is the product of a Hankel contour in $\rho$ direction and a Pochhammer cycle in $(u,v)$ direction. Note that the divisor $\{ z_3+z_4x+z_5xy=0\}\subset(\C^\times)^2$ is encircled by $\Gamma_{125,0}$. The constructions of $\Gamma_{145,0}$ and $\Gamma_{235,0}$ are similar.
On the other hand, if we consider a simplex $\s=345$, the corresponding degeneration of the integrand is $ e^{z_1x+z_2y}(z_3+z_4x+z_5xy)^{-\gamma}x^{c_1}y^{c_2}\rightarrow (z_3+z_4x+z_5xy)^{-\gamma}x^{c_1}y^{c_2}.$ The change of coordinate $p(x,y)=(\xi,\eta)$ of the torus $(\C^\times)^2$ that we discussed in general fashion in this section, is explicitly given by $\xi=-\frac{z_4}{z_3}x, \eta=-\frac{z_5}{z_3}xy$. This change of coordinate can be seen as a part of blow-up coordinate of $Bl_{(0,0)}(\C^2)$. Thus, the cycle $\Gamma_{345,0}$ is constructed as in Figure \ref{Cycle3450H4}.
\begin{comment}
\begin{figure}[h]
\begin{center}
\begin{tikzpicture
\draw (0,0) node[anchor=north east]{O};
\draw[->] (-2,0)--(2,0);
\draw[->] (0,-2)--(0,4);
\node at (2.2,0) {$x$};
\node at (0,4.2) {$y$};
\draw[-,domain=0.2:2] plot (\x,-1+1/\x);
\draw[-,thin] (0.1,4) -- (0.1,0);
\draw[-,thin] (0.2,4) -- (0.2,0);
\draw[-,thin] (0.3,7/3) -- (0.3,0);
\draw[-,thin] (0.4,3/2) -- (0.4,0);
\draw[-,thin] (0.5,1) -- (0.5,0);
\draw[-,thin] (0.6,2/3) -- (0.6,0);
\draw[-,thin] (0.7,3/7) -- (0.7,0);
\draw[-,thin] (0.8,1/4) -- (0.8,0);
\draw[-,thin] (0.9,1/9) -- (0.9,0);
\node at (2.2,-1) {$\{ z_3+z_4x+z_5xy=0\}$};
\end{tikzpicture}
\caption{$\Gamma_{345,0}$}
\end{center}
\end{figure}
\end{comment}
\begin{comment}
On the other hand, if we consider a simplex $\s=145$, the corresponding degeneration of the integrand is $ e^{z_1x+z_2y}(z_3+z_4x+z_5xy)^{-\gamma}x^{c_1}y^{c_2}\rightarrow e^{z_1x}(z_4+z_5y)^{-\gamma}x^{c_1-\gamma}y^{c_2}.$ Thus, the cycle $\Gamma_{145,0}$ is a product of a Hankel contour in $x$ direction and a Pochhammer cycle in $y$ plane connecting the origin and $\zeta=-\frac{z_4}{z_5}$.
\end{comment}
\end{exa}
\section{A formula for intersection numbers}\label{IntersectionNumbers}
In the following, we fix an $(n+k)$-simplex $\s$ such that the corresponding series $\varphi_{\s,{\bf k}}(z;\delta)$ is convergent. We assume the parameter $\delta$ is generic so that it is non-resonant, $\gamma_l\notin\Z$, and very generic with respect to $\s$. In the previous section, for any given convergent regular triangulation $T$, we constructed a basis of $\Homo_{n,z}^{r.d.}$ at each $z\in U_T$. In this section, we show that they behave well with respect to homology intersection pairing. Under the notation of \S\ref{SectionRapidDecay}, we set $\check{\Homo}_{n,z}^{r.g.}=\Homo_n^{r.g.}\left( \pi^{-1}(z);\nabla_z\right)$. Recall that there is a canonical morphism ${\rm can}:\check{\Homo}_{n,z}^{r.d.}\rightarrow\check{\Homo}_{n,z}^{r.g.}$ which appeared in (\ref{RDCommutativity}). We are interested in the intersection number $\langle \Gamma_{\s_1,\tilde{\bf k}_1},{\rm can} (\check{\Gamma}_{\s_2,\tilde{\bf k}_2})\rangle_h$.
Firstly, we observe that the open set $U_T$ is invariant by $z_j\mapsto e^{\pi\ii\theta_j}z_j$ for any $j$ and $\theta_j\in\R$. Let us consider a path $\gamma_j(\theta)$ $(0\leq\theta\leq 1)$ given by $\gamma_j(\theta)=(z_1,\dots,e^{2\pi\ii\theta}z_j,\dots,z_N)$ where $z=(z_1,\dots,z_N)$ is any point of $U_T$. From the explicit expression of $\Gamma$-series, we see that the analytic continuation $\gamma_{j*}\varphi_{\s,{\bf k}}(z;\delta)$ of $\varphi_{\s,{\bf k}}(z;\delta)$ along $\gamma_j$ satisfies $\gamma_{j*}\varphi_{\s,{\bf k}}(z;\delta)=e^{-2\pi\ii {}^t{\bf e}_jA_\s^{-1}(\delta+A_{\bs}{\bf k})}\varphi_{\s,{\bf k}}(z;\delta)$ if $j\in\s$ and $\gamma_{j*}\varphi_{\s,{\bf k}}(z;\delta)=\varphi_{\s,{\bf k}}(z;\delta)$ if $j\in\bs$. Since the morphism (\ref{Integration}) preserves monodromy, we see from \cref{thm:fundamentalthm3} that $\Gamma_{\s,\tilde{\bf k}}$ is a sum of eigenvectors with eigenvalues $e^{-2\pi\ii {}^t{\bf e}_jA_\s^{-1}(\delta+A_{\bs}{\bf k})}$ if $j\in\s$ or is itself an eigenvector with eigenvalue $1$. Therefore, we have the following proposition in view of the fact that homology intersection pairing is monodromy invariant.
\begin{prop}
If $\s_1\neq \s_2$, then $\langle \Gamma_{\s_1,\tilde{\bf k}_1},{\rm can}(\check{\Gamma}_{\s_2,\tilde{\bf k}_2})\rangle_h=0$.
\end{prop}
\begin{rem}
When there is no risk of confusion, the intersection number $\langle \Gamma_{\s_1,\tilde{\bf k}_1},{\rm can}(\check{\Gamma}_{\s_2,\tilde{\bf k}_2})\rangle_h$ is simply denoted by $\langle \Gamma_{\s_1,\tilde{\bf k}_1},\check{\Gamma}_{\s_2,\tilde{\bf k}_2}\rangle_h$.
\end{rem}
Thus, it remains to compute $\langle \Gamma_{\s,\tilde{\bf k}_1},\check{\Gamma}_{\s,\tilde{\bf k}_2}\rangle_h$. We compute this quantity when the regular triangulation $T$ is unimodular, i.e., when $\det A_\s=\pm 1$ for any simplex $\s\in T$. The computation is based on the basic formula of the intersection numbers of Pochhammer cycles and that of Hankel contours. For complex numbers $\alpha_1,\dots,\alpha_{n+1}$, let us put $X=\C^n_x\setminus\{ x_1\cdots x_n(1-x_1-\dots-x_n)=0\}$, $\mathcal{L}=\C x_1^{\alpha_1}\cdots x_n^{\alpha_n}(1-x_1-\dots-x_n)^{\alpha_{n+1}}$, $x_i=e^{-\pi\ii}\frac{\tau_i}{\tau_0}$ $(i=1,\dots,n)$, and $\alpha_0=-\alpha_1-\dots-\alpha_{n+1}$. Under this notation, we have $X=\mathbb{P}^n_\tau\setminus\{ \tau_0\cdots\tau_n(\tau_0+\dots+\tau_n)=0\}$. The local system $\mathcal{L}$ is symbolically denoted by $\mathcal{L}=\underline{\C}\tau_0^{\alpha_0}\cdots\tau_n^{\alpha_n}(\tau_0+\dots+\tau_n)^{\alpha_{n+1}}$.
\begin{prop}\label{prop:PochhammerIntersection}
If $P_\tau\in\Homo_n(X,\mathcal{L})$ and $\check{P}_\tau\in\Homo_n(X,\mathcal{L}^\vee)$ denote the $n$-dimensional Pochhammer cycles with coefficients in $\mathcal{L}$ and $\mathcal{L}^\vee$ respectively, one has a formula
\begin{equation}
\langle P_\tau,\check{P}_\tau\rangle_h=\prod_{i=0}^{n+1}(1-e^{-2\pi\ii\alpha_i})=(2\ii)^{n+2}\prod_{i=0}^{n+1}\sin\pi\alpha_i.
\end{equation}
\end{prop}
\noindent
The proof of this proposition will be given in the appendix. In $\rho$ direction, we also have a formula of the intersection number of the Hankel contour $C_0$ and the dual Hankel contour $C_0^\vee$. We set $\nabla_\alpha=d\rho+\alpha\frac{d\rho}{\rho}\wedge+d\rho\wedge$. The following proposition is an immediate consequence of \cite[THEOREM4.3]{MajimaMatsumotoTakayama}.
\begin{prop}\label{prop:HankelIntersection}
If $C_0\in\Homo_1^{r.d.}\left((\Gm)_\rho,\nabla_\alpha^\vee\right)$ and $C_0^\vee\in\Homo_1^{r.d.}\left((\Gm)_\rho,\nabla_\alpha\right)$ denote the Hankel contour and the dual Hankel contour respectively, one has a formula
\begin{equation}
\langle C_0,C_0^\vee\rangle_h=1-e^{-2\pi\ii\alpha}.
\end{equation}
\end{prop}
Now we apply \cref{prop:PochhammerIntersection} and \cref{prop:HankelIntersection} to integration cycles constructed in the previous section. In the following computations, we may assume that $z\approx z_\infty^\s$ since $\langle\bullet,\bullet\rangle_h$ is invariant under parallel transport. Let us recall the identity
\begin{equation}
{}^t{\bf e}_l={}^t{\bf e}_lA_\s A_\s^{-1}=\sum_{i\in\s^{(l)}}{}^t{\bf e}_iA_\s^{-1}.
\end{equation}
In particular, if $|\s^{(l)}|=1$ and $\s^{(l)}=\{ i_l\}$, we have ${}^t{\bf e}_l={}^t{\bf e}_{i_l}A_\s^{-1}$ which implies $\gamma_l={}^t{\bf e}_{i_l}A_\s^{-1}\delta$. Thus, we can factorize the integrand as follows:
\begin{align}
&\prod_{l=1}^k\left(\sum_{i\in\s^{(l)}}\tau_i+\sum_{j\in\bar{\s}^{(l)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}\right)^{-\gamma_l}\times\nonumber\\
&\exp\left\{
\rho+\sum_{j\in\bs^{(0)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j
\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}
(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}
\right\}
\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}\delta}
(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}\delta}
\nonumber\\
=&\prod_{l:l\geq 1, |\s^{(l)}|>1}\left\{\left(\sum_{i\in\s^{(l)}}\tau_i+\sum_{j\in\bar{\s}^{(l)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}\right)^{-\gamma_l}\prod_{i\in\s^{(l)}}\tau_i^{{}^t{\bf e}_iA_\s^{-1}\delta}\right\}\times\nonumber\\
&\prod_{l:l\geq 1, |\s^{(l)}|=1}\left(1+\tau_{i_l}^{-1}\sum_{j\in\bar{\s}^{(l)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}\right)^{-\gamma_l}\prod_{i\in\s^{(0)}}u_i^{{}^t{\bf e}_iA_\s^{-1}\delta}\times\nonumber\\
&\exp\left\{
\rho+\sum_{j\in\bs^{(0)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j
\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}
(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}
\right\}
\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}\delta}.
\end{align}
Thus, on a neighborhood of the cycle $\Gamma_{\s,0}$, the factor
\begin{equation}\label{Factor1}
\prod_{l:l\geq 1, |\s^{(l)}|=1}\left(1+\tau_{i_l}^{-1}\sum_{j\in\bar{\s}^{(l)}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}\right)^{-\gamma_l}
\end{equation}
is holomorphic since $z_\s^{-A_\s^{-1}{\bf a}(j)}z_j$ are very small complex numbers and $\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)\leq 0$. By the formula $\displaystyle\sum_{i\in\s^{(l)}}{}^t{\bf e}_iA_\s^{-1}\delta=\gamma_l$, the assumption of the \cref{prop:PochhammerIntersection} is satisfied.
On the other hand, if we take an open neighbourhood $\widetilde{V}\subset\widetilde{X_z}$ so that its slice in $\rho$-space is a small neighbourhood of both the Hankel contour and the dual Hankel contour, (cf. Figure \ref{TheSlice}) and its slice in $(u_{\s^{(0)}},\tau_\s)$-space is a small neighbourhood of $\Gamma_0=P_{u_\s^{(0)}}\times\displaystyle\prod_{l=1}^kP_{\tau_{\s^{(l)}}}$, we see that the factor
\begin{equation}\label{Factor2}
\exp\left\{
\displaystyle\sum_{\substack{j\in\bs^{(0)}\\ \sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)\leq 0}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j
\rho^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)}
(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}
\right\}
\end{equation}
is bounded on $\widetilde{V}$. The remaining exponential factor is
\begin{equation}
\exp\left\{ \left(1+
\displaystyle\sum_{\substack{j\in\bs^{(0)}\\ \sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)=1}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j
(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}\right)\rho
\right\}.
\end{equation}
We introduce a new coordinate $\tilde{\rho}$ by setting $\tilde{\rho}=\left(1+
\displaystyle\sum_{\substack{j\in\bs^{(0)}\\ \sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)=1}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j
(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}\right)\rho
$. Since $\displaystyle\sum_{\substack{j\in\bs^{(0)}\\ \sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}{\bf a}(j)=1}}z_\s^{-A_\s^{-1}{\bf a}(j)}z_j
(u_{\s^{(0)}},\tau_\s)^{A_\s^{-1}{\bf a}(j)}$ remains very small when $(u_{\s^{(0)}},\tau_{\s})$ runs over our contour, this change of coordinate still gives the Hankel contour in $\tilde{\rho}$ coordinate.
\begin{figure}[H]
\begin{center}
\begin{tikzpicture}
\filldraw[fill=lightgray,draw=black!0] (-5,1.5) -- (5,1.5) -- (5,-1.5) -- (-5,-1.5) -- cycle;
\filldraw[domain=0:360, fill=white] plot ({(0.6)*cos(\x)}, {(0.6)*sin(\x)});
\node at (0,0){$\cdot$};
\draw (0,0) node[below right]{O};
\draw[->-=.5,domain=-90:90] plot ({cos(\x)}, {sin(\x)});
\draw[-<-=.5] ({cos(-90)},{sin(-90)}) -- (-5, {sin(-90)});
\draw[->-=.5] ({cos(90)},{sin(90)}) -- (-5, {sin(90)});
\draw[->-=.5,domain=90:270] plot ({cos(\x)}, {sin(\x)});
\draw[-<-=.5] ({cos(90)},{sin(90)}) -- (5, {sin(90)});
\draw[->-=.5] ({cos(270)},{sin(270)}) -- (5, {sin(270)});
\filldraw[fill=white, draw=black!0] (-2,0.5) -- (-5,0.5) -- (-5, -0.5) -- (-2,-0.5) -- cycle;
\filldraw[fill=white, draw=black!0] (2,0.5) -- (5,0.5) -- (5, -0.5) -- (2,-0.5) -- cycle;
\node at (-5.5,1){$C_0$};
\node at (5.5,1){$C_0^\vee$};
\end{tikzpicture}
\caption{The slice of $\widetilde{V}$ in $\rho$-space (the gray zone)}
\label{TheSlice}
\end{center}
\end{figure}
Below, we apply \cref{prop:TheLocalizationFormula} and \cref{prop:TheProductFormula} to $\langle \Gamma_{\s,{\bf 0}},\check{\Gamma}_{\s,{\bf 0}}\rangle_h$. We put
\begin{align}
\Psi_{0}=&e^{\tilde{\rho}}\tilde{\rho}^{\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}\delta}\\
\Psi_{1}=&\prod_{i\in\s^{(0)}}u_i^{{}^t{\bf e}_iA_\s^{-1}\delta}\\
\Psi_{2}=&\prod_{l:l\geq 1, |\s^{(l)}|>1}\left\{\left(\sum_{i\in\s^{(l)}}\tau_i\right)^{-\gamma_l}\prod_{i\in\s^{(l)}}\tau_i^{{}^t{\bf e}_iA_\s^{-1}\delta}\right\}.
\end{align}
Note that these functions are multivalued functions on $\C^*_{\tilde{\rho}}$, $W_1=\left\{ u_{\s^{(0)}}\in(\C^*)^{|\s^{(0)}|}\mid \displaystyle\sum_{i\in\s^{(0)}}u_i=1\right\}$, and on $W_2=\displaystyle\prod_{l=1}^k\left(\PP^{|\s^{(l)}|-1}_{\tau_{\s^{(l)}}}\setminus\bigcup_{i\in\s^{(l)}}\{ \tau_i=0\}\cup\left\{ \sum_{i\in\s^{(l)}}\tau_i=0\right\}\right)$ respectively. Since $\widetilde{V}$ is a neighbourhood of both $\Gamma_{\s,{\bf 0}}$ and $\check{\Gamma}_{\s,{\bf 0}}$, we see that there exist cycles $\gamma\in\Homo^{\rm r.d.}_n\left( V,\nabla_z^\vee\right)$ and $\gamma^\vee\in\Homo^{\rm r.d.}_n\left( V,\nabla_z\right)$ such that $\Gamma_{\s,{\bf 0}}=\iota_{\widetilde{X}\widetilde{V}!}\gamma$ and $\check{\Gamma}_{\s,{\bf 0}}=\iota_{\widetilde{X}\widetilde{V}!}\gamma^\vee$. By \cref{prop:TheLocalizationFormula}, we obtain $\langle \Gamma_{\s,{\bf 0}},\check{\Gamma}_{\s,{\bf 0}}\rangle_h=\langle \gamma,\gamma^\vee\rangle_h$. Since (\ref{Factor1}) and (\ref{Factor2}) are both holomorphic and bounded on $\widetilde{V}$, we see that the connection $\nabla_z$ is equivalent to $\nabla_{\rm red}=\nabla_0+\nabla_1+\nabla_2$ where $\nabla_0=\Psi^{-1}_0\circ d_{\tilde{\rho}}\circ\Psi_0$, $\nabla_1=\Psi^{-1}_1\circ d_{u_{\s^{(0)}}}\circ\Psi_1$, and $\nabla_2=\Psi^{-1}_2\circ d_{\tau_\s}\circ\Psi_2$. Now, we consider a Cartesian diagram
\begin{equation}
\xymatrix{
V \ar@{^{(}->}[r] \ar@{^{(}->}[d]&(\C^*)_{\tilde{\rho}}\times W_1\times W_2\ar@{^{(}->}[d]\\
\widetilde{V^\prime} \ar@{^{(}->}[r]^{\iota_{\widetilde{Y}\widetilde{V^\prime}!}} &\widetilde{Y},
}
\end{equation}
where $\widetilde{Y}$ is a real oriented blow-up of a good compactification of $(\C^*)_{\tilde{\rho}}\times W_1\times W_2$ with respect to the connection $(\mathcal{O}_{(\C^*)_{\tilde{\rho}}\times W_1\times W_2},\nabla_{\rm red})$ and $\widetilde{V^\prime}$ is an open neighbourhood of the cycle $\gamma$ and $\gamma^\vee$ in $\widetilde{Y}$. In our setting, $Y$ is nothing but a product $\PP_{\tilde{\rho}}\times Y_{12}$ where $Y_{12}$ is a product of projective spaces in $u_{\s^{(0)}}$ and $\tau_{\s^{(l)}}$ coordinates. Note that our cycles $\Gamma_{\s,{\bf 0}}$ and $\check{\Gamma}_{\s,{\bf 0}}$ (hence $\gamma$ and $\gamma^\vee$) are defined by taking closures of semi-algebraic cycles (see the discussion right before \cref{thm:fundamentalthm3}) and therefore, do not depend on the choice of the compactification. Applying \cref{prop:TheLocalizationFormula} to the morphism $\iota_{\widetilde{Y}\widetilde{V^\prime}!}:\Homo_n^{\rm r.d.}\left( V,\nabla_{\rm red}\right)\rightarrow\Homo_n^{\rm r.d.}\left( (\C^*)_{\tilde{\rho}}\times W_1\times W_2,\nabla_{\rm red}\right)$ once again, we obtain $\langle\gamma,\gamma^\vee\rangle_h=\langle \iota_{\widetilde{Y}\widetilde{V^\prime}!}\gamma,\iota_{\widetilde{Y}\widetilde{V^\prime}!}\gamma^\vee\rangle_h$. By our construction of cycles $\Gamma_{\s,{\bf 0}}$ and $\check{\Gamma}_{\s,{\bf 0}}$ in \S\ref{SectionEuler}, we see that $\iota_{\widetilde{Y}\widetilde{V^\prime}!}\gamma$ and $\iota_{\widetilde{Y}\widetilde{V^\prime}!}\gamma^\vee$ are cross products of the forms $\iota_{\widetilde{Y}\widetilde{V^\prime}!}\gamma=C_0\times P_{u_{\s^{(0)}}}\times P_{\tau_\s}$ $\iota_{\widetilde{Y}\widetilde{V^\prime}!}\gamma^\vee=C_0^\vee\times P_{u_{\s^{(0)}}}^\vee\times P_{\tau_\s}^\vee$. Applying \cref{prop:TheProductFormula}, we obtain $\langle \iota_{\widetilde{Y}\widetilde{V^\prime}!}\gamma,\iota_{\widetilde{Y}\widetilde{V^\prime}!}\gamma^\vee\rangle_h=\langle C_0,C_0^\vee\rangle_h \langle P_{u_{\s^{(0)}}},P_{u_{\s^{(0)}}}^\vee\rangle_h\langle P_{\tau_\s},P_{\tau_\s}^\vee\rangle_h$.
If $\s^{(0)}\neq\varnothing$, \cref{prop:HankelIntersection} implies that
\begin{equation}
\langle C_0,C_0^\vee\rangle_h=\left( 1-e^{-2\pi\ii\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}\delta}\right).
\end{equation}
If $|\s^{(0)}|\geq 2$, \cref{prop:PochhammerIntersection} implies that
\begin{equation}
\langle P_{u_{\s^{(0)}}},P_{u_{\s^{(0)}}}^\vee\rangle_h=\left( 1-e^{2\pi\ii\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}\delta}\right)\prod_{i\in\s^{(0)}}\left( 1-e^{-2\pi\ii{}^t{\bf e}_iA_\s^{-1}\delta}\right).
\end{equation}
Finally, \cref{prop:PochhammerIntersection} also implies that
\begin{equation}
\langle P_{\tau_\s},P_{\tau_\s}^\vee\rangle_h=\prod_{l:|\s^{(l)}|>1}(1-e^{2\pi\ii\gamma_l})\prod_{i\in\s^{(l)}}\left(1-e^{-2\pi\ii{}^t{\bf e}_iA_\s^{-1}\delta}\right).
\end{equation}
Summing up all the arguments above, we obtain a
\begin{thm}\label{thm:IntersectionNumber}
We decompose $\s$ as $\s=\s^{(0)}\cup\dots\cup\s^{(k)}$ and set $\gamma_0=\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}\delta$. If $\det A_\s=\pm 1$, then,
\begin{equation}
\langle \Gamma_{\s,0},\check{\Gamma}_{\s,0}\rangle_h =
\begin{cases}
\displaystyle\prod_{l:|\s^{(l)}|>1}\left\{(1-e^{2\pi\ii\gamma_l})\prod_{i\in\s^{(l)}}\left(1-e^{-2\pi\ii{}^t{\bf e}_iA_\s^{-1}\delta}\right)\right\}&(\s^{(0)}=\varnothing)\\
\displaystyle\left( 1-e^{-2\pi\ii\gamma_0}\right)\prod_{l:|\s^{(l)}|>1}\left\{(1-e^{2\pi\ii\gamma_l})\prod_{i\in\s^{(l)}}\left(1-e^{-2\pi\ii{}^t{\bf e}_iA_\s^{-1}\delta}\right)\right\}&(\s^{(0)}\neq\varnothing).
\end{cases}
\end{equation}
\end{thm}
\section{Twisted period relations for $\Gamma$-series}\label{QuadraticRelationsForGammaSeries}
In this section, we derive a quadratic relation for $\Gamma$-series associated to a unimodular regular triangulation. For any complex numbers $\alpha,\beta$ such that $\alpha+\beta\notin\Z_{\leq 0}$, we put $(\alpha)_\beta=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)}$. In general, for any vectors ${\bf \alpha}=(\alpha_1,\dots,\alpha_s),{\bf \beta}=(\beta_1,\dots,\beta_s)\in\C^s$, we put $({\bf \alpha})_{\bf \beta}=\prod_{i=1}^s(\alpha_i)_{\beta_i}$. Combining the results of \S 4 and \S 5, we obtain the main result of this section.
\begin{thm}\label{thm:QuadraticRelation}
Suppose that four vectors ${\bf a},{\bf a}^\prime\in\Z^{n\times 1},{\bf b},{\bf b}^\prime\in\Z^{k\times 1}$ and a convergent unimodular regular triangulation T are given. If the parameter $d$ is generic so that $d$ is non-resonant, $\gamma_l\notin\Z$ for any $l=1,\dots,k$, and $
\begin{pmatrix}
\gamma-{\bf b}\\
c+{\bf a}
\end{pmatrix}
$ and
$
\begin{pmatrix}
\gamma+{\bf b}^\prime\\
c-{\bf a}^\prime
\end{pmatrix}
$ are very generic, then, for any $z\in U_T,$ one has an identity
\begin{align}
&(-1)^{|{\bf b}|+|{\bf b}^\prime|}\gamma_1\cdots\gamma_k(\gamma-{\bf b})_{\bf b}(-\gamma-{\bf b}^\prime)_{{\bf b}^\prime}
\sum_{\s\in T}\frac{\pi^{n+k}}{\sin\pi A_\s^{-1}d}\varphi_{\s,0}\left(z;
\begin{pmatrix}
\gamma-{\bf b}\\
c+{\bf a}
\end{pmatrix}
\right)\varphi_{\s,0}^\vee\left(z;
\begin{pmatrix}
\gamma+{\bf b}^\prime\\
c-{\bf a}^\prime
\end{pmatrix}
\right)\nonumber\\
=&\frac{\langle x^{\bf a}h^{\bf b}\frac{dx}{x},x^{{\bf a}^\prime}h^{{\bf b}^\prime}\frac{dx}{x}\rangle_{ch}}{(2\pi\ii)^n}.
\end{align}
\end{thm}
\begin{proof}
We put $\varphi=x^{{\bf a}^\prime}h^{{\bf b}^\prime}\frac{dx}{x}\in\Homo^n_{dR}\left( \pi^{-1}(z),\nabla_z\right)$, $\psi=x^{\bf a}h^{\bf b}\frac{dx}{x}\in\Homo^n_{dR}\left( \pi^{-1}(z),\nabla_z^\vee\right)$. First of all, let us confirm that $\langle\varphi,\psi\rangle_h$ is well-defined. Observe that the canonical morphism
\begin{equation}\label{HomologyAndLFHomology}
{\rm can}:\Homo_n^{r.d.}\left( \pi^{-1}(z)^{an},\nabla_z\right)\rightarrow\Homo_n^{r.g.}\left( \pi^{-1}(z)^{an},\nabla_z\right)
\end{equation}
is an isomorphism. Indeed, by Poincare duality, \cref{thm:mainDresult}, and the fact that $z\notin{\rm Sing} M_A(\delta)$, both sides of (\ref{HomologyAndLFHomology}) have the same dimension. Since the canonical morphism (\ref{HomologyAndLFHomology}) is compatible with intersection pairing $\langle\bullet,\bullet\rangle_h$ and the intersection matrix $\left( \langle\Gamma_{\s,0},\check{\Gamma}_{\s,0}\rangle_h\right)_{\s\in T}$ is invertible by \cref{thm:IntersectionNumber}, we can verify that (\ref{HomologyAndLFHomology}) is an isomorphism. By taking the dual of (\ref{HomologyAndLFHomology}), the canonical morphism
\begin{equation}
{\rm can}:\Homo^n_{r.d.}\left( \pi^{-1}(z)^{an},\nabla_z^\vee\right)\rightarrow\Homo^n_{dR}\left( \pi^{-1}(z),\nabla_z^\vee\right)
\end{equation}
is also an isomorphism. Thus, the cohomology intersection number $\langle\varphi,\psi\rangle_{ch}$ is well-defined as $\langle\varphi,{\rm can}^{-1}(\psi)\rangle_{ch}$. Then, by \cref{thm:fundamentalthm3} we have
\begin{align}
\int_{\Gamma_{\s,0}}\Phi\varphi=&(2\pi\ii)^{n+k} f_{\s,0}\left(z;
\begin{pmatrix}
\gamma-{\bf b}\\
c+{\bf a}
\end{pmatrix}
\right)\nonumber\\
=&(2\pi\ii)^{n+k}\frac{{\rm sgn}(A,\s)\displaystyle\prod_{l:|\s^{(l)}|>1}e^{-\pi\ii(1-\gamma_l+b_l)}\displaystyle\prod_{l:|\s^{(l)}|=1}e^{-\pi\ii(\gamma_l-b_l)}}{\det A_\s\Gamma(\gamma_1-b_1)\dots\Gamma(\gamma_k-b_k)\displaystyle\prod_{l:|\s^{(l)}|=1}(1-e^{-2\pi\ii\gamma_l})}\times\nonumber\\
&\varepsilon_\s\left(\begin{pmatrix}
\gamma-{\bf b}\\
c+{\bf a}
\end{pmatrix},{\bf 0}\right)
\varphi_{\s,{\bf 0}}\left(z;
\begin{pmatrix}
\gamma-{\bf b}\\
c+{\bf a}
\end{pmatrix}
\right).
\end{align}
\noindent
and
\begin{align}
\int_{\check{\Gamma}_{\s,0}}\Phi^{-1}\psi=&(2\pi\ii)^{n+k}f_{\s,0}^\vee\left( z;
\begin{pmatrix}
\gamma+{\bf b}^\prime\\
c-{\bf a}^\prime
\end{pmatrix}
\right)\nonumber\\
=&(2\pi\ii)^{n+k}
\exp\left\{ -\pi\ii\sum_{i\in\s^{(0)}}{}^t{\bf e}_iA_\s^{-1}
\begin{pmatrix}
\gamma+{\bf b}^\prime\\
c-{\bf a}^\prime
\end{pmatrix}
\right\}\times\nonumber\\
&\frac{{\rm sgn}(A,\s)\displaystyle\prod_{l:|\s^{(l)}|>1}e^{-\pi\ii(1+\gamma_l+b_l^\prime)}\displaystyle\prod_{l:|\s^{(l)}|=1}e^{-\pi\ii(\gamma_l+b_l^\prime)}}{\det A_\s\Gamma(-\gamma_1-b_1^\prime)\dots\Gamma(-\gamma_k-b_k^\prime)\displaystyle\prod_{l:|\s^{(l)}|=1}(1-e^{2\pi\ii\gamma_l})}\times\nonumber\\
&\varepsilon_\s\left(
\begin{pmatrix}
-\gamma-{\bf b}^\prime\\
-c+{\bf a}^\prime
\end{pmatrix},
{\bf 0}
\right)
\varphi_{\s,{\bf 0}}^\vee\left(z;
\begin{pmatrix}
\gamma+{\bf b}^\prime\\
c-{\bf a}^\prime
\end{pmatrix}
\right).
\end{align}
\noindent
In view of these formulae, we can conclude that $\varphi$ and $\psi$ are non-zero as cohomology classes. We can take a basis $\{ \varphi_j\}_{j=1}^L$ (resp. $\{\psi_j\}_{j=1}^L$) of the cohomology group $\Homo^{n}_{dR}\left(\pi^{-1}(z),\nabla_z\right)$ (resp. $\Homo^{n}_{dR}\left(\pi^{-1}(z),\nabla_z^\vee\right)$) so that $\varphi_1=\varphi$ and $\psi_1=\psi$. We also take a basis $\{ \Gamma_{\s,0}\}_{\s\in T}$ (resp. $\{ \check{\Gamma}_{\s,0}\}_{\s\in T}$) of the homology group $\Homo_n^{r.d.}\left(\pi^{-1}(z)^{an},\nabla_z^\vee\right)$ (resp. $\Homo_n^{r.d.}\left(\pi^{-1}(z)^{an},\nabla_z\right)$). Then, $(1,1)$ entry of the general quadratic relation is
\begin{equation}\label{Formula73}
\sum_{\s\in T}\langle \Gamma_{\s,0},\check{\Gamma}_{\s,0}\rangle_h^{-1}\left( \int_{\Gamma_{\s,0}}\Phi\varphi\right)\left( \int_{\check{\Gamma}_{\s,0}}\Phi^{-1}\psi\right)=\langle\varphi,\psi\rangle_{ch}.
\end{equation}
Formula (\ref{Formula73}) combined with \cref{thm:IntersectionNumber} will lead to the desired formula. Note that we have $\varepsilon_\s\left(\begin{pmatrix}
\gamma-{\bf b}\\
c+{\bf a}
\end{pmatrix},{\bf 0}\right)=\varepsilon_\s(\delta,{\bf 0})$ and $\varepsilon_\s\left(
\begin{pmatrix}
-\gamma-{\bf b}^\prime\\
-c+{\bf a}^\prime
\end{pmatrix},
{\bf 0}
\right)=\varepsilon_\s(-\delta,{\bf 0})$ by our assumption $\det A_\s=\pm 1$.
\end{proof}
\begin{rem}
It is a folklore that the cohomology intersection number $
\langle x^{\bf a}h^{\bf b}\frac{dx}{x},x^{{\bf a}^\prime}h^{{\bf b}^\prime}\frac{dx}{x}\rangle_{ch}
$ is a rational function in $z$. This is proved only when the GKZ system is regular holonomic. See \cite{MatsubaraTakayamaCohomologyIntersection}.
\end{rem}
\begin{exa}{\bf (Appell's $F_1$-series)}
We consider a one dimensional integral $f_\Gamma(z)=\int_\Gamma(z_1+z_4x)^{-c_1}(z_2+z_5x)^{-c_2}(z_3+z_6x)^{-c_3}x^{c_4}\frac{dx}{x}$. In this case, the $A$ matrix is given by $
A=
\begin{pmatrix}
1&0&0&1&0&0\\
0&1&0&0&1&0\\
0&0&1&0&0&1\\
0&0&0&1&1&1
\end{pmatrix}
$ and the parameter vector is
$
c=
\begin{pmatrix}
c_1\\
c_2\\
c_3\\
c_4
\end{pmatrix}
$. The associated GKZ system $M_A(c)$ is related to the differential equations satisfied by Appell's $F_1$ functions. As a regular triangulation, we can take $T=\{ 1234,2346,2456\}$. The local system in question is associated to the multivalued function $\Phi=(z_1+z_4x)^{-c_1}(z_2+z_5x)^{-c_2}(z_3+z_6x)^{-c_3}x^{c_4}$. By \cite{MatsumotoIntersection}, if we take $\varphi=\frac{dx}{x}\in\Homo^1_{dR}(\Gm\setminus\{ -\frac{z_1}{z_4},-\frac{z_2}{z_5},-\frac{z_3}{z_6}\};\nabla_z)$ and $\psi=\frac{dx}{x}\in\Homo^1_{dR}(\Gm\setminus\{ -\frac{z_1}{z_4},-\frac{z_2}{z_5},-\frac{z_3}{z_6}\};\nabla_z^\vee)$, we have a formula $\langle\varphi,\psi\rangle_{ch}=2\pi\ii\frac{c_1+c_2+c_3}{c_4(c_1+c_2+c_3-c_4)}$. Applying \cref{thm:QuadraticRelation} and taking a restriction to $z_2=z_3=z_4=z_6=1$, we obtain a new identity for Appell's $F_1$-series:
\begin{align}
&\frac{c_1}{c_4(c_1-c_4)}F_1\left(\substack{c_4,c_2,c_3\\ 1+c_4-c_1};z_1z_5,z_1\right)F_1\left(\substack{-c_4,-c_2,-c_3 \\1-c_4+c_1};z_1z_5,z_1\right) \nonumber\\
&+\frac{c_3}{(c_1+c_3-c_4)(c_4-c_1)}G_2(c_1,c_2,c_4-c_1,c_1+c_3-c_4;-z_1,-z_5)G_2(-c_1,-c_2,c_1-c_4,c_4-c_1-c_3;-z_1,-z_5) \nonumber\\
&+\frac{c_2}{(c_1+c_2+c_3-c_4)(c_4-c_1-c_3)}F_1\left(\substack{c_1+c_2+c_3-c_4,c_1,c_3 \\1+c_1+c_3-c_4};z_1z_5,z_5\right)F_1\left(\substack{c_4-c_1-c_2-c_3,-c_1,-c_3 \\1+c_4-c_1-c_3};z_1z_5,z_5\right) \nonumber\\
=&\frac{c_1+c_2+c_3}{c_4(c_1+c_2+c_3-c_4)}
\end{align}
\noindent
Here, we have put
\begin{equation}
F_1\left(\substack{a,b,b^\prime\\ c};x,y\right)=\sum_{m,n\geq 0}\frac{(a)_{m+n}(b)_m(b^\prime)_n}{(c)_{m+n}m!n!}x^my^n
\end{equation}
and
\begin{equation}
G_2(a,a^\prime,b,b^\prime;x,y)=\sum_{m,n\geq 0}\frac{(a)_m(a^\prime)_n(b)_{n-m}(b^\prime)_{m-n}}{m!n!}x^my^n.
\end{equation}
\end{exa}
\begin{exa}{\bf (Horn's $\Phi_1$-series)}
We consider a one dimensional integral $f_\Gamma(z)=\int_\Gamma e^{z_1x}(z_2+z_3x)^{-\gamma_1}(z_4+z_5x)^{-\gamma_2}x^c\frac{dx}{x}$. In this case, the $A$ matrix is given by
$
A=
\begin{pmatrix}
0&1&1&0&0\\
0&0&0&1&1\\
1&0&1&0&1
\end{pmatrix}
$ and the associated GKZ system $M_A(\delta)$ is related to the differential equations satisfied by Horn's $\Phi_1$-series. As a convergent regular triangulation, we take $T=\{ 135,234,345\}$. By \cite{MajimaMatsumotoTakayama}, if we take $\varphi=\frac{dx}{x}\in\Homo^1_{dR}(\Gm\setminus\{ -\frac{z_2}{z_3},-\frac{z_4}{z_5}\};\nabla_z)$ and $\psi=\frac{dx}{x}\in\Homo^1_{dR}(\Gm\setminus\{ -\frac{z_2}{z_3},-\frac{z_4}{z_5}\};\nabla_z^\vee)$, we have a formula $\langle\varphi,\psi\rangle_{ch}=\frac{2\pi\ii}{c}$. Applying \cref{thm:QuadraticRelation}, we obtain a new identity for Horn's $\Phi_1$-series:
\begin{align}
&(c-\gamma_1-\gamma_2)(\gamma_1-c)\nonumber\\
=&c(\gamma_1-c)\Phi_2\left(\substack{\gamma_1,\gamma_2\\ 1+\gamma_1+\gamma_2-c};-zw,-w\right)\Phi_2\left(\substack{-\gamma_1,-\gamma_2\\ 1-\gamma_1-\gamma_2+c};zw,w\right)\nonumber\\
&+\gamma_1(c-\gamma_1-\gamma_2)\Phi_1\left(\substack{c,\gamma_2\\ 1+c-\gamma_1};z,-zw\right)\Phi_1\left(\substack{-c,-\gamma_2\\ 1-c+\gamma_1};z,zw\right)\nonumber\\
&+c\gamma_2\Gamma_1\left(\gamma_1,c-\gamma_1,\gamma_1+\gamma_2-c;-z,w\right)\Gamma_1\left(-\gamma_1,-c+\gamma_1,-\gamma_1-\gamma_2+c;-z,-w\right).
\end{align}
\noindent
Here, the series $\Phi_1\left(\substack{\alpha,\beta\\ \gamma};x,y\right),$ $\Phi_2\left(\substack{\beta_1,\beta_2\\ \gamma};x,y\right)$, and $\Gamma_1\left(\alpha,\beta_1,\beta_2;x,y\right)$ are given by
\begin{equation}
\Phi_1\left(\substack{\alpha,\beta\\ \gamma};x,y\right)
=\sum_{m,n=0}^\infty\frac{(\alpha)_{m+n}(\beta)_m}{(\gamma)_{m+n}m!n!}x^my^n,
\end{equation}
\begin{equation}
\Phi_2\left(\substack{\beta_1,\beta_2\\ \gamma};x,y\right)
=\sum_{m,n=0}^\infty\frac{(\beta_1)_{m}(\beta_2)_n}{(\gamma)_{m+n}m!n!}x^my^n,
\end{equation}
and
\begin{equation}
\Gamma_1\left(\alpha,\beta_1,\beta_2;x,y\right)
=\sum_{m,n=0}^\infty\frac{(\alpha)_{m}(\beta_1)_{n-m}(\beta_2)_{m-n}}{m!n!}x^my^n.
\end{equation}
\end{exa}
\section{Quadratic relations for Aomoto-Gelfand system}\label{QuadraticRelationsForGrassman}
In this section, we apply \cref{thm:QuadraticRelation} to the so-called Aomoto-Gelfand hypergeometric functions (\cite{AomotoKita}, \cite{GelfandGraevRetakh}). This class enjoys a special combinatorial structure. Firstly, we revise the general result on this class of hypergeometric functions based on \cite{GelfandGraevRetakh}. Let $k\leq n$ be two natural numbers. We consider the following integral
\begin{equation}
f_\Gamma(z)=\int_\Gamma\prod_{j=0}^nl_j(x;z)^{\alpha_j}\omega(x)=\int_\Gamma\prod_{j=0}^n(z_{0j}x_0+\dots+z_{kj}x_k)^{\alpha_j}\omega(x)
\end{equation}
where $\omega(x)=\displaystyle\sum_{i=0}^k(-1)^ix_idx_{\hat{i}}\in\Gamma(\mathbb{P}^k,\Omega^k_{\mathbb{P}^k}(k+1))$ and $z=(z_{ij})_{\substack{i=0,\dots,k\\ j=0,\dots,n}}\in Z_{k+1,n+1}.$ Here, we denote by $Z_{k+1,n+1}$ the space of all $(k+1)\times (n+1)$ matrices with entries in $\C$. The Aomoto-Gelfand system $E(k+1,n+1)$ is defined, with the aid of parameters $\alpha_0,\dots,\alpha_n\in\C$ such that $\alpha_0+\dots+\alpha_n=-(k+1)$ by the formula
\begin{equation}
E(k+1,n+1):
\begin{cases}
\displaystyle\sum_{i=0}^kz_{ij}\frac{\partial f}{\partial z_{ij}}=\alpha_jf&(j=0,\dots,n)\\
\displaystyle\sum_{j=0}^nz_{ij}\frac{\partial f}{\partial z_{pj}}=-\delta_{ip}f&(i,p=0,1,\dots,k)\\
\frac{\partial^2 f}{\partial z_{ij}\partial z_{pq}}=\frac{\partial^2 f}{\partial z_{pj}\partial z_{iq}}& (i,p=0,1,\dots,k,\quad j,q=0,\dots, n).
\end{cases}
\end{equation}
\noindent
If we take a restriction to $z=
\begin{pmatrix}
1& & &z_{0k+1}&\cdots&z_{0n}\\
&\ddots& & \vdots &\ddots&\vdots\\
& &1&z_{kk+1}&\cdots&z_{kn}
\end{pmatrix}
$ and $x_0=1$, our integral $f_\Gamma(z)$ becomes
\begin{equation}
f_\Gamma(z)=\int_\Gamma\prod_{j=k+1}^nl_j(x;z)^{\alpha_j}x_1^{\alpha_1}\dots x_k^{\alpha_k}dx.
\end{equation}
If we put $c={}^t(
\alpha_0+1,\dots,\alpha_k+1,-\alpha_{k+1},\dots,-\alpha_n)$, and put ${\bf a}(i,j)={\bf e}(i)+{\bf e}(j)$ $(i=0,1,\dots,k,j=k+1,\dots,n)$, where ${\bf e}(s)$ is the standard basis of $\Z^{(n+1)\times 1}$, $f_\Gamma(z)$ is a solution of $M_A(c)$ with $A=({\bf a}(i,j))_{\substack{i=0,\dots,k\\ j=k+1,\dots,n}}$. The system $M_A(c)$ is explicitly given by
\begin{equation}
M_A(c):
\begin{cases}
\displaystyle\sum_{i=0}^kz_{ij}\frac{\partial f}{\partial z_{ij}}=-c_jf&(j=k+1,\dots,n)\\
\displaystyle\sum_{j=k+1}^nz_{ij}\frac{\partial f}{\partial z_{ij}}=-c_if&(i=0,1,\dots,k)\\
\frac{\partial^2 f}{\partial z_{ij}\partial z_{pq}}=\frac{\partial^2 f}{\partial z_{pj}\partial z_{iq}}& (i,p=0,1,\dots,k,\quad j,q=k+1,\dots, n).
\end{cases}
\end{equation}
\noindent
We also put $\tilde{\bf a}(i,j)=-{\bf e}(i)+{\bf e}(j)$ $(i=0,1,\dots,k,j=k+1,\dots,n)$ and $\tilde{A}=(\tilde{\bf a}(i,j))_{\substack{i=0,\dots,k\\ j=k+1,\dots,n}}$. Note that this configuration is equivalent to ${\bf a}(i,j)$ via the isomorphism of the lattice $\Z^{(n+1)\times 1}$ given by ${}^t(m_0,\dots,m_n)\mapsto {}^t(-m_0,\dots,-m_k,m_{k+1},\dots,m_n)$. We should also be aware that $\tilde{A}$ does not generate the ambient lattice $\Z^{(n+1)\times 1}$ hence neither does $A$. However, since the quotient $\Z^{(n+1)\times 1}/\Z A$ is torsion free, we can apply the previous result by, for example, considering a projection $p:\Z^{(n+1)\times 1}\rightarrow\Z^{n\times 1}$ which sends ${\bf e}(0)$ to $0$ and keeps other standard basis ${\bf e}(s)$ ($s=1,\dots,n$). Thus, if we define the projected matrix $A^\prime=pA$ and a projected parameter $c^\prime=p(c)$, it can readily be seen that the GKZ system $M_A(c)$ is equivalent to the reduced GKZ system $M_{A^\prime}(c^\prime)$.
We consider the special regular triangulation called ``staircase triangulation'' (\cite[\S 6.2.]{DeLoeraRambauSantos},\cite[\S 8.2.]{GelfandGraevRetakh}). A subset $I\subset\{ 1,\dots,k\}\times\{ k+1,\dots,n\}$ is called a ladder if $|I|=n$ and if we write $I=\{ (i_1,j_1),\dots,(i_n,,j_n)\}$, we have $(i_1,j_1)=(k,k+1)$ and $(i_n,,j_n)=(0,n)$ and $(i_{p+1},j_{p+1})=(i_{p}+1,,j_p)$ or $(i_p,j_p+1)$. It can readily be seen that any ladder $I$ is a simplex. Moreover, the collection of all ladders $T=\{ I\mid I:\text{ladder}\}$ forms a regular triangulation. This regular triangulation $T$ is called the staircase triangulation. It is also known that staircase triangulation $T$ is unimodular. For any ladder $I\in T$, we consider the equation $Av^I=-c$ such that $v^I_{ij}=0\quad ((i,j)\notin I)$. Defining $\tilde{c}_l=
\begin{cases}
c_l&(l=0,\dots, k)\\
-c_l&(l=k+1,\dots,n),
\end{cases}
$ it is equivalent to the system $\tilde{A}v^I=\tilde{c}$. This equation can be solved in a unique way. We can even obtain an explicit formula for $v^I$ by means of graph theory. For each ladder $I$, we can associate a tree $G_I$ of a complete bipatite graph $K_{k+1,n-k}$. Recall that the complete bipartite graph $K_{k+1,n-k}$ consists of the set of vertices $V(K_{k+1,n-k})=\{ 0,\dots,n\}$ and the set of edges $E(K_{k+1,n-k})=\left\{ (i,j)\mid \substack{i=0,\dots,k\\ j=k+1,\dots,n}\right\}$. For a given ladder $I=\{ (i_1,j_1),\dots,(i_n,,j_n)\}$, we associate a tree $G_I$ so that edges are $E(G_I)=\{ (i_s,j_s)\}_{s=1}^n$ and vertices are $V(G_I)=\{ 0,\dots,n\}$. Let us introduce the dual basis $\phi(l)$ $(l=0,\dots,n)$ to ${\bf e}(l)$. For any edge $(i,j)\in G_I$, we can easily confirm that $G_I\setminus (i,j)$ has exactly two connected components. The connected component which contains $i$ (resp. $j$) is denoted by $C_i(i,j)$ (resp. $C_j(i,j)$). For each $(i,j)\in G_I$, we put
\begin{equation}
\varphi(ij)=\sum_{l\in V(C_j(i,j))}\phi(l).
\end{equation}
\begin{figure}[H]
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{tikzpicture}
\draw[-] (0,0) -- (0,4);
\draw[-] (0,0) -- (5,0);
\draw[-] (5,0) -- (5,4);
\draw[-] (0,4) -- (5,4);
\draw[-] (1,0) -- (1,4);
\draw[-] (2,0) -- (2,4);
\draw[-] (3,0) -- (3,4);
\draw[-] (4,0) -- (4,4);
\draw[-] (0,1) -- (5,1);
\draw[-] (0,2) -- (5,2);
\draw[-] (0,3) -- (5,3);
\node at (0.5,3.5) {$\bullet$};
\node at (1.5,3.5) {$\bullet$};
\node at (1.5,2.5) {$\bullet$};
\node at (2.5,2.5) {$\bullet$};
\node at (2.5,1.5) {$\bullet$};
\node at (3.5,1.5) {$\bullet$};
\node at (4.5,1.5) {$\bullet$};
\node at (4.5,0.5) {$\bullet$};
\node at (-0.5,3.5) {$3$};
\node at (-0.5,2.5) {$2$};
\node at (-0.5,1.5) {$1$};
\node at (-0.5,0.5) {$0$};
\node at (0.5,-0.5) {$4$};
\node at (1.5,-0.5) {$5$};
\node at (2.5,-0.5) {$6$};
\node at (3.5,-0.5) {$7$};
\node at (4.5,-0.5) {$8$};
\end{tikzpicture}
\caption{ladder}
\end{center}
\end{minipage}
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{tikzpicture}
\node at (-1.5,3.5) {$3$};
\node at (-1.5,2.5) {$2$};
\node at (-1.5,1.5) {$1$};
\node at (-1.5,0.5) {$0$};
\node at (1.5,4) {$4$};
\node at (1.5,3) {$5$};
\node at (1.5,2) {$6$};
\node at (1.5,1) {$7$};
\node at (1.5,0) {$8$};
\draw[-] (-1,3.5) -- (1,4);
\draw[-] (-1,3.5) -- (1,3);
\draw[-] (-1,2.5) -- (1,3);
\draw[-] (-1,2.5) -- (1,2);
\draw[-] (-1,1.5) -- (1,2);
\draw[-] (-1,1.5) -- (1,1);
\draw[-] (-1,1.5) -- (1,0);
\draw[-] (-1,0.5) -- (1,0);
\end{tikzpicture}
\caption{spanning tree corresponding to the ladder}
\end{center}
\end{minipage}
\end{figure}
\begin{figure}[H]
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{tikzpicture}
\node at (-1.5,3.5) {$3$};
\node at (1.5,4) {$4$};
\node at (1.5,3) {$5$};
\draw[-] (-1,3.5) -- (1,4);
\draw[-] (-1,3.5) -- (1,3);
\end{tikzpicture}
\caption{connected component $C_5(2,5)$}
\end{center}
\end{minipage}
\begin{minipage}{0.5\hsize}
\begin{center}
\begin{tikzpicture}
\node at (-1.5,2.5) {$2$};
\node at (-1.5,1.5) {$1$};
\node at (-1.5,0.5) {$0$};
\node at (1.5,2) {$6$};
\node at (1.5,1) {$7$};
\node at (1.5,0) {$8$};
\draw[-] (-1,2.5) -- (1,2);
\draw[-] (-1,1.5) -- (1,2);
\draw[-] (-1,1.5) -- (1,1);
\draw[-] (-1,1.5) -- (1,0);
\draw[-] (-1,0.5) -- (1,0);
\end{tikzpicture}
\caption{connected component $C_2(2,5)$}
\end{center}
\end{minipage}
\end{figure}
\begin{prop}\label{prop:GraphicalInverse}
For $(i,j),(i^\prime,j^\prime)\in I$, we have
\begin{equation}
\langle \varphi(ij),\tilde{\bf a}(i^\prime,j^\prime)\rangle=
\begin{cases}
1&((i,j)=(i^\prime,j^\prime))\\
0&(otherwise).
\end{cases}
\end{equation}
\end{prop}
\begin{proof}
Suppose $(i^\prime,j^\prime)\in C_i(i,j)$. Then we have $\langle \varphi(ij),\tilde{\bf a}(i^\prime,j^\prime)\rangle=0.$ On the other hand, if $(i^\prime,j^\prime)\in C_j(i,j)$, we see $\langle \varphi(ij),\tilde{\bf a}(i^\prime,j^\prime)\rangle=\langle \phi(i^\prime)+\phi(j^\prime),\tilde{\bf a}(i^\prime,j^\prime)\rangle=0.$ Finally, since $i\notin V(C_j(i,j))$ and $j\in V(C_j(i,j))$, we have $\langle \varphi(ij),\tilde{\bf a}(i,j)\rangle=1$.
\end{proof}
\noindent
Therefore, we obtain a
\begin{cor}
Under the notation above, one has
\begin{equation}
v^I_{ij}=\sum_{l\in V(C_j(i,j))}\tilde{c}_l.
\end{equation}
\end{cor}
\noindent
Substitution of this formula to $\Gamma$-series yields the formula
\begin{equation}
\varphi_{v^I}(z)=z_I^{v^I}\sum_{u_{\bar{I}}\in\Z^{\bar{I}}_{\geq 0}}\frac{\left(z_I^{-\langle \varphi(I),\tilde{A}_{\bar{I}}\rangle}z_{\bar{I}}\right)^{u_{\bar{I}}}}{\displaystyle\prod_{(i,j)\in I}\Gamma(1+v_{ij}^I-\langle\varphi(ij),\tilde{A}_{\bar{I}}u_{\bar{I}}\rangle)u_{\bar{I}}!}.
\end{equation}
Since this series is defined by means of a ladder $I$ and a parameter $\alpha$, we also denote it by $f_I(z;\alpha)$.
Next, we consider the de Rham cohomology group $\Homo_{\rm dR}^k\left( \PP_x^k\setminus\bigcup_{j=0}^n\{ l_{j}(x;z)=0\},\nabla_z\right)$ with $\nabla=d_x+\sum_{j=0}^n\tilde{c}_jd_x\log l_{j}(x;z)\wedge$. Note that we identify the set of rational differential forms on $\PP^k_x$ of homogeneous degree $0$ with that on $\{ l_0(x;z)\neq 0\}\simeq\A^k$. As a convenient basis of the twisted cohomology group, we take the one of \cite{GotoMatsumotoContiguity}. We consider matrix variables $z=
\begin{pmatrix}
1& & &z_{0k+1}&\cdots&z_{0n}\\
&\ddots& & \vdots &\ddots&\vdots\\
& &1&z_{kk+1}&\cdots&z_{kn}
\end{pmatrix}.
$ For any subset $J=\{ j_0,\dots,j_k\}\subset\{0,\dots,n\}$ with cardinality $k+1$, we denote by $z_J$ the submatrix of $z$ consisting of column vectors indexed by $J$. We always assume $j_0<\dots<j_k$. We put
\begin{equation}
\omega_J(z;x)=d_x\log\left(\frac{l_{j_1}(x;z)}{l_{j_0}(x;z)}\right)\wedge\dots\wedge d_x\log\left(\frac{l_{j_k}(x;z)}{l_{j_0}(x;z)}\right).
\end{equation}
By a simple computation, we see that $\omega_J(x;z)=\displaystyle\sum_{p=0}^k(-1)^pl_{j_p}(x;z)\frac{d_xl_{j_0}\wedge\dots\wedge \widehat{d_xl_{j_p}}\wedge\dots\wedge d_xl_{j_k}}{l_{j_0}(x;z)\cdots l_{j_k}(x;z)}.
$
As in \cite[Fact 2.5]{GotoMatsumotoContiguity}, we also see that
$
\displaystyle\sum_{p=0}^k(-1)^pl_{j_p}(x;z)d_xl_{j_0}\wedge\dots\wedge \widehat{d_xl_{j_p}}\wedge\dots\wedge d_xl_{j_k}=\det(z_J)\omega(x).
$
Therefore, we have $\omega_J(x;z)=\det(z_J)\frac{\omega(x)}{l_{j_0}(x;z)\cdots l_{j_k}(x;z)}.$ We set $\mathcal{J}\overset{def}{=}\llbracket 0,n\rrbracket\overset{def}{=}\{ 0,\dots,n\}$. Then, for any distinct elements $p,q\in\mathcal{J}$, we set ${}_q\mathcal{J}_p=\{ J\subset\mathcal{J}\mid |J|=k, q\notin J, p\in J\}$. \cite[Proposition 3.3]{GotoMatsumotoContiguity} tells us that the set $\{ \omega_J\}_{J\in{}_q\mathcal{J}_p}$ is a basis of $\Homo_{\rm dR}^k\left( \PP_x^k\setminus\bigcup_{j=0}^n\{ l_{j}(x;z)=0\},\nabla_z\right)$.
Now we are going to derive a quadratic relation for $f_I(z;\alpha)$. We take any pair of subsets $J,J^\prime\subset\{0,\dots,n\}$ with cardinality $k+1$. Let us put $J_a=J\cap\{ 1,\dots,k\}$, $J_a^\prime=J^\prime\cap\{ 1,\dots,k\}$, $J_b=J\cap\{ k+1,\dots,n\}$, and $J_b^\prime=J^\prime\cap\{ k+1,\dots,n\}$. We denote by ${\bf 1}_{J_a}$ (resp. ${\bf 1}_{J_b}$) the vector $\sum_{j\in J_a}{\bf e}(j)$ (resp. $\sum_{j\in J_b}{\bf e}(j)$). If we write $\alpha$ as $\displaystyle\sum_{j=0}^n\alpha_j{\bf e}(j)$, we also put $\alpha_a=\displaystyle\sum_{j=1}^k\alpha_j{\bf e}(j)$ and $\alpha_{b}=\displaystyle\sum_{j=k+1}^n\alpha_j{\bf e}(j)$. We can readily confirm the identities
\begin{equation}
\frac{\omega_J(x;z)}{\det(z_J)}=\frac{\omega(x)}{l_{j_0}(x;z)\cdots l_{j_k}(x;z)}=\frac{x_0\dots x_k}{l_{j_0}(x;z)\cdots l_{j_k}(x;z)}\frac{\omega(x)}{x_0\dots x_k}=x^{{\bf 1}-{\bf 1}_{J_a}}l^{-{\bf 1}_{J_b}}\frac{dx}{x}.
\end{equation}
Setting ${\bf 1}_{\llbracket 0,k\rrbracket}=\sum_{j=0}^k{\bf e}(j)$, the quadratic relation leads to the form
\begin{align}
&(-1)^{|J_b|+|J_b^\prime|}(-\alpha_{k+1})\cdots (-\alpha_n)(-\alpha_b+{\bf 1}_{J_b})_{-{\bf 1}_{J_b}}(\alpha_b+{\bf 1}_{J^\prime_b})_{-{\bf 1}_{J^\prime_b}}\times\nonumber\\
&\sum_{I:\text{ladder}}\frac{\pi^n}{\displaystyle\prod_{(i,j)\in I}\sin\pi (-v_{ij}^I)}f_I(z;\alpha+{\bf 1}_{\llbracket 0,k\rrbracket}-{\bf 1}_J)f_I^\vee(z;\alpha-{\bf 1}_{\llbracket 0,k\rrbracket}+{\bf 1}_{J^\prime})\nonumber\\
=&\det(z_J)^{-1}\det(z_{J^\prime})^{-1}\frac{\langle \omega_{J}(x;z),\omega_{J^\prime}(x;z)\rangle_{ch}}{(2\pi\ii)^k}.
\end{align}
On the other hand, by \cite{MatsumotoIntersection}, we know
\begin{equation}\label{MatsumotoFormula}
\frac{\langle \omega_{J}(x;z),\omega_{J^\prime}(x;z)\rangle_{ch}}{(2\pi\ii)^k}=
\begin{cases}
\frac{\sum_{j\in J}\tilde{c}_j}{\prod_{j\in J}\tilde{c}_j}& (J=J^\prime)\\
\frac{\text{sgn} (J^\prime,J)}{\prod_{j\in J\cap J^\prime}\tilde{c}_j}& (\sharp(J\cap J^\prime)=k)\\
0&(otherwise)
\end{cases}.
\end{equation}
Here, sgn$(J,J^\prime)$ is defined to be $(-1)^{p+q}$ where $p$ and $q$ are chosen so that $J^\prime\setminus\{ j^\prime_p\}=J\setminus\{ j_q\}$. In sum, we obtain the general quadratic relation of Aomoto-Gelfand hypergeometric functions:
\begin{thm}\label{thm:QuadraticRelationsForAomotoGelfand}
Under the notation as above, for any $z\in U_T$, we have an identity
\begin{align}
&(-1)^{|J_b|+|J^\prime_b|+k}\alpha_{k+1}\dots\alpha_n(-\alpha_b+{\bf 1}_{J_b})_{-{\bf 1}_{J_b}}(\alpha_b+{\bf 1}_{J^\prime_b})_{-{\bf 1}_{J_b^\prime}}\times\nonumber\\
&\sum_{I:\text{ladder}}\frac{\pi^n}{\displaystyle\prod_{(i,j)\in I}\sin\pi v_{ij}^I}f_I(z;\alpha+{\bf 1}_{\llbracket 0,k\rrbracket}-{\bf 1}_J)f_I^\vee(z;\alpha-{\bf 1}_{\llbracket 0,k\rrbracket}+{\bf 1}_{J^\prime})\nonumber\\
=&
\det(z_J)^{-1}\det(z_{J^\prime})^{-1}\frac{\langle \omega_{J}(x;z),\omega_{J^\prime}(x;z)\rangle_{ch}}{(2\pi\ii)^k}.\label{EknQuadraticRelation}
\end{align}
Here, the right hand side is explicitly determined by (\ref{MatsumotoFormula}).
\end{thm}
\begin{rem}
Since the right-hand side (\ref{EknQuadraticRelation}) is a rational function in the parameters $\alpha_j$, (\ref{EknQuadraticRelation}) holds without any restriction on the parameters $\alpha_j$.
\end{rem}
\begin{exa}{\bf (Gau\ss' hypergeometric series)}
The simplest case is $E(2,4)$. This amounts to the classical Gau\ss' hypergeometric functions. By computing the cohomology intersection number $\langle\frac{dx}{x},\frac{dx}{x}\rangle_{ch}$, we obtain a quadratic relation (\ref{QRGauss}) in the introduction. Note in particular that this identity implies a series of combinatorial identities
\begin{align}
&(1-\gamma+\alpha)(1-\gamma+\beta)\sum_{l+m=n}\frac{(\alpha)_l(\beta)_l}{(\gamma)_l(1)_l}\frac{(-\alpha)_m(-\beta)_m}{(2-\gamma)_m(1)_m}\nonumber\\
=&\alpha\beta\sum_{l+m=n}\frac{(\gamma-\alpha-1)_l(\gamma-\beta-1)_l}{(\gamma)_l(1)_l}\frac{(1-\gamma+\alpha)_m(1-\gamma+\beta)_m}{(2-\gamma)_m(1)_m}
\end{align}
where $n$ is a positive integer.
\end{exa}
\begin{exa}{\bf (Hypergeometric function of type $E(3,6)$)}\label{exa:AomotoGelfand}
This type of hypergeometric series was discussed by several authors (cf. \cite{MSY},\cite{MSTY}). The integral we consider is $f_\Gamma(z)=\int_\Gamma\prod_{j=3}^5(z_{0j}+z_{1j}x_1+z_{2j}x_2)^{-c_j}x_1^{c_1}x_2^{c_2}\frac{dx_1\wedge dx_2}{x_1x_2}$. The (reduced) $A$ matrix is given by
$
A^\prime=
\bordermatrix{
&z_{03}&z_{04}&z_{05}&z_{13}&z_{14}&z_{15}&z_{23}&z_{24}&z_{25}\cr
c_3&1&0&0&1&0&0&1&0&0\cr
c_4&0&1&0&0&1&0&0&1&0\cr
c_5&0&0&1&0&0&1&0&0&1\cr
c_1&0&0&0&1&1&1&0&0&0\cr
c_2&0&0&0&0&0&0&1&1&1
}.
$ The associated arrangement of hyperplanes is described as in Figure \ref{fig:AomotoGelfandConfiguration}.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture
\draw (0,0) node[below right]{O};
\draw[->] (-6,0)--(6,0);
\draw (6,0) node[right]{$x_1$};
\draw[thick, ->] (0,-3)--(0,3);
\draw (0,3) node[above]{$x_2$};
\draw[-] (-5,2.5)--(6,-1);
\draw[-] (-4,3)--(2,-3);
\draw[-] (-3,-3.5)--(6,1);
\draw (6,-1) node[right]{$H_3=\{l_3=0\}$};
\draw (2,-3) node[right]{$H_4=\{l_4=0\}$};
\draw (6,1) node[above]{$H_5=\{l_5=0\}$};
\draw[-] (0.2,-1.4) to [out=70,in=100] (2,1);
\draw[dashed] (0.2,-1.4) to [out=20,in=-80] (2,1);
\draw[-] (0.1,-1.5) to [out=130,in=70] (-0.4,-1.5);
\draw[dashed] (0.1,-1.5) to [out=-110,in=-30] (-0.4,-1.5);
\draw[-] (0.25,-1.675) to [out=-50,in=20] (0.45,-2.075);
\draw[dashed] (0.25,-1.675) to [out=-100,in=-100] (0.45,-2.075);
\draw (0.1,-1.3)--(0.4,-1.6);
\draw (0.4,-1.6)--(0.1,-1.75);
\draw (0.1,-1.75)--(0.1,-1.3);
\end{tikzpicture}
\caption{Arrangement of hyperplanes and the cycle corresponding to the ladder $\{ 23,24,25,15,05\}$}
\label{fig:AomotoGelfandConfiguration}
\end{center}
\end{figure}
Let us put $H_j=\{ x\in\C^2\mid l_j(x;z)=0\}$ for ($j=1,\dots,5$). We also denote by $H_0$ the hyperplane at infinity $H_0=\mathbb{P}^2\setminus\C^2$. As was clarified in \S\ref{SectionEuler}, each ladder ($=$simplex) induces a degeneration of arrangements. The rule is simple: for each ladder $I$, we let variables $z_{\bar{I}}$ corresponding to the complement of $I$ go to $0$ while we keep variables $z_I$ corresponding to $I$ fixed. For example, if we take a ladder $\{ 23,24,25,15,05\}$, the induced degeneration is $z_{13},z_{14},z_{03},z_{04}\rightarrow 0$. By taking this limit the hyperplanes $H_3$ and $H_4$ both tend to the hyperplane $H_2$ ($x_1$ axis) which is simply denoted by $\substack{ H_3\rightarrow H_2\\ H_4\rightarrow H_2}$. Therefore, there only remain 3 hyperplanes after this limit: $H_1,H_2$ and $H_5$. Restricted to the real domain they form a chamber when variables $z_{ij}$ are all real and generic. We consider the Pochhammer cycle associated to this bounded chamber. The important point of this construction is that, unlike the usual Pohhammer cycle, we have to go around several divisors at once. In this case, $H_3$ and $H_4$ should be regarded as a perturbation of $H_2$. Therefore, they are linked as in Figure \ref{fig:AomotoGelfandConfiguration}. We call such a cycle ``linked cycle'' (or ``Erd\'elyi cycle'' after the pioneering work of Erd\'elyi \cite{ErdelyiActa} where this type of cycle is called ``double circit'' in the cases of Appell's $F_1$ and its relatives). We summerize the correspondence between ladders and degenerations in the following table.
\begin{equation*}
\begin{array}{c|c|c|c|c|c|c}
\text{ladder}
&\begin{matrix}
\bullet&\bullet&\bullet\\
& &\bullet\\
& &\bullet
\end{matrix}
&\begin{matrix}
\bullet&\bullet& \\
&\bullet&\bullet\\
& &\bullet
\end{matrix}
&\begin{matrix}
\bullet&\bullet& \\
&\bullet& \\
&\bullet&\bullet
\end{matrix}
&\begin{matrix}
\bullet& & \\
\bullet&\bullet&\bullet\\
& &\bullet
\end{matrix}
&\begin{matrix}
\bullet& & \\
\bullet&\bullet& \\
&\bullet&\bullet
\end{matrix}
&\begin{matrix}
\bullet& & \\
\bullet& & \\
\bullet&\bullet&\bullet
\end{matrix}
\\ \hline
\text{degeneration} &\substack{H_3\rightarrow H_2\\ H_4\rightarrow H_2}& H_3\rightarrow H_2& \substack{H_3\rightarrow H_2\\ H_5\rightarrow H_0}&H_4\rightarrow H_1& H_5\rightarrow H_0&\substack{ H_4\rightarrow H_0\\ H_5\rightarrow H_0}
\end{array}
\end{equation*}
Now the quadratic relation with respect to the cohomology intersection number $\langle\frac{dx_1\wedge dx_2}{x_1x_2},\frac{dx_1\wedge dx_2}{x_1x_2}\rangle_{ch}$ is explicitly given by
\begin{equation}
c_0c_1c_2c_3c_4c_5\sum_{i=1}^6\frac{\pi^5}{\sin\pi(-v_i)}\varphi_i(z;c)\varphi_i(z;-c)=c_3+c_4+c_5
\end{equation}
where parameters $c_0,\dots,c_5$ satisfy a linear relation
\begin{equation}
c_0+c_1+c_2-c_3-c_4-c_5=0
\end{equation}
and vectors $v_i$ are given by
\begin{align}
v_1&={}^t(-c_3,-c_4,c_0+c_1-c_5,-c_1,-c_0)\\
v_2&={}^t(-c_3,-c_2+c_3,-c_0-c_1+c_5,c_0-c_5,-c_0)\\
v_3&={}^t(-c_3,-c_2+c_3,-c_1,c_5-c_0,-c_5)\\
v_4&={}^t(-c_2,c_2-c_3,-c_4,c_0-c_5,-c_0)\\
v_5&={}^t(-c_2,c_2-c_3,c_0-c_4-c_5,c_5-c_0,-c_5)\\
v_6&={}^t(-c_2,-c_1,-c_0+c_4+c_5,-c_4,-c_5).
\end{align}
Below, we list the explicit formulae of $\Gamma$-series $\varphi_i(z;c)$:
\begin{align}
\varphi_1(z;c)&=z_{23}^{-c_3}z_{24}^{-c_4}z_{25}^{c_0+c_1-c_5}z_{15}^{-c_1}z_{05}^{-c_0}\nonumber\\
&\sum_{u_{13},u_{14},u_{03},u_{04}\geq 0}\frac{1}{\Gamma(1-c_3-u_{13}-u_{03})\Gamma(1-c_4-u_{14}-u_{04})\Gamma(1+c_0+c_1-c_5+u_{13}+u_{14}+u_{03}+u_{04})}\nonumber\\
&\frac{(z_{23}^{-1}z_{25}z_{15}^{-1}z_{13})^{u_{13}}(z_{24}^{-1}z_{25}z_{15}^{-1}z_{14})^{u_{14}}(z_{23}^{-1}z_{25}z_{05}^{-1}z_{03})^{u_{03}}(z_{24}^{-1}z_{25}z_{05}^{-1}z_{04})^{u_{04}}}{\Gamma(1-c_1-u_{13}-u_{14})\Gamma(1-c_0-u_{03}-u_{04})u_{13}!u_{14}!u_{03}!u_{04}!}
\end{align}
\begin{align}
&\varphi_2(z;c)\nonumber\\
&=z_{23}^{-c_3}z_{24}^{-c_2+c_3}z_{14}^{-c_0-c_1+c_5}z_{15}^{c_0-c_5}z_{05}^{-c_0}\nonumber\\
&\sum_{\substack{u_{25},u_{13},\\ u_{03},u_{04}\geq 0}}\frac{1}{\Gamma(1-c_3-u_{13}-u_{03})\Gamma(1-c_2+c_3-u_{25}+u_{13}+u_{03})\Gamma(1-c_0-c_1+c_5+u_{25}-u_{13}-u_{03}-u_{04})}\nonumber\\
&\frac{(z_{24}^{-1}z_{14}z_{15}^{-1}z_{25})^{u_{25}}(z_{23}^{-1}z_{24}z_{14}^{-1}z_{13})^{u_13}(z_{23}^{-1}z_{24}z_{14}^{-1}z_{15}z_{05}^{-1}z_{03})^{u_{03}}(z_{14}^{-1}z_{15}z_{05}^{-1}z_{04})^{u_{04}}}{\Gamma(1+c_0-c_5-u_{25}+u_{03}+u_{04})\Gamma(1-c_0-u_{03}-u_{04})u_{25}!u_{13}!u_{03}!u_{04}!}
\end{align}
\begin{align}
\varphi_3(z;c)&=z_{23}^{-c_3}z_{24}^{-c_2+c_3}z_{14}^{-c_1}z_{04}^{c_5-c_0}z_{05}^{-c_5}\nonumber\\
&\sum_{u_{25},u_{15},u_{13},u_{03}\geq 0}\frac{1}{\Gamma(1-c_3-u_{13}-u_{03})\Gamma(1-c_2+c_3-u_{25}+u_{13}+u_{03})\Gamma(1-c_1-u_{15}-u_{13})}\nonumber\\
&\frac{(z_{24}^{-1}z_{04}z_{05}^{-1}z_{25})^{u_{25}}(z_{14}^{-1}z_{04}z_{05}^{-1}z_{15})^{u_{15}}(z_{23}^{-1}z_{24}z_{14}^{-1}z_{13})^{u_{13}}(z_{23}^{-1}z_{24}z_{04}^{-1}z_{03})^{u_{03}}}{\Gamma(1+c_5-c_0+u_{25}+u_{15}-u_{03})\Gamma(1-c_5-u_{25}-u_{15})u_{25}!u_{15}!u_{13}!u_{03}!}
\end{align}
\begin{align}
\varphi_4(z;c)&=z_{23}^{-c_2}z_{13}^{c_2-c_3}z_{14}^{-c_4}z_{15}^{c_0-c_5}z_{05}^{-c_0}\nonumber\\
&\sum_{u_{24},u_{25},u_{03},u_{04}\geq 0}\frac{1}{\Gamma(1-c_2-u_{24}-u_{25})\Gamma(1+c_2-c_3+u_{24}+u_{25}-u_{03})\Gamma(1-c_4-u_{24}-u_{04})}\nonumber\\
&\frac{(z_{23}^{-1}z_{13}z_{14}^{-1}z_{24})^{u_{24}}(z_{23}^{-1}z_{13}z_{15}^{-1}z_{25})^{u_{25}}(z_{13}^{-1}z_{15}z_{05}^{-1}z_{03})^{u_{03}}(z_{14}^{-1}z_{15}z_{05}^{-1}z_{04})^{u_{04}}}{\Gamma(1+c_0-c_5-u_{25}+u_{03}+u_{04})\Gamma(1-c_0-u_{03}-u_{04})u_{24}!u_{25}!u_{03}!u_{04}!}
\end{align}
\begin{align}
\varphi_5(z;c)&=z_{23}^{-c_2}z_{13}^{c_2-c_3}z_{14}^{c_0-c_4-c_5}z_{04}^{c_5-c_0}z_{05}^{-c_5}\nonumber\\
&\sum_{u_{24},u_{25},u_{15},u_{03}\geq 0}\frac{1}{\Gamma(1-c_2-u_{24}-u_{25})\Gamma(1+c_2-c_3+u_{24}+u_{25}-u_{03})}\nonumber\\
&\frac{1}{\Gamma(1+c_0-c_4-c_5-u_{24}-u_{25}-u_{15}+u_{03})\Gamma(1+c_5-c_0+u_{25}+u_{15}-u_{03})\Gamma(1-c_5-u_{25}-u_{15})}\nonumber\\
&\frac{(z_{23}^{-1}z_{13}z_{14}^{-1}z_{24})^{u_{24}}(z_{23}^{-1}z_{13}z_{14}^{-1}z_{04}z_{05}^{-1}z_{25})^{u_{25}}(z_{14}^{-1}z_{04}z_{05}^{-1}z_{15})^{u_{15}}(z_{13}^{-1}z_{14}z_{04}^{-1}z_{03})^{u_{03}}}{u_{24}!u_{25}!u_{15}!u_{03}!}
\end{align}
\begin{align}
\varphi_6(z;c)&=z_{23}^{-c_2}z_{13}^{-c_1}z_{03}^{-c_0+c_4+c_5}z_{04}^{-c_4}z_{05}^{-c_5}\nonumber\\
&\sum_{u_{24},u_{25},u_{14},u_{15}\geq 0}\frac{1}{\Gamma(1-c_2-u_{24}-u_{25})\Gamma(1-c_1-u_{14}-u_{15})\Gamma(1-c_0+c_4+c_5+u_{24}+u_{25}+u_{14}+u_{15})}\nonumber\\
&\frac{(z_{23}^{-1}z_{03}z_{04}^{-1}z_{24})^{u_{24}}(z_{23}^{-1}z_{03}z_{05}^{-1}z_{25})^{u_{25}}(z_{13}^{-1}z_{03}z_{04}^{-1}z_{14})^{u_{14}}(z_{13}^{-1}z_{03}z_{05}^{-1}z_{15})^{u_{15}}}{\Gamma(1-c_4-u_{24}-u_{14})\Gamma(1-c_5-u_{25}-u_{15})u_{24}!u_{25}!u_{14}!u_{15}!}.
\end{align}
\noindent
Note that if we substitute
\begin{equation}
\begin{pmatrix}
z_{03}&z_{04}&z_{05}\\
z_{13}&z_{14}&z_{15}\\
z_{23}&z_{24}&z_{25}
\end{pmatrix}
=
\begin{pmatrix}
1&1&1\\
1&\zeta_1&\zeta_1\zeta_2\\
1&\zeta_1\zeta_3&\zeta_1\zeta_2\zeta_3\zeta_{4}
\end{pmatrix},
\end{equation}
\noindent
all the Laurent series $\varphi_{i}(z;c)$ above become power series, i. e., they do not contain any negative power in $\zeta_1,\dots,\zeta_4$.
\end{exa}
\section{Quadratic relations for a confluence of Aomoto-Gelfand system}\label{E(21...1)}
In this section, we consider the generalized confluent hypergeometric system $M_{(2,1^{n-1})}$ in the sense of \cite{KimuraHaraokaTakano}. The system $M_{(2,1^{n-1})}$ can be obtained by taking a ``confluence'' of the system $E(k+1,n+1)$. We use the same notation as \S\ref{QuadraticRelationsForGrassman}. The general solution of the system $M_{(2,1^{n-1})}$ has an integral representation of the form
\begin{equation}
f_\Gamma(z)=\int_{\Gamma}\prod_{j=0}^{n-1}l_j(x;z)^{\alpha_j}e^{\frac{l_n(x;z)}{l_0(x;z)}}\omega(x)
\end{equation}
with $z=(z_{ij})_{\substack{i=0,\dots,k\\ j=0,\dots,n}}\in Z_{k+1,n+1}.$ The parameters $\alpha_0,\dots,\alpha_{n-1}\in\C$ are subject to the constraint $\alpha_0+\dots+\alpha_{n-1}=-(k+1)$. The system $M_{(2,1^{n-1})}$ in our setting is given by
\begin{equation}
M_{(2,1^{n-1})}:
\begin{cases}
\displaystyle\sum_{i=0}^k\left( z_{i0}\frac{\partial}{\partial z_{i0}}+z_{in}\frac{\partial}{\partial z_{in}}\right)f=\alpha_0 f\\
\displaystyle\sum_{i=0}^kz_{ij}\frac{\partial f}{\partial z_{ij}}=\alpha_jf&(j=1,\dots,n-1)\\
\displaystyle\sum_{i=0}^kz_{i0}\frac{\partial f}{\partial z_{in}}=f\\
\displaystyle\sum_{j=0}^nz_{ij}\frac{\partial f}{\partial z_{pj}}=-\delta_{ip}f&(i,p=0,1,\dots,k)\\
\frac{\partial^2 f}{\partial z_{ij}\partial z_{pq}}=\frac{\partial^2 f}{\partial z_{pj}\partial z_{iq}}& (i,p=0,1,\dots,k,\quad j,q=0,\dots, n).
\end{cases}
\end{equation}
\noindent
If we take a restriction to $z=
\begin{pmatrix}
1& & &z_{0k+1}&\cdots&z_{0n}\\
&\ddots& & \vdots &\ddots&\vdots\\
& &1&z_{kk+1}&\cdots&z_{kn}
\end{pmatrix}
$ and $x_0=1$, our integral $f_\Gamma(z)$ becomes
\begin{equation}
f_\Gamma(z)=\int_\Gamma e^{z_{0n}+z_{1n}x_1+\cdots+z_{kn}x_k}\prod_{j=k+1}^{n-1}l_j(x;z)^{\alpha_j}x_1^{\alpha_1}\dots x_k^{\alpha_k}dx.
\end{equation}
If we put $c={}^t(
\alpha_0+1,\dots,\alpha_k+1,-\alpha_{k+1},\dots,-\alpha_{n-1})$, and put ${\bf a}(i,j)={\bf e}(i)+{\bf e}(j)$ $(i=0,1,\dots,k,j=k+1,\dots,n-1)$ and ${\bf a}(in)=-{\bf e}(0)+{\bf e}(i)$ $(i=1,\dots,k)$, $g_{\Gamma}(z)=e^{-z_{0n}}f_\Gamma(z)$ is a solution of $M_A(c)$ with $A=({\bf a}(i,j))_{(i,j)\in\llbracket 0,k\rrbracket\times\llbracket k+1,n\rrbracket\setminus\{ (0,n)\}}$. As in \S\ref{QuadraticRelationsForGrassman}, we also put $\tilde{\bf a}(i,j)=-{\bf e}(i)+{\bf e}(j)$ $(i=0,1,\dots,k,j=k+1,\dots,n-1)$, $\tilde{\bf a}(i,j)={\bf e}(0)-{\bf e}(i)$ and $\tilde{A}=(\tilde{\bf a}(i,j))_{(i,j)\in\llbracket 0,k\rrbracket\times\llbracket k+1,n\rrbracket\setminus\{ (0,n)\}}$. Since the variable $z_{0n}$ does not appear in $g_{\Gamma}(z)$, we write $z=
\begin{pmatrix}
1&& & &z_{0k+1}&\cdots&z_{0n-1}&*\\
&1& & &z_{1k+1}&\cdots&z_{1n-1}&z_{1n}\\
&&\ddots& & \vdots &\ddots&\vdots&\vdots\\
&& &1&z_{kk+1}&\cdots&z_{kn-1}&z_{kn}
\end{pmatrix}
$ by abuse of notation.
We consider a ``confluence'' of the staircase triangulation. Namely, for a ladder $J\subset\llbracket 1,\dots,k\rrbracket\times\llbracket k+1,\dots,n\rrbracket$, we associate a simplex $I=J\setminus\{ (0,n)\}$ of $A$. The collection of all such simplices $T=\{ I=J\setminus\{ (0,n)\}\mid J:\text{ladder}\}$ forms a convergent regular triangulation. Since $\vol_{\Z A}(\Delta_A)=\begin{pmatrix} n-1\\ k\end{pmatrix}$ and $|T|=\begin{pmatrix} n-1\\ k\end{pmatrix}$, $T$ is unimodular. For any simplex $I\in T$, we consider the equation $Av^I=-c$ such that $v^I_{ij}=0\quad ((i,j)\notin I)$. Defining $\tilde{c}_l=
\begin{cases}
c_l&(l=0,\dots, k)\\
-c_l&(l=k+1,\dots,n-1),
\end{cases}
$ it is equivalent to the system $\tilde{A}v^I=\tilde{c}$. Let us introduce the dual basis $\phi(l)$ $(l=0,\dots,n-1)$ to ${\bf e}(l)$. For any edge $(i,j)\in G_I$, we can easily confirm that $G_I\setminus (i,j)$ has exactly two connected components. The connected component which contains $i$ (resp. $j$) is denoted by $C_i(i,j)$ (resp. $C_j(i,j)$). For each $(i,j)\in G_I$, we put
\begin{equation}
\varphi(ij)=\sum_{l\in V(C_j(i,j))\setminus\{ n\}}\phi(l).
\end{equation}
\begin{prop}
For $(i,j),(i^\prime,j^\prime)\in I$, we have
\begin{equation}
\langle \varphi(ij),\tilde{\bf a}(i^\prime,j^\prime)\rangle=
\begin{cases}
1&((i,j)=(i^\prime,j^\prime))\\
0&(otherwise).
\end{cases}
\end{equation}
\end{prop}
\begin{proof}
We set $S=\llbracket 0,k\rrbracket\times\llbracket k+1,n\rrbracket$, $S^\prime=S\setminus\{ (0,n)\}$. Let $A_S:\Z^{S\times 1}\rightarrow \Z^{\llbracket 0,n\rrbracket\times 1}$ be the $\Z$-linear map defined by $X=(X_{ij})\mapsto \sum_{(i,j)\in S}X_{ij}(-{\bf e}(i)+{\bf e}(j))$ and let $A_{S^\prime}:\Z^{S^\prime\times 1}\rightarrow \Z^{\llbracket 0,n-1\rrbracket\times 1}$ be the $\Z$-linear map defined by $X\mapsto \sum_{(i,j)\in S^\prime}X_{ij}\tilde{\bf a}(i,j)$. Let $\pi_1:\Z^{S\times 1}\rightarrow\Z^{S^\prime \times 1}$ be the canonical projection and $\pi_2\Z^{\llbracket 0,n\rrbracket}\rightarrow \Z^{\llbracket 0,n-1\rrbracket}$ be the $\Z$-linear map given by $(x_0,\dots,x_n)\mapsto (x_0+x_n,x_1,\dots,x_{n-1})$. It is easy to check the identity $A_{S^\prime}\circ \pi_1=\pi_2\circ A_S.$ Let us denote by $A_J$ (resp. $A_I$) the restriction of the map $A_S$ (resp. $A_{S^\prime}$) to the submodule $M_J=\{ X\in\Z^{S\times 1}\mid X_{ij}=0\text{ if }(i,j)\notin J\}$ (resp. $M_I=\{ X\in\Z^{S^\prime\times 1}\mid X_{ij}=0\text{ if }(i,j)\notin I\}$). Then, we have the commutative diagram
\begin{equation}
\xymatrix{
\{ X\in M_J\mid X_{0n}=0\} \ar[r]^-{A_J} \ar[d]^{\pi_1}&\{ x\in\Z^{\llbracket 0,n\rrbracket\times 1}\mid x_0=x_0+\dots+x_n=0\}\ar[d]^{\pi_2}\\
M_I \ar[r]^-{A_I} &\{ x\in\Z^{\llbracket 0,n-1\rrbracket\times 1}\mid x_0+\dots+x_{n-1}=0\}.
}
\end{equation}
Since $A_J$ and the two vertical morphisms are isomorphisms, we see that $A_I^{-1}$ is given by $\pi_1\circ A_J^{-1}\circ\pi_2^{-1}$. In view of \cref{prop:GraphicalInverse}, we obtain the proposition.
\end{proof}
\noindent
Therefore, we obtain a
\begin{cor}
Under the notation above, one has
\begin{equation}
v^I_{ij}=\sum_{l\in V(C_j(i,j))}\tilde{c}_l.
\end{equation}
\end{cor}
\noindent
Substitution of this formula to $\Gamma$-series yields the formula
\begin{equation}
\varphi_{v^I}(z)=z_I^{v^I}\sum_{u_{\bar{I}}\in\Z^{\bar{I}}_{\geq 0}}\frac{\left(z_I^{-\langle \varphi(I),\tilde{A}_{\bar{I}}\rangle}z_{\bar{I}}\right)^{u_{\bar{I}}}}{\displaystyle\prod_{(i,j)\in I}\Gamma(1+v_{ij}^I-\langle\varphi(ij),\tilde{A}_{\bar{I}}u_{\bar{I}}\rangle)u_{\bar{I}}!}.
\end{equation}
Since this series is defined by means of a ladder $I$ and a parameter $\alpha$, we also denote it by $g_I(z;\alpha)$. Similarly, setting $\bar{I}_{irr}=\bar{I}\cap\{ (1,n),\dots,(k,n)\}$, we obtain the formula for the dual series
\begin{equation}
\varphi_{v^I}^\vee(z)=z_I^{v^I}\sum_{u_{\bar{I}}\in\Z^{\bar{I}}_{\geq 0}}\frac{(-1)^{|u_{\bar{I}_{irr}}|+\sum_{(i,n)\in I}\langle \varphi(in),\tilde{A}u_{\bar{I}}\rangle}\left(z_I^{-\langle \varphi(I),\tilde{A}_{\bar{I}}\rangle}z_{\bar{I}}\right)^{u_{\bar{I}}}}{\displaystyle\prod_{(i,j)\in I}\Gamma(1+v_{ij}^I-\langle\varphi(ij),\tilde{A}_{\bar{I}}u_{\bar{I}}\rangle)u_{\bar{I}}!}.
\end{equation}
This series is denoted by $g_I^\vee(z;\alpha)$.
We consider the de Rham cohomology group $\Homo^k_{dR}\left( \PP^k_x\setminus\bigcup_{j=0}^{n-1}\{ l_j(x;z)\},\nabla_z\right)$ with $\nabla_z=d_x+\sum_{j=0}^{n-1}d_x\log l_j(x;z)\wedge+d_x\left( \frac{l_n(x;z)}{l_0(x;z)}\right)\wedge$. For any subset $J=\{ j_1,\dots,j_k\}\subset\{1,\dots,n-1\}$ with cardinality $k$, we denote by $z_J$ the submatrix of $z$ consisting of column vectors indexed by $\{ 0\}\cup J$. We always assume $j_1<\dots<j_k$. We put
\begin{equation}
\omega_J(z;x)=d_x\log\left(\frac{l_{j_1}(x;z)}{l_{0}(x;z)}\right)\wedge\dots\wedge d_x\log\left(\frac{l_{j_k}(x;z)}{l_{0}(x;z)}\right).
\end{equation}
By a simple computation, we see that $\omega_J(x;z)=\displaystyle\sum_{p=0}^k(-1)^pl_{j_p}(x;z)\frac{d_xl_{j_0}\wedge\dots\wedge \widehat{d_xl_{j_p}}\wedge\dots\wedge d_xl_{j_k}}{l_{j_0}(x;z)\cdots l_{j_k}(x;z)}$. Here, we have put $j_0=0$.
As in \cite[Fact 2.5]{GotoMatsumotoContiguity}, we also see that
$
\displaystyle\sum_{p=0}^k(-1)^pl_{j_p}(x;z)d_xl_{j_0}\wedge\dots\wedge \widehat{d_xl_{j_p}}\wedge\dots\wedge d_xl_{j_k}=\det(z_J)\omega(x).
$
Therefore, we have $\omega_J(x;z)=\det(z_J)\frac{\omega(x)}{l_{j_0}(x;z)\cdots l_{j_k}(x;z)}.$ Moreover, the set $\{ \omega_J(x;z)\}_{J}$ forms a basis of the algebraic de Rham cohomology group $\Homo_{\rm dR}^k\left( \mathbb{P}^k_x\setminus\displaystyle\bigcup_{j=0}^{n-1}\{ l_j(x;z)=0\};\nabla_z\right)$ (\cite{AomotoKitaOrlikTerao}, \cite{KimuraBasisOfCohomology}). Now we are going to derive a quadratic relation for $g_I(z;\alpha)$. We take any pair of subsets $J,J^\prime\subset\{1,\dots,n-1\}$ with cardinality $k$. Let us put $J_a=J\cap\{ 1,\dots,k\}$, $J_a^\prime=J^\prime\cap\{ 1,\dots,k\}$, $J_b=J\cap\{ k+1,\dots,n-1\}$, and $J_b^\prime=J^\prime\cap\{ k+1,\dots,n-1\}$. We denote by ${\bf 1}_{J_a}$ (resp. ${\bf 1}_{J_b}$) the vector $\sum_{j\in J_a}{\bf e}(j)$ (resp. $\sum_{j\in J_b}{\bf e}(j)$). If we write $\alpha$ as $\displaystyle\sum_{j=0}^{n-1}\alpha_j{\bf e}(j)$, we also put $\alpha_a=\displaystyle\sum_{j=1}^k\alpha_j{\bf e}(j)$ and $\alpha_{b}=\displaystyle\sum_{j=k+1}^{n-1}\alpha_j{\bf e}(j)$. We can readily confirm the identities
\begin{equation}
\frac{\omega_J(x;z)}{\det(z_J)}=\frac{\omega(x)}{l_{j_0}(x;z)\cdots l_{j_k}(x;z)}=\frac{x_0\dots x_k}{l_{j_0}(x;z)\cdots l_{j_k}(x;z)}\frac{\omega(x)}{x_0\dots x_k}=x^{{\bf 1}-{\bf 1}_{J_a}}l^{-{\bf 1}_{J_b}}\frac{dx}{x}.
\end{equation}
The quadratic relation leads to the form
\begin{align}
&(-1)^{|J_b|+|J_b^\prime|}(-\alpha_{k+1})\cdots (-\alpha_{n-1})(-\alpha_b+{\bf 1}_{J_b})_{-{\bf 1}_{J_b}}(\alpha_b+{\bf 1}_{J^\prime_b})_{-{\bf 1}_{J^\prime_b}}\times\nonumber\\
&\sum_{I:\text{ladder}}\frac{\pi^{n-1}}{\displaystyle\prod_{(i,j)\in I}\sin\pi (-v_{ij}^I)}g_I(z;\alpha+{\bf 1}_{\llbracket 0,k\rrbracket}-{\bf 1}_J)g_I^\vee(z;\alpha-{\bf 1}_{\llbracket 0,k\rrbracket}+{\bf 1}_{J^\prime})\nonumber\\
=&\det(z_J)^{-1}\det(z_{J^\prime})^{-1}\frac{\langle \omega_{J}(x;z),\omega_{J^\prime}(x;z)\rangle_{ch}}{(2\pi\ii)^k}.
\end{align}
By the same argument as \cite{MatsumotoIntersection}, we obtain a formula of the cohomology intersection numbers $\langle \omega_{J}(x;z),\omega_{J^\prime}(x;z)\rangle_{ch}$.
\begin{prop}\label{prop:E(21...1)c.i.n.}
Under the assumption $\tilde{c_j}\notin\Z$, one has
\begin{equation}\label{MatsumotoFormula2}
\frac{\langle \omega_{J}(x;z),\omega_{J^\prime}(x;z)\rangle_{ch}}{(2\pi\ii)^k}=
\begin{cases}
\frac{1}{\prod_{j\in J}\tilde{c}_j}& (J=J^\prime)\\
0&(J\neq J^\prime)
\end{cases}.
\end{equation}
\end{prop}
\begin{proof}
The result immediately follows from the computation of \cite{MatsumotoIntersection} after a minor modification. For readers' convenience, we give necessary modifications. Let $X$ be the projective variety obtained by blowing-up $\mathbb{P}^k$ along the 2-codimensional linear subvariety $\{ l_0(x;z)=l_n(x;z)=0\}$. It is readily seen that there is a well defined regular map $f:X\rightarrow\mathbb{P}^1$ whose restriction to the subspace $\{ l_0(x;z)\neq 0\}$ is identical with $\frac{l_n(x;z)}{l_0(x;z)}$. We denote by $L_j$ the proper transformation of the divisor $\{ l_j(x;z)=0\}\subset\mathbb{P}^k$ ($j=0,\dots,n-1$) and denote by $L_n$ the exceptional divisor of the blowing-up. For any pair of $k$-tuples $P=(p_1,\dots,p_m)$ and $Q=(q_1,\dots,q_m)$ $(0\leq p_1<\dots< p_m\leq n-1, \ \ 0\leq q_1<\dots<q_m\leq n-1)$, we set
\begin{equation}
\delta(P,Q)=
\begin{cases}
1&(\text{if }P=Q)\\
0&(\text{if }P\neq Q)
\end{cases}.
\end{equation}
For each $j=0,\dots,n$, we denote by $V_j$ a tubular neighbourhood of $L_j$ and by $h_j$ a smooth function on $X$ which is equal to $1$ near $L_j$ and $0$ outside $V_j$. For each multi-index $P=(p_1,\dots,p_k)$, we denote by $w_P=(w_1,\dots,w_k)=(l_{p_i})$ the local coordinate around $\displaystyle\bigcap_{i=1}^kL_{p_i}$ so that $L_{p_i}=\{ w_i=0\}$ and we set $D_P=\displaystyle\bigcap_{i=1}^kV_{p_i}$. By solving the equation $\nabla^\vee \xi=\omega_{J^\prime}(x;z)$ locally as in \cite[Lemma 5.1, 5.2 and Lemma 6.1]{MatsumotoIntersection}, we can find a smooth $k$-form $\psi_{J^\prime}(x)$ on $X$ such that the following properties 1-5 hold.
\begin{enumerate}
\item $\psi_{J^\prime}$ has a compact support in $X\setminus\displaystyle \bigcup_{j=0}^{n}L_j$ and cohomologous to $\omega_{J^\prime}$ in $\Homo_{\rm dR}^k\left( \mathbb{P}^k_x\setminus\displaystyle\bigcup_{j=0}^{n-1}\{ l_j(x;z)=0\};\nabla_z^\vee\right)$.
\item The $k$-form $\psi_{J^\prime}-\omega_{J^\prime}$ vanishes outside a small tubular neighbourhood of the divisor $\displaystyle \bigcup_{j=0}^{n}L_j$.
\item On each $V_j\setminus D_P$, $\psi_{J^\prime}$ is of the form $\psi_{J^\prime}=\sum_{i}\xi_i\wedge\eta_i$ where $\xi_i$ are smooth forms and $\eta_i$ are rational differential forms of degree greater or equal to one.
\item For each multi-index $P=(p_1,\dots,p_k)$ such that $p_1=0$, we have $\psi_{J^\prime}=f_P(w_P)dh_{p_1}\wedge\dots\wedge dh_{p_k}$ where $f_P(w_P)$ is a holomorphic function such that $f_P(0,w_2,\dots,w_k)\equiv 0$.
\item For each multi-index $P=(p_1,\dots,p_k)$ such that $0\notin P$, we have $\psi_{J^\prime}=f_P(w_P)dh_{p_1}\wedge\dots\wedge dh_{p_k}$ where $f_P(w_P)$ is a holomorphic function with $f_P({\bf O})=(-1)^{k}\frac{\delta(P,J^\prime)}{\prod_{j=1}^k\tilde{c}_{p_j}}$.
\end{enumerate}
The property 4 is due to the fact that the term $d\left( \frac{l_n(x;z)}{l_0(x;z)}\right)\wedge$ in $\nabla^\vee_z=d-\sum_{j=1}^{n-1}\tilde{c}_jd\log l_j(x;z)\wedge-d\left( \frac{l_n(x;z)}{l_0(x;z)}\right)\wedge$ gives rise to the term $-\frac{dw_1}{w_1^2}\wedge$. If we denote by $\iota_z$ the inverse of the canonical isomorphism
\begin{equation}
{\rm can}:\Homo_{r.d.}^k\left( \mathbb{P}^k_x\setminus\displaystyle\bigcup_{j=0}^{n-1}\{ l_j(x;z)=0\};\nabla_z^\vee\right)\tilde{\rightarrow}\Homo_{\rm dR}^k\left( \mathbb{P}^k_x\setminus\displaystyle\bigcup_{j=0}^{n-1}\{ l_j(x;z)=0\};\nabla_z^\vee\right),
\end{equation}
it is clear that $\psi_{J^\prime}$ is a representative of $\iota_z\omega_{J^\prime}$ in $\Homo_{r.d.}^k\left( \mathbb{P}^k_x\setminus\displaystyle\bigcup_{j=0}^{n-1}\{ l_j(x;z)=0\};\nabla_z^\vee\right)$. Therefore, by properties 3,4, and 5, we obtain
\begin{align}
\langle \omega_{J}(x;z),\omega_{J^\prime}(x;z)\rangle_{ch}&=\int \omega_{J}(x;z)\wedge\psi_{J^\prime}\\
&=\sum_{P}\int_{D_P}\omega_{J}(x;z)\wedge\psi_{J^\prime}\\
&=(2\pi\ii)^k\sum_{P}\underset{w_k=0}{\rm Res}\left( \underset{w_{k-1}=0}{\rm Res}\left( \cdots \underset{w_1=0}{\rm Res}\left((-1)^kf_{P}\omega_{J}(x;z)\right)\cdots\right)\right)\\
&=(2\pi\ii)^k\frac{\delta(J,J^\prime)}{\prod_{j\in J}\tilde{c}_{j}}.
\end{align}
\end{proof}
In sum, we obtain the general quadratic relation of a confluence of Aomoto-Gelfand hypergeometric functions:
\begin{thm}\label{thm:QuadraticRelationsForAomotoGelfand2}
Under the notation as above, for any $z\in U_T$, we have an identity
\begin{align}
&(-1)^{|J_b|+|J^\prime_b|+k}\alpha_{k+1}\dots\alpha_n(-\alpha_b+{\bf 1}_{J_b})_{-{\bf 1}_{J_b}}(\alpha_b+{\bf 1}_{J^\prime_b})_{-{\bf 1}_{J_b^\prime}}\times\nonumber\\
&\sum_{I:\text{ladder}}\frac{\pi^{n-1}}{\displaystyle\prod_{(i,j)\in I}\sin\pi v_{ij}^I}g_I(z;\alpha+{\bf 1}_{\llbracket 0,k\rrbracket}-{\bf 1}_J)g_I^\vee(z;\alpha-{\bf 1}_{\llbracket 0,k\rrbracket}+{\bf 1}_{J^\prime})\nonumber\\
=&
\det(z_J)^{-1}\det(z_{J^\prime})^{-1}\frac{\langle \omega_{J}(x;z),\omega_{J^\prime}(x;z)\rangle_{ch}}{(2\pi\ii)^k}.
\end{align}
Here, the right hand side is explicitly determined by (\ref{MatsumotoFormula}).
\end{thm}
\begin{exa}{\bf (Kummer's hypergeometric series)}
The simplest case is $k=1$ and $n=3$. This case is known as Kummer's hypergeometric equation. By computing the cohomology intersection number $\langle \frac{dx}{x},\frac{dx}{x}\rangle_{ch}$, we obtain a quadratic relation
We have a relation
\begin{equation}
(\gamma-\alpha-1){}_1F_1\left(\substack{\alpha\\ \gamma};z\right){}_1F_1\left(\substack{-\alpha\\ 2-\gamma};-z\right)+\alpha{}_1F_1\left(\substack{1+\alpha-\gamma\\ 2-\gamma};z\right){}_1F_1\left(\substack{\gamma-\alpha-1\\ \gamma};-z\right)=\gamma-1,
\end{equation}
where the series ${}_1F_1\left(\substack{\alpha\\ \gamma};z\right)$ is Kummer's hypergeometric series
\begin{equation}
{}_1F_1\left(\substack{\alpha\\ \gamma};z\right)=\sum_{n=0}^\infty\frac{(\alpha)_n}{(\gamma)_nn!}z^n.
\end{equation}
This identity implies a series of combinatorial identities
\begin{equation}
(\gamma-\alpha-1)\sum_{l+m=n}(-1)^m\frac{(\alpha)_l(-\alpha)_{m}}{(\gamma)_l(1)_l(2-\gamma)_m(1)_m}
+\alpha\sum_{l+m=n}(-1)^m\frac{(1+\alpha-\gamma)_l(\gamma-\alpha-1)_m}{(2-\gamma)_l(1)_l(\gamma)_m(1)_m}=0,
\end{equation}
where $n$ is a positive integer.
\end{exa}
\begin{exa}{\bf (A confluence of $E(3,6)$)}
This is a confluence of \cref{exa:AomotoGelfand}. The integral in question takes the form $f_\Gamma(z)=\int_\Gamma\prod_{j=3}^4(z_{0j}+z_{1j}x_1+z_{2j}x_2)^{-c_j}e^{z_{1j}x_1+z_{2j}x_2}x_1^{c_1}x_2^{c_2}\frac{dx_1\wedge dx_2}{x_1x_2}$. The quadratic relation with respect to the cohomology intersection number $\langle\frac{dx_1\wedge dx_2}{x_1x_2},\frac{dx_1\wedge dx_2}{x_1x_2}\rangle_{ch}$ is given by
\begin{equation}
c_1c_2c_3c_4\sum_{i=1}^6\frac{\pi^4}{\sin\pi(v_i)}\varphi_i(z;c)\varphi_i^\vee(z;c)=1,
\end{equation}
where $v_i$ are given by
\begin{align}
v_1&={}^t(-c_3,-c_4,c_0+c_1,-c_1)\\
v_2&={}^t(-c_3,-c_2+c_3,-c_0-c_1,c_0)\\
v_3&={}^t(-c_3,-c_2+c_3,-c_1,-c_0)\\
v_4&={}^t(-c_2,c_2-c_3,-c_4,c_0)\\
v_5&={}^t(-c_2,c_2-c_3,c_0-c_4,-c_0)\\
v_6&={}^t(-c_2,-c_1,-c_0+c_4,-c_4).
\end{align}
Note that we have a relation $c_0+c_1+c_2-c_3-c_4=0$. The series $\varphi_i(z;c)$ are given by the following series.
\begin{align}
\varphi_1(z;c)&=z_{23}^{-c_3}z_{24}^{-c_4}z_{25}^{c_0+c_1}z_{15}^{-c_1}\nonumber\\
&\sum_{u_{13},u_{14},u_{03},u_{04}\geq 0}\frac{1}{\Gamma(1-c_3-u_{13}-u_{03})\Gamma(1-c_4-u_{14}-u_{04})\Gamma(1+c_0+c_1+u_{13}+u_{14}+u_{03}+u_{04})}\nonumber\\
&\frac{(z_{23}^{-1}z_{25}z_{15}^{-1}z_{13})^{u_{13}}(z_{24}^{-1}z_{25}z_{15}^{-1}z_{14})^{u_{14}}(z_{23}^{-1}z_{25}z_{03})^{u_{03}}(z_{24}^{-1}z_{25}z_{04})^{u_{04}}}{\Gamma(1-c_1-u_{13}-u_{14})u_{13}!u_{14}!u_{03}!u_{04}!}
\end{align}
\begin{align}
\varphi_2(z;c)&=z_{23}^{-c_3}z_{24}^{-c_2+c_3}z_{14}^{-c_0-c_1}z_{15}^{c_0}\nonumber\\
&\sum_{u_{25},u_{13},u_{03},u_{04}\geq 0}\frac{1}{\Gamma(1-c_3-u_{13}-u_{03})\Gamma(1-c_2+c_3-u_{25}+u_{13}+u_{03})}\nonumber\\
&\frac{1}{\Gamma(1-c_0-c_1+u_{25}-u_{13}-u_{03}-u_{04})\Gamma(1+c_0-u_{25}+u_{03}+u_{04})}\nonumber\\
&\frac{(z_{24}^{-1}z_{14}z_{15}^{-1}z_{25})^{u_{25}}(z_{23}^{-1}z_{24}z_{14}^{-1}z_{13})^{u_{13}}(z_{23}^{-1}z_{24}z_{14}^{-1}z_{15}z_{03})^{u_{03}}(z_{14}^{-1}z_{15}z_{04})^{u_{04}}}{u_{25}!u_{13}!u_{03}!u_{04}!}
\end{align}
\begin{align}
\varphi_3(z;c)&=z_{23}^{-c_3}z_{24}^{-c_2+c_3}z_{14}^{-c_1}z_{04}^{-c_0}\nonumber\\
&\sum_{u_{25},u_{15},u_{13},u_{03}\geq 0}\frac{1}{\Gamma(1-c_3-u_{13}-u_{03})\Gamma(1-c_2+c_3-u_{25}+u_{13}+u_{03})\Gamma(1-c_1-u_{15}-u_{13})}\nonumber\\
&\frac{(z_{24}^{-1}z_{04}z_{25})^{u_{25}}(z_{14}^{-1}z_{04}z_{15})^{u_{15}}(z_{23}^{-1}z_{24}z_{14}^{-1}z_{13})^{u_{13}}(z_{23}^{-1}z_{24}z_{04}^{-1}z_{03})^{u_{03}}}{\Gamma(1-c_0+u_{25}+u_{15}-u_{03})u_{25}!u_{15}!u_{13}!u_{03}!}
\end{align}
\begin{align}
\varphi_4(z;c)&=z_{23}^{-c_2}z_{13}^{c_2-c_3}z_{14}^{-c_4}z_{15}^{c_0}\nonumber\\
&\sum_{u_{24},u_{25},u_{03},u_{04}\geq 0}\frac{1}{\Gamma(1-c_2-u_{24}-u_{25})\Gamma(1+c_2-c_3+u_{24}+u_{25}-u_{03})\Gamma(1-c_4-u_{24}-u_{04})}\nonumber\\
&\frac{(z_{23}^{-1}z_{13}z_{14}^{-1}z_{24})^{u_{24}}(z_{23}^{-1}z_{13}z_{15}^{-1}z_{25})^{u_{25}}(z_{13}^{-1}z_{15}z_{03})^{u_{03}}(z_{14}^{-1}z_{15}z_{04})^{u_{04}}}{\Gamma(1+c_0-u_{25}+u_{03}+u_{04})u_{24}!u_{25}!u_{03}!u_{04}!}
\end{align}
\begin{align}
\varphi_5(z;c)&=z_{23}^{-c_2}z_{13}^{c_2-c_3}z_{14}^{c_0-c_4}z_{04}^{-c_0}\nonumber\\
&\sum_{u_{24},u_{23},u_{15},u_{03}\geq 0}\frac{1}{\Gamma(1-c_2-u_{24}-u_{25})\Gamma(1+c_2-c_3+u_{24}+u_{25}-u_{03})}\nonumber\\
&\frac{1}{\Gamma(1+c_0-c_4-u_{24}-u_{25}-u_{15}+u_{03})\Gamma(1-c_0+u_{25}+u_{15}-u_{03})}\nonumber\\
&\frac{(z_{23}^{-1}z_{13}z_{14}^{-1}z_{24})^{u_{24}}(z_{23}^{-1}z_{13}z_{14}^{-1}z_{04}z_{25})^{u_{25}}(z_{14}^{-1}z_{04}z_{15})^{u_{15}}(z_{13}^{-1}z_{14}z_{04}^{-1}z_{03})^{u_{03}}}{u_{24}!u_{25}!u_{15}!u_{03}!}
\end{align}
\begin{align}
\varphi_6(z;c)&=z_{23}^{-c_2}z_{13}^{-c_1}z_{03}^{-c_0+c_4}z_{04}^{-c_4}\nonumber\\
&\sum_{u_{24},u_{25},u_{14},u_{15}\geq 0}\frac{1}{\Gamma(1-c_2-u_{24}-u_{25})\Gamma(1-c_1-u_{14}-u_{15})\Gamma(1-c_0+c_4+u_{24}+u_{25}+u_{14}+u_{15})}\nonumber\\
&\frac{(z_{23}^{-1}z_{03}z_{04}^{-1}z_{24})^{u_{24}}(z_{23}^{-1}z_{03}z_{25})^{u_{25}}(z_{13}^{-1}z_{03}z_{04}^{-1}z_{14})^{u_{14}}(z_{13}^{-1}z_{03}z_{15})^{u_{15}}}{\Gamma(1-c_4-u_{24}-u_{14})u_{24}!u_{25}!u_{14}!u_{15}!}
\end{align}
\begin{comment}
\begin{align}
\varphi_1^\vee(z;c)&=z_{23}^{c_3}z_{24}^{c_4}z_{25}^{-c_0-c_1}z_{15}^{c_1}\nonumber\\
&\sum_{u_{13},u_{14},u_{03},u_{04}\geq 0}\frac{1}{\Gamma(1+c_3-u_{13}-u_{03})\Gamma(1+c_4-u_{14}-u_{04})\Gamma(1-c_0-c_1+u_{13}+u_{14}+u_{03}+u_{04})}\nonumber\\
&\frac{(z_{23}^{-1}z_{25}z_{15}^{-1}z_{13})^{u_{13}}(z_{24}^{-1}z_{25}z_{15}^{-1}z_{14})^{u_{14}}(-z_{23}^{-1}z_{25}z_{03})^{u_{03}}(-z_{24}^{-1}z_{25}z_{04})^{u_{04}}}{\Gamma(1+c_1-u_{13}-u_{14})u_{13}!u_{14}!u_{03}!u_{04}!}
\end{align}
\begin{align}
\varphi_2^\vee(z;c)&=z_{23}^{c_3}z_{24}^{c_2-c_3}z_{14}^{c_0+c_1}z_{15}^{-c_0}\nonumber\\
&\sum_{u_{25},u_{13},u_{03},u_{04}\geq 0}\frac{1}{\Gamma(1-c_3-u_{13}-u_{03})\Gamma(1-c_2+c_3-u_{25}+u_{13}+u_{03})}\nonumber\\
&\frac{1}{\Gamma(1-c_0-c_1+u_{25}-u_{13}-u_{03}-u_{04})\Gamma(1+c_0-u_{25}+u_{03}+u_{04})}\nonumber\\
&\frac{(z_{24}^{-1}z_{14}z_{15}^{-1}z_{25})^{u_{25}}(z_{23}^{-1}z_{24}z_{14}^{-1}z_{13})^{u_{13}}(-z_{23}^{-1}z_{24}z_{14}^{-1}z_{15}z_{03})^{u_{03}}(-z_{14}^{-1}z_{15}z_{04})^{u_{04}}}{u_{25}!u_{13}!u_{03}!u_{04}!}
\end{align}
\begin{align}
\varphi_3^\vee(z;c)&=z_{23}^{c_3}z_{24}^{c_2-c_3}z_{14}^{c_1}z_{04}^{c_0}\nonumber\\
&\sum_{u_{25},u_{15},u_{13},u_{03}\geq 0}\frac{1}{\Gamma(1+c_3-u_{13}-u_{03})\Gamma(1+c_2-c_3-u_{25}+u_{13}+u_{03})\Gamma(1+c_1-u_{15}-u_{13})}\nonumber\\
&\frac{(-z_{24}^{-1}z_{04}z_{25})^{u_{25}}(-z_{14}^{-1}z_{04}z_{15})^{u_{15}}(z_{23}^{-1}z_{24}z_{14}^{-1}z_{13})^{u_{13}}(z_{23}^{-1}z_{24}z_{04}^{-1}z_{03})^{u_{03}}}{\Gamma(1+c_0+u_{25}+u_{15}-u_{03})u_{25}!u_{15}!u_{13}!u_{03}!}
\end{align}
\begin{align}
\varphi_4^\vee(z;c)&=z_{23}^{c_2}z_{13}^{-c_2+c_3}z_{14}^{c_4}z_{15}^{-c_0}\nonumber\\
&\sum_{u_{24},u_{25},u_{03},u_{04}\geq 0}\frac{1}{\Gamma(1+c_2-u_{24}-u_{25})\Gamma(1-c_2+c_3+u_{24}+u_{25}-u_{03})\Gamma(1+c_4-u_{24}-u_{04})}\nonumber\\
&\frac{(z_{23}^{-1}z_{13}z_{14}^{-1}z_{24})^{u_{24}}(z_{23}^{-1}z_{13}z_{15}^{-1}z_{25})^{u_{25}}(-z_{13}^{-1}z_{15}z_{03})^{u_{03}}(-z_{14}^{-1}z_{15}z_{04})^{u_{04}}}{\Gamma(1-c_0-u_{25}+u_{03}+u_{04})u_{24}!u_{25}!u_{03}!u_{04}!}
\end{align}
\begin{align}
\varphi_5^\vee(z;c)&=z_{23}^{c_2}z_{13}^{-c_2+c_3}z_{14}^{-c_0+c_4}z_{04}^{c_0}\nonumber\\
&\sum_{u_{24},u_{23},u_{15},u_{03}\geq 0}\frac{1}{\Gamma(1+c_2-u_{24}-u_{25})\Gamma(1-c_2+c_3+u_{24}+u_{25}-u_{03})}\nonumber\\
&\frac{1}{\Gamma(1-c_0+c_4-u_{24}-u_{25}-u_{15}+u_{03})\Gamma(1+c_0+u_{25}+u_{15}-u_{03})}\nonumber\\
&\frac{(z_{23}^{-1}z_{13}z_{14}^{-1}z_{24})^{u_{24}}(-z_{23}^{-1}z_{13}z_{14}^{-1}z_{04}z_{25})^{u_{25}}(-z_{14}^{-1}z_{04}z_{15})^{u_{15}}(z_{13}^{-1}z_{14}z_{04}^{-1}z_{03})^{u_{03}}}{u_{24}!u_{25}!u_{15}!u_{03}!}
\end{align}
\begin{align}
\varphi_6^\vee(z;c)&=z_{23}^{c_2}z_{13}^{c_1}z_{03}^{c_0-c_4}z_{04}^{c_4}\nonumber\\
&\sum_{u_{24},u_{25},u_{14},u_{15}\geq 0}\frac{1}{\Gamma(1+c_2-u_{24}-u_{25})\Gamma(1+c_1-u_{14}-u_{15})\Gamma(1+c_0-c_4+u_{24}+u_{25}+u_{14}+u_{15})}\nonumber\\
&\frac{(z_{23}^{-1}z_{03}z_{04}^{-1}z_{24})^{u_{24}}(-z_{23}^{-1}z_{03}z_{25})^{u_{25}}(z_{13}^{-1}z_{03}z_{04}^{-1}z_{14})^{u_{14}}(-z_{13}^{-1}z_{03}z_{15})^{u_{15}}}{\Gamma(1+c_4-u_{24}-u_{14})u_{24}!u_{25}!u_{14}!u_{15}!}
\end{align}
\end{comment}
The series $\varphi_i^\vee(z;c)$ is obtained from $\varphi_i(z;c)$ by replacing $c_i$ by $-c_i$ and $z_{15},z_{25}$ by $-z_{15},-z_{25}$ in the summand. Note that if we substitute
\begin{equation}
\begin{pmatrix}
z_{03}&z_{04}&*\\
z_{13}&z_{14}&z_{15}\\
z_{23}&z_{24}&z_{25}
\end{pmatrix}
=
\begin{pmatrix}
1&1&*\\
1&\zeta_1&\zeta_1\zeta_2\\
1&\zeta_1\zeta_3&\zeta_1\zeta_2\zeta_3\zeta_{4}
\end{pmatrix},
\end{equation}
\noindent
all the Laurent series $\varphi_{i}(z;c)$ and $\varphi_{i}^\vee(z;c)$ above become power series, i. e., they do not contain any negative power.
\end{exa}
\section*{Appendix 1: A lemma on holonomic dual}
In this appendix, we prove \cref{lem:Duality}. Let $\Delta_X:X\hookrightarrow X\times X$ be the diagonal embedding. We also denote its image by $\Delta_X$. Since ${\rm Ch}(M\boxtimes N)={\rm Ch}(M)\times {\rm Ch}(N)$ and ${\rm Ch}(M)\cap {\rm Ch}(N)\subset T^*_XX$ by the assumption of \cref{lem:Duality}, we obtain the inclusion
\begin{equation}
T_{\Delta_X}(X\times X)\cap {\rm Ch}(M\boxtimes N)=\{ (x,\xi;x,\xi)\in T^*X\times T^*X\mid (x,\xi)\in {\rm Ch}(M)\cap {\rm Ch}(N)\}\subset T^*_{X\times X}X\times X.
\end{equation}
Therefore, $M\boxtimes N$ is non-characteristic with respect to the morphism $\Delta_X$. By \cite[Theorem 2.7.1.]{HTT}, we have the commutativity $\D_X (\mathbb{L}\Delta^*_X(M\boxtimes N))\simeq\mathbb{L}\Delta^*_X\D_{X\times X}(M\boxtimes N)$. Therefore, we have quasi-isomorphisms
\begin{align}
\D_X (M\overset{\D}{\otimes}N)&=\D_X (\mathbb{L}\Delta^*_X(M\boxtimes N))\\
&\simeq\mathbb{L}\Delta^*_X(\D_XM\boxtimes \D_XN)\\
&\simeq \D_XM\overset{\D}{\otimes} \D_XN.
\end{align}
\section*{Appendix 2: Proof of \cref{prop:PochhammerIntersection}}
We apply the twisted period relation to $\Homo_n\left( X, \mathcal{L}\right)$, where $X=\C^n_x\setminus\{ x_1\cdots x_n(1-x_1-\dots-x_n)=0\}$ $\mathcal{L}=\C x_1^{\alpha_1}\cdots x_n^{\alpha_n}(1-x_1-\dots-x_n)^{\alpha_{n+1}}$. We take a basis $\frac{dx}{x}=\frac{dx_1\wedge \dots\wedge dx_n}{x_1\dots x_n}$ of twisted cohomology group $\Homo^n\left( X, \mathcal{L}\right)$ and of $\Homo^n\left( X, \mathcal{L}^\vee\right).$ By \cite{MatsumotoIntersection}, we have $\langle\frac{dx}{x},\frac{dx}{x}\rangle_{ch}=(2\pi\ii)^n\frac{\alpha_0+\dots+\alpha_n}{\alpha_0\dots\alpha_n}.$ On the other hand, we have
\begin{equation}
\int_{P_\tau}x_1^{\alpha_1}\cdots x_n^{\alpha_n}(1-x_1-\dots-x_n)^{\alpha_{n+1}}\frac{dx}{x}=\prod_{i=1}^{n+1}(1-e^{-2\pi\ii\alpha_i})\frac{\Gamma(\alpha_1)\dots\Gamma(\alpha_n)\Gamma(\alpha_{n+1}+1)}{\Gamma(1-\alpha_0)}
\end{equation}
and
\begin{equation}
\int_{\check{P}_\tau}x_1^{-\alpha_1}\cdots x_n^{-\alpha_n}(1-x_1-\dots-x_n)^{-\alpha_{n+1}}\frac{dx}{x}=\prod_{i=1}^{n+1}(1-e^{2\pi\ii\alpha_i})\frac{\Gamma(-\alpha_1)\dots\Gamma(-\alpha_n)\Gamma(1-\alpha_{n+1})}{\Gamma(1+\alpha_0)}.
\end{equation}
\noindent
Therefore, we have
\begin{align}
&\langle P_\tau,\check{P}_\tau\rangle_{h}\nonumber\\
=&\left(\int_{\check{P}_n}x_1^{-\alpha_1}\cdots x_n^{-\alpha_n}(1-x_1-\dots-x_n)^{-\alpha_{n+1}}\frac{dx}{x}\right)\langle\frac{dx}{x},\frac{dx}{x}\rangle_{ch}^{-1}\left(\int_{P_n}x_1^{\alpha_1}\cdots x_n^{\alpha_n}(1-x_1-\dots-x_n)^{\alpha_{n+1}}\frac{dx}{x}\right)\\
=&\prod_{i=0}^{n+1}(1-e^{-2\pi\ii\alpha_i})\\
=&(2\ii)^{n+2}\prod_{i=0}^{n+1}\sin\pi\alpha_i.
\end{align}
\section*{Appendix 3: Construction of a lift of a Pochhammer cycle}
In this appendix, we summarize the construction of Pochhammer cycles following \cite[\S 6]{Beukers} and construct its lift by a covering map.
We consider a hyperplane $H$ in $\C^{n+1}$ defined by $\{ t_0+\cdots+t_n=1\}$. Let $\varepsilon$ be a small real positive number. We consider a polytope $F$ in $\R^{n+1}$ defined by
\begin{equation}
|x_{i_1}|+\cdots+|x_{i_k}|\leq 1-(n+1-k)\varepsilon
\end{equation}
for all $k=1,\dots, n+1$ and all $0\leq i_1 <i_2<\dots <i_k\leq n$. The faces of this polytope can be labeled by vectors $\mu\in\{0,\pm 1\}^n\setminus\{ 0\}^n$. We define $|\mu|=\displaystyle\sum_{i=0}^n|\mu_i|$. The face $F_\mu$ corresponding to $\mu$ is defined by
\begin{equation}
\mu_0 x_0+\mu_1 x_1+\dots+\mu_nx_n=1-(n+1-|\mu|)\varepsilon,\; \mu_jx_j\geq\varepsilon\text{ whenever }\mu_j\neq 0,\; |x_j|\leq\varepsilon\text{ whenever }\mu_j=0.
\end{equation}
\noindent
The number of faces of $F$ is $3^n-1$ and each $F_\mu$ is isomorphic to $\Delta_{|\mu|-1}\times {\rm I}^{n+1-|\mu|}$ where ${\rm I}$ is a closed interval. The vertices of $F$ are points with one coordinate $\pm(1-n\varepsilon)$ and all other coordinates $\pm \varepsilon$. Therefore, the number of vertices is $(n+1)2^{n+1}$. Define a continuous piecewise smooth map $P:\cup_\mu F_\mu\rightarrow H$ by
\begin{equation}
P(x_0,\dots ,x_n)=\frac{1}{\tilde{y}_0+\dots+\tilde{y}_n}(y_0,\dots,y_n)
\end{equation}
where
\begin{equation}\label{argument}
y_j=
\begin{cases}
x_j & (x_j\geq\varepsilon)\\
e^{-2\pi\ii}|x_j| & (x_j\leq-\varepsilon)\\
\varepsilon e^{-\pi\ii(1-\frac{x_j}{\varepsilon})} & (|x_j|\leq\varepsilon).
\end{cases}
\end{equation}
\begin{equation}\label{argument2}
\tilde{y}_j=
\begin{cases}
|x_j| & (|x_j|\geq\varepsilon)\\
\varepsilon e^{-\pi\ii(1-\frac{x_j}{\varepsilon})} & (|x_j|\leq\varepsilon).
\end{cases}
\end{equation}
Let us denote by $\pi:H\rightarrow \C^n$ be the projection $\pi(t_0,\dots,t_n)=(t_1,\dots,t_n).$ By definition, the image of the map $\pi\circ P$ is contained in the complement of a divisor $\{ 1=t_1+\dots+t_n\}$ in the torus $(\C^\times)^n\subset\C^n$. On each face $F_\mu$, the branch of a multivalued function $t_1^{\beta_1-1}\dots t_n^{\beta_n-1}(1-t_1-\dots -t_n)^{\beta_{0}-1}$ on $\pi\circ P (F_\mu)$ is defined by
\begin{equation}\label{TheBranch}
t_1^{\beta_1-1}\dots t_n^{\beta_n-1}(1-t_1-\cdots -t_n)^{\beta_{0}-1}=\prod_{\mu_j\neq 0}|x_j|^{\beta_j-1}e^{\pi\ii(\mu_j-1)\beta_j}\prod_{\mu_k=0}\varepsilon^{\beta_k-1}e^{\pi\ii(\frac{x_k}{\varepsilon}-1)(\beta_k-1)}.
\end{equation}
Thus, we can define a multi-dimensional Pochhammer cycle $P_n$ as a cycle with local system coefficients.
Now we consider a (covering) map between tori $p:(\C^\times)^n_\tau\rightarrow (\C^\times)^n_t$ defined by $p(\tau)=\tau^A$ where $A=({\bf a}(1)|\dots|{\bf a}(n))$ is an invertible $n$ by $n$ matrix with integer entries. We put $\beta^\prime={}^t(\beta_1,\dots,\beta_n).$
\begin{prop}
There exists a twisted cycle $P_n^\prime$ in $\Homo_n\left((\C^\times)^n_\tau\setminus\left\{ 1=\displaystyle\sum_{i=1}^n\tau^{{\bf a}(i)}\right\};\underline{\C}\left(1-\displaystyle\sum_{i=1}^n\tau^{{\bf a}(i)}\right)^{\beta_0}\tau^{A\beta^\prime}\right)$ such that the identity $p_*(P^\prime_n)=P_n$ holds.
\end{prop}
\begin{proof}
Let us put $\pi\circ P(x)=(q_1(x),\dots, q_n(x)).$ Define a map $P^\prime:\cup_\mu F_\mu\rightarrow (\C^\times)^n_\tau\setminus\left\{ 1=\displaystyle\sum_{i=1}^n\tau^{{\bf a}(i)}\right\}$ by
\begin{equation}
P^\prime(x)=(q_1(x),\dots, q_n(x))^{A^{-1}}.
\end{equation}
Note that this is a well-defined continuous map in view of (\ref{argument}) and (\ref{argument2}). The branch of a multivalued function $\left(1-\displaystyle\sum_{i=1}^n\tau^{{\bf a}(i)}\right)^{\beta_0}\tau^{A\beta^\prime}$ on the face $F_\mu$ is therefore defined by the formula
\begin{equation}
\left(1-\sum_{i=1}^n\tau^{{\bf a}(i)}\right)^{\beta_0}\tau^{A\beta^\prime}=\prod_{\mu_j\neq 0}|x_j|^{\beta_j-1}e^{\pi\ii(\mu_j-1)\beta_j}\prod_{\mu_k=0}\varepsilon^{\beta_k-1}e^{\pi\ii(\frac{x_k}{\varepsilon}-1)(\beta_k-1)}.
\end{equation}
Thus, we can define a twisted cycle $P^\prime_n$. It is obvious from the construction that the identity $p_*(P^\prime_n)=P_n$ holds.
\end{proof}
Write $A=(A_1|\cdots|A_k)$, $A_l=({\bf a}^{(l)}(1)|\cdots|{\bf a}^{(l)}(n_l))$
One can easily generalize the result above to the following
\begin{prop}
Suppose $t=(t^{(1)},\dots,t^{(k)})$ and $\beta_i^{(l)}\in\C$ $(l=1,\dots,k, \; i=1,\dots,n_l)$. We put $\mathcal{L}=\displaystyle\prod_{l=1}^k\underline{\C}(1-\sum_{i=1}^{n_l}t^{(l)}_i)^{\beta^{(l)}_0}(t^{(l)}_1)^{\beta_{1}^{(l)}}\cdots (t^{(l)}_{n_l})^{\beta_{n_l}^{(l)}}$. Then, there exists a twisted cycle $P_n^\prime$ in
\newline
$\Homo_n\left(\displaystyle\prod_{l=1}^k\left((\C^\times)^{n_l}_{\tau^{(l)}}\setminus\{ 1=\sum_{i=1}^{n_l}\tau^{{\bf a}^{(l)}(i)}\}\right); p^{-1}\mathcal{L}\right)$ such that the identity $p_*(P^\prime_n)=\displaystyle\prod_{l=1}^kP^{(l)}_{n_l}$ holds.
\end{prop}
\section*{Acknowledgement}
The author would like to thank Yoshiaki Goto, Katsuhisa Mimachi, Kanami Park, Genki Shibukawa, Nobuki Takayama, and Yumiko Takei for valuable comments. The use of triangulations of a semi-analytic set was suggested by Takuro Mochizuki. The suthor would like to thank him. The author is grateful to Francisco-Jesus Castro-Jim\'enez, Maria-Cruz Fern\'andez-Fern\'andez, Michael Granger, and Susumu Tanab\'e for their interest. Finally, the author would like to thank Toshio Oshima and Hidetaka Sakai for their constant encouragement during the preparation of this paper.
This work is supported by JST CREST Grant Number JP19209317 and JSPS KAKENHI Grant Number 19K14554.
\bibliographystyle{alpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,076 |
Blindsight is the ability of people who are cortically blind to respond to visual stimuli that they do not consciously see due to lesions in the primary visual cortex, also known as the striate cortex or Brodmann Area 17. The term was coined by Lawrence Weiskrantz and his colleagues in a paper published in a 1974 issue of Brain. A previous paper studying the discriminatory capacity of a cortically blind patient was published in Nature in 1973.
Type classification
The majority of studies on blindsight are conducted on patients who are hemianopic, i.e. blind in one half of their visual field. Following the destruction of the left or right striate cortex, patients are asked to detect, localize, and discriminate amongst visual stimuli that are presented to their blind side, often in a forced-response or guessing situation, even though they may not consciously recognize the visual stimulus. Research shows that such blind patients may achieve a higher accuracy than would be expected from chance alone.
Type 1 blindsight is the term given to this ability to guess—at levels significantly above chance—aspects of a visual stimulus (such as location or type of movement) without any conscious awareness of any stimuli. Type 2 blindsight occurs when patients claim to have a feeling that there has been a change within their blind area—e.g. movement—but that it was not a visual percept. The re-classification of blindsight into Type 1 and Type 2 was made after it was shown that the most celebrated blindsight patient, "GY", was in fact usually conscious of stimuli presented to his blind field if the stimuli had certain specific characteristics, namely being of high contrast and moving fast (at speeds in excess of 20 degrees per second).
In the aftermath of the First World War, a neurologist, George Riddoch, had described patients who had been blinded by gunshot wounds to V1, who could not see stationary objects but who were, as he reported, "conscious" of seeing moving objects in their blind field. It is for this reason that the phenomenon has more recently also been called the Riddoch syndrome.
Since then it has become apparent that such subjects can also become aware of visual stimuli belonging to other visual domains, such as color and luminance, when presented to their blind fields. The ability of such hemianopic subjects to become consciously aware of stimuli presented to their blind field is also commonly referred to as "residual" or "degraded" vision.
As originally defined, blindsight challenged the common belief that perceptions must enter consciousness to affect our behavior, by showing that our behavior can be guided by sensory information of which we have no conscious awareness. Since demonstration that blind patients can experience some visual stimuli consciously, and the consequent redefinition of blindsight into Type 1 and Type 2, a more nuanced view of the phenomenon has developed. Blindsight may be thought of as a converse of the form of anosognosia known as Anton syndrome, in which there is full cortical blindness along with the confabulation of visual experience.
History
Much of our current understanding of blindsight can be attributed to early experiments on monkeys. One monkey, named Helen, could be considered the "star monkey in visual research" because she was the original blindsight subject. Helen was a macaque monkey that had been decorticated; specifically, her primary visual cortex (V1) was completely removed, blinding her. Nevertheless, under certain specific situations, Helen exhibited sighted behavior. Her pupils would dilate and she would blink at stimuli that threatened her eyes. Furthermore, under certain experimental conditions, she could detect a variety of visual stimuli, such as the presence and location of objects, as well as shape, pattern, orientation, motion, and color. In many cases she was able to navigate her environment and interact with objects as if she were sighted.
A similar phenomenon was also discovered in humans. Subjects who had suffered damage to their visual cortices due to accidents or strokes reported partial or total blindness. Despite this, when prompted they could "guess" the presence and details of objects with above-average accuracy and, much like animal subjects, could catch objects tossed at them. The subjects never developed any kind of confidence in their abilities. Even when told of their successes, they would not begin to spontaneously make "guesses" about objects, but instead still required prompting. Furthermore, blindsight subjects rarely express the amazement about their abilities that sighted people would expect them to express.
Describing blindsight
Patients with blindsight have damage to the system that produces visual perception (the visual cortex of the brain and some of the nerve fibers that bring information to it from the eyes) rather than to the underlying brain system controlling eye movements. The phenomenon was originally thought to show how, after the more complex perception system is damaged, people can use the underlying control system to guide hand movements towards an object even though they cannot see what they are reaching for. Hence, visual information can control behavior without producing a conscious sensation. This ability of those with blindsight to act as if able to see objects that they are unconscious of suggested that consciousness is not a general property of all parts of the brain, but is produced by specialised parts of it.
Blindsight patients show awareness of single visual features, such as edges and motion, but cannot gain a holistic visual percept. This suggests that perceptual awareness is modular and that—in sighted individuals—there is a "binding process that unifies all information into a whole percept", which is interrupted in patients with such conditions as blindsight and visual agnosia. Therefore, object identification and object recognition are thought to be separate processes and occur in different areas of the brain, working independently from one another. The modular theory of object perception and integration would account for the "hidden perception" experienced in blindsight patients. Research has shown that visual stimuli with the single visual features of sharp borders, sharp onset/offset times, motion and low spatial frequency contribute to, but are not strictly necessary for, an object's salience in blindsight.
Cause
There are three theories for the explanation of blindsight. The first states that after damage to area V1, other branches of the optic nerve deliver visual information to the superior colliculus, pulvinar and several other areas, including parts of the cerebral cortex. In turn, these areas might then control the blindsight responses.
Another explanation for the phenomenon of blindsight is that even though the majority of a person's visual cortex may be damaged, tiny islands of functioning tissue remain. These islands are not large enough to provide conscious perception, but nevertheless enough for some unconscious visual perception.
A third theory is that the information required to determine the distance to and velocity of an object in object space is determined by the lateral geniculate nucleus (LGN) before the information is projected to the visual cortex. In a normal subject, these signals are used to merge the information from the eyes into a three-dimensional representation (which includes the position and velocity of individual objects relative to the organism), extract a vergence signal to benefit the precision (previously auxiliary) optical system, and extract a focus control signal for the lenses of the eyes. The stereoscopic information is attached to the object information passed to the visual cortex.
More recently, with the demonstration of a direct input from the LGN to area V5 (MT), which delivers signals from fast moving stimuli at latencies of about 30 ms, another explanation has emerged. This one proposes that the delivery of these signals is sufficient to arouse a conscious experience of fast visual motion, without implying that it is V5 alone that is responsible, since once signals reach V5, they may be propagated to other areas of the brain. The latter account would seem to exclude the possibility that signals are "pre-processed" by V1 or "post-processed" by it (through return connections from V5 back to V1), as has been suggested. The pulvinar nucleus of the thalamus also sends direct, V1 by-passing, signals to V5 but their precise role in generating a conscious visual experience of motion has not yet been determined.
Evidence of blindsight can be indirectly observed in children as young as two months, although there is difficulty in determining the type in a patient who is not old enough to answer questions.
Evidence in animals
In a 1995 experiment, researchers attempted to show that monkeys with lesions in or even wholly removed striate cortexes also experienced blindsight. To study this, they had the monkeys complete tasks similar to those commonly used for human subjects. The monkeys were placed in front of a monitor and taught to indicate whether a stationary object or nothing was present in their visual field when a tone was played. Then the monkeys performed the same task except the stationary objects were presented outside of their visual field. The monkeys performed very similar to human participants and were unable to perceive the presence of stationary objects outside of their visual field.
Another 1995 study by the same group sought to prove that monkeys could also be conscious of movement in their deficit visual field despite not being consciously aware of the presence of an object there. To do this, researchers used another standard test for humans which was similar to the previous study except moving objects were presented in the deficit visual field. Starting from the center of the deficit visual field, the object would either move up, down, or to the right. The monkeys performed identically to humans on the test, getting them right almost every time. This showed that the monkey's ability to detect movement is separate from their ability to consciously detect an object in their deficit visual field, and gave further evidence for the claim that damage to the striate cortex plays a large role in causing the disorder.
Several years later, another study compared and contrasted the data collected from monkeys and that of a specific human patient with blindsight, GY. GY's striate cortical region was damaged through trauma at the age of eight, though for the most part he retained full functionality, GY was not consciously aware of anything in his right visual field. In the monkeys, the striate cortex of the left hemisphere was surgically removed. By comparing the test results of both GY and the monkeys, the researchers concluded that similar patterns of responses to stimuli in the "blind" visual field can be found in both species.
Research
Lawrence Weiskrantz and colleagues showed in the early 1970s that if forced to guess about whether a stimulus is present in their blind field, some observers do better than chance. This ability to detect stimuli that the observer is not conscious of can extend to discrimination of the type of stimulus (for example, whether an 'X' or 'O' has been presented in the blind field).
Electrophysiological evidence from the late 1970s has shown that there is no direct retinal input from S-cones to the superior colliculus, implying that the perception of color information should be impaired. However, more recent evidence point to a pathway from S-cones to the superior colliculus, opposing previous research and supporting the idea that some chromatic processing mechanisms are intact in blindsight.
Patients shown images on their blind side of people expressing emotions correctly guessed the emotion most of the time. The movement of facial muscles used in smiling and frowning were measured and reacted in ways that matched the kind of emotion in the unseen image. Therefore, the emotions were recognized without involving conscious sight.
A 2011 study found that a young woman with a unilateral lesion of area V1 could scale her grasping movement as she reached out to pick up objects of different sizes placed in her blind field, even though she could not report the sizes of the objects. Similarly, another patient with unilateral lesion of area V1 could avoid obstacles placed in his blind field when he reached toward a target that was visible in his intact visual field. Even though he avoided the obstacles, he never reported seeing them.
A study reported in 2008 asked patient GY to misstate where in his visual field a distinctive stimulus was presented. If the stimulus was in the upper part of his visual field, he was to say it was in the lower part, and vice versa. He was able to misstate, as requested, in his left visual field (with normal conscious vision); but he tended to fail in the task—to state the location correctly—when the stimulus was in his blindsight (right) visual field.<ref
name= " exclusion"
></ref>
> This failure rate worsened when the stimulus was clearer,<ref
name= exclusion
/> indicating that failure was not simply due to unreliability of blindsight.
Case studies
Researchers applied the same type of tests that were used to study blindsight in animals to a patient referred to as "DB". The normal techniques used to assess visual acuity in humans involved asking them to verbally describe some visually recognizable aspect of an object or objects. DB was given forced-choice tasks to complete instead. The results of DB's guesses showed that DB was able to determine shape and detect movement at some unconscious level, despite not being visually aware of this. DB themselves chalked up the accuracy of their guesses to be merely coincidental.
The discovery of the condition known as blindsight raised questions about how different types of visual information, even unconscious information, may be affected and sometimes even unaffected by damage to different areas of the visual cortex. Previous studies had already demonstrated that even without conscious awareness of visual stimuli, humans could still determine certain visual features such as presence in the visual field, shape, orientation and movement. But, in a newer study evidence showed that if damage to the visual cortex occurs in areas above the primary visual cortex, the conscious awareness of visual stimuli itself is not damaged. Blindsight shows that even when the primary visual cortex is damaged or removed a person can still perform actions guided by unconscious visual information. Despite damage occurring in the area necessary for conscious awareness of visual information, other functions of the processing of these visual percepts are still available to the individual. The same also goes for damage to other areas of the visual cortex. If an area of the cortex that is responsible for a certain function is damaged, it will only result in the loss of that particular function or aspect, functions that other parts of the visual cortex are responsible for remain intact.
Alexander and Cowey investigated how contrasting stimuli brightness affects blindsight patients' ability to discern movement. Prior studies have already shown that blindsight patients are able to detect motion even though they claim they do not see any visual percepts in their blind fields. The study subjects were two patients who suffered from hemianopsia—blindness in more than half of their visual field. Both subjects had displayed the ability to accurately determine the presence of visual stimuli in their blind hemifields without acknowledging an actual visual percept previously.
To test the effect of brightness on the subject's ability to determine motion they used a white background with a series of colored dots. The contrast of the brightness of the dots compared to the white background was altered in each trial to determine if the participants performed better or worse when there was a larger discrepancy in brightness or not. The subjects focused on the display for two equal length time intervals and where asked whether they thought the dots were moving during the first or the second time interval.
When the contrast in brightness between the background and the dots was higher, both of the subjects could discern motion more accurately than they would have statistically through guesswork. However, one subject was not able to accurately determine whether or not blue dots were moving regardless of the brightness contrast, but he/she was able to do so with every other color dot. When the contrast was highest subjects were able to tell whether or not the dots were moving with very high rates of accuracy. Even when the dots were white, but still of a different brightness from the background, subjects could still determine whether they were moving. But, regardless of the dots' color, subjects could not tell when they were in motion when the white background and the dots were of similar brightness.
Kentridge, Heywood, and Weiskrantz used the phenomenon of blindsight to investigate the connection between visual attention and visual awareness. They wanted to see if their subject—who exhibited blindsight in other studies—could react more quickly when their attention was cued without the ability to be visually aware of it. The researchers aimed to show that being conscious of a stimulus and paying attention to it was not the same thing.
To test the relationship between attention and awareness, they had the participant try to determine where a target was and whether it was oriented horizontally or vertically on a computer screen. The target line would appear at one of two different locations and would be oriented in one of two directions. Before the target would appear an arrow would become visible on the screen, sometimes pointing to the correct position of the target line and less frequently not. This arrow was the cue for the subject. The participant would press a key to indicate whether the line was horizontal or vertical, and could then also indicate to an observer whether or not he/she actually had a feeling that any object was there or not—even if they couldn't see anything. The participant was able to accurately determine the orientation of the line when the target was cued by an arrow before the appearance of the target, even though these visual stimuli did not equal awareness in the subject who had no vision in that area of his/her visual field. The study showed that even without the ability to be visually aware of a stimulus the participant could still focus his/her attention on this object.
In 2003, a patient known as "TN" lost use of his primary visual cortex, area V1. He had two successive strokes, which knocked out the region in both his left and right hemispheres. After his strokes, ordinary tests of TN's sight turned up nothing. He could not even detect large objects moving right in front of his eyes. Researchers eventually began to notice that TN exhibited signs of blindsight and in 2008 decided to test their theory. They took TN into a hallway and asked him to walk through it without using the cane he always carried after having the strokes. TN was not aware at the time, but the researchers had placed various obstacles in the hallway to test if he could avoid them without conscious use of his sight. To the researchers' delight, he moved around every obstacle with ease, at one point even pressing himself up against the wall to squeeze past a trashcan placed in his way. After navigating through the hallway, TN reported that he was just walking the way he wanted to, not because he knew anything was there.
In another case study, a girl brought her grandfather in to see a neuropsychologist. The girl's grandfather, Mr. J., had suffered a stroke that had left him completely blind apart from a tiny spot in the middle of his visual field. The neuropsychologist, Dr. M., performed an exercise with him. The doctor helped Mr. J. to a chair, had him sit down, and then asked to borrow his cane. The doctor then asked, "Mr. J., please look straight ahead. Keep looking that way, and don't move your eyes or turn your head. I know that you can see a little bit straight ahead of you, and I don't want you to use that piece of vision for what I'm going to ask you to do. Fine. Now, I'd like you to reach out with your right hand [and] point to what I'm holding." Mr. J. then replied, "But I don't see anything—I'm blind!" The doctor then said, "I know, but please try, anyway." Mr. J then shrugged and pointed, and was surprised when his finger encountered the end of the cane which the doctor was pointing toward him. After this, Mr. J. said that "it was just luck". The doctor then turned the cane around so that the handle side was pointing towards Mr. J. He then asked for Mr. J. to grab hold of the cane. Mr. J. reached out with an open hand and grabbed hold of the cane. After this, the doctor said, "Good. Now put your hand down, please." The doctor then rotated the cane 90 degrees, so that the handle was oriented vertically. The doctor then asked Mr. J. to reach for the cane again. Mr. J. did this, turning his wrist so that his hand matched the orientation of the handle. This case study shows that, although (on a conscious level) Mr. J. was completely unaware of any visual abilities that he may have had, he was able to orient his grabbing motions as if he had no visual impairments.
Brain regions involved
Visual processing in the brain goes through a series of stages. Destruction of the primary visual cortex leads to blindness in the part of the visual field that corresponds to the damaged cortical representation. The area of blindness – known as a scotoma – is in the visual field opposite the damaged hemisphere and can vary from a small area up to the entire hemifield. Visual processing occurs in the brain in a hierarchical series of stages (with much crosstalk and feedback between areas). The route from the retina through V1 is not the only visual pathway into the cortex, though it is by far the largest; it is commonly thought that the residual performance of people exhibiting blindsight is due to preserved pathways into the extrastriate cortex that bypass V1. However both physiological evidence in monkeys and behavioral and imaging evidence in humans shows that activity in these extrastriate areas, and especially in V5, is apparently sufficient to support visual awareness in the absence of V1.
To put it in a more complex way, recent physiological findings suggest that visual processing takes place along several independent, parallel pathways. One system processes information about shape, one about color, and one about movement, location and spatial organization. This information moves through an area of the brain called the lateral geniculate nucleus, located in the thalamus, and on to be processed in the primary visual cortex, area V1 (also known as the striate cortex because of its striped appearance). People with damage to V1 report no conscious vision, no visual imagery, and no visual images in their dreams. However, some of these people still experience the blindsight phenomenon, though this too is controversial, with some studies showing a limited amount of consciousness without V1 or projections relating to it.
The superior colliculus and prefrontal cortex also have a major role in awareness of a visual stimulus.
Lateral geniculate nucleus
Mosby's Dictionary of Medicine, Nursing & Health Professions defines the LGN as "one of two elevations of the lateral posterior thalamus receiving visual impulses from the retina via the optic nerves and tracts and relaying the impulses to the calcarine (visual) cortex".
What is seen in the left and right visual field is taken in by each eye and brought back to the optic disc via the nerve fibres of the retina. From the optic disc, visual information travels through the optic nerve and into the optic chiasm. Visual information then enters the optic tract and travels to four different areas of the brain including the superior colliculus, pretectum of the mid brain, the suprachiasmatic nucleus of the hypothalamus, and the lateral geniculate nucleus (LGN). Most axons from the LGN will then travel to the primary visual cortex.
Injury to the primary visual cortex, including lesions and other trauma, leads to the loss of visual experience. However, the residual vision that is left cannot be attributed to V1. According to Schmid et al., "thalamic lateral geniculate nucleus has a causal role in V1-independent processing of visual information". This information was found through experiments using fMRI during activation and inactivation of the LGN and the contribution the LGN has on visual experience in monkeys with a V1 lesion. These researchers concluded that the magnocellular system of the LGN is less affected by the removal of V1, which suggests that it is because of this system in the LGN that blindsight occurs. Furthermore, once the LGN was inactivated, virtually all of the extrastriate areas of the brain no longer showed a response on the fMRI. The information leads to a qualitative assessment that included "scotoma stimulation, with the LGN intact had fMRI activation of ~20% of that under normal conditions". This finding agrees with the information obtained from, and fMRI images of, patients with blindsight. The same study also supported the conclusion that the LGN plays a substantial role in blindsight. Specifically, while injury to V1 does create a loss of vision, the LGN is less affected and may result in the residual vision that remains, causing the "sight" in blindsight.
Functional magnetic resonance imaging has launched has also been employed to conduct brain scans in normal, healthy human volunteers to attempt to demonstrate that visual motion can bypass V1, through a connection from the LGN to the human middle temporal complex. Their findings concluded that there was an indeed a connection of visual motion information that went directly from the LGN to the V5/hMT+ bypassing V1 completely. Evidence also suggests that, following a traumatic injury to V1, there is still a direct pathway from the retina through the LGN to the extrastriate visual areas. The extrastriate visual areas include parts of the occipital lobe that surround V1. In non-human primates, these often include V2, V3, and V4.
In a study conducted in primates, after partial ablation of area V1, areas V2 and V3 were still excited by visual stimulus. Other evidence suggests that "the LGN projections that survive V1 removal are relatively sparse in density, but are nevertheless widespread and probably encompass all extrastriate visual areas," including V2, V4, V5 and the inferotemporal cortex region.
See also
Visual agnosia
Riddoch syndrome
References
Further reading
External links
Blind man navigates maze
Blind man avoids obstacles when reaching
Neurology
Neuroscience
Vision
Consciousness studies | {
"redpajama_set_name": "RedPajamaWikipedia"
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#include "org_jitsi_impl_neomedia_quicktime_QTFormatDescription.h"
#import <Foundation/NSAutoreleasePool.h>
#import <Foundation/NSGeometry.h>
#import <Foundation/NSString.h>
#import <Foundation/NSValue.h>
#import <QTKit/QTFormatDescription.h>
#include <stdint.h>
JNIEXPORT jobject JNICALL
Java_org_jitsi_impl_neomedia_quicktime_QTFormatDescription_sizeForKey
(JNIEnv *jniEnv, jclass clazz, jlong ptr, jstring key)
{
const char *cKey;
jobject size = NULL;
cKey = (const char *) (*jniEnv)->GetStringUTFChars(jniEnv, key, NULL);
if (cKey)
{
QTFormatDescription *formatDescription;
NSAutoreleasePool *autoreleasePool;
NSString *oKey;
NSValue *attribute;
formatDescription = (QTFormatDescription *) (intptr_t) ptr;
autoreleasePool = [[NSAutoreleasePool alloc] init];
oKey = [NSString stringWithUTF8String:cKey];
(*jniEnv)->ReleaseStringUTFChars(jniEnv, key, cKey);
attribute = [formatDescription attributeForKey:oKey];
if (attribute)
{
NSSize oSize;
jclass dimensionClass;
oSize = [attribute sizeValue];
dimensionClass = (*jniEnv)->FindClass(jniEnv, "java/awt/Dimension");
if (dimensionClass)
{
jmethodID dimensionCtorMethodID;
dimensionCtorMethodID
= (*jniEnv)
->GetMethodID(
jniEnv,
dimensionClass,
"<init>",
"(II)V");
if (dimensionCtorMethodID)
size
= (*jniEnv)
->NewObject(
jniEnv,
dimensionClass,
dimensionCtorMethodID,
(jint) oSize.width,
(jint) oSize.height);
}
}
[autoreleasePool release];
}
return size;
}
JNIEXPORT jstring JNICALL
Java_org_jitsi_impl_neomedia_quicktime_QTFormatDescription_VideoEncodedPixelsSizeAttribute
(JNIEnv *jniEnv, jclass clazz)
{
NSAutoreleasePool *autoreleasePool;
jstring jstr;
autoreleasePool = [[NSAutoreleasePool alloc] init];
jstr
= (*jniEnv)
->NewStringUTF(
jniEnv,
[QTFormatDescriptionVideoEncodedPixelsSizeAttribute
UTF8String]);
[autoreleasePool release];
return jstr;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,234 |
/*
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*
*/
package com.flaminem.flamy.parsing.hive
import com.flaminem.flamy.parsing.model.TableDependency
import com.flaminem.flamy.utils.collection.immutable.UniqueSeqMapLike
import scala.collection.immutable.TreeMap
/**
* A collection where we store the tabledependencies indexed with their aliases.
* It has the following properties:
* - The aliases are case-insensitive.
* - Inserting the same alias twice will throw a UnicityConstraintException.
* - Iterating over the keys, values, or key-value pairs will preserve the insertion order.
*
* @param entries
* @param map
*/
class AliasMap private (
override val entries: Seq[(String, TableDependency)] = Seq() ,
override val map: Map[String, TableDependency] = TreeMap[String, TableDependency]()(scala.math.Ordering.comparatorToOrdering(String.CASE_INSENSITIVE_ORDER))
) extends UniqueSeqMapLike[String, TableDependency, AliasMap]
{
/**
* Returns a new instance of the collection, containing the specified map
*/
override def copy(entries: Seq[(String, TableDependency)], map: Map[String, TableDependency]): AliasMap = {
new AliasMap(entries, map)
}
def aliases: Iterable[String] = {
keys
}
def tables: Iterable[TableDependency] = {
values
}
}
object AliasMap {
def apply(pairs: (String, TableDependency)*): AliasMap = {
new AliasMap() ++ pairs
}
def apply(pairs: Iterable[(String, TableDependency)]): AliasMap = {
new AliasMap() ++ pairs
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 96 |
//
// This file was generated by the JavaTM Architecture for XML Binding(JAXB) Reference Implementation, v2.2.4-2
// See <a href="http://java.sun.com/xml/jaxb">http://java.sun.com/xml/jaxb</a>
// Any modifications to this file will be lost upon recompilation of the source schema.
// Generated on: 2015.05.18 at 05:13:02 PM MST
//
package eml.ecoinformatics_org.units_2_1;
import javax.xml.bind.annotation.XmlEnum;
import javax.xml.bind.annotation.XmlEnumValue;
import javax.xml.bind.annotation.XmlType;
/**
* <p>Java class for angleUnitType.
*
* <p>The following schema fragment specifies the expected content contained within this class.
* <p>
* <pre>
* <simpleType name="angleUnitType">
* <restriction base="{http://www.w3.org/2001/XMLSchema}string">
* <enumeration value="radian"/>
* <enumeration value="degree"/>
* <enumeration value="grad"/>
* </restriction>
* </simpleType>
* </pre>
*
*/
@XmlType(name = "angleUnitType", namespace = "eml://ecoinformatics.org/units-2.1.1")
@XmlEnum
public enum AngleUnitType {
@XmlEnumValue("radian")
RADIAN("radian"),
@XmlEnumValue("degree")
DEGREE("degree"),
@XmlEnumValue("grad")
GRAD("grad");
private final String value;
AngleUnitType(String v) {
value = v;
}
public String value() {
return value;
}
public static AngleUnitType fromValue(String v) {
for (AngleUnitType c: AngleUnitType.values()) {
if (c.value.equals(v)) {
return c;
}
}
throw new IllegalArgumentException(v);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,568 |
#
L i t t é r a t u r e d ' A m é r i q u e
Collection dirigée par
Isabelle Longpré
#
De la même auteure
Lazy Bird, Montréal, Québec Amérique, 2009.
Mirror Lake, Montréal, Québec Amérique, 2006.
Prix Ringuet de l'Académie des lettres du Québec.
Le Pendu de Trempes, Montréal, Québec Amérique, 2004.
Projections (en collaboration avec la photographe Angela Grauerholz), coll. « L'image amie », Québec, J'ai vu, 2003.
Le Ravissement, Québec, L'instant même, 2001.
Prix littéraire du Gouverneur général 2001, catégorie « romans et nouvelles ».
Prix littéraire des collégiennes et des collégiens 2002 (Collège de Sherbrooke).
Les derniers jours de Noah Eisenbaum, Québec, L'instant même, 1998.
Alias Charlie, Montréal, Leméac, 1994.
Portraits d'après modèles, Montréal, Leméac, 1991.
La Femme de Sath, Montréal, Québec Amérique, 1987.
Rivière Tremblante
#
Catalogage avant publication de Bibliothèque et Archives nationales du Québec et Bibliothèque et Archives Canada
Michaud, Andrée A.
Rivière Tremblante
(Littérature d'Amérique)
ISBN 978-2-7644-0952-7 (Version imprimée)
ISBN 978-2-7644-1280-0 (PDF)
ISBN 978-2-7644-1281-7 (ePub)
I. Titre. II. Collection : Collection Littérature d'Amérique.
PS8576.I217R58 2011 C843'.54 C2010-942337-2
PS9576.I217R58 2011
Nous reconnaissons l'aide financière du gouvernement du Canada par l'entremise du Fonds du livre du Canada pour nos activités d'édition.
Gouvernement du Québec — Programme de crédit d'impôt pour l'édition de livres — Gestion SODEC.
Les Éditions Québec Amérique bénéficient du programme de subvention globale du Conseil des Arts du Canada. Elles tiennent également à remercier la SODEC pour son appui financier.
L'auteure remercie le Conseil des Arts du Canada pour son soutien financier.
Québec Amérique
329, rue de la Commune Ouest, 3e étage
Montréal (Québec) Canada H2Y 2E1
Téléphone : 514 499-3000, télécopieur : 514 499-3010
Dépôt légal : 1er trimestre 2011
Bibliothèque nationale du Québec
Bibliothèque nationale du Canada
Projet dirigé par Isabelle Longpré
Mise en pages : Karine Raymond
Révision linguistique : Claude Frappier et Diane-Monique Daviau
Conception graphique originale : Isabelle Lépine
Adaptation de la grille graphique : Célia Provencher-Galarneau
Illustration en couverture : Lyse Marion
Conversion au format ePub : Studio C1C4
Pour toute question technique au sujet de ce ePub :
service@studioc1c4.com
Tous droits de traduction, de reproduction et d'adaptation réservés
© 2011 Éditions Québec Amérique inc. et Andrée A. Michaud
www.quebec-amerique.com
Andrée A. Michaud
Rivière Tremblante
roman
Québec Amérique
#
Les noms des personnages apparaissant dans ce roman, bien qu'ils soient courants au Québec, désignent des personnages purement fictifs, inventés aux seules fins de l'histoire, à moins qu'il ne s'agisse de figures publiques ou ayant défrayé l'actualité. Quant aux lieux où se déroule principalement le récit, ils ont été construits à partir de divers lieux assemblés par l'auteure.
Quiconque reconnaîtrait Rivière-aux-Trembles ou l'un des personnages de ce roman devrait donc mettre cette parenté sur le compte des hasards de l'invention ou la lier au fait qu'un homme ne peut ressembler qu'à un autre homme et que les paysages, les villages et les forêts d'un même pays ont en commun des couleurs qui nous amènent à les confondre.
#
À tous les enfants
qui ne sont pas rentrés pour le souper
# PREMIÈRE PARTIE
La nuit tombait sur Rivière-aux-Trembles. Dans le cimetière planté d'érables, mon père dormait dans le brouillard soulevé par le redoux des derniers jours, au terme duquel février couvrirait de nouveau le sol d'une couche de glace où se figeraient les cailloux et les bouts de branches sectionnés par le gel. Derrière le cimetière, sur la colline des Loups, stagnait un nuage dont la densité laissait croire qu'il pleuvait sur la colline, seulement là, au milieu des sapins noirs. Les derniers oiseaux du jour finissant lançaient des notes solitaires dans l'air saturé de silence, et moi, je demeurais immobile, à me demander que faire de cette sombre beauté coincée entre la mort et la proche obscurité.
Toute la journée, j'avais marché sur les petites routes boueuses qui menaient au village ou nous en éloignaient, selon qu'on avait envie de rentrer à la maison ou de fuir la tristesse des endroits délaissés. Après des heures d'épuisement, les mains et les pieds gelés, je m'interrogeais toujours, incapable de déterminer quelle direction je désirais prendre. Je n'étais revenue à Rivière-aux-Trembles que pour rendre un dernier hommage à mon père et déposer sur son cercueil, dans la chapelle où il serait enfermé jusqu'au printemps, quelques Mary-Jean, ses roses préférées, dont le rapide étiolement me permettrait d'envisager la mort pour ce qu'elle était, un nouvel état de la matière. Car la mort n'était que cela, une transformation de la chair et du sang, voilà ce que je me répétais pour ne pas penser à la fin de toute chose et de tout homme. Le corps de mon père s'était vidé de sa pensée pour atteindre une nouvelle forme de communion avec le monde, un état où la perception de la lumière, de la chaleur et du froid ne serait entravée ni par la douleur ni par la conscience.
Mais que savais-je de la douleur de la matière et des états d'âme de la pourriture ? J'imaginais de possibles résurrections, des réincarnations excluant la souffrance parce que je refusais d'envisager la mort comme un état définitif et immuable. La disparition de mon père m'obligeait à penser que j'allais disparaître aussi et à me demander ce que je faisais là, arrêtée au milieu d'une vie dont la conclusion serait sans appel. Je m'étais promis de repartir sitôt la cérémonie funèbre terminée et de ne jamais remettre les pieds à Rivière-aux-Trembles, mais le sentiment de n'en avoir plus pour si longtemps bouleversait tous mes plans. Contre toute attente, je ne me sentais pas la force de m'arracher au décor de ce village où s'était brutalement achevée mon enfance.
Accroupie près de la pierre tombale d'une inconnue faisant face au couchant pour l'éternité, j'ai demandé à mon père de me venir en aide, j'ai prié la mère que je n'avais pas connue, morte trop jeune, broyée en plein soleil, j'ai imploré le ciel de m'envoyer un signe, et c'est à ce moment que le cri a retenti, un cri d'effroi pouvant aussi bien venir du creux de la terre que des profondeurs de ma mémoire. Michael, ai-je murmuré d'une voix enrouée par la pluie et le froid, c'est toi Michael ? Mais le silence était retombé sur la campagne, plus compact encore que celui ayant précédé le cri. Alors j'ai couru, j'ai pris un des quatre chemins divisant mon village à la manière d'une croix et me suis enfoncée dans la brunante. Après quelques minutes, je me suis arrêtée, à bout de souffle. J'ai posé mes mains sur mes genoux et, en levant la tête, j'ai aperçu un chevreuil qui broutait près d'un maigre bras de forêt progressant dans les champs, perdu dans le brouillard descendant de la colline des Loups, pareil à ces visions salvatrices éloignant du gouffre les hommes sur le point d'y basculer.
J'ai avancé de quelques pas dans le champ gorgé de neige fondante et, pendant que le chevreuil disparaissait dans la forêt, j'ai laissé couler sur le gouffre refermé les larmes que je retenais depuis le matin. Il était temps, pour moi, d'enfin rentrer à la maison.
## I
LES HISTOIRES
### MARNIE
Il avait douze ans, j'en avais onze, et, comme dans les récits chevaleresques, nous avions juré de ne jamais nous quitter, ignorant que l'éternité du serment peut être brève. Michael Saint-Pierre a néanmoins tenu ses promesses, son souvenir ne m'a jamais quittée, mais les abris que nous construisions au cœur de la forêt et les voyages intergalactiques que nous projetions n'existent désormais que dans ces rêves de plus en plus rares où Michael, grimpé sur la plus haute branche d'un arbre immense, me parle avec ses mots d'enfant de l'infinité de l'univers.
Michael Superman Saint-Pierre, fils de Jeanne Dubé et de Victor Saint-Pierre, a disparu dans les bois de Rivière-aux-Trembles le 7 août 1979. J'ignore ce qui est arrivé dans ces bois, tout le monde l'ignore, sauf Mike et son possible agresseur, mais il m'arrive de croire qu'il est toujours vivant, qu'une force dont je ne peux concevoir la puissance est venue le chercher sur sa branche pour l'emmener sur une planète jumelle de Krypton ou sur un de ces astres lointains que l'homme n'a pas encore découverts. Dans ces moments, je me dis qu'un jour, un message m'arrivera de l'espace, lumineux parmi la myriade d'étoiles que j'observe parfois jusqu'à l'étourdissement, pour m'annoncer que Michael Saint-Pierre prépare son retour sur terre.
La vérité, c'est que Michael est probablement mort au fond des bois, son corps disloqué charrié par la rivière jusque dans la gueule des coyotes et des loups, à moins qu'un prédateur à visage trop humain se soit jeté sur lui pour des motifs que seule la folie peut expliquer. En l'absence de corps, je continue toutefois à espérer que mon ami Mike, prisonnier d'une amnésie engendrée par la foudre, marche toujours quelque part sur le sol moussu d'une forêt lointaine. Si c'est le cas, peut-être ses pas, poussés par quelque mémoire floue de l'enfance, finiront-ils par le mener ici, à Rivière-aux-Trembles, dans ce lieu où, à cause d'un cri dont je ne connais pas l'origine, j'ai décidé de revenir m'installer après vingt-neuf ans d'absence.
Nous avions entendu ce cri à deux reprises au cours du même été, Michael et moi, l'été 79, celui de sa disparition. La première fois, c'était un peu avant la tombée de la nuit, quand l'absence de vent permet aux cris de voyager dans l'air humide. On s'amusait à lancer des cailloux dans le lac aux Barbotes, celui qui rate le cap est une patate, lorsqu'un hurlement à faire frémir l'écorce des bouleaux avait déchiré l'obscurité. Un dernier caillou avait plongé dans l'eau sans vagues et Michael s'était figé sur place, le bras levé, pendant que je laissais retomber la pierre plate que j'avais dénichée sous un tas de bois de plage. Nos regards s'étaient croisés et, sans avoir à se concerter, on s'était précipités dans le sentier menant au lac pour filer à toutes jambes vers le village.
Avant d'arriver au chemin des sœurs Morin, près duquel on avait caché nos bicyclettes chromées, bleue pour Michael, avec des poignées rouges, aux couleurs de Superman, noire pour moi, parce que c'était la dernière en stock le jour où mon père me l'avait achetée, j'avais trébuché sur une racine qui traversait le sentier et effectué une longue chute au ralenti. Mes tresses de squaw, tenues par des élastiques jaune citron, avaient volé devant mes yeux, précédées par ma casquette des Expos, pendant que mes mains tentaient de s'accrocher au vide, puis le pan de ciel où tournoyait ma casquette avait disparu. Le nez dans la terre boueuse, les feuilles pourries et les aiguilles d'épinette, j'avais eu l'impression qu'une main agrippait ma jambe gauche pour me tirer vers l'arrière, vers le lac et ses profondeurs. Il s'agissait d'un premier avertissement, ai-je cru par la suite, du premier signe d'un destin qui nous happerait, Michael et moi, et mettrait un terme à tous nos jeux. Des ongles que j'imaginais craquelés et noirs, pareils aux serres des sorcières de la swamp aux Fantômes, lieu mythique inventé pour éloigner les enfants des tourbières s'étendant à l'est de la colline des Loups, s'enfonçaient déjà dans ma cheville quand les mains de Michael m'avaient saisie sous les épaules pour m'aider à me relever. Grouille-toué, Marnie, vite, ça va nous rattraper, ça s'en vient, je le sens. Mais c'était arrivé, c'était déjà là, cette chose qui rendrait fou le soleil de Rivière-aux-Trembles.
Le lendemain, on a appris que Martin Bouchard, le fils aîné du maire Jos Bouchard, s'était noyé aux environs de huit heures, la veille, en retirant les filets à pêche qu'il avait jetés à l'embouchure de la rivière aux Bleuets, à cinq kilomètres du lac aux Barbotes par la route, à moins de deux kilomètres à vol d'oiseau et à portée de voix dans l'air humide. On a entendu le cri de la mort, Marn, a murmuré Michael en crachant sa gomme sur le trottoir, et on est restés là, figés, les yeux tournés vers la forêt où la mort avait hurlé.
La deuxième fois que ce cri a retenti dans la région serait aussi la dernière, du moins cet été-là. C'était le 7 août, un 7 août torride où le soleil alternait avec des nuages de plus en plus denses annonçant l'orage. Appuyés sur la rampe de la galerie, on essayait de tuer le temps en comptant les mouches agglutinées sur la façade de la maison de Michael, à qui sa mère avait confié la tâche de surveiller sa petite sœur, Émilie, Emmy-Lou, Emmy-Lili, pendant qu'elle se rendait chez madame Tremblay pour lui apprendre une recette de pouding aux fraises dans laquelle les fraises pouvaient être remplacées par des framboises, des bleuets, des mûres, des prunes, des pommes et des gadelles, bref, par tous les petits, moyens ou gros fruits qui poussaient dans le coin.
L'après-midi s'étirait dans une chaleur humide qui semblait affecter jusqu'aux poupées de chiffon d'Emmy-Lili, étendues jambes écartées sur le bois de la galerie. Nos vêtements nous collaient à la peau, les fins cheveux blonds d'Emmy dessinaient sur son front des virgules mouillées et les criquets rassemblés en masse dans le foin jaune avaient fait taire tous les oiseaux. On en voulait à Denise Tremblay, qui s'était mariée en juin dans une robe de magazine et ne savait même pas faire cuire un œuf, et on en voulait à la mère de Michael, qui s'était mis en tête de transformer cette nouvelle mariée en cordon-bleu. En voyant sa vieille Datsun déboucher dans l'allée de la maison, on a couru chercher nos maillots et Michael lui a crié qu'on partait se baigner, sans écouter ses recommandations ni prendre le temps de la saluer.
Derrière nous, Emmy-Lili s'est mise à pleurer. Je veux aller avec toi, Mike. Emmène Mimi, emmène Lili... Ces pleurs n'étaient que d'innocents pleurs de gamine, mais ils allaient s'incruster dans la mémoire que je garderais de ce jour aussi farouchement qu'une nuée de moustiques attirés par le sang frais. Ce sont ces pleurs qui me tireraient de mon sommeil, bien des années plus tard, les pleurs de la petite Emmy-Lou implorant son frère de ne pas l'abandonner, de ne pas la laisser seule sous l'orage, de l'emmener dans ces bois dont certains ne reviennent pas. Les pleurs de la petite Emmy qui avait peut-être deviné, à la couleur du ciel, qu'elle ne reverrait jamais Michael, son frère, son dieu.
En lui tapotant la tête, Michael lui a donné un suçon au caramel qui traînait au fond d'une de ses poches, Mike sera pas parti longtemps, Lili, et on a enfourché nos bicyclettes volantes pour emprunter le chemin du 4, puis un sentier qui nous mènerait là où la rivière formait une cuvette dans laquelle il était possible de plonger sans se blesser, de se dérober aux yeux des monstres de la forêt et de faire la planche en regardant les nuages se déformer à travers les feuilles des arbres. C'était notre lieu secret, notre oasis, loin du monde des grands, de sa rigueur, de sa grisaille. Michael l'avait baptisé le bassin magique et prétendait que c'est du fond de ce bassin que surgirait la forteresse où il abriterait sa solitude lorsqu'il aurait atteint ses dix-huit ans et y lancerait l'aiguille de verre émeraude de Superman.
En attendant qu'émerge de la rivière l'étincelante preuve des origines lointaines de Superman, on s'était construit une cabane en branches sous les sapins, face au bassin. C'est là qu'on se trouvait quand l'orage a éclaté, à classer selon leur taille et leur couleur les pierres qu'on avait pêchées au fond de l'eau. Les premières gouttes ont frappé le toit de la cabane alors que Michael disposait les blanches près du cercle de cailloux noirs dont les pouvoirs nous rendaient invulnérables. Ça va tonner, Marn, a-t-il lancé après avoir jeté un œil à travers l'ouverture servant de porte à la cabane, puis il a couru chercher nos maillots, qui séchaient côte à côte sur un rocher plat.
Il n'aurait dû s'absenter que quelques secondes et, ne le voyant pas revenir, je me suis glissée jusqu'à la porte. Il était debout près du rocher, dos à la cabane, et il se balançait de droite à gauche en soulevant une jambe, puis l'autre, à la manière du robot de métal rouillé dont il avait fini par se débarrasser au début de l'été. Nos maillots pendaient au bout de ses bras écartés et sa tête retombait sur sa poitrine, comme s'il n'avait plus la force de la tenir droite. Le cou rompu, il fixait son running gauche, dont le lacet s'était détaché.
J'ai d'abord cru qu'il s'agissait d'une blague, que Michael voulait me faire marcher, mais quelque chose ne tournait pas rond. Il ressemblait aux zombies de La nuit des morts-vivants, qu'on avait regardé en cachette quelques semaines plus tôt. Il ressemblait aux poupées d'Emmy-Lou, avec leurs membres mous et leur cou cassé. Qu'est-ce que tu fais, Mike ? Je te trouve pas drôle ! Mais il n'a pas réagi. Il a continué à osciller de gauche à droite, puis il s'est mis à reculer à pas prudents, les bras levés, jusqu'à ce qu'un éclair illumine le ciel au-dessus du bassin. Il a alors bondi vers l'arrière, battant cette fois des bras à la manière du robot de Perdus dans l'espace lorsqu'il crie « Danger ! Danger ! ». Mike ne criait cependant pas mais, comme le robot, il avait peur de je ne sais quelle ombre s'avançant sur lui, de je ne sais quelle créature que me masquaient les arbres, son corps, la pluie. J'ai cherché cette ombre, ce mort-vivant qui voulait dévorer Michael, mais je ne distinguais que le mouvement des branches agitées par le vent, derrière lesquelles auraient pu se mouvoir autant de bras puissants et menaçants. J'étais sur le point de percevoir, me semblait-il, le bras velu qui surgirait des arbres, il était là, lacéré de griffures, quand j'ai entendu, à travers le grondement du tonnerre, le cri perçant du noyé de la rivière aux Bleuets, le hurlement qui nous avait annoncé la mort de Martin Bouchard. Un deuxième éclair a immédiatement fendu les nuages et Michael s'est immobilisé, laissant nos maillots tomber à ses pieds dans un geste d'une telle lenteur que j'ai cru que les bouts de tissu mouillés allaient demeurer suspendus à ses mains ouvertes et ne jamais toucher le sol.
La gorge aussi sèche que si j'avais couru du village jusqu'à la mare à Mailloux sans m'arrêter, je lui ai répété que ses simagrées ne m'amusaient pas, arrête de niaiser, Mike, t'es pas drôle pantoute. Quand le vent s'est mis à tournoyer sur le bassin, j'ai essayé de lui dire de rentrer à la cabane, vite, Mike, ça arrive, c'est là, les loups-garous et les sorcières, mais j'étais sans voix, paralysée par la fixité de Mike, par son corps pétrifié sous la pluie. Tu me fais peur, Mike, suis-je parvenue à murmurer ou à pleurnicher, je ne sais plus, mais ma peur n'avait pas le pouvoir de secouer sa torpeur. Après quelques secondes interminables, il s'est enfin tourné vers moi, plus blanc que les draps frais étendus dans les arrière-cours de Rivière-aux-Trembles, tous les matins d'été, par des femmes aux cheveux fous. J'ai alors cru qu'il allait s'évanouir, mais il a prononcé quelques paroles à mon intention ou à l'intention des spectres invisibles postés devant son regard vide. Dans le vacarme du vent faisant ployer les arbres, je n'ai pas entendu ces paroles, j'ai seulement vu sa bouche qui s'ouvrait mécaniquement, ses lèvres qui s'arrondissaient autour de sons étranges, mauvais temps, madame, mauvais temps, ne plie pas le jour. Puis, comme s'il était soudain sorti de sa transe, il m'a souri d'un sourire empreint de la plus infinie désolation, a discrètement levé la main en signe d'au revoir et s'est rué entre les arbres, poussé par les bras se multipliant dans la tempête. J'aurais dû le rattraper, suivre le couloir d'obscurité où il s'enfonçait, mais mes jambes étaient aussi molles que celles des poupées d'Emmy-Lili, qui devaient se réjouir, là-bas où la peur n'était pas encore arrivée, de la bienheureuse fraîcheur de la pluie.
Je suis sortie de la cabane à quatre pattes, ne songeant qu'aux paroles absurdes prononcées par Mike, mauvais temps, madame, mauvais temps, puis à son running détaché, tout ça mélangé, tout ça dans le désordre, à ce running usé qui claquait sur son talon nu, resterait coincé entre deux racines et le ferait trébucher. Quand j'ai réussi à me mettre debout, l'esprit aussi confus qu'au sortir d'un cauchemar, la tache claire du t-shirt rouge de Mike s'était déjà perdue dans l'enchevêtrement de la forêt. Un autre éclair a illuminé le ciel noirci de nuages catapultés par le vent et je suis demeurée seule, à crier inutilement le nom de Michael Saint-Pierre.
Je l'ai attendu dans la cabane jusqu'à la fin de l'orage, emmurée par l'écho du hurlement qui avait brisé le ciel d'un éclair mauve, puis je suis rentrée à la maison les genoux tremblants sur ma bicyclette chromée, luttant à chaque coup de pédale contre la force inconnue qui voulait me tirer vers l'arrière.
Une heure plus tard, une trentaine d'hommes, guidés par une enfant secouée de spasmes, prenaient le chemin du bassin magique, à partir duquel ils se sont séparés pour sillonner la forêt en suivant les branches d'une étoile dont le centre se situait là où Michael avait échappé nos maillots trempés. Dans la cabane, les cercles de cailloux s'étaient brisés.
### BILL
J'ai échoué à Rivière-aux-Trembles comme une pierre ballottée par le courant. J'aurais pu finir à Chicago, Sept-Îles, Tombouctou ou Maniwaki, mais c'est ici que ma dégringolade m'a mené. C'est exactement ce dont j'avais besoin, d'un trou perdu où Lucy-Ann ne viendrait pas me harceler de ses accusations et où je pourrais macérer dans mon malheur en racontant à Billie toutes les histoires que cette chienne de vie ne m'a pas permis de terminer.
J'ai tellement raconté d'histoires à cette enfant que j'avais parfois l'impression que la réalité n'existait plus, qu'elle n'était qu'un pâle succédané du film se déroulant dans l'esprit un peu tordu des gars de mon espèce. On s'assoyait bien au chaud dans la lumière de la véranda, je choisissais un livre au hasard, Moby Dick, l'Odyssée, Cendrillon ou Le Petit Poucet, et je réinventais pour elle, ma fille, le périple du capitaine Achab, les virées nocturnes de Cendrillon ou les aventures d'Ulysse au pays des cyclopes. Pour elle, je mélangeais l'Histoire et ses milliers d'histoires et n'hésitais pas à troubler la mémoire des livres en faisant monter Pinocchio à la proue du Pequod et en mariant le Petit Poucet à la princesse au petit pois, parce qu'elle aimait ça, tout simplement, parce que ça la faisait rire et rêver et que les rêves des petites filles sont parmi les plus belles choses du monde, personne ne m'enlèvera ça de l'esprit.
Ses histoires préférées demeuraient toutefois les miennes, Ronie le crapaud volant, La dynastie de Noutes, Alice et les sept chats, des contes de quelques pages qui avaient fait ma fortune et m'avaient amené jusqu'en Islande. Ronie avait été traduit en dix-sept langues et adapté en dessin animé dans les années quatre-vingt-dix, si bien que, quand j'avais connu L.A., Lucy-Ann, j'avais pu nous payer une maison cossue dans un quartier fréquenté par des gens propres et bien éduqués, qui ne crachaient pas sur les trottoirs et ne se soûlaient qu'au vin haut de gamme.
À quarante ans, j'aurais pu vivre de mes rentes, mais je m'acharnais à multiplier mes histoires de Noutes et de crapauds polyglottes, tout en essayant d'inculquer à quelques étudiants pour la plupart blasés des théories qu'ils tenteraient d'appliquer à la fiction avant de se rendre compte que ça n'intéressait personne.
Ma vie tournait autour de ces trois pôles, mes étudiants revenus de tout ce qui leur échappait, mes carnets de notes et Billie-Billienoute, l'unique princesse de mon royaume, qui sautait sur mes genoux à la première occasion pour que je lui invente une nouvelle histoire de crapaud, d'alligator ou de harfang des neiges, jusqu'à ce qu'elle découvre Harry Potter, le jeune Indiana Jones, Fifi Brindacier et toute une ribambelle d'héroïnes et de héros n'ayant ni l'allure ni la gueule d'un crapaud droit sorti d'un marais enchanté. À compter de ce jour, mes batraciens, Homère et les frères Grimm ont eu de la concurrence.
J'aurais pu être jaloux et en vouloir à Billie de préférer à l'univers féerique où évoluait mon bestiaire parlant les aventures d'un apprenti sorcier ou d'une fillette capable de soulever un cheval en chantant en suédois, mais comment en vouloir à une princesse qui vous transforme chaque jour en prince charmant. Bisous, papanoute, criait-elle en sautant sur mon lit, et mes cauchemars refluaient sous les draps en même temps que mes angoisses. La soudaine attirance de Billie pour ces histoires où l'enfance affrontait la bêtise humaine m'indiquait qu'elle grandissait, ce que je préférais ignorer tant qu'elle ne brûlerait pas ses robes de princesse avec des mégots de cigarette, mais elle me prouvait aussi qu'il était temps pour moi de me recycler. Or, la perspective d'abandonner mes crapauds pour des cow-boys frayant avec Sigmund Freud et Gertrud Stein ou pour des créatures métalliques communiquant en se tapant dans le front m'enchantait presque autant que de devenir fossoyeur en Azerbaïdjan. Mon désir de susciter l'admiration de Billie étant toutefois plus fort que mes inclinations littéraires, j'avais conçu le projet de lui concocter pour son prochain anniversaire une aventure de mon cru mettant en scène Indie le jeune et Billie Brindacier, car j'avais un certain penchant pour Indy, je l'avoue, de même que pour cette Fifi qui avait traversé les époques sans trop d'égratignures, un peu comme Jeanne d'Arc, mais à la manière punk. L'anniversaire en question n'est cependant jamais venu. Je l'ai attendu, Dieu sait que je l'ai attendu, persuadé que Billie ne pouvait pas rater son gâteau aux cerises ni ses neuf chandelles rouges, qu'il était impossible qu'elle ne descende pas l'escalier en courant, son chapeau pointu posé de travers sur ses cheveux en bataille.
J'ai attendu pour rien et n'ai jamais écrit mon Billie and the Lost Ark, ce qui vaut sans doute mieux, car je me serais probablement planté en essayant de me mesurer à George Lucas et Astrid Lindgren d'un seul souffle. La dernière histoire qu'elle avait voulu que je lui raconte appartenait à la série des Potter. Je m'en souviens parce que j'avais laissé traîner le livre, qui est tombé du guéridon lorsque Lucy-Ann est entrée en trombe en criant que Billie n'avait pas assisté à son cours de ballet, qu'elle avait quitté l'école à quinze heures trente, selon son habitude, mais ne s'était pas présentée chez mademoiselle Lenoir, sa prof de danse. Elle hurlait et gesticulait de façon hystérique et j'ai dû lui saisir les deux bras avant qu'elle s'étrangle avec le foulard qu'elle tentait d'arracher de son cou.
Quand j'ai compris ce qu'elle essayait de me dire, je me suis précipité dans la chambre de Billie, au cas où elle s'y serait cachée pour s'éviter la corvée que représentaient pour elle les cours de la Lenoir. Après avoir fouillé tout le deuxième, j'ai dévalé l'escalier quatre à quatre et j'ai aperçu le foulard de Lucy-Ann, qui gisait dans la neige fondue sur le plancher, à côté du roman piétiné. J'ai repoussé le foulard et j'ai ramassé le livre en essuyant frénétiquement la couverture, sur laquelle un des talons de Lucy-Ann avait écrasé le visage de Potter et ses lunettes du même coup. Pendant un instant, j'ai eu l'impression que le jeune Harry, derrière ses lunettes cassées, voulait me jeter un mauvais sort et faire de moi une des victimes de la malédiction pétrificatoire qui frappait dans ce roman. Je n'avais pas tout à fait tort, car je demeurais là, paralysé, pendant que Lucy-Ann gueulait derrière moi pour que je me bouge le cul et lui retrouve sa fille.
C'est Lucy-Ann qui avait insisté pour inscrire Billie à des cours de ballet, malgré les protestations de la petite. Elle aurait dû savoir qu'on n'oblige pas une enfant qui fait de l'embonpoint à affronter un peloton de fillettes filiformes prêtes à ouvrir le feu sur quiconque n'entre pas dans leur moule. C'est ainsi que les armées fonctionnent, que les guerres font leurs premières victimes. Mais L.A. n'avait rien voulu entendre. L.A. avait toujours été mince, L.A. était aussi bouchée que toutes les femmes minces question image de soi et n'avait aucune idée du calvaire des rondes. Résultat, après deux mois de danse, Billie s'était plus ou moins mise aux carottes et au jus de légumes. Alors le jour où le petit Dumas s'était fait renverser par une voiture, j'avais mis le poing sur la table et j'avais dit c'est fini, c'est assez, on ne va pas obliger une enfant qui risque à tout moment de rouler sous un dix tonnes à se ridiculiser dans un tutu qui pique pour n'avoir droit qu'à un rôle de souris minable dans une production miteuse de Casse-Noisette !
L.A. m'avait claqué la porte au nez et Billie avait poursuivi ses cours et son régime minceur jusqu'au jour où elle avait craqué devant un éclair au chocolat et où je l'avais vivement encouragée en m'en enfilant trois de suite dans le gosier. Elle était trop fragile, Billie, trop douce, trop rieuse, pour se retrouver engoncée à huit ans à peine dans le corset de papier glacé fabriqué par des magazines qui, depuis trois ou quatre décennies, modelaient l'image de la femme supposément idéale sur des filles à peine pubères ayant atteint le dernier stade de l'anorexie.
Le combat éclairs versus carottes, au terme duquel la puissance calorifique du chocolat mettrait knock-out les vertus de la vitamine C, a eu lieu quelques jours avant la disparition de la puce. L.A. recevait des copines à bruncher et on en a profité, Billie et moi, pour se tirer en douce et aller se balader en ville. On a hésité quelques instants entre le Jardin botanique et le cinéma, puis Billie a décrété qu'il faisait trop froid pour chercher des pistes d'extraterrestres dans les allées enneigées du Jardin, ainsi qu'on l'avait prévu. Ça remontait à loin, cette histoire de pistes d'extraterrestres. Billie devait avoir quatre ans et, grimpée sur un des tabourets de la cuisine, elle observait les marques creusées dans la neige fraîche par la neige tombée des arbres entourant la maison. Ces empreintes qu'elle n'avait jamais remarquées la fascinaient et je lui avais expliqué qu'il s'agissait de pistes de Martiens, version moderne des pistes de lutin de mon enfance.
Il y a des parents qui tentent de tout expliquer de façon rationnelle à leurs enfants dès que ceux-ci percent leur première dent, qui leur détaillent la loi de la relativité à quatre ans, les empêchent d'écouter Télétoon et refusent de les laisser s'évader dans ces mondes magiques remplis de créatures insaisissables, monstres ou fées, ogres ou farfadets se nichant dans les rêves pour vous apprendre que le réel n'est pas toujours ce qu'il paraît être et qu'il existe des univers où les arbres sont rouges, où des fleurs de la taille d'un cachalot sillonnent les océans. L.A., malheureusement, faisait partie de la faction parentale ne jurant que par les jouets éducatifs, lapins carrés comptant jusqu'à cent, réveille-matin s'activant sur des airs d'opéra, blocs de cire à oreilles récitant les fables de La Fontaine. Voyant qu'elle s'apprêtait à expliquer à Billie d'où venaient ces marques dans la neige, je lui ai lancé un de mes regards les plus assassins, préférant blesser l'orgueil de L.A. plutôt que d'empêcher l'imagination de Billie de s'emballer pour se créer un monde qui n'appartiendrait qu'à elle et dont elle se souviendrait, des années plus tard, comme d'une des plus belles régions de la terre.
La recherche de pistes d'extraterrestres étant exclue, Billie s'est mise à sautiller autour de moi pour qu'on retourne voir L'ère de glace, même si on avait vu ce film la semaine d'avant. Je n'ai jamais compris pourquoi les enfants ressentent le besoin de toujours se faire raconter les mêmes histoires. Si j'interrogeais Billie à ce sujet, elle demeurait bouche bée, puis me donnait la réponse la plus logique qui soit : ben... parce que je les aime, ces histoires-là, papanoute. J'ouvrais alors l'album ou le livre qu'elle m'avait désigné en me disant que ça devait la sécuriser de savoir dès le départ que le grand méchant loup allait être neutralisé, que le génie d'Indy démolirait son adversaire, quoi qu'il advienne, et qu'aucun véritable danger ne pouvait se glisser sous son lit tant que le bien triomphait du mal. J'imaginais que ce rituel fondé sur la répétition lui donnait le sentiment d'une forme de stabilité dont elle pressentait déjà qu'elle n'existait pas.
C'est ce désir de vivre aussi longtemps que possible dans une illusion les protégeant de l'immensité du monde et de l'incohérence du temps qui pousse les enfants à réclamer Les trois petits cochons quatre soirs de suite. Enfin, je suppose. On oublie tellement de choses essentielles quand on vieillit qu'on se demande parfois ce que ça donne d'avoir été petit si on est trop con pour se souvenir de quoi peut être constituée la joie, la vraie joie, celle qui éclate devant un cornet de crème glacée à trois boules surmonté d'une cerise siliconée. Avec Billie, je m'en tirais en mélangeant les histoires qui finissaient bien ou en modifiant la finale de celles qui n'avaient aucun sens aux yeux d'une enfant qui s'endormirait en serrant contre sa poitrine un ourson de peluche. De cette façon, je lui racontais toujours la même et jamais la même, et j'évitais d'emmerder l'adulte un peu con que j'étais devenu.
Comment ça se fait, papanoute, que Blanche-Neige a dormi dans la maison de Goofy ? Elle me sortait son arsenal de questions et je lui répondais que Blanche-Neige était tannée de son château humide, perdu au fond de la forêt de Nulle Part, et qu'elle avait décidé de prendre l'avion pour aller se faire bronzer sur les plages de la Floride avec Goofy et ses amis. Elle éclatait alors de son petit rire espiègle ou se glissait dans un rêve en technicolor lui donnant accès à la maison de Goofy et aux longues étendues sablonneuses longeant les côtes floridiennes.
Ce jour-là, on a donc revu L'ère de glace. Billie a ri aux mêmes endroits que la semaine précédente et moi aussi, sauf qu'on ne riait pas toujours en même temps. Chaque fois que je m'esclaffais devant une blague qui la laissait aussi indifférente que les autres enfants enfoncés dans l'obscurité de la salle, elle me regardait comme si j'étais idiot, ce qui me faisait rire de plus belle et confirmait du même coup sa crainte : son père pouvait parfois se comporter en imbécile, à l'image de la majorité des adultes. J'ai tellement ri durant cette représentation que j'ai failli m'étouffer, au grand dam de Billie, qui s'est empressée de me pincer une cuisse en me jetant un regard à la Mantan Moreland, un vieux comique américain qui avait des yeux plus grands que leurs orbites.
J'ignore si Billie avait honte de son crétin de père, mais je m'en foutais, parce que Billie m'aimait, je le savais, et parce que je me découvrais soudain une affection sans bornes pour les mammouths et les paresseux. Si je revoyais L'ère de glace une troisième fois, la compassion que j'éprouvais pour les espèces menacées m'inciterait peut-être à modifier quelques-unes des habitudes stupides que la plupart de mes contemporains et moi nous entêtions à maintenir en nous enfonçant la tête dans le sable. C'était ça qui clochait avec les adultes. Les histoires qui se tenaient debout, ils les oubliaient plutôt que de se les faire raconter jusqu'à ce que les passages ayant de l'allure leur entrent dans le crâne.
À la fin de la séance, j'avais une faim de loup et j'ai entraîné Billie dans une pâtisserie où l'on servait aussi des soupes, des salades et des sandwichs. J'ai commandé le spécial du jour, une crème d'asperges accompagnée d'un pain Kaiser débordant de fromage fondu, de luzerne, de concombre râpé et d'une sauce onctueuse spécialement conçue pour vous couler sur le menton et tacher vos jeans du dimanche. Je voulais que Billie prenne la même chose, mais elle a commandé une misérable salade de carottes avec ce petit air buté signifiant que si je m'en mêlais, elle piquait une crise. J'ai fermé ma gueule et j'ai avalé mon Kaiser de travers, parce que je savais que Billie était aussi affamée que moi.
Elle a vidé son bol de salade avec l'avidité d'un écureuil préhistorique au bord de l'inanition venant soudainement de découvrir une réserve de glands géants sous un glacier. Quand je lui ai demandé si elle voulait autre chose, elle a fait signe que non, les lèvres serrées, en essuyant rapidement les deux petites larmes qui avaient réussi à s'échapper de ses yeux noisette. En voyant ces larmes minuscules et sous-alimentées, c'est moi qui ai piqué ma crise, tout doucement, une petite crise feluette, parce qu'un adulte doit savoir se contrôler dans un lieu public. J'ai murmuré ça va faire, Billie, j'en peux plus, tu manges, et j'ai demandé à notre serveuse de nous apporter deux éclairs au chocolat, les deux plus gros qu'elle pouvait dénicher dans son comptoir à cochonneries.
D'autres larmes ont coulé sur les joues de Billie, des larmes du genre à faire éclater tous les cœurs de père du monde, mais elle a mangé son éclair et moi aussi. J'ai même enfourné les deux autres qui ont suivi avec l'empressement d'un père prêt à se faire péter la panse pour que sa fille lui sourie. Et puis c'est arrivé. La bouche pleine de chocolat, Billie m'a souri, je lui ai souri à mon tour, on a éclaté de rire en se cachant derrière nos serviettes de table et j'ai juré que Billie ne mangerait plus jamais de carottes, sous quelque forme que ce soit. Ma réaction était idiote, mais j'étais heureux.
C'est à ce fou rire que j'ai pensé quand j'ai téléphoné à la police pour signaler la disparition de Billie, un fou rire si franc que l'absence momentanée de cette enfant ne pouvait être qu'une monumentale erreur, une blague de clowns se lançant des tartes à la crème au visage ou s'écrasant des éclairs au chocolat dans le cou.
Après avoir raccroché, j'ai forcé L.A. à s'asseoir et lui ai servi un double scotch en lui répétant ce que tous les gens normalement constitués se disent lorsqu'une bombe s'écrase sur leur maison ou qu'un croquemort se pointe au milieu du souper pour leur annoncer que le monde compte désormais quelques milliards d'habitants moins un. Je lui ai saisi la main et lui ai juré dans toutes les langues que je connaissais que c'était impossible, que le croquemort n'existait pas, qu'il suffisait de passer une main dessus pour l'effacer et supprimer du même coup les paroles de sans-dessein déformant sa maudite grande face enfarinée. Le croquemort appartenait à un cauchemar, point à la ligne, à un hostie de cauchemar débile que la sonnerie du réveil allait interrompre avec les premiers rayons du soleil.
Une heure plus tard, deux agents de police déguisés en apprentis croquemorts frappaient à notre porte, une blonde décolorée dont la queue de cheval dérivait vers la gauche et un grand rouquin auquel il manquait une dent sur le devant. Ils nous ont posé un tas de questions sur Billie, sur ses habitudes, ses amis, notre entourage, puis ils sont repartis avec la dernière photo de la puce, une photo sur laquelle elle portait le manteau rose à capuchon que Lucy-Ann lui avait acheté au début de l'hiver, le manteau qu'elle avait enfilé ce matin-là pour se rendre à l'école. Et elle souriait, de son petit sourire de fillette espiègle plus retroussé d'un côté que de l'autre, de ses petits yeux noisette sous lesquels s'envolaient quelques taches de son, quelques picots de chouchounette.
### MARNIE
Moins de vingt-quatre heures plus tard, les chiens sont arrivés, trois bergers allemands nommés Rex, Chet et Lucy, entraînés pour renifler la peur, la mort, le sang, pour suivre le dispersement d'odeurs si infimes qu'on a peine à croire qu'elles aient subsisté dans la puanteur des tourbières. Dès qu'on les a fait sortir du fourgon de la police, le village a été envahi de jappements et de hurlements exprimant mieux que n'importe quel cri humain la pesanteur du drame ayant déplié ses ombres sur la légèreté de l'été. Même quand ils cessaient de hurler, on continuait à entendre l'écho de leurs plaintes multipliées. Plusieurs jours après qu'ils furent repartis, leurs jappements résonnaient encore dans les sentiers piétinés où aucun enfant n'avait plus le droit de s'aventurer, chacun confiné à l'étroite bande de lumière entourant les fenêtres d'où on pouvait le surveiller, braquer son regard anxieux sur la possible incursion du mal et tirer à bout portant, sans avertissement, sur quiconque perturbait la familiarité des lieux.
La première journée, on a lâché les chiens près du bassin magique, à partir du centre de l'étoile qu'avaient dessinée la veille les hommes brusquement tirés de la tranquillité de leur maison par le père de Michael. Quant à Jeanne Dubé, la mère de Mike, elle refusait de quitter la maison, au cas où le petit arriverait, au cas où il aurait faim, ça mange tout le temps à cet âge-là. Pendant que les hommes marchaient dans la terre humide, les femmes de Rivière-aux-Trembles guettaient leur bout de trottoir, au cas où, au cas où... Quelques-unes s'étaient rassemblées dans la maison de Jeanne, croyant pouvoir l'aider, mais celle-ci se rendait à peine compte de leur présence. Elle préparait des sandwichs aux œufs, les préférés de Mike, lui confectionnait des gâteaux, répétant sans cesse que le petit serait affamé quand il rentrerait. Elle tamisait la farine en marmonnant, essuyait ses mains sur son tablier, réglait la température du four, certaine qu'elle entendrait bientôt les pneus du vélo de Mike crisser dans la cour. Il faut que tout soit prêt, disait-elle à Emmy-Lou, va, ma puce, va voir s'il s'en vient. Et Emmy-Lou sortait sur la galerie, sa poupée favorite entre les bras, Charlotte, son suçon collant au fond de la poche de sa robe jaune.
Les recherches ont duré jusqu'au coucher du soleil, mais les chiens, chaque fois qu'on les lançait dans une nouvelle direction, rebroussaient chemin pour suivre un tracé tortueux menant à la rivière, près de laquelle ils s'arrêtaient en frémissant et en gémissant. Toujours au même endroit, à deux kilomètres du bassin. Au début de l'après-midi, on a apporté un canot et on les a fait traverser la rivière avec le maître-chien. Une fois de l'autre côté, les chiens se sont comportés de la même façon, regardant la rivière, accusant la rivière, aboyant que Michael était là, au milieu des eaux calmes. Une équipe a immédiatement été chargée de draguer le cours d'eau, car la conclusion qu'on pouvait tirer de la connaissance innée des chiens était claire, la piste de Michael Saint-Pierre, avec ses odeurs d'enfant, se perdait dans cette eau aux apparences inoffensives.
Pendant qu'on s'affairait sur le terrain, deux policiers, accompagnés du père de Michael, sont venus sonner à notre porte. Ils se sont entretenus à voix basse avec mon père, puis celui-ci m'a demandé de descendre. Ces messieurs désirent te poser quelques questions, Marnie, a-t-il déclamé, empruntant sa réplique à un roman de gare ou à une série télévisée des années cinquante. On s'est assis autour de la table de la cuisine, le gros inspecteur avec une moustache à ma gauche, le plus petit à ma droite et mon père devant moi. Quant au père de Michael, il est resté debout dans l'encadrement de la porte, à triturer sa casquette comme si, à force d'en écraser les rebords et de la retourner dans tous les sens, elle allait lui donner la réponse qui empêcherait son cœur d'éclater sous la pression de l'angoisse.
Sa présence m'intimidait, car je savais que sa douleur, qu'il tentait de dissimuler en baissant la tête, me tenait pour responsable de la disparition de son fils. Chaque fois qu'il levait les yeux vers moi, je détournais rapidement les miens, parce que cet homme au visage ravagé par la peur avait raison. Si je n'avais pas été là, si Michael et moi n'avions pas été inséparables et si notre amitié ne l'avait entraîné à ma suite au bord de la rivière, rien de tout cela ne serait arrivé. J'étais coupable de mon trop grand attachement envers Michael, j'étais coupable de ne l'avoir pas rattrapé et j'étais coupable, surtout, de n'avoir pas disparu en même temps que lui. Si je m'étais évanouie dans la nature pour ne laisser derrière moi que le ruban nouant mes cheveux, jamais on n'aurait songé à m'accuser à demi-mot d'être ressortie de la forêt sans tenir dans la mienne la main crottée de Michael Saint-Pierre. J'aurais été une victime, au même titre que lui, et on aurait accroché ma photo près de la sienne, dans le grand hall de l'école, au-dessus d'une vignette vantant mes mérites scolaires ou sportifs. On me refusait le statut de victime sous le simple motif que j'étais vivante, et l'enfant effarée en moi donnait raison à ceux qui m'en voulaient presque, dans leur douleur ou à cause de leur insondable bêtise, d'avoir échappé au mal qui avait emporté Michael. J'étais la rescapée, celle qui n'avait droit ni à la compassion ni à la chaleur des bras tendus. C'est ainsi qu'une vie d'enfant se termine, dans l'apprentissage de la culpabilité, dans la honte du survivant.
J'ai répondu aux questions des policiers machinalement, parce qu'on m'avait déjà posé les mêmes trois ou quatre fois au moins. J'ai décrit l'orage, j'ai décrit le balancement puis l'immobilité de Michael, la pluie mouillant son t-shirt rouge, la lenteur des maillots tombant à ses pieds, puis, alors que je revoyais mon maillot s'écraser au sol, je me suis souvenu du running gauche de Mike, qui s'était dénoué et trempait dans la boue. J'avais oublié ce détail, qui me semblait soudain d'une importance capitale, parce qu'on ne peut pas courir avec un soulier détaché. J'ai tenté d'expliquer aux policiers que Michael avait couru pareil, qu'il s'était enfoncé à toute vitesse dans les arbres emmêlés après avoir murmuré quelques paroles à mon intention, mais ils se souciaient peu du lacet défait, tout ce qu'ils voulaient savoir, c'est ce que Michael m'avait dit. Quelles paroles, Marnie ? demandait le gros policier à la voix douce. Quelles paroles ? Essaie de te rappeler.
Sur le coup, je ne me souvenais plus de rien. Les quatre paires d'yeux braqués sur moi espéraient un miracle, une formule magique capable de ramener Mike, et leur fixité me donnait la frousse. J'avais beau m'efforcer de lire sur les lèvres de Mike, je n'y déchiffrais qu'une suite de mots sans queue ni tête qu'absorbaient les souffles suspendus autour de moi.
Puis Victor Saint-Pierre s'est frappé la poitrine comme on le fait dans les églises, mea culpa, et les mots sont apparus, si invraisemblables que j'avais honte de les répéter. Ça voulait rien dire, ai-je murmuré, ça voulait rien dire et j'ai pas bien compris. On veut quand même savoir, Marnie, c'est important, a murmuré à son tour le gros policier qui sentait la menthe, son visage si proche du mien que j'ai reculé en laissant échapper mauvais temps, madame, mauvais temps, c'est ça que Mike a dit, mauvais temps, ne plie pas le jour. J'ai pas bien compris.
Tout le monde est demeuré silencieux, le gros agent s'est raclé la gorge, puis il m'a demandé de me concentrer très fort et d'essayer d'entendre mieux. J'ai essayé, mais c'était inutile. J'ai rien entendu, ai-je fini par avouer, j'ai juste vu des mots, ces mots-là, mauvais temps, madame.
Le gros inspecteur a soupiré pendant que le petit prenait des notes et revenait à la charge pour savoir si on avait construit d'autres cabanes, si on connaissait plusieurs cachettes et si on jouait parfois dans des endroits qui nous étaient interdits. Après quelques moments d'hésitation, j'ai mentionné le lac aux Barbotes. Je n'aurais pas dû. Je n'aurais pas dû parce que cette histoire est alors devenue plus incompréhensible, plus irréelle qu'elle ne l'était déjà, parce que c'est là, sur les rives broussailleuses d'un lac depuis longtemps vidé de ses barbotes, qu'on a découvert le lendemain le running gauche de Michael, avec son lacet détaché.
C'est Lucy, la chienne dont les hurlements avaient déchiré la nuit de Rivière-aux-Trembles, qui a trouvé ce running après qu'on l'eut lâchée dans le sentier que Mike et moi empruntions pour nous rendre au lac. Elle s'est arrêtée près de la petite chaussure bleue, a poussé un long gémissement de chienne soudainement blessée par les armes silencieuses des hommes, puis a attendu que son maître la rejoigne. Pendant que celui-ci la récompensait et la soulageait de son invisible blessure en lui donnant des biscuits en forme d'os et en lui lançant une balle, les policiers ont mis la chaussure dans un sac et l'ont apportée au père de Michael pour qu'il l'identifie. En voyant la chaussure enfermée dans un sac semblable à ceux qu'on utilise pour congeler de la viande, Victor Saint-Pierre a eu un mouvement de recul, comme si les flics lui tendaient un serpent venimeux. Il a ensuite arraché le sac des mains de l'agent Desmarais et l'a serré contre sa poitrine, puis il s'est plié en deux et s'est affaissé au pied d'un arbre en poussant des cris mêlés de paroles insensées.
Agenouillée sur la banquette de la voiture de police dans laquelle on m'avait fait monter avec mon père, j'ai vu le corps énorme de Victor Saint-Pierre se plier, s'effondrer puis se balancer au pied de l'arbre, d'avant en arrière, d'avant en arrière, aussi mécaniquement que Mike au bord de la rivière. Il ressemblait à un cheval ruant dans ses brancards ou dans sa stalle, acculé aux murs d'un brasier lui dévorant la gorge. Je me suis collée contre mon père et j'ai aperçu, entre les bras puissants de Victor Saint-Pierre, le running bleu Superman que Mike avait perdu dans sa fuite, c'était prévu, parce qu'aucun enfant ne peut courir longtemps avec un soulier défait. Alors, dans ma tête, les mots et les images se sont mis à osciller, d'avant en arrière. Le torrent bouillonnant de la rivière aux Trembles a déferlé sur la langueur du lac aux Barbotes, le soleil a plongé au nord de l'horizon et j'ai regardé mon père pour qu'il m'explique comment le running de Michael avait abouti là. Puis tout est devenu blanc, du blanc de la neige et de la poudrerie, de la pâleur des draps frais battant au vent.
Je me suis éveillée sous les hautes branches d'un arbre penché, lovée dans les bras de mon père qui fredonnait sans s'en rendre compte Lili Marnie, Lili Marleen, une chanson inventée aussi vieille que moi, aussi vieille que la voix de ma mère penchée sur mon berceau, Lili Marnie fera dodo, Lili Marleen a fait dodo, goodbye hello Lili Marnie. Il y avait des larmes dans ses yeux, il y en avait aussi, plus rouges, dans les yeux de Victor Saint-Pierre, qui se tenait maintenant la tête à deux mains, prêt à éclater, prêt à laisser se disperser les morceaux sanguinolents de son cerveau dans le foin bordant la route, et il y en avait aussi, je l'aurais juré, dans les yeux gris de l'agent Desmarais, le gros flic à la voix douce.
Je ne m'étais évanouie qu'une petite minute mais, durant cette minute, le monde avait changé. Dans le monde d'avant, Michael portait deux chaussures et il courait. Dans ce monde-là, l'ouest se situait derrière la colline des Loups et le soleil s'y couchait tous les soirs, prêt pas prêt j'y vas pareil. Dans le monde de maintenant, Michael ne courait plus, Michael s'était arrêté, Michael était tombé dans un trou où le soleil s'était engouffré à sa suite, indifférent à la géographie des lieux, qu'un phénomène excédant la science des hommes avait modifiée durant la nuit pour détourner le chemin de Mike. Il n'y avait pas d'autre explication possible, le ciel avait parlé au diable, sinon Michael aurait abouti chez lui, pour la simple raison qu'on ne pouvait se rendre du bassin magique aux rives du lac aux Barbotes sans effectuer un détour par le village, puis par cette rue où la petite Emmy-Lili guettait le retour de Michael, son frère, son dieu.
Quant aux hommes assemblés sous les arbres amollis par la moiteur d'août, ils ne comprenaient pas que la piste de Mike ait pu se perdre au bord de la rivière et l'enfant aboutir ici. Victor Saint-Pierre ne comprenait pas non plus, ni le maître-chien, ni les chiens reniflant le sable où fuyaient les sauterelles. Dans mon esprit, les choses étaient cependant tout à fait claires. Quelqu'un avait menti, quelqu'un mentait encore et ce n'était pas moi. Alors ça ne pouvait être que la rivière et le soleil. Ça ne pouvait être que la chaussure de Mike, et Michael lui-même.
### BILL
Lucy-Ann a passé la nuit à donner des coups de fil, à réveiller nos amis, nos voisins, nos connaissances proches ou lointaines, à hurler aux parents des rares amis de Billie d'aller sortir leur môme du lit et de le secouer jusqu'à ce qu'il leur révèle où était sa fille, sans autre résultat que de foutre la pagaille et d'alimenter les cauchemars des petits. Quant à moi, par moins 20 degrés, j'ai arpenté les rues avoisinant notre maison, puis celles entourant l'école de Billie et l'édifice abritant son école de danse en hurlant son nom assez fort pour faire surgir aux fenêtres un paquet de visages blêmes alertés par la furie de l'énergumène qui perturbait la quiétude de leur nuit. Je lui ai crié que je l'aimais, que je l'aimais comme c'était pas possible, en grimpant sur les bancs de neige, les balcons, les clôtures, sur tout ce qui pouvait s'escalader, après quoi j'ai montré sa photo à tous les passants et commis de dépanneurs du quartier, j'ai apostrophé quelques hommes sortant d'un bar en titubant, la larme à l'œil, jusqu'à ce qu'un gyrophare me force à protéger mes yeux de la lumière rouge et bleue tourbillonnant dans l'air glacial et à placer ma main gauche, celle qui tenait la photo de Billie, devant mon visage de père éperdu.
Ce n'est qu'à ce moment que j'ai enfin senti la chaleur des larmes sur ma peau, le doux soulagement que cette chaleur procure à la tension des muscles. Pleurer n'atténue en rien la souffrance, mais ça lui donne le droit de se laisser aller un peu, de se répandre en petites rigoles salées sur la peau froide, pendant que le cerceau de métal enserrant le plexus solaire se relâche d'un cran ou deux. Les quelques larmes qui ont franchi la barrière de ma colère ont donné du lest à ma douleur et je me suis assis sur le trottoir glacé, sous la lumière du gyrophare, d'où j'ai montré la photo de Billie au policier d'à peine trente ans qui se tenait devant moi, une main sur sa matraque et l'autre tendue vers la photo que je refusais de lâcher.
On va vous conduire chez vous, monsieur, a-t-il dit en me saisissant par une épaule, et j'ai cessé de pleurer net. Je me suis levé d'un bond et j'ai crié que c'était hors de question, que je ne pouvais pas rentrer chez moi alors que ma fille, ma toute petite fille de huit ans, rose, avec des yeux noisette, regardez sa photo, était perdue quelque part dans le noir. Allez, venez, monsieur, elle doit être rentrée à l'heure qu'il est, a-t-il ajouté en faisant signe à l'agent demeuré de l'autre côté de la voiture clignotante de le rejoindre. Mais je savais que Billie n'était pas rentrée. Mon intuition de père et ma chienne de peur de père tremblant de la tête aux pieds me disaient exactement le contraire. Avant qu'il me pousse dans la voiture, j'ai tenté de lui foutre mon poing de père enragé sur la gueule mais il l'a esquivé. L'autre flic, un petit nerveux, a rappliqué en dirigeant sa main droite vers son holster. Craignant qu'il s'énerve davantage et essaie de jouer de la gâchette, j'ai abandonné, j'ai baissé les bras parce que Billie n'avait pas besoin d'un père au cœur percé d'un trou de balle.
Quinze minutes plus tard, je contemplais le bout de mes Kodiak derrière les barreaux d'une cellule de détention provisoire, genre communautaire, en compagnie de deux poivrots et d'un gars portant un costume d'agent d'assurances. La cellule voisine, réservée aux femmes, n'abritait qu'une seule pensionnaire, une rousse permanentée qui, à en juger par son allure, devait faire le trottoir depuis si longtemps que ses talons aiguilles s'étaient enfoncés dans le bitume et s'y étaient fossilisés. Qui sait, d'ailleurs, si quelque archéologue du futur ne se penchera pas un jour sur ces artefacts pour essayer de comprendre les mœurs étranges de cette époque où on perchait les femmes sur des instruments de torture. Elle ne portait qu'une paire de bas déchirés laissant voir le vernis écarlate de ses ongles et elle gueulait aussi, mais pour des raisons différentes des miennes. Elle tempêtait contre la police, contre les hommes, contre l'injustice du monde qui avait créé les filles de son espèce et n'était même pas foutu de leur fournir des chaussures et de les traiter avec un peu de décence.
Je n'aime pas brusquer les femmes, encore moins les femmes brisées, mais j'étais à bout de nerfs et sa voix nasillarde créait des interférences avec celle de Billie, que j'essayais d'entendre par-delà les murs. Ses vociférations me rentraient dedans comme une vrille munie d'une mèche à béton dans un deux par quatre et j'ai fini par lui crier de fermer son hostie de grande gueule. Un silence de mort s'est abattu sur les deux cellules, les coins des lèvres de la fille se sont mis à tressauter et elle est allée pleurer dans un coin. En d'autres circonstances, j'aurais tenté de la consoler, parce que je me sentais complètement nul, aussi macho qu'un pimp enragé, mais en ce moment, je n'avais pas la force de tendre les bras à travers les barreaux pour caresser les joues mouillées d'une femme qui avait passé sa putain de vie à haïr le premier homme qui l'avait touchée, puis tous les autres ensuite. Je me suis contenté de bafouiller de vagues excuses et me suis écrasé à côté d'un banc de métal en attendant que Lucy-Ann rapplique pour me sortir de là. J'aurais voulu pleurer aussi mais, devant le spectacle sordide du poivrot numéro un qui se bavait dessus, de l'agent d'assurances boutonné de travers et de la fille aux bas troués, j'ai refoulé mes larmes en appelant à la rescousse ce qu'il me restait d'orgueil. Je me suis improvisé un passe-montagne avec mon chandail et me suis enfoncé la tête entre les genoux, là où tout était noir.
On croit toujours que ça n'arrive qu'aux autres, ces histoires d'horreur. On compatit avec les parents éplorés quand on voit le visage d'un gamin ou d'une gamine reproduit dans le journal ou placardé sur un mur, mais on n'imagine pas que ce visage puisse un jour être celui de notre enfant. Si cette idée nous traverse l'esprit, on la repousse immédiatement en se disant que ça ne peut pas se produire, qu'on est là pour protéger le petit, la petite, que rien ne peut l'atteindre tant qu'on monte la garde. On plaint les pauvres types qui se promènent avec la photo de première communion de leur fillette ou avec celle de son dernier anniversaire, sur laquelle l'enfant gonfle les joues devant un gâteau en forme de lapin surmonté de sept ou huit bougies. On plaint les femmes aux yeux cernés abruties par les somnifères, les antidépresseurs, le gros gin caché dans l'armoire de la cuisine, à portée de la main, au cas où la conscience du temps se ferait trop présente, mais on n'arrive pas à s'associer réellement à leur détresse. Ce type de fatalité ne frappe que ceux dont le destin est programmé pour le malheur et on ne fait pas partie de ces gens. C'est ce qu'on croit.
Et puis, par une nuit glaciale de janvier, on se ramasse en taule, dans l'odeur de sueur rance, à se demander si on reverra la puce, si elle grandira, si elle ramènera un mongol boutonneux à la maison le jour de ses quatorze ans et si elle finira par nous traiter de vieux con. À ce moment, c'est ce qu'on souhaite le plus ardemment au monde, que notre fille ait un jour la possibilité de nous traiter de crétin sénile, qu'elle ramène soixante mongols à la maison si ça lui chante, qu'elle devienne bassiste dans un groupe punk, se convertisse à l'islam, se fasse coudre une rangée d'anneaux dans le front ou s'engage dans la Légion étrangère. Ce qu'elle fera de sa vie, on s'en fout royalement. Tout ce qu'on veut, c'est pouvoir la regarder respirer et vieillir, la regarder pleurer devant un film d'amour des années cinquante, le visage plein de rimmel et les cheveux en bataille. Tout ce qu'on espère, c'est avoir un mal de chien devant sa tristesse et s'arracher les tripes pour la rendre heureuse. Rien d'autre n'importe, et tant pis si elle ne ressemble pas à l'adulte qu'on avait imaginée.
Les jours où l'archéologie et la sorcellerie perdaient de leur attrait, Billie déclarait qu'elle voulait devenir astronaute. Elle rêvait d'aller dans l'espace et de découvrir dans l'apesanteur des vaccins pour guérir les arbres et les chats malades, les chiens galeux, les crocodiles. Il y avait de la mère Teresa en elle, ou du saint François d'Assise, un foutu bon fond, et elle serait peut-être devenue la grande protectrice universelle des faibles, des pas beaux et des laissés-pour-compte si cette maudite vie ne lui avait coupé l'herbe sous le pied.
Elle avait recueilli un chat, l'année d'avant, un matou de gouttière plutôt mal en point auquel elle avait donné un nom de souris, Pixie, s'inspirant du dessin animé d'Hanna-Barbera, Pixie & Dixie & Mr. Jinks, que je lui avais fait découvrir peu de temps auparavant sur Internet. Tout de suite, Billie était tombée amoureuse de Pixie, la souris portant un nœud papillon bleu, et c'est sans hésiter qu'elle avait donné ce nom à son chat. Comme ça, popinouche, y mangera pas de souris, car Billie aimait les souris, les alligators, les chevreuils, les rats et tout ce qui avait quatre pattes d'un amour démesuré lui faisant oublier que la plupart des quadrupèdes ne pouvaient survivre qu'en bouffant du quadrupède ou du bipède, et vice versa. Pixie devait être couché sur son lit, à l'heure qu'il était, à attendre anxieusement son retour, à dresser ses oreilles inquiètes chaque fois que Lucy-Ann ouvrait la porte pour hurler le nom de Billie dans l'obscurité. J'étais en train de dire à Pixie de tenir bon, t'en fais pas, Pixie, elle va revenir, c'est juste un cauchemar, quand un flic m'a fait sortir de la cellule pour me conduire à ce qu'il a nommé le bureau d'accueil. J'ignore s'il s'essayait à l'ironie ou voulait me servir un échantillon d'humour policier, mais je n'avais pas le cœur à rire, et Lucy-Ann non plus, qui m'attendait sur un banc de cuirette trop étroit en se rongeant les ongles.
Je ne l'avais jamais vue se déchiqueter les ongles à coups de dents. Dans la famille, c'est Billie qui se rongeait les ongles, habitude dont L.A. était presque parvenue à la débarrasser en lui enduisant le bout des doigts d'un produit qui sentait le diable et goûtait pareil. Je n'étais pas trop chaud à l'idée de laisser Billie sucer cette saloperie dont on ignorait les effets secondaires, qui pouvait donner le cancer ou la maladie de Tourette, qui sait, mais en cette matière, L.A. avait invariablement le dernier mot.
Depuis la fin de l'après-midi, aucune règle ne tenait cependant plus pour L.A. Elle mordillait l'ongle de son index droit comme si elle avait voulu se l'arracher et s'excuser du même coup auprès de Billie de l'avoir asticotée pour des niaiseries et des bouts de peau morte. Elle faisait pitié à voir, avec ses yeux gonflés, son manteau de travers, ses cheveux décoiffés, mais je ne pouvais rien pour elle, ni pour moi, ni pour Billie. J'étais acculé à ce type d'impuissance qui rend complètement fou, qui soulève en vous une violence que vous n'aviez jamais cru pouvoir abriter. Si j'avais eu les moyens de convertir cette violence en énergie, j'aurais inventé une machine à remonter le temps et j'aurais ramené Billie sur-le-champ, je l'aurais attrapée à sa sortie de l'école, avec toutes les autres petites Billie, Cédrika, Victoria et Ashley, avec tous les jeunes David dont le sourire s'était effacé au coin d'une rue déserte, mais je ne l'aurais fait ni pour L.A. ni pour moi. Je l'aurais fait pour Cédrika, Victoria, Ashley et David, je l'aurais fait pour Billie, pour qu'elle puisse grandir et voyager dans la pureté des espaces intersidéraux.
Lucy-Ann a attendu que je récupère mes affaires, puis elle s'est levée, je l'ai suivie, et on n'a pas prononcé une parole entre le poste de police et la voiture. Ce n'est qu'en quittant le stationnement qu'elle a éclaté. Elle a effectué un virage en U pour éviter un sens unique et elle a brusquement appliqué les freins.
L.A. avait l'habitude de s'exprimer dans une langue soignée, histoire de donner le bon exemple à Billie, et se laissait rarement aller à des écarts de langage, mais quand le barrage sautait, tous les charretiers du monde pouvaient se rhabiller. Je connaissais suffisamment L.A. pour savoir quand elle allait entrer dans ce que je considérais comme une forme de combustion langagière. Avant même qu'elle ouvre la bouche, j'ai vu une bande de charretiers tout nus ravaler leur gomme, puis le barrage a sauté dans une explosion de tabarnak. En résumé, j'étais un tabarnak d'inconscient d'hostie de malade d'irresponsable qui réussissait à foutre la merde le jour même où sa fille disparaissait. J'étais un enculé, un enfoiré, un trou-de-cul, un imbécile et un salaud qui ne méritait tout simplement pas d'avoir une fille aussi brillante que Billie. Si c'était à recommencer, m'a-t-elle craché à la figure, je lui trouverais un autre putain de géniteur. Tu vaux rien en tant que père, Bill Richard !
En matière de coup bas, on peut difficilement inventer mieux, et celui-là m'a coupé le souffle aussi efficacement qu'une savate entre les jambes. Je n'étais pas un père parfait, mais je défiais n'importe quel putain de géniteur de me prouver qu'il aurait pu se débrouiller mieux que moi pour assurer le bien-être de Billie. Je me suis mordu les joues pour ne pas sortir de la voiture en arrachant la portière, et un autre tonnerre de jurons a déferlé dans l'habitacle surchauffé, en provenance de mon siège, cette fois, le siège du moribond, que la première déflagration n'avait pas tout à fait achevé malgré sa virulence. Ce n'est que lorsque je me suis rendu compte que Lucy-Ann pleurait et que les vitres étaient complètement embuées que je me suis arrêté.
Ce n'était pas à moi de m'excuser, mais je l'ai fait quand même, pour Billie, qui n'avait pas davantage besoin d'un père et d'une mère prêts à s'arracher les yeux que d'un père au cœur percé d'un trou de neuf millimètres, mais de deux adultes qui se serraient les coudes et vireraient le monde à l'envers pour la retrouver. J'ai tendu le bras vers Lucy-Ann, qui n'a pas bronché lorsque je l'ai posé sur son épaule pour masser sa nuque, tout doucement, avec le restant de tendresse que j'éprouvais pour cette femme qui ne m'aimait plus. Elle a laissé tomber sa tête sur le volant et je lui ai tendu un paquet de Kleenex dont elle a déchiqueté le premier avant de remplir les trois autres et de les envoyer valser sur la banquette arrière. Après ça, on est rentrés à la maison à fond de train, parce qu'il s'agissait du premier endroit vers lequel Billie se dirigerait si elle arrivait à fuir l'épaisseur de la nuit.
Pixie était là, qui guettait le retour de Billie à l'une des fenêtres de la véranda, couché sur le dossier du fauteuil d'enfant où Billie s'assoyait parfois pour regarder tomber la neige. En l'apercevant, Lucy-Ann s'est remise à pleurer. Elle a couru vers lui et l'a saisi dans ses bras en gémissant pauvre Pixie, pauvre chaton, pauvre bébé Pixie. Elle n'associait pas sa douleur à la mienne, mais à celle de ce chat pelé qui se laisserait mourir à petit feu si Billie ne rentrait pas pour le caresser et lui raconter comment le Chat botté et la petite fille aux allumettes s'étaient rencontrés. Si L.A. ne m'avait devancé, c'est ce que j'aurais fait aussi. J'aurais pris ce vieux pouilleux dans mes bras et je l'aurais bercé à la manière de Billie, en lui donnant des becs de nez, ben plus le fun que des becs de bec, papanoute, disait-elle en collant son nez sur celui de Pixie.
L.A. n'avait jamais aimé ce chat mais, cette nuit-là, il était le seul être qui la reliait à Billie, le seul être vivant. Moi, je n'existais plus. Moi, je n'étais qu'un homme, un parmi les milliers de salauds qui s'approchaient des petites filles en leur offrant des fleurs et des bonbons, des fleurs toutes simples et des bonbons colorés, de ceux qu'aiment les enfants et qu'ils traînent dans leurs poches. Pendant un moment, j'ai détesté Lucy-Ann d'avoir immédiatement fait de cet animal qu'elle n'avait jamais caressé que du bout des doigts le complice de son malheur, puis j'ai pensé que Billie serait heureuse de constater que sa mère s'occupait enfin de Pixie.
Je me suis alors rendu compte que je réagissais comme si Billie n'allait pas revenir, que ma peur de ne plus tenir cette enfant dans mes bras était telle que j'anticipais le pire au lieu de me grouiller le cul. Avant de tout casser dans la baraque, j'ai remis ma tuque et j'ai dit à Lucy-Ann que je repartais à la recherche de Billie.
À mon retour, Lucy-Ann dormait, à demi affalée sur la table de la cuisine, ma bouteille de Jack Daniel's Gold Medal 1 981 à ses côtés. J'ai enlevé mes bottes et je suis monté me coucher sur le lit de Billie, avec ses poupées, avec ses oursons et avec Pixie, dont les grands yeux verts savaient déjà que notre maison s'était écroulée.
### MARNIE
Je n'aurais jamais dû parler du lac aux Barbotes. Si j'avais préservé notre secret, on n'aurait peut-être pas retrouvé le running de Mike et les points cardinaux seraient demeurés à leur place. En moins de temps qu'il n'en faut à une goutte de pluie pour toucher le sol, je suis devenue une intrigante et une menteuse, et la délicatesse qu'on avait manifestée à mon égard a disparu d'un coup. La voix douce de l'agent Desmarais s'est durcie, le regard du père de Michael est devenu pareil à celui de Lex Luthor quand il s'aperçoit que Superman a encore déjoué ses plans machiavéliques, et les chuchotements des hommes venus prêter main-forte à la police se sont transformés en une rumeur hostile. La petite a menti, murmurait la rumeur, la petite a tout inventé, va savoir pourquoi, la petite a commis une gaffe et elle a voulu se couvrir.
Dans l'esprit de tous ces hommes, la fillette qui sursautait au moindre bruit comme un petit écureuil nerveux avait forgé de toutes pièces l'histoire abracadabrante du bassin magique pour n'avoir pas à révéler qu'elle s'était rendue près du lac aux Barbotes avec le fils de Victor, et parce qu'elle refusait d'avouer quel malheur avait frappé le gamin. Seul mon père, qui ne comprenait pas plus que moi ce qui s'était produit, continuait à me croire. Y a sûrement une explication logique, Victor. Je connais Marnie. Elle a pas pu mentir à propos de Michael. Y sont pareils à des siamois, ces deux-là, tu le sais aussi bien que moi.
Mais Victor Saint-Pierre refusait d'entendre les arguments de mon père. À ses yeux, celui-ci ne cherchait qu'à protéger sa fille et on ne pouvait pas lui accorder sa confiance. En quelques heures, son attitude s'était radicalement transformée, au point que même ses amis ne le reconnaissaient plus. Cet homme d'ordinaire si calme qu'on entendait fredonner de vieux airs des Beatles dans son atelier le samedi matin, qui venait nous ébouriffer les cheveux en traitant affectueusement Michael de baswell, n'était plus que hargne et colère. Après qu'on lui eut confisqué la chaussure de son fils, je l'ai vu repousser violemment mon père pour se diriger vers moi à grandes enjambées. Instinctivement, je me suis collée contre la voiture des flics, dont la tôle brûlante sous le soleil d'août m'a arraché un petit cri, et j'ai fermé les yeux, certaine que Victor Saint-Pierre allait me flanquer une volée, mais quelqu'un l'a arrêté avant qu'il arrive à moi. Vire pas fou, Victor, c'est tout ce que j'ai entendu, puis le son produit par les grosses mains calleuses de mon père s'abattant sur le capot de l'auto. Mais mon père n'a rien dit. Il s'est contenté de regarder le père de Michael dans le blanc des yeux et il m'a prise par la main pour m'entraîner sur le chemin des sœurs Morin. On n'avait pas fait dix pas que Desmarais a rappliqué. Ils ne pouvaient pas nous laisser partir. Ils avaient des questions à me poser, des questions graves, des questions à dix piastres, alors que je n'avais en poche que les quelques sous qui devaient me servir à payer un Sprite à Michael à notre retour du bassin magique, la veille.
C'était notre rituel, cet été-là. Un jour sur deux, avant le souper, on se rendait au dépanneur d'Yvonne Leclerc s'acheter un Sprite ou une orangeade qu'on buvait sur la galerie. On s'assoyait sur le vieux frigidaire Pepsi Cola qui ronronnait dans un coin, on décapsulait nos bouteilles en imitant les hommes assoiffés de l'hôtel Plazza, en face du bureau de poste, et on calait quelques gorgées pétillantes qui nous remontaient dans le nez. Après, on parlait de notre journée, de ce qu'on allait faire dans la soirée, de nos projets pour le lendemain, jamais à court d'idées, de rêves ni de plans de nègres. On était comme deux moitiés d'une même personne, Michael Saint-Pierre et moi, comme des siamois, mon père l'avait dit, comme un bolo et sa boule, un vélo et ses roues, et les policiers auraient dû comprendre qu'on ne peut pas mentir quand on vient de se faire arracher une partie de la tête, mais les adultes ne comprennent plus rien aux liens de l'enfance, à ces amours plus fortes que la chair et le sang, poussant les petites mains sales à se tendre l'une vers l'autre lorsque le soleil décline.
Pendant qu'une dizaine d'hommes s'affairaient à draguer la rivière au nord-est du village, une équipe venue de la ville ratissait les rives du lac aux Barbotes sous la supervision de l'agent Desmarais, qui avait installé une espèce de quartier général sur le bord de la route avec une table à pique-nique et des chaises pliantes qu'un ami de Victor Saint-Pierre traînait dans le coffre de sa voiture. C'est là que Desmarais et son coéquipier, McCullough, je crois, un nom anglais, m'ont emmenée pour m'interroger. Puisqu'il n'y avait que deux chaises, McCullough et mon père sont restés debout, au soleil, et on m'a fait asseoir à côté de l'agent Desmarais.
Il devait être près de midi car les ombres étaient courtes, ramassées au pied des arbres et des hommes. Au départ, je n'écoutais pas vraiment l'agent Desmarais. J'étais fascinée par l'ombre ratatinée de McCullough, qui rentrait la tête dans le cou chaque fois que celui-ci reculait de quelques pas pour s'éloigner de la guêpe qui lui tournait autour, attirée par la puissante odeur de sa lotion après-rasage. Je cherchais aussi l'ombre de la guêpe, qui se perdait parmi les ombres plus vastes. Si elle avait butiné au milieu d'un champ, on aurait pu voir sa minuscule silhouette sur une fleur d'épervière des prés ou de chicorée sauvage, mais là, le soleil niait son existence. McCullough battait des bras quand Desmarais a fini par lui demander d'aller chercher un truc dans la voiture, après quoi il a pris ma main dans sa grosse patte d'ours pour que je le regarde bien en face.
De la sueur coulait sur son front, qu'il essuyait avec un mouchoir sur lequel étaient brodées deux lettres de la même couleur que ses yeux, d'un bleu qui m'avait semblé inoffensif, le jour d'avant, pareil au bleu d'un ciel qui s'ennuage lentement, mais qui avait maintenant un éclat de métal froid dans son visage rougi par la chaleur. Desmarais avait deux paires d'yeux, en fait. Une paire qu'il gardait pour sa femme et ses enfants et qu'il s'enfonçait d'un revers de la main dans les globes oculaires s'il devait s'adresser à des victimes, à des pauvres gens, puis sa paire d'yeux de flic, réservée aux assassins, aux bandits, aux voleurs et aux menteuses. Et ces yeux-là attendaient, brûlés par la sueur et le manque de sommeil, que je leur raconte ce qui s'était vraiment passé après que Michael et moi eûmes abandonné Émilie sur la galerie des Saint-Pierre.
Que pouvais-je dire à cet homme pour le convaincre que notre seule destination avait été le bassin magique, que c'est l'orage qui avait tout déclenché, précédé du cri de la mort ? J'ai cherché appui dans le regard de mon père, mais il fixait la forme tronquée de son ombre à ses pieds, se demandant par quel détour il était arrivé là et s'il allait parvenir à s'extirper de cette zone dans laquelle son corps était réduit à l'impuissance, comprimé par le soleil ainsi que l'était son ombre.
J'avais souvent vu mon père dans cet état, lorsqu'il s'arrêtait brusquement au milieu du jardin, cherchant autour de lui quelle voix l'avait appelé, puis baissant tristement la tête et restant là, sans bouger, jusqu'à ce que son ombre s'allonge. Parfois, il traînait les pieds jusqu'au buisson de Mary-Jean, laissait tomber son râteau ou sa truelle, puis s'agenouillait près du buisson. Je crois alors qu'il priait ou qu'il pleurait, implorant les Mary-Jean de lui répondre.
Il y avait de nombreuses roses nommées Mary dans notre jardin, Mary-Jean, Mary Kay, Mary Mackillop, mais les Mary-Jean étaient les fleurs de ma mère, qui préférait les roses abricot à celles arborant la couleur de leur nom, trop limpides, trop évidentes, trop pareilles à ce qu'on attendait d'elles. Elle aimait que les fleurs la surprennent, que l'apparence des hommes ne soit pas identique à leur nom. Elle prétendait que mon père, par exemple, n'avait absolument pas la tête d'un Alex et que son charme venait de là, de ce que son nom et lui n'avaient rien en commun. Si tes parents t'avaient baptisé Paul ou Jean, j'aurais jamais accepté de t'épouser, disait-elle, puis elle éclatait de rire et plongeait son seau dans le baril d'eau de pluie recueillie pour arroser les Mary-Jean.
La plupart des choses que je savais de ma mère, Marie Beaupré, c'est mon père qui me les avait racontées. Le reste, je l'avais deviné, je l'avais inventé ou je l'avais appris dans les quelques lettres qu'elle avait écrites à mon père avant de le suivre jusqu'à l'autel dans des parfums d'abricot. Mais la mort l'avait emportée depuis si longtemps qu'elle ne possédait à mes yeux aucune forme de réalité. Elle n'était qu'une ombre dans le jardin, près de laquelle s'agenouillait mon père quand une voix d'autrefois l'appelait, une voix de l'au-delà. Inutile, alors, d'essayer de le rejoindre. Il parcourait le royaume des morts au bras de Marie Beaupré.
C'est là que se trouvait mon père sous le soleil de midi, prisonnier de son ombre et de celle de Marie Beaupré, pendant que l'agent Desmarais attendait patiemment que je passe aux aveux. Ne pouvant espérer aucune aide de mon père et n'ayant aucune confession à livrer, je me suis rabattue sur ce qui me semblait le plus logique. Peut-être qu'il a volé, ai-je murmuré en rougissant malgré moi, peut-être qu'il a trouvé une cape et qu'il a plané jusqu'au lac, comme Superman.
Devant le silence de l'agent Desmarais, je me suis instinctivement raidie et me suis enfoncée dans ma chaise, consciente que ma logique d'enfant venait de frapper un mur. Mon hypothèse ne plaisait pas à Desmarais et une colère qu'il avait de la difficulté à contenir s'est étalée sur son visage en partant de son cou, pareille à une marée rouge poussée par les pulsations de son cœur. Il a fermé les yeux, a soulevé sa casquette pour s'essuyer le crâne et, au passage, il a accidentellement renversé le verre d'eau tiède qu'il m'avait servi quelques minutes plus tôt. En entendant le bruit du verre contre le métal, mon père a brusquement réintégré le monde des vivants, l'agent McCullough a choisi ce moment pour revenir de la voiture en pestant contre la guêpe qui l'avait adopté et j'ai éclaté en sanglots. Je m'étais juré de ne pas pleurer, parce que dans les aventures de Superman, Lois Lane ne pleure jamais, et parce que si Superman avait disparu, ce ne pouvait être que pour accomplir une mission dont ni Victor Saint-Pierre, ni l'agent Desmarais, ni même mon père ne pouvait saisir l'importance. Mais il reviendrait, j'en avais la conviction, pour leur dire qu'ils se trompaient, tous, que Lois Lane ne mentait pas.
Pendant que je ravalais mes larmes et essuyais mon visage avec le mouchoir de McCullough, mon père a avisé Desmarais que les interrogatoires, c'était fini, que plus aucun flic ne m'approcherait, puis il m'a pris la main et on est partis à pied vers le village, sous le soleil de plomb.
Derrière nous, près du lac aux Barbotes, croissait la rumeur qui s'était élevée un peu plus tôt. Nous l'avons entendue gonfler sous les feuillages, s'enrouler aux arbres puis monter à leur cime. Après, le vent l'a emportée vers les maisons de Rivière-aux-Trembles pour la faire glisser par les fenêtres ouvertes jusqu'aux oreilles des hommes et des femmes qui tenaient férocement leurs enfants contre leur ventre. Le soir même, tout Rivière-aux-Trembles avait décrété que derrière la disparition de Michael Saint-Pierre, rôdait la vilaine petite Marnie Duchamp.
### BILL
On a cherché pendant des semaines, on a harcelé la police, on a même engagé un détective privé, qu'on a foutu à la porte pour en embaucher un autre, mais la piste de Billie se perdait à la sortie de son école, à croire qu'elle s'était volatilisée dans les dernières lueurs de ce jour de janvier, sans rien laisser derrière elle, pas même une petite mitaine sale ou une de ces barrettes en forme de papillon qu'elle perdait sans cesse.
Toutes les fois que le téléphone sonnait ou qu'un cri d'enfant résonnait dans la rue, Lucy-Ann se ruait vers l'appareil ou vers la porte. Moi, je restais dans mon coin, le cœur battant, espérant voir Lucy-Ann défaillir, ses genoux fléchir et ses mains s'accrocher au comptoir de la cuisine ou au chambranle de la porte. Je me disais que si L.A. tombait dans les pommes, ça serait le signe indiscutable que Billie était de retour, que notre Billie s'élançait vers la maison les bras tendus ou qu'elle sanglotait maman au bout du fil, maman, c'est moi, Billie... Mais Lucy-Ann ne s'évanouissait jamais. La lueur d'espoir qui avait fouetté son regard quelques instants plus tôt s'effaçait d'un coup et son visage reprenait ses allures de masque mortuaire, chaque fois plus dur, plus impénétrable. La peur et la douleur l'enlaidissaient et je la haïssais presque de ne pas faire l'effort de se coiffer et de porter jour après jour les mêmes vêtements sales et froissés. J'essayais de lui dire que ce n'était pas en se laissant dépérir qu'on allait ramener la puce, mais elle m'envoyait promener comme si je ne valais pas plus que les culottes souillées traînant à côté du lit.
Au départ, on avait eu l'aide de quelques amis, on avait été entourés de chaleur, d'affection, de sourires qui disaient je comprends sans comprendre mais qui étaient là, entièrement présents. Des mains chaudes s'étaient posées sur nos épaules, nous avaient tendu des verres d'alcool remplis à ras bord, tiens, bois ça avant de te momifier, nous avaient cuisiné des plats qui sentent les maisons vivantes, puis, peu à peu, les mains s'étaient refroidies et la maison s'était vidée. Les bonnes âmes avaient battu en retraite et je n'arrivais pas à leur en vouloir d'avoir cherché à se protéger de notre malheur.
Seules Régine, la mère de Lucy-Ann, et Patricia, sa meilleure amie d'enfance, avaient persisté en dépit de l'attitude pour le moins hostile de L.A. Elles faisaient régulièrement irruption dans notre maison, les bras chargés de nourriture qui finirait à la poubelle, de livres et de magazines qui ne seraient jamais ouverts, et elles essayaient de mettre un peu d'ordre dans le bordel qui se formait autour de nous. Une fois sur deux, Lucy-Ann les jetait dehors en hurlant qu'elle n'avait pas besoin d'une femme de ménage, puis elle s'effondrait dès la porte refermée, et c'est moi qui devais la ramasser. J'aurais eu besoin qu'on me ramasse, moi aussi, que Lucy-Ann me prenne parfois dans ses bras en me chuchotant des choses douces à l'oreille, pleure pas, Bill, elle va revenir la chouchounette, je le sens, mais Lucy-Ann n'avait de force que pour sa propre douleur et ne s'imaginait pas que je puisse avoir mal à vouloir m'en arracher les cheveux et la peau du crâne avec. À ses yeux, j'aurais pu crever que ça n'aurait pas fait de différence, parce qu'elle s'était mis en tête que, peu importe ce qui était arrivé à Billie, c'était moi le responsable.
Elle prétendait que j'aurais dû être là, à la sortie de l'école, pour cueillir Billie et l'amener à son cours de danse, que j'aurais dû l'attendre sur place parce que c'est ce que font les pères normaux, ceux qui aiment leur fille et n'ont rien d'autre à foutre de leurs journées que d'imaginer comment ils pourraient faire voler une hostie de grenouille. Crapaud, la corrigeais-je en évitant le crayon, la tasse, le livre ou l'éléphant de porcelaine qui volaient alors dans ma direction. Je protestais pour la forme, pour ne pas sombrer dans la mare d'eau croupie qui s'élargissait autour de moi et ne pas me détester davantage, mais je n'en menais pas large, car je m'adressais les mêmes reproches, et avec plus de véhémence que Lucy-Ann, si la chose était possible. J'aurais dû être là, je le savais, beau temps mauvais temps, les pieds dans la slush ou dans la boue, avec mon parapluie me dégoulinant dans le dos ou mon t-shirt mouillé de sueur, pour guetter l'apparition du petit sourire en coin qui faisait chavirer mon cœur de père. Mais après des heures de discussion, on avait décidé du contraire, L.A. et moi. Puisque le studio de la Lenoir était situé immédiatement à côté de l'école, on s'était rangés aux arguments de Billie, qui rechignait chaque fois qu'on venait lui prendre la main pour la conduire à la porte d'à côté. Chus capable de marcher, vous avez pas besoin de me tenir. On avait accepté que Billie s'y rende seule, désormais. C'était une façon de lui prouver qu'on ne la considérait plus comme un bébé et qu'on lui accordait notre confiance. On avait pris cette décision stupide ensemble, mais L.A. semblait l'avoir oublié. Probablement se sentait-elle aussi coupable que moi, alors elle déformait la réalité, elle en gommait des bouts pour se protéger, pour écraser la voix qui massacrait sa conscience.
J'étais donc le responsable, celui qui aurait dû tenir la main de Billie. Elle m'a envoyé ça en plein nez à notre retour du poste de police, la première nuit, et ses accusations se sont aggravées dans les jours qui ont suivi, après que les flics nous eurent soumis tous deux à des interrogatoires musclés. En l'absence d'une véritable piste, on soupçonne d'abord les parents, c'est classique, puis on braque le projecteur sur le père, sur celui qui est le plus susceptible de violer ou de blesser à mort. Au départ, j'ai eu envie d'étrangler ces types qui croyaient que j'avais pu m'attaquer à ma propre fille, je suis passé à deux doigts de leur sauter à la gorge et de leur mettre mon pied au cul, puis je me suis calmé. Ces gars accomplissaient leur boulot et leur seul objectif était aussi le mien, retrouver Billie, même s'il fallait pour cela passer la ville entière au bulldozer.
L'affaire avait été confiée à une équipe dirigée par deux enquêteurs, Michel Dubois et Gilles Ménard, qui n'en étaient pas à leur première disparition d'enfant. Ça s'entendait quand ils vous ordonnaient gentiment mais fermement de ne répondre qu'à leurs fichues questions. C'est clair, monsieur Richard ? Les questions, c'est nous qui les posons. Ça se voyait aussi quand leurs gros doigts déplaçaient délicatement les photos de l'enfant rassemblées sur la table de métal autour de laquelle avait lieu l'interrogatoire. Ces hommes en avaient vu d'autres et leur seul souci était le sort de l'enfant. Votre souffrance, à la limite, ils s'en moquaient.
Ils m'ont bombardé de questions durant deux bonnes heures le lendemain de la disparition de Billie, revenant sans cesse sur les mêmes détails et surveillant le moment où je me contredirais, le mot qui allait me trahir et leur permettre de m'envoyer au tapis. Par vingt fois, ils m'ont demandé où je me trouvais lorsque Billie était sortie de l'école, et par vingt fois, je leur ai répondu chez Jack et Jack, un café minable que je fréquentais depuis des années parce que je m'y sentais bien et que j'y avais ma table attitrée, dans un coin, derrière une colonne, où je pouvais réfléchir tranquille sans qu'un duo de flics s'acharne sur moi au lieu de chercher le salaud qui s'en était pris à mon enfant. Ils pouvaient vérifier auprès de la serveuse, elle confirmerait mes dires. Il y avait aussi deux ou trois habitués qui m'avaient vu dans ce café ce jour-là, ils n'avaient qu'à s'arranger pour les retracer. Soyez tranquille, monsieur Richard, c'est ce que nous comptons faire, a grogné Ménard en me signifiant que notre entretien était terminé. Avant que je quitte la salle d'interrogatoire, il a noté sur un bout de papier les trois numéros où je pouvais les joindre, Dubois et lui, si un élément auquel je n'avais pas pensé me revenait.
Dès le lendemain, cependant, Ménard a débarqué chez moi, accompagné du silencieux et stoïque Dubois, pour m'annoncer qu'il avait vérifié mon alibi et que personne ne se souvenait de m'avoir vu chez Jack et Jack le 20 janvier, aux environs de quinze heures. Conséquemment, lui et son collègue avaient d'autres questions à me poser. Malgré mes protestations, ils m'ont ramené au poste, m'ont de nouveau interrogé et m'ont laissé poireauter pendant trois heures dans une salle minable qui sentait le désinfectant ou quelque chose se rapprochant de la gomme Clorets.
À la fin de l'après-midi, ils sont revenus à la charge, mais je n'ai pas craqué car je n'avais aucune raison de craquer. J'étais au café des deux Jack et si personne ne se rappelait m'y avoir vu, c'est que j'étais dans mon coin, bien tranquille, que je faisais en quelque sorte partie du décor, ou parce que quelqu'un mentait ou confondait les dates. On n'avait qu'à y retourner ensemble. En me voyant en chair et en os, la serveuse me replacerait. Ménard avait eu une autre idée. Il avait convoqué la fille pendant que je comptais les trous dans les tuiles du plafond. Elle était dans la pièce d'à côté, derrière le faux miroir, et elle avait reconnu mon foulard rouge, semblable à celui que sa grand-mère lui avait tricoté deux ans plus tôt. Du foulard, elle était remonté jusqu'à l'anonymat de ma gueule et avait fini par certifier que j'étais bel et bien chez Jack et Jack le 20 janvier, à la table du fond, la 12 ou la 17, elle mélangeait les chiffres.
Pour Ménard et Dubois, le témoignage de cette fille ne valait pas grand-chose, puisqu'elle m'avait replacé à l'aide d'un foulard qui ne ressemblait en rien aux crémones que nous tricotaient nos grands-mères. Dubois avait insisté là-dessus, les grands-mères tricotaient des motifs, des mailles à l'endroit, des mailles à l'envers, pas des trucs qu'on peut acheter dans n'importe quelle boutique. Preuve que la chance n'aimait pas mon style, en plus de mélanger ses chiffres, la fille ne connaissait aucun des habitués du café. Elle remplaçait la serveuse habituelle, ce jour-là, une copine qui devait préparer un examen, et elle ne se souvenait même plus de l'allure du gars qui l'avait engueulée parce qu'elle lui avait servi un mocha à la place d'un cappuccino. Ce témoignage n'étant pas fiable, ils m'ont proposé le polygraphe, persuadés qu'ils allaient me coincer en examinant l'activité de mon cerveau. Manque de pot, mes ondes cérébrales étaient d'accord avec moi. Je ne mentais pas. C'est ce détecteur de mensonge qui a représenté mon billet de sortie, sinon je serais encore là, dans la salle empestant le désinfectant, à essayer de convaincre deux hommes qui en avaient vu d'autres que je leur disais la stricte vérité.
Quand je suis rentré chez moi, L.A. descendait l'escalier menant à l'étage en tenant de travers une tasse de café qui menaçait de se répandre sur le tapis à soixante piastres le mètre. Elle s'est arrêtée net au milieu de l'escalier pour me regarder comme si j'étais le pire des monstres que la terre ait jamais enfanté. Qu'est-ce que t'as fait à Billie ? a-t-elle murmuré en s'accrochant à la rampe. Qu'est-ce que t'as fait à notre fille, crisse de malade ? Puis ses genoux ont ployé sous elle, sa tête a basculé, la tasse de café a dévalé les marches dans un bruit étouffé par le tapis et L.A. s'est évanouie, mais aucune Billie n'est apparue derrière elle dans son pyjama décoré d'oursons. Personne n'est apparu derrière L.A., même pas un petit fantôme qui m'aurait envoyé un baiser du bout des doigts pour me dire de ne plus m'inquiéter.
J'ai ramassé L.A. pour la porter sur le divan du salon, j'ai ramassé ses dégâts, puis j'ai essuyé ses pleurs alimentés au café cognac en même temps que les reproches acrimonieux qui ont fusé dès qu'elle a rouvert les yeux. L.A. savait aussi bien que moi que je n'avais pas touché à Billie, mais elle avait besoin d'un coupable, de quelqu'un sur qui taper, et elle avait laissé le doute de l'inspecteur Ménard s'insinuer en elle.
Son attitude était carrément grotesque. Je n'arrivais pas à croire que ma propre femme, la mère de ma fille, qui ne m'avait jamais vu lever la main sur Billie et pouvait savoir si je mentais rien qu'au timbre de ma voix dans le combiné d'un téléphone situé à trois mille kilomètres de distance, puisse avoir des doutes à mon sujet quand on parlait de l'enfant pour qui je me serais tranché la gorge et les couilles avec s'il l'avait fallu. Calvaire, L.A., réveille-toué, ai-je hurlé au moment où elle me servait un autre de ses arguments débiles, puis j'ai piqué une colère noire, une colère qui aveugle et vire au rouge lorsque le mal de tête vous prend. J'ai même renversé une chaise, que j'ai ensuite lancée au bout de mes bras, confirmant du même coup la violence dont j'étais capable aux yeux de L.A.
Pendant des jours, j'ai plaidé mon innocence sur tous les tons, voyons, L.A., bordel, L.A., hostie, L.A., tu peux tout de même pas croire que j'ai fait du mal à ma puce ? Peine perdue, mon plaidoyer ne valait pas de la schnoute. Dans l'esprit de Lucy-Ann Morency aussi bien que dans celui de l'inspecteur Gilles Ménard, Bill Richard demeurait jusqu'à nouvel ordre le seul et impossible suspect dans la disparition de sa fille, Billie Richard, huit ans et neuf mois.
Durant les semaines qui ont suivi, je me suis servi des dizaines de fois des numéros de téléphone que Ménard avait griffonnés pour moi sur un bout de papier, au point que la voix bien lisse de Dubois au bout du fil a fini par monter d'un cran et qu'il s'est laissé aller à quelques démonstrations d'impatience, mais je me moquais de ses humeurs de flic, de ses brûlures d'estomac, de ses problèmes de prostate ou autres, tout ce que je voulais, c'est que lui et son équipe d'enculés accomplissent un miracle.
Quant à Ménard, il se pointait à la maison à l'improviste à tout bout de champ, espérant peut-être me surprendre en train de cacher la petite dans une valise ou d'escamoter des preuves compromettantes. Il arrivait avec son air à la Colombo, ne prenait pas la peine d'enlever son manteau et s'installait à la table de la cuisine sans y avoir été invité. Il dessinait ensuite des schémas ou construisait avec les objets s'accumulant sur la table des ensembles n'ayant de sens que dans son esprit. Il plaçait un couteau sur une assiette, déposait une fourchette sur le couteau, perpendiculairement, et lui imprimait un mouvement de balancement au-dessus des restes de pizza séchés. Puis, de but en blanc, il se raclait la gorge et demandait si Billie aimait la pizza et les hamburgers et si on l'emmenait parfois au McDo. Sa fille à lui adorait le McDo, ajoutait-il, et ça lui faisait plaisir d'y aller avec elle, de la regarder commander son wrap au fromage toute seule, comme une grande, même s'il détestait ces décors de plastique éclairés par une lumière trop crue ne lui donnant aucune chance de cacher à sa fille que son père était vieux.
Les enfants ont un instinct de survie plus fort que le nôtre, les laisser manger ce qu'ils aiment de temps en temps peut pas leur faire de mal, murmurait-il en ajoutant une deuxième fourchette à sa bascule improvisée, mais dans un autre angle, pour former une étoile, et je revoyais Billie partir à ses cours de ballet les épaules courbées, avec son goûter santé dans sa petite boîte à lunch rose. J'aurais tout donné, dans ces moments, pour pouvoir remonter le temps et allumer un énorme feu de camp dans lequel Billie et moi on aurait jeté ses collants roses, son tutu rose et ses putains de chaussons de ballet tout en se faisant griller un plein sac de guimauves jumbo.
S'il constatait que L.A. ou moi on n'en pouvait plus, Ménard sortait son arsenal de questions. Où ? Quand ? Comment ? Pourquoi ? Plus fort, j'entends pas ! Puis il repartait comme il était venu, sans nous laisser le moindre espoir. Parfois, il s'arrêtait devant les dessins de Billie affichés sur le réfrigérateur et demandait si la maison verte apparaissant sur deux de ces griffonnages d'enfant correspondait à la maison d'un parent ou d'un ami, ou si le personnage de géant barbouillé sur un autre dessin évoquait quelqu'un de notre connaissance, un type dont nous aurait parlé Billie, par exemple.
Je lui répondais que la maison verte représentait le chalet du père de L.A., en Mauricie, et que le géant était un personnage hybride imaginé par Billie à partir du Géant égoïste, d'Oscar Wilde, et d'un géant apparaissant dans un de mes contes, le géant bleu, qui vivait en partie dans l'océan et en partie sur la terre ferme. Un gentil géant, mi-homme mi-poisson. Ménard fronçait les sourcils, suivait les contours du géant avec son index maculé d'encre, puis marmonnait que ce géant était peut-être un homme qui effrayait Billie, un étranger dont elle aurait tenté de nous parler. Prenez le temps d'y réfléchir.
Après le départ de Ménard, L.A. pleurait pendant quelques minutes, puis elle m'engueulait en me demandant où j'avais été pêcher mon crisse de géant, pour ensuite courir s'enfermer dans notre chambre, qui était maintenant la sienne et où je ne pénétrais que pour chercher des vêtements propres et ramasser les culottes sales qu'elle avait lancées dans un coin. Elle n'entrait jamais dans la chambre de Billie, elle ne pouvait pas, et elle faisait maintenant tout pour éviter Pixie, qui lui rappelait trop la puce. Alors Pixie était devenu mon ami, mon confident à moi, le seul être auprès de qui je pouvais pleurer tranquille.
La dernière fois que Ménard est venu à la maison, j'étais assis avec Pixie dans la véranda et je lui racontais une de ces histoires qui faisaient rire Billie, aussi pitoyable qu'un vieillard parlant au cadavre de sa défunte. Selon son habitude, Ménard est allé directement à la table de la cuisine et a attendu que je l'y rejoigne. Il a ensuite pris une enveloppe déchirée et l'a pliée en deux pour se confectionner une petite tente sous laquelle il a glissé un crayon de cire orange, un des crayons de Billie que ni L.A. ni moi n'avions eu le courage de ranger dans sa boîte, puis il a dit qu'il ne reviendrait plus, à moins qu'il y ait de nouveaux éléments dans l'enquête. Billie était maintenant disparue depuis trois mois et les recherches tournaient en rond. Il ne bouclait pas le dossier mais, n'ayant aucune nouvelle piste à explorer, il devait s'en remettre au temps. Je suis désolé, a-t-il murmuré, puis il s'est dirigé vers la porte.
J'ai bondi derrière lui en criant qu'il ne pouvait pas baisser les bras, qu'il n'avait pas le droit d'abandonner Billie, que c'était son boulot de s'occuper d'elle, qu'il était payé pour ça et que j'irais jusqu'en haut lieu si l'affaire était mise sur une tablette. Il a doucement levé un bras pour m'arrêter et m'a renvoyé une de ses remarques tordues : Alors dites-moi où vous étiez réellement, monsieur Richard, le 20 janvier dernier ?
### MARNIE
Je me souviens d'un conte dans lequel le personnage principal, Lucas, il me semble, tente désespérément d'effacer une tache de vin couvrant la moitié gauche de son visage d'un nuage violacé ayant la forme d'une gueule de loup ouverte sur son œil droit. C'est la marque du diable, avait murmuré la mère de Lucas sur son berceau, conférant ainsi à son fils des pouvoirs sataniques dont on l'accuserait injustement jusqu'à ce qu'il se précipite dans un puits, y entraînant avec lui la bête s'agrippant à son visage depuis sa naissance.
C'est ce qui m'est arrivé à moi aussi. La disparition de Michael a imprimé sur mon visage la marque du diable et je suis dès lors devenue une paria, une intouchable, une enfant dont il ne fallait pas s'approcher. Les conclusions de l'enquête n'avaient mené nulle part. Les hypothèses étaient trop nombreuses, trop invraisemblables. Le corps de Mike n'était pas dans le lac, les plongeurs étaient formels. Il n'était pas dans la rivière non plus. Compte tenu de la faiblesse du courant, s'il s'y était noyé, on l'aurait forcément retrouvé, quelque part sur un fond vaseux ou près des bosquets bordant le rivage. Si les chiens avaient raison, et comment pouvait-on douter des chiens, Mike avait marché dans l'eau, ou on l'y avait fait marcher pour l'emmener ensuite on ne savait pourquoi jusqu'au lac aux Barbotes. Ce qui n'avait pas de sens.
L'hypothèse retenue par la majorité des gens soutenait que les chiens avaient simplement suivi une ancienne piste de Mike. Le garçon venait souvent jouer à cet endroit, son odeur était partout. Il n'avait pas marché dans l'eau. Il avait parcouru les deux kilomètres reniflés par les chiens un autre jour, une autre fois, puis avait rebroussé chemin. Moi je disais que non, qu'il n'y avait pas de vrai sentier à cet endroit, que Mike n'allait jamais si loin, mais on ne m'écoutait qu'à demi, se demandant pourquoi je refusais d'admettre que l'orage avait eu lieu près du lac aux Barbotes, où quelqu'un avait enlevé Mike, si bien que son odeur se perdait au milieu de nulle part. On retournait la question dans tous les sens, on fronçait les sourcils, puis on me fixait en soupirant : la petite ment, Marnie cache quelque chose. Mon père croyait que la rumeur finirait par se taire, que les gens recouvreraient leurs esprits et comprendraient que je n'étais que l'innocente victime d'un mystère que le temps éclaircirait, mais c'était ignorer que la marque du diable est indélébile et qu'un mystère, par définition, est voué à laisser ses survivants dans l'ignorance.
On a tenu un peu moins d'un an à Rivière-aux-Trembles, papa et moi, en faisant l'impossible pour ignorer les chuchotements qui s'effilochaient en longues traînées baveuses sur notre passage, les regards pleins de sous-entendus, les conversations qui s'interrompaient brusquement quand mon père poussait la porte du bureau de poste ou de la quincaillerie de Jos Bouchard, qui s'esquivait chaque fois dans l'atelier attenant au magasin, comme s'il me croyait également responsable de la mort de Martin, son fils, emporté par le cri de la rivière aux Bleuets. Des flopées de mots méchants se faufilaient derrière moi, crachés ou à peine audibles, mais celui qui m'effrayait le plus était le mot « sorcière », que seuls les enfants prononçaient, puisé dans ces contes où les princesses se piquent aux épines de fleurs empoisonnées, Betsy Rose ou Black Velvet ayant l'exacte couleur du sang qu'elles font couler.
Les sorcières, on brûle ça dans le feu, en pleine nuite, m'avait crié Germain Leblanc en sortant de l'école. Depuis ce temps, je m'éveillais en sueur au milieu de la nuit, persuadée qu'au dehors, des hommes au visage noirci de suie préparaient un bûcher à mon intention. J'entendais crépiter l'autel de branches qu'on avait élevé sous de longs piquets appuyés l'un sur l'autre, puis j'imaginais l'odeur de ma chair rôtie se mêler à celle des églantiers sous la rosée. Je ne disais toutefois rien de ces cauchemars à mon père. J'attendais que le matin arrive en guettant derrière les rideaux les lueurs vacillantes où l'on me jetterait vive. Puis, par une nuit de la fin mai, un orage a éclaté, illuminant ma chambre de la couleur des bûchers et me ramenant au bord du bassin magique, près duquel le grondement du tonnerre se mêlait au cri de Martin Bouchard. En arrière-plan, très loin derrière la pluie, Michael se balançait mécaniquement, son visage blanchi s'effaçant peu à peu, trait par trait, puis formant dans le brouillard un disque lisse où se répercutait la foudre.
J'ai dû crier en voyant ce disque blanc, car mon père est entré en trombe dans ma chambre quelques instants plus tard pour me prendre dans ses bras. À travers mes pleurs et mes tremblements, je lui ai alors parlé des sorcières, de Lucas, du diable et des roulements du tonnerre, des orages qui, depuis la disparition de Michael, grondaient de fureur parce que j'avais échappé aux volontés du ciel. Les éclairs ont pris Mike, pop, c'est ma faute, les nuages vont venir m'attraper. Je m'étais dérobée à la foudre et, si j'évitais le bûcher, ce ne serait que pour être happée dans une déchirure du ciel.
Trois semaines plus tard, un camion sur lequel était inscrit « Bob Déménagements » en énormes lettres noires s'est arrêté devant la maison. Si mon père ne pouvait me protéger de l'orage, il pouvait au moins m'éloigner des chuchotements. Les Mary-Jean et les Sparkling Scarlet allaient devoir fleurir, pour les six années à venir, sans les soins d'Alex Duchamp.
Je n'ai pas pleuré quand les lourdes portes du camion se sont refermées dans un claquement définitif sur les dizaines de boîtes empilées devant nos meubles, ni quand mon père a verrouillé la clôture entourant le jardin, ni lorsque notre vieille Oldsmobile a dépassé le panneau clamant « Rivière-aux-Trembles vous salue », car ce village ne représentait désormais pour moi que le souvenir d'une enfance qui s'était achevée au bord d'un bassin maléfique. Les milliers d'images que j'en conservais n'étaient plus que des moments de vie figés sur du papier glacé, des milliers d'instants de bonheur enfermés dans un album dont la dernière page portait la date du 7 août 79. Après cette date, le papier se craquelait, les couleurs désertaient les images et mes souvenirs n'étaient plus qu'une série de clichés en noir et blanc semblables aux illustrations d'un livre d'école ne contenant que des schémas ennuyeux. La lumière avait quitté Rivière-aux-Trembles en même temps que le soleil avait chuté dans le ciel du nord avec Michael et ce lieu s'était vidé de tout ce qui permet au temps de suivre son cours. Les arbres n'y poussaient plus, les enfants n'y couraient plus dans les sentiers bordés d'aulnes, de fougères et d'herbe à poux, dans les champs semés de seigle et de bouses de vaches. Rivière-aux-Trembles s'était enfermée dans un passé ayant épuisé la richesse d'une terre maintenant stérile.
Nous avons roulé pendant plus de quatre heures, mon père et moi, sans presque prononcer un mot, puis nous avons traversé un pont immense pour pénétrer dans le vrombissement continu qui servirait de refuge à notre mémoire pour quelques années.
Au bout de ces quelques années, mon père étant devenu aussi gris que le béton qui nous entourait, j'ai fouillé dans ses tiroirs pour trouver les clés de notre maison, que j'ai balancées sur la table de la cuisine en lui ordonnant de retourner là-bas avant de se fossiliser dans le carré d'asphalte sur lequel notre immeuble étendait son ombre. J'avais dix-huit ans et j'avais envie de lever les voiles, moi aussi, de partir aussi loin que possible de cet appartement où on n'avait pas réussi, malgré tous nos efforts, à oublier Rivière-aux-Trembles.
Pendant que mon père roulerait vers le nord pour s'occuper d'un jardin en friche, je m'orienterais vers le sud pour suivre la route de Canucks jusqu'à la frontière des États-Unis. De là, je descendrais le long de la ligne des Appalaches, du Maine jusqu'en Alabama, au cas où je croiserais quelque part un vaste champ de blé mûr où j'aurais envie de planter mes racines. Mon périple a duré un an, une longue année durant laquelle j'ai tenté de m'illusionner quant aux vertus libératrices de la misère et du dépaysement, puis je suis remontée vers New York, voir si la Big Apple était aussi pourrie que le Big Apricot de Jerry Siegel et Joe Shuster, les pères de Superman, et si je ne distinguerais pas, dans les couloirs rectilignes formés par les édifices, la silhouette fuyante de Michael Superman Saint-Pierre.
Il m'est arrivé de l'entrevoir, qui descendait d'un autobus avec son t-shirt rouge ou jouait de l'harmonica au coin d'une rue en comptant les pièces de monnaie qui rebondissaient dans sa casquette sale. Je l'ai aussi aperçu au Metropolitan Opera, dans des cafés de Greenwich Village ou dans des culs-de-sac poussiéreux, mais chaque fois que je m'approchais, ses yeux changeaient de forme ou de couleur et son visage s'effaçait derrière le regard hostile d'un ado dégingandé qui me lançait fuck you man avec un accent du Queens.
Avec le temps, son t-shirt a rétréci et j'ai cessé de le voir. Je suis néanmoins restée à New York pendant vingt-trois ans, à vendre des glaïeuls, des roses et des marguerites aux clients chics de Tribeca, jusqu'à ce que la mort de mon père me ramène à Rivière-aux-Trembles pour m'apprendre que ma fuite était un leurre, que ce n'était pas en contemplant les rives polluées de l'Hudson que je verrais surgir des eaux le cadavre de Mike. C'était ici, sur les lieux du drame, que devait se résoudre l'histoire de Marnie Duchamp et de Michael Saint-Pierre.
### BILL
Ça fait mal de penser qu'on est seul, que même la femme à qui on a si souvent dit je t'aime ne nous considère plus que comme un tas de merde. Il est vrai que notre couple battait de l'aile depuis un certain temps, que Lucy-Ann et moi, on ne renvoyait pas exactement l'image de l'union idéale, de la paire de tourtereaux en perpétuel voyage de noces. Les soupers aux chandelles, les mains qui glissent furtivement sous le chandail, les sourires entendus d'un coin à l'autre de la pièce où une demi-douzaine de génies politiques débitent leurs conneries avec un sérieux de pape, tout ça, pour nous, appartenait au passé. On continuait à baiser, parce qu'on avait la baise dans le sang, parce que cet affrontement charnel nous permettait de posséder l'autre tout en lui reprochant d'exister, avec juste assez de violence pour nous sentir soulagés de la tendresse perdue, mais je ne me rappelle plus la dernière fois que Lucy-Ann m'a dit quelque chose de gentil, de vraiment gentil, du genre t'es pas mal beau à matin, Bill, tu sais, ou du style ta petite couette de gamin est encore retroussée, avec un sourire amusé, avec un doigt effleurant doucement la mèche rebelle.
Si on était encore ensemble, c'était pour Billie, pour qu'elle n'ait pas à remplir sa valise Mickey Mouse ou son sac à dos Snoopy tous les vendredis soir en se demandant quel ourson elle allait fourrer dedans et quel autre elle allait devoir abandonner dans la tristesse de sa chambre vide jusqu'au lundi. L.A. avait connu cette double vie obligeant les enfants à se fendre en quatre pour satisfaire ce qu'elle appelait l'égoïsme de leurs parents et elle ne voulait pas de ça pour Billie. Moi non plus, je ne voulais pas de ce type de déchirement pour ma chouchoune. Je considérais toutefois qu'il valait mieux, dans certains cas, que les petits trimbalent derrière eux leur sac à dos plutôt que d'avoir à entendre les couteaux voler tout en subissant l'insipide et désolant spectacle de la haine conjugale. Dans tous les cas, cependant, les petits devaient payer une partie de la note, de plus en plus élevée à mesure que la passion matrimoniale s'effilochait. Avec eux, finies les folies. Les nuits d'amour entrecoupées de biberons cédaient invariablement la place aux biberons et ils devaient se montrer particulièrement géniaux pour compenser les virées dans les bars, le cinéma de fin de soirée et les allers-retours à New York ou à Los Angeles que se permettaient leurs parents sur un coup de tête avant leur naissance. Ils devaient équilibrer le déficit et la perte d'insouciance à coup d'éclats de rire, de mots de deux syllabes et de petits pas tout croches suscitant les cris d'enthousiasme et les applaudissements de leurs vieux cernés jusqu'au menton.
C'est peu après l'arrivée de Billie qu'on a commencé à s'éloigner, L.A. et moi, pour reporter le plus gros de notre potentiel amoureux sur cette petite face joufflue tachée de compote de pêches. Au fil des mois, on s'est rendu compte qu'on ne s'aimait peut-être pas assez pour que l'équation deux plus un décuple la tendresse qu'on éprouvait l'un pour l'autre. L'éloignement s'est produit doucement, sans véritable déchirure, sans marques de bagarres, de sang sur les murs blancs, si bien qu'on ne comprenait toujours pas pourquoi l'autre nous semblait si terne, pourquoi sa voix nous agaçait soudain, pourquoi le lit était devenu trop étroit. Il a fallu des mois pour que la vérité nous saute en plein visage : notre lune de miel s'était déroulée dans un décor préfabriqué, et l'illusion ne s'était maintenue que parce qu'elle s'entourait des artifices d'un désir fondé sur l'image du bonheur frelaté auquel devaient aspirer tous ceux qui ne voulaient pas être rejetés par le courant. En nous révélant le mensonge sur lequel reposait notre union, Billie en avait tout de même colmaté les brèches, et c'est grâce à cette enfant qu'on n'en était pas encore à s'envoyer des lettres d'avocats ou des messages fielleux par amis communs interposés.
On avait établi une espèce de statu quo qui nous arrangeait tous les deux, tu fais tes affaires, je m'occupe des miennes et tout va bien. De cette façon, on arrivait à se supporter et à se regarder en face, on parvenait même à rire et à s'amuser avec la puce, comme une vraie famille, et il aurait suffi de pas grand-chose pour qu'on recommence à s'aimer et à se tenir par la taille, pour qu'on fasse de nouveau équipe devant l'imbécillité du monde. Il aurait suffi que L.A. dilue son vin et moi mon scotch, ma boisson d'homme, mais maintenant que Billie n'était plus là, c'était la débandade. Au lieu de se tendre la main, on s'éloignait à une vitesse supersonique. On s'envoyait promener pour un oui pour un non, on pétait nos vieux boutons pleins de pus, on laissait sortir le méchant, mais on ne s'en sentait pas pour autant soulagés, bien au contraire. Chaque fois que je passais devant un miroir après une dispute et que je voyais des restes de venin me sécher sur le bord des babines, j'avais envie de vomir, et je suis certain que L.A. ressentait le même écœurement.
Lorsqu'il a été clair, dans l'esprit de L.A., que j'étais le seul et unique responsable de la disparition de Billie, j'ai vidé la place avant que le doute qui m'attaquait parfois en traître se transforme en névrose. Je refusais de devenir pareil à ces innocents poussés à bout, à ces hommes fragiles qui finissent par se mettre la corde au cou pour qu'on les laisse tranquilles et qu'on cesse de les torturer avec des questions dont ils ne connaissent pas les réponses. Parce que le doute, bien sûr, entrait parfois par la porte de derrière, au moment où je m'y attendais le moins, pour me sauter dans le dos, me prendre la tête à deux mains et la secouer jusqu'à ce que je hurle que c'était pas moi, que ça pouvait pas être moi qui avais fait disparaître Billie dans la brume de janvier.
Depuis que Ménard avait entrepris de me bombarder de questions à ce point absurdes qu'il m'arrivait de ne plus me souvenir du nom de ma propre mère, j'avais passé des nuits entières à m'interroger sur ma santé mentale et à me demander s'il ne détenait pas des informations qui m'auraient causé un infarctus s'il me les avait révélées. J'avais tellement peur que ce crétin ait raison quand il m'accusait à demi-mot de cacher moi-même certains renseignements qui nous auraient permis de nous rapprocher de Billie que je devenais dingue. Ma mémoire s'entourait d'une espèce de voile flou derrière lequel je confondais les jours et les années, le manteau rose de la puce avec son imperméable à pois. La méthode de harcèlement de Ménard était si efficace qu'elle vous enlevait toute certitude et déformait du même coup ce qui était la pure vérité. Ce flic était dangereux. J'ai même failli lui proposer un jour de m'injecter une dose de penthotal et qu'on en finisse, puis j'ai recouvré mes esprits. Ménard déraillait et allait me faire quitter les rails avec lui si je ne me ressaisissais pas. Je n'avais rien fait à la puce et je n'étais pas cinglé.
Le détecteur de mensonge avait confirmé mes dires, la petite serveuse avait reconnu mon foulard, un foulard rouge, oui, c'est ça, rouge sang, Billie, de la couleur des roses dont les épines empoisonnées blessent la Belle au bois dormant. J'aimais ma fille et je corrigeais des copies nulles chez Jack et Jack, à la table du fond, en sirotant un cappuccino dont la mousse me collait à la lèvre supérieure quand le monstre qui avait enlevé ma fille avait fait démarrer sa voiture pour se diriger vers des routes ennuitées.
Bien malgré moi, j'étais en effet arrivé à la conclusion que Billie avait été enlevée, qu'elle n'avait pas pu se désintégrer ni être recueillie à bord de quelque vaisseau spatial à la recherche des spécimens les plus réussis de l'espèce humaine. L'enlèvement demeurait l'hypothèse la plus plausible, même si elle était la plus cruellement inadmissible. Je retournais donc l'hypothèse dans ma tête et la première image qui me venait chaque fois à l'esprit était celle d'un homme, d'un homme seul et sans visage, gigantesque aux côtés de Billie, qui se penchait pour l'enfermer dans l'ombre de son manteau sombre et y retenir ses cris. J'imaginais un pardessus, un anorak ou un large imperméable déroulant ses plis jusqu'aux genoux, puis une voiture sombre conduite par l'homme sombre et filant dans la nuit. La morsure d'un brouillard glacé s'écrasait sur le pare-brise, s'y accumulait jusqu'à masquer l'horizon, et je respirais l'odeur de la sueur de l'homme pendant que ses mains larges et rugueuses actionnaient les essuie-glaces. C'était là qu'on avait porté Billie, dans l'habitacle de cette voiture dont le silence était traversé d'un souffle rauque se mêlant aux battements réguliers des essuie-glaces.
Quant à moi, j'étais dans ce café de malheur, et si personne ne se rappelait m'y avoir vu, c'est que l'endroit n'était fréquenté que par des épaves qui commençaient à noyer leur désillusion dans le vin bon marché dès dix heures du matin. Je n'avais rien oublié. J'étais aussi sain d'esprit qu'un homme désespéré peut l'être et j'ai quitté L.A. pour le demeurer. Si je restais ne fut-ce qu'un mois de plus dans sa zone d'influence, je finirais par céder devant ses accusations, pour ensuite entériner les propos d'un Ménard dont l'acharnement tournait à l'obsession. Ce flic qui se prenait pour le bon Dieu m'aurait à l'usure et me ferait avouer un crime que je n'avais pas commis, que jamais au grand jamais je n'aurais pu commettre.
Je me suis loué un petit appartement pas trop loin de chez nous, pas trop loin de l'école de Billie, pas trop loin des lieux où elle pouvait réapparaître, et j'ai emmené Pixie avec moi. Ce n'est jamais une bonne idée de déraciner un chat, mais notre maison était en train de sortir de ses fondations, de toute façon, et le pauvre vieux serait mort de faim si je l'avais laissé avec L.A., qui ne semblait même plus savoir qu'un être vivant doit se nourrir. Si elle n'avait pas reporté sur moi tout ce qu'il y avait de rancœur en elle, j'aurais pu essayer de l'aider, de remettre un peu de chair sous cette peau autrefois si ferme, mais c'était au-dessus des compétences qu'elle m'accordait. J'ai bouclé mes valises, j'ai confié à Régine et à Patricia, la mère et l'amie fidèle, le soin de s'occuper de ce qu'il restait de L.A., et j'ai enfermé Pixie dans la cabane de voyage que lui avait fabriquée Billie avec une caisse de soupe aux tomates Campbell évoquant les tableaux d'Andy Warhol.
Une fois installé dans mon nouvel appartement, j'ai donné ma démission à l'université, où je n'avais mis les pieds qu'à quelques reprises depuis la disparition de Billie, j'ai continué à harceler Ménard et Dubois, qui se sont respectivement transformés en boîte vocale et en courant d'air, et j'ai poursuivi mes recherches, parcourant une à une les rues de la ville, quartier par quartier, épiant toutes les fenêtres allumées, écumant les bars pour discuter avec des gars que je ne connaissais ni d'Ève ni d'Adam en leur montrant des photos de la petite. Le soir, quand j'avais fini ma tournée Billie, je m'assoyais avec Pixie sur les genoux dans mon salon de fortune et je lui racontais une histoire en écoutant John Coltrane, Django Reinhardt ou Billie Holiday sur mon vieux lecteur CD.
J'ai maintenu ce régime pendant presque trois années, tournée des bars et des rues, Coltrane et Holiday, au terme desquelles je connaissais le décor d'à peu près tous les appartements et maisons de la ville et où les quelques personnes que je fréquentais de loin en loin, si fréquenter est le terme juste, avaient décrété que j'étais fou.
Le 20 janvier 2009, jour du troisième anniversaire de la disparition de Billie, j'ai téléphoné à Lucy-Ann pour lui annoncer que j'allais bientôt quitter la ville. Elle m'a souhaité bonne chance en reniflant, comme si elle avait cessé d'espérer et de me haïr du même coup. L'abdication de L.A. aurait dû susciter ma compassion, mais tous les mots pour décrire ma désolation, j'aurais pu t'aimer encore, L.A., j'aurais pu tenir ta main dans la mienne pour un autre bon bout de chemin, avaient été emportés dans la tourmente. Depuis que la barbarie du monde avait déferlé sur notre maison avec la fureur d'un ouragan de force 5 sur l'échelle de Saffir-Simpson, toutes les paroles d'encouragement possibles s'écrasaient dans les décombres. Dans le silence suivant la tempête, j'ai vainement cherché quelques mots qui auraient résisté au désastre, puis j'ai bêtement répondu à Lucy-Ann de prendre soin d'elle. Pas une allusion à Billie, pas une remarque sur le funeste anniversaire que nous célébrions ce jour-là. Ni L.A. ni moi n'étions capables d'évoquer cet anniversaire, car le 20 janvier resterait toujours dans notre esprit un jour de deuil, un jour de trop, une date qu'il aurait fallu effacer du calendrier, éradiquer comme une épidémie de peste, un de ces jours noirs qui voient s'ouvrir les entrailles de l'enfer. Non, ni L.A. ni moi n'avions la force d'évoquer la musique funèbre qui aurait convenu à cette journée.
Après avoir raccroché, je me suis rasé et me suis rendu dans une agence immobilière. Le type qui s'est occupé de moi a paru surpris par mon manque d'enthousiasme et mon peu d'exigences, mais je lui offrais une vente facile, alors il n'allait pas rechigner. Il m'a montré des photos de quatre ou cinq maisons dont personne ne voulait, jusqu'à ce que je jette mon dévolu sur la quatrième cambuse, après avoir reçu l'assurance que la région de Rivière-aux-Trembles était assez reculée pour que même les ours s'y égarent. La maison serait libre dans une semaine ou deux et je pourrais y emménager quand je voudrais à partir de ce moment. Je lui ai donné mon accord sans discuter le prix ni demander à visiter d'abord, puis j'ai traîné mes savates jusqu'à un bistro où, dans l'odeur de bière refroidie, je pourrais ruminer les pensées désabusées de l'homme en partance pour un nowhere.
À la télé accrochée au-dessus du bar, Barack Obama allait dans quelques secondes prononcer son discours d'investiture. Moi qui me passionnais d'ordinaire pour la politique états-unienne, j'avais à peine suivi la dernière campagne présidentielle, mais je savais que l'Amérique et le monde entier vivaient un grand jour, pas seulement parce qu'Obama était noir, mais parce que ce n'était pas souvent que les U.S.A. mettaient au pouvoir un homme qui savait prononcer le mot « pauvre » sans s'étonner que la moitié du pays lève la main.
« We, the People », a clamé Barack peu de temps après avoir entamé son discours, reprenant en cela les premiers mots d'une des plus vieilles constitutions du monde et rappelant à la planète entière que l'Amérique de Jefferson et de Washington avait fondé ses lois sur une démocratie qui, selon certains, tenait encore sur ses deux jambes tatouées de signes de dollars. Les applaudissements fusaient, tous ceux et celles qui possédaient une télé ou une radio, de l'Afghanistan jusqu'au Zimbabwe en passant par l'Australie tendaient l'oreille et, pendant un instant, j'ai eu l'impression d'appartenir à ce peuple, à cette grande famille nord-américaine dont Obama énumérait les couleurs du haut de sa tribune. En apercevant dans le miroir du bar le sourire imbécile qui illuminait mon visage, j'ai eu envie d'y projeter mon verre. La seule famille qu'il me restait jusqu'à récemment n'existait plus, toutes mes idées sur la famille avaient sacré le camp, et j'aurais beau me nourrir de junk food, de dérivés de la pomme de terre ou de la viande hachée en écoutant des séries hollywoodiennes jusqu'à la fin des temps, je n'appartiendrais plus jamais à ce qu'on nomme une famille, qu'elle soit nord-américaine ou pas. Finie la famille, kaputt, pulvérisée dans une collision d'atomes aussi imprévisible que la prochaine explosion du supervolcan de Yellowstone.
J'ai calé mon verre et j'en ai commandé un autre, pour la route, pour oublier que je respirais l'air d'une planète qui permettait que respirent aussi les tueurs d'enfants. Quand mon verre est apparu devant moi, je l'ai levé en direction d'Obama, qui souriait de toutes ses grandes dents blanches, et j'ai lancé un « Yes, you can ! » débile qui m'a valu le regard réprobateur du serveur, un jeune con, de même qu'un grognement pouvant aussi bien exprimer la solidarité que le désaccord de la part du seul autre client de la place, qui examinait distraitement une grille de mots croisés posée à côté d'une Molson Dry dont il arrachait l'étiquette lanière par lanière, l'esprit visiblement à des kilomètres du premier mot de la ligne 12 de sa grille, horizontalement, ouverture aménagée à l'avant d'un navire ou première épouse de l'empereur Qin Shi Huang. J'ai été tenté d'aller lui demander s'il avait perdu un enfant, lui aussi, et n'arrivait plus à se concentrer sur quoi que ce soit d'autre que son souvenir, mais son allure peu engageante m'en a dissuadé. Qu'il se soûle tout seul. J'avais déjà l'immensité d'une Billie à pleurer, je ne pouvais pas y ajouter celle d'un gamin dont le père avait oublié à quoi ressemble un sourire.
Trente secondes après, je me suis rendu compte que je pleurais sur mes frites mayonnaise. Je me suis mouché dans ma serviette de table et j'en ai demandé une autre, parce que ça n'arrêtait pas de couler, parce que les vannes avaient lâché et qu'il y avait derrière le barrage que j'avais érigé entre le monde et moi de quoi inonder une partie de la ville et noyer au passage le jeune barman toujours aussi con qui ne savait pas comment se comporter devant le grand type qui braillait au bout de son comptoir, un autre taré, un autre qu'on devrait enfermer. Je lui ai piqué un paquet de serviettes propres et, en guise d'excuse, je lui ai appris que ma fille disparue depuis trois ans était officiellement morte aujourd'hui, 20 janvier 2009, jour d'investiture du 44e président des États-Unis sous les acclamations confiantes de la foule ayant déferlé sur Washington. Pour lui prouver que Billie avait existé, j'ai glissé vers lui une photo de la petite arborant son plus beau sourire, que j'ai laissée là, à côté des serviettes chiffonnées, jusqu'à ce que ce crétin se fende d'une formule de sympathie. Quelques jours plus tard, je quittais la ville qui avait tué Billie.
## II
LES NOMS
### WILLIAM, BILLIE ET LUCY-ANN
J'ai pris possession de la maison de bardeaux numéro 4 de l'agent immobilier Max Lapointe le 9 février, alors que des feux de broussailles dévastaient le sud-est de l'Australie sous des chaleurs caniculaires que je n'avais aucune peine à imaginer malgré le froid frappant le Québec, car j'étais moi-même ravagé par une fièvre qui teintait de rouge et d'orange le décor tout en courbes et pentes raides entourant Rivière-aux-Trembles. La grippe m'avait attaqué de front le vendredi, faisant presque bouillonner le mercure dans mon thermomètre, pendant que l'Australie se tapait un record de 46,4 degrés. Le mercure avait un peu chuté le dimanche, mais ni moi ni l'Australie, en ce lundi matin, n'étions sortis de « l'enfer dans toute sa fureur », ainsi que le premier ministre Kevin Rudd avait qualifié le brasier qui rasait une partie de son pays.
J'avais peine à tenir sur mes jambes lorsque j'ai pris la route au volant du camion de location que quelques samaritains désœuvrés m'avaient aidé à charger. Le premier crétin venu m'aurait recommandé de sortir mon matelas du camion et de retourner me coucher, mais rester une minute de plus dans cet appartement vide m'apparaissait aussi insensé que d'essayer de respirer sous un tas de farine. Depuis que j'avais enfin admis que si Billie refaisait surface, ce ne serait ni dans la cour de son école ni dans sa chambre, la ville n'était plus qu'un organisme hostile cherchant à m'expulser, une machine à combustion me bousculant vers ses canaux d'éjection. Si j'avais tenté de résister à cette force de répulsion, on m'aurait probablement repêché dans les égouts. J'aurais pu voir dans cette poussée de fièvre inattendue chez un homme qui n'avait jamais avalé une aspirine de sa vie un signe que la ville avait changé d'avis et voulait maintenant me retenir quelque part entre ses intestins et son foie bilieux au lieu de me recracher, mais j'avais depuis longtemps franchi cette étape où l'espoir me poussait à associer le moindre chant d'oiseau un peu trop aigu à un message envoyé par Billie des limbes indistincts où son corps et son esprit flottaient. J'imaginais ainsi le lieu où Billie se trouvait, comme un espace nébuleux situé dans une dimension parallèle à la nôtre, où le temps n'avait plus cours et où Billie s'était arrêtée aussi, dans l'attente que quelqu'un perce le mystère de cet au-delà et la délivre de l'emprise du temps qui stagne. Je refusais de me laisser envahir par l'image d'une Billie pleurant, d'une Billie hurlant, d'une Billie blessée ou gisant sous la blancheur de la neige, par toutes ces images de journaux à potins dont votre mémoire de l'horreur vous bombarde dès qu'un enfant s'éclipse dans le noir, car c'est bien ce qui se produit quand un enfant disparaît, quand son corps est soudainement emporté loin de votre champ de vision. Il est propulsé hors du temps et de l'espace, vers un inconnu dont les nuées l'enveloppent, minuscule dans l'immensité du froid. Je me représentais Billie dans un endroit de cette sorte, où le mal ne pouvait l'atteindre mortellement et d'où elle pouvait m'appeler à travers le chant d'une mésange ou le pépiement d'une hirondelle ayant effleuré la froideur des nuages.
Pendant plus d'un an, à demi lucide et à demi aveugle, j'ai cru qu'elle m'envoyait des signaux, jusqu'au jour où, à genoux sur un trottoir crotté, je me suis rendu compte que je devenais carrément cinglé, que l'irrationalité prenait le dessus et que le vase était sur le point de déborder.
C'était une journée particulièrement douce, le soleil brillait, le printemps chauffait la couenne des badauds. J'avais rapidement expédié mon déjeuner, constitué d'un croissant sec et de deux cafés sirupeux qui auraient tué sur-le-champ n'importe quel diabétique, pour ensuite aller me perdre dans la foule des promeneurs. Il y avait dans l'air une de ces odeurs que vous n'arrivez pas à fixer, à associer à un souvenir, mais qui vous rappelle néanmoins un instant de lointaine pureté, quand la vie n'était pas si compliquée, quand la lumière du matin se laissait prendre à bras-le-corps et que la joie avait un putain de sens.
Je me promenais dans une rue bondée lorsque j'ai aperçu sur le sol la pochette poussiéreuse et à moitié déchirée d'un CD de Billie Holiday. Je me suis penché pour la ramasser et il y avait une flèche, sous la pochette, une de ces flèches orange tracées en vue de délimiter des zones de travaux. Je savais à quoi servaient ces flèches, ce qui ne m'a pas empêché de suivre celle-là jusqu'à un tas de cochonneries que le vent avait poussées dans le renfoncement d'une porte condamnée. À quatre pattes sur le sol crasseux, je fouillais avec l'énergie du désespoir dans l'amoncellement de papiers gras, de pages de journaux déchiquetées et de mégots de cigarettes, à la recherche d'une trace, d'un indice, d'un signe, quand trois pièces de monnaie ont atterri près de ma main gauche dans un bruit assourdi par les ordures.
C'était le signe que j'attendais, le gong annonçant la fin du match et m'apprenant que j'avais perdu la tête. J'ai ramassé les trois trente sous en vue de les encadrer ou de les couler dans le béton, question de me rappeler que je dérapais ferme, et je suis rentré écouter Lady in Autumn, un album de Billie Holiday que je considérais comme une relique, car c'est entre autres à cause de cette chanteuse sublime, d'un film de Billy Wilder et de la Billie Jean de Michael Jackson que Lucy-Ann et moi on avait décidé d'appeler notre fille Billie. C'est aussi à cause de mon surnom, bien sûr, qui a éclipsé mon prénom dès que je suis entré à l'école et que Jacques Lacroix, mon premier véritable ami, s'est mis à m'appeler Bill pendant que je lui donnais du Jack. Depuis ce temps, Bill me colle à la peau, à tel point que si quelqu'un m'appelle William, j'ai l'impression qu'il s'adresse au fantôme d'un enfant de six ans ou à un gars que je ne connais pas. William a été enfoui au fin fond d'un de ces terrains de jeux où les gamins se surnomment Alex, Chris, Bill, Bob ou Jos s'ils ont de la chance, Bouboule ou Face de rat s'ils en ont moins. Il n'y a que sur les papiers officiels que ce nom figure encore, même s'il ne définit en rien qui je suis. Mes livres, je les signe également du prénom de Bill, parce que je ne vois pas comment il est possible de signer William après Shakespeare, Faulkner et Blake.
Mais Billie, c'était le vrai prénom de la puce, qu'elle portait comme une reine, et qu'on ne vienne pas me faire chier parce que je parle d'elle au passé. Aucun temps mieux que l'imparfait ne peut exprimer l'absurdité d'une vie qui doit pivoter vers l'arrière si elle ne veut pas se prendre un mur de briques en pleine face. Ma vie à moi penche de ce côté-là, du côté d'hier, du côté de Billie, que je conjugue en des temps se déroulant à l'imparfait pour ne pas m'enfoncer dans l'illusion et devenir aussi pathétique que ces mères attendant le retour de guerre du fils qu'on leur a pourtant expédié en soixante-quinze morceaux dans un cercueil enveloppé du drapeau de son pays, hymne national inclus. Notre Billie arborait son nom aussi magnifiquement que l'éternelle Lady Day, et quand on a vu le petit visage plissé de ce poupon, sa petite gueule de Bill au féminin, on a tout de suite su qu'on ne s'était pas trompés en choisissant ce prénom. Parce qu'elle me ressemblait, Billie, parce qu'elle avait hérité de mon nez, de mon front, de mes lèvres, mais en cent fois plus beau.
À l'époque, on s'aimait comme des malades, L.A. et moi, comme des tourtereaux aveugles, comme des inséparables se bécotant à longueur de jour dans une cage de trente centimètres carrés, preuve que notre union était fondée sur un malentendu. Quoi qu'il en soit, Lucy-Ann éprouvait une fierté de paon de compétition à l'idée que sa fille porterait le nom de l'homme qu'elle fixait parfois avec de grands yeux étonnés, l'air de se demander comment elle avait pu tomber sur un type aussi génial. C'est d'ailleurs elle qui a proposé ce nom, question d'établir clairement la filiation de Billie. Elle aurait pu vouloir que cette filiation s'ancre dans la lignée maternelle et j'aurais compris, je n'aurais eu aucune objection à ce que la petite s'appelle Anna, par exemple, ou Marie-Lucie, mais pour Lucy-Ann, c'était hors de question. Elle avait eu suffisamment de problèmes avec son prénom, elle n'allait pas le refiler stupidement à une enfant innocente.
Le vrai nom de Lucy-Ann était Lucienne, et elle n'avait jamais pardonné à ses parents de l'avoir affublée d'un nom de vieille tante de l'Abitibi. Elle avait raison, je ne sais pas à quoi ils avaient pensé de donner un nom pareil à une fille née dans les années soixante. Fatalement, ça l'avait marquée. On s'était moqué d'elle à l'école, à la piscine, au camp de vacances pourri, pas loin de Shawinigan, où elle avait failli perdre sa virginité avant terme. À dix-huit ans, elle en avait eu plein le casque et avait entrepris des démarches pour changer officiellement son nom en Lucie-Anne, histoire de ne pas trop dépayser son entourage et de ne pas se créer de problèmes d'identité supplémentaires. Se transformer en Marie-Christine quand on a passé sa vie dans la peau d'une Lucienne peut être assez perturbant, j'imagine, et on peut comprendre qu'elle ait opté pour Lucie-Anne, même si ça représentait probablement la plus grosse erreur de son existence en matière identitaire. Elle non plus, je ne sais pas à quoi elle a pensé. Les gens qui la connaissaient ont continué à l'appeler Lucienne, ont transmis ça à ceux qui ne la connaissaient pas ou qui étaient durs d'oreille, avec pour résultat qu'elle était encore une Lucienne.
Avant d'entrer à l'université, elle a enfin décrété que désormais, on devrait prononcer son nom à l'américaine, Loussy-Ann, et l'épeler dans une langue que même les Papous comprenaient, Lucy-Ann. Cette décision a réglé une partie de ses problèmes, d'autant plus que sa colocataire la surnommait L.A., comme la ville de Los Angeles, conférant à cette Lucy-Ann bronzée à l'année un soupçon d'exotisme qui lui allait comme un gant de dentelle à une courtisane.
Elle s'appelait Lucy-Ann depuis des années quand je l'ai rencontrée dans un bar à proximité de la tour à bureaux où elle travaillait. Après trois verres de tequila sunrise, elle m'avait dit de l'appeler L.A. en inclinant vers moi un décolleté évoquant la rondeur des montagnes californiennes, puis elle m'avait débité un tas de conneries qui m'avaient alors paru brillantes sur les ressemblances qu'on pouvait établir entre la ville de Los Angeles et sa propre topographie interne. Elle prétendait que le nom qu'on se choisissait reflétait notre nature profonde, qu'il marquait notre destin aussi sûrement que le passage plus ou moins réussi de notre phase œdipienne et nous définissait mieux que n'importe quelle biographie. Je n'avais pas connu son époque Lucienne et le parallèle qu'elle traçait entre elle et la cité des anges me paraissait logique, d'autant plus que L.A. était une fille chaude, capable de donner à son corps pourtant frêle des dimensions de mégalopole. Dès le premier soir, j'ai pu constater que l'intense activité sismique de la faille de San Andreas n'était pas une légende et qu'on pouvait avoir envie de se transformer en sismographe rien que pour se rapprocher du soleil de L.A.
On a passé de bons moments ensemble, je dirais même de très belles années. Si L.A. n'était pas entrée dans une période de changements climatiques après la naissance de Billie, on aurait pu continuer à sentir la terre trembler sous nos pieds pendant que les édifices s'écroulaient autour de nous dans des nuages de poussière rouge. Mais la température s'est peu à peu refroidie et la fille dont les seules initiales parvenaient à réchauffer mon atmosphère s'est mise à marcher vers le nord, tête baissée, pour enfin déposer ses bagages aux environs de Whitehorse.
J'ai également ma part de responsabilités dans l'apparition de ce qu'on pourrait considérer comme notre ère préglaciaire. Je revendique le privilège d'être aussi bouché que la plupart des hommes en matière de relations de couple, de diplomatie conjugale et d'harmonie postcoïtale, mais Dieu que j'ai regretté la Lucy-Ann d'autrefois, celle que j'appelais L.A. quand j'avais envie de me coller à son corps qui riait aux éclats dans la moiteur de la nuit. Je ne sais pas exactement à quel moment la véritable rupture s'est amorcée, mais le nom de la femme que je croyais pouvoir aimer jusqu'à la fonte du dernier glacier de la galaxie s'est tout à coup vidé de son sens. Quand je l'appelais L.A., je ne sentais plus la douceur des anges ni l'odeur du Pacifique se glisser sous ma langue. L et A n'étaient plus que deux lettres mortes me permettant d'établir la communication avec la femme vivant à mes côtés. Avec le temps et le durcissement du froid dans la lumière de notre hiver, ces lettres se sont chargées de tout ce que je désapprouvais chez Lucy-Ann, à tort ou à raison, si bien que je ne l'appelais plus L.A. qu'avec un soupçon de mépris n'ayant d'autre but que de lui reprocher de n'avoir pas tenu ses promesses, de n'avoir pas été capable de demeurer à la hauteur des feux faisant étinceler Los Angeles. Lucy-Ann était une ville dont je m'éloignais le cœur déçu en me rappelant la beauté des premiers jours du voyage, quand la tequila sunrise avait l'exacte couleur du soleil se couchant au large des plages de Santa Monica.
Le soleil enflammait encore la ligne d'horizon, pareil à une grosse orange sanguine coupée en deux, le jour où Lucy-Ann m'a annoncé qu'elle était enceinte. On s'était donné rendez-vous pour l'apéro sur une terrasse aménagée dans une arrière-cour où étaient exposées quelques œuvres d'art postmacramé imitant le mobilier urbain, bornes-fontaines, lampadaires, feux de circulation, tout pour nous donner l'illusion qu'on vivait en ville. L.A. était arrivée avant moi et elle déballait ses achats de l'après-midi sur la table en forme d'essieu où trônait un pichet de sangria. Elle s'était procuré une petite camisole blanche qui frôlerait sur son corps les limites de l'indécence, une affreuse paire de sandales jaunes, pour porter avec ma robe tournesol, tu sais, celle avec les grosses fleurs jaunes, un DVD de Sunset Boulevard, le chef-d'œuvre de Billy Wilder, se déroulant dans une des villas du mythique boulevard traversant Los Angeles de Hollywood jusqu'à Pacific Palisades, puis un album double de Billie Holiday, Lady in Autumn, qui allait m'échoir quelques années plus tard.
À part le Motown et quelques vieux airs de country, L.A. ne connaissait rien rayon musical. Elle détestait le jazz, croyait que Miles Davis était le frère de Sammy Davis Jr. et confondait invariablement les trompettes et les sax. On était toutefois arrivés à une entente. Je fermais ma gueule et me tapais du Motown à l'occasion, en échange de quoi je pouvais écouter du Mingus ou du Parker sans que L.A. râle aussitôt. Son dernier achat ne cadrant pas avec les sandales jaunes, j'ai pensé qu'elle avait une gaffe à se faire pardonner lorsqu'elle a glissé vers moi l'album de Billie, dans les tons de noir et d'orange, de la couleur d'une nuit d'été cernée de flammes. À nous deux, baby, a-t-elle dit en frappant son verre de sangria contre le mien, avec plein d'étincelles dans les yeux, de belles petites étincelles de la couleur un peu dorée, indéfinissable, de ses iris los angeliens. À nous deux, ai-je répondu en omettant le baby, que je rangeais dans la catégorie des mots réservés à l'intimité des espaces clos où aucune oreille vicieuse ne peut se moquer de votre façon de prononcer honey.
Il y a une chose que je t'ai pas encore montrée, s'est-elle exclamée en fouillant dans son immense sac à main imitation léopard. Elle en a sorti un paquet de mouchoirs froissés, de tubes de rouge à lèvres et autres babioles sans lesquelles aucune femme, paraît-il, ne peut sortir de chez elle sans avoir peur de tomber ce jour-là, précisément ce jour-là, sur George Clooney ou Clive Owen. Elle a enfin trouvé ce qu'elle cherchait et a brandi devant moi une petite suce rose à laquelle était attaché un bout de ruban bleu. À nous trois, baby, a-t-elle chuchoté en déposant la suce sur la table, et j'ai mis un certain temps à comprendre ce qu'elle essayait de me dire. Quand ses yeux ont répondu oui à la question que je n'arrivais pas à poser, j'ai eu l'impression que la terrasse basculait, emportant L.A. tête première vers la cime des arbres et des lampadaires de tôle galvanisée.
J'ai calé mon verre de sangria et m'en suis servi un autre en murmurant papa avec ce qui devait ressembler au sourire d'un débile découvrant sa queue. J'aurais voulu sauter au cou de L.A., mais on ne peut pas se déplacer aussi rapidement avec le poids d'un tout petit enfant sur les épaules. Finalement, c'est L.A. qui s'est tapé le rôle du gars. Elle est venue près de moi, s'est appuyée contre notre borne-fontaine et a pris mes mains dans les siennes pour les embrasser. Plein de questions se bousculaient dans ma tête, plein de questions heureuses entourées d'étincelles qui se bousculaient aussi et empêchaient les mots de sortir. À travers le ventre à peine rebondi de L.A., je voyais déjà une petite face ronde et plissée qui flottait dans l'obscurité des eaux amniotiques en attendant de se diriger vers la sortie. Un enfant prenait forme dans ce ventre et je me demandais si on allait devoir lui acheter une bicyclette rose ou bleue, des poupées ou des camions, et si cet enfant m'aimerait autant que j'aimais la femme qui m'enduisait en ce moment les mains de rouge à lèvres à saveur de cerise. Comment on va l'appeler ? ai-je fini par lâcher en engloutissant mon troisième verre de sangria avec un sourire si large qu'il m'aurait fallu boire dans une chaudière pour que ça en vaille la peine.
J'ai compris que L.A. avait déjà son idée là-dessus quand elle a désigné la pochette de Lady in Autumn, sur lequel le visage de Billie Holiday surgit de la noirceur d'une nuit qui l'enveloppera jusqu'à sa mort. C'est un joli nom, Billie, et Billy n'est pas mal non plus, a-t-elle ajouté en examinant la pochette du film de Wilder. Billie Richard, ça sonne bien, tu trouves pas ? Le message était clair. L.A. avait choisi le nom de notre enfant et j'aurais un mal de chien à la faire revenir sur sa décision si je m'y risquais. Mais je n'avais pas l'intention de m'opposer à son choix, parce que c'était celui d'une femme qui m'aimait assez pour vouloir que son enfant porte à la fois mon nom et mon prénom.
T'as raison, L.A., c'est un sapré beau nom, presque aussi beau que Lucienne, ai-je lancé en esquivant la cerise au marasquin expédiée depuis l'autre côté de la table où éclatait une autre fois le rire de Lucy-Ann, le rire si franc de L.A., qui se scléroserait quelques années plus tard, lorsque les cerises seraient remplacées par des couteaux tranchants.
Je ramassais la cerise dans l'intention de la lui renvoyer quand la voix de Michael Jackson chantant Billie Jean a retenti dans les haut-parleurs installés aux quatre coins de la terrasse. C'était un signe du destin, il n'y avait aucun doute, les augures s'adressaient à nous par la voix d'un dieu de la pop. On s'est regardés dans le blanc des yeux jusqu'à ce que Jackson termine sa chanson, aussi heureux qu'Adam et Ève avant l'épisode du pommier et totalement indifférents à l'aspect kitsch des augures qui faisaient branler nos chaises au rythme de Billie Jean. Pendant ce temps, dans le ventre de L.A., le petit amas de cellules qui deviendrait notre Billie à nous continuait à se multiplier.
Ç'a été aussi simple que ça. Qu'il s'agisse d'un garçon ou d'une fille, ce serait Billie ou Billy. J'ai levé mon verre à la santé de Billie-Billy et L.A. a fait de même en précisant que ce serait son dernier verre pour les mois à venir, régime sec jusqu'en avril, jusqu'aux premiers vagissements de l'enfant dont je ne sentais plus seulement le poids sur mes épaules, mais également la chaleur, la douce moiteur des poupons emmaillotés de laine.
Le 20 avril de cette année, Billie aurait eu douze ans et je me demande souvent ce qu'elle aurait souhaité pour cet anniversaire. Probablement un autre chat, car le vieux batêche de Pixie a cru bon de rendre l'âme quelques jours avant mon déménagement à Rivière-aux-Trembles. Peut-être qu'il n'avait pas envie d'avoir à s'acclimater à un nouveau lieu, qu'il ne se sentait plus la force d'affronter le voyage, la bourrasque, les odeurs inconnues, mais je crois plutôt qu'il a compris que Billie ne serait pas là où nous allions et qu'il a cessé de respirer en même temps que d'espérer. Il ne s'était maintenu en vie que pour pouvoir dire adieu à Billie et lui donner un dernier bec de nez, ben plus le fun que les becs de bec, papanoute, mais les jours et les mois passant, l'ennui avait transformé son désir de sentir sur son ventre les mains de sa petite maîtresse en cette forme d'amertume brisant ceux qui perdent la foi. J'avais bien remarqué qu'il déclinait depuis un certain temps, qu'il ne dressait plus les oreilles chaque fois que des pas provenant du corridor pouvaient être confondus avec des pas de princesse, mais je lui avais strictement interdit de mourir. Si tu crèves, Pixie Richard, je te sers aux pigeons en boulettes de chat haché. Mais les menaces ne peuvent rien contre la tristesse.
Je l'ai trouvé un matin dans sa caisse de soupe aux tomates Campbell. Il gisait sur la petite couverture que Billie avait taillée pour lui dans un de mes vieux pyjamas de flanelle, les yeux fermés, la gueule à demi ouverte. Quand j'ai constaté que sa grosse bedaine de goinfre à la Garfield ne se soulevait plus au rythme de sa respiration, je suis devenu fou. Je me suis mis à crier, à hurler qu'il n'avait pas le droit de me faire ça. Je lui ai ordonné de se réveiller, câlisse, Pixie, respire, puis je me suis écrasé à côté de sa cabane et j'ai pleuré jusqu'à ce que l'envie de pisser m'oblige à me relever. Si je n'avais pas eu de vessie, je pense que je serais encore là, à le supplier de revenir, d'ouvrir ses maudits grands yeux verts et d'aller voir la surprise que je lui réservais à la cuisine, une belle boîte de thon Clover Leaf, Pixie, tu peux pas lever le nez là-dessus, viarge !
En disparaissant, ce chat m'enlevait tout ce qu'il me restait de Billie et anéantissait mes derniers espoirs de la voir réapparaître au coin de la rue un beau matin, un matin si beau, en fait, que toutes mes tentatives pour le décrire demeureraient vaines, même si je m'y employais jusqu'à ce qu'on ensevelisse ma dépouille de vieux schnock au pied d'un érable gris. Si Pixie avait décidé de saprer son camp, c'est qu'il savait toute attente désormais inutile. Son intelligence de la vie, de la mort, de la douleur et du simple bien-être lui avait fait comprendre que plus jamais la puce ne viendrait lui caresser les oreilles. Pixie savait ce que savent tous les animaux quand ils consentent à fermer les yeux, que la fin de leur monde est arrivée et qu'il ne sert à rien de s'y opposer. Si j'avais été aussi intelligent que lui, je me serais procuré une caisse de soupe aux tomates, moi aussi, où mon cœur aurait tranquillement cessé de battre, mais une part de ma stupidité d'homme voulait encore croire aux miracles, aux contes dans lesquels, contre toute logique, les princesses ressuscitent et réapparaissent dans la beauté du matin les bras chargés de fleurs des champs.
Je me souviens du jour, pas longtemps après l'accident du petit Dumas, où j'avais fait rouler Pinocchio sous un train de marchandises, mettant Gepetto au défi de le rafistoler. À la fin de l'histoire, Billie m'avait considéré avec de grands yeux ronds, puis un petit sourire avait retroussé le coin de ses lèvres et elle m'avait donné un coup de poing sur l'épaule en s'exclamant ça se peut pas, papa, t'inventes encore, Pinocchio peut pas mourir, voyons donc. Sans attendre ma réaction, elle était sortie jouer dans la cour avec Pixie, certaine que les personnages de contes ne pouvaient pas davantage mourir que les héros et que son père lui avait servi une mauvaise blague. En lui racontant cette histoire, j'avais stupidement voulu lui donner ce qu'on appelle une leçon de vie, lui faire comprendre que rien ni personne n'était immortel et qu'elle devait assurer ses arrières, mais c'était elle qui avait raison, les héros ne peuvent pas plus mourir que les petites filles, j'en ai maintenant la preuve chaque fois que je regarde une photo de Billie et la vois reprendre vie, faire son premier nœud de bottine ou esquisser ses premiers pas sur l'herbe, aussi immortelle que la mémoire des hommes.
Les chats non plus ne peuvent pas mourir, surtout pas les Pixie, c'est ce que je me disais le matin où je l'ai trouvé dans sa cabane en me demandant ce que j'allais faire de lui. Le seul animal domestique dont j'avais jamais partagé l'existence était un poisson rouge répondant au nom de Conrad, ce qui est une façon de parler, car si Conrad a un jour réagi à son nom, personne ne s'en est rendu compte. Cette indifférence ne signifiait pas pour autant que Conrad était idiot. Elle était plutôt la preuve que les hommes ne comprennent pas les poissons ou que les poissons ne peuvent tout simplement pas s'appeler Conrad. Quand Conrad a viré sur le capot, selon l'expression de mon père, qui n'aurait jamais pu envisager une carrière dans la diplomatie, c'est ma mère qui s'en est occupée, me privant ainsi de l'expérience fondamentale qui aurait consisté à enterrer ou à flusher Conrad moi-même. Devant le corps inerte de Pixie, je me sentais complètement idiot, parce qu'on ne m'avait pas permis d'apprendre comment on doit s'occuper de la dépouille des animaux morts.
J'ai fini par téléphoner à un vétérinaire et, parmi toutes les possibilités qu'on m'a offertes, j'ai opté pour la congélation, car je ne pouvais pas supporter l'idée que Pixie soit jeté dans un four crématoire avec des dizaines de bêtes qu'il n'avait pas connues et aurait détestées à mort s'il les avait rencontrées. Pas question, donc, que je l'expédie dans un enfer où il croupirait pêle-mêle avec une bande de matous réduits en poudre. J'aurais aimé l'enterrer derrière notre maison, sous la fenêtre de la chambre de Billie, mais la neige était trop épaisse, le sol trop gelé, l'été trop lointain pour que Pixie résiste d'ici là à l'invasion des asticots et autres bestioles nécrophiles. La congélation m'a donc paru la solution idéale en ce qu'elle m'épargnait d'avoir à prendre une décision trop rapide quant au sort que je réservais à la carcasse de Pixie. Le vieux batêche ne serait sûrement pas content s'il savait que j'ai stoppé son processus de décomposition et retardé ainsi sa cinquième ou sixième vie, mais c'était le chat de Billie et je me sentais l'obligation de lui réserver des obsèques dignes de l'amour que lui portait ma fille.
Pixie a donc fait le voyage avec moi jusqu'à Rivière-aux-Trembles, enfermé dans un caisson cryogénique, façon Walt Disney, à la différence que Pixie ne se réveillera jamais dans un monde du futur rempli de fées Clochette décongelées. Si on découvre un jour un remède à la mort, ni lui ni moi ne serons plus là pour avaler la potion méphistophélique qui foutra le bordel parmi les immortels. Dès mon arrivée, j'ai remisé le caisson à la cave en attendant le printemps, et il m'arrive de penser que le vrai Pixie n'est plus là, mais quelque part avec Billie, dans ce qu'on nomme l'éden ou le paradis. Quand je me sens d'humeur bucolique et que j'ai envie de brailler un bon coup parce que j'ai mal aux reins de retenir tous ces torrents d'eau salée en moi, je les imagine dans un décor pareil à celui de What Dreams May Come, Au-delà de nos rêves, un film avec Robin Williams dans lequel les gens qui s'aiment sont réunis après leur mort dans un paradis plus beau que nature, avec tout ce qu'il faut d'arbres, de fleurs et de paysages enchanteurs pour vous donner envie de crever tout de suite. L.A. a dû voir ce film cent fois durant la semaine précédant mon départ de la maison, la larme à l'œil et le mouchoir au nez, arrêtant le déroulement du film à tout bout de champ pour en brailler un bon coup, mêler ses larmes aux rires que suscite le spectacle de la joie, puis replongeant, le cœur à l'envers, dans des jardins qui niaient la mort. Je lui reprochais de s'accrocher ainsi à la représentation d'un au-delà auquel elle ne croyait pas, de se torturer avec des chimères qui lui péteraient au visage comme des ballons trop gonflés, mais, secrètement, je faisais pareil, de même que je persiste à me vautrer dans de naïves allégories du bonheur éternel alors que j'ai enterré Dieu depuis longtemps.
Les hommes et les femmes qui perdent un enfant n'ont d'autre choix que de croire au paradis, sinon ils sont forcés de croire à l'enfer et d'imaginer les flammes dévorantes où se sont élevés les hurlements de leur petit. En ce qui me concerne, j'ai mes jours paradis et mes jours limbes. J'exclus d'emblée le purgatoire de ces régions inaccessibles, puisque les enfants n'ont rien à expier que les fautes des autres. Je préfère néanmoins mes jours limbes, car je sais qu'on peut revenir des limbes, à plus forte raison depuis que le Vatican les a abolis, rendant plus imprécise cette zone à mi-chemin entre le bien et le mal que je ne me sens pas encore prêt, pour ma part, à rayer de ma géographie des cieux. Je sais qu'on peut revenir des limbes, mais pas du paradis. Qui aurait envie, de toute façon, de quitter l'allégresse pour se retaper l'enfer ? Il n'y a que dans les films de Wim Wenders que les anges sont assez cons pour envier la bonne vieille souffrance quotidienne de l'homme. Je préfère donc ces jours où je parviens à me représenter Billie dans le décor vaporeux d'un entre-deux-mondes d'où elle ressurgira un beau matin, oui, un beau matin, pour m'apprendre que ni l'enfer ni le paradis n'ont jamais existé.
### MARNIE, SQUOUIRÈLE, OWL ET SUPERMAN
Après l'enterrement de mon père, j'ai fait un saut à New York pour régler mes affaires, puis j'ai quitté la ville dans l'urgence et la précipitation, certaine que si je réfléchissais calmement à ma décision de m'installer à Rivière-aux-Trembles, je percevrais immédiatement l'absurdité de ce retour aux sources. J'ai tout arrangé en moins de deux semaines. Pour un prix dérisoire, j'ai vendu à mon assistante A Rose Is A Rose, ma boutique de fleuriste, j'ai pris rendez-vous avec mes rares amis pour leur faire mes adieux et j'ai téléphoné à Harry Fields, un sauvage perdu au cœur de New York dans les bras de qui je me consolais parfois de mon propre exil. Je lui ai annoncé la nouvelle de but en blanc, car Harry ne me regretterait pas plus que Gregory Holmes, Dave Cohen et Jake Danowski, les amants plus ou moins amoureux que j'avais fait poireauter sur le pas de ma porte jusqu'à ce qu'ils m'abandonnent dans le rai de lumière filtrant de l'extérieur. Ces hommes n'avaient laissé que quelques traces plus ou moins profondes sur le tapis de l'entrée et Harry Fields, avec les autres, ne résisterait pas au coup de balai que je m'apprêtais à passer dans mon existence. Je l'ai embrassé une dernière fois et j'ai fermé ma porte.
C'était il y a quarante-huit heures et, pendant que les lourdes neiges de la fin février abattent leur silence sur la campagne, j'arpente les pièces de la maison en me demandant ce que je fabrique là, au milieu des boîtes, des souvenirs et des fleurs fanées. Après mon départ pour les États-Unis, je n'avais remis les pieds à Rivière-aux-Trembles qu'une ou deux fois par année, car ce village était devenu pour moi une terre étrangère. Seule la maison de mon père conservait une certaine réalité au milieu des forêts dénudées. J'y revenais généralement pour la fête de Noël, préférant éviter les mois d'été et tout ce qui pouvait me rappeler le vrombissement de l'orage ayant emporté Michael.
Mon père ressortait pour moi les guirlandes, les boules de verre et les angelots remisés dans le grenier. Pendant que la dinde rôtissait dans le four, on décorait le sapin en buvant du kir et en écoutant Bing Crosby interpréter I'll Be Home for Christmas. Il arrivait à mon père d'inviter quelques amis mais, la plupart du temps, on célébrait Noël rien que nous deux, dans la chaleur des bougies découpant nos ombres sur les murs, comme un père et une fille dont le malheur a depuis longtemps scellé l'amitié. Je lui parlais de ma boutique, lui décrivais ma dernière visite au musée d'histoire naturelle, au Guggenheim ou au zoo de Central Park, y a même des lions, pop, des pauvres vieux lions qui n'ont jamais reniflé l'odeur de la savane et qui croient probablement que New York est au cœur de l'Afrique, et lui me parlait du dernier livre qu'il avait lu, un essai de Stephen Jay Gould ou un recueil de poèmes de Renaud Longchamps, des lectures qui raffermissaient son lien avec la terre et l'aidaient à prendre sa mesure d'homme. Puis il quittait la falaise de Miguasha de Longchamps, gagnait le Saint-Laurent par la baie des Chaleurs, avant de dériver lentement vers ses rosiers, ses arbres et les plans de sa nouvelle serre.
À notre dernier Noël, je lui avais détaillé mon expédition au Hayden Planetarium du Rose Center for Art and Space. Pendant que je tentais de lui expliquer le vertige que j'avais éprouvé devant les collisions cosmiques et les impacts supersoniques narrés par la voix envoûtante de Robert Redford, j'avais tout à coup eu l'impression qu'il me quittait, que son esprit s'envolait par la fenêtre vers le lointain où se perdait la voix de Redford, alors que son corps s'effondrait dans son vieux fauteuil de cuir craquelé. Quelques minutes plus tard, le visage tourné vers la nuit, il m'annonçait tranquillement sa mort prochaine.
Deux mois, avait-il murmuré, peut-être un peu plus, peut-être un peu moins, pendant que Bing Crosby terminait Holy Night. Puis sa voix m'exhortant à ne pas pleurer, c'est normal, Marnie, y a rien de plus normal que la mort, sa voix venue de la nuit étoilée, s'était perdue dans l'engourdissement qui gagnait tous mes membres devant le choc de cette mort annoncée.
Si je ne lui avais pas parlé de ces histoires de marée gravitationnelle, de galaxies cannibales et de nébuleuses s'entrechoquant, il aurait peut-être tenu sous silence la maladie qui le rongeait, mais la conscience de sa petitesse devant l'immensité l'avait poussé à la confidence. Il avait dû se dire que si j'étais capable d'admettre que des galaxies s'entredévorent pour recréer de nouveaux mondes et faire apparaître la vie dans une explosion d'atomes, je pourrais sûrement m'incliner devant les mouvements qui, à une échelle infinitésimale, entraînaient la collision des galaxies se déplaçant dans son organisme malade.
Il avait raison. Il est plus facile d'admettre la mort quand on observe les étoiles. Cette sagesse venue de l'infini se dissipe toutefois dans le lever du jour, lorsqu'on aperçoit son père chanceler dans les premières lueurs du matin, avec à la main une tasse de café qui tremble. On se rend alors compte qu'il a déjà commencé à disparaître, que ses mots se font plus rares, ses pas moins amples, et qu'en cela il a renoncé aux gestes qui poussent les corps vers l'avenir. Un avenir sans lendemain s'était déjà installé dans le corps de mon père et celui-ci ne se mouvait plus que dans le cercle restreint d'un futur proche auquel il refusait que je prenne part. Pendant des heures, je l'ai supplié de me laisser rester auprès de lui, mais Alex Duchamp s'opposait catégoriquement à ce que j'assiste à sa déchéance. Il voulait mourir seul, sans personne pour le voir tomber.
Lorsque la sonnerie du téléphone a retenti, le matin du 6 février, dans l'appartement que j'occupais en plein cœur de Brooklyn, je n'ai pas décroché, car je savais déjà que mon père n'était plus. J'avais entendu le cri, durant la nuit, le cri de la rivière aux Bleuets, le hurlement de la mort. Emmitouflée dans de lourdes couvertures, je rêvais que je marchais dans un vaste champ couvert de neige duquel émergeaient ici et là des amas de longues tiges séchées semblables aux plantes bordant les rivages. Ce n'était ni le jour ni la nuit, ni le matin ni le soir. Je m'avançais dans un temps figé entre deux lumières où même la neige n'était pas froide, dans un large couloir privé de vent, de bruit, de vie. Puis l'ombre d'un oiseau s'est imprimée sur la neige, là où s'enfuyaient quelques pistes de lièvre, et elle est demeurée là, alors qu'aucun oiseau ne volait au-dessus de cette tache sombre. Quand l'ombre s'est enfin abattue sur le champ de neige, le cri a retenti, cri d'homme et couinement de bête à la fois, et je me suis réveillée.
De quatre heures à sept heures du matin, je suis restée assise dans mon lit, jusqu'à ce que la sonnerie du téléphone m'annonce que mon rêve n'en était pas un. J'ai laissé le répondeur prendre le message, c'est Phil, Marnie, rappelle-moi, a murmuré la voix enrouée du meilleur ami de mon père, et j'ai enfilé mon manteau pour me rendre au Rose Center for Art and Space. Grelottant sous la pluie, j'ai attendu l'ouverture des portes, puis, assise dans l'obscurité du Hayden Planetarium, j'ai regardé mon père s'envoler parmi les étoiles, ainsi que s'était autrefois envolé Michael Superman Saint-Pierre, happé par la force d'attraction d'un destin qui l'avait entraîné vers le soleil rouge de Krypton. Quelques heures plus tard, je prenais l'avion pour aller faire mes adieux à un homme dont l'esprit voguait parmi le temps des galaxies.
Je te laisse tout, Marnie, la maison, les serres, les terrains, tu en feras ce que tu voudras, m'avait-il dit après m'avoir annoncé qu'il s'en allait. Dans mon esprit, les choses étaient cependant claires. Je refusais de vivre au milieu des fantômes. Je mettrais la maison en vente dès que mon père n'y habiterait plus. Il avait pourtant suffi du cri d'un spectre, peu de temps après, pour que je décide de prendre la direction du passé. Et me revoilà ici, à Rivière-aux-Trembles, à me demander ce que je fous dans une maison que seule la présence de mon père pouvait réchauffer.
En arrivant, j'avais pris la résolution de ne pas me laisser piéger par la nostalgie, mais je n'ai rien trouvé de mieux à faire depuis que de me vautrer dans les souvenirs de mon père. Avant-hier, j'ai découvert que le dernier film qu'il avait regardé était le Marnie d'Alfred Hitchcock. Le DVD était encore dans le lecteur et le boîtier traînait, à demi ouvert, sous la table basse du salon. Alors je l'ai regardé à mon tour, à la recherche de mes origines, car c'est à cause de Tippi Hedren, d'Alfred Hitchcock et d'un roman de Winston Graham que je m'appelle Marnie, « a pet name, easy to remember », affirme Sean Connery en ramenant Marnie à la maison au milieu du film, « a pet name », un nom de petite chose, de petit animal insouciant néanmoins poursuivi par la peur.
J'ai été conçue, paraît-il, après une représentation de ce film, avec l'entière bénédiction d'Alfred Hitchcock et de Phil Morisset, qui avait présenté ma mère à mon père et vice versa, Marie, Alex, Alex, Marie, sous la marquise d'un cinéma de répertoire où les flèches de Cupidon avaient attisé les foudres d'Éros, électrocutant du même coup Marie Beaupré et Alex Duchamp. Je me serais peut-être appelée Scarlett ou Willard, qui sait, si un autre film avait été à l'affiche, mais il semble que le hasard, en ces années, avait une prédilection pour le cinéma d'Hitchcock.
C'est pour cette raison que mes parents ont décidé de m'appeler Marnie, mais aussi parce que ma mère ressemblait à Tippi Hedren et que sa beauté sciait les jambes de mon père. N'empêche, je n'ai jamais vraiment compris pourquoi ils ont choisi de me donner le nom d'une fille qui était folle. Il est possible que ma mère, qui prétendait qu'on ne devait pas ressembler à son nom, ait voulu conjurer le sort et s'assurer ainsi que je n'aurais rien de la Marnie de Hitchcock. Je ne sais pas, mais j'aurais préféré qu'ils m'appellent Marion, Mary, Margaret ou Martha, qu'ils me fassent porter un des nombreux pseudonymes de Marnie Edgar. Ça n'aurait rien changé, il se serait toujours agi d'un des nombreux noms de la folie de Marnie, mais l'identification aurait été moins directe et je ne me demanderais pas, chaque fois que je m'assois devant ce foutu film, si mes parents ne m'ont pas transmis les obsessions de cette fille pendant qu'ils me fabriquaient.
J'ai vu pour la première fois la version française de Marnie, intitulée avec beaucoup d'économie Pas de printemps pour Marnie, dans les années quatre-vingt, peu de temps après l'apparition du magnétoscope dans nos existences. Je devais avoir à peu près seize ans et mon père m'avait détaillé quelques jours plus tôt les circonstances de sa rencontre avec ma mère. Depuis, je brûlais de connaître cette Marnie qui m'avait en quelque sorte conféré mon identité et que je considérais comme une sœur jumelle dont on m'aurait séparée à la naissance. J'avais arpenté les allées de trois clubs vidéo avant de trouver une copie du film et, de retour à la maison, je m'étais installée comme si j'allais assister à la grande première du plus grand chef-d'œuvre de tous les temps. J'avais fermé les rideaux du salon, les doubles portes séparant cette pièce de la salle à manger puis, quand la musique de Bernard Herrmann avait éclaté sur la première image du générique, j'avais avalé une poignée de pop-corn encore chaud en me calant dans mon fauteuil.
Au bout de quelques minutes à peine, le pop-corn refroidi ne passait plus. Ma sœur jumelle était une voleuse, une menteuse, une criminelle sans scrupules qui changeait d'identité en même temps qu'elle changeait de sac à main et de couleur de cheveux. Les nombreux noms de Marnie tournaient autour de moi comme autant de mensonges, Marion, Mary, Margaret, Martha... Mon nom reposait-il aussi sur une imposture ou ne servait-il qu'à désigner autrement ce qui n'a pas de vrai visage ni de réelle identité, ce qui gravite machinalement autour d'un pôle aussi lisse que l'oubli ?
Même si un tas de pop-corn m'était resté coincé dans la gorge, j'avais continué à regarder le film, pour le stopper net au milieu de la scène durant laquelle Marnie pète les plombs et devient hystérique devant l'orage illuminant de rouge les rideaux du bureau de son patron, Mr. Rutland, son futur époux. J'avais même crié avec Marnie quand un arbre foudroyé avait fracassé la fenêtre près de laquelle elle se tenait. Et peut-être avais-je été envahie, comme elle, par une marée rouge, rouge Marnie et rouge sang. Mais personne n'était là pour me prendre dans ses bras et m'embrasser passionnément, ni Rutland, ni Superman, ni quelque autre sauveur.
J'avais fixé un certain temps les rideaux du salon, certaine qu'un éclair allait bientôt les embraser, même si nous étions en plein hiver, puis j'avais fondu en larmes. Quelque chose m'échappait, qui avait un lien avec le temps, l'antériorité, la postériorité, avec l'imprévue conjonction des destins. Mes parents savaient-ils qu'en vieillissant, je serais affectée de la même phobie que Marnie, de la même hantise de la foudre ? Si oui, comment avaient-ils deviné que ma vie basculerait dans un orage et que le moindre grondement de tonnerre me jetterait par la suite dans un état de catatonie semblable à celui qui avait forcé Michael à s'immobiliser sous la pluie ? C'était impossible, ils ne pouvaient avoir anticipé cet orage ni l'avoir aperçu dans quelque rêve prémonitoire ayant imprimé le nom de Marnie dans leur subconscient. Une seule explication m'apparaissait plausible. Mon nom avait lui-même scellé mon destin. À la vie, à la mort, il était lié à celui de Marnie Edgar.
J'avais attendu le lendemain pour écouter la suite du film. Quand j'avais demandé à mon père pourquoi maman et lui avaient choisi pour moi le nom d'une fille dérangée, il avait éclaté de rire, ne comprenant visiblement pas mon malaise. Dans son esprit, Marnie n'était qu'une autre façon de nommer la force du sentiment qu'il avait immédiatement éprouvé pour ma mère, ce fameux coup de foudre qui me tombait aujourd'hui dessus avec la puissance d'une tonne de T.N.T. J'avais eu envie de lui renvoyer qu'ils auraient pu m'appeler Alfred, tant qu'à y être, mais je n'avais pas insisté. Mon père n'avait aucune idée de l'héritage que maman et lui m'avaient légué. Je m'étais procuré le roman de Winston Graham qui avait inspiré Hitchcock, au cas où l'esprit tordu du vieux Hitch aurait réinventé Marnie Edgar de la tête aux pieds. J'aurais été soulagée d'apprendre que la Marnie d'origine n'était pas aussi fêlée que son clone hitchcockien, mais ce n'était pas le cas. L'intrigue de ce roman, comme celle du film, reposait sur la fragilité de Marnie et sur l'amnésie ayant rayé de sa mémoire les événements traumatisants de son enfance. Hitch et sa scénariste avaient fait du bon boulot. Aucune déformation des faits notable, aucun Zoulou de service dans le décor, aucun Prince rebaptisé Pitou. La classe.
Pendant des mois, je me suis ensuite demandé si l'agent Desmarais ne pouvait pas avoir eu raison, de même que les voix chuchotant sur mon passage. Était-il possible que j'aie tout oublié et que j'aie été à l'origine de la disparition de Michael, ainsi que le supposait Desmarais ? Était-il possible que l'horreur que la tempête inspirait à Marnie Edgar se soit révélée à moi ce jour-là, près du bassin magique, me poussant à blesser Michael, à enfouir mes ongles dans sa chair sous le coup de la folie d'une autre, pour ensuite aller jeter sa chaussure sur les rives du lac aux Barbotes, loin du lieu du crime de Marnie Duchamp ? Mais alors, où aurais-je mis le corps ? Dans quelle crevasse ou grotte une enfant de onze ans aurait-elle pu dérober un corps à l'odorat des chiens et à la douleur d'un père enragé ?
Pendant des mois, je me suis torturée, seule sous les nuages obscurcissant le ciel d'orage. J'ai revécu mille fois la scène où Michael se fige, où il se tourne lentement vers moi, me chuchote son secret d'un air égaré, mauvais temps, madame, mauvais temps, puis s'enfuit à toutes jambes dans la forêt, là où il n'y a ni sentier, ni clairière, ni chemin de traverse. J'ai tenté de reconstituer la scène sous tous les angles, mais l'angle le plus juste était le mien, celui de Marnie Duchamp, la petite fille qui avait vu son meilleur ami céder à la pression d'un effroi qu'il lui avait transmis sans lui en expliquer la cause. Je n'avais rien oublié, voilà de quoi je suis parvenue à me convaincre. Je n'avais rien oublié de ce qui s'était passé avant, pendant et après cette scène, et pourtant, près de trente ans plus tard, j'avais senti le besoin de revenir sur les lieux du drame pour m'assurer qu'il ne s'agissait pas des lieux d'un crime.
Lorsque j'ai trouvé le DVD de Marnie dans le lecteur de mon père, me sont revenus en mémoire tous ces mois durant lesquels ma crainte de l'amnésie m'avait presque poussée à avouer un meurtre dont j'ignorais tout. Alors j'ai fermé les rideaux, j'ai déniché dans le garde-manger un sachet de pop-corn Orville Redenbacher, je me suis ouvert une bière et je me suis tapé une petite séance de cinéma maison.
À la fin du film, je ne savais plus qui j'étais. Les doutes que j'avais entretenus des années plus tôt ont reflué en masse, accompagnés de leur armada de questions pièges. Qu'est-ce que t'as raté, Marn ? Qu'est-ce qui t'a échappé, bordel ? La voix de la conscience coupable de la vilaine petite Marnie Duchamp était de retour, se frappant aux murs, qu'est-ce que tu caches ? qu'est-ce que t'oublies ? cherchant à y ouvrir des lézardes par où je serais aspirée vers le passé pour découvrir enfin, dans le film un peu jauni se dévidant derrière les murs fissurés, quelle main avait frappé Michael Saint-Pierre. Une angoisse sourde se mêlait à la colère que j'éprouvais contre Michael, contre la rivière, contre ce lieu maudit et contre Alfred Hitchcock. Sans réfléchir, j'ai retiré le DVD du lecteur et l'ai piétiné jusqu'à ce qu'aucune image de Marnie Edgar ne puisse y subsister. Je venais de détruire une des reliques de mon père et je m'en contrebalançais. Ma santé mentale méritait bien quelque infidélité à la mémoire des morts. J'ai enfilé mon manteau et mes bottes et j'ai claqué la porte derrière moi. Mon intention était de me rendre au bassin magique en vue d'y interroger tous les maudits arbres qui avaient vu Michael entrer en transe, mais la neige, le froid et l'obscurité m'en ont empêchée. Alors j'ai marché dans les rues de Rivière-aux-Trembles en sacrant des coups de pied sur toutes les mottes de glace qui avaient le malheur d'encombrer mon chemin. J'ai fait le tour du village trois fois, refrappant les mêmes mottes débiles, pour enfin m'arrêter devant la maison de Michael, où l'ombre immense de Victor Saint-Pierre se courbait derrière les rideaux. La maison n'avait pratiquement pas changé et l'immense galerie sous laquelle nous nous abritions du soleil ou de la pluie était toujours là, de même que l'atelier de Victor Saint-Pierre, derrière la maison.
C'est là que Mike, l'été de ses dix ans, avait découvert une boîte contenant une douzaine d'albums de Superman. Ces albums étant en anglais, son père avait accepté, devant l'excitation de Michael, de lui traduire les grandes lignes de A City Goes Dark le jour même. Dix minutes après la fin du récit de son père, Mike me téléphonait pour me dire qu'il venait de découvrir un superhéros encore plus fort que Batman et l'incroyable Hulk réunis. Viens-t'en, Marn, faut que je te montre ça. C'est ainsi que notre dévotion à un univers où les superlatifs comblaient les carences des mortels avait pris naissance. Au fil de l'été, son père lui avait sommairement traduit les autres albums qui s'entassaient dans la boîte, Michael m'avait à son tour raconté les incroyables aventures de l'enfant de Krypton en y ajoutant des détails de son cru, et notre existence en avait été transformée. Mike s'était immédiatement identifié à Superman, « the greatest exponent of justice the world has ever known », et j'étais devenue l'impétueuse et courageuse Lois Lane, la super-reporter qui suivait Superman à la trace et que celui-ci protégeait de tous les dangers. En l'espace de quelques jours, l'écureuille et le hibou que nous avions été jusque-là avaient disparu.
Phil a été le premier, les yeux enfoncés sous sa calotte du Vermont, à prétendre que je ressemblais à un petit écureuil, toujours à courir à gauche et à droite, à sautiller et à grimper partout. Juillet battait son plein et il était assis avec mon père sous la pergola, où ils buvaient une bière en se protégeant du soleil. Quant à moi, je faisais des pirouettes dans la cour. Après une culbute particulièrement audacieuse qui m'avait valu les applaudissements enthousiastes de papa et de Phil, j'avais demandé à ce dernier de refaire ma queue de cheval. Ma couette est démanchée, Phil, veux-tu me la rattacher ? Il avait lissé mes cheveux vers l'arrière, les avait noués avec un de ces élastiques multicolores que je perdais sans cesse, et il m'avait dit que ma couette rousse était pareille à la queue d'un écureuil. Pas vrai qu'elle ressemble à un écureuil, Alex, avec ses petits yeux brillants, à un petit squirrel espiègle ?
Papa avait acquiescé en me lançant un de ces sourires si pleins d'amour qu'ils vous obligent à vous sauver, au risque d'y être englouti. À partir de ce moment et de ce sourire, j'étais devenue son petit écureuil. Quelques jours plus tard, il avait emprunté le mot utilisé par Phil et s'était mis à me surnommer Squirrel, qu'il prononçait maladroitement Squouirèle, et ce surnom m'était resté. Toute sa vie, il avait continué à m'appeler Squouirèle avec un sourire en coin, quand l'émotion l'étreignait ou qu'il revoyait en moi l'enfant insouciante que j'avais été avant la disparition de Mike.
Peu de temps après ma première pirouette d'écureuil patenté, j'avais fait la connaissance de Michael. Son père était passé à la maison chercher une douzaine de roses pour sa femme, dont c'était l'anniversaire, tu me donnes tes plus belles, Alex, et il avait amené Michael avec lui. Pendant que papa faisait le tour du jardin et des serres avec Victor Saint-Pierre, j'avais entraîné Michael dans l'atelier pour lui montrer les cabanes d'oiseaux que mon père construisait, puis, de but en blanc, je lui avais révélé que j'étais une écureuille. T'es quoi, toi ? Il m'avait regardée avec de grands yeux ronds, pas trop certain du sens de ma question, puis il avait lâché un hibou, moi je suis un hibou.
Un hibou ? Pourquoi ça, un hibou ?
Parce qu'un hibou ça voit dans le noir, pis parce que ça voit tout, Marnie, absolument tout.
Ça voit aussi les écureuils, Michael, pis ça les mange...
Il était demeuré silencieux quelques instants, cherchant probablement le nom d'une bestiole qui ne mangeait pas d'écureuil mais avait un peu plus de gueule qu'un moineau ou une souris, puis il m'avait répondu que son espèce de hibou à lui mangeait seulement des frites. Je m'étais contentée de cette explication, puis, assis en Indien sous l'établi, on avait inventé notre première histoire d'écureuille et de hibou, une histoire se déroulant au milieu de la nuit parmi les arbustes crochus qui entouraient la mare à Mailloux, à l'heure où les hiboux voient tout.
Notre amitié avait pris naissance de cette façon, par la rencontre d'une écureuille et d'un hibou amateur de frites. Dans les coassements nocturnes s'élevant de la swamp de Gustave Mailloux, j'étais devenue Squouirèle, son amie pour toujours, et lui mon ami Owl, croix de bois, croix de fer, si je mens je vais en enfer, jusqu'à ce qu'on découvre Superman, Clark Kent, Lois Lane et Jimmy Olsen et qu'on oublie à l'orée de la forêt l'écureuille et le hibou qui avaient grimpé pendant plus de trois ans dans à peu près tous les arbres et poteaux de Rivière-aux-Trembles.
Je ne me souviens plus exactement quand Michael a commencé à m'appeler Lois, mais lorsqu'il se glissait dans la peau de Superman et qu'on jouait à sauver la terre des innombrables périls menaçant son équilibre, je n'étais plus ni Marn, ni Marnie, ni Squouirèle, mais l'incomparable Lois Lane. Quant à Mike, il ne répondait plus qu'au nom de Superman, que j'abrégeais généralement en Sup quand la tension de certaines situations exigeait qu'on fasse preuve de laconisme et de rapidité. Personne d'autre ne m'appelait Lois et personne d'autre n'appelait Mike Superman. C'étaient nos noms secrets, qu'on s'était promis de ne jamais révéler, même sous la torture, au risque de voir nos missions compromises ou d'éveiller la suspicion des invisibles ennemis de Superman disséminés sur la planète entière, y compris et surtout à Rivière-aux-Trembles, nouvelle Metropolis constituant le centre des activités interlopes de la racaille internationale.
Dès que je me transformais en Lois Lane, je n'étais plus une petite fille mais une héroïne sans peur et sans reproche s'avançant dans le large sillage de Superman, the Man of Steel. On feignait toutefois d'ignorer l'amour de Lois pour Superman et vice versa, que notre pudeur d'enfants trouvait stupide, sauf quand l'homme de fer me prenait la main pour franchir les bouillonnants rapides du ruisseau d'Alex Morin, un homme immense qu'on associait au redoutable Lex Luthor, ou lorsqu'on devait se serrer l'un contre l'autre dans un espace étroit pour éviter d'être découverts par nos poursuivants. Je redevenais alors une fillette aux joues trop rouges se demandant si c'était son cœur qui battait si fort ou celui du garçon dont elle sentait près d'elle l'odeur de shampoing Baby's Own. Il m'arrivait de regretter, dans ces moments-là, de n'avoir pas choisi de m'identifier à Supergirl, la fille au corps protoplasmique qui pouvait se métamorphoser à volonté en lampadaire ou en lapin de garenne.
À neuf ans j'étais déjà amoureuse, à ma façon, de mon ami Michael Saint-Pierre. Combien de temps cet amour exclusif aurait-il duré ? Je l'ignore, mais il est fort probable que l'un de nous deux se serait lentement détaché pour s'intéresser à une fille qui faisait du patin à roulettes, à un gars qui possédait une collection de cartes de baseball ou à une gang de tripeux qui écoutaient du Jethro Tull. On aurait pris nos distances, le cœur un peu gros, la conscience un peu coupable, pour considérer cette période de notre existence comme un lointain éden où notre seul souci était de nous lever assez tôt pour ne rien rater, ni l'éveil des oiseaux, ni l'arrivée du soleil derrière la colline surplombant la mare à Mailloux, ni le dernier chant des merles. On se serait éloignés et, les beaux soirs d'été, Michael aurait parlé à sa femme de son amie d'enfance, Marnie Duchamp, une petite rousse aux gestes nerveux d'écureuil avec qui il regardait chuter les Perséides en faisant des paris.
On aurait continué à se voir de loin en loin, j'imagine, à se croiser à Rivière-aux-Trembles à l'occasion des vacances, des fêtes de famille, des enterrements. On se serait maladroitement embrassés sur les joues à la sortie de l'église ou de l'épicerie, un peu anxieux, un peu timides devant cette personne avec qui on avait partagé jusqu'à notre dernière gomme baloune, dont on avait juré de ne jamais se séparer et dont on ne savait pratiquement plus rien. Puis un jour, Mike m'aurait appris qu'il venait d'avoir une fille. On l'a appelée Lois, m'aurait-il avoué en rougissant devant ce flagrant délit de nostalgie. Quant à moi, je l'aurais félicité en posant des questions sur la petite, quel âge, combien de dents, quelles couleurs de cheveux et d'yeux, des détails banals qu'il m'aurait fournis avec son sourire de jeune père convaincu que son enfant est la première et seule merveille du monde. Je serais ensuite retournée vers la maison de mon père en essayant de stopper les maudites larmes provoquées par la simple mention du nom d'une héroïne de bande dessinée, d'un personnage de fiction, d'une petite fille morte et enterrée dans la lumière du passé.
Il m'arrive cependant d'imaginer des scénarios dans lesquels Mike et moi on ne se serait jamais quittés, parce qu'on ne s'arrache pas un membre aussi facilement, parce qu'on ne sépare pas l'eau de la terre sans provoquer la sécheresse, des histoires dans lesquelles on aurait grandi ensemble, abandonnant peu à peu l'univers de Superman pour le cinéma de Huston ou de Leone, pour les randonnées dans les Adirondacks et le camping sauvage en Gaspésie.
Dans ces scénarios, on est pareils à ces vieux couples heureux n'ayant connu qu'un seul amour, un amour d'enfance, plus fort qu'un barrage de diamants de soixante-dix-huit carats, plus vrai que la vérité nue, et on a une flopée d'enfants dont aucun ne se prénomme Lois, Clark, Marnie, Superman, Squouirèle ou Owl, parce qu'on n'a aucun regret, parce que la nostalgie, c'est seulement bon pour ceux que la vie a expulsés du paradis.
Dans ces scénarios, Michael est vivant et il a une barbe, un pick-up rouge, une collection de pierres volcaniques et deux ou trois cents CD de musique country. Quant à moi, j'ai toujours l'allure d'une écureuille qui aurait trop grandi, et même si je sursaute au moindre bruit, prête à grimper à toute vitesse au sommet de mon arbre, je n'ai peur ni de l'orage ni de l'oubli. Je m'appelle Marnie et je m'en fous. Je me demande encore à quoi mes parents ont pensé en me donnant ce nom, mais ça n'a pas de réelle importance, parce que je ne me sens aucun lien avec la créature de Winston Graham et d'Alfred Hitchcock, si ce n'est que j'aime aussi les chevaux, mais qui n'aime pas les chevaux ?
# DEUXIÈME PARTIE
Un convoi funèbre descendait la rue Principale de Rivière-aux-Trembles, le 9 février 2009, quand Bill Richard était arrivé au village au volant de son U-Haul deux tonnes, la tête alourdie par ce qu'il nommerait dans son souvenir sa fièvre australienne. Il avait pris l'autoroute à la sortie de la ville, y avait roulé pendant près de deux heures sur la voie de droite, ne dépassant aucun véhicule, à l'exception d'un camion-citerne dont le chauffeur avait dû être contaminé, comme lui, par les feux de l'Australie, puis il avait pris un embranchement et s'était arrêté pour consulter le plan sur lequel il avait tracé, à l'aide d'un feutre orange emprunté à Billie, le trajet qui le mènerait vers la maison où il allait tenter de se refaire un semblant de vie.
À plus d'une reprise, il avait cru que la douleur qui lui fendait le crâne l'obligerait à s'arrêter dans un motel minable où la tentation de s'ouvrir les veines mettrait fin à la carrière de la femme de chambre qui découvrirait le nom de Billie en lettres de sang sur les murs tapissés de fleurs jaunes, boutons d'or ou tournesols depuis longtemps asphyxiés par l'absence de ce qu'il appelait de la lumière propre, de la lumière de maison heureuse, aux carreaux étincelants et aux rideaux légers. Il fallait être fou pour prendre la route alors que la réalité se dissolvait au pourtour du regard fiévreux, mais il fallait l'être encore plus pour s'imaginer qu'un total isolement était susceptible de combler le vide de l'absence.
Après sa première peine d'amour, une rupture inattendue qui lui avait donné un aperçu de la misère des romantiques, il avait affiché derrière la porte de sa chambre un vers de Lamartine : « Un seul être vous manque, et tout est dépeuplé. » Il avait collé le bout de papier au-dessus du jeu de fléchettes sur lequel il avait punaisé la photo de la fille qui avait massacré ses illusions et où il exerçait parfois son tir avant de se coucher. Tandis qu'il se morfondait dans ses draps froids, il s'était souvent interrogé sur la justesse des mots de Lamartine. La perte entraînait-elle le dépeuplement ou, au contraire, était-elle à l'origine d'un insoutenable surpeuplement ?
Depuis la disparition de Billie, la question avait refait surface, toujours aussi insoluble, lui rappelant amèrement que la solitude de l'endeuillé se définit aussi bien à l'aune du néant que de la multitude. Certains jours, il lui semblait que le monde était incroyablement encombré, saturé de visages et de corps sans substance lui masquant celui de sa fille. À d'autres moments, Billie se trouvait partout, dans toutes les images et toutes les pensées, à tous les coins de rue, et c'était elle qui surpeuplait la terre. Mais cette surpopulation, il le savait, était le signe d'une forme de dépeuplement mettant en lumière l'inutilité du surnombre. Il ne savait donc pas, ce matin-là, s'il se dirigeait vers une maison dont les couloirs seraient hantés par la multiplication du même visage ou s'il aboutirait dans un lieu ayant des allures d'après cataclysme, quand un seul survivant promène son ombre sur la désertification des villes.
Lorsqu'il avait vu descendre vers lui le corbillard rayonnant de propreté sinistre, il avait pensé qu'un autre monde se dépeuplait. Cet accueil n'augurait rien de bon. S'il avait su qu'il mettrait les pieds en territoire inconnu en même temps qu'y défilait la mort, il aurait peut-être choisi le motel, où il se serait abîmé dans le tambourinement de la pluie de février sur les carreaux salis en écoutant à la radio des chansons country remplies d'âmes errantes et solitaires. Il aurait attendu là, dans ce monde imbibé d'alcool et d'hommes avalant la poussière des chemins, que la fièvre s'atténue et que l'esprit du mort quitte la zone d'attraction des vivants.
Puisqu'il n'était pas encore tombé assez bas pour se comporter en minable, il s'était garé près du trottoir le temps que la procession passe et en avait profité pour sortir prendre un peu d'air. Appuyé à l'arrière de son U-Haul, il comptait des cailloux blanchis par le gros sel quand l'étincelante Lexus suivant le corbillard était arrivée à sa hauteur. Instinctivement, il avait levé la tête et avait aperçu la femme, à l'arrière, la fille aux yeux rouges qui ne regardait rien et semblait implorer ce rien de briser le sort la réduisant à l'immobilité. Puis leurs regards s'étaient croisés et une brève douleur à la poitrine, aussi rapide qu'un coup de poignard porté par une main experte, s'était mêlée aux frissons qui couvraient son corps.
Pendant un instant, il avait eu la certitude de connaître cette femme. Puis le poignard s'était retiré de sa poitrine et il avait compris que c'était la douleur qu'il reconnaissait, la douleur et les yeux rougis qui l'imploraient maintenant, au cœur du paysage dévasté, de faire bouger le temps désespéré du deuil. Il reconnaissait la douleur, puis le regard, semblable à celui de L.A. exigeant de lui l'impossible : fais quelque chose, Bill, ramène-moi ma fille, ramène-moi ma petite... Puis il avait vu ses propres yeux, cercles froids et vides figés dans le miroir comme des billes au fond d'un étang, incapables d'ordonner à son corps de se remuer un peu. Il était parfois resté là mille ans, debout devant le miroir, rien que debout et vide. Les paupières rougies de la femme plongeaient le monde dans la même léthargie et il lui semblait que le corbillard n'avançait plus, que le vent arrêté maintenait entre ciel et terre une feuille arrachée au sol. Le sentiment, alors, que cette femme enterrait son fils, son gamin riant dans le lointain de l'éternel rire des enfants disparus, l'avait paralysé, car qui d'autre qu'un enfant pouvait arrêter le vent ?
Quittant un instant la feuille immobile, il s'était tourné vers les occupants des autres voitures, y cherchant des petites têtes blondes ou rousses, des copains de classe, des garçons et des filles en costume de scout ou de jeannette, et il avait ressenti un intense soulagement quand il avait constaté qu'aucun enfant ne participait à cette procession et que le défunt, selon toute évidence, avait atteint un âge susceptible de voiler l'indécence de la mort. À ce moment, le charme s'était rompu et il était remonté dans son camion en mettant sur le compte de la mouche folle du kangourou ou de quelque virus directement importé d'Australie le trouble qui lui avait percé la poitrine avec la froide efficacité d'une arme blanche.
Dans la voiture suivant le corbillard, Marnie Duchamp avait suivi de ses yeux rouges cet homme qu'elle surnommerait l'homme fiévreux. C'était cette fièvre qui l'avait frappée, pareille à la sienne, incrustée sous les paupières. Une fièvre longue qui ne quitterait jamais l'homme et laisserait en lui le souvenir du corps luttant contre les mirages. Quand il avait disparu de son champ de vision, il lui avait semblé voir l'aura de chaleur ceinturant son front, jaune et semée d'épines rouges, comme un champ d'herbe brûlé par la chaleur d'août dans un tableau de Van Gogh. Puis l'image de son père, Alex Duchamp, avait repris sa place dans l'enfilade d'arbres ternes emmurant le cimetière, jusqu'à ce que le corbillard s'immobilise et que deux hommes gantés de blanc fassent glisser sur un tréteau le cercueil de bronze qu'aucun soleil ne faisait briller. Elle s'était alors penchée sur ce cercueil, l'avait touché, mouillé de ses larmes et embrassé, transmettant sa chaleur au mort à travers le métal froid, love you forever, pop, puis Phil Morisset, l'ami de toujours, avait tendu son bras sur ses épaules. À travers les larmes, les paupières sombres de l'homme fiévreux lui étaient apparues un instant, flottant au milieu du rassemblement d'hommes gris venus rendre un dernier hommage au jardinier de Rivière-aux-Trembles, puis elle l'avait oublié lorsque les portes de la chapelle où le cercueil serait remisé jusqu'au printemps s'étaient refermées dans un grincement définitif.
Quelques pétales de Mary-Jean avaient volé dans le grincement des portes, des têtes s'étaient baissées en se signant et les corps gris s'étaient dispersés. Marnie avait pour sa part erré dans les rues du village, puis sur la route longeant la colline des Loups, déchirée entre la force d'attraction et de répulsion d'un paysage dont les couleurs contrastées, arbres noirs sur fond blanc, reflétaient en quelque sorte l'univers incolore dans lequel elle avait l'impression de s'avancer, se demandant depuis toujours si elle méritait la beauté du monde. Un peu plus tard, alors que la lumière ne subsistait plus qu'en vagues traînées bleues derrière les nuages, un cri venu d'elle ne savait où avait réveillé l'enfance et ses couleurs vives et elle avait décidé qu'elle reviendrait vivre à Rivière-aux-Trembles, là où tout avait commencé et où tout devait se terminer. Si la beauté du monde l'attendait quelque part, ça ne pouvait être qu'à l'endroit où on la lui avait enlevée. En repassant devant le cimetière, l'image de l'homme aperçu plus tôt avait effleuré son esprit, mais elle l'avait rapidement chassée, de même que le frisson qui l'avait poussée à hâter le pas et à resserrer sur elle les pans de son manteau noir.
## I
BILL
Février-Avril 2009
Sur le toit du hangar chambranlant jouxtant ma nouvelle maison, Ronie le crapaud volant expérimente la cape de supercrapaud que j'ai dessinée pour lui aux environs de minuit, la nuit dernière, à cette heure fatidique où tout peut arriver, y compris le sommeil. Le sommeil refusant toutefois d'empiéter sur le terrain encombré où mes pensées se bousculaient, je me suis relevé, poursuivi par cette idée débile de supercrapaud. Si je ne m'occupais pas de ce batracien, ses coassements se mêleraient au silence et m'empêcheraient de dormir jusqu'au matin. Plutôt que d'écrire le début de ses aventures, j'ai préféré dessiner d'abord Ronie avec sa cape, me disant que l'histoire me viendrait plus facilement si j'avais mon héros sous les yeux. J'aurais voulu lui faire porter une cape pareille à celle de Batman, mais je me suis emmêlé dans les plis, alors j'ai écrit « cape » sur la cape en question, au cas où mes exégètes voudraient savoir ce qui pend dans le dos de ce crapaud. Le résultat aurait sûrement été aussi minable si je n'avais pas raté mes plis, mais le but de l'opération n'était pas de gagner un concours de dessin. Je voulais simplement m'occuper l'esprit et meubler d'une façon ou d'une autre le silence qui s'abat sur ce trou perdu dès que vous fermez la télé ou la radio, un silence d'une telle intensité qu'il vous emmure et vous pousse à vous demander si vous ne seriez pas par hasard le seul rescapé d'une catastrophe que n'aurait pas annoncée le présentateur de nouvelles. Inutile d'essayer de chanter pour déjouer ce calme oppressant, parce que le silence demeure là, tout autour, qui donne un son métallique à votre voix, qui comprime la misérable chanson et la réduit à une dimension presque nulle tout en accentuant l'absence qui vous entoure.
Je ne m'étais jamais imaginé, avant d'emménager dans cette maison, que le silence puisse empêcher un homme de fermer l'œil et le tenir à l'affût du moindre craquement s'élevant du plancher ou de l'étage au-dessous, et pourtant je tendais l'oreille, attentif à tout ce qui traversait et rompait le silence, à ces froissements furtifs de la matière, de la vie animale, de la vie des choses.
Nuit après nuit, je découvrais que le silence était un révélateur, une forme d'absence quasi palpable grâce à laquelle il m'était possible de percevoir ce qui se déplaçait sous la surface. Le silence m'apprenait que les lieux respiraient, que la forêt était habitée de présences invisibles à l'œil de l'homme quand il fait noir, et que je n'étais jamais seul dans le désert que j'avais tenté de créer au sein de l'hiver. À Rivière-aux-Trembles, je constatais que tous les vides étaient pleins et que le monde bougeait incessamment. L'agitation de la ville ne m'avait pas préparé à affronter la vie dans ses manifestations les plus calmes et les plus lentes, et j'apprenais en cela que si un homme peut s'isoler des autres hommes, il ne peut se couper de la vie.
J'avais presque oublié ce silence pour ne l'avoir connu qu'autrefois, dans les camps de vacances où mes parents m'envoyaient ruminer ma tristesse d'enfant solitaire sous la voûte étoilée de l'hémisphère nord pendant qu'ils se payaient un voyage aux antipodes. Durant ma jeunesse entière, j'avais été persuadé que l'expression « camp de vacances » était synonyme de l'hypocrisie des adultes, qui vous faisaient croire qu'ils vous offraient des vacances, alors que cette réclusion forcée au royaume des feux de camp et des chansons de bivouac leur permettait de se la couler douce sans vous avoir dans les pattes. Dans la splendeur de juillet, le mot « vacances » s'appliquait aux parents, et j'avais par la suite juré que jamais je n'enverrais Billie, ma fille, dans une de ces colonies d'enfants abandonnés aux punaises, aux ours, aux sangsues et autres bestioles s'épanouissant dans l'air pur et vivifiant de la campagne. J'avais tenu ma promesse, Billie n'avait jamais eu à entonner « valderi, valdera » à cinq heures du matin, ce dont le citadin en moi n'était pas peu fier quand il se remémorait la détresse qu'il avait éprouvée dans les lueurs de l'aube soulevant au-dessus du lac à la Truite ou du lac aux Perchaudes un brouillard duquel s'élevait le chant déchirant des huards. J'avais six ans, sept ans, huit ans, et me demandais si les huards pleuraient aussi, si leurs larmes d'oiseau augmentaient la salinité des lacs. Mes séjours dans l'humidité de cabanes en bois rond ou en planches ne m'avaient appris que la détresse. Et voilà que je me retrouvais là, au milieu de nulle part et d'un silence plus déroutant que celui des aubes froides du lac aux Perchaudes, à réitérer la promesse faite à Billie en des temps si lointains qu'ils me semblaient appartenir à une autre vie : Ben non, puceronne, ben non, penses-tu que je t'enverrais te faire manger par les maringouins ?
Il n'y a pas de maringouins à Rivière-aux-Trembles en ce matin de février, mais il y a une enfant, une petite Billie dont la présence prend des proportions à ce point envahissantes qu'il me faut me réfugier auprès d'un crapaud harnaché d'une cape que seul un ouragan pourrait soulever pour ne pas étouffer dans l'exiguïté de cette maison trop pleine de Billie appelant à l'aide. Tiens bon, Billie, tiens bon, crie le crapaud volant, Batcrap is coming, mais Batcrap ne vaut pas mieux que de la schnout, aussi merdique que Superdad, qui n'a pas été fichu, durant trois années entières de recherche, de mettre la main sur la barrette en forme de papillon qui l'aurait guidé vers l'antre de l'homme gigantesque ayant enlevé sa fille.
C'était la raison pour laquelle j'avais mis le cap sur Rivière-aux-Trembles, pour tenter de fuir l'image de cet homme, pour tuer dans l'œuf le doute qui me rongeait quand je devenais fou, quand Ménard venait essuyer ses grosses bottes crottées jusque dans mes rêves ou que L.A. crachait son fiel sur mes mains blanches. Ma réaction était des plus normales. Tous les animaux fuient devant le feu et la mitraille, tous les êtres vivants, et les arbres fuient aussi, à leur manière, en s'éloignant des zones désertiques, des marais rongeant leurs membres. Puisque j'étais vivant et pas encore totalement cinglé, j'avais pris mes jambes à mon cou, inconscient que la bête que je tentais de semer avait fait son nid dans mes entrailles, que l'homme est un putain de cheval de Troie transportant dans ses tripes tout ce dont il a besoin pour s'autodétruire et s'empoisonner la vie, à commencer par l'attirail de souvenirs tranchants qui lui lacèrent les flancs à chaque faux pas. On ne peut rien contre cette tumeur qui prolonge ses métastases du cerveau jusqu'au ventre. La seule façon de fuir sa mémoire, c'est de se faire lobotomiser. Je n'en étais pas encore là, mais il m'arrivait d'envisager cette option lorsque les heures s'étiraient dans tous les sens et que le cafard, avec sa flopée de pensées visqueuses, profitait de cet instant de stagnation universelle pour me sauter dessus. La solitude a beau tenter de s'occuper les mains, elle finit toujours par se ramasser au milieu de ce vide sans lequel elle n'existe pas, de cette vaste plaine silencieuse où elle s'étale, toute nue dans sa vérité, face à face avec elle-même.
Je ne regrettais pas de m'être réfugié à Rivière-aux-Trembles. Le total dépaysement que m'offrait ce lieu me permettait parfois d'oublier que je m'appelais Bill Richard et que ma fille avait fait la une des journaux à potins. Je n'irais pas jusqu'à dire que j'y avais déniché la sacro-sainte paix à laquelle aucune forme de vie ne peut aspirer, mais il m'arrivait d'y rêver crapaud et de redevenir un écrivain pour quelques heures. C'était déjà ça de pris, quelques heures sans la douleur de Billie, mais quand ma montre s'arrêtait, j'aurais préféré me retrouver n'importe où ailleurs, là où il y avait du bruit, des concerts de klaxons, des tirs de marteaux-piqueurs, des feux d'artifice et des explosions de trompettes, parce qu'au lieu de se calmer, la pensée choisit toujours ces moments pour plonger dans ce qu'elle trouve de plus noir et de plus visqueux. Quand l'aiguille des secondes s'immobilisait au cadran de l'horloge, je piquais aussi une tête pour refaire surface pas loin d'un plan d'eau dormante, avec une petite Billie dans les bras, une petite Billie flasque et mouillée, une enfant dont la peau froide ne respirait plus. Et je me mettais à brailler. Qu'est-ce que t'as fait à ma puce, mon hostie de sale ? Pourquoi tu t'en es pris à elle, crisse de malade ? Pourquoi ? Pourquoi ?
Cette question, je me l'étais posée des dizaines de fois et je continuais à me la poser. Pourquoi Billie ? Pourquoi ma fille ? Invariablement, toutes les sous-questions inavouables qu'elle supposait rappliquaient en masse, car si le mal ne s'était pas attaqué à Billie, aurait-il abattu sa grande cape noire sur une autre fillette ? Sur Anna-Sophia Smith, dix ans, vue pour la dernière fois dans sa classe de gymnastique, portant un t-shirt blanc parsemé d'étoiles rouges quand elle avait dit au revoir à ses amies ? Sur Juliette Masson, neuf ans, disparue à l'arrêt de l'autobus scolaire, où sa petite botte de caoutchouc maculée de neige sale avait été retrouvée par sa mère hystérique deux heures plus tard ? Sur Mathilde Dumas-Benoît, surnommée Mattie, huit ans, qui s'était absentée pour aller aux toilettes et n'était jamais revenue ?
Quand je m'engageais dans ce type de réflexion, je cherchais désespérément autour de moi un objet susceptible de m'offrir une forme de diversion et d'effacer du même coup les noms des fillettes, de faire taire leurs cris et de m'empêcher de suffoquer, parce qu'il me semblait inadmissible d'imaginer qu'une autre enfant, une autre petite fille vêtue d'un parka rose ou blanc comme on en voit tant, aurait pu être la proie de l'agresseur de Billie.
Et pourtant, ces pensées me traversaient l'esprit. Que serait-il arrivé si Billie avait été malade, si elle était demeurée bien au chaud à la maison le jour où cette pourriture avait décidé d'enfiler ses gants d'assassin ? Que se serait-il produit si elle avait quitté l'école un peu plus tôt ou un peu plus tard ? Anna-Sophia Smith se serait-elle volatilisée dans le froid de janvier à la place de Billie ? L'homme qui veillait dans l'habitacle humide de sa voiture aurait-il attendu le lendemain ou se serait-il jeté sur la première gamine qui passait par là, Juliette ou Mathilde, perdue dans ses pensées et fredonnant une comptine qu'elle déclinerait tout à l'heure devant son chien, son chat, son poisson rouge ou sa poupée préférée ? Billie faisait toujours ça. En rentrant de l'école, elle récitait ses leçons à Pixie. Elle lui apprenait la table des multiplications et lui montrait sur le globe terrestre où se situaient la Chine et le Groenland. Là, Pixie, y a des ours polaires, tout blancs, OK ? Pis là y a des pandas. Ça mange du bambou. Il ne devait pas y avoir de chat plus instruit dans toute l'Amérique du Nord, et si la fatalité dans ce qu'elle a de plus minable n'avait pas emporté la puce loin de Pixie, ce vieux batêche serait sûrement devenu prof de maths ou anthropologue.
Mais le destin a frappé et jamais personne ne saura si Billie se trouvait seulement au mauvais endroit au mauvais moment ou si elle avait été choisie par son agresseur, si elle avait été l'élue de la folie d'un homme qui conservait de son enfance des souvenirs de petits manteaux roses, de cheveux fins flottant au vent, d'objets qu'il lui fallait à tout prix posséder pour tuer l'angoisse entourant la course si douce des fillettes d'autrefois, le frôlement de leurs jupes sur ses cuisses meurtries. On a beau mettre les enfants en garde sur tous les tons et dans toutes les langues, ça ne suffit pas, ils sont trop confiants pour déceler la puanteur du mensonge. C'est cette pureté qui perd ceux qui se perdent. Pour soupçonner le mal, il faut avoir la capacité de le concevoir, et de cela, Billie était incapable, sinon je ne serais pas là à échafauder toutes ces hypothèses avec lesquelles j'affûte mes propres instruments de torture.
L.A. aussi était passée maître en la matière. Pas une journée ne s'écoulait sans qu'elle se demande ce qui serait arrivé si j'avais, si elle avait, si nous avions, si la terre entière avait agi autrement et tourné dans le sens contraire, quitte à rendre fous ceux qui n'avaient pas ajusté leur montre et croyaient que la Chine roupillait pendant qu'ils avalaient leur bol de Corn Flakes. Elle se déchirait à grands coups de griffes et de dents en mettant le passé au conditionnel et en s'accusant de tous les maux parce qu'elle n'avait pas le pouvoir de remonter le temps et d'abolir le conditionnel.
Sa mère l'avait traînée, deux ou trois mois après la disparition de Billie, à la réunion mensuelle d'une association regroupant des proches de personnes disparues, mais L.A. était dans un état à ce point catastrophique à ce moment-là qu'elle ne pouvait admettre le moindre contact avec des gens prétendant avoir connu une souffrance pareille à la sienne.
Régine avait ensuite tenté sa chance avec moi, au cas où je pourrais sortir Lucy-Ann du trou en me frottant à la douleur des autres, mais j'avais gentiment repoussé son invitation. Non pas que je refusais d'aider L.A., non pas que je mettais en doute la pertinence d'une telle association, mais j'étais du genre cavalier solitaire, du genre qui préfère souffrir tranquille dans son coin plutôt que de vider ses tripes devant une assemblée en buvant du café pas buvable dans des verres de polystyrène. Chacun son rituel. Chacun sa façon de lécher sa plaie. Contrairement à L.A., je n'avais cependant aucune difficulté à croire que d'autres malheurs puissent se comparer au mien. Je préférais juste saigner tout seul, sans l'aide de personne.
De toute façon, le malheur des autres, vous n'aviez pas besoin de vous y frotter pour le recevoir en pleine face. Il entrait chez vous par la télé, par la radio, au gré des milliards d'images charriées sur Internet, il se glissait dans votre cuisine avec le livreur de journaux, et la limite acceptable se situait là, entre l'image et la réalité. En ce qui me concernait, je ne pouvais me rapprocher davantage de la souffrance universelle. Autrement, tu crèves, tu vires fou ou tu te transformes en missionnaire, et je n'avais pas la fibre du gars qui décide de tout plaquer un beau matin dans l'intention d'éradiquer le sida de l'Afrique. Billie aurait pu faire ça, Billie aurait eu ce courage, cette grandeur d'âme, mais moi, j'étais taillé pour macérer dans mon petit étang de merde, où rebondissaient parfois quelques cailloux venus de l'extérieur, plus ou moins gros selon les jours.
Au cours des deux premières semaines passées à Rivière-aux-Trembles, on aurait d'ailleurs dit que tous les petits maudits cailloux de la planète s'étaient ligués pour sauter en même temps dans mon trou de bouette et m'éclabousser autant qu'ils le pouvaient. Je n'avais qu'à ouvrir le journal pour tomber sur la photo d'un adolescent disparu, d'un pauvre gamin au visage doux aperçu une dernière fois au moment où il montait dans un véhicule sombre, pour voir étalé devant moi le compte rendu des funérailles de deux tout-petits, deux tout petits anges, assassinés par leur propre géniteur, pour entendre les hurlements des deux poupons, encore des anges, qu'un tueur fou avait sauvagement poignardés, pour entendre les cris de dénégation de leurs mères, de leurs pères, de leurs frères. Je m'installais devant la télé et j'étais bombardé par la même violence absurde : « Fusillade à Cleveland », mort de trois enfants et de deux femmes, « Carnage dans un collège allemand », neuf élèves tués par balle, leur sang mêlé à celui de six autres innocents.
Je fermais la télé en jurant et une photo de la puce se superposait à l'écran noir, inévitablement, la photo où elle retroussait son sourire espiègle vers sa pomette gauche, publiée dans tous les journaux de la province, expédiée à tous les corps de police du continent, affichée sur tous les poteaux, babillards et feuilles de plywood accotées sur les murs borgnes que j'avais croisés sur mon chemin. Billie Richard. Huit ans et neuf mois. Cheveux bruns. Yeux noisette. Portait au moment de sa disparition un manteau rose à capuchon, un papillon dans ses cheveux, deux papillons sur ses chaussettes. Ressemblait aussi à un ange. Forte récompense promise pour tout papillon rose capturé dans le sillage d'un ange.
Il était impossible que personne ne l'ait aperçue, et pourtant, aucune piste ne nous avait menés jusqu'à elle. Tous les appels reçus par la police venaient de timbrés, de minables alléchés par l'odeur de l'argent, de pauvres bonnes femmes qui essayaient de combler le vide de leur existence en s'inventant des chimères, de gens à ce point rongés par la culpabilité, par la peur viscérale de ne pas avoir agi quand il le fallait qu'ils voyaient Billie partout. Dans tous les cas, il s'avérait que l'enfant aperçue n'était pas Billie, mais une des dizaines de gamines portant des manteaux de petite fille, de la couleur des bonbons, des fleurs, des petits fruits trempés dans le lait.
Certaines préféraient le jaune ou le bleu, mais Billie étirait sa période rose depuis qu'elle savait prononcer ce mot. L'été, on allait au marché ensemble et on achetait de gros paniers de fraises de l'île d'Orléans, bien rouges et bien juteuses, que Billie s'empressait d'écraser dans le yogourt nature ou la crème glacée à la vanille dès qu'on était rentrés à la maison. Regarde comme ça fait une belle couleur, popinouche, et elle éclatait de rire parce qu'elle m'avait encore appelé popinouche, alors que je m'évertuais à lui répéter que mon vrai nom de papa, c'était papanoute, empereur de la dynastie des Noutes, à laquelle n'appartenaient que les gentils : mamanoute, papanoute, Billienoute, Pixienoute...
Elle aimait le rose comme on peut aimer l'odeur de la lavande ou des lilas, notre Billie, et comme on peut vouloir s'en entourer. Elle aimait le rose qui éclatait dès juin dans les jardins de la ville, le rose qui s'étalait dans le ciel le soir après souper et celui qu'elle savait créer avec quelques fruits frais, mais, après le 20 janvier 2006, aucun des petits manteaux roses à avoir sillonné les rues de la ville merdique qui me l'avait enlevée n'était celui de Billie. Pourtant, j'ai bien cru l'apercevoir à plusieurs reprises, moi aussi. Il me suffisait de distinguer une petite tache rose qui sautillait dans la foule pour me ruer entre les passants en criant son nom. Chaque fois, je me retrouvais en nage devant un père prêt à me mettre son poing sur la gueule ou une mère décidée à m'arracher les yeux avec ses ongles si je ne dégageais pas illico. Alors je souriais à la petite et je retournais à ma place, nulle part, dans la ville surpeuplée, incompréhensible sans Billie.
J'ai continué à traquer les petits manteaux roses pendant trois ans, même en été, même au plus fort des chaleurs de juillet, mais aucune Billie ne se cachait derrière ces éclairs de couleur fendant la grisaille. Billie avait disparu. Billie s'était volatilisée et personne ne l'avait revue, personne ne connaissait l'endroit où elle se trouvait, sauf l'homme aux mains rugueuses, l'homme de la voiture sombre, qui seul savait si elle avait continué à grandir et si son petit manteau avait dû être remplacé. Cet homme savait, ce fumier avait touché ses mains potelées, peut-être son visage, ses cheveux d'enfant et leur odeur de shampoing à la pêche, peut-être, et cette perspective me rendait fou. J'aurais préféré qu'un camion la fauche, comme le petit Dumas, qu'un soûlard la renverse après une partie de poker clandestine dans un bouge du Centre-Sud ou qu'une de ces maudites maladies dont nous tentions de la protéger, L.A. et moi, l'emporte avec la neige. Au moins, j'aurais su. Au moins, je n'aurais pas été forcé d'imaginer l'inimaginable ni de m'éveiller avec l'image d'une crisse de grosse main sale caressant les cheveux d'enfant de ma Billie, morte ou vive.
Mais Billie n'était plus là, c'était la seule vérité qui vaille, et le misérable semblant de survie que ma mémoire lui offrait ne valait pas cinq cennes. J'aurais beau m'imaginer berçant Billie pendant cent ans, aucune Billie ne sentirait la chaleur de mes bras fatigués. La survie des disparus dans le cœur de ceux qui les chérissent, c'est de la bullshit.
De la bullshit, calvaire ! ai-je hurlé en renversant la chaise sur laquelle j'étais assis depuis le matin, à surveiller Ronie le crapaud en attendant que la minuterie de l'horloge réglant mes jours se remette en marche. Tant qu'à y être, j'ai aussi renversé la table, qui a entraîné deux autres chaises dans sa chute, meublant ainsi le silence du fracas tant espéré. Le temps avait enfin repris son cours et j'avais survécu à cette plongée en apnée dans ses couloirs les plus étroits.
Je me suis dirigé vers l'évier pour m'y plonger la tête sous l'eau froide et faire cesser le tremblement de mes mains, puis j'ai évalué autour de moi l'étendue des dégâts. Sur le sol, les deux rôties que je n'avais pas touchées trempaient dans le café froid, à côté de morceaux de vaisselle fracassée, de tessons de verre et d'un litre de lait dont le contenu s'écoulait lentement vers la flaque de café noir. J'en aurais pour une bonne heure à remettre la cuisine en ordre et à laver le plancher. C'était déjà ça de pris et je n'allais pas me plaindre de cette diversion. Je ne savais pas encore comment j'occuperais le reste de cette journée, mais je trouverais un moyen, il le fallait, pour oublier que partout sur terre, même à Rivière-aux-Trembles, des hommes gigantesques pouvaient surveiller les fillettes fredonnant machinalement en rentrant de l'école.
Je me suis éveillé ce matin en plein milieu d'un rêve où l'Australie brûlait, faisant fuir tous les animaux, parmi lesquels trottinaient de minuscules koalas cherchant à s'accrocher à mes jambes, pareils à ceux dont j'ai vu défiler les yeux tristes sur Internet, hier soir, entre quelques rainettes aux yeux rouges et l'image de Billie pénétrant la totalité des images. J'avais réussi à tromper la lenteur de la journée en nettoyant la cuisine et en traînant ma moppe dans le reste de la maison, jusqu'à ce que le soir tombe et que je me rabatte sur un de ces écrans qui vous montrent le malheur des autres, espérant que le monde merveilleux de Google me distrairait de ma propre douleur et du monde abyssal se développant dans le creux de mon nombril.
Après quelques recherches sur le comportement et l'habitat des grenouilles tropicales, de la grenouille túngara au phyllobate tricolore, dont je prévoyais inclure quelques spécimens dans les aventures de Ronie le crapaud, question d'ajouter un peu de couleur au monde relativement verdâtre des amphibiens, je suis tombé sur un site reproduisant des photos de koalas prises pendant les chaleurs atroces qui s'étaient récemment abattues sur l'Australie. J'ai fait défiler les photos une dizaine de fois et je suis allé me chercher un immense verre d'eau froide en remerciant le ciel d'être né dans un pays qui ne connaissait pas la sécheresse, la vraie, celle qui pousse des animaux sauvages, conçus pour résister aux pénuries d'eau, à implorer des hommes de leur donner à boire.
Les deux premières photos montraient un de ces petits bonshommes agrippé aux jambes d'un cycliste le faisant boire à même sa bouteille. Sur les autres, figurait l'un de ses frères ou sœurs en train de prendre un bain sous le porche d'une maison, dans une cuvette où il s'était réfugié pour se protéger des 44 degrés cuisant la terre australienne depuis des jours. Il regardait l'objectif avec ses petits yeux ronds, trop assoiffé, trop exténué de chaleur pour avoir peur. Encore une fois, j'ai pensé à Billie, qui pouvait parfois vous regarder avec des yeux pareils, des yeux désemparés qui disaient je suis trop petite pour comprendre, popinouche, trop petite pour affronter ça toute seule, la vie, la faim, la soif, le noir. J'ai pensé à ses animaux de peluche encore alignés sur une tablette, dans sa chambre, parce que rien ne ressemble plus à un animal de peluche qu'un koala. J'ai pensé au vieux Pixie, mort d'avoir été privé de Billie, et j'ai refermé l'ordinateur en pestant contre l'Australie. Je ne savais pas ce que me voulait ce pays, ce que me voulait ce continent, mais j'en avais assez de ses feux, de sa sécheresse, de ses koalas assoiffés et de ses kangourous, pourquoi pas, qui devaient eux aussi sautiller sous un soleil de plomb à la recherche d'un filet d'eau.
L'Australie me poursuivait alors que je n'y avais jamais mis les pieds et ne connaissais de cette région du monde que les images glanées sur Internet, à la télé ou dans le National Geographic que je feuilletais parfois chez le dentiste ou chez le pédiatre de Billie, à qui j'avais expliqué un jour les particularités du développement embryonnaire des marsupiaux. Elle m'avait talonné durant une semaine, par la suite, pour que je lui dessine des mamans kangourous. Des roses, papanoute, avec des poches roses, pas des grises, c'est pas beau, si bien que mon Australie à moi est une Australie peuplée de kangourous roses, moitié chimérique moitié folklorique, une Australie dont les paysages incertains empruntent une part de leur relief et de leur végétation à l'Afrique et au continent sud-américain. Dans mon cauchemar, le feu dévorait avec la même furie une improbable forêt d'eucalyptus et de palmiers d'où s'enfuyaient des troupeaux d'animaux effrayés. Et je courais avec eux, asphyxié par la proximité des flammes, trébuchant sur des souches grouillant de fourmis qui quittaient le navire, et courant encore, dérouté par ce pays qui me forçait à fuir mes rêves.
Le jour se levait quand j'ai tiré les rideaux du salon, mais c'était un jour incertain, assombri de nuages qui apporteraient assurément de la neige. La présentatrice du canal météo, juchée sur ses talons d'été, avait d'ailleurs annoncé la veille qu'une grande partie de la province recevrait quelques flocons. Les régions plus à l'est, quant à elles, risquaient d'essuyer le passage d'un tempête qui blanchirait le sol croûté de pas moins de vingt centimètres de neige fraîche. Cette perspective me réjouissait, car il n'était pratiquement rien tombé depuis mon arrivée à Rivière-aux-Trembles, si ce n'est un peu de pluie d'hiver, et cette absence de précipitations, après la sécheresse de l'Australie, me faisait entrevoir des paysages apocalyptiques étouffant sous une poussière grise. Je guettais l'arrivée des nuages en me demandant ce que deviendraient les écureuils, les renards, les mouffettes et les crapauds si l'eau venait à manquer. L'homme se débrouillerait bien pour piller quelque source enfouie, mais comment survivraient les animaux ?
J'espérais donc les nuages et la neige comme on espère le printemps après des mois à se les geler sous la barre d'un zéro figé dans le frimas. Or il semblait que la terre de Rivière-aux-Trembles, en ce jour de la fin février, ne serait abreuvée que par la violence d'une tempête qui ferait aussi fuir les animaux. À mesure que les heures passaient, le jour se faisait plus sombre, mais j'ai dû attendre le début de l'après-midi pour voir venir les premiers flocons. J'ai allumé une lampe près du vieux fauteuil de cuir poussé contre la fenêtre du boudoir, d'où j'observerais les arbres ployer sous la fureur du vent, puis leurs branches s'effacer dans la poudrerie pour resurgir à la faveur d'un fléchissement momentané de la fureur.
Afin de souligner ma première tempête en rase campagne, je me suis ouvert une bouteille de chianti sur laquelle j'ai tracé une marque en plein milieu. C'était la limite à ne pas franchir, la frontière entre le plaisir, le mal de tête et les divagations d'homme seul. Je m'étais imposé cette règle longtemps avant et entendais la maintenir à Rivière-aux-Trembles, une demi-bouteille de temps en temps, pas plus, parce que je n'avais pas envie de devenir le vieil alcoolique du chemin du 4, l'épave qui sirotait sa bière en camisole sale ou en bedaine sur sa galerie branlante et passait ses journées à roter du houblon en essuyant ses grosses lèvres molles avec son bras tatoué.
Après le départ de la puce, alors que j'arpentais les bars avec sa photo, j'étais passé à deux doigts de me jeter pour de bon dans l'alcool. J'avais aligné quelques cuites et je m'étais rendu compte un matin, en trouvant mes bas dans le frigidaire, que j'étais en passe de donner raison aux gens qui me croyaient fou. Il fallait que ça cesse. De toute façon, je ne portais pas l'alcool et détestais l'inconscience pâteuse des lendemains de veille autant que je méprisais les hommes qui braillaient dans leur whisky cheap en racontant leur vie à des barmaids qui n'avaient d'autre envie que de leur mettre leur pied au cul. Des relents de nausée m'incendiaient la glotte rien qu'à imaginer tous ces hommes qui gueulaient en postillonnant sur des bars poisseux, ceux qui se pissaient dessus, se vomissaient et s'oubliaient. Je refusais de devenir une de ces loques au nez couperosé qui avaient pourtant été des maudits bons gars, jadis, avant le malheur, avant la peur, avant de se mettre à trembler et à perdre la mémoire dans leurs divagations d'ivrogne. Quand mon miroir a commencé à me toiser avec des yeux vitreux et mon frigidaire à sentir le chausson sale, j'ai fait marche arrière en quatrième vitesse, parce que Billie méritait mieux qu'un père ne se souvenant qu'à demi de l'éblouissante et déchirante beauté de son enfant. Je me suis mis à tracer des lignes sur les bouteilles de vin avec un crayon-feutre emprunté à Billie, pour que le message soit encore plus clair, et si j'avais le malheur de dépasser la dose prescrite, je me forçais à vomir jusqu'à ce que mes tripes me remontent dans la gorge.
Je n'avais pas avalé deux gorgées de mon chianti, remerciant le souvenir de Billie de m'avoir sauvé de la déchéance et de ses séquelles, que, déjà, la neige s'intensifiait, poussée à l'oblique par le vent qui se levait. Au bout d'une demi-heure, le bois de la galerie, le toit de mon automobile et l'allée de gravier menant à la maison avaient pris la teinte des nuages se déversant sur Rivière-aux-Trembles, la teinte de l'atmosphère. L'espace sans nom dans lequel hommes et animaux se déplacent normalement s'emplissait de cette matière blanche empêchant le regard de traverser les creux, interstices et ouvertures dans lesquels il se meut par temps clair. Soudainement, le monde était plein. J'ai achevé mon deuxième verre de vin dans cet état de semi-hypnose où vous plonge l'observation des vagues ou de la pluie tombant drue devant la fenêtre, perdu au cœur du vent et de la poudrerie. Je n'étais plus assis dans mon boudoir, bien au chaud, là où l'espace comportait encore des vides, mais au sein de cette masse compacte où toute forme de contact avec ce qui m'entourait ne pouvait être qu'immédiat, sans horizon et sans avenir. La ville et ses ciels coupés ne possédait pas de véritable horizon non plus, mais l'enfermement prenait ici une autre dimension. Je découvrais soudain qu'on pouvait être confiné même au sein des plus vastes étendues.
J'aurais pu mourir sur le moment, me regarder mourir dans cette inaction et cet enfermement, et les vides de mon existence en auraient également été comblés. Tous les trous par où se faufilaient le malheur et la douleur auraient disparu, colmatés par le vent, et nul regret n'aurait pu s'y glisser. Il me suffisait d'arrêter de respirer, d'ouvrir la bouche pour y laisser entrer la neige et devenir une statue de givre, un gisant au corps lisse. J'étais sur le point de m'abandonner au sort des mourants immobiles quand la sonnerie du téléphone a franchi cet espace que je croyais sans interstices. Je me suis brusquement redressé, j'ai renversé mon verre de vin, et une tache est apparue sur mon chandail, rouge sang, Billie, pareille au doigt blessé par le bouquet de la sorcière.
Le téléphone ne sonnait jamais, dans cette maison. La dernière personne à qui j'avais parlé, à part Ronie le crapaud, était mon agent d'assurances, et c'est moi qui l'avais appelé. À Rivière-aux-Trembles, cet appareil muet fonctionnait à sens unique, de l'intérieur vers l'extérieur, jamais l'inverse, et il ne savait pas parler. Qu'il s'adresse à moi me semblait aussi absurde que si j'avais tenté de joindre le 911 en tapant sur un piquet de clôture.
J'ai saisi le combiné, persuadé que la voix que j'entendrais au bout du fil ne m'apporterait que des mauvaises nouvelles, inévitablement liées à Billie, qu'on avait aperçue près d'un étang, petit ange rose surgissant du brouillard matinal en piétinant le bouquet qui l'avait jeté dans le sommeil de la forêt magique.
Oui, ai-je simplement soufflé dans l'appareil. Oui, je suis là. Oui, j'écoute, conscient que ce simple mot de trois lettres, prononcé dans certaines circonstances, vous met à la merci entière de l'autre, qu'il vous place dans la position de celui qui est prêt à tout recevoir, y compris un coup sur la gueule. Il suffit de dire oui, parfois, pour que le cours du destin s'incurve. Trois lettres et vous vous ramassez au pied de l'autel, dans le lit du patron ou derrière des barreaux de prison. J'ai quand même dit oui, je suis là, j'écoute.
Bill ? a murmuré une faible voix à des kilomètres de là.
Oui, ai-je répété. Oui, qu'on en finisse.
C'était Régine, la mère de L.A., le nez congestionné à force d'avoir trop pleuré et trop reniflé. Si elle avait été à côté de moi, j'aurais pu voir la peau qui se fendillait autour de ses narines et le mouchoir chiffonné qu'elle tripotait d'une main tremblante. J'imaginais sa main manucurée sur sa cuisse, qui tripotait le mouchoir, qui tripotait sa jupe, qui échappait le mouchoir et s'élevait parfois dans un mouvement d'impuissance. Que peut-on y faire ? disait la main, et je n'avais de réponse à cette question que le silence béat provoqué par les vagues d'engourdissement parcourant mon corps immobile.
Quelques heures plus tôt, alors que l'Australie flambait, L.A. s'était ouvert les veines avec un couteau de cuisine. Régine dormait dans la chambre d'amis lorsque le couteau avait transpercé la chair amaigrie de L.A. La peau et les os, Bill, il ne lui restait plus que ça. Son instinct de mère l'avait réveillée en sursaut, elle avait entendu l'eau du bain clapoter faiblement et elle avait défoncé la porte de la salle de bain avec une pioche. Le temps qu'elle descende au sous-sol chercher la pioche, sa robe de nuit s'emmêlant dans ses longues enjambées, ses pleurs butant contre ses cris de dénégation, l'eau du bain avait pris la teinte d'un désespoir contre lequel personne ne pouvait plus lutter.
Pendant que Régine, avec des petits hoquets hystériques, me décrivait la couleur de l'eau, rouge pivoine, Billie, rouge coquelicot et couleur de fleurs broyées, je pensais à la pulsation des artères poussant hors de la déchirure le sang destiné aux battements du cœur, et je voulais crier à Régine de se taire, de lancer son téléphone contre le mur et de se précipiter dans la salle de bain pour faire couler de l'eau brûlante sur le corps flasque de L.A. Si le sang demeurait chaud, peut-être serait-il possible de ressusciter L.A., de faire de nouveau pénétrer ce sang chaud dans ses veines. Cours, Régine, cours, disait la voix qui m'étranglait, mais Régine, ses vêtements couverts de neige, arpentait maintenant les couloirs silencieux de l'hôpital où L.A. avait été emmenée, troquant l'horreur du rouge pour la froideur du blanc. Des murs blancs, Bill, des draps blancs, des uniformes. Le monde entier vidé de son sang.
J'ai froid, a murmuré Régine.
Cette petite phrase de rien du tout exprimait à son tour la misère de Régine, qui ne songeait même pas à recouvrir d'un chandail ses épaules tremblantes, affaissées sous le poids de l'étrange et terrible dépaysement des mères cherchant le cri de leur petit sous les arbres tombés. Pendant un instant, je n'ai plus entendu que sa respiration souffrante et froide qui chuchotait j'ai mal, Bill, j'ai mal, et moi, impuissant, je recevais ce souffle froid comme une autre des rafales cinglant Rivière-aux-Trembles. Il m'aurait fallu être là, tout près, pour déposer un chandail sur ses épaules courbées, pour la prendre dans mes bras et saisir sa main, sa petite main nerveuse et rouge, qu'elle cesse enfin de tripoter son maudit mouchoir. Mais j'avais choisi l'exil, j'avais déserté les champs de bataille et un blizzard de quatre cents kilomètres me séparait du lieu où mon impuissance aurait au moins pu réchauffer la main de Régine. J'ai cherché les paroles susceptibles de produire à distance cette chaleur réclamée par l'anéantissement de Régine et, n'en trouvant aucune qui puisse atténuer le froid, je lui ai promis que je viendrais dès la réouverture des routes et la fin de la tempête. Une tempête à faire peur, Régine, puis j'ai raccroché.
Dans la maison, tout n'était que silence, mais dehors, des voix hurlaient, des voix qui n'étaient pas humaines, ramassées par le vent sous les branches essayant de se frotter aux troncs blanchis. J'ai saisi la bouteille de chianti et j'ai rempli mon verre sans me soucier de la ligne tracée au feutre orange. À la tienne, L.A., ai-je bredouillé en me laissant tomber dans un fauteuil avec la lenteur des soldats ne comprenant pas qu'ils viennent de se prendre une balle de Ruger AC-556 en pleine poitrine et que la faiblesse soudaine de leurs jambes est causée par ce trou rougeoyant près du cœur. Pas beau, le trou, bordé de chairs sanguinolentes et de bouts de tissu trempé.
La nouvelle que venait de m'annoncer Régine n'avait pas encore creusé son nid dans le réel. La mort de L.A. faisait partie de ces abstractions que l'esprit ne peut immédiatement saisir, essayant en vain d'associer la mort et son immobilité aux traits d'un visage qu'il n'a connu que vivant. J'ai porté mes mains à mon visage, échappant sans m'en rendre compte mon verre à demi vide sur le tapis tressé, et me suis balancé d'avant en arrière jusqu'à ce que les mots exprimant l'irrémédiable éclatent dans la semi-pénombre de la tempête. L.A. est morte, morte, ai-je murmuré en pesant sur les mots, un à un, pour en extraire le sens, puis quelques larmes ont coulé, des larmes de colère que j'ai tenté de ravaler jusqu'à avoir l'impression que la charge de cris morveux qui me déboulaient dans la gorge allaient m'étouffer. Pourquoi, L.A. ? Pourquoi ? ai-je crié en crachant ma morve, mais cette fois, je connaissais la réponse à la question que je répétais au rythme de mon balancement. Parce que ça faisait trop mal, je sais, L.A., je sais, parce que c'était plus possible d'imaginer le corps de la puce déchiré par la crisse de grosse queue poilue d'un malade à qui on aurait dû l'arracher à coups de dents le jour de sa première érection, parce qu'une mère a le droit de se reposer à un maudit moment donné, bien sûr, L.A., et d'aller voir si l'herbe est plus verte de l'autre côté, les fleurs plus grosses, les enfants plus heureux.
Incapable de remplir mon verre maintenant vide, archi-vide, j'ai bu à même le goulot de la bouteille pendant que défilaient sur les murs qui tanguaient quelques images disparates de ma vie d'homme marié. J'ai laissé de côté les plus sombres, les clichés ratés, pour extraire du fouillis ceux qu'on dit les plus beaux et les plus douloureux, ceux qui me rappelaient les bons moments partagés avec L.A., nos vacances à Cape Cod et notre voyage à Los Angeles, inévitable et formidable, nos concours de grimaces, nos baignades from coast to coast dans l'eau froide du Pacifique ou glacée de l'Atlantique, qui se réchauffait toujours de quelques degrés lorsque y plongeait le rire de L.A. se répercutant sur les vagues. La bouteille pressée contre la poitrine, je me suis remémoré la peau luisante à contre-jour, blonde et pailletée d'eau de mer, la peau blanchie de sel que je léchais de la cheville aux tempes, des oreilles au nombril, encore, au risque de devoir subir une greffe du rein avant mes quarante ans. Après la peau, j'ai tenté de ressusciter les mains, les seins, la cicatrice au bas du dos, mais le corps livide de L.A., sur fond d'émail où stagnait une eau rouge, rouge sang, Billie, se substituait invariablement au corps chaud, et je me demandais, comme un imbécile malheureux, si elle avait versé dans le bain un bouchon de son huile préférée, Désir d'Orient, je crois, et si le parfum subtil de la fleur d'oranger, mêlé à celui du sang, avait empesté la pièce d'une odeur proche de celle du sexe, d'une puanteur trop intime et primitive pour que Régine, découvrant le corps, ne soit pas contrainte de hurler son horreur en même temps qu'elle éructait ses vomissures.
J'ai fini le chianti en pensant aux morceaux visqueux de vomi sur le carrelage, pourquoi t'as fait ça à ta mère, L.A. ? encore incapable de l'engueuler, de lui dire ma vraie façon de penser, crisse d'égoïste, crisse de sans-cœur, parce qu'elle était trop petite, là, trop pâle et vulnérable dans les reflets du couteau tranchant, et parce que je me sentais soudain atrocement coupable de ne pas l'avoir aimée suffisamment pour empêcher ça, le couteau et le sang, mais comment aimer une mère à ce point aimante qu'elle en était venue à détester tout ce qui lui rappelait son enfant, soit la totalité des êtres et des choses ?
Aussi engourdi que si j'avais fumé trois joints en ligne sans me permettre de respirer entre chaque bouffée, je me suis relevé en titubant et, après avoir effectué un détour par ma boîte de Scotties, je suis descendu à la cave annoncer la nouvelle à Pixie. C'est ce qu'on fait quand une personne meurt, on le dit à ceux qui l'ont connue. Le vieux batêche devait déjà être au courant, mais j'avais besoin de parler, de formuler la mort de L.A. à voix haute, même si mon interlocuteur était aussi sourd que peut l'être un macchabée frigorifié. Il dormait dans son caisson, que pouvait-il faire d'autre, étendu derrière la paroi sur laquelle j'avais collé une photo de lui avec une fausse souris dans la gueule, une souris à pois roses, choisie par Billie, qui aimait le rose, les souris et les pois. La souris déchiquetée reposait désormais avec Pixie dans le caisson, accompagnée de la balle de tennis vert fluo, du lacet échiffé et du baluchon d'herbe à chat constituant le trésor du vieux batêche. J'avais placé ces objets dans le caisson parce que c'est ce qu'aurait souhaité Billie, qui croyait que les chats, à l'instar des pharaons, continuaient à vivre dans l'au-delà.
Je me suis assis dans le fauteuil miteux laissé là par l'ancien propriétaire et j'ai demandé à Pixie-Toutankhamon si c'était vrai, s'il existait un au-delà où on se réveillait avec la même tête qu'avant, avec les mêmes cheveux, les mêmes souliers, le même bouton dans le dos. Si c'était vrai, il roupillait peut-être en ce moment même dans un décor enchanteur à la What Dreams May Come, près de Billie et de L.A., tous trois divinement heureux, les pattes et les pieds au chaud dans le sable blanc pendant que je me gelais les fesses entre le quarante-septième et le quarante-huitième parallèle de la planète Terre.
S'il y avait quelqu'un, dans cette maison, qui pouvait répondre à ma question, c'était ce sapré chat congelé. Les yeux rivés sur son caisson, j'attendais qu'il me parle, j'espérais un signe de la mort, sachant pertinemment que la mort ne parle pas, qu'elle ne vous dit rien que nous n'ayez vous-même imaginé. Je voulais que la mort me mente, qu'elle me raconte une bonne blague de Newfies de l'enfer ou me transmette en tapant sur les murs un message que je pourrais décoder avec quelques rudiments de morse, quelque pouvoir extrasensoriel que j'irais pêcher dans mon désir d'entendre sa voix d'outre-tombe. L.A. avait dû espérer la même chose, un signe de l'au-delà, une phrase échappée des nuages. Épuisée d'attendre, elle s'était inventé un mensonge. À bout de forces, elle avait décidé d'aller vérifier elle-même si le paradis existait. Elle s'était emparée du couteau qu'elle lorgnait depuis des jours et était allée chercher Billie dans le seul endroit que les flics ne pouvaient ratisser, me laissant seul de l'autre côté de la clôture et m'obligeant à y rester, car il fallait bien qu'un de nous deux continue à attendre Billie sur ce versant-ci du monde. L.A. est morte, ai-je murmuré, puis j'ai laissé Pixie digérer la nouvelle.
Le soir allait tomber lorsque j'ai refermé derrière moi la porte de la cave. Dehors, la tempête faisait toujours rage, traversée par des voix qui n'étaient pas humaines. J'allais fermer les rideaux quand, par la fenêtre du boudoir, j'ai aperçu une ombre surgir de la forêt. Mon cœur n'a fait qu'un bond et j'ai lâché un juron, persuadé que cette apparition parlait au diable. Pendant un moment, j'ai cru que cette ombre venue de l'enchevêtrement des arbres représentait le mensonge attendu de l'au-delà, puis j'ai vu le foulard rouge enserrant le cou de l'ombre, trop concret, d'une couleur trop vive pour avoir voyagé depuis l'enfer ou le royaume des cieux sans s'altérer. Cette ombre était vivante, comme toutes les ombres recouvrant la terre.
Je l'ai observée quelques instants, qui s'avançait dans la poudrerie et luttait contre le vent, me demandant quel imbécile avait délaissé la chaleur de son salon pour se jeter au cœur de la tourmente. Il va se faire ensevelir, ai-je murmuré, puis j'ai changé de pronom, elle va se faire ensevelir, car l'ombre s'éloignant de la forêt était trop frêle pour porter le corps d'un homme. Cette ombre était l'ombre d'une femme courbée sur sa douleur. La femme du cortège funèbre, ai-je pensé, convaincu, à sa posture et à son allure, qu'il s'agissait de la même femme, pleureuse aux yeux rougis ne se montrant qu'avec la mort. J'ai entrebâillé la porte pour lui crier, l'aviser que je voyais déjà la neige recouvrir son foulard rouge d'une cascade qui la plaquerait au sol et ne laisserait poindre sur la blancheur désertique des champs, au terme de la tempête, que quelques franges colorées évoquant l'herbe d'un roman de Boris Vian, mais mes avertissements ont été inutiles, emportés par le vent dans le sens contraire à la progression de l'ombre. Alors j'ai fermé les rideaux, laissant la silhouette arquée assurer sa propre survie, et me suis écroulé dans mon fauteuil. Quelques larmes ont d'abord brûlé mes yeux, puis leur flot a emporté la brûlure, pendant que le sourire de L.A., le sourire ravageur du temps où nous marchions pieds nus dans les ressacs du Pacifique ou de l'Atlantique, commençait son travail de sape.
L.A. refusait la crémation. Dans une lettre exposant ses dernières volontés, elle avait également exprimé le souhait d'être enterrée sans embaumement, sans cercueil, pour que son corps puisse courir intact au-devant de celui de Billie, inconsciente que l'intégrité physique à laquelle elle aspirait n'était possible qu'au prix de la momification. Elle craignait que, soumises au feu et réduites en poussière, ses mains ne puissent plus caresser le front de sa fille, qu'elle allait rejoindre enfin parmi les jardins de ses hypothétiques paradis.
Régine avait accepté de confier à la terre le corps de Lucy-Ann, même si elle ne croyait pas plus en la réincarnation qu'en l'existence d'un Dieu de bonté, mais il y avait des règles qu'elle ne pouvait ignorer, et le corps de L.A. avait été embaumé. Malgré toute sa bonne volonté, Régine n'était pas de taille à lutter contre les lois régissant nos vies jusqu'à ce que celles-ci se résument en pourriture. J'ai essayé, Bill, mais ils m'ont flanqué leurs paperasses sous le nez, leurs clauses hygiéniques, leurs histoires de charognards creusant les tombes. Je me fous de tout ça, mais qu'est-ce que je pouvais faire ? Rien. Régine ne pouvait rien, au risque d'entendre résonner dans son dos le maillet d'un juge qui n'avait pas le choix non plus. De toute façon, L.A. s'était elle-même vidée de son sang, entamant ainsi le processus dont elle voulait épargner son corps. Cette inconséquence ne lui ressemblait pas, mais qui peut demeurer logique avec un couteau de boucherie sur la gorge.
Au fond de la salle à demi vide où résonnaient les chuchotements, elle était étendue dans le cercueil d'un blanc virginal choisi pour elle par Régine. On lui avait croisé les doigts en signe de piété, alors qu'elle n'avait découvert les relatives vertus de la prière qu'après la disparition de Billie, s'accrochant à ce dernier recours en implorant puis fustigeant Dieu tout à la fois. Faites, mon Dieu... S'il vous plaît, mon Dieu... J'en peux plus, crisse... Autant de paroles vides adressées à un Dieu en qui elle ne croyait pas, mais dont l'inexistence pouvait être mise entre parenthèses en cas de détresse. Tous les athées élevés dans la foi ont le même réflexe, ils ont recours à Dieu quand la peur les prend aux tripes ou que le souffle vient à leur manquer. C'est une autre façon d'appeler au secours la douceur rassurante de l'enfance.
Aucun Dieu n'avait toutefois répondu aux appels de L.A., dont le visage émacié évoquait celui des martyrs s'astreignant au jeûne et à la mortification pour le salut des âmes perdues. Je ne l'avais jamais vue si pâle, en dépit de la poudre rose appliquée sur ses joues par le maquilleur des croquemorts. Sous la poudre, la peau cireuse était d'une blancheur exténuée et j'avais la certitude que L.A. ne profitait pas du repos promis aux morts, de cette béatitude éternelle à laquelle s'accrochaient quelques vivants pour tenter de se consoler de l'immobilité des cadavres. L.A. ne reposait pas, elle ne dormait pas et n'éprouvait pas davantage la paix du silence que celle de l'inconscience. L.A. était morte. L.A. n'était plus. Dans un certain sens, ce n'était même pas son corps qu'on avait allongé sur le satin après l'avoir vêtu d'un chemisier trop strict acheté pour la circonstance, mais une dépouille anonyme. La forme rigide et froide étendue devant nous n'était plus qu'un objet autrefois femme auquel je ne trouvais rien à dire.
Depuis que Régine m'avait annoncé sa mort, j'avais pourtant parlé à L.A. plus souvent que je ne l'avais fait durant nos dernières années de vie commune. Après m'être apitoyé sur sa fragilité, je l'avais engueulée des dizaines de fois, me servant des mots impitoyables qu'on réserve aux suicidés, aux traîtres, aux lâches, aux pisseuses, aux maudites égoïstes, aux lâcheuses et à tous ceux qui nous placent devant le fait accompli et nous laissent nous arranger avec leur douleur muette. Je m'étais confié à la seule femme que j'avais réellement aimée, mais je n'avais rien à dire au semblant de L.A. couché devant moi, avec ses mains jointes et son chemisier boutonné jusqu'au cou. Les seuls contacts que j'aurais désormais avec L.A. impliqueraient que je croie aux fantômes ou que je m'abîme dans ma mémoire heureuse.
Adieu, L.A., ai-je quand même dit à la dépouille de mon ex-femme avant qu'on referme sur elle le couvercle du cercueil, puis j'ai passé mon bras autour des épaules tremblantes de Régine, si petite dans sa robe noire, et nous avons marché ensemble jusqu'à la voiture réservée pour nous par les pompes funèbres, une voiture qui sentait les fleurs mortes et les couronnes de glaïeuls flétris. La procession s'est ensuite ébranlée vers le cimetière et j'ai eu le sentiment de revivre la scène que j'avais vécue à mon arrivée à Rivière-aux-Trembles, à la différence que c'est moi qui me trouvais cette fois dans le cortège. Élément indispensable à la perfection de cette coïncidence où les rôles étaient inversés, une jeune femme, debout sur le trottoir, attendait le passage des voitures fraîchement lavées pour poursuivre son chemin. À peu de détails près, c'était la même femme que celle aperçue dans la Lexus descendant la rue Principale de Rivière-aux-Trembles, le même type de femme, se reconnaissant à la détresse contenue dans leur silence, avec le même regard de veuve ou d'orpheline, la même raideur hiératique. Il ne manquait que la pluie de février pour donner à l'impossible matérialisation de l'inconnue de Rivière-aux-Trembles la place lui revenant dans ce tableau où la froideur avait la grâce de la tristesse.
Le vent semblait avoir redoublé d'ardeur quand nous sommes descendus de la voiture après avoir roulé dans les allées du cimetière jusqu'à la chapelle où le cercueil de L.A. serait remisé jusqu'à ce que la terre meuble ne résiste plus aux coups de pioche et de pelle. Plus fragile que jamais, Régine tenait son chapeau d'une main pour l'empêcher de s'envoler, serrant dans son autre main le bouquet de roses jaunes et blanches destiné à L.A. Après que le cercueil eut été glissé sur un tréteau, Régine y a déposé les roses en collant sa joue contre le métal froid, qu'elle a ensuite embrassé en y laissant l'empreinte de ses lèvres, rouge cerise, Billie, rouge sang de maman sous la peau et les os. Derrière nous, les sanglots de Patricia, l'amie fidèle et éplorée, se sont élevés dans le sifflement du vent, opposant à la mort un dernier refus, puis nous avons quitté le cimetière et ses arbres gris.
Dans les rues animées de la ville, je n'avais qu'une idée, retourner en quatrième vitesse à Rivière-aux-Trembles sans avoir à me taper la réception que Patricia avait organisée dans un café de l'ouest de la ville avec le lointain concours de Régine. Je n'avais pas envie de subir les regards condescendants des anciens amis du couple que nous avions formé, L.A. et moi, d'essuyer leurs sous-entendus ni de m'incliner devant leur mine atterrée. Je ne voulais pas toucher de nouveau toutes ces mains qui avaient hésité à se tendre vers la mienne au salon funéraire, se demandant si je faisais légitimement partie de la famille endeuillée, puis joignant leurs murmures hypocrites afin d'évaluer ma part de responsabilité dans la mort de L.A., à qui j'avais peut-être fourni de ma lointaine retraite le couteau tranchant avec lequel elle tracerait deux lignes définitives sur sa vie passée.
Ces gens-là ne savaient rien. Ils ne savaient pas ce qu'il en coûte d'étouffer ses larmes sur le ventre d'un ourson pelé sentant les joues de petite fille, de parler à un chat mort, de frôler l'évanouissement chaque fois que le téléphone sonne et de ramasser des culottes sales en maudissant la terre entière. Ces gens-là ne comprenaient rien.
Par affection pour Régine et pour Pat, j'ai conservé mon masque d'homme civilisé et me suis rendu à la réception. Après une coupe de sauternes et un feuilleté au crabe qui ne passait pas, je me suis esquivé en promettant à Régine de garder le contact. T'es tout ce qu'il me reste d'elles, Bill, m'a-t-elle confié en prononçant le nom de Billie et de Lucy-Ann dans la langue la plus déchirante du monde, probablement inventée par Ève quand Abel avait rendu l'âme. Je me suis arraché à son regard de mère avant qu'il m'engloutisse dans son désarroi et j'ai quitté les lieux sans saluer la douzaine de faces blêmes qui se prétendaient encore les amis de L.A. alors qu'ils ne passaient à la maison qu'un dimanche après-midi sur dix, à l'heure où on visite les vieux et les malades.
Avant de prendre la route, je me suis permis un crochet par le café de Jack et Jack afin de tester ma mémoire et de vérifier si l'homme au foulard rouge qui sirotait un cappuccino pendant que sa fille quittait son école pour la dernière fois occupait toujours la banquette du fond. J'ai imaginé autour de lui des gueules de janvier, un peu de givre aux fenêtres, et j'ai constaté qu'il était encore là, le visage grave, inconscient que la mousse de son cappuccino lui avait dessiné une bouche de clown dont la sévérité de son visage accentuait le ridicule. J'ai levé la main dans sa direction, son image s'est volatilisée, et j'ai regagné ma voiture. Cet homme était un homme innocent.
Sur le chemin du retour, j'ai étouffé mes pensées noires en appuyant sur l'accélérateur, puis je me suis calmé quand la voix de L.A., venue de nos treize années de vie commune, m'a ordonné de ralentir. Si tu le fais pas pour toi, fais-le pour Billie. T'es tout ce qu'il lui reste. L.A. avait raison. Régine avait raison. Je n'étais désormais qu'un restant d'homme, j'étais le peu sur lequel pouvaient compter ceux qui n'avaient plus rien, une ombre, ce qui persiste du souvenir des morts alors que tout s'est envolé et qu'il faut bien un abruti pour garder le fort.
J'ai cligné des yeux et la portion de l'autoroute filant vers le nord s'est divisée en quatre voies sinueuses et floues. Avant de m'abîmer dans le décor ou de faucher quelques innocents revenant d'un souper de famille, j'ai bifurqué à droite, traversant trois des voies qui serpentaient dans la lumière des phares, et me suis garé sur l'accotement. La tête appuyée contre le volant, j'ai pleuré tout ce que je n'avais pas osé pleurer devant Régine, qui avait besoin que le peu de moi qui subsistait ne s'effondre pas sous ses yeux. Lorsque j'ai repris la route, il n'y avait que deux voies devant moi, deux voies bordées d'ombres et de spectres surgissant de l'obscurité.
De retour à Rivière-aux-Trembles, le silence parfait de cette nuit d'hiver m'est retombé dessus comme une enclume dévalant d'un toit où on se demande bien ce qu'elle fichait. J'étais seul et c'est à ça que ressemblerait désormais ma vie, à la vie d'un père sans enfant, d'un mari sans épouse, d'un homme sans désirs d'homme, un automate, tout au plus, un être vide dont le cœur ne battait que parce qu'il était conçu pour ça, pomper, ramener vers le cerveau et les organes vitaux un sang qui entretenait le sentiment de perte et de vacuité.
J'ai été tenté de fermer les rideaux afin d'ériger une frontière entre les deux silences qui m'entouraient, celui des champs noirs et celui qui créait entre les murs nus sa propre résonance, mais il vaut toujours mieux que le silence circule, qu'il vous montre son étendue et sa capacité de se mouvoir si vous voulez tenter de lui échapper. N'ayant ni faim ni sommeil, j'ai traîné mes savates jusqu'au fauteuil d'où j'avais été tiré de ma léthargie par la sonnerie du téléphone, cinq jours plus tôt. Les pourtours de la fenêtre jouxtant le fauteuil avaient été blanchis par la tempête, qui avait laissé au bas de la vitre froide un ruban sinueux ayant la forme d'une chaîne de montagnes lunaires, d'où s'élevaient quelques pics dont les sommets se perdaient dans une nébuleuse de givre. À travers cet écran blanc, la campagne était d'un noir compact et immuable, et j'avais l'impression de naviguer à l'intérieur d'un vaisseau progressant sans bruit dans la nuit des galaxies.
L.A. avait peut-être échoué là, avec Billie et Pixie, dérivant dans un autre silence. Je les ai imaginés en état d'apesanteur, allégés du poids de la terre, puis les ai transportés sur les montagnes lunaires se découpant dans la nuit. Au moment où l'ombre de la terre couvrait la lune, des hurlements se sont élevés de la forêt, derrière la maison. Des hurlements de pleine ou de nouvelle lune. Pendant un moment, encore troublé par l'odeur des glaïeuls flétris et l'alignement des pierres tombales enneigées, j'ai été tenté de voir dans ces hurlements un appel de L.A., dont l'âme m'avait peut-être suivi sur la route se dédoublant, mi-ange, mi-mensonge, pour m'aviser que l'au-delà n'était qu'un leurre, que le seul au-delà possible consistait en cet espace intermédiaire où l'esprit voguait le temps que se refroidisse le lit du disparu.
Mais je ne croyais plus à ces foutaises auxquelles s'agrippe la solitude. Ces hurlements n'avaient pu être poussés que par quelques coyotes, loups ou chiens errants rassemblés pour la chasse. J'ignorais s'il y avait des chiens errants à Rivière-aux-Trembles, mais j'ai écarté cette possibilité. Dans mon esprit, ces meutes d'animaux affamés ne pouvaient être associées qu'aux étendues sauvages du Nevada, du Texas ou de je ne sais quel État où l'homo americanus vivait encore à l'âge du cow-boy. En fait, je préférais qu'il s'agisse de loups ou de coyotes, moins dangereux, moins tristes que les chiens de berger, huskies, braques ou labradors abandonnés à leur sort du jour au lendemain, sans nourriture, sans caresses, sans voix guidant leurs pas : assis, couché, reste. Je n'avais pas grandi dans une de ces familles dont les liens se cimentent autour du nom d'un chien dont la photographie trône sur le manteau de la cheminée, près de celles des enfants et des grands-parents, ce qui ne m'empêchait pas d'avoir envie de vomir devant la bêtise et la cruauté qui poussaient certains hommes à larguer les bêtes qu'ils avaient d'abord recueillies. Je ne comprenais pas qu'on puisse rouler vers la forêt, ouvrir la portière de sa voiture, siffler pour que le chien en descende, come on, boy, allez hop, Jeff, puis repartir en trombe pendant que le pauvre animal se demande ce qui se passe, croit à un jeu, peut-être, puis s'affole et court dans la poussière, court jusqu'à en perdre haleine et ne devient plus, dans le rétroviseur, qu'un minuscule point sombre dans la poussière retombée.
C'est Billie, avant Dieu, Jésus, Greenpeace et tous leurs saints, qui m'a appris le respect du vivant. C'est sa voix effrayée m'empêchant d'écrabouiller les bestioles qui croisaient innocemment mon chemin. Stop ! papanoute... criait-elle quand l'ombre de mon pied menaçait de s'abattre sur une araignée ou une fourmi, et mon pied changeait de cap, et l'araignée se poussait. C'est son petit visage endormi sur l'oreiller, exprimant toute la vulnérabilité de celui qui ignore ce qu'est un loup. On ne peut pas vivre aux côtés d'une enfant qui attrape des coccinelles pour les mettre à l'abri de la pluie, d'un petit être dont la survie tient à notre vigilance, sans récrire certaines des règles de la loi du plus fort, à savoir que le plus fort n'est pas toujours là pour manger l'autre ni pour aller le perdre dans les bois, mais pour le protéger et le nourrir, au péril, s'il le faut, de sa propre survie. Je préférais donc que les hurlements franchissant les sommets givrés où L.A. tenait la main de Billie soient ceux d'une bande de coyotes devinant les reflets de la lune sous les nuages.
Je me suis approché de la fenêtre, au cas où j'apercevrais, à la lisière des bois, l'ombre au dos courbé de deux ou trois spécimens de l'espèce des canidés, mais je n'ai vu ni loup, ni coyote, ni chien errant. Dans l'opacité de la nuit, je n'ai pu distinguer que la silhouette des arbres alignés derrière la maison, telle une rangée de sentinelles me protégeant des esprits de la forêt ou gardant la forêt, qui sait, de l'intrusion des hommes. Pendant un moment, j'ai pensé aux ormes gardant L'île des morts, un tableau de Böklin dont j'avais accroché une reproduction dans mon bureau, plusieurs années auparavant, pour faire contrepoids aux couvertures de mes livres affichées sur les murs. Chaque fois que je m'attardais devant cette reproduction, je sentais peser sur moi le poids d'une angoisse aussi blanche qu'un drap d'agonisant, et c'est ce que je ressentais aussi devant ces arbres m'interdisant l'accès à la forêt. J'ai allumé une lampe, comme on le fait pour anéantir les monstres dissimulés sous les lits des enfants, et je suis monté me coucher. Dans la forêt, les hurlements s'étaient tus, laissant place aux bruits dérobés formant la matière du silence.
J'ai mis longtemps à m'endormir, perturbé par le chuintement de ma propre respiration, qui s'étirait en un lent sifflement quand j'étais sur le point de sombrer. Plutôt que de compter des moutons ou des koalas assoiffés, je me suis concentré sur les ténus craquements provenant du mur, juste derrière mon lit, essayant de déceler une logique dans la succession de ces bruits, une formule mathématique ou algébrique reliant les trois points d'où originaient apparemment les craquements et qui me permettrait d'en déterminer le rythme. J'étais parvenu à un modèle musical à deux temps pour triangle et orchestre lorsque, enfin, mes muscles se sont détendus.
À mon réveil, ayant encore à l'esprit les hurlements de la veille, j'ai avalé un rapide déjeuner avant de sortir, emmitouflé jusqu'aux oreilles, en vue de chercher des pistes à la lisière de la forêt, pistes d'animaux ou d'extraterrestres ayant capté les hurlements des loups des fins fonds de l'espace et ayant cru qu'il s'agissait là de la langue des hommes. La neige bordant les bois étant immaculée, je me suis enfoncé de quelques centaines de mètres sous le couvert des arbres, rassuré que mes pas s'impriment dans la neige pour m'indiquer le chemin du retour, mais je n'ai rien trouvé qu'un bidon d'huile à moteur 5W30 reposant sur une branche de sapin près d'une vieille chaussette brune, que j'ai ramassés en vue de les mettre aux ordures. Les produits de l'activité humaine ne connaissant pas de frontières, je ne me suis pas étonné de la présence insolite de ces objets là où personne n'avait besoin d'huile ni de chaussette trouée. Je suis d'ailleurs persuadé que si l'homme pose un jour les pieds sur une autre planète, il y découvrira des emballages de chips Humpty Dumpty, des boutons de chemise et des chaussettes orphelines, de quelque couleur qu'elles soient, pareilles aux milliers de chaussettes esseulées parsemant les trottoirs mouillés, les parcs, les bords de route, comme si l'homme était un animal unijambiste prenant plaisir à se déchausser en pleine rue pour ensuite y oublier son bas.
Le bidon à la main et la chaussette dans une des poches de mon manteau, j'ai poursuivi mon chemin entre les arbres enchevêtrés, pour enfin aboutir à la rivière donnant son nom au village. Quelqu'un était passé par là il n'y avait pas longtemps, un homme ou une femme dont les pistes n'avaient pas été entièrement recouvertes par la dernière tempête. Les traces longeaient la rivière, pénétraient dans la forêt sur quelques mètres, puis redescendaient près de l'anfractuosité où la rivière devait former une cuvette quand elle n'était pas gelée. À cet endroit, les traces s'élargissaient, montrant que la personne s'était couchée par terre ou avait glissé sur la rivière, balayant dans sa chute une partie de la neige accumulée.
Dans le ciel tout à l'heure d'un bleu intense, des nuages s'amoncelaient. À travers les branches des arbres dénudés se refermant sur la rivière, je les voyais s'avancer doucement. Un pan de bleu, un pan de gris, un pan de blanc constitué d'infimes particules tournoyant sur elles-mêmes. Hypnotisé par ce mouvement, je me suis aussi couché par terre et j'ai commencé à battre des bras et des jambes, comme nous le faisions enfants, comme je le faisais avec Billie dans la cour arrière de la maison. On appelait ce jeu le vol de l'ange ou du hibou, selon qu'on se sentait plus près des cieux que de la terre. Penses-tu qu'on pourrait s'envoler dans la neige ? m'avait un jour demandé Billie, et je nous avais imaginés planant au ras du sol, traversant les bancs de neige et les pulvérisant à grands coups d'ailes.
C'était ce qui m'arrivait en ce moment. Je volais dans la neige, mi-ange, mi-hibou, soulevant des nuages de poudreuse qui retombaient sur mon visage. Je volais sur le dos, adoptant le point de vue du hibou et de l'ange rasant la blancheur du sol quand un cri a retenti dans la forêt, un hurlement de bête déchirée, aussi définitif que la mort en personne. Je me suis arrêté net. Un écureuil venait de rendre l'âme, un lièvre, attrapé par l'ange ou le hibou. Rien ne bougeait plus. Le cri avait absorbé toute autre forme d'appel ou de mouvement. Son écho s'inscrivait lentement dans la mémoire des lièvres, un nouveau cri dans le répertoire de la peur assurant la survie des faibles. Assis sur la rivière gelée, j'ai constaté que mes mains tremblaient et que j'attendais le cri suivant. Je me suis prudemment relevé, craignant de perturber le silence, et j'ai regardé autour de moi. Aucun ange, aucun hibou ne volait ni ne s'enfuyait près de la rivière. Et pourtant, des dizaines d'yeux me surveillaient, pétrifiés par le cri et attendant que la forêt respire de nouveau.
J'ai ramassé mon bidon d'huile et me suis enfoncé dans la forêt en me guidant sur mes pistes entre les arbres, contrarié par la vague mais tenace impression de n'être pas seul, d'être filé par quelque animal que ma présence dérangeait. De ténus craquements me suivaient, qui se taisaient chaque fois que je m'immobilisais pour me retourner. Devant l'exactitude de mon poursuivant, qui ne faisait jamais un pas de plus que moi, j'ai fini par conclure que ces craquements étaient provoqués par mes propres pas, par le voyagement des bruits entre les arbres. Et pourtant l'impression persistait. Cet endroit était malsain. Cette rivière n'était pas nette. Quand je suis ressorti des bois, un amas de nuages sombres s'avançait derrière la maison. Un pan de gris, un pan de noir. Une nouvelle tempête allait s'abattre sur Rivière-aux-Trembles. Si le pire ne m'était déjà arrivé, j'y aurais vu un mauvais présage, mais j'ai laissé filer. Je me suis assuré que la porte du hangar ne battrait pas au vent et je suis rentré.
Le bidon toujours à la main, j'ai gagné la cuisine, j'ai jeté le bidon à la poubelle et j'ai inutilement cherché la chaussette, qui avait dû tomber de ma poche quand, dans la forêt silencieuse, j'étais poursuivi par quelque animal voulant s'assurer que je quittais bel et bien les bois. Pendant que les nuages s'amoncelaient au nord, je me suis confectionné un sandwich moutarde, laitue et triple fromage que j'irais manger au sous-sol, où m'attendait une pile de planches destinée à se transformer en établi. Ce travail n'était pas urgent, mais j'avais besoin de m'occuper les mains pour atténuer le malaise que le cri de la forêt avait provoqué.
Mon sandwich avalé, je me suis mis au boulot en racontant à Pixie, qui attendait toujours le printemps dans son caisson, une histoire mettant en vedette le Chat du Cheshire et le Chat botté. Je les ai placés tous les deux dans une forêt enchantée, conte oblige, avec des fées qui n'avaient pas d'allure, unijambistes ou danseuses de baladi, pour passer le temps, puis j'ai décrit les bottes du botté, des bottes de prince, à ne pas confondre avec les bottes de sept lieues, pour m'attarder ensuite au sourire flottant du Chat du Cheshire, pas trop rassurant pour un sourire, et j'ai interrompu mon histoire au moment où ce dernier miaulait une espèce de ouf ! indigné. C'était ça ou l'assassiner. Ce chat m'avait toujours tapé sur les nerfs et je n'allais pas attendre le prochain déluge avant de le rayer de mon paysage mental. Sans transition, j'ai enchaîné avec Ronie le supercrapaud, question de tester la métamorphose de Ronie sur un public, et j'ai poursuivi ainsi de chat en ogre et d'ogre en superbatracien jusqu'à ce que mon tas de planches ressemble à un établi.
La nuit était tombée quand je suis remonté. Un cercle de clarté hésitante, écrasé par les nuages, s'évaporait dans la lourdeur du ciel. À peine ai-je eu le temps de m'approcher de la fenêtre qu'il s'effaçait dans l'obscurité des cimes. Aucune lumière ne subsistait plus sur la campagne, hormis le faible halo projeté sur la galerie par l'ampoule installée près de l'entrée. J'ai allongé le bras pour l'éteindre et j'ai observé la nuit, que rien de vivant, apparemment, n'habitait. Et pourtant, je savais qu'il y avait des loups sur la colline, des coyotes ou des loups, et que des rapaces, perchés sur quelque branche, guettaient les petites bêtes qui filaient sur la neige. On entendait leur cri, puis tout se taisait de nouveau. C'était la vie dans ce qu'elle avait de plus authentique et de plus cruel, et il n'y avait en son sein ni Chat botté ni monde féerique dans lequel biches et loups dormaient flanc contre flanc. La vérité résidait là, au creux de cette noirceur muette, exempte du mensonge dont l'homme entourait son existence.
La tempête dont j'avais pressenti la venue a été encore plus violente que la précédente, une tempête de printemps mêlant neige et orage et transformant la cour en une immense mare dont le vent poussait avec fureur les vaguelettes désorientées sur l'amoncellement de slush granuleuse servant de rivage à ce nouvel étang. En observant cette mare secouée par le vent, dans les jours suivant la tempête, je pensais à la rivière, dont les eaux devaient aussi s'agiter sous les glaces de plus en plus minces, je pensais au cri entendu, aux pas furtifs s'arrêtant derrière moi, et il m'arrivait de frémir avec la mare de slush, rien de sérieux, un petit frisson qui me plissait le front et me rabattait les sourcils sur les paupières, me donnant l'air du gars qui réfléchit à des choses graves. Mais la plupart du temps, je me traitais de pisseux de la ville qui ne savait même pas faire la différence entre un cri de castor et un marmottement de marmotte.
J'ai tout de même mis un certain temps avant de retourner à la rivière, prétextant que je n'avais pas envie de patauger jusqu'aux genoux dans l'eau glacée ni de m'asseoir devant une talle d'épinettes stoïques qui attendraient que je parle le premier. Puis, un matin du début d'avril, je venais de mettre le point final à une nouvelle aventure de Ronie le supercrapaud quand j'ai pris conscience que mes craintes étaient aussi insignifiantes qu'un bouton de chaleur au regard des raz-de-marée climatiques secouant la planète. CNN venait de m'annoncer que des pluies torrentielles avaient fait sortir de leur lit plusieurs cours d'eau de la Namibie et de la Zambie, exposant les sinistrés aux attaques des crocodiles et des hippopotames, aux morsures de serpents, aux fièvres bilieuses du paludisme et aux crampes de la diarrhée, en comparaison de quoi la menace représentée par les lièvres et les écureuils de Rivière-aux-Trembles ne pesait pas lourd.
Le temps s'adoucissait et une pluie fine tombait sur la campagne. Depuis la tempête, les bancs de neige reculaient de jour en jour, laissant apparaître le foin jauni. Dans quelques semaines, je pourrais m'asseoir sur ma galerie et regarder pousser les fleurs sauvages en me faisant manger par les maringouins. En observant les rigoles d'eau de pluie et de neige fondue charriant le gravier de la cour, je me suis demandé si la rivière était dégelée et si elle allait aussi envahir ses rives. La crainte stupide que ses eaux ne débordent jusqu'au village et qu'une famille de castors s'établisse sur la rue Principale m'a traversé l'esprit, provoquant un de ces fous rires dont je ne me croyais plus capable, un fou rire de Billie la tête en bas, un fou rire d'une franchise de première communiante, qui vous nettoie le cerveau des conneries l'encombrant et vous donne de sérieuses quoique honnêtes crampes dans le ventre.
Trois quarts d'heure plus tard, j'étais accroupi près de la rivière, au bord de la cuvette. Dans le tumulte de la crue charriant les déchets de l'hiver, je n'entendais ni mon cœur battre ni les lointains cris montant de la forêt. Tout n'était que bruit d'eau. J'ai cherché autour de moi une bûche ou un tronc d'arbre sur lequel je pourrais m'asseoir pour écouter la crue, puis j'ai aperçu, à l'entrée d'un sentier, une petite croix de bois comme on en voit le long des routes, près des courbes mortelles où la fatalité a accompli son œuvre. Des petites croix discrètes auxquelles demeure accrochée une couronne de fleurs artificielles aux couleurs délavées ou une photo plastifiée de celui ou de celle qui s'est envolé dans le décor.
Sur la croix qu'ombrageaient ici les arbres, quelqu'un avait gravé le nom de Michael avec un canif ou un burin. Aucune fleur ne l'ornait cependant. Au lieu de ça, on y avait suspendu un petit hibou de paille et d'écorce qui se balançait dans le vent. La corde fichée dans sa tête s'était enroulée autour de son cou et il ressemblait à un pendu qui se serait trompé de longueur de corde et aurait été obligé de se coller la tête au plafond pour ne pas se rater. Ce hibou était carrément sinistre avec ses petits yeux jaunes qui semblaient capter autour de lui le moindre mouvement, la moindre anomalie dans le paysage. Quand il s'est immobilisé face à moi, j'ai eu l'impression qu'il me regardait, tentant d'évaluer si une anomalie telle qu'un homme au milieu de la forêt méritait qu'on s'y attarde, puis le vent l'a fait pivoter en même temps qu'un frisson courait sur la rivière.
Je me suis approché de la croix pour vérifier si elle portait d'autres inscriptions, mais seul le nom de Michael figurait sur le bois érodé, rongé par les années et les larves d'insectes. Aucune date, aucune épitaphe, aucune mention de l'âge de l'homme ou de l'enfant nommé Michael. Je penchais toutefois pour un enfant, un garçon disparu dans la rivière. Je m'imaginais mal un adulte venir se noyer dans une cuvette qui ne devait pas contenir plus d'un mètre d'eau en été. Je voyais plutôt un enfant qui se précipitait dans la cuvette en criant, en ressortait les cheveux plaqués contre le crâne et recommençait, recommençait jusqu'à ce qu'une crampe lui torde l'estomac et qu'un homme muni d'une masse, le visage ravagé par l'alcool, la colère, les larmes, par tout ce qui dévaste un homme qui pisse le sang de son fils par chacun de ses pores, vienne planter une croix près de l'hostie de rivière qui avait étouffé son petit, opposant ainsi Dieu à sa création et l'outrageant de ses blasphèmes.
Ce père semblable à moi, s'accrochant à une croix faute de pouvoir s'accrocher à un dieu, me faisait à la fois honte et pitié, et je pensais à Billie, qui n'aurait jamais sa croix ni son cercueil à elle, parce qu'elle était partout, Billie, parce qu'elle n'était nulle part et qu'il m'aurait fallu semer la terre entière de petites croix roses pour m'assurer qu'il y avait quelque part, sous cette plantation macabre, une Billie semblable à celle que j'avais connue, grande architecte des surpeuplements et dépeuplements consécutifs de mon univers. Les enfants disparus n'ont droit à aucune véritable sépulture. Ils n'ont droit qu'à un trou, qu'à un coin de dépotoir, qu'au rivage d'un marais glauque planté de quenouilles. L'homme aux mains rugueuses, l'homme de janvier, savait où avait été creusé ce trou, où stagnait l'eau de ce marais, mais il n'avait sûrement planté aucune croix parmi les quenouilles, parce qu'un salaud pareil ne pouvait croire en Dieu. Je n'y croyais pas non plus, mais pour d'autres motifs, assurément, et ne voyais pas comment un objet sur lequel avait coulé le sang d'un martyr pouvait servir de réconfort à qui que ce soit, pas plus à un criminel qu'au pauvre innocent qui pleurait sur un corps absent. La croix du Christ n'était qu'un symbole morbide et celle du petit Michael était pareille, aussi macabre que le hibou qu'on y avait pendu, qui continuait à me fixer avec ses yeux exorbités, identiques à ceux de tous les pendus, lui sortant du crâne et accusant je ne sais qui de lui avoir passé la corde au cou.
Je me demandais qui était assez tordu pour avoir eu la géniale et épatante idée de pendre un oiseau. La mère du gamin, peut-être, qui avait pété les plombs, ou son père aussi soûl que fou, ou sa sœur psychopathe, que sais-je, et je m'en foutais. J'ai déroulé la corde qui étranglait l'oiseau et j'ai fait un semblant de prière à l'intention de l'enfant nommé Michael, dis bonjour à Billie, Mike, embrasse-la pour moi, take care. Ma première impression était la bonne. Ce lieu était malsain. Cette rivière hantée par l'esprit d'un enfant n'était pas nette.
La pluie tombait toujours, fine et glacée, mes vêtements étaient trempés et je grelottais comme si j'étais tombé au creux de la rivière. J'ai fait un signe d'adieu à Michael X, Y ou Z, où qu'il soit, et j'ai sacré mon camp sans attendre la prochaine glaciation. Poursuivi par de ténus craquements, j'ai couru à travers les arbres, incertain quant au chemin à suivre et trébuchant sur mes lacets de bottine, fidèle à ma nature de maudit pisseux de la ville n'arrivant à se sentir à l'aise qu'au milieu de quelques tonnes de béton.
Il m'avait été facile de suivre mes pistes dans la neige fraîche la première fois que je m'étais aventuré près de la rivière, mais aujourd'hui, le sol n'était plus recouvert que de boue, de feuilles mortes et de plaques de neige éparses. De temps à autre, je pouvais distinguer l'empreinte partielle d'un pas, mais je ne reconnaissais plus les arbres ni aucun des repères que j'avais croisés plus tôt. Je suis enfin parvenu à sortir du bois en suivant des marques de pas que j'avais cru être les miens, mais qui se sont révélés être des pas d'enfants, de deux ou trois enfants portant des chaussures aux semelles différentes, les unes striées de barres obliques, les autres de traits horizontaux. Que faisaient des enfants au cœur d'une forêt hantée par le souvenir d'un gamin ? Peut-être connaissaient-ils le petit Michael ? Peut-être étaient-ils venus se recueillir au pied de sa croix en lançant des cailloux dans l'eau, ne craignant aucunement les craquements de leurs pas et considérant la forêt comme un asile les tenant à distance des hommes qu'ils deviendraient. Les enfants élevés sous les arbres savent que les bois ne sont hostiles qu'à ceux qui n'ont jamais appris à distinguer leur ombre au creux du noir. Je n'avais pas à m'inquiéter pour eux. Si les fantômes faisaient partie de leurs jeux, ce n'était qu'en vertu d'un mensonge qu'ils protégeaient farouchement. Quand j'ai enfin débouché dans un champ, j'ai constaté que j'avais marché dans la mauvaise direction. J'avais abouti à l'autre bout du rang, à environ deux kilomètres de chez moi.
J'ai parcouru ces deux kilomètres sous la pluie, convaincu que j'attraperais la crève. En posant le pied sur ma galerie, j'ai cru entendre un hululement étouffé à l'orée du bois, semblable à un cri d'enfant, mais ce n'était que le vent, que le vent et la fièvre qui chauffait déjà mon front.
Pendant trois jours, j'ai dû subir de nouveau les affres de la grippe australienne. Je m'éveillais dans des draps mouillés de sueur et de larmes destinées à Billie, destinées à L.A., destinées à tous les affligés du monde, et ne quittais le lit que pour me traîner aux toilettes, où le miroir me renvoyait l'image d'une homme d'une pâleur cadavérique que j'avais peine à reconnaître. Bill ? C'est toi, vieux schnock ? À tout coup, mon reflet tentait vainement de me rassurer en esquissant un sourire qui n'était pas le mien, mais celui d'un homme qui s'effaçait derrière la brume d'hiver entrée chez lui avec la fièvre. J'avalais deux Tylenol et je retournais me coucher en me demandant si je n'étais pas le rêve d'un esprit de la forêt manquant cruellement d'imagination.
J'avais peut-être avalé mon extrait de baptême, qui sait, au volant du camion de location qui m'emmenait en divaguant vers Rivière-aux-Trembles, deux mois auparavant, ratant le dernier virage, celui qu'on ne voit jamais venir, pour m'engager dans cette portion de route qui bifurque au dernier moment, se transforme en incontournable dead end et vous envoie valser dans l'infini. Si c'était le cas, le convoi funèbre que j'avais vu défiler dans la rue Principale de Rivière-aux-Trembles était peut-être le mien, conduit par la femme éplorée que j'aurais pu aimer si mon système immunitaire n'avait pas déclaré la guerre à l'Australie. Qui sait d'ailleurs si je n'étais pas plutôt le rêve de cette femme, un rêve tragique à la fin duquel elle me tendait la main pendant que mon U-Haul s'envolait dans un bruit de ferraille. Du fond de l'espace nébuleux où je voguais, il m'arrivait d'entendre ses pleurs et ses cris de dénégation, pendant qu'elle embrassait mon front couvert de sang dans le cliquetis du moteur se refroidissant.
J'ignorais ce qui pouvait se passer dans la tête d'un gars qui était tout à coup catapulté au cœur d'un rêve. Si je me fiais aux personnages que je croisais dans mes propres rêves, ils n'avaient pas l'air de penser grand-chose. Mais qu'est-ce que j'en savais ? Jusque-là, j'avais toujours cru que le facteur qui sonnait à ma porte en plein milieu d'un de mes cauchemars n'était même pas conscient de son existence, mais s'il en allait autrement ? Ce facteur éprouvait peut-être des sentiments, des peurs, des angoisses. Le pauvre imbécile réfléchissait peut-être à la théorie des nombres premiers avant de dégringoler en bas d'un escalier, son sac de courrier en bandoulière, pour être propulsé dans le royaume des rêves sans l'avoir demandé.
Ce n'est qu'au moment où le mercure a chuté en bas des 40 degrés au niveau desquels il se maintenait depuis soixante-douze heures que j'ai recouvré peu à peu mes esprits pour décréter que je n'étais le rêve de personne et que le facteur effectuant sa tournée de nuit dans mon subconscient n'était actionné que par mon cerveau. Cette fièvre avait tout de même laissé ses marques et je ne pouvais m'empêcher de penser au garçon nommé Michael qui s'en trouvait à l'origine. Un garçon dont la mort avait démoli une famille entière et peut-être poussé sa mère sous les essieux du train qui stoppait tous les lundis et jeudis soir à Saint-Alban, la ville la plus proche de Rivière-aux-Trembles, à une vingtaine de kilomètres au nord-ouest. On avait déjà vu ça, des pères ou des mères qui se faisaient sauter la cervelle ou se saignaient à blanc, n'est-ce pas, L.A., parce que mourir est moins douloureux que respirer, parce que ça dure moins longtemps, surtout.
Tout le village, sauf moi, devait savoir ce qui était arrivé à ce garçon et à sa famille. La région entière devait avoir entendu parler du corps écrabouillé par le train ou de celui du garçon, méconnaissable, qu'un pêcheur avait vu flotter en aval de la cuvette. Après trois jours d'insomnie entrecoupée de cauchemars, j'avais besoin de savoir aussi, ne serait-ce que pour sortir ce gamin de mes rêves et empêcher le hibou moribond qui veillait sur lui de s'y faufiler à sa suite. Mon placard à hantises était déjà occupé et il n'y restait assez de place ni pour un noyé ni pour un pendu. Mais je me voyais mal me pointer à l'épicerie pour questionner Max le boucher à propos d'un gamin dont la mort devait remonter à plusieurs années, d'autant plus que le comité d'accueil de Rivière-aux-Trembles ne m'avait pas organisé une réception de bienvenue avec fanfare et allocution du maire quand j'avais débarqué dans le coin sans m'annoncer.
J'étais l'étranger, le type bizarre qui avait récemment emménagé dans l'ancienne maison d'un type nommé Lucien Ménard, « Entretien de pelouses en tous genres », qui avait passé sa vie à essayer de tuer des pissenlits qu'il combattait aujourd'hui par la racine, comme quoi les obsessions ont la couenne dure. Si les gens se montraient polis en ma présence, j'entendais leurs chuchotements dans mon dos chaque fois que je refermais la porte de la quincaillerie ou de l'épicerie. La rumeur qui s'élevait des petits conciliabules réunissant deux ou trois clientes devant le comptoir à viande de Max le boucher n'avait rien à voir avec mon sex-appeal ni avec le désir de l'une d'entre elles de m'attraper au lasso pour me traîner dans son lit. C'était la rumeur excitée de la méfiance, qui me collerait au cul jusqu'à ce que je marie une fille de la place, éclaireuse ou kamikaze qui témoignerait de mon innocence en achetant sa saucisse.
On devait m'avoir inventé mille malheurs pour expliquer ma venue dans ce trou perdu ou m'avoir attribué des origines douteuses, des activités louches justifiant ma réclusion. Par-dessus tout, on devait avoir mis les enfants et les adolescentes en garde. Les hommes seuls n'ont pas la cote et on se les imagine aisément dans la peau d'un satyre ou d'un pédophile. Mon statut d'homme seul faisait de moi un être marginal dont il valait mieux se méfier. J'aurais débarqué avec femme et enfants, on m'aurait plus facilement entrouvert les bras, mais ma solitude me rendait suspect et on ne m'accorderait aucun crédit tant que je n'aurais pas prouvé que je n'étais pas le petit-fils de Ted Bundy ou d'un autre putain de tueur en série, ce qui me prendrait probablement une vingtaine d'années si j'écartais l'option du mariage.
N'espérant pas recueillir les informations que je cherchais au village, je me suis rabattu sur Internet, mais, sans date ni nom de famille, autant chercher un bouton de manchette dans le Sahara. De toutes façons, l'événement était trop ancien. Personne ne remontait dans le passé pour archiver les morts ordinaires, celles qui n'avaient provoqué aucun tapage et n'avaient pas éclaboussé la une des tabloïds de bouts de cervelle. Je finirais bien par glaner une information ici et là et par expulser ce gamin de mes pensées. En attendant, je me suis concentré sur les pluies d'avril et sur Myrtle the Turtle, le nouveau personnage que m'inspiraient ces pluies. Ce nom n'avait pas de sens, mais j'en aimais la sonorité. Myrtle the Turtle, the Genious of the Magic Bottle. J'écrivais n'importe quoi, pour tuer le temps et oublier que les marguerites n'étaient pas près d'éclore.
Le matin du 12 avril, jour de la résurrection du Christ, j'ai téléphoné à Régine, qui n'en menait pas large et attendait comme moi l'arrivée de la chaleur à défaut de croire à l'avènement du Messie, espérant que le soleil réchaufferait enfin ses mains continuellement froides. On a parlé de tout et de rien, évitant le sujet de L.A. et de Billie, même si on n'avait que ça à l'esprit, même si on pensait tous deux aux œufs colorés que L.A. cachait dans le jardin pour Billie, beau temps mauvais temps, chaque dimanche de Pâques. Elle peignait des poussins sur ces œufs, des fleurs ou de simples bandes de couleur à la Mark Rothko ou à la Kenneth Noland. Billie devait posséder pas loin d'une centaine d'œufs signés L.A., dont elle conservait les plus beaux dans un panier d'osier teint en jaune pour s'harmoniser avec la couleur des poussins de peluche que L.A. plaçait parfois près d'autres œufs artistiquement craquelés.
J'ignorais ce qu'était devenu ce panier. C'est Régine qui s'occupait des affaires de L.A. Elle procédait à un ménage en règle dans les paperasses, les tiroirs et les garde-robes, le nez dans la poussière et les souvenirs qui tuent. Quand tous les tiroirs seraient vides, elle mettrait la maison en vente, puisque j'avais tout laissé à L.A., la propriété, les meubles, les jouets de Billie, me débarrassant de mes responsabilités en même temps que des objets dont la vue m'était insupportable. J'avais lâchement abandonné L.A. au milieu des oursons et des jupettes roses et n'avais gardé de la puce que le cadavre de Pixie, que je retournerais bientôt à la terre. Je ne voulais pas de ses poupées autour de moi, de ses animaux de peluche aux yeux mouillés qui me suivraient de pièce en pièce et s'accrocheraient à mes jambes pour me parler d'elle. Alors j'avais laissé aux autres le soin de se démerder avec cet héritage empoisonné. Mais aujourd'hui, pendant que la voix de Régine se perdait dans le son des cloches revenues de leur pèlerinage à Rome, j'aurais aimé avoir avec moi la collection d'œufs de Billie, qui m'aurait aussi parlé de L.A. J'ai fini par rompre le silence dont nous entourions L.A. pour demander à Régine si elle savait ce qu'étaient devenus ces œufs. Ils sont sur la table, devant moi, m'a-t-elle répondu, et j'ai raccroché après qu'elle m'eut décrit l'un des derniers qu'avait peints L.A., marqué d'un point cerclé de jaune évoquant l'œil d'un hibou.
Je n'avais jamais vu cet œuf, mais je me le suis immédiatement représenté avec l'allure du hibou pendu près de la rivière, flanqué de deux yeux jaunes et exorbités qui lisaient vos pensées. J'ai repoussé l'image de cet œuf à tête de hibou avant de me mettre à créer des liens stupides entre L.A. et les mystères de Rivière-aux-Trembles. Plutôt que de délirer en ruminant la douleur trop palpable de Régine le reste de la journée, je me suis imposé une petite heure de marche rapide, trois kilomètres au nord, trois kilomètres au sud, aller-retour dans la grisaille. À mon retour, un chat faisait la sieste sur la galerie, un mâle jaune et blanc qui s'est mis à miauler dès que j'ai posé les pieds sur les marches, un vieux maudit matou qui ressemblait à Pixie. Je l'ai contourné en lui ordonnant de retourner chez lui. Je n'avais aucune envie d'adopter un chat, à plus forte raison un chat qui aurait pu se faire passer pour la réincarnation de Pixie si on lui avait enlevé la tache en forme de cirrostratus qui lui descendait sur le front. J'ai refermé la porte sur ses miaulements mais, de l'intérieur, ceux-ci me parvenaient assourdis.
Ce chat avait flairé quelle espèce de chiffe molle j'étais, du genre à se laisser attendrir, du genre qu'on a à l'usure, et il s'entêterait tant qu'il n'aurait pas obtenu ce qu'il désirait. Des images de l'époque où Pixie me donnait des ordres en tapant de la queue ont défilé sur l'horizon embrumé de ce morne jour de Pâques, des images de Billie berçant Pixie et flattant sa grosse bedaine de chat gavé de Whiskas à saveur de fruits de mer, et j'ai compris que j'étais foutu.
J'ai déniché dans le garde-manger une boîte de thon Clover Leaf qu'en d'autres temps j'aurais réservée à Pixie et j'ai servi ça à mon visiteur sur la galerie, dans un contenant Tupperware datant de mon ancienne vie. Pendant qu'il s'empiffrait, j'ai constaté que le vieux schnock ressemblait au vieux batêche jusque dans sa façon de manger et de vous jeter des coups d'œil en coin pour s'assurer que vous n'alliez pas lui enlever les fruits de son épuisante pêche. Si j'avais été ne serait-ce qu'un peu soûl, ne serait-ce qu'un peu plus fou, je me serais précipité à la cave pour vérifier si Pixie avait eu la géniale idée de sortir de son caisson pour venir me hanter. J'aurais présumé que le cirrostratus ne constituait qu'un des multiples effets de la congélation ou de la métempsychose et j'aurais séquestré ce chat pour le surveiller du coin de l'œil vingt-quatre heures sur vingt-quatre, au cas où il déciderait de se métamorphoser en magicien ou en chaman pendant que je balayais le plancher. Mais je n'étais ni soûl ni fou. Ce chat n'était que le sosie de Pixie, une espèce de frère, de quasi-jumeau ayant hérité du même air bête et suscitant chez moi le même élan de sympathie. Pixie et Dixie. Tout en caressant la tête de bum du jumeau de Pixie, je lui ai annoncé que je l'appellerais Dixie, en souvenir de Pixie, au cas où on se reverrait, et ne voulais pas l'entendre râler qu'il s'agissait d'un putain de nom de souris. Si ce n'était pas Dixie, ce serait Pixie Deux, et personne n'aime s'appeler Deux.
Il n'a pas écouté un mot de ce que je lui racontais. Il a fini son assiette et s'est renversé sur le dos pour que je lui flatte le ventre pendant que ses ronronnements emplissaient l'air embrumé du bruit des animaux satisfaits. Écoute, popinouche, me disait Billie quand Pixie, au seuil de l'extase, roucoulait comme un pigeon le soir de ses noces, écoute, y fait son bruit de chat content. Billie ne disait pas heureux, mais content, parce que la notion du bonheur est une notion qui appartient aux adultes, à ceux qui ont perdu le plaisir simple de l'enfance et qui espèrent un inaccessible nirvana au lieu de se contenter d'être contents. Le bonheur est un concept trop complexe pour que les enfants s'en embarrassent. Ils rient, ils jouent, ils sont et ne passent pas leur temps à se demander s'ils ne pourraient pas rire davantage ou s'esclaffer sous un éclairage plus conforme à leur idée du rire. Écoute, popinouche, y fait son bruit de chat content. Ça, c'était le summum. Pixie ne pouvait se sentir mieux.
Je me disais parfois qu'il aurait fallu établir la nomenclature des sons exprimant le contentement, des bruits produits par les animaux heureux, du gazouillis du moineau en passant par le grognement d'aise du porc devant son auge. Je me disais qu'il aurait fallu effectuer le décompte de ces sons et les enregistrer en vue de donner des cours de rattrapage aux millions de déprimés de la planète, à commencer par moi. Mais je n'avais aucune intention de me taper des cours du soir avec un chat sur les genoux. Ce vieux matou qui puait de la gueule n'avait rien à faire chez moi. Je ne serais pas sa prochaine victime, le prochain imbécile qui se posterait sous les rideaux pour surveiller son retour pendant que l'autre se ferait nourrir au saumon du Pacifique ou au thon du Nicaragua dans la maison d'un inconnu. Pas question que je t'adopte, ai-je annoncé au tas de poils qui ne m'écoutait toujours pas, et je l'ai laissé se les geler sur le bois de la galerie. Billie n'aurait pas approuvé ma conduite, elle aurait voulu que j'ouvre grand la porte à ce matou miteux pour ne plus la refermer, mais je refusais de donner asile à une créature qui deviendrait le centre de mon existence d'homme seul et me ferait le coup de disparaître au moment où je m'y attendrais le moins. Le chat, comprenant mon dilemme, m'a regardé marmonner derrière la fenêtre, puis il m'a répondu d'un sincère bâillement. Trois minutes plus tard, Dixie descendait la galerie en s'étirant pour retourner d'où il était venu. Je l'ai vu traverser la route de sa démarche chaloupée, puis il a disparu derrière un bosquet d'aulnes.
Quant à moi, j'ai quitté mon poste d'observation et me suis replongé dans les aventures de Myrtle the Turtle, pour oublier, pour tuer ce qui n'est pas tuable et m'illusionner quant au dispersement de l'ennui dans le défilement du temps. À la tombée de la nuit, quelques feulements de chat s'ajoutaient aux hurlements des loups, des enfants et des hiboux habitant les forêts de Rivière-aux-Trembles.
Aujourd'hui, Ronie le crapaud et Myrtle the Turtle ont mis leur chapeau de fête. Je les entends chanter « joyeux anniversaire, Billie » sous la couverture du cahier que j'ai vivement refermé lorsque Ronie s'est essayé à quelques vocalises à la mode tyrolienne.
Je me suis pourtant efforcé de faire taire ces voix, mais quand même j'étranglerais Ronie et expédierais Myrtle sur une île du Pacifique Sud grouillant de prédateurs, elles continueraient à s'élever dans la douceur de ce 20 avril, car aujourd'hui, Billie aurait eu douze ans et aurait descendu l'escalier en courant, selon son habitude, pour voir ce qui l'attendait sur la table de la cuisine. Un Dixie, probablement, un Pixie Deux avec une boucle bleue, illustre descendant d'une dynastie de chats pharaoniques convertis en chats de ruelle. À part ça, nous lui aurions peut-être offert un iPod ou des CD de groupes punk. Les goûts et les intérêts des filles de cet âge me plongent dans le noir le plus total. C'est Billie qui me guidait d'année en année et c'est elle qui m'aurait encore guidé dans cette noirceur en m'apprenant qu'elle ne voulait plus de poupées, de chandails roses ni de films mettant en vedette des animaux parlants.
Ça m'aurait fait un mal de chien de constater que ma fille grandissait et refusait désormais que je la prenne sur mes genoux pour lui raconter comment le capitaine Achab défie la foudre, mais j'aurais ravalé ma gomme et tenté de la suivre dans l'étourdissant tango qui, à cet âge, vous oblige à reculer de deux pas pour chaque pas effectué vers l'avant. J'aurais même essayé de fermer ma grande gueule devant les revendications débiles qui poussent tous les ados et pré-ados à réclamer une justice ne s'appliquant qu'à eux. J'aurais esquivé les tirs à vue de ma fille et on s'en serait sortis indemnes. On aurait quitté le champ de bataille en brandissant nos drapeaux blancs et on serait allés voir ensemble le dernier Woody Allen en se rappelant l'époque où je me tapais Fourmiz et Monstres, Inc. le même après-midi.
Voilà à quoi m'acculaient les voix qui s'égosillaient sur un pathétique Happy Birthday, à imaginer un avenir dont l'avènement reposait sur celui d'un miracle. La rengaine de Ronie et Myrtle m'obligeait à me remémorer les bons moments du passé, lorsque la petite avait mangé la moitié de sa chandelle le jour de son premier anniversaire, quand elle avait piqué une crise pour que Giroflée la girafe ait une part de gâteau, après quoi Giroflée avait passé deux heures dans le bain en état d'apnée et en était ressortie avec les taches à côté des trous, provoquant une nouvelle crise et m'obligeant à faire le tour de la ville sous la pluie battante pour dénicher une autre girafe répondant au ridicule nom de Giroflée. Même la contravention qu'une zélée en veste orange m'avait collée ce jour-là parce que ma voiture empiétait légèrement sur un stationnement pour handicapés absents appartenait à mes bons souvenirs. La fille finissait de rédiger sa contravention quand j'avais dévalé le trottoir avec une girafe heureuse dans les bras et un sourire auquel n'aurait pu résister le plus aguerri des pessimistes. Elle avait pourtant résisté, preuve que notre société recrute bel et bien certains de ses fonctionnaires en fonction de l'atrophie de leurs muscles zygomatiques. La casquette enfoncée jusqu'aux yeux et le stylo brandi, elle était demeurée insensible à mes arguments, que j'invoque les pleurs de Billie ou le fait qu'il n'y avait aucun handicapé dans le coin.
Devant son stoïcisme professionnel, j'avais claqué ma portière en lui faisant une grimace de girafe, regrettant que Giroflée ne soit pas un lama, puis j'avais dû m'arrêter sur la prochaine place de stationnement pour handicapés en vue, parce que ma farce de lama appartenait à ce répertoire de blagues totalement insignifiantes qui déclenchent néanmoins en vous un incontrôlable accès d'hilarité. On se trouve drôle, tout à coup, et on a envie d'en profiter un peu.
Ma bonne humeur m'avait suivi jusqu'à la maison, où j'avais agité ma contravention sous le nez de L.A. en riant comme un malade. La contravention n'avait pas amusé L.A., ma joke de lama non plus, ce qui ne m'avait pas empêché de placer ma rencontre avec les forces constabulaires parallèles parmi mes bons souvenirs.
J'en avais des dizaines, de ces bons souvenirs, des centaines, la plupart associés à Billie, j'en avais pour toute une vie, parce qu'un enfant vous comble une existence en criant lapin, mais j'aurais préféré devenir amnésique sur-le-champ plutôt que d'en être réduit à me rappeler que j'avais déjà été heureux. Il n'y a rien de plus triste qu'un bon souvenir quand la fillette avec qui vous devriez le partager n'est plus là pour que vous lui rafraîchissiez la mémoire, ni pour vous demander ce qu'était devenue Giroflée Première après sa catastrophique plongée en apnée.
Ronie et Myrtle refusant toujours de se taire, je suis descendu me chercher une bouteille de rouge à la cave en évitant de regarder le caisson de Pixie, qui devait lui aussi s'abîmer dans les bons souvenirs, et j'ai avisé Billie que j'avais besoin d'un remède de cheval et ne tracerais aujourd'hui aucune marque au feutre orange au milieu de la bouteille. Il y a des circonstances où toutes les règles tombent, quand votre unique fille n'est pas présente à son anniversaire, par exemple, ou lorsque la vie de la femme que vous avez aimée prend le chemin des égouts avec l'eau du bain et que vous êtes tenté de faire pareil, histoire de noyer le crapaud qui chante sous votre crâne.
J'ai dévissé le bouchon de mon Cigare volant à cinquante dollars au milieu de l'escalier et me suis envoyé une rasade d'anniversaire derrière la cravate en chantant « whiskey is the devil in his fucking liquid form ». Je ne me souvenais pas si c'était les frères Bailes, les frères Baileys ou les frères Blues qui avaient interprété Whiskey Is the Devil (in Liquid Form) et ça n'avait aucune importance. Bales ou Baileys, les frères parlaient des mêmes lendemains de veille. Leur mise en garde avait du sens, l'alcool n'était qu'une des représentations des démons qui vous rongent, mais je repenserais à ce sage aphorisme plus tard, quand l'alcool à forte dose ne m'apparaîtrait plus comme la seule façon de traverser ce funeste jour d'anniversaire. Pour le moment, j'avais envie de me soûler la gueule et d'oublier mes bons souvenirs en écoutant mes démons chanter du blues. Mon objectif était simple, me sentir un peu plus lézardesque à chaque gorgée, car j'avais une certaine admiration pour le comportement des lézards, qui ne coupent pas les cheveux en quatre, se contentent de tirer la langue s'ils ont faim et dorment carrément au plafond si les murs sont occupés.
N'ayant rien mangé depuis le matin, j'ai avalé mon premier verre de rouge avec un Jos Louis dont j'ai dédié la première bouchée à Billie. Une bouchée pour Billie, une bouchée pour papa, une bouchée pour maman. La première bouchée a bien descendu, mais les deux autres me sont restées coincées dans l'œsophage, puis quelques larmes ont mouillé le bord de mes paupières et les bouchées dédiées à l'enfant appelée Billie ont poursuivi leur chemin dans un bruit de déglutition plus ou moins ragoûtant que personne n'a entendu, alors qu'est-ce qu'on s'en sacre.
En d'autres circonstances, la tristesse aurait entamé mon appétit, mais puisque je mangeais pour Billie, j'ai englouti le reste du Jos Louis en quelques secondes. J'ai même léché l'emballage, où un peu de chocolat avait adhéré, en me disant que Billie aurait sûrement aimé les demi-lunes et les Jos Louis si L.A. avait consenti à en acheter, puis j'ai vidé mon verre en fredonnant un de ces petits airs sans queue ni tête n'ayant d'autre but que de faire taire les voix qu'on refuse d'entendre, en l'occurrence, en ce moment précis et en ce qui me concernait, celle de Lucy-Ann Morency. Je n'avais pas envie d'essuyer les remontrances outre-tombales de Lucy-Ann ni de penser que mon ex était une foutue bourgeoise dont les goûts de luxe avaient privé notre fille des trésors de la pâtisserie industrielle québécoise. Si on lui avait permis d'en manger, Billie aurait sûrement adoré les demi-lunes et les Jos Louis, tous les enfants adorent ça. Il faudrait toujours agir avec eux comme s'ils allaient disparaître dans l'instant, comme si un tsunami allait les emporter, comme si les monstres faisant le guet dans leur voiture étaient toujours sur le point d'ouvrir leur portière pour tendre la main vers la première fillette passant par là. Si j'avais su que Billie s'envolerait sans crier gare, j'aurais aboli toutes ces interdictions idiotes qui contrarient les désirs simples des enfants et lui aurais permis de manger de la tarte au citron au déjeuner et de se balader dans le salon en patins à glace, si ça pouvait lui faire plaisir.
Tu dérailles, Richard, a soufflé dans mon cou la voix de Lucy-Ann, t'es complètement à côté de la track, puis elle s'est mise à pleurer, son haleine sur mon cou brûlant, et j'ai repris ma rengaine en me bouchant les oreilles, « je l'ai connue la la, en dansant le yaya, ah ah (ah ah) », mais la voix de L.A. refusait de me ficher la paix, tu dérailles, my love, faisant à son tour resurgir une flopée de souvenirs goûtant le bord de mer et la peau salée. Constatant que je n'avais aucune chance de repousser les fantômes qui me hantaient, je me suis laissé pleurer quelques instants, mêlant mes larmes à celles de L.A., qui se sont enfin diluées dans mon deuxième verre de rouge, dont je n'ai dédié aucune gorgée à personne d'autre que moi.
J'étais en plein dans mon devenir lézard, à deux pas de l'indifférence reptilienne à laquelle j'aspirais, quand une auto a freiné dans la cour. Deux portières ont ensuite claqué et j'ai vu apparaître deux types à la fenêtre de la porte d'entrée, deux hommes qui avaient la gueule de Ménard et Dubois, les inspecteurs qui s'étaient occupés de l'affaire Billie Richard. Il ne s'agissait pas de Ménard et Dubois, mais les deux gars qui scrutaient l'intérieur de ma maison en plissant les yeux avaient été coulés dans le même moule et arboraient cet air de flic sinistre et renfrogné n'augurant rien de bon.
Spontanément, j'ai pensé qu'ils venaient m'apprendre la mort officielle de Billie, qu'ils avaient été délégués par Ménard et Dubois pour se taper le sale boulot et m'annoncer qu'un couple d'amoureux ou une quinquagénaire promenant son doberman dans un sous-bois plein de chaussettes orphelines avait découvert un squelette qu'on avait identifié à l'aide des quelques boutons roses logés dans le sternum. J'aurais pu penser le contraire et croire qu'ils étaient porteurs d'une bonne nouvelle, mais des gueules aussi froides, sans la moindre lueur de joie dans l'œil, ne pouvaient transporter de bonnes nouvelles.
Depuis trois ans, j'attendais désespérément ce type de visite, mais là, je ne voulais pas savoir, pas en ce jour de fête, pas alors que Billie venait à peine d'avoir douze ans et que Ronie le supercrapaud soulignait l'événement en compagnie d'une tortue sortie de sa léthargie pour la circonstance. J'ai ignoré le tambourinement de leurs poings contre la porte et me suis calé dans mon fauteuil, comme s'il avait le pouvoir de me rendre invisible. Les coups ont toutefois redoublé et un des deux hommes m'a crié qu'ils avaient un mandat et enfonceraient ma porte si je n'ouvrais pas.
À compter de cet avertissement, je n'ai plus rien compris. Pourquoi ces deux policiers avaient-ils besoin d'un ordre du tribunal pour m'annoncer qu'un ou deux jours plus tôt, une femme était ressortie d'un sous-bois en poussant des hurlements hystériques ou en hoquetant, le souffle coupé, après avoir vomi près de ce qui avait jadis ressemblé à un enfant ? Pourquoi leur fallait-il un tel document pour m'apprendre que j'étais désormais et officiellement un père sans fille, une espèce d'orphelin à l'envers, puisqu'il n'existe aucun terme pour désigner les pères et les mères qui perdent un enfant ? Ces parents-là ne sont ni orphelins, ni veufs, ni expatriés. Ils ne sont rien. Des pères et des mères à vie, sans mot pour décrire leur nouveau statut sur leurs déclarations de revenus. Des êtres à jamais anonymes, se promenant avec un trou béant dans le ventre et ne bénéficiant du secours d'aucun mot pour attester leur identité trouée.
J'ai titubé jusqu'à la porte, suivant l'ondulation d'un rayon de soleil s'étalant à l'oblique sur le plancher de lattes, dont la solidité me semblait soudainement douteuse. J'ai ouvert la porte en m'appuyant au chambranle et les deux flics m'ont montré leur plaque, Marchessault et Doyon, de la Sûreté du Québec. J'attendais qu'ils me fassent part du motif de leur visite, mais ils se sont dirigés vers la cuisine, où ils m'ont invité à m'asseoir avec eux. Les flics ont une prédilection pour les cuisines, à moins qu'il ne s'agisse d'une forme d'atavisme bien québécois, d'une espèce de nostalgie des réunions de cuisine du temps des fêtes atteignant plus profondément les membres des forces policières. Je ne sais pas. Quand Marchessault a fait grincer les pattes de sa chaise sur le plancher, j'ai eu l'impression de revivre une des innombrables scènes auxquelles m'avait contraint Ménard en se pointant chez moi sans s'annoncer et j'ai attendu que l'un des deux hommes assis à ma table se décide à parler.
Michael Faber, ça vous dit quelque chose ? m'a demandé Marchessault, le plus grand, celui qui allait mener l'interrogatoire en se donnant des airs de dur et en faisant passer l'autre pour un mou, alors que le mou en question noterait tout dans sa grosse tête chauve et me tomberait sur le paletot à la première occasion. Je connaissais la chanson. J'ai hésité un instant avant de répondre, n'ayant à l'esprit que le nom de l'enfant gravé sur la croix de bois plantée près de la rivière. Pouvait-il s'agir du même Michael, du garçon que je croyais mort et enterré depuis longtemps ? Dans ce cas, que lui était-il arrivé ? J'ai essayé de leur tirer les vers du nez, mais ils sont demeurés de glace. Ils m'observaient, stoïques et silencieux, pour voir si mes paupières allaient frémir ou ma voix se mettre à trembler quand je répondrais à leur question, alors je leur ai affirmé que non, je ne connaissais pas de Michael Faber, sans bégayer. Parce que je ne mentais pas. Je ne connaissais aucun Michael, y compris celui qui s'était ou non noyé dans la rivière. J'avais pourtant la certitude qu'ils étaient là pour me raconter l'histoire de ce garçon et comment elle était liée à la mienne. Je les ai donc laissés aller en redoutant ce qu'ils me révéleraient sur cet enfant.
Ils m'ont d'abord posé quelques questions sur mon emploi du temps en lorgnant sur le comptoir la bouteille de rouge dont le niveau avait dépassé le seuil de l'invisible ligne orange et en faisant totalement fi de mon désir de savoir pourquoi j'étais soumis à cet interrogatoire. Chaque chose en son temps, grognait Doyon, le chauve, en lissant la moustache qui lui servait à prouver que son système pileux n'avait pas davantage rendu l'âme que sa virilité. Vous répondez d'abord à nos questions, c'est clair ?
Je ne pouvais pas leur dire grand-chose sur mon emploi du temps, parce que le temps, je le tuais, jour après jour et sans relâche, tel un assassin qui voit sa victime ressusciter chaque fois que le soleil se lève et devient obsédé par la lumière. Je leur ai toutefois parlé de Ronie et de Myrtle, de ce qui me permettait de gagner ma vie, parce que la vie a un prix, tout le monde l'apprend un jour ou l'autre. Je leur ai parlé du crapaud volant qui m'avait donné les moyens de payer une maison à une femme qui avait repeint la salle de bain en rouge, une maison qu'il faudrait maintenant vendre à rabais, personne n'aimant particulièrement le rouge qu'avait choisi L.A., rouge sang, Billie, rouge qui coule dans les veines.
Mon gagne-pain ne les intéressant pas plus que la façon dont L.A. et moi avions dépensé mon argent, ils ont glissé sur le sujet et leurs questions sont devenues plus précises. Qu'avez-vous fait, monsieur Richard, dans la journée du 18 avril ? Le 18 avril, c'était l'avant-veille. En principe, j'aurais dû me souvenir clairement de la façon dont j'avais occupé ce samedi mais, mes journées se ressemblant toutes, celle-ci se perdait dans la grisaille immuable de la répétition. Fournissez un petit effort, c'est pas si loin, a soufflé Marchessault en appuyant ses coudes sur la table et en plaçant son visage assez près du mien pour que je devine ce qu'il avait mangé au déjeuner. Le parfum fétide d'œufs frits à moitié digérés m'a forcé à reculer et, en me balançant sur ma chaise, j'ai fait marche arrière jusqu'au samedi 18 avril, heure par heure, en prenant le temps à rebours, pour enfin me rappeler que le matin du 18 avril, j'avais appris en ouvrant un journal vieux de deux ou trois jours que la petite Ashley Tara Bravo Gonis allait enfin pouvoir sauter dans les bras de son père après deux ans d'absence. Cette nouvelle m'avait fait trembler le menton, je m'en souvenais, parce qu'elle prouvait qu'on pouvait demeurer vivant dans l'obscurité. Je suivais cette affaire de près, comme tous les pères dans ma situation, je suppose, solidaires de leurs frères meurtris. Après avoir appris que Tara avait été kidnappée par sa mère, je m'étais mis à espérer que Billie ait été enlevée par une folle qui plaçait sa stérilité au-dessus de toute forme de morale. Frustrée dans son désir de maternité et rejetant l'un après l'autre les ovules que son impotent de mari n'arrivait pas à féconder, la folle nous aurait volé Billie pour l'amener au Mexique ou en Argentine et se faire appeler mama. Dans mes rêves les plus désespérés, Billie déjouait la vigilance de mama et s'évadait en sautant par une fenêtre. Le reste de l'histoire comportait plusieurs variantes, mais se terminait invariablement par un happy end digne de la période la plus rose de Hollywood.
J'ai dépensé ma salive pour rien car la libération d'Ashley Tara ne passionnait pas non plus Marchessault et Doyon. Ce qui les chicotait, par contre, c'était mon intérêt pour les enfants disparus. Devant leurs insinuations, j'ai abattu mon poing sur la table en exigeant qu'ils me disent enfin si leur présence avait quelque chose à voir avec ma fille et si ce Michael Faber était lié à sa disparition. Chaque chose en son temps, a répété Doyon, visiblement peu impressionné par ma colère, décrivez-nous d'abord votre journée. Vous aurez tout le temps de massacrer votre mobilier quand vous aurez clarifié certains points avec nous.
J'y répondrai le 18 avril prochain à tes questions, connard, ai-je pensé en me cousant les lèvres avec une fermeture éclair imaginaire, mais Marchessault a pris la relève avec son haleine d'œufs frits et j'ai compris que je m'épuisais inutilement à ne rien dire. Je me suis donc creusé les méninges et me suis rappelé qu'après ma lecture du journal, j'avais pris une douche avant de me rendre au supermarché de Saint-Alban. Quand je me suis dézippé les lèvres pour prononcer le nom de Saint-Alban, ils se sont tout à coup énervés. Immédiatement, ils ont voulu savoir à quelle heure j'y étais allé, à quelle heure j'étais revenu, combien de personnes m'avaient vu là-bas et si quelqu'un pouvait confirmer l'heure de mon retour. Alors je me suis énervé aussi, pour la deuxième et dernière fois. Je voulais des explications tout de suite, sinon je n'ouvrais plus la bouche que pour appeler un foutu avocat à trois cents dollars de l'heure.
Marchessault et Doyon se sont regardés dans le blanc des yeux, comme des mecs qui se comprennent à un cillement des paupières, Doyon a fait un signe du menton en direction de Marchessault et ce dernier s'est lancé dans un récit abracadabrant relatant la disparition d'un gamin de douze ans, Michael Faber, vu pour la dernière fois le samedi 18 avril près du passage des Grottes, à mi-chemin entre Saint-Alban et Rivière-aux-Trembles, vers les treize heures, alors qu'il roulait sur la bicyclette toute neuve qu'on lui avait offerte pour son anniversaire. Une bicyclette qu'il réclamait depuis Noël, anticipant les beaux jours, et qu'il ne quittait que pour dormir ou franchir la porte de son école. Bleue, la bicyclette, avec un klaxon en forme de poire. D'après ce que j'ai pu comprendre, l'enfant se dirigeait vers Rivière-aux-Trembles. Le vélo avait été découvert à quatre kilomètres du village, dans le fossé bordant la route. Le duo Marchessault et Doyon ratissait donc les alentours et interrogeait toute personne ayant un lien plus ou moins étroit avec une affaire de disparition, ce qui était mon cas, puisque le dossier de Billie n'avait jamais été classé. Jusqu'à nouvel ordre, il n'y avait que moi et un autre suspect qui répondions à ce critère dans toute la région, alors j'avais intérêt à coopérer et à leur dire si une ou plusieurs personnes pouvaient témoigner de mes allées et venues le 18 avril.
J'étais sonné et plus aucun mot ne parvenait à sortir de ma bouche. Ces hommes étaient cinglés, encore plus cinglés que Ménard et Dubois, qui ne m'avaient jamais traité de psychopathe. J'ai exigé un temps mort pour aller au salon chercher mon verre de vin, que j'ai avalé cul sec. J'aurais pu leur dire de débarrasser le plancher et de me convoquer au poste, où je me rendrais dans la BMW de mon avocat véreux, mais je n'avais rien à me reprocher. Malheureusement, je n'avais aucun témoin à leur fournir non plus, à part Myrtle the Turtle, dont j'avais bouclé une aventure dans l'après-midi du 18. La date et l'heure inscrites sur mon ordinateur le leur prouveraient. Quant aux gens qui m'avaient vu dans l'allée des légumes ou des produits de nettoyage du supermarché, ils étaient des dizaines, des femmes en majorité, ce n'est pas moi qui allais leur mentir sur la durabilité de certains phénomènes sociaux et sur l'échec partiel de la révolution féministe. Ils pouvaient aussi s'adresser à la caissière aux cheveux orange, qui se rappellerait sûrement avoir vu un homme avec un foulard rouge, vers les dix heures trente, un homme correspondant à ma description. Ils n'avaient qu'à le lui demander.
Dès qu'on en aura fini ici, a grogné Marchessault, puis, muni de son mandat, il a fouillé la maison de fond en comble avec Doyon, au cas où ils y trouveraient la casquette de Michael Faber, les poignées fluo qu'il avait installées sur son vélo flambant neuf ou, pire encore, son cadavre enveloppé dans du papier ciré ou du Saran Wrap. Ces imbéciles ont même forcé le caisson de Pixie, décrétant du même coup que j'étais fou à lier. Le chat du petit Faber avait disparu en même temps que lui, paraît-il, et Pixie répondait à son signalement. J'ai eu beau leur expliquer qu'il s'agissait du chat de ma fille, auquel je devais des funérailles dignes de ce nom, ils m'ont immédiatement classé dans la catégorie des malades qui torturent les animaux et se les congèlent pour le déjeuner. Ce n'était pas moi, le malade, mais eux, qui avaient profané le cercueil temporaire de Pixie et mettaient leurs grosses pattes sales partout.
Quand Marchessault m'a dit qu'il embarquait Pixie en tant que pièce à conviction, j'ai serré les poings, prêt à me jeter dessus, mais Doyon, prévoyant mon geste, m'a intercepté pour me plaquer au mur avant que j'aie le temps d'écrabouiller l'autre. Trente secondes plus tard, j'étais menotté. Voie de fait sur un agent de la paix, a glapi le chauve, puis il m'a lu mes droits en me recommandant de me servir du coup de fil auquel j'avais droit pour appeler le meilleur criminaliste de la province. Selon lui, j'allais en avoir besoin. Son histoire de voie de fait ne tenait pas debout, puisque je n'avais pas effleuré un cheveu de Marchessault, mais c'est lui qui tenait le gros bout du bâton. Sachant pertinemment qu'en cas de disparition d'enfant, les flics n'ont pas le cœur à rire ni à se laisser emmerder par des abrutis, j'ai fermé ma gueule et on est repartis tous les cinq, Marchessault, Doyon, Pixie, mon ordinateur et moi, Marchessault et Doyon avec cet air inquiet que j'avais observé tant de fois sur le visage de Ménard, Pixie et moi avec des têtes d'enterrement. Quand Doyon a fermé la porte de la maison, Ronie et Myrtle s'étaient tus. Happy birthday Billie, ai-je murmuré en m'assoyant à l'arrière de la voiture de police, d'où j'ai aperçu Dixie le chat, qui venait chercher sa ration de thon. J'ai tapé dans la vitre pare-balles me séparant des deux gars qui discutaient à l'avant pour attirer leur attention sur Dixie, qui répondait également au signalement du chat disparu, mais autant crier à l'aide derrière un régiment de sourds. Marchessault et Doyon avaient entendu tellement de gars taper dans cette vitre au cours de leur carrière qu'ils ne s'en occupaient plus.
Les cloches d'une église invisible sonnaient midi lorsque la voiture s'est engagée dans le stationnement du poste local de la Sûreté du Québec de Saint-Alban. À travers les vitres fermées, cette musique évoquait la joie des dimanches en famille, les robes soulevées par un vent de printemps. J'étais soûl et Billie avait douze ans.
## II
MARNIE
Février-Avril 2009
Le hibou de Saint-Alban se balance devant la fenêtre noire au coin de laquelle s'écorchent les dernières rafales de la tempête, mêlées de flocons glissant sans bruit sur la poussière d'automne que les pluies de février n'ont pas lavée. Et il me parle encore, il me dit qu'au milieu de la forêt enneigée, des petits animaux disparaissent, des petits Mike et des petites Marnie, certains ne laissant derrière eux qu'une traînée de sang, d'autres ne laissant rien, pas même quelques pistes égarées ni quelques battements d'ailes.
C'est mon père qui m'a offert ce hibou, que j'ai secrètement baptisé Mister Holy Owl, alias Mister Holy Crappy Owl, parce qu'il est né le jour de Noël, en quelque sorte, et parce que je déteste les saloperies de hiboux qui vous fixent dans l'obscurité et vous parlent au lever du jour. J'avais aperçu cet oiseau dans une boutique de souvenirs de Saint-Alban, l'avant-veille de Noël, pendant que mon père et moi faisions des courses dans la joyeuse fébrilité du temps des fêtes. J'avais tout de suite eu l'impression que ce petit animal inerte me regardait, moi et personne d'autre, et qu'il m'appelait à lui. Chaque fois que la porte de la boutique s'ouvrait, il oscillait au bout de sa cordelette et tournait ses petits yeux jaunes dans ma direction. Chaque fois. Je m'en étais approchée mine de rien, puis les yeux exorbités de Michael m'affirmant que les hiboux voient tout, absolument tout, Marnie, s'étaient plaqués sur ceux de l'oiseau. L'espace d'un instant, je m'étais engouffrée dans ce regard et j'avais été projetée ailleurs, en un autre temps et dans un autre lieu, seule avec le hibou dans une région de ma mémoire où le ciel s'ennuageait. Les murs de la boutique avaient disparu avec les clients pressés, avec la musique de Noël, le neige virevoltant derrière la vitrine, pour ne laisser place qu'à cela, un hibou écarquillant les yeux devant un ciel d'orage. Michael, c'est toi, Michael... ?
Un frisson m'avait zigzagué dans le dos quand mon père avait posé sa main sur mon épaule et j'avais mis sur le compte de la chaleur régnant dans la boutique la rougeur envahissant mon visage. Faut que je prenne l'air, pop, je vais à la pâtisserie chercher la bûche, puis j'avais plaqué mon père devant un étalage de gnomes et de lutins.
L'époque où Michael et moi prétendions être un hibou des neiges et une écureuille rousse était à ce point lointaine qu'il m'arrivait d'oublier les hululements maladroits de Mike lorsqu'il venait me chercher après le souper. Trois ou quatre hou-hou traversaient la fenêtre ouverte et je répondais à l'appel du hibou en me précipitant dehors aussi vite qu'un écureuil dévalant un toit de tôle. Ce jeu avait duré un peu plus de trois ans, mais j'avais depuis longtemps rangé le hibou simplement nommé Owl dans une boîte que je n'ouvrais jamais, préférant imaginer Michael sous l'apparence d'un superhéros envolé vers quelque destination secrète. La vue de ce petit oiseau d'écorce m'obligeait à rouvrir la boîte et à me rappeler qu'avant les robots, l'interdimensionalité et la génétique futuriste, il y avait eu la forêt, les arbres et les animaux, qu'avant que Michael soit investi des superpouvoirs de Superman, ce sont les pouvoirs d'un oiseau de nuit qui guidaient son imagination d'enfant.
Avant d'arriver à la pâtisserie, j'avais été tentée de rebrousser chemin pour aller chercher ce maudit hibou, puis je m'étais ravisée. Si tu fais ça, Marnie, t'es foutue. Tu vas te mettre à parler aux poignées de porte et à croire qu'elles te répondent.
J'étais entrée à la pâtisserie en nage, malgré le froid, et avais été forcée de m'asseoir pour reprendre mon souffle. Heureusement, il n'y avait ni hibou ni écureuil dans l'établissement, seulement des têtes de père Noël au visage joufflu, des branches de gui et des guirlandes. Je m'étais concentrée sur le père Noël numéro deux, celui qui me faisait face, défiant sa grosse tête décapitée de se mettre à parler, puis, le bonhomme demeurant de glace, je m'étais enfin détendue et avait été prise d'un fou rire qui m'avait obligée à me ruer vers les toilettes du café attenant à la pâtisserie. Une vraie folle, c'est comme ça qu'on dit, et c'est ce que je m'étais répété dans le miroir du lavabo : Vire pas folle, bordel !
Le reste de la journée, de même que le lendemain, j'avais à peine pensé au hibou. Le souvenir de sa petite tête hérissée de fausses plumes me traversait parfois l'esprit, mais je la chassais aussitôt. À l'heure qu'il était, quelqu'un l'avait probablement acheté pour le suspendre à son arbre de Noël, inconscient des révélations que cet oiseau allait lui susurrer à l'oreille entre la dinde et la tarte au sucre.
Je m'étais consacrée à papa, à nos retrouvailles, à la décoration du salon et à la confection d'une meringue aux fraises que j'avais façonnée en forme de sapins. C'est ce que permet la magie de Noël, de s'entourer de sapins roses, de rennes harnachés de diamants de pacotille, de petites fées vêtues de mousseline. J'avais ensuite bu du kir avec papa, nous avions écouté Bing Crosby puis, aux douze coups de minuit, le 24, j'avais tendu son premier cadeau à mon père. Les papiers multicolores s'étaient ensuite accumulés autour de nous, dans le joyeux désordre dont aime à s'entourer la nostalgie. Au moment où j'appliquais derrière mes oreilles une ou deux gouttes du parfum que papa m'avait offert, il avait glissé vers moi la petite boîte contenant le hibou. Quand j'avais soulevé le papier de soie le recouvrant et aperçu, dardées sur moi, les petites pastilles jaunes et noires enfoncées dans la tête de l'oiseau, mon cœur n'avait fait qu'un bond et j'avais levé vers mon père un visage stupéfait.
Où est-ce que t'as pris ce hibou ?
À Saint-Alban, avant-hier, il avait l'air de te plaire. Tu l'aimes pas ?
Oui, oui, bien sûr. C'est juste que je m'attendais pas du tout à ça.
Je n'avais jamais si bien dit, mais il n'était pas question que je gâche la joie de mon père avec mes chimères. Toute la nuit, cependant, ce maudit hibou m'avait observée, guettant le moindre de mes mouvements, saisissant chacune de mes paroles, et c'est avec un immense soulagement que je l'avais remis dans sa boîte, deux jours plus tard, en préparant mes bagages.
Il est resté dans cette boîte pendant deux ans et, chaque fois que j'ouvrais l'armoire où je l'avais rangée, il me semblait entendre Holy chuchoter et gratter les parois de carton en poussant un de ces faibles hululements que j'entendais autrefois se glisser sous les rideaux battant au vent. Le plus simple aurait été de m'en débarrasser, mais je n'ai jamais pu m'y résoudre, me disant qu'une parcelle de l'esprit, de l'âme ou du corps de Michael s'était peut-être logée dans cet oiseau, que le petit Michael Saint-Pierre, une trentaine d'années plus tôt, avait peut-être frôlé l'écorce dont était constitué ce hibou, qu'il s'était peut-être appuyé à l'arbre d'où il provenait, y laissant l'empreinte de son souffle ou de son sang, et que c'est cette empreinte qui animait aujourd'hui le regard du hibou de Saint-Alban.
J'avais toujours la certitude, contre toute logique, que cet objet était porteur d'un message et que même si je l'expédiais en Afghanistan, il reviendrait tôt ou tard frapper son petit bec pointu contre ma fenêtre en battant des ailes. Le premier psy venu aurait interprété mon attitude comme une autre des manifestations de ma culpabilité refoulée, mais je m'entêtais à croire que le malaise que j'éprouvais devant cet animal ne venait aucunement de mes hantises. Ce sont les seules raisons qui m'ont empêchée de jeter ce volatile aux ordures ou d'aller le perdre dans les allées de Central Park.
Ce sont les mêmes raisons qui m'ont poussée à le sortir de sa boîte pour le suspendre à la fenêtre de la cuisine. Et voilà qu'aujourd'hui, alors que la neige se transformait en poudrerie, Holy Crappy Owl s'est mis à me parler, à osciller au bout de sa cordelette en m'indiquant de ses petits yeux lumineux la direction à suivre, là-bas, près des eaux glacées de la rivière.
J'achevais mon café du midi lorsque les premiers flocons se sont mis à tournoyer et que j'ai entendu l'appel du hibou, ce lointain appel qui m'incitait jadis à me ruer dans la cour fleurie de Mary-Jean et de Golden Girls. Avachie dans mon fauteuil, je feuilletais tranquillement le journal quand il a retenti, venant de la cuisine, de la fenêtre de Holy Owl. Ça devait être le vent, ça ne pouvait être que le vent faisant grincer la porte du hangar ou la poulie accrochée au coin de la maison, faisant claquer les tuiles de bardeau du toit. Et pourtant aucun de ces bruits, je le savais, ne pouvait s'apparenter à l'appel du hibou.
Cette fois, je ne me suis pas précipitée. Je me suis lentement dirigée vers la fenêtre de la cuisine en murmurant what the fuck, Holy ? what the holy fuck ? avant de saisir cet oiseau de malheur par la queue pour le placer face à moi. Mais cette tête de pioche résistait et, d'un mouvement saccadé, pareil à celui d'un oiseau sautillant sur une branche, il a de nouveau pivoté vers l'extérieur, le regard tourné vers les eaux invisibles de la rivière aux Trembles, dont un des bras passait derrière la forêt, tout près.
J'ignore combien de temps je suis demeurée devant la fenêtre. Tout ce dont je me souviens, c'est que la neige a redoublé d'intensité avec le vent et qu'en baissant les yeux, j'ai aperçu à mes pieds la tasse de porcelaine héritée de ma mère, en trois ou quatre morceaux gisant dans le café froid. J'ai marché sur l'un des morceaux, dont j'ai senti la pointe traverser ma pantoufle pour se ficher dans ma chair tendre, puis je suis montée m'habiller sans prendre la peine de panser la blessure qui avait laissé quelques gouttes colorées dans l'escalier. Une dizaine de minutes plus tard, j'étais dehors, devant la maison, chaussée des raquettes de mon père. Dans mon dos, Mister Owl veillait, son regard dardé sur ma nuque et sur les bois de Rivière-aux-Trembles.
Je n'avais pas chaussé de raquettes depuis une bonne dizaine d'années et j'ai dû gueuler pendant un demi-kilomètre contre l'inventeur de cette saloperie, puis j'ai fini par retrouver le rythme, la façon dont il fallait allonger les jambes, se tenir le dos droit et assurer son équilibre. Il était toutefois difficile de ne pas courber l'échine devant la violence du vent, mais je lui faisais face et j'avançais, j'avançais, parce que je n'avais pas d'autre choix. Si Phil avait su où je me rendais, il serait sûrement venu me chercher pour me ramener à la maison en me traitant d'écervelée, car Phil Morisset, depuis la mort de mon père, veillait sur moi comme si j'étais une petite chose fragile, une Marnie à la puissance dix qu'il fallait protéger de la bêtise ambiante. C'est ce que font les hommes de la trempe de Phil, ils s'occupent des trésors des morts sans les piller, et j'avais une sacrée chance de pouvoir compter sur lui, car celui que j'appelais autrefois mononcle Phil était le seul être vivant capable de m'empêcher de dérailler. Sans ce pont jeté entre mon passé et mon présent, Rivière-aux-Trembles aurait perdu sa consistance. J'avais besoin de l'intelligence rugueuse de Phil Morisset pour m'accrocher à une réalité que ma fuite avait disloquée.
Durant mes années New Yorkaises, j'étais pourtant parvenue à me débrouiller seule et à faire la paix avec mon passé. Je n'avais pas oublié Mike, mais son visage était devenu aussi flou que certaines images de Rivière-aux-Trembles, perdues entre neige et brouillard. Il avait peu à peu quitté mes nuits et j'avais pu dormir sans être oppressée par le sentiment que je n'avais pas droit au sommeil. La culpabilité que j'avais ressentie autrefois ne m'appartenait plus. Elle appartenait à la fillette que certains nommaient sorcière, bitch, Marnie the Witch, et je savais désormais que les sorcières n'existaient pas. Si j'étais restée là-bas, les sorcières auraient continué à ne pas exister, les hiboux seraient demeurés dans la forêt et j'aurais mené la vie d'une femme semblable à toutes les autres, avec son drame à elle et sa peur de l'orage. Mon retour à Rivière-aux-Trembles a mis un terme à cette insouciance fabriquée de toutes pièces. En l'espace de quelques jours, plus de vingt années de ma vie ont disparu, le doute a refait surface avec la petite Marnie et la culpabilité a pris l'apparence d'un hibou de paille et d'écorce me soufflant à l'oreille que ma vie n'a été qu'un leurre. Je devais déjouer ce leurre et Phil, dans ce jeu, ne pouvait m'être d'aucun secours. C'était à moi et à moi seule d'esquiver la feinte du hibou, de suivre la trace des petites pattes étoilées dans la neige afin d'y distinguer les vraies pistes des fausses. Alors j'avançais, ma crémone rouge sang alourdie par les petits mottons de glace créés par la rencontre de mon souffle et de la neige.
Je n'avais qu'un but, me rendre au bassin magique et m'y rendre rapidement, parce que c'est là que tout commençait, que tout se terminait, et que tout allait recommencer. Je ne cherchais pas à saisir la nature de mon empressement ni à en expliquer le comment et le pourquoi. Je lui obéissais, c'est tout, persuadée que le sentiment d'urgence qui me poussait ne pouvait me mener que vers quelque révélation.
Arrivée près de la maison de Lucien Ménard, j'ai hésité un peu devant la force de la poudrerie balayant les champs. C'était toujours par là qu'on piquait, Michael et moi, guettant Ménard qui travaillait dehors, autour de sa maison. Dans ses bons jours, il nous envoyait la main en nous recommandant de nous méfier de la crue de la rivière, des ourses qui trimballaient leurs petits, des nids de guêpes ou des nuages d'orage, mais quand il avait forcé sur le gros gin la veille, une traînée de jurons nous accompagnait jusqu'à ce que le vieux Lulu s'étouffe avec son jus de pipe.
Je me suis engagée dans le champ en dépit des rafales cinglant le vide auquel elles se frappaient, face contre le vent et les souvenirs d'enfance, et me suis dirigée vers la forêt. J'aurais dû repérer le sentier rapidement, mais en trente ans, tout avait changé. Certains arbres n'étaient plus là, d'autres avaient poussé, le pommier perdu produisant des fruits durs et pas mangeables dont nous nous gavions jusqu'à en avoir mal au ventre avait disparu et je n'avais plus aucun repère qu'une rangée d'arbres semblables à ceux de mon enfance, et pourtant en tous points différents. J'ai néanmoins fini par déceler entre deux sapins une ouverture qui pouvait ressembler à l'entrée d'un sentier et m'y suis engouffrée.
Je n'avais pas fait dix pas sous le couvert du bois que, déjà, la tempête s'atténuait. Le vent grondait à la cime des arbres, que je voyais ployer en tous sens au-dessus de ma tête, quelques rafales de neige venues de la hauteur du monde s'écroulaient du sommet des pruches et des épinettes, mais la forêt était d'un calme inouï auprès de la tourmente se déchaînant au dehors. J'étais dans un sanctuaire secoué par la furie du nordet s'écorchant à ses voûtes, mais le tumulte demeurait à l'extérieur, au-delà des murs sombres enfermant la lumière de la neige. Au creux du sanctuaire, régnait l'inaliénable silence des bois.
J'ai fermé les yeux quelques instants pour mieux sentir la paix qui m'enveloppait, m'accueillait dans son asile, et j'ai été propulsée à des années de là, quand Michael et moi nous arrêtions au milieu d'un sentier pour entendre ce qui ne s'entend pas, le souffle des arbres, le froissement des branches contre la lumière, la chute des flocons de neige. Écoute, Marnie, on entend la neige tomber... Et nous l'entendions, qui frappait doucement la terre blanchie. Comment avais-je pu m'éloigner de tout cela ? Comment avais-je pu croire qu'il m'était possible de tracer mon chemin dans des lieux nommés Forty Second Street ou Fifth Avenue, alors que tout ce qui me constituait résidait dans le bruit de la neige ensevelissant Rivière-aux-Trembles ? Tout était là, dans ce moment sous les arbres, toutes les saisons et tous les lieux. J'aurais pu mourir sur-le-champ et ma vie aurait été entière. Qui sait d'ailleurs si je n'aurais pas compris, en tendant l'oreille vers la neige, ce qui était arrivé à Michael, mais j'ai poursuivi mon chemin. J'ai rouvert les yeux et j'ai marché.
Ce que j'avais pris pour un sentier n'en étant pas un, j'ai dû effectuer de nombreux détours, m'accroupir sous les branches, contourner des groupes d'arbres si serrés qu'il était impossible de se faufiler entre eux, puis j'ai enfin abouti à la rivière, qui ressemblait à une route enneigée creusée au milieu de nulle part, sans réverbères, sans panneaux de signalisation, sans voyageurs susceptibles de vous prêter main-forte au cœur de la tempête. Là aussi, tous mes repères avaient disparu et j'étais seule. Le bassin magique se situait tout près, il ne pouvait en être autrement, mais il se dérobait à mon regard, ainsi que tout ce qu'il m'aurait fallu percevoir, et je devrais probablement patienter jusqu'au printemps pour le situer correctement.
J'ai observé les longues traînées de poudrerie s'enroulant sur elles-mêmes dans l'allée ouverte par la rivière et je me suis abritée en attendant que les eaux gelées me racontent le début de mon histoire. Accroupie contre un arbre, je contemplais les nuages catapultés par le vent depuis l'horizon, perdue dans le souvenir d'un soir lointain où on s'était allongés sous les arbres, Mike et moi, pour compter les étoiles à travers les branches. Ça devait être en août, car le ciel était à ce point chargé qu'il semblait vouloir nous aspirer. On était longtemps demeurés silencieux, fascinés par ce grouillement de lumières au-dessus de nos têtes, puis Michael avait pointé un doigt vers la Grande Ourse en disant les écureuils, Marnie, c'est ça qui doivent voir quand y sont dans leur nid, pis les hiboux avec.
Par la suite, chaque fois qu'on s'étendait sur le sol pour regarder les nuages, les étoiles ou la neige dardant ses flocons droit sur nous, on pensait aux écureuils, blottis dans leur nid au sommet des arbres, qui devaient apercevoir de là des choses qui nous échappaient. Si on pouvait grimper dans un nid, Marn, je suis sûr qu'on verrait des nouvelles étoiles, pis des sortes de hiboux qu'on connaît pas, pis des soucoupes volantes, chuchotait Michael en chassant les moustiques ou les brins de neige tournoyant autour de son visage.
Le temps était venu pour moi de grimper enfin, d'adopter le point de vue de l'écureuil ou du hibou si je voulais apercevoir les étoiles vers lesquelles s'était dirigé Mike et deviner ce qu'il avait vu sous l'orage du bassin magique. Remplie de souvenirs d'autres hivers, j'ai quitté mon abri, j'ai enlevé mes raquettes et me suis couchée sur la rivière, où j'ai battu des bras et des jambes, ainsi que le font depuis toujours les enfants élevés dans la neige, pour imprimer au sol la silhouette d'un ange ou d'un oiseau. Michael et moi, on appelait ça faire le hibou, parce que c'est ainsi qu'on disait à Rivière-aux-Trembles, faire le hibou, qu'on soit ou non un hibou.
Les yeux rivés sur la tempête, j'ai déployé mes ailes dans la poudreuse, le visage offert aux milliers de flocons quittant l'attraction du ciel pour s'abattre sur moi, pareils aux milliers d'étoiles veillant sur le sommeil des écureuils. En les imaginant qui filaient dans la froideur de l'atmosphère, j'ai fait le vœu que Michael et moi, un jour, puissions nous étendre ensemble dans la neige pour couvrir la rivière gelée d'une enfilade de hiboux jumeaux qui réuniraient enfin le passé et le présent.
J'ai allongé un bras et j'ai senti au bout de ma mitaine la mitaine de Michael, sa vieille mitaine de nylon rouge se refermant sur ma mitaine de laine. Pleure pas, Marnie, pleure pas, m'exhortait la voix de Michael, si tu pleures, tu vas te transformer en statue de glace, et la voix avait raison. Si j'ouvrais les vannes, mon regard se figerait dans l'eau de la rivière et le hibou censé déchiffrer les signes accumulés dans la forêt ne serait plus qu'un hibou aveugle. Mais la mitaine de Mike, sa maudite mitaine trouée, se refermait sur ma gorge, et j'ai senti la chaleur des larmes mouiller mes joues puis glisser vers mes oreilles en imbibant les bords de ma tuque.
J'aurais pu rester là et personne n'en aurait vraiment été affecté, sauf ce bon vieux Phil. Il aurait maudit tous les saints du ciel, auxquels il ne s'adressait que pour les engueuler, il se serait maudit et m'aurait vouée aux gémonies, puis il serait allé pleurer sur la tombe de mon père en s'excusant de n'être qu'un vieux sacrament d'imbécile. Tous les ans, durant la pire tempête de février, il aurait apporté des roses sur la rivière, des Beauty Marnie, que la bourrasque aurait ensevelies sous les hululements des hiboux, jusqu'au jour où il aurait décidé de se coucher avec moi parmi les roses enneigées. Phil m'en aurait voulu à mort de saccager ainsi ses dernières années, mais il aurait fini par comprendre, Phil comprenait tout, à commencer par la tristesse inconsolable des enfants. Quant aux gens du village, ils auraient conclu que les histoires circulant autrefois à mon sujet étaient fondées, que la petite Marnie Duchamp était carrément folle et qu'elle avait fini par rejoindre Michael Saint-Pierre, sa première victime, Dieu ait son âme, parce que sa culpabilité l'étranglait ou qu'il y avait une justice divine, c'est ce qu'auraient clamé les bigotes en se signant d'une main tremblante avant d'aller piquer quelque autre enfant innocent de leur langue fourchue. Les ragots auraient repris de plus belle et je serais devenue une légende, une histoire de sorcières et de mauvais esprits qu'on raconte aux enfants dissipés, aux mauvaises graines, pour qu'ils ne soient pas tentés d'entrer dans la demeure de l'ogre.
Couchée sur la rivière hurlante, je me moquais éperdument des racontars et du caquetage des culs-bénits. J'étais dans l'œil du cyclone, au centre même de la tourmente, là où il n'existe pas de paix plus entière ni de silence plus parfait. Toutes mes forces me quittaient, mes ailes ne battaient plus, rompues par la trop lourde charge dont je les avais lestées, et je n'avais soudain qu'une envie, m'endormir dans mes larmes et oublier enfin que Michael Superman Saint-Pierre aurait peut-être sauvé la cité de Kandor si la vilaine petite Marnie Duchamp n'avait pas existé. J'ai fermé les yeux pour mieux sentir les flocons s'abattre sur mes paupières et sur mon front, et j'ai imaginé des araignées, des fourmis blanches dont les pattes se liquéfiaient au contact de ma peau. Dans l'irisation de mon sang traversé par une faible lumière, se promenaient des paysages sans aspérités, des lunes éclatant sous des entrelacements de lianes, des montagnes roulant vers des rivières houleuses. La paix se trouvait là, au flanc de ces montagnes couvertes d'araignées blanches. Dormir était la solution, dormir enfin. Sous mes paupières maintenant trop froides pour boire les pattes des insectes, la lumière baissait, l'irisation prenait des tons de rouille, la nuit venait lentement.
J'étais sur le point de m'abandonner au doux engourdissement grimpant le long de mes jambes et enveloppant lentement mon torse quand un grondement de tonnerre a surgi des nuages, au-delà des milliers de flocons chutant sur mon corps immobile. Un frisson venu du creux de la rivière a glissé le long de mon cou, là où quelques larmes maintenant glacées avaient coulé, et je me suis instinctivement roulée en boule. Ce n'était pas la première fois que j'entendais gronder le tonnerre d'hiver, cela arrivait parfois lorsque le ciel était si chargé qu'il se comportait comme un ciel d'été balayé par des vents du nord, mais ce coup de tonnerre ne venait pas de l'égarement des cieux. Il venait de la fureur de Kandor, il venait du mois d'août 1979 ou de ce qu'on nomme la providence, mais je n'étais pas disposée à me lancer dans une réflexion sur l'inexistence du hasard. J'avais été tentée de lâcher prise, là, sur la rivière, sans même essayer de repousser les mains qui me tenaient rivée au sol, enfoncées dans les mitaines de nylon rouge de Michael Saint-Pierre. Deux ou trois minutes de plus et j'aurais laissé la douceur des mains m'entraîner dans une blancheur plus infinie que celle tombant du ciel. Je me serais endormie dans un déferlement de lunes rousses et j'aurais rêvé de galaxies blanches jusqu'à ce que mon corps perde la faculté de rêver.
Jamais je n'avais souhaité mourir, même quand tout allait de travers, même au creux des nuits d'insomnie où les voix accusatrices de Rivière-aux-Trembles, bien après la disparition de Michael, continuaient à me harceler. Je ne m'étais jamais sentie appelée par ces gouffres s'ouvrant entre deux rails de métro filant dans la noirceur invitante du vide. Il avait pourtant suffi de quelques flocons de neige pour que je sois tentée de baisser les bras et de me précipiter tête première au fond de l'abîme. Mes pensées se bousculaient, plus insensées les unes que les autres, mais j'avais d'autres préoccupations pour le moment. Je devais me grouiller, faire circuler le sang dans mes veines et réchauffer mes pieds gelés. Après m'être péniblement relevée, je me suis mise à sauter comme un crapaud infirme, à faire des flexions des jambes, à me couvrir le corps de coups de poing, puis je me suis assise, j'ai enlevé mes bottes et j'ai massé mes pieds jusqu'à sentir des dizaines de picotements brûler ma peau blanchie. Je me suis ensuite relevée et j'ai sautillé encore, j'ai battu l'air de mes membres ankylosés, pareille à un boxeur dont la survie tient au nombre de crochets du gauche qu'il va enfoncer dans le gros ventre blindé de son adversaire, puis mes pieds ont glissé et je me suis retrouvée cul par-dessus tête.
Près de moi, les ailes du hibou que j'avais dessiné plus tôt avaient découvert de larges pans de glace. J'ai vivement balayé la rivière autour de moi, pour constater enfin que j'étais en plein milieu du bassin magique et que ce foutu bassin avait essayé de m'emporter ainsi qu'il avait emporté Michael. J'ai reculé jusqu'à la rive en m'écorchant les genoux sur les arêtes de glace striant le bassin, puis j'ai attendu que la terre tremble et que le couvert de glace se fende sous la pression de l'eau. Mais la terre est demeurée stable, le bassin lisse et blanc. Après avoir jeté un dernier coup d'œil au hibou reposant ailes écartées sur la rivière, j'ai chaussé mes raquettes en vitesse et me suis enfoncée dans le bois, là même où Michael avait disparu trente ans plus tôt, puis j'ai suivi mes traces, que la neige n'avait pas complètement recouvertes, en me disant que ces traces étaient peut-être celles de Mike, qui sourdaient lentement de la terre et de ma mémoire.
À l'orée de la forêt, j'ai cru entendre un hibou mêler son hululement aux plaintes du nordet, un filet de voix m'exhortant à la prudence, mais je n'ai pas demandé à Michael si c'était lui qui m'appelait. J'ai plutôt murmuré ça va, Mike, ça va, je commence à t'entendre, puis j'ai repris ma route. La nuit était tombée quand je suis arrivée à la maison et Mr. Holy Owl m'attendait calmement dans le noir, hypnotisé par le déchaînement de la tempête.
Je n'ai pas dormi cette nuit-là. Je me suis enfoncée dans le vieux Chesterfield de mon père et j'ai écouté la tempête battre les champs de Rivière-aux-Trembles, fascinée par les motifs découpant la fine couche de neige que le vent avait collée aux vitres. Quand la grande horloge du boudoir a sonné trois heures du matin, je me suis levée et me suis rendue à la cuisine pour demander à Holy Owl ce qu'il pensait du pouvoir d'attraction morbide de certains lieux et s'il lui était déjà arrivé, au cours d'une de ses innombrables nuits de guet, de se sentir aspiré dans le couloir du temps. J'ai saisi son petit corps rugueux et j'ai attendu qu'il cligne des yeux, que ses pupilles se rétractent au souvenir d'une lueur trop claire ou que son bec s'entrouvre pour laisser parler tous les vents qu'il avait connus. Puis j'ai pris conscience du fait que j'attendais réellement une réponse de Holy Owl. Je l'ai laissé retomber et j'ai reculé d'un pas, ainsi que doivent reculer ceux qui découvrent dans le miroir un visage qui n'est pas le leur. Il a pivoté vers la gauche, vers la droite, vers la gauche, le cou enfoncé dans les épaules, et s'est enfin immobilisé face à la nuit agitée de bourrasques. Tu dérapes, Marnie, me suis-je dit en reculant à l'extérieur de la pièce, tu dérapes ferme. Après avoir vérifié que la porte d'entrée était bien verrouillée, au cas où d'autres Holy voudraient venir se réchauffer près de Crappy, je suis montée me réfugier dans mon lit, évitant de croiser mon reflet dans le miroir en pied qu'une faible lueur traversait au fond de ma chambre.
Jusqu'aux petites heures, j'ai réfléchi à la possibilité que Michael, trente ans plus tôt, ait tout comme moi été happé par la lumière d'un gouffre à laquelle il n'avait pu résister. Je me suis demandé si j'avais seulement imaginé les mitaines de Mike ou si ma main, à un certain moment, avait bel et bien effleuré le tissu troué. Puis je me suis remémoré toutes les histoires de revenants que j'avais lues ou entendues au cours de ma vie, essayant d'en trouver une qui racontait comment les fantômes viennent parfois chercher les vivants, pour conclure enfin que je délirais et que ni Michael ni quelque autre esprit n'avait essayé de s'emparer de ma volonté près du bassin magique. T'as juste perdu les pédales, Marnie Duchamp.
Quand le soleil, malgré sa froideur d'hiver, a enfin surgi de derrière les arbres, j'ai remercié le ciel de ne pas vivre sous ces latitudes plongeant chaque année dans la nuit polaire. Après une douche brûlante, j'ai décidé qu'il était temps pour moi de rendre visite à Victor Saint-Pierre. Depuis mon départ de Rivière-aux-Trembles, je ne l'avais revu qu'à l'enterrement de mon père, au milieu des hommes en complet foncé réunis dans l'odeur de tabac dominant le porche du salon funéraire. Une douzaine d'hommes aux cheveux gris qui avaient connu l'époque des beatniks et parlaient d'hier sous un nuage de fumée blanche remplaçant celle de la mari. Victor Saint-Pierre avait affreusement vieilli depuis le temps où il chantait Back in the U.S.S.R. à tue-tête, le samedi matin, sans se douter qu'un vent de catastrophe s'avançait dans le ciel bleu. Pourtant, malgré ses épaules voûtées, c'était toujours l'homme qui avait vu l'arrivée du fléau, le cheval se cabrant devant le rougeoiement du brasier, en proie à une douleur lui donnant des allures de géant. Quand il m'avait tendu la main, j'avais instinctivement reculé, pareille à l'enfant craignant que ne s'abatte sur elle la fureur de cet homme luttant contre une procession d'images ensanglantées. Puis nos regards s'étaient croisés et j'avais compris que Victor Saint-Pierre n'était qu'un cheval blessé, un père meurtri qui attendait toujours le retour de son fils. Ce vieil homme ne pouvait me faire de mal qu'en me montrant sa blessure, une large entaille le déchirant du cou jusqu'au bas du ventre. Après quelques secondes d'hésitation, j'avais saisi la main qu'il me tendait en cherchant les mots qui résumeraient le passé, mais c'était inutile, ni lui ni moi n'avions besoin de mots. Il m'avait montré sa blessure et s'en était retourné chez lui.
Aujourd'hui, il me semblait cependant que Victor Saint-Pierre pourrait m'aider en me parlant du Mike que lui seul avait connu, de la duplicité des tempêtes s'abattant sur les rivières, des ombres et des esprits qui hantent les vivants. Mais par-dessus tout, j'avais besoin de son absolution. J'ai passé la journée à me demander comment j'aborderais cet homme trop silencieux puis, peu avant la brunante, j'ai enfilé mes bottes et mon manteau.
Une lampe était allumée, au rez-de-chaussée, qui découpait la silhouette arquée de Victor Saint-Pierre sur les rideaux, pareille à un pantin de carton manipulé par ces mains invisibles s'agitant derrière des voiles de papier de riz. J'ai pris une grande inspiration et suis montée sur la galerie, où les poupées de la petite Emmy-Lili, affalées de tout leur long, absorbaient la lumière du dernier mois d'août.
Il a répondu à mon premier coup de sonnette, comme s'il n'attendait que ma visite. Il m'a tout de suite reconnue sous ma capuche et m'a fait signe d'entrer. Il m'a ensuite conduite au salon et est allé préparer du café pendant que je m'installais. Pratiquement rien n'avait changé depuis trente ans, ni les couleurs, ni les lustres, ni les coussins fleuris, à l'exception des photos de Michael apparues sur tous les murs et tous les meubles. Cette profusion de photos donnait à la pièce une allure de sanctuaire et ne parvenait qu'à accentuer l'absence de Mike. Là où une seule photo aurait ravivé le souvenir vivant de Michael, l'accumulation des images ne désignait que sa disparition irrémédiable. Où que vous posiez le regard, un enfant était là, pâle et sans consistance, qui vous parlait de la mort, et encore de la mort, qui vous rappelait que son sourire n'existait plus et qu'il n'était jamais devenu grand. Vous sentiez sa peau froide, son souffle froid sur votre nuque, son odeur fade et âcre de petit condamné et n'aviez d'autre envie que de retourner face aux murs toutes ces photos où plus personne ne vous regardait. Ce salon figé dans le temps empestait les fleurs fanées et aucune présence autre que celle d'un garçon devenu anonyme au milieu des cadres dorés ne le hantait.
C'est Jeanne, ma femme, qui a installé toutes ces photos, a murmuré Victor Saint-Pierre dans mon dos. Après sa mort, j'ai pas eu le cœur de les enlever. J'aurais eu l'impression de trahir Mike et Jeanne à la fois.
J'avais appris par mon père que Jeanne Dubé avait succombé à une longue maladie quelques années plus tôt, un mal que les médecins avaient été incapables de nommer mais qui s'appelait simplement Mike, le mal de Mike, un mal de mère qui n'en peut plus d'attendre et qui finit par lâcher prise. Toute sa vie, Jeanne Dubé avait attendu, préparant des gâteaux pour Mike, au cas où, au cas où, époussetant les cadres des photographies puis en ajoutant de nouvelles quand elle tombait sur de vieux clichés oubliés au fond d'un tiroir ou entre les pages d'un livre de prières. Puis elle était morte de ça, de l'attente, de l'incurable maladie de Mike, sous les yeux impuissants de Victor Saint-Pierre, qui n'avait su ni comment la guérir ni comment cesser de l'aimer, pour que ça fasse moins mal, pour ne pas s'accrocher à chacun de ses sourires comme à une bouée de sauvetage qui lui crèverait dans les bras.
C'est terrible, mais il y a des jours où je les vois plus, a murmuré Victor Saint-Pierre en replaçant une photo et en m'invitant à m'asseoir. Au début, toutes ces images de Mike me rendaient carrément fou, puis elles ont fini par se confondre à la tapisserie. C'est ce qu'est devenu Michael, Marnie, un motif douloureux dans la tapisserie.
Il a avalé une gorgée de café et m'a demandé si j'arrivais à m'acclimater de nouveau à Rivière-aux-Trembles. Je lui ai menti, à quoi bon lui révéler que la forêt me voulait morte aussi, puis nous avons parlé d'Émilie, qui était sortie de l'École nationale de police de Nicolet avec une mention spéciale et occupait depuis quelques années un poste lui permettant de traquer les tueurs, les fous, les kidnappeurs d'enfants. Elle non plus ne s'était jamais remise de la disparition de Mike, de l'odeur des gâteaux incrustée jusque dans la peinture, mais au lieu de s'enliser dans le chagrin, elle avait foncé dans le tas. Elle passait aujourd'hui les menottes aux petits violeurs de merde qui avaient le malheur de croiser son chemin et leur faisait cracher le morceau dans une salle d'interrogatoire où elle s'enfermait seule avec eux et avec sa haine. Elle faisait payer les salauds et n'éprouvait aucune compassion pour ceux que le système tentait parfois de transformer en victimes. Son cas était classique. Elle se substituait à ceux qui n'avaient pas été foutus de retrouver son frère et mettait la main au collet des tarés qui empoisonnaient le monde des petites Emmy-Lili. Elle vengeait Mike, jour après jour, à peine consciente que sa vie entière était guidée par la main de son frère, Michael, son dieu.
Elle va venir à Pâques, a ajouté Victor Saint-Pierre après m'avoir décrit la maison qu'Émilie venait d'acheter pour y vivre seule avec ses tueurs et leurs cadavres. Tu passeras prendre un verre. L'invitation était sincère, mais je n'étais pas certaine de vouloir revoir Emmy Saint-Pierre après toutes ces années. Le souvenir que je gardais d'elle était celui d'une enfant brisée par la perte et la colère de qui ne comprend pas que certains contes puissent se terminer dans le ventre de l'ogre. Je préférais ne pas me coller à cette colère. Emmy avait grandi dans mes rêves et dans mes cauchemars et cela me suffisait. Nous n'avions rien à nous dire. Les rescapés d'un naufrage n'ont jamais rien à se raconter que l'autre ne sache déjà. Seul Mike nous liait et ce lien avait été rompu par ma faute, c'est ce qu'avait dû penser la petite Emmy-Lou en déchirant ses poupées et en les lançant contre le mur noirci de mouches. Elle ne devait pas être chaude non plus, contrairement à ce que semblait croire Victor Saint-Pierre, à l'idée de regarder en pleine face celle qui avait entraîné son frère sur sa bicyclette volante, regarde, Lili, on vole, et ne l'avait jamais ramené. J'ai remercié Victor Saint-Pierre en lui disant que j'allais y penser, que Pâques était encore loin.
Tu peux m'appeler Victor, a-t-il répondu, on est des vieux copains de tranchée, toi et moi.
Il avait raison, on avait fait la guerre ensemble, on était liés par les liens du sang qui coule. En négligeant le fait que j'avais d'abord été l'ennemie, Victor Saint-Pierre venait en quelque sorte de me donner l'absolution que j'espérais. Nos positions antagonistes avaient cependant creusé un fossé entre nous, une tranchée au fond de laquelle nous n'avions pas toujours partagé nos vivres et qui m'interdisait de trop me rapprocher.
Il n'a pas attendu ma réponse, sachant très bien qu'on ne se défait pas si rapidement des habitudes d'une vie. Il m'a plutôt demandé comment s'était déroulée ma vie à New York et m'a raconté qu'il avait visité cette ville quelques années plus tôt en souvenir de Mike et de sa fascination pour Superman. Tu te souviens, Marnie, vos jeux de fous avec vos capes... ? Il a souri à l'évocation de ces souvenirs, du jour où Mike avait déchiré sa cape sur une clôture de broche piquante, puis, sans transition, il m'a demandé pourquoi j'étais revenue vivre à Rivière-aux-Trembles.
Parce que c'est chez moi, parce que je veux comprendre.
S'il y avait quelque chose à comprendre, Marnie, on l'aurait compris depuis longtemps. La forêt a pris Michael, c'est tout.
Qu'est-ce que vous entendez par là ?
Rien de plus que ce que j'ai dit. Mike s'est enfui dans le bois et on l'a plus revu. Pas de signe d'enlèvement ni de violence, pas de traces, pas de suspects, pas d'indices, à part ce maudit soulier abandonné dans la vase du lac aux Barbotes.
Il est demeuré silencieux assez longtemps pour que je me demande si je ne devais pas partir, puis il s'est raclé la gorge et m'a regardée comme on regarde un petit animal ayant oublié que c'est lui qui a crevé le ballon qu'il réclame. J'ai immédiatement pressenti ce qu'allait me dire Victor Saint-Pierre mais, au lieu de me sauver à toutes jambes, je me suis enfoncée dans mon fauteuil et j'ai attendu. J'étais ici pour que Victor Saint-Pierre me révèle ce qui m'échappait. Fuir ne m'apporterait rien.
Prends-le pas mal, Marnie, a-t-il fini par lâcher, je sais que t'as rien fait à Mike, mais s'il y a quelqu'un qui peut expliquer pourquoi son soulier trempait dans le lac aux Barbotes, c'est toi. T'as oublié des choses, ma fille. C'est pas de ta faute. T'as eu une peur bleue, pis t'as oublié. C'est la seule explication possible. La clé du mystère, tu l'as perdue dans ta mémoire.
Pendant que Victor Saint-Pierre évoquait la folie d'une autre Marnie, une Marnie de cinéma et de fiction inventée de toutes pièces, le sourire de petit mort de Mike, se multipliant autour de moi, s'ouvrait mécaniquement pour me livrer son secret : mauvais temps, madame, mauvais temps. Toutes les années passées n'avaient rien changé. Le temps était toujours à la tempête et j'étais toujours la détentrice du secret qui aurait expliqué la vie et la mort de Mike Saint-Pierre. Stormy weather, monsieur, bad weather, ai-je pensé alors qu'un rideau de pluie drue s'installait entre Victor Saint-Pierre et moi, une muraille infranchissable derrière laquelle il ne pouvait m'entendre.
Je sais pas, ai-je simplement répondu, espérant que quelques-unes de mes paroles parviendraient à couvrir le bruit de la pluie. J'ai peut-être oublié, monsieur Saint-Pierre, mais j'ignore comment on prend conscience de l'oubli. J'ai conclu là-dessus, me disant que jamais je n'appellerais Victor Saint-Pierre par son prénom. On n'appelle pas Victor un homme qui attend que vous lui rendiez son fils. Victor était un nom d'ami, et le seul véritable ami que j'aie jamais connu s'appelait Mike. Des Vic, je n'en connaissais pas.
Après que Victor Saint-Pierre m'eut reconduite à la porte en insistant pour que je revienne discuter avec lui, je suis demeurée quelques instants sur la galerie avec les poupées d'Emmy-Lou, dont les yeux de plastique et de porcelaine m'accusaient aussi. Je me suis excusée, pardon, Emmy, shame on me, puis j'ai traîné les pieds jusqu'au trottoir, pendant que s'éteignait derrière moi la lumière brillant au rez-de-chaussée de la maison de Victor Saint-Pierre, dont la silhouette s'est éclipsée derrière l'opacité des rideaux.
J'ai ensuite marché jusqu'au cimetière, jusqu'à la chapelle où mon père attendait d'être mis en terre, en écorchant une chanson stupide où il était question d'amours terminus et en donnant des coups de pied sur tous les mottons de glace qui me barraient encore le chemin. Jurer me calmait, varger me soulageait, alors je jurais et je vargeais comme une enragée. Pleure pas, Marnie, sinon tu vas te transformer en statue de glace, me chuchotait la voix de Mike, mais les larmes coulaient, accompagnées de plaintes, de jurons, de mots débiles empruntés à une chanson qui élevait la rime au rang de la connerie. Comment ça se fait que je me rappelle plus ? Comment ça se fait ? Chus pas folle, viarge ! Après m'être agenouillée quelques instants devant la tombe de Jeanne Dubé, m'excusant pitoyablement de tous les maux du monde, j'ai martelé à deux poings la porte de la chapelle, réponds-moi, pop, réponds, je t'en prie, pleurant de plus belle et me transformant peu à peu en statue de glace et de givre. Autour de moi, d'autres statues fixaient l'horizon, certaines orientées vers l'est et le renouveau, d'autres vers le froid du nord et de l'enfer, des Vierge, des colombes, des anges aux ailes cassées, des Christ pas foutus de me répondre non plus, leurs yeux de pierre fermés sur les secrets de la mort et de la résurrection. J'ai laissé mon père tranquille et j'ai gueulé contre elles, les statues, contre les Christ sales aux pieds couverts de mousse, puis, une fois ma crise passée, je me suis machinalement dirigée vers la maison de Phil, où je savais qu'il ferait chaud.
Je devais faire peur quand j'ai frappé à sa porte, car Phil m'a entraînée dans le salon avec mes bottes pleines de slush, où il m'a forcée à m'asseoir avant de me servir de son meilleur cognac dans un verre à eau. La chaleur de l'alcool a eu un effet immédiat. Mes membres se sont détendus, mes tremblements se sont atténués, mais l'effarante impression d'être habitée par la folie d'une autre ne m'a pas quittée. L'autre était là, derrière moi, qui prenait alternativement les voix de Mike et de Victor Saint-Pierre pour me chuchoter à l'oreille des mots dont je ne comprenais pas le sens, ne plie pas le jour, madame.
Assis sur le bord d'un fauteuil qu'il avait tiré face au mien, Phil attendait que je parle. Mais comment parler avec des voix plein la tête ? J'ai calé mon cognac et me suis concentrée sur les photos de castors, de marmottes et de chevreuils qui garnissaient les murs de la pièce. Depuis le temps que ces animaux avaient été photographiés, ils devaient tous être morts, mais leur présence sur les murs ne donnait pas le sentiment de la mort. Il s'agissait d'animaux vivants dans une maison vivante. Ce lieu avait perdu quelques occupants, lui aussi, mais il n'en avait gardé que la trace heureuse. Contrairement à la maison de Victor Saint-Pierre, dont les murs étaient imprégnés d'une humidité brumeuse d'où pouvaient surgir à tout moment des images étouffées par le remords, la maison de Phil ne retenait pas ses disparus dans une atmosphère de culpabilité et de regret.
T'as ajouté des nouvelles photos à ta collection, mais ces trois-là faisaient déjà partie du décor quand j'étais petite, ai-je dit à Phil d'une voix presque éteinte en désignant les trois photos en noir et blanc sur lesquelles figuraient deux oiseaux de proie et un jeune ours. Tu leur avais même donné des noms, tu te souviens ? Harvey, Hervé et Irving.
Phil se souvenait. Il se rappelait aussi que je n'arrêtais pas de le tanner pour qu'il me raconte l'histoire d'Irving, le bébé ours, puis celle de Jésuite et Récollet, les deux ratons laveurs qui ravageaient son jardin depuis des générations, aussi tenaces qu'un jésuite et ne changeant jamais de nom. Il était une fois Jésuite et Récollet, ai-je murmuré, puis j'ai souri à Phil, j'ai pris sa main chaude dans ma main froide et me suis lancée tête baissée en vue de lui demander s'il avait déjà oublié des choses, des choses importantes, Phil, comme si t'étais tombé dans un trou noir, des choses qui auraient changé ta vie si le trou les avait pas absorbées ? J'ai attendu quelques instants une réponse qui n'est pas venue, puis j'ai continué, des choses cruciales, qui auraient modifié le cours de ton existence ? Réponds-moi, Phil, c'est pas si compliqué que ça. Mais le regard de Phil n'était pas celui du gars qui a envie de vous parler de l'oubli. C'était le regard d'un homme qui n'en peut plus de voir souffrir la fille de son meilleur ami. T'en as pas encore fini avec la disparition du petit Saint-Pierre, hein ? Sacrament, Marnie, c'est arrivé y a presque trente ans !
Phil avait été témoin des événements de l'été 79, de la disparition de Mike et du rejet dont j'avais été victime. Malgré le choc qui avait secoué le village, il n'avait jamais compris comment des adultes supposément sensés avaient pu s'en prendre à moi au lieu d'essayer de m'aider, comment ils avaient pu m'empoisonner la vie au point de forcer mon père à m'emmener loin de Rivière-aux-Trembles. Devant l'imbécillité qui s'affichait sur les visages comme une éruption de boutons purulents, il avait bien failli sacrer son camp lui aussi, mais il avait été incapable de s'y résoudre. Il avait passé sa vie à Rivière-aux-Trembles et il avait sa forêt, ses lacs et ses rivières dans le sang. Quitter ce coin de pays aurait équivalu pour lui à un suicide. Ce drame l'avait cependant profondément marqué et sa colère resurgissait chaque fois qu'il en était question et que le temps se contractait, le forçant à plisser les yeux devant le soleil empoisonné de l'été 79. Le temps s'était récemment comprimé autour de moi aussi, écrasant New York, dont l'horizon n'était plus que poussière grise, et réduisant les années que j'y avais passées à un amas de décombres n'ayant mis que quelques secondes ou quelques jours à devenir décombres. Entre hier et l'enfance, il n'y avait que ce ciel poussiéreux.
Qu'est-ce qui s'est passé, Marnie ? Quel crisse d'imbécile t'a encore achalée avec ça ?
Personne, Phil, personne. C'est juste moi. J'essaie de comprendre pis je me dis que les gens avaient peut-être raison en affirmant que je cachais des choses, sauf que je les ai si bien cachées que j'arrive plus à les retrouver.
Je ne voulais pas lui parler de Victor Saint-Pierre, de cet homme qui avait passé sa vie à creuser le sol de Rivière-aux-Trembles dans l'espoir d'en exhumer les ossements de son fils. Victor Saint-Pierre n'était pas en cause. Il ne survivait que pour savoir. Je ne pouvais lui en vouloir de me considérer comme la seule personne susceptible de lui offrir une mort paisible.
Arrête de te torturer, Marn, m'a imploré Phil. On se souvient davantage de ses malheurs qu'on les oublie. Si t'en avais perdu des bouts, ta mémoire les aurait recrachés depuis longtemps. Ça serait remonté à la surface, sinon t'aurais craqué, tu serais devenue folle à lier.
C'est justement ça que je me demande, Phil, si je deviens pas folle.
Mais Phil ne voulait pas entendre parler de démence, de névrose, de psychose ou de quoi que ce soit qui s'en rapproche. Dans son esprit, la folie n'existait que dans les livres, que dans les guerres et les massacres, que dans les familles où des enfants se faisaient casser les jambes avant d'apprendre à marcher, mais pas chez nous, pas dans nos maisons. La petite Marnie devenue grande ne pouvait être atteinte d'aucune forme de folie. Il m'a pris la main à son tour et m'a conduite à la cuisine, où il nous a préparé des pâtes pendant que je dressais la table. On va manger, ça va nous remettre les yeux en face des trous. Pis ouvre-nous une bouteille, Marnie, un chianti, dans l'armoire du fond. Va aussi chercher le chandelier du salon. On a trois ou quatre affaires à oublier, pis on va le faire en grand.
J'ai évité le sujet de Michael pendant une bonne partie du repas, même si sa voix demeurait dans mon dos, avec celle de son père, qui me parlaient du mauvais temps et du jour qu'il ne fallait pas plier, de deux images inconciliables, car comment maintenir le jour dans l'axe de la lumière quand la tempête fait rage ? À l'heure du café, le vin et la fatigue aidant, j'ai laissé les voix parler, parce que je n'arrivais plus à les retenir. Puisque ni les morts, ni les Christ et leurs colombes ne voulaient me répondre, il fallait bien que je me rabatte sur les vivants. Je voulais simplement savoir s'il était déjà arrivé à Phil de se sentir aspiré par les bois de Rivière-aux-Trembles, de sentir sa volonté fléchir et l'abandonner là, près de la rivière ou au bord d'un sentier. Il avait vécu trop longtemps à Rivière-aux-Trembles pour ne pas avoir entendu la forêt l'appeler.
Je voulais qu'il me décrive cet appel, qu'il me parle de ces moments où on a envie de s'enfoncer dans un banc de neige, ou alors de courir pour ne plus jamais s'arrêter, au diable le passé, la vie, la mort et l'au-delà, au diable la mascarade, la maudite marde qui colle aux semelles, mais je m'embrouillais dans mes pensées, je m'enfargeais dans les branches cassées du sentier de Mike, pendant que les saprés grands yeux de beagle de Phil s'embrumaient de l'autre côté de la table, derrière les chandelles éteintes dont il ne restait plus que quelques amas de cire difformes cherchant à planter leurs racines dans le métal du chandelier.
Je te suis pas, Marnie, je te suis plus pantoute, murmurait-il en triturant sa serviette de table tachée de sauce rouge, mais il me suivait à cent milles à l'heure, c'est ce que ma petite tête de pioche me disait, il me suivait aussi bien que Holy Owl, sinon son visage ne se serait pas décomposé comme si je venais de lui annoncer que Barack Obama avait déclaré la guerre à la Saskatchewan. Puisque les branches du sentier de Mike me barraient le passage, je me suis plantée là, devant le sentier, et j'ai dit à Phil que, selon moi, Mike pouvait s'être enlevé tout seul. C'est ce que j'essaie de t'expliquer, que la forêt a peut-être enlevé Mike et qu'il s'est laissé faire.
Phil a déposé sa serviette maculée de sauce, qui ressemblait à un canard écrasé, un pauvre canard à moitié rouge et qui ne coincouinerait plus jamais à force d'avoir été chiffonné, et il s'est levé pour venir me caresser les cheveux. Mets-toi pas d'idées de même dans la tête, Squouirèle, murmurait la voix de Phil dans mes cheveux, sa voix comme un vent chaud qui sentait les vignes et le raisin, je t'en prie, accuse pas les arbres des crimes des hommes. C'est un homme qui a enlevé Mike. Y a rien qu'un homme qui peut commettre une folie pareille, un monstre qui avait une tumeur à la place du cerveau, pis les monstres, ça vit pas dans l'écorce des arbres.
Et pourtant, j'avais longtemps pensé le contraire. Quand un orage, où que je sois, me forçait à me réfugier dans des musiques rivalisant avec le bruit du ciel, il m'arrivait de revoir les bras velus voulant s'arracher des arbres, près du bassin magique, et de croire que l'un d'eux y était parvenu, qu'il avait quitté son arbre pour s'enrouler autour du cou de Mike. Après l'orage, cependant, après la musique tonitruante, les bras prenaient la forme de bras humains et je décrétais, à la suite de Phil, que seul un homme avait pu rendre le visage de Mike aussi blanc. Mais aucun œil, aucune bouche haletante ne surgissait de mes souvenirs affolés. C'est peut-être cet œil, trop semblable à celui d'un homme, que ma mémoire avait occulté.
C'est possible, Phil, tout est possible, ai-je murmuré, et Phil m'a tenue dans ses bras d'homme, ses bras amis, pareils à de vieux ceps de vigne imprégnés d'odeurs de bois et de terre, jusqu'à ce que les larmes provoquées par le mot Squouirèle dans toute sa tendresse cessent enfin de me brûler les yeux. Il a ensuite lavé la vaisselle en silence pendant que, hypnotisée par les miettes éparpillées sur la nappe de plastique, j'essayais de respirer calmement. Une à une, j'ai avalé les miettes en les collant au bout de mon index, une bouchée pour papa, une bouchée pour Squouirèle, puis, en voyant les épaules un peu trop courbées de Phil, j'ai conclu qu'il était temps de faire le clown.
J'avais mis cette stratégie au point lorsque j'étais enfant et j'y avais recours chaque fois que le regard de mon père s'embuait du souvenir de Marie Beaupré. Je tournais autour de lui en sautillant et j'exécutais une série de pirouettes de squouirèle. En général, ça fonctionnait. Marie Beaupré s'éclipsait devant mes pirouettes en m'envoyant un baiser du bout des doigts, heureuse de voir la grimace de joie qui retroussait lentement le visage de mon père, juste assez pour lui retrousser le cœur en même temps.
J'avais toujours détesté les clowns et leur nez bouché, leurs petites maudites oreilles rouges, leur gueule d'Albert Einstein grimé jusqu'au fond de la calvitie, comme si Einstein méritait ça, mais dès que j'étais trop désemparée pour agir intelligemment, dès que ma peur de me coller le front à la réalité déclenchait un branle-bas de combat parmi mes neurones épouvantés, j'appelais les clowns en renfort et je faisais l'innocente. J'ai donc ressorti mes vieilles blagues de clown qui ne s'aimait pas, mais ni Phil ni moi n'avions le cœur à rire. On savait depuis trop longtemps, tous les deux, que les clowns tristes ne sont pas des vrais clowns, qu'il s'agit de pauvres filles, de pauvres gars tombés du ciel dans une arène de cirque alors qu'ils ne sont pas taillés pour ce métier stupide.
Phil a éteint le plafonnier de la cuisine, dans laquelle une goutte frappait le fond de l'évier toutes les trois secondes, aussi obsédante qu'une horloge détraquée ou qu'une miette de pain oubliée sur une nappe, et il est allé préparer mon lit. Ce soir j'allais dormir chez lui, c'était un ordre. Il m'a installée dans la chambre verte, de la couleur du foin frais coupé, des feuilles de mai à peine écloses. Avant de m'endormir, j'ai compté des canards, des canards rouges et des moutons blancs, et me suis remémoré l'histoire d'Irving l'ourson, qui s'était égaré dans la forêt pour y cueillir du miel. Selon les jours, Phil modifiait autrefois la finale de cette histoire. Parfois, Irving était sauvé par sa mère, d'autres fois par un garde-chasse ou par la petite Marnie se promenant dans les bois. Ce soir, ce serait Marnie qui sauverait Irving, mais il fallait pour cela qu'elle se souvienne de l'orage qui l'avait emporté, qu'elle retrouve l'œil trop bleu ou la formule magique qui ouvrirait la porte de l'antichambre où était enfermé l'ourson. C'était simple. Il fallait que Marnie se souvienne. Il fallait qu'Irving soit sauvé.
Il fallait que Marnie se souvienne... Pendant plusieurs jours, ces mots m'ont obsédée. J'ouvrais le robinet de la douche, je me brossais les dents, je nettoyais la cafetière et ils me traversaient l'esprit avec la rapidité de l'éclair, comme si l'obligation de se souvenir appartenait à une autre que moi, à une fillette née dans un conte étrange dont la trame m'échappait. Ce n'était pas moi, mais cette petite Marnie en salopette de garçon qui avait la tâche de retrouver Michael Saint-Pierre. Pas moi, mais l'enfant du conte enveloppé de brume. Si je parvenais à secouer cette fillette et à la placer dans le sentier par où s'était enfui Irving l'ourson, le passé s'éclairerait. Quand la phrase s'imposait à moi, je m'arrêtais en plein milieu d'un geste et plongeais dans une forme de rêverie ayant des allures de transe. Le temps s'arrêtait, les contours des objets s'estompaient dans le flou du temps, et j'étais propulsée dans la forêt de la fillette, que je tentais inutilement de suivre, car celle-ci s'éclipsait invariablement derrière un arbre ou dans un banc de brouillard où son corps se désintégrait, ne laissant derrière elle que son rire enfantin. J'en étais presque venue à me dire que cette enfant n'avait aucun lien avec moi lorsque enfin, dans le balancement d'une mer captive, je suis parvenue à situer les éclats de son rire.
J'avais dû me rendre chez la notaire de mon père, dans cette ville où nous avions vécu pendant six ans, afin de régler quelques détails relatifs à la succession. Très en avance sur l'heure de mon rendez-vous, j'examinais la salle d'attente, une pièce claire meublée de façon moderne et presque spartiate. À part les diplômes accrochés au mur donnant à l'est et les quelques photos disposées sur le bureau de la réceptionniste, le seul objet ornant le lieu était un petit arceau de bois animé d'un mouvement perpétuel. À l'intérieur de l'arceau, était encastré un bloc de verre rectangulaire à demi rempli d'eau colorée ou d'un liquide ayant une densité se rapprochant de celle de l'eau. Le balancement de l'arceau produisait des vagues dans le bloc de verre, qui allaient se frapper à ses parois, se repliaient sur elles-mêmes puis s'élevaient de nouveau.
À quarante-deux ans, je ne connaissais pas la mer. J'avais vu l'Atlantique bordant New York, mais cette portion d'océan gris se jetant sur des plages polluées par l'odeur des hot-dogs ne correspondait pas à mon idée de la mer, de la vraie. Ma mer à moi, je l'avais fantasmée et j'en avais rêvé, et c'est à cela qu'elle ressemblait dans mon esprit, à une masse d'eau perpétuellement bleue dont les mouvements répondaient à un désordre calculé. Je repoussais les mers sauvages, les rochers escarpés, pour me confiner dans la représentation d'une mer se brisant sur elle-même. Pendant longtemps, j'avais essayé d'imaginer l'odeur bleue de cette mer rêvée, mais il me manquait des éléments comme le vent et la putrescence des algues. J'aurais pourtant eu de nombreuses occasions de séjourner près de la mer et d'y découvrir son odeur, mais je m'étais toujours refusée à la voir telle qu'elle était ou telle qu'on pouvait la percevoir. Je préférais celle qui ressemblait à mes lacs et à mes rivières et ne risquait pas de supplanter la fascination qu'exerçaient sur moi les eaux douces. La mer ne me manquait pas.
Je pensais à des mots pouvant exprimer le parfum iodé du large quand la réceptionniste s'est penchée vers moi pour m'annoncer d'une voix douce, s'accordant à l'atmosphère feutrée du lieu, que maître Legendre me recevrait dans une dizaine de minutes. Je lui ai souri, puis j'ai reporté mon attention sur le bibelot, où la mer projetait ses écumes et ses bleus contre les murs de verre enfermant l'univers. Envoûtée par le mouvement de l'objet, j'ai fermé les yeux en songeant qu'il fallait que Marnie se souvienne, puis la mer s'est retirée à l'embouchure de la rivière aux Trembles, calme et paisible sous les nuages d'août. Michael et moi y nagions dans une couleur irréelle ressemblant aux couleurs des rêves. Michael riait de je ne sais quelle blague, puis il plongeait au fond de l'eau cueillir des cailloux que nous disposerions en deux groupes, le premier réservé au trésor qu'il cachait au creux d'un arbre dans une pochette contenant aussi des billes, de vieux bijoux, des pièces de monnaie étrangère, le second destiné à ces jeux dans lesquels nous invoquions les pouvoirs secrets de la rivière.
Les cailloux luisaient et, autour de ma taille, la jupette de mon maillot de bain rayé flottait, pareille à la corolle d'une fleur étrange. On dirait un nénuphar jaune et rouge, Mike, regarde, un nénuphar en Jell-O mou. Et Mike riait, plongeait, puis ressortait de l'eau avec des pierres, des mauves et des blanches, qu'il déposait sur le rivage.
Puis soudain, sans transition, nous sommes dans notre cabane. Vingt-cinq cailloux sont disposés en cercles devant la porte, selon leur taille et leur couleur. Au total, il y a cinq cercles, cinq cercles parfaits formant un autre cercle décuplant leurs pouvoirs. Il s'agit de cercles magiques, ensorcelés par les esprits de la rivière permettant aux enfants de voler, de sauter du haut des montagnes. Autour de nous, l'air est alourdi par l'odeur du sapinage dont les aiguilles amollies impriment sur nos genoux des dessins de fossiles, de mille-pattes, des petits squelettes d'insectes disparus. Aucun bruit ne nous parvient du dehors, les oiseaux se sont tus. Mike et moi connaissons ce silence, le mauvais temps s'en vient. J'entends Mike chuchoter mauvais temps, ça va tonner, Marn, puis il me montre les nuages qui s'avancent en roulant sur la rivière. Bouge pas, je vais chercher nos maillots. En sortant il trébuche, brisant le cercle de pierres ocre. Un éclair venu du sommet des arbres blanchit alors le ciel, nos maillots se balancent au bout des bras de Mike, mais je ne vois que le cercle brisé. C'est un mauvais présage. Il ne faut pas bousculer l'ordre des pierres ocre. Je tente désespérément de reformer le cercle quand un nouvel éclair se répand à l'entrée de la cabane. Dehors, Michael s'incline et se balance en fixant ses runnings. Nos maillots sont tombés sur le sol. Sous la pluie forte et drue, il aligne des mots absurdes d'une voix monocorde, puis il m'ordonne de ne pas bouger. Approche pas ! Avec des yeux exorbités. Va-t'en, Marn, va-t'en ! Après quoi un cri retentit au milieu de l'orage, le cri de la mort et du noyé, qui obscurcit le ciel, et puis tout devient noir. Les nuages ont chuté sur Mike dans l'enchevêtrement de la forêt. Je rampe et je crie aussi sous la pluie, je hurle, j'attrape une pierre et me relève, prête à frapper l'ennemi, l'homme aux yeux de cristal, puis une voix s'insinue dans le vacarme qui m'entoure. Je respire et la mer réapparaît, projetant ses eaux dans la lumière. Le visage de maître Legendre est là, éclairé par la mer, qui me demande si tout va bien.
J'ai mis quelques secondes à recouvrer mes esprits, à reconnaître cette femme dont les verres reflétaient un pan de l'océan, et j'ai demandé de l'eau, de l'eau douce, en me remémorant les paroles prononcées par Mike, des paroles effrayées sur lesquelles s'abattait la pluie. Va-t'en, Marn ! Mauvais temps, va-t'en ! J'ai chassé les images qui culbutaient sur les mots de Mike et j'ai suivi Suzanne Irène Legendre dans son bureau sans entendre la moitié de ce qu'elle me disait. J'ai ensuite signé les papiers qu'elle me tendait, j'ai répondu machinalement à quelques questions, puis je me suis sauvée en claquant derrière moi la porte de la réception, où une mer se retirait dans l'obscurité de ses profondeurs.
Dehors, la nuit allait bientôt tomber et l'humidité de mars vous transperçait la peau. J'ai couru jusqu'à ma voiture et j'ai verrouillé les portières derrière moi. La vérité était là, au seuil de ma conscience, l'homme ou la bête qui s'était abattu sur Mike en même temps que les nuages, pendant que les paroles muettes m'exhortant à ne pas m'approcher s'entremêlaient dans mon esprit. Les lèvres de Mike ne disaient plus mauvais temps, madame, mais va-t'en, Marn, sacre ton camp, comme s'il voulait me protéger, m'éloigner de l'ennemi sans visage qui l'avait transformé en poupée mécanique. J'ai essuyé la sueur qui couvrait mon front et j'ai ouvert la radio en poussant le volume au maximum, puis j'ai quitté la ville qui s'illuminait. La route qui me mènerait à Rivière-aux-Trembles était par contraste d'une égale noirceur. Sur l'écran de cette noirceur, défilait en accéléré le paysage de la rivière, ses arbres se confondant à ceux que balayaient les phares des voitures. À la radio, un homme parlait de la guerre d'Irak, et moi, me prenant pour la grande Dietrich, je fredonnais Lili Marnie, Lili Marleen, Lili Marnie a fait dodo, Lili Marleen fera dodo, goodbye hello Lili Marnie, inlassablement, d'une voix monocorde, les yeux rivés sur le ruban blanc allongeant la route dans l'obscurité.
Après avoir roulé pendant environ une heure trente, j'ai dû m'arrêter pour prendre un café. J'ai pris la sortie menant au restaurant bar nommé Le Madrid et l'univers s'est métamorphosé. La cour du restaurant était peuplée de dinosaures de plastique ou de fibre de verre qui dardaient sur vous leurs yeux injectés de sang avec l'air de se demander ce qu'ils fichaient dans ce coin perdu alors qu'ils auraient pu figurer dans un film de Steven Spielberg. La banalité de ce décor qui semblait dater du jurassique, combinée à l'artificialité ambiante, était exactement ce qui me convenait pour le moment. J'avais besoin de kitsch et de fiction.
J'ai commandé une frite que j'ai à peine touchée et un café noir, que la serveuse me réchauffait chaque fois que j'avalais une gorgée, pendant qu'un type dans la cinquantaine, assis au bout du comptoir en forme de fer à cheval, plongeait le nez dans un verre de bière qu'il ne touchait pas non plus. Ce type s'ennuyait, ça sautait aux yeux, il s'ennuyait de sa femme, de ses enfants, il s'ennuyait de son chien et de toutes les femmes à la fois. Il aurait suffi d'un mot gentil, d'une main sur son épaule, pour qu'il se mette à brailler tout ce qui empêchait sa bière de passer. J'ai détourné le regard, mon café non plus ne passait pas, va-t'en, Marn, et j'ai saisi le journal qui traînait à côté de mon assiette. « Découverte macabre », titrait la une du tabloïd sous un encadré montrant la photo d'une jolie jeune femme qui souriait à l'objectif de la même manière que Mike dans le salon de Victor Saint-Pierre, de la même manière que tous les disparus de la terre, que tous les morts qu'on force à sourire pour oublier qu'ils ont aussi pleuré. Mélinda X, la femme qui faisait l'objet de cette couverture tapageuse, avait disparu trois mois plus tôt en quittant son travail et on ne l'avait plus revue. La veille, la voiture de Mélinda, avec Mélinda à bord, avait été trouvée près d'un lac, à demi ensevelie dans la neige. On avait d'abord cru qu'elle dormait, les bras repliés sur la tête. Mais Mélinda ne dormait pas, Mélinda ne souriait pas. Quelqu'un avait pris son sourire et son sommeil, un homme, présumait-on, un fou qui voulait la beauté de Mélinda.
En refermant le journal, mon regard a croisé celui de la serveuse, Teresa, disait l'étiquette cousue sur son uniforme, qui observait aussi la photo de Mélinda en essuyant ses mains avec un chiffon. Pauvre enfant, a-t-elle murmuré, cherchant ce qu'elle pouvait dire de plus sans trahir à coups de clichés la souffrance et la peur atténuées par l'encre grise. Je lui ai souri faiblement et j'ai baissé la tête. Il n'était pas nécessaire de rajouter quoi que ce soit. Nous étions des femmes, pareilles à Mélinda, et un seul regard suffisait à exprimer la compassion que nous éprouvions par contagion, en quelque sorte, du seul fait d'être des femmes, en vertu de cette parenté chromosomique qui nous avait appris à nous méfier de la nuit, des stationnements déserts, des impasses et des escaliers plongés dans le noir. Teresa ne comprenait pas cette violence qu'excitait le parfum des femmes et des enfants et je ne la comprenais pas non plus. Va-t'en, Marn ! J'ai placé le journal à l'envers, pour que Mélinda se repose un peu de l'indécence des mains froissant son sourire, et j'ai quitté le restaurant.
Dans le stationnement désert où veillaient quelques descendants d'une préhistoire revisitée par Hollywood, une fine neige tombait du ciel opaque, une neige à hiboux, qui frappait le sol dans un bruit si près du soupir qu'il fallait retenir son propre souffle pour saisir la chute démultipliée des flocons expirant au contact du sol. Écoute, Mike, on l'entend soupirer... Je plongeais avec la neige dans une mélancolie familière quand un grincement de porte m'a fait sursauter. L'homme qui s'ennuyait, épaules courbées, quittait le restaurant. J'ai regagné ma voiture en écrasant la neige morte et j'ai démarré.
Dans mon rétroviseur, les dinosaures se sont amenuisés avec l'homme voûté, le restaurant tout droit sorti de mauvaises photos de Las Vegas a disparu et j'ai regagné la noirceur de la route, au plus profond de laquelle filaient les dégénérés qui étranglaient les Mélinda et les Michael. J'ai fermé la radio, qui diffusait une vieille chanson d'Enrico Macias et, pour la millième fois peut-être, j'ai répété à voix basse les paroles de Mike, va-t'en, Marn, fuis, fuis vite le mauvais temps, essayant de me représenter ce qui avait effrayé Mike à ce point qu'il en était devenu méconnaissable, car ce n'était pas Mike qui se balançait sous la pluie, mais le corps de Mike vidé de Mike, l'enfant au visage blanc qui se dématérialiserait dans le fouillis des branches. Dans l'habitacle surchauffé, Lili Marnie s'était endormie avec Lili Marleen, mais la Marnie qui tenait le volant à deux mains avait les yeux grands ouverts, rivés sur un passé qui lui pourrissait l'existence. Parmi les hiboux, les sourires et les cailloux qui jonchaient la route, elle cherchait l'homme dont lui avait parlé Phil, elle cherchait le monstre et la rédemption, la main tremblante et qui avait fauté.
Sur la dernière portion de route, dans les courbes bordées de sapins qu'illuminaient les phares, je ne voyais pratiquement plus rien. L'effet de la caféine avait disparu et c'est dans un état de quasi-hypnose que j'ai aperçu les réverbères de Rivière-aux-Trembles. En arrivant à la maison, j'étais aussi épuisée que si j'avais pleuré mille morts et ne songeait qu'à m'endormir devant une vieille comédie mettant en vedette un gentil clown à la Charlie Chaplin qui me ferait oublier tout ce qui concernait Michael Saint-Pierre. J'ai quand même salué Holy Owl, qui observait les trois érables gris se découpant du côté gauche de la maison, n'ayant d'autre désir que d'aller s'y percher. La captivité n'était pas pour Holy, je le savais, elle n'était pour aucun oiseau, mais Holy avait eu le tort de naître avec de toutes petites ailes qui ne savaient pas voler. J'ai ouvert la fenêtre pour lui donner l'illusion que sa liberté n'appartenait qu'à lui, mais ses petites ailes sont demeurées rivées à son corps d'écorce et de bois. Le vent s'est engouffré dans ses plumes rêches et il a simplement suivi le mouvement du nordet, tournoyant et frappant avec son bec ou avec sa queue les carreaux salis de poussière d'hiver.
Dans la cuisine que n'éclairait qu'un réverbère posté au bord de la route, ces coups de bec accentuaient l'intensité du silence. J'ai tenté de redresser Holy, mais il a continué à battre la mesure du vent, inconstante et d'une perfection troublante. Le vent est comme la mer, il est l'achèvement du désordre, comme la mer et l'orage, ai-je pensé pendant que les rideaux effleuraient mon visage. J'ai écouté son chuintement à la cime des arbres, d'une beauté qui enfante les fous, et j'ai refermé la fenêtre avant que cette beauté m'incite à me précipiter au bassin magique.
Je n'étais pas retournée là-bas depuis cette tempête que je nommais la tempête du hibou. Tous les jours, je chaussais mes bottes avec l'intention de m'y rendre, et tous les jours, la crainte d'être séduite par l'étrange paix de la forêt me poussait à remettre cette expédition au lendemain. J'irais quand les vents seraient moins violents, quand la neige se diluerait en centaines de minuscules étangs et formerait des rigoles s'enfonçant dans la boue. La glace du bassin magique, alors, se serait séparée de ses rives et je ne risquerais plus de m'y coucher.
J'ai réchauffé la pizza congelée que j'avais achetée quelques minutes plus tôt au dépanneur du village et j'ai passé la soirée à m'abrutir devant la télé. De temps à autre, il me semblait entendre le bec de Holy Owl frapper les carreaux, mais c'était les volets qui claquaient, les branches dénudées des buissons de Mary Matthews se frottant au bois de la maison. Vers vingt-trois heures, je me suis endormie sur l'image d'un soleil traversé de nuages, engourdie par la voix soporifique d'une présentatrice météo faisant état d'un avertissement de vents violents dans l'Est de la province.
Toutes les conditions étaient réunies pour que je rêve que le vent m'emportait. Mais il n'y a jamais de vent dans mes rêves, jamais de bruissement évoquant celui des vagues. Ils se déroulent tous dans un univers sans climat où il m'est impossible de sentir la chaleur du sable ou de l'eau recouvrant mes pieds. Au lieu de rêver au vent qui secouait la maison, j'ai rêvé à la mer emprisonnée dans le cabinet de S. I. Legendre, qui ballottait entre ses murs de verre sans que le vent l'effleure, sans que le bruit du ressac m'atteigne. Portée par un tic-tac d'une absurde régularité, elle se déversait sans bruit dans la rivière. Le monde était entouré de cloisons hermétiques étouffant jusqu'au bruit de mes pas et de ma respiration. Je rêvais de la mer et du silence.
À mon réveil, le vent était tombé, mais la mer ondulait toujours aux quatre coins de la planète, de même que l'eau devait onduler sous les glaces de la rivière. Il faisait froid dans la maison et le jour se languissait derrière les nuages. Pour tromper l'ennui qui tentait de s'infiltrer en moi avec le froid, j'ai rempli la baignoire d'eau brûlante et m'y suis plongée après y avoir versé deux bouchons de Mr. Bubblelou. J'adorais les bains remplis de mousse à ras bord, lorsque j'étais enfant, et mon père gardait toujours une bouteille de Mr. Bubblelou pour moi, pour mes rares visites, car il savait que j'avais aimé mon enfance. Avec la Caramilk, le Cheez Whiz et les Rice Crispies, le bain moussant Mr. Bubblelou faisait partie des rares produits qui avaient survécu au temps sans altération. Les autres avaient été emportés par la grande marée d'objets de consommation interchangeables dont l'homme occidental ne peut apparemment se passer.
La bouteille de Mr. Bubblelou avait la forme d'un ours ou d'un panda, ce n'était pas clair, que papa et moi rebaptisions chaque soir. Certains soirs, il s'appelait John, John Bubblelou. D'autres soirs il se nommait Martin, Clément ou Dominique, et il avait toujours une histoire à me raconter en fonction des formes que je sculptais dans la mousse. Aujourd'hui, l'ourson de plastique s'appelait Michael, mais il refusait de me raconter l'histoire de la rivière dont j'avais creusé le lit à la surface de l'eau mousseuse. J'ai tenté de le réveiller en secouant son petit corps translucide, qui a fait un bruit mouillé pendant que l'huile de bain adhérait à ses parois, puis je l'ai plongé sous l'eau et l'y ai maintenu, me disant qu'il serait peut-être plus enclin à parler s'il craignait de terminer ses jours dans cette étendue d'eau sans vie. Quelques remous ont agité l'écume, Michael Bubblelou a battu des pieds, et je l'ai délivré en lui demandant s'il était maintenant prêt à me révéler son secret, celui que gardait précieusement le cercle brisé des pierres ocre depuis trois décennies, et depuis plusieurs siècles, peut-être. Devant son mutisme, je l'ai immergé de nouveau, un peu plus profondément, un peu plus longuement. La face de plastique de Mr. Bubblelou continuait à sourire, pendant que ses yeux s'agrandissaient, que ses traits se diluaient dans la lenteur des vagues transportant leur écume. Mauvais temps, madame, mauvais temps, a hoqueté Bubblelou. Très loin sous la surface bleutée, les lèvres également bleues de Mike disaient va-t'en, Marn, va-t'en, en parfaite synchronie avec les lèvres de plastique de Bubblelou. J'ai brusquement lâché la bouteille. Mr. Bubblelou a bondi à la surface, les yeux remplis de mousse, et il a lentement vogué vers l'extrémité de la baignoire.
J'ai observé mes mains tremblantes, puis j'ai saisi Mr. Bubblelou pour le lancer contre le mur. Pendant un instant, ces mains avaient enserré le cou de Mike et l'avaient immergé dans la rivière que réchauffait le soleil d'août. À travers celui de Bubblelou, j'avais vu le sourire figé de Mike Saint-Pierre, va-t'en, Marn, mauvais temps, entouré des trois petites bulles d'une rondeur parfaite qui s'étaient échappées de la bouteille.
C'est pas moi, calvaire ! ai-je hurlé en battant des bras, me frappant les coudes aux parois du bain, au mur de tuiles jaunes et blanches, éclaboussant le plancher d'eau savonneuse puis glissant sur l'émail doux, au risque de m'assommer ou de me casser un membre. C'est pas moi ! C'est pas moi ! C'est pas moi... Mon acharnement à faire surgir la lumière des hypothétiques zones d'ombre couvrant mon souvenir du 7 août 79 tournait à l'obsession. Si je ne me ressaisissais pas, Phil serait bientôt contraint d'admettre que la folie pouvait aussi frapper les petites Marnie. Je me suis forcée à respirer calmement, puis j'ai regardé le plafond, d'une blancheur immaculée. Du blanc, rien que du blanc.
Et s'il n'y avait pas d'images manquantes, ai-je murmuré à l'intention de Bubblelou, si le bras du monstre frappant Mike avait été invisible, ses griffes dissimulées parmi le foisonnement de la verdure ? Personne, sauf Mike, n'avait peut-être vu le bras de l'homme. Personne, sauf Mélinda.
Je suis sortie du bain en grelottant, j'ai enfilé une robe de chambre et des bas de laine, puis j'ai téléphoné à Phil pour qu'il m'appelle Squouirèle et me rappelle que certains mystères ne pouvaient être résolus.
La pluie avait presque cessé, hier soir, quand je suis rentrée d'une balade en voiture n'ayant d'autre but que de me distraire de l'ordinaire de mes soirées, qui se déroulent habituellement entre un bol de pop-corn et une télé qui s'ennuie. Un épais brouillard s'élevait au-dessus des champs enneigés, les maisons de Rivière-aux-Trembles disparaissaient dans de lourdes vapeurs, légèrement vacillantes, et le moindre objet, poubelle ou banc de bois, perdait sa stabilité derrière les nuages. La lumière des réverbères formait des halos jaunâtres suspendus entre ciel et terre, semblables aux lueurs floues des lampes torches qui balaient la cime des arbres, aux abords des villages, quand un enfant n'est pas rentré pour le souper.
J'avais vu ces lueurs monter des bois durant la nuit du 7 au 8 août 79, dessinant autour de Rivière-aux-Trembles un périmètre irrégulier traversé de la voix des hommes appelant Michael. L'écho des cris lancés vers l'ouest allait mourir sur la colline des Loups, qui se taisaient avec les coyotes devant l'agitation des bois, ceux lancés vers l'est s'éteignaient d'eux-mêmes dans l'air humide, ne rencontrant que des obstacles qui en absorbaient la détresse. Le nom crié de Mike était partout, dans les champs, dans les maisons, dans le ciel ennuité. Puis le nom était emporté au loin pour n'être bientôt plus qu'un souffle se frappant aux frontières indistinctes où le vent va mourir. Mike se trouvait là, exactement là où son nom se taisait, dans la zone d'extinction des vents. Lorsque j'entendais les échos tomber au sommet des collines, je m'enfonçais sous mes couvertures en tentant une brève prière et j'attendais, j'attendais que les arbres se courbent de nouveau sous la poussée du vent, espérant que le nom de Mike s'accroche à une rafale qui irait cingler son visage.
J'avais aussi vu ces halos jaunâtres au cinéma, traquant des hommes en fuite ou se dirigeant vers l'une des chaussures boueuses de la fillette qu'on avait envoyée chercher du lait au coin de la rue. Parfois, la caméra nous montrait la chaussure en gros plan, et on pouvait constater qu'elle était rose ou blanche, de ces couleurs qui alourdissent le poids du drame. Ces images me troublaient un peu plus chaque fois, non seulement parce qu'elles se superposaient au running de Michael, mais parce qu'elles étaient toujours boueuses et que c'est toujours cet objet qu'on découvrait en premier, des chaussures sales indiquant que la course de l'enfant s'était arrêtée là où s'imprimait une dernière fois l'empreinte de son pied. Mon père ne comprenait pas pourquoi je demeurais rivée devant la télé quand une chaîne diffusait un film relatant la disparition d'un enfant, va donc te coucher, Marnie, ça te donne quoi de regarder ça, mais c'était plus fort que moi, il me fallait voir, il me fallait savoir, il me fallait comprendre. Tous ces films ne m'ont cependant appris qu'une amère vérité : les enfants disparus ne reviennent pas souper à la maison. On découvre leurs chaussures, parfois leur corps, parfois leurs vêtements ensanglantés, mais rarement un gamin ou une gamine qui a faim et qui pleure.
Le brouillard enveloppant Rivière-aux-Trembles me rappelait les soirs de battue de l'été 79, quand la confusion régnait sur le village, que les contours s'estompaient et que rien ne m'apparaissait plus qu'à travers une forme de nébulosité atteignant aussi le cours du temps, plus lent, s'écoulant dans l'épaisseur de la matière impalpable entourant les visages et les choses. Pendant cinq nuits, les lumières projetées dans les sous-bois avaient cherché Michael, puis il n'y avait plus eu qu'une unique lampe, celle de Victor Saint-Pierre, qui criait plus fort que celle de tous les autres hommes.
J'ai traversé le village embrumé dans le vague engourdissement entourant ces souvenirs, puis je me suis garée devant la maison, où j'ai écouté le crépitement irrégulier des gouttes de pluie tombant des arbres sur le toit de la voiture. Bercée par ce bruit, je me suis remémoré à quel point j'avais aimé la pluie, toutes les pluies, avant que ma vie se disloque dans le déferlement d'un orage. Pourquoi tu me fiches pas la paix, Michael Saint-Pierre ? Pourquoi tu me laisses pas tranquille ? Pourquoi ? Puis les larmes s'en sont mêlées, qui dégoulinaient lentement au bout de mon menton, ralenties par la matière visqueuse formée de bave et de morve qui leur faisait obstacle. Abrutie de fatigue, j'ai pleuré jusqu'à ce que mon engourdissement se double de vertiges et m'oblige à ouvrir la fenêtre pour reprendre mon souffle et m'oxygéner le cerveau. Dehors, les gouttières remplies de feuilles d'automne débordaient, quelques rigoles heureuses multipliaient leurs branches dans le gravier et l'air sentait bon la fin de l'hiver, la neige pourrie, la boue de feuilles piétinées. J'ai profité de cette accalmie pour m'essuyer le visage et, tranquillement, je suis montée vers la maison en demandant à Mike pourquoi je n'étais plus capable d'aimer ni de sentir la pluie.
Dans la cuisine, Holy Owl m'attendait, ainsi qu'il le ferait jusqu'à la fin des temps si je le laissais suspendu devant sa fenêtre. Soudain, je n'avais plus envie que Holy m'attende et me rappelle Mike jusqu'au jugement dernier. Je suis descendue à la cave chercher le projecteur portatif de mon père, son gros spotlight, celui-là même qu'il avait utilisé pour partir à la recherche de Mike pendant que Julie Lacroix me gardait et me bombardait de questions, les yeux aussi brillants qu'exorbités, pour pouvoir raconter à ses copines qu'elle avait interrogé Marnie Duchamp, la folle, qui lui avait révélé plein de détails sordides sur la disparition du petit Saint-Pierre. Julie Lacroix, une maudite menteuse qui, plus que tous les autres, avait empoisonné mon existence. Je l'ai envoyée chier en souhaitant qu'elle se soit fait arracher la langue par un train et je suis remontée au rez-de-chaussée, où j'ai enfilé un imperméable et des bottes étanches avant d'aller chercher Holy Owl dans la cuisine.
C'est cette nuit que tu t'envoles, Crappy, je te rends ta liberté, ai-je chuchoté en collant ma bouche contre sa petite tête de fausses plumes, puis je l'ai fourré dans une des poches de mon imperméable. J'avais tort d'accuser Mike. Ce n'était pas lui qui empêchait la pluie de me tomber dessus, mais le fantôme que je nourrissais, jour après jour, avec une mauvaise conscience capable d'alimenter tout un régiment de revenants. Mon problème ne s'appelait pas Mike, mais Marnie Duchamp, alias Marnie Hitchcock, alias Squouirèle. Si je voulais que Mike me sacre patience, il me fallait d'abord l'expulser de cette zone fantôme où, sans m'en rendre compte, je l'avais confiné. J'avais toujours été fascinée, d'une fascination proche de l'effroi, par ces enclaves carcérales, ces phantom zones issues de l'univers futuriste de Superman. La séquestration éternelle à laquelle étaient condamnés les hors-la-loi enfermés dans ces espaces dérivant dans l'infini me semblait la forme de châtiment la plus atroce qui soit, pire que tous les enfers jusque-là imaginés par l'homme, et voilà que j'y avais emprisonné Mike. Dans mon désir de garder son souvenir vivant, j'avais conçu pour lui un nouvel enfer, non plus réservé aux criminels, mais aux êtres aimés devenus fantômes, captifs de cet amour les vouant à une insensée survivance. Je devais le sortir de là et me libérer du même coup.
Avant de partir pour la rivière, j'ai téléphoné à Phil afin de lui éviter une nuit blanche s'il me rendait une de ses visites surprises et trouvait la maison vide. J'ai cloué le bec de Holy Owl qui s'agitait dans ma poche et j'ai dit à Phil que j'allais me coucher tôt pour écouter la pluie. On souperait ensemble le lendemain et c'est moi qui régalerais. Cinq heures, Phil, bonne nuit.
La pluie avait reflué vers l'est pendant que je réunissais mon barda, mais le brouillard était toujours à couper au couteau. J'ai traversé le village en pataugeant dans les flaques de neige fondue, puis j'ai pris le chemin du 4. Arrivée devant les champs de Lucien Ménard, je me suis enfoncée dans la forêt.
À la lumière du projecteur de mon père, les arbres prenaient une autre dimension. Quelque chose de trop humain semblait les animer, la nuit pervertissait leur nature diurne et je ne reconnaissais plus le semblant de sentier. Je découvrais des arbres que je n'avais jamais vus, leurs branches torses barrant l'accès à d'autres sentiers, des sentiers secrets et sinueux, interdits aux promeneurs ou appartenant à des enfants qui n'existaient plus. Une brume légère courait parfois au ras du sol, là où la terre était plus chaude, mais la forêt repoussait le brouillard qui l'enserrait. Des formes invisibles à la clarté du jour se découpaient dans les troncs humides, figures totémiques ou esprits cherchant à s'extirper de la matière ayant avalé leur âme. L'éclairage artificiel que je promenais devant moi ou braquais sur quelque creux d'ombre surprenait la nuit dans son intimité avec les arbres. À cette heure tardive, l'homme qui rompait les conciliabules et les chuchotements s'élevant de la brume n'était pas le bienvenu. Cette heure était le royaume des contes et des démons, des histoires de loups-garous et des croyances engendrant la peur. C'est ici qu'étaient nés l'ogre et le loup.
Michael se trouvait peut-être là, dans l'une des excroissances prenant l'apparence de corps torturés cherchant à s'échapper des troncs rugueux. Il avait peut-être disparu ainsi que disparaissent les enfants des contes, capturés par les arbres, séquestrés par quelque sorcière se nourrissant de sang vierge. Effrayé par l'orage, il avait pu trébucher sur les jambes décharnées de la sorcière de Rivière-aux-Trembles, étalées sur le sol telles des racines exsudant une sève âcre, pour tomber ensuite au cœur d'une forêt métamorphosée ne comportant d'autre issue qu'une éternité solitaire.
Certains avaient cru que j'étais cette sorcière, me poussant à me demander si je n'étais pas le produit des fornications de quelques incubes ayant visité le lit de mes parents pour donner naissance à Marnie the Witch, la sorcière de Rivière-aux-Trembles. L'absurdité de cette hypothèse aurait pu me valoir la camisole de force si je n'avais fini par me rentrer dans le crâne que les sorcières étaient des êtres purement chimériques ne sévissant que dans les contes. Et ma vie n'était pas un conte. Pourtant, pendant que j'avançais dans l'irréalité des arbres, rien ne m'apparaissait plus impossible. Il existait peut-être des dimensions dont nous ignorions la profondeur, des forces surnaturelles ou maléfiques frappant hommes et enfants sans distinction. Le brouillard courant à mes pieds pouvait aussi m'envoûter, mais je continuais d'avancer. Si la forêt m'avait voulue, elle m'aurait prise bien avant cela, et si elle me voulait aujourd'hui, mes efforts pour contrecarrer ses plans ne parviendraient qu'à m'épuiser, et je m'enfoncerais, tête basse, dans ces enclaves ne s'ouvrant que vers l'intérieur, le bas du monde.
Après avoir contourné un groupe de cèdres si serrés qu'ils s'étouffaient les uns les autres, j'ai enfin entendu la rivière, coulant sous les glaces dont la pluie avait érodé les bords. Le sentier incertain qui serpentait avec la brume entre les troncs tordus m'avait menée là où je voulais aller. Au lieu de m'égarer ainsi qu'elle avait égaré Mike, la forêt s'écartait devant moi. Aucun mauvais esprit ne prendrait mon âme aujourd'hui. J'ai suivi le bruit de l'eau et, deux minutes plus tard, j'arrivais près du bassin magique.
J'ai promené le faisceau du projecteur sur les arbres bordant la rivière, parmi lesquels on n'apercevait aucun tremble. Des sapins, des bouleaux, des épinettes, mais aucun spécimen de l'espèce ayant donné son nom à ce cours d'eau. Les peupliers faux-trembles, apparemment, avaient disparu de ses rives après qu'on l'eut baptisé, à moins que les hommes l'ayant nommé se soient trompés, confondant bouleaux et peupliers. Il faudrait que je demande à Phil s'il avait une idée du sort qui avait frappé les trembles de la rivière.
J'ai continué à balayer l'obscurité de la nuit sans lune et la lumière de ma lampe a effleuré l'extrémité équarrie d'un bout de bois noirci d'eau que je ne me rappelais pas avoir vu là auparavant. Lentement, j'ai redirigé le faisceau de la lampe vers l'endroit où j'avais cru percevoir le bout de bois et me suis immobilisée. À quelques pieds du bassin magique, une croix dont l'aspect funeste se mariait parfaitement à l'atmosphère de cette nuit lugubre émergeait de la neige, une petite croix toute simple comme on en voit dans les virages mortels, dans les films où l'horreur s'entoure de symboles sacrés. Je me suis approchée en prenant soin de ne pas faire de bruit, craignant d'éveiller la mort qui dormait sous cette croix, puis j'ai vu qu'on y avait gravé un nom, Michael, sans date ni autre inscription.
Un seul mot, Michael, dont la lourdeur faisait pencher la croix vers les ténèbres. J'ai reculé de quelques pas, fixant cette croix qui me contraignait à imaginer le corps de Mike six pieds sous la couverture du sol, la bouche et les yeux remplis de toutes ces petites bêtes habitant la terre molle. Et pourtant, je savais que j'aurais beau gratter le sol jusqu'à m'en arracher les ongles, jamais mes mains ne rencontreraient le visage de Mike, parce que personne, ni Victor Saint-Pierre, ni Dieu, ni la police, n'avait jamais obtenu la moindre preuve de sa mort.
Pendant un moment, j'ai eu envie d'arracher cette croix officialisant une mort non confirmée et donnant aux sorcières de Rivière-aux-Trembles l'occasion de se livrer à des rituels païens autour de cet emblème de la résurrection. C'est Victor Saint-Pierre qui avait dû la planter là par un jour de printemps ou d'été, peut-être en compagnie de sa femme et d'Emmy-Lou. Il avait dû graver le nom de Mike dans son atelier, s'appliquant à tracer des lettres droites et d'égale longueur, sans fioritures, aussi sèches que la mort, puis se munir d'une masse pour venir enfoncer le nom de Mike dans la terre noire. Victor Saint-Pierre avait baissé les bras et confié la mémoire de Michael à un objet dont l'aspect insolite, en ce lieu précis, ne pouvait susciter qu'un profond malaise. S'il n'en avait tenu qu'à moi, je l'aurais fait disparaître, mais je n'avais aucun droit de détruire cette croix près de laquelle le vieux Victor venait sûrement se recueillir tous les 7 août.
En l'observant de plus près, j'ai constaté que le bois de la croix pourrissait et qu'elle avait probablement été plantée il y avait très longtemps, quand Emmy jouait encore avec ses poupées déchirées. Pendant toutes ces années, j'avais ignoré l'existence de ce gibet dont l'ombre, les jours de grand soleil, devait s'étaler à l'entrée du sentier par où Michael avait fui. Compte tenu de la configuration des lieux, l'ombre de cette croix pourrie ne pouvait s'étendre que dans cette direction. Son unique fonction était d'indiquer l'absence, la fuite en direction des arbres.
En quittant la maison, j'avais l'intention de venir accrocher Holy à une branche surplombant la rivière pour l'abandonner à un sort dont se chargeraient les intempéries, mais en examinant la croix, j'ai eu l'idée de l'enterrer avec la dépouille absente de Mike. Sa place était là, sous cette croix branlante. J'ai relevé mes manches en jurant que Holy Crappy Owl allait rejoindre Mike, puis j'ai enfoncé mes mains nues dans la neige et j'ai creusé, creusé jusqu'à toucher la terre gelée. Incapable d'entamer cette surface glacée, j'ai couru chercher une pierre tranchante près de la rivière, puis j'ai tenté de creuser encore, le front en sueur et les mains raidies par le froid, mais l'arête de la pierre s'est brisée et je l'ai jetée derrière moi en maudissant tous les saints du ciel. Le bruit de la pierre tombant dans l'eau, si semblable au bruit de toutes les pierres que Mike et moi avions lancées dans le lac aux Barbotes et la mare à Mailloux, a percé d'un son clair le silence de la nuit. Accroupie dans la neige fondue, j'ai regardé mes mains écorchées, mes ongles noirs, j'ai entendu mes halètements et j'ai eu peur. Vire pas folle, Marnie !
Les mains toujours tremblantes, j'ai replacé la neige, la terre et les feuilles mouillées que j'avais dispersées au pied de la croix et j'ai sorti Holy Owl de ma poche. Aucun oiseau n'est conçu pour la captivité. Aucun oiseau n'est conçu pour la terre. J'ai posé mes lèvres sur le petit bec de Holy, good luck, Crappy, et j'ai noué sa cordelette sur un des bras de la croix, pour que son ombre suive le trajet emprunté par celle de Mike et qu'un jour, peut-être, il aille se percher au sommet d'un arbre. C'est là que devaient vivre les oiseaux et c'est là qu'ils devaient mourir, dans les arbres ou dans les nuages. Après m'être assurée que Holy ne tomberait pas au premier coup de vent, j'ai reculé de quelques pas. Dans la lumière projetée par la lampe que je tenais au bout de mes bras, l'ombre de la croix s'imprimait sur la neige, accompagnée de l'ombre d'un petit pendu se balançant doucement. Arrivederci, Crappy, on se reverra peut-être. Salue Mike de ma part si tu le croises, ai-je ajouté, puis je suis partie sans me retourner.
Pendant que le bruit de la rivière s'amenuisait derrière moi, la pluie s'est remise à tomber, amplifiant les chuchotements qui couraient entre les branches. De temps à autre, un craquement me faisait sursauter, mais je continuais ma route en reniflant l'odeur de neige fondue et de bois trempé qui montait du sol. J'étais seule, Holy surveillait le bassin magique et, autour de moi, les rapaces et les loups soupiraient avec les sorcières. J'ai voulu hâter le pas, mais j'ai buté contre une de ces racines ne montant du sol que la nuit venue et, le visage dans la neige granuleuse, j'ai frénétiquement cherché la lampe que j'avais échappée dans ma chute. J'ai tâté autour de moi, ne rencontrant que des objets visqueux ou putréfiés. Alors qu'une main aux ongles crochus, je le savais, s'approchait de ma jambe pour me tirer vers l'arrière, ça arrive, Marn, c'est là, j'ai aperçu la lueur de la lampe vacillant derrière une souche. J'ai rampé sur les coudes, certaine que la main allait planter ses ongles dans ma chair d'une seconde à l'autre, puis j'ai attrapé le projecteur pour en diriger la lumière derrière moi. Tout était immobile. La main qui m'avait poursuivie avait réintégré la terre.
J'étais à bout de souffle quand j'ai débouché dans la clairière de Lucien Ménard, mais ne m'y suis pas attardée. J'ai couru jusqu'à la route sans me soucier de l'eau qui glissait dans mon cou ni des douleurs provoquées par ma chute et ma tentative d'inhumer Holy Owl encore vivant. Peut-être un hibou hululait-il au loin, mais je ne l'ai pas entendu, pas plus que je n'ai entendu le cri monté autrefois de la rivière aux Bleuets.
Nanamiu-shipu, rivière Tremblante, c'est probablement de ce mot innu que vient le nom de la rivière aux Trembles, m'a expliqué Phil à l'heure du café, peu avant que les bougies s'éteignent. On avait d'abord dû l'appeler rivière Qui-Tremble, puis le nom avait peu à peu été déformé pour se transformer en rivière aux Trembles, alors qu'aucun peuplier faux-tremble ne bordait ni n'ombrageait cette rivière.
Nous étions attablés dans la salle à manger et discutions à la lueur d'une lampe murale et de trois bougies qui se consumaient lentement dans les petits plats de verre où, faute d'avoir pu mettre la main sur les chandeliers, je les avais fixées. Phil avait frappé à ma porte vers les dix-sept heures et nous avions préparé le souper ensemble en buvant de la Belle Gueule blonde. Phil râpait le fromage, moi les zucchinis, pendant que Miles Davis s'occupait de la musique, poussant jusqu'au sublime la bande sonore d'Ascenseur pour l'échafaud. Devant le vide laissé par Holy Owl à la fenêtre, je tentais d'oublier la croix de bois où j'avais suspendu ce volatile aux yeux vitreux. Je dansais sur L'assassinat de Carala, une pièce conçue pour l'échafaud, pas pour la danse ou l'insouciance, ce dont je me foutais royalement, car j'éprouvais un urgent besoin de faire le clown, d'arborer mon innocent sourire de Bozo pendant que Jeanne Moreau cherchait désespérément Julien Tavernier, son amant, qui venait d'accomplir le crime presque parfait. Alors je dansais, je louvoyais autour de l'échafaud et me prenais les pieds dans la langueur de la trompette. Pour oublier. Pour faire le clown.
À mon retour de la rivière, la veille, je m'étais allongée sur mon lit, espérant m'endormir dans le tambourinement de la pluie, mais j'étais restée éveillée jusqu'aux petites heures, à observer l'ombre de la croix se dessinant sur un des murs de ma chambre puis se reflétant dans le miroir en pied, où gisait la silhouette de mon corps allongé. À l'instar des marâtres des contes de fée, j'avais emmené Holy dans les bois pour l'y perdre et je craignais maintenant qu'il retrace son chemin en suivant les cailloux éparpillés dans les sentiers. Mon désir d'égarer Holy constituait ma première tentative pour me débarrasser de Mike, pour retrouver ma vie et sentir de nouveau la pluie, mais ses chances de réussite m'apparaissaient encore très aléatoires. Voilà pourquoi je dansais, aussi langoureuse qu'une carpe après le dégel, aussi ridicule qu'une ado se prenant pour Madona ou Elvira Madigan, essayant de me persuader que la croix de Mike représentait un signe du ciel et non de l'enfer, qu'il me suffisait de confier Mike à Dieu pour enfin être affranchie de mes tourments. Je n'avais en somme qu'à me réfugier dans une vérité toute simple. Mike était mort. Carala était mort. Marnie était vivante.
Qu'est-ce qui t'arrive, Marn, ça fait longtemps que je t'ai pas vue d'aussi bonne humeur ? me demandait Phil, et je lui répondais que c'était le printemps, le printemps et la pluie, les perce-neige dans trois semaines, Phil, tu te rends compte, la chaleur qui ferait éclore les bourgeons et les œufs d'hirondelle. Phil était heureux de me voir danser et, malgré toute sa perspicacité, il ne pouvait distinguer l'affreux nez de clown que j'avais plaqué sur mon masque de plastique craquelé.
Durant le souper, portés par ma bonne humeur de clown épuisé et les effets du Château Timberlay offert par Phil, nous avons parlé de tout et de rien, de la pluie, encore, de l'impossible température québécoise, tout en commentant le nombre record d'apostasies consécutives aux propos de Joseph Ratzinger sur l'usage du condom, le tremblement de terre qui avait provoqué cent cinquante morts et privé cinquante mille personnes de leur foyer dans les Abruzzes, en Italie, pendant que je m'avançais dans une forêt où seules mes mains tremblaient.
Ce n'est qu'à la fin du repas que j'ai annoncé à Phil que Mike était mort et que c'est sur sa tombe, en quelque sorte, que je dansais plus tôt. Je mentais, un mince filet d'air s'échappait encore de la gorge de Mike, mais je voulais prendre Phil à témoin de sa mort prochaine. Il était temps que tu le laisses enfin mourir, a-t-il soupiré, ça se fatigue, les vieux fantômes, ça peut devenir méchant. Le sourire de Phil était presque heureux et j'avais honte, soudain, de lui avoir annoncé la mort d'un moribond. Je me sentais pareille à ces rédacteurs chargés d'écrire la rubrique nécrologique de gens qui respiraient encore, mais dont la mort imminente était mise en boîte quelques semaines à l'avance, histoire d'être prêt à imprimer quand le gars ou la fille serait débranché. Ça va se tasser, Marn, tu sais aussi bien que moi qu'on finit par s'habituer à la mort, a ajouté Phil avant de se lever pour aller préparer le café.
Dans les petits plats de verre où la cire fondue formait des amas de larve rouge, les bougies vacillaient. Je passais mes mains sur les flammes en pensant à l'été, en espérant l'été comme on espère enfin renaître, jeter ses vieux runnings et s'acheter des sandales à deux cents dollars pour se rendre compte au bout de la ligne qu'il vaut mieux courir pieds nus, sans rien qui vous rattache à quoi que ce soit que l'herbe à la terre. J'installais des tuteurs près des bosquets de Mary-Jean, pieds nus, tête nue, quand Phil est revenu avec le café.
Tu penses à quoi, Marn ?
Aux rosiers de pop, il va falloir que je m'en occupe.
Puis, de Mary Hope en Mary Kay, j'ai abouti près de la rivière, incapable de passer une heure sans que ce maudit cours d'eau déferle dans mon champ de vision. Je me suis hâtée de dépasser la croix de Mike et d'enjamber le bassin magique pour demander à Phil ce qu'il savait des trembles de la rivière. Une question simple, supposant une réponse simple. Nanamiu-shipu, rivière Tremblante, a murmuré Phil dans le vacillement des bougies qui allaient bientôt s'éteindre. Selon lui, les Indiens avaient probablement donné ce nom à la rivière à la suite du tremblement de terre qui avait violemment frappé la région en octobre 1860, une secousse dont les effets avaient été ressentis de la Nouvelle-Écosse au New Jersey. J'ai lu quelque part que la rivière aurait été légèrement déviée par la formation d'une faille, a poursuivi Phil, si bien que les Innus lui auraient aussi donné le nom de Nana-shipu, rivière disparue.
Certains experts prétendaient au contraire que le véritable nom de cette rivière était Nanamassiu-shipu, rivière de la foudre. Les premiers colons blancs de la région, confondant nana, nanamiu et nanamassiu, avaient donné différents noms à la rivière, hésitation qui, après une cinquantaine d'années, avait conduit à rivière aux Trembles.
Phil parlait, accumulant des mots qui m'étaient étrangers, les bougies vacillaient, Miles rejouait L'assassinat de Carala, et moi, je n'entendais que nana, nanamiu, nanamassiu : disparition, tremblement, foudre, Michael et Marnie sous l'orage. Les mots des anciens Innus qui avaient nommé cette rivière en fonction de ses humeurs, de sa révolte contre le ciel puis de sa chute dans les failles de la terre ne pouvaient mentir. Ce sont les trembles qui mentaient, les mots blancs venus de la méprise, mais pas nana, pas nanamiu, pas nanamassiu. Tremblement, orage, éblouissement et ravissement. La sueur couvrait de nouveau mon front, son odeur rance imprégnait mon chandail sous les aisselles, nanamiu, nanamiu-shipu... Je répétais ces mots que personne ne m'avait appris alors qu'ils encerclaient ma destinée comme le feu encercle les villes assiégées. Ils révélaient la véritable nature de la rivière, Nanamassiu-shipu, qui avait voulu m'emporter durant la dernière tempête. Phil avait tort lorsqu'il prétendait qu'on ne pouvait accuser les arbres et les rivières de certains crimes. La disparition de Mike échappait à l'entendement parce qu'elle avait été causée par la résurgence des lointains tremblements de la rivière devant la foudre, parce que Michael, avec son soulier défait, était tombé dans l'une de ces failles que l'eau avait ensuite recouvertes.
Qu'est-ce que t'as, Marnie ? m'a demandé la voix inquiète de Phil. Rien, lui ai-je répondu, j'ai marché sous la pluie, hier, ce doit être la fièvre. Il m'a touché le front : brûlant. La fièvre était bel et bien installée. Va te coucher, m'a ordonné Phil, je m'occupe de la vaisselle, mais je suis demeurée à table, à regarder les bougies s'éteindre dans les formations de lave rouge figées dans les plats de verre, m'interrogeant sur les rivières des Abruzzes, sur la possibilité que quelques-unes d'entres elles aient quitté leur lit pour emporter les enfants disparus. J'ai réuni les miettes éparpillées sur la nappe, puis je les ai classées selon leur forme et leur grosseur pour ensuite les disposer en cercles, cinq petits cercles évoquant ces ronds de sorcière au centre desquels les démentes et les condamnées appâtaient les démons de leurs danses frénétiques. Cinq petits cercles rappelant les pierres pêchées par Mike dans la rivière Tremblante, on met les blanches avec les blanches, Marn, avant le crépitement de la pluie, avant la foudre, avant que les enfants crient tous sorcière, bitch, witch, Marnie the Witch !
À onze heures pile, la maison était en ordre et Miles Davis avait regagné ses quartiers, dans ces inconfortables coulisses situées entre la mort et l'immortalité. Quant à Phil, il avait compris que je venais de ressortir Mike de sa tombe.
La fièvre a duré plusieurs jours durant lesquels, dans mon quasi-délire, je ressuscitais Mike pour ensuite retourner son corps à la rivière, où ses cheveux blonds caressaient ses joues au rythme du courant. Je suivais la lenteur de ses mouvements dans l'eau douce et me demandais pourquoi aucun peintre n'avait songé à représenter la figure tranquille du noyé sous les traits d'un homme ou d'un garçon. Il existait sûrement, dans quelque galerie ou musée, des visages d'hommes torturés surgissant d'algues noires mais, à ma connaissance, aucune paisible Ophélie voguant parmi les nénuphars ne portait le nom d'un homme.
Quand je ne l'imaginais pas au fond d'une rivière ou d'un ravin, je voyais Mike arriver par la côte Croche, que nous nommions ainsi parce que son tracé contournait autrefois le flanc ouest de la colline qu'elle traverse aujourd'hui. Il avait les cheveux sales, la barbe longue, et traînait sur son épaule une poche informe contenant toutes ses possessions, qui se résumaient à quelques vêtements et à quelques livres. C'était le Mike ressuscité, l'enfant prodigue qui avait mis trente ans à regagner son village. Si Victor Saint-Pierre ne mourait pas, c'est parce qu'il attendait ce Mike-là, celui qui appartenait à cette catégorie d'enfants prolongeant malgré eux l'existence des morts-vivants. On les attendait, on ne mourait pas, puis, de sursis en sursis, on finissait par croire que l'immortalité n'était qu'une version camouflée de l'attente. Dans les moments où je ressuscitais Mike, j'avais parfois envie de courir chez Victor Saint-Pierre pour lui dire de ne pas mourir avant d'avoir vu la silhouette vacillante d'un homme se profiler au sommet de la côte Croche dans la lumière d'été, mais la fièvre m'en empêchait, qui me faisait trembler des pieds à la tête pendant que mon corps se couvrait de sueur. C'était la fièvre tremblante, nanamitshiu-akushu, provoquée par la pluie montée des rivières disparues.
Phil se manifestait tous les jours, en fin d'après-midi, m'apportant des oranges, de la soupe au poulet Campbell ou de la mélasse noire, mais seule la ponce de gin au miel qu'il me préparait après le souper me soulageait. Il n'y a rien de tel que les remèdes d'enfance pour forcer la nostalgie à sortir son arsenal d'effets placebos. C'est ce que je disais à Phil en lui tendant mon verre pour qu'il me serve une deuxième rasade de gin. Au milieu de la soirée, après un bain brûlant auquel je ne conviais plus Mr. Bubblelou, je m'endormais dans des odeurs de genièvre pendant que Bones classait des fémurs à la télé.
Puis la fièvre s'en est allée aussi soudainement qu'elle était venue, comme un traître s'esquivant par la porte de derrière après vous avoir planté un couteau dans le dos. Je me suis traîné les pieds pendant une bonne semaine encore, ignorant si je souffrais des symptômes de la convalescence ou du deuil, puis la vie a repris son cours. J'essayais de ne pas penser à Mike en me concentrant sur les cahiers dans lesquels mon père avait consigné des notes relatives à l'entretien de différents types de rosiers, je remplissais des commandes de semis et faisais le ménage dans l'atelier et dans les serres.
Pendant que la fièvre tremblante faisait ses ravages, j'avais sérieusement considéré la possibilité de quitter Rivière-aux-Trembles, puis j'avais compris, en regardant la neige fondre dans le jardin, que mon enfance m'avait trop manqué pour que j'y renonce maintenant que le souvenir de Mike s'apaisait et que je pouvais espérer redevenir peu à peu la Marnie qui sifflait parfois As Time Goes by sous les cieux New Yorkais. Puisque j'en avais hérité, j'allais m'occuper de la petite entreprise de mon père et reprendre mon enfance là où je l'avais laissée, sous la pergola en fleurs. Je vivrais désormais ici, dans l'odeur des rosiers et du foin sec, de la sueur et des sapinages exsudant leur sève amère dans les étés de Rivière-aux-Trembles. J'évoquerais ces étés, le foin qui crisse, le sable se collant aux mains mouillées et je m'étendrais près des vagues parfois immenses roulant sur les plages où reposaient les chaudières à châteaux. J'enlèverais toutes les moustiquaires et laisserais les couleurs de l'enfance entrer dans la maison de mon père. Seule la couleur de la pluie me résisterait, car personne n'a jamais pu saisir la couleur de la pluie d'été. Ceux qui croient que la pluie est grise n'ont jamais vraiment vu la pluie. Ils n'ont vu que le sol trempé et les nuages.
J'ai donc attendu que la fièvre se calme et je me suis plongée dans le travail avec un seul but en tête, reconstituer certains états bienheureux de l'enfance. J'étais dans la serre des hybrides, m'y activant depuis le matin, quand deux portières de voiture ont claqué dans la cour. Phil était pratiquement la seule personne à me rendre visite, mais ça ne pouvait pas être lui. Phil ne claquait pas sa portière deux fois, Phil ne crachait pas sur les araignées et ne sautait pas par-dessus les rainures des trottoirs. Phil était aussi rationnel qu'un polygone, et s'il ne passait pas sous les échelles, c'était pour éviter de faire tomber le gars qui peinturait sa corniche en espérant qu'aucun imbécile ne s'aventurerait sous son échelle. La normalité du temps allait aujourd'hui devoir faire face à l'instabilité menaçant les milliards de quotidiens se croyant à l'abri du désordre.
J'ai déposé mon sécateur et me suis dirigée vers l'entrée de la serre. Les portières qui avaient claqué étaient accrochées à une voiture de la Sûreté du Québec, et les hommes qui les avaient claquées montaient deux à deux les marches menant à l'entrée de la maison. Instinctivement, je me suis réfugiée au fond de la serre. Ces hommes en uniforme venaient m'arrêter. Ils avaient découvert le chaînon manquant, le détail qui m'incriminait et que j'avais vainement cherché en lisant Winston Graham et en me laissant bercer par les flots d'une mer inexistante. Appuyée contre l'établi, je les entendais frapper à la porte en criant mon nom, madame Duchamp, Marnie Duchamp... Ouvrez, s'il vous plaît !
J'ai fermé les yeux et j'ai compté jusqu'à cent. S'ils étaient encore là lorsque j'ouvrirais les yeux, ce serait le signe qu'ils étaient réels. J'arrivais à soixante-quinze quand une voix a résonné à l'entrée de la serre. Ouvre pas les yeux, Marn, m'implorait la voix de Mike, faut que tu te rendes jusqu'à cent si tu veux qu'ils disparaissent. Mais les voix se rapprochaient, accompagnées de bruits de pas. Madame Duchamp ? Inspecteurs Marchessault et Doyon, on aimerait vous parler. J'ai ouvert les yeux à quatre-vingt-dix, me privant ainsi de toute possibilité de savoir si ces inspecteurs étaient réels. Devant moi, les deux policiers attendaient que je confirme mon identité. Le plus grand avait des cheveux, le plus petit était chauve et portait la moustache pour compenser. À part cette différence, ils avaient la même tête. On aurait pu poser la moustache du petit sous le nez du grand et implanter les cheveux de ce dernier au chauve, ça n'aurait rien changé, la froideur de leur attitude ne se serait pas réchauffée d'un degré.
Après m'avoir montré leurs plaques, ils m'ont demandé si on pouvait discuter dans un lieu plus approprié. La serre ne me semblait pas moins appropriée que la chapelle Sixtine pour ce qu'ils avaient à m'annoncer, mais je n'ai pas protesté. Je les ai guidés jusqu'à la maison et leur ai offert de s'asseoir au salon, mais ils préféraient la cuisine, comme Desmarais et McCullough la première fois qu'ils m'avaient interrogée à propos de la disparition de Mike. Il ne manquait que mon père, devant moi, dont la place était occupée par le chauve, et la silhouette de Victor Saint-Pierre dans l'encadrement de la porte. Pour me donner contenance, j'ai cherché des miettes sur la nappe de plastique et me suis rendu compte que mes mains tremblaient, ce qui n'a pas échappé à Doyon, qui a fait un signe du menton à Marchessault. J'avais beau essayer de me les représenter sous les traits de deux imbéciles pour contrôler la panique qui me nouait les intestins, ces deux flics étaient tout sauf des nuls. Ils connaissaient leur métier et pouvaient flairer l'odeur de la peur jusque sous un putain de scaphandre.
J'attendais qu'ils me révèlent quelle preuve irréfutable ils avaient dénichée pour m'inculper de la disparition de Mike, mais Marchessault s'est lancé dans une description de la cuisine de sa grand-mère, où il y avait des armoires tout en lattes, pareilles aux miennes, puis le garde-manger dans lequel sa grand-mère conservait d'énormes biscuits à l'orange dans une jarre de grès. Il se demandait si les biscuits étaient vraiment énormes ou s'il les voyait alors ainsi parce qu'il était petit. Il a fini par conclure que sa taille d'enfant n'était pas en cause et qu'il s'agissait bel et bien de grosses galettes sèches qui fondaient dans la bouche et laissaient un peu de poudre orange au bout des doigts.
J'ignorais s'il s'adressait à Doyon ou à moi, s'il cherchait seulement à m'énerver ou s'il soliloquait, perdu dans la splendeur d'une enfance où il n'avait pas à se taper de coéquipier chauve ni d'arrestations le lundi après-midi. Les flics avaient recours aux méthodes les plus invraisemblables pour vous faire avouer, lesquelles incluaient les vieilles dames en tablier fleuri qui les avaient gavés de galettes jusqu'à l'âge de douze ans. Il ne fallait pas se fier à eux ni à leurs souvenirs attendris. Quand il a eu fini de s'extasier sur la saveur des biscuits à l'orange parsemés de grains de sucre, des biscuits comme on n'en faisait plus, il s'est tourné vers moi et m'a demandé comment j'avais occupé ma journée du samedi 18 avril.
Je suis demeurée bouche bée. En quoi le 18 avril avait-il un rapport avec la disparition de Mike ? Pourquoi le 18 avril et pas le 25 décembre 1940 ? Je n'avais rien foutu le 18 avril qui pouvait avoir un lien quelconque avec Mike. Je n'étais pas allée me recueillir près de sa croix ni près du lac aux Barbotes, je n'avais quitté la maison que pour me rendre à une quincaillerie de Saint-Alban qui vendait un type d'engrais spécial dont j'avais besoin pour mes roses. C'est ce que je leur ai raconté, que j'avais passé la journée dans les serres avant d'aller faire des courses à Saint-Alban. Le fait que je me sois rendue à Saint-Alban a paru les intéresser. Doyon a donné un autre coup de menton en direction de Marchessault, qui s'est penché vers moi pour me demander si j'aimais les enfants, les petits garçons. Ses grosses mains poilues touchaient presque les miennes et son visage était si proche du mien que je pouvais voir des perles de sueur se former sur son front, pareilles à celles qui coulaient du front de l'agent Desmarais sous le soleil du mois d'août. Marchessault pompait l'air de mon espace vital et cette proximité me donnait la nausée. Les petits garçons ? a-t-il répété en inclinant un peu plus son visage vers le mien, et je me suis levée d'un bond.
De quoi est-ce que vous parlez, bordel ? Je comprends rien à ce que vous dites. Qu'est-ce que vous me voulez ? Je criais et je reculais lentement vers la porte, attendant que mon père apparaisse pour mettre son poing sur la table et faire cesser cet interrogatoire. Mon attitude était celle d'une coupable, d'une maudite folle pas capable de se contrôler, je le savais, et pourtant je continuais à reculer. Ma peur n'a pas échappé aux deux hommes assis à ma table, dont les regards, devant mon affolement, brillaient comme s'ils avaient confondu l'assassin de JFK.
Rassoyez-vous, m'a ordonné Doyon.
Pas avant que vous m'ayez expliqué ce que vous faites chez moi, ai-je répondu en saisissant un Scott Towel pour me moucher.
Marchessault a attendu que j'aie fini de renifler, puis il a repris la parole, confinant l'autre au rôle du muet, pour me détailler le pourquoi de leur présence. Sa voix était calme, presque chaude. Le flot ininterrompu de ses paroles emplissait la cuisine d'une espèce de bourdonnement dans lequel s'immisçait sans cesse le prénom d'un garçon, Michael, dont la bicyclette toute neuve, bleue, bleu Superman, était tombée dans un fossé. Mais Marchessault racontait n'importe quoi. Il mélangeait les faits, les dates, les saisons. La bicyclette de Mike n'était pas tombée dans un fossé. Il l'y avait cachée avec la mienne et il ne portait pas de veste rouge, il faisait trop chaud, personne ne porte de veste rouge en plein mois d'août. Ça ne s'est pas passé de cette façon, vous vous trompez, ai-je affirmé aux deux hommes qui m'observaient en retenant leur souffle, pareils à deux chats accroupis près d'une flaque d'eau de pluie où s'abreuvent quelques innocents moineaux. J'ai lissé la nappe avec mes mains moites et leur ai expliqué comment Michael s'était sauvé pendant que j'étais dans la cabane, à genoux dans un cercle de pierres brisé. Je leur ai décrit la force soudaine du vent, les mots chuchotés par Mike, qui ne parlaient pas du mauvais temps, non monsieur, mais me disaient de m'enfuir, va-t'en, Marn, va-t'en. Ils se trompaient sur toute la ligne. J'étais dans la cabane, Michael près de la rivière, un orage entre nous deux.
La nappe était maintenant bien lisse et les deux inspecteurs me fixaient, leurs yeux de chat écarquillés. Les poings crispés, Doyon me demandait où était le garçon, où j'avais séquestré Michael Faber. Ce nom avait traversé le bourdonnement, tout à l'heure, mais je ne m'y étais pas attardée, il avait glissé sur le nom de Michael pour tomber dans ce vaste espace où les mots négligés se ramassent pêle-mêle, inutiles et impuissants. Pendant que Marchessault insistait à son tour pour que je lui révèle où j'avais caché le corps de Michael Faber, j'ai compris qu'ils parlaient d'un autre garçon, d'un autre Michael.
Le bourdonnement a brusquement cessé, what the fuck, Holy, et j'ai vu que des petites perles de sueur brillaient maintenant sur la moustache de Doyon, pareilles à celles couvrant le front de Marchessault. Je sais pas, ai-je bafouillé en m'essuyant la bouche, agacée par la moustache mouillée révélant des poils rêches qui bifurquaient vers les commissures des lèvres, je connais pas de Michael Faber, j'ai séquestré personne et je suis pas folle. Je suis pas folle, m'entendez-vous ? C'est la forêt qui a pris Mike. Mais Doyon et Marchessault refusaient d'entendre mes explications autant que mes justifications. De leur point de vue, je venais plus ou moins de confesser l'enlèvement d'un garçon nommé Michael Faber. L'un d'eux, je ne sais plus lequel, m'a passé les menottes en me récitant mes droits et m'a conduite à la voiture stationnée devant l'entrée malgré mes cris et mes protestations. J'ai tenté de me débattre, de leur expliquer que leur histoire n'avait pas d'allure, mais celui qui m'emmenait m'a délicatement poussée dans la voiture alors que je me tortillais comme un ver enragé au bout d'un hameçon. Il a ensuite refermé la portière, certain que je ne pourrais me sauver, puisqu'il n'y a jamais de foutue poignée à l'intérieur des portières qui se referment aussi sec qu'un piège à rats sur les petites Marnie, et il est allé rejoindre l'autre à l'intérieur de la maison.
J'ignore combien de temps ils y sont restés, mais j'avais l'impression qu'ils n'en ressortiraient jamais. La porte s'est enfin ouverte alors que je m'étais résignée à ne plus les attendre, à suffoquer dans cet habitacle qu'on avait vaporisé d'essence de sapin pour masquer l'odeur de vomissure des poivrots ramassés sur le bord de la route le samedi soir, et Marchessault est apparu sur le perron, suivi de Doyon, qui tenait à la main un sac de plastique dans lequel j'ai cru distinguer la casquette des Yankees que je m'étais procurée à New York, longtemps après que ma casquette des Expos avait rendu l'âme. Là encore, les policiers se trompaient et confondaient tout, le passé et le présent, le bleu roi et le bleu marine, les Expos et les Yankees, qui ne tenaient aucun rôle dans cette histoire. Doyon a déposé le sac dans l'escalier, puis il a entrepris l'inspection du jardin avec Marchessault. Ils ont scruté chaque centimètre carré du terrain, soulevé tout ce qu'ils pouvaient soulever, déplacé tout ce qui se déplaçait, puis ils ont disparu dans les serres, Marchessault dans celle des hybrides, Doyon dans l'autre. Là encore, j'ai pensé qu'ils allaient passer leur vie là-dedans, puis Marchessault est ressorti en appelant Doyon. Ils ont discuté quelques instants en gesticulant, puis se sont dirigés vers l'atelier.
Recroquevillée sur le siège arrière de la voiture, j'essayais de comprendre ce qui m'arrivait. Un enfant avait été enlevé et c'est moi qu'on était venu chercher. Ils étaient remontés jusqu'à Mike et avaient conclu que j'étais deux fois coupable, à cause des prénoms et des bicyclettes bleues, de la couleur du t-shirt de Mike détrempé par la pluie. La méchante Marnie avait refait surface et on s'était empressé de l'arrêter avant qu'elle abandonne d'autres victimes dans le sentier piétiné. Tout ça n'avait aucun sens. Je n'avais pas touché à Mike, je n'avais enlevé aucun Michael en veste rouge, mais une petite voix, en moi, me chuchotait que je n'en savais rien, que la mémoire est oublieuse, parfois, et qu'il arrive qu'on ne se souvienne pas d'où vient la tache de sang incrustée dans les mailles de son chandail.
Quand Doyon et Marchessault sont montés dans la voiture, je n'étais plus sûre de rien et ne savais toujours pas si ces deux flics étaient réels. Je voyais deux bicyclettes bleues, bleu Superman, rouler côte à côte sur la route du 4. J'entendais des rires. Je voyais un soulier trempé de boue et une casquette de baseball portant les couleurs des Yankees de New York, réunis dans un sac de plastique. Je voyais deux Michael, l'un souriant et l'autre sans visage, qui pénétraient dans la bouteille de verre contenant la cité prisonnière de Kandor.
À l'avant de la voiture, Marchessault poursuivait son soliloque. À cause de la vitre me séparant des sièges avant, je n'entendais pas ce qu'il racontait ni si ça concernait encore sa grand-mère, mais je savais qu'il bourdonnait, que Doyon était enveloppé de la vibration de ce bourdonnement et de la rumeur qui, bientôt, allait s'échapper de la voiture pour annoncer à tout Rivière-aux-Trembles que la vilaine petite Marnie Duchamp était de retour.
# TROISIÈME PARTIE
## I
L'ENQUÊTE
Dès mon arrivée au poste de police, j'ai commencé à me ronger les ongles, comme Billie, comme Lucy-Ann après Billie, parce que la situation me dépassait et que j'avais les nerfs en boule, expression qu'on ne peut comprendre que quand ça nous arrive, ça ne s'explique pas, mais au lieu de m'apitoyer sur mon sort, je songeais à celui de Pixie, m'imaginant qu'ils l'expédieraient dans un laboratoire pour lui faire subir des tests d'ADN, relever ses empreintes dentaires ou je ne sais quoi, mais la rapidité avec laquelle ils ont conclu que Pixie n'était pas Nuage, le chat du gamin disparu, excluait cette possibilité. Ces crétins ont dû se contenter d'ouvrir son caisson pour demander au père effaré de se pencher dessus afin d'identifier la carcasse du chat qu'un imbécile y avait enfermé. Si le gars n'était pas déjà démoli, la vue de ce chat raide mort a dû l'achever. Je parle du père parce que c'est en général devant les pères qu'on soulève le drap blanc recouvrant le visage aux paupières noircies des enfants morts. On n'ose pas infliger ce spectacle à la mère, alors on part du principe que le gars est plus fort et qu'il va s'en sortir, qu'il ne piquera pas une crise qui vous obligera à le maîtriser et à appeler le médecin de service pour qu'il enfonce une seringue dans le bras que deux costauds auront peine à maintenir le long du corps.
Une fois rassuré quant au sort de Pixie, l'image qui me trottait dans la tête était donc celle d'un homme aux yeux cernés et veinés de rouge, bouffis par l'insomnie, mais rien ne prouvait que la mère n'avait pas insisté pour voir Pixie, menaçant Doyon de lui replanter la moustache dans le front s'il lui barrait le chemin. Avec Lucy-Ann, j'avais eu un sérieux aperçu de la férocité des mères appelant leurs petits et j'avais appris que rien ne pouvait les empêcher de soulever une tonne de briques si elles pensaient que leur enfant gisait sous les décombres. La prétendue force des hommes, au regard de la violence née du ventre des femmes, ne pesait pas très lourd. Ça me faisait donc deux visages au-dessus du caisson, deux visages anéantis qui continueraient à chercher Nuage et à crier son nom en espérant que leur fils apparaîtrait en courant derrière le chat.
Marchessault et Doyon n'ont rien trouvé de compromettant non plus dans mon ordinateur, qu'ils ont toutefois décidé de conserver pendant quelques jours pour le soumettre à l'examen d'un spécialiste, au cas où je l'aurais trafiqué en vue d'y dissimuler des images d'enfants mutilés derrière des photos de Billie. J'ai tenté de plaider ma cause, cet ordinateur était mon instrument de travail, mais Marchessault est demeuré sourd à mes arguments. Il a quitté la salle agrémentée d'un miroir sans tain et j'ai tué le temps en imaginant la gueule du gars, flic ou psy, qui se tâtait le menton derrière le miroir en essayant de déterminer si mon attitude et ma nervosité étaient celles d'un coupable.
Marchessault m'a laissé poireauter là pendant deux heures, probablement occupé à asticoter l'autre suspect dans cette affaire, avant de revenir m'annoncer que j'étais libre de partir, visiblement contrarié de devoir me laisser filer. Il aurait pu invoquer la supposée voie de fait qui avait poussé Doyon à me sauter dessus un peu plus tôt et me garder ainsi à l'ombre une nuit ou deux, mais il a dû conclure que la paperasserie n'en valait pas le coup. Aucune preuve ne lui permettant de me retenir, il n'avait d'autre choix que de renvoyer chez lui l'animal qui congelait ses chats. On vous tient cependant à l'œil, Richard. Vous ne quittez pas la région, on va se revoir bientôt. Après un bref conciliabule avec la fille qui faisait le pion dans le corridor, Marchessault s'est éclipsé et la fille m'a conduit à la réception, où on m'a remis le caisson de Pixie en me détaillant encore une fois comme si j'étais le pire taré que la terre ait jamais enfanté. Puis je me suis retrouvé sur le trottoir mouillé, à compter les voitures sales qui ramenaient leurs conducteurs chez eux après leur première journée de travail de la semaine. L'esprit complètement vide, j'ai marché sous la pluie douce jusqu'à une rue commerçante où j'ai attrapé un taxi qui a accepté de me conduire à Rivière-aux-Trembles moyennant un tarif usuraire payable avant la course.
Le chauffeur faisant partie de cette catégorie de gens ayant une opinion sur tout et n'importe quoi, de même qu'une solution à tous les fléaux s'abattant sur la planète, je l'ai laissé débiter ses conneries en lui renvoyant ici et là un grognement qui pouvait passer pour un acquiescement. M'opposer aux vérités que son gros bon sens m'assenait aurait nécessité que je démolisse du revers de la main la droite nord-américaine au grand complet, et j'avais en quelque sorte d'autres chats à fouetter.
Pendant que défilait à ma droite un paysage où arbres et champs mouillés se recueillaient dans le même silence, je songeais au fait que je n'avais pas révélé les visites de Dixie. Pour me déculpabiliser, je passais aux rayons X la beauté de l'inspectrice qui était venue remplacer Doyon durant l'interrogatoire. Rien ne clochait dans le visage de cette fille, mais c'est son regard qui vous attaquait de front. Elle avait des yeux immenses dont la démesure vous enveloppait comme de grandes ailes d'oiseau de quelque paradis perdu et vous aviez l'impression que rien de ce que vous pouviez confesser pendant que vous vous teniez dans l'ombre bleutée des ailes n'était susceptible de vous expédier derrière des barreaux de prison. La présence de cette femme sous la lumière aveuglante des néons atténuait ma fatigue et j'étais presque heureux de pouvoir me confier à sa voix chaude. Je pénétrais dans la zone d'ombre azurée entourant son regard, confiant, quand une sonnette d'alarme avait retenti sous mon crâne. Je devais me tenir sur mes gardes et ne surtout pas oublier que cette femme était flic, même si rien dans son apparence ne laissait supposer que, sous le masque, elle avait une trogne identique à celles de Marchessault et de Doyon.
Elle m'a d'abord posé les mêmes questions que ces derniers, au cas où j'ajouterais à ma déposition un détail qui lui permettrait de me coincer, ce qui ne se produirait pas, puisque je n'avais rien à me reprocher. Après avoir noté mon emploi du temps du 18 avril et m'avoir demandé de lui expliquer qui était Ronie le crapaud, elle m'a sorti une photo en gros plan de Michael Faber, sur laquelle le gamin riait à gorge déployée, comme si quelqu'un s'était agenouillé en bas du cadre pour lui chatouiller le nombril. Cette photo a eu sur moi l'effet d'une gifle, car je savais en mon for intérieur que Michael Faber ne riait probablement plus. Elle a ensuite poussé vers moi une autre photo du garçon prise à côté de la bicyclette qu'on lui avait offerte pour ses douze ans, une CCM dix vitesses au bleu étincelant sur le devant de laquelle Michael avait installé un panier pour son chat Nuage. Regardez bien cet enfant, monsieur Richard, m'a dit la femme, prenez tout votre temps et dites-moi ce que vous ressentez à l'idée qu'on l'a enlevé.
La cruauté de cette remarque m'a coupé le souffle et j'ai senti mes joues s'enflammer. Cette femme savait aussi bien que Marchessault et Doyon que ma fille avait disparu. Par conséquent, elle devait bien se douter que tout ce qui était susceptible de me rappeler cette disparition me déchirait les entrailles, mais elle enfonçait le couteau, elle fouillait dans la chair encore vive, elle se servait de Billie pour me déstabiliser. Je lui ai répondu que ça faisait mal, bien plus mal qu'elle ne pouvait se l'imaginer, puis j'ai fermé ma gueule, fin de la réponse, car la voix douce, je le constatais amèrement, était un piège tendu à seule fin de tromper l'imbécile qui se laissait porter par ses inflexions.
Voyant que je ne formulerais aucun autre commentaire quant à mes états d'âme devant la disparition de Michael Faber, elle a changé d'angle d'attaque. Elle a retiré d'une enveloppe une dernière photo du garçon en compagnie de son chat pour me faire remarquer à quel point ce chat ressemblait à Pixie, qu'elle avait également examiné dans son caisson. Cette ressemblance ne démontrait absolument rien et si elle croyait que je perdais mon temps à congeler les chats qui me rappelaient Pixie, elle était carrément cinglée. Je plaignais l'innocent qui serait chargé de s'occuper de ses obsessions le jour où elle péterait une coche.
J'ai tout de même attrapé la photo qu'elle me tendait pour jeter un coup d'œil au garçon, qui portait un jeans troué, un chandail des Canadiens et des runnings pourvus d'une lumière clignotante à l'arrière du talon. Sur la photo, la lumière ne clignotait pas, mais j'entrevoyais les brefs éclairs dont elle devait strier la brunante quand Michael enfourchait sa bicyclette après le souper, juste avant le match des Canadiens. Un garçon ordinaire, en apparence, pareil à tous les autres, mais dont le sourire vous broyait le cœur dès que vous saviez que son destin n'était pas celui des autres.
J'ai laissé le petit Michael tranquille pour me concentrer sur le chat, dont le front était marqué d'une tache en forme de cirrostratus et qui dardait ses maudits grands yeux verts en direction de l'objectif, des maudits grands yeux pareils à ceux qui attendaient leur ration de thon derrière ma porte d'entrée. Jusque-là, je n'étais pas certain que le chat de Michael Faber était celui qui venait bouffer sur ma galerie, mais mes derniers doutes se sont dissipés. Dixie était Nuage, Nuage était Dixie, inutile de s'étendre plus longuement sur ce sujet. Je ne voyais toujours pas où l'inspectrice voulait en venir avec ce chat, mais je n'avais aucune raison de lui mentir. J'étais sur le point de lui révéler la vérité quand j'ai vu son regard rivé sur moi, dont le bleu n'évoquait plus la clarté des ciels d'été, mais cette couleur grise et métallique annonçant l'imparable venue du froid. Cette fille me tend un piège, ai-je tout de suite pensé, un foutu piège à cons dans lequel je me précipite comme un rat affamé. Je n'avais aucune idée non plus de la nature de ce piège, mais une bouffée de méfiance mêlée de paranoïa m'a incité à rebrousser chemin. J'ai repoussé la photo en prétendant que ce chat pourrait tout au plus passer pour un cousin par alliance de Pixie. Si je parlais de Dixie et de sa mine de chat perdu, j'étais persuadé que deux paires de gros bras débarqueraient illico dans la salle d'interrogatoire pour m'entraîner dans une cellule avec vue sur les richesses du sous-sol québécois ou sur une cour bétonnée, remplie d'ombres courtes, où d'autres gros bras éteindraient leurs rouleuses dans la saignée de leurs coudes en admirant l'effet de la chair brûlée au milieu des tatouages.
Maintenant qu'apparaissaient au loin les premières maisons de Rivière-aux-Trembles, je n'étais plus certain d'avoir eu raison de cacher l'arrivée récente de Dixie dans mon entourage. Le piège à cons, c'était moi qui me l'étais tendu. Si les flics venaient à découvrir les visites de ce chat, mon silence me rendrait doublement suspect. J'avais voulu m'éviter d'autres interrogatoires tordus et ne menant nulle part, repousser l'image de Billie dont les traits enfantins se confondaient avec ceux de Michael Faber, mais j'avais en même temps dissimulé un élément qui aurait pu permettre de retrouver un enfant disparu. Je m'étais non seulement comporté lâchement, mais de manière criminelle, parce que je connaissais mieux que quiconque l'importance du moindre détail lorsque la vie d'un enfant était en jeu. Si un crétin s'était comporté aussi bêtement pendant qu'il demeurait encore un espoir pour Billie, j'aurais été le premier à lui sauter à la gorge et à lui enfoncer mes pouces dans la trachée jusqu'à ce qu'il crache le morceau.
J'étais aujourd'hui ce crétin fini, cet homme sans scrupules qui avait pris ses jambes à son cou plutôt que d'essayer d'attraper la main d'un gamin égaré dans la nébuleuse où s'était enfoncée Billie. À travers les vociférations du chauffeur qui venait tout à coup de décréter qu'il fallait réformer le Parlement de fond en comble et réviser à la baisse le salaire de nos élus, j'entendais la voix de Billie, au loin, sa petite voix qui refusait d'admettre que son père était un putain de salaud. C'est pas vrai, papanoute, hein ? C'est pas vrai ? Et je demeurais silencieux, aussi silencieux que les arbres immobiles, que les pierres tombales alignées dans le cimetière de Rivière-aux-Trembles qui venait de disparaître derrière moi, avec ses morts gelés sous la terre froide et ses milliers d'ossements noircis.
Au moment où le chauffeur intarissable s'engageait dans l'allée de ma maison, j'ai été tenté de lui ordonner de faire demi-tour, de regagner Saint-Alban à cent quarante à l'heure, puis j'ai aperçu Dixie sur la galerie, qui avait sûrement faim et soif. Alors j'ai ouvert la portière, comme un lâche, Billie, comme un homme justifiant ses actes en se dissimulant derrière des mensonges qui l'auraient fait vomir il y avait quelques heures à peine, et je me suis enfoncé avec le caisson de Pixie dans le brouillard entourant la maison.
Une enfilade de pistes grises maculaient le plancher jusqu'au sofa où Dixie roupillait. Tiraillé par ma mauvaise conscience, je l'avais laissé entrer, le temps de déterminer si j'allais ou non appeler les flics et offrir ainsi ma tête à l'indifférence d'un bourreau qui ne se gênerait pas pour m'entailler la jugulaire. Assis dans le fauteuil à bascule faisant face au sofa, j'observais le sommeil de Dixie, me demandant si le petit Michael Faber l'avait nommé Nuage à cause de la légèreté qu'évoquaient les longs poils blancs qui commençaient à former des nœuds près de la peau, ou à cause de cette tache couleur caramel dessinant sur le front de l'animal un cirrostratus dont les bords s'effilochaient vers les oreilles. J'ai prononcé son nom, Nuage, pour voir s'il réagirait, mais Nuage voyageait au cœur du pays des rêves, le nez enfoui entre ses pattes avant, le corps agité de spasmes quand il rencontrait un autre chat, peut-être, dans une ruelle auréolée de lumière de nuit.
À onze heures pile, je me suis levé, j'ai décroché le téléphone et j'ai laissé un message au poste de police de Saint-Alban à l'intention de Marchessault et Doyon, dans lequel je leur annonçais que j'hébergeais Nuage. Cette révélation équivalait plus ou moins à un suicide, mais je pourrais au moins me diriger vers la potence la tête haute et ne plus craindre la honte de Billie devant son père. J'ai déposé un bol d'eau fraîche près du sofa et je suis monté me coucher. Je savais que je ne dormirais pas, mais je tentais ma chance, au cas où le sentiment d'avoir agi pour le bien d'un enfant qui, à un cirrostratus près, aimait le même chat que Billie, couvrirait mon corps de ce bienheureux engourdissement permettant aux âmes tourmentées d'oublier leur vie pour quelques heures. De toute façon, je n'avais rien de mieux à faire en attendant que Marchessault et Doyon débarquent pour me ramener au poste.
Après avoir éteint ma lampe, j'ai laissé mes yeux s'habituer à l'obscurité et j'ai fixé le plafond, où les nœuds parsemant les lattes de bois brut formaient un réseau de sombres constellations enroulant leur spirale autour d'un noyau qui les empêchait de se disperser. L'équilibre de l'univers tenait à ces pôles d'attraction autour desquels gravitaient des milliards de mondes qui se percuteraient peut-être un jour pour recréer le chaos originel. Au milieu de ces mondes, si tant est qu'on puisse parler du milieu de l'infini, progressaient la Terre et son soleil, vivaient et mouraient des hommes tournoyant autour d'infimes pôles d'attraction les empêchant aussi de disperser leur existence, de penser à son insignifiance, et les poussant à se croire, le temps d'une vie, le milieu d'un infini qu'ils réduisaient à leurs proportions.
Billie était mon centre, le milieu des milliards de mondes dont elle m'épargnait le vertige, et voilà qu'à travers un enfant inconnu, on m'accusait de nouveau d'avoir été à l'origine de l'effondrement de mon univers. Mais comment une planète pouvait-elle détruire son soleil ? Ceux qui m'accusaient d'avoir enfoui sous terre la chaleur et la lumière dont dépendait ma survie ne se rendaient pas compte que cette accusation allait à l'encontre de toutes les lois physiques.
Harcelé de questions toute la journée, je n'avais pas eu le temps de réfléchir calmement à la nouvelle déflagration que la fragilité des enfants provoquait dans l'étroitesse de ma vie. La nuit m'apportait ce calme dont j'avais besoin pour constater que l'histoire se répétait et qu'à moins de coincer le véritable agresseur de Billie, on me considérerait jusqu'à ma mort comme un homme dangereux. Je n'avais pas touché à un cheveu de ce Michael Faber mais, en cachant l'existence de Dixie-Nuage, je m'étais foutu dans un sérieux pétrin. Du point de vue des forces de l'ordre, cette faute par omission constituerait un autre indice de ma culpabilité. Seul l'aveu de ma lâcheté serait susceptible de me disculper.
Couvert d'une sueur âcre évoquant l'odeur de la maladie et de la peur, j'ai repoussé mes couvertures et suis allé ressasser mes remords sous le jet brûlant de la douche, où j'ai vainement essayé, en massacrant Guantanamera, de faire taire les voix qui instituaient mon procès. Il y avait des siècles que je n'avais pas fredonné cette chanson que mon père aimait tant, il y avait des mois que je n'avais pas pensé à mon père, que nous entendions siffloter à travers les bruits d'eau pendant qu'il se rasait ou prenait son bain. L'intervalle entre les moments que j'accordais au souvenir de mon père s'élargissait à mesure que sa mort s'éloignait de moi et il m'arrivait de me dire qu'un jour, peut-être, cet homme que j'avais adulé déserterait ma pensée. Et pourtant, il avait encore la capacité, après des mois d'absence, d'apparaître subitement sous ma douche en plein milieu de la nuit. Ça devait être sa façon à lui de me donner une grande tape dans le dos, décourage-toi pas, mon gars, de me dire que je n'avais aucune chance de l'oublier, parce que les morts, ça s'oublie pas, fils, pas plus que les disparus. Je pouvais ignorer mes craintes, chantait mon père, l'image de Billie ne s'effacerait jamais de mon esprit, elle se ferait seulement plus rare, mais aussi plus joyeuse, pareille à celle de l'homme heureux enduisant son visage de mousse devant le miroir embué.
J'ai remercié mon père pour sa visite inespérée, et c'est sur l'air de Guantanamera que j'ai attendu jusqu'au matin de voir filer sur la route du 4e Rang, depuis la courbe que me masquait en partie une rangée d'arbres, la voiture conduite par un Marchessault dont l'haleine fétide empesterait ma journée.
Même si Doyon ne me l'avait pas présentée, je l'aurais immédiatement reconnue. Emmy, Emmy-Lili, Émilie Saint-Pierre, qui semblait n'avoir grandi que pour mieux ressembler à son frère. Seule la couleur des yeux différait, mais pas la forme, pas le regard, pareil à celui d'un oiseau de nuit auquel rien n'échappait. Emmy Saint-Pierre avait si longtemps observé la rivière que la couleur de celle-ci s'était imprimée sur son œil. À jamais la rivière coulait en elle.
C'est le lieutenant Saint-Pierre, a marmonné Doyon après les présentations, qui prendra la direction de l'enquête, puis il a ramassé son stylo et m'a laissée seule avec cette femme qui ne disait rien, les mains enfoncées dans les poches de son pantalon. Appuyée contre un mur, Emmy-Lou Saint-Pierre me jaugeait, Emmy-Lou Saint-Pierre m'évaluait en se demandant probablement ce que je faisais encore dans ses pattes et si elle parviendrait à me casser.
Ça fait un bail, a-t-elle fini par lâcher, puis elle est venue s'asseoir face à moi. Elle travaillait pour le Bureau régional d'enquêtes de la Sûreté du Québec et avait expressément demandé à être affectée à ce dossier. Elle considérait la disparition de Michael Faber comme une affaire personnelle et n'avait pas l'intention qu'on lui mette des bâtons dans les roues. Cette enquête, elle la mènerait jusqu'au bout. Je sais pas vraiment ce que t'as fabriqué au cours des trente dernières années, a-t-elle ajouté, mais moi j'ai fait qu'une chose, chercher Mike, alors je veux que ce soit bien clair, toi et moi, on se connaît pas, et c'est pas parce que tu t'es prise pour Supergirl que je vais t'accorder un traitement de faveur.
La situation était on ne peut plus limpide. Emmy Saint-Pierre me détestait, contrairement à ce qu'avait prétendu son père, et elle continuerait à me détester quoi que je dise ou fasse, parce que rien n'avait bougé dans la tête de la fillette qui avait vu Mike s'envoler avec moi sur nos bicyclettes chromées. Une image s'y était fixée, dans laquelle je lui enlevais à jamais son frère et ravissais du même coup sa mère, Jeanne Dubé, qui ne l'avait plus aimée qu'à travers le souvenir de Mike. Sa haine venait de cette image et rien ne saurait l'altérer. Je croyais être blindée contre le mépris et la colère, mais j'avais l'impression de me ratatiner devant la froideur d'Emmy. Son regard réveillait ma culpabilité et je me détestais à mon tour de céder aux accusations d'une fille qui se permettait de me juger alors qu'elle ne savait rien de ma putain de vie et ignorait ce qu'il en coûtait de grandir avec un grand trou noir dans le cerveau.
J'ignore pourquoi je suis ici, Emmy, mais vas-y, pose-moi tes questions et qu'on en finisse, ai-je rétorqué en saisissant le verre d'eau qu'on avait posé devant moi et qui tremblait soudain comme si un séisme avait secoué la salle où j'étais confinée.
Où étais-tu exactement le 18 avril ? m'a-t-elle demandé. Il reste encore quelques imprécisions dans ta déposition. Décris-moi en détail ton emploi du temps.
J'avais déjà répondu à cette question en long et en large, mais j'avais prévu le coup. On me la poserait sur tous les tons, encore et encore, jusqu'à ce que je me contredise, que je mélange les heures et ne me souvienne plus si j'avais ce jour-là mangé un sous-marin chez Mikes, à la sortie de Saint-Alban, ou une pizza quatre fromages à la pizzeria du centre-ville. Dans le silence à peine perturbé par les pas provenant du corridor, qui allaient et venaient à intervalles irréguliers, j'ai donc parlé des rosiers de mon père, Alex Duchamp, emporté par un cancer qui avait grugé jusqu'à sa voix. J'ai mentionné l'engrais que je devais me procurer, le soleil qui chauffait la voiture et donnait le sentiment qu'on était en plein été. J'ai aussi parlé du bonheur que je ressentais à l'idée que je m'écorcherais bientôt les doigts aux rosiers et verrais le sang perler sur ma peau salie de terre sèche. Puis j'ai décrit la coiffure de la serveuse de la pizzeria, deux tresses rousses semblables à celles de la Schtroumpfette nommée Sassette, ça se remarque, les tresses et les Schtroumpfs ayant fait leur temps.
Concentrée sur le verre d'eau où mes doigts avaient laissé des empreintes poisseuses trahissant ma nervosité, je n'oubliais aucun détail, la mozzarella caoutchouteuse de la pizza, le doigt blessé du commis de la quincaillerie, le sac d'engrais trop lourd. À la fin de mon récit, j'avais la gorge aussi râpeuse que du papier sablé et ma salive épaisse produisait des bruits de succion gênants que j'essayais de camoufler en articulant exagérément certaines syllabes. J'avais besoin d'eau mais n'osais pas toucher au verre, de peur que mes mains ne le fassent éclater pour masquer leur tremblement. Autour du verre, la table était immaculée, pas une miette, pas une poussière, de l'acier gris frotté à l'aide d'un produit qui en faisait reluire la surface et les angles. Compter des miettes, des taches ou des brins de poussière m'aurait calmée, alors j'ai cherché des égratignures sur le métal froid, des lignes que je pourrais relier pour en faire des dessins, puis j'ai éclaté : est-ce que quelqu'un peut me dire ce que je fous ici, bordel ?
Ma voix a résonné sur les murs blancs, trop haut perchée, les pas qui traversaient le corridor se sont arrêtés et j'ai saisi le verre d'eau qui s'agitait de plus belle. Deux ou trois gorgées ont coulé sur mon menton, puis sur la vieille salopette que j'avais enfilée pour travailler dans les serres, et je me suis rendu compte que j'étais sale, que ma queue d'écureuille pendait de travers, que mes ongles étaient bordés de noir.
Faut que j'aille aux toilettes, ai-je murmuré.
Au bout du corridor, à droite, a répondu Emmy. Je te donne cinq minutes. Je suppose que t'es assez intelligente pour revenir.
Je suis sortie tête basse, honteuse d'avoir exhibé mes ongles noirs devant la parfaite Emmy, dont pas un poil ne retroussait, puis devant les inconnus qui m'observaient de derrière le miroir ornant un des murs de la salle d'interrogatoire. Ces ongles noirs me plaçaient en position de faiblesse et s'ajoutaient à la liste des éléments m'accusant. Les meurtriers ont les mains sales, ai-je pensé en ralentissant le pas, ne me souvenant plus où j'avais pêché cette phrase, puis la petite voix qui me servait de conscience s'en est emparée, la répétant machinalement, les meurtriers ont les mains sales, les meurtriers ont les mains sales, les mains souillées, insistait-elle pendant que je tentais frénétiquement de récurer mes ongles sous l'eau bouillante coulant dans le lavabo et m'y plongeais le visage après l'avoir enduit d'une couche de savon à mains qui aurait pu servir de décapant. L'image que m'a renvoyée le miroir lorsque j'ai relevé la tête était celle d'une femme au bord de l'hystérie. J'ai failli fracasser ce miroir et me suis ressaisie. Faut pas que tu craques, Marn, m'ordonnait maintenant la voix, faut pas. J'ai pris une grande inspiration et j'ai tenté de me recoiffer, lissant mes cheveux avec mes doigts rougis par l'eau trop chaude, tortillant l'élastique qui servait à les maintenir attachés, puis je me suis mordu les lèvres, faut pas que tu craques, et je suis retournée à la salle d'interrogatoire en laissant derrière moi une odeur de savon bon marché.
Six minutes trente secondes, Marnie, m'a annoncé Emmy en tapotant sa montre pendant que je me rassoyais devant elle. Je n'ai pas relevé, je n'avais aucun moyen de rattraper les quelques secondes s'opposant à la rectitude quasi militaire d'Emmy Saint-Pierre. J'ai attendu qu'elle poursuive en fixant le verre d'eau qu'une âme charitable ou sadique était allée remplir. Les empreintes de mes doigts poisseux y étaient toujours visibles, qui donnaient l'impression que l'eau claire était souillée. Emmy Saint-Pierre fixait pour sa part mon visage, marqué de plaques rougeâtres dues au savon. Ne me quittant pas des yeux, elle s'est avancée, a posé ses bras sur la table et m'a sorti une autre de ces remarques conçues pour déstabiliser les pauvres idiots qui piétinent déjà au bord d'une fosse au fond de laquelle tournoient nerveusement quelques lions affamés.
Tu prétends que t'as pas vu le gamin, Marnie, alors je comprends pas comment tu pouvais connaître la couleur de son vélo et de son chandail. Peux-tu m'expliquer ça ?
Je n'avais jamais mentionné le vélo du gamin, ni son chandail, ni ses foutues chaussures. Marchessault et Doyon avaient eux-mêmes glissé ces éléments dans la conversation pour ensuite prétendre que j'avais confirmé leurs dires, mais ces crétins se trompaient et mélangeaient tout, le vélo de Mike et celui de Michael, le t-shirt de l'un et le chandail de l'autre, mais comment expliquer à Emmy Saint-Pierre que j'avais pour ma part confondu Mike et l'autre Michael ? Si j'avouais à cette fille qu'il m'arrivait de me croire coupable de la disparition de Mike, j'en étais quitte pour la pendaison ou l'injection létale. Et pourtant, Mike Superman Saint-Pierre était ma seule chance de m'en sortir.
J'étais sous le choc, ai-je répliqué, j'ignorais de quoi on essayait de m'accuser et j'ai pensé à Mike, à son vélo bleu, à son t-shirt rouge. T'es pas la seule, Emmy, à pas avoir oublié Mike, et c'est pas ma faute si tous les enfants aiment les mêmes couleurs.
Quand je me suis tue, le visage de l'imperturbable Emmy s'était teinté de rouge, comme le mien, des petits cercles de chaleur maculaient ses joues, et j'ai cru un instant qu'elle allait perdre son sang-froid et me sauter dessus.
Mêle pas Mike à ça, m'a-t-elle prévenue. Et pourtant, la seule raison de ma présence en ce lieu était Mike, personne d'autre que Mike, ce que je lui ai fait remarquer en baissant les yeux sur mes mains, sur mes ongles que je n'avais pas réussi à nettoyer correctement, où j'avais parfois enfoncé la crasse jusqu'au morceau de chair tendre auquel ils adhéraient. Puis le besoin impérieux de me laver, de me savonner jusqu'à m'en arracher la peau m'a de nouveau saisie. Il fallait que je quitte cette salle immédiatement et me plonge dans un bain sentant les fruits sucrés et débordant d'écume de Bubblelou.
Sans attendre la prochaine attaque d'Emmy, je me suis levée et lui ai annoncé que je m'en allais. Elle ne possédait aucun élément assez sérieux pour me retenir, aucune preuve concrète, aucune empreinte. J'étais libre de sacrer mon camp si ça me chantait.
Je te le conseille pas, a-t-elle rétorqué, mais je l'ai ignorée et me suis dirigée vers la porte. Si je n'avais pas enfilé cette salopette informe, elle aurait constaté que mes rotules tressautaient et qu'il aurait suffi qu'elle hausse la voix d'un ton ou deux pour que je m'écrase et regagne docilement ma chaise. Mais j'ai marché droit, j'ai ouvert la porte et j'ai descendu le corridor jusqu'à la sortie sans me retourner, certaine qu'un de ses sbires allait m'attraper par le collet pour me ramener dans l'antre d'Emmy.
Quand j'ai enfin posé les pieds sur le parvis du bâtiment, j'avais l'impression de marcher dans un rêve. Le jour avait fait place à la nuit sans que je voie le ciel s'obscurcir derrière la colline des Loups. En mon absence, le temps s'était replié sur la lumière, ainsi qu'il l'avait fait près de la rivière Tremblante. La lueur des réverbères se reflétant sur la chaussée luisante semblait irréelle, peinte à l'aide d'une laque qui ralentissait les soubresauts de la pluie au contact du sol. J'étais égarée dans cette noirceur subite, observant la pluie qui mouillait ma peau sans que j'en sente la fraîcheur. Son odeur de feuilles pourries se perdait dans l'odeur huileuse rejetée par les hottes d'un snack à patates frites et j'étais privée du simple plaisir que peut procurer cette eau venue du ciel à ceux qui aiment l'approche des nuages.
Depuis trente ans, la pluie ne m'atteignait plus que si elle était projetée par un vent de tempête, et Emmy Saint-Pierre venait de lui donner une autre raison de tomber à côté de moi, si belle, pourtant, si mélodieuse. D'un pas de somnambule, j'ai gagné la rue et me suis assise sur un banc mouillé. Le cauchemar recommençait. Les sorcières allaient de nouveau devoir fuir les fantômes.
Mr. Bubblelou a regagné les rangs des amis de la famille. Debout sur le rebord du bain, il m'offre son sourire du temps jadis, quand nous nous racontions des histoires qui finissaient toujours bien. J'en invente une autre pour lui, dans laquelle il n'y a ni enfants disparus, ni ours ou pandas noyés, mais une petite fille qui s'appelle Squouirèle, avec un accent grave, un l et un e, Squouirèle, que la main magique de Superman vient sauver des périls de la rivière pour l'emmener dans un château érigé au sommet d'un mont de glace. Pour Bubblelou je sculpte le château, la forteresse où rien ne peut m'atteindre, puis je façonne le chien Krypto, fidèle allié de Superman. Je m'enfonce ensuite sous l'eau, laissant le château dériver vers mon visage, puis je remonte à la surface en tenant dans mes mains les ruines d'une forteresse pourtant indestructible loin de laquelle, sur un frêle esquif de matière inconnue sur terre, vogue le chien Krypto.
Sur le tabouret d'osier, ma montre m'indique qu'il y a une heure quatre minutes, pas une de plus ni de moins, Emmy-Lili, que je trempe dans la soupe mousseuse de Mr. Bubblelou au parfum de fraise, mon préféré depuis toujours, depuis que Michael Superman Saint-Pierre, accroupi dans une talle de fraises aussi grande que le Liechtenstein, m'a appris la prédilection de Lois Lane pour ce petit fruit. En une heure, j'ai eu tout le loisir de raconter un tas d'histoires à Bubblelou, qui s'étonne de me voir dans de si bonnes dispositions à son égard et de m'entendre chanter, au milieu des histoires, des airs appartenant à un passé que lui ont enseigné ses prédécesseurs : Trois fois passera, la dernière, la dernière, trois fois passera, la dernière y res-te-ra ! Et je referme mes deux mains sur la dernière, tant pis pour elle, les dernières seront toujours les dernières, j'écrase son petit corps de mousse, qui se réincarnera tantôt dans celui d'un castor ou d'un lapin aux oreilles scintillantes de bulles multicolores dont la courbure reflétera les violets et les bleus pervenche d'un coucher de soleil de conte de fées.
Toutes les dix minutes, j'ouvre la bonde pour que s'écoule une partie de l'eau, que je remplace par de l'eau si chaude que des volutes de vapeur s'élèvent du cercle d'où s'enfuient les bulles sous la pression de l'eau. Je rajoute alors un peu de Bubblelou, rien qu'un peu, le quart d'un petit bouchon, pour construire d'autres histoires et faire fondre la crasse qui s'est collée à ma peau au poste de police de Saint-Alban. Mes ongles sont propres, maintenant, mes cheveux aussi, la peau rougie de mes mains et de mes pieds est aussi plissée que celle d'un éléphant, mais il reste encore de la saleté, je le sens, qui ne partira qu'à la fin de ma dernière histoire, celle de Mike et Marn à la rivière, dont j'écrirai bien la conclusion un jour. Celle-là, je la chanterai à tue-tête et Mr. Bubblelou battra des mains pendant que je ferai voler la mousse jusqu'au plafond. Pour le moment, Bubblelou a froid, Marn est fatiguée, Marn voudrait dormir, mais elle sait que c'est impossible, qu'on ne peut dormir quand le passé rejoint le présent et que tous deux se liguent pour vous assaillir de questions, comme à l'école, les jours où vous avez oublié vos leçons. Qui a découvert l'Amérique, Marnie ? Qui a inventé la roue ? Qui a dit ceci, cela, pourquoi ? Qui, qui, qui ? Où étais-tu le 18 avril, Marn ? Et qu'as-tu fait à Mike ? Quelle était la couleur de la bicyclette bleue de Superman ? Que caches-tu, Marn ? Quoi ? Quoi ? Quoi ?
Avant la fin de la dernière question, Mr. Bubblelou vole une autre fois contre le mur, tant pis pour lui, les derniers seront les derniers. J'ouvre la bonde et l'eau s'écoule. À mesure que le niveau baisse, mon corps se couvre de mousse irisée. Il ressemble à un de ces vieux arbres envahis de lichens dont les troncs avalent les âmes perdues, confinées dans un des cercles de l'enfer. C'est là qu'Emmy Saint-Pierre m'a renvoyée, au cœur d'une forêt silencieuse retenant les cris des damnés et où seule résonne ma voix appelant Mike.
Je ne m'attendais pas à ce qu'Emmy brigue la présidence de mon fan club, mais j'espérais au moins trouver en elle une alliée, quelqu'un qui me tendrait la main dans la forêt, au lieu de quoi Emmy me hait, au lieu de quoi Emmy m'accuse, camouflant à peine le mot « sorcière » dans son discours supposément fondé sur l'analyse rationnelle des faits et des indices.
J'ai jamais vu Michael Faber, Bubblelou, tu m'entends ? J'ai jamais touché à son vélo et je l'ai jamais poussé dans le fossé. Tu m'entends ? Mais Bubblelou ne m'entend pas plus qu'Emmy Saint-Pierre. Coincé entre le calorifère et le panier à linge, il fredonne Trois fois passera en scandant le rythme de la chanson avec ses petites pattes figées dans le plastique. Il n'est pas content, ça se comprend, et il va bouder jusqu'à ce que je m'excuse. On verra ça plus tard. En attendant, je dois rincer la crasse, toute la crasse, arracher le lichen et quitter la forêt. La dernière y res-te-ra, continue Bubblelou, la dernière... la dernière... puis il claque des mains au moment où mon corps de mousse croit atteindre l'orée de la forêt et je tombe, je tombe sur le carrelage glacé, pendant que la voix étouffée de Mike traverse un amas d'étoiles rouges, une nuée de naines clignotantes, pour me chuchoter à l'oreille ne crie pas, cours ! Dans le brouillard qui m'envahit, les mots qui me torturent depuis trente ans s'effacent, « ne plie pas le jour » s'embrume, pendant que Mike s'acharne à me répéter ne crie pas, cours, ne crie pas, cours... Et je comprends alors que Mike m'exhortait simplement à courir silencieusement loin du cercle infernal s'ouvrant près du bassin magique. Ne crie pas, cours ! Je comprends que la main que craignait Mike n'était pas celle de la vilaine petite Marnie.
Marchessault ne participait pas à l'opération Nuage. Quand la voiture s'est immobilisée devant ma galerie, j'en ai d'abord vu descendre Doyon, qui avait coiffé son crâne chauve d'une casquette à la Sherlock Holmes pour se protéger de la pluie et tentait maladroitement d'enfiler un imperméable dont une des manches était retournée vers l'intérieur. Pendant un instant, je me suis cru dans un film mettant en vedette l'inspecteur Clouseau. Je m'attendais à voir Marchessault extirper péniblement sa carcasse de la voiture en gueulant contre le mauvais temps, mais c'est la fille qui en est descendue, impeccable dans son imper ciré et ses bottes de cuir marron. Immédiatement, l'atmosphère s'est modifiée et le scénario a changé de registre. Clouseau a plié son imper sur son bras gauche et il a sorti une cage à chat du coffre, pour ensuite suivre la fille en évitant scrupuleusement les flaques d'eau accumulées dans la cour. Le punch final était raté. Posté à la fenêtre de ma chambre, j'ai attendu qu'ils frappent avant de descendre leur ouvrir. J'ai pris tout mon temps, conscient que ces moments de liberté étaient peut-être mes derniers pour une période indéterminée.
Où est le chat ? a demandé Doyon sans prendre la peine de me saluer. La dernière fois que j'avais aperçu ce vieux schnock, il lavait sa queue à grands coups de langue sur le sofa, mais il avait dû courir se cacher en entendant le moteur de la voiture ou les coups frappés contre la porte. Les chats n'aiment pas les flics.
Je me suis donc mis en quête de Nuage en l'appelant Dixie, Doyon sur les talons, qui ne s'est pas donné la peine non plus d'enlever ses bottes, ajoutant aux délicates pistes de Dixie d'énormes empreintes d'homme qui m'obligeraient à laver les planchers au grand complet si Dixie ne se montrait pas rapidement. L'inspectrice, quant à elle, demeurait près de la porte, d'où son œil avisé détaillait mon salon centimètre par centimètre, y cherchant probablement d'autres empreintes, d'autres indices dont un imbécile de ma trempe n'aurait pas soupçonné qu'ils puissent l'expédier en taule avant même qu'il ait le temps d'appeler sa mère.
J'ai enfin repéré le bout de la queue de Dixie dans la cuisine, qui s'entortillait autour d'un des pieds du buffet sous lequel il s'était réfugié. Ç'a été la croix et la bannière pour le sortir de là, mais j'y suis parvenu avec l'aide de Doyon, qui soulevait le buffet pendant que j'attrapais Dixie. On était de nouveau dans la catégorie Panthère rose, musique en moins, dont j'ai presque eu envie de fredonner un bout, histoire de dérider Doyon, mais celui-ci m'a ordonné d'enfermer Dixie dans la cage sitôt que j'ai pu attraper un autre bout de son anatomie que la queue. On rirait une autre fois. Dixie tout contre moi, j'ai d'abord eu droit à un coup de griffes bien assené qui a marqué mon bras gauche de deux longues entailles où perlait un sang frais, rouge, Billie, de la couleur des pommes à la fin août, puis à un regard assassin où s'imprimait en lettres majuscules une injure empruntée au dialecte des chats de ruelle. J'avais honte d'avoir ainsi piégé Dixie, mais je n'ai même pas essayé de m'excuser, sachant pertinemment qu'il me recracherait mes excuses au visage. Les chats n'aiment pas les traîtres.
J'ai dit à Doyon et à la fille de m'attendre pendant que je nettoyais mes égratignures. À mon retour, je les ai trouvés en train d'examiner des photos de Nuage pour les comparer à Dixie, qui gardait obstinément la tête enfouie entre ses pattes. J'aurais pu leur faciliter la tâche en leur certifiant que Dixie était Nuage, mais j'estimais avoir suffisamment coopéré jusqu'à maintenant.
Vous venez avec nous au poste, a décrété la fille en décrochant mon manteau de la patère. Je leur ai plutôt proposé qu'on s'installe dans la cuisine, les flics aiment les cuisines, mais la fille n'en appréciait ni le mobilier ni les couleurs, je suppose, car elle m'a lancé mon manteau en faisant signe à Doyon de s'occuper de la cage.
Trois quarts d'heure plus tard, j'étais de nouveau assis sur l'inconfortable chaise de métal qui s'empoussiérait depuis la veille dans la salle d'interrogatoire du poste de la SQ de Saint-Alban, à attendre l'arrivée de l'avocat que j'avais embauché quelques heures plus tôt, alors que j'arpentais mon salon sur l'air de Guantanamera. Préférant assurer mes arrières, j'avais sorti mon agent littéraire du lit avant l'aube pour qu'il me déniche illico un avocat de droit criminel qui aimait le grand air. Au bout de quarante-cinq petites minutes, Béchard me rappelait pour m'annoncer qu'un certain maître Jean-Pierre Maheux acceptait de s'occuper de mon cas moyennant des honoraires que personne n'oserait prononcer à voix haute. J'ai rapidement conclu qu'un bandit serait sûrement plus qualifié pour me défendre qu'un homme honnête et j'ai demandé ses coordonnées à Béchard. Je l'appellerais directement, sans intermédiaire qui gonflerait ma facture de dix pour cent. Après que je lui eus expliqué la situation, Maheux m'a confirmé qu'il serait à Saint-Alban avant midi et j'ai pu me remettre à respirer.
Je l'attendais donc seul dans la salle d'interrogatoire pendant que Doyon et l'inspectrice vaquaient à leurs occupations. Question de galanterie ou de machisme, tout dépendant du nom que vous donnez au savoir-vivre dont essaient de faire preuve certains attardés, je n'arrivais pas à évoquer Émilie Saint-Pierre à l'aide de son simple patronyme, alors que j'appelais Doyon Doyon sans me fendre la tête en quatre. Quoi qu'il en soit, le nom d'Émilie Saint-Pierre ne cadrait pas du tout avec ses fonctions et aurait mieux collé à une actrice de séries télévisées pour enfants. Mon impression venait probablement de ce que l'inconscient crée ses propres clichés, mais un nom comme Jane Adamsberg ou Harriet Bosch lui aurait mieux convenu, de même que Doyon aurait été plus crédible en Marlowe. Émilie était un prénom trop doux, trop candide pour qu'on l'emmène frayer dans les arènes du crime. À son allure, j'étais certain que la petite Émilie avait dû jouer du coude et mordre une couple de fanfarons avant que ses coéquipiers la prennent au sérieux. Si elle avait grimpé aussi rapidement dans la hiérarchie, c'est qu'elle savait se battre et donner quelques coups en bas de la ceinture s'il le fallait, histoire d'écraser les couillons qui se plaçaient en travers de son chemin. N'empêche, je l'imaginais plus facilement en Blanche-Neige.
Étant donné qu'on avait refusé de me procurer de la lecture pour tuer le temps, de crainte que j'essaie de m'étouffer en bouffant du papier journal ou que je m'ouvre les veines avec les agrafes d'une brochure sur l'éradication des gangs de rue, j'inventais des scénarios sur la vie d'Émilie Saint-Pierre, auxquels je mêlais Marchessault et Doyon, leur accordant alternativement le bon et le mauvais rôle, de préférence le mauvais. Mon animosité envers Marchessault et Doyon n'avait rien de personnel. Elle tenait à nos positions, à nos rôles respectifs. On n'était pas du même côté de la clôture, alors je tirais à vue dès que je voyais l'un d'eux s'approcher de la frontière qui nous séparait. J'aurais pu profiter de cette attente pour préparer ma défense au lieu de fabuler sur d'honnêtes travailleurs, mais le compteur de mon avocat étant déjà en marche, ça me semblait une dépense d'énergie inutile. J'aurais aussi pu essayer de réfléchir à la façon dont Michael Faber s'était volatilisé, au cas où une idée géniale m'aurait permis de créer un lien entre la disparition du petit et l'apparition de Dixie, car j'espérais de toute mon âme qu'on retrouve ce garçon, mais me le représenter au fond d'une grange ou d'un silo à grains était au-dessus de mes forces. J'avais déjà arpenté avec Billie la galerie des horreurs où pendouillaient des tableaux plus atroces les uns que les autres et je ne tenais pas à me taper une nouvelle visite guidée. Ce fardeau revenait maintenant aux parents de Michael Faber, et même si je ne souhaitais pas ça à mon pire ennemi, rien ne pourrait les empêcher de pousser la porte de ce musée sordide. S'ils n'en ressortaient pas complètement fous, c'est qu'ils auraient instinctivement relégué les pires images dans une zone de leur cerveau se fermant hermétiquement à l'approche de la folie.
Émilie Saint-Pierre, coiffée d'un voile de fée, poursuivait un revendeur de drogue à cent cinquante à l'heure sur une autoroute enneigée, accompagnée d'un Marchessault suant plus vrai que nature, quand la porte de la salle d'interrogatoire s'est enfin ouverte. Émilie Saint-Pierre a pénétré dans la pièce, visiblement peu affectée par l'épreuve à laquelle je l'avais soumise, suivie de Doyon et d'un petit homme en complet-veston qui ne pouvait être que Jean-Pierre Maheux, mon avocat.
Compte tenu du tarif demandé, j'aurais apprécié que Béchard m'en déniche un plus grand, que j'aie au moins le sentiment d'en obtenir pour mon argent. Maheux ne souffrait pas de nanisme, mais de mon point de vue d'homme assis, il me paraissait suffisamment court pour traverser la pièce en passant sous la table sans se frapper la tête. Cette particularité physique me contrariait, car qui peut prendre au sérieux quelqu'un qui se marche sur la cravate et doit demander à ses clients de rester assis s'il désire leur parler face à face. Or il s'avéra rapidement que maître Maheux connaissait son métier et compensait sa petite taille, à la suite d'Astérix et de Napoléon, par une férocité n'ayant d'égale que sa perspicacité.
Après les formalités d'usage, Émilie Saint-Pierre nous a annoncé que les parents de Michael Faber avaient identifié le chat que j'avais prétendument séquestré et que ce dernier était bel et bien Nuage. Expliquez-nous alors de quelle manière ce chat a abouti chez vous, monsieur Richard.
J'ai répondu que je n'en avais aucune idée, que ce chat était apparu sur ma galerie par un bel après-midi d'avril, que je l'avais nourri et qu'il était revenu. C'était la stricte vérité, mais ni Doyon ni Saint-Pierre ne semblaient disposés à me croire.
Quel bel après-midi d'avril ? s'est enquis Doyon en mâchouillant un crayon de bois qui avait déjà subi l'assaut d'autres dents que j'espérais être les siennes, question d'hygiène.
Le 12, ai-je répondu sans avoir à réfléchir. Je m'en souvenais parce que c'était le jour de Pâques, que j'avais un peu plus tôt parlé à Régine et que j'étais bouleversé. Après le coup de fil, j'étais sorti marcher, et à mon retour, le chat m'attendait.
J'aurais donné une gifle à Doyon qu'il n'aurait pas rougi davantage. Quant à Émilie Saint-Pierre, elle a échappé son stylo et a fait déferler sur moi les vagues de ses yeux océaniques, cherchant manifestement à m'engloutir dans une lame de fond. Monsieur Richard, ce chat a disparu en même temps que le gamin, soit le 18 avril, a-t-elle précisé en insistant sur la date, alors soit vous vous trompez, soit vous nous prenez pour des cons.
Je ne me trompais pas, je ne les considérais pas comme des imbéciles, même si j'aimais bien rester dans ce champ syntagmatique quand je pensais à eux, et ne voyais qu'une solution au problème : contre toute apparence, Dixie n'était pas Nuage.
Lorsque j'ai lâché cette vérité flagrante, Doyon a répété ce que Saint-Pierre venait à peine de dire, à savoir que je les prenais réellement pour des cons. Un lourd silence s'est installé, puis Émilie Saint-Pierre a repris le contrôle de l'interrogatoire. Ses questions se succédaient, claires, précises, directes, mais chaque fois que je m'apprêtais à ouvrir la bouche, Maheux m'intimait de me taire ou prétendait carrément que l'inspectrice dépassait les bornes. N'ajoutez pas un mot, Richard. Mon client n'a pas à répondre à de telles questions. Vous êtes hors sujet, inspectrice, et autres formules du genre qui me donnaient carrément l'air coupable, à l'image de tous ceux qui se font représenter par un avocat, mais Maheux avait raison, si j'avais tenté de répondre à ces questions absurdes, je me serais forcément contredit parce que les allégations d'Émilie Saint-Pierre n'avaient pas de sens.
Quand les deux aiguilles de l'horloge murale se sont arrêtées sur le 12 en produisant un léger clic qui m'a fait sursauter, Maheux a ramassé ses paperasses et a décrété que cet interrogatoire était terminé.
J'avais révélé l'existence de Dixie en vue d'aider Michael Faber et, au total, je n'étais parvenu qu'à mélanger les cartes pour me ramasser dans le rôle du deux de pique, de l'abruti qui glisse sur la peau de banane dont il n'avait pas prévu la présence sur la pente lisse où il croyait s'engager. J'espérais sortir de cette pièce le cœur content, l'esprit allégé, l'âme en paix avec le monde, mais Doyon et Émilie Saint-Pierre m'avaient balancé une poche de deux cents kilos sur les épaules. Avant de franchir le seuil de la porte, je me suis retourné vers eux pour leur rappeler que dans ce cas précis, j'étais de leur côté, que je désirais autant qu'eux voir surgir Michael Faber des bois sain et sauf, que son foutu chat avait peut-être un don d'ubiquité et que j'étais tout sauf coupable. En m'entendant prononcer le mot « coupable », Maheux a tenté de m'arrêter, mais j'étais lancé. J'étais prêt à me soumettre à tous leurs interrogatoires ridicules pour sauver cet enfant et je voulais qu'ils le sachent.
Alors dites-nous la vérité, monsieur Richard, a rétorqué Émilie Saint-Pierre, puis une lame de fond m'a frappé et j'ai dû m'appuyer au chambranle.
On m'avait déjà servi une réplique semblable il y avait longtemps. Un autre flic, les mains posées sur ma table, m'avait déjà accusé de mentir. L'histoire se répétait, aussi incompréhensible que dans sa première version. J'avais perdu ma fille, ma toute petite fille aux yeux noisette, et j'étais du coup devenu un monstre. Je vais vous reconduire chez vous, a murmuré Maheux en me prenant le bras, mais j'ai répondu que j'avais besoin d'air et je l'ai planté là, avec son compteur qui tournait.
Lorsque j'ai rouvert les yeux, mon environnement n'était qu'un amas de couleurs, de formes et d'angles auxquels je ne comprenais rien. Si j'avais eu cette capacité, j'aurais pu croire que j'étais morte, mais les notions de vie, de mort et de survie se situaient au-delà de mon entendement. Je ne ressentais strictement rien et mon cerveau était vide de toute pensée. Je n'étais que deux yeux sans conscience effleurant des objets sans nom, car il n'y avait rien derrière les yeux, aucune âme, aucun passé, aucune mémoire susceptible de nommer la réalité. Ce qui regardait ne connaissait ni le mot « je » ni son sens. Ce qui regardait s'ignorait. Puis, peu à peu, des mots exprimant mon appartenance au monde ont refait surface et je me suis souvenue. J'étais une femme, je m'appelais Marnie, j'étais tombée. Alors les images ont reflué, Bubblelou, la mousse, Michael, l'eau brûlante et le chien Krypto. Ne crie pas, cours ! Puis j'ai senti le froid. Mes pieds gelés ont remué, une douleur lancinante m'a fendu le crâne et je me suis glissée jusqu'au mur pour saisir la serviette suspendue à un crochet. Luttant contre la douleur, j'ai réussi à m'asseoir et à m'envelopper dans la serviette, la moitié des fesses sur le plancher mouillé.
Je tremblais comme une enfant qu'on a arrachée des eaux d'un fleuve glacé, quelques larmes coulaient sur mes joues, mais j'ignore si je pleurais parce que j'avais échappé de peu à une mort stupide ou parce que j'étais encore vivante. J'imagine que ceux qui se réveillent après avoir sauté d'un pont ou d'un quatrième étage doivent ressentir la même chose, une profonde compréhension du vide, soudain, doublée du sentiment que c'est foutu, qu'il faut maintenant continuer à vivre, recommencer à laver ses vêtements, à mettre ses chaussures, à sortir le chien et à répondre au téléphone.
Si ma tête avait heurté le bain, je serais peut-être encore là, raide morte sur le carrelage froid, couverte de mousse séchée, ma peau rougie par l'épanchement du sang sur tous les points en contact avec le sol. On m'aurait découverte ainsi, les yeux exorbités, la main tendue vers Bubblelou, ou alors dans la calme attitude des morts n'ayant rien vu venir, soudain frappés de noirceur en plein milieu d'une phrase.
J'ai frémi à l'idée que la mort puisse être si subite et imprévue. Il suffisait d'une surface mouillée, d'un tapis qui béait en haut d'un escalier, d'un moment d'inattention pendant que le souvenir d'un jour heureux vous distrayait, et soudain, vous n'étiez plus là. Votre vie s'effaçait d'un coup et il ne restait qu'un corps muet pour en témoigner. De ce point de vue, la survie tenait du miracle. Les assassins connaissaient cette fragilité. Peut-être même ne tuaient-ils que pour l'éprouver. Un coup de marteau et hop, la victime n'existait plus qu'en tant que victime au crâne réduit en bouillie. À ma manière, j'avais aussi expérimenté la précarité du vivant quand, avec Michael, je piétinais le sable d'où s'envolaient les sauterelles. Sous le pied, l'insecte produisait un craquement sec, semblable à un bruit d'os broyés, et hop, la bestiole n'existait plus qu'en tant que bestiole réduite en bouillie.
Je n'étais pas une criminelle pour autant. Mike me l'avait dit avant que je m'endorme sur le carrelage glacé, Mike me l'avait chuchoté, ne crie pas, cours, pendant qu'il balançait les bras pour faire diversion, attirer loin de moi l'ennemi, pareil à un oiseau feignant d'être blessé pour protéger ses petits. Mike n'avait été frappé d'aucune forme de folie durant l'orage, d'aucune espèce de stupeur ayant atteint sa lucidité. Il avait vu ce que les arbres m'empêchaient de percevoir, puis il avait étendu au sol son aile cassée afin de s'offrir en pâture à la créature, homme ou animal, qu'éclairaient les reflets de l'orage. L'espace d'un instant, il était devenu oiseau et avait transformé sa danse d'animal meurtri en un rituel sacrificiel n'ayant d'autre but que d'éloigner de moi le mal qui s'avançait dans la forêt. Tous les éléments du casse-tête que je tentais de reconstituer depuis des années s'étaient mis en place durant le bref moment qui avait précédé mon évanouissement, tous, sauf un, celui sur lequel s'imprimait le visage de la créature, homme ou animal, qui avait poussé Mike à mimer l'effarement des fous sous la pluie drue pour sauver la peau de la petite Marnie.
Devais-je me réjouir de cette révélation ou, au contraire, accuser de stupidité l'enfant du conte, la fillette en salopette de garçon qui n'avait pas su reconnaître la feinte de l'oiseau sous la pâleur de son ami ? Mais la question n'avait plus lieu d'être posée, car il était trop tard pour les examens de conscience. L'enfant du conte ne pouvait en réinventer l'intrigue. Seul importait désormais le fait que l'autre enfant avait appelé le loup et l'avait entraîné dans la forêt afin que sa fureur épargne la fillette.
J'ai tenté de me relever, mais une brûlure mordante a irradié ma hanche gauche, m'obligeant à changer de position, à coller mes genoux contre mon torse et à basculer sur le côté. J'ai attendu que la brûlure s'apaise en songeant que Mike m'avait sauvé la vie avec quelques mots qui auraient constitué mon arrêt de mort si je les avais entendus. Si la peur ne m'avait en effet paralysée et si j'avais compris qu'il fallait courir, j'aurais tenté d'entraîner Mike avec moi, c'est ce que font les enfants, c'est ce que font les meilleurs amis, qui se tiennent par la main à l'approche de la noirceur, et la chose nous aurait rattrapés tous les deux, près du lac aux Barbotes ou ailleurs, cette chose qui allait désorienter le soleil de Rivière-aux-Trembles. Grâce à Mike, j'étais vivante, bien vivante, et la douleur, encore, me contraignait à me mouvoir. J'ai agrippé le bord de la baignoire et, en m'appuyant au mur, dont les carreaux de céramique colorée vibraient sous l'intensité de la lumière, je suis parvenue à me relever.
J'ai avalé deux Atasol sans oser me regarder dans le miroir et me suis traînée jusqu'à mon lit. Malgré la lourdeur de mes couvertures d'hiver, j'ai mis longtemps à me réchauffer et me suis endormie d'épuisement à l'instant où Michael, près du lac aux Barbotes, perdait son running bleu sous le ciel lavé par l'orage. Derrière la colline des Loups, se couchait le dernier soleil. Je t'entends, Mike, je t'entends enfin.
Mon mal de tête m'a réveillée quelques heures plus tard. Derrière l'église, se levait une aube sombre, traversée d'une lumière étirant la nuit dans le bleu foncé des nuages. En tâtant ma nuque, j'ai compris à quoi étaient dus les battements sourds qui résonnaient entre mes deux oreilles. Une énorme protubérance, couverte d'une croûte de sang séché, avait poussé durant la nuit derrière mon crâne. Une douche s'imposait, que j'ai prise assise au fond de la baignoire. Je finissais de sécher mes cheveux quand le téléphone a sonné. J'ai tout de suite pensé qu'il devait s'agir d'Émilie Saint-Pierre, qui m'avait entendue hurler les mots prononcés par son frère. Au déclenchement du vieux répondeur de mon père, j'ai tendu l'oreille. La voix provenant du salon était une voix d'homme, difficile à saisir, qui chuchotait et retenait son souffle. Absorbée par les murs, les tapis et les meubles, elle n'était qu'une forme de chuintement lointain, pareille au murmure d'une colonie d'insectes rongeant de l'intérieur le bois d'une porte.
Au signal indiquant la fin du message, je suis descendue au salon, une serviette enroulée autour du corps. J'ai fait défiler le ruban du répondeur en marche arrière et j'ai appuyé sur la touche « Play ». Michael Saint-Pierre, c'était pas assez pour toi, hein ? a chuchoté la voix rauque. Y t'en fallait un autre. Un autre enfant innocent. Mais ça va s'arrêter là. Si la police t'enferme pas, on va se charger de ton cas tout seuls. On veut plus de toi à Rivière-aux-Trembles, Marnie Duchamp !
Appuyée sur le dossier d'un fauteuil, je me sentais défaillir à mesure que les paroles défilaient, davantage crachées que chuchotées. Cet homme n'avait rien compris, rien de rien. Il ignorait que mon seul crime était de n'avoir pas entendu Mike. Replié dans cette ignorance crasse, il ne pouvait mesurer le poids de mon innocence. T'as rien compris, ai-je murmuré, t'as rien saisi, pauvre con, puis le signal indiquant la fin du message a résonné, éveillant les sauterelles écrasées, méchante, vilaine petite Marnie, dont le chant s'est mêlé au grattement des insectes xylophages, capricornes ou termites dont les mandibules creusaient et rongeaient le bois de la porte se refermant sur moi.
Appelle Marchessault, Marn, appelle tout de suite, laisse pas les commérages briser ta petite maudite vie une deuxième fois, m'intimait la voix, en moi, qui avait remplacé celle de l'homme rauque et se glissait parmi les grignotements étouffés des capricornes. Appelle avant de virer folle ! Je saisissais le combiné du téléphone quand, par la grande fenêtre du salon, j'ai vu une ambulance passer en trombe devant la maison, suivie des aboiements des chiens du deuxième voisin. J'ai subitement été ramenée en arrière, dans la forêt chauffée de soleil d'où montaient les jappements de Rex, Chet et Lucy, une série d'appels excités m'indiquant qu'ils cherchaient un enfant et qu'ils traquaient son agresseur : assassin, ravisseur ou sorcière. Les membres engourdis, je me suis dirigée vers la fenêtre de la cuisine, certaine qu'un groupe d'hommes armés de bâtons et de fusils préparaient mon bûcher dans le jardin. Mais la cour était vide, les hommes étaient cachés, n'attendant qu'un faux pas de ma part pour surgir de l'intérieur des serres ou de derrière les arbres.
Au loin, en direction de la rivière, les chiens hurlaient toujours. Rex, Chet et Lucy. J'ai aspergé mon visage d'eau froide et suis retournée m'asseoir au salon. L'horloge marquait neuf heures dix-huit quand un silence de mort s'est abattu sur la maison et que l'ambulance est repassée en sens inverse. On venait de découvrir le corps de Michael Faber.
Après avoir largué Maheux au poste de police, j'ai erré sans but, me demandant ce que je foutais dans cette ville morne où les passants me dévisageaient comme si j'avais la rage. Je me suis arrêté devant une vitrine pour examiner ma gueule, voir si j'avais les cheveux hirsutes et si un filet de bave me coulait sur le menton. J'avais une mine normale, me semblait-il, si tant est qu'un homme déambulant au rythme lent de qui ne sait plus où il va puisse se fondre dans l'agitation du midi sans soulever un soupçon d'inquiétude. C'est ce que devaient percevoir les gens, cette absence de but et de foi vidant le regard. J'avais pourtant un but, oublier toute cette histoire. Mais comment oublier ce qui imprègne chaque particule de l'air que vous respirez ? Si j'avais croisé un pont suffisamment haut à ce moment-là, je me serais jeté en bas, je crois, puisque je ne voyais pas d'autre moyen d'oublier que d'interrompre la chute libre de mes pensées dans un grandiose dispersement de matière cervicale. Bras écartés, j'aurais frappé le vent. La surface houleuse serpentant sous le pont serait devenue ma destination et on aurait pu voir la sérénité d'une foi brève sur mon visage. Au-delà du paisible miroitement dont s'entoure la mort, c'est probablement ce qui avait poussé L.A. à sortir son couteau de cuisine, la foi, la certitude de rencontrer Billie dans le long couloir menant au paradis.
Immobilisé au croisement des deux principales artères de Saint-Alban, je songeais que la mort était la seule vérité qui vaille, la seule infaillible certitude, quand une voiture de police s'est arrêtée au feu rouge, avec à son bord une femme qui semblait également effondrée dans une absence d'espoir, fixant les passants indifférenciés qui se fondaient dans l'anonymat d'un décor dont le pourquoi lui échappait. Puis un coup de klaxon l'a fait sursauter et elle a tourné la tête dans ma direction. Je connaissais ce visage, cet ovale aux yeux rougis figé dans la pâleur des statues. Il appartenait à la femme du convoi funéraire qui m'avait accueilli à Rivière-aux-Trembles, à la veuve ou à l'orpheline que la vie traînait contre son gré, de même qu'il appartenait à toutes les femmes vêtues de noir qui se postaient à l'aube devant l'horizon grugé par la mer ou les montagnes. Depuis le défilement du convoi, rien n'avait changé dans l'allure de cette femme, ni la peur, ni la tristesse, ni l'incompréhension qui l'écrasaient.
J'ai machinalement levé la main, ainsi qu'on le fait pour saluer une amie, puis le voile qui couvrait son regard a disparu un moment. Le feu tournait au vert quand elle a plaqué une main contre la vitre de la voiture et s'est brusquement retournée, me reconnaissant aussi, puis elle a prononcé un mot, le même, à plusieurs reprises, cours, disaient ses lèvres, cours, puis son visage a été emporté par le flux de la circulation. Alors j'ai couru, je me suis stupidement précipité derrière la voiture, ignorant pourquoi je courais et si j'allais dans la bonne direction, puis me suis arrêté, à bout de souffle, tandis que l'auto s'engageait dans une rue qui cherchait à se donner des allures de boulevard.
Mon cœur battait encore contre ma poitrine lorsque je suis entré dans le McDonald's dont la gigantesque enseigne dominait les édifices s'alignant de ce côté de la rue. Je n'avais pas faim, mais j'ai commandé un chausson avec mon café pour rassurer la caissière, qui devait penser que je souffrais d'asthme chronique ou venais d'échapper à une bande de poursuivants armés. Pendant qu'elle versait mon café, j'ai parcouru des yeux la salle à demi vide. Curieusement, il y avait très peu d'enfants en train de colorier des Ronald ou de s'emplir la bouche de frites. La clientèle était majoritairement composée de travailleurs qui avalaient distraitement leur Big Mac ou leur Quart de livre, de retraités qui feuilletaient le journal et discutaient de leurs problèmes de prostate au-dessus de plateaux remplis de papiers froissés et de verres jetables.
J'ai choisi une table isolée, près d'une fenêtre sur laquelle quelqu'un avait dessiné un bonhomme sourire avec du ketchup. Pas besoin de savoir dessiner pour reproduire un bonhomme sourire, mais celui-là avait du style, un peu comme si on l'avait peint avec une spatule et de l'acrylique. La présence de cette tête sans pattes avait probablement pour but de faire sourire ceux qui s'assoyaient à cette table, mais il était inutile d'espérer de ma part un semblant de joie alors que le destin avait remis les compteurs à zéro et recommencerait chaque fois qu'un malheur frapperait dans un rayon de trente kilomètres autour de moi. Il en serait toujours ainsi. Au moment où je croirais avoir une chance de m'en sortir, le destin incurverait le temps et toutes les horloges, montres et cadrans de la planète marqueraient quinze heures trente, pendant que les pages des calendriers se replieraient à toute vitesse jusqu'au 20 janvier 2006. Il en serait toujours ainsi. La disparition de l'amour de ma vie m'avait marqué au fer rouge. Nulle part on ne me pardonnerait cette disparition.
Tout en versant quelques sachets de sucre dans mon café, j'ai observé la bruine, intarissable depuis la veille, en essayant de chasser l'image de l'infinie spirale qui s'ouvrait devant moi, puis je me suis demandé ce qui m'avait pris de courir ainsi après une inconnue. La folie, peut-être, la violence qui bouleversait de nouveau ma vie et m'incitait à m'accrocher au moindre signe de sympathie ou de reconnaissance. Je ne savais rien de cette femme, sinon qu'elle était pâle, mais j'avais le sentiment qu'elle et moi étions dans le même bateau, ballottés par le courant, sans rames pour redresser l'embarcation qui tourbillonnait à l'approche des rapides. D'ailleurs, que faisait-elle dans cette voiture de police ? L'avait-on interceptée alors qu'elle s'apprêtait à lancer sa propre voiture contre un bloc de béton, une photo de sa famille collée sur le tableau de bord ? Avait-elle troublé l'ordre public, fracassé une vitrine, volé un stock de romans d'amour pour se prouver qu'elle était encore vivante ? Elle n'avait pourtant la gueule ni d'une voleuse ni d'une hystérique. Je l'aurais plutôt rangée dans la catégorie des filles capables de s'enfiler un vingt-six onces de gros gin pour être certaine de n'avoir plus la force de se jeter par la fenêtre. Une fille qui avait toutes les raisons de sauter par-dessus bord mais qui s'accrochait à l'idée que la mer était belle.
Je retournais toutes ces hypothèses dans ma tête, inutilement, car mon instinct me soufflait que cette femme avait été arrêtée pour les mêmes motifs que moi. Je n'avais pas prêté attention aux flics qui l'accompagnaient, mais j'aurais mis ma main au feu que l'un des deux se nommait Doyon ou Marchessault. L'autre suspecte, c'était elle. Sa fille ou son gamin avait dû s'égarer dans les bois, des années auparavant, sans laisser de traces, sans qu'on réussisse à épingler l'ordure à l'origine du drame, si bien qu'on avait décidé de donner un nom à cette ordure, le sien, pour ne pas demeurer dans le noir. On lui avait posé une étiquette sur le front, tenir éloignée des enfants, on l'avait montrée du doigt, lui enlevant ainsi la possibilité de pleurer son petit tranquille, et on la soupçonnait chaque fois qu'un gamin perdait la notion du temps et rentrait à la maison plus tard que prévu, la tête pleine d'exploits qu'il renonçait à raconter devant l'atmosphère chargée de reproches pesant sur la cuisine où le souper refroidi séchait dans les assiettes. À peu de choses près, son histoire devait ressembler à ça, c'est-à-dire à la mienne.
J'ai jeté mon café et mon chausson à la poubelle, au diable les scrupules, et me suis dirigé vers la sortie. À une table pas loin de la porte, un jeune père s'empiffrait en compagnie de sa fillette, six ou sept ans, une des deux seules gamines à illuminer ce repaire de vieux. Pendant que le père reluquait une fille qui passait sur le trottoir, l'enfant terminait un casse-tête réunissant la plupart des animaux de la ferme, de la poule au cochon en passant par le bœuf dont le père mastiquait un des rejetons. À côté du casse-tête, la une du tabloïd de la région reproduisait en gros plan le visage de Michael Faber, souriant de son grand sourire d'enfant content. J'ai eu envie d'attraper le père par la chemise pour lui crier au visage de ne jamais quitter sa fille des yeux, jamais, mais je me suis contenté de lui demander poliment si je pouvais lui emprunter le journal. La bouche pleine de hamburger à demi mastiqué, il m'a fait signe que oui, et j'ai plié le journal dans la poche de mon manteau. On ajouterait ce larcin aux autres crimes qu'on voulait me coller sur le dos.
Incertain quant à la façon dont j'occuperais les prochaines heures, j'ai fait les cent pas devant le McDo, pour enfin pointer mes chaussures en direction du poste de police. Si je voulais porter secours à l'enfant qui souriait au fond de ma poche, je ne pourrais y arriver qu'aux côtés de Doyon, de Marchessault et d'Émilie Saint-Pierre, et tant pis s'ils me passaient les menotes et me coupaient le sifflet du même coup. Avec un peu de chance, ma voix parviendrait peut-être à franchir les barreaux de la prison, malgré leur étanchéité notoire aux cris des hommes entravés. Avec un peu de chance... De toute façon, si on avait arrêté la femme de la procession funèbre, c'est qu'il y avait du nouveau dans l'affaire Michael Faber. Inutile de me payer un taxi jusqu'à Rivière-aux-Trembles pour effectuer immédiatement le trajet inverse à bord de la voiture aux gyrophares éteints qui m'attendrait dans ma cour. Autant en finir tout de suite et épargner aux contribuables le paiement de quelques litres d'essence supplémentaires.
J'ai demandé à parler à Marchessault ou à Doyon. Je ne me fiais pas à l'agressivité contenue d'Émilie Saint-Pierre, qui pouvait vous exploser au visage si vous aviez le malheur d'appuyer sur le détonateur qu'elle cachait sous sa chemise boutonnée aux trois quarts. Marchessault et Doyon présentaient l'avantage d'être clairs et directs, sans nuances. Si vous leur tapiez sur les nerfs, la réaction était immédiate. Vous receviez leur mauvaise humeur en pleine gueule et compreniez que vous n'aviez qu'une chance sur deux de garder vos couilles intactes si vous vous entêtiez à faire le clown.
Occupés, m'a répondu l'agent Tremblay, qui classait des formulaires derrière le comptoir de la réception, puis il m'a désigné la rangée de chaises appuyées contre le mur. Je pouvais les attendre là ou revenir un peu plus tard. J'ai choisi la rangée de chaises et j'ai déplié le journal que j'avais piqué au McDo. Un des deux articles consacrés à la disparition de Michael Faber décrivait les habitudes du gamin et rapportait les commentaires des parents éplorés, des voisins, des camarades de classe de l'enfant, de sa maîtresse d'école, de l'épicier du coin, du gars qui distribuait les publisacs et de la cousine par alliance de ce dernier. S'il avait eu l'occasion d'interviewer le chat du petit, le pseudo-journaliste qui signait l'article aurait sûrement trouvé le moyen de lui attribuer deux ou trois conneries.
En prime, le lecteur avait droit à une photo de la bicyclette abandonnée, soulignée d'une vignette dégoulinant de fausse compassion. Au souvenir de la bande de charognards qui s'étaient emparés des photos de Billie afin d'augmenter le tirage de leur feuille de chou et nous avaient harcelés, L.A. et moi, pour qu'on verse des larmes dans leur micro et qu'on se mouche dans leur torchon, j'ai eu envie de foutre le feu à ce ramassis de bêtises ou de le déchirer en soixante-quinze mille morceaux destinés à la cuvette la plus merdique des toilettes pour hommes, dont la porte, agrémentée de la représentation asexuée d'un bonhomme qui n'avait jamais pissé de sa vie, s'ouvrait au bout du corridor prolongeant la rangée de chaises. Mais je voulais lire l'autre article, consacré celui-là à l'enquête policière.
On y citait d'abord le sergent Gilles Marchessault, bien connu à Saint-Alban, qui prétendait suivre quelques pistes prometteuses, puis on passait ensuite la parole au lieutenant Émilie Saint-Pierre, originaire du comté et spécialement affectée à cette affaire depuis les bureaux régionaux de la SQ. Selon le lieutenant Saint-Pierre, la police tenait déjà deux suspects dans sa ligne de mire et comptait clore ce dossier très rapidement. On avait encore espoir, ajoutait-elle, de retrouver Michael Faber sain et sauf. Elle lançait à cet effet un appel à toute la population pour prêter main-forte à la police et lui révéler le moindre détail pouvant avoir un lien avec cette disparition.
Fidèle à sa nature de fossoyeur, le journaliste déballait ensuite tout un pan de la vie privée d'Émilie Saint-Pierre pour apprendre au lecteur que le grand frère du lieutenant avait disparu une trentaine d'années plus tôt et qu'elle était considérée parmi les forces policières comme l'enquêtrice la plus féroce lorsqu'il s'agissait de coincer des agresseurs d'enfants. Et voilà, tout s'expliquait. L'attitude d'Émilie Saint-Pierre plongeait ses racines dans une enfance traversée de colère et de pleurs. On vous enlevait votre enfance et c'était foutu. Vous cherchiez l'enfant assassiné d'un coup de matraque le reste de votre vie. C'est à cette quête qu'Émilie Saint-Pierre se consacrait. Elle cherchait qui l'avait assassinée.
Je pouvais maintenant déchirer le journal. Il ne m'apprendrait plus rien. Je l'ai emporté avec moi aux toilettes, mais je me suis contenté de le jeter dans une poubelle. Un bruit de métal creux a résonné dans la pièce étroite, rappelant le son définitif et morbide de la première cloche annonçant le glas, et j'ai récupéré le journal en prenant soin de ne pas toucher aux parois gommeuses de la poubelle. Je ne pouvais pas mettre aux ordures la photo d'un enfant qui pleurait peut-être, en ce moment même, pour que son père vienne le délivrer. J'ai délicatement déchiré la première page de L'Écho de Saint-Alban, que j'ai pliée en quatre pour la ranger dans ma poche. Elle irait avec les photos de Billie, avec toutes ces photos qui tachaient les doigts et dont je n'arrivais pas à me débarrasser, parce qu'on ne peut tout simplement pas enfouir le visage d'un enfant parmi des objets sans mémoire, tachés de boue, de sueur ou de café refroidi.
Mon visage, au contraire, se serait senti tel un poisson dans l'eau au milieu des mouchoirs sales. J'avais le teint gris, les yeux cernés, les deux coupures que je m'étais faites en me rasant le matin avaient saigné et j'aurais eu besoin d'un deuxième rasage avec des lames non émoussées. Il n'y avait qu'une trentaine d'heures que tout ce cirque avait démarré et j'avais l'impression qu'il s'était écoulé une semaine, une longue semaine durant laquelle je n'avais ni dormi ni mangé. Je me suis passé le visage à l'eau froide, je me suis lissé les cheveux et je suis retourné sur ma rangée de chaises.
À la réception, un homme à qui j'aurais donné dans les soixante-dix ans, mais qui semblait taillé pour atteindre son centenaire en fendant son bois de chauffage, engueulait l'agent Tremblay et menaçait d'intenter un procès à la police pour abus de pouvoir et procédés douteux. Tremblay demeurait calme, il en avait vu d'autres, et répétait à l'homme ce qu'il m'avait dit plus tôt, à savoir qu'il pouvait lire son journal et attendre tranquillement qu'un des inspecteurs chargés de l'enquête qui l'intéressait se libère, ou revenir plus tard. Conscient qu'il ne tirerait rien de Tremblay, l'homme a pris la même décision que moi, il a choisi à contrecœur la rangée de chaises. Il a tapé du pied quelques instants puis il s'est relevé pour faire les cent pas de la porte d'entrée à celle des toilettes pour hommes.
La colère de cet homme était à ce point palpable qu'elle créait autour de lui une espèce de champ magnétique sur lequel devaient rebondir tous ceux qui s'approchaient de lui sans y avoir été invités. Il se déplaçait à l'intérieur d'une aura minérale qui lui donnait une certaine ressemblance avec Clint Eastwood dans ses mauvais jours, et je n'aurais pas été étonné de le voir dégainer si quelqu'un lui avait tapé dans le dos. Je plaignais le gars ou la fille qui devrait négocier avec lui. Après trois allers et retours d'un bout à l'autre du corridor, il est revenu s'asseoir et m'a demandé si j'avais des cigarettes. Négatif. Il ne sentait pas le tabac et ne fumait probablement plus depuis des années, mais la situation qui lui donnait envie de casser l'agent Tremblay en deux avait réveillé le goût de la nicotine et il s'en serait bien fumé deux ou trois de suite si j'avais eu de quoi tenter le diable. Il s'est planté une allumette de bois entre les lèvres, je me suis poussé un peu pour ne pas être happé par l'attraction de son champ magnétique, et le silence s'est installé, perturbé ici et là par le bruit métallique des tiroirs de classeurs qu'ouvrait et refermait Tremblay derrière son comptoir.
Une quinzaine de minutes plus tard, une porte s'est ouverte, une autre a claqué, peut-être la même, et Émilie Saint-Pierre a descendu le corridor en compagnie de la femme du convoi funèbre, qui avait le visage de ceux qui viennent de se taper une procession de revenants. L'homme assis à ma droite s'est levé d'un bond pour se précipiter vers elle en criant Marnie, bon sang, veux-tu m'expliquer ce qui se passe ? J'ai tout de suite pensé que Marnie était un prénom qui convenait parfaitement à cette femme, un prénom qui avait perdu son espièglerie quand le monde lui avait dégringolé dessus. Phil, a-t-elle murmuré, mais Émilie Saint-Pierre s'est interposée entre elle et l'homme nommé Phil. Vous n'avez pas le droit de lui parler, monsieur Morisset, Marnie vient d'être placée en garde à vue, a-t-elle dit en ouvrant la porte d'une pièce sans fenêtre où elle a ordonné à la femme d'entrer.
C'est quoi ces maudites histoires de fous-là, Emmy ? a tonné Phil Morisset.
C'est confidentiel, monsieur Morisset, je peux rien vous dire.
Pendant un instant, j'ai cru que Morisset allait sauter sur l'inspectrice. Il a serré les poings à s'en faire blanchir les phalanges, aussi raide qu'une barre de fer, puis un de ses bras s'est détendu et il a tapé dans le mur assez fort pour y laisser une marque à côté d'un interrupteur. Si t'étais pas une femme, Emmy Saint-Pierre, je te casserais la gueule. Il a ensuite dit à la fille nommée Marnie de ne pas s'inquiéter, qu'il se chargeait de lui trouver un avocat tout de suite et de la sortir de là. Au même moment, Marchessault a surgi de je sais plus quelle pièce, a déboulé jusqu'à la rangée de chaises et s'est arrêté face à moi. Qu'est-ce que vous foutez encore ici, Richard ? En entendant mon nom, la fille nommée Marnie a levé la tête dans ma direction. J'ai pensé qu'elle allait m'ordonner de courir, de prendre mes jambes à mon cou et de quitter cette ville de fous sans me retourner, mais elle est demeurée muette, me fixant de ses yeux de statue immobile. J'ignorais quel traitement Émilie Saint-Pierre lui avait réservé dans la salle d'interrogatoire, mais si quelque chose pouvait encore être cassé dans cette fille, elle s'y était sûrement appliquée à coups de matraque.
Avant que la porte se referme sur elle, Marnie m'a lancé un regard qui pouvait aussi bien ressembler à un appel à l'aide qu'à un aveu d'impuissance, et son visage a été remplacé par celui de Marchessault, qui voulait que je lui explique c'était quoi, ce bordel ?
L'homme de l'enterrement était là, l'homme qui avait la fièvre, l'oiseau de malheur qui surgit chaque fois que je me traîne derrière un mort. Il m'attendait au coin d'une rue, ce midi, puis au poste de police, pour me rappeler qu'on est pareils, que le ciel nous est tombé dessus par un jour autrement semblable à tous les autres et qu'il n'y a rien de plus effrayant, pour les gens qui n'ont pas été frappés, qu'un homme ou une femme s'avançant au milieu d'un ciel détruit. Personne ne m'a dit ce qu'il fabriquait à Saint-Alban, mais c'est facile à deviner. Il s'y trouve pour la même raison que moi, parce qu'il continue à payer le crime de la pourriture qui a massacré son existence. C'est de là que vient sa fièvre et son allure de Crappy Owl. Richard, c'est le nom que lui a donné Marchessault. Richard Richard, Crappy Richard, Holy Richard. Avec un peu de chance, j'apprendrai peut-être son véritable prénom avec l'aide d'Emmy-Lili, qui me bombarde de questions auxquelles j'ai déjà répondu tout à l'heure, la plupart du temps en lui répétant que je ne savais pas, que je ne me rappelais plus, que j'ignorais, bordel, ce que je devais savoir ou pas.
Quand je suis entrée dans la salle d'interrogatoire, un peu plus tôt, Emmy avait disposé quelques objets sur la table, tous enveloppés dans des sacs de plastique. Elle voulait analyser ma réaction devant ces objets, se faire une idée de mon degré de culpabilité en fonction de la nervosité ou de la panique qui me ferait monter le rouge aux joues.
Entre une casquette des Yankees de New York tachée de boue et une pochette ayant contenu des billes et des pierres de diverses couleurs étiquetées en tant que pièces à conviction, Holy Owl gisait sur le dos, asphyxié par le sac dans lequel on l'avait enfermé. Holy Crappy Owl, ai-je murmuré en tendant la main vers lui, mais Emmy m'a interdit de le toucher, de même qu'elle m'a empêchée de m'emparer de la pochette ayant renfermé le trésor de Mike, des billes, des pierres, des bijoux colorés. Il était impossible que ces objets aient abouti au cœur de cette enquête. Quelqu'un se trompait encore, qui mélangeait les dates et les enfants. Ils nous appartenaient, à Mike et à moi, et ne pouvaient être tombés par hasard entre les mains de Michael Faber. À part Mike et moi, seul l'agresseur du petit Faber pouvait les avoir réunis, à supposer qu'il ait connu les secrets de Mike.
Tu reconnais ces objets ? m'a demandé Emmy, et j'ai fait signe que oui, car aucun son ne sortait de ma bouche. Quel que fût son objectif, Emmy avait visé juste. Mes joues étaient brûlantes et je comptais les secondes tombant mollement de l'horloge avant de m'évanouir, certaine que je m'éveillerais au fond d'un cachot, avec les rats, privée de toute possibilité d'annoncer à Emmy que Mike m'avait sauvé la vie. Je voyais le piège d'Emmy-Lili se refermer sur moi quand je me suis rendu compte que la casquette emballée dans le troisième sac de plastique n'était pas celle que Doyon avait trouvée en fouillant ma maison. Cette casquette ne m'appartenait pas. Sur la mienne, un des jambages du N s'était décollé et les fils de la broderie s'effilochaient. C'est pas ma casquette, ai-je murmuré en la montrant du doigt. Mais Emmy le savait. Elle voulait vérifier si j'avais déjà vu celle-là. Regarde, le bouton est arraché sur le dessus.
Le bouton était arraché, en effet. Ce n'était pas ma casquette. C'était tout ce que je savais. Constatant que je n'allais pas faire de coup de théâtre et lui révéler que cette calotte appartenait à Tiger Woods ou à l'entraîneur des Yankees, Emmy l'a mise de côté pour un moment et elle a poussé Holy Owl vers moi, juste sous mes yeux, à côté de ma main.
Que peux-tu me dire à propos de ce hibou, Marnie ? a-t-elle enchaîné en tapotant la tête de Holy Owl du bout du doigt par-dessus le plastique. Ses ongles laqués étaient courts, bien taillés, propres, des ongles de fille qui se manucure toutes les semaines en écoutant Grey's Anatomy ou Desperate Housewives, ainsi que je le faisais au cours de cette période New Yorkaise durant laquelle j'avais perdu contact avec l'odeur de la terre, mais aussi avec celle de la mort. Comment Emmy Saint-Pierre s'y prenait-elle pour jouer au milieu des fosses et des dépotoirs tout en gardant des ongles propres ?
J'ai observé mes ongles à moi, encore salis de terre, et une vague de haine comme je n'en avais pas ressenti depuis longtemps a ajouté une deuxième couche de rouge à celle qui me brûlait déjà les joues. La sensation d'engourdissement qui m'avait privée de parole un peu plus tôt avait disparu, emportée par la colère, et mon sang circulait de nouveau. Cette fille aurait dû être en train de creuser le sol de la forêt avec moi, à quatre pattes dans la boue, nos mains exhumant parfois un fémur noirci et se rencontrant dans une fébrile prière pour que cet os ne soit pas humain, au lieu de quoi elle m'accusait d'avoir les mains sales des meurtriers et des fous, car les fous non plus ne se lavent pas. Ils étalent leur merde en fresques cauchemardesques sur les murs de leur chambre et se rongent les ongles d'orteil en se balançant au rythme d'une ancienne comptine.
J'ai regardé Emmy en plein dans les yeux et je lui ai répondu que ce hibou m'appartenait, avant de lui demander si elle se souvenait de l'histoire du Petit Poucet. J'ai fait la même chose que les parents du Petit Poucet, Emmy. J'ai raconté une histoire à dormir debout à Crappy Owl et je l'ai emmené en plein milieu du bois. Mais je me suis pas contentée de l'égarer, je l'ai pendu à la croix de ton frère, en plein milieu du bois. Les roches, c'est lui qui les a semées dans mon dos pour tracer son chemin, mais les roches lui ont servi à rien, pauvre Crappy. C'est-tu assez clair pour toi ? Pis si tu veux savoir si j'ai procédé de la même façon avec Mike, t'as rien qu'à trouver d'autres roches.
Je ne criais plus, je hurlais, et la porte s'est brusquement ouverte sur Doyon, qui m'a neutralisée afin de m'empêcher de sauter à la gorge d'Emmy pour ensuite lui arracher ses maudits ongles roses. Mais je n'étais pas la seule à vouloir étriper l'autre. J'avais mentionné la croix de Mike, la triste croix de bois plantée près de la rivière par les mains écorchées de Victor Saint-Pierre, franchissant ainsi une frontière au-delà de laquelle la petite sœur de Mike était prête à tuer.
Tu vas me reprendre ton histoire tranquillement, Marnie Duchamp, tu vas m'expliquer où t'as emmené le Petit Poucet. Tout de suite ! s'est-elle écriée en frappant sans broncher la table du plat de ses deux mains. Elle a attendu, le souffle court, le visage en sueur, et j'ai eu l'impression de retomber dans un cauchemar dont toutes les issues étaient encombrées d'objets disparates que mes mains sales n'arrivaient pas à dégager.
Alors j'ai abdiqué. J'ai dit à Emmy que je ne comprenais plus rien, que tout s'embrouillait, que Holy Crappy Owl n'était pas méchant et que je voulais retourner chez moi, dans le passé, en plein cœur de la forêt d'août.
Je vais t'aider à comprendre, Marnie, a-t-elle murmuré, et elle m'a raconté une histoire à son tour, qui pouvait ressembler à celle du Petit Poucet, de Hansel et Gretel, du Chaperon rouge, aussi sanglante et animale, mêlée de hurlements de loups, d'enfants transportant des pierres dans leur ventre, des galettes et des petits pots de beurre dans leur panier, des galettes à l'orange, pareilles à celles de la grand-mère de Marchessault. Je n'ai toutefois commencé à saisir son histoire que quand elle a parlé de ce qui avait fait taire les chiens, Rex, Chet et Lucy.
À neuf heures dix-huit ce matin-là, on avait découvert Michael Faber, l'enfant à la bicyclette bleu Superman, noyé dans les eaux froides de la rivière aux Trembles, sa peau glacée et son petit visage tout boursouflé. Autour de son cou, était accroché un hibou de paille et d'écorce, Holy Owl, et de la poche de son chandail, pendait une pochette de toile rouge contenant les cailloux magiques de Michael Superman Saint-Pierre, une pochette qu'Emmy avait cherchée pendant des années. Jusque dans le jardin de ton père, Marn...
J'ai entendu la respiration de Doyon, qui n'avait pas quitté la pièce, puis un cœur qui battait à tout rompre, et je crois que j'ai dit non, quelqu'un ment, le petit mort ment. Puis les issues du cauchemar sont devenues blanches. Il me semble qu'ensuite, Emmy a demandé à Doyon de s'occuper des formalités pour une garde à vue, puis elle m'a laissée seule dans la salle d'interrogatoire, j'ignore combien de temps puisque le temps ne bougeait plus. Quand j'ai descendu le corridor, Phil m'attendait, de même que l'homme des funérailles, Richard Richard, pareil à moi, celui qui souffrait de la fièvre.
Silencieux depuis quelques instants, Marchessault dessinait des flèches sur une feuille de papier ligné. Je lui avais juré sur la tête de Billie que je ne connaissais pas la femme nommée Marnie, Marnie Duchamp, c'est ce qu'il avait dit, mais ça ne lui suffisait pas. Dans son esprit, les serments d'un père sur qui les soupçons n'avaient jamais cessé de peser ne valaient pas grand-chose. Qu'il aille au diable ! Billie était la seule personne au monde sur la tête de qui il m'aurait été impossible de mentir ou de maquiller la vérité, et si Marchessault était trop con pour se rendre compte que je lui offrais la meilleure garantie de ma bonne foi en invoquant le souvenir de ma fille, je n'allais pas me traîner à genoux pour qu'un éclair d'intelligence traverse les ténèbres où sa bêtise s'était confortablement installée.
Je me suis fermé comme un coffre-fort à sept combinaisons et j'ai pensé à appeler Maheux, le gnome qui logeait à mes frais dans le seul hôtel recommandable de Saint-Alban, mais je ne voyais pas en quoi il pouvait m'être utile pour le moment. J'ai compté les secondes, Marchessault a déchiré sa feuille fléchée puis, tel un magicien, il a prestement retiré une photographie de sa manche de chemise pour me la flanquer sous le nez. Je n'ai jamais aimé les magiciens, pas plus que les funambules et les jongleurs, particulièrement lorsqu'ils enlèvent leur pantalon bouffant pour se travestir en flic, et mon cœur s'est immédiatement mis à pomper à toute vitesse mon sang jusque-là relativement tranquille.
La photo qu'avait fait apparaître Marchessault en claquant des doigts aurait pu figurer dans une anthologie de l'horreur. On y voyait un garçon couché près de la berge d'une rivière, regardant de ses yeux blancs le ciel livide, au milieu duquel subsistait peut-être le sillon d'air pâle remué par l'envol de ce qu'on nomme l'âme ou l'esprit, forme d'énergie ou ombre nue créant le rire et la tristesse. Son visage, par contraste, avait pris la teinte de cieux violents tachés de nuages d'orage, ses cheveux se prolongeaient d'épines givrées qui fondaient au soleil et une large entaille creusait jusqu'à l'os la chair maigre du front, une entaille qui ressemblait à un éclair lie-de-vin dans le ciel nuageux. Je n'aurais cependant su dire si c'étaient les yeux qui vous blessaient le plus, ou les petites mains gonflées, dont l'une, flottant sous l'eau, semblait vouloir recueillir la rivière dans sa paume ouverte.
Tout ce que j'espérais, c'est qu'on n'allait pas montrer aux parents cette photo qui éclipserait toutes les autres et les condamnerait à ne plus voir de leur fils que cette main tendue, que cette entaille surmontant le regard globuleux, pareil à celui des poissons échoués sur la grève. L'horreur est cent fois plus forte que l'innocence et c'est toujours elle qui imprime ses marques dans la mémoire blessée. On se souvient du sourire, mais le rictus amer prend le dessus et c'est lui qui demeure, abolissant tous les sourires. Après la disparition de Billie, j'avais craint pendant des mois que l'inspecteur Ménard débarque dans ma cuisine pour glisser une photo de ce genre au milieu du fouillis, près d'une fourchette à bascule improvisée ou d'une petite maison de papier confectionnée par ses gros doigts aux phalanges poilues, et voilà que cette photo s'étalait devant moi, reproduisant l'image parfaite et terrifiante de la mort.
C'est le petit Faber ? ai-je stupidement demandé à Marchessault tout en repoussant délicatement vers lui la photo de l'enfant, de peur de briser davantage ce corps meurtri.
Qui voulez-vous que ce soit ? a grogné Marchessault avant de m'apprendre qu'on avait retrouvé l'enfant quelques heures plus tôt, qui gisait dans la rivière aux Trembles, à un demi-kilomètre de chez moi en ligne droite. Avec le chat, ça vous place en mauvaise posture, Richard.
Cette précision était inutile. Dès que je l'avais entendu prononcer le nom de la rivière, j'avais senti mes jambes s'amollir, j'avais pensé aux pistes d'enfants qui m'avaient guidé hors de la forêt le jour où j'avais découvert la croix de Michael X, Y ou Z, puis je m'étais demandé si les semelles striées de barres horizontales étaient celles de Michael Faber. J'ai ramené la photo vers moi en vue de déterminer si je reconnaissais la portion de la rivière où il s'était échoué, et j'ai aperçu le hibou, celui qu'un cinglé avait pendu à la croix de l'autre Michael. Il pendait maintenant au cou de l'enfant, entortillé au bout de sa cordelette. Si je ne l'avais pas remarqué plus tôt, c'est que le courant de la rivière l'avait poussé entre le corps et le bras droit de Michael Faber, où il était resté coincé. Il n'y avait cependant aucun doute, ce petit tas de plumes rêches piquant du nez dans la rivière était bien le hibou accroché auparavant à la croix de bois plantée sous les arbres. Était-ce Michael qui l'y avait pendu pour le reprendre ensuite ?
J'ai pointé la photo du doigt avec l'intention de dire à Marchessault que j'avais déjà vu ce hibou, mais les mots ont culbuté dans ma gorge sèche, créant un embâcle stoppant la circulation de l'air, et j'ai lancé à la ronde un regard implorant. J'allais mourir asphyxié, de la même manière que Michael Faber, étranglé par le hibou ou étouffé par la rivière. Le temps qu'un des gars dissimulés derrière le miroir m'apporte un verre d'eau, j'ai cru que j'allais voir défiler ma vie au ralenti, mais je n'ai aperçu que le moment présent, composé de formes et de couleurs d'une netteté saisissante, au centre desquelles la gueule énorme de Marchessault criait que j'allais lui crever dans les mains.
Quand j'ai enfin retrouvé mon souffle, je tremblais de tous mes membres. Cette réaction du corps ayant lutté pour sa survie était normale, le choc irradiait les muscles et les nerfs auparavant tendus, mais je tremblais surtout parce que j'avais échappé à l'asphyxie qui avait violacé le visage enfantin de Michael Faber.
De l'autre côté de la table, Marchessault s'épongeait le front en attendant que je sois de nouveau en mesure de parler. J'ai calé mon verre d'eau en en répandant la moitié sur moi, puis je lui ai raconté ce que je savais du hibou et des pistes d'enfants. Holy shit ! a juré Marchessault en se lissant les cheveux derrière le crâne, puis il a exigé des détails sur la croix, sur son état, sur ce qu'on y avait gravé.
Vous m'attendez quelques instants, a-t-il marmonné avant de quitter la pièce. Il est revenu avec trois sacs protégeant des pièces à conviction. Il m'a demandé d'identifier le hibou et de lui dire si les autres objets m'étaient familiers. La casquette de baseball ne me disait rien, mais la pochette rouge, si. Billie en avait une semblable, dans laquelle elle rangeait ses billes. Regarde, papanoute, celle-là ressemble à un œuf de moineau. Elle les répandait sur le tapis du salon pendant que j'écoutais les Myth Busters ou un documentaire sur les difficultés de reproduction d'une espèce menacée. Elle les classait ensuite selon leur style et leurs couleurs, séparant les billes unies des chamarrées et des marbrées, pour enfin les remettre toutes ensemble dans la pochette. Chaque fois, elle s'émerveillait de les redécouvrir, ne se lassant jamais d'observer la lumière à travers les billes translucides lui donnant une vision déformée du monde. T'as toute la face croche, popinouche...
Billie en avait une semblable, ai-je murmuré en désignant la pochette, me tirant du même coup dans le pied. Immédiatement, Marchessault s'est mis à me bombarder de questions à propos de cette pochette. J'ai eu beau lui expliquer que ça ne pouvait pas être celle de Billie, il était trop tard, il ne m'écoutait plus. À travers les pierres que contenait la pochette enfouie dans une des poches du pantalon de Michael Faber, se trouvaient aussi quelques billes, des bleues et des émeraude, des translucides, des chamarrées et des tachetées, pareilles à des œufs d'hirondelle.
Je me suis tourné vers l'horloge. Seize heures quarante-huit. Il était temps de sortir maître Jean-Pierre Maheux de sa chambre d'hôtel avant qu'il descende s'enfiler quelques scotchs au bar et parte découvrir les charmes de Saint-Alban by night au bras d'une fille qui devrait se coucher par terre pour lui tailler une pipe.
Maheux m'a fait sortir de là avant qu'ils aient la géniale idée de me confiner en garde à vue à la suite de Marnie Duchamp, coupable de je ne sais quel manquement aux règles immuables régissant les petites communautés. On n'avait qu'à regarder cette femme pour se rendre compte qu'elle n'avait rien d'une criminelle et n'était qu'une autre des victimes de l'hypocrisie des hommes.
Puisqu'elle et moi étions seuls dans le même bateau, j'essayais de me convaincre que je ne partageais pas ma cabine avec une psychopathe, mais je pouvais être carrément à côté de la plaque et me laisser abuser par une pâleur qui n'était qu'un masque, un attribut sous lequel Marnie Duchamp dissimulait sa froideur et sa cruauté. Les assassins n'ont pas toujours des gueules d'assassins, de même que les pédophiles n'ont pas le mot « pédé » écrit dans le front, ce qui leur permet de frayer tranquilles et d'accumuler les cadavres dans le sous-sol de leur bungalow de banlieue ou le walk-in de leur penthouse. Cette Marnie demeurait pour moi un mystère, que j'associais à la profondeur parfois déconcertante des forêts de Rivière-aux-Trembles, remplies de secrets peut-être inavouables.
Contrairement à Marnie Duchamp, j'étais cependant libre pour le moment. Encore sur la bateau, mais libéré des chaînes qui entravaient mes pas. Sitôt débarqué au poste avec sa cravate à pois de travers, preuve que j'avais eu du flair concernant les activités apéritives de maître Jean-Pierre Maheux, celui-ci avait sorti sa voix de prétoire, balayé du revers de la main les arguments de Marchessault, puis il m'avait reconduit chez moi en m'interdisant de parler à qui que ce soit en dehors de sa présence, probablement parce que lui non plus n'était pas persuadé de mon innocence. Depuis mon retour, je remuais tout ça en examinant ma tasse de café refroidi comme si le marc se collant aux parois pouvait me livrer la vérité. J'essayais de me représenter les dernières heures du petit Michael, si jeune qu'il ne devait même pas savoir qu'il mourrait un jour, je pensais à la fragilité de Marnie Duchamp, qui cachait peut-être une Mrs. Hyde, puis au chat nommé Nuage, qui avait abouti chez moi par un triste jour de Pâques où il aurait dû roupiller dans la maison de son petit maître.
Ce chat n'avait passé que quelques heures avec moi, mais son départ avait creusé un trou béant sur le fauteuil qu'il avait adopté. La pièce semblait soudain privée de vie, de même que la maison entière, engourdie dans une forme d'ennui qui suintait des murs. Je n'étais cependant pas dupe, cette odeur de vieille sacristie imprégnant l'air ambiant ne venait pas de l'absence d'un chat qui avait à peine eu le temps de laisser quelques poils sur un coussin râpé. C'était moi qui suintais, qui puais le renfermé et n'avais plus assez d'énergie pour croire qu'il valait le coup de se botter le cul, de préparer le souper en chantant Y a d'la joie ! et de se tricoter des pantoufles au coin du feu quand des enfants mouraient dans la rivière coulant derrière chez soi.
La vie me sortait par tous les trous qu'elle s'était acharnée à me creuser dans le ventre et parfois dans le dos, en véritable traîtresse, et je ne voyais pas trop comment je pouvais donner le coup de grâce aux couleurs moribondes qui teintaient l'atmosphère. Même si j'avais peint les murs en fuchsia, les couleurs seraient demeurées là, en suspension dans l'air qui stagnait autour de ma carcasse trouée.
J'ai trouvé la force de monter à ma chambre, au cas où je serais encore assez vivant pour m'endormir, mais j'ai à peine eu le temps de replier les couvertures qu'un fracas de verre brisé est monté du rez-de-chaussée, suivi d'un crissement de pneus sur le gravier. Je suis resté figé quelques instants et me suis précipité à la fenêtre, à demi ressuscité par la surprise et le bruit, mais je n'ai vu que deux feux arrière au bout de l'allée, qui se sont rapidement évanouis derrière la courbe menant au village. Plus ou moins rassuré, j'ai descendu l'escalier en tendant l'oreille. Aucun bruit ne parvenait d'en bas, sinon celui du vent, dont la fraîcheur atteignait mes pieds nus. En allumant le plafonnier du salon, j'ai constaté que les rideaux ondulaient lentement devant la fenêtre, ainsi qu'ils le font par les belles nuits d'été où l'air est trop doux pour qu'on imagine quelque spectre coulant son corps immatériel dans les plis du tissu, entre nuit et lumière. Le froid évoquait cependant ce spectre, dont les pas invisibles s'enfonçaient dans le tapis jonché d'éclats de verre, au milieu desquels reposait une pierre emmaillotée dans un bout de papier retenu par une corde de jute, comme si j'avais besoin de ça, comme si mon salon n'était pas assez moche.
J'ai ramassé la pierre, effleurant au passage la fuite du spectre, qui a quitté la scène en aspirant au dehors un des pans des rideaux. Le message qu'on venait de m'expédier était on ne peut plus clair : « Retourne chez vous, maudit malade ! On veut pas de maniaque dans ton genre dans le coin. » Le tout était écrit en rouge, probablement pour accentuer l'effet dramatique, et le papier avait été souillé de ces diverses substances qu'on associe aux ordures de ma trempe.
J'ai pensé appeler les flics, mais personne ne se déplacerait en pleine nuit pour s'occuper d'un salaud de ma trempe, qui ne récoltait en somme que ce qu'il méritait. Je mettrais Maheux là-dessus le lendemain matin, qu'il gagne au moins une partie de ses honoraires. J'ai remisé le message dans un sac à pain vide, avec la pierre et la corde, puis j'ai passé l'heure suivante à ramasser les dégâts et à boucher ma fenêtre avec des morceaux de carton. Une fois ce travail terminé, je suis demeuré devant la télé jusqu'au matin, à penser à Marnie Duchamp, à Michael Faber, et à me demander si je ne devais pas me tirer en vitesse, ainsi qu'on m'y avait gentiment invité, ou alors foutre le feu à cette baraque et m'y immoler, pour le plus grand bien de l'humanité.
Les trois jours qui ont suivi ont été relativement calmes. Deux agents en uniforme sont venus rédiger un constat d'effraction avant de repartir avec ma roche et mon message, j'ai renvoyé Maheux en ville jusqu'à nouvel ordre, j'ai contacté mes assurances et j'ai appelé un vitrier. Ces quelques activités ont secoué mon apathie et j'ai failli avoir une réaction presque normale, proche de l'émotion, quand le soleil a enfin percé la couche nuageuse dans la journée du 22 avril. Si le soleil maintenait ses efforts, il m'arracherait peut-être un authentique sourire d'ici la floraison des lilas.
J'en ai aussi profité pour récupérer mon ordinateur, qui n'avait révélé aucun secret honteux ni fourni d'élément incriminant aux enquêteurs. Puis, ne sachant que faire de ma peau, je me suis occupé de celle de Ronie. Je l'ai sorti du trou de vase où il avait choisi d'hiberner pendant que je lui fichais la paix et je l'ai expédié sous les palétuviers des Everglades, histoire de lui réchauffer un peu la couenne et de m'éloigner du même coup des paysages de Rivière-aux-Trembles.
Une fois Ronie acclimaté à son nouvel environnement, je l'ai laissé macérer, parce que je ne savais absolument pas quoi faire de lui. J'aurais pu le placer sur la route d'une lapine des marais, près de laquelle il aurait allègrement sautillé dans les herbes hautes au soleil couchant, donnant à mon illustrateur l'occasion de verser dans le romantisme animal. J'aurais pu pousser la lapine à le demander en mariage et à lui donner une flopée de crapauds à oreilles, mais le cœur n'y était pas. Chaque fois que j'incitais Ronie à s'aventurer dans sa swamp, je l'imaginais en train de se faire bouffer par un crocodile ou un alligator, et qu'on en finisse.
Le visage balafré et violacé du petit Faber ne cessait de s'interposer entre Ronie et moi. De la rivière, il glissait jusqu'aux marécages d'eau douce de la Floride, où il voguait sans hâte parmi les aigrettes et les ibis. Parfois Billie nageait avec lui, ses cheveux d'enfant couronnés d'orchidées, et leurs mains se rejoignaient par-dessus la lenteur des algues. Le plus grand rassurait la petite sous la pâleur d'un soleil qui ne blessait pas et la nuit ne venait jamais, car la nuit ne pouvait atteindre ces corps libérés de la douleur. C'était ma nouvelle version de l'enfer et du paradis réunis, une forme de vision idyllique issue des œuvres du malin. J'affichais des photos de mangroves sur l'écran de mon ordinateur et j'y superposais celles de Michael sur sa bicyclette et de Billie dans sa robe à pois, leurs sourires éclatant parmi les ombres des cyprès. Si j'en avais eu la possibilité, je me serais procuré un aller simple pour les Everglades en vue d'y chercher les portes du ciel et de l'enfer, mais j'étais en quelque sorte assigné à résidence. Émilie Saint-Pierre m'avait avisé qu'il m'était interdit de quitter la région avant la fin de l'enquête, et tant pis si je me faisais lapider en repassant mes chemises.
Depuis la nuit du mardi au mercredi, personne n'était venu fracasser mes fenêtres ni pendre d'épouvantail sur ma galerie, je n'avais reçu aucune lettre de menaces, aucun téléphone haineux, mais je craignais toujours, lorsque le soir tombait, de m'éveiller dans le tonnerre d'une explosion ou de voir se profiler dans l'obscurité de ma chambre deux ou trois silhouettes armées de bâtons de baseball. Je n'osais pas me rendre au village non plus, de peur d'avoir à affronter la franche hostilité de la caissière de l'épicerie ou de Max le boucher, qui aurait peut-être l'idée de farcir mon bœuf haché avec de la strychnine. Je vivais comme un criminel, alors que le seul reproche qu'on pouvait m'adresser était de n'avoir pas surveillé ma fille vingt-quatre heures sur vingt-quatre, au risque de la rendre folle ou de la mener tout droit à la délinquance.
La sonnerie du téléphone m'a tiré de ma léthargie alors que je barbotais dans une mare imaginaire avec Ronie, aussi las qu'un ornithorynque qui vient de se taper le Tour de France. C'était Marchessault, dont la voix traînante indiquait qu'il manquait également de sommeil. Je m'attendais à une convocation, mais Marchessault voulait seulement vérifier quelques détails de ma déposition concernant les visites de Dixie-Nuage. S'il n'était pas venu me chercher par le collet, c'est qu'il y avait du nouveau et que je ne faisais plus partie de la liste des suspects. Je lui ai demandé s'ils avaient arrêté le coupable, mais il m'a envoyé sur les roses. Vous le saurez en lisant les journaux de demain, Richard, puis il a raccroché avant de me dire si j'étais libre de mes mouvements, si je pouvais mettre mon maillot de bain dans une valise et m'envoler pour les terrains palustres de la Floride. J'ai été tenté de le rappeler, mais j'ai conclu que Maheux parviendrait plus facilement que moi à obtenir des informations. J'ai composé son numéro et sa secrétaire m'a appris qu'il était au tribunal pour la journée, mais qu'elle lui transmettrait mon message dès qu'elle parviendrait à le joindre. N'ayant d'autre choix que l'attente, j'ai poussé Ronie du coude et me suis assis près de lui, les pieds dans l'eau fangeuse, en pensant à Michael Faber, à qui l'arrestation d'un coupable ne rendrait jamais la vie, pas plus qu'elle ne mettrait fin au cauchemar de ses parents. Le seul avantage qu'ils auraient sur moi, c'est qu'ils sauraient à quel endroit, dans la tête de quel fou furieux ce cauchemar avait pris forme, et je les enviais presque de pouvoir mettre un nom et un visage sur leur malheur, de pouvoir élaborer des scénarios aboutissant à la vengeance sanglante de leur fils. C'est ce genre d'inavouable sentiment qu'éprouvent ceux qu'on place devant le vide et qui doivent se démerder avec l'infinité des possibles susceptibles de donner quelque consistance à leur ignorance.
Maheux a rappelé vers les quinze heures, alors que je m'étais enfin décidé à bouger et que deux œufs de poules élevées en liberté crépitaient dans ma poêle T-fal. Je lui ai expliqué la situation pendant qu'un bruit de va-et-vient autour de lui assourdissait ses paroles, et je l'ai entendu marmonner qu'il me rappelait dans quelques instants. Je jetais mes œufs enrobés de beurre noirci à la poubelle quand le téléphone a de nouveau sonné. C'était re-Maheux, fier comme un paon de pouvoir m'annoncer que j'étais libre et que sa facture suivrait sous peu. Pour le reste, il n'avait pu obtenir aucun détail. Je l'ai remercié et j'ai raccroché, incapable de partager son enthousiasme. J'étais libre, soit, mais la terre n'en continuerait pas moins d'enfanter les centaines et les milliers de fumiers qui massacraient la beauté du monde.
Emmy m'a appris que le garçon qui accompagnait Michael Faber à la rivière s'appelait Martin, un nom banal qui pouvait toutefois se transformer en méchant, vilain petit Martie. Après plusieurs jours de silence, Martie avait craqué. Ses parents croyaient que sa fièvre était causée par la disparition de son meilleur ami, mais l'origine de sa maladie était plus compliquée. La fièvre de Martie était une fièvre d'après débâcle, provoquée par la luisance des minces couches de glace couvrant les arbres tombés au milieu des rivières en crue. Le virus de cette fièvre vous frappait d'un coup, sans que vous l'ayez senti venir, et vous plongeait dans un délire hallucinatoire rempli de morts impossibles et de cadavres grotesques.
Martie Jacob avait attrapé ce virus aux alentours de quinze heures, le samedi 18 avril, au moment où le soleil frôlait la cime des plus grands arbres. Un cri perçant avait retenti, il avait vu un corps flotter sur la rivière dans une position étrange, pas normale, pas celle d'un corps qui nage, puis il s'était mis à courir à gauche et à droite, pareil à un animal cherchant une issue dans l'enclos étouffant où on l'a enfermé. Il avait perdu les pédales, hurlant le nom de son ami, Michael, Mike, puis l'accompagnant de mots incompréhensibles qu'aucune oreille, sinon la sienne, n'entendait cependant plus. Il avait ensuite sauté dans la rivière, butant contre les pierres et tombant à genoux, puis, en apercevant le tronc ensanglanté, il avait vomi dans l'eau claire un liquide brunâtre chargé de grumeaux qui s'effilochaient en chutant vers le fond sablonneux où étincelaient quelques cailloux blancs.
Réveille-toué, Mike, avait-il crié à s'en arracher les poumons tout en tirant le bras flasque de son ami, puis il avait culbuté. Pendant un instant, la rivière avait coulé par-dessus ses yeux, s'interposant entre le ciel et lui, elle avait tenté de pénétrer dans sa bouche et il s'était relevé dans un râle goûtant le vomi et le poisson cru. Après, il ne se souvenait plus. Il avait vainement appelé Nuage, s'accrochant à l'image de ce chat comme à une preuve que rien de ce qu'il avait vu n'était vrai, et il avait encore couru. À l'orée de la forêt, il avait attrapé son vélo, les mains gelées, les pieds gelés, déjà fiévreux et coupé des bruits du monde par les sons caverneux de la rivière dans ses oreilles.
C'est la froideur du vent qui l'avait poussé jusque chez lui, où il s'était réfugié dans sa chambre, sous ses couvertures de laine, après avoir caché ses vêtements trempés sous sa commode. Il rêvait quand sa mère avait posé une main sur son front, affolée par la chaleur émanant du corps de son fils, pour lui dire que la mère de Mike avait téléphoné, que la mère de Mike cherchait son fils, qu'Anna Dickson et Charles Faber, les parents de l'enfant, avaient une peur du diable. Dans son esprit, tout de suite, une alarme avait sonné. La mère de Mike ne devait pas savoir. Il fallait protéger ses yeux des larmes et du tronc d'arbre ensanglanté. Mike n'était pas mort. Mike allait revenir avec Nuage, poussé par le vent chaud, par l'été qui ne peut pas blesser. C'est lui qui avait glissé sur le tronc d'arbre, lui, Martie, et s'était ouvert le crâne aux éclisses d'une branche cassée. Ça ne pouvait être que lui, puisqu'il avait vu le ciel, le bleu découpant les nuages, à travers la rivière qui luttait pour ouvrir sa bouche.
J'ai effectué le trajet jusqu'à Saint-Alban pour me procurer des journaux que j'aurais pu acheter au village, mais je préférais me taper quelques kilomètres de route plutôt que d'entendre les portes me claquer au nez dans un sifflement d'insultes. J'avais mal au crâne à force d'avoir attendu l'explosion qui pulvériserait ma maison et me sentais incapable d'affronter les sous-entendus et les menaces de quelques péquenauds assoiffés de sang qui voudraient me faire la peau sur la place publique. Je me jetterais dans la gueule du loup quand j'aurais retrouvé mes moyens et révisé mes Trois petits cochons à fond. J'ai filé sur les routes encore embrumées par la fraîcheur du matin et me suis arrêté au premier dépanneur en vue, chez Mamie Berthe, dont le sourire invitant agrémentait l'enseigne qui grinçait doucement à l'entrée de la rue fuyant au cœur de Saint-Alban, aussi sinistre qu'un décor de ville fantôme en cette heure matinale.
J'ai poussé la porte du dépanneur, qui a grincé de concert avec l'enseigne, et suis tombé face à face avec le visage en gros plan de Michael Faber, qui occupait la une de trois des journaux alignés sur les présentoirs. Je me suis figé quelques instants devant L'Écho de Saint-Alban, où le petit était assis avec Nuage devant un arbre de Noël si semblable à tous les arbres de Noël qu'il ma semblé voir Billie avec Pixie le jour où L.A. lui avait offert une maison de poupées. Je suis sorti de ma torpeur quand la porte a grincé de nouveau et j'ai déposé un exemplaire de chaque journal sur le comptoir en m'interdisant d'en lire immédiatement les titres. Je ne voulais pas découvrir dans l'atmosphère humide de ce dépanneur, entre les étalages de chips et de produits ménagers, que Michael Faber avait été assommé puis étranglé par son entraîneur de hockey ou par le cousin germain de sa mère. Je me suis servi un café couleur de thé à la cafetière installée près du frigidaire à bière et y ai ajouté quelques généreuses cuillerées de sucre. Il goûterait l'eau de vaisselle sirupeuse et je m'en foutais, j'avais besoin d'un liquide chaud pour me donner du cœur au ventre et dégourdir mes mains ankylosées pour être demeurées crispées sur le volant de chez moi à Saint-Alban.
Pauvre enfant, ç'a-tu du bon sens, a murmuré celle qui devait être Mamie Berthe, sourire en moins, en désignant les journaux d'une main abîmée, dont l'annulaire était si enflé que sa bague semblait s'être incrustée dans la peau. Les années avaient rattrapé Mamie Berthe depuis qu'elle avait servi de modèle à l'enseigne de son commerce. La jeune femme avenante devant qui s'ouvrait un avenir rempli de promesses avait rapidement compris que l'avenir est un ciel dont l'horizon se rétracte et s'était probablement mise à enfler pour le repousser. Elle avait le visage bouffi de ces chiens qui ont toujours la larme à l'œil et se déplaçait péniblement dans l'étroit espace qui s'était refermé sur son corps obèse. Ç'a-tu du bon sens, a-t-elle répété en me tendant ma monnaie, son œil de pékinois attendant une réaction de ma part. J'ai marmonné une imbécillité sur le non-sens de l'existence et me suis esquivé, de crainte d'en avoir pour dix minutes à l'entendre gémir sur le sort des générations laissées à elles-mêmes, sur l'absence des mères, des pères, sur l'innocence des petits qui attendaient anxieusement dans la cour de l'école et tendaient la main au premier venu, se dopaient à onze ans et perdaient leur virginité deux mois plus tard. Au cours des trois dernières années, on m'avait servi et resservi tous les lieux communs associés au malheur de l'enfance et j'avais aligné suffisamment d'indigestions pour être devenu allergique aux propos englués de compassion ou chargés de reproches à peine voilés que les gens se sentaient obligés de formuler alors que la meilleure chose à faire était de fermer sa gueule.
De retour à ma voiture, j'ai déposé les journaux sur le siège du mort, sans penser que nul autre siège ne convenait plus à l'enfant dont le sourire était maquillé par l'encre d'imprimerie. Avant de reprendre la route, j'ai monté le chauffage à fond, transi par l'humidité qui s'étendait en nappes légères au ras du sol, et j'ai cherché un endroit tranquille où me stationner. À la sortie de la ville, je me suis engagé dans l'allée boueuse d'une aire de repos encore privée de ses tables à pique-nique, aussi désolée qu'une piste de cirque sans ses ballerines et ses chevaux, et j'ai entrepris ma lecture.
Michael Faber n'avait été ni assommé ni étranglé. Contrairement à ce qu'on avait d'abord cru, tout permettait d'affirmer, selon les dernières informations, que la mort du petit Faber était une mort accidentelle. On attendait encore les résultats de l'autopsie, qui serait effectuée le jour même, mais le témoignage d'un ami de Michael, dont le nom ne pouvait être divulgué compte tenu de son jeune âge, ne laissait planer presque aucun doute quant aux circonstances du drame. L'ami en question, semblait-il, avait été témoin de la noyade, mais l'enfant avait refoulé l'horreur de cette scène jusqu'à la découverte du corps. À l'heure actuelle, la police se refusait à révéler d'autres détails, mais la chasse aux sorcières était terminée. Aucune main inconnue, sinon la main d'un destin que certains appelaient Dieu, n'avait fracassé le crâne de Michael Faber lorsqu'il était tombé de l'arbre qui barrait la rivière.
À peu de choses près, les trois journaux rapportaient la même version des événements. Michael Faber était mort parce qu'il aimait la vie. S'il avait préféré s'écraser devant la télé plutôt que de partir à l'assaut des rivières, il aurait continué à vivre dans un univers constitué de milliards de pixels pendant que les rivières coulaient et que son vélo rouillait contre le mur d'un hangar. Le hasard avait frappé, le croque-mitaine n'existait pas et j'étais désormais libre de mes mouvements. Les soupçons qui pesaient sur moi avaient été balayés par un vent de printemps, mais il m'était impossible de me réjouir. Un enfant était mort et ma liberté perdait son sens dans l'écho lugubre du glas qui résonnait maintenant au loin.
J'ai laissé tomber sur mes genoux le journal sur lequel Michael posait avec Nuage devant un arbre de Noël et j'ai pensé que Billie était peut-être morte, elle aussi, d'avoir trop aimé la vie, d'avoir saisi la main gantée qui lui promettait fleuves et rivières et de l'avoir docilement suivie dans l'espoir d'aboutir à un vaste delta s'ouvrant sur une mer tissée de rubans roses et bleus. Dehors, le brouillard se dissipait, quelques mésanges pépiaient et un merle, annonciateur de temps plus doux, sautillait sur le gravier à l'ombre des ballerines et des chevaux enfuis. Billie, dis-moi que c'est toi Billie ? Le merle a fait un bond, les mésanges se sont envolées, mais le ciel est resté muet.
En arrivant à la maison, j'ai téléphoné à Marchessault, qui me devait bien quelques explications après m'avoir accusé de tous les crimes commis dans la région au cours du dernier siècle et transformé du même coup mon absence de réputation en réputation de gibier de potence. Il m'a fourni les explications demandées d'une voix lasse, brisée par l'impuissance, et j'ai compris à quel point la vie de flic pouvait vous bousiller un homme. Mais la voix lasse a fait son boulot, elle n'existait que pour cette raison, tenter d'empêcher la violence et la mort, et se fendre ensuite en quatre en vue d'expliquer pourquoi la mort avait encore gagné. C'est ainsi que j'ai appris pour quelle raison ce vieux schnock de Dixie avait abouti chez moi le jour de Pâques. L'explication était si simple qu'elle ne méritait pas qu'on emprunte cinquante détours pour l'obtenir. Tout ce que m'a raconté Marchessault à ce sujet, je le savais déjà en substance. Je l'avais deviné en observant les photos de Michael Faber publiées dans les journaux et en apprenant qu'il avait coulé dans la rivière sourire aux lèvres.
Le tout se résumait en quelques mots. Michael Faber adorait son chat, Michael Faber aimait fendre le vent au volant de sa bicyclette et il aimait par-dessus tout lancer des cailloux dans les rivières, construire des barrages avec des bouts de branches et attraper des menés après la saison du frai. Dès que ses parents lui avaient offert son vélo, il s'était mis à explorer la région avec Nuage, s'aventurant chaque fois un peu plus loin et entraînant avec lui son meilleur ami, le seul avec qui il acceptait de partager ses Smarties et son Play Station, celui-là même qui l'avait vu se noyer et ne remettrait jamais plus les pieds dans une rivière.
Le 12 avril, jour de Pâques, Michael Faber, Nuage et l'ami en question avaient poussé leur expédition jusqu'au coude que forme la rivière aux Trembles à un demi-kilomètre de ma maison, enfonçant leurs pieds dans la boue des sentiers et permettant ainsi aux innocents de la ville de ne pas se ramasser dans le Grand Nord alors qu'ils croyaient se diriger d'un pas tranquille vers leur maison. C'est l'ami qui avait fourni ces détails aux policiers en précisant que, ce jour-là, Nuage avait pris un mulot en chasse et n'était revenu chez Michael que le lendemain. C'est dans ces circonstances que le vieux schnock était venu frapper à ma porte et avait pris goût au thon de l'Atlantique. C'était aussi simple que ça. Les enfants jouaient, les chats chassaient, le monde était presque en ordre.
Malgré cet ordre auquel ne dérogeaient que peu d'enfants et encore moins de chats, je demeurais troublé à l'idée que Michael Faber s'était noyé derrière chez moi. Si j'avais marché vers la rivière, le jour de Pâques, au lieu de m'embourber dans le chemin du 4e Rang pour fuir la voix de Régine, j'aurais pu le croiser et, qui sait, modifier le cours de son existence à la manière du papillon d'Edward Lorenz. Un battement d'ailes, une branche craquant sous mes pas, et sa vie entière se serait déplacée des quelques millimètres nécessaires pour que son pied ne glisse pas six jours plus tard.
À plus d'une reprise, ce garçon avait joué derrière chez moi, courant dans les bois où j'avais failli me perdre ou pataugeant dans la rivière avec les bottes de caoutchouc striées de barres obliques qui m'avaient évité une expédition à la baie James et m'avaient peut-être sauvé la peau. Nous avions reniflé le même vent, la même maudite pluie de printemps chargée d'odeurs de pourriture, et je me trouvais encore là le jour de sa mort, tout près de lui, à tenter d'accoupler Ronie avec une rainette alors qu'il m'aurait fallu le sauver à mon tour. Un battement d'ailes et l'ami qui se frappait la tête à deux poings aurait continué à dessiner dans la vase de la rivière le chemin menant à une île aux trésors, Nuage serait rentré chez lui dans son panier et Billie, Billie-Billienoute, aurait effectué quelques pointes devant le large miroir traversant le studio de danse de mademoiselle Lenoir.
J'ai beau me répéter que mes ailes ont été brûlées depuis longtemps et que je ne suis en rien responsable de l'accident de Michael Faber, pas une heure ne s'est écoulée, depuis que Marchessault m'a montré la photo de son cadavre, sans que je me reproche de n'avoir pas suivi plus longtemps ses pistes, de n'avoir pas été là, près de la rivière, à surveiller la mort. Je suis l'homme immobile qui, à une seconde près, aurait pu permettre à Michael Faber de se relever de sa chute avec un simple bras cassé ou une bonne poque sur le front. Je suis celui qui n'a pas réussi à stopper la folie de l'inconnu de janvier, de l'homme invisible dont seule une petite fille nommée Billie connaît le véritable visage.
C'est tout ce que m'a apporté Rivière-aux-Trembles, une conscience doublement coupable qu'accentue le silence. Puisque rien de bon ne semble m'attendre ici et que les paysages entourant la rivière ressemblent trop à ceux d'un purgatoire, je vais plier bagages, mettre la clé dans la porte et partir pour la Floride, l'Australie ou la forêt amazonienne, en espérant ne pas y croiser une rivière près de laquelle je m'agenouillerai en me demandant si un enfant n'est pas en train de se noyer dans le bras méandreux filant au creux des arbres. Et personne, absolument personne ne regrettera le départ de l'homme qui a peut-être orchestré la disparition de sa fille et ne songe qu'à se précipiter dans le premier confessionnal venu pour se marteler la poitrine à coups de mea culpa en s'accusant de n'avoir pas battu des ailes dans la bonne direction, comme le papillon de Lorenz repoussant un ouragan au large de l'océan, puis attrapant au vol un gamin perdant pied.
Près d'un sapin de Noël décoré d'angelots et de boules multicolores, certaines givrées de minuscules paillettes évoquant la neige ou l'or des mages, Michael Faber tient dans ses bras un chat nommé Nuage. C'est ce que nous apprend la vignette soulignant la photo, Michael et Nuage, Noël 2008, au centre d'un article m'apprenant aussi que je n'ai pas tué ce garçon.
Étant parvenue à la même conclusion, Emmy Saint-Pierre m'a signifié mon congé il y a quelques heures, t'es libre Marn, tu peux partir, ce qui n'empêche pas Phil de pester contre elle. Il s'active dans la cuisine depuis une vingtaine de minutes en claquant les portes d'armoires et en battant du lait chaud comme s'il voulait le rachever. Je l'entends bougonner à travers le vacarme du fouet qui frappe les parois du bol de métal, tête de linotte, tête de pioche, une vraie Saint-Pierre, comme si Marnie pouvait, même pas capable de tuer une mouche, maudite police, mais la colère de Phil s'éteint au contact des images se bousculant dans ma tête. Quelques mots me rejoignent, privés de leur véritable sens, pendant que je pense à Michael Faber glissant sur le givre, à Martie Jacob culbutant sur les roches gluantes, l'un et l'autre condamnés par la rivière. Je pense au trésor enfoui de Mike, exhumé par Michael du tronc d'un arbre creux, quelques billes, quelques pierres transformées en kryptonite par la magie de la foi, à l'aide de formules empruntées à la mystérieuse alchimie de Superman. Je pense à la petite main blanche soupesant le trésor. Viens voir ça, Mart, j'ai trouvé des diamants ! Le reste est facile à imaginer, parce que Michael et Martie sont pareils à moi, pareils à Mike et à tous les enfants, inconscients de la mort, libres d'imaginer le monde.
Je suis libre aussi, semble-t-il, libre de grimper aux arbres et de marcher tête haute, mais depuis qu'Emmy-Lili, une autre fois trahie par la rivière aux Trembles, a consenti à me raconter ce qu'elle savait de l'histoire des deux enfants de Saint-Alban, ma liberté se double d'une nouvelle vie, celle de Michael et Martie, qui se fond à la mienne, celle de Mike et Marn. J'envisage ces deux vies et leurs morceaux s'imbriquent lentement, à la manière des pans de glace dont la dérive n'a d'autre but que de créer de nouveaux continents. Je suis ce nouveau continent, au milieu duquel, en équilibre précaire, j'avance sur un tronc tombé au-dessus de Nanamiu-shipu, la rivière Tremblante. Un petit hibou d'écorce se balance à mon cou, Holy Owl, Crappy Owl, very Crappy Holy Owl, les reflets de Nanamiu-shipu dans ses yeux jaunes et qui voient tout.
J'ignore comment Michael et Martie ont découvert ce hibou et comment il s'est retrouvé au cou de Michael, mais j'imagine la frousse du gamin tombant par hasard sur une croix lugubre portant un nom, le sien, sept lettres qui lui font l'effet d'un coup de massue, d'un message inquiétant du destin. Cibole, Martie, viens voir ça ! Et Martie arrive en courant, échappe un juron, puis s'agenouille près de Michael en lui désignant le hibou aux yeux jaunes. C'est un signe, Michael, faut que tu le mettes autour de ton cou, y va te protéger. Avec Martie, j'invente alors l'histoire de l'autre Michael, celui à qui est destinée la croix, une histoire de vampires et de terre profonde, de démons osseux sortant la nuit de leurs cercueils pour dévorer les loups. Depuis hier, je ne suis plus une mais quatre, et je n'essaie pas de comprendre. La forêt a ses lois, la rivière ses règles, l'une a voulu Mike, l'autre a pris Michael.
Bois ça, m'ordonne Phil, et je tends les mains vers le café couvert d'une mousse écumeuse me rappelant les paysages créés par Bubblelou. Il a peur, Phil, peur que je pète les plombs, que je confonde les deux Michael et n'aille me coucher au fond de la rivière, là où Martie a vu le bleu diffracté du ciel à travers le courant de la crue. Mais Phil se trompe, je ne peux confondre Mike et Michael puisqu'ils ne font qu'un, un seul enfant pareil à tous les autres, qui enterrent des trésors et les redécouvrent des mois ou des années plus tard, aussi brillants que les bijoux volés par les corneilles et scintillant au creux des nids, aussi fascinants que les diamants étincelant aux doigts des femmes.
Je déguste la mousse blanche que Phil a généreusement saupoudrée de cannelle et lui révèle que la pochette de pierres dénichée par Michael Faber appartenait à Mike. On jurerait que Mike a transmis son héritage à ce garçon, Phil, signé avec lui un pacte de sang. Et pourtant, la mort de Michael ne peut avoir été causée par Mike, pas plus qu'elle ne peut avoir été provoquée par Holy Owl, dont le poids trompeur des plumes d'écorce aurait fait basculer l'enfant. La mort est le fruit du hasard, d'une fatalité s'obstinant à cacher sur les troncs d'arbres morts des cercles de glace noire dont les miroitements s'éclipsent sous les nuages. C'est ainsi, toujours, la mort se conjugue au hasard et aux eaux vives. C'est la rivière qui a tué Mike, dis-je à Phil entre deux gorgées de café et, pendant que je parle, un quartier d'ombre obstrue peu à peu le soleil, à la place exacte où s'est formée la pellicule de glace qui a été fatale à Michael Faber.
Qu'est-ce qui t'arrive, Marn ? demande la voix de Phil du fond d'un interminable tunnel. Je lui réponds qu'on ne peut plus rien, que ni Rex, ni Lucy, ni le garde-chasse ne retrouveront jamais Irving l'ourson, qu'il est trop tard pour qu'Irving soit sauvé. Jusqu'à la nuit je parle à Phil des arbres déformés, des membres presque humains se détachant des troncs pluvieux, dont je ne saurai jamais s'ils étaient bien réels ou le simple fruit de ma peur. Je lui décris mon ignorance, contournant le pourtour du trou où certains croient que ma mémoire s'est effondrée, alors qu'il n'y a peut-être qu'un grand vide au fond du trou, que des images inexistantes. Et jusqu'à la nuit Phil écoute, jurant entre ses dents et cherchant à cacher la peur qu'il éprouve à me voir trembler. Mais Phil n'a rien à craindre. La folie est loin derrière moi, éclipsée par la destinée et les eaux vives. Il suffisait d'un peu de glace. Il suffisait de quelques mots : va-t'en, Marn ! Cours ! Il suffisait que mes mains n'aient pas tué Michael Faber.
## II
LA FIN
Puisque j'allais quitter sous peu Rivière-aux-Trembles et qu'on ne pouvait plus m'accuser de la disparition de Michael Faber, je ne me souciais pas vraiment de l'accueil qu'on me réserverait au village. La caissière de l'épicerie, Max le boucher, le commis du dépanneur et celui de la quincaillerie ne seraient bientôt que des souvenirs flous dans ma mémoire de Rivière-aux-Trembles, où n'auraient réussi à s'incruster que quelques arbres et une rivière mal nommée qu'on rebaptiserait peut-être un jour rivière des Deux Michael, rivière à la Croix, rivière de la Mort. Que Max, sa fille, sa tante ou sa maîtresse me méprisent ne m'importait guère. Leur version de la vérité était celle de gens qui s'ennuient et que la médisance excite. La banalité de leur ennui les absolvait presque de leur bêtise.
Nous étions le 30 avril, le temps était magnifique et j'en avais assez de me casser la tête. J'avais besoin de quelques babioles pour fermer la maison et c'est à Rivière-aux-Trembles que je me les procurerais. J'ai crié à Ronie, qui servait alternativement d'oreille et de punching-ball à ma solitude, sorte d'ami imaginaire et de souffre-douleur, que je me rendais au village pour une heure ou deux. J'ai chaussé mes verres fumés de crooner et j'ai parcouru les deux kilomètres me séparant de la rue Principale vitres baissées, attrapant ici et là une odeur de verdure annonçant les couleurs de l'été. En entrant à la quincaillerie, j'ai joyeusement salué le commis qui étiquetait des tournevis fraîchement sortis de leur boîte et me suis engouffré dans une allée sans attendre qu'il me lance un bonjour aussi enjoué que le mien.
Je comparais deux marques de ruban à masquer quand je l'ai aperçue, Marnie Duchamp, qui se débattait avec une poche de terre lui glissant des bras. J'avais plusieurs fois songé à la possibilité que cette rencontre ait lieu un jour, sans corbillard ni agent de police pour s'interposer entre nous, mais me retrouver face à cette femme qu'on avait menottée pour je ne sais quelle raison me jetait dans un profond malaise. En réalité, j'avais peur de m'approcher d'elle. Dans mon esprit, elle n'était qu'un regard marqué par une forme de fatalité qui m'effrayait parce que trop pareille au destin qui m'avait moi-même frappé. Je ne voulais pas qu'elle me révèle le nom du mort qu'elle suivait au cimetière, qu'elle me parle de tombes creuses ni de cet enfant disparu qui avait donné à Marchessault et à Doyon un motif pour débarquer chez elle sans s'annoncer et la soumettre à leurs interrogatoires absurdes. J'aurais pu faire semblant de ne pas l'avoir vue et me sauver en courant, mais il aurait fallu pour cela que je prétende avoir été frappé d'une cécité soudaine et me déniche illico une canne blanche ou un chien d'aveugle derrière la pyramide de masking tape qui menaçait de s'effondrer.
J'ai risqué un timide sourire en espérant que les choses en resteraient là, mais c'était m'illusionner quant à la nature du rapport qui s'était établi entre nous. Cette femme avait reconnu en moi je ne sais quel trait de parenté assez visible pour que son regard m'implore à deux ou trois reprises de me grouiller le cul et de lui tendre la main, et je ne m'en tirerais pas avec un sourire idiot. Que dire cependant à une femme qui s'imagine qu'on peut l'aider à sortir de son trou alors qu'on croupit soi-même au fond d'une fosse dans laquelle on s'ingénie à faire dégringoler quotidiennement quelques tas de terre humide ? La loi de la gravité m'a donné un coup de pouce, se manifestant comme toujours quand on a un pied dans le vide, et j'ai attrapé la poche de terre prise entre deux forces adverses et d'inégale puissance, l'attraction de la terre et les bras épuisés de Marnie Duchamp. Elle a murmuré un rapide merci en ramenant derrière son oreille gauche une mèche de cheveux rebelles, laissant du même coup sur sa joue une traînée de poussière brune empruntant la forme d'une queue de comète projetant son voile dans l'infini, puis elle m'a souri.
C'était la première fois que je la voyais sourire et je me suis tout de suite dit que ça lui allait bien. Quelque chose en elle s'était métamorphosé depuis que je l'avais croisée au poste de police. Son corps s'était allégé, aurait-on dit, pas au point de lui permettre de s'envoler, mais suffisamment pour qu'elle ait la possibilité d'admirer la Voie lactée sans se casser les vertèbres cervicales parce qu'elle avait trop longtemps piqué du menton vers le sol. Je devais avoir l'air complètement con à la fixer ainsi, car elle a éclaté de rire, autre signe de sa transformation, et s'est présentée : Marnie. Je sais, ai-je été tenté de lui dire, mais j'ai simplement répondu Bill, Bill Richard. Sans se faire prier, elle a ensuite accepté que je l'aide à transporter ses achats en me désignant un vieux pick-up rouge stationné devant la grande vitrine du magasin. Je déposais une quatrième poche de terre dans la boîte déjà encombrée du pick-up quand elle m'a invité à prendre un café. Je pense qu'on pourrait avoir des choses à se dire, ça passe mieux devant un café. Cette perspective me donnait toujours la frousse, mais j'étais d'accord, on avait sûrement quelques histoires à se raconter. On a donc quitté la quincaillerie ensemble, sous l'œil méfiant de l'homme aux tournevis, qui se demandait probablement s'il ne laissait pas filer les successeurs de Paul Bernardo et de Karla Homolka, les nouveaux Ken et Barbie de la diaspora pédophilique.
Le regard rivé sur la table, il m'a appris qu'il avait une fille, Billie, qui n'était jamais rentrée de l'école. Ça expliquait la fièvre. Ça expliquait pourquoi il était assis là, en face de moi, à tourner sa tasse de café dans tous les sens en cherchant une explication à son cauchemar. On était dans la même intenable situation, lui et moi, on ne savait pas ce qu'il était advenu de nos disparus, ça nous tuait, et on essayait de vivre malgré tout. Le seul avantage que j'avais sur lui était le temps écoulé depuis la disparition de Mike, mais était-ce réellement un avantage ? Dans une trentaine d'années, Bill Richard pourrait peut-être répondre à cette question et me dire si l'espoir est préférable à l'usure de la foi. Moi, je ne savais plus. Quand il a replacé la photo écornée de sa fille dans son portefeuille, j'en ai profité pour sortir la bouteille de cognac. Le café goûtait la cendre et on avait tous les deux besoin d'un remontant. J'ai cogné mon verre contre le sien et on a fait cul sec, sans dire un mot, à la mémoire de Mike et de Billie et à la santé des survivants, des rescapés de Nanamiu-shipu et autres rivières tremblantes allongeant leurs bras jusqu'au cœur des villes.
Dans le silence qui s'appesantissait, il a repoussé sa tasse et a repris son manège avec son verre, le retournant d'un côté puis de l'autre sous les reflets du soleil obliquant à la fenêtre. Il faisait tournoyer au fond du verre la petite goutte d'alcool qui avait échappé à notre cul sec, en suivant le sens de la rotation de la terre, puis il changeait de direction, prenait le temps à rebours et recommençait. Peut-être se disait-il que si le verre pivotait assez longtemps vers l'arrière, il pourrait remonter vers le passé et obliger la terre à replier ses nuits les unes après les autres. Il aurait pourtant dû savoir qu'il ne faut pas davantage plier la nuit que le jour, qu'il faut au contraire courir et retenir ses cris, que c'est l'unique façon de sauver sa peau.
Je m'apprêtais à lui servir un deuxième cognac lorsqu'il a de nouveau prononcé le nom de sa fille, Billie, dans un murmure à peine audible. C'était la première fois, depuis des mois, qu'il pouvait prononcer ce nom sans qu'il se frappe au silence des murs. Ça devait le soulager de n'être plus seul à l'entendre et à savoir ce qu'il signifiait. Alors il répétait le nom, Billie, l'accompagnant d'un sourire triste, un sourire d'à quoi bon me remerciant néanmoins d'être là et de n'être pas sourde. Il m'a ensuite parlé de Pixie, le chat de Billie, de Ronie le crapaud et de toute la ménagerie qui copulait et se reproduisait à qui mieux mieux dans la mémoire de son vieux Mac, pareils à une bande de microbes heureux dans une bouse de vache se prélassant au soleil. Il se livrait à ces accouplements transgéniques afin que les animaux parlants ne disparaissent jamais de l'univers des petits. Il n'y avait rien de plus triste, selon lui, qu'un enfant qui ignore que les arbres chantent et que les oiseaux récitent des fables en grignotant du fromage. Il gagnait sa vie avec ces bestioles colorées, principalement avec Ronie le crapaud, son plus vieil et plus grand ami, inventé un jour de pluie, bien avant Billie. S'il avait su, il aurait inventé Billie sous la même pluie que Ronie et lui aurait conféré des superpouvoirs afin qu'elle puisse arracher les griffes des méchants et leur crever le ventre avec.
Un paquet de souvenirs désordonnés ont déboulé devant moi quand il a mentionné Batcrap, une de ses dernières créations, et j'ai quitté l'espace-temps où sa voix tentait de créer une indestructible Billie pour me retrouver à l'ombre de Superman. Lorsque j'ai réintégré la cuisine, il parlait d'immortalité, la pire chose qui soit, l'immortalité sur terre. Il aurait voulu que Billie vive, qu'elle casse les prédateurs d'enfant en quatre et les balance dans des bennes à ordures, mais pas au prix d'une condamnation à cette vie éternelle qu'on essayait d'insuffler à des héros vite dépassés, qui finissaient par crever de solitude et d'ennui. Il avait raison, l'éternité était un châtiment que ne méritaient pas les innocents, et pourtant, j'avais maintes fois souhaité que Mike soit parti pour la gloire et l'infini, alors qu'il m'aurait fallu espérer pour lui une simple vie, qui aurait duré le temps d'une vie normale et ne se serait terminée qu'avec la rupture des os fragiles et l'éclatement du cœur fatigué.
De fil en aiguille, on est passés de l'éternité à la mère de la petite, Lucy-Ann, noyée dans son sang, partie pour les jardins éternels où elle croyait que Billie cultivait des fleurs translucides. Tout en observant les motifs de la nappe à travers son verre, il m'a demandé si les fleurs translucides existaient. Je ne m'étais jamais posé la question mais, que je sache, il n'existait aucune fleur qui puisse se perdre ainsi dans le paysage, aucune fleur dont les pétales n'étaient que pure transparence, de la couleur quasi absente du verre et de l'eau. Si Lucy-Ann avait raison, cela signifiait que seule Billie, dans le monde fantasmatique des hommes, cultivait des fleurs proches de l'invisibilité, a-t-il conclu, puis il s'est excusé d'avoir pris le plancher pour me décrire les paradis que se forgent les mères mortes. Parlez-moi plutôt de Mike.
Je ne savais pas quoi lui raconter. En lui décrivant l'orage, un peu plus tôt, il me semblait avoir tout dit. Je suis donc repassée par le bassin magique pour lui parler des histoires que nous nous inventions, des hiboux fritomanes aux héros invincibles, jamais à court d'idées, de rêves ni de plans de nègres. On se prenait pour Lois Lane et Superman, ai-je lâché en repoussant un rire qui sonnait faux, puis j'ai écrasé le petit maudit tas de miettes que j'avais réunies sur la nappe pendant que je tentais de définir la transparence du bassin magique, des forteresses de glace, de l'œil du hibou, jaune et laissant couler la nuit sur le sol des forêts. J'écrasais une miette sur une des cerises vertes égarées dans les motifs de la nappe quand j'ai constaté que Bill Richard était perdu dans un nuage qui devait dater de cinq ou six ans, peut-être sept, peut-être huit, un de ces vieux nuages increvables stagnant toujours au-dessus des mêmes têtes et imprégnant l'atmosphère d'une odeur de moisi que seuls respirent ceux qui ne sont pas plongés dans le nuage. Se rendant compte que je l'observais, il a ramené vers moi son nuage pour me dire que Billie aussi aimait les histoires, comme toutes les Billie, comme tous les enfants. Il lui en avait tellement raconté qu'il se demandait parfois si elle n'avait pas confondu son agresseur avec quelque personnage qu'il avait imaginé pour elle. Si ça se trouve, elle est entrée dans un conte, a-t-il murmuré, une fable où elle a peut-être croisé Mike, car les enfants se rencontraient toujours dans des histoires où ils devenaient écureuils ou fées, où ils portaient des capes et des souliers magiques, déguisés en mensonges inoffensifs. C'est ainsi qu'ils apprivoisaient le monde et le touchaient, en coloriant la surface des miroirs.
Il a posé un doigt sur la goutte de cognac renversée, puis il a tracé un cercle avec, lentement, si lentement que le cercle a séché, et il a dit c'est ça, les enfants disparus deviennent des Hansel, des Gretel, des personnages de contes et de légendes. Le doigt toujours posé sur le cercle gommeux, il s'est alors mis à réinventer l'histoire de Billie et de Mike, mêlant son supercrapaud à leurs aventures, et je lui ai imposé un autre cul sec, espérant que l'ivresse le déleste un peu du poids de ces fictions qu'on s'invente pour ne pas crever, mais qui finissent par nous bouffer tout rond, comme les enfants et les rêves enfuis.
Les disparitions de Mike et de Billie appartenaient au monde incompréhensible du hasard et de l'inattendu. Chercher à les expliquer aurait été aussi vain que d'essayer de comprendre pourquoi Dieu jouait aux dés avec des adversaires archi-nuls. Coincé entre Ronie et les sept nains, Bill Richard continuait pourtant à chercher une issue à son ignorance, puis il s'est mis à marmonner des paroles incompréhensibles à propos du chant des oiseaux, de la voix de Billie tentant de se frayer un chemin à travers le pépiement des mésanges qui venaient picorer dans sa main, si petites et confiantes, alors qu'il suffisait d'une pression de la main pour fermer à jamais leurs yeux noisette. Bill Richard s'enfonçait, régressait jusqu'aux premiers jours de douleur, promenant son ivresse là où ça fait le plus mal. Il était temps de créer une diversion avant qu'il s'étende sur la vieille nappe échiffée, avec les morts que nous y avions couchés dans la lumière tombante. J'ai fait rouler le bouchon de la bouteille de cognac dans sa direction en lui disant que s'il associait tout ce qui lui échappait à quelque croyance ésotérique, il se ramasserait entouré de boules de cristal et de tables parlantes qui lui ordonneraient de peindre ses murs en noir et mauve.
J'ignore ce qui s'est produit à ce moment-là ni sur quel bouton j'avais pesé, mais il a éclaté d'un fou rire incontrôlable, reprenant un soliloque où il était question de trente sous coulés dans le plexiglas. Pendant qu'il allait se moucher aux toilettes en répétant putains de trente sous, j'ai préparé un bol de pop-corn, ne trouvant rien d'autre qui se cuisine aussi facilement en trois minutes, et je suis retournée m'asseoir en face de lui, à la table sur laquelle il faisait glisser la photo de Billie, qu'il avait ressortie de son portefeuille pour se prouver qu'il n'était pas fou.
Dans le silence retombé, on s'est rempli la bouche de pop-corn en faisant craquer les grains mal éclatés entre nos molaires plombées, puis il a été question de départ, de voyage, de fuite. On cherchait ce qu'on pouvait encore avoir à se dire quand, lentement, j'ai glissé ma main sur la nappe, ma main gauche, celle qui n'était pas salie de beurre et de sel. Je pourrais prétendre que j'ai agi machinalement, sans y penser, mais ce serait totalement faux. J'avais envie de toucher cet homme qui se préparait à partir alors qu'il venait à peine de débarquer, j'avais envie de laisser en lui une marque indélébile, qu'il emporterait avec la bombe à retardement qui clignotait dans sa poitrine et pouvait le réduire en charpie aussi bien à Honolulu qu'à Rivière-aux-Trembles. J'avais envie, en fait, de lui donner quelques regrets. Mais Bill Richard avait suffisamment de regrets. Il n'avait pas besoin de s'encombrer des souvenirs d'une femme qui pendait les hiboux et s'empêtrait dans tout ce qui lui rappelait son enfance. Il s'est levé de table, un peu chancelant, puis je l'ai vu traverser la rue, tête basse, pendant que Lili Marnie, assise au bord de la nuit, construisait un château avec quelques morceaux de maïs éclaté.
Il y a de l'orage dans l'air. Le temps immobile est chargé d'odeurs de pluie et de lourds nuages arrivent lentement de l'ouest. Ce sera mon premier et dernier orage à Rivière-aux-Trembles, car je pars demain, 10 mai, pour rendre visite à Régine, après quoi je m'envolerai pour l'Australie. Mes valises sont prêtes, le frigo est vide, et j'ai réussi à embaucher un homme du village pas trop regardant sur ses principes quand il y a du fric à la clé, afin qu'il s'occupe de la maison durant mon absence. Cette maison sera mon lieu de recueillement, mon pied-à-terre, au cas où la nostalgie du pays me prendrait et où j'aurais envie de me reposer près d'une rivière disparue.
C'est Marnie Duchamp qui m'a parlé du nom de la rivière. Nous étions assis à sa table, avec Billie et Mike, quand elle a murmuré Nana-shipu, Nanamiu-shipu, Nanamassiu-shipu, me racontant comment les tremblements de la terre avaient pris la forme d'arbres inexistants. C'était au début de notre conversation, avant que Superman atterrisse dans les miettes de pain sec, qu'elle glisse sa main vers la mienne et que je ressente soudain le besoin pressant de prendre l'air. Elle me parlait à ce moment-là de ces images d'infini qu'elle disait n'appartenir qu'aux habitants des cimes, oiseaux ou écureuils possédant le privilège de vivre plus près du vent. Plus près du bleu, disait-elle également en parlant de cette couleur aussi insaisissable que la tranquillité qu'elle devait en principe vous inspirer. Avec Mike, elle avait essayé de toucher le bleu du lac aux Barbotes et de la rivière aux Trembles, qui se fondait dans la transparence des eaux dès que vous vous penchiez sur elles. Le bleu était une illusion dont elle désirait s'approcher et je pariais qu'elle y réussirait, qu'elle toucherait l'illusion du bout des doigts et se libérerait ainsi du poids des rêves. Elle avait une maison, des fleurs, des arbres et un ami nommé Phil sur qui elle pouvait s'appuyer si jamais la vue du lointain l'étourdissait. À Rivière-aux-Trembles, Marnie Duchamp avait prise sur le sol, ce qui lui permettait d'observer le ciel sans y tomber. Je ne possédais pas de telles assises, raison pour laquelle il me fallait partir et tenter de trouver mon équilibre à moi dans le mouvement.
La lumière commençait à descendre quand je lui ai annoncé mon départ prochain. On était un peu soûls, complètement épuisés, et on ne comprenait pas vraiment ce qui nous avait poussés à vider notre sac sur une nappe collante, alors qu'on savait très bien que, dès qu'il n'y aurait plus rien à dire, chacun ramasserait ses vieux souvenirs coupables et repartirait avec son sac sous le bras. C'est peut-être cette conscience de la piètre utilité de la confidence qui a incité Marnie Duchamp à avancer une de ses mains vers les miennes, brisant au passage le petit tas de miettes qu'elle éparpillait puis reconstituait sans cesse. Là où une comète avait auparavant laissé son empreinte, un peu de rose a chauffé ses joues, rose rose, Billie, rose Purple Cloud et Rainbow's End, de la couleur des arcs-en-ciel, des nuages de fin de journée et des fleurs alignées sur les murs de la maison de Marnie Duchamp dans des cadres de bois. Sur le moment, je n'ai pas su si ce geste était sa façon à elle de me souhaiter bon voyage ou s'il s'agissait au contraire d'une timide tentative pour me retenir, afin qu'on conjugue nos passés pourris et qu'on se fabrique un semblant d'avenir sur des ruines branlantes.
J'ai avancé une main à mon tour, juste assez pour que le bout de nos doigts se rejoignent et brisent le charme. Ce geste n'a pas rompu l'enchantement et, pendant un instant, j'ai eu envie de sauter sur cette femme, de la renverser sur la table et de m'étendre sur elle, haletant, le pantalon à moitié défait et la tête pleine de mots désespérés. On avait tant en commun, tant de nuits blanches à s'arracher les cheveux et à se ronger les ongles, que la chose la plus naturelle au monde aurait été de me précipiter sur le corps de cette femme triste et d'enfouir ma main dans ses cheveux pour qu'elle m'inonde d'injures, me morde, me frappe et m'accuse de ne pas valoir davantage que tous ceux qui avaient essayé de la soulager en la montant comme des cow-boys de rodéo. Je n'avais pas baisé depuis trois ans parce que j'étais persuadé que si je touchais à une femme, ma douleur et ma rage l'écrabouilleraient avant qu'elle ait eu le temps d'écarter les cuisses, mais avec Marnie Duchamp, la situation était totalement différente. On était sur un pied d'égalité et on avait pris assez de coups pour être capables d'en essuyer d'autres et d'arracher quelques bouts de chair si ça pouvait nous débarrasser de la tension qui nous oppressait jour et nuit.
J'essayais de me persuader qu'une partie de jambes en l'air était probablement la seule activité intelligente à laquelle pouvaient se livrer un homme et une femme qui se tendaient la main par-dessus un précipice couvert d'une nappe de plastique où les doigts nerveux de la femme érigeaient des monticules de miettes qu'ils détruisaient aussitôt, mais une force s'apparentant à une forme d'instinct de survie me tenait rivé à ma chaise. Si je me ruais sur cette femme, soit je ne quitterais jamais Rivière-aux-Trembles, soit je partirais la queue entre les jambes, tourmenté par le sentiment d'avoir sali la seule personne qui comprenait le nom de Billie.
J'ai reculé ma main pour la diriger vers la photo de la puce que j'avais ressortie de mon portefeuille afin d'oublier les oiseaux, tous les oiseaux qui avaient pris la voix de Billie à l'époque où j'espérais encore un signe du ciel et de l'enfer, et j'ai annoncé à Marnie Duchamp que je devais partir. Elle n'a pas protesté. Elle a ramené sa main vers elle, un peu hésitante, un peu tremblante, puis elle a érigé un nouveau monticule avec des morceaux de pop-corn. Quand je suis sorti, elle était toujours assise à la table, fredonnant tête basse une chanson empruntant l'air de Lili Marleen. Dans le ciel noir de Rivière-aux-Trembles, quelques étoiles brillaient, mais pas un pan de bleu n'adoucissait l'horizon.
Je ne l'ai pas revue depuis cette rencontre qui a failli se terminer dans les halètements et je n'ai pas cherché à la revoir. Tout ce qu'on avait à se dire a été dit. Si on se croise de nouveau, on sera forcés de s'inventer un présent et de s'y mettre en scène, avec tout ce que cela comportera de mensonges et d'inutiles tentatives pour faire avaler à l'autre qu'on a une vie qui se tient debout.
De toute façon, je pars demain et n'ai plus le moindre intérêt pour quelque présent que ce soit. Dans les semaines, les mois et les années à venir, ma vie ne sera qu'une forme de passage entre deux lieux. C'est la raison pour laquelle j'ai d'abord choisi l'Australie, qui me permettra de changer de décor à volonté, de visiter à la fois un pays et un continent, de voyager de la Grande Barrière de corail au désert des Pinnacles, à cheval sur des frontières dessinées dans la matière du temps. Si l'Australie a un message à me transmettre, je le découvrirai peut-être là, dans les forêts d'eucalyptus où survivent les quelques milliers de koalas ayant échappé à l'expansion des villes, aux feux de forêt et à la vogue des toques ou des cols de fourrure. Je n'emporte rien avec moi, sinon quelques photos de Billie et ce vieil ordinateur dans lequel se multiplie Ronie le crapaud. Si j'en ai l'occasion, je l'initierai aux mœurs des marsupiaux, avec des poches roses, papanoute, pas des grises, c'est pas beau, et inventerai pour Billie des histoires où le bonheur aura les yeux d'un ourson de peluche dormant serré contre le tronc d'un arbre. Il est bien entendu possible que la grande fièvre australasienne me saute sur le paletot dès que j'aurai mis les pieds sur le continent, mais je cours le risque, au cas où un chercheur rattaché à l'Université de Sydney aurait découvert un antidote à ce mal des profondeurs ne s'attaquant qu'à ceux qui dégringolent de ravins en abysses.
Il me restait une seule chose à faire avant de partir, donner à Pixie l'enterrement qu'il méritait et lui permettre ainsi d'accéder à sa septième ou à sa huitième vie. Je l'ai sorti avant-hier du caisson dans lequel il se les gelait depuis trop longtemps et l'ai amené près de l'érable ombrageant ce qui ressemble à un ancien jardin, derrière la maison. La terre y était meuble, amollie par les dernières pluies, et je ne risquais pas d'y tomber sur un fond de roc ou de terre gelée. À cet endroit, il pourrait profiter du soleil du matin, de l'ombre de l'érable quand la lumière se ferait trop intense et sortir chasser le mulot si ça lui chantait.
J'ai travaillé moitié à genoux, moitié courbé en récitant des prières impies qui m'auraient valu l'asile ou l'excommunication si un psy ou un pape s'était baladé dans le coin. Quand j'ai eu fini de prier et de creuser, j'ai déposé la couverture de flanelle confectionnée par Billie au fond de la minuscule fosse et j'y ai délicatement couché le vieux batêche avec les quelques jouets qui lui permettraient de vivre correctement sa vie d'éternel pharaon. Puis j'ai braillé dors bien Pixie, adieu mon grand, mais j'aurais pu brailler dors bien Billie, adieu ma grande, ma toute grande, ma toute belle, que ça n'aurait rien changé. J'enterrais Billie, la vie de Billie, les jouets de Billie en même temps que Pixie. Je mettais en terre ma toute belle, ma toute grande, dont j'ai déposé une photo sur le ventre encore froid de Pixie, sa photo au sourire espiègle. Après, je me suis hâté de le recouvrir avant d'avoir envie de m'étendre à côté de lui jusqu'à la fin de ma neuvième vie, la photo de Billie contre ma joue. J'ai ensuite écrit le nom de ce foutu chat sur la largeur du jardin à l'aide d'un piquet que j'enfonçais dans la terre aussi profondément que possible, afin qu'il soit visible d'une de ces étoiles d'où Billie observe peut-être mes piètres efforts pour apprendre à marcher sans elle. Je n'ai cependant pas planté de croix en bordure du jardin, puisque Pixie ne croyait pas plus en Dieu que moi et que je suis excommunié d'office.
La meilleure chose à faire après cet enterrement aurait été d'aller me laver, mais j'ai décidé de m'offrir ce que certains appelleraient un pèlerinage. J'ai pris le semblant de sentier menant à la rivière et suis allé m'asseoir près du bassin magique de Marnie Duchamp et de Michael Superman Saint-Pierre, peut-être creusé par un lointain tremblement de terre ou une minuscule météorite ayant voyagé depuis l'un des corps célestes se mouvant dans la galaxie kryptonienne.
Je m'attendais presque à trouver une deuxième croix de bois non loin de la rivière, au pied de laquelle un bouquet de fleurs blanches se serait étiolé, près d'une photo de Michael Faber tenant une des pattes de Nuage inclinée en un geste d'adieu. Peut-être le père du deuxième Mike planterait-il cette croix un jour, quand l'envie d'aller se jeter sous le train passant tous les lundis et jeudis soir à Saint-Alban l'aurait quitté, à moins qu'il soit devenu comme moi et ne croie plus que le fait d'invoquer cet hypothétique être céleste se déplaçant dans le corps de l'univers puisse apporter quelque secours à son enfant.
Pour l'instant, seule la croix du premier Michael témoignait des drames survenus près de Nanamiu-shipu, une simple croix ne mentionnant ni l'âge du disparu, ni le jour de sa disparition, devenant ainsi la croix des deux Michael, qui ne formaient désormais qu'un seul Michael parmi tous les enfants disparus à l'orée des forêts tremblantes. Là où un bouquet de fleurs étiolées fleurirait peut-être un jour, j'ai tracé le nom de Billie, l'imaginant qui s'avançait dans la rivière, main dans la main avec les deux garçons.
Au retour de mon pèlerinage, j'ai lancé mes vêtements dans un coin de ma chambre et me suis payé une douche qui aurait fait hurler quiconque se préoccupe un tant soit peu des réserves d'eau de la planète, à commencer par le gars qui, il n'y a pas longtemps, s'inquiétait de la sécheresse tarissant l'Australie. Ce gars était crevé et ne désirait qu'une chose, perdre toute conscience du temps, de l'espace, des déluges et de la guerre que livreraient bientôt les pays arides aux pays baignés d'eau douce.
J'y suis parvenu tant bien que mal, mais aujourd'hui que l'heure du départ approche, les souvenirs refluent comme à l'heure des bilans. Le seul bilan que je peux cependant dresser est que je suis désormais seul avec Ronie, sans attaches ni espoirs. S'il n'y avait cet orage qui s'avance, je partirais tout de suite, mais le vent s'est déjà levé et des éclairs strient le ciel, au-dessus de la forêt, qui feront bientôt trembler le sol et la rivière pendant que des giclées de pluie furieuse se frapperont aux carreaux des fenêtres.
Ce sera mon dernier orage avant ceux qui feront rugir les démons de Tasmanie et ployer les eucalyptus. Ce sera mon dernier souvenir de Rivière-aux-Trembles, celui d'un coin de pays irrigué par la fureur et la lumière. Si j'y reviens un jour, des arbres auront poussé, d'autres seront morts, calcinés par la foudre ou rongés par les vers, Marnie Duchamp aura vieilli et un chat au pelage blanc et roux m'attendra peut-être sur la galerie, fidèle réincarnation d'un vieux schnock auquel une enfant nommée Billie racontait autrefois l'histoire du monde.
J'étais à ma fenêtre lorsque les premiers éclairs ont bariolé le ciel derrière la colline des Loups. Il n'y a pas longtemps, j'aurais vivement fermé les rideaux et poussé à fond le volume de la chaîne stéréo, dans laquelle j'aurais auparavant inséré un CD de Jethro Tull ou de Led Zeppelin en prévision du vacarme à venir. Genoux repliés contre la poitrine, je me serais balancée dans mon fauteuil en massacrant Stairway to Heaven jusqu'à la fin de l'orage, puis je me serais jetée dans le ménage ou dans toute autre activité susceptible de m'occuper les mains et de m'empêcher de penser.
Cette fois-ci, je suis demeurée à la fenêtre, laissant la violence de la lumière frapper ma rétine et les tremblements du tonnerre rouler sur ma peau moite pendant que les battements de mon cœur s'accéléraient. Dans ma tête, défilaient les paroles de Stairway, des anneaux de fumée, des « rings of smoke », s'enroulaient dans les arbres, mais ceux-ci m'indiquaient maintenant qu'il n'était pas trop tard pour changer de route. Page et Plant avaient raison, « there's still time to change the road you're on », et c'est précisément ce à quoi je m'appliquais, assise devant cet orage dont je tentais enfin de percevoir la beauté.
Je commence à sentir la pluie, Mike, ai-je murmuré tandis que quelques branches de Mary Delahunty, pareilles à des tumbleweeds poussés par les vents du désert, s'envolaient sous le ciel blanc, virevoltaient quelques instants puis s'écrasaient contre la serre des hybrides, dont les murs de verre s'illuminaient toutes les dix ou quinze secondes. La peur me rivait encore à ma chaise, mais elle se mêlait à la joie fragile de qui réapprend à marcher. Si le ciel m'avait voulue, il m'aurait prise bien avant cela... Lorsque les grondements se sont amenuisés pour rouler vers l'est, j'ai enfilé mon imperméable et je suis sortie m'asseoir dans le jardin, sous la pergola. Le vent était tombé, une fine pluie de fin d'orage martelait doucement le toit de la maison et la nature se dépliait lentement après l'assaut de la tempête. C'est dans ce calme que je voulais désormais vivre, dans ce soupir à peine audible succédant à la tourmente.
Il y aura toujours quelque imbécile, bien sûr, pour venir lancer des pierres dans mon jardin ou me téléphoner en pleine nuit, quelque enfant dressé dans la stupidité pour me crier maudite sorcière et s'enfuir rapidement, l'ignorance a la vie longue, mais si la rivière a su résister aux tremblements de la terre, probablement puis-je affronter les secousses de plus en plus rares du cataclysme qui a ruiné une partie de ma vie et dont personne n'a jamais pu situer l'origine, moi pas plus que les autres. Tout ce que je sais de cette histoire, c'est où et quand elle a commencé.
Il était une fois deux enfants et une rivière, disent les premiers mots de l'histoire, il était une fois un orage, puis il fut ensuite une enfant, seule au milieu de la forêt perdue, une Marnie égarée devant quelques cailloux. Il était une fois deux enfants, puis il n'en fut plus qu'un. Il n'y a pas d'autre véritable histoire. Il est possible, je sais, qu'un troisième personnage se soit glissé entre les enfants et que, dans mon affolement, j'aie fait disparaître la main poilue étranglant Mike pour l'enduire d'une épaisse couche de peinture magique et la rendre invisible, tout est possible, mais je ne le saurai que si le temps parvient à tromper la magie pour laver la peinture. Je sais par contre que la suite de mon histoire commence ici, à quelques semaines d'un été dont j'attends la chaleur suffocante et les nuits belles à en brailler. Ce sera mon premier été depuis l'été 79, trop beau pour être vrai, si beau que le mois d'août a perdu les pédales et renversé le ciel dans la rivière Tremblante. Mon histoire commence ici et maintenant, et je sais aussi que plus rien, ni ragots ni bêtise, ne me fera fuir cette maison où je cultiverai, nouvel été après nouvel été, les rosiers d'Alex Duchamp et de Marie Beaupré.
Je m'appelle Marnie et je m'en fous. Je continue à me demander à quoi mes parents ont pensé en me donnant ce nom, mais ça n'a pas de réelle importance, car je m'appelle aussi Squouirèle et ne crains plus l'orage.
Aucun convoi funèbre ne descendait la rue Principale de Rivière-aux-Trembles lorsque Bill Richard a quitté le village, le 10 mai 2009, au volant de sa Volvo 2005 S40, avec pour tout bagage un ordinateur, une valise Samsonite et un album de photos de sa fille. L'averse du matin avait cessé, mais la pluie reprendrait bientôt, qui rendrait son trajet morose et l'empêcherait de poser son coude sur la vitre baissée pour chanter à tue-tête n'importe quel succès ayant marqué ces années lointaines dont il ne pouvait alors mesurer l'insouciance. En attendant, il fredonnait « Hit the road, Bill, don't you come back no more ». Mais il ne savait pas s'il reviendrait. Il partait, c'est tout. Il bougeait. Il s'en allait là où il ouvrirait tous les jours les yeux sur des décors anonymes ou des paysages grandioses au milieu desquels il aurait peine à reconnaître son ombre et où il espérait parvenir à oublier son propre nom. Il n'était ni amer ni triste, seulement soulagé d'avoir enfin compris qu'il ne pouvait vivre sous un même toit des mois durant sans que ce toit menace de s'écrouler par suite de quelque phénomène ayant un lien avec la pression négative de l'ennui.
Quand la maison de Marnie Duchamp est apparue au loin, le premier souvenir qu'il conservait de cette femme s'est matérialisé devant les essuie-glaces, frêle et noir fantôme voulant l'ensorceler. Ne regarde pas, s'est-il dit en appuyant sur l'accélérateur, ne regarde surtout pas en arrière, mais il a malgré lui tourné la tête, à l'instar de cette femme subitement transformée en statue de sel sur un chemin de sable. En apercevant Marnie qui bêchait la terre au fond de son jardin, il a brusquement freiné pour se garer sur l'accotement. La peur lui tordait les tripes, une sueur épaisse mouillait sa chemise, mais il n'avait d'autre envie que d'ouvrir à toute volée la portière de sa Volvo pour courir vers elle, lui sauter dessus et la renverser dans le jardin trempé. S'il n'avait connu le pouvoir des femmes tristes et redouté que les paysages de l'Australie s'effacent derrière ceux que lui montrerait le regard insondable de Marnie Duchamp, rempli de roses, de rivières et d'écureuils grimpant aux arbres verdoyants, il aurait arraché cette foutue portière pour aller s'enrouler dans les longs cheveux roux sentant la terre. Mais il connaissait les femmes tristes.
Il s'apprêtait à repartir, ses mains moites glissant sur le volant, ne regarde plus, quand Marnie s'est retournée. Après un moment d'hésitation, elle a laissé tomber sa bêche et s'est dirigée vers lui. Arrivée près de la route, elle s'est arrêtée pour s'appuyer au hayon de son pick-up, ramenant derrière son oreille la mèche de cheveux qui lui chatouillait la joue. Elle n'irait pas plus loin. Elle a levé la main gauche, celle qui avait laissé sur son visage l'empreinte d'un nuage sombre, l'a agitée en signe d'adieu ou d'au revoir, puis la main est retombée au moment où l'averse reprenait de plus belle. Il a levé la main aussi, en signe d'adieu, et il a repris la route sous la pluie battante, mais l'image de Marnie Duchamp dans son imperméable jaune, jaune citron, Billie, jaune bouton d'or et bouquets échevelés de pissenlits, les fesses appuyées sur son vieux pick-up rouge et les cheveux mouillés, l'a poursuivie durant tout le trajet qui l'éloignait de Rivière-aux-Trembles. Elle le poursuivrait aussi longtemps après lorsque, debout devant les vagues écumeuses de l'océan Indien, il se demanderait s'il n'avait pas raté une chance, ce jour-là, de se refaire une véritable vie dans les bras d'une femme qui dessinait des nuages sur ses joues, des nuages ou des comètes filant vers son regard d'écureuille rousse.
Marnie Duchamp, quant à elle, a regardé la Volvo s'éloigner jusqu'à ce qu'elle disparaisse derrière la côte Croche, puis elle est retournée à son jardin où, toute la journée, elle s'est éreintée malgré l'averse à bêcher, à sarcler et à tailler des buissons, demandant à la pluie de la toucher et souriant quand l'odeur de l'eau fraîche, parfois, frôlait ses mains tachées. En fin d'après-midi, elle s'est dirigée d'un pas tranquille vers la rivière aux Trembles, où elle n'était pas retournée depuis le jour où elle y avait emmené Holy Owl. En passant devant les anciens champs de Lucien Ménard, au milieu desquels se dressait la maison fermée de Bill Richard, elle a regretté de n'avoir pas marché jusqu'à la Volvo, quelques heures plus tôt, pour embrasser Bill Richard à pleine bouche, de gré ou de force, et lui donner une bonne raison de faire demi-tour. Puis elle s'est dit que si celui-ci voulait l'embrasser, il reviendrait un jour ou l'autre, un peu plus vieux, bien sûr, mais guéri de cette fièvre qui cernait son regard d'ombres malades.
Elle s'est ensuite engagée dans le sentier menant à la rivière, près de laquelle elle est demeurée assise jusqu'après la brunante, à côté d'une croix de bois où le nom de Michael, peu à peu, a disparu dans l'obscurité.
### REMERCIEMENTS
Puisqu'il est là, juste à côté de moi, je remercie d'abord Pierre, mon editor maison, pour ses multiples lectures, ses conseils avisés et sa solide épaule d'ours.
Je tiens bien entendu à remercier tout le personnel de Québec Amérique, à commencer par Jacques Fortin, pour sa confiance et son soutien, et parce qu'il ne s'épargne aucun effort pour élargir mon lectorat. Je m'en voudrais ici d'oublier l'infatigable Isabelle Longpré, ma gentille et dévouée directrice littéraire, dite GDDL au long cours, à qui rien ou presque ne parvient à échapper, de même que Luc Roberge, Rita Biscotti, Roxane Dumas-Noël, Sandrine Donkers, Anne-Marie Fortin, Mylaine Lemire, Louise Piché, Lyne Trudel et tous ceux et celles avec qui j'ai eu l'occasion de travailler chez QA.
Mes remerciements vont également à Joséphine Bacon et à Rita Mestokosho, qui ont généreusement accepté de traduire pour moi en innu-aimun quelques termes qui m'ont permis de mieux comprendre le véritable nom de la rivière aux Trembles. Tshinashkumitin, les filles.
Je remercie aussi mon amie Yvette, qui a soulagé quelques-unes de mes angoisses linguistiques, Claude Frappier et Diane-Monique Daviau, qui ont révisé les dernières versions de ce roman, et Gilles Mitchell, de la Sûreté du Québec, qui a gentiment accepté de répondre à mes questions. J'espère qu'il ne m'en voudra pas d'avoir parfois placé ses collègues fictifs dans des situations où, inévitablement, ils n'auraient pas le beau rôle. Je me reprendrai dans un autre roman.
Un gros merci à ma sœur Viviane, encore, pour ses photos de koalas assoiffés. C'est à elle que Bill Richard, en quelque sorte, doit sa fièvre australienne. Et, comme à la télé, je remercie les autres membres de mon irremplaçable famille, maman (mamanoute), Odette, Louise et Tommy, de même que mes neveux, mes nièces, mes beaux-frères et mes chats, ma gang de Noutes, quoi, pour leurs constants encouragements mais, surtout, surtout, parce qu'ils parviennent à me faire croire que je suis pas si pire, en tant qu'écrivaine.
Un dernier merci, enfin, aux membres du fonds Gabrielle-Roy, dont la confiance m'a permis de terminer ce roman dans un lieu pour moi paradisiaque, ainsi qu'au Conseil des Arts du Canada et aux membres du jury qui m'ont accordé une bourse sans laquelle je pataugerais encore dans les eaux de la rivière Tremblante.
### NOTE
. Pour connaître le sens de ce mot, il faut lire le roman...
#
| {
"redpajama_set_name": "RedPajamaBook"
} | 7,993 |
Q: sql max(val1,val2)? is there some way to determine the max value out of two values in SQL?
I use the mod function:
MOD(cnt, cnt/100)
This yields a division by 0 error when cnt is smaller than 100. I therefore would like something like this:
MOD(cnt, MAX(cnt/100, 1))
A: You can use greatest
SELECT greatest(a, b, c) FROM your_table;
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,459 |
\section{Introduction}
Many complex real-world problems of physical and engineering interests have multiple scales and uncertainties, for example, the subsurface flow models in highly heterogeneous porous media. Due to the measurement noise and the lack of knowledge about the physical properties, the model inputs usually contain some uncertainties. The uncertainties are often
parameterized by random variables. Thus, parameterized partial differential equations (PPDEs) are used to characterized the models with uncertainties.
One of the great challenges in these models is to efficiently and accurately solve the PPDEs with high-contrast multiple scales to predict the model's outputs and estimate the model's parameters. Both the multiple scales and uncertainties have great challenges for numerical simulation. To reduce computation complexity and improve simulation efficiency, model reduction methods are desirable to the complex multiscale models.
In recent years, many numerical methods have been proposed to solve the complex PPDEs using multiscale methods and model reduction methods \cite{clc11, pde08, jpl12, ncn08,rhp08, aej08, arbogast2007multiscale, efendiev2013generalized, jiang2010stochastic}.
In many practical applications such as multiphase flow in porous media, local mass conservation is necessary for a numerical method to solve the problems.
The mixed finite element methods retain local conservation of mass and have been found to be
useful for solving these problems.
To illustrate the main idea of our approach, we consider the following mixed formulation of an elliptic PDE with random inputs,
\begin{eqnarray}
\label{mix-eq}
\begin{cases}
\begin{split}
k(x,\mu)^{-1}v(\mu)+\nabla p(\mu)&=0 ~~\text{in} ~~D, \\
\nabla\cdot v(\mu)&=f(x) ~~\text{in}~~D,
\end{split}
\end{cases}
\end{eqnarray}
where $D \in R^2$ is a bounded spatial domain, $f(x)$ is a source term, and $k(x,\mu)$ is a random permeability field, which may be oscillating with respect to the random parameter $\mu$ and highly heterogeneous with respect to the spatial variable $x$. To take account of the heterogeneities, multiscale methods \cite{abgr05, arbogast2007multiscale, ee03, hughes98, pde08, efendiev2009multiscale}
can be used to solve the problem. In this work, our interest is to use mixed multiscale finite element methods \cite{chen2003mixed, cel15}.
The main idea of mixed multiscale finite element methods is to compute multiscale basis functions for each interface supported in the coarse blocks sharing the common interface.
In order to accurately capture complex multiscale features, a few multiscale basis functions may be required for each coarse block. This is the idea of Generalized Multiscale Finite Element Method (GMsFEM) \cite{cel15, efendiev2013generalized}.
We note that the multiscale basis functions can be computed overhead and used repeatedly for the model with different source terms, boundary conditions and the coefficients with similar multiscale structures \cite{efendiev2009multiscale}.
Because the coefficient $k(x,\mu)$ in equation (\ref{mix-eq}) varies with the realizations of random parameter $\mu$, the GMsFE basis functions usually depend on the parameters.
This significantly effects on the computation efficiency. In order to get the multiscale basis functions independent of the parameter, we build multiscale basis functions based on a set of samples in the parameter space. This will result in a high-dimensional GMsFE space for approximation and bring great challenge for simulation. To alleviate the high dimensionality of the GMsFEM space, we
identify a set of optimal reduced basis functions from the high dimensional GMsFE space and develop reduced mixed GMsFE basis methods. Thus
a reduced order multiscale model can be obtained by projecting the original full order model onto the reduced multiscale finite element space.
In this work, we focus on mixed GMsFEM and present reduced mixed GMsFE basis methods to solve parameterized elliptic PDEs. Reduced basis method is one of effective model order reduction methods and has been used to solve PPDEs in a low-dimensional manifold \cite{ buffa2012priori, chen2010certified, ncn08, rhp08}. The main idea of the reduced basis method is to construct a small set of basis functions based on a set of snapshots, which are the solutions of the PPDEs corresponding to a set of optimal parameter samples selected by some sampling strategies. An offline-online computational decomposition is achieved to improve efficiency. In offline stage, snapshots are computed and reduced basis functions are generated. In online stage, the output is computed by the resultant reduced model for many instances of parameters and the influence of the uncertainty is estimated.
When the model contains multiple scales and high-contrast information, the computation of the snapshots may be quite expensive since we are required to resolve all scales of the model in a very fine mesh. To this end, we employ mixed GMsFEM to compute snapshots in a coarse grid and develop reduced mixed GMsFE basis methods. To get a set of optimal reduced multiscale basis functions from the snapshots, we propose two sampling strategies: basis-oriented cross-validation (BOCV) and proper orthogonal decomposition (POD). For the two approaches, we select some optimal samples from a training set by a greedy algorithm \cite{gerner2012certified} based on mixed GMsFEM. Then we
construct the snapshots for the reduced mixed GMsFE basis functions based on the optimal sample set. The optimal reduced mixed GMsFEM basis functions are identified by an incremental constructive manner from the snapshots using BOCV and POD. In the BOCV method, the optimal basis functions are searched with a minimum average error for a validation set. The POD is devoted to finding a low rank approximation to the space spanned by snapshots, and
has been widely used in model reduction. We use the two different approaches to generate a set of optimal reduced basis functions, and construct a reduced order model by projecting the full order model to the space spanned by the optimal reduced basis functions. We carefully compare BOCV with POD and find that BOCV can achieve better accuracy and robustness than POD in the reduced mixed GMsFE basis methods.
Although the reduced mixed GMsFE basis model needs much less computation effort than original full order model, it may be not a very small-scale problem because the reduced order model still involves many unknowns.
This is still not desirable for the online computation in a many-query situation.
In order to significantly improve the online computation, we want to get a reduced model representation for the solution and model outputs, which can accurately express the reduced model and can be used to estimate the influence of the uncertainty directly. To get the representation, we propose two methods: least-squares method of snapshots (LSMOS) and sparse tensor approximation (STA). They attempt to construct
the representation with a form of variable-separation, which has been explored in many applications \cite{beylkin2005algorithms, gonzalez2010recent, doostan2009least, doostan2007least}.
STA is particularly useful when the model
suffers from the difficulty of high-dimensionality.
LSMOS is based on Karhunen-Lo\`{e}ve expansion (KLE) \cite{loeve1978probability} and can extract most important information from a set of snapshots. When it is difficult to get the knowledge of a covariance function for the
model's ouputs, we can use method of snapshots \cite{handler2006karhunen} to construct a KL expansion for the model's outputs.
In LSMOS, the coefficient functions of random variables in the KLE are the functions of the random parameters of inputs.
We can use orthogonal polynomial basis functions to express the coefficient functions of the random variables. To determine the coordinates of the orthogonal polynomial basis functions,
we take a few snapshots and use least-squares methods. If the dimension of the random parameters is high, a large number of snapshots are required for the least-squares methods. Thus,
the computation may be expensive. To overcome the difficulty,
we here propose a sparse tensor approximation method to obtain a sparse representation of the model's outputs instead of using LSMOS.
The sparse tensor approximation exploits the inherent sparsity of the model and the reduced GMsFE basis.
The number of the effective basis functions for many practical models is often small. We use the optimization methods from compressive sensing \cite{rish2014sparse} to extract the effective basis functions and obtain
a sparse representation. There are roughly two classes of approaches to obtain the sparse representation: optimization based on $l_0$-norm and convex optimization \cite{tropp2007signal, elad2009sparse, rish2014sparse}. The typical methods of the $l_0$ optimization include orthogonal matching pursuit (OMP) and iterative hard thresholding. The convex optimization based on $l_1$-norm includes least angle regression, coordinate descent and proximal methods. In this work, we get the sparse tensor approximation using orthogonal matching pursuit, i.e., STAOMP.
We integrate the reduced mixed GMsFE basis methods and STAOMP together to get the model's sparse representation in coarse scale.
The proposed model's sparse representation method can significantly enhance the computation efficiency for the multiscale models
with high-dimensional random inputs. The Figure \ref{Schema} illustrates the procedure to get the model's sparse representation based on reduced mixed GMsFE basis methods.
\begin{figure}[htbp]
\centering
\includegraphics[width=5in, height=3in]{diagram4}
\caption{Schema of model's sparse representation based on reduced mixed GMsFE basis methods}
\label{Schema}
\end{figure}
The paper is organized as follows. In the next section, we give some preliminaries and notations for the paper. Section \ref{ssec:PGDDF-LSMOS-STAOMP} is devoted to developing LSMOS and STAOPM to represent a generic random field.
In Section \ref{ssec:Re-MIXGMsFEM-RB-method}, we present the construction of the mixed GMsFE basis functions and introduce the reduced mixed GMsFE basis methods. Section \ref{ssec:sampling algorithm} is to present two approaches for constructing the optimal reduced basis: BOCV and POD. In Section \ref{ssec:Online-online}, we combine STAOMP with the reduced mixed GMsFE basis methods together and get the model's sparse representation. A few numerical examples are provided in Section \ref{ssec:Numerical result} to illustrate the performance of all the methods developed above. In the last section, we make some comments and conclusions.
\section{Preliminaries and notations}
\label{ssec:prelim}
In this section, we present some preliminaries and notations for the rest of paper. Let $L^2(D)$ be the space of square integrable functions over a domain $D$ with the $L^2$-norm $\|\cdot\|_{L^2(D)}^2=(\cdot, \cdot)$.
We define the space
\[
V:=H(\text{div},D)=\big\{u\in [L^2(D)]^d: \text{div}(u)\in L^2(D)\big\}.
\]
The space $V$ is a Hilbert space with the norm given by $\|u\|_{H(\text{div},D)}^2=\|u\|_{L^2(D)}^2+\|\text{div}(u)\|_{L^2(D)}^2$.
We consider parameterized elliptic PDEs in mixed formulation (\ref{mix-eq})
with the Neumann boundary condition $k\nabla p\cdot n=g$ on $\partial D$, where $n$ is the outward unit-normal vector on $\partial D$.
For a simplicity of notation, we denote $L^2(D)$ by $Q$ and
\[
V^0=H_0(\text{div},D):=\big\{u\in H(\text{div},D): u\cdot n=0 ~\text{on}~ \partial D\big\}.
\]
The problem (\ref{mix-eq}) leads to the variational problem: for any parameter $\mu\in \Gamma\subset \mathbb{R}^p$, we find $\{v(\mu),p(\mu)\}\in V \times Q$ such that $v(\mu) \cdot n =g$ on $\partial D$,
\begin{eqnarray}
\label{mix-eq-weak}
\begin{cases}
\begin{split}
a\big(v(\mu),u; \mu\big)-b\big(u, p(\mu); \mu\big)&=0 ~~~\forall ~u \in V^0, \\
b\big( v(\mu), q; \mu\big)&=l(q) ~~~\forall ~q \in Q.
\end{split}
\end{cases}
\end{eqnarray}
where
\[
a(v,w):=\int_{D}k(x,\mu)^{-1}(v\cdot w )dx, ~~b(v,w):=\int_{D}\nabla\cdot(v)qdx,~~\text{and}~~ l(q):=\int_{D}f qdx.
\]
For any parameter $\mu\in \Gamma$, $a(\cdot,\cdot;\mu):V\times V\longrightarrow \mathbb{R}$ is a symmetric bilinear form , $b(\cdot,\cdot;\mu):V \times Q\longrightarrow \mathbb{R}$ is bilinear form, and $l(\cdot;\mu)$ is a bounded linear functional over $Q$.
Given $\mu\in \Gamma$, we evaluate the output $G(x,\mu)$ of the model (\ref{mix-eq-weak}),
\begin{eqnarray*}
\label{evaluate-eq}
G(x,\mu)=L\big(v(x,\mu)\big),
\end{eqnarray*}
where $L$ is a operator on $V$.
For well-posedness of (\ref{mix-eq-weak}), we assume that $a(\cdot , \cdot; \mu)$ is continuous and coercive over $V$ for all $\mu \in \Gamma$, i.e., there exist $\gamma>0$ and $\alpha >0$ such that
\begin{eqnarray}
\label{bounded-a}
\gamma(\mu):=\sup_{u \in V}\sup_{v \in V}\frac{a(u,v;\mu)}{\|u\|_V \|v\|_V}<\infty,~~~ \forall \mu\in \Gamma,
\end{eqnarray}
\begin{eqnarray}
\label{coercive-a}
\alpha(\mu):=\inf_{u \in V}\frac{a(u,u;\mu)}{\|u\|_V^2}> 0,~~~ \forall \mu\in \Gamma,
\end{eqnarray}
and $b(\cdot , \cdot; \mu)$ is continuous
\begin{eqnarray*}
\label{bounded-b}
b(u,q)\leq\gamma_b\|u\|_V \|q\|_Q, \quad \forall u \in V,~ q \in Q, ~~\forall \mu\in \Gamma,
\end{eqnarray*}
and satisfies the inf-sup condition
\begin{eqnarray}
\label{inf-sup condition}
\beta(\mu):=\inf_{q\in Q}\sup_{u\in V}\frac{b(u,q ;\mu)}{\|u\|_V\|q\|_Q}> 0, ~\forall \mu\in \Gamma.
\end{eqnarray}
To fulfill offline-online computation, we assume that the parametric bilinear form $a(\cdot, \cdot; \mu)$ is affine with respect to $\mu$, i.e.,
\begin{eqnarray}
\label{affinely-ag}
a(u,v;\mu)&=\sum_{i=1}^{m_{a}}k^{i}(\mu)a^{i}(u,v), \quad \forall u,v\in V,\quad \forall \mu \in \Gamma,
\end{eqnarray}
where $k^{i}: \Gamma\longrightarrow \mathbb{R}$ is a $\mu$-dependent function and $a^{i}: V \times V\longrightarrow \mathbb{R}$ is a symmetric bilinear form independent of $\mu$, for each $i= 1,\cdots , m_{a}$. The affine assumption (\ref{affinely-ag}) is crucial to achieve the decomposition of offline-online computation for many queries to model's outputs. When $a(\cdot, \cdot; \mu)$ is not affine with regard to $\mu$, such an expansion can be obtained by using some variable separation methods for $a(\cdot, \cdot; \mu)$, e.g., EIM \cite{eftang2010posteriori}, LSMOS and STAOMP. The LSMOS and STAOMP will be introduced in Section \ref{ssec:PGDDF-LSMOS-STAOMP}.
Let $V_h\times Q_h$ be the pair of standard Raviart-Thomas space for the approximation of (\ref{mix-eq-weak}) on a fine grid $\mathcal{K}^h$.
Then the standard mixed FEM of problem (\ref{mix-eq-weak}) reads: given any $\mu \in \Gamma$, we find $\{v_h(\mu),p_h(\mu)\}\in V_h \times Q_h$ such that
\begin{eqnarray}
\label{mix-FEM-dis}
\begin{cases}
\begin{split}
a(v_h(\mu),u; \mu)-b(u, p_h(\mu); \mu)&=0 ~~~\forall ~u \in V_h^0, \\
b( v_h(\mu), q; \mu)&=l(q) ~~~\forall ~q \in Q_h,
\end{split}
\end{cases}
\end{eqnarray}
where $V_h^0=V_h\bigcap\{v\in V_h: v\cdot n=0 ~\text{on}~\partial D \}$.
We remark that the fine-grid solution is considered as a reference solution in the paper.
\section{ Variable-separation strategies}
\label{ssec:PGDDF-LSMOS-STAOMP}
In order to achieve offline-online computation, we want to represent model's inputs (e.g., coefficients and source terms) and outputs
by a form of variable-separation. Let $G(x,\mu)$ be a generic function parameterized by $\mu$.
In this section, we introduce two strategies, i.e., LSMOS and STAOMP, to construct an approximation in the form
\begin{eqnarray}
\label{separating presentation}
G(x,\mu)\approx G_N(x,\mu):=\sum_{i=1}^{N}\zeta_i(\mu)g_i(x),
\end{eqnarray}
where $\zeta_i(\mu)$ only depends on $\mu$ and $g_i(x)$ only depends on $x$.
Let $L_2(\Gamma)$ be the space of square integrable functions over the parameter space $\Gamma$. We consider the approximation in the tensor space $\mathcal{H}\otimes L_2(\Gamma)$, where
$\mathcal{H}$ is a Hilbert space with regard to the spatial space.
We define an inner product on $\mathcal{H}\otimes L_2(\Gamma)$ by
\begin{eqnarray*}
\label{TENSOR PRODUCT}
(w,u)_{\mathcal{H}\otimes L_2(\Gamma)}=E[(w,u)_\mathcal{H}]:=\int_{\Gamma}(w,u)_\mathcal{H} \rho(\mu)d\mu,
\end{eqnarray*}
where $\rho(\mu)$ is the density function for the random parameter $\mu$. The norm is defined by
\[
\|u\|_{\mathcal{H}\otimes L_2(\Gamma)}^2:=(u,u)_{\mathcal{H}\otimes L_2(\Gamma)}.
\]
We assume that $S_N \subset \mathcal{H}\otimes L_2(\Gamma)$ is a finite dimensional subspace space, and $\{\Psi_i\}_{i=1}^N$ is a set of basis functions for $S_N $.
We want to find an approximation of $G(x, \mu)$ in $S_N$ such that
\begin{eqnarray}
\label{output-approximation}
\|G(x,\mu)-\sum_{j=1}^{N} c_{j}\Psi_j(x,\mu) \|_{\mathcal{H}\otimes L_2(\Gamma)}\leq \delta,
\end{eqnarray}
where $\delta$ is a given threshold.
\subsection{Least-squares method of snapshots}
In this subsection, we introduce a least-squares method of snapshots to get the approximation (\ref{output-approximation}).
Let $\Xi_{t}$ be a collection of a finite number of samples in $\Gamma$ and the cardinality $|\Xi_{t}|=n_t$.
For $\forall ~\mu\in \Xi_{t}$, we can split $G(x,\mu)$ into two parts, i.e.,
\[
G(x,\mu)=\bar{G}(x)+\tilde{G}(x,\mu),
\]
where $\bar{G}(x):=E[G(x,\cdot)]=\frac{1}{n_{t}}\sum_{i=1}^{n_{t}}G(x,\mu_i)$ is the mean, and $\tilde{G}(x,\mu)=G(x,\mu)-\bar{G}(x)$ is a random fluctuating part.
To obtain $\tilde{G}(x,\mu)$, we take a set of snapshots $\{\tilde{G}(x,\mu_i)\}_{i=1}^{n_{t}}$ and compute a covariance matrixes $\textbf{C}$, whose entries can be defined by
\[
\textbf{C}_{n,m}:=\frac{1}{n_{t}}\bigg(\tilde{G}(x,\mu_n),\tilde{G}(x,\mu_m)\bigg)_\mathcal{H}.
\]
Let $\{\hat{\lambda}_k, \textbf{e}_k\}$ be the eigen-pairs (normalized) of $\textbf{C}$, $1\leq k\leq n_{t}$. Set $(\textbf{e}_k)_j=e_k^{j}$, we define the functions
\begin{eqnarray}
\label{KLE basis}
g_k(x):=\frac{1}{\sqrt{\hat{\lambda}_k n_{t}}}\sum_{j=1}^{n_{t}}e_k^{j}\tilde{G}(x,\mu_j).
\end{eqnarray}
It is easy to get $(g_k,g_l)_\mathcal{H}=\delta_{k,l}$, $1\leq k$, $l\leq n_{t}$. Then it holds that
\[
\tilde{G}(x,\mu)\approx \sum_{i=1}^{ n_{t}} \sqrt{\hat{\lambda}_i} \zeta_i(\mu) g_i(x),
\]
where $\{\zeta_i(\mu)\}_{i=1}^{n_{t}}$ are given by
\begin{eqnarray}
\label{KLE parameter}
\begin{split}
\zeta_i(\mu):= \frac{1}{\sqrt{\hat{\lambda}_i}} \big(\tilde{G}(\cdot ,\mu),g_i\big)_\mathcal{H}.
\end{split}
\end{eqnarray}
Thus we get the decomposition
\begin{eqnarray}
\label{KLE Snapshots}
G(x,\mu)\approx \bar{G}(x)+\sum_{i=1}^{ M} \sqrt{\hat{\lambda}_i} \zeta_i(\mu) g_i(x).
\end{eqnarray}
Because the equation (\ref{KLE parameter}) involves $\tilde{G}(\cdot,\mu)$, it can not be directly used to compute the functions $\{\zeta_i(\mu)\}_{i=1}^{n_{t}}$ for arbitrary $\mu$.
We will apply least-squares methods to approximate $\{\zeta_i(\mu)\}_{i=1}^{n_{t}}$ based on orthogonal polynomials.
Let $\{p_i(\mu)\}_{i=1}^{M_{g}}$ be the set of orthogonal polynomials basis functions with total degree less than $N_g$ regarding to the parameter variable $\mu$. We rearrange the set of basis functions from the first to the last one and place them in the following row vector,
\[
\big[p_1(\mu), p_2(\mu),...,p_{M_{g}}(\mu)\big], ~~~~\text{where} ~~M_{g}=\left( \begin{array}{cc} N_g+p\\ N_g \\\end{array}\right).
\]
For the sample date $\Xi_{t}$, we compute $[p_1(\mu_j), p_2(\mu_j),\cdots,p_{M_{g}}(\mu_j)]$ and $\zeta_i(\mu_j)=\frac{1}{\sqrt{\hat{\lambda}_i}} (\tilde{G}(\cdot,\mu_j),g_i)_\mathcal{H}$ ($j = 1,\cdots,n_{t}$). They are putted in the following matrix $\textbf{A}$ and vector $\textbf{F}$ , respectively,
\begin{eqnarray}
\label{least square A}
\textbf{A}:=\left[ \begin{array}{ccc} p_1(\mu_1)& \ldots & p_{M_{g}}(\mu_1)\\ \vdots &\ddots &\vdots\\ p_1(\mu_{n_{t}})& \ldots & p_{M_{g}}(\mu_{n_{t}})\\\end{array}\right] ,
\end{eqnarray}
\begin{eqnarray}
\label{least square F}
\textbf{F}:=[\zeta_i(\mu_1) \cdots \zeta_i(\mu_{n_{t}})]^{T}.
\end{eqnarray}
We obtain the approximation of the parameter functions $\zeta_i(\mu)$ by solving the following least square problem,
\begin{eqnarray}
\label{least square problem}
\begin{split}
\textbf{d}=\arg\min_{\alpha}\|\textbf{A}\alpha-\textbf{F}\|_2.
\end{split}
\end{eqnarray}
Thus we get $\zeta_i(\mu)\approx \sum_{i=1}^{M_{g}}d_ip_i(\mu)$, and $d_i=(\textbf{d})_i$.
\subsection{ Sparse tensor approximation}
In this subsection, we are devote to seeking the optimal solution $\textbf{c}$ for (\ref{output-approximation}) with the minimum number of non-zero terms. This can be formulated as the optimization problem:
\begin{eqnarray}
\label{P0-PROBLEM}
\arg\min_{\textbf{c}}\|\textbf{c}\|_0 ~~ \text{subject to} ~~\|G(x,\mu)-\sum_{j=1}^{N} c_{j}\Psi_j(x,\mu) \|_{\mathcal{H}\otimes L_2(\Gamma)}\leq \delta,
\end{eqnarray}
where
\[
\textbf{c}=(c_1, \cdots, c_N), \quad \text{and} \quad \|\textbf{c}\|_0=\sharp\{j:c_{j}\neq 0\}.
\]
Thus the sparse solution can be constructed as follows.\\
$\bullet$ \textit{Step 1: Find the optimal $N$-dimensional subspace $\mathcal{H}_N\subseteq \mathcal{H}$}
With the snapshots $\{G(x,\mu_i)\}_{i=1}^{n_{t}}$, we can use methods of snapshots or POD to construct the optimal $N$-dimensional subspace
\[
\mathcal{H}_N=\text{span}\big\{g_j(x): 1\leq j\leq N\big\}.
\]
$\bullet$ \textit{Step 2: Choose orthogonal polynomials for $L_2(\Gamma)$ and construct the finite dimensional approximation space $S_{N\times M}\subseteq \mathcal{H}\otimes L_2(\Gamma)$}
For the parameter, we introduce appoximation spaces $X_M$ for $L_2(\Gamma)$,
\[
X_M=\text{span}\big\{p_i(\mu): 1\leq i\leq M\big\}.
\]
A finite dimensional approximation space $S_{N\times M}\subseteq \mathcal{H}\otimes L_2(\Gamma)$ is then obtained by
\[
S_{N\times M}:=\mathcal{H}_N\otimes X_M=\text{span}\big\{p_i(\mu)g_j(x): 1\leq i\leq M,1\leq j \leq N\big\}.
\]
To simplify notation, we use the following single-index notation
\[
S_{N\times M}=\big\{w(x,\mu)=\sum_{i\in I}w_i\Psi_i(x,\mu); ~w_i\in \mathbb{R}\big\},
\]
where $I=\{1,\cdots,N\}\times \{1,\cdots,M\}$ and $\Psi_i(x,\mu)=p_{i_1}(\mu)u_{i_2}(x)$.\\
$\bullet$ \textit{Step 3: Construct the sparse solution based on Orthogonal-Matching-Pursuit}
In general, the optimization problem (\ref{P0-PROBLEM}) is an NP-hard problem. We attempt to seek efficient algorithms to approximately solve (\ref{P0-PROBLEM}).
There are a few approaches \cite{rish2014sparse} to solve the problem (\ref{P0-PROBLEM}). In the paper, we focus on Orthogonal Matching Pursuit (OMP) algorithm \cite{tropp2007signal} to get a sparse solution.
We place the basis functions $\{\Psi_i(x,\mu)\}_{i=1}^{M\times N} \subseteq S_{N\times M}$ in the following row vector,
\[
\Psi:=[\Psi_1, \Psi_2,\cdots,\Psi_{M\times N}].
\]
Assume that $n=n_x\times n_{\mu}$ sample data $\{(x_i,\mu_j)\}\subseteq D \times \Gamma$ ($1\leq i\leq n_x$, $1\leq j\leq n_{\mu}$) are chosen to solve the optimization problem (\ref{P0-PROBLEM}). For the sample date, we compute $\Psi(x_i,\mu_j)$ and $G(x_i,\mu_j)$, and they
are putted in the following matrix $\Pi\in\mathbb{R}^{n\times (M\times N)}$ and vector $\textbf{b}\in\mathbb{R}^{n\times 1}$, respectively,
\begin{eqnarray}
\label{Pi-b-matrix}
\Pi:=\left[ \begin{array}{ccc} \Psi_1(x_1,\mu_1)& \ldots & \Psi_{M\times N}(x_1,\mu_1)\\\vdots &\ddots &\vdots\\ \Psi_1(x_{n_x},\mu_{n_{\mu}})& \ldots &\Psi_{M\times N}(x_{n_x},\mu_{n_{\mu}})\\\end{array}\right], ~~~~\textbf{b}:=\left[ \begin{array}{ccc} G(x_1,\mu_1)\\\vdots\\ G(x_{n_x},\mu_{n_{\mu}}) \\\end{array}\right].
\end{eqnarray}
The coefficient vector can be solved by the following optimization problem,
\begin{eqnarray}
\label{P0-PROBLEM-discrete}
\arg\min_{\textbf{c}}\|\textbf{c}\|_0 ~~ \text{subject to} ~~\|\textbf{b}-\Pi \textbf{c}\|_2\leq \delta.
\end{eqnarray}
The main idea of OMP is to pick columns in a greedy manner. At each iteration, we choose the column of $\Pi$ that is most strongly correlated with the remaining part of $\textbf{b}$, i.e., the residual $\textbf{r}_k$ in Algorithm \ref{algorithm-OMP}. Then we subtract off the contribution to $\textbf{b}$ and iterate with regard to the updated residual. After $k$ iterations, the algorithm can identify the correct set of columns.
The OMP is described in Algorithm \ref{algorithm-OMP}.
\begin{algorithm}
\caption{ Orthogonal-Matching-Pursuit}
\textbf{Input}: A matrix $\Pi$, the vector $\textbf{b}$ and the error tolerance $\varepsilon$\\
\textbf{Output}: The sparse solution $\textbf{c}$ and the solution support $ \emph{I}:=supp(\textbf{c})$ \\
~1:~~Initialize the residual $\textbf{r}_0=\textbf{b}$, the index set $\emph{I}^0={\O}$, the iteration counter $k=1$, and \\
$~~~~~\Pi_0$ is an empty matrix;\\
~2:~~Find the index $j_0$ that solves the easy optimization problem:\\
$~~~~~j_0=\arg\max_{j=1,...,M\times N}|\langle \textbf{r}_k, \Psi_j\rangle|$;\\
~3:~~Update the index set $\emph{I}^k=\emph{I}^{k-1}\cup \{j_0\}$, the matrix $\Pi_k=[\Pi_{k-1} ~\Psi_{j_0}]$; \\
~4:~~Solve a least-squares problem to obtain a new estimate: \\
$~~~~~\textbf{c}^k=\arg\min_{\textbf{c}}\|\textbf{b}-\Pi_k \textbf{c}\|_2$;\\
~5:~~Calculate the new residual: $\textbf{r}_k=\textbf{b}-\Pi_k \textbf{c}^k$; \\
~6:~~$k\rightarrow k+1$, return to Step 2 if $\frac{\|\textbf{r}_k\|_2}{\|\textbf{b}_k\|_2}\geq \varepsilon$, otherwise \textbf{terminate} .\\
~7:~~$\text{Mt}=k$, $\emph{I}=\emph{I}^k$, and $\textbf{c}=\textbf{c}^k$.
\label{algorithm-OMP}
\end{algorithm}
The residual is always orthogonal to the columns that have been selected. In fact, we can get the conclusion from step $4$ in Algorithm \ref {algorithm-OMP} by vanishing the derivative of $\|\textbf{b}-\Pi_k \textbf{c}\|_2$, i.e.,
\[
-\Pi_k^{T}(\textbf{b}-\Pi_k \textbf{c})=-\Pi_k^{T}\textbf{r}_k=0.
\]
Thus, OMP never selects the same column twice. Provided that the residual is nonzero, the algorithm selects a new atom at each iteration and the matrix $\Pi_k$ has full column rank.
\begin{rem} For the classical least-squares method, it is required that the number of parameter sample scales quadratically with the number of unknowns \cite{chevreuil2015least}. However, STAOMP can provide an accurate approximation by a much fewer number of samples.
\end{rem}
\section{Reduced mixed GMsFE basis method}
\label{ssec:Re-MIXGMsFEM-RB-method}
Let $\mathcal{K}^H$ be a conforming coarse partition for the computational domain $D$, where $H$ is the coarse mesh size. Each coarse-grid block is further partitioned into a connected union of fine-grid blocks, we get the fine grid partition $\mathcal{K}^h$. Let $\varepsilon^H:=\bigcup_{i=1}^{N_e}\{E_i\}$ be the set of all edges/interfaces of coarse mesh $\mathcal{K}^H$ and $N_e$ the number of coarse edges. The coarse neighborhood $w_i$ corresponding to the coarse edge $E_i$ is defined by
\[
w_i=\bigcup\{K_j\in\mathcal{K}^H; E_i\in\partial K_j\}.
\]
In the mixed GMsFEM, the velocity field is approximated by using mixed GMsFE basis functions, while piecewise constant functions over $\mathcal{K}^H$ are used to approximate the pressure field. Let $Q_H$ be the space of piecewise constant functions over the coarse grid $\mathcal{K}^H$. Let $\{\phi_j\}_{j=1}^{L_i}$ be the set of GMsFE basis functions corresponding to edge $E_i$. We define the GMsFE space for the velocity field as the linear span of all local basis functions, i.e.,
\[
V_H=\bigoplus_{\varepsilon^H}\{\phi_j\}_{j=1}^{L_i}.
\]
Let $V_H^0=V_H\bigcap\{u\in V_H: u\cdot n=0 ~\text{on}~\partial D \}$ be a subspace of $V_H$. Thus the mixed GMsFEM is to find $\{v_H(\mu), p_H(\mu)\}\in V_H\times Q_H$ such that
\begin{eqnarray}
\label{discre-mix-eq}
\begin{cases}
\begin{split}
a(v_H(\mu),u; \mu)-b(u, p_H(\mu); \mu)&=0 ~~~\forall~ u \in V_H^0, \\
b(v_H(\mu), q; \mu)&=l(q) ~~~\forall ~q \in Q_H,\\
\end{split}
\end{cases}
\end{eqnarray}
where $v_H(\mu)\cdot n=g_H ~\text{on} ~ \partial D$, and for each coarse edge $E_i \in \partial D$,
\[
\int_{E_i} (g_H-g) \phi_j \cdot n=0, ~ j=1,\cdots, L_i.
\]
It is easy to see that $Q_H\subset Q_h$ and $V_H\subset V_h$ in the mixed GMsFEM. We will briefly present the mixed GMsFEM and
introduce reduced mixed GMsFE basis methods.
\subsection{Mixed GMsFE space}
In this section, we follow mixed GMsFEM \cite{cel15} and present the construction of the GMsFE space $V_H$ for the approximation of the velocity field. We first t generate the snapshot space and
then use spectral decomposition to obtain a lower-dimensional offline space.
Let $E_i\in \varepsilon^H$ be a coarse edge and $e_j\subset E_i$ a fine edge, and define a piecewise constant function $\delta_j^i$ on $E_i$ as
\begin{eqnarray*}
\delta_{j}^{i}&=
\begin{cases}
\begin{split}
1,& ~\text{on} ~ e_{j}, \\
0,&~ \text{on other fine edges of}~ E_i.
\end{split}
\end{cases}
\end{eqnarray*}
We solve the following problem on the coarse neighborhood $w_i$ corresponding to the edge $E_i$,
\begin{eqnarray}
\label{local-mix-eq-snap}
\begin{cases}
\begin{split}
k(x,\mu)^{-1}v_j^i(\mu)+\nabla p_j^i(\mu)&=0 ~~\text{in} ~~w_i, \\
\nabla\cdot v_j^i(\mu)&=\alpha_j^i ~\text{in} ~~w_i, \\
v_j^i(\mu)\cdot n_i&=0 ~~\text{on} ~~\partial w_i,
\end{split}
\end{cases}
\end{eqnarray}
where the constant $\alpha_j^i$ satisfies the compatibility condition $\int_{K_n}\alpha_j^i=\int_{E_i}\delta_j^i$ for all $K_n\subset w_i$, and $n_i$ denotes the outward unit normal vector on $\partial w_i$. The local problem (\ref{local-mix-eq-snap}) is solved separately in the coarse-grid blocks of $w_i$. We need an extra boundary condition on $E_i$ for well-posedness. Let $J_i$ be the total number of fine-grid edges on $E_i$ and $E_i=\cup_{j=1}^{J_i}e_j$, where $e_j$ denotes a fine-grid edge.
The remaining boundary condition on the coarse edge $E_i$ for the local problem (\ref{local-mix-eq-snap}) is taken as
\begin{eqnarray*}
\label{extra boundary}
v_j^i(\mu)\cdot m_i=\delta_j^i ~~\text{on} ~~E_i ,
\end{eqnarray*}
where $m_i$ is a fixed unit-normal vector on $E_i$. See Figure \ref{fig1-coarseblock} for illustration of a coarse neighborhood.
Then we define the snapshot space $V_{snap}$ space by
\[
V_{snap}=\text{span}\{v_j^i(\mu):1\leq j\leq J_i,1\leq i\leq N_e\}.
\]
\begin{figure}[htbp]
\centering
\includegraphics[width=3.5in, height=3in]{mixed-grid}
\caption{Illustration of a neighborhood $\omega_i = K_1^i\cup K_2^i$.}
\label{fig1-coarseblock}
\end{figure}
After the construction of the snapshot space, the offline space is constructed by performing some local spectral problem on the snapshot space.
Let $V_{snap}^i$ be the snapshot space corresponding to the coarse edge $E_i$, i.e.,
\[
V_{snap}^i=\text{span}\{v_j^i(\mu):1\leq j\leq J_i\}.
\]
We consider the local spectral problem: find eigenpair $\{\lambda, v\}$ ($v\in V_{snap}^i$) such that
\begin{eqnarray}
\label{tem-spectral-problem}
a_i(v,w)=\lambda s_i(v,w)~~~~\forall~ w \in V_{snap}^i,
\end{eqnarray}
where
\begin{eqnarray*}
\label{bilinear-form}
\begin{split}
a_i(v,w)&=\int_{E_i}k(x,\mu)^{-1}(v\cdot m_i)(w\cdot m_i)dx,\\
s_i(v,w)&=\int_{w_i}k(x,\mu)^{-1}v\cdot wdx+\int_{w_i}(\nabla\cdot v) (\nabla\cdot w)dx.
\end{split}
\end{eqnarray*}
Suppose that the eigenvalues of (\ref{tem-spectral-problem}) are arranged in increasing order, the corresponding eigenvectors are denoted by $z_r^i$.
We take the eigenfunctions $\{\phi_r^i\}_{r=1}^{l_i(\mu)}$ corresponding to the first $l_i(\mu)$ eigenvalues to form the local reduced snapshot space,
\[
V_{\text{off}}^i=\text{span}\{\phi_j^i(\mu):1\leq j\leq l_i(\mu)\}.
\]
Then the offline space is
\[
V_{\text{off}}=\text{span}\{\phi_j^i(\mu):1\leq j\leq l_i(\mu),1\leq i\leq N_e\}.
\]
To simplify notation, we use the following single-index notation
\begin{eqnarray*}
\label{local-snapshot-space}
V_{\text{off}}=\text{span}\{\phi_k(\mu):1\leq k \leq n_{\text{off}}\},
\end{eqnarray*}
where $n_{\text{off}}=\sum_{i=1}^{N_e}l_i(\mu)$.
\subsection{Reduced mixed GMsFE space }
In the reduced basis method, we construct a set of reduced basis for arbitrary parameter $\mu$ based on a set of snapshots.
If the snapshots have strong multiscale features, we have to use a very fine mesh to resolve the features in all scales. This computation may be very expensive. To overcome the difficulty, we use mixed GMsFEM to compute the snapshots.
Let $\Xi_{op}$ be an optimal parameter set, which is a collection of a finite number of samples in $\Gamma$. In the paper, we will use a greedy algorithm to identify $\Xi_{op}$, which will be discussed in Section \ref{greedy selection}.
The reduced mixed GMsFEM is devoted to approximating the solution $\{v(\mu), p(\mu)\}$ of the problem (\ref{mix-eq-weak}) by a set of pre-computed basis functions $\{\psi_j^i:1\leq j\leq N,1\leq i\leq N_e\}$, which are selected from
\[
\Sigma:=\{\phi_j^i(\mu):\mu\in\Xi_{op}, 1\leq j\leq l_i,1\leq i \leq N_e\}
\]
with some optimal strategies. Let $V_H^N$ be an $(N \times N_e)$-dimensional subspace of $V$ and define
\[
\mathcal{F}:=\{v(\mu)\in V_h:\mu\in \Gamma\}.
\]
To assess approximation property, it is natural to compare the subspace $V_H^N$ with the best $N \times N_e$-dimensional subspace spanned by some elements of $\Sigma$, which minimizes the projection error for the $\Sigma$ over all $N \times N_e$-dimensional subspaces of V. This minimal error can be described by the Kolmogorov width
\begin{eqnarray*}
\label{Kolmogorov width}
d_{N \times N_e}(Y_{N \times N_e}, \mathcal{F}):=\inf\{ E(\mathcal{F};Y_{N \times N_e}): Y_{N \times N_e} \text{ is an $N \times N_e$-dimensional subspace of } V \},
\end{eqnarray*}
where $E(\mathcal{F};Y_{N \times N_e})$ is the angle between $\mathcal{F}$ and $Y_{N \times N_e}$ under a metric. We construct a finite dimensional space, which is spanned by elements of $\Sigma$ with good approximation. The procedure is described as follows:\\
$\bullet$ $V_H^1=\mathop{\arg\min}\limits_{\mathop{Y_{1\times N_e}\subset V}\limits_{dim (Y_{1\times N_e)}=N_e}} d_{1\times N_e}(Y_{1\times N_e}, \mathcal{F})$,\\
$\bullet$ Assume that $V_H^{N-1}$ have been constructed. Then $V_H^{N}=\mathop{\arg\min}\limits_{\mathop{V_H^{N-1}\subset Y_{N \times N_e}\subset V}\limits_{dim (Y_{N \times N_e)}=N \times N_e}} d_{1\times N_e}(Y_{N \times N_e}, \mathcal{F})$.
A sequence of reduced GMsFE approximation spaces for velocity are obtained as follow,
\[
V_H^1\subset V_H^2\subset,. . .,\subset V_H^{N} \subset V_h,
\]
and a set of basis functions $\{\varphi_{n}^{i}: 1\leq i\leq N_e, 1\leq n\leq N\}$ are available.
To obtain a set of $(\cdot,\cdot)_V$-orthonormal basis functions, we apply the Gram-Schmidt process to $\{\varphi_{n}^{i}: 1\leq i\leq N_e, 1\leq n\leq N\}$ in the $(\cdot,\cdot)_V$ inner product. The set of orthonormal basis functions is denoted by
\begin{eqnarray}
\label{MsRB-basis}
\{\psi_{i}: 1\leq i\leq N \times N_e\}.
\end{eqnarray}
We note that the support of each basis function $\psi_{i}$ in (\ref{MsRB-basis}) is on a coarse block.
We apply Galerkin projection to construct a reduced model with the reduced multiscale basis functions defined in (\ref{MsRB-basis}).
Let $\{v_H^N(\mu), p_H^N(\mu)\}\in V_H^N\times Q_H^N$ solve
\begin{eqnarray}
\label{RB-discre-mix-eq}
\begin{cases}
\begin{split}
a(v_H^N(\mu),u; \mu)-b(u, p_H^N(\mu); \mu)&=0 ~~~\forall~ u \in V_H^N, \\
b(v_H^N(\mu), q; \mu)&=l(q) ~~~\forall ~q \in Q_H,
\end{split}
\end{cases}
\end{eqnarray}
where $v_H^N(\mu)\cdot n=g_H ~\text{on} ~ \partial D$.
Suppose that $\{\psi_{i}\}_{i=1}^{N_e \times N}$ is the set of basis functions for $V_H^N$, and $\{\eta_{r}\}_{r=1}^{N_{\text{el}}}$ is the basis functions for $Q_H$, where $N_{\text{el}}$ is the number of the coarse element. Then solutions $\{v_H^N(\mu), p_H^N(\mu)\}$ can be represented by
\begin{eqnarray}
\label{GMsRB solution}
v_H^N(\mu)=\sum_{i=1}^{N_e \times N}v_{i}^N(\mu) \psi_{i}, ~~~p_H^N(\mu)=\sum_{r=1}^{N_{\text{el}}}p_r^N(\mu) \eta_{r}.
\end{eqnarray}
By plugging $u =\psi_{j}$, $1\leq j\leq N \times N_e$ and $q =\eta_{n}$, $1\leq n\leq N_{\text{el}}$ into (\ref{RB-discre-mix-eq}), we have
\begin{eqnarray}
\label{RB-stiffness-equations}
\begin{cases}
\begin{split}
\sum_{i=1}^{N_e \times N}a(\psi_{i},\psi_{j}; \mu)v_i^N(\mu)-\sum_{r=1}^{N_{\text{el}}} b(\psi_{j}, \eta_{r})p_r^N(\mu)&=0, ~~~\quad 1\leq j\leq N \times N_e,\\
\sum_{i=1}^{N_e \times N}b(\psi_{i}, \eta_{n})v_i^N(\mu)&=l(\eta_{n}),~\quad 1\leq n \leq N_{\text{el}}.
\end{split}
\end{cases}
\end{eqnarray}
Then the output of the model can be evaluated by
\begin{eqnarray*}
\label{evaluate output}
G_H^N(\mu)&=L\big(v_H^N(\mu)\big).
\end{eqnarray*}
\subsection{Offline-online computation decomposition for reduced mixed GMsFE basis method}
\label{procedure}
The equation (\ref{RB-stiffness-equations}) implies a linear algebraic system with $N \times N_e+N_{\text{el}}$ unknowns. It involves the computation of inner products with entities $\{\psi_{i}\}_{i=1}^{N \times N_e}$ and $\{\eta_{n}\}_{n=1}^{N_{\text{el}}}$, each of which is represented by fine grid finite element basis functions of $V_h \times Q_h$.
This will lead to substantial computation for the input-output evaluation $\mu\longrightarrow G_H^N(\mu)$. With the assumption (\ref{affinely-ag}) of affine decomposition, the equation (\ref{RB-stiffness-equations}) can be rewritten by
\begin{eqnarray*}
\label{OffOnline matrix equations-GMsRB}
\begin{cases}
\begin{split}
\sum_{i=1}^{N_e \times N}\sum_{q=1}^{m_{a}}k^{q}(\mu)a^q(\psi_{i},\psi_{j})v_i^N(\mu)-\sum_{r=1}^{N_{\text{el}}} b(\psi_{j}, \eta_{r})p_r^N(\mu)=0, ~~~\quad 1\leq j\leq N \times N_e, \\
\sum_{i=1}^{N_e \times N}b(\psi_{i}, \eta_{n})v_i^N(\mu)=l(\eta_{n}),~\quad 1\leq n \leq N_{\text{el}}.
\end{split}
\end{cases}
\end{eqnarray*}
This gives rise to the matrix form
\begin{eqnarray}
\label{RB-G-matrix-form}
\begin{cases}
\begin{split}
\sum_{q=1}^{m_{a}}k^{q}(\mu)M_{N}^q\textbf{v}_N+B_N^T\textbf{p}_N=0,\\
B_N\textbf{v}_N=F_N,
\end{split}
\end{cases}
\end{eqnarray}
where
\[
(M_{N}^q)_{ij}=a^{q}(\psi_{i},\psi_{j}), \quad (B_N)_{in}=b(\psi_{i}, \eta_{n}), \quad (F_N)_{n}=l(\eta_{n}),\quad (\textbf{v}_N)_{i}=v_i^N,\quad (\textbf{p}_N)_{r}=p_r^N,
\]
\[
1\leq i,j\leq N \times N_e, ~~1\leq n\leq N_{\text{el}},~~1\leq r\leq N_{\text{el}}.
\]
Because basis functions $\{\psi_{i}, \eta_{n}\}$ belong to the standard Raviart-Thomas space $V_h\times Q_h$ ,
we can express them by
\[
\psi_{i}=\sum_{k=1}^{N_e^f}Z_{ik}\xi_{k}, 1\leq i\leq N \times N_e, ~~ \eta_{n}=\sum_{k=1}^{N_{\text{el}}^f}I_{ik}\gamma_{k}, 1\leq n\leq N_{\text{el}},
\]
where $N_e^f$ is the number of fine edges and $N_{\text{el}}^f$ is the number of fine elements.
Let $(\mathcal{Z})_{ki}=Z_{ki}$ and $(\mathcal{I})_{ki}=I_{ki}$. Then we get
\[
\mathbf{M}_{N}^{q}=\mathcal{Z}^{T}\mathcal{M}_{N_f}^{q}\mathcal{Z}, \quad \mathbf{B}_{N}=\mathcal{I}^{T}\mathcal{B}_{N_f}\mathcal{Z}, \quad \mathbf{F}_{N}=\mathcal{I}^{T}\mathcal{F}_{N_f},
\]
where $(\mathcal{A}_{N_f}^{q})_{ij}=a^{q}(\xi_{j}, \xi_{i})$, $(\mathcal{B}_{N_f})_{ij}=b(\xi_{i},\gamma_{j})$, $(\mathcal{F}_{N_f}^{q})_{i}=l(\gamma_{i})$.
The matrixes $\mathcal{A}_{N_f}^{q},~\mathcal{B}_{N_f}$ and the vectors $\mathcal{F}_{N_f}^{q}$ are independent of parameter $\mu$, and their computation is once and in offline phase. The online computation
is to solve equation (\ref{OffOnline matrix equations-GMsRB}) and evaluate the output $G_H^N(\mu)$ for any $\mu\in \Gamma$. The system (\ref{RB-G-matrix-form}) involves $N \times N_e+N_{\text{el}}$ unknowns, where $N \times N_e+N_{\text{el}}< N_e^f+N_{\text{el}}^f$.
We will present details of the two sampling strategies used for constructing the reduced mixed GMsFE basis: basis-oriented cross-validation and POD.
\section{Strategies for constructing reduced mixed GMsFE basis functions}
\label{ssec:sampling algorithm}
We construct the set of snapshots for reduced basis by computing the mixed GMsFE basis functions for each $\mu\in\Xi_{op}$, and denote them by
\[
\Sigma^{i}=\{\phi_j^i(\mu_k):1\leq j\leq l_i(\mu_k), \forall \mu_k \in \Xi_{op}\}, ~~1\leq i\leq N_e.
\]
To simplify notation, we use the following single-index notation
\[
\Sigma^{i}=\{\phi_n^i:1\leq n\leq n_{\text{snap}}^i\}, ~~1\leq i\leq N_e,
\]
where $n_{\text{snap}}^i=\sum_{j=1}^{|\Xi_{op}|}l_i(\mu_j),~ \mu_j\in \Xi_{op}$. Let
$\Sigma=\{\Sigma^{i}\}_{i=1}^{N_e}$
be the set of the snapshots for reduced mixed GMsFE basis and $n_{\text{snap}}=\min_{1\leq i\leq N_e}\{n_{\text{snap}}^i\}$.
\subsection{Basis-oriented cross-validation method for reduced mixed GMsFE basis}
\label{BOCV sampling algorithm}
The cross-validation method is devoted to selecting the optimal parameters for multiscale basis in \cite{jiang2016reduced}. In the paper, we will extend
the idea to identify reduced mixed GMsFE basis functions from the snapshots $\Sigma$. Thus we call it as basis-oriented cross-validation (BOCV).
Let $\Xi_{\text{validate}} \subset \Gamma$ be a given validation set, and $\varepsilon^{*}$ a tolerance for the stopping criterion for the greedy algorithm. The main steps of the basis-oriented cross-validation method are as follows:\\
$\bullet$ Split the snapshots $\Sigma$ into $n_{\text{snap}}$ disjoint subset $\digamma^n=\{\phi_n^i: 1\leq i\leq N_e\},$ $n=1,...,n_{\text{snap}};$ \\
$\bullet$ For each subset $\digamma^n$, we solve the equation (\ref{RB-G-matrix-form}) in the corresponding space $\text{span}\{\phi_n^i: 1\leq i\leq N_e\}$ for each $\mu \in \Xi_{\text{validate}}$, and compute the mean errors for the validation set;\\
$\bullet$ Detect the subset $\digamma^n$ with the minimum mean error, and update the reduced pace with $V_H^{N,CV}=V_H^{N-1,CV}\cup \text{span}\{\phi: \phi\in \digamma^n\}$. Here we initialize the $V_H^{0,CV}={\O}$;\\
$\bullet$ Repeat the process until the prescribed error tolerance $\varepsilon^{*}$ is satisfied.
We describe the details of BOCV in Algorithm \ref{algorithm-BOCV}. Here $M_C$ is the number of local basis functions for velocity.
\begin{algorithm}
\caption{ Basis-oriented cross-validation for reduced mixed GMsFEM}
\textbf{Input}: Some snapshots $\Sigma=\{\digamma^n\}_{n=1}^{n_{\text{snap}}}$, a validating set $\Xi_{\text{validate}} \subset \Gamma$, $n_{\text{snap}}$ and \\a tolerance $\varepsilon^{*}$\\
\textbf{Output}: the reduced mixed GMsB space: $ V_H^{N,CV}$ \\
~1:~$~$\textbf{Initialization}: $N=1$, $V_H^{0,CV}={\O}$; \\
~2:~$~$\textbf{for} $j=1:n_{\text{snap}}$\\
~3: ~~~~~$e^{j}(\mu)=v_h(\mu)-v_H^{N,j}(\mu),$ \text{where}~ $v_H^{N,j}(\mu) ~ \text{solves} ~(\ref{RB-discre-mix-eq})~ \forall ~u \in V_H^{N,j} $, \\
$~~~~~~~~~$-\text{and} $ V_H^{N,j}=V_H^{N-1,CV}\cup \text{span}\{\phi:\phi\in \digamma^j\};$\\
~4:$~~~~~\mathcal{E}_{\text{mean}}^N(j)=\text{mean}_{\mu\in\Xi_{\text{validate}}}\|e^{j}(\mu)\|_{V}$; \\
~5:~$~$\textbf{end for}\\
~6:~~$~\text{index}=\arg\min_j \mathcal{E}_{\text{mean}}^N(j);$ \\
~7:~~$~V_H^{N,CV}=V_H^{N-1,CV}\cup V_H^{N,\text{index}}:=\text{span}\{\phi_j^i, ~1\leq j\leq N,~1\leq i\leq N_e\};$ \\
~8:~~$~\varepsilon=\max_j \mathcal{E}_{\text{mean}}^N(j);$\\
~9:$~~~$update $\Sigma$ with $\Sigma=\Sigma \setminus \digamma^{\text{index}};$\\
10:~~$n_{\text{snap}}\leftarrow n_{\text{snap}}-1$; \\
11:~~$N\leftarrow N+1$; \\
12:~~\textbf{if} $\varepsilon \leq \varepsilon^{*} $\\
13:$~~~~~~$Go back to step 2;\\
14:~~\textbf{end if} \\
15:~~$M_C=N$;
\label{algorithm-BOCV}
\end{algorithm}
According to Algorithm \ref{algorithm-BOCV}, we get the reduced mixed GMsB space using BOCV
\[
V_H^{M_C,CV}:=\text{span}\{\varphi_{i}^n: 1\leq i\leq N_e,1\leq n\leq N_{M_C}\}.
\]
To obtain a set of $(\cdot,\cdot)_V$-orthonormal basis functions, we apply the Gram-Schmidt process to the set $\{\varphi_{i}^n: 1\leq i\leq N_e,1\leq n\leq N_{M_C}\}$ in the $(\cdot,\cdot)_V$ inner product, denote them by
\begin{eqnarray}
\label{MsRB-basis-cv}
\big\{\psi_{i}: 1\leq i\leq M_C \times N_e\big\}.
\end{eqnarray}
\subsection{POD method for reduced mixed GMsFE basis}
\label{POD-MsRB}
POD can be used to construct a low rank approximation for a Hilbert space (ref. \cite{sv08}).
Now we consider to use POD for sampling reduced mixed GMsFE basis functions. The POD multiscale basis can be constructed based on the given local snapshots $\Sigma^i$.
For each $1\leq i\leq N_e$, we consider a set of snapshots: $\Sigma^i=\{\phi_n^i:1\leq n\leq n_{\text{snap}}^i\}$, which can be written as
\begin{eqnarray*}
\label{eigen-eq}
\phi_n^i=\sum_{k=1}^{N_e^{f,i}}y_{ki}^n \xi_{k}=\vec{\xi}\vec{y}_{i}^n, ~1\leq n\leq n_{\text{snap}}^i,
\end{eqnarray*}
where $\vec{\xi}=[\xi_{1},\cdot\cdot\cdot,\xi_{N_e^{f,i}}]$ is the set of mixed FE basis functions for velocity supporting on $w_i$, $N_e^{f,i}$ is the number of the corresponding basis functions, and $\vec{y}_{i}^n=[y_{1i}^n,\cdot\cdot\cdot,y_{N_e^{f,i} i}^n]^T\in\mathbb{R}^{N_e^{f,i}}$. The matrix of the snapshot coefficients is defined by
\[
Y_i=(y_{ki}^n)=[\vec{y}_{i}^1, \cdots \vec{y}_{i}^n, \cdots \vec{y}_i^{n_{\text{snap}}^i}]
\in\mathbb{R}^{N_e^{f,i}\times n_{\text{snap}}^i},\quad 1\leq n\leq n_{\text{snap}}^i,\quad 1\leq k\leq N_e^{f,i}.
\]
The POD method for sampling reduced mixed GMsFE basis is described in Algorithm \ref{algorithm-POD}.
\begin{algorithm}
\caption{POD method for sampling reduced mixed GMsFE basis}
\textbf{Input}: Some snapshot $\Sigma=\{\Sigma^n\}_{n=1}^{N_e}$, and $M_{P}$\\
\textbf{Output}: the reduced mixed GMsB space: $V_H^{N,POD}$ \\
$\diamond$:~~For each $1\leq i\leq N_e$, based on the snapshots $\Sigma=\{\Sigma^n\}_{n=1}^{N_e}$, the POD basis functions\\
$~~~~~$can be constructed as follows.\\
~~~~~$\triangleright$:~Construct a matrix $\aleph$ using the inner product of the snapshots.\\
$~~~~~\aleph=((\phi_m^i,\phi_n^i)_V)=Y_i^{T}M_h^i Y_i\in \mathbb{R}^{n_{\text{snap}}^i\times n_{\text{snap}}^i}.$ where $M_h^i=[(\xi_{k},\xi_{k'})_X]\in\mathbb{R}^{N_e^{f,i}\times N_e^{f,i}}$.\\
~~~~~$\triangleright$:~Compute $M_{P}$ eigenvectors of $\aleph$ corresponding the first $M_{P}$ largest eigenvalues, i.e.,\\
$~~~~~\aleph z_j=z_j\lambda_j, j=1,\cdot\cdot\cdot,M_{P},$ where $z_j\in\mathbb{R}^{n_{\text{snap}}^i}$ and $\lambda_1\geq \lambda_2,\cdot\cdot\cdot,\lambda_{M_{P}}$.\\
~~~~~$\triangleright$:~For any $1\leq j\leq M_P$, the POD basis functions $\psi_{i}^j$ are given by:\\
$~~~~~\psi_{i}^j(x)=\frac{1}{\sqrt{\lambda_j}}\sum_{n=1}^{n_{\text{snap}}^i}(z_j)_{n}\phi_n^i.$\\
$\diamond$:~~We get the reduced mixed GMsB space using POD:~$V_H^{M_P,POD}:=\text{span}\{\psi_{i}^j: 1\leq i\leq N_e,1\leq j\leq M_P\}.$
\label{algorithm-POD}
\end{algorithm}
\subsection{Selecting $\Xi_{op}$ by a greedy algorithm}
\label{greedy selection}
Let $\Xi_{\text{train}}$ be a training set, which is a collection of a finite number of samples in $\Gamma$. Typically the training set is chosen by Monte Carlo methods. It is required that the samples in $\Xi_{\text{train}}$ are sufficiently scattered in the domain $\Gamma$. We apply a greedy algorithm to get a few optimal samples from the training set, and denote the set of the optimal samples by $\Xi_{op}$.
We use $\Xi_{op}$ instead of $\Xi_{\text{train}}$ to construct the set of snapshots for reduced mixed GMsFEM, which can significantly improve computation efficiency since $|\Xi_{op}|\ll |\Xi_{\text{train}}|$. We consider an error bound for mixed GMsFEM. There are two basic ingredients of the error bound: residual error and stability information of the corresponding bilinear form $a(\cdot,\cdot;\mu)$ and $b(\cdot,\cdot;\mu)$.
First we consider the residual error for mixed GMsFEM, which is important for posteriori analysis.
Let
\[
e_v(\mu): = v_h(\mu)-v_H^N (\mu), \quad e_p(\mu): = p_h(\mu)-p_H^N (\mu).
\]
By equation (\ref{mix-FEM-dis}), we get
\begin{eqnarray*}
\label{deduce-equation}
\begin{cases}
\begin{split}
a\big(e_v(\mu),u; \mu\big)-b\big(u, e_p(\mu); \mu\big)&=-a\big(v_H^N (\mu),u; \mu\big)+b\big(u,p_H^N (\mu); \mu\big), ~~~\forall ~u \in V_h, \\
b\big(e_v(\mu), q; \mu\big)&=l(q)-b\big(v_H^N (\mu), q; \mu\big), ~~~\forall ~q \in Q_h.
\end{split}
\end{cases}
\end{eqnarray*}
Let $r_1(u;\mu)\in V_h^{*}$ (the dual space of $ V_h$) and $r_2(q;\mu)\in Q_h^{*}$ be the residual
\begin{eqnarray*}
\label{Rfa-weak-eq}
\begin{cases}
\begin{split}
r_1(u;\mu):&=-a\big(v_H^N (\mu),u; \mu\big)+b\big(u,p_H^N (\mu); \mu\big), ~~~\forall ~u \in V_h,\\
r_2(q;\mu):&=l(q)-b\big(v_H^N (\mu), q; \mu\big), ~~~\forall ~q \in Q_h.
\end{split}
\end{cases}
\end{eqnarray*}
Then we get
\begin{eqnarray}
\label{error-residual-eq}
\begin{cases}
\begin{split}
a\big(e_v(\mu),u; \mu)-b\big(u,e_p(\mu); \mu\big)&=r_1(u;\mu), ~~~\forall ~u \in V_h,\\
l(q)-b\big(e_v(\mu), q; \mu\big)&=r_2(q;\mu), ~~~\forall ~q \in Q_h.
\end{split}
\end{cases}
\end{eqnarray}
By Riesz representation theory, there exist $\hat{e}_v(\mu)\in V_h$ and $\hat{e}_p(\mu)\in Q_h$ such that
\begin{eqnarray}
\label{Riesz-weak-eq}
\begin{cases}
\begin{split}
\big(\hat{e}_v(\mu),u\big)_{V}&=r_1(u;\mu), \quad \forall u\in V_h,\\
\big(\hat{e}_p(\mu),q\big)_{Q}&=r_2(q;\mu), \quad \forall q\in Q_h.
\end{split}
\end{cases}
\end{eqnarray}
Then we can rewrite the error residual equation (\ref{error-residual-eq}) as
\begin{eqnarray*}
\label{Riesz error residual equation}
\begin{cases}
\begin{split}
a\big(e_v(\mu),u; \mu)-b\big(u,e_p(\mu); \mu\big)&=\big(\hat{e}_v(\mu),u\big)_{V}, \quad \forall u\in V_h,\\
l(q)-b\big(e_v(\mu), q; \mu\big)&=\big(\hat{e}_p(\mu),q\big)_{Q}, \quad \forall q\in Q_h.
\end{split}
\end{cases}
\end{eqnarray*}
Consequently, the dual norm of the residual $r_1(u; \mu)$ and $r_2(q; \mu)$ can be evaluated through the Riesz representation,
\begin{eqnarray}
\label{dual-norm-residual}
\begin{cases}
\begin{split}
\|r_1(u;\mu)\|_{V_h^{*}}:=\sup_{u\in V_h}\frac{r_1(u;\mu)}{\|u\|_{V}}=\|\hat{e}_v(\mu)\|_{V},\\
\|r_2(q;\mu)\|_{Q_h^{*}}:=\sup_{q\in Q_h}\frac{r_2(q;\mu)}{\|q\|_{Q}}=\|\hat{e}_p(\mu)\|_{Q}.
\end{split}
\end{cases}
\end{eqnarray}
Secondly, we need a lower bound and an upper bound for the continuity constant in (\ref{bounded-a}) and coercivity constant in (\ref{coercive-a}),
\[
0<\gamma_{LB}(\mu)\leq\gamma(\mu)\leq\gamma_{UB}(\mu), ~~~ \forall \mu \in \Gamma,
\]
\[
0<\alpha_{LB}(\mu)\leq\alpha(\mu)\leq\alpha_{UB}(\mu), ~~~ \forall \mu \in \Gamma,
\]
and for the inf-sup constant in (\ref{inf-sup condition}),
\[
0<\beta_{LB}(\mu)\leq\beta(\mu)\leq\beta_{UB}(\mu), ~~~ \forall \mu \in \Gamma.
\]
Based on these bounds, we define an error estimator \cite{ gerner2012certified, rhp08} for the solution of equation (\ref{error-residual-eq}) by
\begin{eqnarray}
\label{erroe estimator-v}
\Delta_{N}^v(\mu):=\frac{\|\hat{e}_v(\mu)\|_{V}}{{\alpha_{LB}(\mu)}}+\bigg(1+\frac{\gamma_{UB}(\mu)}{\alpha_{LB}(\mu)}\bigg)\frac{\|\hat{e}_p(\mu)\|_{Q}}{{\beta_{LB}(\mu)}},
\end{eqnarray}
\[
\Delta_{N}^p(\mu):=\frac{\|\hat{e}_v(\mu)\|_{V}}{{\beta_{LB}(\mu)}}+\frac{\gamma_{UB}(\mu)}{\beta_{LB}(\mu)}\Delta_{N}^v(\mu).
\]
An efficient method for computing $\alpha_{LB}(\mu)$, $\gamma_{UB}(\mu)$ and $\beta_{LB}(\mu)$ is the Successive Constraint Method (ref. \cite{ gerner2012certified, pgsa07, rhp08}).
Note that $\Delta_{N}^v(\mu)$ and $\Delta_{N}^p(\mu)$ are the upper bounds for the errors $\|e_v(\mu)\|_{V}$ and $\|e_p(\mu)\|_{Q}$ such that
\[
\|e_v(\mu)\|_{V}\leq \Delta_{N}^v(\mu), ~~~\|e_p(\mu)\|_{Q}\leq \Delta_{N}^p(\mu), ~~~ \forall \mu \in \Gamma.
\]
In this paper, the velocity is what we are particularly interested in. Thus we introduce the associated effectivity with the error estimator $\Delta_{N}^v(\mu)$,
\[
\eta_{N}^v(\mu):=\frac{\Delta_{N}^v(\mu)}{\|e_v(\mu)\|_{V}}.
\]
The effectivity is a measure of the quality of the proposed estimator. Following the references \cite{gerner2012certified, quarteroni2011certified, rhp08}, the effectivity satisfies
\[
1\leq \eta_{N}^v(\mu)\leq \frac{\gamma(\mu)}{\alpha_{LB}(\mu)}+\frac{\gamma_b}{\alpha_{LB}(\mu)}\frac{\|e_p(\mu)\|_{Q}}{\|e_v(\mu)\|_{V}}+\bigg(1+\frac{\gamma_{UB}(\mu)}{\alpha_{LB}(\mu)}\bigg)\frac{\gamma_b}{{\beta_{LB}(\mu)}}, ~~~\forall \mu\in \Gamma.
\]
In the process of iteration, we just update the reduced mixed GMsFEM basis for velocity, the pressure space is constructed by piecewise constant functions on the coarse grid. Thus the inequality $\|e_v(\mu)\|_{V}\leq\|e_p(\mu)\|_{Q}$ holds in general.
Then the associated effectivity satisfies
\[
1\leq \eta_{N}^v(\mu)\leq \frac{\gamma(\mu)+\gamma_b}{\alpha_{LB}(\mu)}+\bigg(1+\frac{\gamma_{UB}(\mu)}{\alpha_{LB}(\mu)}\bigg)\frac{\gamma_b}{{\beta_{LB}(\mu)}}, ~~~\forall \mu\in \Gamma.
\]
We note that the $\hat{e}(\mu)$ is related to $r(v;\mu)$ by the equation (\ref{dual-norm-residual}).
By (\ref{GMsRB solution}) and (\ref{affinely-ag}), the residual can be expressed by
\begin{eqnarray}
\label{Qresidual-eq}
\begin{split}
r_1(u;\mu)&=-a\big(v_H^N (\mu),u; \mu\big)+b\big(u,p_H^N (\mu); \mu\big) \\
&=-\sum_{i=1}^{N_e \times N}\sum_{j=1}^{m_{a}}k^{j}(\mu)v_i^N(\mu)a^j(\psi_{i},u)+\sum_{r=1}^{N_{\text{el}}} p_r^N(\mu) b(u, \eta_{r}), \quad \forall u\in V_h,\\
r_2(q;\mu)&=l(q)-\sum_{i=1}^{N_e \times N}v_i^N(\mu)b(\psi_{i}, q) ~~~\forall ~q \in Q_h.
\end{split}
\end{eqnarray}
By (\ref{Qresidual-eq}) and (\ref{Riesz-weak-eq}), we have
\begin{eqnarray*}
\label{Riesz Qerror estimators}
\begin{cases}
\begin{split}
\big(\hat{e}_v(\mu),u\big)_{V}&=-\sum_{i=1}^{N_e \times N}\sum_{j=1}^{m_{a}}k^{j}(\mu)v_i^N(\mu)a^j(\psi_{i},u)+\sum_{r=1}^{N_{\text{el}}}p_r^N(\mu) b(u, \eta_{r}), \quad \forall u\in V_h,\\
\big(\hat{e}_p(\mu),q\big)_{Q}&=l(q)-\sum_{i=1}^{N_e \times N}v_i^N(\mu)b(\psi_{i}, q) ~~~\forall ~q \in Q_h.
\end{split}
\end{cases}
\end{eqnarray*}
This implies that
\begin{eqnarray}
\label{Riesz-error-estimators1}
\hat{e}_v(\mu)=\sum_{i=1}^{N_e \times N}\sum_{j=1}^{m_{a}}k^{j}(\mu)v_i^N(\mu)\mathcal{L}_{i}^{p}+\sum_{r=1}^{N_{\text{el}}}p_r^N(\mu)\mathcal{X}^r,
\end{eqnarray}
\begin{eqnarray}
\label{Riesz-error-estimators2}
\hat{e}_p(\mu)=\mathcal{C}+\sum_{i=1}^{N_e \times N}v_i^N(\mu)\mathcal{X}_i,
\end{eqnarray}
where $\mathcal{L}_{i}^{j}$ is the Riesz representation of $a^{j}(\psi_{i},u)$, i.e., $(\mathcal{L}_{i}^{j},u)_{V}=-a^{j}(\psi_{i},u)$ for any $u\in V_h$, $\mathcal{X}^{r}$ is the Riesz representation of $b(u, \eta_{r})$, i.e., $(\mathcal{X}^{r},u)_{V}=b(u, \eta_{r})$ for any $u\in V_h$, $\mathcal{X}_{i}$ is the Riesz representation of $b(\psi_{i}, q)$, i.e., $(\mathcal{X}_{r},q)_{Q}=-b(\psi_{i}, q)$ for any $q\in Q_h$, and $\mathcal{C}$ is the Riesz representation of $l$, i.e., $(\mathcal{C},q)_{X}=l(q)$ for any $q\in Q_h$, where $1\leq j\leq m_{a}$, $1\leq i\leq N\times N_e$, and $1\leq r\leq N_{\text{el}}$.
The equations (\ref{Riesz-error-estimators1}) and (\ref{Riesz-error-estimators2}) give rise to
\begin{eqnarray}
\label{error-norm-est}
\begin{split}
\|\hat{e}_v(\mu)\|_{V}^2=&\sum_{r=1}^{N_{\text{el}}}\sum_{r'=1}^{N_{\text{el}}}p_r^N(\mu)p_{r'}^N(\mu)(\mathcal{X}^r,\mathcal{X}^{r'})_{V}+\sum_{i=1}^{N_e \times N}\sum_{j=1}^{m_{a}}k^{j}(\mu)v_i^N(\mu)\\
&\times\bigg\{2\sum_{r=1}^{N_{\text{el}}}p_r^N(\mu)(\mathcal{X}^r,\mathcal{L}_{i}^{j})_{V}+\sum_{i'=1}^{N_e \times N}\sum_{j'=1}^{m_{a}}k^{j'}(\mu)v_{i'}^N(\mu)
(\mathcal{L}_{i}^{j},\mathcal{L}_{i'}^{j'})_{V}\bigg\},\\
\|\hat{e}_p(\mu)\|_{Q}^2=&(\mathcal{C},\mathcal{C})_{Q}+2\sum_{i=1}^{N\times N_e}v_i^N(\mu)(\mathcal{C},\mathcal{X}_i)_{Q}+\sum_{i=1}^{N\times N_e}\sum_{i'=1}^{N\times N_e}v_{i'}^N(\mu)v_i^N(\mu)(\mathcal{X}_i,\mathcal{X}_{i'})_{Q}.
\end{split}
\end{eqnarray}
To efficiently compute $\|\hat{e}_v(\mu)\|_{V}$ and $\|\hat{e}_p(\mu)\|_{Q}$, we apply an offline-online procedure. In the offline stage we compute and store the parameter-independent quantities. In particular, we compute $\mathcal{C}$, $\mathcal{X}^{r}$, $\mathcal{X}_{i}$ and $\mathcal{L}_{i}^{j}$, and store $(\mathcal{X}^r,\mathcal{X}^{r'})_{V}$, $(\mathcal{X}^r,\mathcal{L}_{i}^{j})_{V}$, $(\mathcal{L}_{i}^{j},\mathcal{L}_{i'}^{j'})_{V}$, $(\mathcal{C},\mathcal{C})_{Q}$, $(\mathcal{C},\mathcal{X}_i)_{Q}$ and $(\mathcal{X}_i,\mathcal{X}_{i'})_{Q}$, where $1\leq i,i'\leq N\times N_c$, $1\leq j,j'\leq m_{a}$, $1\leq r,r'\leq N_{\text{el}}$.
In the online stage, for any $\mu$, we compute $v_i^N(\mu)$ ($1\leq i\leq N\times N_e$) and use (\ref{error-norm-est}) to compute $\|\hat{e}_v(\mu)\|_{V}$ and $\|\hat{e}_p(\mu)\|_{Q}$.
In summary, we describe the greedy algorithm for selecting $\Xi_{op}$ in Algorithm \ref{algorithm-Greedy}.
\begin{algorithm}[hbtp]
\caption{ Greedy algorithm for selecting $\Xi_{op}$}
\textbf{Input}: A training set $\Xi_{\text{train}} \subset \Gamma$, $\mu_1\in \Xi_{\text{train}}$, and $N_p$ \\
\textbf{Output}: The optimal set $\Xi_{op}\in \Xi_{\text{train}}$\\
~1:~~Initialize $N=1$, $\Xi_{op}=\{\mu_1\}$ and $\mu_{N}=\mu_{1}$;\\
~2:~~Construct the snapshots $\Sigma$ for reduced mixed GMsFEM by computing the mixed \\
$~~~~~$ GMsFE basis for each $\mu\in\Xi_{op}$, and construct reduced basis space $V_H^{N_p}$ by POD;\\
~3:~~Update $\Xi_{\text{train}}$ with $\Xi_{\text{train}}=\Xi_{\text{train}}\setminus \mu_{N};$\\
~4:~~For each $\mu\in \Xi_{\text{train}}$, compute the reduced basis approximation $v_H^{N_p}$ by (\ref{RB-G-matrix-form}) in $V_H^{N_p}$.\\
~5:~~For each $\mu\in \Xi_{\text{train}}$, evaluate the error estimator $\Delta_{N_p}^v(\mu)$ by (\ref{erroe estimator-v}).\\
~6:~~Choose $\mu_{N+1}=\arg\max_{\mu\in\Xi_{\text{train}}}\Delta_{N_p}^v(\mu)$, and set $\varepsilon_N=\max_{\mu\in\Xi_{\text{train}}}\Delta_{N_p}^v(\mu)$;\\
~7:~~Update $\Xi_{op}$ with $\Xi_{op}=\Xi_{op}\bigcup\{\mu_{N}\}$;\\
~8:~~$N\leftarrow N+1$\\
~9:~~Return to Step 2 if $\varepsilon_N \leq \varepsilon_{N-1}$, otherwise \textbf{terminate}.
\label{algorithm-Greedy}
\end{algorithm}
\begin{rem}
We can use the cross-validation method in \cite{jiang2016reduced} to choose the first optimal sample $\mu_1$ from the training set $\Xi_{\text{train}}$.
This choice can improve the accuracy.
\end{rem}
\section{Reduced model representation method}
\label{ssec:Online-online}
The reduced order model defined in (\ref{RB-G-matrix-form}) for mixed GMsFEM is a linear algebraic system with $N \times N_e+N_{\text{el}}$ unknowns. Although the reduced
model needs much less computation effort than original full order model, it may be not a very small-scale problem because the reduced order model defined in (\ref{RB-G-matrix-form}) for mixed GMsFEM involves $N \times N_e+N_{\text{el}}$ unknowns.
In order to significantly improve the online computation efficiency,
we want to get a representation like the form (\ref{separating presentation}) for the solution and model outputs.
We call this by reduced model representation method, which allows online computation by direct evaluation.
We employ LSMOS and STAOMP presented in Section \ref{ssec:PGDDF-LSMOS-STAOMP} to get the reduced model representation. Let $\Xi_{t}$ be a collection of a finite number of samples in $\Gamma$, and $\{v(x,\mu_i), p(x,\mu_i)\}_{i=1}^{n_t}$ are the snapshots solved by the reduced model (\ref{RB-G-matrix-form}), where $n_t=|\Xi_{t}|$.
Algorithm \ref{algorithm-LSMOS} outlines the reduced model representation method by LSMOS.
\begin{algorithm}[hbtp]
\caption{ Least-squares method of snapshots for reduced model representation}
\textbf{Input}: A training set $\Xi_{t} \subset \Gamma$ and a tolerance $\varepsilon^\text{on}$\\
\textbf{Output}: Reduced model representation $G(x,\mu)\approx \sum_{i=1}^{M} c_i \zeta_i(\mu)v_i(x)$\\
~1:~~Compute the snapshots $\{v(x,\mu_i)\}_{i=1}^{n_{n_t}}$ by (\ref{RB-G-matrix-form}) for all $\mu_i\in \Xi_{t}$ and construct \\
$~~~~~$the covariance
matrix \textbf{C};\\
~2:~~Solve the eigenvalue problem and determine $M$ such that $\frac{\sum _{i=1}^{M}\hat{\lambda}_i}{\sum _{i=1}^{n_t}\hat{\lambda}_i}\leq\varepsilon^\text{on};$\\
~3:~~Assemble $\textbf{A}$ based on GPC basis functions and $\Xi_{t}$ by (\ref{least square A});\\
~4:~~Construct the functions $\{g_i(x)\}_{i=1}^{M}$ by (\ref{KLE basis}), for each $i=1,..,M$, assemble \textbf{F}\\
$~~~~~$ by (\ref{least square F});\\
~5:~~For each $i=1,..,M$, solve problem (\ref{least square problem}) by least square procedure to obtain $\textbf{d}$,\\
$~~~~~$ and then get $\zeta_i(\mu)\approx\sum_{i=1}^{M_{g}}d_ip_i(\mu)$;\\
~6:~~Return the representation $G(x,\mu)\approx \sum_{i=1}^{M} \sqrt{\hat{\lambda}_i} \zeta_i(\mu)g_i(x)\approx\sum_{i=1}^{M}\sum_{j=1}^{M_{g}} \sqrt{\hat{\lambda}_i}d_jp_j(\mu) g_i(x)$.
\label{algorithm-LSMOS}
\end{algorithm}
Algorithm \ref{algorithm-STAOMP} combines reduced mixed GMsFE basis methods and STAOMP together and provides a sparse representation for the model.
STAOMP needs much fewer samples than LSMOS. Thus,
the cardinality of the training set in Algorithm \ref{algorithm-STAOMP} is much less than the cardinality of the training set in Algorithm \ref{algorithm-LSMOS}.
The whole process for STAOMP based on reduced mixed GMsFE basis methods is illustrated in Figure \ref{Schema}.
\begin{algorithm}[hbtp]
\caption{ Sparse tensor approximation based on Orthogonal-Matching-Pursuit for reduced model representation}
\textbf{Input}: A training set $\Xi_{t} \subset \Gamma$, the sample data $\{(x_i,\mu_j)\}\subseteq D \times \Gamma$ ($1\leq i\leq n_x$, $1\leq j\leq n_{\mu}$),
the number of the optimal basis functions $M$ and a tolerance $\varepsilon^\text{on}$\\
\textbf{Output}: Reduced model representation $G(x,\mu)\approx \sum_{i=1}^{\text{Mt}} \textbf{c}(\emph{I}(i)) \Psi_{\emph{I}(i)}(\mu, x);$\\
~1:~~Compute the snapshots $\{v(x,\mu_i)\}_{i=1}^{n_t}$ by (\ref{RB-G-matrix-form}) for all $\mu_i\in \Xi_{t}$ and construct the\\
$~~~~~$optimal basis functions $\{v_j(x)\}_{j=1}^{N}$ for $\mathcal{H}_N\subseteq \mathcal{H}$;\\
~2:~~Choose orthogonal polynomials basis functions $\{p_i(\mu)\}_{i=1}^M$ for $L_2(\Gamma)$, and construct the
$~~~~~$finite dimensional approximation space $S_{N\times M}=\{\sum_{i\in I}w_i\Psi_i(x,\mu); ~w_i\in \mathbb{R}\}$\\
~3:~~Assemble matrix $\Pi$ and vector $\textbf{b}$ by (\ref{Pi-b-matrix});\\
~4:~~Solve the optimization problem (\ref{P0-PROBLEM-discrete}) by Algorithm \ref{algorithm-OMP} with $\varepsilon=\varepsilon^\text{on}$ and get the sparse \\
$~~~~~$solution $\textbf{c}$ and the solution support $\emph{I}$;\\
~5:~~Return the representation $G(x,\mu)\approx \sum_{i=1}^{\text{Mt}} \textbf{c}(\emph{I}(i)) \Psi_{\emph{I}(i)}(\mu, x).$
\label{algorithm-STAOMP}
\end{algorithm}
\section{Numerical results}
\label{ssec:Numerical result}
In this section, we present a few examples to illustrate the performance of the proposed reduced mixed GMsFE basis methods
and make comparisons for different strategies for the model reduction. In Section \ref{Numerical-result1}, we consider an example to illustrate performance of the different reduced mixed GMsFE basis methods for elliptic PDEs with one-dimensional parameters. In Section \ref{Numerical-result2}, we study reduced model representation method for multiscale
elliptic PDEs with multivariate parameters.
In Section \ref{Numerical-result3}, we consider a two-phase flow problem in random porous media. We apply reduced mixed GMsFE basis methods to the flow equation and integrate STAOMP into the model outputs (water saturation and water-cut) for uncertainty quantification.
For the numerical examples, the models are defined in the spatial domain $D=(0,1)^2$. Let $v(x,\mu_i)$ and $p(x,\mu_i)$ be the reference solutions for velocity and pressure,
respectively, which are solved by mixed FEM on a fine grid. Let $v_{H}(x,\mu_i)$ and $ p_{H}(x,\mu_i)$ be solved by the reduced mixed GMsFE basis methods.
Then the relative mean errors for velocity and pressure are defined, respectively, by
\begin{eqnarray}
\label{errors-velocity}
\varepsilon_v=\frac{1}{N}\sum_{i=1}^N\frac{\|v(x,\mu_i)-v_{H}(x,\mu_i)\|_{L^2(D)}}{\|v(x,\mu_i)\|_{L^2(D)}},
\end{eqnarray}
\begin{eqnarray}
\label{errors-pressure}
\varepsilon_p=\frac{1}{N}\sum_{i=1}^N\frac{\|p(x,\mu_i)-p_{H}(x,\mu_i)\|_{L^2(D)}}{\|p(x,\mu_i)\|_{L^2(D)}}.
\end{eqnarray}
In all of the examples, we will consider the models with high-contrast fields, which are depicted in Figure \ref{fig-exam}.
\begin{figure}[htbp]
\centering
\subfigure[$\kappa_1$ ]{
\label{fig:subfig:a}
\includegraphics[width=3in, height=2.4in]{khc0}}
\subfigure[$\kappa_2$]{
\label{fig:subfig:b}
\includegraphics[width=3in, height=2.4in]{khc1}}
\caption{High-contrast fields in the numerical experiments.}
\label{fig-exam}
\end{figure}
\subsection{Numerical results for reduced mixed GMsFE basis methods}
\label{Numerical-result1}
In this subsection, we consider the elliptic equation in mixed formulation (\ref{mix-eq}), where the coefficient function $k(x, \mu)$ is defined by
\[
k(x,\mu)=\frac{10000}{10\sin(20\mu+x_1x_2)+\big(\cos(\mu)+1.2\big)\kappa_1+25},
\]
where $\kappa_1$ is a high-contrast field and depicted in Figure \ref{fig-exam} (left), and the source function $f(x, \mu)$ is
\[
f(x, \mu)=(x_2-0.5)\cos\big(\pi(x_1-0.5)\big).
\]
Here $x:=(x_1, x_2)\in D$ and the random parameter $\mu\sim U(-1,1)$. The diffusion coefficient $k(x, \mu)$ is oscillating with respect to the random parameter $\mu$ and highly heterogeneous with respect to the spatial variable $x$. We apply EIM for $k^{-1}(x, \mu)$ to achieve offline-online computation decomposition. For the discretization of the spatial domain, we use $80\times80$ uniform fine grid, where the reference solution is computed by the mixed FEM with the lowest Raviart-Thomas FE space. The mixed GMsFEM is implemented on the $8\times8$ coarse grid. We choose $|\Xi_{\text{train}}|=200$ parameter values, and select $|\Xi_{op}|=10$ optimal samples from the training set $\Xi_{\text{train}}$ by the greedy algorithm (Algorithm \ref{algorithm-Greedy}). The mixed GMsFEM is used to compute the snapshots, we take $l_i(\mu)=5$ for each coarse block $\omega_i$ and each $\mu\in \Xi_{op}$, thus the number of the snapshot is $n_{\text{snap}}^i=\sum_{j=1}^{n_t}l_i(\mu_j)=50$ for each coarse block $\omega_i$.
To show the efficacy of greedy algorithm for selecting the optimal sample set $\Xi_{op}$, we use two methods to select $\Xi_{op}$: the greedy algorithm and random selection, and then
compare the results.
For the numerical example, we consider the four different reduced GMsFE basis methods: reduced mixed GMsFE basis method using greedy algorithm and BOCV (RmGMsB-GBOCV), reduced mixed GMsFE basis method using greedy algorithm and POD (RmGMsB-GPOD), reduced mixed GMsFE basis method using random selection and BOCV (RmGMsB-RBOCV), and reduced mixed GMsFE basis method using random selection and POD (RmGMsB-RPOD). The first two methods use the greedy algorithm to select $\Xi_{op}$, while the last two methods are performed by randomly choosing $10$ parameter samples from $\Xi_{\text{train}}$.
To compare the approximation accuracy of the four reduced mixed GMsFE basis methods, we randomly choose $1000$ samples from the parameter space and compute the average of relative error defined in (\ref{errors-velocity}) and (\ref{errors-pressure}) for the four methods.
In Figure \ref{fig1-exam1}, we depict average relative velocity error versus number of local basis functions for the four reduced mixed GMsFE basis methods. By the figure we have three observations: (1) as the number of local basis functions increases, the approximation becomes more accurate for all methods; (2) RmGMsB-GBOCV method always achieves better approximation than RmGMsB-RBOCV, and RmGMsB-GPOD gives better approximation than RmGMsB-RPOD when the number of local GMsFE basis functions is more than $3$; (3) BOCV approach renders better accuracy than POD approach.
To visualize the individual errors of the first $100$ samples, we plot the relative errors for the four methods in Figure \ref{fig2-exam1}, which shows that: (1) the error of RmGMsB-GBOCV is much less sensitive to the parameter samples compared with RmGMsB-RBOCV, and the POD approaches have the same situation; (2) BOCV approaches generally gives better approximation than POD approaches. Figure \ref{fig4-exam1} shows the velocity solution profile of the example for the three methods: mixed FEM on fine grid (reference solution), RmGMsB-GBOCV and RmGMsB-GPOD. The figure shows all the velocity profiles have very good agreement and the reduced mixed GMsFE basis methods provide accurate approximation to the original fine scale model.
\begin{figure}[htbp]
\centering
\includegraphics[width=5.5in, height=2.5in]{errorcorveg1}
\caption{The average relative error for velocity versus number of local basis functions for RmGMsB-GBOCV, RmGMsB-GPOD, RmGMsB-RBOCV and RmGMsB-RPOD, $80\times 80$ fine grid, and $8\times 8$ coarse grid. }
\label{fig1-exam1}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=5in, height=3in]{errorcv}\\
\includegraphics[width=5in, height=3in]{errorpod}
\caption{The relative error for $100$ samples by RmGMsB-GBOCV, RmGMsB-GPOD, RmGMsB-RBOCV and RmGMsB-RPOD, $80\times 80$ fine grid, and $8\times 8$ coarse grid.}
\label{fig2-exam1}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=6in, height=3.5in]{velocity}
\caption{The mean of velocity profile, $5$ local GMsFE basis functions on each coarse edge, the first row are the velocity in x-axis direction and the second row are the
velocity in y-axis direction.}
\label{fig4-exam1}
\end{figure}
\subsection{Numerical results for reduced model's representation method}
\label{Numerical-result2}
In this section, we consider two examples to illustrate the performance of reduced model representation using LSMOS and STAOMP. In the first example,
we use a multivariate function to highlight the differences between LSMOS and STAOMP. For the second example, we consider an elliptic PDE with multi-dimensional parameters and apply LSMOS and STAOMP to obtain a reduced model representation for solution.
\subsubsection{Numerical example I: a multivariate function}
Let us consider the function
\[
u(x,\mu):=\sum_{i=1}^2x_i\mu_i+\sin\big(\frac{\pi}{4} (x_1+\frac{1}{3}\prod_{i=1}^3 \mu_i)\big)+\cos\big(\frac{\pi}{4} (x_2+\frac{1}{3}\prod_{i=4}^6 \mu_i)\big),
\]
For the discretization of the spatial domain, we use $50\times50$ uniform grid and will consider the two variable-separation methods: LSMOS and STAOMP. With regard to LSMOS, we take $\varepsilon^\text{on}=1.3\times 10^{-3}$ in Algorithm \ref{algorithm-LSMOS} and take $900$ parameter samples for snapshots, the representation (\ref{KLE Snapshots}) can be obtained by Algorithm \ref{algorithm-LSMOS} with $M=6$. We use Legendre polynomials with degree up to $N_g=5$ to approximate $\{\zeta_i(\mu)\}_{i=1}^{M}$ based on least square methods. Hence, the number of the total orthogonal polynomial basis is $M_g=462$.
For STAOMP, we take $160$ parameter samples for snapshots, i.e., $n_{t}=160$, and construct $10$ optimal basis functions $\{v_j(x)\}_{j=1}^{6}$ for spatial space, and set $\varepsilon^\text{on}=1.3\times 10^{-3}$ in Algorithm \ref{algorithm-STAOMP}, $160$ parameter samples and $100$ spatial coordinates are selected for snapshots, then we apply Algorithm \ref{algorithm-STAOMP} to construct the representation $u(x,\mu)\approx \sum_{i=1}^{\text{Mt}} \textbf{c}(\emph{I}(i)) \Psi_{\emph{I}(i)}(\mu, x)$ with $\text{Mt}=41$. Based on these representations, we choose $1000$ samples and compute the average relative error, which is defined as follows,
\begin{eqnarray*}
\label{errors-u}
\varepsilon_u=\frac{1}{N}\sum_{i=1}^N\frac{\|u(x,\mu_i)-\tilde{u}(x,\mu_i)\|_{L^2}}{\|u(x,\mu_i)\|_{L^2}},
\end{eqnarray*}
where $N=1000$ and $\tilde{u}(x,\mu_i)$ is the solution by LSMOS or STAOMP. We list the average relative errors in Table \ref{tab1-exam2} along with the average online CPU time. From the table, we find that: (1) for LSMOS, the approximation error decays fast when the number of terms increases; (2) using much fewer terms, STAOMP can achieve the same accuracy as LSMOS; (3) the magnitude of CPU time for STAOMP is much smaller than that of LSMOS.
\begin{table}[hbtp]
\small
\centering
\caption{ Comparison of relative mean errors $\varepsilon$ and the average online CPU time for different approaches. $\text{Mt}$ is the number of terms retained in reduced model representation.}
\begin{tabular}{c|c|c|c}
\Xhline{1pt}
Strategies & $\text{Mt}$ & $\varepsilon_u$ & CPU time\\
\hline
\multirow{5}{*}{LSMOS}
& $1\times 462$ & $1.10\times10^{-1}$ & $4.12\times10^{-4}s$ \\
& $2\times 462$ & $1.33\times10^{-2}$ & $7.80\times10^{-4}s$ \\
& $3\times 462$ & $1.87\times10^{-3}$ & $1.50\times10^{-3}s$ \\
& $4\times 462$ & $1.80\times10^{-3}$ & $2.10\times10^{-3}s$ \\
& $5\times 462$ & $1.70\times10^{-3}$ & $2.30\times10^{-3}s$ \\
\hline
\multirow{1}{*}{STAOMP} & 41 & $1.20\times10^{-3}$ & $9.86\times10^{-5}s$ \\
\Xhline{1pt}
\end{tabular}
\label{tab1-exam2}
\end{table}
\subsubsection{Numerical example II: an elliptic PDE with multi-dimensional parameters }
In this subsection, we consider the elliptic equation (\ref{mix-eq}) with the source term
\[
f(x)=(x_1+1)\cos(\pi x_2), ~ ~x\in (0, 1)^2.
\]
Let $a(x,\mu)$ be a random field, which is characterized by a two point exponential covariance function $\text{cov}[a]$, i.e.,
\begin{eqnarray}
\label{covariance function}
\text{cov}[a](x_1, y_1; x_2, y_2)= \sigma^2 \exp\bigg(-\frac{|x_1-x_2|^2}{2l_x^2}-\frac{|y_1-y_2|^2}{2l_y^2}\bigg),
\end{eqnarray}
where $(x_i, y_i)$ ($i=1,2$) is the spatial coordinate in $D$. Here the variance $\sigma^2=1$, correlation length $l_x=l_y=0.2$. The random coefficient $a(x, \mu)$ is obtained by truncating a Karhunen-Lo\`{e}ve expansion, i.e.,
\begin{eqnarray}
\label{KLE-TRUNC1}
a(x,\mu):= E[a]+ \sum_{i=1}^{N} \sqrt{\gamma_i}b_i(x) \mu_i,
\end{eqnarray}
where $E[a]=6$ and the random vector $ \mu:=(\mu_1, \mu_2, ...,\mu_{N})\in \mathbb{R}^{N}$. Each $\mu_i$ ($i=1,\cdots, N$) is uniformly distributed in the interval $(-1,1)$.
The diffusion coefficient $k(x, \mu)$ in (\ref{mix-eq}) is defined as follows,
\[
k(x,\mu)=a(x,\mu)k_{hc},
\]
where $k_{hc}=\frac{10^4}{\kappa_1}$ and $\kappa_1$ is depicted in Figure \ref{fig-exam} (left), $a(x,\mu)$ take the first thirteen terms of $a(x,\mu)$, i.e., $N=12$ in KLE (\ref{KLE-TRUNC1}).
We note that $k^{-1}$ is not affine with respect to the parameter $\mu$. Here we apply STAOMP to $k^{-1}$ to get affine expression and achieve offline-online computation decomposition. The reference solutions are computed by the mixed FEM on $60\times60$ fine grid. We compute the mixed GMsFE basis on $10\times10$ coarse grid.
For the example, $500$ parameter samples are selected for the training set $\Xi_{\text{train}}$. Then we use the greedy algorithm (Algorithm \ref{algorithm-Greedy}) to select $|\Xi_{op}|=40$ optimal samples from the training set $\Xi_{\text{train}}$, and take $l_i(\mu)=5$ for each coarse block $\omega_i$ and each $\mu\in \Xi_{op}$, so the number of the snapshots is $n_{\text{snap}}^i=200$ for each coarse block $\omega_i$. Then we use RmGMsB-GBOCV to get the reduced multiscae basis and apply LSMOS and STAOMP to the reduced model representation.
For LSMOS, we randomly choose $2000$ parameter samples for the snapshots, which are computed by RmGMsB-GBOCV model with $7$ local basis functions.
For LSMOS, we take the first $M$ KLE terms in (\ref{KLE Snapshots}). Here we take $M=3,4,5$, respectively. They are corresponding to the third, forth and fifth row in Table \ref{tab2-exam2}.
Then use Legendre polynomials with total degree up to $N_g=4$ to approximate the KLE coefficients $\{\zeta_i(\mu)\}_{i=1}^{M}$, and the total number of the Legendre polynomial basis functions is $M_g=1820$.
For STAOMP, we use POD to construct $6$ optimal global basis functions for spatial space, and choose
$70$ parameter samples and $100$ spatial coordinates to get the reduced model representation by Algorithm \ref{algorithm-STAOMP}, where the tolerance error $\varepsilon_\text{on}=1\times 10^{-4}$ for both velocity and pressure. To assess the approximation by the model reduction methods,
we randomly choose $1000$ samples and compute the average relative errors defined as (\ref{errors-velocity}), (\ref{errors-pressure}). The results are listed in Table \ref{tab2-exam2} along with the average online CPU time per sample. In the table, $\varepsilon_v^1$ and $\varepsilon_p^1$ are the relative errors based on the reference solutions computed by mixed FEM on fine grid, while $\varepsilon_v^2$ and $\varepsilon_p^2$ are the relative errors based
on the reference solution computed by RmGMsB-GBOCV. From Table \ref{tab2-exam2}, we can see: (1) from the forth and sixth column of the table, we can see that the approximation by the three approaches is almost identical, which means that RmGMsB-GBOCV model can be accurately expressed by the reduced model representation using LSMOS or STAOMP; (2) the average CPU time per sample by STAOMP is the smallest among all of the approaches. The numerical example shows that the STAOMP method achieves a good trade-off in both approximation accuracy and computation efficiency.
We note that the average online CPU time per sample is $21.4444$ seconds using mixed FEM on fine grid. This time is much larger than the reduced model methods.
\begin{table}[hbtp]
\small
\centering
\caption{ Comparison of relative mean errors and the average online CPU time for different approaches. $\text{Mt}_v$ is the number of terms of reduced model representation for velocity, and $\text{Mt}_p$ is the number of terms of reduced model representation for pressure}
\begin{tabular}{c|c|c|c|c|c|c|c}
\Xhline{1pt}
Strategies & $\text{Mt}_v$ & $\text{Mt}_p$ & $\varepsilon_v^1$ &$\varepsilon_v^2$ &$\varepsilon_p^1$ &$\varepsilon_p^2$ & CPU time\\
\hline
\multirow{1}{*}{RmGMsB}
& & & $0.63 \times10^{-2}$ & &$2.76 \times10^{-2}$ & & $0.3841 s$\\
\hline
\multirow{3}{*}{LSMOS}
& 5460 & 5460 & $0.64 \times10^{-2}$ &$7.52\times10^{-5} $&$2.77\times10^{-2} $&$2.50\times10^{-3}$& $1.02\times10^{-2} s$\\
\cline{2-8}
&7280& 7280 & $0.63 \times10^{-2}$ &$6.27\times10^{-6}$&$2.76 \times10^{-2}$& $1.70\times10^{-3} $&$1.68\times10^{-2} s$\\
\cline{2-8}
& 9100& 9100 & $0.63 \times10^{-2}$ &$5.98 \times10^{-7} $&$2.76 \times10^{-2} $&$6.60\times10^{-4} $& $1.84\times10^{-2} s$\\
\hline
\multirow{1}{*}{STAOMP} &36 & 40 & $0.63 \times10^{-2}$ & $9.45\times10^{-6}$ &$2.76\times10^{-2}$& $1.40\times10^{-3} $ & $2.02\times10^{-4} s$\\
\Xhline{1pt}
\end{tabular}
\label{tab2-exam2}
\end{table}
\subsection{Example of two-phase flow in random porous media}
\label{Numerical-result3}
In this numerical example, we consider the following two-phase flow problem (in the
absence of gravity and capillary effects) with zero Neumann boundary condition
\begin{eqnarray}
\label{two-phase-1}
\begin{cases}
\begin{split}
&v=\eta(S) k(x,\mu) \nabla p ~~\text{in} ~~D, \\
&-\nabla \cdot v=q ~~\text{in} ~~D,\\
&\frac{\partial S}{\partial t}+\nabla \cdot \big(f_w(S) v\big)=q_s ~~\text{in}~~ D,
\end{split}
\end{cases}
\end{eqnarray}
where the total mobility $\eta(S)$ is defined by $\eta(S)=S^2/\mu_w+(1-S)^2/\mu_o$ and $v$ refers to total velocity. Here $\mu_w/\mu_o=0.1$ is the ratio
between viscosity of water and oil, the fractional flow function $f_w(S)$ is given by
\begin{eqnarray*}
\label{flux}
f_w(S)=\frac{S^2}{S^2+\mu_w/\mu_o(1-S)^2},
\end{eqnarray*}
where $S$ is the saturation of water and $q$ is the source term for pressure equation and the source term for the saturation equation is expressed by
\[
q_s=\max(q,0)+f_w(S)\min(q,0).
\]
We consider a logarithmic random field,
\[
k(x,\mu):=\exp\big(\kappa_2+a(x,\mu)\big),
\]
where $\kappa_2$ is depicted in Figure \ref{fig-exam} (right), and $a(x,\mu)$ is a stochastic field, which is characterized by a two point exponential covariance function $\text{cov}[a]$ in (\ref{covariance function}). Here the variance $\sigma^2=1$, correlation length $l_x=l_y=0.02$. The random field $a(x, \mu)$ is obtained by truncating Karhunen-Lo\`{e}ve expansion (\ref{KLE-TRUNC1}),
where $E[a]=0$ and the random vector $ \mu:=(\mu_1, \mu_2, ...,\mu_N)\in \mathbb{R}^{N}$ and $\mu_i\sim N(0, 1)$ ($i=1,\cdots, N$), i.e, normal distribution with zero mean and unit variance. We truncate the KLE (\ref{KLE-TRUNC1}) after the first $20$ terms to represent the random field $a(x, \mu)$.
To fulfil offline-online computation decomposition, we use EIM for $k^{-1}(x, \mu)$ and get an affine expression.
The reference solution is computed on a $56 \times 56$ fine grid. The mixed GMsFEM is performed on a $7 \times 7$ coarse grid.
We solve the flow equation on the coarse grid using RmGMsB-GBOCV first and then reconstruct the fine-scale velocity field as a superposition of the GMsFE basis functions. The reconstructed velocity field is used to solve the saturation equation with a finite volume method on the fine grid. We use an Implicit Pressure Explicit
Saturation (IMPES) formulation for the coupled system (\ref{two-phase-1}).
Let $S_i^n$ be the saturation at the $i_{th}$ fine element $K_i^h$ at time $t_n=t_0+n \Delta t$, where $t_0$ is the initial time. Then the saturation equation can be discretized by
\[
|K_i^h|\frac{S_i^{n+1}-S_i^{n}}{\Delta t}+\int_{\partial K_i^h} f_w(\hat{S}^n)(v\cdot n)=|K_i^h|(q_s)_i,
\]
where $(q_s)_i$ is the average value of $q_w$ on $K_i^h$, and $\hat{S}^n$ is the upwind flux. For numerical test, we take $q$ to be zero except for the top-left and bottom-right coarse-grid elements, where $q$ takes the values 1 and $-1$, respectively. This is a traditional
two-spot problem,
in which the water is injected at the top-left corner and oil is
produced at the bottom-right corner.
Moreover, we set the initial staturation to be zero.
For the pressure equation in (\ref{two-phase-1}), we construct the multiscale finite element space $V_H^{M_C,CV}$ by RmGMsB-GBOCV at initial time, and use it for all the time levels. We randomly choose $n_t=|\Xi_{\text{train}}|=200$ parameter samples and select $|\Xi_{op}|=40$ optimal samples by greedy algorithm to compute the snapshots. For each coarse block $\omega_i$ and each $\mu\in \Xi_{op}$, we take $l_i(\mu)=7$ local basis functions to construct snapshots ($n_{\text{snap}}^i=\sum_{j=1}^{n_t}l_i(\mu_j)=280$ for each coarse block $\omega_i$) for the reduced GMsFE basis. Then we employ RmGMsB-GBOCV to simulate the flow equation and get the following weak formulation,
\begin{eqnarray}
\label{two-phase-dis}
\begin{cases}
\begin{split}
&a(v_H^N(\mu),u; \mu)-b(u, p_H^N(\mu); \mu)=0 ~~~\forall~ u \in V_H^{M_C,CV}, \\
&b(v_H^N(\mu), q; \mu)=l(q) ~~~\forall ~q \in Q_H,\\
&|K_i^h|\frac{S_i^{n+1}-S_i^{n}}{\Delta t}+\int_{\partial K_i^h} f_w(\hat{S}^n)(v\cdot n)=|K_i^h|(q_s)_i.
\end{split}
\end{cases}
\end{eqnarray}
\begin{figure}[htbp]
\centering
\includegraphics[width=2.1in, height=1.9in]{FEM5}
\includegraphics[width=2.1in, height=1.9in]{rb5}
\includegraphics[width=2.1in, height=1.9in]{omp5}\\
\includegraphics[width=2.1in, height=1.9in]{FEM10}
\includegraphics[width=2.1in, height=1.9in]{rb10}
\includegraphics[width=2.1in, height=1.9in]{omp10}
\caption{The mean of water saturation profiles at $t=600$ (the first row) and $t=1000$ (the second row), $56\times 56$ fine grid, and $7\times 7$ coarse grid, the number of local multiscale basis functions is $4$.}
\label{fig1-exam3}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=2.1in, height=1.9in]{FEMvar5}
\includegraphics[width=2.1in, height=1.9in]{rbvar5}
\includegraphics[width=2.1in, height=1.9in]{ompvar5}\\
\includegraphics[width=2.1in, height=1.9in]{FEMvar10}
\includegraphics[width=2.1in, height=1.9in]{rbvar10}
\includegraphics[width=2.1in, height=1.9in]{ompvar10}
\caption{The variance of water saturation profiles at $t=600$ (the first row) and $t=1000$ (the second row), $56\times 56$ fine grid, $7\times 7$ coarse grid, the number of local multiscale basis functions is $4$.}
\label{fig2-exam3}
\end{figure}
\begin{table}[hbtp]
\small
\centering
\caption{ Comparison of the average online CPU time for water saturation per sample at different time levels for the two approaches: STAOMP and RmGMsB-GBOCV, the number of local multiscale basis functions is $4$.}
\begin{tabular}{c|c|c|c|c}
\Xhline{1pt}
Strategies & $t=500$ & $t=1000$ & $t=1500$ & $t=2000$\\
\hline
STAOMP & $1.60\times 10^{-3} s$ & $1.50\times10^{-3} s$ & $1.40\times10^{-3} s$ & $1.50\times10^{-3} s$\\
\hline
RmGMsB-GBOCV & $1.7837$ & $5.0227$ & $6.6340$ & $7.8311$\\
\Xhline{1pt}
\end{tabular}
\label{table3b-exam3}
\end{table}
Water saturation is an important quantity for the model. We utilize STAOMP to represent the water saturation $S(x,t,\mu)$ of the reduced model (\ref{two-phase-dis}).
To this end, we choose the set of parameter samples $\Xi_{t}=\{\mu_i\}_{i=1}^{600}$ and apply POD to the snapshots associated with $\Xi_{t}$ and get the optimal global basis functions $\{\hat{S_i}(x,t)\}_{i=1}^{4}$.
To approximate random parameter space, we use Hermite polynomial basis functions with total degree up to $N_g=3$ and denote the basis set by
$\{p_i(\mu)\}_{i=1}^{1771}$.
A finite dimensional approximation space for $\{S(x,t,\mu_i); \mu_i\in \Gamma\}$ is then obtained by
\[
\textbf{S}_{4\times 1771}:=\text{span}\{p_i(\mu)\hat{S_j}(x,t): 1\leq i\leq 1771,1\leq j \leq 4\}.
\]
To get a model reduction representation using STAOPM, we randomly choose $600$ parameter samples, $64$ spatial points and 18 time levels for snapshots, and then use Algorithm \ref{algorithm-STAOMP} to construct
a sparse representation $S(x,t,\mu)\approx \sum_{i=1}^{\text{Mt}} \textbf{c}(\emph{I}(i)) \Psi_{\emph{I}(i)}(x,t,\mu)$, where $\text{Mt}=160$,
$I=\{1,\cdots,1771\}\times \{1,\cdots,4\}$ and $\Psi_i(t,\mu)=p_{i_1}(\mu)\hat{S_{i_2}}(x,t)$.
To calculate the approximation errors by RmGMsB-GBOCV and STAOPM, we randomly choose $1000$ samples and compute the RmGMsB-GBOCV model (\ref{two-phase-dis}) and evaluate the reduced model representation by STAOMP for each sample. The relative mean errors for saturation $S$ is defined by
\begin{eqnarray}
\label{errors-saturation}
\varepsilon_s(t)=\frac{1}{N}\sum_{i=1}^N\frac{\|S(x,t,\mu_i)-S_{H}(x,t,\mu_i)\|_{L^1(D)}}{\|S(x,t,\mu_i)\|_{L^1(D)}},
\end{eqnarray}
where $N=1000$, $S(x,t,\mu_i)$ is the solution of fine scale model, and $S_{H}(x,t,\mu_i)$ is the solution of (\ref{two-phase-dis}) by RmGMsB-GBOCV or STAOPM.
\begin{figure}[htbp]
\centering
\includegraphics[width=4.5 in, height=2.5in]{errorcurver}
\caption{The average relative error for water saturation versus different time levels. the number of local multiscale basis functions is $4$.}
\label{fig3-exam3}
\end{figure}
Figure \ref{fig1-exam3} shows the mean of water saturations at $t=600$ and $t=1000$ for the different methods. Here the reference solution is computed by mixed FEM on fine grid. By the figure, we see that
the model reduction methods provide accurate approximation to the reference saturation in the average sense.
Figure \ref{fig2-exam3} demonstrates the variance of water saturation at $t=600$ and $t=1000$ for the different methods. By the figure, we can find: (1) the largest variance occurs along the advancing water front; (2)
there is no clear difference for the variance profiles between the reference solution and RmGMsB-GBOCV solution; (3) compared with reference solution, the STAOMP approach renders some clear difference for the variance approximation around the water front.
Table \ref{table3b-exam3} lists the average online CPU time per sample for the water saturation by the different approaches. We find that the STAOMP method significantly improves computation efficiency especially for the later time levels. Moreover, the online CPU time of STAOMP method is almost independent of the time level.
This is an important advantage of the STAOMP method for two-phase flow simulation. Figure \ref{fig3-exam3} shows the average relative error for water saturation defined by (\ref{errors-saturation}) versus different time levels for the different methods. By the figure, we find that
the error by STAOMP decays after the water break-through time and then tends to be stable.
\begin{figure}[htbp]
\centering
\includegraphics[width=3in, height=2.5in]{cutmean1}
\includegraphics[width=3in, height=2.5in]{cutvar1}
\caption{The expectation (left) and variance (right) of water-cut curves}
\label{fig4-exam3}
\end{figure}
Water-cut is also an important quantity for the water-oil two-phase flow. The water-cut is defined as the fraction of water in
the produced fluid and is given by $q_w/q_t$, where $q_t=q_o+q_w$, with $q_o$ and $q_w$ being the flow rates of oil
and water at the production edge of the model. In particular, $q_w=\int_{\partial D^{out}} f_w(S){v}\cdot { n} ds$,
$q_t=\int_{\partial \Omega^{out}} {v}\cdot { n} ds$,
where ${\partial D^{out}}$ is the out-flow boundary.
For the water-cut, we utilize STAOMP to approximate the water-cut of the reduced model (\ref{two-phase-dis}).
We use the set of parameter samples $\Xi_{t}=\{\mu_i\}_{i=1}^{600}$ and apply POD to the snapshots associated with $\Xi_{t}$ to obtain $6$ optimal global basis functions $\{\hat{w_i}(t)\}_{i=1}^{6}$. We use the same polynomial basis functions as the water saturation, and get a finite dimensional approximation space
\[
\textbf{W}_{6\times 1771}:=\text{span}\big\{p_i(\mu)\hat{w_j}(t): 1\leq i\leq 1771,1\leq j \leq 6\big\}.
\]
To get a model reduction representation using STAOPM, we randomly choose $600$ parameter samples and $10$ time levels for snapshots, and then use Algorithm \ref{algorithm-STAOMP} to construct
a sparse representation $W(t,\mu)\approx \sum_{i=1}^{\text{Mt}} \textbf{c}(\emph{I}(i)) \Psi_{\emph{I}(i)}(t,\mu)$ with $\text{Mt}=195$,
where $I=\{1,\cdots,1771\}\times \{1,\cdots,6\}$ and $\Psi_i(t,\mu)=p_{i_1}(\mu)\hat{w_{i_2}}(t)$.
Figure \ref{fig4-exam3} shows the expectation (left) and variance (right) of the water-cut curves with the different approaches. The figure shows that the expectation curves of the water-cut are all nearly identical. This implies a good mean approximation for water-cut using RmGMsB-GBOCV and STAOMP; The variance of water-cut rises rapidly at the break-through time, and then decays fast after a while. The error of water-cut variance by
STAOMP is slightly larger around the water break-through time than other time instances.
\section{Conclusions}
\label{ssec:Conclusions}
In the paper, we presented two variable-separation representations for random fields: LSMOS and STAOMP. The LSMOS is devoted to constructing a KL expansion and building a relation between the random inputs and the stochastic basis functions based on least-squares methods. The STAOMP provides a tensor product for the random field and achieves a sparse approximation by using orthogonal-matching-pursuit method.
The two methods can be applied to get a variable-separation approximation for any generic functions with random inputs.
For the multiscale problems with high-dimensional random inputs, the computation would be prohibitive if we directly solve the problems for a many-query situation. To overcome the difficulty,
we developed reduced mixed GMsFE basis methods to improve the computation efficiency. To this end, we have chosen a few optimal parameters from the parameter space by a greedy algorithm. Then we use the optimization methods BOCV and POD to obtain reduced multiscale basis functions and get a reduced order model. In the reduced multiscale basis methods, the whole computation admits offline-online decomposition. In the online phase, a reduced model
is solved for each parameter sample.
Although the offline computation may be expensive, the online computation is efficient. We have carefully compared the performance of the different reduced mixed GMsFE basis methods, and found that BOCV may lead to the reduced model with better accuracy and robustness than POD does. The dimension of the space spanned by the set of reduced multiscale basis functions is much smaller than the dimension of the original full order model, but it depends on the size of coarse grid. When the size of coarse grid is large, the computation of the reduced order model may be still expensive.
To further improve the efficiency of the online computation, we construct a sparse representation for model's outputs by combining the reduced mixed GMsFE basis methods and STAOMP.
This is very desirable for predicting the model's outputs for arbitrary parameter values and quantifying the uncertainty propagation. We applied the proposed reduced mixed GMsFE basis methods and the model's sparse representation method to a few numerical models with multiscale and random inputs. Careful numerical analysis is carried out for these computational models.
Although the proposed methods significantly reduces the online computation, it takes much effort in offline computation stage. In future, we will explore novel methods
to substantially decrease the offline computation burden as well.
\section*{Acknowledgments}
We would like to thank the helpful discussion with Prof. Yalchin Efendiev for the work. We acknowledge the support of Chinese NSF 11471107.
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{"url":"https:\/\/nullren.com\/blog\/2018\/11\/02\/ctfp-13-free-monoids.html","text":"## Category Theory for Programmers Chapter 13: Free Monoids\n\nChallenges\n\n1. Show that an isomorphism between monoids that preserves multiplication must automatically preserve unit.\n\nLet $A$ and $B$ be monoids and $A \\cong B$. Then there exists $f:A \\to B$ and $f^{-1}:B \\to A$ such that $f \\circ f^{-1} = \\mathrm{Id}_B$ and $f^{-1} \\circ f = \\mathrm{Id}_A$. And because $f$ and $f^{-1}$ preserve multiplication, we have also that $f(a a\u2019) = f(a) f(a\u2019)$ for all $a, a\u2019 \\in A$ and $f^{-1}(b b\u2019) = f^{-1}(b) f^{-1}(b\u2019)$ for all $b,b\u2019 \\in B$. Also, because $A$ and $B$ are monoids, there exists $e_A \\in A$ such that $a e_A = e_A a = a$ for all $a \\in A$ and similarly $e_B \\in B$ such that $b e_B = e_B b = b$ for all $b \\in B$.\n\nLet $f(a e_A) = f(a) f(e_A) = f(a)$ for all $a \\in A$, then $f(e_A)$ is a candidate for identity in $B$ because for all $b \\in B$, we have $b f(e_A) = f(f^{-1}(b f(e_A)))$ $= f(f^{-1}(b) f^{-1}(f(e_A)))$ $= f(f^{-1}(b) e_A)$ $= f(f^{-1}(b))$ $= b.$\n\nThen we have that $e_B f(e_A) = e_B$ by above and $e_B f(e_A) = f(e_A)$ by definition of $e_B$ and so we have that $e_B f(e_A) = e_B = f(e_A).$\n\nSimilarly, we can show $e_A f^{-1}(e_B) = e_A = f^{-1}(e_B).$ $\\blacksquare$\n\n2. Consider homomorphism from ([Integer], [], ++) to $(\\mathbb{Z}, 1, \\times)$.\n\n1. What is image of []?\n\nSince [] is the identity in [Integer], the homomorphism preserves structure and so would be mapped to the identity, $1$.\n\n2. Assume all singleton lists are mapped to their integers, [x] $\\mapsto x$. What\u2019s the image of [1, 2, 3, 4]?\n\nBecause we don\u2019t need to worry about associativity (property of monoids), we can say that [1, 2, 3, 4] = [1] ++ [2] ++ [3] ++ [4]. Then using the assumption above, this maps to $1 \\times 2 \\times 3 \\times 4 = 24$.\n\n3. How many lists map to $12$?\n\nThe prime factors of $12$ are $2 \\times 2 \\times 3$ and so we can enumerate all groups of integers that have product 12:\n\n\u2022 $2, 2, 3$\n\u2022 $4, 3$\n\u2022 $2, 6$\n\u2022 $12$\n\u2022 $-4, -3$\n\u2022 $-2, -6$\n\u2022 $-2, -2, 3$\n\u2022 $-2, 2, -3$\n\nIncluding identities, we could have infinitely many.\n\n3. What is the free monoid generated by a one-element set?\n\n$(\\mathbb{N}, 0, +)$ where the isomorphism given by the length of the list.","date":"2022-01-22 19:59:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9933063387870789, \"perplexity\": 180.68545493464083}, \"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-2022-05\/segments\/1642320303884.44\/warc\/CC-MAIN-20220122194730-20220122224730-00633.warc.gz\"}"} | null | null |
Erik Sjöberg (January 14, 1794 - March 4, 1828) was a Swedish poet. He wrote under the pseudonym Vitalis (which roughly means "life is a battle" in Latin).
Although he grew up in poor circumstances he studied at Uppsala University and got a Master of Arts degree. He mostly worked as private tutor while he was writing. His poetry was characterized by the ideals of romanticism, i.e. melancholy and spiritual ecstasy, but he also wrote many comical and satirical poems. Among his most widely known poems are Enslingens sång i den stora öknen ("The lonely man's song in the great desert") and Komiska fantasier ("Comical fantasies"). He is also represented in the 1986 official hymnbook of the Church of Sweden, with one hymn (no. 573). Sjöberg made translations of Thomas Moore and Washington Irving.
Bibliography
- Utg. af Carl Adolf Forselius, med företal af Erik Gustaf Geijer, och lefnadsteckning öfver förf. af utgifvaren.
- Utg. av Algot Werin.
Hymns
Vi skulle jag ej möta glad min plåga (1937 nr 387). Bearbetad av Olle Nivenius Jag ber om hjälp till stillhet i min plåga (1986 nr 573)
Min vän är ljuv, min vän är mild (1921 nr 586). Dikten Den bästa vännen skriven 1824
Further reading
External links
Erik Sjöberg, författarpresentation på Litteraturbanken
Swedish poets
Swedish male poets
Swedish male writers
1794 births
1828 deaths
19th-century Swedish poets
19th-century male writers | {
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<?php
namespace PayPal\Rest;
use PayPal\Core\PPConfigManager;
use PayPal\Core\PPCredentialManager;
/**
* Class ApiContext
*
* Call level parameters such as request id, credentials etc
*
* @package PayPal\Rest
*/
class ApiContext
{
/**
* Unique request id to be used for this call
* The user can either generate one as per application
* needs or let the SDK generate one
*
* @var null|string $requestId
*/
private $requestId;
/**
* This is a placeholder for holding credential for the request
* If the value is not set, it would get the value from @\PayPal\Core\PPCredentialManager
*
* @var \Paypal\Auth\OAuthTokenCredential
*/
private $credential;
/**
* Construct
*
* @param \PayPal\Auth\OAuthTokenCredential $credential
* @param string|null $requestId
*/
public function __construct($credential = null, $requestId = null)
{
$this->requestId = $requestId;
$this->credential = $credential;
}
/**
* Get Credential
*
* @return \PayPal\Auth\OAuthTokenCredential
*/
public function getCredential()
{
if ($this->credential == null) {
return PPCredentialManager::getInstance()->getCredentialObject();
}
return $this->credential;
}
/**
* Get Request ID
*
* @return string
*/
public function getrequestId()
{
if ($this->requestId == null) {
$this->requestId = $this->generateRequestId();
}
return $this->requestId;
}
/**
* Resets the requestId that can be used to set the PayPal-request-id
* header used for idempotency. In cases where you need to make multiple create calls
* using the same ApiContext object, you need to reset request Id.
*
* @return string
*/
public function resetRequestId()
{
$this->requestId = $this->generateRequestId();
return $this->getrequestId();
}
/**
* Sets Config
*
* @param array $config SDK configuration parameters
*/
public function setConfig(array $config)
{
PPConfigManager::getInstance()->addConfigs($config);
}
/**
* Gets Configurations
*
* @return array
*/
public function getConfig()
{
return PPConfigManager::getInstance()->getConfigHashmap();
}
/**
* Gets a specific configuration from key
*
* @param $searchKey
* @return mixed
*/
public function get($searchKey)
{
return PPConfigManager::getInstance()->get($searchKey);
}
/**
* Generates a unique per request id that
* can be used to set the PayPal-Request-Id header
* that is used for idempotency
*
* @return string
*/
private function generateRequestId()
{
static $pid = -1;
static $addr = -1;
if ($pid == -1) {
$pid = getmypid();
}
if ($addr == -1) {
if (array_key_exists('SERVER_ADDR', $_SERVER)) {
$addr = ip2long($_SERVER['SERVER_ADDR']);
} else {
$addr = php_uname('n');
}
}
return $addr . $pid . $_SERVER['REQUEST_TIME'] . mt_rand(0, 0xffff);
}
}
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\section{Introduction}
Measurements of the top quark mass with highest possible precision
are useful for constraining physics beyond the Standard Model through
precision electroweak tests. In spite of spectacular Tevatron
results \cite{TevatronCDF,TevatronD0},
the motivation to continue top quark measurements will
remain strong even at the era of the
LHC since one would like to see a consistency between
direct and indirect evidences for New Physics. Absence of
such a consistency will be a strong indication that our understanding
of emergent New Physics is incomplete.
ATLAS and CMS plan to employ a variety of methods
for measuring the top quark mass
\cite{tdrs,Beneke:2000hk,rev1, rev2, rev3,Bernreuther:2008ju}.
These methods can be divided into
two classes. The first class includes so-called matrix element methods.
The idea is to fit the top quark mass by adjusting
its value, to best describe
various kinematic features of $t \bar t$ events, using squared leading
order matrix elements as probability density functions.
Such methods typically
lead to very small uncertainties in the top quark mass because
nearly all the information about the events is utilized. However,
the drawback of these methods is that it is hard
to estimate the theoretical uncertainty in the top quark mass
obtained in that way.
The second class includes determination of the top quark mass from
kinematic distributions that are sensitive to the value of $m_t$.
It is interesting that, up to now,
all analyses of such distributions including their sensitivity to the
top quark mass and the theoretical uncertainty in $m_t$ have been
performed \cite{tdrs,Beneke:2000hk,rev1, rev2, rev3,Bernreuther:2008ju}
using parton shower event generators,
such as HERWIG \cite{hw} and PYTHIA\footnote{Even determination of the top quark
mass from the total cross-section for top quark pair production requires computation
of the acceptance since the total cross-section is never measured.
Such acceptances are routinely computed using parton shower event generators.}
\cite{pt}.
As the result
of such studies, it is often claimed that high precision in
the top quark mass measurement can be achieved. Unfortunately,
it is not clear how reliable such conclusions are since,
by construction,
parton showers {\it can not guarantee} high precision for
a generic observable due to their approximate nature. It is therefore
important to find alternative ways to estimate theoretical
uncertainties
in the description of relevant kinematic distributions,
since those uncertainties impact directly
the precision of $m_t$ which can be expected from those measurements.
In this regard, we point out that some kinematic
distributions that are expected to be used for the top quark mass
measurements,
can be computed in perturbative QCD. Typically, those distributions
involve top quark decay products; their computation at leading
order in perturbative QCD is straightforward.
On the other hand, precision requirements on $m_t$
make it necessary to go to higher orders in the perturbative expansion
and, in spite of the fact that NLO QCD corrections to $t \bar t$ pair
production have been known for about twenty years
\cite{nde,bnmss}, only recently NLO QCD
corrections to top quark pair production {\it and} decay with all the spin
correlations included
became available \cite{ms,bernsi,mcfm}.
The availability of NLO QCD corrections
to realistic final states and observables is a necessary
pre-requisite for the high-precision
analyses of kinematic distributions relevant for the top quark mass
determination. Higher-order corrections are also
important from a more theoretical viewpoint
since they allow us to distinguish between the mass parameters
defined in different renormalization schemes. In the context of
$t \bar t$ production at the Tevatron, this issue was recently discussed
in Ref.~\cite{moch}. Throughout this paper, we employ the pole
mass of the top quark, for convenience.
The goal of this paper is to use the computation
reported in Ref.~\cite{ms} as a starting point
to study some observables relevant
for the top quark mass determination
at leading and next-to-leading order in QCD perturbation theory
and to investigate
their sensitivity to input parameters. We will study four
observables in this paper.
In Section~\ref{sect5} we
discuss the invariant mass
distribution of a lepton and a $B$-meson from the top quark decay; this
is a simplified version of an observable suggested in
Ref.~\cite{kharch} for the top quark mass measurement. Analysis of
$m_{Bl}$ distribution with high precision requires computation of
NLO QCD radiative corrections to the {\it exclusive} decay $t \to B + X+l^+
+\nu$ which we report in Section~\ref{sect5}.
In Section~\ref{dilepton}
we investigate the
invariant mass distribution of the $b$-jet and the charged lepton, and
the distributions of the sum of energies of the
two leptons and the sum of energies of the two
$b$-jets in the dilepton channel.
Calculation of those distributions is performed
using results reported in Ref.~\cite{ms}. We conclude in Section~\ref{conc}.
\section{Measurement of $m_t$ in top
decays to the final state with identified $B$-meson}
\label{sect5}
It was pointed out in Ref.~\cite{kharch} that the top quark mass
can be accurately measured by studying top quark decays to
an exclusive hadronic state. For example,
one may consider the process
$pp \to (t \to W^++b \to W^+ + J/\psi) + (\bar t \to W^- + \bar b)$
and require that the $W^-$ decays hadronically, $W^+$ decays leptonically, $J/\psi$ decays into
a pair of leptons and $ \bar b$ decays into a lepton (inside the jet)
as well. Then,
one may use the invariant mass distribution
of a $J/\psi$ and an isolated lepton
to determine the top quark mass. The requirement of a
large number of leptonic decays reduces the rate significantly.
However, it also
reduces the combinatorial background from
the incorrect pairing of the $J/\psi$ and a lepton. In addition,
since no jet measurements are involved, the measurement is
insensitive to jet energy scale uncertainties.
As the result, a very accurate
reconstruction of the invariant mass $m_{J/\psi l}$ and the
measurement of the top quark mass become possible.
It is expected \cite{kharch,cms,cmes}
that ${ \cal O }(1~{\rm GeV})$
error on the top quark mass
can be achieved in such measurement. The only (serious)
drawback of this method
is that very large luminosity -- about $100~{\rm fb}^{-1}$ -- is required
since the rate is suppressed due to all
(semi)leptonic branching fractions involved. However, it was pointed
out in Ref.~\cite{cmsnote} that the situation can be improved by
giving up the requirement of the leptonic decay of the
$\bar b$-quark. In this case, combinatorial background increases but remains
manageable, and the luminosity needed to reach uncertainty
of about $1.5~{\rm GeV}$ in the top quark mass is reduced to about
$20~{\rm fb}^{-1}$ \cite{cmsnote}.
The small uncertainty in the top quark mass that can, potentially,
be achieved in those measurements is very attractive. It also
sets the bar for other methods
of the top quark mass measurement, planned at the LHC.
It is therefore peculiar that the analyses
in Refs.~\cite{kharch,cms,cmes} are performed by using parton shower
event generators to describe production of top quarks and their decays.
The uncertainty in the extracted value of $m_t$ is estimated
in those references
by comparing results obtained using different parton shower event
generators, such as PYTHIA and HERWIG \cite{cmes}
or even different versions of HERWIG
\cite{cms}. It is possible that parton shower event generators
give reasonable description of the required mass distribution
and that the resulting error estimates of the top quark mass
are trustworthy. However, it is important to check this,
given the potential importance
of the top quark mass measurement. To this end, it is
useful to look for alternative ways to describe
the process $pp \to (t \to W^++b \to W^+ + J/\psi)
+ (\bar t \to W^- + \bar b)$, to ensure that the current understanding
of top quark decays and $b \to B$ fragmentation is consistent
with a very small error on $m_t$ that is claimed to be
achievable through $m_{J/\psi l}$ measurement.
One way to achieve that is to avoid using parton showers and,
instead, to compute the $m_{J/\psi l}$ invariant mass distribution
in the process
$pp \to (t \to W^++b \to J/\psi)
+ (\bar t \to W^- + \bar b)$
in fixed-order perturbative QCD.
We describe how this can be done
in this Section. To simplify the problem,
we follow Refs.~\cite{cms,cmes} in that we do not include
the decay of a $B$-meson to a $J/\psi$ meson, but only consider
a $b$-quark fragmentation into a $B$-meson. This is a reasonable
first step because decays of $B$-mesons to $J/\psi$-mesons are
well-studied at $B$-factories.
The energy spectrum of $B$-mesons in top decays can be
computed using the $b \to B$ fragmentation function formalism
\cite{nasonmele} which allows systematic inclusion of higher-order
QCD effects. The observable that we study in this Section
is the invariant mass distribution
of the $B$-meson and the lepton from the associated $W$-decay.
The NLO QCD calculation of the
$B$-meson energy spectrum in top quark decays
was performed in Ref.~\cite{cm}
within the $b \to B$
fragmentation function formalism. However, the
results of that reference can not be used directly for our
purpose since leptons from $W$-decays were integrated over. Because
the primary object of our study is the invariant mass of a lepton from the
$W$ decay and a $B$-meson from the $b$-fragmentation and since we would
like to be able to impose kinematic
constraints on top quark decay products,
we require a calculation of the NLO QCD corrections
that is exclusive inasmuch as the top quark decay products are concerned.
To perform such a calculation, we employ the dipole subtraction formalism
of Ref.~\cite{catani}. We point out, however, that we have an identified
hadron in the final state. Hence, care is required
when the dipole subtraction
formalism is applied. In principle, Ref.~\cite{catani} does describe the
construction of the subtraction terms for such a situation,
but since we deal here with decay kinematics and since massive particles are
involved, we adopt a slightly different approach.
As our starting point,
we take subtraction terms constructed specifically for
top quark
decays in Ref.~\cite{ekt}. We modify the subtraction procedure
slightly, to allow for the identified hadron in the final state, and
obtain a fully differential description of the decay
$t \to { l^+ \nu} + B + X$ through NLO QCD.
\subsection{Calculation of radiative corrections
to $t \to l^+ \nu + B + X$.}
In this Section, computation of the radiative corrections to the
decay $t \to (W^+ \to l^+ \nu) + B + X$ is described. We assume that
the $B$-meson is produced by the fragmentation of the massless $b$-quark. The
$W$ boson is on the mass shell.
We denote by $x$
the fraction of energy carried away by the $B$-meson
in the top quark rest frame\footnote{Except for the $B$-meson,
we denote particles and their momenta by the same label. We hope that
this fact does not cause any confusion.}
\be
x = \frac{2t p_{B}}{m_t^2(1-r^2)},\;\;\;\; r^2= \frac{m_W^2}{m_t^2}.
\ee
The differential decay rate reads
\be
\frac{{\rm d} \Gamma_{B}}{{\rm d} x} =
\int \limits_{x}^{1}
\frac{{\rm d} \xi}{\xi}\; \frac{{\rm d} \Gamma_{b}}{{\rm d} \xi} \;
D\left ( \frac{x}{\xi} \right ),
\label{eq4_1}
\ee
where $\xi = 2tb/(m_t^2(1-r^2))$, ${\rm d}\Gamma_{b}/{\rm d} \xi$
is the differential decay rate for the partonic decay
$t \to W + b +X$ and $D(x)$ is the fragmentation function
for $b \to B$. One can restore the dependence on other partonic
variables in Eq.~(\ref{eq4_1}) because collinear fragmentation does not
affect them. Therefore,
Eq.~(\ref{eq4_1}) is a starting point for the computation
of various kinematic distributions for top quark
decays to final states with jets and an identified $B$-meson.
Our goal is to compute these distributions
through next-to-leading order in perturbative QCD.
To this end, the partonic decay width ${\rm d} \Gamma$
is expanded in series of the strong coupling constant
$\alpha_s$
\be
{\rm d} \Gamma_b = {\rm d} \Gamma_b^{(0)} + {\rm d} \Gamma_b^{(\mathrm{V})}
+ {\rm d} \Gamma_b^{(\mathrm{R})} + {\cal O}(\alpha_s^2),
\ee
where the three terms refer to leading order decay rate and
virtual and real contributions to the NLO decay rate, respectively.
Because ${\rm d} \Gamma_b^{(0)}$ and ${\rm d} \Gamma_b^{(\mathrm{V})}$
have two-body final states\footnote{We count decay
products of a $W$ boson as a single particle.},
the $b$-quark in that decay has maximal energy.
This implies that ${\rm d}\Gamma_b^{(0,\mathrm{V})}$ are
proportional to a delta-function of $\xi$
\be
{\rm d} \Gamma_b^{(0)} \sim {\rm d} \Gamma_b^{(\mathrm{V})} \sim \delta (1-\xi),
\ee
and we can write
\ba
\Gamma_{B} &=&
\int {\rm d} x
\left \{ {\rm d} \Gamma_b^{(0)} + {\rm d}\Gamma_b^{(\mathrm{V})} \right \}
D(x) F_{J,2}(\{p\})
\nonumber \\
&&
+ \int {\rm d} x \;
{\rm d} \Gamma_b^{(\mathrm{R})}\; \frac{m_t^2(1-r^2)}{2bt}\;
D \left ( \frac{x m_t^2(1-r^2)}{2bt} \right )
F_{J,3}(\{p\}).
\ea
Note that we introduced the ``measurement function'' $F_{J,n}$ to indicate
external constraints that are applied to a $n$-particle final state.
The measurement function depends on the momenta of final state
particles, including the momentum of the $B$ meson. It
is assumed to satisfy the usual requirements of infra-red and collinear safety.
Since virtual and real corrections are separately infra-red and collinear
divergent and since the measurement function is arbitrary,
we need to set up a calculation where all divergences
in ${\rm d}\Gamma^{(\mathrm{R})}$ and in ${\rm d}\Gamma^{(\mathrm{V})}$ are regulated
separately. We construct the necessary subtraction term below
following Ref.~\cite{ekt} closely.
We begin by considering the matrix element that describes the real emission
process $t \to W+b + g$
\be
{\rm d} \Gamma_b^{(\mathrm{R})} \propto |{\cal M}_3|^2~{\rm d}\Phi^{(3)},
\ee
where ${\rm d}\Phi^{(3)}$ is the phase-space element for $W,b$ and $g$,
and introduce variables $z,y$ to parametrize the scalar products that
involve the gluon momentum
\be
bg = \frac{m_t^2}{2} (1-r)^2 y;\;\;\; tg = \frac{m_t^2}{2} (1-r^2) (1-z) .
\ee
The soft limit $g \to 0$ requires $z \to 1,\;\; y \to 0$. The collinear
limit $g \ne 0,\; gb \to 0$ corresponds to $y \to 0$.
We express the fraction of maximal energy (in the top quark rest frame)
carried by the $b$ quark in $t \to b +g +W$ decay
through $z$ and $y$ variables
\be
\frac{E_b}{E_{b,\rm max}}
= \frac{2tb}{m_t^2(1-r^2)} = f(z,y) =
z + \frac{(1-r)}{(1+r)}y.
\label{eq_ww}
\ee
Using Eq.~(\ref{eq_ww}), we find that $f(z,y) \to 1$ and
$f(z,y) \to z$ in the soft and collinear limits, respectively.
The soft and collinear limits motivate the
construction of subtraction counter-terms
for real emission corrections. Taking the difference of the real
emission correction and the subtraction counter-term, we obtain
an integrable expression
\be
\frac{ |{\cal M}_3|^2}{f(z,y)} \; F_{J,3}(\{p \},p_{B})
D \left ( \frac{x}{f(z,y)}\right )
-
\frac{|\tilde { \cal M}_3(\tilde p)|^2}{z}\;
F_{J,2}\left (\{ \tilde p\}, \tilde p_{B} \right )
D \left ( \frac{x}{z} \right ),
\label{eq671}
\ee
which explicitly involves the fragmentation function.
We emphasize that,
as with any subtraction method, the counter-term is evaluated
for values of momenta
that differ from the momenta used in the evaluation of the
matrix element. In particular, the $B$-meson
momenta are related to a particular
$b$-quark momenta in the following way
\be
p_{B} = \frac{x b}{f(z,y)},\;\;\;\;\;\;\;
\tilde p_{B} = x \tilde b.
\label{eq1010}
\ee
In the soft limit $f(z,y) \to 1$, $z \to 1$ and $b \to \tilde b$,
whereas in the collinear limit, $f(z,y) \to z$, $b \sim z t$ and
$\tilde b \sim t$. It follows from Eq.~(\ref{eq1010}) that
$p_{B}$ and ${\tilde p}_B$ coincide in both limits; of course, this is
an important condition for the proper work of the
subtraction counter-term.
For the subtraction matrix element $|\tilde {\cal M}_3|^2$, we employ
\cite{ekt}
\be
|\tilde { \cal M}_3|^2 = |\tilde {\cal M}_2|^2 \, {\rm Dip}(z,y),
\ee
where
\be
{\rm Dip}(z,y) = C_\mathrm{F} g_s^2 \mu^{2\epsilon}
\left ( \frac{1}{bg} \left ( \frac{2}{1-z} - 1 -z - \eta \epsilon (1-z)
\right ) - \frac{m_t^2}{(tg)^2}
\right ),
\ee
and $\tilde {\cal M}_2$ is the matrix element for $t \to b + W$.
The term proportional to $\eta$ distinguishes the
t'Hooft-Veltman ($\eta =1$) and the four-dimensional
helicity ($\eta = 0$) regularization schemes. We need to employ
the t'Hooft-Veltman scheme
in our calculation since this is the scheme (combined with
the ${\overline {\rm MS}}$ subtraction) in which fragmentation functions
are extracted from the $e^+e^-$ data in Ref.~\cite{cm}.
The required
momentum mapping is constructed in Ref.~\cite{ekt}; we summarize
it here for completeness. We need to map a three particle final
state ($ t \to W+b+g$) onto two particle final state ($ \tilde t \to \tilde W
+ \tilde b$).
We require that the top momentum
does not change, $\tilde t = t$, so that
\be
t = \tilde W + \tilde b.
\ee
Since $W^2 = {\tilde W}^2 = m_W^2$, $W$ is a valid candidate
to be the four-momentum of the $W$ boson after mapping but
it has wrong energy for the two-body
decay. To correct for that, we can make a Lorentz transformation
\be
\tilde W^{\mu} = \Lambda^{\mu}_{\nu} W^{\nu}.
\ee
The matrix $\Lambda^{\mu \nu}$ reads \cite{ekt}
\ba
\Lambda^{\mu \nu} && = g^{\mu \nu}
+ \frac{\sinh(x)}{\sqrt{(tW)^2-m_t^2 m_W^2}}\left (t^\mu W^\nu
- W^\mu t^\nu \right) \\
&& + \frac{\cosh(x) - 1}{(tW)^2 - m_W^2 m_t^2}
\left (tW(t^\mu W^\nu + W^\mu t^\nu) - m_W^2 t^\mu t^\nu - m_t^2 W^\mu W^\nu
\right ),
\ea
where
\be
\sinh(x) = \frac{1}{2m_t^2 m_W^2}
\left ( - (m_t^2 - m_W^2) tW + (m_t^2 + m_W^2)\sqrt{ (tW)^2 - m_W^2 m_t^2 }
\right ).
\ee
Applying the Lorentz transformation to $W$, we obtain a simple expression
\be
\tilde W = \alpha \left ( W - \frac{tW}{m_t^2}\; t \right )
+ \beta t,
\ee
where
\be
\alpha = \frac{\sqrt{(m_t^2-m_W^2)^2 - 4 m_W^2 m_t^2}}{2\sqrt{(tW)^2 -
m_W^2 m_t^2}},\;\;\;
\beta = \frac{(m_t^2 - m_W^2)}{2 m_t^2}.
\ee
Explicit knowledge of the matrix
$\Lambda^{\mu \nu}$ is required to account for momenta
changes of the decay products of the $W$ boson.
We are now in position to discuss the integration of the subtraction term
over the unresolved phase-space. We use the phase-space factorization,
described in Ref.~\cite{ekt}, ${\rm d} \Phi^{(3)}
= {\rm d} \tilde \Phi^{(2)} {\rm d} \tilde g$, where
\be
\int {\rm d} \tilde g =
\frac{(1-r)^2 m_t^{2-2\ep}}{16\pi^2} \frac{(4\pi)^\ep}{\Gamma(1-\ep)}
\left (
\frac{1+r}{1-r}
\right)^{2\ep}
\int \limits_{0}^{1} {\rm d} z
\left ( r^2 + z(1-r^2) \right )^{-\ep} \int \limits_{0}^{y_{\rm max}} y^{-\ep}
(y_{\rm max} - y)^{-\ep}.
\label{eq567}
\ee
In Eq.~(\ref{eq567}), $\epsilon$ is the dimensional
regularization parameter and $ y_{\rm max} = (1+r)^2 z(1-z)/(z+r^2(1-z))$.
We need to calculate
\be
I_{\rm dip} = \int {\rm d}{ \tilde g} \;
{\rm Dip}(z,y)\; z^{-1}\; D \left ( \frac{x}{z} \right ).
\ee
The two integrals over $y$ that we need (through an appropriate
order in $\epsilon $) are
$$
\int \limits_{0}^{y_{\rm max}}
\frac{{\rm d} y}{y} y^{-\ep}(y_{\rm max} - y)^{-\ep}
= y_{\rm max}^{-2\ep} \left ( -\frac{1}{\ep} + \ep \frac{\pi^2}{6} \right),
\;\;\;\;\int \limits_{0}^{y_{\rm max}}
{\rm d} y y^{-\ep} (y_{\rm max} - y)^{-\ep} =
y_{\rm max}^{1-2\ep} ( 1 + 2\ep).
$$
Upon integrating over $y$, we obtain
\ba
I_{\rm dip} = &&
\frac{2g_s^2 \mu^{2\ep} C_\mathrm{F} m_t^{-2\ep}}{(4 \pi)^{2-2\ep} \Gamma(1-\ep)
(1-r^2)^{2\ep}} \int \limits_{0}^{1}
\frac{{\rm d} z}{z} D \left ( \frac{x}{z} \right ) (r^2 + z(1-r^2) )^{\ep}
z^{-2\ep} (1-z)^{-2\ep}
\nonumber \\
&& \times
\left \{ \left ( \frac{2}{1-z} - 1- z\right )
\left (-\frac{1}{\ep} + \ep \frac{\pi^2}{6} \right )
- \frac{2 z(1+2\ep)}{(z+r^2(1-z))(1-z)}
\right \}.
\label{fgz}
\ea
Because the $z$-dependent fragmentation function $D$ is present in
Eq.~(\ref{fgz}), we can not integrate over $z$ analytically.
The best we can do is to extract infra-red and collinear
divergences.
We find
\ba
I_{\rm dip} =
&& \frac{C_\mathrm{F} \alpha_s (1-r^2)^{-2\ep} }{2 \pi \Gamma(1-\ep)}
\left ( \frac{4\pi \mu^2}{m_t^2} \right )^\ep
\Bigg[ \frac{D(x) }{\ep^2}
- \frac{1}{\ep} \int \frac{{\rm d} z}{z}
D \left ( \frac{x}{z} \right )
\left ( \frac{2}{(1-z)_+} - (1+z) - \delta(1-z) \right )
\nonumber \\
&& \left.
+D(x) \left (2- \frac{\pi^2}{6} \right )
+\int \frac{{\rm d} z}{ z} D \left ( \frac{x}{z} \right )
\Bigg\{ 4 \left [ \frac{\ln(1-z)}{1-z} \right ]_+
- 2 (1+z) \ln (1-z)
\right. \nonumber \\
&&
- \left ( \frac{2}{(1-z)}
- (1+z) \right ) \ln \left( \frac{r^2+z(1-r^2)}{z^2} \right)
- \frac{2}{(1-z)_+} \frac{z}{r^2+z(1-r^2) }
\Bigg\}
\Bigg].
\label{eq_intdip}
\ea
The infra-red and collinear divergences explicit
in the result for the integrated dipole Eq.~(\ref{eq_intdip})
must cancel with the virtual corrections
{\it and} the ${\overline {\rm MS}}$ renormalization of
the fragmentation function. The sum of the leading
order decay rate and the one-loop virtual
correction reads
\be
{\rm d} \Gamma_{B}^{(0)} +
{\rm d} \Gamma_{B}^{(V)} \sim |\mathcal{M}_2|^2 \; I_{\rm virt}
\; {\rm d} \Phi_2,
\ee
where
\be
I_{\rm virt} = D(x) \left [ 1
+ \frac{\alpha_s C_\mathrm{F} (1-r^2)^{-2\ep} }{2\pi \Gamma(1-\ep)}
\left ( \frac{4\pi \mu^2}{m_t^2} \right )^{\ep}
\left ( C_0 + \frac{C_1}{2} \frac{1-r^2}{1+2 r^2} \right )
\right ].
\ee
The functions $C_{0,1}$ read
\cite{ekt}
\ba
&& C_0 = -\frac{1}{\ep^2} - \frac{5}{2\ep}
- \frac{11+\eta}{2} - \frac{\pi^2}{6} - 2{\rm Li}_2(r^2)
- 2\ln (1-r^2) - \frac{\ln (1-r^2)}{r^2}
\nonumber \\
&& C_1 = \frac{2}{r^2} \ln(1-r^2).
\ea
Taking the sum of $I_{\rm dip}$ and
$I_{\rm virt}$, we obtain
\ba
&& I_{\rm dip} + I_{\rm virt}
= D(x) + \frac{\alpha_s C_\mathrm{F}}{2\pi \Gamma(1-\ep)}
\left ( \frac{4\pi \mu^2}{m_t^2} \right )^{\ep} (1-r^2)^{-\ep}
\Bigg [ D(x) V(r)
\nonumber \\
&&
- \frac{1}{\ep} \int \frac{{\rm d} z}{z}
D \left ( \frac{x}{z} \right ) \tilde P_{qq}(z)
+\int \frac{{\rm d} z}{ z} D \left ( \frac{x}{z} \right )
\Bigg\{ 4 \left [ \frac{\ln(1-z)}{1-z} \right ]_+
- 2(1+z) \ln (1-z)
\ \nonumber \\
&&
- \left ( \frac{2}{(1-z)}
- (1+z) \right ) \ln \left( \frac{r^2+z(1-r^2)}{z^2} \right)
- \frac{2}{(1-z)_+} \frac{z}{r^2+z(1-r^2) }
\Bigg\}
\Bigg ],
\label{eq_dip_vv}
\ea
where $\tilde P_{qq} = 2/(1-z)_+ - (1+z) + 3/2\;\delta(1-z)$ and,
for $\eta = 1$,
\be
V(r) = -\frac{7}{2} - \frac{\pi^2}{3} - 2 {\rm Li_2}(r^2)
-\frac{5+4r^2}{1+2r^2}\ln(1-r^2).
\ee
We observe that Eq.~(\ref{eq_dip_vv}) contains collinear divergences.
To remove them, the fragmentation function $D(x)$ needs to be renormalized.
By convention, we use the $\overline {\rm MS}$ scheme. We obtain
\ba
&& \overline{I_{\rm virt}}+\overline{I_{\rm dip}}
= D(\mu, x) + \frac{\alpha_s(\mu) C_\mathrm{F}}{2\pi}
\left ( \; D(\mu,x) V(r)
- \ln \left (\frac{\mu^2}{m_t^2(1-r^2)} \right )
\int \frac{{\rm d} z}{z}
D \left (\mu, \frac{x}{z} \right ) \tilde P_{qq}(z)
\right. \nonumber \\
&& \left.
+\int \frac{{\rm d} z}{ z} D \left ( \mu,\frac{x}{z} \right )
\left \{ 4 \left [ \frac{\ln(1-z)}{1-z} \right ]_+
- 2(1+z) \ln (1-z)
\right. \right. \nonumber \\
&& \left. \left.
- \left ( \frac{2}{(1-z)}
- (1+z) \right ) \ln \frac{r^2+z(1-r^2)}{z^2}
- \frac{2}{(1-z)_+} \frac{z}{r^2+z(1-r^2) }
\right \}
\right).
\label{eq_v_int}
\ea
Equation~(\ref{eq_v_int}) contains everything that is needed to
compute the contribution of the virtual corrections and the integrated
dipoles to the decay rate $t \to l^+ \nu + B +X$.
These results should be supplemented with the
contribution of the real emission matrix elements, described by
Eq.~(\ref{eq671}). Combining Eq.~(\ref{eq_v_int}) and Eq.~(\ref{eq671}),
we can compute ${\cal O}(\alpha_s)$ correction to the
fully differential rate for $t \to l^+ \nu + B + X$. We then
interface the corrections to the decay, that we just described,
with the production process, in the spirit of Ref.~\cite{ms}. This
allows us to get a description of
$pp \to (t \to W^++b \to W^+ + J/\psi)
+ (\bar t \to W^- + \bar b)$
at leading and next-to-leading order, including the possibility to
apply kinematic cuts to the final state particles.
\subsection{The fragmentation function}
The NLO QCD calculation described in the previous Section
leads to radiative corrections enhanced by the logarithm
of the ratio of the top quark mass and the factorization scale $\mu$.
We can choose $\mu \sim m_t$
to get rid of the logarithmically enhanced terms in the short-distance
partonic decay rate (cf. Eq.~(\ref{eq_v_int})).
However, by doing that, we face the challenge
of evaluating the fragmentation function $D(\mu,x)$ at a high value of
the factorization scale in spite of the fact that $b \to B$ fragmentation
is, intrinsically, the low-scale phenomenon.
The standard way to deal with the problem is to use the Altarelli-Parisi (AP)
equation
\be
\mu^2 \frac{\partial D_{b \to B}(x,\mu) }{\partial \mu^2}
= \sum_j
\int \frac{{\rm d} z}{z} P_{bj}\left ( \frac{x}{z},\alpha_s(\mu) \right )
D_{j \to B}(z,\mu),
\label{ap_eq}
\ee
to evolve the fragmentation function
to the required values of the factorization
scale $\mu \sim m_t$.
For the purpose of the NLO calculation, we
include ${\cal O}(\alpha_s)$ and ${\cal O}(\alpha_s^2)$ contributions
to the AP evolution kernel which leads to a resummation
of the leading and next-to-leading logarithms of the ratio of the factorization
scale and the $b$-quark mass.
Similar to what was done in the
previous studies, we neglect all off-diagonal contributions to the evolution
equation Eq.~(\ref{ap_eq}) and only keep there terms proportional
to $P_{bb}$ splitting function.
Solution of the AP equation requires an
initial condition, which is to say that $D_{b \to B}$ needs to be known
for some value of the factorization scale $\mu_0$. Traditionally,
this is accomplished by fitting the fragmentation function at the
scale
$\mu_0 \sim m_b$ to data on $e^+e^- \to b \bar b$
\cite{aleph,sld}.
Since $\mu_0 \sim m_b$ is a perturbative scale,
we may attempt
to completely factorize perturbative and non-perturbative contributions,
by writing the heavy quark fragmentation function as a convolution of the
perturbative fragmentation function $D_b(\mu,x)$ and the ``non-perturbative''
fragmentation function $D_{\rm np}(x)$ \cite{nasonmele}
\be
D_{b \to B} (\mu,x) = \int \limits_{x}^{1} \frac{{\rm d} \xi}{\xi}
D_b(\mu,\xi) D_{\rm np}\left ( \frac{x}{\xi} \right ).
\ee
The perturbative fragmentation function receives contributions
from momenta comparable to the $b$-quark mass and
is therefore computable in perturbation theory.
At NLO QCD, the result reads \cite{nasonmele}
\be
D_b(\mu,x) = \delta(1-x) + \frac{\alpha_s(\mu)C_\mathrm{F} }{2\pi}
\left [\frac{1+x^2}{1-x} \ln\left( \frac{\mu^2}{m_b^2} \right)- 2 \log (1 - x) -1
\right ]_+ + {\cal O}(\alpha_s^2).
\label{eq:incond}
\ee
Note that the expansion parameter in Eq.~(\ref{eq:incond})
is $\alpha_s \log \mu/m_b$. This observation
makes it clear that $D_b(\mu,x)$ has to be evaluated
at the scale $\mu \sim m_b$.
In Ref.~\cite{cm} $D_{\rm np}(x)$ was determined from fits to the
$e^+e^- \to b \bar b$ data by ALEPH and SLD collaborations
\cite{aleph,sld}.
For numerical
calculations, we use results reported in that reference. We restrict
our attention to two types of non-perturbative
fragmentation functions
\be
D_{\rm np} =
\left \{
\begin{array}{c}
x^{\alpha} (1-x)^{\beta}/B(\alpha+1,\beta+1),\;\;\; \\
(1+\delta)(2+\delta) (1-x) x^{\delta}.
\end{array}
\right.
\label{eq:ff_np}
\ee
It was shown in Ref.~\cite{cm} that the following values of
the parameters
\be
\alpha = 0.66 \pm 0.13,\;\;
\beta = 12.39 \pm 1.04,\; \delta = 14.97 \pm 0.44,
\label{eq:ff_par}
\ee
lead to a good fit to the ALEPH data provided that {\it no soft gluon
resummation} is applied to the fragmentation function\footnote{
If soft gluon resummation in the perturbative fragmentation function
is employed, the preferred values of $\alpha, \beta$ and $\delta$ change,
see Ref.~\cite{cm} for details.}. We will use the
range of parameters shown above to estimate the sensitivity of
the extracted value of the top quark mass to the employed model
of the heavy quark fragmentation function.
We solve the AP evolution equation in a standard way
by applying the Mellin transform since
the AP equation
becomes ordinary differential equation in the Mellin space.
The results that are required to perform the Mellin
transform can be found in Refs.~\cite{nasonmele,cm}.
\subsection{Results: $m_{Bl}$ distribution in top quark decays}
In this Section, we discuss our results for the $m_{Bl}$ spectrum,
as obtained within the perturbative fragmentation function
framework. We ignore all the
subtleties associated with the heavy quark production
mechanism and study the invariant mass of the lepton and the
$B$-meson as produced by the top quark decaying in isolation.
We quote results
at leading and next-to-leading order, but we need to clarify what
we mean by that. Indeed, a fragmentation function, is extracted
from data on $e^+e^-$ annihilation to $B$-hadrons, using
a short-distance function for $e^+e^- \to b \bar b$, computed through
a {\it particular} order in perturbative QCD.
Therefore, if we change the short-distance
function by truncating it to leading order, we
get a different fragmentation function.
This phenomenon is well-known from studies of parton
distribution functions that {\it do} change from one order in perturbation
theory to the other. Unfortunately, information on how non-perturbative
fragmentation functions change when perturbative predictions
for $e^+e^- \to b \bar b $ are truncated
at leading order are not available to us, so that for our leading
order calculation we use the same non-perturbative
fragmentation function $D_{\rm np}$, Eq.~(\ref{eq:ff_np}), as in next-to-leading
order computation. However,
for leading order computations, we neglect all the ${\cal O}(\alpha_s)$
corrections to partonic decay rate of the top quark and the initial condition
$D_b$, Eq.~(\ref{eq:incond}), and we
solve the AP evolution equation and compute
the evolution of the strong coupling constant in the leading
logarithmic approximation.
To obtain numerical results reported below, we use
$\alpha_s(M_Z) = 0.130$ and $\alpha_s(M_Z) = 0.118$
for leading and next-to-leading computations, respectively.
In Tables~1~and~2, we show average values of
the invariant mass of the $B$-meson and the lepton
$\langle m_{Bl} \rangle$ and the dispersion $\sigma_{m_{Bl}}$ of the
$m_{Bl}$ distribution at leading and
next-to-leading order in perturbative
QCD. To arrive at those results, we calculate $\langle m_{Bl} \rangle $
for three values of the renormalization and factorization scales
$m_t,m_t/2,m_t/4$, changing them independently. We also use two
different fragmentation functions, as explained in the previous Section.
For each parameter that one needs to describe the fragmentation
function, we do a calculation for its central value and for the central
value shifted by plus/minus the error quoted for that parameter.
As the result, we obtain $108$ values of $\langle m_{Bl} \rangle $
and $\sigma_{m_{Bl}}$
for each of the input values of the top quark mass. We calculate
the mean and the error from these samples of $108$ numbers for
both $\langle m_{Bl} \rangle $ and $\sigma_{m_{Bl}}$ at leading
and next-to-leading order. In Tables~1~and~2
those results are shown; the difference between the two Tables is
that an additional constraint $m_{Bl} > 50~{\rm GeV}$ is
employed to obtain results in Table~2.
There are two immediate comments that one can make about
those results. First, we observe that
NLO QCD corrections to $\langle m_{Bl} \rangle $
strongly depend on the applied cut on the invariant mass.
For example, if no such cut is applied, the
shift from leading to next-to-leading order in $\langle m_{Bl} \rangle $ is about $2.5~{\rm GeV}$, whereas
if a $50~{\rm GeV}$ cut is applied, $\langle m_{Bl} \rangle $
shifts by $-0.3~{\rm GeV}$.
Second, the uncertainty in
$\langle m_{Bl} \rangle $ decreases by a factor between two and three,
when NLO QCD effects are included, indicating their importance for
the high-precision top quark mass measurement.
It is interesting to compare the results of the computation reported
in this paper with the previous analysis where parton shower
event generators were employed \cite{cms,cmes}. We note that
results of those two references are not consistent; the reason is explained in
Ref.~\cite{cmes}. We will therefore compare to the results in
Ref.~\cite{cmes}, where $\langle m_{Bl} \rangle $ and higher
moments of $Bl$ invariant mass distribution
are computed using HERWIG and PYTHIA. The $B \to b$ fragmentation
functions were fitted in Ref.~\cite{cmes} to reproduce $B$-meson energy spectra
in $e^+e^-$ annihilation. Systematic ${\cal O}(1~{\rm GeV})$
differences
in values of $\langle m_{Bl} \rangle$ obtained with PYTHIA and HERWIG
were observed in \cite{cmes}, with PYTHIA results being lower.
\begin{table}[t]
\label{Jtab}
\vspace{0.1cm}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$m_t$ &
$\langle m_{Bl}\rangle,\;{\rm LO} $ &
$\langle m_{Bl}\rangle,\;{\rm NLO}$ &
$\sigma_{m_{Bl}},\;{\rm LO}$ &
$\sigma_{m_{Bl}},\;{\rm NLO}$ \\ \hline \hline
$171$ & $73.51 \pm 1.87$&
$76.03 \pm 0.61$&
$31.46 \pm 0.12$&
$29.21 \pm 0.29$\\ \hline
$173$ & $74.71 \pm 1.90$&
$77.24 \pm 0.62$&
$31.92 \pm 0.12$ &
$29.63 \pm 0.29$\\ \hline
$175$ &
$75.91 \pm 1.93$ &
$78.44 \pm 0.63$ &
$32.37 \pm 0.13$ &
$30.04\pm 0.30$ \\ \hline
$177$ &
$77.10 \pm 1.95$ &
$79.64 \pm 0.63$ &
$32.82 \pm 0.13$ &
$30.46\pm 0.30$ \\ \hline
$179$ &
$78.29 \pm 1.98$ &
$80.84 \pm 0.64$ &
$33.26 \pm 0.13$ &
$30.87\pm 0.30$
\\ \hline
\hline
\end{tabular}
\caption{The estimate of the
average value of the $B$-meson-lepton invariant
mass and its dispersion at leading and next-to-leading order, in
dependence of the top quark mass. The top quark mass and all the results
are in GeV. Decay of an isolated top quark is considered.}
\vspace{-0.1cm}
\end{center}
\label{table:n1}
\end{table}
\begin{table}[t!]
\label{Jtabb}
\vspace{0.1cm}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$m_t$ &
$\langle m_{Bl}\rangle,\;{\rm LO}$ &
$\langle m_{Bl}\rangle,\;{\rm NLO}$ &
$\sigma_{m_{Bl}},\;{\rm LO}$ &
$\sigma_{m_{Bl}},\;{\rm NLO}$ \\ \hline \hline
$171$ &
$87.51 \pm 1.04$&
$87.20\pm 0.43$&
$22.17 \pm 0.23$ &
$21.28\pm 0.17$ \\ \hline
$173$ &
$88.53 \pm 1.07$ &
$88.22\pm 0.43$ &
$22.68 \pm 0.24$ &
$21.77\pm 0.17$ \\ \hline
$175$ &
$89.56 \pm 1.10$ &
$89.25 \pm 0.44$ &
$23.19\pm 0.24$ &
$22.25\pm 0.18$ \\ \hline
$177$ &
$90.58 \pm 1.13$ &
$90.29 \pm 0.45$ &
$23.69 \pm 0.24$ &
$22.73\pm 0.18$ \\ \hline
$179$ &
$91.61 \pm 1.15$ &
$91.32 \pm 0.46$ &
$24.20 \pm 0.24$ &
$23.22\pm 0.18$
\\ \hline
\hline
\end{tabular}
\caption{The average value of the
invariant mass $\langle m_{Bl} \rangle$ and its dispersion,
evaluated with the cut
on the invariant mass $m_{Bl}>50~{\rm GeV}$.
The top quark masses and all the results are in GeV.
Decay of an isolated
top quark is considered.}
\vspace{-0.1cm}
\end{center}
\label{table:n2}
\end{table}
We find that the NLO QCD result for $\langle m_{Bl} \rangle $
and $\sigma_{m_{Bl}}$ are close to the results obtained
with parton showers. Nevertheless, the difference is
not negligible, given the expected precision of the top quark
mass measurement. By comparing our results with that of
Ref.~\cite{cmes}, we find that the average values of
$\langle m_{Bl} \rangle $ computed through NLO QCD is about
$2.4~{\rm GeV}$ lower than $\langle m_{Bl} \rangle $
obtained with HERWIG and only $1.1~{\rm GeV}$
lower than $\langle m_{Bl} \rangle $ obtained PYTHIA.
On the other hand, the dispersion $\sigma_{Bl}$
that we compute through NLO QCD, differs
by $1~{\rm GeV}$ from PYTHIA
and by $0.5~{\rm GeV}$ from HERWIG results.
The results for $\langle m_{Bl} \rangle$
displayed in Tables~1~and~2 can be described by a linear function
of the top quark mass. We present the results of such
a fit for the two cases -- with and without
a cut on $m_{Bl}$ in Fig.~\ref{fig1}. We find
\ba
&& \langle m_{Bl} \rangle ^{\rm NLO} = 0.601 m_t - 26.7~{\rm GeV},
\;\;\;\delta_{\rm rms} = 0.004;
\label{eq431}
\\
&&
\langle m_{Bl} \rangle_{m_{Bl} > 50~{\rm GeV}}^{\rm NLO}
= 0.516 m_t - 0.96~{\rm GeV},\;\;\;\delta_{\rm rms} = 0.006,
\label{eq432}
\ea
where $\delta_{\rm rms}$
is the root mean square (rms) of the residuals of the linear
fit. It is clear from the value of $\delta_{\rm rms}$
that the linear fit works
very well. It is straightforward to translate the results of the linear
fit shown in Eqs.~(\ref{eq431},\ref{eq432}) to an {\it estimate}
of the error on the top quark mass. Indeed, suppose that a typical
uncertainty of the measured value of $\langle m_{Bl}
\rangle $ is $0.4~{\rm GeV}$.
The slopes in Eqs.~(\ref{eq431},\ref{eq432}) then imply that
the corresponding error in the top quark mass $m_t$ is about
$0.8~{\rm GeV}$. On the other hand, assuming perfect measurement
of $\langle m_{Bl} \rangle $, we find that theoretical
uncertainties in $\langle m_{Bl} \rangle $ shown in Table~1 and the value
of the slope of the linear fit lead to a $1~{\rm GeV}$
uncertainty in the extracted value of the top quark mass.
The errors on the top quark mass that follow from the NLO QCD computation
are similar to differences between PYTHIA, HERWIG and NLO QCD.
To show this, we quote results of a
fit to $\langle m_{Bl} \rangle $
obtained with PYTHIA and HERWIG in Ref. \cite{cmes}
\be
\langle m_{Bl} \rangle_{\rm Pythia} = 0.59~m_t - 24.11~{\rm GeV},
\;\;\;\;
\langle m_{Bl} \rangle_{\rm Herwig} = 0.61~m_t - 25.31~{\rm GeV}.
\label{eq351}
\ee
It is clear from the comparison of the fits Eqs.~(\ref{eq351},\ref{eq431})
that NLO QCD results
and parton shower results are close but not identical and these
differences are essential. Indeed, we note that
a slope difference between parton showers and NLO QCD is
about $0.01$. Although such slope difference may look insignificant,
it leads to ${\cal O}(3~{\rm GeV})$
shift in the reconstructed value of the top quark. Hence, parton showers are
insufficient for measurements of the top quark mass with a precision
higher than a few GeV. On the contrary, it follows from
Eqs.~(\ref{eq431},\ref{eq432})
that NLO QCD computations lead to results with small uncertainties
that can be estimated in a systematic way and, perhaps, be
even further improved.
\begin{figure}[t!]
\begin{center}
\scalebox{0.45}{\input{MLB_mt_1.tex}} \hspace{5mm}
\scalebox{0.45}{\input{MLB_mt_2.tex}}
\vspace{3mm}
\caption{Results of the linear fit to $\langle m_{Bl} \rangle^{\rm NLO}$
are shown. Left panel -- no cut on $m_{Bl}$ is applied. Right panel --
$m_{Bl} > 50~{\rm GeV}$ cut is applied. In both cases, decays
of isolated top quarks are considered.
}
\label{fig1}
\end{center}
\end{figure}
\subsection{Results: $m_{Bl}$ distribution
in $pp \to (t \to W^++b \to W^+ + B) + ( \bar t \to W^- + \bar b)$
}
In this Section, we consider production of $B$-mesons
through the fragmentation of $b$-quarks
in top decays but, in contrast to the previous Section,
we include the full production process through next-to-leading order
in perturbative QCD. To claim that the NLO QCD computation can do a good
job in describing $\langle m_{Bl} \rangle $ in {\it reality},
it is very important to have
full production and decay chain included.
To this end, we consider top quark pair production in
$pp$ collisions at $\sqrt{s} = 14~{\rm TeV}$ and focus on
the lepton + jets decay channel.
We note that NLO QCD corrections to the decay
$W \to q \bar q'$ need to be interfaced with $pp \to t \bar t$ production
process, to describe lepton + jets channel through NLO QCD.
We require that there are at least
four jets in the event. We include the $B$-meson
{\it and} the non-$B$-meson remnant of the fragmenting $b$-quark
into a list of proto-jets that are passed to the jet reconstruction algorithm.
We employ $k_\perp$ jet algorithm with $R = 0.5$ and the
four-momentum recombination scheme. All reconstructed jets
and the positron from the $W^+$ decay are required to have transverse momenta
in excess of $20~{\rm GeV}$\footnote{A standard argument \cite{cms,cmes}
that $\langle m_{Bl} \rangle$
involves a Lorentz invariant product of the two
four-vector and, therefore, does not depend on the production mechanism
is not applicable once cuts on the transverse momenta are applied. Such
cuts are only invariant under restricted class of Lorentz transformations -
boosts along the collision axis.}. The
scalar sum of the transverse momenta of all jets
in the event should exceed $100~{\rm GeV}$ \cite{cmsnote}. For the sake
of simplicity, we do not
impose any other kinematic constraints including cuts on the missing
energy and the lepton isolation cuts. In addition, we do not consider
combinatorial backgrounds, assuming that the correct
pairing between a lepton and a $B$-meson can be established.
Finally, similar to what was done in Ref.~\cite{ms}, throughout this paper
we consider
intermediate top quarks to be on the mass-shell and we do not
include the so-called non-factorizable corrections~\cite{my}.
For observables that we study in this paper, this is a good approximation
since we, effectively, integrate over the invariant masses of each of the
top quarks.
In contrast to the previous Section, we do not change the parameters
of the fragmentation functions, fixing them to their central
values, see Eq.~(\ref{eq:ff_par}). We use CTEQ parton distribution
functions \cite{Pumplin:2002vw,Nadolsky:2008zw} in the analysis.
For each input value of $m_t$
we compute $\langle m_{Bl} \rangle $
for three values of the renormalization and
(pdf)-factorization scales $\mu_R = \mu_F = [m_t/4,m_t/2,m_t]$ and
for three values of the factorization scale in the $b \to B$ fragmentation
function $[m_t/4,m_t/2,m_t]$ and for two different types of fragmentation
functions Eq.~(\ref{eq:ff_par}). As the result, for each value of $m_t$ we have
eighteen values of $\langle m_{Bl} \rangle$ and $\sigma_{m_{Bl}}$. Although
this is not an extensive scan of the parameter space, it gives a sense
of theoretical uncertainties in $\langle m_{Bl} \rangle $ provided
that realistic production mechanism is employed\footnote{We have checked
that if we only change parameters that are related
to the decay process, we find the
${\cal O}(0.5~{\rm GeV})$ uncertainty in $\langle m_{Bl} \rangle$, similar
to Tables 1,2.}.
Computing
the mean and the error, we arrive at
the results shown in Table~3. We see that the NLO QCD effects
in this case are quite small and {\it negative},
which is similar to the case of $m_{Bl} > 50~{\rm GeV}$ cut
discussed earlier. It is also clear that the uncertainty in $m_{Bl}$
decreases when NLO QCD corrections are included.
By comparing results in Table~1 and in Table~3,
we see that effects of kinematic cuts on
$\langle m_{Bl} \rangle $ are more important at leading order, where
the average value of $m_{Bl}$ can shift by $3.5~{\rm GeV}$. On the other
hand, at NLO, a typical shift is of the order of $0.8~{\rm GeV}$
and, therefore, is less dramatic. Comparison of Tables~1 and~3 shows that
the uncertainty of $\langle m_{Bl} \rangle $ at NLO nearly
doubles if production
mechanism is taken into account. Performing the fit, we obtain (see Fig.\ref{fig25})
\be
\langle m_{Bl} \rangle^{\rm NLO}_{\rm prod} =
0.6365~m_t -32.12~{\rm GeV},\;\;\;\delta_{\rm rms} = 0.053.
\label{eq821}
\ee
Comparing this result with Eq.~(\ref{eq431}), we find a significant
change in both the slope and the constant part. This demonstrates
that $\langle m_{Bl} \rangle $ depends in a non-trivial way on the production
mechanism, because of kinematic cuts applied to top quark decay products
and additional jets in the production process. It follows
from Eq.~(\ref{eq821}) and the uncertainties of $\langle m_{Bl} \rangle $
shown in Table~3 that the theoretical error on the extracted value of $m_t$
is close to $1.5~{\rm GeV}$.
\begin{table}[t]
\label{Jtabb}
\vspace{0.1cm}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$m_t$ &
$\langle m_{Bl}\rangle,\;{\rm LO}$ &
$\langle m_{Bl}\rangle,\;{\rm NLO}$ & $\sigma_{m_{Bl}},\; {\rm LO}$
& $\sigma_{m_{Bl}},\; {\rm NLO}$ \\ \hline \hline
$171$ &
$77.07 \pm 1.92$&
$76.75 \pm 1.12$ &
$30.60 \pm 0.13$ &
$28.41 \pm 0.36$
\\ \hline
$173$ &
$78.34 \pm 1.93$ &
$77.92 \pm 1.09$ &
$31.01 \pm 0.14$ &
$28.72 \pm 0.31 $
\\ \hline
$175$ &
$79.63 \pm 1.98$ &
$79.31 \pm 1.04$ &
$31.46 \pm 0.14$ &
$29.12 \pm 0.18$
\\ \hline
$177$ &
$80.91 \pm 2.03$ &
$80.55 \pm 1.05$ &
$31.83 \pm 0.15$ &
$29.48 \pm 0.13$
\\ \hline
$179$ &
$82.16 \pm 2.04$ &
$81.80 \pm 1.04$ &
$32.24 \pm 0.16$ &
$29.83 \pm 0.13$
\\ \hline
\hline
\end{tabular}
\caption{The average values of the
invariant mass $\langle m_{Bl} \rangle$ and the dispersion
in case where all the cuts on the final state particles are applied.
The top quark masses and all the results are in GeV.
See text for details.}
\vspace{-0.1cm}
\end{center}
\label{table:n3}
\end{table}
\begin{figure}[t!]
\begin{center}
\scalebox{0.45}{\input{MLB_mt_cut.tex}}
\vspace{3mm}
\caption{Result of the linear fit to $\langle m_{Bl} \rangle^{\rm NLO}$
is shown, {\it with all kinematic cuts on the final state particles
applied}. See text for details.
}
\label{fig25}
\end{center}
\end{figure}
\section{Dilepton channel}
\label{dilepton}
In the previous Section we saw that top quark decays to
final states with identified hadrons
provide an interesting way to determine the top quark mass.
In this Section we study inclusive final states.
We focus on the case where the top and the anti-top quarks
decay semileptonically, e.g. $t \to W^+ b \to l^+ \nu b$. We
study the kinematic distribution of an invariant mass
of a $b$-jet and a lepton, and the distributions of the sum of
energies of the two leptons and the two
$b$-jets. We employ
the NLO QCD corrections
to top quark pair production and decay, as computed in Ref.~\cite{ms}.
Throughout this Section, the center-of-mass energy of proton collisions
is $14~{\rm TeV}$.
We begin by summarizing the kinematic cuts that are employed
to identify dilepton $t \bar t$ events \cite{Beneke:2000hk}.
Leptons are required to be central
$|\eta^{l}| < 2.5$ and have large transverse momentum
$p_\perp^{l} > 25~{\rm GeV}$. There should be missing
energy in the event, $E_\perp^{\rm miss} > 40~{\rm GeV}$.
The jet transverse momentum cut is $p_{\perp, j}> 25~{\rm GeV}$.
We employ the $k_\perp$ jet algorithm
with $R = 0.4$.
\subsection{Invariant mass of a lepton and a $b$-jet}
It is pointed out in Ref.~\cite{Beneke:2000hk} that
an average value of the invariant mass squared of a $b$-jet and a
lepton $m_{lb}^2$ and an average value of the
the angle between the lepton and the $b$-jet in the $W$ boson rest frame,
can be used to construct an estimator of the top quark mass.
The estimator reads
\be
M_{\rm est}^2 = m_W^2 + \frac{ 2\langle m_{lb}^2 \rangle}{1
- \langle \cos \theta_{lb} \rangle }.
\label{eq01}
\ee
To see that this is a good estimator, we note that
for the top quark decay computed
at leading order in perturbative QCD
and without
any restrictions on the final state $M_{\rm est}$ equals to $m_t$
\be
\langle m_{lb}^2 \rangle = \frac{m_t^2-m_W^2}{2}
\left ( 1 - \langle \cos \theta_{lb} \rangle \right ),\;\;\;
\langle \cos \theta_{lb} \rangle = \frac{m_W^2}{m_t^2+ 2m_W^2}\;\;\;\;
\Rightarrow M_{\rm est}^2 = m_t^2.
\ee
In reality $M_{\rm est}$ is not equal to $m_t$ for a variety
of reasons including i) kinematic
cuts required to identify the dilepton events; ii) effects of higher
order QCD corrections; iii) impossibility to choose the ``correct''
pair of a lepton and a $b$-jet and iv) the experimental issues with
$b$-jet misidentification and the jet energy resolution. The computation
reported in Ref.~\cite{ms} allows us to calculate $M_{\rm est}^2$
within the framework of perturbative QCD, accounting for
the points i)-iii) {\it exactly}.
\begin{figure}[t]
\begin{center}
\scalebox{0.45}{\input{MLBScale.tex}} \hspace{5mm}
\scalebox{0.45}{\input{MLB_mass.tex}}
\vspace{3mm}
\caption{The invariant mass distribution of the lepton
and the $b$-jet. Note that the lepton and the $b$-jet do not necessarily
come from the decay of the same top quark, see text.
The left panel shows the scale uncertainty bands
for $\mu_R = \mu_F = [0.5m_t, 0.75m_t,m_t,1.25 m_t]$. The
right panel shows two NLO {\it normalized}
$m_{lb}$ distributions for $m_t = 171~{\rm GeV}$ and $m_t = 179~{\rm GeV}$.
}
\label{fig2}
\end{center}
\end{figure}
We point out that the computation of NLO QCD corrections to
$pp \to t \bar t$ process reported in \cite{ms} includes exact
spin correlations, one-loop effects in top quark decays and
allows arbitrary constraints to be imposed on top quark decay products.
These features are crucial for reproducing experimental procedures.
Indeed, experimentally, it is not possible to determine the
charge of the $b$-jet. Hence, it is unclear
which of the two $b$-jets should be combined with the
chosen, definite-sign, lepton.
For the purpose of $m_{lb}$ reconstruction,
one pairs the lepton with the $b$-jet
that gives the smallest $m_{lb}$ value \cite{Beneke:2000hk}.
The parameter $\langle \cos \theta_{lb} \rangle $ in Eq.~(\ref{eq01})
is not measured and should be estimated theoretically. We have
also chosen to calculate $\langle \cos \theta_{lb} \rangle $ for the
$b$-jet that minimizes the invariant mass $m_{lb}$ since in this
case, there is a partial compensation of
incorrect assignments between the numerator and the denominator
in Eq.~(\ref{eq01}). As the result,
$M_{\rm est}$ becomes closer to the input value $m_t$ as compared
to the case when ``correct'' pairing of the $b$-jet and the lepton
is chosen to calculate $\langle \cos \theta_{lb} \rangle $
in Eq.~(\ref{eq01}).
It is argued in Ref.~\cite{Beneke:2000hk} that with $10~{\rm fb}^{-1}$
integrated luminosity, the statistical and systematic uncertainties
in the top quark mass
of about $1~{\rm GeV}$ each can be achieved from
$\langle m_{lb}^2 \rangle $ measurement.
To assess how realistic those uncertainties are, we
consider five different values of the top quark
mass $m_t = [171,173,175,177,179]\;\;{\rm GeV}$. For each of these
$m_t$ values, we compute $M_{\rm est}$ for
four values of the renormalization and
the factorization scales
$\mu_R = \mu_F = [0.5m_t, 0.75m_t,m_t,1.25 m_t]$ and
for two sets of parton distribution functions CTEQ
\cite{Pumplin:2002vw,Nadolsky:2008zw}
and MRST \cite{Martin:2002aw}.
We use the mean value and the standard deviation
of these eight values to compute central value of
$M_{\rm est}$ and its error. Clearly, by no means
this is an exhaustive scan through the parameter
space\footnote{For example, one can and perhaps should
use different renormalization scales to compute
numerator and denominator in Eq.~(\ref{eq01}), to get a better
idea of the scale uncertainties in $M_{\rm est}$.}
but it gives us an idea of the uncertainties on
the theoretical side. Examples of $m_{lb}$ distributions and the results
of the calculation are shown
in Figs.~\ref{fig2},\ref{fig3}. The uncertainties on $M_{\rm est}$ do not depend
on $m_t$ in significant way; they are $0.1~(0.2)~{\rm GeV}$ at leading
and next-to-leading order, respectively.
Performing the linear fit, we find
\be
M_{\rm est}^{\rm LO}\; =\; 0.8262 m_t + 23.22\;\mathrm{GeV},\;\;\;\;
M_{\rm est}^{\rm NLO} = 0.7850 m_t + 28.70\;\mathrm{GeV}.
\label{eq010}
\ee
The quality of the
linear fit is very good; for example, the root mean square of the residuals
of the NLO fit is $\delta_{\rm rms} = 0.032$.
It is instructive that the analysis of this observable at leading
order shows stronger correlation between $m_t$ and $M_{\rm est}$
than at next-to-leading order. In addition, the theoretical
uncertainty in $M_{\rm est}$ increases when NLO QCD corrections are included.
The primary reason for the increased
uncertainty is stronger dependence of $M_{\rm est}$ on the
renormalization and factorization scales at NLO. This feature can be understood
by considering the situation where no phase-space cuts are applied and
where all the assignments of a lepton and a $b$-jet are done correctly.
In this case, as follows from the discussion at the beginning of this Section,
the estimator equals to the top quark mass regardless of the renormalization and
factorization scales and the chosen parton distribution functions.
At next-to-leading order, this is not true anymore because
of the gluon radiation in top decay
that is sensitive to the value of the strong coupling constant
and, hence, to the renormalization scale.
We note that we observe a very weak dependence
of $M_{\rm est}$ on parton distribution functions which implies
that even with the phase-space cuts and incorrect pairing,
this variable is primarily sensitive to top quark decays rather
than to top quark production mechanism.
Finally, we can use Eq.~(\ref{eq010}) and Fig.~\ref{fig3} to estimate
uncertainty in $m_t$ that can be achieved by measuring
$M_{\rm est}$ with infinite precision.
Since, as follows from Fig.~\ref{fig3},
the uncertainty in $M_{\rm est}$ is $0.2~{\rm GeV}$ at NLO and given
the slope of about $0.8$ in Eq.~(\ref{eq010}), we find
the minimal uncertainty in the extracted value of
$m_t$ to be close to $0.25~{\rm GeV}$.
We note that this result does not include
the $b$-quark fragmentation uncertainty
and the jet scale uncertainty, estimated to be
$0.7$ and $0.6~{\rm GeV}$, respectively, in Ref.~\cite{Beneke:2000hk}.
\begin{figure}[t!]
\begin{center}
\scalebox{0.5}{\input{Mest_mt_1.tex}}
\vspace{3mm}
\caption{Results of a linear fit to $M_{\rm est}$, Eq.~(\ref{eq01}),
at leading and next-to-leading order in perturbative QCD.
}
\label{fig3}
\end{center}
\end{figure}
\subsection{Sum of energies of the two leptons from top quark decays}
Another observable that we consider is the sum of the energies
of the two leptons, $ E_{l_1}+E_{l_2} $, in the laboratory frame. Lepton energies
in the laboratory frame can be easily measured and they are free
from jet energy scale uncertainties that are important sources
of errors, if the top quark mass is reconstructed from hadronic
final states. The important question is whether
or not the average value\footnote{One can ask the
same question about the shape of the distribution but such discussion
is outside the scope of this paper.}
of the sum of lepton energies is correlated
with the top quark mass at the parton level and how well such correlation
can be described by perturbative QCD.
The corresponding distributions are shown in Fig.~\ref{fig4}. There we display
$E_{l_1}+E_{l_2}$ computed through leading and next-to-leading order
in perturbative QCD for $m_t = 175~{\rm GeV}$, as well the NLO QCD
distributions in $E_{l_1}+E_{l_2}$ for $m_t = 171$~GeV and $m_t =179~{\rm GeV}$.
To compute the mean value
of $\langle E_{l_1}+E_{l_2} \rangle $ we consider
the same range of the renormalization and factorization scales
and the two sets of parton distribution functions, as in the previous
Section. The results of the calculation are shown in Fig.~\ref{fig5}. Performing a
linear fit, we find
\ba
&& \langle E_{l_1}+E_{l_2} \rangle_{\rm LO}\;\;
= 0.645 m_t + 120.6\;\mathrm{GeV},\;\;\;\; \delta_{\rm rms} = 0.08;
\nonumber \\
&& \langle E_{l_1}+E_{l_2} \rangle_{\rm NLO}
= 0.670 m_t + 114.4\;\mathrm{GeV}, \;\;\;\; \delta_{\rm rms} = 0.07.
\label{eq1456}
\ea
The results of the linear fit are displayed in Fig.~\ref{fig5}.
Theoretical errors on
$\langle E_{l_1}+E_{l_2} \rangle$ are independent of the top mass;
they are $1.7~{\rm GeV}$ at leading order
and $1~{\rm GeV}$ at next-to-leading
order. Combining information about the slope in Eq.~(\ref{eq1456}) with
the theoretical uncertainty on
$\langle E_{l_1}+E_{l_2} \rangle $, we conclude that the ultimate uncertainty
in $m_t$ that can be achieved by studying this observable is close to
$1.5~{\rm GeV}$.
It is interesting to point out that, in this case, both the scale
dependence of the NLO result and the difference between MRST and CTEQ
parton distribution
functions are the two important sources of the uncertainty.
\begin{figure}[t]
\begin{center}
\scalebox{0.45}{\input{ELep.tex}} \hspace{5mm}
\scalebox{0.45}{\input{ELep_mass.tex}}
\vspace{3mm}
\caption{Left panel: normalized distribution of
the sum of lepton energies at leading
and next-to-leading order calculated for
$m_t = 175~{\rm GeV}$. The renormalization and factorization
scales are set to $m_t$
and the MRST (left panel) and CTEQ (right panel)
parton distribution functions set is used.
Note a shift in the position of the maximum of this distribution.
Right panel: normalized distributions of the sum of lepton energies
at next-to-leading order, for $m_t = 171~{\rm GeV}$ and
$m_t = 179~{\rm GeV}$.}
\label{fig4}
\end{center}
\end{figure}
\begin{figure}[t!]
\begin{center}
\scalebox{0.45}{\input{ELep_mt.tex}}
\vspace{3mm}
\caption{Results of a linear fit to the sum of the average
energy of the two leptons
at leading and next-to-leading order.
}
\label{fig5}
\end{center}
\end{figure}
\subsection{Sum of jet energies}
Another observable that was discussed \cite{Beneke:2000hk}
in connection with the top quark mass measurement is the sum of energies of
the two hardest jets in the laboratory frame.
Similar to the lepton energies just discussed, the shape of the distribution
is an observable that is to be fitted; this is beyond the scope
of the present paper. Here, we limit ourselves to the discussion
of average values. Instead of considering the
two hardest jets in the event, we found
it more useful to take the sum of energies of the two $b$-jets.
\begin{figure}[t]
\begin{center}
\scalebox{0.55}{\input{bjet_mass.tex}}
\vspace{3mm}
\caption{Normalized distribution of
the sum of energies of two $b$-jets
$E_{\rm bjet} = E_{bj1}+E_{bj2}$
at next-to-leading order, calculated for
$m_t = 171~{\rm GeV}$ and
$m_t = 179~{\rm GeV}$. We set renormalization and factorization
scales to $m_t$ and use CTEQ parton distribution functions. }
\label{fig6}
\end{center}
\end{figure}
We consider the distribution of
the sum of energies of the two $b$-jets. We expect that
this distribution is strongly correlated with the top quark mass, since
$b$-quarks originate directly from top decays. The results
of the calculation are shown in Figs.~\ref{fig6}.
Performing a linear
fit, we obtain the correlation between the average value of the
two $b$-jets and the top quark mass
\ba
&& \langle E_{bj1}+E_{bj2} \rangle_{\rm LO}
= 2.18 m_t -42.2\;\; {\rm GeV},\;\;\; \delta_{\rm rms} =0.02;
\nonumber \\
&& \langle E_{bj1}+E_{bj2} \rangle_{\rm NLO}
= 2.09 m_t -29.2\;\; {\rm GeV},\;\;\; \delta_{\rm rms} = 0.05.
\ea
The results of the linear fit together with theoretical uncertainties
in $\langle E_{bj1}+E_{bj2} \rangle $ are shown in Fig.~\ref{fig7}.
These uncertainties are $2.6~{\rm GeV}$ at LO and $2.4~{\rm GeV}$
at NLO; they do not exhibit a strong dependence on the top quark mass.
Interestingly, inclusion of NLO QCD corrections makes the
correlation between $\langle E_{bj1}+E_{bj2} \rangle_{\rm LO}$
and $m_t$ weaker.
However, the correlation is still quite strong. If we assume
that energies of $b$-jets can be measured infinitely accurately,
the irreducible uncertainty on the determination of the
top quark mass from $\langle E_{bj1}+E_{bj2} \rangle$
becomes only $1.2~{\rm GeV}$.
Of course, the main issue here is to understand how well
$b$-jet energies can actually be measured;
this issue will be at the center of the experimental studies at the LHC.
\begin{figure}[t!]
\begin{center}
\scalebox{0.5}{\input{bjet_mt.tex}}
\vspace{3mm}
\caption{Results of a linear fit to the sum of the average
energy of the two $b$-jets
at leading and next-to-leading order.
}
\label{fig7}
\end{center}
\end{figure}
\section{Conclusion}
\label{conc}
Determination of the top quark mass with high precision is
an important part of the top quark physics program
at the LHC. It is expected, that a
variety of methods will be employed by
ATLAS and CMS collaborations to measure the top quark mass.
Some of those methods involve studies of the correlations
between the value of the top quark mass and the kinematics of the top quark
decay products. As the results of those studies,
it is often claimed that the top quark mass can be determined
with ${\cal O}(1\%)$ uncertainty at the LHC, but it is not clear
whether or not these uncertainty estimates can be trusted.
Indeed, all such studies employ PYTHIA or HERWIG parton shower event
generators to describe top quark pair production and decay; however,
no parton
shower is designed to handle this level of precision.
An interesting question therefore is to estimate,
in a parton-shower-independent way, the uncertainty on $m_t$ that
can be achieved in forthcoming LHC experiments.
The goal of this paper is to address this question by computing
correlations between the top quark mass and the values of some
kinematic observables through NLO QCD. Among other things,
we consider NLO QCD corrections to the invariant mass of a $B$-meson
and a lepton from top quark decays which is considered to be one of the
most accurate ways to determine the top quark mass. Such computation
is rather unusual in the context of NLO QCD calculations since it refers
to the final state with an identified hadron.
In general, we find that parton shower event generators do a good
job in estimating both the central value and the uncertainty in the top quark
mass that can be achieved. However, as can be seen from the discussion
of the average value of the invariant mass of the $B$-meson and the lepton,
NLO QCD computations give both, a more accurate central values and an
estimate of the uncertainty that can be trusted. Both of these features
are important if we want to use the measured value of
the top quark mass with confidence, to constrain physics beyond the Standard Model
through precision measurements.\\
{\bf Acknowledgments}
K.M. would like to acknowledge conversations with A.~Mitov that
triggered this investigation and useful discussions with Z.~Kunszt.
This research is supported by the NSF under grant
PHY-0855365 and by the start up
funds provided by Johns Hopkins University.
Calculations reported in this paper were performed on the Homewood
High Performance Cluster of Johns Hopkins University.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,000 |
Q: How to collect RSS feed from Twitter Search? to the point. I need collect Rss Feed from Twitter Search. Usually it can appear like this:
http://search.twitter.com/search.rss?q=KEYWORD
I Have try googling for 3 days and the result is same. It can appear with this script
Hashtag search: http://search.twitter.com/search.rss?q=%23YourSearch
User mention search: http://search.twitter.com/search.rss?q=%40UserName
To a User: http://search.twitter.com/search.rss?q=to%3AUserName
From a User: http://search.twitter.com/search.rss?q=from%3AUserName
But.. It can run anymore. it have an error like this
<errors>
<error code="68">
The Twitter REST API v1 is no longer active. Please migrate to API v1.1. https://dev.twitter.com/docs/api/1.1/overview.
</error>
</errors>
seems twitter has change the API.
Can anyone help me for this problem? I need the Feed for built the twitter bot.
is there any other way to get the feed from twitter?
Im Very Appreciated your Advice.
Thanks
A: Twitter does not support RSS anymore.
A: But still you can generate a twitter RSS feed with third party application.
https://queryfeed.net
It does exactly this however, the website does have a bit of downntime i observed in last 2 days.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,286 |
Q: Object Cast Error In My AddProduct Method there is something wrong with my code. I get a message saying Object cannot be cast from DBNull to other Types when I test my code. Anybody have any ideas where this problem is coming from?
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Data;
using System.Data.SqlClient;
namespace ProductMaintenance
{
class ProductDB
{
public static Product GetProduct(string code)
{
SqlConnection connection = MMABooksDB.GetConnection();
string select = "SELECT ProductCode, Description, UnitPrice "
+ "FROM Products "
+ "WHERE ProductCode = @ProductCode";
SqlCommand selectCommand = new SqlCommand(select, connection);
selectCommand.Parameters.AddWithValue("@ProductCode", code);
try
{
connection.Open();
SqlDataReader prodReader = selectCommand.ExecuteReader(CommandBehavior.SingleRow);
if (prodReader.Read())
{
Product product = new Product();
product.Code = prodReader["ProductCode"].ToString(); ;
product.Description = prodReader["Description"].ToString();
product.Price = ((decimal)prodReader["Price"]);
return product;
}
else
{
return null;
}
}
catch (SqlException ex)
{
throw ex;
}
finally
{
connection.Close();
}
}
public static bool UpdateProduct(Product oldProduct, Product newProduct)
{
SqlConnection connection = MMABooksDB.GetConnection();
string updateStatement = "UPDATE Products SET " + "Description = @NewDescription, " + "UnitPrice = @NewUnitPrice, " + "WHERE ProductCode = @oldProductCode " + "AND Description = @OldDescription " + "AND UnitPrice = @OldUnitPrice";
SqlCommand updateCommand =
new SqlCommand(updateStatement, connection);
updateCommand.Parameters.AddWithValue(
"@NewDescription", newProduct.Description);
updateCommand.Parameters.AddWithValue(
"@NewUnitPrice", newProduct.Price);
updateCommand.Parameters.AddWithValue(
"@OldProductCode", oldProduct.Code);
updateCommand.Parameters.AddWithValue(
"@OldDescription", oldProduct.Description);
updateCommand.Parameters.AddWithValue(
"@OldUnitPrice", oldProduct.Price);
try
{
connection.Open();
int count = updateCommand.ExecuteNonQuery();
if(count > 0)
return true;
else
return false;
}
catch(SqlException ex)
{
throw ex;
}
finally
{
connection.Close();
}
}
public static int AddProduct(Product product)
{
SqlConnection connection = MMABooksDB.GetConnection();
string insertStatement = "INSERT Products " + "(ProductCode, Description, UnitPrice) " + "VALUES (@ProductCode, @Description, @UnitPrice)";
SqlCommand insertCommand = new SqlCommand(insertStatement, connection);
insertCommand.Parameters.AddWithValue("@ProductCode", product.Code);
insertCommand.Parameters.AddWithValue("@Description", product.Description);
insertCommand.Parameters.AddWithValue("@UnitPrice", product.Price);
try
{
connection.Open();
insertCommand.ExecuteNonQuery();
string selectStatement = "SELECT IDENT_CURRENT('Products') FROM Products";
SqlCommand selectCommand = new SqlCommand(selectStatement, connection);
int productC = Convert.ToInt32(selectCommand.ExecuteScalar());
return productC;
}
catch (SqlException ex)
{
throw ex;
}
finally
{
connection.Close();
}
}
}
}
A: You're selecting IDENT_CURRENT which can possibly return null.
See this documentation.
Returns NULL on error or if a caller does not have permission to view
the object.
In SQL Server, a user can only view the metadata of
securables that the user owns or on which the user has been granted
permission. This means that metadata-emitting, built-in functions such
as IDENT_CURRENT may return NULL if the user does not have any
permission on the object.
Based on the error you're describing, this appears to be where it's failing. Your AddProduct method wouldn't be trying to cast a null value when it does the INSERT so it is likely a null value when you try to retrieve the identity at
int productC = Convert.ToInt32(selectCommand.ExecuteScalar());
Since the call to IDENT_CURRENT is the only statement in that SQL command, that all but rules out a syntax or other error in the SQL, which would narrow this down to permissions.
There's a bit of conjecture and deduction there but I suspect that's where it's at.
(If the error is thrown from AddProduct method there's no need to post everything else. Many people - including myself - are going to jump to the conclusion that the error is when you access the SqlDataReader since you aren't checking for nulls there. But as you said, that's not the method throwing the exception. It's just easy to miss when there's so much irrelevant code.)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,790 |
The Coln Valley Annual Tennis Mixed Doubles Tournament on Saturday 14 June 2003, saw 80 competitors playing sizzling tennis in similar temperatures at Chedworth and other courts kindly loaned for the day in surrounding villages. The tournament culminated in a tenaciously-fought and nail-biting final with Simon Sheppard and Alison Armitage of Ashton Keynes narrowly defeating Chedworth's mother and son duo Sue and Jamie Francis to take home the Speke Challenge Cup.
New this year to the 21-year old event was an additional competition for a silver plate, enabling those who did not qualify in the first round of the main tournament to continue playing competitively throughout the afternoon. The plate was won by Mark Hall and Georgina Seel from Fifield. Commenting after the presentation, Hill and Valley LTC Chairman Stephen Scott Smith said: "If there has been any criticism at all of the tournament in the past it has been that for some people there was not enough tennis. The plate competition has addressed this and proved so popular that we will make it a permanent part of the fixture in the future. It should help us attract even more participants next year for what is an extremely sociable and enjoyable event".
Now that 16 year-old Jamie Francis has shown himself to be such a rising star you may want to see him in action next year! Details of next year's tournament on June 12th, which raises money for the club and for the churches of Chedworth, Coln St Dennis and Coln Rogers, will be posted on this website.
Hill and Valley LTC is putting considerable effort into encouraging young talent including holding annual tennis factories with the Gloucestershire LTA and starting a programme of junior and youth coaching. John Robson, the club's coach, is currently undertaking his LTA Development Coaching award. The club was runner-up in the 2002 Gloucestershire LTA 'Club of the Year' competition in recognition of its contribution to developing youth tennis in the area. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,522 |
\section*{\noindent Editor\hfill}
David Garfinkle\\
\smallskip
Department of Physics
Oakland University
Rochester, MI 48309\\
Phone: (248) 370-3411\\
Internet:
\htmladdnormallink{\protect {\tt{garfinkl-at-oakland.edu}}}
{mailto:garfinkl@oakland.edu}\\
WWW: \htmladdnormallink
{\protect {\tt{http://www.oakland.edu/?id=10223\&sid=249\#garfinkle}}}
{http://www.oakland.edu/?id=10223&sid=249\#garfinkle}\\
\section*{\noindent Associate Editor\hfill}
Greg Comer\\
\smallskip
Department of Physics and Center for Fluids at All Scales,\\
St. Louis University,
St. Louis, MO 63103\\
Phone: (314) 977-8432\\
Internet:
\htmladdnormallink{\protect {\tt{comergl-at-slu.edu}}}
{mailto:comergl@slu.edu}\\
WWW: \htmladdnormallink{\protect {\tt{http://www.slu.edu/colleges/AS/physics/profs/comer.html}}}
{http://www.slu.edu//colleges/AS/physics/profs/comer.html}\\
\bigskip
\hfill ISSN: 1527-3431
\bigskip
DISCLAIMER: The opinions expressed in the articles of this newsletter represent
the views of the authors and are not necessarily the views of APS.
The articles in this newsletter are not peer reviewed.
\begin{rawhtml}
<P>
<BR><HR><P>
\end{rawhtml}
\end{flushleft}
\pagebreak
\section*{Editorial}
Matters of Gravity has adopted a new publication schedule: it will appear in December and June. The
purpose of this change is so that a preliminary description of the GGR sessions of each upcoming April APS meeting can be shown to the GGR membership before the deadline for submission of an abstract for the April meeting.
The next newsletter is due June 2015. This and all subsequent
issues will be available on the web at
\htmladdnormallink
{\protect {\tt {https://files.oakland.edu/users/garfinkl/web/mog/}}}
{https://files.oakland.edu/users/garfinkl/web/mog/}
All issues before number {\bf 28} are available at
\htmladdnormallink {\protect {\tt {http://www.phys.lsu.edu/mog}}}
{http://www.phys.lsu.edu/mog}
Any ideas for topics
that should be covered by the newsletter, should be emailed to me, or
Greg Comer, or
the relevant correspondent. Any comments/questions/complaints
about the newsletter should be emailed to me.
A hardcopy of the newsletter is distributed free of charge to the
members of the APS Topical Group on Gravitation upon request (the
default distribution form is via the web) to the secretary of the
Topical Group. It is considered a lack of etiquette to ask me to mail
you hard copies of the newsletter unless you have exhausted all your
resources to get your copy otherwise.
\hfill David Garfinkle
\bigbreak
\vspace{-0.8cm}
\parskip=0pt
\section*{Correspondents of Matters of Gravity}
\begin{itemize}
\setlength{\itemsep}{-5pt}
\setlength{\parsep}{0pt}
\item Daniel Holz: Relativistic Astrophysics,
\item Bei-Lok Hu: Quantum Cosmology and Related Topics
\item Veronika Hubeny: String Theory
\item Pedro Marronetti: News from NSF
\item Luis Lehner: Numerical Relativity
\item Jim Isenberg: Mathematical Relativity
\item Katherine Freese: Cosmology
\item Lee Smolin: Quantum Gravity
\item Cliff Will: Confrontation of Theory with Experiment
\item Peter Bender: Space Experiments
\item Jens Gundlach: Laboratory Experiments
\item Warren Johnson: Resonant Mass Gravitational Wave Detectors
\item David Shoemaker: LIGO Project
\item Stan Whitcomb: Gravitational Wave detection
\item Peter Saulson and Jorge Pullin: former editors, correspondents at large.
\end{itemize}
\section*{Topical Group in Gravitation (GGR) Authorities}
Chair: Beverly Berger; Chair-Elect:
Deirdre Shoemaker; Vice-Chair: Laura Cadonati.
Secretary-Treasurer: Thomas Baumgarte; Past Chair: Daniel Holz;
Members-at-large:
Curt Cutler, Christian Ott,
Andrea Lommen, Jocelyn Read,
Kimberly Boddy, Steven Drasco,
Sarah Gossan, Tiffany Summerscales.
\parskip=10pt
\vfill
\eject
\vfill\eject
\section*{\centerline
{we hear that \dots}}
\addtocontents{toc}{\protect\medskip}
\addtocontents{toc}{\bf GGR News:}
\addcontentsline{toc}{subsubsection}{
\it we hear that \dots , by David Garfinkle}
\parskip=3pt
\begin{center}
David Garfinkle, Oakland University
\htmladdnormallink{garfinkl-at-oakland.edu}
{mailto:garfinkl@oakland.edu}
\end{center}
Jacob Bekenstein has been awarded the APS Einstein Prize.
Stanley Deser and Charles Misner have been awarded the Einstein Medal of the Albert Einstein Society.
Duncan Brown, Guido Mueller, Maria Alessandra Papa, and Robert Schofield have been elected APS Fellows.
Hearty Congratulations!
\section*{\centerline
{Centenial of General Relativity Speakers Bureau}}
\addtocontents{toc}{\protect\medskip}
\addcontentsline{toc}{subsubsection}{
\it GR Centenial Speakers Bureau, by Deirdre Shoemaker}
\parskip=3pt
\begin{center}
Deirdre Shoemaker, Georgia Institute of Technology
\htmladdnormallink{deirdre-at-gatech.edu}
{mailto:deirdre@gatech.edu}
\end{center}
2015 marks the centennial of Albert Einstein's lectures first describing his theory of general relativity. The American Physical Society Topical Group in Gravitation is organizing the Centennial of General Relativity Speakers Bureau to provide opportunities for experts in astronomy, cosmology, and general relativity to visit colleges, universities, schools, and communities to give public talks, lectures, meet with students and faculty, or participate in other events. Assistance with travel funding for the speaker is available through this program, especially for minority serving institutions and for schools with little or no research activity in physics and astronomy.
For more information and to request a speaker please visit the website:
\url{http://apsggr.org/?page_id=24}
\section*{\centerline
{GGR $\to$ DGR}}
\addtocontents{toc}{\protect\medskip}
\addcontentsline{toc}{subsubsection}{
{\it GGR} $\to$ \it DGR, by Nicolas Yunes}
\parskip=3pt
\begin{center}
Nicolas Yunes, Montana State University
\htmladdnormallink{nyunes-at-physics.montana.edu}
{mailto:nyunes@physics.montana.edu}
\end{center}
Many of you probably already know about the American Physical Society (the APS) -- one of the most important physics organizations, representing over 50,000 physicists from all over. The APS sponsors meetings to promote the exchange of new results in physics, the most relevant of which for the gravity community is the April APS Meeting. The APS also publishes some of the most prestigious journals in our field, including the Physical Review D and Physical Review Letters. Moreover, the APS also advocates for physics education and science education in general, and in particular, in front of Congress, promoting funding for physics research at NSF, DOE, and NASA.
What many of you may be less familiar with is the role of ``units'' within the APS. When physicists are left to interact freely, they will natural interact and coalesce to form ``units'', which in the APS can be classified as follows:
\begin{itemize}
\item {\bf{Sections}}: regional groups that foster a spirit of community.
\item {\bf{Forums}}: national groups built to address broad issues, such as graduate student education or international cooperation.
\item {\bf{Technical Units}}: national groups that coalesce around a common interest in physics. Technical units are of two types: {\emph{Divisions}} and {\emph{Topical Groups}}, the main difference being primarily one of size -- divisions have a total number of members that exceeds 3\% of total APS membership, while topical groups do not. Their mission, however, is similar: to bring together scientists with overlapping interests in physics and foster cooperation and communication.
\end{itemize}
But size matters. Divisions play a much more important role in the APS governance. They have the right to a permanent seat in the APS council, with greatly enhanced opportunities to advocate for their fields. Divisions also have a higher likelihood of getting plenary speakers for the April Meeting, giving much broader visibility to their field. In addition, divisions receive a larger budget allocation from the APS and offer \$600 in student travel support, compared to the \$300 that Topical Groups offer. A larger budget allocation also means that they usually provide food and wine at their business meetings, during the April APS meeting, which are open to all members.
Our community, gravitational physics, is represented by the Topical Group in GRavitation (GGR) in APS. Their members specialize in wide range of disciplines, including analytical and computational studies of general relativity, mathematical and numerical relativity, tests of general relativity and experimental relativity as a whole, modified theories of gravity, relativistic astrophysics, cosmology, quantum gravity, and gravitational wave detection, to name a few. GGR was established in 1995, through an effort spearheaded by Beverly Berger with the invaluable assistance of Stan Whitcomb, Neil Ashby, and others. From the beginning, LIGO has been a key part of GGR, but by no means is GGR only a LIGO community. Both GGR and the LSC owe their existence to the growth of interest in gravitation which the past two decades have seen.
GGR provides a broadly-based but external voice to allow the gravitational physics community to advocate for its interests and, as a member of GGR, you benefit directly from the activities the GGR executive committee organizes. GGR organizes a number of sessions in the APS April meeting, including talks on quantum gravity, cosmology, relativistic astrophysics, and gravitational waves. This meeting, in fact, has become one of our main forums for releasing new results. GGR sponsors and selects the winner of the APS Einstein Prize, awarded to Jacob Bekenstein (2015), Irwin Shapiro (2013), Ted Newman (2011), Jim Hartle (2009), Rai Weiss and Ron Drever (2007), Bryce DeWitt (2005), and John Wheeler and Peter Bergmann (2003). GGR has named over sixty APS fellows, an important and distinct honor signifying recognition by one's professional peers in the physics community. And, of course, GGR provides the much needed student travel support in these days of difficult funding.
These activities don't just happen -- it is the GGR executive officers and the support for GGR membership that make them happen. Growth in GGR membership is one of the best arguments we have to leverage more benefits for all of us, including increased funding in physics and gravitation and more visibility at conferences. Joining GGR was your way of standing up to be counted in the community of gravitational physicists and you have done just that. GGR is the largest topical group in the APS, but now it has grown and matured to the point that we will be able to petition APS to become a Division. Your membership did that.
But we are not there yet. To become a division, we must maintain our membership numbers above 3\% of total APS membership for two consecutive years. (at this time we are either just slightly below or just slightly above the 3\% threshold). I have no doubt that this can be achieved, if we all work together to retain old members and make new ones. Thank you once more for your continuous support. Achieving this milestone makes me immensely proud to be a member of this community.
Nicolas Yunes
Membership Czar 2012-2014
\section*{\centerline
{GGR program at the APS meeting in Baltimore, MD}}
\addtocontents{toc}{\protect\medskip}
\addcontentsline{toc}{subsubsection}{
\it GGR program at the APS meeting in Baltimore, MD, by David Garfinkle}
\parskip=3pt
\begin{center}
David Garfinkle, Oakland University
\htmladdnormallink{garfinkl-at-oakland.edu}
{mailto:garfinkl@oakland.edu}
\end{center}
We have a very exicting GGR related program at the upcoming APS April meeting in Baltimore, MD, in 2015 the Centenial of General Relativity. Our Chair-Elect, Deirdre Shoemaker, did
an excellent job of putting together this program.
\vskip0.25truein
{\bf Note that the deadline for submitting an abstract for this meeting is Friday, January 9, 2015 at 5:00 pm EST}
\vskip0.10truein
{\centerline {abstracts can be submitted at}}
\url{http://www.aps.org/meetings/april/scientific/index.cfm}
\vskip0.25truein
At the APS meeting three of the plenary talks will be devoted to gravity:
\vskip0.2truein
Clifford Will: Precision Tests of the Theory of General Relativity\\
Stuart Shapiro: Sources and Detection of Gravitational Waves\\
James Hartle: Quantum Gravity and Cosmology\\
There will be several invited sessions of talks sponsored by the Topical Group in Gravitation as follows:\\
Innovative Computing in Relativity\\
(Peter Diener, Zachariah Etienne, Tyson Littenberg)\\
with DCOMP\\
Neutron Stars as Laboratories for Neutrino, Nuclear and Gravitational Physics\\
(Benjamin Lackey, Evan O'Connor, Jorge Piekarewicz)\\
with DNP\\
Detecting GWs from the Ground and in Space\\
(Shane Larson, Jason Hogan, David Shoemaker)\\
with DAP\\
Astrophysical Black holes on all mass scales\\
with DAP\\
Quantum Gravity in the 100th Anniversary of General Relativity\\
(Joseph Polchinski, Thomas Faulkner, Walter Goldberger)\\
with DPF\\
100 Years of GR, 20 Years of GGR Looking back and looking forward (panel session)\\
(Rainer Weiss, Gabriela Gonzalez, James Hartle and Jorge Pullin)\\
Precision experimental measurements of gravitation\\
(Michael Hohensee, JamesFaller, Charles Hagedorn)\\
with GPMFC\\
History of Relativity\\
(Saul Teukolsky, Diana Kormos-Buchwald )\\
with FHP\\
GGR Prize Session\\
\vfill\eject
\section*{\centerline
{New Frontiers in Dynamical Gravity}}
\addtocontents{toc}{\protect\medskip}
\addtocontents{toc}{\bf Conference reports:}
\addcontentsline{toc}{subsubsection}{
\it New Frontiers in Dynamical Gravity,
by Helvi Witek}
\parskip=3pt
\begin{center}
Helvi Witek, DAMTP, University of Cambridge
\htmladdnormallink{h.witek-at-damtp.cam.ac.uk}
{mailto:h.witek@damtp.cam.ac.uk}
\end{center}
In March 2014 we hosted the workshop ``New frontiers in dynamical gravity'' or, in short,
``Gauge/gravity duality
meets Numerical Relativity meets fundamental math'',
at the futuristic site of DAMTP at the University of Cambridge,
organised by P.~Figueras, H.~Reall, U.~Sperhake and myself.
The gauge/gravity correspondence provides a powerful tool to understand strongly coupled
conformal field theories in $D-1$ dimensions
by investigating gravity in asymptotically anti-de Sitter (AdS) spacetimes in $D$ dimensions and vice versa.
Nowadays, the duality is available in many different flavours,
employing calculations in GR to explore vastly different fields such as hydrodynamics and
condensed matter physics.
On the gravity side, which was the main focus of our workshop,
this requires finding solutions in AdS and understanding their properties and stability
as well as their dynamical evolution in time.
Many of these issues cannot be tackled by ``pen and paper'' calculations and demand a numerical treatment using both
``soft'' and ``hard'' numerics~\footnote{These labels have been introduced by R. Emparan
at yet another fantastic workshop on ``Numerical Relativity and High Energy Physics'' held at Madeira in 2011 to distinguish
between solving ODEs or simplified PDEs using computer algebra tools like Mathematica on a desktop and
the more complex computations of BH dynamics
(in at least 2+1 dimensions)
requiring typically hundreds of computer cores.}.
In this workshop we brought together leading experts in these fields.
The schedule of the conference -- typically two one-hour long overview talks in the morning
and four half-hour talks in the afternoon --
left plenty of time for fruitful discussions, the exchange of ideas and the launching of new collaborations.
The slides as well as the group photo are available on the conference's website
\url{http://www.ctc.cam.ac.uk/activities/adsgrav2014/}.
While the main focus of our workshop was gravity in AdS, there are still many open questions concerning
the stability of black holes (BHs) even in four dimensional, asymptotically flat spacetimes.
This topic has been in the spotlight of a number of talks,
kicking off with M.~Dafermos who discussed the nature of BH singularities -- we have
learned that the generic BH singularity
might not be space-like after all -- and its importance for the
(strong) cosmic censorship conjecture.
This is closely related to still open questions about the linear and non-linear stability of BHs
which is an active field of research.
In his talk, S.~Hollands discussed the thermodynamic stability of black objects
(in four and higher dimensional spacetimes)
and its implications for their dynamical properties.
Although the ``standard'' lore states that BHs in four-dimensional, asymptotically flat spacetimes are stable
this, in fact, only refers to mode stability which excludes a vast number of possibly growing solutions.
G.~Holzegel presented a very pedagogical summary of the state-of-the-art
of the mathematical understanding of the
stability of BHs: while it has been proven that Schwarzschild BHs are linearly stable, he reminded us
that the linear stability in a strict mathematical sense even of the Kerr BH is still a completely open question.
Indeed, it has recently been shown that extremal BHs do suffer from an instability,
which failed to show up in a mode analysis.
Nevertheless, such a mode analysis giving the characteristic response of a BH towards perturbations
provides substantial insight into BH phenomenology.
C.~Warnick discussed the construction of quasi-normal modes (QNMs) for AdS BHs.
As an illustrative example he presented computations of excitations of the Schwarzschild-AdS BH induced by scalar perturbations.
Despite the importance and beauty of QNMs
they do not form a complete basis for AdS BH spacetimes, thus hinting at more physics that
await to be uncovered.
Another novel result which may make us revise our understanding of
classical BHs was presented by C. Herdeiro
in the form of hairy rotating BH solutions. Inspired by recent studies
on the superradiant instability of Kerr BHs in the presence of massive fields
they constructed a family of rotating solutions with (complex) scalar hair
which branches off at the onset of the superradiant instability and can evade the ``no-hair'' theorem.
The specific properties of these solutions such as their stability are still under investigation.
D. Hilditch discussed new results concerning critical phenomena in asymptotically flat spacetimes.
These phenomena, marking a phase-transition between BH formation versus dispersion
during scalar field collapse, are well understood in spherical symmetry.
However, the axisymmetric case still lacks a complete understanding. In the literature one can find
various, sometimes contradictary results, which have been revisited with
the new spectral (Numerical Relativity) code {\textsc{BAM-PS}}.
Ongoing, challenging numerical simulations probe the phase-space near the onset of criticality.
We are looking forward to hear more about the final results.
A couple of years ago investigations of critical phenomena in spherically symmetric
AdS spacetimes brought forward a surprising outcome,
which has been summarized in P. Biz\'{o}n's talk.
While the asympotically flat case provided a clear-cut transition between
collapse and dispersion this picture changes dramatically in AdS.
If the scalar field fails to form a BH upon its first interaction,
in subsequent reflections at the AdS (timelike) boundary the field is focused more and more,
thus eventually yielding collapse.
Given that this mechanism generically involves a cascade towards higher energies
and a transition from ``pure'' AdS to an AdS BH spacetime
this phenomenon has been termed ``turbulent instability''.
However, the understanding of this instability is still in its infancy.
Although pure AdS appears to be generically non-linearly unstable
there do exist ``islands of stability'' -- a fine-tuned set in the parameter space of perturbations
which evade this fate.
Both M. Maliborski and S. Liebling gave us an update on their ongoing work
attempting to reveal the underlying mechanisms.
These outcomes inspired investigations of gravitational turbulence in
AdS BH spacetimes as L. Lehner demonstrated in his talk. Indeed, it has been
observed in perturbative calculations
that the interplay between the AdS BH and perturbations provokes a non-linear mode-coupling and
cascade towards higher energies in certain regimes
defined by the gravitational analog of the Reynolds number.
Additionally, these studies motivated numerical simulations re-visiting turbulence in hydrodynamics.
Features such as the formation and annihilation of vortices have been illustrated in
beautiful animations.
A further application of the fluid/gravity duality has been discussed in a series of talks by
by R. Janik, M. Heller, P. Romatschke and K. Balasubramanian.
They focused on shock-wave collisions in AdS spacetimes which
can be interpreted as interactions between a strongly coupled plasma
and have specifically been employed to model heavy-ion collisions at RHIC or LHC.
Before starting to attack interesting physics' questions one has to solve a number
of technical (numerical) challenges including
(i) the representation of AdS spacetimes (i.e. Poincar\'{e} versus global AdS),
(ii) the specific space+time formulation of Einstein's equations, which needs to be a well-posed
initial boundary value problem,
and (iii) means to extract relevant physical information from the boundary stress-energy tensor.
These challenges and novelties for Numerical Relativity have recently been tackled.
Simulations of the collisions of shockwaves in (2+1) AdS indeed allowed to represent
the strong dynamics of the collisions of heavy ions in the field theory counterpart.
It is exciting to note, that a proposed model using
both GR simulations for the dynamical collision part followed by a hydrodynamic description
after the equilibration
shows excellent agreement with real-world particle collision.
For example, particle spectra resulting from Pb-Pb collisions at the LHC
have been fitted well with this new model
while pure hydrodynamical models have been off by an order of magnitude.
T. Wiseman reported on the latest news concerning plasma flows with space-dependent temperature profiles.
On the gravity side these fluids can be represented by recently discovered AdS BHs with a non-Killing horizon.
He gave a predagogical summary of the numerical methods to construct these stationary solutions
which requires solving a set of PDEs with mixed elliptic and hyperbolic characteristics.
Via the correspondence, information about the surface gravity and (linear) velocity of the horizon
of these BHs provides insight into physical parameters of the flowing plasma such as its (local) velocity.
For a certain region of the phase-space, this plasma velocity seems to diverge
hinting at an instability, possibly related to turbulent behaviour.
A further exciting application of the gauge/gravity duality represents itself in
the correspondence between gravity and condensed matter physics.
In particular, J. Gauntlett, O. Dias and J. Santos discussed the holographic duals
of metals, insulators and superconductors.
A complete understanding of their phase transitions in condensed matter physics is still lacking.
Via the correspondence these materials can be modelled in the context of gravity in AdS
coupled to a Maxwell and, possibly, scalar fields.
Using a periodic potential or constructing AdS BHs with non-Killing, ``floppy'' horizons
allow to mimic so-called Q-lattices with broken translational invariance.
Computations on the gravity side recover, e.g., metal/insulator transitions and
the (DC) conductivity known from the solid-state physics side.
Moreover, recent investigations predict the existence of new insulator and metal phases and it
will be interesting to see whether they can be discovered in real materials.
A.~Ishibashi addressed the instability of AdS and its connection with singularity theorems in AdS
by studying
Bianchi black branes.
The key requirements of known rigidity theorems (in asymptotically flat spacetimes)
include the weak energy condition and the compactness and analyticity of BH horizons.
While the model presented in this talk still satisfies the weak energy condition it is possible to construct
solutions which have either non-smooth or non-compact horizons, such as stationary solutions with
non-Killing horizons, and thus evade the rigidity theorem.
So far we have seen a number of applications of the gauge/gravity duality which exploit well-understood physics
on the gravity side to learn about hard-to-tackle problems in the dual field theory.
However, we can also employ the duality to understand fundamental issues in
quantum gravity which should emerge near a BH singularity.
Bearing in mind that bulk BHs can be described as thermal states in the boundary field theory
this opens up the exciting possibility to learn about quantum gravity in AdS by investigating the gauge theory.
G.~Horowitz presented a holographic model which employs a conformal field theory,
specifically $\mathcal{N}=4$ super Yang-Mills theory
in an anisotropic generalization of de Sitter spacetimes,
to explore the nature of (quantum) gravity near singularities in the bulk theory.
V. Hubeny discussed how extremal surfaces and geodesics in Vaidya-AdS spacetimes,
modelling BH formation through gravitational collapse,
can be used to probe the spacetime region close to the singularity.
In contrast to static spacetimes they can penetrate the (event) horizon in the dynamical setting
at hand and, furthermore, exhibit a strikingly rich structure.
The properties of these geodesics and surfaces can be interpreted in terms of
thermodynamic quantities such as entanglement entropy.
In a related talk M. Taylor described how holography might help to better understand
the nature of the BH interior. One of the key concerns are related to our
still poor comprehension of the horizon and the information loss paradox,
driven by (semi-) classical pictures.
She gave a short review on recent proposals such as firewalls and fuzzballs
to resolve these issues,
and focused on a solution using BH microstates.
These states represent horizonless ``stringy'' geometries, thus
escaping the ``teething troubles'' of our classical models.
Through a coarse-graining over these geometries we can recover the familiar classical BH picture.
C. Pope gave a talk reviewing the thermodynamics of BHs in asymptotically AdS spacetimes.
M. Rangamani presented his ongoing work on computing the entanglement entropy in a boundary field theory
using minimal surfaces in the bulk. While this construction in principle is well understood,
ensuring the causality of these solutions is fundamental.
In fact, causality can be violated in the presence of time-like singularities in the bulk.
He discussed a number of examples including
negative-mass Schwarzschild-AdS solutions or charged scalar solitons with positive boundary energy.
The requirement that causality should hold therefore yields non-trivial constraints on these extremal surfaces.
J. Armas illustrated how membranes in hydrodynamics can be mimicked using blackfolds.
In particular, he discussed how this approach in higher order perturbation theory can be employed
to compute transport coefficients for surfaces or (mem-) branes in hydrodynamics.
K. Skenderis discussed the dynamics of non-equilibrium solutions using the fluid/gravity duality. In particular,
it is possible to capture the dynamics in the long wavelength regime, when the field theory is close
to a thermal equilibrium, using a hydrodynamic description. This regime can be modelled in the gravity dual
by constructing solutions in a gradient expansion giving a good prescription at long distances
and late times.
One such solution is the Robinson-Trautman metric, which can be seen as the dynamical (non-linear)
version of algebraically special perturbations of the Schwarzschild-AdS solution,
allowing us to investigate the effect of non-linearities and the approach to equilibrium.
A further focus of this meeting were the properties of BHs in higher dimensional,
but asymptotically flat spacetimes.
R. Emparan gave a pedagogical review about his work on BHs in the large-D limit.
In a nutshell, one can treat the spacetime dimension $D$ simply as a further parameter.
Then, in the limit that the dimension becomes very large the equations of motion
decouple into a far region (essentially described by flat spacetime),
and a near region, in which the gravitional field is concentrated in a thin shell around the BH.
Employing this approach facilitates an analytic treatment of generically rather complicated
problems, such as the computation of QNMs or the understanding of the
stability of higher-dimensional BHs.
In order to study the non-linear stability properties of singly-spinning Myers-Perry solutions
in finite spacetime dimensions a fully numerical treatment is mandatory.
M. Shibata presented new results of evolutions of Myers-Perry BHs in five dimensions
obtained with the improved {\textsc{Sacra-5D}} code which uses the constraint damping mechanism
facilitated in the so-called Z4c formulation of Einstein's equations.
These new numerical simulations are in excellent agreement
with a perturbative calculation published earlier this year.
Because this type of ``hard'' numerical evolutions are extremely demanding in terms of computational resources,
they have not yet covered the entire parameter space and we are looking forward to
read and hear about the latest results.
Wednesday night we all gathered for a feast at the beautiful Trinity college.
After indulging in excellent food and wine H.~Reall gave an inspiring speech recapturing the first days of our conference
and recalled some amazing anecdotes about I.~Newton. Apparently, once upon a time science could come with
a large amount of suffering and physical pain.
One of the highlights of this conference was the visit to the {\textsc{Cosmos}} supercomputer
which is part of the DiRAC HPC Facility funded by STFC and BIS. Thank you for the tour, Juha and James!
On Thursday we had a special lunch to celebrate the recent creation by the University of Cambridge of a
Stephen~W.~Hawking Professorship in Cosmology. This chair was endowed by a donation from Dennis~Avery and
Sally~Wong~Avery. Sadly, Dennis died in 2012 but we were delighted that Mrs~Avery and members of her family
were able to join the workshop participants for this celebration.
G.~Horowitz announced that J.~Santos has been awarded the 2014 General Relativity and Gravitation Young
Scientist Prize by the International Union of Pure and Applied Physics. Congratulations, Jorge!
\noindent{\bf{Acknowledgements:}}
We thank all the participants for their invalueable input in many illuminating talks and discussions
making this workshop a great success.
We acknowledge financial support for this conference from
the Institute of Physics/ Gravitational Physics Group, STFC,
the {\it ERC-2011-StG 279363--HiDGR} ERC Starting Grant and
Intel through the Centre for Theoretical Cosmology in Cambridge.
\vfill\eject
\section*{\centerline
{Frontiers of Neutron Star Astrophysics}}
\addtocontents{toc}{\protect\medskip}
\addcontentsline{toc}{subsubsection}{
\it Frontiers of Neutron Star Astrophysics,
by David Nichols}
\parskip=3pt
\begin{center}
David Nichols, Cornell University
\htmladdnormallink{david.nichols-at-cornell.edu}
{mailto:david.nichols@cornell.edu}
\end{center}
On May 29 and 30, 2014, a meeting called ``Frontiers of Neutron Star
Astrophysics'' was held at Cornell University to review open problems in
neutron-star astrophysics and discuss future directions for their
solution.
It was also an opportunity to celebrate the 65$^{\rm th}$ birthday of Jim
Cordes and the 60$^{\rm th}$ birthday of Ira Wasserman at a conference
banquet on the evening of the 29$^{\rm th}$.
Over the two days, a wide range of topics were covered in seventeen invited
talks, eleven contributed talks, and a concluding panel discussion.
These areas include mechanisms of supernova explosions; properties of radio
pulsars, magnetars, and accreting neutron stars; evolution of magnetic
fields; models of neutron star interiors and equation of state; binary
neutron-star mergers and their accompanying gravitational waves and
electromagnetic counterparts; and tests of general relativity using pulsars.
Many of the slides from the talks are available on the conference website
at \url{http://www.astro.cornell.edu/nsfrontiers/}.
The conference would not have been possible without the work of the chair
of the scientific committee (Lars Bildsten) and its members (Phil Arras,
David Chernoff, Jim Cordes, \'Eanna Flanagan, Dong Lai, Saul Teukolsky,
and Ira Wasserman), and support from the Cornell Department of Astronomy
and Center for Radiophysics and Space Research.
The conference started off with a bang with Adam Burrows' talk on the status
of simulations of core collapse supernovae and the mechanisms that drive the
collapse.
Lars Bildsten's presentation followed along a similar note by discussing
the link between superluminous supernovae and the birth of magnetars.
This first session concluded with a lecture by Duncan Lorimer on the
populations of radio pulsars, focusing on how the current catalog of known
pulsars can help understand the formation and evolution of neutron stars in
a variety of environments.
Vicki Kaspi began the next session with a talk entitled
``Magnetars and their ilk.''
David Kaplan described the properties of nearby neutron stars that are
emitting thermally, with an emphasis on the optical and x-ray emission from
isolated neutron stars and what this emission can reveal about their physics.
Andrew Cumming then gave the last talk of the section with a presentation on
the evolution of magnetic fields in the crust of neutron stars.
The third session of the day began with talks from the birthday honorees.
Jim Cordes discussed the detection of fast radio bursts and commented on
whether these are likely to be local or cosmological sources.
Ira Wasserman elaborated on the effects of superconductivity on the
behavior of neutron-star magnetic fields and how this could affect
pulsar-timing measurements.
The third presentation was given by Armen Sedrakian who described
superfluidity and pair-breaking processes in baryonic matter and exotic
cooling mechanisms in neutron stars.
The final talk of the session was by Andrzej Szary, and he presented
a model of a partially screened gap in pulsars that helped to explain
properties of the radio and x-ray emission.
The last group of talks on the first day commenced with Anatoly Spitkovsky
on the computational modeling of pulsar magnetospheres.
Next was Konstantinos Gourgouliatos who commented on the role of Hall drift
in determining the braking indices of young pulsars.
Describing a model for radio emission from magnetars was George Melikidze,
and ending the session was Wojciech Lewandowski, who discussed the emission
from gigahertz-peaked pulsar spectra.
Beginning the second day was Andrei Beloborodov, who reviewed the mechanisms
responsible for the activity in pulsars and magnetars.
David Tsang presented a mechanism by which tidal gravitational fields in a
compact binary containing a neutron star could resonantly shatter the star's
crust leading to a precursor to short gamma-ray bursts.
Next, Scott Ransom gave a survey of several accurately timed pulsars in
binaries (and hierarchical triples) that allow for measurements of the
systems' post-Keplerian parameters and mass-radius relationship.
The session concluded with Sebastien Guillot describing how measurements of
the spectra of low-mass x-ray binaries can be modeled to determine the
mass-radius relationship, and the neutron stars' equation of state,
accordingly.
Michael Kramer started the next section of talks, in which he reviewed the
ability of the most relativistic binary pulsars to test the predictions of
general relativity in the strong-field regime and to put constraints on
modified gravitational theories.
Following this talk, Maura McLaughlin reviewed the prospects for using
a pulsar-timing array to detect gravitational waves from several types of
low-frequency sources.
The last speaker of the session was Anna Watts, who described recent progress
in understanding x-ray bursts and burst oscillations in neutron stars.
The penultimate session began with a talk by Edo Berger explaining the
observations supporting compact-object mergers being the progenitors
of short gamma-ray bursts.
Ben Lackey then discussed the prospects for measuring information about
neutron-star equation of state from the gravitational waves from binary
neutron-star mergers in interferometric gravitational-wave detectors.
James Clark described how burst gravitational-wave searches might be able
to detect gravitational waves emitted after the merger of binary neutron
stars, when the merger forms a hypermassive neutron-star.
Finally, Brian Metzger gave the last talk in this part on kilonovae,
powered by the decay of r-process elements, being an important
electromagnetic counterpart to the gravitational-wave signal from binary
neutron-star mergers.
In the final session of talks, Francois Foucart discussed the status of
numerical-relativity simulations of black-hole--neutron-star binaries and
binary neutron-star mergers.
Marc Favata described the systematic errors that arise when incomplete
gravitational-wave models are used to measure the neutron-star equation of
state from binary neutron-star mergers.
The last talk of the conference was delivered by Sinead Walsh, who reviewed
the status of gravitational-wave searches for unknown isolated neutron stars.
Before the conference ended, several of the invited speakers served on a
panel discussion about where the field of neutron-star astrophysics will be
headed in the next decade.
While there were a range of opinions about the specific results that would be
found, the general consensus was that the prospect for new discoveries is
good.
\vfill\eject
\section*{\centerline
{Quantum Information in Quantum Gravity}}
\addtocontents{toc}{\protect\medskip}
\addcontentsline{toc}{subsubsection}{
\it Quantum Information in Quantum Gravity,
by Mark Van Raamsdonk}
\parskip=3pt
\begin{center}
Mark Van Raamsdonk, University of British Columbia
\htmladdnormallink{mav-at-phas.ubc.ca}
{mailto:mav@phas.ubc.ca}
\end{center}
During recent years, a truly remarkable connection between the physics
of spacetime/ gravitation and the physics of quantum information has
emerged, largely via the AdS/CFT correspondence in string theory.
While surely still far from being understood completely, there is now
intriguing evidence that the structure and geometry of spacetime in
these examples is related directly and quantitatively to the structure
of entanglement of the fundamental degrees of freedom of the
theory. Further, even the dynamics of spacetime, at least in the limit
of weak curvature, has been understood to emerge from fundamental
constraints obeyed by entanglement. The new ideas have presented
challenges to some long-held beliefs about gravitational physics,
famously including the smoothness of spacetime at black hole horizons.
In order to present and discuss the latest work on these exciting
developments, the conference "Quantum Information in Quantum Gravity"
was held during the week of August 18-22, 2014 in Vancouver, Canada.
The setting for much of the recent work presented at the conference
is the AdS/CFT correspondence, by which the states of certain
conformal quantum field theories are in one-to-one correspondence
with the states of some corresponding quantum theory of gravity.
Several of the talks related to a conjecture by Ryu and Takayanagi
(and its covariant generalization by Hubeny, Rangamani, and
Takayanagi), which suggests that the entanglement entropy (a measure
of quantum entanglement) for some spatial subset of degrees of
freedom in the field theory is directly proportional to the area of a
certain surface in the corresponding spacetime geometry. Specific
topics included: understanding how this proposal can be used to
extract dual spacetime geometry from a CFT state (Myers, Sully),
understanding how the proposal generalizes to include quantum and
higher curvature corrections (Dong, Wall), understanding the quantum
information-theoretic interpretation of more general geometrical
observables in the gravity theory (Hayden), and deriving the proposal
for the case of 2D CFTs (Hartman) with special properties.
Several of the other talks (Karch, Mathur, Berkooz) concerned a
more general proposal for relating entanglement and geometry, which
suggests that entangling non-interacting subsets of the fundamental
degrees of freedom, amounts to creating a wormhole in spacetime
between two distant (or disconnected) parts of spacetime. This
proposal has recently been dubbed "ER=EPR" by Maldacena and
Susskind.
Connections between entanglement entropy and gravitational physics
provide a generalization and refinement of the now famous connections
between gravitational quantities and thermodynamic quantities; in
particular the entropy-area connection for black holes. Entanglement
entropy has the property that it can be evaluated for any quantum
state without assumptions about equilibrium. Thus, certain previous
conjectures involving entropy in the gravitational context can be
made more general and precise if the entropy is interpreted as
entanglement entropy (either directly in the gravitational theory, or
for some dual degrees of freedom). Talks by Bousso (on a proof of his
covariant entropy bound in certain contexts) and Marolf (on a version
of the generalized second law) related to these entanglement entropy
generalizations of conjectures about gravitational thermodynamics.
A subject of great debate in the recent quantum gravity literature is
the "firewall paradox" of Almheri, Marolf, Polchinski, and Sully.
These authors have argued that the maximal entanglement of an old
black hole with its Hawking radiation forbids the local entanglement
of quantum fields across the black hole horizon that would be
necessary to ensure a smooth spacetime there. Thus, according to the
argument, such a black hole must have a singularity (or "firewall")
at its would-be horizon. Talks by Giddings, Harlow, Verlinde and
Silverstein dealt with various aspects of this and the closely
related black hole information paradox.
The conference also featured talks aimed at better understanding
entanglement in a purely field theory context, including understanding
entanglement entropies in gapped theories (Nishioka), and understanding
the evolution of entanglement/density matrices corresponding to a long
wavelength subset of degrees of freedom in field theory (Lawrence). In
discrete field theory systems, there is a useful representation of
quantum states that makes the spatial entanglement structure manifest.
This is known as the Multiscale Entanglement Renormalization Ansatz or
MERA. It was conjectured by Brian Swingle that the MERA representation of
a quantum state for a field theory system with gravity dual may be
directly related to how the dual spacetime is encoded. A number of talks
(Swingle, Takayanagi) presented new results relating to this MERA
description of field theory states.
Overall, the activities of the conference reinforced the impression that
connections between gravity and quantum information represents an
extremely interesting frontier in gravitational research.
Note: slides for many of the talks can be found at:
\url{http://www.maths.dur.ac.uk/~dma0mr/qiqg-ubc/programme.html}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,487 |
Q: Activating godmode in Super Star Wars for SNES? I tried entering A A A A X B B B B Y X X X X A Y Y Y Y B like it says, but nothing happens. The below video shows it going to a character select screen afterwards when you start the game, but this doesn't happen to me. I have the controller in port 1 with no controller in port 2. Am I missing something about this?
A: Per this Gamefaqs Cheat guide, you are not entering in the code during the game play which is required to be done by the 2nd controller.
(Emphasis my own)
Press A, A, A, A, X, B, B, B, B, Y, X, X, X, X, A, Y, Y, Y, Y, B at the title screen. During game, press A+B+X+Y+Select on controller 2
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,333 |
Q: Plotting heading between two points in x-y plane I'm trying to implement a small radar that plots targets based on latitude and longitude coordinates similar to the radar in the Layar AR iPhone app. I have the compass and locationManager working to get the lat/lon, heading and distance between two points. However I'm having trouble plotting the points onto the x-y plane. Could you point me in the right direction(so-to-speak)?
This is the method that I am using to plot but the results are not correct:
-(void) addTargetIndicatorWithHeading:(float)heading andDistance:(float)distance{
//draw target indicators
//need to convert radians and distance to cartesian coordinates
float radius = 50;
float x0 = 0.0;
float y0 = 0.0;
//convert heading from radians to degrees
float angle = heading * (180/M_PI);
//x-y coordinates
float x1 = (x0 + radius * sin(angle));
float y1 = (y0 + radius * cos(angle));
TargetIndicator *ti = [[TargetIndicator alloc] initWithFrame:CGRectMake(x1, y1, 5, 5)];
[self addSubview:ti];
[ti release];
}
A: I guess the problem lies within the present view's origin coordinate not being added to ur coordinate.
just modify your x1 and y1 by adding the origin.x and origin.y of the current view to which you add ti as a subview.
A: I figured out what was wrong but I'm don't know the reasoning behind it. First I should not have been converting the radians to degrees. This gives me the correct positioning but it is rotated 180 degrees. So to fix it, I subtract the radians from PI.
Here is the solution:
-(void) addTargetIndicatorWithHeading:(float)heading andDistance:(float)distance{
//draw target indicators
//need to convert radians and distance to cartesian coordinates
float radius = 50;
//origin offset
float x0 = 50.0;
float y0 = 50.0;
//convert heading from radians to degrees and rotate by 180 deg
float angle = M_PI - heading;
float x1 = (x0 + radius * sin(angle));
float y1 = (y0 + radius * cos(angle));
TargetIndicator *ti = [[TargetIndicator alloc] initWithFrame:CGRectMake(x1, y1, 5, 5)];
[self addSubview:ti];
[ti release];
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,231 |
Henry Kuttner
Two-handed Engine: The Selected Stories of Henry Kuttner & C.L. Moore (by HK & C. L. Moore) • collection • Centipede Press
• Locus Awards 2006 — collection — 19th place
Cordwainer Smith Rediscovery Award 2004 — winner
Clash by Night and Other Stories (by HK & C. L. Moore) • collection • Hamlyn, 1980
• 2006: 100 Must-Read Science Fiction Novels, Stephen E. Andrews & Nick Rennison
The Best of Henry Kuttner • collection • Nelson Doubleday/SFBC
• Locus Awards 1976 — single author collection — 6th place
• 2004: Anatomy of Wonder: A Critical Guide to Science Fiction, 5th edition, Neil Barron
• 1996: Age of Wonders, David G. Hartwell
• 1994: The Ultimate Guide to Science Fiction: Second Edition, David Pringle
• 1988: A Basic Science Fiction Library, James Gunn
• 1976: The World of Science Fiction: 1925-1976, Lester del Rey
The Mask of Circe • novel • Ace, 1971
• 1996: Magill's Guide to Science Fiction and Fantasy Literature, T. A. Shippey & A. J. Sobczak
The Dark World • novel • Ace, 1965
• 2009: 100 Must-Read Fantasy Novels, Stephen E. Andrews & Nick Rennison
• 1999: Fantasy and Horror, Neil Barron
• 1988: Fantasy: The 100 Best Books, James Cawthorn & Michael Moorcock
• 1979: A Reader's Guide to Science Fiction, Baird Searles & Martin Last, Beth Meacham, Michael Franklin
Valley of the Flame • novel • Ace, 1964
Earth's Last Citadel (by C. L. Moore & HK) • novel • Ace, 1964
Destination Infinity • novel • Avon, 1958
"A Cross of Centuries " • Star Science Fiction Stories
anthologies:
• A Cross of Centuries: Twenty-five Imaginative Tales About the Christ, ed. Michael Bishop, Thunder's Mouth 2007
• Other Worlds, Other Gods, ed. Mayo Mohs, Doubleday 1971
• Star of Stars, ed. Frederik Pohl, Doubleday 1960
• Star Science Fiction Stories #4, ed. Frederik Pohl, Ballantine 1958
Chessboard Planet • novel • Galaxy, 1956
"Rite of Passage" (by HK & C. L. Moore) • F&SF May 1956
• Isaac Asimov Presents the Great SF Stories: 18 (1956), ed. Isaac Asimov & Martin H. Greenberg, DAW 1988
"Two-Handed Engine" (by HK & C. L. Moore) • F&SF Aug 1955
• Criminal Justice Through Science Fiction, ed. Joseph D. Olander & Martin H. Greenberg, Franklin Watts 1977
• Decade the 1950s, ed. Brian W. Aldiss & Harry Harrison, Macmillan UK 1976
• The Metal Smile, ed. Damon Knight, Belmont 1968
"Home There's No Returning" (by HK & C. L. Moore) • short fiction • No Boundaries Ballantine 1955
• Hugo Awards 1956 — novelette — nomination
• SF: The Year's Greatest Science Fiction and Fantasy, ed. Judith Merril, Gnome 1956
Mutant • novel • Gnome Press, 1953
• 2003: The 100 science fiction books you just have to read!, Keith Olexa
"Or Else" (by HK & C. L. Moore) • Amazing Aug/Sep 1953
• A Century of Science Fiction 1950-1959, ed. Robert Silverberg & Martin H. Greenberg, MJF Books 1996
• Best SF, ed. Edmund Crispin, Faber and Faber 1955
"Home Is the Hunter" • Galaxy Jul 1953
• Robert Silverberg's Worlds of Wonder, ed. Robert Silverberg, Warner 1987
"A Wild Surmise" (by HK & C. L. Moore) • Star Science Fiction Stories
Robots Have No Tails • collection • Gnome Press, 1952
• 2005: Essential SF: A Concise Guide, Jonathan Cowie & Tony Chester
Tomorrow and Tomorrow and the Fairy Chessmen • collection • Gnome Press, 1951
Tomorrow and Tomorrow & The Fairy Chessmen • collection • Gnome, 1951
Fury • novel • Grosset & Dunlap, 1950
• 1974: Modern Science Fiction, Norman Spinrad
"See You Later" • Thrilling Wonder Stories Jun 1949
• The Oxford Book of Fantasy Stories, ed. Tom Shippey, Oxford University Press 1994
"Private Eye" (by HK & C. L. Moore) • Astounding Jan 1949
• Science Fiction: The Science Fiction Research Association Anthology, ed. Patricia S. Warrick, Charles G. Waugh & Martin H. Greenberg, Harper & Row 1988
• The Arbor House Treasury of Modern Science Fiction, ed. Robert Silverberg & Martin H. Greenberg, Arbor House 1980
• Tomorrow, and Tomorrow, and Tomorrow..., ed. Bonnie L. Heintz, Frank Herbert, Donald A. Joos & Jane Agorn McGee, Holt, Rinehart & Winston 1974
• The Mirror of Infinity, ed. Robert Silverberg, Harper & Row 1970
• Science Fiction Inventions, ed. Damon Knight, Lancer 1967
• The Best Science-Fiction Stories: 1950, ed. Everett F. Bleiler & T. E. Dikty, Fredrick Fell 1950
"Cold War" • Thrilling Wonder Stories Oct 1949
• Science Fiction Adventures in Mutation, ed. Groff Conklin, Vanguard 1955
"Happy Ending " • Thrilling Wonder Stories Aug 1948
• Beyond Tomorrow, ed. Damon Knight, Harper & Row 1965
"Don't Look Now" • Startling Stories Mar 1948
• Science Fiction of the 40's, ed. Frederik Pohl, Martin H. Greenberg & Joseph D. Olander, Avon 1978
• My Best Science Fiction Story, ed. Leo Margulies & Oscar J. Friend, Merlin Press 1949
"Jesting Pilot " • Astounding May 1947
• The City: 2000 A.D., ed. Ralph S. Clem, Martin H. Greenberg & Joseph D. Olander, Fawcett Crest 1976
• Cities of Wonder, ed. Damon Knight, Doubleday 1966
"Dream's End" • Startling Stories Jul 1947
• Perchance to Dream, ed. Damon Knight, Doubleday 1972
"The Cure" (by HK & C. L. Moore) • Astounding May 1946
• The Best of Omni Science Fiction, No. 3, ed. Ben Bova & Don Myrus, Omni Society 1982
• A Science Fiction Argosy, ed. Damon Knight, Simon & Schuster 1972
• The Other Side of the Moon, ed. August Derleth, Pellegrini Cudahy 1949
"Call Him Demon" • Thrilling Wonder Stories Fll 1946
• A Treasury of Modern Fantasy, ed. Terry Carr & Martin H. Greenberg, Avon 1981
• Strange Ports of Call, ed. August Derleth, Pellegrini Cudahy 1948
"Absalom " • Startling Stories Fll 1946
• The Great SF Stories 8 (1946), ed. Isaac Asimov & Martin H. Greenberg, DAW 1982
• Tomorrow, the Stars, ed. Robert A. Heinlein, Doubleday 1951
"What You Need " • Astounding Oct 1945
• Omnibus of Science Fiction, ed. Groff Conklin, Crown 1952
"The Piper's Son" • Astounding Feb 1945
• The Oxford Book of Science Fiction Stories, ed. Tom Shippey, Oxford University Press 1992
• Children of Wonder, ed. William Tenn, Simon & Schuster 1953
• Best of Science Fiction, ed. Groff Conklin, Crown 1946
"Camouflage" • Astounding Sep 1945
• Best SF 6, ed. Edmund Crispin, Faber and Faber 1966
"When the Bough Breaks " (by HK & C. L. Moore) • Astounding Nov 1944
• Tomorrow's Children, ed. Isaac Asimov, Doubleday 1966
• The Astounding Science Fiction Anthology, ed. John W. Campbell, Jr., Simon & Schuster 1952
• Beyond Time and Space, ed. August Derleth, Pellegrini Cudahy 1950
"The Children's Hour" (by HK & C. L. Moore) • Astounding Mar 1944
• Four for the Future, ed. Groff Conklin, Pyramid 1959
• A Treasury of Great Science Fiction, Vol. 1, ed. Anthony Boucher, Doubleday 1959
"Housing Problem " • Charm Oct 1944
• Little People!, ed. Jack M. Dann & Gardner R. Dozois, Ace 1991
• Timeless Stories for Today and Tomorrow, ed. Ray Bradbury, Bantam 1952
"A God Named Kroo" • short fiction • Thrilling Wonder Stories Win 1944
• Retro Hugo Awards 2020 — novella — nomination
The Proud Robot • collection • Hamlyn
• Retro Hugo Awards 2019 — novelette — nomination
Earth's Last Citadel (by C. L. Moore & HK) • novel • Ace
• Retro Hugo Awards 2019 — novel — nomination
"Time Locker" • Astounding Jan 1943
• The Best Time Travel Stories of the 20th Century, ed. Harry Turtledove & Martin H. Greenberg, Ballantine Del Rey 2005
• Adventures in Time and Space, ed. Raymond J. Healy & J. Francis McComas, Random House 1946
"The Proud Robot" • Astounding Oct 1943
• The World Treasury of Science Fiction, ed. David G. Hartwell, Little, Brown 1989
"Shock " • Astounding Mar 1943
• The Outer Reaches, ed. August Derleth, Pellegrini Cudahy 1951
"Mimsy Were the Borogoves" (by C. L. Moore & HK) • short fiction • Astounding Science Fiction Feb 1943
• Retro Hugo Awards 2019 — novelette — winner
• Science Fiction: Stories and Contexts, ed. Heather Masari, St. Martin's/Bedford 2009
• The Ascent of Wonder, ed. David G. Hartwell & Kathryn Cramer, Tor 1994
• The Road to Science Fiction #3, ed. James E. Gunn, Mentor 1979
• Science Fiction Hall of Fame Volume 1, ed. Robert Silverberg, Doubleday 1970
• A Treasury of Science Fiction, ed. Groff Conklin, Crown 1948
• The Night Side, ed. August Derleth, Rinehart 1947
"Clash by Night" (by C. L. Moore & HK) • short fiction • Astounding Science Fiction Mar 1943
• The Astounding-Analog Reader, Volume One, ed. Harry Harrison & Brian W. Aldiss, Doubleday 1972
"We Guard the Black Planet! " • Super Science Stories Nov 1942
• Planets of Wonder, ed. Terry Carr, Thomas Nelson 1976
• Modern Masterpieces of Science Fiction, ed. Sam Moskowitz, World 1965
"The Twonky" (by C. L. Moore & HK) • short fiction • Astounding Science Fiction Sep 1942
• Retro Hugo Awards 2018 — short story — winner
• Men and Machines, ed. Robert Silverberg, Meredith Press 1968
"Piggy Bank " • Astounding Dec 1942
"Masquerade " • Weird Tales May 1942
• The Sleeping and the Dead, ed. August Derleth, Pellegrini Cudahy 1947
"A Gnome There Was" (by HK & C. L. Moore) • Unknown Oct 1941
• The Fantasy Hall of Fame, ed. Robert Silverberg & Martin H. Greenberg, Arbor House 1983
• Beyond Human Ken, ed. Judith Merril, Random 1952
"Threshold " • Unknown Dec 1940
• Circus of Dr. Lao and Other Improbable Stories, ed. Ray Bradbury, Bantam 1956
"Beauty and the Beast " • Thrilling Wonder Stories Apr 1940
• The Baen Big Book of Monsters, ed. Hank Davis, Baen 2014
• Creatures from Beyond, ed. Terry Carr, SFBC 1975
"The Misguided Halo " • Unknown Aug 1939
"The Time Trap " • short fiction • Marvel Science Stories Nov 1938
• Evil Earths, ed. Brian W. Aldiss, Weidenfeld & Nicolson 1975
"Hollywood on the Moon" • short fiction • Thrilling Wonder Stories Apr 1938
"We Are the Dead " • Weird Tales Apr 1937
• The American Fantasy Tradition, ed. Brian M. Thomsen & Martin H. Greenberg, Tor 2002
"The Graveyard Rats " • Weird Tales Mar 1936
• The Horror Hall of Fame, ed. Robert Silverberg & Martin H. Greenberg, Carroll & Graf 1991
• The Graveyard Reader, ed. Groff Conklin, Ballantine 1958
This page last updated Saturday 27 Nov 2021 at 11:12 PT | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,953 |
{"url":"https:\/\/mechaelephant.com\/dev\/Halting-Problem\/","text":"# Halting Problem\n\nThe Halting Problem asks whether there exists a program that takes other programs as input and determines whether they loop forever or halt.\n\nAssume such a program exists and call it HP.\n\nThe program HP(P,X) assumes:\n\n\u2022 HP is a finite program\n\u2022 HP stops in finite time\n\u2022 HP takes as input a program, P, with input, X, both of finite length\n\u2022 HP can access arbitrarily long memory such as a tape in a Turing Machine model.\n\nThough a bit far afield from the current topic, there also needs to be constraints on the time it takes to access distant memory so as not to 'hide' computation in memory access. For example, assuming memory is a linear tape and the time to reach a distance, d, from the current location takes time proportional to d.\n\nConsider the program SPITE:\n\nSPITE(P) {\n\nif (HP(P,P) reports P halts with P as input) {\nwhile (true) {} \/\/ loop forever\n}\n\nelse if (HP(P,P) reports P loops forever with P as input) {\nhalt \/\/ return\n}\n\n}\n\n\nWhen we run SPITE(SPITE) (SPITE with itself as input), there are two cases:\n\n\u2022 SPITE(SPITE) halts, in which case the first condition would have been hit, contradicting the subsequent action of looping forever.\n\u2022 SPITE(SPITE) loops forever, in which case the second condition would have been hit, contradicting the subsequent action of halting.\n\nNo matter the path we take, we get a contradiction, proving the impossibility of HP existing with our given assumptions.","date":"2018-12-12 10:31:24","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\": 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\": 0.4674167037010193, \"perplexity\": 2465.508335766725}, \"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-2018-51\/segments\/1544376823817.62\/warc\/CC-MAIN-20181212091014-20181212112514-00525.warc.gz\"}"} | null | null |
Панамська плита — тектонічна мікроплита. Має площу — 0,00674 стерадіан. Зазвичай розглядається у складі Карибської плити.
Розташована на теренах Панами і Коста-Рики.
Має конвергентну границю з Карибською плитою. Трансформаційну границю з плитою Наска й Північноандською плитою. Зону субдукції з плитою Кокос.
Посилання
Peter Bird, An updated digital model of plate boundaries, Geochemistry Geophysics Geosystems, 2003
Тектонічні плити | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,021 |
\section{Introduction}
Quantum annealing (QA) has been one of the flagship procedures for performing meaningful calculations on Noisy Intermediate Scale Quantum (NISQ) Devices. QA, introduced in \cite{kadowaki98, apolloni89, finnila94}, aims to find the ground state of a target quantum Hamiltonian from a more trivial starting Hamiltonian by smoothly interpolating between the two. A regime of interest for QA is Adiabatic Quantum Computation (AQC) \cite{farhi00}, which corresponds to the limit of slow transition between the initial and target Hamiltonian. In AQC, qubits remains in the instantaneous ground state of the evolving Hamiltonian, thereby ending in the ground state of the final Hamiltonian. Recently, quantum annealers from D-Wave systems have become commercially available \cite{johnson11}.
Any problem that can be formulated as a specific form of a binary objective function has the potential to be run on quantum annealers, and for many problems there exists efficient methods to convert to this form \cite{date19}. Previous research has been able to formulate and solve important problem instances from a wide array of scientific fields, such as combinatorial optimization \cite{chapuis18,santra14,farhi12,mcandrew20,martonak04}, physics \cite{harris18,brooke99,santoro02}, biology \cite{perdomo12,mulligan20}, economics \cite{orus19,rosenberg16,ding19}, and operations research \cite{neukart17}. Problem instances that are natively cast as binary objective functions, however, are a natural choice for the application of QA. Previous research into such problems has been conducted, benchmarking the performance of QA against classical heuristics and algorithms \cite{trevisan12,goemans95,dunning18, sahni76}. This research has shown proven quantum speedups for a specific graph optimization problem \cite{somma12}, as well as evidence of a scaling advantage of QA over classical simulated annealing for specific classes of more general problems \cite{albash18, denchev16}. However, the investigation of quantum speedups over more general problem instance sets has led to more inconclusive or instance-dependent results \cite{ronnow14, hamerly19}. For the purposes of this paper, a comparison to classical methods is not of great importance, but rather the paper focuses on the quantum annealing method and simply compares to random groupings in order to demonstrate that the annealers are not simply returning random results for problems of the size analyzed.
While there has been recent research investigating applications of graph theory and classical graph optimization to reduce disease spread \cite{crnkovic21, alguliyev21, bhapkar20}, this paper specifically applies the Maximum-Cut graph problem (Max-Cut) to epidemiology. Max-Cut is a combinatorial optimization problem that serves to find a partition of a given graph such that the number of edges crossing the partition is maximized. Max-Cut has been proven to be NP-Complete \cite{karp72}, and has been widely studied in both classical and quantum optimization literature \cite{trevisan12,goemans95,dunning18, farhi14}. The problem has a classical objective function which simply counts the number of edges that cross the chosen cut:
\begin{equation}
f(x) = \sum_{(i,j) \in E} x_i \oplus x_j \label{maxcut_classical}
\end{equation}
Where $E$ represents the edge set of the graph and $x_i$'s are binary variables representing which of the two groups a given node has been assigned to. This cost function can easily be cast into a quantum Ising Hamiltonian, using the relation $x = \frac{1}{2}(I-Z)$
\begin{equation}
\frac{1}{2} (I -\sum_{(i,j) \in E} Z_iZ_j) \label{maxcut_quantum}
\end{equation}
Where $Z_i Z_j$ refers to a Pauli matrix $Z$ acting on qubits $i$ and $j$.
Conveniently, quantum annealers from D-Wave Systems natively implement Quadratic Unconstrained Binary Optimization (QUBO) problems, to which Ising Hamiltonians can be trivially cast. Thus, Max-Cut defined on an arbitrary graph can be efficiently run on a D-Wave quantum annealer to obtain sample solutions, after mapping the nodes of the graph to qubits on the quantum annealer \cite{dwavesolver}.
This paper, motivated by the recent COVID-19 pandemic, investigates the recursive application of QA-assisted Max-Cut to group students into physically separated cohorts, with respect to networks specified by the physical interactions of students on a college campus. The efficacy of this method is tested via numerical simulations of disease outbreak. We hypothesize that this method can reduce the spread of an infectious disease compared to random groupings of students, based on the intuition that "cutting" the most edges from a network reduces the total number of physical interactions between students, thus reducing the opportunities for the spread of disease.
This paper is organized as follows: in Sec \ref{networks}, we describe our method for creating course enrollment networks (CENs) and student interaction networks (SINs), which encapsulate physical interactions on university campuses. Sec \ref{annealing} fleshes out details on our recursive QA-assisted Max-Cut algorithm. Sec \ref{epidemics} describes our methods for simulating and testing disease spread. In Sec \ref{results}, we numerically simulate our approach and compare disease spread to the case of random groupings. Discussions and further directions are presented in Sec \ref{discussion}.
\section{Methods}\label{methods}
\subsection{Generating Networks}\label{networks}
Social networks, commonly used in psychology, sociology, and other fields, are a natural tool for describing a university population, as they can encapsulate a given student body and possible interactions within it. In the networks we generated, nodes represent students and bidirectional edges represent possible physical interactions (and thereby possible disease spread) between two nodes/students.
We first consider the case where two students share an edge only if they share a physical class in the same room at the same time, and refer to these networks as Course Enrollment Networks (CENs). To construct these CENs, we consulted published statistics of a social network based on the 2020 course enrollment statistics of Cornell University's Ithaca campus given in \cite{weeden20}. This reference specifies key network parameters listed in Table \ref{tab1}, and we found that, after toying with parameters, a Watts-Strogatz "small-world" graph \cite{watts98} generated using the NetworkX package \cite{networkx} produces a network with closest fit to these parameters. A sample CEN generated using this procedure can be found in Fig.~\ref{CEN}.
In addition to CENs, we analyzed enhanced networks that more fully encapsulate the actual physical environment of a campus. To accomplish this, we added inter-cohort and intra-cohort interactions to our network with specified probabilities. We label these enhanced networks Student Interaction Networks (SINs). For our analysis of SINs, we considered a slightly smaller campus, modeled after Harvey Mudd College in Claremont, California. This college was chosen because approximately 99\% of the students live on campus \cite{harveymudd}, so we can attempt to more closely capture all possible interactions between students without considering off-campus influences. For the SINs, we assume that each cohort assigned by our algorithm lives in a separate designated dormitory hall or floor. Harvey Mudd College houses 800 students into 8 dorms, and each dorm has two floors on average \cite{harveymudd}. Thus, in order to create SINs with inter-cohort and intra-cohort interactions, we first generate a CEN of an 800 student campus, then divide the students into 16 groups (8 dorms and 2 floors), and add random edges between floors, dorms and dormitories with tunable probabilities. We label these edges floor, dorm, and campus interaction edges. We then run a disease model on the imperfectly separated cohorts. A sample SIN generated using this procedure can be found in Fig.~\ref{SIN}.
\begin{table}[htb]
\caption{Network parameters for CENs}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\textbf{}&\textbf{\textit{Cornell}}& \textbf{\textit{NetworkX}} \\
\cline{2-3}
\hline
\textbf{Number of nodes}&\textbf{\textit{3800}}& \textbf{\textit{3800}} \\
\hline
\textbf{Network density}&\textbf{\textit{0.040}}& \textbf{\textit{0.040}} \\
\hline
\textbf{Clustering coefficient}&\textbf{\textit{0.480}}& \textbf{\textit{0.465}} \\
\hline
\textbf{Mean geodesic distance}&\textbf{\textit{2.233}}& \textbf{\textit{3.110}} \\
\hline
\end{tabular}
\label{tab1}
\end{center}
\end{table}
\begin{figure}[htb]
\centerline{\includegraphics[width=5.5cm]{cornell.png}}
\caption{A sample Course Enrollment Network (CEN) based off data collected from Cornell University's Ithica Campus.}
\label{CEN}
\end{figure}
\begin{figure}[htb]
\centerline{\includegraphics[width=5.5cm]{harvey.png}}
\caption{A sample Student Interaction Network (SIN) based on a returned assignment of 800 student to 16 cohorts (8 dormitories with 2 rooms each), with added floor, dorm, and campus interaction edges after partitioning.}
\label{SIN}
\end{figure}
\subsection{Network Partitioning}\label{annealing}
Once a CEN or SIN is defined, we can run a partitioning algorithm in order to divide students into cohorts. We outline Algorithm \ref{alg1}, which perform these partitions using a quantum annealer for the black-box subroutine \emph{performMaxcut}, and Algorithm \ref{alg2}, which randomly assigns each student to a cohort. The quantum algorithm can be made recursive or iterative, but is written iteratively for simplicity. The \emph{performMaxcut} subroutine is implemented on a D-Wave Systems hybrid solver \cite{dwavehybrid}. Once a cohort assignment has completed, we can remove edges between students of different cohorts from the network, as we assume that students in separate cohorts can no longer transmit disease through a shared class.
\begin{algorithm}[htb]
\SetAlgoLined
G = Network Graph\;
N = Target number of cohorts (power of 2) \;
currCohorts = [G]\;
\While{currCohorts.length $<$ N}{
newCohorts = []\;
\For{c in currCohorts} {
c1, c2 = performMaxcut(c)\;
newCohorts.append(c1)\;
newCohorts.append(c2)\;
}
currCohorts = newCohorts\;
}
\caption{Quarantine Partitioning Algorithm: iteratively perform max-cut until we reach the desired number of cohorts}
\label{alg1}
\end{algorithm}
\begin{algorithm}[htb]
\SetAlgoLined
G = Network Graph\;
N = Target number of cohorts (power of 2) \;
currCohorts = [N arrays]\;
\For{i in G.nodes} {
cohort\_idx = random.choice([1,2,...N])\;
currCohorts[cohort\_idx].append(i)\;
}
\caption{Random Partitioning Algorithm: assign each individual to a cohort with uniform probability}
\label{alg2}
\end{algorithm}
\subsection{Epidemic Modelling}\label{epidemics}
Once a network has been set, a disease outbreak can be simulated using the Susceptible-Infected-Recovered (SIR) model, derived from the Kermack-McKendrick theory of disease spread \cite{kermack27}. We implemented this model with the Python library NDlib \cite{ndlib}. In order to fully specify the disease model, we needed to define the following parameters:
\begin{itemize}
\item $r_r$ (recovery rate): the probability that an infected individual will recover in any given day.
\item $r_i$ (infection rate): the probability that a healthy individual will be infected in any given day.
\end{itemize}
Estimates vary, but show for COVID-19, the average recovery time $T_r$ is around 10 days \cite{recovery}. Since $r_r$ denotes a constant probability of recovering in a single day, the probability of an individual to recover after multiple days follows a geometric distribution, with an average recovery time that should equal $T_r$. Thus given $T_r$, we can calculate $r_r = 1/T_r$.
To calculate $r_i$, we note that in epidemiology, $R_0$ quantifies the average number of people that an individual can infect during the course of an illness. The exact $R_0$ value for COVID-19 depends on a wide array of variables, but has been reported to range from 1.5 to 6.7 \cite{achaiah20}. Using a chosen $R_0$ value, we can estimate $r_i$ via the equation below:
\begin{equation} \label{eu_eqn}
R_0 = avg\_deg(G)*r_i/r_r
\end{equation}
The reasoning for this is as follows: each infected individual will interact with $avg\_deg(G)$ individuals on average per day, and each interaction may create a new infection with probability $r_i$. Thus, at each time step, the expected number of infections caused by an infected individual in a day is avg\_deg(G)*$r_i$. Moreover, since the infected individual is expected to be infected for 1/$r_r$ days, the total expected number of infections needs to be further multiplied by 1/$r_r$.
Having specified $r_i$, $r_r$, and a CEN or SIN, we can run the SIR disease model and calculate two valuable data points, the \emph{total infected percentage}, representing the percent of the population that contracts the disease at some point by the end of the simulation, and the \emph{peak infected percentage}, which is the percent of infected students at the peak of the outbreak.
\section{Results}\label{results}
\subsection{Course Enrollment Networks}\label{cen}
\begin{figure}[htb]
\centerline{\includegraphics[width=8.5cm]{cornell_sir.png}}
\caption{Disease progression comparison between random (left) and QA-assisted (right) student groupings for 3800-student CEN based on Cornell University Ithica Campus. The susceptible, infected, and removed curves represent percentages of students that have not yet gotten the disease, currently have the disease, and have already had the disease, respectively}
\label{cornell_sir}
\end{figure}
For the 3800-student CEN, modeled after Cornell Ithaca campus's coursework enrollment network, our algorithm was able to reduce the total infected percentage by over 50\% compared to random quarantine assignment with four groups. Moreover, the infection curve steadily declines as opposed to the random quarantine assignment, which produces an infection curve that remains constant for a few weeks before descending. This disease progression is shown in Fig \ref{cornell_sir}. For this simulation, we assume 5\% of students are initially infected, an $R_0$ value of 6, and a recovery time of 10 days. In addition to this simulation, we tested other initial infected student percentages and number of cohorts. In these simulations, the total infections were reduced between 10-50\%, depending on the two variables mentioned.
\subsection{Student Interaction Networks}\label{sin}
\begin{figure}[htb]
\centerline{\includegraphics[width=9.5cm]{harvey_sir.png}}
\caption{Disease progression comparison between random (left) and QA-assisted (right) student groupings for 800-student SIN based on Harvey-Mudd College Campus. The susceptible, infected, and removed curves represent percentages of students that have not yet gotten the disease, currently have the disease, and have already had the disease, respectively}
\label{harvey_sir}
\end{figure}
For the 800-student SIN, with students grouped into dormitories according to Harvey-Mudd campus housing, our algorithm was able to reduce the total infected percentage by over 20\% compared to random quarantine assignment. This disease progression is shown in Fig \ref{harvey_sir}. In this simulation we assume 1\% of students are initially infected, an $R_0$ value of 2, a recovery time of 10 days, and added floor, dorm, and campus interaction edges with probability .2, .005, and .0001 respectively.
In order to more thoroughly test a wider variety of parameters (initial infection rate, infection rate, and interaction edge rates), we repeated the comparisons between quantum and random groupings with each parameter ranging between lower and upper bounds given by Table \ref{tab2}. Using parameters from these ranges, we found that on average we were able to reduce the peak and total infected percentages by only 2.3\% and 1.7\%, respectively, over random assignments. Furthermore, the quantum groupings provided better infection percentages in only 53\% of the settings. Upon closer analysis, we realized that in the majority of these parameter settings, there was uncontrolled spread in both the random and quantum cohort case, which accounted for the minuscule improvement. We thus hypothesized that quantum groupings would be more effective in parameter regimes corresponding to values of the listed parameters. Motivated by this idea, we set our parameters according to Table \ref{tab3}, ran multiple simulations with randomly added edges, and found that in this case, we were able to reduce the peak and total infected percentages by 4.9\% and 16.7\% compared to random groupings. Additionally, quantum groupings performed better than random groupings in 9 out of the 10 simulations.
\begin{table}[htb]
\caption{Parameter Variation for SINs}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\textbf{}&\textbf{\textit{lower bound}}& \textbf{\textit{upper bound}} \\
\hline
\textbf{initial infection}&\textbf{\textit{0.05}}& \textbf{\textit{0.5}} \\
\hline
\textbf{infection rate}&\textbf{\textit{0.01}}& \textbf{\textit{0.1}} \\
\hline
\textbf{floor interaction edge rate}&\textbf{\textit{0.1}}& \textbf{\textit{0.6}} \\
\hline
\textbf{dorm interaction edge rate}&\textbf{\textit{0.001}}& \textbf{\textit{0.016}} \\
\hline
\textbf{campus interaction edge rate}&\textbf{\textit{0.00005}}& \textbf{\textit{0.0005}} \\
\hline
\end{tabular}
\label{tab2}
\end{center}
\end{table}
\begin{table}[htb]
\caption{Parameters for low-density SINs}
\begin{center}
\begin{tabular}{|c|c|}
\hline
\textbf{}&\textbf{\textit{value}}\\
\hline
\textbf{initial infection}&\textbf{\textit{0.05}} \\
\hline
\textbf{infection rate}&\textbf{\textit{0.02}} \\
\hline
\textbf{floor interaction edge rate}&\textbf{\textit{0.2}} \\
\hline
\textbf{dorm interaction edge rate}&\textbf{\textit{0.005}} \\
\hline
\textbf{campus interaction edge rate}&\textbf{\textit{0.0001}} \\
\hline
\end{tabular}
\label{tab3}
\end{center}
\end{table}
\section{Discussion}\label{discussion}
The aim of this work was to introduce a technique that leverages the computational power of quantum annealing to group students on college campuses into physically separated cohorts in order to reduce the spread of COVID-19. This was achieved via the recursive application of Quantum Annealing (QA) to course enrollment networks (CENs) and more general student interaction networks (SINs), both of which capture physical interactions between students. We found that for both CENs and SINs, with certain assumptions on disease spread, the proposed method was able to reduce the total and peak infected percentages under the SIR epidemic model compared to random cohort assignment. A severe limitation to our method, however, is that our method is not effective if the combination of initial infected student percentage, infection rate, and added interaction edges in a SIN is large enough that there is uncontrolled spread in both the quantum and random grouping case. We also note that we had to estimate values of floor, dorm, and campus interaction edges for the SIR model, as we were not able to find publicly available estimates of these values. Additionally, there may be more interactions in a SIN that cannot be accounted for, such as interactions with individuals living off-campus.
As mentioned, the advantages and possible speedups of QA over classical Max-Cut algorithms and heuristics remain widely unknown for arbitrary graphs. Thus, one could immediately build off this work by viewing the recursive Max-Cut step as a black box, and deciding groupings using purely classical, quantum gate-based, or hybrid quantum-classical devices for optimization. This work could then be relevant outside of quantum annealing literature. One could additionally replace recursive Max-Cut with Max k-Cut, wherein one divides the problem graph into $k$ cohorts in a single optimization procedure. In order to demonstrate quantum advantage, one must be able to regularly show that the groupings returned from the quantum-assisted method outperform groupings returned by the best possible classical algorithms or heuristics for an array of CENs or SINs. This would most likely require Max-Cut solved on a quantum annealer to outperform any classical method, and current research into this comparison is inconclusive. As stated in the introduction, our goal was not to perform this extensive benchmarking, but rather introduce a novel application of QA.
This approach can likewise be extended to any environment that can be described by a mostly self-contained network representing individuals and shared physical environments, such as office spaces, gyms, and primary/secondary schools.
\section*{Acknowledgment}
J.S. and V.L. would like to thank enlightening and helpful discussions from Dr. Hossein Sadeghi, as well as D-Wave Systems, INC, for free COVID-19 research access to D-Wave Leap, providing cloud-accessed quantum annealers and quantum/classical hybrid solvers. J.S. and V.L. would also like to thank the Quantum Computing @ Berkeley Club, and Dr. Bjoern Hartmann, our mentor for the 2020 Jacobs Institute Of Design Innovation Covid-19 Design Challenge\cite{jacobs}. J.S. would also like to thank Maya Chan Morales for research consultation and editing. All code used in this project can be found at https://github.com/qcberkeley/optimization
| {
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{"url":"https:\/\/proofwiki.org\/wiki\/Not_every_Non-Empty_Subset_of_Natural_Numbers_has_Greatest_Element","text":"# Not every Non-Empty Subset of Natural Numbers has Greatest Element\n\n## Theorem\n\nLet $S \\subseteq \\N_{>0}$.\n\nThen, despite Non-Empty Subset of Initial Segment of Natural Numbers has Greatest Element, it is not necessarily the case that $S$ has a greatest element.\n\n## Proof\n\nLet:\n\n$S = \\left\\{{x \\in \\N_{>0}: x > 1}\\right\\}$\n\nThen $S \\subseteq \\N_{>0}$.\n\nSuppose $S$ has a greatest element.\n\nLet the greatest element of $S$ be $k$.\n\nBut $\\N_{>0}$ is an inductive set.\n\nTherefore $k + 1 \\in \\N_{>0}$.\n\nBy definition of $S$:\n\n$k + 1 \\in S$\n\nTherefore $k$ cannot be the greatest element of $S$.\n\nBy Proof by Contradiction it follows that $S$ has no greatest element.\n\nIt is not possible to apply Non-Empty Subset of Initial Segment of Natural Numbers has Greatest Element, because $S$ is not a subset of an initial segment of $\\N_{>0}$.\n\n$\\blacksquare$","date":"2019-12-06 17:45:57","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\": 0.6272780895233154, \"perplexity\": 136.44807460328136}, \"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\/1575540490743.16\/warc\/CC-MAIN-20191206173152-20191206201152-00333.warc.gz\"}"} | null | null |
LAACU's Spring Happy Hour Kickoff!
Join us! Unwind, relax, catch up and expand your network! Come to Kingston Hall and enjoy Happy Hour specials, including: BOGO on all beers and well drinks.
There is no cost for the event, but please consider welcoming our graduates to LAACU by donating a graduation stole for la Raza Graduation. | {
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Moon Hyung-in est un chercheur en biologie médicale, membre du département de biotechnologie médicale et professeur à l'université Dong-A à Busan en Corée du Sud, auteur de critiques frauduleuses de ses propres publications.
Éléments biographiques
Moon Hyung-in a obtenu son doctorat au département de pharmacie de l'université Sungkyunkwan en 2001. En 2012, il est accusé d'avoir publié un certain nombre d'articles dans des journaux scientifiques à comité de lecture en en rédigeant lui-même les critiques. Il soumettait ses manuscrits à des journaux qui acceptent les experts suggérés par l'auteur ; il utilisait de faux noms et des adresses mail qu'il contrôlait pour proposer des critiques. Un éditeur a découvert le pot aux roses après avoir reçu un compte-rendu 24 heures après avoir envoyé l'article à l'« expert ». Moon Hyung-in aurait aussi reconnu falsifier les données dans ses articles.
Liste des publications retirées
Pharmaceutical Biology, 2011, 49,2, ;
Journal of Enzyme Inhibition and Medicinal Chemistry, 2012, doi: 10.3109/14756366.2011.641014, publication électronique ; 2012, doi: 10.3109/14756366.2011.615746, publication électronique ; 2010, 25,5, ; 2010, 25,3, ;
International Journal of Food Sciences and Nutrition, 2012, 63,5, , doi: 10.3109/09637486.2011.607801 ; 2011, 62,2, , doi: 10.3109/09637486.2010.513682 ; 2011, 62,3, , doi: 10.3109/09637486.2010.503187 ;
The Journal of Ethnopharmacology, 2005, 97,3,21, , doi: 10.1016/j.jep.2005.01.006 ;
FEBS Letters, 2006, 580,3, , doi: 10.1016/j.febslet.2005.12.094 ;
Phytotherapy Research ; 2006, 20,8, , doi 10.1002/ptr.1941 ; 2005, 19,3, , doi 10.1002/ptr.1682.
Vingt autres articles, tous publiés dans Immunopharmacology and Immunotoxicology, ont également été retirés, avec les excuses de l'éditeur qui assure que toutes les mesures ont été prises afin que ce problème ne puisse pas se reproduire. Au total, ce sont 35 publications qui ont été retirées.
Notes et références
Voir aussi
Bibliographie
.
Articles connexes
Fraude scientifique
Liens externes
.
Scientifique sud-coréen
Faussaire
Communication scientifique
Personnalité impliquée dans une affaire de fraude scientifique | {
"redpajama_set_name": "RedPajamaWikipedia"
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Q: How do I add data to http request Here is my code:
cj = cookielib.CookieJar()
opener = urllib2.build_opener(urllib2.HTTPCookieProcessor(cj))
urllib2.install_opener(opener)
opener.add_data(json_data) #runtime error on this line
r = None
try:
r = opener.open(config.get_url())
except urllib2.HTTPError as e:
print "..."
I get a runtime error: opener.add_data(get_json)
AttributeError: OpenerDirector instance has no attribute 'add_data'
How can I add request json data to this request?
A: If all you're doing is sending a requests I highly recommend using requests:
import requests
data = {"param": value}
url = your_url
cookie = your_cookie
requests.get(your_url, data=data, cookie=cookie)
you can access the ._content attribute of the response to get the raw http response.
See this for more requests documentation.
This gist should give you a good idea as to why you should use requests instead of urllib: https://gist.github.com/kennethreitz/973705
A: Give it as an argument to the open method (it needs to be url-encoded first):
import urllib
try:
r = opener.open(config.get_url(), data=urllib.urlencode(json_data))
except urllib2.HTTPError as e:
print "..."
See the documentation of the open method:
OpenerDirector.open(url[, data][, timeout])
Open the given url (which can be a request object or a string), optionally passing the given data.
You can find the documentation by noting that opener is a urllib2.OpenDirector instance:
>>> opener
<urllib2.OpenerDirector instance at 0x1011acef0>
Or by doing help(opener), which shows:
class OpenerDirector
| Methods defined here:
|
| __init__(self)
|
| add_handler(self, handler)
|
| close(self)
|
| error(self, proto, *args)
|
| open(self, fullurl, data=None, timeout=<object object>)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,825 |
Splavnění Labe z Chvaletic do Pardubic je projekt, který si za cíl klade prodloužení splavnosti Labe o 24 km z Chvaletic do Pardubic. Setkal se s kritikou části veřejnosti a nevládních organizací, kteří považují plánovaný plavební kanál za nešetrný k životnímu prostředí.
Historie projektu
Labe, tedy především Dolní Labe od Mělníka po proudu, bylo dopravní tepnou již od počátku historie. První archeologické důkazy o využití řeky k dopravě kamenných bloků jsou datovány do doby římské Jako živá dopravní tepna spojující Čechy s mořem a severní Evropou sloužilo a slouží dodnes, přičemž největšího rozkvětu dosáhla labská plavba v druhé polovině 19. století. Na základě Labských plavebních akt, podepsaných roku 1821, a novelizovaných roku 1922, je Labe od soutoku s Vltavou po ústí do moře svobodnou mezinárodní vodní cestou, přístupnou bez omezení plavidlům všech národností
Pro zlepšení plavebních poměrů byla počátkem 20. století na Dolním Labi po Střekov a Vltavě vybudována souvislá kaskáda zdymadel. V 60. letech pak ze stejného důvodu přibylo v Německu nad Hamburkem vodní dílo Geesthacht . V současnosti probíhá, na základě Česko-Německého memoranda, zlepšení plavebních parametrů zbylého úseku řeky úpravou dřívější regulace a plánovanou výstavbou vodního díla Děčín, na minimální zajištěný ponor 140 cm po 365 dní v roce a 220 cm po 180 dní v roce.. Proti tomuto záměru a především proti výstavbě vodního díla Děčín, se však dlouhodobě staví nevládní organizace, například Arnika.
Naproti tomu Střední Labe bylo ve středověku plavebně využíváno pouze k plavbě vorů z podkrkonoší do Kolína, pro potřeby Kutnohorských dolů. S úpravami Středního Labe od Mělníka po Pardubice výstavbou kaskády zdymadel, pro účely plavby, zavlažování a ochrany proti povodním, se začalo počátkem 20. století na základě říšského vodocestného zákona z r. 1901. Práce probíhaly s několika přestávkami po celé délce řeky a do 70. let 20. století se podařilo vybudovat souvislou vodní cestu z Mělníka do Chvaletic a izolovaný splavný úsek od Přelouče po Pardubice.
V rámci současných splavňovacích prací bylo dokončeno splavnění mezi Chvaleticemi a začátkem Labských hrčáků v pl.km 112,430 nedaleko Přelouče. Dokončením posledních 2 km vodní cesty k zdymadlu Přelouč v pl.km 114,535 by se souvislá Labská vodní cesta prodloužila celkem o 24 km a dosáhla svého konečného cíle v koncovém přístavu Pardubice, včetně místa eventuálního zaústění labské větve Kanálu Dunaj-Odra-Labe nad Pardubicemi.
Časová osa
1996 – Vláda rozhodla svým usnesením č.635/96, o zařazení splavnění do Pardubic mezi priority rozvoje vodních cest.
1999 – Začalo se s úpravou koryta mezi Chvaleticemi a Přeloučí..
2000 – projekt splavnění do Pardubic získává kladné hodnocení EIA
2003 – Na základě kladného hodnocení EIA ministr životního prostředí Libor Ambrozek vydává výjimky ze zákona 114/1992. Výjimky jsou podmíněny realizací mnoha kompenzačních opatření a přesunu ohrožených druhů živočichů a rostlin na náhradní lokality. Na těchto pracích začíná pracovat tým ekologických odborníků. Studie vypracovaná v tomto období bývalým ministrem životního prostředí Ivanem Dejmalem například uvádí, že stavbou způsobená ekologická újma bude plně kompenzována do 5 let po plánované výsadbě nové zeleně a do 25 let se ekologická hodnota nových biotopů zvýší o 40 procent oproti současné ekologické hodnotě území.
2005 – 31. 12. projekt získal platné územní rozhodnutí.
2006 – městský soud v Praze ruší na základě žalob ekologických sdružení udělené výjimky ze zákona 114/1992, z formálních důvodů, konkrétně z důvodu chybějícího odůvodnění rozhodnutí. Ministerstvo životního prostředí podalo proti rozhodnutí kasační stížnost, ke které se připojilo i Ředitelství Vodních cest.
2007 – v lednu zvítězila ve výběrovém řízení na stavu průplavu firma Skanska. Plánuje začít stavbu již na jaře 2007. Ministr životního prostředí Martin Bursík však odmítá zachovat právní kontinuitu a odstranit formální nedostatky předešlé výjimky. Realizace projektu je pozastavena.
2008 – v květnu Nejvyšší správní soud potvrdil předešlé rozhodnutí městského soudu v Praze a zrušil výjimku k zásahu do biotopů chráněných druhů organizmů bývalého ministra životního prostředí Libora Ambrozka s odůvodněním, že ministerstvo se nezabývalo ochranou přírody a stavbu kanálu, jehož výstavba měla stát 2 miliardy Kč, tak zablokoval . (v rozsudku stojí, že v odůvodnění udělení výjimky nebyl dostatečně popsán veřejný zájem převažující nad veřejným zájmem ochrany přírody. M. Šefara ve svém článku (viz výše u roku 2003) označil toto za "Lajdáckou práci úředníků MŽP")
2009
leden – nové vedení Pardubického kraje vzešlé z krajských voleb, plánuje na ministra životného prostředí Martina Bursíka podat žalobu pro porušení zákona. Podle názoru vedení kraje pro odmítnutí odstranit formální nedostatky v předešlé výjimce, která byla založena na kladném posouzení projektu v procesu EIA a na jejímž základě bylo již proinvestováno 750 mil. Kč, nebyl žádný reálný důvod.
únor – Ředitelství vodních cest podalo stížnost k Ústavnímu soudu ve věci zamítnutí kasační stížnosti proti předešlému rozhodnutí Nejvyššího správního soudu.
2010
únor – Nejvyšší správní soud vyhověl kasačním stížnostem Ředitelství vodních cest ČR ve sporu o stavbu vodního díla na Labi u Přelouče. Zrušil skupinu verdiktů pardubického krajského soudu, kterými pardubičtí soudci zneplatnili různá rozhodnutí správních orgánů, jež se týkala sporného vodního díla. Pardubický soud musí kauzu prozkoumat znovu.
Projekt
Prodloužení splavnosti z Chvaletic do Pardubic s sebou nese spoustu nákladů, které dohromady činí přes 2,5 miliardy korun. Investice by pravděpodobně byla částečně kryta z prostředků Evropské unie.
Do Přelouče by se ještě musela řeka prohloubit, na tom se již v současnosti pracuje. Ale tam se totiž nachází jediný úsek Labe s bystřinným prouděním Hrčáky. Tuto unikátní lokalitu není možné zničit. Proto je zde jako kompromis mezi zájmy ekologů a plavby uvažována na pravém břehu Labe stavba kanálu, který tuto lokalitu obejde a na koryto Labe se napojí až nad jezem Přelouč. Celkově by kanál dosáhl délky 3150 m. V plavební komoře by lodě překonávaly výšku 8,4 m. V Srnojedech by se ještě musel modernizovat stávající jez. Na okraji Pardubic by vznikl velký přístav, sloužící pro nakládání a vykládání zboží až ze čtyř lodí zároveň.
Klady
Ředitelství vodních cest ČR udává, že by po splavnění Labe bude možné po této řece přepravovat zejména hromadné substráty – tedy stavební materiály, zemědělské produkty a hnojiva. Předpokládá se také přeprava hutního zboží, kontejnerů a nadrozměrných nákladů. Svedení části nákladu ze silnic na vodu by zlepšilo dopravní situaci v regionu.
Především by se zhruba o 100 km zkrátila cesta nadměrných nákladů, mířících z ostravských hutí přes Pardubice do přístavu Mělník k naložení místními těžkými jeřáby do lodí k exportu do zahraničí. Tím by se nejen výrazně zlevnila doprava výrobků do místa určení, ale především zabránilo nadměrnému opotřebení polabských komunikací a obtěžování místních obyvatel.
Součástí projektu výstavby pravobřežního průplavu jsou i rozsáhlá kompenzační opatření, díky kterým by se již za 5 let od výstavby ekologická hodnota území vrátila na původní úroveň a do 25 let by se zvýšila oproti současnému stavu o 40%.
Zápory
Na první pohled šalamounské řešení, spočívající ve výstavbě kanálu kolem cenných partií Labe, má své stinné stránky. Nový plavební kanál by protínal biocentrum Slavíkovy ostrovy, kde se vyskytují vzácné, celoevropsky chráněné druhy Natura 2000.
Česká zemědělská univerzita shrnula výsledky následovně. Stavbou nebude zničeno naleziště žádné vzácné rostliny (česnek hranatý (Allium angulosum), uváděný z této lokality, nebyl nalezen). Stavba vážně ohrozí populaci modrásků očkovaných (Maculinea teleius) a modrásků bahenních (Maculinea nausithous). V rámci České republiky i Evropy jsou to velmi ohrožené druhy, kterých kriticky ubylo v posledních 40 letech. V poslední době však na loukách s rozumným obhospodařováním začíná přibývat (a to i v Přelouči). Modrásci jsou svým výskytem vázáni na rostlinu krvavec toten (Sanguisorba officinalis) a současně na mravence rodu Myrmica, v jejichž mraveništích prodělávají larvy druhou fázi vývoje.
V okolí Přelouče se vyskytuje velmi životaschopná populace modrásků na několika loukách. Mezi těmito lokalitami modrásci velmi čile migrují. Jedna velmi bohatá populace by byla ze dvou třetin zničena a modrásci by se museli přesunout jinam. Je otázka, zda by to udělali, ale vědci se snaží připravit jim v okolí dobré podmínky. Slibněji prý vypadá šíření modrásků bahenních.
Na Slavíkových ostrovech byli několikrát nalezeni i brouci lesák rumělkový (Cucujus cinnaberinus) a páchník hnědý (Osmoderma eremita), oba jsou to druhy celoevropsky chráněné, podobně jako zmínění modrásci.
Projekt obsahuje rozsáhlá kompenzační opatření na kompenzaci možných zásahů do chráněných biotopů a bylo na něj i vydáno kladné stanovisko v procesu EIA
Reference
Související články
Seznam sledovaných vodních cest v Česku
Splavnění střední Vltavy
Externí odkazy
Přibližná poloha plánovaného plavebního kanálu na Wikimapia
Vodní cesty a plavba 3-4/2006 - Číslo časopisu věnované plavebnímu stupni Přelouč II.
Radní se bojí o splavnění Labe: https://web.archive.org/web/20071012095050/http://www.prelouc.com/cz/modules.php?name=News&file=article&sid=1181
Nový plavební kanál – pro a proti: https://web.archive.org/web/20071012094748/http://www.prelouc.com/cz/modules.php?name=News&file=article&sid=1187
Ředitelství vodních cest ČR: https://web.archive.org/web/20070708214646/http://www.rvccr.cz/pardubice.htm
Postoj NGO – Přátelé přírody: http://www.prateleprirody.cz/clanek-179
Protesty ekologů a krajinných inženýrů proti projektu Splavnění Labe do Pardubic nejsou založeny "jen" na ochraně Slavíkových ostrovů: https://web.archive.org/web/20111003181527/http://krajinazp.ic.cz/splavneni.htm
Reportáž Radiožurnálu ze dne 29.4.2010: http://www.rozhlas.cz/zpravy/volby/_zprava/726303
Vodní doprava na Labi
Vodní cesty v Česku
Záměry dopravních staveb v Česku
Záměry vodních staveb
Dopravní stavby v okrese Pardubice
Doprava v Přelouči
Dopravní stavby v Pardubicích
Vodní dopravní stavby v Pardubickém kraji | {
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// Copyright 2014 The Chromium Authors. All rights reserved.
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.
/**
* Icon of the video player.
* TODO(yoshiki): Consider providing an exact size icon, instead of relying
* on downsampling by ash.
*
* @type {string}
* @const
*/
var ICON_IMAGE = 'images/icon/video-player-64.png';
/**
* Configuration of the video player panel.
* @type {Object}
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var windowCreateOptions = {
frame: 'none',
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/**
* Backgound object. This is necessary for AppWindowWrapper.
* @type {BackgroundBase}
*/
var background = new BackgroundBase();
/**
* Queue to serialize initialization.
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var initializeQueue = new AsyncUtil.Queue();
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initializeQueue.run(function(fulfill) {
chrome.fileManagerPrivate.getStrings(function(stringData) {
loadTimeData.data = stringData;
fulfill();
}.wrap());
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// Initializes the volume manager. This needs for isolated entries.
initializeQueue.run(function(fulfill) {
VolumeManager.getInstance(fulfill);
}.wrap());
// Registers the handlers.
chrome.app.runtime.onLaunched.addListener(onLaunched);
/**
* Called when an app is launched.
* @param {Object} launchData Launch data.
*/
function onLaunched(launchData) {
if (!launchData || !launchData.items || launchData.items.length == 0)
return;
var playlist = {};
initializeQueue.run(function(fulfill) {
var isolatedEntries = launchData.items.map(function(item) {
return item.entry;
}.wrap());
chrome.fileManagerPrivate.resolveIsolatedEntries(isolatedEntries,
function(externalEntries) {
var urls = util.entriesToURLs(externalEntries);
playlist = {items: urls, position: 0};
fulfill();
}.wrap());
}.wrap());
initializeQueue.run(function(fulfill) {
openVideoPlayerWindow(playlist, false);
fulfill();
}.wrap());
}
var generateWindowId = (function() {
var seq = 0;
return function() {
return 'VIDEO_PLAYER_APP_' + seq++;
}.wrap();
}.wrap())();
/**
* Opens player window.
* @param {Object} playlist List of videos to play and index to start playing.
* @param {boolean} reopen True if reopen, false otherwise.
* @return {Promise} Promise to be fulfilled on success, or rejected on error.
*/
function openVideoPlayerWindow(playlist, reopen) {
var items = playlist.items;
var position = playlist.position;
var startUrl = (position < items.length) ? items[position] : '';
var windowId = null;
return new Promise(function(fulfill, reject) {
util.URLsToEntries(items).then(function(result) {
fulfill(result.entries);
}.wrap()).catch(reject);
}.wrap()).then(function(entries) {
if (entries.length === 0)
return Promise.reject('No file to open.');
// Adjusts the position to start playing.
var maybePosition = util.entriesToURLs(entries).indexOf(startUrl);
if (maybePosition !== -1)
position = maybePosition;
windowId = generateWindowId();
// Opens the video player window.
return new Promise(function(fulfill, reject) {
var urls = util.entriesToURLs(entries);
var videoPlayer = new AppWindowWrapper('video_player.html',
windowId,
windowCreateOptions);
videoPlayer.launch(
{items: urls, position: position},
reopen,
fulfill.bind(null, videoPlayer));
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{"url":"http:\/\/chiesaevangelicapresbiteriana.it\/isbo\/matlab-plotting-curl.html","text":"Plot vc(t) using MATLAB Fig 1 This circuit is excited by a DC (constant) voltage source, an AC (sinusoidal) voltage source, and an AC current source of different frequency. Find the values of the following double FOR loops: 2-D height plot Draw a 2-D function in the following domain space: x between [-3, 3], with step size of 0. Tools like herald use the Telegram Bot API to interact with channels through bots. If the eld is. Therefore, we have included it in the Matlab Option Tree tarball. streamribbon(vertices,X,Y,Z,cav,speed) assumes precomputed streamline vertices, curl angular velocity, and flow speed. 75 Object-Oriented Programming. A simple example showing how to plot a vector field (quiver) with matplotlib. 9\u00b0 less for the incline curl. Make sure the Matlab binary path is added to $PATH. Vector of dimensions, specified as a positive integer vector. For instance you could add a path name to Matlab's internal path and run fname. MATLAB can evaluate and plot most of the common vector calculus operations that we have previously discussed. The Mathematica Programs page for the User Sites Site on the USNA Website. Both the mean and maximum elbow flexion moments were significantly higher for the incline curl. A 3-D array, for example, uses three subscripts. Python Matplotlib is a plotting library for the Python programming language and its numerical mathematics extension NumPy. compute the Laplacian of functions from R^n to R. Matplotlib provides a function, streamplot, to create a plot of streamlines representing a vector field. I am trying to convert some code from Matlab into Python and having a few troubles. There are two main archives to be downloaded. aria2 is a lightweight multi-protocol & multi-source command-line download utility. You can visualize a vector field by plotting vectors on a regular grid, by plotting a selection of streamlines, or by using a gradient color scheme to illustrate vector and streamline densities.$\\begingroup$Nope there isn't a matlab stackexchange. 0: FELICITY now includes a stand-alone 3-D tetrahedral mesh generator for iso-surfaces. A bot is a Telegram entity which can send messages to a channel and manage them. This is convenient for interactive work, but for programming it is recommended that the namespaces be kept separate, e. The program performs plotting of functions, matrix manipulations. curl, we \ufb01nd that the vorticity vector is identically zero. \" This tab may be used with the. While it runs, the Simulink model uses a Level-2 MATLAB S-function to redraw the position and orientation of every stone in play. curl() to calculate the curl of a vector field. where [m,n,p] = size(U). MATLAB's ability to evaluate functions of vectors elementwise, for example sin(2*pi*x), is incredibly useful for plotting functions. When displaced to an initial angle and released, the pendulum will swing back and forth with periodic motion. While there are examples of making changes in which one method is clearly preferred, for the most part the choice is a matter of personal comfort. S-1 Supporting Information Quantitative Analysis of Nanorod Aggregation and Morphology from Scanning Electron Micrographs using SEMseg Rashad Baiyasi,a Miranda J. I am trying to convert some code from Matlab into Python and having a few troubles. gnuplot is a command line driven graph plotter tools for us to generate graphs. Description. Shibboleth Authentication Request Note: Since your browser does not support JavaScript, you must press the Continue button once to proceed to the authentication service. Bibliography Includes bibliographical references and index. Vector Magnetic Potential Page 2 According to the curl de nition we have made, r r A = 0 and we have satis ed Maxwell's equations. Then, if we have a grid like the one above, we can systematically pick points on the grid at which to plot the corresponding vector. The 3D version of plot, takes three vectors, one for x, one for y, and one for z surf(Z) creates a 3D shaded surface, where the column index is the x value and the row index is the y value. Matlab Code Example offers you codes for all your requirements; it can be your Matlab projects, assignments or thesis etc. The following files were compiled for MATLAB version 5. 6(b) is a least-squares fit of eqn (1) to. Either way, without a little creative storytelling, a few unexpected twists to the plot, or breaks in the tail, there is no story to tell. Anyway I used my expression for##\\vec{B}##, used the curl function in matlab, multiplied by the appropriate constant, took the cross product, plotted the vector field and got this. But all of this in MATLAB. Create a script file and type the following code \u2212 x = [0:5:100]; y = x; plot(x, y) When you run the file, MATLAB displays the following plot \u2212 Let us take one more example to plot the function y = x 2. This submission contains VERT2LCON and LCON2VERT, which will find the linear constraints defining a bounded polyhedron in R^n, given its vertices, or vice versa. ^2); plot(x,y) and observe what happens. (d) Plot a contour of the vorticity eld for the third case above using the Matlab \ufb01contourf\ufb02 function; add a color bar as well. The only thing that needs to be in the eval('') is the MATLAB commands for plotting, which is typically just one line of code anyway. curl(V,X) returns the curl of the vector field V with respect to the vector X. You can also plot a vector field from a list of vectors as opposed to a mapping. Matlab GUI for Elementary Flows as an Educational Tool Gabriel A. Download OSL. Toggle or set the \"hold\" state of the plotting engine which determines whether new graphic objects are added to the plot or replace the existing objects. Plotting the full data set first can help you select. 3 for finding r(t), the distance, and plotting the trajectory. Each element of vecdim represents a dimension of the input array X. The facilities include built-in functions for plotting wire-frame objects, space-curves, shaded surfaces. For methods \"wget\"and \"curl\"a system call is made to the tool given by method, and the respective program must be installed on your system and be in the search path for executables. Here is when I am analyzing the noissin signal in Matlab Wavelet Toolbox. m 14,969 lineintv. It has syntactical similarities to C, C++, and Java. 5 2 Vector Calculus Examples Using MATLAB MATLAB can evaluate and plot most of the common vector calculus operations that we have previously discussed. X and Y must have the same number of elements, as if produced by meshgrid. 9\u00b0 for the standing bicep curl and 35. C = imfuse(A,B) creates a composite image from two images, A and B. See Examples for 13. A lagged difference is defined by:. Spirals by Polar Equations top Archimedean Spiral top You can make a spiral by two motions of a point: There is a uniform motion in a fixed direction and a motion in a circle with constant speed. For example: A Galilean transformation for the linear wave equation is. Plotting Symbolic Function. Text in R is represented by character vectors. In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to. The package has been tested on Ubuntu 16 where it works. The following two techniques, illustrated by examples, are probably the most useful ones. To exemplify visualization of scalar and vector \ufb01elds with various tools, we use a common set of examples. Plotting with Matlab plot. If you are using Matlab, you must use the same architecture for Cantera and Matlab. Use InfluxDB to capture, analyze, and store millions of points per second and much more. Anomalies are available. It should be noted that newer versions of GNU Plot are available; however, we found that only 4. An Introduction to MATLAB A Session on MATLAB The Operations *, \/ , and ^ Defining and Plotting Functions in MATLAB 3-Dimensional Plotting M-files Loops and Iterations in MATLAB Conditional Statements in MATLAB Fourier Series in MATLAB Solving Differential Equations Concluding Remarks. As I said earlier, matlab's 'curl' function works only for cartesian coordinates, so you cannot use it directly on the u, v, w components you have described. The vector field V and the vector X are both three-dimensional. Gradient, Divergence, and Curl. MATLAB Central contributions by YT. 5 the model is evenly split between competing species and predator-prey. I let this program run for some weeks, and I now have hourly data from May, June and July. Galilean invariance. In the previous lesson, we evaluated line integrals of vector fields F along curves. com is really the right place to pay a visit to!. The contour plot of Fig. Mathematica Programs. streamslice For plotting streamlines on the cross-sections streamribbon For the display of flow with information about curl streamtube For the display of flow with information about div. pdf), Text File (. The fact that both calculations gave such complicated exact answers, made me draw the two 3D plots below that at least confirm that the signs of the two line integrals are correct (positive: field and line element vectors tend to be \"more parallel\" along the curve; negative: field and line element vectors tend to be \"more anti-parallel\" along. The package has been tested on Ubuntu 16 where it works. This leads to a small but significant ~6 \u00c5 difference in average ligand-shell thickness between the two faces of the sheet, as measured by Grazing-incidence X-ray small-angle scattering (GISAXS) at beamline Sector 8-ID-E, the Advanced Photon Source (see the paper by Jiang et al, for. Please be kind to other ERDDAP users: run just one script or curl command at a time. We begin by parametrizing the circle. gradient(f,v) finds the gradient vector of the scalar function f with respect to vector v in Cartesian coordinates. MATLAB is yet another application and programming language I've added to my skill set this year. In computer programs this function is called atan2(y,x) and gives angles in (-pi,pi]. coolwarm, rstride=1, cstride=1) # add the parametric curve. The column and row indices of Z are the x and y coordinates in the plane, respectively. Create a script file and type the following code \u2212 x = [0:5:100]; y = x; plot(x, y) When you run the file, MATLAB displays the following plot \u2212 Let us take one more example to plot the function y = x 2. The Mathematica Programs page for the User Sites Site on the USNA Website. However, as you mentioned in your question that you need to plot several sets of data in the same figure but with different axes you cannot use this solution unless you scale your data to have the same plotting amplitudes for each data set. Plotting the full data set first can help you select. Result: Installation. When used in conjunction with the curl function, streamribbon is useful for displaying the curl angular velocity of a vector field. 0: FELICITY now includes a stand-alone 3-D tetrahedral mesh generator for iso-surfaces. MATLAB Graphics User Guide. It has syntactical similarities to C, C++, and Java. U \uc640 V \uc5d0 \ub300\ud55c \uc88c\ud45c\ub97c \uc815\uc758\ud558\ub294 \ubc30\uc5f4 X \uc640 Y \ub294 \ub2e8\uc870(Monotonic) \ubc30\uc5f4\uc774\uc5b4\uc57c \ud558\ub098, \uac04\uaca9\uc774 \uade0\uc77c\ud560 \ud544\uc694\ub294 \uc5c6\uc2b5\ub2c8\ub2e4. Function Used to Create mesh, surf Surface plot meshc, surfc Surface plot with contour plot beneath it meshz Surface plot with curtain plot (reference plane) pcolor Flat surface plot (value is proportional only to color). If r is now our distance, h ypotenuse we find. Symbolic Math in Matlab. Comment on your results. Books that make me think--about the world, about life, about myself, about others. matlab_cmd can be the name of a Matlab function, or a command string, exactly as you would type interactively. For monthly, plot composite maps of the 20CR ensemble spread (uncertainty). It reshapes and permutes a 2D matrix into a 3D cube. A d escription of Curl. In case that you have to have guidance on solving equations or arithmetic, Rational-equations. A simple pendulum consists of a mass m hanging from a string of length L and fixed at a pivot point P. At a point$(x,y)$, we plot the value of$\\dlvf(x,y)$as a vector with tail anchored at$(x,y)$, such as in the following figure. An example of plotting a continuous data flow is shown here: The package description provides more information how to customize the plot. People are working on it. We have world class developers and experts with us working on Matlab for the past 10 years. Matlab has a powerful symbolic math ability. For methods \"wget\"and \"curl\"a system call is made to the tool given by method, and the respective program must be installed on your system and be in the search path for executables. mask3 applies a mask to all levels of 3D matrix corresponding to a 2D mask. syms x y z real F = [ cos(x+2*y), sin(x-2*y) ]; vectorfield(F,-2:.$\\endgroup$\u2013 Steve Hatcher Aug 11 '14 at 7:40. ; rect2cube is the complement of cube2rect. Note that this is quite different from the built-in MATLAB function curl, which won't help you much. 15: 10\/1\/18: 10\/10\/18 Review - Momentum equation - differential form Read Ch. 4 Curl Curl describes the rotation (curling) of the eld. In Cartesian In Cylindrical In Spherical. In fact norm of the difference = 0. A vector is a one-dimensional array of numbers. Please be kind to other ERDDAP users: run just one script or curl command at a time. I meant to put the negative charges on the y-axis. cube2rect reshapes a 3D matrix for use with standard Matlab functions. Let us plot the simple function y = x for the range of values for x from 0 to 100, with an increment of 5. If nothing happens, download GitHub Desktop and try again. The program performs plotting of functions, matrix manipulations. 5 2 Vector Calculus Examples Using MATLAB MATLAB can evaluate and plot most of the common vector calculus operations that we have previously discussed. Creating Stream Particle Animations. curl() to calculate the curl of a vector field. Then, users are encouraged to visualize the curl of the vector field plot (Fig ure 4) before the plot is generated (Figure 5). Matlab May 2, 2010 Displaying Curl with Stream Ribbons 2019-05-12T01:50:52+00:00 Examples What Stream Ribbons Can Show Stream ribbons illustrate direction of flow, similar to stream lines, but can also show rotation about the flow axis by twisting the ribbon-shaped flow line. Discover what MATLAB. The main purpuse of this is to write a two dimensional function which replicates the inflow parameters of a studied domain. However, as you mentioned in your question that you need to plot several sets of data in the same figure but with different axes you cannot use this solution unless you scale your data to have the same plotting amplitudes for each data set. MATLAB - Vector Dot Product - Dot product of two vectors a = (a1, a2, \u00e2 \u00a6, an) and b = (b1, b2, \u00e2 \u00a6, bn) is given by \u2212. Published with MATLAB\u00ae R2013b. You can visualize a vector field by plotting vectors on a regular grid, by plotting a selection of streamlines, or by using a gradient color scheme to illustrate vector and streamline densities. Note that this is quite different from the built-in MATLAB function curl, which won't help you much. This applet allows you to visualize vector fields and their divergence and curl, as well as work done by a field. , null_A' * null_A = eye(4). (matplotlib. We have world class developers and experts with us working on Matlab for the past 10 years. As for multi line analysis, you can do all of that using MATLAB code. Learn the protocol to send this data to the server and to return a response to the client. However, as you mentioned in your question that you need to plot several sets of data in the same figure but with different axes you cannot use this solution unless you scale your data to have the same plotting amplitudes for each data set. m: % % Filename: example6. Curl, being the deviation of a sheet from its flat surface, is a prevalent occurrence in paper materials and often Hence, the Matlab\u00ae generated contour plots of Figs. Because there I used a hold on call. I don't think it is possible to show curl in this way, except in the rare cases that the curl is parallel to the flow or normal to the flow (would show direction only); if the twisting. Published with MATLAB\u00ae 7. Artinya adanya titik singularitas ini mengakibatkan Teorema Stokes yang menghubungkan integral lintasan dengan curl vektor tidak bisa diterapkan pada medan vektor yang bersangkutan. (b) Plot the velocity vectors as function of (x;y) using Matlab\u2019s quiver function. 12 FD-based MATLAB code \u2013 direct solution. This submission contains VERT2LCON and LCON2VERT, which will find the linear constraints defining a bounded polyhedron in R^n, given its vertices, or vice versa. Awarded to Dhanashree Mohite on 04 May 2019. Neither of them gives me nonzero plot. If the field is curl- and divergence-free, it\u2019s a laplacian (harmonic) vector field. By plotting this field of arrows, we see that the vector field$\\dlvf(x,y) = (y, -x)$appears to rotate in a clockwise direction. MATLAB (short for MATrix LABoratory) was invented by Cleve Moler in the 1970's and was upgraded and is now sold by The Mathworks company. When used in conjunction with the curl function, streamribbon is useful for displaying the curl angular velocity of a vector field. Example 2: Plot exp(sin(x)) on the. If potential cannot verify that V is a gradient field, it returns NaN. View Terence Barrett\u2019s profile on LinkedIn, the world's largest professional community. Quiver Plots in Python How to make a quiver plot in Python. October 2008 Online only Revised for MATLAB 7. The way one being trying to get the curl of matrices. curl(V,X) returns the curl of the vector field V with respect to the vector X. xml'); This should take around 30 seconds for larger data files (500,000 to 1,000,000 voxels with a few substrates, and around 250,000 cells). curl, we \ufb01nd that the vorticity vector is identically zero. Plotting a vector field: quiver\u00b6. A-10 Appendix 3: List of MATLAB Exercises ME 2. Creating Stream Particle Animations. The WFDB Toolbox for MATLAB is a collection of applications for reading, writing, manipulating, and plotting PhysioBank data, implemented as functions in MATLAB. In the text that follows, any line that starts with two greater than signs (>>) is used to denote the matlab command line. Author: Juan Carlos Ponce Campuzano. The arrays X , Y define the coordinates for U , V and must be monotonic and 2-D plaid (as if produced by meshgrid ). However, you can still use the PLOT and YYAXIS functions in your MATLAB Analysis app and use the suggestion as above. gnuplot is a command line driven graph plotter tools for us to generate graphs. Examining trend with autocorrelation in time series data. Find more Mathematics widgets in Wolfram|Alpha. 2020-06-21 \u2013 Matlab + Octave unit tests quick-start 2020-06-20 \u2013 pre-commit check website markdown internal links 2020-06-20 \u2013 Global PEP8 Git pre-commit check hook. (3b) Find the flux integral by using Green's theorem. In fact norm of the difference = 0. Make sure the Matlab binary path is added to$PATH. When the first argument to genplot is a two dimensional vector field, genplot will plot the vector field for ranges given in the next two arguments. ch6_cycloid3dmApp. This can be seen in the abundance of scientific tooling written in Julia, such as the state-of-the-art differential equations ecosystem (DifferentialEquations. Due to concern about the COVID-19 pandemic, all training sessions are pure Zoom online events from 9:00AM to 11:00AM. X , Y , Z , cav , and speed are 3-D arrays. 01: MATLAB M-FILE FOR PLOTTING TRUNCATED FOURIER SERIES AND ITS SPECTRA MATLAB M-File example6. streamribbon(vertices,X,Y,Z,cav,speed) assumes precomputed streamline vertices, curl angular velocity, and flow speed. Hence, the Matlab\u00ae generated contour plots of Figs. In order to take a look at the trend of time series data, we first need to remove the seasonality. For example, to plot ve ctor field defined by sin(x) and cos(y) in the region x \u00ce (0, )p and y \u00ce (0, )p , a matrix needs to be. Provide details and share your research! But avoid \u2026 Asking for help, clarification, or responding to other answers. Running CURL - Post method to Thingspeak Receiving multiple math errors while plotting data with Matlab AQI Calculator Example. We begin with the planar case. Select a Subset of Data to Plot. Compute the curl of A~ using Matlab curl command. The following two techniques, illustrated by examples, are probably the most useful ones. degrees in electrical engineering from the University of Belgrade, Belgrade, Yugoslavia, in 1988, 1992, and 1995, respectively. Animation may take some time to load. Neither of them gives me nonzero plot. But you WILL need the boundary conditions for Psi to solve this. 0 Mile Run (Min). Active 4 years, 6 months ago. In Matlab I can just plug it in as follows: [curlJ_x,curlJ_y,curlJ_z]=curl(Jx,Jy,Jz); And I do not believe Python has anything like that. 5 Evaluation of Capacitances of Capacitors and Transmission Lines. When you open the documentation for quiver, have a look on the left-hand side of the page: there you will see the Contents. From 1996 to 1998, he was an Assistant Professor in the Department of Electrical Engineering at the University of Belgrade, and before that, from 1989 to 1996, a. streamribbon(vertices,X,Y,Z,cav,speed) assumes precomputed streamline vertices, curl angular velocity, and flow speed. Simpli\ufb01ed derivation of delta function identities 7 x y x Figure 2: The \ufb01gures on the left derive from (7),and show \u03b4 representations of ascending derivatives of \u03b4(y \u2212 x). Making statements based on opinion; back them up with references or personal experience. If the number of matrix columns equals the vector length, then the function plots each matrix row versus the vector. Center for Subsurface Sensing and Imaging Systems (CenSSIS) A National Science Foundation Engineering Research Center Research and Industrial Collaboration Conference November 13-15, 2000 This work was supported in part by the Engineering Research Center Program of the National ScienceFoundation under award number EEC-9986821. MATLAB Graphics User Guide. 685 x 685 10 0 10 0. m 15,048 elect. Before we evaluate the integral, let us plot the vector field F together with the unit circle. I am trying to convert some code from Matlab into Python and having a few troubles. Associated differential equation. The plot function in MATLAB usually takes two arguments, the first is the X values of the points to plot, and the second is the Y value of the points to plot. Plot the vector field together with C. A character vector is \u2014 you guessed it! \u2014 a vector consisting of characters. This table lists the various forms. C = imfuse(A,B) creates a composite image from two images, A and B. Principios de \u00d3ptica y F\u00edsica Moderna Jos\u00e9 Luis Escamilla Reyes 1 D. The WFDB Toolbox for MATLAB is a collection of applications for reading, writing, manipulating, and plotting PhysioBank data, implemented as functions in MATLAB. integral of contour plot. Problem 1: See Examples for 12. In your description, Rizwana, you haven't made it clear what components of velocity the variables u, v, and w represent. m: % % Filename: example6. But you WILL need the boundary conditions for Psi to solve this. In particular, the Edit Plot button, which has the icon \u2b01, enables you to double-click or right-click virtually any graphical object to change its attributes. Plus easy-to-understand solutions written by experts for thousands of other textbooks. Lagged differencing is a simple transformation method that can be used to remove the seasonal component of the series. To access gradient, divergence, and curl first input <') is the MATLAB commands for plotting, which is typically just one line of code anyway. In case that you have to have guidance on solving equations or arithmetic, Rational-equations. nfluxDB open source time series database, purpose-built by InfluxData for monitoring metrics and events, provides real-time visibility into stacks, sensors, and systems. org) This means you have to have a working python installation, including development headers. Free simple curl download - simple curl script - Top 4 Download - Top4Download. There are different ways of plotting in MATLAB. The arrays X , Y define the coordinates for U , V and must be monotonic and 2-D plaid (as if produced by meshgrid ). How do i get the radiation pattern polar plot Learn more about radiation pattern, polar plot MATLAB, MATLAB and Simulink Student Suite. Thanks, Elham Hi In the end I've figured out, after many trials, what was the problem. the plot between distance and time is written using plot and hold on command. When you click the Go button, the MATLAB App starts the Simulink model using the sim command. \" This tab may be used with the. ; rect2cube is the complement of cube2rect. Several functions will be used with the emphesas on a couple of important commands such as Meshgrid and camlight, zoom, etc. So in order to get historical data (a time series essentially), I created a data logger. it provides a description of curl tying together the mathematical operation and co ncept with the physical meaning (Figure 3). TUTORIAL [folder ME2 13(GUI) on IR] ME 2. This is convenient for interactive work, but for programming it is recommended that the namespaces be kept separate, e. MATLAB (short for MATrix LABoratory) was invented by Cleve Moler in the 1970's and was upgraded and is now sold by The Mathworks company. Sometimes however, it is not desired to give each curve its own individual legend, but rather group a number of curves into a common legend entry. A blog about PHP, HTML, CSS, JavaScript, jQuery programming and coding, Internet technology from Jiansen Lu. Maybe we crave more excitement, maybe we want to forget the fact that our electric bill tripled this month. TUTORIAL (ME2 12. Learn more about curl. Plotting 3 variables of a single function. When using upward an Y-axis (mode axis xy , for function representation), Matlab re-orders the Y vector, so the Y-axis is wrong. $\\begingroup$ Yes, by using the plane wave equation, I would like to have a vector field showing that if we take the curl of E, we get a flipped out of phase cos wave. Use the squeeze function to obtain a 2D array. In particular, the Edit Plot button, which has the icon \u2b01, enables you to double-click or right-click virtually any graphical object to change its attributes. matlab_cmd can be the name of a Matlab function, or a command string, exactly as you would type interactively. We include information about both freely-available and licensed (commercial) software that can be used with netCDF data. Seaborn provides an API on top of Matplotlib that offers sane choices for plot style and color defaults, defines simple high-level functions for common statistical plot types, and integrates with the. [curlx,curly,curlz,cav] = curl(X,Y,Z,U,V,W)computes the curl and angular velocity perpendicular to the flow (in radians per time unit) of a 3-D vector field U, V, W. txt or javaclasspath. 32-bit: Download the most recent \u201cWindows x86 MSI Installer\u201d for. Create a script file and type the following code \u2212 x = [0:5:100]; y = x; plot(x, y) When you run the file, MATLAB displays the following plot \u2212 Let us take one more example to plot the function y = x 2. This is convenient for interactive work, but for programming it is recommended that the namespaces be kept separate, e. a bit difficult. Both the mean and maximum elbow flexion moments were significantly higher for the incline curl. MATLAB Central contributions by Sven Koerner. Both the mean and maximum elbow flexion moments were significantly higher for the incline curl. The graph of the vector-valued function then shows each of these parametric plots as their orthogonal projections onto each of the coordinate planes. The 3D version of plot, takes three vectors, one for x, one for y, and one for z surf(Z) creates a 3D shaded surface, where the column index is the x value and the row index is the y value. the equation properties remain unchanged under a Galilean transformation. Artinya adanya titik singularitas ini mengakibatkan Teorema Stokes yang menghubungkan integral lintasan dengan curl vektor tidak bisa diterapkan pada medan vektor yang bersangkutan. Comment on your results. Graph them upon arrival. The following example illustrates this technique. The contour plot of Fig. coolwarm, rstride=1, cstride=1) # add the parametric curve. Books that make me think--about the world, about life, about myself, about others. 3 had consistent performance (with Octave). Our interactive demo allows you to enter any function you like for $$g(x,y)$$ and $$h(x,y)$$. 3d Streamline Matlab. A simple pendulum consists of a mass m hanging from a string of length L and fixed at a pivot point P. When displaced to an initial angle and released, the pendulum will swing back and forth with periodic motion. So in order to get historical data (a time series essentially), I created a data logger. (If you prefer to use the MATLAB built-in function for plotting vector fields, see the help for quiver. Advanced Engineering Mathematics with MATLAB. 0: Fixed a few. This variable will form an axis of the plot, more then likely the horizontal or x-axis. An Introduction to MATLAB A Session on MATLAB The Operations *, \/ , and ^ Defining and Plotting Functions in MATLAB 3-Dimensional Plotting M-files Loops and Iterations in MATLAB Conditional Statements in MATLAB Fourier Series in MATLAB Solving Differential Equations Concluding Remarks. I am able to plot vector fields using a couple nested lists, but this makes adjusting the density of arrows, etc. learn more View Options. Here you should be sure to be the super user. 5 2 Vector Calculus Examples Using MATLAB MATLAB can evaluate and plot most of the common vector calculus operations that we have previously discussed. ch6_parabolaApp. Recently i have been asked to compute curl of three velocity components and plot i. Terence has 9 jobs listed on their profile. txt or javaclasspath. While it runs, the Simulink model uses a Level-2 MATLAB S-function to redraw the position and orientation of every stone in play. Workshop Vector eld interpolation and decomposition Grady Wright In this problem you will look at interpolating and decomposing a vector eld u that is tangent to the sphere using the RBF method described in the lecture notes (which comes from the paper [1]). Problem 1: See Examples for 12. openpowernet. They will block all other activity on the Rprocess until they complete: this may make a GUI unresponsive. Matlab : Deep Learning Toolbox : FeedForwardnet : 1 Hidden Layer : Example : (x-a)(x-b)(x-c) Matlab : Deep Learning Toolbox : Patternnet : 1 Hidden Layer : Example : (x-a)(x-b)(x-c) Matlab : Deep Learning Toolbox : FeedForwardnet : Example : Digit Classifier. Show transcribed image text. Find the values of the following double FOR loops: 2-D height plot Draw a 2-D function in the following domain space: x between [-3, 3], with step size of 0. 14 coneplot Plot velocity vectors as cones in 3-D vector field 1. I let this program run for some weeks, and I now have hourly data from May, June and July. pi help display help text at command line. Recently i have been asked to compute curl of three velocity components and plot its contour w. Visit for free, full and secured software\u2019s. txt), slightly modified by me to include an optional classname input arg. Then, users are encouraged to visualize the curl of the vector field plot (Fig ure 4) before the plot is generated (Figure 5). The common graphs that we are looking forward to present the resource performance per seconds, hours, days, weeks or months are usually plot graphs, which it consist of lines and dots. U m v i Rsn V 11. TUTORIAL (ME2 12. 9 for a Lagrange multiplier example. Before we evaluate the integral, let us plot the vector field F together with the unit circle. MATLAB Tutorial; MATLAB Bode Plots; Algebraic Transfer Functions. Each row of the table represents an iris flower, including its species and dimensions of its botanical parts, sepal and petal, in centimeters. $\\endgroup$ \u2013 Steve Hatcher Aug 11 '14 at 7:40. m 14,860 divergence. Searles,b Qingfeng Zhang,b,c Stephan Link,a,b,c Christy F. A vector is a one-dimensional array of numbers. MATLAB Tutorial; MATLAB Bode Plots; Algebraic Transfer Functions. 000000000000000. People are working on it. On Windows, to get to a command line, click on \"Start\" and type \"cmd\" into the search text field. Plot the vector field A and its divergence. Rankine Cycle: The Rankine cycle is a model used to predict the performance of steam turbine systems. , no matrix multiplication lMATLAB's class functionality enables users to create their own objects lThe tensorclass extends the MDA capabilities to. Refer to Matlab's help page for curl function, it's a good reference for both computing vorticity and slicing a 3D array. Div, grad and curl in polar coordinates We will need to express the operators grad, div and curl in terms of polar coordinates. And then integrating that gets us to our E-M plot. The facilities include built-in functions for plotting wire-frame objects, space-curves, shaded surfaces. counter variable i8s initiated as 1. You can also plot a vector field from a list of vectors as opposed to a mapping. Originally published at UCI Machine Learning Repository: Iris Data Set, this small dataset from 1936 is often used for testing out machine learning algorithms and visualizations (for example, Scatter Plot). 2020-06-21 \u2013 Matlab + Octave unit tests quick-start 2020-06-20 \u2013 pre-commit check website markdown internal links 2020-06-20 \u2013 Global PEP8 Git pre-commit check hook. It should be noted that newer versions of GNU Plot are available; however, we found that only 4. t circumferential angle along x-axis and radius along y-axis. Department of Mathematics. A stream particle animation is useful for visualizing the flow direction and speed of a vector field. But you WILL need the boundary conditions for Psi to solve this. Active 4 years, 6 months ago. 2) Plotting the corresponding T-S and H-S plots for the given set of inputs. Workshop Vector eld interpolation and decomposition Grady Wright In this problem you will look at interpolating and decomposing a vector eld u that is tangent to the sphere using the RBF method described in the lecture notes (which comes from the paper [1]). ContentsSet the StageChanging the Order of IntegrationSingularitiesCan You Take. McCarthy,b Emily K. You can also plot a vector field from a list of vectors as opposed to a mapping. Landes*a,b,c,d aDepartment of Electrical and Computer Engineering, Rice University, Houston, Texas 77005,. doc display HTML docs in the Help browser. If the eld is. Compression levels -3, -6, and -9 are also improved. The arrays X, Y, and Z, which define the coordinates for U, V, and W, must be monotonic, but do not need to be uniformly spaced. Introduction to excel; Excel for Numerical Integration; Visual Basic; Visual Basic Syntax; Visual Basic Examples; Regression; VBA Control Button; Excel - Formulas and Plotting; excel 2007 solution of Laplace's equation; Amortization; Excel - Formulas and Plotting. Notaro\u02c7s (from now on, referred to as \"the book\"), provides an extremely large and comprehensive collection of MATLAB computer exercises and projects, strongly coupled to the book material, both the theory and the R. ch6_cycloid2dApp. Even simple three-dimensional vector fields like the linear ones here can produce quite complicated pictures. Lagged differencing is a simple transformation method that can be used to remove the seasonal component of the series. There are different ways of plotting in MATLAB. streamslice For plotting streamlines on the cross-sections streamribbon For the display of flow with information about curl streamtube For the display of flow with information about div. Hi I am trying to find the curl of a 3D vector field using the Matlab function curl. txt) or read online for free. Description. In this paper, we present an e cient implementation of problem (4){(5) in Matlab leading to the setup of system (7). 4 mm to any other point. The following links will help you review linear algebra. To determine the gradient of f(x,y) = (x^2+y^2)^(1\/2): Grad[Sqrt[x^2+y^2]] To plot the contours of f: To make the picture smoother: Divergence. GRANDVkLII'Y Stai i. Compares free fall with and without drag. The Plotly JSON chart schema is the complete, human-readable text format that describes all physical attributes of any scientific chart. ch6_cycloid3dmApp. X , Y , Z , cav , and speed are 3-D arrays. Other readers will always be interested in your opinion of the books you've read. ch6_cycloid2dApp. Several functions will be used with the emphesas on a couple of important commands such as Meshgrid and camlight, zoom, etc. The PIVMat Toolbox for Matlab contains a set of command-line functions to import, post-process and analyse 2- and 3-components vector fields from PIV (particle image velocimetry), stereo-PIV, DIC (digital image correlation) SS (synthetic schlieren) or BOS (background-oriented schlieren) applications. Gradient, Divergence, and Curl. Finally, the flexion angle at which the maximum flexion moment occurred was 71. For negative x-values, the graph gets very close to the x-axis, but doesn't touch it. It was also used to study the performance of reciprocating steam engines. To read the data, just do this: MCDS = read_MultiCellDS_xml ('sample. Anomalies are available. This is an exponential growth curve, where the y-value increases and the slope of the curve increases as x increases. compute the Laplacian of functions from R^n to R. Anomalies are available. Appendix A shows two examples on how to write new functions to plot vector fields in 2D and 3D in cylindrical coordinates. help me on plotting of curl. how to compute curl for velocity components expressed interms of spherical coordinates(r,alpha,beta). (a)For any two-dimensional scalar eld f (expressed as a function of r and ) we have r(f) = grad(f) = f r e r + r 1f e : (b)For any 2-dimensional vector eld u = me r + pe (where m and p are expressed as functions of r. OSL consists of a set of repositories available from the OHBA Analysis on GitHub: osl-core and ohba-external contain the main Matlab scripts, and external dependencies; The toolboxes HMM-MAR, GLEAN and MEG-ROI-nets provide additional analysis tools. When the page first loads, these functions are set to. When making matlab plots with multiple curves, it is usually nice to include a legend, describing the nature of the different curves. Design and simulate a fully functioning 8 bit computer , instruction set , assembly language , assembler , debugger. Matlab Code Example offers you codes for all your requirements; it can be your Matlab projects, assignments or thesis etc. Matlab : Deep Learning Toolbox : FeedForwardnet : 1 Hidden Layer : Example : (x-a)(x-b)(x-c) Matlab : Deep Learning Toolbox : Patternnet : 1 Hidden Layer : Example : (x-a)(x-b)(x-c) Matlab : Deep Learning Toolbox : FeedForwardnet : Example : Digit Classifier. 4 through 6 are based on approximately 50,000 input data values. elbow was 11. It reshapes and permutes a 2D matrix into a 3D cube. 3 for finding r(t), the distance, and plotting the trajectory. 15 curl Compute curl and angular velocity of vector field 1. MATLAB\u00ae 7 3-D Visualization. Compares free fall with and without drag. Concerning the study setup, we first compute only the magnetic fields problem. $\\endgroup$ \u2013 Fixed Point Mar 5 '13 at 9:15. The arrays X, Y, and Z, which define the coordinates for U, V, and W, must be monotonic, but do not need to be uniformly spaced. Gradient, Divergence, and Curl. I don't think it is possible to show curl in this way, except in the rare cases that the curl is parallel to the flow or normal to the flow (would show direction only); if the twisting. We include information about both freely-available and licensed (commercial) software that can be used with netCDF data. This MATLAB function returns the sample skewness of X. In case that you have to have guidance on solving equations or arithmetic, Rational-equations. Open MATLAB. ; rect2cube is the complement of cube2rect. MATLAB Central contributions by Sven Koerner. Image Dimension: 685 x 685 File size: 89. This can be seen in the abundance of scientific tooling written in Julia, such as the state-of-the-art differential equations ecosystem (DifferentialEquations. Tips on plotting additional derived quantities (such as velocity gradient) In order to plot additional derived quantities such as the gradient, divergence or curl of the velocity field, you may find these quantities in a pop-up window in CFD post. 2-D \u21d2 v z = 0 3. ch6_projectileApp. I meant to put the negative charges on the y-axis. 14 RG-55\/U coaxial cable. txt), slightly modified by me to include an optional classname input arg. The plot is formed by joining adjacent points with straight lines. The arrays X, Y, and Z, which define the coordinates for U, V, and W, must be monotonic, but do not need to be uniformly spaced. coolwarm, rstride=1, cstride=1) # add the parametric curve. mask3 applies a mask to all levels of 3D matrix corresponding to a 2D mask. If A is a multidimensional array, then sum(A) operates along the first array dimension whose size does not equal 1, treating the elements as vectors. To this point (in both Calculus I and Calculus II) we've looked almost exclusively at functions in the form $$y = f\\left( x \\right)$$ or $$x = h\\left( y \\right)$$ and almost all of the formulas that we've developed require that functions be in one of these two forms. Toggle or set the \"hold\" state of the plotting engine which determines whether new graphic objects are added to the plot or replace the existing objects. You can learn a lot by removing semicolons in someone else's Mathematica code. However, you can still use the PLOT and YYAXIS functions in your MATLAB Analysis app and use the suggestion as above. In our recent post, Mike Hosea and I talked about adjusting both the absolute and relative tolerances for getting more accurate results when calculating a double integral. Compute the curl of A~ using Matlab curl command. Divergence is a measure of source or sink at a particular. HTML plots allow easy sharing of interactive data plots to any web browser. Expertly curated help for Advanced Engineering Mathematics with MATLAB. The semicolons after each line suppress printout. X, Y, and Z must have the same number of elements, as if produced by meshgrid. plot_surface(xv, yv, hv, cmap=cm. I grew up with MATLAB, where extracting a subset of a vector V was easy as feeding a vector of indices into a vector. We conclude with a brief overview of some general aspects relating to linear and nonlinear waves. The software also has an extra option to interface with the Maple software to allow for simplification of easy to complex math functions. Problem 1: See Examples for 12. I programmed a simple MATLAB script that queries the status of the bike stations once every hour using cURL, and saves the data into a text file. 9\u00b0 less for the incline curl. ContentsSet the StageChanging the Order of IntegrationSingularitiesCan You Take. verifying orthonormality, i. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. You can learn a lot by removing semicolons in someone else's Mathematica code. Show transcribed image text. The following links will help you review linear algebra. In plot the first variable is the \"x-axis. Are there simple technique or simple coding to remove or detect these fluctuations during training? What I've tried only is to reduce step size but I had to compensate that with increasing number of iterations. 4 Curl Curl describes the rotation (curling) of the eld. This table lists the various forms. $\\begingroup$ Yes, by using the plane wave equation, I would like to have a vector field showing that if we take the curl of E, we get a flipped out of phase cos wave. The following files were compiled for MATLAB version 5. txt) or read online for free. m 7,173 fluxint 14,926 grad. If Y is a scalar, then potential expands it into a vector of the same length as X with all elements equal. Plots parabolas with various curvatures. There are different ways of plotting in MATLAB. Quiver3 Matlab Quiver3 Matlab. In the world of computer programming, text often is referred to as a string. The plot is formed by joining adjacent points with straight lines. 64-bit: Download the most recent \u201cWindows X86-64 MSI Installer\u201d for Python X. The code to calculate the vector field curl is:. In this example, we will draw two graphs with the same function, but in second time, we will reduce the value of increment. This example shows how to combine a contour plot and a quiver plot using the hold function. When making matlab plots with multiple curves, it is usually nice to include a legend, describing the nature of the different curves. 2) Plotting the corresponding T-S and H-S plots for the given set of inputs. GCOS\/WGSP: Can plot composite maps of sea level pressure from plotting pages on monthly timescales. To work with MATLAB codes for solving the 1D heat equation, you should be familiar with some basic concepts from linear algebra and MATLAB programming. Spirals by Polar Equations top Archimedean Spiral top You can make a spiral by two motions of a point: There is a uniform motion in a fixed direction and a motion in a circle with constant speed. Creating Stream Particle Animations. ch6_cycloid2dApp. It reshapes and permutes a 2D matrix into a 3D cube. Expert Answer. Please I need Full steps of doing the cur of this field. This leads to a small but significant ~6 \u00c5 difference in average ligand-shell thickness between the two faces of the sheet, as measured by Grazing-incidence X-ray small-angle scattering (GISAXS) at beamline Sector 8-ID-E, the Advanced Photon Source (see the paper by Jiang et al, for. Note that this is quite different from the built-in MATLAB function curl, which won't help you much. X , Y , Z , cav , and speed are 3-D arrays. If potential cannot verify that V is a gradient field, it returns NaN. Let us plot the simple function y = x for the range of values for x from 0 to 100, with an increment of 5. TUTORIAL (ME2 12. That said, we think that, at least in principle, an open source implementation would be preferred. Go to python. Compares free fall with and without drag. 15 curl Compute curl and angular velocity of vector field 1. computes the directional derivative of a scalar field in the direction given by a vector. The vector field V and the vector X are both three-dimensional. Section 3-1 : Parametric Equations and Curves. You can visualize a vector field by plotting vectors on a regular grid, by plotting a selection of streamlines, or by using a gradient color scheme to illustrate vector and streamline densities. U \uc640 V \uc5d0 \ub300\ud55c \uc88c\ud45c\ub97c \uc815\uc758\ud558\ub294 \ubc30\uc5f4 X \uc640 Y \ub294 \ub2e8\uc870(Monotonic) \ubc30\uc5f4\uc774\uc5b4\uc57c \ud558\ub098, \uac04\uaca9\uc774 \uade0\uc77c\ud560 \ud544\uc694\ub294 \uc5c6\uc2b5\ub2c8\ub2e4. You can also plot a vector field from a list of vectors as opposed to a mapping. NOAA PSL: Can plot composite maps from plotting pages on monthly, daily and sub-daily timescales for 20CR. 5 Evaluation of Capacitances of Capacitors and Transmission Lines. Also updated TIGER meshing interface, as well as the meshing tutorials on the Google-Code page. Make sure Matlab is installed in the same machine running your web server. streamribbon(X,Y,Z,U,V,W,startx,starty,startz) draws stream ribbons from vector volume data U, V, W. 2020-06-21 \u2013 Matlab + Octave unit tests quick-start 2020-06-20 \u2013 pre-commit check website markdown internal links 2020-06-20 \u2013 Global PEP8 Git pre-commit check hook. nfluxDB open source time series database, purpose-built by InfluxData for monitoring metrics and events, provides real-time visibility into stacks, sensors, and systems. When using upward an Y-axis (mode axis xy , for function representation), Matlab re-orders the Y vector, so the Y-axis is wrong. If the number of matrix rows equals the vector length, then the plot function plots each matrix column versus the vector. Type the variable name, in this case 'A', and the equals sign ( = ). Use MathJax to format equations. Thingiverse-- Free 3D models, primarily intended for 3D printing. Gradient, Divergence, and Curl. If Y is a scalar, then potential expands it into a vector of the same length as X with all elements equal to Y. In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to. Iam beginner to matlab. The currently opened section is MATLAB -> Graphics -> Vector Fields -> Functions, and you can also see some other functions for plotting vectors: these include feather, compass. The wind data set contains the arrays u , v , and w that specify the vector components and the arrays x , y , and z that specify the coordinates. 1 and are used in this course packet: PROGRAM FILES M-File Bytes animlaplace. Hence, the Matlab\u00ae generated contour plots of Figs. Use InfluxDB to capture, analyze, and store millions of points per second and much more. Workshop Vector eld interpolation and decomposition Grady Wright In this problem you will look at interpolating and decomposing a vector eld u that is tangent to the sphere using the RBF method described in the lecture notes (which comes from the paper [1]). streamribbon(X,Y,Z,U,V,W,startx,starty,startz) draws stream ribbons from vector volume data U, V, W. streamslice For plotting streamlines on the cross-sections streamribbon For the display of flow with information about curl streamtube For the display of flow with information about div. You can use sympy. MATLAB-Based Electromagnetics. Then, if we have a grid like the one above, we can systematically pick points on the grid at which to plot the corresponding vector. Animation of Radiation by Matlab in 3D. However, as you mentioned in your question that you need to plot several sets of data in the same figure but with different axes you cannot use this solution unless you scale your data to have the same plotting amplitudes for each data set. From The Divergence of a Vector Field and The Curl of a Vector Field pages we gave formulas for the divergence and for the curl of a vector field $\\mathbf{F}(x, y, z) = P(x, y, z) \\vec{i} + Q(x, y, z) \\vec{j} + R(x, y, z) \\vec{k}$ on $\\mathbb{R}^3$ given by the following formulas: (1). This makes liberal use of the functions sublist_indices() , lambda() , and several functions from the stringproc package: sequal() , eval_string() , and simplode(). Once you unzip and open the directory that you downloaded, run the installer. coolwarm, rstride=1, cstride=1) # add the parametric curve. Topic: Vectors. $\\begingroup$ Yes, by using the plane wave equation, I would like to have a vector field showing that if we take the curl of E, we get a flipped out of phase cos wave. Text in R is represented by character vectors. m 7,281 laplace. The facilities include built-in functions for plotting wire-frame objects, space-curves, shaded surfaces. 5 for plotting a surface over a vertically simple region using ezsurfvs , and plotting a surface over a region given in polar coordinates using ezsurfpol. In order to get an intelligible plot, the step size must be taken relatively large. m % % Description: This M-file plots the truncated Fourier Series % representation of a square wave as well as its % amplitude and phase spectrum. To compute the curl, you can use the mfile curl. A simple example showing how to plot a vector field (quiver) with matplotlib. Matrix Algebra Vectors and Matrices Vector Operations. 1 and are used in this course packet: PROGRAM FILES M-File Bytes animlaplace. MATLAB can evaluate and plot most of the common vector calculus operations that we have previously discussed. I will start with the obvious 1. I programmed a simple MATLAB script that queries the status of the bike stations once every hour using cURL, and saves the data into a text file. learn more View Options. This is a bit descriptive so read on :- By definition Curl(Psi)=V so Curl(curl(Psi))=Curl(V) -> L(Psi)=-Curl(V) where L is the laplacian operator ~ This is a Poisson equation and can be easily solved by writing the Laplacian in CD2 and solving the resulting sparse matrix equation. The other dimension lengths are the same for X and y. Software for Manipulating or Displaying NetCDF Data This document provides references to software packages that may be used for manipulating or displaying netCDF data. R Preface to MATLAB Exercises R MATLAB Exercises in Electromagnetics, an e-supplement to Electromagnetics by Branislav M. NOAA PSL: Can plot composite maps from plotting pages on monthly, daily and sub-daily timescales for 20CR. We use cookies for various purposes including analytics. I am able to plot vector fields using a couple nested lists, but this makes adjusting the density of arrows, etc. Matlab has a powerful symbolic math ability. Read on: 1. The arrays X and Y, which define the coordinates for U and V, must be monotonic, but do not need to be uniformly spaced. You can write a book review and share your experiences. [curlz,cav]= curl(X,Y,U,V) computes the curl z-component and the angular velocity perpendicular to z (in radians per time unit) of a 2-D vector field U, and V. 34 MATLAB Functions. For example entries i thru j could be had with V(i:j) and an more complicated index scheme could be accomplished with a vector of indices i_index and then V(i_index). The currently opened section is MATLAB -> Graphics -> Vector Fields -> Functions, and you can also see some other functions for plotting vectors: these include feather, compass. 6(a) and (b) demonstrates how well maps. Show transcribed image text. An example of plotting a continuous data flow is shown here: The package description provides more information how to customize the plot.","date":"2020-09-22 15:01:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.5601763129234314, \"perplexity\": 1429.9426854018145}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600400206133.46\/warc\/CC-MAIN-20200922125920-20200922155920-00779.warc.gz\"}"} | null | null |
{"url":"http:\/\/cust-serv@ams.org\/bookstore-getitem\/item=CMIP-4","text":"New Titles \u00a0|\u00a0 FAQ \u00a0|\u00a0 Keep Informed \u00a0|\u00a0 Review Cart \u00a0|\u00a0 Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education\nHarmonic Analysis, the Trace Formula, and Shimura Varieties\nEdited by: James Arthur, University of Toronto, ON, Canada, David Ellwood, Clay Mathematics Institute, Cambridge, MA, and Robert Kottwitz, University of Chicago, IL\nA co-publication of the AMS and Clay Mathematics Institute.\n SEARCH THIS BOOK:\nClay Mathematics Proceedings\n2005; 689 pp; softcover\nVolume: 4\nISBN-10: 0-8218-3844-X\nISBN-13: 978-0-8218-3844-0\nList Price: US$131 Member Price: US$104.80\nOrder Code: CMIP\/4\n\nOn Certain $$L$$-Functions - James Arthur, James W Cogdell, Steve Gelbart, David Goldberg, Dinakar Ramakrishnan and Jiu-Kang Yu\n\nThe modern theory of automorphic forms, embodied in what has come to be known as the Langlands program, is an extraordinary unifying force in mathematics. It proposes fundamental relations that tie arithmetic information from number theory and algebraic geometry with analytic information from harmonic analysis and group representations. These \"reciprocity laws\", conjectured by Langlands, are still largely unproved. However, their capacity to unite large areas of mathematics insures that they will be a central area of study for years to come.\n\nThe goal of this volume is to provide an entry point into this exciting and challenging field. It is directed, on the one hand, at graduate students and professional mathematicians who would like to work in the area. The longer articles in particular represent an attempt to enable a reader to master some of the more difficult techniques. On the other hand, the book will also be useful to mathematicians who would like simply to understand something of the subject. They will be able to consult the expository portions of the various articles.\n\nThe volume is centered around the trace formula and Shimura varieties. These areas are at the heart of the subject, but they have been especially difficult to learn because of a lack of expository material. The volume aims to rectify the problem. It is based on the courses given at the 2003 Clay Mathematics Institute Summer School. However, many of the articles have been expanded into comprehensive introductions, either to the trace formula or the theory of Shimura varieties, or to some aspect of the interplay and application of the two areas.\n\nThis book is suitable for independent study.\n\nTitles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).\n\nGraduate students and research mathematicians interested in number theory, automorphic forms, and group representations.\n\n\u2022 J. Arthur -- An introduction to the trace formula\n\u2022 J. S. Milne -- Introduction to Shimura varieties\n\u2022 F. Murnaghan -- Linear algebraic groups\n\u2022 R. E. Kottwitz -- Harmonic analysis on reductive $$p$$-adic groups and Lie algebras\n\u2022 S. DeBacker -- Homogeneity for reductive $$p$$-adic groups: An introduction\n\u2022 M. Goresky -- Compactifications and cohomology of modular varieties\n\u2022 T. J. Haines -- Introduction to Shimura varieties with bad reduction of parahoric type\n\u2022 T. C. Hales -- A statement of the fundamental lemma\n\u2022 P. Sarnak -- Notes on the generalized Ramanujan conjectures","date":"2014-08-28 13:40:03","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\": 0.3282501697540283, \"perplexity\": 1727.8360619491573}, \"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-2014-35\/segments\/1408500830834.3\/warc\/CC-MAIN-20140820021350-00396-ip-10-180-136-8.ec2.internal.warc.gz\"}"} | null | null |
Apartments of Hotel Madrid in Torrevieja. Official Website.
The "El Abuelo" Apartments are located in Torrevieja and have air conditioning, 4 terraces, (2 on the bottom and 2 on the first floor), Solarium and swimming pool.
This accommodation is 10' by walking from the beach and it is surrounded by all services. Every apartment has 3 bedrooms, kitchen and 2 bathrooms with shower, fridge, washing machine, oven, microwave, coffee maker, toaster and water heater, television 50 ", etc.
In addition, you will find a wide range of services, such as a 24-hour reception, airport shuttle service and a bicycle rental service.
Every apartment has the dimensions of 100 m2 plus 70m2 of terraces. Total 170 m2 approx. The ideal place for your vacations.
** Note: "El Abuelo" apartments are located at number 4 of the Paris Street, at 1.5 km from Hotel Madrid. Guest should pick up the keys at the Hotel Madrid.
This accommodation is a 4-minute by walking from the beach. Offering a shared swimming pool, Apartamentos Madrid is located in Torrevieja, 500 meters from the beach.They have air-conditioned apartments with free Wi-Fi and a private terrace.
All 3-bedroom apartments include a living room with a sofa and TV, a kitchen with an oven, microwave and washing machine, and 2 bathrooms.
The property has a terrace with sun loungers by the pool and offers a wide range of services, such as a 24-hour reception, airport shuttle service and a bicycle rental service.
The apartments are less than 5 minutes 'walk from many bars and restaurants, 1.5 km from a shopping center and 45 minutes' drive from Alicante Airport.
** Note: The apartments are located at Avenida Diego Ramírez number 29, at 50 meters from the Hotel Madrid, where guest should pick up the keys.
Capacity: from 1 to 6 people. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,308 |
> Aproximemo-nos com toda a confiança, do trono da graça.
**Leitura da Carta aos Hebreus 4,12-16**
Irmãos:
12 A Palavra de Deus é viva,
eficaz e mais cortante do que qualquer espada de dois gumes.
Penetra até dividir alma e espírito,
articulações e medulas.
Ela julga os pensamentos e as intenções do coração.
13 E não há criatura que possa ocultar-se diante dela.
Tudo está nu e descoberto aos seus olhos,
e é a ela que devemos prestar contas.
14 Temos um sumo sacerdote eminente, que entrou no céu,
Jesus, o Filho de Deus.
Por isso, permaneçamos firmes na fé que professamos.
15 Com efeito, temos um sumo sacerdote
capaz de se compadecer de nossas fraquezas,
pois ele mesmo foi provado em tudo como nós,
com exceção do pecado.
16 Aproximemo-nos então, com toda a confiança,
do trono da graça,
para conseguirmos misericórdia
e alcançarmos a graça de um auxílio no momento oportuno.
Palavra do Senhor.
## Salmo - Sl 18,8. 9. 10. 15 (R. Cf.Jo 6,63c)
`R. Vossas palavras são espírito, são vida,tendes palavras, ó Senhor, de vida eterna.`
8 A lei do Senhor Deus é perfeita,*
conforto para a alma!
O testemunho do Senhor é fiel,*
sabedoria dos humildes. R.
9 Os preceitos do Senhor são precisos,*
alegria ao coração.
O mandamento do Senhor é brilhante,*
para os olhos é uma luz. R.
10 É puro o temor do Senhor,*
imutável para sempre.
Os julgamentos do Senhor são corretos*
e justos igualmente. R.
15 Que vos agrade o cantar dos meus lábios*
e a voz da minha alma;
que ela chegue até vós, ó Senhor,*
meu Rochedo e Redentor! R.
## Evangelho - Mc 2,13-17
> Eu não vim para chamar justos, mas sim pecadores.'
**+ Proclamação do Evangelho de Jesus Cristo segundo São Marcos 2,13-17**
Naquele tempo:
13 Jesus saiu de novo para a beira do mar.
Toda a multidão ia ao seu encontro
e Jesus os ensinava.
14 Enquanto passava,
Jesus viu Levi, o filho de Alfeu,
sentado na coletoria de impostos,
e disse-lhe: 'Segue-me!'
Levi se levantou e o seguiu.
15 E aconteceu que, estando à mesa na casa de Levi,
muitos cobradores de impostos e pecadores
também estavam à mesa com Jesus e seus discípulos.
Com efeito, eram muitos os que o seguiam.
16 Alguns doutores da Lei, que eram fariseus,
viram que Jesus estava comendo
com pecadores e cobradores de impostos.
Então eles perguntaram aos discípulos:
'Por que ele come
com os cobradores de impostos e pecadores?'
17 Tendo ouvido, Jesus respondeu-lhes:
'Não são as pessoas sadias que precisam de médico,
mas as doentes.
Eu não vim para chamar justos, mas sim pecadores.'
Palavra da Salvação.
## Reflexão - Mc 2, 13-17
Ser coletor de impostos na época de Jesus era ser um pecador profissional. Por isso, a escolha de Levi, ou Mateus, para ser discípulo de Jesus e ir comer na casa dele com os outros cobradores de impostos e pecadores, significava que Jesus comungava com eles, o que era muito grave. No entanto, esse fato nos mostra que Jesus veio para nos mostrar o amor misericordioso de Deus, que havia dito pelo profeta que não quer a morte do pecador, mas que se converta e viva e que Deus quer que todas as pessoas participem do banquete do Reino definitivo.
Veja mais no [Liturgia Diária - CNBB](http://liturgiadiaria.cnbb.org.br/app/user/user/UserView.php?ano=2017&mes=1&dia=14) | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,499 |
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A New Way to Play: Super Nintendo World Opens February 2023
Let's-a go! SUPER NINTENDO WORLD™, the innovative, immersive and highly-anticipated themed land officially opens at Universal Studios Hollywood on Friday, February 17, 2023. So, get ready; this exciting debut marks SUPER NINTENDO WORLD's official arrival to the United States. This all-new, dynamic land will be a visual spectacle of vibrant colors and architectural ingenuity located within a newly expanded area of Universal Studios Hollywood. SUPER NINTENDO WORLD will feature the groundbreaking "Mario Kart™: Bowser's Challenge" ride, as well as many thrilling and interactive activities designed to engage the entire family within the captivating Mushroom Kingdom. Along with themed dining at the Toadstool Cafe™ and shopping at the 1-UP Factory™ retail store, SUPER NINTENDO WORLD will be an inspiring, game-changing addition to the world-famous theme park.Bowser statue - courtesy of universalstudioshollywood.com From the moment guests pass through the iconic green pipe, a journey filled with exploration, discovery and play awaits that is entirely unlike anything they've experienced before. Their journey begins as they enter Peach's Castle before venturing further into the colorful Mushroom Kingdom. SUPER NINTENDO WORLD will sweep guests into the 360-degree world of Mario, Luigi and Princess Peach where they will become an integral part of their exhilarating universe. Plus, with purchase of a Power-Up Band™ – wearable, interactive wristbands – guests will be able to deepen their interactions across the land.Super Nintendo World - courtesy of universalstudioshollywood.com As the countdown to the grand opening begins, here's a sneak peek of what to expect: "Mario Kart: Bowser's Challenge:" This all-new, technologically-advanced ride, inspired by the popular Mario Kart™ video game series, will seamlessly fuse cutting-edge augmented reality (AR) with projection mapping technology and actual set pieces along a moving ride track. Set against a multi-sensory backdrop of color, sound and movement, guests will be seated in stadium-style, four-seat vehicles as they navigate familiar courses through the creative use and integration of head mounted AR goggles. This unique feature is a key point of differentiation that distinguishes this ride from other theme park attractions. The premise of "Mario Kart: Bowser's Challenge" is simple but equally inspiring and challenging, appealing to guests of all ages regardless of gaming experience. As part of Team Mario, guests will steer through underwater courses and courses in the clouds to compete for the Golden Cup while collecting coins to defeat Team Bowser and win. "Mario Kart: Bowser's Challenge" raises the stakes for guests as an intriguing and repeatable ride with a variety of outcomes. Mount Beanpole: This towering landmark sits at the heart of the land and also serves as the entry queue for "Mario Kart: Bowser's Challenge." Bowser's Castle: Adorned with a breathtaking sculpture of a large and powerful Bowser, this structure serves as a key centerpiece of the land, home to SUPER NINTENDO WORLD's signature ride, "Mario Kart: Bowser's Challenge." As guests navigate the winding corridors and pass through the hall of medallions and trophies of the ominous castle towards the "Mario Kart: Bowser's Challenge" ride, they will learn more about Bowser's calculated plans to defeat Team Mario for the coveted Golden Cup. Power-Up Band™: These wearable, state-of-the-art wristbands sync with Universal Studios Hollywood's free downloadable app to level-up the guest experience within the land and enhance its many interactive elements. These include, but are not limited to, keeping individual and team scores, collecting digital coins and obtaining keys after winning challenges throughout the land. Power-Up Bands will also invite guests to enjoy extra-special interactions with Mario, Luigi and Princess Peach. They will come in six design options with character themes and will be available for purchase both within the land and at SUPER NINTENDO WORLD retail shops located in the theme park and on CityWalk. Interactive Gameplay: Within the land, guests will become fully engulfed in an engaging world of real-life gameplay, including punching ? Blocks to collect digital coins and a variety of interactives. They will discover a new dimension of SUPER NINTENDO WORLD via the interactive binoculars employing augmented reality technology positioned within the land. Guests can beat the four Key Challenges and collect keys from Goomba, Koopa Troopa, Piranha Plant and Thwomp interactive activities to ultimately allow access to the culminating boss battle with Bowser Jr. These energetic, familiar and fun interactive games will further enhance the entire kinetic experience within the land and immerse guests into the unique world of Super Mario. Toadstool Cafe™: The land's signature restaurant Toadstool Cafe will serve a gourmet menu perfected by Chef Toad where he will greet guests upon entry. Menu items prepared fresh daily include Toadstool Cheesy Garlic Knots, Super Mushroom Soup, Piranha Plant Caprese, Mario Bacon Cheeseburger, Luigi Pesto Chicken Burger, ? Block Tiramisu and Princess Peach Cupcake. 1-UP Factory: What better way to commemorate a visit to SUPER NINTENDO WORLD than to visit the 1-UP Factory retail shop to purchase collectable memorabilia. The 1-UP Factory will offer an extensive selection of merchandise from Mario and Luigi themed apparel to iconic character hats and an array of plush characters, including Mario, Luigi, Yoshi and Bowser. SUPER NINTENDO WORLD is conceived in partnership with Nintendo and the visionaries at Universal Creative to deliver exhilarating entertainment with innovative technological achievements inspired by characters and video games that have appealed to Nintendo fans for generations. Click here for a sneak peak and visit www.UniversalStudiosHollywood.com for theme park updates. 1-UP Factory Store - courtesy of universalstudioshollywood.com
Universal Orlando Opens New Escape Room Experience
Universal Orlando's world-renowned craft of creating innovative, multi-sensory themed entertainment comes to life in two distinct escape adventures. Guests can now book epic escape adventures inspired by Universal Pictures and Amblin Entertainment's iconic Jurassic World and Back to the Future films by purchasing tickets for Universal's Great Movie Escape – comprised of two next-level escape room experiences, opening today on December 9 at Universal CityWalk. Jurassic World: Escape - courtesy of universalorlando.com Universal's Great Movie Escape invites guests of all ages to solve their way through a pair of first-of-its-kind escape rooms based on two of the biggest film franchises in history. Universal's entertainment team partnered closely with filmmakers and escape room industry experts around the country to create innovative experiences that go beyond the typical escape room – blending state-of-the-art challenges, detailed sets and original storylines to immerse guests in story-driven adventures that allow them to escape into the movies like never before. Another unique feature of Universal's Great Movie Escape is the variety of challenges guests must work together to complete – each one specifically crafted so that the experience is enjoyable for novices and escape room-aficionados alike. Each adventure boasts eight incredibly detailed environments featuring state-of-the-art activities that are: randomized – so the solution can vary with each visitcustomizable depending on party size – ensuring everyone gets to participatedesigned to adjust based on skill level and complexity – so there's always something to solve As soon as guests enter the art deco-style, theater-inspired venue, they'll immediately be swept into one of the following adventures: Jurassic World: Escape Jurassic World: Escape - courtesy of universalorlando.com Guests are new geneticists at a secret lab on Isla Nublar, where they are undergoing training that includes the completion of various day-to-day tasks that range from feeding ferocious dinosaurs to splicing dinosaur DNA to create new genetic codes…until they learn an apex predator has broken free and now they must work together to avoid becoming its prey. Back to the Future: OUTATIME Back to the Future: OUTATIME - courtesy of universalorlando.com Guests are transported to a museum in 1993, where they learn Back to the Future's antagonist, Biff, is up to his antics and has stolen Doc Brown's newest time travel device to sabotage the space-time continuum. Guests must work together to uncover Doc Brown's clues as they travel through time to track down Biff and save the past and future before they run "outatime."Tickets for Universal's Great Movie Escape are on sale now and start at $49.99 per person, plus tax for access to either Jurassic World: Escape or Back to the Future: OUTATIME (experience includes up to six guests per group and may include multiple parties). Pricing varies depending on date and time of visit. Guests may also book a private experience exclusive for their party starting at $300 per group, plus tax (up to eight people). Tickets can be purchased at www.universalorlando.com. For discounted Universal tickets be sure to check out Budget Travel's Universal page. For more information about Universal Orlando Resort, visit www.universalorlando.com.
InspirationFamilyTheme Parks
Best Holiday Celebrations at US Theme Parks
Smoky Mountain Christmas at Dollywood Pigeon Forge, Tennessee - Nov. 5 - Jan 1, 2023 With several new lighting displays across its 160 acres, Dollywood theme park—the 14-time winner for Best Theme Park Christmas Event—now boasts more than 6 million shining, shimmering lights as part of the popular Smoky Mountain Christmas presented by Humana. "I believe each season in the Smokies is a precious gift from above, but I also know there is no better time than Christmas to experience the magic of this special place," Dolly Parton explained. "Christmas is a time for new memories to be created, cherished family traditions to be celebrated and the love of the holidays to warm us all." Guests will find hundreds of lighted trees throughout the park this year, as well as more new décor elements and eight different light zones to provide each area of Dollywood its own unique look and feel. Glacier Ridge fills Dollywood's Timber Canyon, Wildwood Grove, Wilderness Pass and Upper Craftsman'sValley with an expansive arctic experience sure to inspire and delight guests. Wilderness Pass serves as one of the highlights of Glacier Ridge thanks to the Wonderful Christmas! Plaza Tree Show, which features a 50-ft. tall animated tree sparkling to life to present a fully-synchronized, dazzling light show throughout the evening. Each show is capped by an immersive snow finale! Dollywood's Smoky Mountain Christmas - Courtesy of dollywood.com Additionally, Dollywood features indoor and outdoor stages with special shows. The festival's headliner, "Christmas in the Smokies," has been a must-see for families every Christmas season since it began in 1990. Many other Dollywood Christmas classics return including "'Twas the Night Before Christmas," "O' Holy Night," "Christmas with the Kingdom Heirs," "Heart of the Holidays," "Candlelight Carolers," "The Mistletones," "Appalachian Christmas with the Smoky Mountain String Band," "Holiday Wild Roots" and more. Returning this year on Friday and Saturday evenings is the "Merry & Bright!" fireworks display, which provides a sparkling ending to the day. And for those looking to celebrate the holidays with the perfect family meal, Dollywood's team of chefs have a menu full of culinary surprises sure to delight. From herb-roasted turkey breast and citrus-glazed carved ham to chicken pot pie in a bread cone, there are a number of savory items to satisfy every appetite. A number of unique items abound including eggnog cupcakes, gingerbread-dusted funnel cake, holiday limeade and more. An Olde Time Christmas at Silver Dollar City Christmas lights in Silver Dollar City - Courtesy of silverdollarcity.com Branson, Missouri - Nov. 5 - Dec. 30 Bright lights line streets, buildings, pathways and trees during Silver Dollar City's An Old Time Christmas. This season marks the debut of a new production show Coming Home For Christmas, along with over 6.5 million lights, an elaborate 8-story animated Christmas tree, a light parade, holiday foods and more. Throughout the streets of The City carolers sing and stroll, while the centerpiece of Joy On Town Square, an 8-story Christmas tree, and surrounding lights glow spectacularly. Even brighter is Christmas in Midtown with its remarkable display of light tunnels, wreath portals, flying angels and special effects 9-stories tall. For thrill seekers, rides soar under the Ozark Mountain starlight with all the Christmas lights visible below. Plus, craftsman in the Christmas spirit demonstrate glass-blowing wood carving and pottery, creating one-of-a-kind heirlooms. The City's stages offer more than 30 shows each day and night including the longtime favorite production, A Dickens' Christmas Carol. The all new show, Coming Home for Christmas, features a live band and a cast of 14 singers and dancers presenting holiday music, stories and family traditions. Other productions include The Living Nativity and the Saloon Frontier Fa-La-La Follies. Rudolph's Holly Jolly™ Christmas Light Parade winds through The City each evening with lighted floats, characters, dancers and performers. To add to the festivities, menus offer a variety of seasonal treats such as a Holiday Dinner with smoked turkey, ham, prime rib and trimmings; specialty soups like potato leek; Silver Dollar City's Miner's Beef Stew; plus, a variety of sweets such as hot chocolate and wassail, s'mores or warm apple dumplings with homemade cinnamon ice cream. Tasting Passports let guests sample their way through The City. WinterFest at Kings Island WinterFest Wonderland Parade at Kings Island - Courtesy of visitkingsisland.com Mason, Ohio - Nov. 25 - Dec. 31 WinterFest is an unforgettable holiday experience where Kings Island transforms into 11 enchanting winter wonderlands. Voted in 2021 as one of the top Theme Park Holiday Events in the country by USA TODAY readers, the annual event features ice skating on the Royal Fountain, the Eiffel Tower turned into a magical Christmas tree, the WinterFest Wonderland Parade, more than five million lights and live entertainment throughout the park. Plus, experience up to 20 rides including Mystic Timbers and Kings Mills Antique Autos. Cincinnati's premier holiday event is open select nights in November and December. Guests will be treated to a grandiose showcase of dazzling displays and extravagantly decorated floats with the WinterFest Wonderland Parade filled with classic holiday themes such as toy boxes, gingerbread houses, choo-choo trains, and more. This year, special live entertainment acts include the all-new Swingin' Into Christmas, plus Tinker's Toy Factory, Jingle Jazz, Cool Yule Christmas and so much more. Visitors can participate in holiday fun like skating on Snow Flake Lake, decorating cookies with Mrs. Claus and getting a family photo with the big man in red himself! Christmas Town at Busch Gardens Busch Gardens Williamsburg - Courtesy of buschgardens.com Williamsburg, Virginia - Nov. 11 - Jan 8, 2023 The World's Most Beautiful Theme Park is once again transformed with over ten million twinkling lights, one of the largest holiday displays in North America. Enjoy heart-warming holiday shows and classics such as Santa's Workshop, the Christmas Town Express and 'Twas that Night ice skating show. Over 20 rides and coasters pair thrilling moments with holiday cheer. During "Santa's Fireside Feast" presented by Coca-Cola, guests gather around the majestic castle as Santa recounts a classic Christmas story while elves prepare a scrumptious all-you-care-to-eat meal and Mrs. Claus dazzles the dining room with holiday cheer. Visitors can also find a number of seasonal characters in the park's different sections, which are themed after various countries. In England's Kidsington Palace, families can snap a photo with Father Christmas and enjoy a drink in front of the fireplace. In Festa Italia, Rudolph the Red-Nosed Reindeer and Clarice can be found in Rudolph's Winter Wonderland, while Frosty the Snowman & Gingy the Gingerbread Man can be found in France. Families are invited to participate in an all-new holiday scavenger hunt celebrating traditions around the world including Christmas, Hanukkah, and Kwanzaa. Pick up a clue sheet in England to identify holiday symbols throughout the park and redeem a sweet treat at Emporium once all have been found. Throughout the park, festive treats, holiday shopping and special photo opportunities complete this holiday tradition for friends and families. Christmas Town at Busch Gardens Christmas on Ice - Courtesy of buschgardens.comTampa, Florida - Nov. 14 - Jan. 9, 2023 Just like its Virginia counterpart, Busch Gardens Tampa Bay puts on its own spectacular Christmas festival. Visitors can experience the joy of the season up close with animal encounters, thrilling rides, holiday inspired culinary delights and festive shows. The park's longest running classic holiday show, Christmas on Ice, returns this year to the Moroccan Palace Theater. This inspiring skating production transforms the stage with incredible feats of skating athleticism for a truly unmissable show set to holiday classic songs. At night, enjoy classic holiday tunes and watch festive bursts of Christmas colors as they cover the sky in a dazzling fireworks and fountain display (offered on the Festival Field every Saturday and select nights of the event). Visitors can ride the Holly Jolly Express and celebrate the most wonderful time of the year with favorite songs of the season and wild views of Busch Gardens' 65-acre veldt, home to giraffes, rhinos, zebras and more. In the Pantopia section, Rudolph's Winter Wonderland features the famous reindeer and all his friends. And, in Santa's North Pole Experience, jolly elves guide guests to see Santa in his workshop as he prepares for the holiday season. Great for Christmas photo opportunities, this fun and festive activity is perfect for making memories with your family and friends. Other memorable shows include Storytime with Mrs. Claus, Three Kings Journey, and Elmo's Christmas Wish. Mickey's Very Merry Christmas Party at Walt Disney World Magic Kingdom's Main Street at Walt Disney World - Courtesy of wdwnt.com Orlando, Florida - Nov. 8 - Dec. 22 Holiday cheer will fill the Magic Kingdom as Mickey's Very Merry Christmas Party returns this year on 24 select nights in November and December. A 65-foot-tall Christmas tree, adorned with dazzling lights and ornaments, welcomes guests to the park where they'll discover favorite attractions featuring special holiday overlays, Disney characters dressed in the spirit of the season and seasonal entertainment, seasonal nighttime spectaculars, transformations of favorite attractions, special food and beverages and more. All the standard entertainment is back this year. First off, Mickey Mouse and his pals appear on the Cinderella Castle stage for the fifth year of the Christmas extravaganza "Mickey's Most Merriest Celebration." The show features singers, dancers and surprises all choreographed to nostalgic and modern medleys of holiday music. And every night, Minnie Mouse orchestrates a sparkling nighttime spectacular with dazzling fireworks, castle projections and seasonal songs in "Minnie's Wonderful Christmastime Fireworks." Of course, don't miss Santa Claus as he joins Mickey Mouse and friends in the classic "Mickey's Once Upon a Christmastime Parade." As well, partygoers can indulge in complimentary cookies and hot cocoa at locations throughout Magic Kingdom. Select Magic Kingdom attractions will get a jolly holiday overlay with Jungle Cruise transforming into Jingle Cruise once again, and special holiday makeovers of Space Mountain, Tomorrowland Speedway, Monsters Inc. Laugh Floor, and Mad Tea Party returning for the first time since 2019! New sweet treats this year include Texas-sized Sweet Potato Pie with Marshmallow Meringue and Candied Pecans found at Pecos Bill Tall Tale Inn & Café to go along with returning favorites like the Mickey-shaped Cinnamon Roll with White Icing and Holiday Sprinkles at Main Street Bakery.
Budget Travel ListsTheme Parks
Mermaids at Sea Life Aquarium Mermaids Dive - Courtesy of Sea Life Aquarium Kansas City, Missouri Now – November 13th These seven magical maidens are a fanta-sea come true! Watch as the mermaids dive into the Ocean Exhibit with sea turtles, sharks, and other aquatic friends. Come see these beautiful mermaids dive with their friends. The dives take place Friday – Sundays at various times. Be sure to look for them on land for some Meet & Greets between 9:30am – 5 pm. Las Vegas Days Rodeo at CORE Arena Las Vegas Days Rodeo - Courtesy of the Las Vegas Days Rodeo Las Vegas, Nevada - November 11 and 12th The rodeo will feature bull riding, barrel racing, team roping, tie-down roping, steer wrestling, saddle bronc riding, breakaway, and bareback riding, with prize purses totaling $40,000 and custom belt buckles for champions. The event will also feature a free Whiskerino contest, Mutton Busting, historical exhibits, and coloring contest for Southern Nevadan children in grades kindergarten through six. The Whiskerino contest returns to the rodeo this year. Free to enter, the contest offers prizes in multiple categories including Longest Beard, Longest Mustache, Best Natural Beard, Worst Beard, and Most Unique. Participants are encouraged to dress in Western attire or early 1900's period costumes. Registration is open until Oct. 28 on the Plaza's website. Participants must be present at the rodeo to win. If your child wants to become a cowboy/girl make sure to sign them up for the Mutton Busting! The sheep riding competition is open to children ages 4-10 who weigh less than 60 pounds. The winner gets a commemorative buckle. Registration and more details can be found here. Both nights start at 6 p.m. (gates open at 5 p.m.). Admission is free but RSVP is recommended online. Universal's Holiday Tour at Universal Orlando Grinchmas Courtesy of Universal Orlando Orlando, Florida - November 12 through January 1, 2023 Embarking on an exclusive guided tour - Universal's Holiday Tour and encounter the mean, green one himself, the Grinch, while enjoying a delicious breakfast during the popular The Grinch & Friends Character Breakfast. Enjoy a meet-and-greet and photo opp with Santa Claus and cap off the night in The Wizarding World of Harry Potter – Hogsmeade with an exclusive, after-hours showing of "The Magic of Christmas at Hogwarts Castle" – a stunning projection of holiday spirit and Christmas moments inspired by the beloved "Harry Potter" stories brought to life on the majestic Hogwarts castle.Tickets begin at $79.99 (plus tax) and separate theme park admission with a 2-Park 1-Day Park-to-Park ticket is required. Illumination: Tree Lights at The Morton Arboretum Human and Nature sculpture Hallow - Courtesy of Morton Arboretum Chicago, Illinois - November 19 through January 7, 2023 You will explore 18 different features along a tree-lined one-mile walking trail immersed in color, light and sound created exclusively for The Morton Arboretum. Within this array, six new exhibits will join returning favorites such as Treemagination and Hug-a-Tree, including a spectacular finale in the Arboretum's new Grand Garden. This year is the addition of Electric Illumination, an 18 and older late night event, 9:30-11 p.m., Dec. 16, 23 and 30, featuring an eclectic, DJ-curated mix of Trance and Progressive House music synchronized to the breathtaking light displays. IllumiBrew returns for a special pre-opening event Nov. 17 and 18 for guests 21 and older to enjoy craft beer tastings along the Illumination trail. The annual light exhibition highlights the beauty and form of trees in winter. Warm concession tents offer beverages and snacks such as s'mores kits to enjoy by crackling bonfires along the route. In addition to the new Grand Garden spectacular, a traditional favorite, Meadow Lake Magic, will be completely reimagined to include 15-foot-high mirrored towers floating in the water with light beams shooting from atop each into the sky. Among the interactive features will be Shadow Play, a new touch-free wall where guests can express themselves while casting colorful shadows. Two of artist Daniel Popper's Human+Nature exhibition sculptures will be featured in this year's exhibition. Illumination tickets are on sale now, ticket prices range from $10 to $35 per person. Children under age 2 will be free of charge. It will open each evening at 4:30 p.m., with the last entry at 8:30 p.m., and lights will go out at 9:30 p.m. with the exception of Electric Illumination late-night hours from 9:30 to 11 p.m..
FamilyBudget Travel ListsTheme Parks
Most affordable water parks across the U.S.
Just in time for National Waterpark Day (July 28), vacation rental marketplace HomeToGo is making a splash by releasing its 2022 Water Park Index to help travelers identify the top water parks to visit this year. From Florida to California, from pulse-pounding to placid, they've done a deep dive into the best water parks that America has to offer. Most Affordable States to Make a Splash: Water parks in Texas, Louisiana, Florida, California and Illinois climb to the affordable peak of this year's ranking.Best States for Water Park Enthusiasts: California reigns supreme with the most water parks on this year's ranking (11), followed by Texas (10) and Florida (9). The average total cost per day among the featured water parks in California is $138.06, compared to $132.96 in Texas and $139.16 in Florida.Average Entry Prices: This year's average entry cost is $39.82. The water parks with the cheapest entry costs are Buccaneer Bay and Sun-N-Fun Lagoon both in Florida, which both cost $13 per adult.The tallest waterside in the country is Thrillagascar at DreamWorks Water Park in New Jersey (Ranked #32), which stands at 142 feet tall and reaches free-fall speeds of +60 mph and features the world's biggest indoor wave poolThe largest water park in the U.S. is Noah's Ark Water Park in Wisconsin Dells (#44), with a total of 51 waterslides, 2 wave pools, 2 lazy rivers, and 6 additional attractions. Rankings are based on cost of parking, entry, locker and nightly accommodations. 3. Buccaneer Bay - $91.02Weeki Wachee, Florida Buccaneer Bay - Courtesy of floridastateparks.org Buccaneer Bay, located within Weeki Wachee Springs State Park, is the perfect place for families to spend a hot summer day. This water park is Florida's only spring-fed water park and is home to a variety of different attractions including waterslides, sandy beach area, lazy river float ride, beach volleyball court and legendary mermaid shows in a submerged theater. On top of this, the water park is surrounded by beautiful natural scenery due to its unique location. This water park offers food and beverage options for guests who need to refuel and hydrate, as well as covered picnic pavilions for those who would like to bring their own snacks. 2. Splash Kingdom Oasis - $90.46Shreveport, Louisiana Splash Kingdom Oasis - Courtesy of hometogo Splash Kingdom Waterpark in Shreveport, Louisiana is a water park for the entire family to enjoy and embrace the family tradition of taking trips together. With more than 15 rides and attractions to choose from, people of all ages can come to this fun oasis and never worry about running out of things to do. On top of this, guests can enjoy food and drinks for whenever they get hungry or need to quench their thirst after a long day. Guests can also expect a multitude of different aspects of entertainment, with events being held that showcase music, movies and more. 1. NRH2O Family Water Park - $87.99North Richland Hills, Texas Thunder - Courtesy of NRH2O Family Water Park NRH20 is a 17-acre water park filled with plenty of fun and excitement for guests to enjoy. This water park is home to 14 attractions along with food and beverage amenities. With a great balance between thrilling waterslides and kid-friendly attractions, this water park is welcome to all ages looking to let off some steam and just have fun. NRH20 is known for its friendly environment, relaxing atmosphere, and reasonable prices. Located in North Richland Hills, Texas, the water park's popularity has flourished within the Dallas/Fort Worth area. Click here to see the complete list of top affordable water parks and the methodology for the rankings.
FamilyTheme Parks
Did you know - Land of Oz!
Autumn at Oz September 9th-11th, 16th-18th, & 23rd-25th at Land of Oz - Beech Mountain, NCCourtesy of Land of Oz Travel Over the Rainbow to Beech Mountain, North Carolina and celebrate Land of Oz's annual Autumn at Oz Festival - one of the world's largest Wonderful Wizard of Oz events! Experience the magic of Oz during this three-weekend event where the Scarecrow, Tin Woodman, and Cowardly Lion are ready to perform for you on your journey to meet the Wizard! Keep a lookout for the Wicked Witch and her Winged Monkeys, plus other favorite characters from this classic story! Visit the Gale farm in Kansas, experience the twister, meet Glinda in Munchkinland, and follow the real yellow brick road to Emerald City where the Wizard grants everyones' wishes - right before your eyes! Courtesy of Land of Oz Shop unique craft & memorabilia vendors, enjoy delicious varieties of food and beverages, get your face painted, and watch live performances. Plus, there are plenty of photo-ops along the way including our Over the Rainbow Observation Deck - one mile high in the sky! Enjoy the magical Land of Oz on the site of the original 1970's theme park where the dreams that you dare to dream really do come true. Courtesy of Land of OzIn the Land of Oz:Don't wear your high heel ruby slippers - Oz has uneven paths, ramps and stairs and therefore is not wheelchair or stroller friendly. The weather in Oz (Beech Mountain) can be cold and blustery, so be prepared for the weather by dressing in layers and wearing comfortable shoes. It is also suggested that guests bring a jacket and an umbrella just in case.You can get to Oz by bus or scenic lift ride (additional cost). The lift ride from Beech Mountain Ski resort will give you amazing views of the Blue Ridge Mountains. General Admission Tickets: $55.00 + Taxes & Processing Fees Children 2 & Under are free.
Most Affordable Amusement Parks
HomeToGo revealed the results of its 2022 Theme Park Index, ranking 30 theme parks in the U.S. based on affordability. Taking into account factors such as the cost of an admission ticket, parking and nearby vacation rentals, they compared many of the most popular theme parks across the country to uncover both budget-friendly and top dollar options for those planning upcoming vacations. While ticket prices have seen an overall increase, the index revealed that some of the best theme parks across the country are accessible for a fraction of the price compared to those in states like Florida and California. Amusement parks in Alabama, Connecticut, Georgia, Missouri and New York climb to the affordable peak of this year's ranking, whereas world-renowned parks in Florida and California are on the pricier side. Based on the cost of a one-day adult admission ticket, parking and data for nearby accommodations, here's a snapshot of the top rankings from the 2022 Theme Park Index: Top 5 1. Tropic Falls at OWA - Alabama Ticket Price: $22.49 Parking: Free Courtesy of visitowa.com 2. Quassy Amusement Park - Connecticut Ticket Price: $39.99 Parking: $10 Courtesy of quassy.com 3. Wild Adventures Theme Park - Georgia Ticket Price: $49.99 Parking: $15 Courtesy of wildadventures.com 4. Worlds of Fun - Montana Ticket Price: $34.99 Parking: $2 Courtesy of worldsoffun.com 5. Six Flags Darien Lake - New York Ticket Price: $34.99 Parking: $20 Courtesy of sixflags.com For a complete breakdown of the methodology and all 30 destinations in this year's Theme Park Index, read the full report at: https://www.hometogo.com/united-states/amusement-parks/#theme-park-index
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Q: Unity C# Process.GetProcesses() yields different results in editor vs runtime I'm trying to kill "explorer" when my Unity app is running.
In the editor, by using System.Diagnostics.Process.GetProcessesByName("explorer"); I'm able to find and to kill the process without issue.
However, using the same code in the compiled app returns an empty array. Printing out all the available processes with Process.GetProcesses() I notice that in the editor I have access to many more processes then I do in the compiled app.
I've tried running the compiled app as admin and I've also tried running the app from C:\Program Files with the same result.
My workaround for the time being is to run a .bat file from my unity app that kills windows explorer.
I'm not sure what the issue. Any help is greatly appreciated.
A: Solution: In the build settings I changed the app to 64bit (x86_64). Once I did this the complied app had access to all the processes including explorer.
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{"url":"https:\/\/tex.stackexchange.com\/questions\/430349\/provoke-or-avoid-ligatures-in-micro-typography","text":"# Provoke or avoid ligatures in micro typography\n\nIn German language there exists a word like Wachstube.\n\nIt has different meanings when split into syllables:\n\n1. Wachs-tube\n2. Wach-stube\n\nNo problem, when the word crosses a line -- one can fix like above with a forced line break.\n\nBut how to set it up correctly without any line break to make it read best to get its correct meaning? It is about to provoke or avoid some kind of ligatures ...\n\nOf course it might depend on the font one chooses and what ligatures are provided within that font. What is the correct way to make the reader get the correct meaning -- when the word won't cross a line?\n\nAny hints welcome!\n\n\u2022 if your font has a st-ligature you should break it in the first case, but apart from this you should simply expect your readers to get the meaning from the context, like with all other words that have more than one meaning. \u2013\u00a0Ulrike Fischer May 6 '18 at 21:43\n\u2022 @Ulrike Fischer you are of course right to get the meaning by the surrounded context ... That shouldn't be a problem ... it is more about the st-ligature to be broken or not -- if existant within the font. \u2013\u00a0user151328 May 6 '18 at 21:49\n\u2022 @marmot ... either\/or ... I'd like it more subtle, like from case to case if necessary ... btw. it is not about that word only -- just for general purposes that behave quite equally ... \u2013\u00a0user151328 May 6 '18 at 21:53\n\u2022 @marmot ... it is more how to setup the word without crossing a line and therefore set it up to provoke or avoid a st-ligature if provided by the font \u2013\u00a0user151328 May 6 '18 at 22:04\n\nYou can use the selnolig package. It provides an interface to suppress ligatures in certain patterns.\n\nRequires fontspec and LuaTeX.\n\n\\documentclass{article}\n\\usepackage{fontspec}\n\\setmainfont[Ligatures=Historic]{Linux Libertine O}\n\\usepackage{selnolig}\n\n\\begin{document}\n\n\\nolig{st}{s|t}\nWachstube\n\n\\keeplig{st}\nWachstube\n\n\\end{document}\n\n\nselnolig breaks ligatures by inserting a whatit in the node list. Unfortunately this also inhibits kerning between the letters taking part in the broken ligature. If you have to adapt the kerning between them (for whatever reason) you have to remove the ligature from the font itself. This, and adding some custom kerning, is feasible with LuaTeX and fonts.handlers.otf.addfeature.\n\n\\documentclass{article}\n\\usepackage{fontspec}\n\n\\directlua{\n\nfonts.handlers.otf.addfeature({\nname = \"kernst\",\ntype = \"kern\",\ndata = {\n[\"s\"] = { [\"t\"] = 200 },\n},\n})\nname = \"ligast\",\ntype = \"multiple\",\ndata = {\n[\"s_t\"] = { \"s\", \"t\" },\n},\n})\n\n}\n\n\\setmainfont[Ligatures=Historic]{Linux Libertine O}\n\n\\begin{document}\n\n{","date":"2019-04-19 06:34:34","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\": 0.6394728422164917, \"perplexity\": 2300.4345595776354}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-18\/segments\/1555578527148.46\/warc\/CC-MAIN-20190419061412-20190419083412-00132.warc.gz\"}"} | null | null |
\section{Introduction and Review}
\label{s:Introduction}
The response function is used to convert the electronic output of a
gravitational-wave detector into the measured gravitational-wave
signal. This response function is determined experimentally by
performing a series of measurements when the detector is offline, and
then monitoring the output of the working, resonant detector (in a
time and frequency dependent way) as it reacts to inputs designed to
simulate its interaction with gravitational
waves~\cite{LIGOS4Calibration}. The calibration procedure produces a
response function that is known therefore only to the level of
accuracy with which these various measurements are performed, and only
to the extent the state of the detector changes predictably between
calibration measurements. This paper evaluates the effects of these
response-function errors on the subsequent gravitational-wave
data-analysis process, and from this determines the optimal levels for
calibration accuracy.
Inaccuracies in the response function degrade the ability to detect
gravitational-wave signals in the noisy data stream; and once
detected, they also reduce the ability to measure the physical
properties of the gravitational-wave source that produced the signal.
Errors in the gravitational-waveform models used in the data-analysis
process also degrade the detection and measurement procedures in a
very similar way. An earlier discussion of the role of calibration
error on these data-analysis functions, cf. Ref.~\cite{Lindblom2008},
adopted the viewpoint that the level of calibration error was fixed.
The analysis there focused on determining the point at which further
reduction of waveform-modeling errors would be made irrelevant by the
presence of calibration error. A more proactive viewpoint is adopted
here: that both the calibration error and the waveform-modeling error
levels can (in principle) be set to any desired level. This paper
determines the optimal levels for the combined calibration and
waveform-modeling errors needed to perform detections and also to
perform measurements on any detected gravitational-wave signals.
These error levels are optimal in the sense that lower accuracy levels
would reduce the quantity and quality of the scientific information
extracted from the data; while higher accuracy would be made
irrelevant by the intrinsic noise level of the detector.
Let us begin by discussing briefly some of what is already known about
the effects of calibration error. To that end, let us first establish
some notation. Let $v(f)$ denote the direct electronic output of the
detector, and $R(f)$ the response function used to convert this raw
output to the inferred gravitational-wave signal $h(f)$:
\begin{eqnarray}
h(f)=R(f)v(f).
\end{eqnarray}
For simplicity, the discussion here is expressed in terms of the
frequency-domain representations of the various quantities. For
example the frequency-domain waveform, $h(f)$, is related to its
time-domain analog, $h(t)$, by the Fourier transform:
\begin{eqnarray}
h(f)=\int_{-\infty}^{\infty} h(t)e^{-2\pi i f t} dt.
\end{eqnarray}
This transform follows the convention of the LIGO Scientific
Collaboration~\cite{T010095} (and the signal-processing community) by
using the phase factor $e^{-2\pi i f t}$, while most of the early
gravitational-wave literature and essentially all other computational
physics literature use $e^{2\pi i f t}$. This choice does not affect
any of the subsequent equations in this paper.
Let us assume that the measured response function $R(f)$ differs from
the correct exact function $R_e(f)$ by $\delta R(f) =
R(f)-R_e(f)$. This error in the response function will affect
measurements in two ways. The response of the detector to a
gravitational-wave signal $h_e$ will produce an electronic output
$v_e$. So the first effect of using the measured response function
$R$, is to interpret the signal as the waveform
$h=Rv_e=h_ee^{\delta\chi_R+i\delta\Phi_R}$, where the logarithmic
response function amplitude $\delta \chi_R$ and phase $\delta\Phi_R$
errors are defined by
\begin{eqnarray}
R= R_e+\delta R = R_e e^{\delta\chi_R+i\delta\Phi_R}.
\end{eqnarray}
This will produce a waveform error,
\begin{eqnarray}
\delta
h_R=h_ee^{\delta\chi_R+i\delta\Phi_R}-h_e
\approx h_e(\delta\chi_R+i\delta\Phi_R),
\end{eqnarray}
caused by the calibration error of the detector.
The second effect of calibration error on measurements made with the
detector are errors in understanding the characteristics of the
detector noise. In particular, the measured power spectral density of
the noise $S_n$ will differ from the exact $S_e$ due to the
calibration error $\delta R$. The measured power spectral density of
the noise $S_n$ is related to $S_e$ by
\begin{eqnarray}
S_n(f) = S_e(f)\,e^{2\delta\chi_R}.
\label{e:MeasuredSh}
\end{eqnarray}
Both the detection and the measurement of a gravitational wave's
properties are adversely affected by response-function induced errors
in the waveform, $\delta h_R$, and the measured noise spectrum,
$S_n(f)$. Similar adverse effects are caused by errors in the
waveform models used as part of the gravitational-wave data-analysis
procedure. Let $\delta h_m(f) = h_m(f)-h_e(f)$ denote the difference
between a model gravitational waveform $h_m$, (e.g., one produced by a
numerical-relativity simulation) and the exact waveform $h_e$. Both
types of waveform error, $\delta h_R$ and $\delta h_m$, cause
reductions in the signal-to-noise ratio, $\rho_m$, obtained when
a signal is projected onto the model waveform. Keeping terms
through second-order in $\delta h_R$ and $\delta h_m$, it was shown
previously~\cite{Lindblom2008} that the resulting
measured signal-to-noise ratio, $\rho_m$, is related to the optimal
signal-to-noise ratio, $\rho$, by the expression:
\begin{eqnarray}
\rho_m = \rho-\frac{1}{2\rho}\langle(\delta h_m-\delta h_R)_\perp|
(\delta h_m-\delta h_R)_\perp\rangle,
\label{e:modelpluscalibrationerror}
\end{eqnarray}
where the quantity $(\delta h_m - \delta h_R)_\perp$ is the projection of
$\delta h_m - \delta h_R$ orthogonal to the exact waveform,
\begin{eqnarray}
(\delta h_m - \delta h_R)_\perp &=&
\delta h_m -\delta h_R - h_e \frac{\langle \delta h_m - \delta h_R|h_e\rangle}
{\langle h_e|h_e\rangle}.\quad
\label{e:delta_h_perp_def}
\end{eqnarray}
The noise-weighted inner products, e.g., $\langle \delta h_m| \delta
h_R\rangle$, used in these expressions are defined with respect to the
measured power spectral density of the noise $S_n(f)$:
\begin{eqnarray}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\langle \delta h_m| \delta h_R\rangle=\nonumber\\
&&\quad2\int_{0}^\infty \frac{
\delta h_m(f)\delta h^*_R(f)+\delta h_m^*(f)\delta h_R(f)}{S_n(f)}df.
\end{eqnarray}
The derivations of these expressions are given in some detail in
Sec.~III of Ref.~\cite{Lindblom2008}.
\section{Calibration Accuracy Standards}
\label{s:CalibrationAccuracyStandards}
The expression for the difference between the measured and optimal
signal-to-noise ratios $\delta\rho=\rho_m-\rho$ in
Eq.~(\ref{e:modelpluscalibrationerror}) is remarkably simple,
depending only on the difference between the waveform errors,
$\delta\rho =\delta\rho(\delta h_m-\delta h_R)$. At the most basic
level, the waveform-accuracy standards developed in
Ref.~\cite{Lindblom2008} were obtained by limiting the size of the
waveform errors to those producing acceptably small changes in
$\delta\rho$. Since $\delta\rho$ depends only on the difference in
waveform errors, $\delta h_m-\delta h_R$, from
Eq.~(\ref{e:modelpluscalibrationerror}), it follows that the
ideal-detector waveform accuracy standards can be extended to the
realistic-detector case ($\delta h_R\neq 0$) simply by replacing
$\delta h_m$ with $\delta h_m-\delta h_R$ in those ideal-detector
standards. Thus the optimal accuracy requirement on the combined
(calibration plus modeling) waveform errors that ensures no loss of
scientific information during the measurement process is,
\begin{eqnarray}
\langle \delta h_m-\delta h_R | \delta h_m-\delta h_R\rangle < 1.
\label{e:measurmentoptimal}
\end{eqnarray}
This is the generalization of the ideal-detector condition derived as
Eq.~(5) of Ref.~\cite{Lindblom2008}. Similarly the optimal accuracy
requirement on the combined waveform errors that ensures no
significant reduction in the rate of detections is,
\begin{eqnarray}
\langle (\delta h_m-\delta h_R)_\perp | (\delta h_m-\delta
h_R)_\perp\rangle < 2\rho^2\epsilon_\mathrm{max}.
\end{eqnarray}
The parameter $\epsilon_\mathrm{max}$ determines the fraction of
detections that will be missed as a result of calibration and modeling
errors, as discussed in some detail in Ref.~\cite{Lindblom2008}. As in
the ideal-detector case, it is more convenient to convert this optimal
accuracy requirement for detection into the slightly stronger
sufficient condition,
\begin{eqnarray}
\langle \delta h_m-\delta h_R | \delta h_m-\delta
h_R\rangle < 2\rho^2\epsilon_\mathrm{max},
\label{e:detctionbasic}
\end{eqnarray}
which does not require a knowledge of the projection $(\delta
h_m-\delta h_R)_\perp$. This simpler expression is the generalization
of the ideal-detector condition derived as Eq.~(15) of
Ref.~\cite{Lindblom2008}.
Both of these accuracy standards, Eqs.~(\ref{e:measurmentoptimal}) and
(\ref{e:detctionbasic}), on the combined (calibration plus modeling)
waveform errors are conditions on the noise-weighted norm of $\delta
h_m-\delta h_R$. The waveform-modeling errors, $\delta h_m$, have an
entirely different source and are therefore completely uncorrelated
with the calibration errors, $\delta h_R$. It is useful therefore to
express these accuracy standards in a form that isolates each type of
error. This can be done with a simple application of the Schwarz
inequality, cf. Eq.~(44) of Ref.~\cite{Lindblom2008}:
\begin{eqnarray}
&&\langle \delta h_m - \delta h_R | \delta h_m - \delta h_R\rangle
\leq \nonumber\\
&&\qquad\qquad\qquad
\left(\sqrt{\langle\delta h_m|\delta h_m\rangle}
+\sqrt{\langle\delta h_R|\delta h_R\rangle}\right)^2.
\qquad
\end{eqnarray}
This inequality is fairly tight, in the sense that equality is
actually achieved for the case $\delta h_m = - \delta h_R$. Based on
this inequality, the following slightly stronger, sufficient, versions
of the accuracy requirements can be constructed for measurement,
\begin{eqnarray}
\sqrt{\langle \delta h_m| \delta h_m\rangle}
+\sqrt{\langle \delta h_R| \delta h_R\rangle}
< 1,
\label{e:measurementlimit}
\end{eqnarray}
and for detection,
\begin{eqnarray}
\sqrt{\langle \delta h_m| \delta h_m\rangle}
+\sqrt{\langle \delta h_R| \delta h_R\rangle}
< \sqrt{2\epsilon_\mathrm{max}}\,\rho.
\label{e:detectionlimit}
\end{eqnarray}
These conditions reduce to the model-waveform accuracy standards
derived in Ref.~\cite{Lindblom2008} for the ideal-detector case
($\delta h_R=0$). These accuracy standards place more stringent
conditions, however, on the waveform-modeling error when there is a
non-negligible level of calibration error.
The allowed error levels, due to calibration and waveform
modeling, can be apportioned between the two error sources in any way
that is consistent with Eqs.~(\ref{e:measurementlimit}) and
(\ref{e:detectionlimit}). Determining the most efficient way to do
this would require an analysis of the relative costs of improving the
accuracies of each error source. It seems likely that adopting
standards with comparable requirements for each type of error will be
close to optimal. Let us explore in some detail then what the
resulting calibration and modeling accuracy requirements would be in
that case. From Eqs.~(\ref{e:measurementlimit}) and
(\ref{e:detectionlimit}) it follows that the appropriate limits
become
\begin{eqnarray}
\langle \delta h_m| \delta h_m\rangle=
\langle \delta h_R| \delta h_R\rangle
< \frac{1}{4},
\label{e:measurementlimit1}
\end{eqnarray}
for measurement, and
\begin{eqnarray}
\langle \delta h_m| \delta h_m\rangle=
\langle \delta h_R| \delta h_R\rangle
< \frac{\epsilon_\mathrm{max}}{2}\rho^2,
\label{e:detectionlimit1}
\end{eqnarray}
for detection. These waveform-modeling standards are stricter by a
factor of two than those derived in Ref.~\cite{Lindblom2008} for the
ideal-detector case.
It is useful to translate these accuracy requirements into a more familiar
language, by noting that (to lowest order) the waveform error can
be written in terms of logarithmic amplitude and phase errors:
$\delta h_m \approx h_e(\delta \chi_m+i\delta\Phi_m)$.
It follows that the norm of the waveform-modeling error can be
expressed in the form,
\begin{eqnarray}
\langle \delta h_m|\delta h_m\rangle =
\rho^2\left(\overline{\delta\chi_m}^{\,2} +\overline{ \delta\Phi_m}^{\,2}
\right),
\end{eqnarray}
where the signal and noise weighted averages of the amplitude
and phase errors are defined by,
\begin{eqnarray}
\overline{\delta\chi_m}^{\,2}&=&\int_{0}^\infty
\left(\delta\chi_m\right)^2
\frac{4|h_e|^2}{\rho^2S_n(f)}df,\\
\overline{\delta\Phi_m}^{\,2}&=&\int_{0}^\infty
\left(\delta\Phi_m\right)^2
\frac{4|h_e|^2}{\rho^2S_n(f)}df.
\end{eqnarray}
Note that the weight term, $4|h_e|^2/\rho^2S_n$, which appears in
these definitions has integral one; so these are true (signal and
noise weighted) averages of $\delta \chi_m$ and $\delta \Phi_m$. The
averages of the calibration amplitude and phase errors, are defined
analogously. In terms of these averages then, the waveform-accuracy
standards of Eqs.~(\ref{e:measurementlimit1}) and
(\ref{e:detectionlimit1}) become
\begin{eqnarray}
\sqrt{\overline{\delta\chi_m}^{\,2} +\overline{ \delta\Phi_m}^{\,2}}=
\sqrt{\overline{\delta\chi_R}^{\,2} +\overline{ \delta\Phi_R}^{\,2}}
< \frac{1}{2\rho},
\label{e:measurementlimit2}
\end{eqnarray}
for measurement and
\begin{eqnarray}
\sqrt{\overline{\delta\chi_m}^{\,2} +\overline{ \delta\Phi_m}^{\,2}}=
\sqrt{\overline{\delta\chi_R}^{\,2} +\overline{ \delta\Phi_R}^{\,2}}
< \sqrt{\frac{\epsilon_\mathrm{max}}{2}},
\label{e:detectionlimit2}
\end{eqnarray}
for detection.
For Advanced LIGO the maximum signal-to-noise ratio for a binary
black-hole signal may be as large as about 100, so the resulting
accuracy requirements sufficient for measurement, from
Eq.~(\ref{e:measurementlimit2}), for such events are
\begin{eqnarray}
\sqrt{\overline{\delta\chi_m}^{\,2} +\overline{ \delta\Phi_m}^{\,2}}=
\sqrt{\overline{\delta\chi_R}^{\,2} +\overline{ \delta\Phi_R}^{\,2}}
\lesssim 0.005.
\label{e:measurementLIGO}
\end{eqnarray}
Thus the averages of the frequency-domain amplitude and phase errors
must be at about the 0.35\% and the 0.0035 radian levels respectively
for measurement. If the Advanced LIGO search template banks are
constructed (as in Initial LIGO) with waveform templates spaced so
that no point in the template subspace has a mismatch larger than 0.03
from some element in the bank, then the maximum mismatch
$\epsilon_\mathrm{max}$ must be chosen to be 0.005 to ensure the
signal loss rate is about 10\%, cf. Ref.~\cite{Lindblom2008}. In this
case the resulting accuracy requirements for both waveform and
calibration errors sufficient for detection from
Eq.~(\ref{e:detectionlimit2}) are
\begin{eqnarray}
\sqrt{\overline{\delta\chi_m}^{\,2} +\overline{ \delta\Phi_m}^{\,2}}=
\sqrt{\overline{\delta\chi_R}^{\,2} +\overline{ \delta\Phi_R}^{\,2}}
\lesssim 0.05.
\label{e:detectionLIGO}
\end{eqnarray}
Thus the accuracy requirements for detection are an order of magnitude
less stringent than those needed for measurement of the strongest
likely sources in Advanced LIGO. The required averages of the
frequency-domain amplitude and phase errors must be at about the 3.5\%
and the 0.035 radian levels respectively for detection.
It is easy to imagine how two different sets of model waveforms could
be designed to accomplish the two distinct data-analysis tasks. One
set could be prepared for use in searches of gravitational-wave
signals using the lower accuracy standards needed for detection. And
a second set could be prepared with the higher accuracy standards
needed for measurement, but only in the very small portion of waveform
parameter space where they are needed for measurements on previously
detected signals. At present it seems unlikely that it will be
possible to perform detector calibrations in a way that provides the
lower level of calibration accuracy needed for detections at all
times, and only calibrates to the higher accuracy standards needed for
measurements {\it{ex post facto}} in those data segments where
detections have been made. So it seems likely that it will be
necessary to calibrate the detectors at the level needed for
measurements, e.g., according to the standards of
Eq.~(\ref{e:measurementlimit2}), whenever data is collected. In this
case, the accuracy standard for detections for waveform-modeling error
could be relaxed somewhat to the level
\begin{eqnarray}
\sqrt{\overline{\delta\chi_m}^{\,2} +\overline{ \delta\Phi_m}^{\,2}}
< \sqrt{2\epsilon_\mathrm{max}}-\frac{1}{2\rho_\mathrm{max}}\approx
0.095,
\label{e:detectionLIGO1}
\end{eqnarray}
which is almost identical to the ideal-detector requirement derived in
Ref.~\cite{Lindblom2008}.
\section{User Friendly Standards}
\label{s:UserFriendlyStandards}
The accuracy standards derived in
Sec.~\ref{s:CalibrationAccuracyStandards} for the combined
(calibration plus modeling) waveform errors are not easily applied
using the basic expressions given in Eqs.~(\ref{e:measurementlimit2})
and (\ref{e:detectionlimit2}). Model waveforms are generally
constructed in the time domain (e.g., by performing numerical
simulations), so verifying the basic frequency-domain standards using
estimates of the time-domain errors is not
straightforward~\cite{Lindblom2009b}. Neither can the basic
expressions for the standards on the calibration errors be enforced in
a straightforward way. While good estimates of the frequency-domain
response-function errors are generally
available~\cite{LIGOS4Calibration}, the accuracy standards of
Eqs.~(\ref{e:measurementlimit2}) and (\ref{e:detectionlimit2}) require
computing their averages weighted by the gravitational waveform
$h_e$. What waveform should be used when applying these standards?
This section transforms the basic accuracy standards of
Eqs.~(\ref{e:measurementlimit}) and (\ref{e:detectionlimit}) into
forms that are more easily used by those responsible for calibrating
the detector, and by those responsible for constructing and verifying
the accuracy of model waveforms as well.
\begin{figure}
\centerline{\includegraphics[width=3in]{Fig1.eps}}
\caption{\label{f:CSNratio} Curves illustrate $C$, the ratio of the
standard signal-to-noise measure $\rho$ to a non-standard measure
defined in Eq.~(\ref{e:SNRatioRatio}), as a function of the total mass
for non-spinning equal-mass binary black-hole waveforms. Dashed curve
is based on the Initial LIGO noise spectrum~\cite{InitialLIGONoise};
solid curve is based on an Advanced LIGO noise
curve~\cite{AdvancedLIGONoise}.}
\end{figure}
The norm of the waveform-modeling error $\langle \delta h_m|\delta
h_m\rangle$, which appears in the accuracy standards of
Eqs.~(\ref{e:measurementlimit}) and (\ref{e:detectionlimit}), is
constructed from the frequency-domain estimates of those errors. It
is not straightforward to determine useful estimates of these
frequency-domain errors from the time-domain waveform errors that are
directly accessible to the waveform-modeling community. It is
useful therefore to transform the expression for the limits on the
modeling error into ones based directly on time-domain estimates of
the errors. This can be done, following the argument in Sec.~II.C of
Ref.~\cite{Lindblom2008}, using an application of Parseval's theorem:
\begin{eqnarray}
\langle\delta h_m|\delta h_m\rangle \leq
\frac{\rho^2}{C^{\,2}}\frac{||\delta h_m(t)||^2}{||h_e(t)||^2},
\label{e:time-domain-limit}
\end{eqnarray}
where $||\delta h_m(t)||$ is the $L^2$ norm of $\delta h_m(t)$, defined
as
\begin{eqnarray}
||\delta h_m(t)||^2=\int_{-\infty}^\infty |\delta h_m(t)|^2 dt,
\end{eqnarray}
and where $C$ is the ratio of the standard signal-to-noise measure
$\rho$ to a non-standard measure:
\begin{eqnarray}
C^{\,2}=\rho^2\left(\frac{2||h_e(t)||^2}{\mathrm{min}\, S_n(f)}\right)^{-1}.
\label{e:SNRatioRatio}
\end{eqnarray}
Figure~\ref{f:CSNratio} illustrates $C$ for non-spinning equal-mass
binary black-hole waveforms (cf. Fig.~4 of Ref.~\cite{Lindblom2008}).
This quantity can be evaluated in a straightforward way when any class
of model waveforms is computed. The right side of
Eq.~(\ref{e:time-domain-limit}) is always larger than the
noise-weighted norm $\langle \delta h_m|\delta h_m\rangle$ that
appears on the left. Sufficient conditions for model-waveform
accuracy based on the time-domain $L^2$ norm $||\delta h_m(t)||$ can
be obtained therefore by replacing $\langle \delta h_m|\delta
h_m\rangle$ with the right side of Eq.~(\ref{e:time-domain-limit})
wherever it appears in the accuracy standards of
Eqs.~(\ref{e:measurementlimit}) and (\ref{e:detectionlimit}).
The norm of the waveform error caused by detector calibration errors,
$\langle \delta h_R|\delta h_R\rangle$, is also difficult to evaluate
because it depends on the gravitational waveform $h_e$ in addition to
the purely detector-based errors $\delta \chi_R$ and $\delta \Phi_R$.
The detector calibration-error terms in this norm can be isolated
from the gravitational-waveform terms in a straightforward way
using the inequality
\begin{eqnarray}
\langle \delta h_R | \delta h_R\rangle &=&
\int_0^\infty \left(\delta\chi_R^2+\delta\Phi_R^2\right)
\frac{4|h_e|^2}{S_n}df,\nonumber\\
&\leq& \rho^2\,\mathrm{max}\left(\delta\chi_R^2+\delta\Phi_R^2\right).
\label{e:calibrationerror3}
\end{eqnarray}
The right side of Eq.~(\ref{e:calibrationerror3}) is very easy to
evaluate, and is always larger than the noise-weighted norm $\langle
\delta h_R|\delta h_R\rangle$ that appears on the left. Sufficient
conditions for calibration accuracy based on
$\mathrm{max}\left(\delta\chi_R^2+\delta\Phi_R^2\right)$ can be
obtained therefore by replacing $\langle \delta h_R|\delta h_R\rangle$
with the right side of Eq.~(\ref{e:calibrationerror3}) wherever it
appears in the accuracy standards of Eqs.~(\ref{e:measurementlimit})
and (\ref{e:detectionlimit}).
In some circumstances the calibration-accuracy standards obtained
using $\mathrm{max}\left(\delta\chi_R^2+\delta\Phi_R^2\right)$ may be
much stronger than necessary; for example when
$\delta\chi_R^2+\delta\Phi_R^2$ is sharply peaked at frequencies where
the detector noise is large. In this case it may be advantageous to
employ a different simplification of the accuracy standards. The
detector calibration-error terms in the noise-weighted norm $\langle
\delta h_R|\delta h_R\rangle$ can also be isolated from the
gravitational-waveform terms with an application of the Schwarz
inequality:
\begin{eqnarray}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\langle \delta h_R | \delta h_R\rangle =
\int_0^\infty \left(\delta\chi_R^2+\delta\Phi_R^2\right)
\frac{4|h_e|^2}{S_n}df,\nonumber\\
&&\!\!\!\!\!\!\!\!\!\!
\leq \sqrt{\int_0^\infty \frac{4|h_e|^4}{S_n}df}\times\nonumber\\
&&\quad
\left(\sqrt{\int_0^\infty \frac{4\delta\chi_R^4}{S_n}df}
+\sqrt{\int_0^\infty \frac{4\delta\Phi_R^4}{S_n}df}\right).
\label{e:calibrationerror0}
\end{eqnarray}
This inequality can be re-written in the more compact form,
\begin{eqnarray}
\langle\delta h_R | \delta h_R\rangle \leq \frac{\rho^2}{\widetilde{C}^2}
\left(\widetilde{\delta \chi_R}^2+\widetilde{\delta \Phi_R}^2\right),
\label{e:calibrationerror}
\end{eqnarray}
by defining a few useful quantities. The noise-weighted averages of
$\widetilde{\delta\chi_R}$ and $\widetilde{\delta\Phi_R}$ are defined as
\begin{eqnarray}
\widetilde{\delta\chi_R}^4 &=& \int_0^\infty \delta\chi_R^4
\frac{4\bar n^2}{S_n}df,
\label{e:noiseweightedaverageChi}\\
\widetilde{\delta\Phi_R}^4 &=& \int_0^\infty \delta\Phi_R^4
\frac{4\bar n^2}{S_n}df,
\label{e:noiseweightedaveragePhi}
\end{eqnarray}
where the total detector noise, $\bar n$, is defined as
\begin{eqnarray}
\frac{1}{\bar n^2}=\int_0^\infty \frac{4}{S_n}df.
\end{eqnarray}
Note that the weight, $4 \bar n^2/S_n$, which appears in
Eqs.~(\ref{e:noiseweightedaverageChi}) and
(\ref{e:noiseweightedaveragePhi}) has integral one; so these are true
(noise-weighted) averages of $\delta\chi_R$ and $\delta\Phi_R$. Note
the unusual fourth power of the averaged quantity which appears in
these definitions. This is required because that is the power of the
error terms which appear on the right side of
Eq.~(\ref{e:calibrationerror0}). Finally, the quantity
$\widetilde{C}$ that appears in Eq.~(\ref{e:calibrationerror}) is the
ratio of the standard signal-to-noise measure $\rho$ to another
non-standard measure:
\begin{eqnarray}
\widetilde{C}^{4}
= \rho^4\left(\int_0^\infty \frac{4|h_e|^4}{\bar n^2 S_n}df\right)^{-1}.
\label{e:ctilde}
\end{eqnarray}
Figure~\ref{f:CtildeSNratio} illustrates $\widetilde{C}$ for
non-spinning equal-mass binary black-hole waveforms using standard
Initial and Advanced LIGO noise
curves~\cite{InitialLIGONoise,AdvancedLIGONoise}. This quantity can be
evaluated in a straightforward way when any class of model waveforms
is computed. It isn't completely clear why the Advanced LIGO version
of this curve is almost a factor of two smaller than the Initial LIGO
curve. This may be due to the fact that the integral of $|h_e(f)|^4$
in $\widetilde{C}$, Eq.~(\ref{e:ctilde}), is dominated by its low
frequency contributions where $h_e(f)$ is largest. The Advanced LIGO
noise curve is significantly smaller in this low-frequency range, so
these contributions are much more significant in that case.
\begin{figure}
\centerline{\includegraphics[width=3in]{Fig2.eps}}
\caption{\label{f:CtildeSNratio} Curves illustrate $\widetilde{C}$, the
ratio of the standard signal-to-noise measure $\rho$ to another
non-standard measure defined in Eq.~(\ref{e:ctilde}), as a function of
the total mass for non-spinning equal-mass binary black-hole
waveforms. Dashed curve is based on the Initial LIGO noise spectrum;
solid curve is based on an Advanced LIGO noise curve.}
\end{figure}
The maximum calibration-error,
$\mathrm{max}\left(\delta\chi_R^2+\delta\Phi_R^2\right)$, and the
noise-weighted averages, $\widetilde{\delta\chi_R}$ and
$\widetilde{\delta\Phi_R}$, which appear in
Eqs.~(\ref{e:calibrationerror3}) and (\ref{e:calibrationerror})
depend only on information that pertains to the detector itself. All
of the dependence on the waveform $h_e$ in the original norm,
$\langle\delta h_R|\delta h_R\rangle$, has been moved into the
signal-to-noise ratio $\rho$ and the quantity $\widetilde{C}$. Thus
the right sides of Eqs.~(\ref{e:calibrationerror3}) and
(\ref{e:calibrationerror}) should be much easier for those performing
detector calibrations to evaluate. The right sides of
Eqs.~(\ref{e:calibrationerror3}) and (\ref{e:calibrationerror}) are
always larger than the noise-weighted norm $\langle \delta h_R|\delta
h_R\rangle$ that appears on the left. Sufficient conditions for
model-waveform accuracy based on the maximum calibration error
$\mathrm{max}\left(\delta\chi_R^2+\delta\Phi_R^2\right)$ (or the
noise-weighted averages of the calibration errors
$\widetilde{\delta\chi_R}$ and $\widetilde{\delta\Phi_R}$) can be
obtained therefore by replacing $\langle \delta h_R|\delta h_R\rangle$
with the right side of Eq.~(\ref{e:calibrationerror3}) or
(\ref{e:calibrationerror}) wherever it appears in the accuracy
standards of Eqs.~(\ref{e:measurementlimit}) and
(\ref{e:detectionlimit}). The resulting accuracy standards based
on the maximum calibration error [and using
the re-written norm of the waveform-modeling error from
Eq.~(\ref{e:time-domain-limit})] become,
\begin{eqnarray}
\frac{1}{C}\frac{||\delta h_m(t)||}{||h_e(t)||}
+\sqrt{\mathrm{max}\left(\delta\chi_R^2+\delta\Phi_R^2\right)} \leq
\frac{1}{\rho},
\label{e:measurementstandard}
\end{eqnarray}
for measurement, and
\begin{eqnarray}
\frac{1}{C}\frac{||\delta h_m(t)||}{||h_e(t)||}
+\sqrt{\mathrm{max}\left(\delta\chi_R^2+\delta\Phi_R^2\right)}
\leq \sqrt{2\epsilon_\mathrm{max}},
\label{e:detectionstandard}
\end{eqnarray}
for detection. Analogous versions of the standards based on the
noise-weighted averages of the calibration errors,
$\widetilde{\delta\chi_R}$ and $\widetilde{\delta\Phi_R}$, can be
obtained in a similar way. Either version of the standards is
sufficient to guarantee the needed level of accuracy for
gravitational-wave data analysis. The most efficient choice for a
particular detector, and for a particular type of source, will be
determined by whether
$\mathrm{max}\left(\delta\chi_R^2+\delta\Phi_R^2\right)$ or
$(\widetilde{\delta\chi_R}^2+\widetilde{\delta\Phi_R}^2)/\widetilde{C}^2$
is smaller.
\section{Discussion}
\label{s:Discussion}
A new set of accuracy standards have been developed for the calibration
and modeling errors of the waveforms used for gravitational-wave data
analysis. The basic standards, Eqs.~(\ref{e:measurementlimit}) and
(\ref{e:detectionlimit}), are expressed most naturally in terms of the
noise-weighted inner products commonly used in gravitational-wave data
analysis. These basic expressions are not very convenient for
actually applying the standards, however. So the basic standards have
been transformed into expressions that are easier to apply:
Eqs.~(\ref{e:measurementstandard}) and (\ref{e:detectionstandard}).
These new representations of the accuracy standards are slightly
stronger, and if satisfied are sufficient to ensure the original
standards are satisfied.
The new accuracy standards, Eqs.~(\ref{e:measurementstandard}) and
(\ref{e:detectionstandard}), prescribe a maximum for the combined
calibration and modeling errors of the gravitational waveforms, not
for each type of error separately. This means that an increased
accuracy in one allows a somewhat weaker requirement on the other.
Determining how to aportion the accuracy between the two error sources
in an optimal way would require a careful analysis of the costs
involved in reducing the error from each source. It seems reasonable
to expect that requiring approximately equal accuracy for the
calibration and modeling errors will be close to optimal. In this
case the transformed expressions for the new accuracy requirements are
\begin{eqnarray}
\frac{1}{C}\frac{||\delta h_m(t)||}{||h_e(t)||}
\approx
\sqrt{\mathrm{max}\left(\delta\chi_R^2+\delta\Phi_R^2\right)}
\lesssim \frac{1}{2\,\rho_\mathrm{max}},
\label{e:measurementstandard2}
\end{eqnarray}
for measurement, and
\begin{eqnarray}
\frac{1}{C}\frac{||\delta h_m(t)||}{||h_e(t)||}
\approx
\sqrt{\mathrm{max}\left(\delta\chi_R^2+\delta\Phi_R^2\right)}
\lesssim \sqrt{\frac{\epsilon_\mathrm{max}}{2}},
\label{e:detectionstandard2}
\end{eqnarray}
for detection; the constant $\rho_\mathrm{max}$ represents the
signal-to-noise ratio of the strongest detected source. If the
calibration of the instrument must be maintained at the level needed
for accurate measurements of the strongest anticipated sources during
the entire data collection period, then the accuracy requirements on
the waveform-modeling error can be relaxed somewhat for detection:
\begin{eqnarray}
\frac{1}{C}\frac{||\delta h_m(t)||}{||h_e(t)||}
\lesssim \sqrt{2\epsilon_\mathrm{max}}-\frac{1}{2\,
\rho_\mathrm{max}}.
\label{e:detectionstandard3}
\end{eqnarray}
These new accuracy standards should be applicable for essentially any
gravitational-wave detector and any type of model waveform used in the
data analysis process. To apply the standards for each particular
case, the quantities $\rho_\mathrm{max}$, $\epsilon_\mathrm{max}$, $C$
and (perhaps) $\widetilde{C}$ must be evaluated for the particular
family of model waveforms, using the noise spectrum of the particular
detector. Some insight can be gained into what the standards will
actually look like by examining the case of binary black-hole
inspiral-merger-ringdown waveforms using the Advanced LIGO noise
curve. The quantities $C$ and $\widetilde{C}$ have been computed for
this case using an equal-mass non-spinning binary black-hole waveform
obtained by matching together numerical and post-Newtonian
waveforms~\cite{Scheel2008,Boyle2008b}. The results are depicted in
Figs.~\ref{f:CSNratio} and \ref{f:CtildeSNratio} for binary systems
with total masses in the range $4-400M_\odot$. From these graphs we
see that $C\gtrsim0.019$ and $\widetilde{C}\gtrsim0.23$ for these
waveforms and the Advanced LIGO noise curve. The strongest binary
black-hole signals in Advanced LIGO are expected to have
signal-to-noise ratios that may be as large as
$\rho_\mathrm{max}\approx 100$. Assuming the template banks of model
waveforms are constructed in the same way as those for Initial LIGO,
the maximum mismatch compatible with a 10\% event loss rate is
$\epsilon_\mathrm{max}=0.005$. Substituting these values into the
accuracy standards of Eqs.~(\ref{e:measurementstandard2}) and
(\ref{e:detectionstandard2}) results in the following calibration and
waveform modeling accuracy requirements for Advanced LIGO:
\begin{eqnarray}
53\frac{||\delta h_m(t)||}{||h_e(t)||}
\approx
\sqrt{\mathrm{max}\left(\delta\chi_R^2+\delta\Phi_R^2\right)}
\lesssim 0.005
\label{e:LIGOmeasurement}
\end{eqnarray}
for measurement, and
\begin{eqnarray}
53\frac{||\delta h_m(t)||}{||h_e(t)||}
\approx
\sqrt{\mathrm{max}\left(\delta\chi_R^2+\delta\Phi_R^2\right)}
\lesssim 0.05
\label{e:LIGOdetection}
\end{eqnarray}
for detection. If the calibration accuracy is fixed at the higher
level needed for measurements in the strongest sources for the entire
period in which data is collected, then the standard on
waveform-modeling error for detection can be relaxed to
\begin{eqnarray}
53\frac{||\delta h_m(t)||}{||h_e(t)||}
\lesssim 0.095.
\label{e:LIGOaltdetection}
\end{eqnarray}
A somewhat troubling feature of these conditions is the rather large
coefficient $1/{C}\approx 53$ that multiplies the $L^2$ norms of
$\delta h_m(t)$ in these expressions. This is really just an artifact
of the extremely long model waveform (containing about 1000 wave
cycles) used here when evaluating $C$~\cite{Lindblom2009b}. The
quantity $C$ contains the $L^2$ norm of the waveform $h_e(t)$ in its
denominator, and this norm becomes quite large when it is estimated
using model waveforms $h_m$ with many wave cycles. This issue is
discussed at some length in Ref.~\cite{Lindblom2009b} and will not be
addressed further here, since it does not bear directly on the main
focus of this paper: deriving the optimal levels of calibration error.
\acknowledgments I thank Michael Landry for stimulating my interest in
the questions addressed here and for comments and suggestions for
improving an earlier draft of this paper. I also think Benjamin Owen
and Yanbei Chen for several helpful discussions on these issues. This
research was supported in part by a grant from the Sherman Fairchild
Foundation, by NSF grants DMS-0553302, PHY-0601459, and PHY-0652995,
and by NASA grant NNX09AF97G.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,510 |
#
HARVEST HOUSE PUBLISHERS
EUGENE, OREGON
Cover by Koechel Peterson & Associates, Inc., Minneapolis, Minnesota
Cover photos © Koechel Peterson & Associates / iStockphoto / Thinkstock
Backcover author photo Ruettgers Photography
A HOME AT TRAIL'S END
Copyright © 2013 by Melody A. Carlson
Published by Harvest House Publishers
Eugene, Oregon 97402
www.harvesthousepublishers.com
Library of Congress Cataloging-in-Publication Data
Carlson, Melody.
A home at trail's end / Melody Carlson.
pages cm—(Homeward on the Oregon Trail Series; Book 3)
ISBN 978-0-7369-4875-3 (pbk.)
ISBN 978-0-7369-4876-0 (eBook)
1. Widows—Fiction. 2. Women pioneers—Fiction. 3. Oregon Territory—History—Fiction. I. Title.
PS3553.A73257H56 2013
813'.54—dc23
2012044765
All rights reserved. No part of this electronic publication may be reproduced, stored in a retrieval system, distributed, or transmitted in any form or by any means—electronic, mechanical, digital, photocopy, recording, or any other—without the prior written permission of publisher. The authorized purchaser has been granted a non-transferable, non-exclusive, and non-commercial right to access and view this electronic publication and agrees to do so only in accordance with the terms of use under which it was purchased or transmitted. Participation in or encouragement of piracy of copyrighted materials in violation of author's and publisher's rights is strictly prohibited.
Contents
Primary Returning Characters from A Dream for Tomorrow
Chapter One
Chapter Two
Chapter Three
Chapter Four
Chapter Five
Chapter Six
Chapter Seven
Chapter Eight
Chapter Nine
Chapter Ten
Chapter Eleven
Chapter Twelve
Chapter Thirteen
Chapter Fourteen
Chapter Fifteen
Chapter Sixteen
Chapter Seventeen
Chapter Eighteen
Chapter Nineteen
Chapter Twenty
Chapter Twenty-One
Chapter Twenty-Two
Chapter Twenty-Three
Chapter Twenty-Four
Chapter Twenty-Five
Chapter Twenty-Six
Chapter Twenty-Seven
Chapter Twenty-Eight
Chapter Twenty-Nine
Chapter Thirty
Chapter Thirty-One
Chapter Thirty-Two
Discussion Questions
About the Publisher
Primary Returning Characters from
A Dream for Tomorrow
Elizabeth Anne Martin
JT (12) and Ruth Anne (8)
Eli Kincaid, former wagon train scout
Asa and Clara Dawson, Elizabeth's parents
Matthew and Jess Dawson, Elizabeth's brother and sister-in-law
Brady, Elizabeth's farmhand and a freed slave
Malinda Martin, Elizabeth's widowed sister-in-law
Todd (15), Emily (13), Bart (12), and Susannah (9)
William Bramford, a widowed lawyer from Boston
Jeremiah (18), Belinda (17), and Amelia (16)
Hugh and Lavinia Prescott, friends of William Bramford, also from Boston
Julius (19), Evelyn (16), and Augustus (13)
Bert and Florence Flanders
Mahala (18), Ezra (16), Hannah (13), Walter (11), and Tillie (8)
Jane Taylor, fellow emigrant whose husband was killed on the wagon train
Chapter One
October 1857
Elizabeth felt a spring in her step as she guided Eli around the borders of her property. At the corner, she paused to point out the land adjoining hers. "That section combined with my acreage equals a whole parcel. But because I'm single, I could only file for a half parcel."
"I see." He surveyed the meadow that ran clear down to the river and then up into the wooded hills behind it.
"John and Malinda had been saving a whole parcel for us..." She stopped herself. "I mean, for James and me."
He nodded. "I understand."
"And of course, both John and James are gone now." She didn't want to dwell too much on this sad fact. It seemed more important to go forward than to look back. "So I claimed this half when we arrived." She waved her hand toward her land. "Because it was closer to Malinda as well as my family's parcels. But the other half is still available."
He smiled at her. "So you don't mind if I file a claim on it."
She laughed. "Being that you've filed a claim on my heart, it seems you are entitled to the land as well."
"This is fine land, Elizabeth. I can see why you and your family felt it was worth the effort to come out this way."
"Father thinks the meadows along the river are perfect for grazing."
"Yes." Eli's brow creased. "It's hard to believe the government is selling this land for so cheap. But I know they want it settled." He glanced over to the river. "Are there any Indians around here? I know that's the Coquille River, and I heard the Coquille tribe is friendly. This looks like the kind of place Indians would be quite comfortable in."
"To be honest, I haven't seen a single Indian since we left Empire City," she told him. "But Malinda wrote to me, saying there had been trouble in these parts. Perhaps the Indians have moved on." For some reason this wasn't a topic that many of the settlers spoke of...whether it was because it made them uncomfortable or because the Indians were not a problem was unclear.
"I know the ocean isn't too far away. It's possible they've gone over there to fish for salmon or collect clams."
"I'm looking forward to seeing the ocean," she told him.
He grinned. "Yes, we'll have to do that together." He looked intently into her eyes. "Now that you've agreed to become my bride, is there anyone I should see to get permission? Asa perhaps?"
She laughed. "No, I do not need my father's permission to marry, but I would like my parents' blessing. However, there is someone you should ask. Rather, two someones. And school let out a bit ago, so I'm sure they'll be here soon."
"I can't wait to see them."
Now she told him about JT helping to drive the livestock along the Columbia River. "It was almost as if he left a boy and grew into a man while he was gone. He's still talking about the adventures they had."
"Did he take his Bowie knife?"
"He certainly did. And I let him take your canteen as well."
Eli chuckled. "I'm glad you did."
"And Ruth is turning into quite an artist. You'll have to ask her to show you the drawings she did while we traveled down the Columbia River. She says she's saving them for her children." She laughed. "Can you imagine? Ruth planning for her own children?"
"I think we would make fine grandparents."
Elizabeth was touched but didn't respond as she pointed to where her horse Molly, with JT and Ruth on her back, was coming toward them. "There they are." She clutched her hands together anxiously. What if the children were unhappy about her engagement to Eli? Perhaps this was a conversation she should have privately with them. And yet she knew they needed to hear this news now—she had promised them that they would be the first to know.
"Oh, dear," she said quietly to Eli as she smiled and waved to Ruth and JT. "I'm feeling extremely nervous right now."
"So am I," Eli admitted.
"Eli!" JT yelled from the horse, nudging Molly to go faster.
Eli went to meet them, helping Ruth from the horse as JT hopped down. Both children hugged Eli. "When did you get here?" JT asked with enthusiasm.
"Just now," Eli told him.
"How long will you stay?" Ruth asked.
"That depends," he told her.
"On what?" JT peered curiously at him.
"Well...your mother and I want to ask you both about something." He tossed her an uneasy glance.
"That's right," she added. "Eli has asked me a very important question." She looked back at him, hoping he'd take it from here.
"I came here to ask your mother to become my wife," he said bravely.
JT looked stunned, but Ruth just grinned. "I knew it!" she exclaimed.
"But your mother said she can't marry without her children's approval. What do you think about me marrying your mother...and hopefully becoming your father?" He cleared his throat. "Someday..."
"I like it!" Ruth declared. "I wanted Mama to marry you a long time ago, back on the wagon train. Didn't I, Mama?"
Elizabeth just chuckled and nodded. But her eyes were fixed on JT. He was being awfully quiet just now. And his expression was very somber and hard to read. What was he thinking?
"I appreciate you giving your consent, Ruth. But we need the approval of both of you." Eli looked at JT now. "What do you think about this, JT? I realize you've been the man of the house." Eli glanced over to the stone foundation. "Well, there's not much house yet, but you've been playing the role of the man in this family. And doing an excellent job of it too." He grinned. "Your ma told me about how you helped drive the livestock through the Columbia Gorge. Takes a man to do that."
JT nodded proudly. "I know."
"So, JT," Elizabeth began, "what do you think? And please be honest."
JT looked from Eli to Elizabeth, and then a slow smile broke over his face. "I think it's a good idea. I approve." He stuck out his hand to Eli, and they shook on it.
"I appreciate that," Eli told him. "You had me worried there for a bit."
JT chuckled. "Sorry. I just wanted to give it my careful consideration."
Elizabeth laughed, patting him on the back. "I appreciate that, son."
"Does anyone else know about this?" Ruth asked. "Grandma and Grandpa or—"
"Nobody but you two," Elizabeth assured her. "Remember what I told you—you and JT would be the first to know."
"Can we go tell them now?" Ruth asked.
Elizabeth glanced at Eli, and he just shrugged. "Don't see why not."
"Yes, but let's ask them to keep it to themselves until after Malinda and Will's party tomorrow. I don't want to steal their thunder."
"Good thinking," Eli agreed.
And so they all headed over to share the good news with Asa and Clara and Matthew and Jess and Brady. Naturally, everyone was nearly as happy as Elizabeth and Eli, and they all promised to keep it under their hats.
"We don't see anyone out here much anyway," Jess assured her. "Besides our own family, that is."
"And now our family is getting bigger," Ruth said proudly.
"In more ways than one," Elizabeth added mysteriously. She glanced at Jess, who was expecting her first baby in the spring. However, other than immediate family, Jess was keeping this to herself. "How are you doing?"
Jess grinned. "I've never felt better."
"And the house is coming along nicely?" Elizabeth asked. Since Matthew and Jess' house was farther away, she hadn't been out there in a few weeks.
"Oh, yes! Matthew and Pa and Brady are quite the carpenters."
"So when is the big date?" Matthew asked Elizabeth. "I'll need to warm up my fiddle for the dancing afterward."
"I'm not sure," Elizabeth admitted.
"We haven't really discussed that yet," Eli told them. "But I say the sooner the better." He looked at Elizabeth again. "But I don't want to rush you."
She laughed nervously. "I don't really know. Maybe we should think on that a bit. And let's not forget that we don't even have a place to live yet."
"I s'pect Lizzie told you we were making her wait until last for her house," Asa told Eli. "We hadn't really planned it that way. We thought she was going to live with Malinda."
"And I don't know why you and the children don't do that," Clara told Elizabeth. "Seems you could stay up there—at least until the wedding."
"Except that Malinda will have a full house before long. I heard that Will's girls are going to move in with her soon. Will and Jeremiah will keep working on the cabin on his unit...although it sounds like that house might go to Jeremiah now that he and Mahala are planning to wed."
"So many weddings." Clara rubbed her hands together. "Won't it be fun!"
Elizabeth looked at Eli. "Maybe we shouldn't make plans for our wedding until after we get our cabin built."
He just nodded. "That makes sense."
She was glad he agreed, but she also wondered how good a plan it really was. Having a cabin finished sounded like a long way down the road. Still, she wasn't going to worry about it today.
"Where will you stay?" Asa asked Eli.
Eli scratched his chin. "I hadn't given it much thought. But I can camp most anywhere."
Clara made a concerned frown. "Anywhere...except on Elizabeth's land."
"But we have lots of room," Ruth injected.
Asa tweaked one of her braids and then grinned. "Why don't you stay here with us?" he asked Eli.
"I'd be much obliged."
"Watch out," Elizabeth teasingly warned Eli. "They'll probably try to put you to work if you're living here."
"No," Asa assured her. "I'll bet Eli will want to get to work on your place as soon as possible."
"Elizabeth has already made a nice start with that stone foundation," Eli said with pride. "It looks good and square."
"'Ceptin' that she made it a might too big," Matthew teased. "Did you notice?"
Eli shook his head. "I thought it was a good size for a family—with room to grow."
######
For the past couple of weeks, Elizabeth had capitalized on the good weather by sprouting seeds. As a result she now had dozens of tiny delicate plants—apples, peaches, pears, plums, and berries that were nearly ready to be planted in soil. JT was helping his uncle today, but Ruth was on hand to help Elizabeth with the task of planting. Their first task was to finish the stick fence they'd been building to protect the plants from foraging critters. Elizabeth was well aware of the damage deer and rabbits could do to young plants.
"Is this going to be high enough to keep deer out?" Ruth asked as they pounded sticks into the ground.
"No," Elizabeth admitted. "But I plan to put tall poles every few feet. I'll string wire about this tall." She held her hand up as high as she could reach. "And then I'll tie colorful rags that will flutter in the wind. Hopefully they will dissuade the deer from jumping the fence."
"And Flax will help too. He's a good watchdog."
As she and Ruth worked, Elizabeth was surprised that Eli didn't come by to say hello. In fact, as she strung the wire, she realized how much she would have appreciated his help. Even so, they were making good progress. And by late afternoon, not only was the seedling fortress fairly secure, but she and Ruth had planted most of the young plants as well. How many would survive remained to be seen. But hopefully by next fall, she would have enough little trees to transplant into a small fruit orchard.
Just as she and Ruth were finishing, JT came home from helping with the men. "We started putting up Uncle Matthew's roof," he told her as he came over to see the progress in the seedling garden. "Eli was there to help, and that made it go lots faster."
"So that's where Eli was today." She stood up straight and brushed the dirt from her hands.
"He said he'll be here around six to take us to the barn dance at Aunt Malinda's," JT told her as he dipped a cupful of water from one of the buckets she'd been using to water the seedlings.
"Yes." She rubbed the sore spot in her back and nodded. "He told me he would." She pointed to the water buckets. "Why don't you run down to the creek and refill those. Then you can help us water the seedlings and we'll call it a day."
"And then I'll jump in the creek and take a bath," he called out as he grabbed both buckets.
"Me too," Ruth called out.
Elizabeth looked up at the clear sky overhead. "You might as well make the most of this summerlike weather while we have it."
When JT returned, the three of them finished watering the seedlings. "Just enough to moisten the soil," Elizabeth reminded JT. "We don't want to drown them."
"Eli had to quit working early in order to go to town and buy some tools from the Prescotts," JT explained as he dropped the water cup into the nearly empty bucket. "I reckon he got tired of having to borrow tools from everybody."
"The Prescotts aren't open for business yet," Elizabeth said as she dripped some water on the last tiny seedling. "I hope Hugh will have some tools available to sell to him."
"Oh, I'm sure they do," JT assured her. "According to Augustus they have a little bit of everything."
"Wait until you see their mercantile," Ruth said as she wiped her dirty hands onto her work dress. "We've been watching it get bigger and bigger every day on our way to school. It looks just like a real building."
"A real building?" Elizabeth frowned.
"I mean with real wood," Ruth explained. "Not just logs, like Grandpa and Uncle Matthew are using."
"Augustus said his pa went to the lumber mill in Empire City and brought back a bunch of milled wood. That's why they were able to build it so fast."
"I'll bet that cost a lot of money," Ruth added.
"You're probably right about that." Elizabeth led the way out of their seedling garden, latching the stick gate behind them. She hoped it would keep the critters out. "Well, I'm happy for the Prescotts. I can't wait to see the mercantile after church tomorrow." And truly, it would be interesting to see a "real" building out here in the frontier, where everything else seemed to be made of logs and twigs and bark.
While the children bathed in the creek, Elizabeth gave herself a bucket bath in the tent shelter that her father and brother had constructed for her and the children to use as a temporary house. The tent was attached to the wagon, and compared to conditions while coming over the Oregon Trail, it was spacious and comfortable, albeit rather rustic. She had no idea what it would be like once the rains came—probably very damp. She was just finished getting dressed when Ruth and JT returned, shivering from their cool dip in the creek. But at least they were clean. And their good clothes were clean and dry and ready for them to wear.
Elizabeth went outside to survey the seedling garden. The poor tiny plants looked a little worse for wear, but that was to be expected. With sunshine and moisture, they would perk up. And with the mild winter in this region, they would probably be several inches tall by spring—and twice that by next fall. The ones that survived.
She returned to the tent to check on the children's progress.
"Here, Mama." Ruth handed her the hairbrush and her blue satin ribbon and then turned around. Elizabeth sat down on the rocker and brushed out Ruth's honey-colored hair. It was still damp from the creek as she separated it into three parts and then smoothly plaited one long braid, tying the ribbon into a nice big bow. "All done."
"There's Eli," JT called from outside.
Elizabeth reached up to her own hair, smoothing the bun she'd made earlier into place.
"You look beautiful, Mama," Ruth assured her.
Elizabeth laughed. "Thank you."
"Are we all ready to go now, Ma?" JT reached for his guitar case, which was resting as usual next to his bedroll.
"Yes." Elizabeth handed Ruth a kerosene lantern. "We'll need this on our way back."
"I wish we could take the wagon," JT told her.
"It's not that far to walk," she reminded him. She wished they could take the wagon too, but that would mean taking down their tent and breaking camp. Besides, they'd have to unload the wagon and load it all back in later. She picked up the basket containing the two berry pies she and Ruth had made the night before. One of them had gotten a little blackened on the edge of the crust, but the blackberries had been tasty, and hopefully no one would complain. Elizabeth tried not to miss her oven back in Kentucky, reminding herself that pies could burn in it too.
"Hello there," Eli called as he came into their camp. Instead of his usual buckskins, he had on his Sunday-go-to-meeting clothes, and Elizabeth could tell by the shine on his chin that he was freshly shaved. "Now, what can I carry for you?"
Elizabeth handed him the basket, and he rewarded her with a kiss on the cheek. Naturally the children giggled—and Elizabeth's cheeks blushed.
"Where's your guitar?" JT asked Eli.
"Back at camp," Eli told him. "Didn't know I was supposed to bring it tonight."
"That's all right," Elizabeth assured him.
"Maybe next time."
"Uncle Matthew will have his fiddle," JT told them. "And the McIntires will be ready to play music too."
"This will be so much fun!" Ruth clapped her hands and skipped along as if she hadn't worked hard all day.
Elizabeth smiled at Eli as she linked her arm into his. "You look mighty handsome tonight, Mr. Kincaid."
He grinned at her. "I was about to tell you that you look pretty as a picture." He nodded to Ruth. "As do you, Miss Ruth."
"What about me?" JT joked. "Do I look pretty too?"
Eli chuckled. "Well, looks to me like you're fixing to break some young girls' hearts tonight."
"Hannah Flanders wants to dance with JT at the party," Ruth teased.
"I've got an idea," Eli said. "How about if we sing our way to the barn dance?"
"Yes! Yes!" Ruth agreed. "Let's sing 'Coming Round the Mountain.'"
And so they did, going through all the choruses and even making up a few new ones until they got to Malinda's place. It was amazing how singing made the trip go much more quickly.
"As soon as we get the pies set out, I want you to meet Malinda," Elizabeth told Eli. "She's my very best friend."
It wasn't long until Elizabeth got the chance to introduce Eli to Malinda. But they were both careful not to mention their engagement. Still, Elizabeth could tell by Malinda's questions that she was suspicious. When Will came over to join them, politely greeting Eli with a curious expression, Elizabeth grew uncomfortable. It was the first time all four of them had been together, and she so wanted them all to be friends—though she knew that was probably unlikely. Will and Eli...they were so different. Even so, they exchanged polite greetings, and Eli congratulated Will and Malinda and made a respectable attempt at small talk.
"I just know something is up with these two," Malinda said suddenly to Will.
"What do you mean?" Will gave her a perplexed frown.
"I mean something is going on here. I can see it in my best friend's eyes." Malinda grasped both Elizabeth's hands and stared intently at her. "What is it? Tell me."
Elizabeth suppressed a nervous giggle and looked away, attempting to avoid Malinda's probing stare.
"I know it!" Malinda declared. "I know exactly what is going on!"
Elizabeth gave her a warning glance, but Malinda ignored it.
"I don't believe in betting, but I would wager that you two have gotten yourselves engaged." Malinda tilted her head to one side. "I just know it. And if I'm wrong, I challenge you to be forthcoming and set me straight."
"All right," Elizabeth said quietly. "You figured us out. But we wanted to keep this a secret for the evening. This is your night, Malinda. Yours and Will's."
"I just knew it!" Malinda clapped her hands. "I'm so happy for you!"
Elizabeth made a nervous smile. "Thank you."
"Congratulations," Will told Eli.
Eli gave him a slightly uneasy smile. "Well, truth be told, I reckon I owe you my gratitude, Will Bramford."
Will looked confused. "How so?"
"I'm obliged to you for not stealing my girl." Eli chuckled. "It took me a while to figure things out. But I'm sure thankful you left her for me."
Will's brow creased, but then he chuckled too. "Now that you mention it, Eli, I can see that I should be equally grateful to you."
"How is that?" Now Eli looked confused.
"I'm obliged that you had already stolen Elizabeth's heart." Will looked at Malinda with real tenderness. "That allowed me to wait for this one."
"And all's well that ends well," Elizabeth proclaimed with relief.
"Oh, no," Eli told her. "This is just the beginning."
Chapter Two
I insist that you and the children come stay at my house until Will and I wed in December," Malinda told Elizabeth as they gathered in front of the small building that functioned as both the school and the church. The service had just ended, but as usual the congregation continued to visit outside in the autumn sunshine.
"But I thought Will's girls were going to be staying with you."
"They will be, starting this week. But Emily and Amelia came up with a delightful idea." Malinda pointed to where the older girls were circled together, chattering among themselves like a flock of chickens. "They suggested that the boys sleep in the loft of the barn like they did a couple of weeks ago. They all claimed it was very comfortable. Then you and I and Ruth will share my room. And Emily, Susannah, Amelia, and Belinda can share the sleeping loft in the house—we already have enough beds up there."
Elizabeth did the arithmetic in her head. "That's ten people," she told Malinda. "Won't that be crowding your little house?"
"Not at all. Besides, it's only temporary. For the next few months, we'll make the most of it and have fun," Malinda assured her.
"That would be wonderful," Elizabeth confessed. "Especially when the weather gets wetter and colder. I'm sure Ruth and JT would enjoy being there, and they'll appreciate being closer to school. If you're certain we won't be a burden..." Elizabeth tried to imagine everyone at mealtime. "Ten hungry people at suppertime might be a challenge."
"We'll all work together." Malinda nudged Elizabeth's elbow, nodding over to where Belinda was talking with one of the Levine boys. "And unless I'm mistaken, it's only a matter of time before Belinda will be living under someone else's roof."
"Really?" Elizabeth glanced over to where Belinda was chatting with the tall, serious-looking young man. "Which one is he anyway?"
"That's the oldest boy, Jacob. He's twenty-three, I believe, and as dependable as the sun. He really stepped up after his father died." Malinda shook her head. "The second son, Charles...well now, he's another story."
"What do you mean?"
Malinda frowned and then lowered her voice. "Charles took an Indian woman for his wife several years ago...although they're not legally married." She made a tsk-tsk sound. "But I think they have a child."
"Oh..." Elizabeth thought of Eli's previous wife. She shuddered to think what Malinda would say if she knew about that. And yet at the same time, she had the strangest urge to just blurt it out.
"Hello, hello!" Lavinia called out as she hurried over to join them. "I missed church this morning because Augustus was feeling under the weather."
"Nothing serious, I hope," Elizabeth said.
"No, I don't think so."
"Because, as you know, my mother is good with herbal medicines." Elizabeth looked over to where Clara and Asa were visiting with a couple about their age.
"Yes, I'll keep that in mind. Evelyn is with him now." Lavinia looked up at the sky. "It's another marvelous day. If this weather keeps up we will have the mercantile completely finished soon."
"I saw it on my way to church this morning," Elizabeth said. "It's just like Ruthie told me—it looks like a real building."
Lavinia laughed. "It is a real building. And we're getting the store all set up downstairs. Now if only we can get the living quarters finished before the rains come. I hear the rain in this region can be relentless."
"It's what makes things grow and keeps it green," Malinda told her.
"And wet," Lavinia added.
"I wonder what the men are discussing so intently," Malinda said with a tinge of concern in her voice. "I hope there isn't any trouble brewing somewhere."
"Trouble?" Lavinia's brows arched. "What sort of trouble?"
"Oh, nothing." Malinda waved her hand dismissively, but Elizabeth could see the fear in her eyes.
"Well, I'll be pleased to take you up on your offer," Elizabeth told Malinda.
"What offer?" Lavinia asked.
So Malinda explained the plan to take in Elizabeth and the children along with Will's daughters.
"Oh, my. You will have a full house," Lavinia said.
"A full house and a full barn," Malinda told her.
"And everyone will help with the work," Elizabeth said. "Many hands make light work."
Malinda laughed. "Yes...perhaps I will become a lady of leisure soon."
"Out here in the West?" Lavinia scowled. "I don't believe any of us will ever experience leisurely living again. Sometimes I dream about life back in Boston. It seems so far away...like a different country."
Malinda grew serious. "Do you regret your emigration?"
Lavinia pursed her lips as she looked around at the young people talking and frolicking about. "No, no...for the sake of my children and my husband...I think we made the right decision." She gave Elizabeth a sheepish smile. "Although I'm sure there were times on the trail when I highly regretted it—many, many a time."
Just as Malinda and Elizabeth were confirming plans to join households later in the day, Eli and Will and Hugh came over to join the women. "I've been promised a sneak peek into the mercantile," Eli told Elizabeth. "Would you like to come with me?"
"Yes," she said eagerly. "Very much so."
"Oh, good," Lavinia told her. "I've been so eager to show it off to my friends." She lowered her voice. "However, we can't make it appear as though we are open for business yet. Especially since it's the Sabbath. But do come and see what we've done."
They called out to the children, explaining where they were going and that they'd be back shortly. And then they all trekked down the street to where the most impressive and most talked-about building was progressing nicely. Like so many buildings in the frontier towns they'd passed through on the Oregon Trail, the mercantile had a false front, which made it seem even taller than its two stories. Hugh unlocked and opened the front doors, and Lavinia hurried in to light some kerosene lamps.
"You even have a glass window," Elizabeth said. "So nice to get that sunlight in here. And it faces south too. Very smart."
"And we hope to put in a few more glass windows next year," Lavinia gushed. "Now if anyone would have told me a year ago that I would be swooning over glass windows, I would have thought they were daft." She laughed. "But, oh my, how times have changed."
Elizabeth looked at the wooden shelves that were being loaded with provisions and supplies. Many crates were still unopened, stacked in the back of the room. "I can see you're getting ready for business," she told Lavinia. "When will you be officially open?"
"We've actually been serving a few customers already," Lavinia confided. "But we plan to have a grand opening ceremony next Saturday. Isn't that so, Hugh?"
"That's right," he called from where he was showing the men something over in the tool section. "I expect to receive another shipment from Empire City midweek. Hopefully we'll have most of the stock unloaded and put out by then."
"It looks like you've got plenty of goods already," Elizabeth said after they'd browsed a bit. "If today wasn't Sunday I'd be doing some shopping."
"So would I," Malinda said eagerly. "It's so wonderful to have a real store in our settlement."
"Well, you two come on over tomorrow afternoon if you like," Lavinia told the women as they were preparing to leave. "Invite Clara to come too. I'll open the doors especially for you folks."
Before long the children were gathered, and Elizabeth and Eli and the children began the walk back to their homesteads. But they hadn't gone far when her parents and Matthew and Jess came along in their wagon, offering a ride. The children, acting as if they were worn out, happily hopped in the back.
"I don't mind walking," Elizabeth called out to her parents.
Asa winked at her and nodded at Eli. "I'm sure you don't."
By now most of the settlement seemed well aware that Elizabeth and Eli were betrothed, which made having some time alone with the man who would one day be her husband even more precious. "I miss having the use of my wagon," she told Eli as they walked down the rutted road. "But I'm still using it to store our goods, and it seems silly to drive it around fully loaded like it is. But maybe once the house is built...maybe this winter we'll have it to use."
"From what I hear it's hard to drive a wagon in these parts once the rains start. It doesn't take long for these roads to turn to mud." He nodded toward the river lazily meandering alongside the road. "Seems the best way to travel might be the river. That's what the coastal Indians do to get around."
"My father has wondered about that very thing," she told him. "He's talked about using the oxen team to pull a barge up the river from the ocean. But as far as I know, no one has tried that yet."
"The Prescotts would benefit from that kind of venture," he said as he reached for her hand. "As it is, they have to go all the way to Empire City to get their shipments, and that will be difficult once the rains come. From the looks of the mercantile, Empire City has been a good resource for them. Hugh tells me there's an enterprising family running a sawmill up there on the bay. Having access to milled lumber has hastened the Prescotts' building process considerably."
Elizabeth nodded. "I was trying not to feel envious at how quickly they've been able to get their store built." She turned to look at him, still feeling slightly amazed that he was here with her now...that they would be married in the not-so-distant future. "But money is not everything."
He laughed as he squeezed her hand warmly in his own. "No, it certainly is not."
"And there are many families with more challenges than ours," she confessed. "I worry that the Flanders household will suffer this winter. They seem so ill prepared. And yet Flo seems utterly unconcerned."
"This land is generous," he said. "Those who are willing to work hard will benefit from the natural resources." Again he pointed to the river. "I hear the fishing is good this time of year."
"That's for certain." She nodded. "Father and Matthew and JT have had some good luck catching salmon. Father has been running his smoker almost continuously."
"He's a wise man. And the woods abound with deer and bear and other sources of meat, if a man is good with a gun."
"Yes." She smiled at him. Eli was an excellent hunter and fisherman. She couldn't even remember how many times he'd shared game with them on the Oregon Trail.
"And Bert has a useful trade with blacksmithing," Eli added. "Surely he can trade for food and provisions for his family. I know I'd be happy to trade with him."
Certain that no one was around to see them now, she stopped walking and turned to look into his face. "I'm so very grateful you're here, Eli." She felt her heart fluttering. "I feel very blessed to know you will be my husband."
His eyes glimmered as blue as the river as he leaned down to tenderly kiss her. "I feel equally blessed, Elizabeth," he said quietly. "More than equally. By marrying you I get a wonderful family as well. Do you know how much that means to me?" He embraced her now, holding her close. Elizabeth couldn't remember the last time she'd felt this incredibly happy...she didn't even want to.
As they continued walking, Elizabeth told him the news that she and the children would be moving in with Malinda. "I have to admit that I won't miss camping," she told him. "I won't miss waking up with everything damp from dew."
"Well, if you're not going to be camping there, maybe I should camp there myself. That way I could work on the house without having to go back and forth to your parents' place. It might speed up the process."
"Yes," she said eagerly. "That's a wonderful idea."
"And I could take care of the livestock if you like."
Elizabeth considered this. "I already arranged to bring them over to Malinda's," she said. "She has fencing for the horses and cows. And the hens are just starting to lay, and I figure with ten people in the house, we could use the eggs."
He nodded. "Yes, that's a good plan. But perhaps I can keep your team to help with the logging."
"Yes, of course," she told him. "You keep the Percherons here and put them to good use."
"That should help speed things along." He peered up at the sky. "As it is, we're getting a mighty late start, Elizabeth."
She nodded somberly. "I know. Believe me, I know."
"I hear that building becomes quite challenging once the rains start. Mud makes everything much more difficult and slow. Some of the men think we're foolish to start building at all this late."
"What do you think?" she asked.
He grinned. "I think I'd better get busy."
She smiled back at him. "We'll use the team and the wagon to get our things moved over to Malinda's this afternoon. And then I'll send them back with JT."
######
JT and Ruth were both eager to relocate to Malinda's. Not only did it mean being with cousins, it also made the trek to school a little shorter. The three of them worked together to break down their camp. Then they packed and loaded their wagon—almost like they used to do each morning while traveling the Oregon Trail. Elizabeth drove the wagon, and JT drove the livestock. The plan was to let their animals share the pasture with Malinda's, and perhaps Goldie would be with calf by the time they moved back to their own property.
At Malinda's they unloaded everything they felt they'd need for their stay. And while JT returned the wagon back to the property so that Eli could have use of the team for logging, Elizabeth helped Malinda to get supper started and Ruth went outside with Bart and Susannah to help get the chickens situated.
"Oh, Malinda," Elizabeth said happily. "It is so lovely to prepare food indoors again." She didn't even mind that Malinda cooked over an open fire or that her kitchen setup was much more rustic than what Elizabeth had left behind in Kentucky. She knew Malinda had made similar sacrifices years earlier.
"I remember that feeling well," Malinda told her. "The first time I cooked inside my house after so many months of cooking outside over a campfire—it was simply wonderful."
"And I don't even have to wipe the grit out of the bowl before I mix up the biscuits." Elizabeth laughed as she peered inside a clean yellow bowl. "What a treat!"
"We decided that Will's children won't move in here until next weekend," Malinda told Elizabeth. "I thought that would give us time to get settled a bit. Plus we can get some things moved around and prepare for our additional household members." Now she explained how she wanted to transform a section of the barn loft into the boys' bedroom. "We'll put clothing hooks on the wall, and I have a rather worn commode we can put up there with a pitcher and basin—although I expect the boys will do most of their washing up on the porch. I thought we could put some crates up there for storage and seating. And I want to make sure there are safe places to hang lanterns. We don't want them burning down the barn."
Elizabeth blinked. "No, we certainly do not."
"Also, I hoped to get some produce put up for winter. And I have a quilt to finish and...well, so many, many things. Trust me, we will have our work cut out for us, my friend. I hope you don't mind."
"I'm so happy to be of any help—and so grateful to have a roof over my head."
Malinda beamed at her. "It's wonderful to have you here. You know, Elizabeth, it reminds me of when we were young women...remember how we helped each other with our trousseaus before our marriages?" She shook her head. "Did you ever dream we would do it all over again?"
"Make trousseaus?" Elizabeth looked up from measuring the flour.
"No, that's not what I meant." Malinda chuckled. "But do you remember how our families thought we were so silly for wanting to imitate Queen Victoria's wedding back in those days?"
"Looking back...I think I understand their perspective now."
Malinda paused from chopping an onion. "But we did have fun, didn't we?"
"We did." Elizabeth sighed to remember what felt like a lifetime ago.
"And it feels like we will have a bit of fun again."
Elizabeth held up the spoon she was using to measure salt. "I suppose fun is similar to salt...a little bit goes a long way...but it does make life tasty."
"You and me, Elizabeth, marrying two fine men...all over again. Did you ever think this would happen?"
"Never in a hundred years." Elizabeth stirred the batter.
They worked quietly together for a while. Then as Elizabeth was rolling out the biscuit dough, Malinda spoke up again.
"God truly is the giver of second chances," she said quietly. "For both you and me. I am so grateful."
Elizabeth nodded. "Yes...so am I." As she used a water glass to cut the biscuits, Elizabeth pondered Malinda's words. It truly was ironic that she and Malinda found themselves in this position again—except now they weren't starry-eyed young girls trying to imitate a royal wedding. Now they were two widowed mothers...frontier women who were preparing to wed men who had both had previous marriages. Strange.
Maybe it was just a fact of life that one never knew what was coming around the next corner. Perhaps the only thing you could fully expect was the unexpected. Anyway, it certainly provided ample opportunity to trust God.
Chapter Three
The next couple of days seemed to fly past. And, as much as the two women accomplished, it seemed that Malinda's chore list seemed to get longer by the end of each day. Elizabeth knew this was partly because Malinda had let so much go after losing John last winter. And there was so much she wanted done before her wedding in mid-December. But Elizabeth enjoyed pitching in. Many hands truly did make light work.
"I think I'll go over to check on my property today," Elizabeth told Malinda on Wednesday morning.
"Don't you mean to check on Eli?" Malinda teased.
Elizabeth grinned. "As a matter of fact, I wouldn't mind saying hello to him."
"Why don't you take him his dinner," Malinda suggested. "I'm sure he'd enjoyed some of that bread you baked yesterday."
"A very good idea." Elizabeth took care as she prepared a basket of food for Eli. Perhaps he would want to have a picnic with her.
Because JT and Ruth had used Molly to ride to school, Elizabeth walked. But it felt good to walk. She could feel autumn in the air, and she noticed more leaves were turning lovely shades of yellow, orange, and red. And although it was cooler, it was still a pleasant sunny day. Perfect for an impromptu picnic. Feeling excited to see what sort of progress Eli was making, she hastened her pace. How many logs would he have cut by now? She doubted any of them would be laid on the foundation yet. But perhaps they would be cut to length and notched. She knew that Brady planned to go over and help Eli once he was ready to start setting the logs in place. It was only a matter of time.
She listened as she walked, wondering if she was hearing the ring of Eli's ax, or perhaps the sound was coming from Matthew's or Father's projects. She did hear birds chirping merrily and the gurgle of the creek flowing to the river. But as she got closer to her property, she heard no sounds of sawing or chopping or anything.
Finally at the homesite, she looked around in dismay. Eli was nowhere to be seen. Neither was his Appaloosa. Not only that, but her beloved Percherons, Beau and Bella, were gone too. Perhaps he'd taken them into the woods to pull some logs. But now she noticed something else. Her wagon was missing from where she normally kept it parked near the house. In its place was a pile that was neatly covered in tarps. On closer inspection, she discovered that this pile was all the goods and supplies she'd been storing in the wagon. Was Eli using the wagon to carry logs?
She walked over to the section of woods that they'd both decided had the best selection of trees to fell and called out Eli's name. No answer. She went a bit farther, calling for him every once in a while, but still no answer. She paused to listen, but again heard only the birds and the creek.
Returning to the homesite, she sat down on the foundation and pondered this. Where was Eli? And where was her team and her wagon? Of the few possessions that she'd been able to bring across the Oregon Trail, none were more valuable than her team and her wagon. And now they were gone...
She opened the basket and broke off a piece of bread, munching on it as she considered the possibilities. Perhaps Eli had taken the team and the wagon over to help Matthew or her father. She stood and smoothed her skirt. Of course, that had to be it. Eli had recently mentioned that the sooner they finished their houses, the more help he and Elizabeth would have to finish their own. Perhaps he was using the wagon to help them in some way. However, as she walked, she couldn't think of how.
Arriving at her parents' homesite, she discovered Matthew and Brady helping her father set the rafters in place for his roof. They were all perched up high, and not wanting to distract them, she decided to seek out her mother instead.
She found Clara over by their wagon and tent, hanging some clothes on the line to dry. Elizabeth greeted her and then inquired about Eli.
"I haven't seen him since Sunday." Clara frowned as she pushed a strand of graying hair back into her bonnet. "Is something wrong?"
Elizabeth forced a smile as she reached for one of her father's shirts, giving it a good shake before she pegged it to the line. "Not really. I just couldn't find him and I thought he might have come over here."
"No. Your father thought maybe he was going to give a hand with the rafters, but he hasn't been by." Clara pegged the last damp towel up.
"Oh..." Elizabeth tried to conceal her concern.
"Are you all settled in at Malinda's?"
"Yes. It's been lovely having a real roof overhead." Elizabeth held up the basket. "Would you folks like some bread? It comes out a little nicer than what you can make in the outdoor oven."
"That would be lovely. Jess is making blackberry jam today. Perhaps we can enjoy some on this. Which reminds me, did Jess tell you that she got a letter from Ruby?"
"No. So how are Ruby and Doris doing?" Elizabeth still missed these two women, but she did understand their decision to remain in Empire City for the winter. Their plan was to open a restaurant, and according to Matthew, they wanted to liquidate some of the merchandise they brought with them from St. Louis. They hoped to make enough money in Empire City to join the others next summer, stake a claim, and build a house.
"It sounds like they're off to a good start," Clara told her. "They've opened their restaurant in a little cabin that's down near the docks, and it sounds like they're making good money."
Now Elizabeth updated her mother on all that she and Malinda had been doing the past couple of days, as well as all the plans that Malinda had for them yet to do. "I think she's trying to catch up with everything she'd let go after John died." She sighed. "I remember how that was." She smiled. "Thank goodness you were nearby when I needed you."
"Well, I'm sure Malinda is happy to have you with her now. But it'll be a busy place with all those children staying there. You two will certainly have your hands full. Although the older girls should be of some help around the house. I assume they're done with their schooling by now."
"Amelia is still in school, and Belinda has been helping Mrs. Taylor with the younger children. So the girls won't be around very much during the weekdays."
"Perhaps that will be nice for you and Malinda. I'm sure you both have plenty to do to prepare for your upcoming nuptials and whatnot." Clara made a happy sigh.
"So when does Father think your cabin will be finished?" Elizabeth asked as a distraction from speaking of weddings. She didn't want her mother to guess she was worried about Eli.
"Your father still thinks we'll be able to move indoors by mid-November. I sure hope he's right. Once the rains we keep hearing about start to come...well, I'm afraid that living in the tent will turn into a rather soggy affair." She scowled at the tent. "I told Asa that I might move into the house whether it's finished or not."
"Poor Mother." Elizabeth put a hand on her shoulder. "You have been very patient through all this."
"I suppose my patience is starting to wear a bit thin." She rubbed the small of her back and sighed. "Or else I'm just getting too old for all this outdoor living."
"You are not too old." Elizabeth thought for a moment. "And I'll tell you what. If the weather changes and you're still not in your cabin, I'm sure Malinda will be glad to take you in."
Clara made an amused smile. "That's sweet, dear. But poor Malinda—her house will be full to overflowing."
"She has a barn too," Elizabeth reminded her. "I'm sure we could make room for you in there. Maybe I'd join you."
Clara laughed. "Well, I suppose if it comes down to that...I might be willing to sleep in a barn."
They went into the tent, making themselves comfortable in the chairs and visiting a while longer, but the whole time Elizabeth's missing betrothed was lodged firmly in the back of her mind. Where was Eli? Why wasn't he working on their house? Where was her team? And what about her wagon?
"Well, I should get back to Malinda's." Elizabeth slowly stood. "She's probably added half a dozen more chores to her list by now."
Her mother chuckled as she hugged Elizabeth. "Thanks for the bread, dear. And don't worry about Eli. He's a good man."
Elizabeth smiled. "I know he is. I just wish I knew where he is."
"Well, if anyone can take care of himself, he can."
"That's true enough."
As they emerged from the tent, Elizabeth could see that the men were still intently getting a rafter into place. "If I do see Eli, I'll encourage him to come over here to help Matthew and Father," she promised.
"And the sooner they get these houses finished, the more help Eli will have for your house." Clara's eyes twinkled. "And the sooner you two can have that wedding."
"I suppose. But getting started so late in the year..." Elizabeth shook her head. "Well, the children and I might be the ones who end up living in Malinda's barn this winter. At this rate, our wedding might not be until the end of next summer."
"Oh, dear." Clara frowned. "I hope not. Let's pray that house goes up quickly, Elizabeth."
"Well, God willing...maybe that will happen."
Elizabeth did pray as she walked back to Malinda's. She prayed that Eli was all right. She knew it was senseless to be worried about him. Her mother was correct about Eli. If anyone could take care of himself, it was Eli. He'd proven that to them on the trail over and over. Not only did he take care of himself, he helped take care of all the travelers as well. Surely, Eli was just fine.
"So how was your little picnic?" Malinda asked when Elizabeth came into the house.
Elizabeth set the empty basket on the table and sighed. "Actually, I didn't see him." She explained that the wagon and team were gone. "So I dropped the bread and other food at my parents'. My mother was most appreciative."
Malinda looked up from where she was sewing on the quilt she hoped to finish before Will's girls arrived. She planned to put it on the bed that Belinda and Amelia would share upstairs. "Where do you suppose he went?"
"Maybe he found a better section of land to get timber for the house." Elizabeth removed her bonnet.
"Yes...that must be it."
Elizabeth sat down beside her and threaded a needle, and for a while the two of them just stitched together in silence. "Malinda...?" Elizabeth began tentatively.
"Yes?" Malinda kept her eyes on her sewing.
"You wrote about the Indian skirmishes...that happened last year. Remember? In the letter I received while we were traveling."
"Yes." She nodded as she tied a knot and then clipped the thread.
"Well...I haven't seen a single Indian since I arrived here. I assumed that they've moved on. Is that right?"
Malinda frowned as she measured of a new length of thread. "Yes...that's mostly right."
"Mostly?"
"It's not something people care to speak of much, Elizabeth. But yes, the Indians have been relocated."
"Relocated?"
"Yes. After the troubles...well, the army came out and rounded them up. They took them up north I believe."
Elizabeth felt a mixture of emotions now. Her first feeling was one of relief. Her fears that Eli had been ambushed by savages who had stolen her horses and wagon seemed to be unfounded. But her next concern was for the Indians. "All of the Indians were rounded up?" she asked. "Even the women and children?" She thought about some of the Indian women she'd seen along the way. She remembered the old woman that she and Ruth had purchased moccasins from. And she remembered Eli telling her about many of them being so peaceful...about them taking him in...about his wife.
"That's what I understand. Well, except for Charles Levine. Remember I told you about him—the one who took up with the Indian woman. From what I've heard, those two are hiding out somewhere. But if the army finds them, she'll probably be taken up north too."
"Do you think that's fair?" Elizabeth asked quietly.
"Fair?" Malinda looked up. "Some of those Indians ambushed settlers, Elizabeth. Do you not recall those stories?"
"Yes...I do recall." Elizabeth nodded. "But that was only some of the Indians, right? And certainly not the women and children. Wouldn't that be like some of our men doing something wrong or illegal and then all of us being rounded up and put in prison with them? Would that be fair?"
Just then the children came bursting into the house, and the question and the conversation became lost in the noise and commotion. Elizabeth tried to console herself that Eli was safe—he hadn't been taken by Indians. But at the same time, she felt disconcerted to think that Indian women and children had been forced off their land. Furthermore, she knew that Eli would feel even more upset to hear of this...or perhaps he already knew. After all, it was an old story in this new country...a very old story.
Chapter Four
On Thursday morning Elizabeth convinced JT that it was time for him and Ruth to start walking to school with their cousins. "It was one thing to ride Molly from our property, but you're much closer to school here at Malinda's," she told them. "If Susanna can make the walk, then it seems Ruth is big enough as well." Of course, her real reason for this was so that she would have the use of Molly. Her plan was to ride to her property, and if Eli wasn't working on the house as she expected he would be, she would ride all around the area to search for him.
After helping Malinda with the morning chores, Elizabeth saddled Molly and set out. She had been unable to sleep the night before, worried that Eli had been injured while felling timber or attacked by a wild animal or possibly even a renegade Indian. In the dark of the night, anything seemed possible. But now, by daylight, she felt more hopeful. In all likelihood she would arrive at her homesite and find that Eli was busily at work on the house. If that was the situation, she would simply act as if she were out for a ride, spend some time with him, and then return to Malinda's.
However, when she arrived on her property, everything looked exactly as it had the previous day. Eli, the team, and the wagon were nowhere to be seen. So she rode Molly all about the property, sometimes calling out for Eli. She even followed the creek to the river, thinking perhaps he'd been fishing and had fallen in. But her searching was in vain. Eli seemed to have vanished into thin air—along with her beloved team and her wagon.
"I don't know what to make of it," she told Malinda after she'd turned Molly out to pasture. "Eli was nowhere to be found."
Malinda looked up from her churning with concerned eyes. "What do you suppose has become of him?"
Elizabeth removed her riding jacket and just shook her head. "I have no idea."
"How well do you know Eli?" Malinda asked.
"How well?" Elizabeth hung her jacket on a peg on the door. "He was our scout coming across the Oregon Trail...I'm sure I mentioned that to you already."
"Yes, I know. But before that. What sort of life had he lived?"
"He'd been a trapper," Elizabeth said absently. "So he definitely knows how to live off the land and survive."
Malinda frowned. "I don't want to sound critical, Elizabeth, but trappers are an unusual sort. Very independent of society. And some of them seem to live by their own rules."
"What do you mean?"
"Is it possible that Eli is a dishonest man?"
"No," Elizabeth said quickly. "Of course not. I trust Eli implicitly."
"Well, your team and your wagon are very valuable around here. Mrs. Levine's husband traded his team of horses for 500 acres of valuable river land several years ago, and I'm sure your lovely Percherons would bring—"
"No." Elizabeth firmly shook her head. "Eli would not do that. I know it for a fact. Eli is a good man."
Malinda focused her attention on the churn, and Elizabeth went outside to check on her livestock. At least that is what she told herself as she left the house. Mostly she wanted to get away from Malinda's suspicious accusations. Malinda did not know Eli the way Elizabeth did. As Elizabeth went out to the chicken coop, she tried not to let Malinda's words do any more damage. Still she felt perplexed. Where was he?
Elizabeth hadn't told Malinda about Eli's previous marriage to an Indian woman. Just hearing Malinda speaking of the local Indians convinced Elizabeth where Malinda's sympathies lay. And if Malinda was suspicious of Eli now, how much more so would she be if she knew everything about him? Which brought Elizabeth back to Malinda's piercing question—how well did she know Eli?
Was it possible she had agreed to marry a man of low character? Had her heart tricked her? Eli was certainly handsome and charming, and she was definitely attracted to him. But really, how well did she know him? And yet she had accepted his proposal of marriage. She had allowed him into her world—and her children's. Was it possible that she'd made an enormous mistake?
No, no, no, she told herself as she worked on mending the bit of chicken-coop fence that Ruth had mentioned to her this morning. Eli was a good man. She knew it. And yet...where was he? Why would he leave—taking her horses and wagon without saying a word to her? It made no sense. Absolutely none.
Elizabeth kept her concerns to herself as the house grew busy and noisy with the children coming home from school. She remained quiet as she helped Malinda with supper preparations. She feigned intent interest as she listened to her children's stories of school. And she pretended to be absorbed in mending a pair of JT's torn trousers as the children headed off to bed.
Finally, it was just Elizabeth and Malinda sitting by the fireplace, neither of them speaking. Elizabeth was just tying off the last thread on her mending when Malinda cleared her throat. "I know you are fretting about Eli," she said quietly. "And I fear I may have overstepped my bounds when I spoke of him earlier."
"I'm convinced it was only because you love me," Elizabeth said quietly. "And you love my children. You are concerned for our welfare."
"Yes," Malinda said eagerly. "That is exactly right. But I realize that you know Eli much better than I do. I feel certain that your instincts about him are correct, Elizabeth. Please, forgive me for casting any shadow of doubt upon him. I'm so sorry."
Her unexpected apology caught Elizabeth off guard, and she felt tears filling her eyes. "Of course I forgive you," Elizabeth said in a tear-choked voice. "But perhaps you were right. It's possible I don't know Eli as well as I'd like." Elizabeth looked up to the open sleeping loft, knowing that it was possible that the older girls could hear her words. But at least Ruth wasn't sleeping up there. She was in Malinda's bedroom with the door closed. Even so, Elizabeth lowered her voice. "I can't imagine he would have taken my wagon and my team," she whispered. "But I would be so devastated if that was so. Not just for my heart's sake, but because that is such a big part of our livelihood, Malinda. I would feel as if I had robbed my own children." She looked at her with tear-filled eyes. "Do you understand what I'm saying?"
Malinda nodded solemnly. "Being a widowed mother of children is an enormous responsibility. Especially out here in the wilderness."
"Do you think I've made a grave mistake?" Elizabeth whispered as she reached into her skirt pocket for her hanky. "In trusting Eli as I have?"
Malinda set her knitting aside. "Oh, Elizabeth, I don't know what to say." She reached over and grasped her hand. "But whatever happens, you know that you have your family nearby to help you through it. That is so important."
"Yes...that is true." Elizabeth wiped her eyes. "If Eli is still missing tomorrow, I will go and speak to my father."
"That would be wise." Malinda squeezed her hand. "In the meantime we will both be praying for Eli's welfare...and his safe return." Malinda released Elizabeth's hand and then pursed her lips. "I know I assured you that all the Indians had been cleared from this area," she said very quietly. "But it is possible that some of the renegades are still at large."
Elizabeth considered this. "But Eli is very smart about Indians. He got along quite well with the plains Indians. Besides, I thought the coastal tribes were peaceful. That's what Eli told me."
"Perhaps that was partly true. But those Rogue warriors were unpredictable. Just two years ago, there was such a bloody battle. Nearly thirty Indians were killed by the gold miners, and the remaining warriors retaliated against settlers with great aggression. We lived in fear of attacks in these parts. It's only been within the past year, since the tribes were removed, that life quieted down. But you never know...you just never know."
"I understand your fear of the Indians," Elizabeth said slowly. "But do you ever feel sympathy for them? Do you ever consider that they were living here...relatively peaceably until the white settlers came? Do you ever wonder how you would feel if we were in their place?"
Malinda made a tired sigh. "Sometimes John and I spoke of this very thing. We even had Joel Palmer as a guest in our home. He was the Oregon Superintendent of Indian Affairs. And John felt Mr. Palmer was a fair man in regard to the Indians. Mr. Palmer opposed the hostility that some of the settlers were demonstrating toward the Indians. But then General Lane came along. Most of the settlers favored General Lane's approach. However, John felt the general's answer to the Indian troubles was extreme and...well, John felt it was unjust."
"I always admired John," Elizabeth declared, "but now I admire him more than ever. I'm sure Eli would have liked him too."
"Yes, that's good and well, but John's views on the Indians were not typical of settlers, Elizabeth. And it did not win him friends. To be honest, he and I often disagreed. But in all fairness, my concerns were for the safety of my family. And the truth is I was greatly relieved when the Indians were removed."
"Even the women and children?" Elizabeth challenged. "Were you fearful of them too?"
Malinda gave Elizabeth a somewhat exasperated look. "If the women and children remained, the men would want to return. Also, who would hunt and provide for the women and children? Have you considered these things?"
Elizabeth admitted she had not, and then both women agreed they were tired and began to get ready for bed. But once again, Elizabeth found herself unable to sleep.
On Friday morning, Elizabeth hurried to finish her morning chores, saddled up Molly, and once again rode over to her property. She didn't even feel surprised when it looked the same as before. And she knew what she had to do, but it was with heavy heart that she rode over to speak to her father. She knew this news would be nearly as disturbing to him as it was to her. Eli was gone. Her team was gone. Her wagon was gone. In all likelihood, she and the children would need to stay with her parents during the winter—all five of them in the small one-room cabin.
She braced herself as she dismounted from her horse. The men were still on the roof, but it appeared that they were nearly finished with the rafters. Instead of disturbing them, she set off to find her mother. Clara and Jess were at their campsite, preparing the midday meal. "Hello," Elizabeth called in a falsely cheery voice.
"Welcome," Clara said as she replaced a lid on the big cast-iron pot.
"You're just in time for dinner," Jess said warmly.
Elizabeth joined them by the fire, feeling again as if she were on the verge of tears. Telling them this bad news was going to be so hard. She prayed God would give her strength not to break down. She needed to be strong.
Now Clara peered curiously at Elizabeth's face. "Is something wrong?"
Elizabeth nodded. "Yes. As a matter of fact—"
"The children?" Clara said urgently. "Are they—"
"Ruth and JT are fine," she assured her. "At school."
"Oh." Clara made a relieved sigh.
"It's Eli," Elizabeth began slowly, wondering how to say this.
"Oh, no," Jess exclaimed. "Has he been hurt—"
"He's gone," Elizabeth said solemnly.
"Gone?" Jess looked confused.
"How long has he been gone?" Clara asked. "Since the last time I saw you?"
"Yes...although it's possible he was gone before that." She shrugged. "I actually have no idea."
"Do you think he went hunting?" Jess asked hopefully. "Matthew and Pa have been wishing they could do some serious hunting, but they need to get the roofs on first."
"No, I doubt he would take my wagon and team to go hunting," Elizabeth told her. "Along with his own horse."
Jess' eyes grew wide. "He took your wagon and team?"
Elizabeth simply nodded, swallowing the lump in her throat.
"Perhaps he's felling timber," Clara suggested. "Would he use the wagon to bring it back?"
"What do you think?"
Clara frowned. "Your father and brother didn't use the wagons..."
"And I searched all over my property. He's not there. He's gone."
"But that makes no sense," Jess said.
"I know." Elizabeth sat on one of the chairs by the fire, putting her elbows on her knees and her head in her hands.
Clara placed a hand on Elizabeth's shoulder but said nothing.
"I just thought you should know," Elizabeth said quietly. "I'd tell Father, but I don't want to interrupt their work."
"They'll be quitting soon...for dinner," Clara told her as she went over to tend the fire.
"There must be some logical explanation for this." Jess sat next to Elizabeth, putting her arm around her shoulders. "Eli wouldn't just take off like that. He loves you, Elizabeth."
Elizabeth turned to look at her. "What makes you so sure?"
"I've seen him looking at you," Jess said. "Dozens of times. Starting out on the trail. And since he got here. He looks at you the same way Matthew looks at me."
The lump in her throat seemed to be growing. "Maybe...but he's still gone."
"I cannot believe he would run off with your wagon and your team," Clara said a bit sharply. "I just cannot believe it."
Elizabeth stood now. She knew if she remained she would break into tears. Mother and Jess didn't need that. Neither did the men. "I just wanted you to know," she said sadly to her mother. "Please tell Father for me. And Matthew and Brady."
"But don't you want to stay for—"
"No thank you." Elizabeth adjusted the brim of James' old hat and stood straight. "I need to get back to help Malinda."
Jess gave her a sympathetic look. "You're all right?"
Elizabeth nodded firmly. "I'm a frontier woman," she declared. "Just like you two. And we all need to be strong. I'll get past this...in time." Then she turned on the heel of her boot and marched over to where she'd tethered Molly to a tree. She quickly climbed into the saddle and nudged the horse with her knees, biting back tears and steadying herself in the saddle as Molly broke into an awkward trot.
What she'd just told Clara and Jess was true...or rather, she wanted it to be true. She would make it true. She was a frontier woman, and she would be strong. With God's help, even with her heart breaking, she would remain strong.
Chapter Five
Elizabeth had just turned onto the road that ran past Malinda's farm when she noticed a wagon in the distance. Few of the settlers had wagons, and that in itself was enough to get her attention. However, it wasn't really the wagon that drew her eye—it was the team of handsome black horses. Beau and Bella!
She kneed Molly to a gallop, hurrying to see who was driving her wagon. But then she pulled back on the reins, slowing the strong horse. What if it was a thief...or a band of renegades? She wasn't armed. She didn't want a confrontation. So she simply walked Molly slowly down the road, narrowing her eyes as she attempted to see who the driver was. But as she drew nearer, she could tell it was Eli. And now her emotions were mixed. On one hand, she was greatly relieved. On the other, she was seriously irked.
"Eli," she said calmly as he slowed the wagon to a stop and she pulled Molly up alongside him. "What are you doing here?"
His face glowed with a huge smile. "Coming home." But now his smile faded. "Is something wrong?"
She took in a slow deep breath. "Where have you been?"
His brow creased. "Didn't you get my note?"
"What note?"
"The note I wrote to you on Sunday night. The night before I left. I weighted it down with a stone. It was right inside the doorway of the house, along with a bunch of wildflowers in a tin cup. I thought for sure you would find my note when you came looking for me."
"Inside the doorway of the house...?" She pressed her lips together. "I didn't even go in the doorway."
Now he looked slightly alarmed. "So you didn't know where I was? Where I went?"
She shook her head. Now she felt a mixture of shame and curiosity. "I had no idea where you were...it seemed you'd vanished into thin air."
"I'm so sorry, Elizabeth. I hope you weren't too concerned."
She blinked and swallowed. Be strong. "To be honest, I was quite worried, Eli. I thought some harm had come to you. I searched all over the property for you. Then I thought perhaps renegade Indians had done something to you...stolen the horses and the wagon."
He reached over and put a hand on her cheek. "Oh, Elizabeth. I'm sorry you were worried. But as you can see, I'm perfectly fine." He patted the seat beside him. "Want to ride with me?"
Feeling guilt for doubting him and relief at finding him...as well as many other emotions, she dismounted and hurriedly tethered Molly to the back of the wagon, just as she had often done on the Oregon Trail. Then she came around, and Eli extended a hand, helping her into the wagon seat. Only as she settled into the seat did she think to peek inside the covered wagon, where, to her surprise, she saw a full load of lumber and some other things. "Eli!" she exclaimed, "where did you get all that wood?"
"If you'd found my letter, you'd know that I was journeying over to Empire City." He released the brake and shook the reins to get the team to move.
"To Empire City?" She frowned. "That's a long trip."
"Don't I know it." He rubbed his back. "Much better on horseback than driving a wagon."
"Speaking of horseback...where is your horse?"
"Back in Empire City."
She frowned. "Why?"
"I traded him for the lumber." He smiled but his eyes looked a little sad.
"Eli?" she exclaimed. "Your horse? You traded him? You love that horse, Eli. How could you—"
"I love you and the children more," he said firmly.
Despite her resolve to be strong, tears filled her eyes—but they were tears of joy this time. She slipped her arm into his and pretended to be looking at something alongside the road. She could hardly believe that when she was questioning his loyalties, he was giving up his horse in order to get lumber to build their home.
"I was starting to cut down trees," Eli told her, "and I realized it would be impossible to get very far on our cabin before the rains set in. Then I remembered the Prescotts and how they bought lumber in Empire City and how much more quickly their building went up. And it seemed clear that was what I needed to do."
Elizabeth thought about the beautiful Appaloosa again. She knew Eli would miss that horse. It had always been his source of livelihood, and in some ways it was his ticket to freedom as well. "Well, anytime you need a horse, Eli, you know that Molly is as much yours as mine. And before we left Kentucky I was starting to break Bella and Beau to saddle too. Some people say you can't have both in a horse, but Percherons are special."
He nodded. "Yes, and if your father gets his way, there will be more horses in time. But what we need most of all right now is a house." He grinned at her. "And a wedding."
If Elizabeth could have her way, she would wed Eli right now. Today. They could live in the tent and the wagon just as her parents and Matthew and Jess were doing. However, it didn't seem like it was her place to bring this up. Besides, there were the children to think of. "How long do you think it will take to build a house?" she asked meekly. "Now that you have lumber?"
"I don't know for sure. But it'll be a lot faster than it would have been if I'd had to log all the timber."
"And Father's house is nearly finished. And it sounds like Matthew's is coming right along too. So I feel certain that we can have Brady's help soon. Especially now that Father's rafters have been raised."
At the homesite, Elizabeth ran inside the house's foundation to discover her dusty note and a wilted bunch of wildflowers. Then she insisted on helping Eli to unload the wagon. And as they carried the planks and boards together, he seemed genuinely grateful to have her help—or perhaps he simply enjoyed her company as she filled him in on the latest happenings in and about the settlement. But after they got all the lumber neatly stacked near the foundation, a couple of boxes still remained in the wagon. He slid the first one out. The wooden crate was big and flat, and Eli seemed to be handling it with extra care. She helped him carry it over to the foundation, where he leaned it on its side.
"What is it?" she asked curiously.
"A window."
She blinked. "A glass window?"
He laughed. "Is there another kind?"
"Well, most of the settlers have wooden windows."
"Yes, and we will have some of those as well. But we will have one glass window in the front of the house."
She threw her arms around him. "Oh, Eli. Thank you—thank you!"
He leaned down and kissed her, pulling her close and holding her tight. "You're so very welcome."
"This is so exciting!" she said as he released her. Of course, she wasn't sure which was more exciting—being in his arms or having a real glass window. "Ruth will be over the moon when she hears that we have a real window."
"Let's just hope it's still in one piece by the time I put it in." Eli returned to the wagon. "Now this next crate is pretty heavy. Why don't you back the wagon up to the house, and I'll attempt to ease it down onto the foundation."
She worked with the team, maneuvering the wagon into a position where the tailgate was right against the front part of the foundation before she pulled on the brake. Then she watched as Eli eased the crate to the back of the wagon and down onto the wall with a clunk. By the time she got back inside of the house, the heavy crate was solidly on the dirt floor.
"What is it?" she asked.
He grinned at her. "What do you think it is?"
She looked at the square, sturdy crate. "Bricks?" she guessed.
"Nope, but you're not far off. Want another guess?"
She shook her head. "I have no idea."
"It's a small cookstove."
Her eyes grew wide as she stared at the crate in wonder. "Really? A cookstove? You found a cookstove in Empire City?"
"It seems someone had ordered it and then changed their mind. It was still sitting on the dock. I actually got a pretty good deal on it."
"A window and a cookstove." She let out a sigh of wonder. "It feels like Christmas, Eli." She smiled happily at him. "But the best part is having you home again."
"And now, since the day is still young, I think I will get to work."
"Why don't I take care of the wagon and the team," she offered.
He nodded. "I'm much obliged."
"And then I'll ride back to Malinda's and fetch you something to eat," she called out. And after that, she would make a run over to her parents and tell them the good news. Eli was home!
######
It was late in the afternoon by the time Elizabeth finally made it back to her parents, where they had already stopped work for the day. As she approached their campfire, she could tell by their glum faces that they were still feeling concerned about her.
"Good news!" she called out as she hurried over. "Eli is safely home."
"Thanks be to God," Clara exclaimed as she hugged her.
Now Elizabeth sat down and told them the whole story—from the lost note to how he'd sacrificed his Appaloosa for lumber—and finally about the window and the cookstove.
"Eli found a cookstove in Empire City?" Clara exclaimed. "Are there more?"
"Now, Clara." Asa shook his head. "We already decided we would wait until next spring to—"
"But were there more?" Clara asked Elizabeth again.
"I don't think so," Elizabeth told her. "It sounded like it was the only one. Someone had ordered it and changed their mind. It was on the dock. And the crate was very heavy."
"Did you open it and see it?" Clara asked eagerly.
"Not yet, Mother."
"Well, I know where we'll be going for Christmas dinner," Clara told her.
Elizabeth laughed. "That's assuming we'll have our house finished by then."
"I'm going to start putting shingles on next," Asa told her. "Brady and I cut down a massive cedar tree last week, and there should be enough shingles for the roofs of all three of our houses."
"That's good to hear. I'll let Eli know."
Asa nodded toward Brady. "And I'll bet Eli would like to have your help now, Brady. Although I'll hate to see you go."
Brady glanced at Elizabeth. "Well, I reckon Miss Elizabeth gets her say on whether I stay or go."
Elizabeth shook her head. "You know you're a free man, Brady. But Eli and I would love having your help. And we already talked about a spot not too far from the creek, but high enough that it should stay dry. And there's enough timber there for you to build a small cabin. Eli wants to help you with it...if you'd like that." She still felt badly that Brady would be unable to claim land as they'd hoped he could. But she also knew that unless he pretended to be their slave, he would be forced to leave. Even with their pretense, according to Oregon law he only had three years to reside here.
But she and Eli had decided to cross that bridge later. Most of all they just wanted to make Brady as comfortable as possible. He was older than her father, and she hoped he could live his final years in peace...and without being uprooted again.
"I'd be much obliged," he told her.
"And we'll section you off a piece of farmland for you to use as you like."
His dark eyes glistened. "I'm most grateful, Miss Elizabeth."
"And I'll help you with your cabin," Matthew offered. "Soon as I finish my house and help Eli and Elizabeth with theirs."
"I'll help too," Asa told Brady. "With all of us working, we'll get it up in no time."
"And we'll share some household goods," Clara told him.
"Ya'll is too kind," he told them. "Better than family."
"We are family," Elizabeth assured him. "And now I better get back to Malinda's before it starts getting dark."
Elizabeth felt unbelievably happy as she rode Molly back to Malinda's farm. It was hard to fathom how she had felt so downhearted and grievous earlier in the day...and now to feel so incredibly hopeful and glad. It was truly amazing. She slowed Molly in order to enjoy the last rays of golden sunlight filtering through the autumn foliage. The air felt so clean and sweet—so full of promise. She was just taking in a deep breath when she heard a rustling off to her right and up ahead.
The hairs on the back of her neck stood up, and she immediately thought of a wild predatory animal. She knew there were bears and wildcats in these parts, but she didn't think they'd want to tangle with a horse. Even so, she nudged Molly to the left side of the road, keeping her eyes trained on the brush where she'd heard the rustling. Just as she was passing, she saw what she was certain was an Indian crouched low in the overgrown brush. Their eyes locked, and Elizabeth's heart pounded with fear. But in the same instant she realized it was a woman. She looked as frightened as Elizabeth felt.
"Hello?" Elizabeth called out timidly. And then, just like that, the woman backed away and disappeared. With her heart still thumping hard, Elizabeth stared into the brush, blinking her eyes in disbelief. Had she imagined the young woman crouched down there like that? Or was that for real? But somehow those big dark eyes, so full of fear, burned into Elizabeth's memory. For a moment, she considered getting down off Molly and investigating more closely. But then she remembered what Malinda had said—if Indian women returned to these parts, the men would probably follow. And if that happened, there would probably be more trouble too.
"Gid-up," she told Molly, squeezing her knees into the horse's sides. "Let's get home." As Molly broke into a smooth gallop, Elizabeth tried to decide how to handle this. As badly as she wanted to rush into the house and tell Malinda, she didn't want to alarm her. Malinda seemed to have an unreasonable amount of fear when it came to the Indians in these parts. Perhaps it was rooted in her concern for her children...or perhaps it was more. But Elizabeth decided she would bite her tongue around Malinda. She would, however, tell Eli. Eli understood the Indians better than anyone. He would know what to do.
Chapter Six
On Saturday morning everyone at Malinda's was bustling about, finishing chores and getting ready to go to town. They were all eager to go to the grand opening of Prescotts' Mercantile. Everyone who lived within ten miles was likely to be there. Well, everyone except for Eli, who was determined to work from dawn until dusk on their house. As much as Elizabeth would miss him, she was grateful for his devotion to their future. Right now, every possible building day seemed precious.
She and Malinda were working in the kitchen when Will's wagon pulled into the yard. "That's right," Elizabeth remembered. "Belinda and Amelia are moving in today."
"Yes." Malinda dried her hands on her apron and smoothed her hair. "Will said he'd drop them by before we headed off to the mercantile." She excused herself and hurried out to meet him. Elizabeth watched from the porch, still marveling at how two people who'd recently been complete strangers now seemed so perfect for each other.
"Hello," Belinda called as she lugged a bulky carpetbag into the house. "We have arrived."
Elizabeth smiled and pointed to the ladder stairway. "Your room awaits."
"I wish I could sleep up there too," Ruth said longingly.
"We already went over that," Elizabeth reminded her. "We're not going to crowd you in with Susannah and Emily—and Amelia and Belinda get the other two beds."
"But I'm crowded in with you and Aunt Malinda," Ruth pointed out.
"Why don't you be an angel and go help Amelia with her things," Elizabeth suggested as she watched Amelia struggling to carry a box and a bag up the porch steps. Fortunately, that distracted Ruth for the time being. It wasn't easy being the youngest of the cousins, but someday Ruth might appreciate it.
While the younger girls helped the older girls get settled in and unpack—or more likely to ogle at the lovely clothes these girls had brought with them from Boston—Elizabeth and Malinda returned to fixing food to take to town. The plan was to have a picnic lunch on the church grounds with family and friends.
"Do you remember when we used to go to town on Saturdays in Kentucky?" Malinda asked Elizabeth.
"I certainly do." Elizabeth tucked a cloth down over the biscuits, which were still warm. "But our town had more than just one store and a church."
"This is just the beginning," Malinda assured her.
"That's true. And the Bible says not to despise small beginnings." Elizabeth smiled. "And I do not. After all those months on the trail, I am thankful for our little settlement."
"And I hear that Bert Flanders is planning for his blacksmith shop. Flo told me they want to build a small house with a lean-to that will function as his business."
"I expect Bert will be kept busy too," Elizabeth said. "With so much building going on there's a need for hardware."
"And didn't you mention that Jessica's aunt and her friend might start a business here too?" Malinda asked. "A boarding house and restaurant?"
"That's their plan, and according to Eli, they were doing a good business when he saw them in Empire City last week. I wouldn't be the least bit surprised if they make enough profits to start a nice boarding house here." Elizabeth smiled to herself to think of the business Ruby used to operate back in St. Louis. Jess had said it was a dance hall, where the whisky ran freely. But during the arduous journey, Ruby and Doris had seemed to reform some of their ideals. Still, she wondered what Malinda would think if she knew about St. Louis. Not that Elizabeth had any intention of telling her. Some things truly were better left unsaid.
Everyone chattered happily as they walked to town together, but eventually the younger children hurried on ahead and only Elizabeth, Malinda, and Will's two daughters were walking together. "I can't get over the feeling that this is like a holiday," Malinda said happily. "Having a store within walking distance of my home. It's just wonderful."
"And it's wonderful the weather is so pleasant." Elizabeth looked up at the clear blue sky, the sunshine filtering through the autumn leaves. "It's such a beautiful day."
"Yes, this time of year is exceptionally nice," Malinda told her.
"So far I haven't noticed all that much difference between the climate here and in Kentucky," Elizabeth said. "Autumn in Kentucky was always lovely too. Remember?"
"That's true," Malinda said. "I love autumn in Kentucky too."
"Well, it's a lot warmer here than it would be in Boston this time of year," Belinda told them. "We were usually wearing our heavy coats in early November."
"Will it be warm like this all winter?" Amelia asked Malinda.
"It will get cooler, and of course, the rains will come. But even so, it's still warmer than winters were back in Kentucky." She grinned at the two girls who would become her stepdaughters in December. "And I can assure you that it's much warmer than a Boston winter."
"Does it ever snow here?" Amelia asked.
"I haven't seen snow once since we settled," Malinda told them. "It did freeze one year, but I don't think that's typical."
"Well, I won't miss the snow or ice," Belinda announced.
"But I will miss some things," Amelia confessed. "Like candy stores and bakeries."
Belinda giggled as she pointed at a well-worn shoe in front of her. "And new shoes."
"And buying ready-made clothes," Amelia added.
"Yes, girls, we do understand." Malinda said in a maternal tone. "It takes some adjusting to get used to some of the deprivations of the frontier. But I believe that someday we will have all those amenities too."
"Really?" Amelia asked hopefully. "Even a candy store?"
Malinda laughed. "Well, perhaps it will be a while before we have a store devoted entirely to candy. But surely the mercantile will carry some sweets."
"Oh, I'm sure Aunt Lavinia will have stocked her candy case by now," Amelia told her. "I can't wait."
"I just hope she's got some of the household items I'll be needing to set up housekeeping," Elizabeth said. "I've got quite a list."
"Lavinia told me they'll be taking orders for merchandise," Malinda said. "Although I'm not sure how long it will take for shipments to arrive."
"Eli predicts that the Coquille River will be used to transport shipped goods from the coast someday." Just bringing Eli's name into the conversation filled Elizabeth with a girlish sort of giddiness.
"John used to say the same thing," Malinda told her.
"Imagine how much time that would save," Elizabeth said wistfully.
They were just coming up to the church, and it was fun to see others milling about what Elizabeth hoped would someday be considered a real town. She estimated at least two dozen people gathered between the church and the mercantile. Certainly, it was not much by other standards, but it gave one hope.
After depositing their food baskets at the church and visiting with some of the women there, they ventured over to the new shopping establishment, pausing to socialize along the way. A freshly painted sign was prominently displayed up high on the false front.
PRESCOTTS' MERCANTILE
EST. 1857
"This is a red-letter day," Malinda said with enthusiasm as they went into the mercantile. "Our very own store."
The place was bustling and smelled of coffee and onions as well as many other pleasant scents. People were chattering happily, and Elizabeth estimated at least a dozen adult customers were perusing the interior of the store. Several children, including her own, were gathered around the candy counter and dry goods, eagerly exploring everything. She had warned JT and Ruth to look but not to touch, and as far as she could see, they were heeding her instructions. However, JT looked as if his fingers were itching to pick up a music book that was in the book section.
"Hello, son," she said quietly from behind him.
"Ma." He turned eagerly, pointing out the book to her. "That book has fifty songs in it."
She nodded. "I see that."
"It's got musical notes for piano, but Mrs. Taylor has been teaching us to read music at school. I think I could learn the songs on my guitar."
She put her hand on his shoulder. "Did you bring enough money for it?"
He nodded, looking up at her with earnest eyes. "But do you think it would be wasteful? I know we need things for our home."
She smiled. "Don't you think music would be good for our home?"
He grinned back at her.
"I trust your judgment, son." She patted him on the back and then went to see what Ruth was admiring.
"Look, Mama," Ruth said when Elizabeth joined her. "There's some really pretty pink calico there."
Elizabeth nodded. "Yes, Ruth. But we brought a fair amount of fabric from home, remember?"
"But none as pretty as that," Ruth told her.
"Perhaps not. But until we use up our fabric, I don't see the need to purchase more." She tweaked Ruth's braid. "Although I'm sure it would look very pretty on you. Maybe you can find a pink hair ribbon instead."
Ruth gave her a surprised look. "I didn't mean for me, Mama. I meant for you. Wouldn't it make a pretty wedding dress for when you and Eli get married?"
Elizabeth laughed. "Oh, my. Well...I'm not sure."
"Oh, it would, Mama. I know it would."
Elizabeth looked down into Ruth's sparkling blue eyes. "That's very sweet of you to think so, dear. But I don't plan on sewing myself a new wedding dress. I would much rather spend my time sewing curtains and such."
Ruth looked truly dismayed.
"But if there is time to sew a dress, I think perhaps I'll just use that green calico I brought from home."
"Oh, no, Mama." Ruth looked appalled. "Not green."
"Why ever not? Green is a nice, sensible color. One of nature's favorite shades."
Ruth firmly shook her head. "Haven't you heard the wedding-dress poem?"
"What?" Elizabeth shook her head.
"It goes like this." Ruth stood straight as if she were reciting in school.
White—chosen right.
Blue—love will be true.
Yellow—ashamed of her fellow.
Red—wish herself dead.
Black—wish herself back.
Gray—travel far away.
Pink—of you he'll always think.
Green—ashamed to be seen.
Elizabeth couldn't help but laugh. "Well then. I certainly won't wear green!"
"Or yellow or red or black." Ruth got a thoughtful look. "And not gray...I don't want you to go far away, Mama."
"So what does that leave me with?" Elizabeth asked.
Ruth held up three fingers. "White, blue, or pink."
Elizabeth thought. "Well, I wore white when I married your father, but I was just a young woman then. Perhaps...blue?"
Ruth's serious expression broke into a smile. "Yes, Mama, blue."
"And I brought some blue calico too," Elizabeth told her.
"But it's dark blue," Ruth reminded her. She pointed back at the shelf holding the bolts of fabric. "How about that one on the end? It's the same color as your eyes."
"What about the blue fabric we already have?" Elizabeth asked.
"It would make pretty curtains and things," Ruth said hopefully. Elizabeth was about to put an end to this discussion and remind Ruth of the practicalities of living on the frontier when Clara came over to join them. Before Elizabeth could set her daughter straight, Ruth was emphatically explaining her plans for Elizabeth to have a beautiful blue wedding dress. "The same color as her eyes. And when a bride wears blue it means her love will be true."
Elizabeth sighed. "Oh, Ruth, my love will be true no matter—"
"You're right," Clara said to Ruth. "Your mother does need a pretty blue dress for her wedding day."
"But we haven't even set a wedding day," Elizabeth reminded her. "For all I know it might not be until next year."
Clara waved her hand at Elizabeth. "Now, I'm sure you have shopping to attend to. Ruth and I will take care of this little matter ourselves."
Elizabeth just shook her head, and chuckling at how her mother and daughter had just joined forces against her, she went off to see what she could find on her list.
The men, wanting to make good use of the first part of the day, arrived later in the afternoon—just in time for the potluck picnic. Asa and Matthew came in the wagon, and Elizabeth almost asked why Brady hadn't come. But then she stopped herself. Of course, she knew. As a colored man, Brady had not been accepted into the community. And although this wasn't surprising to Elizabeth, she still felt badly for Brady's sake. His life, it seemed, would continue to be an isolated one. Sometimes she wondered if he missed the old days, when James had kept numerous slaves to help with the farm. But even thinking of this felt foreign to her now. And wrong.
"I figured you womenfolk would need a wagon to haul all your wares back," Asa teased as they were sitting out in the churchyard to eat.
"I still want to go back for a couple of things," Clara told him.
"So do I," Elizabeth said. "I got to thinking they might run out of sugar before the next shipment arrives."
"Maybe we should have brought two wagons," Asa teased.
Clara gave him a dismissive wave as they walked away.
"Next year will be different," Elizabeth said to Clara as they returned to the mercantile. "Besides eggs and dairy products, we'll have grown our own produce for trading."
"And perhaps we'll have a couple foals by then as well."
"You think so?" Elizabeth wasn't so sure. Matthew's stallion and their only hope of a sire had foundered on the rich grass shortly after they arrived. "Isn't Storm still having trouble with his feet?"
"He's much improved, but his back feet are still very tender," Clara told her. "However, Asa mentioned that the Thompsons have a very nice stallion that he's considering."
"Really?"
"He's already making arrangements with Mr. Thompson to take Penny over for a visit. I'm sure he could take Molly too."
"Has he seen their stallion yet?" Elizabeth trusted her father's sensibility about horses, but she was surprised he'd agree to a sire he'd never seen.
"No. But Mr. Thompson claims he's a handsome horse."
"Well, as much as I'd love a foal by next year, I'd prefer to hear Father's assessment of the Thompsons' stallion after he's seen it."
"That's wise." She chuckled. "Asa might very well get Penny over there and decide to bring her right back home."
"I expect that Storm should be recovered before long."
"And we do know his bloodlines," Clara agreed.
"It would be so wonderful to have foals by next year." Elizabeth was well aware of the value of a good horse in these parts. Selling a yearling could be more profitable than a bumper crop.
They were nearly to the mercantile when Flo came over to speak to them, proudly pointing out where Bert had started to build what would one day be their home and blacksmith shop. "It don't look like much yet, but Bert says it will have a roof by Thanksgiving. And he'll put a tall front on it like the mercantile." She smiled. "Should be right handsome."
"Our town is coming together, isn't it?" Elizabeth said. "The mercantile and the blacksmith...the school and church. We're off to a fine start."
Flo frowned. "Speaking of the church..." She lowered her voice. "Bert flat-out refuses to go no more. Now Ezra won't go neither."
"Oh?" Elizabeth exchanged glances with her mother.
"Why is that?" Clara asked quietly.
"Bert says he doesn't need to give up Sunday mornings just to be yelled at by Reverend Holmes."
Elizabeth pressed her lips together. She was not overly fond of the reverend's fiery sermons either. At first she'd kept her opinion to herself, but eventually she mentioned it to her parents. Asa had reminded her that sometimes preachers went through a season of fire and brimstone, but perhaps it would pass in time. As a result, they'd all decided to wait and see.
"I must agree that the reverend's words can be strong," Clara admitted.
"I defended him at first," Flo told her. "But I'm of a mind to agree with Ezra now. And I hear tell that some other menfolk are quitting the church too."
"Oh, dear." Clara shook her head. "That's a shame."
"The only reason I'm burdening you folks with this is that Bert keeps talking about how much he misses Asa's Sunday services back when we were journeying here."
Elizabeth nodded. "Yes, I miss them too."
Flo looked eagerly at her. "Bert says if Asa was the preacher, he'd gladly come to church."
Clara frowned. "Well, Reverend Holmes is the appointed preacher. Asa can't possibly replace him."
"I know." Flo sighed and then looked over her shoulder as if worried someone was listening. "But we thought maybe Asa could hold church services somewhere else."
"You mean have two churches in our little town?" Clara looked troubled.
"It don't have to be in town. Maybe in somebody's home...or a barn. I recall that party out in Malinda's barn. It could surely house a church meeting, don't you think?"
"Well, that would be up to Malinda," Elizabeth said.
"And Asa," Clara added.
"Well, I'd be much obliged if you folks would give it your consideration."
So before returning to the mercantile, they promised to speak to Asa about the situation. Elizabeth felt sorry for Flo's dilemma, but she wasn't certain that Asa starting a church was the answer. If anything it seemed it would divide the community. She remembered a similar situation on the wagon train. But that had been different—or at least it had seemed so back then. While traveling, they had been a temporary community with numerous Sunday worship services throughout their wagon train. As a result, it hadn't seemed a problem to have more than one group of worshippers in their unit. However, when Mrs. Taylor's husband died so tragically on the trail, Elizabeth regretted not having spent more time getting to know him...or to understand him.
On the ride back to Malinda's, Elizabeth sat in the front of the wagon with her parents. Together they discussed the situation. "I'm well aware that Bert Flanders isn't the only one who's unhappy with the church," Asa quietly told them. "To be honest, I'm not overly fond of anger from the pulpit."
"Why do some preachers feel such a need to shout and carry on like that?" Elizabeth asked him.
He shrugged. "Hard to say. But when I prepare to give a sermon, I usually feel like I'm preaching more at myself than anyone else." He chuckled. "I s'pect if I was in need of chastising I would stand up and yell from the pulpit too."
Clara patted his hand. "I'm most grateful you're not like that."
"So what will you do, Father?"
He adjusted the brim of his hat to block the late-afternoon sunshine. "I reckon I'll pray about it. And I'm sure the good Lord will show us the best way to go. Never hurts to wait on him."
Elizabeth promised that she would pray about it too. And what he'd said made perfect sense. God would continue to direct their paths just as he always did. And rushing forward without God's blessing would only lead to trouble.
Chapter Seven
Autumn had always been a busy time in Kentucky, and it seemed busier than ever here in Oregon. Elizabeth knew that was to be expected. One couldn't settle the new frontier by sitting about. But sometimes she found herself longing for the quiet of winter. She remembered the peaceful days when there'd been snow on the ground, and other than the usual daily chores, life slowed down immensely. Would it ever slow down here or become quiet again?
Of course, Malinda's house had become even noisier since Will's daughters joined their household on Saturday. The four girls, soon to become stepsisters, shared the sleeping loft upstairs, and JT and his male cousins were sleeping in the barn loft. However, most of the time—when the children weren't in school or doing chores—they chose to reside in the house. It was cozy and busy and loud, but Elizabeth was grateful, especially since the weather had turned wet and windy the past few days. Just the same, she was trying not to count the days until she and her children would need to make other living arrangements. The idea of camping in the wind and rain was not appealing.
"Why don't you go out to check on your property?" Malinda suggested. "You haven't been out there for at least a week now."
"Ten days," Elizabeth clarified.
"Well then, it seems you're overdue for a visit." Malinda peered out the window. "And you're in luck because our little autumnal storm seems to have blown over today."
"I am curious to see how Eli's progressing," Elizabeth admitted. "The last time I was there, he and Brady were cutting some logs, and it did give me hope. And Father said that Matthew was going over to help this week. It's possible they've made real progress by now."
"Go on with you," Malinda urged. "Take your men some lunch."
Elizabeth tried not to feel apprehensive as she packed a basket with food. The last time she'd packed a lunch and trekked out to check on his progress, Eli had been missing. However, he'd had good reason to take off, and if she had been more observant, she would have discovered his note. She decided to ride Molly instead of walking. And she would take Flax with her as well. Eli had mentioned that a bear had been sniffing around their property. Maybe with Flax around, the bear would find a more suitable place to live.
She planned to stop by to see her parents as well. She knew they were close to being done with their cabin—at least done enough to start moving inside before long. On Sunday after church, her mother had spoken of little else. And who could blame her? Elizabeth was still sometimes amazed that her parents, especially her mother, had made the incredible trek to this part of the country. Just thinking about it made her feel proud of them...and thankful.
Most of the trees that had recently been bursting with colorful autumn foliage were now showing a fair amount of bare limbs. The recent rain and wind had stripped off many of the leaves. The ground was still a bit soggy from the rain, but the air smelled fresh and clean. Elizabeth pulled the collar of her old barn coat up around her neck to keep the breeze out. Certainly it wasn't very cold, but there was a damp chill in the air.
As the edge of her property came into sight, she felt a thrill of excitement. Flax ran ahead, barking as if he understood this was going to be his home again. She could hardly believe that someday she and Eli and the children would all be living here. There would be a cabin, which could grow larger over the years. And maybe even a barn by next summer. But nothing prepared her for what she saw as she came around the grove of trees—the walls of her house were fully up, logs on the lower part and milled lumber higher up. It was definitely a house, standing tall and straight as if it had every right to be there. And certainly, it did.
She hurried to dismount from Molly, rushing up to where Eli and her father and Matthew and Brady were using ropes to hoist up the first of the rafters. "This is amazing!" she exclaimed. "It looks like a real house."
"It is a real house," Matthew called out. "But you better stay out of the way, sis."
"That's right," Eli warned from where he was pulling the rope hand over fist to get the log up into place. "It's dangerous in here."
She stepped back and made sure Flax stayed out of the way too, just watching as the four men worked together so smoothly. They clearly knew what they were doing. And why wouldn't they after the building experience they'd gained the past couple of months? As she examined their progress, she realized the house wasn't set on her stone foundation the way she'd imagined it would be. She wanted to ask why but didn't want to distract the men. They seemed very intent on getting the big rafter into place. And it did look dangerous.
Instead, she walked around, surveying all sides of her house, which had been built just inside the stone foundation. She tried not to be disappointed even though she'd worked hard to gather and stack all those stones. And she felt it would have been handsome to see a cabin set on a stone foundation.
She looked at the square opening where she assumed the glass window would go and felt slightly dismayed to see it was empty. She hoped the window hadn't been broken during all this construction. Then she noticed the familiar crate leaning against some lumber. Eli was probably waiting to put the window in last to ensure it made it up in one piece. She sat down on a pile of lumber and just watched happily. After all those long weeks of thinking she would be homeless this winter, it seemed that she was really going to have a house after all. Even without a stone foundation, it was a house!
Finally, after the rafter was securely in place, the men came down for a dinner break. She greeted them and opened her basket of food. "I hope I packed enough," she told them. "I didn't know Eli had this much help today or I would have packed more."
"No matter," Asa assured her. "Your mother sent food for us."
"Then you shall have a feast today."
Eli grinned at her. "What do you think of the place?"
"I am truly astonished," she admitted. "I had no idea it would go so quickly."
"That's because I have such excellent help," Eli told her. "I don't know what I'd do without this fine team of experts."
"It's also because you got milled lumber," Matthew pointed out. "I can't even imagine how much time that's saving us. At this rate, we'll have this house finished inside of two weeks."
"Truly?" Elizabeth could hardly believe her ears. "That soon?"
"These good straight boards save us a bundle of time," Asa told her. "Even the wet weather didn't slow us down much."
"It also helps that you men know what you're doing," Eli said. "I might be good at hunting and fishing, but I'm no expert at carpentering. Not like you fellows."
Asa pointed at Elizabeth now. "I know you were worried about being the last one to get your house built, Lizzie. But it's plain to see that you and Eli are benefiting from our recent experiences."
"That's right." Matthew reached for an apple. "We made our learning mistakes on our own houses. Yours should turn out the best of all."
"I am truly grateful," she said.
"Looks like you'll have a right fine house," Brady told her.
"It sure does." She smiled at him and then turned to Eli. "I was a bit surprised that the house isn't sitting on the stone foundation I made..."
Eli nodded as he chewed a piece of bread and jam. "I hope you're not too disappointed 'bout that."
"It was my idea to forgo the foundation," Asa told her. "Without mortar, it just wasn't sturdy enough to hold a house up."
"I got us some mortar in Empire City," Eli explained. "But not enough to do a proper job on the whole foundation."
"Besides that," Matthew added, "it would have taken a week or more just to get the foundation ready to build on."
Brady waved over to the house. "And just think, Miss Elizabeth, we'd hardly have any house built at all right now if'n we were still putting them stones together."
Elizabeth nodded. "Well, that makes perfect sense."
"And all those stones you gathered will be just fine for the fireplace," Eli assured her. "I reckon we have enough mortar to do that job properly."
"I bet you'll have the best-looking fireplace in these parts," Asa told her. "Your ma will be jealous." He elbowed Eli. "Clara's already at me to make a trip to Empire City to find her a cookstove like you got for Elizabeth."
Eli laughed. "Good luck with that, Asa. I'm afraid it was a case of plain good timing that I happened along when I did. Otherwise I'm sure someone else would have snatched that stove up first."
"Well, I promised Clara we'd have the Prescotts order us a nice big cookstove by this time next year. That is, as long as we get some crops in and some livestock birthed by then. But I don't see any reason why we won't. This land looks like it's just bursting to produce goodness."
They all chatted hopefully about the future as they ate. Then, as the meal wound down, Eli excused himself, heading back over to the house as if he was ready to get back to work. Elizabeth, eager to have a moment alone with him, followed. "I brought Flax to keep you company," she said as Flax came alongside her. "And maybe he can help make the bears feel a little less welcome."
"Good idea." He leaned down to scratch Flax behind the ears. "You going to scare that bear away, boy?" Flax wagged his tail.
"I'm truly amazed at what you've accomplished here," she told Eli. "I never expected to see this much progress."
"I couldn't have done it without your family."
She ran her hand over the wood that framed where the window would go. "Getting this milled wood has made everyone's work easier. Thank you for doing that."
"I did it for us, Elizabeth. For you and me and the children."
"Is Matthew right? Is this house truly going to be finished in two weeks?"
"If we stay on course."
She looked down at her feet. "So we could set a wedding date?"
He reached for her hand, squeezing it. "I don't see any reason not to."
Their eyes met, and she felt a warmth rushing through her.
"You pick the date," he said, "and I'll be there."
She laughed. "That's just what I'm going to do."
As the men went back to work, she packed up the food basket and bid them goodbye. Then, instructing Flax to stay with Eli, she happily rode over to see her mother. It was still somewhat amazing to see their little cabin sitting proud and square—inhabiting what had been bare land just months ago.
She found her mother outside, sweeping dirt out the front door. She waved and called out a welcome as Elizabeth slid out of the saddle.
"Well, look at you," Elizabeth said as she walked up to the cabin. "You're in your house!"
"That's right." Clara leaned the broom against the house. "Come on in and look around. I'm still getting things unpacked and put away, but we've slept in here the past four days."
Elizabeth tapped her toe on the carpet that was covering the packed dirt floor. "Aren't you glad you brought this along?"
"Oh, my, yes." Clara shook her head. "It's wonderful to finally have a real roof overhead. And I don't want to complain about the dirt floor, but I do look forward to when Asa can put in a real floor. He ordered some lumber through the Prescotts. Sounds like we should have it in a few weeks. Maybe before Christmas."
"Speaking of Christmas, I need to look at a calendar."
"A calendar?" Clara looked around the room, which was cluttered with crates in various stages of unpacking. "I know I have one somewhere. I'm just not sure where."
"That's all right, Mother. I just wanted to let you know that Eli says we can set our wedding date. The house will be finished in about two weeks, and—"
"Two weeks?" Clara blinked. "Why, it seems they only just started it."
"They're making good time. And the milled lumber is speeding it up some too."
"I'm so pleased to hear that. You and Eli could have your wedding in December. Even before Christmas if you liked."
Elizabeth sighed dreamily. "Oh, it would be so lovely to be in our own house...all of us...by Christmas."
"What about Malinda and Will's wedding?"
Elizabeth returned to the present. "What do you mean?"
"Well, I know they're getting married the third Saturday in December."
Elizabeth nodded. "Yes. Their wedding is just a few days before Christmas. And certainly, Eli and I won't pick that date for our wedding."
"Perhaps you could get married on the first Saturday of December—unless that's too soon."
"That would put two weeks between the two weddings." Elizabeth considered this. "But I'm not sure about having our wedding before Malinda and Will's."
"What in the world would be wrong with that?"
"Wouldn't it be stealing the thunder from their big day?"
Clara laughed. "Goodness no, Lizzie. Remember, this is the frontier. Old social conventions don't necessarily apply here."
"It would be so wonderful to move into the house before Christmas." She touched a finger to her chin. "Not only that, but it would get us out of Malinda's house in time for her to put things back together before her own wedding."
"You see," Clara told her as she opened a box of sewing things. "A perfectly good reason to have your wedding first." She pulled out the pretty blue fabric that Ruth had admired at the mercantile on the day of its grand opening. "Looks like I'll need to get busy with this." She stretched out a length of the shiny fabric. "I thought perhaps I'd put some tucks in the bodice, and I have some lovely pearl buttons packed away somewhere. Perhaps a touch of lace on the collar?"
They discussed dress designs for a while, and then Elizabeth excused herself to return to Malinda's. "I can't wait to tell her the good news," she said. "Hopefully she won't mind that our wedding will precede hers."
"If she has any concerns, you can simply point out that you and Eli have been acquainted longer than she and Will. Then just remind her that your marriage will free up some much-needed space in her house." Clara hugged her goodbye.
"Feel free to tell Jess my good news," Elizabeth said as she went to her horse. "I know she's been looking forward to us setting a date."
"Two weddings in one month—what a busy social calendar we keep out here in the frontier!" Clara reached for her broom. "That reminds me, dear, your father and I want to have everyone over for dinner on Sunday after church. To celebrate our new home. He'll invite Eli."
"Wonderful," Elizabeth called out as she got back onto Molly. "Shall I let Malinda know too?"
"Yes, please do!"
As Elizabeth rode back to Malinda's, she couldn't wait to tell her about the wedding plans. It seemed almost unbelievable that she and Eli could be wed in just a few weeks. But it all made so much sense. And she felt so relieved to know that she and the children would be moved into their own home two weeks before Malinda's wedding. Spending this time with Malinda had been fun, but life would become much simpler for both families after Eli and Elizabeth were married.
Chapter Eight
You're planning to have your wedding before mine?" Malinda looked up from her sewing with a creased brow. "What made you decide to do this?"
As Elizabeth hung her coat on a peg by the door, she regretted bursting in with her good news. She explained that Eli and the men were making amazing progress on the house. "I had no idea it would go so quickly. Having milled lumber has helped immensely." She described what the house had looked like. "It was so exciting to see it, Malinda."
Malinda made what seemed a strained smile. "That's wonderful, Elizabeth. I'm so happy for you. But I still don't understand this rush to have your wedding before mine. What about social conventions and wedding etiquette?"
As Elizabeth warmed her hands on the fire, giving the coals a stir and adding another log, she realized she'd been foolish to dismiss her concerns so quickly. As much as she loved her dear friend, she'd known since childhood that Malinda was accustomed to having her way. Growing up as an only child with elderly parents, Malinda had been a bit spoiled. But surely this demanding life on the frontier would have changed some of those old ways. Or perhaps some things never changed. Whatever the situation, Elizabeth was determined to conclude this difficult conversation before the children came home from school even though it was already getting late in the day.
Elizabeth went back over to where Malinda was sitting in the rocker. "Is this going to be a problem for you?" she asked gently.
Malinda frowned. "Well...I realized you and Eli were going to get married eventually, but I didn't realize it would be so close to our wedding date. The same month even. How do you possibly plan to get everything done on time?"
"Everything?"
"Well, your dress and the food for your guests to start with—but besides that, where do you intend to have your wedding?"
"I thought it would be a simple affair. Saturday morning at the church. Perhaps a potluck dinner afterward. And my mother is making my dress."
"Truly? And she'll have it finished on time? I would think she'd have her hands fairly full housekeeping. Have they moved into their cabin yet?"
Elizabeth filled her in on her parents' news, extending the invitation for Sunday. "But even if Mother doesn't finish that dress for me, I'd be happy to wear my good Sunday dress."
"Oh, Elizabeth." Malinda shook her head. "My best friend needs to set her sights a bit higher than that. This may be the frontier, but we are civilized people. Besides, don't you want your wedding to be a memorable occasion?"
Elizabeth bit the inside of her lip. How could she say this without sounding critical of Malinda's values? "Perhaps...because this is my second marriage...perhaps I'm not as interested in the wedding ceremony so much this time. Certainly I want it to be memorable, but I'm content with a small, quiet affair." She sighed. "To be honest, I think my mother is more excited about the wedding preparations than I am."
"I hope you appreciate how fortunate you are to have your mother around to help you with these things."
Elizabeth considered the insinuation. Besides her children, Malinda had no family here in Oregon. But certainly she couldn't fault Elizabeth for that. "I'm here to help you with your wedding, Malinda. And I know my mother and Jess would help too. You must know that. We're your family too. And you also have Will's daughters. Belinda and Amelia are both so excited about the upcoming wedding."
Malinda made a tolerant smile. "But now you're having your wedding just a short time before mine. It seems rather inconvenient timing and...forgive me for speaking frankly, but...not very considerate."
Elizabeth tried not to feel offended. "Why do you feel that having our weddings two weeks apart is such a problem, Malinda?"
"It's a very small community, Elizabeth. So small that everyone will expect to be invited to both our weddings. Surely you know that. So obviously we will have the same people as guests. Have you considered how they might feel about attending two weddings within two weeks? Doesn't it seem an imposition? And right before Christmas too."
"Oh..." Elizabeth thought perhaps she understood now. "Are you worried about gifts? That there will be too much pressure on guests to bring two gifts within such a short period of time? Because I am happy to let everyone know that Eli and I do not expect wedding gifts. I honestly hadn't even thought about that aspect. But I realize this is the frontier. And so many of our friends—the ones we made from the wagon train—are struggling just to get by. I certainly wouldn't want anyone to feel compelled to bring a gift to my wedding. I will insist on no gifts."
"A wedding with no gifts?"
"Yes," she said firmly. "And I will make sure my mother understands this too. No gifts at this wedding."
Malinda still looked uneasy.
Elizabeth pointed out how much simpler Malinda's life would be with fewer people inhabiting her house before her wedding. "Three less people under your roof. As well as one less dog and numerous livestock and chickens. I should think you would be relieved to see us go."
"Yes, I can understand your rationale. It would be helpful if you and Ruth and JT were settled in your own home by then." She brightened. "But why couldn't you and the children move into your cabin without getting married? If it would help matters, I could invite Eli to bunk in the barn with my boys. He could have JT's bed."
Elizabeth stifled her irritation over what seemed like Malinda's incredibly selfish stubbornness. But even as she resented her friend, she felt like digging in her own heels as well. If she and Eli wanted to get married in early December, why shouldn't they?
"I appreciate your generous offer," she told Malinda in a stiff tone. "But I think I would feel greedy to move the children and myself into our lovely new home—the house that Eli is working so diligently to build for us—and forcing him to come sleep in a barn."
"Sleeping in a barn is a sight better than what we had to put up with when we arrived here," Malinda said a bit sharply. "The rains had already started by then, and it was too late in the year to build much of a cabin. And it didn't help matters that we'd sold our team to afford our passage on the ship. Did you know that we spent our first winter in that horrid little sod house that I now use to store my root vegetables? Can you imagine that tiny space with John and me and three children crammed into it?"
"I'm sure it was difficult." She peered curiously at her friend. "I'm surprised you never wrote me of those deprivations. You never mentioned such things in any of your letters. You always made it sound exciting and wonderful here."
Malinda made a funny little laugh. "If I'd told you of all the hardships, how would I have enticed you and James to move out here, pray tell?"
"Don't get me wrong, Malinda. I loved your letters. And I'm very happy to be here now."
"I hope you appreciate that you have it much better than we did. A school and a church and a mercantile and blacksmith. You should be counting your blessings."
Elizabeth looked at the clock on the mantle and realized she should be counting the seconds because the children would be home any minute now. "So if my wedding is a very simple one," she said quickly, "and if I make it plain to the guests that we expect no gifts, would you be amenable to Eli and me getting married the first Saturday of December?"
"Well, certainly, Elizabeth." Malinda huffed as she reached for her sewing scissors. "You can have your wedding whenever you wish. It is after all a free country, is it not?"
Elizabeth couldn't remember ever being in a position as socially awkward as this. She loved Malinda and wanted to please her. But she loved Eli more. "So it seems obvious that you would be much happier if Eli and I waited to marry?"
"Spring is a lovely time for a wedding, Elizabeth. And it comes early in this part of the country. Everything is so lush and green by March, and wildflowers would be in bloom."
"Spring?" Elizabeth took in a deep breath. How could Malinda be so stubborn? But before they could say another word about it, the children came bursting into the house. Suddenly it was time to hear about the happenings at school and to dole out apples for snacks and start preparations for supper. This uncomfortable conversation would have to wait.
Elizabeth had never liked conflict. She tried to avoid disagreements whenever possible, especially with loved ones. For that reason she was tempted to give in to Malinda without a battle. As she checked on a bowl of bread dough that had been rising, she thought perhaps it would be wise to wait until spring to marry. After all, only a year earlier, she felt she'd never be ready to remarry. James had been gone for nearly three years, and her mother had been pushing available bachelors her way, but Elizabeth had felt certain she could never love again.
She gently removed the risen dough, setting it on a floured board. She thought of Eli and how meeting him had changed everything, even if it had taken a while for her to admit to herself that he really was the one. And then she'd felt she'd lost him for good. As she kneaded the warm dough, she considered the freedoms he'd given up just to come back to her. She thought of all he'd done—including giving up his beloved horse—just to hasten their wedding date. And now she had to tell him to wait until spring?
"Goodness, Elizabeth," Belinda said abruptly. "You look as if you want to beat the living daylights out of that poor lump of dough."
Elizabeth gave Belinda a sheepish look. "I suppose I was overly exuberant."
"Are you disturbed about something?"
Elizabeth glanced around and noticed that Malinda wasn't in the house right now, so she decided to speak freely. After all, Belinda was Will's older daughter, and she'd grown up in Boston, where social conventions were respected. Perhaps she could test Belinda's reaction. "The truth is I'm feeling a bit torn about something."
"What is it?" Belinda asked as she tied on an apron and reached for a paring knife.
Elizabeth quickly told Belinda the good news about her house getting finished sooner than anticipated. "And today Eli encouraged me to pick a wedding date." She placed the pummeled dough back into the bowl, covering it with the towel. "I don't want to have a big wedding. Not like Malinda and your father are planning. And I don't want our guests to bring gifts. But I would like to be married before Christmas so that Eli and the children and I can be settled in our house. So I thought the first Saturday in December might work."
"That sounds wonderful," Belinda exclaimed. "Another wedding. What fun!"
"But do you feel that's unreasonable for your father and Malinda? Just two weeks before their wedding day?"
Belinda pursed her lips as if considering this. "I don't know...maybe it would be odd if we were back in Boston. But life was different there. When I think of the weddings on the trail..." She laughed. "Well, I'd never seen anything like that before. But I loved it. I love frontier weddings. I hope I have one someday too."
"From what I've observed with Jacob Levine and how he's always got his eye on you, I'd wager you'll be planning your own frontier wedding before long."
Belinda made an embarrassed smile as she looked down at the potato in her hand.
"But there's no hurry." Elizabeth set the bowl of dough back over by the fireplace. "Perhaps there is no hurry for Eli and me as well."
"But Eli is building the house for you and the children. It's only natural that you'd want to live there with him." Belinda waved her hand to where Susannah and Emily and Ruth noisily burst into the house. "And it probably won't be as crowded there."
Elizabeth smiled. "Not to mention it will be less crowded here if we go."
Eager to look at something Emily had brought home from school, the giggling girls hurried up to the loft.
"I wonder what your aunt would think about this," Elizabeth said quietly. When she'd first met Lavinia, she'd been taken aback by the Bostonian woman's commitment to propriety even though they were on the trail. Of course, it didn't take long before Lavinia set decorum aside.
Before Belinda could answer, Malinda entered the house. But it was too late for Elizabeth to retract her last question.
"Aunt Lavinia's so busy with the mercantile and trying to get into their house up above the store. I truly doubt she'd have an opinion one way or another. Besides that, she loves weddings as much as I do."
"Weddings?" Malinda came over to where they were working. "What are we discussing about weddings?"
"Elizabeth was just telling me that she was feeling conflicted," Belinda said.
"Conflicted?" Malinda eyed Elizabeth.
"About her wedding date being only two weeks before yours. But surely you don't mind about that, especially since you're best friends." Belinda smiled at her soon-to-be stepmother. "Now I have a beautiful idea!" Belinda set the potato and paring knife down and clapped her hands merrily. "Why don't you two have a double wedding? Can you imagine how fun that would be?"
"Oh, I don't think so," Elizabeth said quickly. She could tell by Malinda's face that she was not in favor of Belinda's idea. "I really wanted a small quiet wedding, and Malinda and your father plan to have quite a party out there in the barn." She looked hopefully at Malinda. "Which I am looking forward to. Besides that, I promised to stand up with you at your wedding. How could I do that if I was getting married at the same time?" She put her hand on Malinda's shoulder. "And I never properly asked you to stand up with me—I suppose I took it for granted—but I do hope you will."
Malinda smiled. "Of course I will."
Elizabeth knew this didn't mean that Malinda was resolved to the two-week gap between their wedding dates, but it seemed a step in the right direction. Perhaps in a day or two Malinda would become more reasonable. Elizabeth could only hope.
Chapter Nine
By Sunday Elizabeth felt slightly more optimistic. Although the two brides-to-be had not discussed any further wedding plans—they had barely conversed at all—she hoped that was only because Malinda had finally accepted that Elizabeth was immovable on this subject. Either that or Malinda had simply swept the whole thing under the rug for the time being. And with all that was going on in Malinda's household—her four children, her two soon-to-be step-daughters, and Elizabeth's own two—it was easy to get sidetracked.
It felt wonderful to sit next to Eli with JT and Ruth on either side of them in church, but Elizabeth once again felt dismayed as the service was ending. She was growing increasingly weary of the negativity of Reverend Holmes' sermons. If this was simply a season, as her father had suggested, she was ready for it to be over. The short wiry man was so obsessed with hellfire and brimstone that he seemed to have completely forgotten about God's grace and love and mercy.
She exchanged glances with Flo as they exited the little church building. As usual, Bert was missing, as was Flo's oldest son. Not only that, but Hugh Prescott and a few other husbands appeared to be absent as well. Was it possible that the reverend was slowly driving all the men away, or at least the recent settlers? Perhaps the ones who'd been here longer were accustomed to this sort of church service.
Out in the churchyard, where the heavy-leaden skies were almost as gloomy as the faces of the congregation, no one mentioned the content of the sermon. However, Elizabeth could see that people were avoiding Reverend and Mrs. Holmes as the older couple stood near the front steps. Even Mrs. Taylor, who'd once embraced fiery sermons, looked uncomfortable as she hurried away with Mrs. Levine.
"We better load up and head out," Asa called. "This storm is threatening to break." He'd brought his wagon, and the plan was for his family to ride with him to his newly built cabin for Sunday dinner. Expecting rain, he'd even put the cover on top of the wagon, and now all of them piled into the back of the wagon and under its shelter. Meanwhile, Malinda and Will and their children would ride over in Will's wagon. Unfortunately for them, Will hadn't had Asa's foresight regarding the weather.
Ruth rode up front with her grandparents, but the rest of them sat in the back, where Matthew had placed layers of fir boughs and covered them with a quilt. He'd done it for Jessica's sake, but the rest of them enjoyed the unexpected comfort.
"I don't like going to church anymore," JT said quietly to Elizabeth.
She frowned, knowing she should probably reprimand him for speaking with disrespect. And yet in all fairness, how could she?
"I don't like going either," Matthew told JT.
JT's eyes lit up. "Really?"
Matthew just nodded.
"Truth be told, I'm not enjoying it too much either," Eli confessed. "I found myself wishing I was back at the cabin, fitting in the floorboards."
"Floorboards?" Elizabeth was surprised. "You mean we're going to have a real floor? Not just packed dirt?"
He grinned. "That's right."
"Back to church," Matthew said in a serious tone. "I don't know how much longer I can endure being hammered on by Reverend Holmes."
"Why does he hate us so much?" JT asked.
"He doesn't hate us," Elizabeth told him. "He just thinks it's his job to keep us from going to—well, from straying from the fold."
"Well, if loud preaching could prevent someone from going astray, we'd all be safe from the fire and brimstone," Matthew said. "But that's not the gospel I believe in."
"That's not how Grandpa preaches," JT added.
Elizabeth peeked ahead to see if her parents were listening, but she discovered they were having a conversation with Ruth about what they would be doing for Thanksgiving, which was still a couple of weeks away.
"Is Reverend Holmes going to officiate your wedding ceremony?" Jess asked Elizabeth and Eli. They exchanged glances, but neither of them answered.
"That'll be one strange wedding if he does," Matthew teased Elizabeth. "I don't want to miss it."
"Maybe that's why Malinda wants to have her wedding in her barn," Elizabeth mused.
"Well, couldn't you have your wedding in her barn too?" JT suggested.
"I don't know...I think it might be better to have it in the church." She looked at Eli. "Unless you have other ideas?"
"You tell me when and where and I'll be there," he told her.
Elizabeth didn't want to mention the friction going on between her and Malinda, but she felt certain it would not improve the situation if Elizabeth wanted to have her wedding in Malinda's barn. It truly did not seem like an option.
"Do you think our house would be big enough to hold a wedding?" she asked Eli.
"Depends on how many folks are coming."
"Mother was making a guest list," Elizabeth told him. "And I suspect she's already invited a few people, like Lavinia and Flo and Mrs. Taylor. But she didn't want anyone to feel left out, and last I heard she thought there would be close to fifty people attending."
"That would be mighty cozy in your house," Matthew told Eli. "Unless you didn't have the bedroom walled off and if you didn't have any furnishing in place. Even then it might be tight."
"The barn would be much better," JT tried again.
She smiled at him. "Yes, I agree. But I don't want to burden Malinda with hosting our wedding just two weeks before hers. I don't think she'd appreciate that."
"Then it looks like we'll be getting married at the church," Eli declared. "Makes no matter to me, as long as we're married."
"But I'd like for it to be a pleasant wedding," Elizabeth said sadly.
"With no yelling," JT added.
"I wish Asa could perform the ceremony," Eli said wistfully.
"Now, there's an idea." Matthew pointed at Elizabeth. "Maybe you should go and have a little talk with the reverend. Explain what kind of ceremony you want, and if he's unable to give it to you, ask him if Pa can do it."
Elizabeth nodded eagerly. "Yes! That is exactly what I'm going to do."
"You're a braver woman than I am," Jess teased.
Elizabeth frowned. "Yes, well, maybe I'll invite Mother to go with me."
They all laughed.
Soon they were unloading at her parents' house. "Why is the tent up?" Elizabeth asked her mother as Brady came out to help Asa with the team and the wagon. "I thought you were all moved into your house by now."
"Asa worried it might be overly crowded in the house, what with all the children and all," Clara explained as the women hurried through the rain to the little cabin. "With Malinda and Will's children, there will be nearly twenty of us." Clara opened the door.
"Oh, Mother, look at how nice it is in here! You've been working hard." Elizabeth went around admiring familiar pieces and shelves and curtains that had been put up since she'd last been here. "Home sweet home."
"Matthew and I just moved into our house this week," Jess told her as they hung their wet coats by the door. "We won't have a party like this, but I'd love to have you come by and visit as soon as we're all settled in. Maybe some sunny day when the children are in school, you and Clara could come for lunch."
"That sounds lovely."
It was fun being in a kitchen with her mother again. Certainly it was nothing like the kitchen she'd left behind in Kentucky, but it was a great improvement over the way they cooked meals on the Oregon Trail. Before long, the men came in along with Malinda and Will and all the children. As Asa had predicted, it was very crowded in the small cabin, and it wasn't long before some of the children opted to go out to the tent to play. But it was plain to see that both Asa and Clara were proud of the house they had built. And in some ways it seemed nothing short of miraculous to Elizabeth.
"God has been good to us," Asa said as they all gathered around for a blessing before the dinner, which would be served buffet style. "For that we are grateful." Then they bowed their heads and prayed. As her father said amen, Elizabeth wished more than anything that he might be allowed to perform her wedding ceremony in December.
After the dinner and a spell of friendly visiting, the sun came out, and it seemed a good time to load up the wagons again. But Elizabeth insisted on staying behind to help her mother clean up, and JT and Ruth went with Eli to see the progress on the house. By the time they returned—with both JT and Ruth suitably impressed with what would become their home in a few weeks—Elizabeth and Clara were just finishing up.
"I'm envious," Elizabeth told Eli. "I haven't seen the house in days."
"Come on." He grabbed her hand. "Let's go see it together."
Asa winked at her. "Off with you. I'll challenge JT to a game of checkers."
"And I'll show Ruth my plans for the wedding dresses."
"Dresses?" Elizabeth looked curiously at her mother.
"One for you and one for Ruthie." Clara grinned. "Didn't I tell you?"
Elizabeth laughed. "No, you did not."
"I get a wedding dress too?" Ruth's eyes grew big.
"Come on," Eli said again. "Let's go while the weather is holding."
It was delightful to be alone with Eli again. It seemed their opportunities were few and far between these days. They had probably spent more time alone together while traveling than since they arrived. He took her hand in his, and they walked blissfully through the cool damp meadow between the two properties. "There's already a path wearing through here," Elizabeth noticed. "That's nice."
"It'll be broken in nicely and I expect it'll be traveled even more regularly once we're all moved in."
"I can hardly wait."
Eli paused in the meadow, and pulling her near, he leaned down and kissed her. "I've been wanting to do this for days now," he declared.
She nodded dreamily. "Me too."
They stayed there for a bit, just enjoying each other, and then Eli frowned up at the dark clouds that were rolling in again. "We better hurry if we don't want to get soaked."
Elizabeth grabbed up the skirt of her Sunday dress, and letting out a whoop, she broke out into a full run. Eli laughed as he hurried to catch her. But they barely reached the house when the sky opened up.
"Come on," he said as he pulled her through the open doorway. "Part of the roof is up on the north side. We can hunker down there and stay dry until this passes."
They went to the back of the house where several stumps were doubling as stools and sat down. "The floor is wonderful," she told him, tapping her toes on the solid board beneath her feet.
"I could easily have the floor done by midweek." He peered up to where the open part of the roof was letting in the rain now. "But I reckon I should make the roof a priority. Your dad and Brady have been making cedar shingles, but when I ran out yesterday, I decided to go to work on the floor instead."
"Well, it's just lovely, Eli. Everything about this house is perfect."
He laughed. "I'm not sure that it's perfectly squared."
"It's a perfect place to bring up a family." She looked down at her wet shoes. They hadn't spoken specifically about the possibility of having more children, but she knew that was likely since they both were young. And being around Jessica today, hearing her talking about the baby that would come in the spring, Elizabeth realized that she would love to have another baby too someday...God willing. But thinking of babies reminded her of something else.
"Eli," she said quietly. "There's something I've wanted to ask you about, but I just never seemed to get the chance."
"What's that?"
She explained about when she'd been riding Molly back to Malinda's just a few days earlier and how she felt certain she'd spied a young Indian woman crouching down in the brush alongside the road. "She looked right at me," she told him. "Straight in the eyes. And then—just like that—she disappeared."
"An Indian woman in these parts?" He picked up a scrap piece of wood, peeling a long sliver from it. "From what I hear, that's not very likely, Elizabeth. I recently heard that all the Indians for miles around were herded to a reservation up north. Fort Umpqua."
She nodded sadly. "Yes, I've heard the same thing."
He looked mildly surprised. "Really? Who did you hear that from?"
"Malinda told me about it."
"Did she also tell you about how inhumane the army has acted toward the Indians these past few years? And how the government broke their treaty? Did she mention how many Indians were forced to walk up the beach for a hundred miles, and how many died just making the journey?" He scowled and grimly shook his head. "And did she happen to say how most of the Indians are starving and dying from sickness on the reservation?"
"No." She stared at him in horror. "Where did you hear all that?"
He reached for her hand. "I'm sorry. I didn't mean to sound irritated at you. But I get so enraged by the unjustness of it." He sighed. "I'd hoped that maybe over here in the far West...well, I thought that perhaps things would be different—that for once there would be enough land for everyone." He made a cynical laugh. "Unfortunately, it seems that wherever the white man goes, the Indians suffer."
She swallowed and then nodded. "I fear you may be right."
"Do you know they won't even allow the Indian men to fish or collect clams on the beach? They won't let the women gather food. They don't even have enough blankets to go around."
"How did you hear all this?"
"Josiah Miller. I picked him up on the road on my way back here from Empire City. Seems he was friends with someone from Indian Affairs, a man by the name of Joel Palmer—"
"Malinda mentioned Joel Palmer. He was friends with James. And it sounds as if he's very sympathetic toward the Indians."
"That's what Josiah said too. But then the army took over the Indian situation and the relocation began, and it seems that no one pays much heed to Joel Palmer anymore." He tossed the piece of wood onto a scrap pile. "Forgive me, my dear Elizabeth, but as you know this is one of my sore spots. As much as I love this country—this brave United States of America—I sometimes cannot comprehend or agree with all of the men in leadership."
"I know," she said quietly. Now she pointed at him. "This country needs leaders like you, Eli. Maybe someday you will hold a political office and straighten them all out."
He laughed heartily, and she felt relieved to see him smiling again.
"Now, back to what you were attempting to tell me before I so rudely jumped on my soapbox and subjected you to my sermon." He chuckled. "One session of fiery words is enough for the day."
"If the reverend's passion had been targeted at the Indian situation, I'm sure I would have listened with much more appreciation." For the second time she described how she'd spotted the Indian woman several days ago. "Her eyes looked so lost and frightened. I really wanted to speak to her. But before I could dismount from Molly, she just seemed to vanish into thin air."
"That sounds typical. She obviously did not want to be seen by anyone."
"But I only wanted to help her."
"Think about it, Elizabeth. Wouldn't you try to hide if someone wanted to herd you and your children off to a death camp?"
"I cannot even imagine that."
"Unfortunately, I can."
The rain had stopped, and the late afternoon sun was glistening through the trees in the west, so Eli suggested it was time to leave. As they walked through the soggy meadow back to Elizabeth's parents' house, she explained what Malinda had told her about Charles Levine offending most of the settlers by taking up with a young Indian woman. "I think she said they've been together a few years. But apparently they're not married."
"Not married by white man's law, which is understandable. But they may consider themselves married."
"Yes...I wondered about that."
"Charles Levine has chosen a difficult path."
"And Malinda said they have a child too."
"Oh..." He let out a long sigh.
"Do you think that was her, the woman I saw that day?" she asked. "And if it was her, do you think there's any way we can help her? Help them? Help the child?"
He shrugged then squeezed her hand. "We can sure try."
"I've been praying for her," she said quietly. "Ever since that day—whenever she comes to mind. Those dark eyes...they were haunting."
He stopped walking and placed his hand gently on her cheek. "You are a good woman, Elizabeth Martin. I am a blessed man."
She smiled up at him but wished she were a better woman. A stronger woman. Because the truth was, even if she could get that Indian girl to trust her enough to accept some help, she had no idea what she could actually do or how much assistance she could truly offer. And what about language? Would she even be able to communicate with the young woman? Furthermore, she feared that her concern for an outcast Indian would be met with severe resistance from the rest of the community. Even her good friend Malinda would probably question the sensibility of that kind of charity. Once again Elizabeth realized her best response to this potentially sticky situation would be to simply pray...to ask God to direct her path.
Chapter Ten
After her mother agreed to pay a visit to Reverend Holmes with her, Elizabeth recruited JT as her postman. She sent a note to school with him on Monday saying she and her mother would like to speak to the reverend about her upcoming wedding. The reverend responded the same day by inviting the two women to tea on Thursday afternoon.
Fortunately the rainy weather gave way to sunshine by midweek, and although the road was muddy, Elizabeth and Clara didn't mind putting on their work boots and trekking to town. "Remember how much we walked every day on the trail?" Elizabeth said. "Sometimes I actually miss it."
Clara chuckled. "You can't be serious."
Elizabeth looked up at the clear blue sky. "I'm not saying I'd like to do the whole trip again," she admitted. "But sometimes I miss the simplicity of having only one goal for the day—getting my family safely down the trail."
"Yes, I suppose that makes some sense."
"Have you ever been in Reverend Holmes' house before?" Elizabeth asked as they came in sight of town—or rather, what was slowly transforming into a town.
"No, but I know they live in that cabin back behind the church," Clara said. "It seems rather small. At first I thought it was a storage shed for the church and school."
"It does seem small," Elizabeth admitted. "And a bit close to the church building. JT said that if the children are too noisy at recess, Reverend Holmes will go out on his doorstep and holler at them to quiet down.
Clara laughed. "Sometimes I'd like to tell the good reverend to quiet down."
They were in sight of the church now, although today it was playing the role of school, and it appeared that class was in session. Elizabeth studied the tiny cabin tucked into the trees back behind. "It's very dark and gloomy back there," she said as they skirted the schoolyard toward it.
"And if I'm not mistaken, it's even smaller than our little cabin," Clara said quietly. "Already I'm planning for Asa to add a lean-to kitchen on the back of it. Wouldn't it be wonderful to have the kitchen housed in its own space?"
Before Elizabeth could respond, Mrs. Holmes opened the door and waved toward them. "Good afternoon, ladies," she said cheerfully. "We've been expecting you."
Soon they were inside the shadowy cabin, and Mrs. Holmes was taking their coats. "We brought slippers for the house," Elizabeth told her as she helped her mother to remove a muddy boot.
"Thank you for your thoughtfulness." Mrs. Holmes waved over to where the reverend was seated in a chair by the smoldering fire. "Please, make yourselves at home while I get our tea ready."
Slightly surprised that Reverend Holmes didn't stand as he greeted them, Elizabeth took the straight-backed chair across from him, and her mother sat down in the rocker to his right.
"Interesting footwear," Reverend Holmes said to Elizabeth.
She looked down at her beaded moccasins and smiled. "My daughter, Ruth, and I got these while traveling the Oregon Trail," she explained. "They make wonderful house slippers."
"Ah, yes...you folks traveled by wagon train. For some reason I thought you came by ship like my wife and I did back in '53."
Elizabeth explained that the last leg of their journey actually was by ship. "It is quite a challenge to get down to this part of the country, don't you think?"
He nodded, pressing his fingertips together as he studied Elizabeth. "So you and your young man are going to pledge your troth to each other. Eli Kincaid, I believe his name is. I've only had the pleasure of his acquaintance once. Does he attend church regularly?"
Elizabeth explained that Eli was relatively new to the area and that he'd made a trip to Empire City for lumber.
"Milled lumber?" The reverend's thin gray brows arched. "That must have cost a pretty penny."
"Yes, Eli traded a wonderful horse for part of it." She smiled sadly. "He felt it was worth the sacrifice to ensure we had a roof over our heads this winter."
"Wise man." The reverend looked around the cramped room. "The parsonage roof was in need of repair last summer. I had been in hopes that the men would enlarge the parsonage as well, but alas, that was not to be." He sighed.
"Maybe next summer?" Elizabeth said with optimism.
"I would be much surprised if that happened." He scowled. "The congregation in this settlement seems to be more concerned for their own welfare than that of their clergy."
"Oh, now, Roland," his wife said as she brought in a wooden tray with some mismatched tea service on it. "Remember what the good book says about—"
"I'm well versed in the good book," he said sharply.
She simply laughed as she set the tray down on a small stool in the center of the chairs. "Roland has very high expectations—both for himself and everyone around him."
"The Lord instructs us to be perfect as he is perfect. I do not intend to settle for anything less."
"The ladies didn't come here for a sermon, Roland." Mrs. Holmes handed a cup of tea to Clara. "I was so pleased to hear that Elizabeth has decided to have her wedding in the church."
"Her first wedding was in a church too," Clara said. "And then we had a big dinner out on our farm. I think there were two hundred people in attendance. We had music and dancing and it lasted late into the night."
"Oh, my." Mrs. Holmes handed a cup to Elizabeth. "That must have been quite a memorable celebration."
"It was," Elizabeth admitted. "But I don't want this wedding to be anything like the other one. I was so young then. And no children. Truly, that feels like another lifetime to me now."
Mrs. Holmes patted her knee. "Well, we're just so pleased you want your wedding to be in the church. Especially after we learned that your sister-in-law is having her wedding in her barn."
"Yes. That was dismaying," the reverend said solemnly. "The Martins have been attending our church since its inception four years ago. I baptized two of the children, and I gave the eulogy for John's funeral. And when I met Malinda's intended, William Bramford, I was quite impressed that he was an attorney of law." He frowned. "But I do not understand why a well-educated man would choose to pledge his marriage vows in the company of farm animals."
"The Lord Jesus was born in the company of farm animals," Clara declared.
"That's true." Elizabeth suppressed the urge to giggle at her mother's boldness. "And we're farmers, so we have the utmost respect for livestock, Reverend."
He cleared his throat. "Yes...I'm sure you do."
"But you do wish to have your wedding in the church, do you not?" Mrs. Holmes looked uneasy as she handed her husband a cup of tea.
"Yes. Eli and I would like to be married in the church. On the first Saturday of December, if that isn't a problem."
"Not at all." Mrs. Holmes smiled at her husband.
"We will schedule it for you," the reverend assured her.
Elizabeth tossed a nervous glance at her mother. They'd done some strategizing as they'd walked to town. The plan was for Clara to take the lead.
"But first we have some questions," Clara said in a firm tone.
"Questions?" He set his cup down and leaned forward. "Please, feel free."
Clara took in a breath. "Elizabeth and Eli both want their wedding to be a joyous event. They have both known sadness and grief in their lives, and we all believe it is a wonderful miracle that they found each other the way they did. They wish for their wedding ceremony to reflect that."
His brow creased. "What are you saying?"
Clara glanced at Elizabeth before she continued. "Sometimes your sermons are...well, sometimes we feel rather heavy and somber after a church service, Reverend. We would not care to have their wedding feel like that."
"But a wedding ceremony is a solemn affair," he told her. "When two people pledge their troth before man and God, it is to be taken seriously."
"We do take it seriously," Elizabeth assured him. "Our vows will be genuine. But we would prefer to make our vows in a positive atmosphere. My children are young and impressionable. I want them to remember this as a happy day."
"I think that's a lovely idea," Mrs. Holmes told her.
"Making a vow to God is not to be taken lightly, ladies." The reverend's voice was getting louder. "If you wish to have a wedding that's glib and silly and childish, you will need to find someone else to officiate it."
"We are not asking for glib and silly and childish," Clara told him.
He waved his hand. "You want me to make light of a serious occasion."
"A funeral is a serious occasion," Clara said firmly. "A wedding should be a joyous one. Don't you think?"
"I agree," Mrs. Holmes said.
Mr. Holmes just glowered at all of them.
Elizabeth wasn't sure what to do now. She had suspected this man would be stalwart, but she had not imagined he would be this resistant. However, she was pleasantly surprised by Mrs. Holmes. She seemed entirely reasonable. How was it possible these two opposite sorts of characters lived peaceably—and in such a small space?
"My husband is willing to officiate the wedding service," Clara said quietly.
"Is he an ordained minister?"
"No. But he led our church services on the Oregon Trail," Elizabeth said boldly. "And everyone in our unit felt he did a wonderful job of it. He is a gifted preacher."
"If he is not ordained and he is not a justice of the peace, he cannot conduct a legal marriage."
"I was told that a marriage was considered legal if the couple had a certificate signed by witnesses and filed within the year at Empire City," Clara told him.
"And to do that, you must make two trips to Empire City," he replied. "One to get the certificate, and one to file it. Traveling that distance this time of year can take weeks."
"But we have marriage certificates," Mrs. Holmes said quietly.
"That is because I am an ordained minister," he said a bit smugly.
The room grew quiet again, and Elizabeth could see that the reverend was feeling victorious. And perhaps it was best to simply give in. Even if the wedding ceremony was a solemn, unhappy affair, they would have time to celebrate afterward. She knew her mother would see to that.
"So..." Elizabeth began slowly. "It seems that I am at your mercy, Reverend Holmes. Eli and I do wish to be married in early December. And we certainly do not have time to make the trips to Empire City and—"
"Who will officiate Malinda and Will's wedding?" Clara asked Elizabeth.
Elizabeth tilted her head to one side. "I don't know."
"As a matter of fact, I will be the officiator," he informed them. "Not here in the church as I would prefer. But I did agree to go out to her farm." He cleared his throat. "For the same fee I would charge here in the church, I will perform a simple ceremony in the barn." He looked slightly disgusted now, as if this was beneath him. "And then I will leave the merrymakers to their folly."
"And the reverend prefers that I will remain here at home for the wedding," Mrs. Holmes said sadly.
Elizabeth pressed her lips together. Perhaps this was the reason Malinda wanted to hold their wedding in the barn—so that Reverend Holmes could conduct a quick ceremony and then depart. It made perfect sense. And yet it seemed all wrong. It was as if everyone in their settlement was being held hostage by an angry preacher. Now she wondered...if the settlers were the ones paying for his services, and she knew that was true, shouldn't they have a say about the quality of those services?
"May I ask you something?" she said to him.
"Certainly." The edge of his lips curled into what might be considered a half smile.
"Do you enjoy your work, Reverend?"
He looked somewhat taken aback by her question. "Do I enjoy my work? Well, that is an interesting question. First of all it is not work. It is a calling. A very serious calling. God called me to be a minister more than twenty-five years ago. To preach the gospel to the wicked—to save the sinners from eternal damnation. I do not take this calling lightly."
"Yes," Elizabeth persisted. "But do you enjoy it?"
He frowned now. "I do not believe God intends for me to enjoy myself as I serve him. Teaching about God's judgment is a sobering responsibility. Not to be taken lightly. Not to be done for pleasure. Exposing sin and corruption, rescuing sinners from their wicked ways...it is not meant to be enjoyable."
"My father's understanding of God is different from yours. When my father preaches, he speaks of God's love and mercy. He talks about how God showed us his forgiveness by sending his beloved Son to atone for our sins," Elizabeth told him. "Even during the hardships of the Oregon Trail, whenever my father finished a sermon, we all felt encouraged and strengthened—ready to face another difficult day." She sighed. "How I miss that."
The room grew uncomfortably quiet again. And now Elizabeth felt certain that she had said too much. Her mother's eyes were large, and poor Mrs. Holmes was staring down at her lap. Elizabeth had obviously stepped over some invisible line. But how could she hold her tongue when everything about this "minister of God" seemed foreign to her? Even with the reverend glaring at her, she was not ready to back down.
She took a deep breath and then continued. "The reason I asked if you enjoy your work is that you strike me as a very unhappy man, Reverend Holmes. And with all due respect, that concerns me. I think if someone is called by God to shepherd a flock, he should do it with gladness and joy. However, you seem to be completely without gladness or joy. You seem to be a very sad man."
"Perhaps I appear sad because I take life and death—heaven and hell—seriously." His eyes narrowed and his mouth was grim. "I take my responsibilities in caring for this flock very seriously."
"Is that because you believe that it's all up to you and you alone?" she asked. "Do you believe that your congregation's spiritual fate is in your hands?"
"God has entrusted me with these people."
"I respect that. But it seems to me that you are not trusting God, Reverend. Perhaps if you trusted God more, if you believed more wholly in his love, if you understood the magnitude of his grace, if you felt the power of his mercy, you would be a much happier person. Perhaps you would even enjoy your work—that is, your calling. And if you were happier, I am sure your sermons would be more encouraging and uplifting. And your congregation would look forward to going to church." She took in a quick breath. "But as it is, your congregation is shrinking. And I'm afraid it will continue to shrink."
The only sound was the ticking of the clock. Elizabeth could tell by the sparkle in Clara's eyes that she was secretly pleased with her daughter's audacity. And although Mrs. Holmes looked somewhat shocked with her hand over her lips, Elizabeth thought she could see a trace of a smile there too.
"For a young woman, you certainly have no problem expressing your opinions," he told her in a condescending way. "Apparently you've not learned to respect your elders during your youth. But perhaps when you grow older, you will grow wiser too."
"I am older," Clara said. "And I agree with every word my daughter just said."
"As do I," Mrs. Holmes said with wide eyes.
"Georgia!" He glowered at his wife.
"I'm sorry, Roland, but it's the truth. And it's what I've been trying to tell you, but you refuse to listen. For the past couple of years, your sermons have gotten louder and meaner. You frighten the children. You offend the men—so much so that many of them are refusing to come to church. The women come only out of habit, and then they cannot seem to get away quickly enough. If you continue like this, I suspect they will give up eventually." She had tears in her eyes now. "I don't like to go to church," she confessed to Clara and Elizabeth. "I always have a stomachache on Sundays."
Clearly enraged, Reverend Holmes stood up with balled fists. His face was red and blotchy now, and the whites of his eyes were showing. But without uttering a word, he marched across the room, snatched up his hat and coat, and exited the tiny house, slamming the door loudly behind him. And now poor Mrs. Holmes burst into sobs.
"Oh, you poor dear," Clara said as she put an arm around her shoulders.
"I am so sorry," Elizabeth said as she patted the woman's back. "I never should have said all that."
"No, no," Mrs. Holmes assured her in a shaky voice. "It needed to be said." She blew her nose on her handkerchief. "Someone needed to come forward and say it."
"I...I don't know." Elizabeth's hands were trembling, and she felt close to tears herself. She could hardly believe what she'd just said to the reverend—where had that come from? "I never meant to say that much to him," she confessed. "All I wanted was a peaceful wedding."
"I've been hoping and praying one of the men would come forward and tell Roland he'd gone too far," Mrs. Holmes told them. "But they are so busy with farming and building and whatnot." She smiled sadly at Elizabeth. "I'm much obliged to you for speaking your mind."
"What will Reverend Holmes do?" Clara looked at the door with concern.
"He'll storm around in the forest, walking for miles I s'pect. He'll rant and rave at God, and in time he will come home and be very quiet." She sighed as if the quiet might be appreciated. "After that...there's no telling what he'll do. He might even vent his wrath from the pulpit on Sunday."
Elizabeth cringed. Because of her the congregation was going to suffer?
"Has he always preached such fiery sermons?" Clara asked her.
"Oh, no. Not at all. Roland used to preach some beautiful sermons. Everyone in the church back in Virginia loved him. But about six or seven years ago, his brother challenged him, saying Roland should become more theological. At first the two of them argued about it. But then his brother got him to read some theology books." She shook her head. "After Roland read those books, he began to see life differently, and he started preaching differently."
"That was back in Virginia?"
"Yes. But the congregation didn't appreciate the change. After a year or so, they decided it was time for Roland to move on." She sadly shook her head. "He was very hurt at first. But then he decided it was God's way of sending us to the Western frontier. We knew the Levine family had settled in these parts, and Roland had been good friends with Wesley Levine when they were growing up."
"Had Mr. Levine passed on yet?" Clara asked.
"Yes, but we hadn't heard about it," she said. "So we sold everything we owned and booked passage to Oregon." She waved her hand. "And here we are."
Clara reached over and grasped Mrs. Holmes' hand. "I'm sure this hasn't been easy for you."
"No...I suppose it's been my cross to bear. But I keep praying that Roland will return to his old way of preaching. I never give up hope."
"Well, I hope we haven't made matters worse for you," Clara said.
"No, don't you worry about me. Roland's bark is much worse than his bite. He truly loves me. And despite his sermons, he loves his congregation too."
"Does the reverend have any close friends? Men he can talk to?"
"I had hoped he and Wesley would rekindle their friendship. We've been here three and a half years, and so far Roland has yet to make a good friend."
"That's probably because the men are so busy trying to scratch out a living, just as you said."
She nodded. "That and they're probably afraid of him."
"Well, there's one man who won't be scared of him." Clara chuckled. "I'm going to ask Asa to come over and visit the reverend. Now that our house is mostly finished, my good husband can afford to make a trip to town now and then."
"That's a wonderful idea," Elizabeth agreed. "No one could have a better friend than Father. The reverend would benefit greatly from getting better acquainted with him."
Elizabeth and Clara visited a bit longer, staying long enough to see Mrs. Holmes return to her merry little self. But as they were leaving, she stopped Elizabeth with a worried brow. "Oh, dear. Does this mean you won't be having your wedding in the church now?"
Elizabeth tied her bonnet strings. "Why don't we just wait and see what happens with Reverend Holmes. And in the meantime, I'll be praying for him."
"Yes, so will I," Clara told her. "Maybe after Asa talks to him, we'll have a better idea of what to do regarding the wedding."
As they walked back toward home, they noticed a new set of dark clouds rolling in from the direction of the ocean and hastened their pace in order to make it back before the sky broke open. As Elizabeth watched the rain pelting down, she wondered if the reverend would still be stomping around in the woods, ranting and raving. And if so, maybe it was for the best. Perhaps a nice chilly autumn shower would cool him off a bit. Maybe God would use it to wash some of the strange notions out of his stubborn head. Then, as promised, Elizabeth said a silent prayer for the reverend.
Chapter Eleven
After the children went to bed, Elizabeth and Malinda remained seated by the fireside, sewing. Malinda was working on her wedding dress. She'd purchased the beautiful silk brocade from the mercantile. Elizabeth had held her tongue when she saw that it was a silvery shade of green. She had no intention of telling Malinda about the wedding dress poem, and she had warned Ruth to keep quiet on the topic. The bride might be wearing green, but Elizabeth knew she would not be "ashamed to be seen."
"Mother and I had an interesting visit with Reverend and Mrs. Holmes today," Elizabeth said quietly.
"Oh, yes, I nearly forgot about that. How did it go?"
Elizabeth slid the needle into the blue gingham shirt that had once belonged to James. She was altering it to fit JT, shortening the sleeve by placing a tuck just underneath the cuff and then putting in similar tucks along the shoulder seams. At the rate JT was growing, she might be letting out the tucks by spring. As she sewed, she told Malinda about their rather interesting afternoon.
"You truly said that to Reverend Holmes?" Malinda looked flabbergasted.
"That and more, I'm afraid." Elizabeth shook her head. "I was just so upset at him. It was as if someone else was talking—not me."
"Well, I would have liked to have seen that." Malinda put the end of thread to her lips.
"I thought my mother was going to be mortified. But when it was all said and done, she told me she was proud of me. Just the same, I wish I hadn't said quite so much. I truly don't like conflict of any kind. But there seemed to be no getting around it."
"When the Levines told us we were getting a minister back in 1852, we were all so excited. That summer, all the settlers worked together to build the church. Of course, we knew it would also suffice as a school, and that alone was very welcome. But everyone really sacrificed to get that building up. The Levines had spoken so highly of Reverend Holmes that we felt extremely fortunate. And I suppose we all had rather high expectations."
"I see." Elizabeth tied off her thread.
"We knew they were to arrive in September of 1853," she continued. "And although we were all busy with our own farms and homes, we decided to go the extra mile and build a parsonage. We knew it was rather small, but there wasn't much time. Besides, it was better than nothing, and we hoped to add onto it the following year."
"But that never happened?"
"By the summer of 1854, I think we were all feeling a bit disappointed in the reverend. No mention was made of enlarging the parsonage. John told me that some of the men were hoping that Reverend and Mrs. Holmes would leave if we didn't improve their home."
"It is extremely small."
"That was about the time we were having the Indian troubles," Malinda continued. "The threat of Indian attacks was an understandable distraction from our clergyman concerns. And I suppose, in light of our fearful circumstances, we were not so opposed to the reverend's style of preaching. Another year passed...and the reverend stayed on. And then we were struck by illness last year...and James died..." She sighed. "I honestly can't remember all of the details of the past couple of years, but Reverend Holmes did not give up. Church attendance fell, and I'm sure the offerings were meager. One would think that alone would have driven them back east. But alas, they are still here. I suppose we all just resolved ourselves to the idea that we would be saddled with Reverend Holmes forever."
"Is he the reason you wanted to have your wedding in the barn?"
Malinda shrugged. "He's still officiating our ceremony. But hopefully we'll all cheer up after he leaves." She gave Elizabeth a sympathetic look. "Are you sure you want to go through with your church wedding now?"
Elizabeth thought about it. "I don't see any way around it. We'll just have to make the best of it."
"If you waited until spring, you could have a very nice wedding in my barn. I would help you with all the details, Elizabeth."
She pursed her lips, considering this offer, but as soon as she thought of Eli and how faithfully he was working to finish their home and how anxious he was to be married and settled—she knew her answer. "Even if Reverend Holmes grumbles and growls and yells throughout our nuptials, I am willing to endure it. For Eli's sake, I will endure it."
######
On Sunday morning, Elizabeth felt nervous about going to church. She knew her father hadn't been able to carve out the time to go visit Reverend Holmes yet, mostly because he and Brady had been helping Eli with her house all week. So when she walked into the building with Eli and the children, she was bracing herself for whatever wrath might come from the pulpit. She had not told Eli about her visit. He seemed to have enough on his mind with finishing the house, especially now that their wedding date was just less than two weeks away.
They took their seats and waited as more families joined them. As usual, the church was not very full. Elizabeth looked up to where Mrs. Holmes normally sat right in front and was surprised to see that she was not there yet.
"Are we early?" Ruth whispered after they'd sat for a spell.
"No, I don't believe we're early," Elizabeth said quietly.
Other members of the congregation began murmuring among themselves, and everyone was obviously wondering when the service would begin. Elizabeth looked over to where her parents were seated near the front just as her mother glanced back with a concerned expression.
"Excuse me," said a female voice from behind the congregation.
Everyone turned around to see Mrs. Holmes standing in the doorway. Her face was creased with worry. "I'm sorry to tell you that Reverend Holmes will be unable to deliver his sermon today," she announced. "He has taken ill."
Now the congregation made sympathetic and concerned noises.
"I'm sorry that all of you made the effort to come today," she continued. "But we didn't know until this morning that he would not be well enough to preach."
"Asa Dawson can take the service," Will Bramford called out.
"Yes," Hugh Prescott agreed. "Let Asa preach to us."
Mrs. Holmes face brightened. "Would you care to preach this morning, Mr. Dawson?" she asked hopefully.
Asa stood and looked around. "I reckon I could."
"We'd be much obliged," Mrs. Holmes said.
Asa held up his Bible. "I came armed with the sword of the Spirit," he said jovially.
As Asa went to the front of the room, Elizabeth noticed Flo whispering to Walter, who hurried out. Flo winked at her, and Elizabeth suspected she'd sent Walter to fetch his dad and older brother.
The room grew quiet as Asa stood behind the podium. Elizabeth could see he was a bit nervous, but she knew their friends from the wagon train would be supportive of whatever he decided to say. However, she didn't know about the others. They were so accustomed to Reverend Holmes, they might be let down.
As usual they began with singing, but the songs Asa selected from the hymnal were more uplifting than usual. He led the singing with enthusiasm, and it wasn't long before the congregation seemed to catch his energy and joy. While they were still singing, Elizabeth was relieved to see Walter return with his dad and brother, taking their seats next to Flo and the girls.
When the singing was done, Asa prayed just as the reverend always did before his sermon. But Asa's prayer was gentle and humble and earnest. He asked that God would speak through him and bless everyone in the room with his holy presence.
"Now as ya'll can imagine, I didn't get up this morning knowing I was going to be standing in front of you now." He chuckled as he opened his Bible. "But I've got a mind to share one of our Lord Jesus' parables with you." He flipped through the pages. "I will be reading from the Gospel of Matthew, chapter seven." He moved his hand down the page. "Yes, here it is." He cleared his throat and then clearly and loudly read the parable of the wise man who built his house upon the rock.
Now he looked back out over the congregation. "I s'pect I chose that parable on account of I've been doing some house building myself of late." He chuckled. "I know that all of you know what I'm talking about since you've all had to build your own homes too. You understand the need to build a house that is solid and sound and strong." And now Asa explained a bit about how important it was to square up the house right from the beginning, how the very first logs or boards were like a foundation.
"If you start out wrong on the ground level, if your foundation is crooked or unstable, your house will end up being crooked or unstable. It might even fall down. That's not what you want in a house. You want a house that can stand up to the wind and the rain. A house that can shelter you from wild animals and storms and thieves in the night. A body needs a good strong house out here in the frontier."
Asa paused as if considering his words. "But Jesus was not speaking of an earthly house when he told that parable. He was talking about a man's life. A woman's life. Jesus was saying that when we listen and obey him, it is as if we have built our house solidly upon him. Jesus is the foundation of our lives." Asa pounded one balled fist onto the other balled fist as if to show this. "Jesus is the rock we want to build our lives upon. When we build our lives securely on him, when we listen to his teaching and practice obedience, we will be strong enough to withstand whatever comes our way. And we all know that hard times come to everyone on this earth. Whether it's death or illness or other kinds of losses, storms do come our way. But if our lives are built sturdily on our Lord Jesus, we can withstand those storms."
He preached awhile longer and even shared from his own trials, telling about when their family was struck down by illness and how some of them died. "The storm came at us and it came hard," he said, "but we had our solid foundation of faith beneath us. That's what got us through then. It's what will get us through anything else that might be coming our way. Build your life on the rock, my friends, the rock that is our Lord and Savior, and when those storms and winds of life come crashing down, you will be safe. You will be safe."
Asa took in a deep breath. "Now let us pray." He said another prayer and led them in a couple more songs. Then, before the service was completely over, he told them he had an announcement to make. "My wife will tan my hide if I forget to say this," he said, causing people to laugh. "Clara made me promise to stand up at the end of the service to make this announcement. But since I already have your attention, I just want to be sure to invite every one of you to a very special event that will be happening right here in a couple of weeks." He waved over to where Elizabeth and Eli were sitting. "You kids stand up," he said. And with some embarrassment, they both stood. "I'm sure everyone's heard the news that Mr. Eli Kincaid and Mrs. Elizabeth Martin are going to get hitched, but you may not have heard all the details. On the first Saturday of December they will be repeating their wedding vows right here at eleven o'clock in the morning." He looked over to his wife. "Is that correct?"
"That's right," she said.
"And I know Eli and Elizabeth will be honored if you will be in attendance. And afterward, you are invited to a dinner which we will have right here as well." Now he pointed to Elizabeth. "Anything else you'd like to add?"
She nodded nervously. "Yes. Eli and I request that you bring no wedding gifts. We simply want to enjoy your company as we celebrate our special day. Thank you."
Although the service was over, instead of everyone scurrying away as they normally did, people lingered and visited among themselves. The general feeling in the room was one of pleasantness. And it felt wonderful to see everyone relaxed and even jovial—so different from the way they usually acted. Elizabeth noticed Mrs. Holmes, who had stayed for the service, coming over to speak to Asa. She shook his hand and thanked him with tear-filled eyes before she turned to leave.
But before Mrs. Holmes could go, Elizabeth and her mother went over to catch up with her, both of them warmly greeting her and inquiring as to her husband's health.
"I hope it's not anything serious," Clara said with concern.
"No, no, I don't think it's serious," Mrs. Holmes assured them. She lowered her voice. "Roland came home looking like a drowned rat several hours after you ladies left. He was soaked to the skin. I wasn't the least bit surprised when he came down with a cold the very next day. But it wasn't until today that we discovered he could barely speak. He was as hoarse as an old bull frog this morning."
"Oh, dear." Elizabeth stifled the urge to giggle.
"The Lord does work in mysterious ways." Mrs. Holmes smiled.
"Do give him our regards," Clara said. "And if there's anything I can do to help, please, let me know."
"My mother is very gifted with herbal medicines," Elizabeth told her.
"And you can let him know that Eli and Elizabeth still wish to have their wedding in the church," Clara added.
Mrs. Holmes grasped Elizabeth's hands. "Oh, dear, I was so pleased to hear that."
"And if the reverend is still under the weather, I'm sure my father would be perfectly happy to do the service."
Mrs. Holmes made a knowing smile. "Yes, well, I've never seen Roland be sick for longer than a week. And with your wedding date being just less than two weeks, I expect he'll be all well by then."
Elizabeth nodded. "Please, don't get me wrong, I am not wishing for your husband to be ill."
Mrs. Holmes chuckled. "No, I did not think that was your intention."
Everyone was in good spirits as they went home from church. Her parents acted as if it was because of the sunshine, but Elizabeth knew that it was because of her father's uplifting sermon. If only he could permanently replace the reverend.
Once again, Elizabeth and Eli and the children went to her parents' cabin for Sunday dinner. But today, with just the six of them, it was less crowded, and they were all able to sit at the table together.
"Asa made these benches last week," Clara told Elizabeth as she was admiring the additions to the cabin. "I told him I wanted enough seating to have my whole family here for Thanksgiving this year—and that means nine of us. Naturally, I've invited Brady to join us."
"It will be snug but lovely," Elizabeth said as she stirred the gravy.
"That reminds me, Jess invited you and me to lunch on Tuesday. I told her I'd let you know. She thought we could spend the afternoon sewing on your wedding dress, and I told her I'd be very grateful for the help. I don't know what inspired me to put so many tucks in that bodice, but it's taking much more time than I imagined." She sighed as she removed a biscuit from the pan. "Or else I'm just getting old."
"You're not old," Ruth told her as she set a jar of jam on the table. "You're just right, Grandma."
As Elizabeth and Clara and Ruth worked to get the food on the table, Elizabeth imagined what it would be like to be doing this in her own house—in just two weeks. It was almost unimaginable. To make it even more unimaginable, Eli had told her that she was not allowed to step foot into the house until after the wedding. When she complained that she would be unable to sew curtains and whatnot, he had promised to write down all the measurements she would need. "But you do not get to see it until we are married," he declared. And so she had agreed, but she was afraid her curiosity would get so strong, she might sneak over there and break her promise. However, she did enjoy the idea of a surprise. Also, she had a hunch—based on various questions her mother had asked recently—that Eli was soliciting help from some of her family members.
Chapter Twelve
To Elizabeth's relief, Malinda had not mentioned anything about the closeness of their wedding dates in the past week. Perhaps she was resolved to it by now. And Malinda had been in church to hear Elizabeth announce they wanted no gifts. That should reassure her that their wedding would be a much smaller affair.
Lavinia had seemed scandalized by this news. "What do you mean by having no gifts?" she'd demanded after church. "What do you intend to do if someone shows up with a gift?" she'd teased. "Will you and Eli throw them out on the street?" Elizabeth had laughed and assured her that no one would be thrown out on the street.
To help keep their relationship moving smoothly forward, Elizabeth was doing all she could to help in the preparations for Malinda's wedding. Every evening, along with Amelia and Belinda, she helped with the sewing on Malinda's dress. At the rate they were going, it would be finished long before the actual date. She and the girls were also helping Malinda with another large quilt, which she assumed was for what would be Malinda and Will's wedding bed. Elizabeth had even donated some of her own favorite fabric scraps to be used in the log cabin design, and as the days and weeks had rolled by, the quilt was turning out to be quite beautiful.
"Are you making a quilt for you and Eli?" Amelia asked Elizabeth on Monday evening.
Elizabeth paused from threading her needle. "I have a nice quilt packed in my wagon that still has lots of wear left in it." She pointed to her sewing basket. "I'd rather put my time into getting those curtains finished," she told her. Certainly it would be lovely to have new things for her new life with Eli, but Elizabeth had no regrets for the way she was going into this marriage. Already she and her children were counting the days until the wedding date.
On Tuesday morning, after finishing with chores, Elizabeth rode Molly over to her mother's house, and from there they walked together to Matthew and Jess' house. "Did you hear about the Thompsons?" Elizabeth asked her mother as they walked.
"The family with the stallion that Asa has spoken of?" Clara said.
"Yes. Their farm is a few miles north of the school," Elizabeth told her. "According to the children, the Thompsons lost their best calf to wild animals on Sunday night."
"Oh, dear. That's too bad. But the stallion is all right?"
"As far as I know."
"I'll tell your father. He may want to bring our stock near the house at night."
"Yes. That's what Malinda and I are doing. She thinks it's a coyote pack."
"All the nights we slept in the tent, I never heard coyotes. Not once," Clara said.
They were on Matthew and Jess' property now. "Oh, look at all they've done," Elizabeth exclaimed as the small cabin came in sight. Built completely of logs, like her parents', it had a stone-lined path and smoke curling from the chimney. "The last time I was out here, the walls weren't even half up. Now it looks like a home!"
"You're here!" Jess called as she came outside to greet them.
"This place is so sweet," Elizabeth said as they exchanged hugs. "I can't wait to see your cabin."
"First I want to show you what I've done outside," Jess said as she pulled on a gray woolen shawl that had once belonged to Clara. She led them around the property, showing where she'd made a good start on some rail fences for livestock as well as a seedling garden similar to Elizabeth's. "And I just planted garlic yesterday." She pointed over to a corner of the garden.
"Wonderful," Clara told her. "It's good for warding off colds."
"And I put onion sets in over there," Jess said proudly.
Elizabeth patted her back. "You're a real farmer now, Jess."
"I can still see you on the trail." Clara chuckled. "Back when I thought you were a boy. Out there riding your horse with your head in your farming book."
Elizabeth laughed. "Yes. I remember how shocked I was to discover you were a girl."
Jess looked slightly embarrassed. "I must have seemed rather pathetic."
"No." Elizabeth shook her head. "You seemed very sweet. A girl who wanted to be a farmer. You won my heart right from the start."
"And Matthew's too," Clara added.
"I still can't believe this is our very own farm." Jess waved her hands. "All this beautiful land. Sometimes I want to pinch myself to see if I'm dreaming."
"I'm sure it's a world different from life in St. Louis," Clara said.
"It will be so exciting to see what kinds of plants will thrive here," Jess said as she led them back to her cabin. "I've read enough to know that this soil might not be as fertile as it looks."
"Asa keeps telling me that our future will be in dairy farming," Clara told them. "But I'm not so sure. With this mild climate, I expect we can grow almost anything."
"It depends on the soil," Jess explained. "I've read about soil amending. And I suspect this soil will require some work."
"I'm already saving ash from our campfires," Clara told them. "Sometimes that's good for soil."
"And I have JT collecting manure to use for fertilizer," Elizabeth said. "But it will take some time to age properly."
"In time, we'll figure it out," Clara proclaimed.
Jess held the door wide open. "Welcome to our home!"
"It's charming," Elizabeth said as she removed her riding hat. "And it smells good too." She recognized one of her mother's rugs over in the sitting area, as well as a rustic pair of chairs that she suspected Matthew and Brady had built. Brady's woodcarving skills were coming in quite handy in the frontier.
"Come see what Jess has done in her kitchen," Clara told Elizabeth. "You won't believe it."
Elizabeth went over to the kitchen area and blinked in surprise. "Wallpaper?" She turned to Jess, who was giggling. "Where on earth did you get wallpaper out here?"
"It's fabric," Jess confessed. "I saw it at the mercantile, and thought it was so pretty, I wanted a way to see it every day. That's when I decided to attach it to the wall. When it gets dirty, I'll just take it down and wash it."
"Very clever." Elizabeth went over to touch the bright red-and-yellow calico. "It's so cheerful."
They visited happily as they ate soup and bread together, and then they sat and sewed on Elizabeth's wedding dress for nearly two hours before Elizabeth knew it was time to go back to Malinda's. "Your house is just perfect," she told Jess as they hugged goodbye. "Tell Matthew his big sister is proud of him."
"And we'll see you on Thanksgiving," Clara reminded her.
There was a chilly breeze blowing as they walked back toward her parents' cabin. "I know we won't have much winter here in Oregon," Elizabeth said as she turned her collar up, "but it does feel like it's getting cooler."
"Your father said it's getting to be perfect hunting weather. But he wants to wait to go out with Eli. And Eli wants to finish your house first. Did you know that they finished the little house for Brady?"
"No, I hadn't heard."
"It's very small. Even smaller than the parsonage. But Brady seemed pleased with it. And he's been building some furnishings for it. I gave him a few things, but with his earnings from the trip, I told him he could probably find a few things at the mercantile to make his place more comfortable." She sighed. "Except I don't think he cares to go into town too much. He's worried that some folks might not be too friendly, if you know what I mean."
"I don't see why Brady should be concerned. He proved himself on the trail. Everyone from the wagon train seemed to be fairly accepting of him by the end of the trip."
"It's the other settlers that concern him." Clara frowned. "No matter how far you go, you can never seem to get away from hatred. Well, that is, until you get to heaven. Brady ought to feel right at home up there."
"I asked him about coming to church a while back, but he told me he holds his own worship services under the trees and the sky."
Clara laughed. "That'd be highly preferable to some of the sermons Reverend Holmes has subjected the rest of us to."
"Well, if you see Brady, tell him that I'm happy to go to the mercantile and do his shopping for him," Elizabeth declared.
"He might be around here," Clara said as they walked toward her cabin. "He's been awfully helpful with Asa lately." She lowered her voice. "And don't tell anyone, but Asa's been teaching Brady to read."
"That's wonderful. But why is it a secret?"
"At first it was because Brady was worried that he was breaking the law. But then he said he just wants to surprise everyone. He hopes that he'll be able to read from the Bible at Christmas."
"I won't tell a soul." Elizabeth spied her father carrying a load of firewood toward the cabin. She greeted him and inquired about Brady.
"I reckon he's working on his own little house today," Asa told her. "Or else he's helping Eli. I'm on my way over there as soon as I fill up the wood box. I promised Eli I'd help him with a door."
"A door," Elizabeth said dreamily. "Just imagine."
"Don't you go imagining too much," Asa warned her. "Might spoil the surprise."
"I'm so anxious to see the house," Elizabeth admitted. "I'm tempted to ride Molly over there right now and sneak a peek."
"Don't you dare!" Asa scolded.
"That's right." Clara shook her finger at her. "Your young man is working as fast as he can. Don't you go and ruin his surprise."
"Yes, yes...I know. Don't worry, I won't break my promise." Elizabeth untied Molly's rein from the hitching post her father had set up in front of their cabin.
"Eli is working so hard." Asa laid his load of firewood in the wood box and then stood straight and looked at her. "I don't know when I've ever seen a man so motivated, Elizabeth."
She smiled. "He's a good man, isn't he, Father?"
"Yep." Now he tipped his hat to them. "And if you'll excuse me, ladies, I will be on my way. I'm sure they're wondering if I'm ever coming back."
"Tell Brady that if he makes a list of things he needs from the mercantile, I'd be happy to shop for him. Also, I have some things I want to give him for his little cabin."
"I will let him know." Asa untied his mare and eased himself up into the saddle.
"And give Eli my best," Elizabeth told him.
"I'll do that." Asa nudged Penny, and as they headed off toward Elizabeth's property, she stood and watched longingly.
"Less than two weeks now," Clara assured Elizabeth.
"Ten more days to be precise." Elizabeth swung into the saddle. "Every morning Ruth reminds me." She laughed. "Almost as if she were counting the days until Christmas."
"I'm sure it'll feel like Christmas for both JT and Ruth—for all of you once you get into your own home."
"Yes. Ruth tries not to complain, but I know she's getting weary of sleeping with Malinda and me. However, JT will probably miss sleeping in the barn with his cousins. He's gotten quite close to Todd and Bart."
"At least they'll be nearby. Speaking of JT and Ruth, that reminds me..." Clara peered up at her. "Would the children like to stay with us for a few days after the wedding?"
"Oh, I don't know." She pulled on her riding gloves. "They'll be so excited about the new house."
"Elizabeth," her mother said firmly. "You and Eli might like some alone time after your wedding. I know you love your children, but you and Eli might be more comfortable if you don't have them with you on your wedding night."
Elizabeth giggled as she pulled her felt hat down more tightly on her forehead. "Yes, Mother, perhaps you're right. I'll ask JT and Ruth if they'd like to visit you after the wedding."
"Or maybe they'd prefer to remain with Malinda and their cousins."
"I'm sure we'll figure it out. See you on Thanksgiving, Mother." Now she kneed Molly, and waving, she was on her way. She really did want to go over and check on Eli and see her house. She was also curious to see where Brady was living and get a list from him. She could think of many excuses to wander on over to her own property. However, she knew she had to keep her word. And so, trying not to feel left out, she turned Molly back toward Malinda's.
As she came to the brushy place where she'd seen the Indian woman before, she decided to slow down. Would she possibly see her again? This spot was close enough to hear the creek bubbling along. And so she slowed Molly to a complete stop and just listened for a spell. She listened to the sound of the wind rustling through the trees, the sound of the creek tumbling along toward the river. She was almost ready to leave when she thought she heard something besides the wind and the water. Or perhaps it was just her imagination.
She held her breath as she strained her ears to listen. Peering into the brushy area where it seemed the rustling sound had come from, she tried to see through the shadows. Perhaps it was simply an animal—a deer or raccoon or rabbit. Or was someone back in there? She was tempted to dismount from her horse, but she worried that she would make a sound and frighten whoever or whatever was there. And so she just waited, quietly stroking Molly's neck lest she make a noise.
Then, just as she was about to give up, the rustling sound grew closer. And suddenly an Indian woman emerged from the shadows. Elizabeth knew it was the same woman, but this time she could see her more clearly. Her long black hair hung limply, and her dark eyes were a mixture of sadness and fear. She wore a dirty threadbare dress, little more than a rag. But it was the small shadow behind the woman that caught Elizabeth's eye. A child perhaps? The Indian woman looked as if she was afraid to move as their eyes locked.
Elizabeth, though startled, attempted to smile and say hello. But the word had barely escaped her lips when the woman backed up and disappeared—just as she'd done before. Elizabeth squinted into the shadows as she tried to decide what to do. Should she dismount the horse and call out to her? Should she attempt to follow her? Or would that only frighten her more? She thought about the small shadow again. She had not imagined it—the woman had a child with her. Elizabeth knew it. But how were they surviving out here? And if this was the woman Malinda had spoken of, where was Charles? And if she wasn't Charles' woman, who was she? Whatever the situation, one thing seemed very clear—the young woman seemed frightened and in need.
Feeling worried for the young woman and her child, as well as frustrated that she'd failed to win the woman's trust, Elizabeth nudged Molly into a gallop. She wasn't comfortable telling Malinda about seeing an Indian nearby. But she really wanted to take something back to share with Indians. And she wanted to get it to them as quickly as possible, before they had a chance to get too far away.
The children were just getting home from school, so after greeting JT and Ruth and seeing that they were off to do their chores, she went to the corner of the barn, where she had stored some of her own goods—items she'd brought along with her when they'd first moved into Malinda's. There were spare blankets and some other things. She removed a blanket and then went into the kitchen, where Amelia and Emily were excitedly telling Malinda about something they'd seen on the way home from school.
"It looked like a cougar had sharpened its claws on the tree," Emily told her mother with wide eyes.
Amelia held her hand high over her head. "But it was this tall."
"So the cougar would have to be really, really big," Emily said.
"And we didn't see those marks there this morning," Amelia explained.
"So it must be around right now," Emily said in a frightened voice.
"Do you think that's what killed the Thompsons' calf?" Malinda asked.
"That's what the boys said," Amelia told her. "They think there's a cougar on the prowl."
As interested as Elizabeth was in this conversation, she was grateful for the distraction because it allowed her to take a bread loaf and tuck it inside the blanket. She'd made three loaves yesterday, and they still had enough for supper tonight. However, this meant she'd need to bake again tomorrow.
"I wonder if one of the boys should take a gun with him to school tomorrow," Elizabeth said to Malinda as she tucked the rolled blanket under her arm.
Malinda's brow creased. "I suppose I could let Todd take the rifle. He's a good shot."
"JT is a good shot too," Elizabeth said. "But our guns are still packed in the wagon."
"Todd's the oldest," Malinda told her. "He should be the one to carry a gun."
Elizabeth nodded. "Yes. That sounds best."
The girls continued telling Malinda about the "giant wild cat" as Elizabeth slipped out the door. She hurried over to where her horse was still tethered by the barn.
"I've got to run an errand," she told JT as he and Bart emerged from the barn, each one with a pail in hand. "I'll be right back."
He just nodded.
She felt guilty for not being fully honest with her son as she nudged Molly into a trot. But at the same time she didn't want to tell her children she was helping an Indian—who knew what kind of problems that could create for them at school? No, it was better this way. And the next time she saw Eli, she would ask him for his counsel.
Before long, she was back at the brushy spot by the creek. She slowed Molly to a walk and then stopped. As she slid down from the saddle, she remembered the girls' fears about the cougar in the area. However, she felt fairly safe with her big horse so close by. Not many cougars would go after a horse.
"Hello?" she called out again, waiting to see if she could hear anything. "I leave this gift in friendship," she said loudly, hoping the woman was listening—and that she understood some English. She knew the woman and child could be far away by now. But maybe they weren't. "I'm sorry I don't have more to share right now," she said as she set the bundle in the exact spot where the woman had stood, looking up with such frightened eyes. "God bless you," she said finally as she went back to her horse.
As she rode back, she prayed for the woman and child. She prayed that God would show her an even better way to help them. For some reason—probably that look in her eyes—they seemed to need help.
Chapter Thirteen
By the time she got back and made herself busy helping with supper preparations, the big talk in the house was about the cougar. "Eli could hunt it down," Ruth was telling her cousins. "He's the best hunter in the whole world. Even my grandpa says so."
"Do you think we should tell Eli about the cougar?" Malinda asked Elizabeth.
"That's not a bad idea," Elizabeth confessed. Now she wished she'd thought of this earlier. She could easily have made it over to her property when she'd taken the bundle to the Indian woman. She could have told him all about it.
"I wouldn't be concerned," Malinda said. "Except that the place where the children spotted the claw marks is less than a mile from here. And that's nearly four miles from where the Thompsons live. It sounds like it's moving this way."
"But we don't know for sure that the cougar killed the calf, do we?" Elizabeth asked.
"No, not for sure."
"Remember when the coyotes killed one of our calves?" Susannah reminded her.
Malinda nodded. "I nearly forgot about that." She turned to Elizabeth. "We lost a calf to a pack of coyotes a couple years ago."
"Pa was so mad," Susannah told her. "He and Todd took turns standing guard at night. On the third night, the coyotes came back. By the light of the moon, Pa could see them almost as plain as day, and he shot two of them."
"And the rest of the pack never came back," Malinda finished for her.
"And Pa saved the coyote skins and Ma made them into fur collars for me and Emily for our winter coats," Susannah proclaimed.
"We need Eli to come shoot that cougar," Ruth declared. "Then I can have a fur collar too."
"He's so big you could probably have a whole fur coat," Susannah told her.
Malinda gave Elizabeth a worried look.
"I'll ride over and let Eli know about this," Elizabeth said quietly. "But I should leave straight away."
"Maybe we should send one of the boys instead," Malinda suggested.
"Except the sun will set before they could get back here. I sure wouldn't want JT riding out by himself in the dark. Especially if there's a cougar on the prowl."
Malinda frowned. "What about you? We don't want you riding around in the dark either."
"I can carry a lantern," Elizabeth told her.
Just now the boys were coming into the house. Like everyone else, they were talking about the cougar as well. "We think we should take turns watching over the livestock tonight," Todd told his mother.
"Aunt Elizabeth wants to ride over to tell Eli," Malinda explained.
"Yes," JT said. "Let's get Eli."
"He's the best hunter in the world," Ruth bragged.
Elizabeth tossed her a warning look. "Not in the world," she corrected her.
"Even so, he could get that cougar," Ruth said with confidence.
"I can ride over and fetch Eli," JT offered.
"I want to go too," Todd said. He was three years older than JT and several inches taller.
"Me too," Bart added.
"You're too young," Malinda told Bart.
"I'm the same age as JT," Bart said defiantly.
"I don't know if I will let JT go," Elizabeth said protectively. "The plan was for me to ride over to fetch Eli. And I should get going now."
"Please, Ma," JT begged. "I rode with the men all the way down the Columbia River and helped drive the livestock to Vancouver. Sometimes we even rode late into the night. You let me do that with them, and this is just a little ride."
She smiled at her man-boy. "That's true, son. Well, I suppose it's all right for you to ride over to speak to Eli. But take a lantern with you. And get our guns while you're there. And don't dillydally along the way. Come straight back here. I don't want to be worrying about you."
After more urging, Malinda let both her boys go as well. Equipped with lanterns and some slices of bread and butter in their pockets, and with Todd carrying a gun, they set out on horseback to get Eli.
"Godspeed," the women and girls called out as the boys rode off into the late-afternoon shadows. Elizabeth looked around the farm, suddenly feeling even more concerned for her chickens and livestock. She turned to Malinda. "I think the boys' idea of guarding the livestock might be wise if that cougar is really that close by. Especially since it's getting dark. Do you have a gun I could use?"
"You're going to stand guard?"
"I most certainly am," Elizabeth declared. "I didn't work that hard to get these animals across the Oregon Trail only to lose them to a cougar attack out here."
Malinda nodded. "You're right. Emily, run and fetch Aunt Elizabeth your father's shotgun from under my bed."
"I'll get my coat and hat," Elizabeth told her.
"And I'll send one of the girls out with your supper," Malinda said as they went inside.
"After the boys get back, we can take shifts throughout the night," Elizabeth told her.
"Just like they used to do on the wagon train," Ruth said in an excited tone. "Except they only let the men and boys do it."
"Well, here in Oregon, we let the women help out too," Elizabeth told Ruth. "As long as they're old enough."
"Amelia and I can do some watches," Belinda said to Elizabeth as she opened the door. "That way everyone will get a little more sleep."
"Are you sure?" Elizabeth asked with uncertainty. "Can you girls even shoot a gun?"
Belinda nodded eagerly. "Our father gave us shooting lessons on the trail. Amelia's an even better shot than I am."
"I'll work out a schedule for everyone," Malinda promised as Elizabeth was closing the door.
The sun was fully down when Elizabeth set up a wooden crate to use as a chair on the backside of the barn. They'd gathered the livestock in on Monday, so she could see most of the animals from this vantage point. Or at least she could see their shadows. But as it grew darker, it became harder to see. She would need to rely more on sound than sight. And if she heard something suspicious and felt the need to shoot the gun, she would probably just aim it into the air. That alone should scare off a cougar. Besides, she didn't want to take the chance of hitting an animal. She would make sure the rest of the watchers would do the same.
After an hour or so, Emily came out with a hot bowl of stew and a chunk of bread for her. "Ma thought you might want a lantern." Emily held out the kerosene lantern.
"No thank you. I'd rather watch from a dark spot. My eyes adjust better that way."
"That makes sense."
She thanked Emily, and then, keeping her ears tuned into the sounds around her, she quietly ate her supper. She was just setting her empty bowl down when she heard something. Silently reaching for the gun, she froze in place, listening carefully. After a few minutes, she realized it was the sound of horses' hooves. And now she could see lantern light as the riders came back into the farmyard.
"Hello," she called out as she went over to see them.
"Elizabeth." Eli slid down from what looked like Beau, although it was hard to tell in the dim light. "What are you doing out here by yourself?"
"Keeping watch," she told him. "We're going to take turns all night."
"Oh." He nodded, smiling at her in the lantern light.
While the boys took the horses into the barn to remove their saddles, she asked Eli if he thought the cougar was very dangerous.
"That all depends. If this is the same cougar that killed the Thompsons' calf, like the boys are claiming, the answer is yes. He could be very dangerous. Once a wild animal goes for livestock, it's time to put it to an end."
Elizabeth felt a shiver go down her back as she looked over to where their animals were penned in near the barn. "I doubt I could shoot it in the dark," she admitted, "but I think I could scare it away."
"I'm sure you could. And hopefully this cougar is just passing through," he told her. "But sharpening his claws on a tree right next to the road where humans travel regularly...well, that seems pretty bold. Even for a cougar."
"Yes, I wondered about that too." She reached up to stroke Beau's mane. She hadn't seen either of her team for some time now. But Beau was still as sleek and handsome as ever. She couldn't imagine how sad she'd be if one of her horses was taken down by a cougar.
"It's possible that this is an older cougar," Eli said. "To be honest, they can be the most dangerous because their hunting skills are rusty. If they're hungry and desperate, they aren't so picky about what they eat. That's when they become a very serious threat."
She sighed. "I told Malinda that after working so hard to get my livestock all the way to Oregon, I don't care to lose them to a cougar now."
"Can't blame you for that. Now I wish I'd brought Flax over here with me. He could help keep watch over these animals."
She thought about the other half of her team. "No, I'm glad you left him there, Eli. Flax can protect Bella if need be."
"That's true enough."
Now the boys emerged from the barn. "There's stew inside," she told them. "I'm sure they're keeping it warm for you." She pointed at Eli. "And I'm certain there's plenty for you as well."
Eli went inside but soon returned with a bowl of stew. "Thought I'd help you keep watch while I ate."
"Thank you."
"I'm going to stick around here for a while," he told her between bites. "Maybe get a little shut-eye. The boys offered me a bed in the barn. Then a couple hours before daybreak, I'll set out and see if I can track that cat down in the predawn light."
Elizabeth thought that sounded dangerous. However, she knew Eli well enough to know that this was a lifestyle he was comfortable with. She was not going to question his skills. "Just be careful out there," she said quietly.
"You know I will."
"I know."
######
With six of them sharing guard duty, no one had missed out on too much sleep by the following morning. Everyone got up in time to do morning chores. And thankfully, all the livestock were safe and sound.
"Eli set out while it was still dark," JT told Elizabeth as she handed him a bowl of oatmeal. "I offered to go with him, but he said no."
"That's on account you have to go to school," Ruth reminded him.
"Hunting a dangerous cougar is more important than school," JT shot back.
"Which is exactly why Eli is out hunting that cougar," Elizabeth assured them both.
"But shouldn't he be back by now?" JT asked. "He told me that cougars sleep during the day."
"Who knows how far he may have tracked it." Elizabeth handed Bart a bowl of oatmeal.
"And maybe he got tired and took a nap," Todd suggested.
Before long it was time for the children to go to school. As planned, Todd carried a gun. "Don't use it unless you really need to," his mother warned as they were heading out.
"I can't imagine a cougar would want to take on a pack of eight kids, half of them mostly grown," Elizabeth said as they watched them trekking off toward school.
"Yes, I do believe there is safety in numbers," Malinda concurred.
Elizabeth looked around the farm. "I do wish Eli would stop by before he goes back to work on the cabin. I'd like to hear if he had any luck or not."
"I'm sure you're worried about him," Malinda said, "and my best cure for worry is to keep busy."
"Yes, and there's plenty to keep us busy around here," Elizabeth agreed. However, as she worked, she longed for the day when she'd be investing her energies into her own farm and her own housekeeping. And even though Ruth had forgotten to give her the day count until the wedding, Elizabeth knew it was nine more days. Just the same, she was grateful for the time they'd been able to stay with Malinda. Certainly, not everything had gone smoothly, and she would be glad when their visit came to an end, but she was still thankful.
It was midmorning when, just as she was finishing churning butter on the porch, she spied a black horse with a rider coming down the road. She knew at once it was Eli, and since the butter was nice and stiff, she hurried out to meet him. Of course, she was not prepared for what she saw slung across the back of her big black Percheron.
"Oh, my!" Her hand flew to her mouth to see the giant lifeless animal behind Eli. "You killed it."
He gave her a tired nod. "I had to, Elizabeth." He extracted himself from the saddle, and once his feet were on the ground, he stretched his arms and back.
"I'll bet you're hungry," she said suddenly.
He grinned. "Now that you mention it."
"Come on in," she told him. "I'll cook you up some eggs, and you can tell us the whole story." She hurried into the house, calling out to Malinda. "Ruth was right," she announced. "Eli is the best hunter in the world."
Elizabeth handed him a cup of coffee, and as she cooked him a late breakfast, he told them about riding along the river until he found what appeared to be cougar signs not far from the Thompsons' ranch. "I wasn't sure if it was from Sunday night or last night. But it made sense that the animal might go back there for another meal. The sun was just coming up by then, so I wasn't feeling too hopeful. I figured the cougar had probably bedded down for the day. But I decided to go check out that ridge that runs alongside the Thompsons' place. Seemed a good spot for a cougar to hide out."
He paused as Elizabeth put a plate with eggs and biscuits and gravy in front of him. "Maybe I ought to go cougar hunting every morning." He hungrily smacked his lips. "This looks delicious."
"Come on," Malinda urged him. "Tell us the rest of the story. You got me on pins and needles here."
He chuckled as he took a big bite, taking his time to savor it. "So I got to the rocky part of the ridge, tethered Beau to a bush, and walked around a bit. I had my rifle loaded and ready. After a while I decided to climb on up and have a look around from a higher spot. And I was just about to scale up the rocks when I heard something above me." He took another bite, and both Elizabeth and Malinda waited anxiously for him to chew and swallow. "When I looked up, there was the cat, perched and ready to pounce." He gave Elizabeth an uneasy look. "Right over Beau."
She grimaced.
"So I had no choice but to take aim and shoot." He stuck his fork back into his eggs. "The cat tumbled down, landing about ten feet away from Beau." He chuckled. "You should have seen your horse, Elizabeth. He just looked at the dead cat and then went right back to munching the grass. Calm as anything."
She laughed. "First of all, he is our horse, Eli. And he obviously knew he was in good hands with you."
"Oh, my." Malinda refilled Eli's coffee cup.
They peppered him with questions as he finished up his breakfast. "Thanks for the meal," he told them.
"The children are going to be so thrilled to hear this story," Malinda said.
"Well, you ladies will have to relay it to them. I've got work to do."
They both walked him outside, pausing to look at the enormous cougar. "Does anyone eat cougar meat?" Elizabeth asked cautiously.
"Most people wouldn't care for it," he told her. "But I've known mountain men who think it tastes like pork. And when the other game sources are scarce, some Indians are known to eat it. I reckon if you're hungry enough you'd eat most anything."
Malinda wrinkled her nose. "I do not care to be that hungry."
Elizabeth cautiously reached over to touch the fur. "The fur's very soft."
"What are you going to do with the hide?" Malinda asked.
He shrugged. "I don't rightly know yet. But you can be sure it won't go to waste."
Malinda excused herself, told Eli goodbye, and then returned to the house.
"Before you go," Elizabeth said quickly, "I want to ask you something." She told him about seeing the Indian woman and the child and taking the bread and blanket. "Was that silly to leave it there like that?"
He smiled. "I think it was very kind."
"I got to thinking...what if Charles is sick or hurt? Or maybe he abandoned her. Or maybe he just can't provide for them. But for some reason I feel like she and her child are in need." She pointed to the cougar carcass. "Do you think they could use any of that meat?"
He shrugged. "I think with the abundance of fish and deer in these parts, we could find her some better food than this."
"Really?" she asked hopefully. "We could share food with her?" Though she contributed from her own staples, Elizabeth was hesitant to take any more food from Malinda's home.
"I don't see why not. I've got plenty of smoked salmon and venison. I'd be happy to share some of that. Why don't you send JT over after school? I'm sure he'll want to see the cougar anyway. I'll send a package of salmon and venison back with him for you to use however you like."
"Thank you." She looked into his eyes, feeling so very grateful that they were in agreement about something like this, something that other people might not understand, might even condemn them for. She was thankful to God that she was about to marry a good and kind and generous man.
Chapter Fourteen
On the afternoon before Thanksgiving, Elizabeth slipped away on her horse without telling Malinda where she was going. Fortunately, Malinda was too caught up in her own ever-growing Thanksgiving plans to pay much heed. First Malinda had invited Will and his family. Then, of course, they decided to include the Prescotts. But the Prescotts had already invited the Flanders family because Julius Prescott was marrying Mahala Flanders after the New Year. Consequently, Malinda felt compelled to invite them as well. As it stood now, Malinda was expecting twenty-one guests to fill her table and give thanks. Because of the size of the crowd, she had decided to set up the dinner in the barn, and she was determined to make it a dinner to remember.
Meanwhile, Elizabeth was thankful that she and her children and Eli would be gathering with Matthew and Jess and Brady in her parents' humble home. And today as she rode Molly toward her property, her plan was to make a quick trip to where she'd left the blanket and bread a few days earlier. If the items were gone, Elizabeth would leave a second blanket, this time wrapped around a parcel of smoked fish and deer meat—the food that Eli had sent by way of JT. She had no idea if the woman and child were still there, but if her first bundle was gone, it would be a good sign. And she figured she could always pick up the bundle tomorrow if it remained there untouched.
When she got close to the brushy spot alongside the road, she slowed Molly to a stop and silently dismounted. Walking quietly, she approached the spot in the brush and was relieved to see that her previous bundle was gone. Like before, she spoke aloud. "I come in friendship," she said clearly. "I want to share this food and blanket with you. Please, know that I am your friend." She stood there for a long moment, wishing that the woman and child would come out and reveal themselves. And then she wondered if they were even within hearing distance. "God bless you," she said as she set the bundle down in the same spot.
Feeling like that was all she could do—besides pray—she got back on Molly and rode back to Malinda's. More than anything, she wanted to just keep on going on to her own property. She wanted to walk around and inspect Eli's work. She wanted to make sure he was watering her seedlings even though the rain came almost every other day now. Mostly she just wanted to see his face. However, she intended on keeping her promise to him. Not until after the wedding.
"Where have you been?" Malinda asked as she walked in to discover Elizabeth removing Molly's saddle in the barn.
"I had a Thanksgiving errand to attend to," Elizabeth said absently.
"Did you sneak over to look at your house?" Malinda asked with suspicion.
Elizabeth made a sheepish smile as she let Molly join the other livestock outside the barn. "I am dying to go take a peek. I can hardly stand it. But a promise is a promise."
Malinda nodded. "That's right. Now, how about if you give me a hand setting up these tables and benches." She pointed to a pile of long boards. "Will brought these over while you were gone. He got the lumber to use for the floor of Julius and Mahala's house, but he thought we could put the boards to good use here first."
Elizabeth examined a board. "This is probably wide enough for a bench—a rather narrow bench. Maybe you could set them on crates or stumps. But wouldn't you have to nail the boards together to make them sturdy enough for a table?"
Malinda showed her the smaller pieces of wood that Will had brought. "He said to attach them to this to make the tables. Maybe the benches too. We'll just leave them fastened together until our wedding."
"Why don't you let JT do the carpentry for you," Elizabeth suggested. "He's helped so much with building houses these past months that he's gotten quite handy with a hammer and saw."
"Good idea. We'll put the boys onto it as soon as they get home." She pointed at Elizabeth. "And you promised to make pies for my feast."
"That's right. I should probably get to it." Elizabeth considered reminding Malinda that the pies were for her family's feast too. After all, she was using her own flour and sugar, and Ruth had picked most of the berries. However, the pumpkins had been grown on Malinda's farm. And she was using Malinda's kitchen and butter. Oh, for the day when she would have her own house again.
Elizabeth was elbow deep in pie filling by the time the children came home from school. And now the house, which had been relatively quiet, suddenly grew noisy, filling as if it were about to burst at the seams as Malinda began giving everyone special jobs to complete before supper. These were in addition to their usual after-school chores. And after some expected complaining, the children shuffled off to their various assignments.
######
Before going to bed, Elizabeth had told JT and Ruth that they would leave for their grandparents' house as soon as their morning chores were finished. And she was pleasantly surprised at how quickly they completed their usual tasks the next day. It wasn't long until they were dressed in their Sunday best and preparing to leave. However, as Elizabeth reached for the basket containing the pies she was taking to her parents' house, she could see that Malinda's household was in a frenzy.
Elizabeth knew that Malinda was trying too hard to impress the Bostonians, but she had no intention of mentioning this. She'd already suggested that Malinda might enjoy the day more if she kept her preparations a bit simpler, but Malinda had not appreciated that advice.
"Have a wonderful Thanksgiving dinner," she told Malinda as she made her way to the door.
"Oh, dear." Malinda wiped a floury hand across her forehead. "I just hope I'll be ready in time."
"I'm sure you will," Elizabeth told her. She knew that if she volunteered to stay and help out, Malinda would not refuse her offer.
Before they could leave, Susannah pointed at JT. "You're taking your guitar with you?" she asked.
He shrugged at his cousin. "Yeah. We like making music together. Uncle Matthew will bring his fiddle, and I hope Eli will have his guitar."
"If he doesn't, we'll make him go back and get it," Ruth declared.
"I wish we could have music at our Thanksgiving dinner," Susannah said to her mother.
"Instead of wishing for what we don't have, come over here and chop these walnuts for me, Susannah Marie."
As Susannah groaned, Elizabeth ushered her two toward the door. "We'll be back before bedtime," she promised as they went out. Her parents had asked if she and the children would like to spend the night, and although it was very tempting, she knew they would be crowded in their little cabin. Besides that, the children had school in the morning.
As they walked down the mucky road, still wet from last night's rain, Ruth led them in cheerful singing. Despite the mud, the morning sun was clear and bright, and Elizabeth soon felt light and happy and free. As much as she loved her sister-in-law, sometimes she grew weary of her high expectations. And sometimes it seemed that Elizabeth's wedding day could not get here quickly enough. Of course, these were thoughts she kept to herself.
As they walked and sang, she kept something else to herself. She slowed her pace when they came to the brushy spot alongside the creek. Without saying a word to the children, she paused to see that the offering she'd left yesterday was now gone. She sighed in relief, feeling certain the Indian woman and her child were the ones who were benefiting from her gift. Just knowing they were a bit more comfortable made her feel much more like celebrating this special day with her family.
"Happy Thanksgiving!" Ruth proclaimed as Clara led them into the house. Hugs were exchanged, and Elizabeth and Ruth were happy to enlist as volunteers in Clara's cheerful kitchen. Meanwhile, JT opted to go outside and find his grandpa and Brady.
"Remember you have on your Sunday clothes," Elizabeth reminded him.
"This is truly a special day," Clara told Elizabeth and Ruth. "Our first Thanksgiving in Oregon. So much to be thankful for."
"Most of all, I'm thankful for my family," Elizabeth said as she hung up Ruth's coat and her own.
"Me too," Ruth chimed in as she removed her bonnet. "We have the best family in the whole world."
Elizabeth chuckled. "Yes, well, I must agree with you on that."
"When will everyone be here?" Ruth asked.
"Matthew and Jess will come around one," Clara told them.
"And Eli too?" Ruth asked.
"Yes, Eli too," Clara assured her.
Ruth sniffed the air. "Are we having turkey, Grandma?"
"Oh, Ruth," Elizabeth said. "You know there are no turkeys in Oregon."
"But she's not too far off," Clara told Elizabeth with a twinkle in her eye.
"What are we having?" Ruth asked.
"Goose." Clara pointed to the cast-iron roaster positioned over the fireplace coals.
Ruth clapped her hands. "A goose!"
"Where in the world did you get a goose?" Elizabeth asked.
"Eli brought it over last night," Clara explained. "Otherwise we would have only had a venison roast. But now I'm fixing both."
"What a feast!" Elizabeth exclaimed. "Our first Thanksgiving, and we get to have roasted goose."
"And Jessica is baking yams with apples."
"And Mama made pies," Ruth told her.
"And now I need someone to peel potatoes," Clara said to Ruth.
Together they worked, but not at the frenzied pace Elizabeth suspected was taking place at Malinda's house. Instead, they enjoyed one another's company—talking and laughing and remembering Thanksgivings past and planning for ones in the future. It was a little past noon when Jess joined them. But her company only made the house merrier. Elizabeth felt truly blessed to be with her family like this, and even more so when Eli came into the house with her father, her brother, her son, and her dear friend Brady. This was truly a Thanksgiving to remember.
Although they were in high spirits, chattering happily as they gathered around the table, the room grew quiet, when Asa held up his hands. "As has always been the tradition of this family—in good times and in bad—we will go around the table expressing gratitude for something we are thankful for." He nodded to Ruth. "As usual we will start with the youngest and end with the oldest."
Ruth smiled, revealing where she'd recently lost a front tooth. "I'm thankful for my school and my teacher, Mrs. Taylor."
Asa looked at JT. "I'm thankful that I'm finally getting to know my cousins Bart and Todd."
"What about Susannah and Emily?" Ruth asked.
"They're girls," JT told her. Everyone laughed.
"I am thankful to be part of this wonderful family." Jess smiled at Matthew. "And thankful that our little family is growing."
"I'm thankful for Jessica," Matthew looked adoringly into her eyes. "And that our cabin is finished."
Jess nodded eagerly. "Yes! Me too."
Elizabeth knew it was her turn now. "I'm thankful for so many things. Where do I begin? I'm exceedingly thankful that we all made it safely to Oregon, thankful to be settling here with my family. I'm thankful for my upcoming marriage to Eli." She laughed. "I could go on and on."
Eli nodded. "I feel thankful for many things too." He looked around the table with genuine appreciation in his eyes. "For the way you've all welcomed me into your family. For all the help Asa, Matthew, Brady, and JT have given me." He turned to Elizabeth. "For my bride-to-be, Elizabeth."
She swallowed and smiled.
"My turn," Clara said. "Well, my list isn't so different. I am very thankful for every one of you at this table. I am thankful and somewhat amazed that an old woman like me made it all the way across the Oregon Trail in one piece." They all laughed. "I'm thankful my little house is finished." She smiled at Asa. "And thankful that God is allowing me to grow old surrounded by my loved ones."
Asa looked at Brady now. "You're a little older than me, Brady, so I reckon I go next." Brady looked relieved. "I'm thankful for all of you. And thankful for our safe passage to our new home in Oregon. I'm thankful for all our friends and neighbors in this settlement. Most of all I am thankful for God's continued goodness to this family."
They all looked at Brady. Sensing he was feeling uncomfortable, Elizabeth smiled. "Your turn, Brady."
He nodded with misty-looking eyes. "I'm thankful for each and every one of you folks. Ya'll been like kin to me. You took me on a journey I won't never forget. I'm thankful for my cabin. And most of all, I'm thankful to be a free man."
Asa nodded. "Now let us bow our heads and give thanks to our generous Father in heaven." He prayed a heartfelt prayer of thanksgiving and gratitude to which everyone added a hearty amen.
######
After dinner some of them went outside to enjoy the afternoon sunshine and to admire the split-rail fence Asa and Brady were building. "Now that Eli is working on the inside of your cabin, he doesn't need our help so much, so we have more time on our hands," Asa told Elizabeth. "At the rate we're going, I s'pect we'll have a couple of acres fenced in by Christmas. Don't you reckon, Brady?"
Brady nodded. "I sure hope so."
"Well, since your fenced pasture is so much bigger and better than ours, we just might be wintering our livestock over here," Elizabeth teased her father.
"That's not a bad idea," he said. "Then when you get your pasture fenced in, we could move them over there and give this grass a chance to replenish itself."
"And come summer, you can move 'em on over to our place if you like," Jess suggested.
Now Asa pointed over to a sunny rise midway between the cabin and the fenced pasture. "I want to put my barn right there," he explained. "As big as I can afford to build it, and with room to grow it even bigger on down the line. I've got this notion in my head that we're going to be a dairy farm." He continued pointing at his imaginary barn. "And there'll be a big sign right there above the door, saying Dawson's Dairy."
"I wouldn't be a bit surprised," Elizabeth told him. "Maybe we'll all want to become dairy farmers someday." She looked at Eli, wondering how he felt about seeing these fences going up.
"The grassland along the river seems just about perfect for cattle," he said. "I'm not opposed to the idea of raising dairy cows." He grinned at her. "I like milk and butter and cheese."
"And my Clara knows how to make some mighty fine cheese." Asa smacked his lips.
"That's true," Elizabeth said. "Mother's cheeses are the best."
"Cheese..." Jess sighed. "That sounds wonderful."
"I'll bet that next Thanksgiving, if not sooner, we'll have cheese at the table," Elizabeth proclaimed.
As Asa, Jess, and Brady continued walking around the fence line, Elizabeth lingered back with Eli, shyly slipping her hand into his. It felt so good to be here with him right now. She could hear the sounds of Matthew's fiddle and JT's guitar as the two warmed up for their after-dinner music. Meanwhile Ruth was helping Clara to serve the dessert. A perfect day.
"Brady was talking to me on our way over here," Eli began. "There's something that I need to talk to you about."
"Oh? Is something wrong?" She studied his profile as he gazed over the pasture, admiring his straight nose and firm chin.
He turned to look at her. "Not really wrong. But Brady would like to live over here...on your parents' property."
"Brady wants to live with my parents?"
"Not with them. But he'd like to have his house near them."
"But he has a house on our property."
Eli nodded. "And he feels badly about that."
"Why?"
"Because he wishes he'd said something sooner. Otherwise I never would have pushed to get his little cabin up. But I had the leftover lumber, and I thought, why not use it to get a roof over Brady's head? I didn't like the idea of him living in your father's tent when the weather gets wetter and colder. He's an old man."
"Yes, that makes perfect sense. But Brady doesn't want his little cabin?"
"He's willing to live in the tent...just to be closer to your parents."
"I wonder why he wants to be closer to them."
"Partly because he feels they could use his help. He talked about how they were all getting older and how they can help each other. But more than that, Brady really admires your father. And they've been turning out some nice-looking pieces of furniture together."
"Yes, I've seen some of it."
"I reckon Brady feels like they could work together more easily if he was living over here."
"It would save him a lot of walking," Elizabeth admitted.
"And he is getting older."
"So are my parents." She nodded. "I suppose it does make sense. Too bad you went ahead and built that little cabin."
He shrugged. "Maybe we'll think of a use for it." He peered at her. "Brady is worried that you'll be hurt if he moves over here."
She considered this. "Well, I might have been hurt if you hadn't explained it. But it does make sense."
"And there's another thing." He lowered his voice. "Nobody's supposed to know, but your pa is teaching Brady to read."
She grinned. "I heard about that already."
He smiled. "Well, keep it under your hat."
She looked to where Asa, Jess, and Brady were coming back toward them now. "So should I tell Brady that I don't mind him moving over here?"
"Not right now. I promised him I'd speak to your parents too. But if you don't mind, I can let Brady know that you understand. I know that will be a comfort to him."
"Yes, of course. And let him know that I appreciate knowing he'll be over here to help my parents if they need it. And after all, like he said at the table, he's a free man. He can live wherever he wants to live."
"Well...not exactly." Eli frowned.
She nodded, remembering Oregon's recently passed legislation—the three-year limit for Negroes to reside here. However, if Brady kept to himself as he'd been doing, if he avoided town and other settlers, perhaps no one would notice. They could only hope.
Chapter Fifteen
I do not understand you," Malinda said to Elizabeth on Saturday evening. As usual, they were sewing by the fire. The children were all in bed except for Belinda and Amelia, who were spending the night with Evelyn and Lavinia in town. The Prescott men had decided to make the most of a recent dry spell by traveling to Empire City to pick up some supplies for the mercantile.
"What is it you don't understand?" Elizabeth asked as she stitched the hem of the curtains she was sewing for the front window of her house. She still hadn't seen the window, but Eli had given her the measurements. She'd found the red and white gingham fabric at the mercantile and felt it would look charming in the cabin. She'd gotten enough fabric for additional curtains for the kitchen cupboards as well as a tablecloth.
"I do not understand how you can appear so calm and nonchalant with your wedding less than a week away."
Elizabeth just laughed. "Oh, is that all?"
"Well, my wedding is three weeks away and I'm a bundle of nerves."
"Perhaps that's because your wedding is a bigger, fancier affair than mine."
Malinda frowned. "When you say it like that it feels as if you are judging me."
"I'm sorry. That's not how I meant it. The truth is I'm really looking forward to your wedding. I think everyone is."
"But you're not looking forward to your own?"
"Of course I am." Elizabeth set her sewing aside. "But maybe I'm looking more forward to the marriage...settling into our house...feeling like a family again."
"Oh." She nodded. "Yes, that makes sense."
"Remember how you felt when you came to Oregon? How you looked forward to putting down roots, moving into a real house?"
"I most certainly do."
Elizabeth returned to hemming the curtain.
"Even so," Malinda persisted. "Are you not a little bit excited about your upcoming wedding?"
Elizabeth laughed again. "I most certainly am."
"But you don't show it." Malinda peered at her.
Elizabeth pursed her lips, unsure of how much to say. "Do you want the truth, Malinda?"
"Yes, of course I do. We're best friends. Why wouldn't you tell me the truth?"
"The truth is I've been trying to play my wedding down for your sake."
"For my sake?" Malinda looked shocked.
"You were so opposed to the date," Elizabeth told her. "You wanted me to wait until spring and—"
"Oh, I'm all over that now." Malinda reached for her scissors. "I don't mind if you get excited over your wedding."
"Thank you," Elizabeth said calmly. "But perhaps I'll wait for a few days before I get overly excited."
Malinda laughed and then held up the quilt top she was working on. "Look, it's all done."
Elizabeth studied the colorful piece. "Oh, Malinda, it's beautiful. Do you think you'll have it finished before your wedding?"
Malinda's brows arched. "Oh, I should say so. I invited a dozen women over here for a quilting bee on Thursday."
"Really?" Elizabeth frowned. "Was I not invited?"
Malinda shrugged. "I thought you'd have your hands full with your own wedding preparations by then."
"Oh." Elizabeth tried not to feel hurt. But missing out on a quilting bee felt like a real slight. "Yes. I suppose I could go to my mother's and help her with—"
"Your mother will be here," Malinda said. "Working on the quilt with me."
"Oh." Elizabeth felt more than slighted now. She felt hurt. "Well, perhaps I'll go visit Jess and—"
"Jess will be here too."
Elizabeth was about to insist that she should participate in the quilting bee as well, but that would be like inviting herself to a party. Bad etiquette. But what sort of manners was it for Malinda to invite Elizabeth's family and leave her out?
"I'm very tired," Elizabeth said quietly. "I think I will turn in."
######
On Sunday, Elizabeth was trying to put her best friend's thoughtlessness behind her. It was silly to be offended by the slight. Besides, she felt certain that if she told Malinda she wished to come, Malinda would probably welcome her with open arms—or at least she'd welcome her help in serving lunch. However, Elizabeth felt just stubborn enough not to mention it. She felt even more stubborn after sitting through Reverend Holmes' sermon about Daniel in the lion's den. To be fair, it was probably the most positive sermon she'd heard him preach. But it was nothing compared to her father's.
Even so, she shook hands and thanked him. "I was greatly relieved to hear you were feeling much better this week," she said. "And that was a fine sermon, Reverend."
He looked slightly surprised. "Thank you, Elizabeth."
Now Eli stepped up to shake his hand, also praising his sermon.
"I look forward to your wedding on Saturday," the reverend told them both.
"We're looking forward to it too," Eli assured him.
Asa and Clara came up to the reverend next as Eli and Elizabeth went outside to where the children were playing in the yard.
Before long, Eli was enticed to go over and visit with some of the men. They wanted to hear his story of how he'd hunted down the cougar. Then Elizabeth, seeing that Lavinia was standing by herself, went over to talk with her. As they chatted about the mercantile, Elizabeth got an idea. "Will you be open for business on Thursday?" she asked.
"Oh, yes, of course. We're always open on Thursdays," Lavinia assured her. "In fact, Hugh and Julius should be back from Empire City by Tuesday or Wednesday at the latest, and I'm sure we'll have the shelves freshly stocked by Thursday."
"Good." Elizabeth nodded. "Then I shall come and do some shopping."
"Do you need things for your new house?"
"Oh, I'm sure that I do. I plan to go through my things this week and make some lists."
"I've heard that Eli won't let you see your house until after the wedding." She chuckled. "I told Hugh that could be a bad sign."
"A bad sign?"
"Yes—that is, if Eli were trying to get you married before showing you the house because it is a complete disaster." She laughed. "Although I'm sure that's not the situation, Elizabeth. But it would be funny, don't you think?"
Elizabeth frowned. "I'm not sure it would be very funny. But I am fairly certain that Eli has built us a fine home. My father and brother and JT have helped him with it, and I cannot imagine—"
"Yes, dear...I'm sorry. You'll have to forgive my humor."
Elizabeth patted her arm. "Don't give it a thought." She sighed. "I suppose with the wedding less than a week away, I'm finally starting to feel a bit nervous."
"That's only natural." Lavinia's cheery countenance faded slightly as she lowered her voice. "Although I still find myself regretting that you're not marrying our Will."
"But you do get on well with Malinda, don't you?"
Lavinia made a small shrug. "I suppose I haven't had the opportunity to get very well acquainted yet, whereas you and I solidified our friendship on the Oregon Trail. I still don't know what I would have done without you and your family."
Elizabeth nodded. "Friendships made on the trail are lasting ones." She was tempted to ask Lavinia if she'd been invited to Malinda's quilting bee, but she knew that was rather meddlesome. "So will you be working at the mercantile on Thursday afternoon?" she asked.
"You can count on that." Lavinia glanced down the street to the imposing building. "And if it's not busy in the store when you come, perhaps you will want to come upstairs and have a tour of our new living quarters. Not everything is finished yet, but we are trying to make ourselves at home."
"I would love to see it, Lavinia."
"Maybe we can have tea."
Elizabeth nodded happily. "I would enjoy that."
Now JT and Ruth, accompanied by Tillie and Walter Flanders, came over to Elizabeth. "Mrs. Flanders invited us to stay in town with them for lunch," Ruth said with bright eyes. "It's Tumbleweed Tillie's birthday today."
Elizabeth laughed. She hadn't heard Tillie's nickname for some time. "Happy Birthday, Tumbleweed," she told her.
"Thank you, Mrs. Martin."
"And I gave her my hair ribbon." Ruth pointed to the blue bow tied around Tillie's wavy blond hair, which had grown out to her shoulders. "Isn't it pretty?"
"Very pretty." Elizabeth tugged Ruth's now-barren braid. Ruth had bought that ribbon with her own money, and it was touching to see her generosity.
"Is it all right, Ma?" JT asked.
"Ma made a cake," Walter said proudly.
"It's just fine," Elizabeth told her children. "Just mind your manners. And I'll expect you home in time for afternoon chores." They agreed to this and then ran off to rejoin their other friends just as Eli came over to join her.
"I see that Hugh and Julius aren't back from Empire City yet," he said to Lavinia.
"No, I don't expect them until midweek."
He nodded.
"Perhaps Elizabeth can pick up your order for you when she comes to town on Thursday," Lavinia told him.
"You're going to town on Thursday?" Eli asked.
"Yes. Lavinia and I might even have tea."
"Good for you. How about if I bring the wagon over to Malinda's on Thursday morning? You can drive it to town and use it to pick up my order."
"What did you order?" she asked curiously.
He just grinned. "Never you mind. But it's heavy enough to need a wagon."
Lavinia laughed. "What I wouldn't give to see your face when you walk into your new home, Elizabeth."
"It won't be long now," Eli assured her.
"And once we get settled, we will invite our friends over," Elizabeth said. "Maybe for a Sunday dinner since the mercantile is closed."
"Let's make it a potluck," Lavinia suggested.
"Wonderful." Elizabeth looked at Eli. "Perhaps a week after our wedding? Or do you think that's too soon?"
"That is entirely up to you."
Elizabeth nodded. "Then we shall plan on it," she told Lavinia. "Feel free to tell any of our friends who come into the store. After church on the second Sunday, the Kincaids will host a potluck at their new house."
"And that gives me an idea," Lavinia said suddenly.
"What's that?"
"Oh...you'll find out." She made a sly smile and waved goodbye as she hurried away.
Elizabeth explained about Tumbleweed Tillie's birthday. "So we can head back whenever we like."
His eyes twinkled. "Just you and me?"
She nodded eagerly as he took her hand and led her down the road. "As much as I love my family and friends, I do enjoy your company," she told him.
He squeezed her fingers in his. "I am counting the days until Saturday," he said. "And the hours too."
She laughed.
"But as much as I'm looking forward to it, I do want to have our house finished. Or nearly."
"I'm certain that it would be fine just as it is right now."
"Oh, sure, it's livable enough."
"Are you living in it?"
He shook his head. "No, my dear, I'm waiting for you."
"Oh, Eli, you can live in it if you like."
"Not without you."
Not for the first time, Elizabeth had to ask herself if this was real or just a delightful dream. But the musty smell of decaying autumn leaves and the sound of the birds twittering in the trees told her it was authentic. And for some reason this reminded her of the Indian woman in the woods. "I didn't have a chance to tell you that the second blanket and parcel of food I left for the Indian woman and child were gone the following day. So I can only assume they got it."
"Yes, if it had been taken by animals, the blanket would probably still be there."
"So it makes me believe they are living somewhere near there."
"At least for now. It's possible they are on their way to somewhere else." He shook his head in a dismal way. "Although, for the life of me, I cannot imagine where that would be. No place is safe for Indians in these parts. Any that are found are rounded up and herded up north like animals."
"I'm sure that's why she looks so frightened," Elizabeth said sadly. "Her eyes looked so empty and hopeless and scared. I wish I could get her to trust me."
"And then what would you do?" He peered at her.
"To be honest, I don't know." She sighed. "But I will keep praying. I believe that God will show me."
"Now, not to detour our conversation, but is there anything I need to know or to do before next Saturday?"
She shook her head. "Not that I can think of."
"I assume you don't want me to show up in my buckskins," he said teasingly.
She laughed. "I would not complain if you did, Eli. That's what you wore the first time I met you, and I happen to think you look very handsome in your buckskins."
He chuckled. "I'm glad to know that because I'm reluctant to give them up."
"However..." She pointed to his Sunday clothes. "What you have on would probably be more appropriate for a church wedding. Don't you think?"
"I am in complete agreement. And I'm content to save my buckskins for other activities like traveling and hunting."
"Speaking of hunting, I know that my father is most eager to go on an elk hunting trip with you."
"Yes. He and Matthew both. I promised them we would go the week before Christmas—if that's agreeable to you."
"Yes, as long as you won't go before Malinda's wedding. I'm sure she'd want to tan my hide if all my menfolk went missing on her big day."
"We'll keep that in mind."
"Speaking of my father, did you have a chance to speak to him about Brady yet? I meant to ask, but we must have gotten distracted."
"As a matter of fact, I did tell him about Brady."
"What did Father say?"
"Both your parents were very touched. And they said that if you were all right with this, they would gladly welcome Brady to live on their land. Brady can continue using the tent, and Asa even offered to help Brady build a new cabin. I expect Brady will move back over there by tomorrow."
Elizabeth had mixed feelings. As much as she wanted Brady to be happy and free and to live where he liked, she would dearly miss having him close by. He'd been in the Martin family for as long as she could remember. And he'd been with her and James from the first days of their marriage, overseeing their farm. Even after she and James decided to free the other slaves, Brady had asked to stay on with them. And then, during the years following James' death, Brady had been as dependable as the sun. Just seeing him about her farm each day, checking on livestock, hitching a wagon, bringing her firewood...it had always been so reassuring.
However, she knew circumstances were different now. This was the Oregon Territory, not the State of Kentucky. Furthermore, she had Eli to depend upon. And really, what more could she ask? Plus, she reminded herself, her parents were getting older. Having Brady around to help her father carry the load was a great comfort. And Brady was getting older too. Why shouldn't he live out the last of his days however and wherever he wished? So truly this was for the best. It was time for her to let Brady go.
Chapter Sixteen
By Thursday morning, Elizabeth harbored no ill feelings about being left out of Malinda's quilting bee. She tried to make herself useful as well as doing her regular chores, but Malinda resembled a small whirlwind as she ran from task to task.
Grateful for an excuse to get out of the house, Elizabeth hurried out to talk to Eli when she saw him arriving with the wagon and her team. He kissed her hello and seemed happy to see her, but she could tell he was eager to get back to the house. For that reason she encouraged him to ride Molly back home.
"It will save you time," she said. "And then you can ride her back when you pick up the wagon at the end of the day."
"I'll see you tonight," he promised. "Have a good day in town."
For the next hour, eager to stay out of Malinda's way, Elizabeth remained in the barn, sorting through crates of goods she'd brought with her to Malinda's. She wanted to have everything ready to load into the wagon before Eli drove it back.
Satisfied that all was organized, she returned to the house only to discover Malinda still scrambling about, trying to get her house into perfect order and her "luncheon" meal prepared. Elizabeth didn't dare to point out that Malinda still hadn't changed out of her work clothes and that her guests would soon arrive. Elizabeth was tempted to offer some additional help, but knowing Lavinia was looking forward to her visit, she simply wished Malinda a good day and made her getaway.
Elizabeth felt relieved to be driving the wagon to town. Her team looked sleek and handsome. Her wagon without the canvas covering was sturdy and strong. Dressed in her Sunday best, she looked forward to getting some much-needed supplies for her first week of married life. She also looked forward to a friendly visit with Lavinia. She had no regrets about missing out on Malinda's gathering and hoped the quilting bee was enjoyable for the ladies, but reminding herself of how much of her spare time was spent sewing, she decided that she'd been lucky to miss out on it.
Besides that, she told herself as town came into view, she would have a quilting bee of her own someday. She would hold it in her own home. And she would be sure to invite Malinda—and anyone else who had the slightest interest in coming.
As she came in sight of the school and mercantile and blacksmith shop, she slowed the team. She wanted to take a good long look at the scene before her. Someday the virtually empty street would be just a memory. In time the town would grow and change. It would become noisy and busy and be transformed into something altogether different. And she and her family could say they remembered when there was nothing here.
She paused to admire the slowly moving river just south of town. She smiled to think that it was one thing that would never change.
She pulled the wagon in front of the mercantile and, removing her driving gloves, held out her skirt as she carefully climbed down. It hadn't been necessary to wear her Sunday dress today, but it had felt good to put it on. And she knew Lavinia would appreciate the effort. As she went up to the store, she looked up at the sky. She'd noticed clouds rolling in from the ocean and suspected that the fair weather they'd enjoyed the past few days was about to disappear. She hoped it wouldn't be wet and gray by tomorrow, although rain on a wedding day was always considered to be good fortune—at least among farmers.
"Hello, hello," Lavinia hailed as Elizabeth entered the store. "I was just telling Mrs. Levine that you were coming to visit me."
"Good day, Elizabeth," Mrs. Levine said. "Are you looking forward to your wedding? Just two days away now."
Elizabeth smiled. "Yes, I can hardly believe it's almost here."
"I was just telling Lavinia that the upcoming year will most likely be filled with even more weddings."
"That's right," Lavinia told Elizabeth. "We predict that Belinda and Jacob will be engaged by Christmas."
Elizabeth nodded. "That would not surprise me."
"And Mahala and Julius have decided to be married in April," Lavinia said with excitement. "That will give me enough time to plan for a proper celebration."
"I hear that springtime is lovely here," Elizabeth said as she perused a basket filled with colorful spools of thread. "Lots of wildflowers to choose from." The three of them chatted congenially for a bit, and finally Mrs. Levine excused herself and, taking her parcel, exited the mercantile.
"Perhaps we should take advantage of this lull and go upstairs," Lavinia said. "I suspect customers will be having their dinners now anyway." She hurried to close the front door, putting the We Will Be Right Back sign on the door. Then she led Elizabeth to a staircase in the back. "Right this way, my friend."
When they emerged from the stairwell, Elizabeth was surprised to see how light and bright the rooms up there were. "Oh, my," she said. "Look at all these windows."
"Yes." Lavinia nodded. "I told Hugh that the last thing I wanted was to be stuck up here in a dreary attic. He made certain that did not happen."
"It's lovely." Elizabeth admired the fir floors in the main room. "And your furnishings!" she exclaimed over the padded settee and chairs. "They obviously came here by ship."
Lavinia laughed. "Well, yes. I had to leave so much furniture behind, but I insisted on a few comforts." She led Elizabeth around, showing her their home, which by frontier standards was quite luxurious.
"I told Hugh that everything we brought for our own use could always be sold in the mercantile should times get hard." She grinned. "It's a bit like living in your storeroom."
"A very lovely storeroom."
Lavinia even showed Elizabeth the bedrooms—all three of them. "One for Augustus and one for Evelyn. Although I imagine Evelyn will not be living her for more than a year or two."
"Has Evelyn found a beau too?"
"I do believe she's got some fellows with their eyes on her."
"I shouldn't wonder."
"When Evelyn marries, we will turn this room into a study."
"Very nice." Elizabeth nodded, trying not to be overly impressed with the advantages that wealth could provide. Even if her own house was rustic and meagerly furnished, she would be wholeheartedly thankful for it. And she knew it would be filled with love. Not that the Prescotts' home was not filled with love. She had no reason to think that. But she also had no reason not to rejoice with Lavinia over her new home. "This is a truly beautiful place to live," Elizabeth said finally. "The Lord has blessed you and your family, and I could not be happier for you."
Lavinia smiled happily. "Oh, that makes me feel so glad to hear that." Her smile faded. "I do worry sometimes that our situation might make others jealous. In fact, Hugh warned me not to invite just anyone up here. And I do understand his rationale. But it is such fun to share it with someone. And I felt that you would be more understanding than most."
"My mother would delight in seeing your home too," Elizabeth told her. "She left a very fine house behind in Kentucky. Although she won't admit it, I know she misses certain things, such as her china dishes and her beautiful bedroom furniture."
"The very sort of items I someday hope to carry in our mercantile."
"That day may be a long way off. At least for most settlers."
"Oh, mark my word, Elizabeth, it's not that far off. This is beautiful country, and I feel certain more settlers are coming. I have no doubts that all of us—or at least most of us—will have very fine houses again someday."
"You already have one," Elizabeth reminded her.
Lavinia shrugged. "Yes, well, this is quite livable for now. But in time, after our town grows and our business prospers, I want a real house built on our land. I want a place where my grandchildren can run and play."
"Yes." Elizabeth nodded. "I understand that."
Lavinia led her to the kitchen area. "As you can see, I am cooking on a real stove now. However, I must confess my cooking skills have not greatly improved since we were on the trail. Thanks to you and your mother, Evelyn has become quite adept at cooking—something, I'm sure, some young man will be most appreciative of someday."
"I hope you're not in a hurry to get Evelyn married off," Elizabeth said.
"Not at all. That girl is too valuable here."
"Isn't she just barely sixteen?"
"She will be seventeen in May. And she is still attending class with Mrs. Taylor. Did you know that Mrs. Taylor not only speaks Latin, but French and German as well? Who would have known?"
"Besides the languages, JT is delighted that she is teaching the children to read music. And my Ruth practically thinks that Mrs. Taylor hung the moon. I had no idea she had so much to offer."
"I suppose it was a blessing in disguise..." Lavina sighed as she checked a pot that was on the stove. "I mean...that she is with us now."
"Most certainly."
"Evelyn insisted on helping me to make us a soup," Lavinia told Elizabeth.
"Oh, my...and I only expected to come for tea."
"Well, certainly you must be hungry. Evelyn was sure you would be."
"As a matter of fact I am," she admitted.
"It's potato and sausage." Lavinia moved the pot to a cooler part of the stove.
"Sounds delicious." Elizabeth looked over to where the kitchen table was set with fine china dishes and silver. "The table looks beautiful too."
"That was my idea," Lavinia declared. "It makes me feel so festive to use my good dishes. But I hope you don't mind eating in the kitchen. It seemed more sensible than the big dining table."
"Not at all. It's very cozy in here."
Soon they were seated, and Lavinia invited Elizabeth to ask the blessing. "Your family seems so much more experienced at that than we are," she said quietly. Elizabeth prayed, and they both began to eat.
"This soup is wonderful," Elizabeth told her.
Lavinia nodded. "It is good, isn't it? I'll be sure to tell Evelyn you liked it. She was so happy to hear you were coming." Lavinia's mouth puckered into a frown. "Although I must confess that we were very disappointed about your insistence that no one bring gifts to your wedding. Why in the world did you do that? And mind you, I am not only asking as a businesswoman, but as a friend."
Elizabeth didn't want to say anything to diminish Malinda in her soon-to-be sister-in-law's eyes. "It just seemed the right thing to do. Most of the settlers can barely feed their families, let alone try to come up with two wedding gifts in a month. Eli and I are not in need of gifts."
"But perhaps your friends and neighbors should have been the ones to decide that."
"Perhaps." Elizabeth buttered her biscuit. "Seeing this butter reminds me, my mother is a very good cheese maker."
"Cheese?" Lavinia's eyes lit up. "Oh, how I miss cheese."
"I wonder if it would sell well in your store."
"Oh, I'm sure it would. I would certainly buy it."
"Well, we have decided that as soon as our cows begin giving milk—mind you, it won't be until a year from now, if we are lucky—my mother and I will begin making cheese."
"That is wonderful. Consider me your first customer. And I will sell it in the mercantile too."
After their meal, they got back downstairs just as Flo came into the store. And while Elizabeth shopped, Lavinia and Flo visited together like old friends. Sometimes Elizabeth found it hard to believe that these two women, who would one day be sharing grandbabies, had come from such completely different economic situations. But the Oregon Trail and the frontier eliminated some of the social barriers that were common in the eastern part of the country. Not all of the barriers, of course, but many. At least for the time being. Elizabeth had no idea how all the various relationships would fare on down the line. But for now they all needed one another. And the tighter their little community remained, the greater their chances for success. Or so it seemed to Elizabeth.
"Your shelves seem better stocked than the last time I was here," Elizabeth called out to Lavinia.
"Yes. Hugh and Julius got back yesterday. We spent the evening unloading and stocking all the merchandise."
"You are making it rather hard on me," Elizabeth admitted.
"What do you mean?" Lavinia came over to where Elizabeth was admiring a coffeepot. Her coffeepot from the trail was dented and stained and lidless. The idea of this shining new red enameled pot was rather appealing. Even so, she did not need it, and it was not on her list. She reluctantly set it back down.
"I made myself a list that I plan to stick to, but some of your lovely items are tempting." She smiled. "And once we start producing cheese and other dairy products, my purse strings will loosen up some." She held up her shopping basket. "As it is, this will do me for today. Well, along with these dry staples." She handed Lavinia her list, waiting as she measured and weighed flour and sugar and salt and coffee and beans and all the other staples necessary to feed a family of four.
While Lavinia was packaging her things, Elizabeth peered outside and noticed that the dark clouds were indeed converging overhead. "Oh, dear," she said. "I hope I make it home before those clouds burst open. I fear the road will turn into a bog soon."
"I'm glad Hugh and Julius made it back." Lavinia frowned at the window. "I told them they were pushing it. They could have easily been wading through mud by now."
Elizabeth hoped she wasn't going to be wading through mud.
"Hello, Elizabeth," Hugh said as he came into the mercantile.
"Good afternoon," she replied.
"I was helping Julius with his house today," he said as he brushed sawdust from his trousers. "Had to quit before the rain came."
"Not in here," Lavinia warned. "I just swept."
He made a sheepish grin. "I'll take this outside."
"And can you help Elizabeth with Eli's order," Lavinia called out the open door. "But don't tell her what it is. Eli said it's a secret."
"Will do." Out on the boardwalk in front of the mercantile, he was busily brushing the sawdust off. "In fact, I'll be happy to load everything up for you, Elizabeth," he called into the store. "And I'll loan you a tarp to keep it dry too. Looks like we're going to have a deluge."
"Thank you," she said appreciatively. "I better get going."
"Take this to JT and Ruth," Lavinia said as she handed Elizabeth a couple of peppermint sticks.
"Thank you." Elizabeth tucked them into her skirt pocket, wondering if she should give the children a ride home, but when she looked at the clock, she realized they would already be well on their way. Perhaps she would pick them up on the road.
"All loaded and ready to roll," Hugh said as he returned to the store. "If I were you, I would roll in haste." He grinned. "And judging by those horses, I'll bet you can too. At least you can go a lot faster than when the wagon was fully loaded on the Oregon Trail."
"That's for sure." She laughed. "And you're right, that team can go fairly well when I let them." She thanked Lavinia again and then, saying goodbye, hurried out to climb into the wagon. The clouds seemed to be getting lower and darker by the moment. She snapped the reins and commanded the team to go, urging them to a faster pace than usual as they got onto the rutted road. It was not a smooth road by any means, but if one kept the wheels in the ruts, and if there was no mud, it was not too terrible.
However, she was only halfway home when the sky opened up and the rain came pouring down, falling so heavily it felt as if it were being dumped by the bucketful. In no time, Elizabeth could feel herself being soaked clear down to her underclothes. She glanced over her shoulder, thankful that Hugh had thought to protect her purchases. Hopefully the dry goods would make it home without being ruined.
"Gid-up," she shouted, urging the team to continue even though the road was quickly turning to gooey mud.
Finally, she realized that it was useless to try to go fast. Beau and Bella were struggling to get their footing in the slippery mud, and the wagon wheels were so coated that the mud was splattering all over the wagon. At one point, she considered just stopping and waiting it out. But who knew how long this downpour would last?
Eventually, she knew it was futile to remain on the road, and seeing a meadow off to one side, she wondered if that might provide a firmer surface. However, the team seemed unable to move. Thinking she could lead them over to the grass, she climbed out of the wagon and immediately slipped in the mud. Now she was not only soaked to the skin, but covered in mud as well. Climbing back to her feet, she went to the horses' heads and grabbed Beau's harness. "Come on," she urged, pulling him toward her.
But their footing, like hers, was slipping. And the more she worked with them, the more she realized she was putting them in danger by asking them to do the impossible. The wagon was so mired down in the mud, it was going nowhere. Now she decided her only goal was to get Beau and Bella safely away from the wagon before one of them fell and injured a leg. In the pouring rain, she spoke calmly to her team, struggling to get them out of the harnesses and finally free from the wagon.
"Easy does it," she said as she slowly led them away from the mud. "Come on, Beau and Bella, let's get you to firmer ground." She finally got them somewhat sheltered under a stand of fir trees. And when she looked back to the road, she was stunned to see that the belly of the wagon was nearly touching the mud. No wonder the team could not budge it.
"Let's see if we can get you home," she said as she led the horses through the grass. The ground was soft and the going was slow, but it wasn't nearly as mucky and sticky as the road had become. She found the going was easier if she kept them moving on the higher ground. But at the same time, she felt worried that she was getting too far from the road. What if she got lost?
She paused beneath another stand of trees, stopping to check her team. She ran her hands down their legs, making sure they were sound. Fortunately, other than being tired, they seemed just fine. She stroked their sleek heads, talking soothingly to them. "You two are worth much more than all the flour and sugar and coffee in the whole mercantile. And I'm sorry I put you at risk just to get my wagon home."
She looked out over the grassy slope she was leading them on. She would prefer walking them on flat land when it was slick like this, but all the flat land seemed to be turning to mud. As it was, she knew if she took it slow and easy, they should make it back all right—as long as she wasn't lost. She wondered what it would feel like to spend the night out here. If she could make a fire—which was doubtful—she might be able to dry out her clothes and make it to morning. She wondered what Eli would say to her. Certainly, he would not allow himself to be caught in a situation like this. What kind of frontier woman was she, anyway?
Chapter Seventeen
In all our days on the Oregon Trail," she said aloud as she trudge along, "I never ever saw anything like this rain and mud." Bella shook her head up and down as if to concur with her. Traveling this off-road route was taking much longer than it would to simply wade through the mud on foot. However, she knew it didn't matter as long as she got her team home in one piece.
Eventually, she found her way back to the road again, and she recognized the land as part of Malinda's property. Feeling somewhat hopeful but exhausted, she encouraged the team to keep going. It was dusky by the time Malinda's farm finally came into sight. Elizabeth couldn't remember when she'd been so happy to see the warm golden light through a window. As she plodded toward the barn, she tried to remember when she'd felt this weary. Probably on the Oregon Trail—although that seemed impossible at the moment.
"Come on," she urged her horses, leading them through the muddy farmyard and into Malinda's pasture. "You're home for now." She watched as they wearily went through the gate. She knew they were all right, and she planned to send JT out to check on them. But in all likelihood, they would recover more quickly than she would. However, they couldn't have possibly pulled that wagon all the way home. And she did not even want to think about her dry goods, which would probably turn to mush by morning.
Feeling as if she could barely plant one foot in front of the other, she trudged over to the house, but when she opened the door and saw the shocked faces of Amelia, Malinda, and Susannah, she knew she couldn't go inside the house like this.
"What happened to you?" Malinda demanded.
Elizabeth held up a tired hand. "Excuse me a moment," she said hoarsely. Then she turned and walked over to where one of the water troughs was overflowing from the rain and, knowing she looked perfectly ridiculous, plunged herself into the trough, washing the caked mud from her outer clothes as if she were bathing. She could hear the girls laughing from the porch. She didn't blame them because she knew she looked foolish, but at the same time they had no idea how cold she was.
Shivering and weak, she climbed from the trough and made her way to the covered porch, where she slowly peeled off her dripping outer clothing, dropping it down on the bench by the door. She would deal with it later.
"Here," Malinda said as she wrapped a blanket around Elizabeth's shoulders. "Get inside before you catch your death of cold."
"Thank you," Elizabeth muttered with chattering teeth, allowing Malinda to guide her toward the fire. There, with the help of the older girls holding blankets as a screen, Elizabeth removed the last of her wet clothing and then wrapped herself up in Malinda's warm woolen robe.
"Sit down," Malinda said as she shoved a straight-backed chair beneath Elizabeth.
"Put these on." Amelia knelt down, pushing Elizabeth's white icy feet into a pair of thick socks.
"And drink this," Malinda handed her a cup of hot coffee with cream.
"What happened, Mama?" Ruth asked her with wide eyes. "Susannah said you jumped in the water trough."
The girls laughed, and Elizabeth couldn't help but giggle despite herself, or maybe it was something in the coffee. And then she explained about going to the mercantile and getting stuck in the mud and how she had to get her team away from the wagon to get them safely home.
"You left our wagon out on the road?" Ruth sounded shocked.
"I had to," she explained as she sipped the odd-tasting coffee. "It was the wagon or the horses. And the wagon will survive the mud. I wasn't so sure about the horses." She turned to Malinda. "Honestly, in all our days and all our hard times on the Oregon Trail, I never ever encountered anything like that horrid mud."
"I should have warned you about downpours like this. You should never take out a wagon this time of year—not unless you've got a good shovel and a bunch of wood planks with you." Now Malinda explained how the men would sometimes dig out the mud and place planks in the road for the wheels to roll onto. "Even that is an enormous undertaking."
"I can imagine."
"You're much better off on foot," she told Elizabeth.
JT and the boys came into the house now. "Ma," he called out, "I noticed our team was here. Are you all right?"
"Yes," she told him. "Did you see to the team?"
He assured her he'd done that. "Did you see Eli on your way here?"
"Eli?" she peered over her coffee mug at JT. His hat was dripping, and he looked nearly as soaked as she'd been. However, seeing he was wearing his father's old waxed barn jacket, she knew his clothing would remain dry underneath.
"He came through here with Grandpa's oxen team about an hour ago."
She blinked. "Eli brought the oxen here?"
"Yeah. He knew you'd need some help to get the wagon home." JT frowned. "Where is the wagon?"
Once again, she explained her recent dilemma. "But where is Eli now?"
"I expect he went on down the road looking for you."
"Oh..." She sighed.
"Don't worry about Eli," JT assured her. "He can take care of himself."
"Did he have a shovel with him?" Malinda asked as JT was heading back out the door.
"Yep. And a couple of planks of wood too," he said as he left.
"See," Malinda told Elizabeth. "Your man's got a good head on his shoulders."
Elizabeth just nodded.
"Okay, we're going to get some soup into her," Malinda told the girls. "And then we'll get her to bed. Susannah, you get those warming stones heated up for her. Amelia, you make her some tea."
Elizabeth watched almost as if she were dreaming as they all scurried around, chattering among themselves. She wasn't sure if Malinda had put something in her coffee, but she felt surprisingly warm and relaxed when Amelia and Susannah finally helped her into bed.
"I'll come to bed and read to you after supper," Ruth promised as she tucked the quilt snugly under Elizabeth's chin.
"Thank you, sweetheart," Elizabeth murmured as she closed her eyes and sighed.
######
"Our bride-to-be is awake," Malinda said in a teasing tone as Elizabeth came out into the front room and looked around.
"Where is everyone?" She buttoned the cuffs of her chore dress.
"Off to school." Malinda handed her a cup of coffee.
Elizabeth peered suspiciously into the coffee. "What's in this?" she asked.
"Cream and a bit of sugar," Malinda said innocently.
"What about last night's coffee?"
"Just a drop or two of medicine," Malinda chuckled. "To ward off a cold or chill."
Elizabeth took a cautious sip and was relieved to discover it tasted normal.
"And it seems to have done the trick," Malinda told her. "You look none the worse for wear."
Elizabeth sighed and sat down at the table. "Thank you, Malinda," she said gratefully. "I'm sure that you and the girls saved my life last night."
Malinda grinned. "Well, we couldn't very well let our bride expire just two days before her wedding. Goodness, if I'd known you were going to go out and try to drown yourself, I would have forced you to stay for the quilting bee."
"How was the quilting bee?" Elizabeth looked down at her coffee.
"Successful. We finished it up in time for the ladies to get home before the rain let loose."
"May I see the finished quilt?"
Malinda frowned. "I'm sorry. It's all wrapped up in paper to keep it clean and safe until the wedding day."
Elizabeth nodded. "I'm sure it's beautiful."
"Eli came by last night to check on you." Malinda set a bowl of oatmeal in front of Elizabeth. "You had already gone to sleep."
"Was he all right?" She poured some milk over her oatmeal.
"Well, other than being soaked, he was fine. He used the oxen team to pull the wagon from the mud. However, he said they were worn out. So he left them and the wagon here. He thought you might want it for tomorrow—if the road dries out by then." She pointed to some bundles sitting on the bench by the fireplace. "He brought those into the house to dry out. I think most of your purchases from yesterday will be just fine." She pointed to a large bowl of damp-looking oatmeal. "Except that one. It must not have been as well covered as the others. But we fixed a big pot of oatmeal for breakfast. And I thought perhaps I could make a big batch of oatmeal and molasses cookies out of the rest of it." She smiled. "I could bring them to the wedding dinner tomorrow."
"That would be nice. Thanks." Elizabeth ate her breakfast as Malinda bustled about the kitchen. But when Elizabeth started to do her normal morning chores, Malinda held up a hand. "Not today, my friend."
"But I want to help."
Malinda just shook her head. "Do you feel well enough to go to your mother's like you planned to do today?"
Elizabeth nodded. "I feel perfectly fine."
"Perhaps you could ride Molly."
"Yes. That might be a good plan. And if the road is too bad, we'll just cut across some fields."
"The rain seems to have moved eastward, but after yesterday's downpour, it's still rather soggy out there."
"I'll be sure to wear my heavy boots and barn coat this time."
"And don't overdo," Malinda warned her.
Elizabeth patted Malinda on the back. "Thank you for caring."
Now Malinda reached out and hugged her tightly. "Of course I care. You are my oldest and dearest friend."
To Elizabeth's surprise, she felt tears welling up. "I'm so glad we're still friends."
Malinda pulled her an arm's length away and looked into her eyes. "I realize I've been a bit harsh at times," she admitted. "I hope you'll forgive me. I've given it some thought, and I suppose I can be a bit self-centered at times."
Elizabeth just shrugged.
"And then, living out here in the frontier, I've also gotten rather self-reliant. It occurred to me last night after you'd gone to bed that being too self-centered and too self-reliant weren't the best qualities in a friend. I want to try harder."
Elizabeth patted Malinda's shoulder. "I suspect we both have much to learn about friendship—especially out here in the frontier," she told her. "But I'm very thankful that I have you, Malinda." She noticed that her wet clothes from yesterday were washed and hanging by the fireplace to dry. "And thank you again for taking such good care of me last night."
"What time do you think you'll be back here?" Malinda asked with an arched brow.
"In time to help prepare supper," she assured her.
"Good. I will expect you then."
######
As Elizabeth rode Molly through a wooded area, she noticed how dirty her fingernails were from yesterday's mud bath. Now if Malinda had allowed her to wash up the breakfast dishes as she usually did, her nails would be clean. Maybe she could offer to wash her mother's dishes or do some laundry, although she was fairly certain that wasn't going to happen. She knew her mother and Jess planned to spend their time sewing and cooking today.
Her mother planned to do a final fitting of the wedding dress and do the finish work on Ruth's dress. At the same time, they wanted to bake some cakes and pies for dessert. Tomorrow, Asa would go early to the church and set up a fire pit to roast venison and salmon. Elizabeth wasn't sure about all the other food details, but she wasn't worried about them either. Her mother had assured her that she was handling them, and Elizabeth trusted her.
She looked up at the sky as she rode. Through the overhead branches, it looked clear and blue. She hoped the fair weather would hold out through tomorrow. Perhaps the road would have a chance to dry out and firm up by then. And if not...well, she expected they would simply make the best of it.
Clara and Jess welcomed her into the snug little cabin, and it wasn't long until Elizabeth was trying on her dress. "That shade of blue is so pretty with your eyes," Clara told her. "Ruth was smart to pick it out."
"It reminds me of the color of the river on a clear day," Jess said. "Very pretty."
Clara stood back to look at Elizabeth. "My dear, you look just beautiful."
"Thank you. And this dress is beautiful too. The prettiest dress I've had since...well, since losing James." Elizabeth sighed as she admired the carefully stitched tucks in the bodice, fingered the delicate lace collar, and smoothed her hands over the silky fabric of the skirt.
"And the fit looks perfect," Jess said.
"I believe you're right. I don't think we need to adjust a thing." Clara nodded with satisfaction.
"It's such a lovely dress that after the wedding, I'll probably be afraid to wear it," Elizabeth confessed.
"Of course you'll wear it," Clara said as she helped her out of it.
"You can wear it to church," Jess said.
"And other weddings."
"But not on a rainy day." Now Elizabeth told them the details of her harrowing trip home from town the day before.
"My goodness!" Clara gasped. "I had no idea. Eli came by here for the team, but I didn't realize you were in such dire straits."
"I learned some important things," she told them. "First of all, we need to be just as prepared out here as we were while traveling the Oregon Trail." She explained what Malinda had told her about carrying shovels and planks. "I also learned never to take this seemingly mild climate for granted. The weather here can change quickly. From now on I will take a very good look at the sky before heading out in the wagon. Getting stuck in the mud like that was a valuable lesson for me. One that I do not care to repeat."
"Couldn't you have simply ridden one of the horses home?"
"Not on the road. They were worn out from trying to pull that wagon. And they were slipping and sliding, and I did not want to risk a broken leg." She shook her head. "Can you imagine how devastating that would be? After getting them safely here from Kentucky only to lose them a couple miles from home?"
"Oh, my." Jess shook her head. "I plan to tell Matthew about the shovels and planks. Not that we'll be taking our wagon out much."
"I can understand now why everyone kept telling us to get our houses built before the rains started."
"Not to mention having roofs over our heads," Clara said. "It rained so hard here last night that I was very concerned about Brady in the tent."
"Oh, dear. I forgot about that. How did he fare?"
"He was a bit damp, I'm afraid." Clara shook her head. "He and Asa and Matthew are working hard to get him a small shelter built, but I doubt it will be as nice as the cabin on your property."
"He is still welcome to it," Elizabeth told her.
"He knows that." Clara smiled. "But he truly wants to be here with us. I have to say, I was very touched. I do hope you don't mind, dear."
"Not at all. I actually believe it is a sensible idea. But I do wish Brady would have spoken up earlier. Then he would have a nice little cabin all built."
"I told him that if it got too wet and cold in the tent, he is welcome to come in here to sleep." Clara pointed to the braided rug by the fireplace. "It's not much, but it would be warm and dry."
The three of them continued working together into the afternoon. By the time they called it quits, they had finished both dresses and baked three cakes and six pies. "And I made two cakes yesterday," Clara assured Elizabeth as she handed her the parcel she'd wrapped, containing the two dresses for tomorrow's wedding.
"And I have two more pies at my house," Jess said.
"And Malinda is making oatmeal and molasses cookies," Elizabeth told them.
"Your father got a big bag of potatoes from the mercantile," Clara told her. "He plans to bake them in the barbecue pit while the venison and salmon cook. Besides that, he has several crates of corn. He asked if the children could come to the church early to help with the husking."
"I'll be sure and tell them," Elizabeth promised as she pulled on her riding hat. "And now I should be on my way because I promised Malinda I would be home in time to help with supper."
Clara glanced at Jess. "Say...before you go, Elizabeth," she said quickly. "Your father wanted you to run down to where they're building Brady's cabin. He wants to ask you something. I believe it has to do with tomorrow. Do you mind, dear?"
"Not at all."
They all hugged goodbye, and Elizabeth tied the precious parcel of dresses onto the back of her saddle and rode on over to the spot her mother had described. A small stand of fir trees were behind what was the beginning of a very small log cabin. "Hello," she called as she slid off her horse and went over to where her father was notching one end of a log and Brady was notching the other.
"Elizabeth," Asa said cheerfully. "What brings you out here—and on the eve of your wedding day?"
"Mother said you wanted to speak to me." She waved to Brady now. "Looks like your cabin is coming along."
Asa's brow creased as he rubbed his chin. "What did she say I wanted to speak to you about?"
She shrugged. "I don't rightly know. I assumed it was something about the wedding."
He still looked slightly confused. He went over the same details her mother had just told her about the venison and salmon. "Brady and I caught the salmon, and Eli and Matthew got three deer this week. We got several good-sized roasts hanging for tomorrow, and the rest is getting smoked." Then he told her about the potatoes and corn. "Oh, that's right, I did want the children to come help husk the corn. Maybe that's why your ma sent you over here."
"But Mother told me that." She frowned. "So there's nothing else to talk about?"
Asa shook his head. "Nothing that I know of." Now his face lit up. "Except that I couldn't be happier for you and Eli. And I'm plum delighted that I get to walk you down the aisle again tomorrow. It will be a grand day."
"Thank you." She turned to Brady again. "And you will be at the wedding too, won't you?"
He made an uncertain smile. "Well, now...I ain't so sure."
"Oh, please, do come, Brady." Then she realized the reason for his hesitation. "I mean, I want you to come if you want to come." She put her hand on his shoulder. "I will understand if you don't want to be in town. But if you don't come, you must promise to join us for a wedding celebration at our new house. That will be just for family."
"Just for family?" He tipped his head to one side.
She smiled. "Yes, and that means you too, Brady. You know that. And you will have to promise to bring your harmonica too."
He grinned. "You can count on it."
"And if you don't make it to the wedding, I'll be sure to bring you home a big plate of food," Asa promised.
"I'd be much obliged."
"But I do want you to know that you are very welcome at the wedding," Elizabeth said as she slipped a boot into the stirrup. "Either way I will understand."
He nodded. "Thank you, Miss Elizabeth."
As she rode back to Malinda's, she wished that the laws were more welcoming to people like Brady and the Indians. She felt certain that someday, with the help of education and good Christian values, these laws would change, both in the Oregon Territory and in the United States of America. But she just didn't know how long it would take. She prayed that this change would occur in Brady's lifetime, but considering his age, she realized that might take a miracle.
Chapter Eighteen
When Elizabeth got back to Malinda's, the house seemed to be more lit up than usual, and a lantern was hanging outside on the porch. Very inviting. She was just slipping down from Molly when she saw JT and Bart hurrying from the barn.
"Hello, Ma," JT called to her. "We'll take care of your horse for you."
"Thank you very much," she said as she handed him the reins.
"And Bart and Todd and me got the wagon and Beau and Bella all cleaned up after school," JT told her.
"They'll look real nice for the wedding tomorrow," Bart told her. "Those are real nice horses, Aunt Elizabeth."
"Well, thank you again," she said as she removed the parcel containing her wedding dress. "And I agree, they are wonderful horses. I felt badly for working them so hard yesterday. That was very thoughtful of you boys to take care of that for me."
"It was Aunt Malinda's idea," JT told her.
"Thank you just the same. I'm sure it was quite a chore."
"And Todd took the oxen team back to your pa," Bart said.
"Well, I am most grateful to all you fine young men," she said happily. "I hardly know what to say."
JT tipped his head toward the well-lit house. "Maybe you ought to go inside now, Ma."
"Yes." She nodded. As she went, she thought she heard the boys giggling, but she didn't care to investigate the reason. She was so thankful for the generous help with the team and so thankful that Malinda had thought of it. She pushed open the door and then blinked to see a room full of females. They were circled around the door as if they expected her.
"Surprise!" they all yelled.
"What?" She looked at the merry faces.
"We're having a bridal shower," Malinda explained as she reached for the parcel and then helped to remove Elizabeth's coat.
"A what?" Elizabeth frowned down at her still dirty nails. "I realize I had a mud bath yesterday, but I did intend to clean up before—"
"No, no..." Lavinia laughed as she came over to hug Elizabeth. "Although I did hear about your misadventures in the mud. But no, that is not what a bridal shower is for. I will explain that later. First you must greet your guests."
"Mother and Jessica—how did you get here before me?" Elizabeth peered curiously at her female relatives. "Ah...so that's why you sent me to speak to Father—so you could beat me back here. No wonder he was confused."
Clara giggled as they hugged. "I felt terrible for tricking you like that."
"That's all right. As it turned out, I wanted to speak to both Father and Brady." She looked around to see not only Belinda, Amelia, Susanna, Emily, and Ruth, but also Evelyn and Flo and Mahala and Hannah and Tillie. Even Mrs. Taylor was there. Elizabeth took time to greet each of them one by one. "What a delightful houseful of women," she declared.
"Since you were opposed to receiving any wedding presents, I decided to make a plan of my own," Lavinia explained to Elizabeth.
"Yes, this was all Lavinia's idea," Malinda told her. "I thought maybe it was her way of drumming up business for the mercantile, but she assured me that had nothing to do with it." They all laughed.
"Now come and sit here, Aunt Elizabeth." Emily led her over to the rocker.
As Elizabeth sat, she looked at Lavinia. "You still haven't told me what a bridal shower is."
Lavinia stood up, folding her hands in front of her as if she were getting ready to recite. "The tradition of a bridal shower comes from Holland," she began. "According to my mother, who was Dutch, there was a young woman who fell in love with a kindhearted miller back in the 1600s. Now the miller was a hard worker, but because of his kind and generous heart, he often gave bags of wheat away to those in need." Lavinia chuckled. "Not very profitable for a businessman to give merchandise away for free."
"Not like our Aunt Lavinia," Amelia teased.
Lavinia smirked at her niece. "So as you can imagine, this kindhearted miller was rather poor. In fact, he was so poor that the young woman's father refused to let her marry him. Both the miller and the girl were devastated." She paused long enough for the women to express their dismay. "To make matters worse, the young woman's father decided his daughter must marry the pig farmer, who was very wealthy but not nearly as nice as the miller."
"Oh, no!" Ruth said. Once again everyone laughed.
"What happened, Aunt Lavinia?" Belinda demanded.
"Well, when the good miller's friends heard his sad news and remembered how generous he had always been to them, they naturally wanted to help him out. They decided they would raise money to provide a dowry for his beloved." She paused as the younger girls cheered.
"However, the miller's friends, like him, were rather poor. Alas, they did not have enough money to make a respectable dowry."
Now the girls expressed their disappointment.
"Then they came up with another idea. Instead of giving them money, they decided to shower the couple with goods. They gave them linens and china and all sorts of things to set up housekeeping. And so the miller and the bride were allowed to marry and lived happily ever after."
Now everyone clapped and cheered.
"So in some places, like Boston, it has become a custom to shower the bride-to-be with gifts, and that is exactly what we are doing."
And now the women and girls came to Elizabeth and began to present her with a variety of wonderful presents. Susannah and Emily gave her a fine set of embroidered bed linens. "We were working on them in secret," the sisters explained.
"They are beautiful," she said as she thanked them.
Next she was presented a tea set from Amelia and Belinda. "We got it at the mercantile," Belinda whispered.
"It's lovely! Thank you."
On the gifts came...a vase from Flo, four teaspoons from Evelyn, a set of kitchen linens from Hannah and Tillie, a pair of knitted slippers from Mahala, a set of pillows from her mother, a kerosene lantern from Mrs. Taylor, a braided rug from Jess, an embroidered wall hanging from Malinda that featured a little cabin and said "Home Sweet Home," and from Lavinia, the enameled red coffeepot that Elizabeth had admired in the mercantile just yesterday. Once again she expressed her sincere gratitude to everyone.
"You're not done yet, Mama." Ruth handed her something wrapped in paper. Elizabeth opened it up to see a hand-drawn picture of a cabin nestled in the trees with a blue river flowing in front of it.
"This is beautiful, sweetie." Elizabeth held it up for all to see. "My daughter is an artist."
"I made the frame too," Ruth said shyly.
"I love it!" Elizabeth pulled Ruth in for a hug. "You are all so wonderful," she told everyone. "I am so very grateful for all my lovely friends, just as the miller surely was for all of his. You are all worth more to me than a dowry." She held up the coffeepot and grinned. "I can assure you that Eli and I will put these items to good use, and we will think of all of our dear friends whenever we do. Thank you all so much!"
"One more thing," Malinda said mysteriously. She nodded to Susannah and Ruth, and the two girls dashed to the bedroom and soon emerged carrying a big bundle between them. It was wrapped in brown paper and tied with string.
"That is from all of us," Malinda proclaimed as the girls set it in Elizabeth's lap.
Elizabeth untied the string and unwrapped the paper and then gasped to see it. "This is your quilt," she said to Malinda, holding the beautiful log cabin design up for everyone to see.
"We were making it for you," Malinda told her. "That was the plan all along."
Elizabeth held it up to her cheek. "It is beautiful. Thank you all so much."
"That's why we didn't want you here when we finished it," Malinda explained.
"Oh, my." Elizabeth just shook her head. "I don't even know what to say."
"That must mean it's time to serve the food," Malinda declared. "But first let's all ask the good Lord's blessing and invite him to bless Elizabeth on her special day tomorrow."
######
Good food, good fun, good friends...it was an occasion to remember always. Elizabeth could not recall the last time she'd had such an enjoyable evening. "A bridal shower is a lovely tradition," she told Lavinia as the women who'd come from town were preparing to leave. They'd all come together in a wagon that the boys had hidden behind the barn.
"I'm not sure that we'll do this for all the upcoming weddings," Lavinia confided. "But when I heard you and Eli were not allowing wedding gifts...well, I knew something had to be done about that."
Elizabeth thanked everyone again. "I'll see you all tomorrow," she called out as the wagon started to move.
Just as it was leaving, another wagon pulled into the farmyard from the opposite direction with lanterns hanging on both sides. Elizabeth was relieved to see that Matthew had come for Clara and Jess.
"How was your shower?" he asked her in a teasing tone. "Did you get all wet?"
"I got wet yesterday," she told him. "Tonight's shower was much more pleasant. Goodness, you should see the lovely gifts these women brought for Eli and me."
"Anything you want us to take in the wagon? I could run it on over to Eli."
"That's a good idea," Clara said. "If you take these gifts up to Eli, she won't need to trouble herself with it tomorrow."
"And I have some crates and things as well." Elizabeth remembered what she'd stowed in the barn. "Would you mind taking them too?"
"Not at all."
With the help of the girls, they soon got the wagon loaded up, and Matthew and Jess and Clara were on their way. "See you tomorrow," Elizabeth called out.
Back in the house, the girls were already cleaning up, and no one would allow Elizabeth or Malinda to help. "This was such a wonderful night," Elizabeth said to Malinda as they sat together. "Thank you so much."
"It really was Lavinia's idea," Malinda said.
"The quilt was your idea," Elizabeth pointed out.
"Well, yes, that's true."
"I love it, Malinda. I don't even know how to thank you."
Malinda smiled. "Just enjoy it."
"Matthew is taking everything up to Eli." Elizabeth shook her head. "I feel a bit envious though. Eli is up there in our house. Do you think he'll put things away?"
"I don't see why. Most men would think of that as women's work."
"I hope he does." Elizabeth sighed. "I am just itching to start keeping house—I mean in my own house."
"Yes, I can imagine." Malinda made a sad sigh. "Although I am going to miss you, Elizabeth. And Ruth and JT too."
Elizabeth laughed. "You'll still have a fairly full house, Malinda. With your four and Will's two girls, you are not hurting for companionship."
"That's true. But it's not the same as having my best friend right here."
Elizabeth tried not to remember the times when she had felt like anything but Malinda's best friend. "And don't forget that in just two weeks, you'll have Will here as well. I expect he'll become your best friend...in time. And, of course, we will just be down the road. I predict our families will have many wonderful times to look forward to in the years to come."
Later that night, after everyone had gone to bed, Elizabeth lay still, listening to the sounds of her daughter's and best friend's even breathing as they peacefully slumbered. And yet, as tired as she was, sleep was evading her tonight. Her mind seemed unwilling to quiet down as she pondered her life and how different everything had been just one year ago. She remembered being awakened in the middle of the night last December, feeling again as if James were leading her. She remembered her recurring dreams about traveling to Oregon. It all seemed so far away now, like a different world. And indeed it was. But it was about to change again. Marrying Eli and moving into their home together—tomorrow! As much as she wanted this, had been waiting for this, she suddenly felt uneasy. The more she thought about it, the more distraught she felt.
If she hadn't been sleeping in the middle of the bed, wedged between Ruth and Malinda, she would have gotten up and slipped out. She would have sat by the fire to think...and think. As it was, she didn't want to disturb them.
Finally, after what felt like hours, she remembered the way her mother taught her to put herself to sleep as a child. She would count her blessings. And so she started with each member of her family and then moved on to friends and even livestock, until finally she was thanking God for every single little thing she could think of—from ladybugs to forget-me-nots to hummingbirds—until eventually she felt herself drifting away.
Chapter Nineteen
When Elizabeth woke up the next morning, she was alone in the bed, but it still seemed dark outside, so she didn't think she'd slept in. But when she poked her head out the bedroom door, it was clear that everyone else was up and busy. And when she saw the clock, she realized it was nearly nine.
"Oh, dear," she exclaimed. "I didn't know it was so late."
"There's our bride," Malinda announced. "You were sleeping so soundly, we thought perhaps you needed it."
"Bless you," Elizabeth said. "I did have a restless night. But now I better get moving."
"Is it time for me to get on my wedding clothes?" Ruth asked hopefully.
"Yes." Elizabeth waved. "Come on in here, and we'll get at least one of us ready."
Ruth entered the room, pushing the muslin curtains aside to let in a little light. "There sure are a lot of dark clouds out there today," she announced sadly. "Do you think it will rain?"
Elizabeth peered out to see that the sky was gray and somber looking—the reason it had seemed dark even though it was late in the morning. "Don't worry, Ruth." She reached for Ruth's new blue dress. "Rain on your wedding is supposed to be good luck." She slid the dress down over Ruth's head and shoulders.
"How can rain be good luck?" Ruth asked as her head popped through the neckline.
"If there's been a dry spell and your crops need water, you consider yourself lucky when it rains. Remember?" She buttoned the back of her dress.
"Yes, but we haven't had a dry spell. And last night when I prayed before bed, I asked God to keep the rain away today," Ruth told her.
"Well, come rain or shine, Eli and I are getting married this morning," Elizabeth declared. "And it will be a very happy day. You can count on it." She tied the sash in back of Ruth's dress, fluffing it out into a big bow before she turned her around. "Ruth Anne, you look lovely. And your grandmother will be relieved to see that the dress fits you just right."
Ruth spun around to make the full skirt flare out. "And it will be good for dancing too!"
"Hold still now, and let's do your hair." Elizabeth reached for the hairbrush.
As she brushed and braided Ruth's hair, she gazed out the bedroom window. Malinda was fortunate that their house had more than just one glass window. Still, Elizabeth would happily move into a house with no glass windows if it meant that she and Eli and the children could be together.
However, her daughter was right about the weather. It did look as if the sky might open up at any moment. Even so, Elizabeth was not going to let that get her down. Today was a happy day regardless of the weather.
"There," she proclaimed as she tied the big blue satin ribbon at the bottom of Ruth's braid. "You are pretty as a picture."
Ruth frowned up at her. "But you are still in your nightie, Mama."
Elizabeth laughed. "Yes—and I can't very well get married in my nightie, can I? You go and remind Aunt Malinda that Grandpa wants the boys to come to the church early to help him, and I will get dressed."
"Yes, Mama!"
With the bedroom to herself, Elizabeth layered her corset over the chemise and bloomers and stockings, and then she put on her best petticoat. Finally she removed the new dress from where she'd hung it last night after she and Malinda had used the sadiron to smooth it out. She slipped the full skirt on, fastening it at the waist. She slid her arms into the long sleeved, fitted bodice, taking her time to button the long row of pearly buttons up the front and along the cuffs of the sleeves.
"Need any help in here?" Malinda peered in. "Oh, my. Don't you look pretty!"
"Thank you." Elizabeth reached for the hairbrush. "I just need to put my hair up and I'll be ready."
"The boys ate breakfast and are just finishing chores, and then they'll head for town to help Asa."
Elizabeth pointed to the window. "The weather looks formidable."
"Yes. In fact, JT asked if you plan to take the wagon to town."
Elizabeth cringed. "Oh, I don't think so. I don't want to go through that again. I suspect that my father will be taking his wagon though. They had so much to carry."
"Well, at least he has his oxen. Will you ride your horse to the church?" Malinda asked. "Or walk?"
Elizabeth looked down at the pretty dress, imagining the hem soaked with mud. "Perhaps I should have waited to put this on in town."
With a finger resting on her chin, Malinda made a thoughtful look. "Here is what I think you should do. Take the wagon into town unless it is already raining. But if it's not, we could all go in the wagon. Then if it's raining after the wedding and dinner, when it's time to go home, you and Eli could simply ride your Percherons." She chuckled. "That would be a lovely picture. The bride and the groom on the beautiful black horses. And the wagon could just remain in town until the weather gets better. I'm sure Lavinia and Hugh wouldn't mind if you left it parked at the mercantile."
"That's a good plan." Elizabeth twisted her long hair into a roll, securing it with pins. "If the rain can hold out long enough, that is just what we'll do."
"So should I tell the boys to hitch up the team before they go?"
"Yes. And tell them to put my saddle in. I guess Eli will have to ride bareback, although perhaps we can borrow a saddle blanket from you."
"I hope you won't need it," Malinda said. "But just in case, I'll tell JT."
Elizabeth looked out the window and shook her head. "I can't imagine that we won't."
"Ruth and Susannah are praying diligently."
Elizabeth checked her image in the small mirror above Malinda's dresser. Her hair seemed to be smoothly in place. "Do you think it looks all right?" she asked Malinda. "In back too?"
Malinda came over and fussed with it a little before she proclaimed it perfect. Then, after she set the brush on her dresser, she opened a small wooden box and extracted a pair of pearl drop earrings. "Do you remember these?" She held them up to the light.
"Yes. From your wedding to John."
"Why don't you wear them today?" Malinda held them up to Elizabeth's ears so she could see herself in the mirror.
"Oh, no, Malinda, I couldn't. They're too precious."
Just then Ruth came back into the room. "Ooh, Mama, you look so pretty."
"How about these earrings?" Malinda asked her.
Ruth's eyes grew wide. "Those are beautiful."
"Have you forgotten the rhyme?" Malinda asked Elizabeth.
"What rhyme?" Ruth asked with her usual interest.
"Something old, something new, something borrowed, something blue, and a silver sixpence in her shoe," Malinda told her.
"What does it mean?" Ruth asked.
"It's to bring good luck to the bride," Malinda explained. "A bride needs to have each of those things. Old and new and borrowed and blue."
"And a sixpence in her shoe," Ruth added.
"That's right."
"Will you do that, Mama? Everything in the rhyme?"
"It was something your aunt and I both did at our first weddings." She stroked Ruth's hair. "With your father and your uncle."
"Do it again today," Ruth urged. "For luck."
Malinda held the earrings out. "These could be something old since they belonged to my mother first."
"Yes, yes," Ruth declared. "Now you have something old, Mama. And you already have something blue—your dress!"
"And it's new too," Elizabeth pointed out.
"Can a dress count as two things?" Ruth asked. "Both new and blue?"
"It has two parts to it," Elizabeth reminded her that the skirt and bodice were separate.
"Good," Ruth said. "Now you just need—something borrowed."
"Do you have a clean handkerchief I can use?" Elizabeth asked Ruth.
"You want to borrow something from me?" Ruth's eyes lit up.
"Yes. I would like that very much."
Ruth hurried to find a clean hanky and pressed it into her mother's hand, and Elizabeth slid it inside her cuff.
"Now, what about a silver sixpence?" Ruth asked. "What is that, and where do we get one?"
Elizabeth smiled at her daughter's persistence. "That's a British coin. I suppose it was a bit like six cents."
Ruth features twisted into a frown. "I have a half dime and a penny. That makes six. You could put the half dime in one shoe and the penny in the other. Will that work, Mama?"
Elizabeth laughed. "Yes, I'm sure that will work just fine."
Ruth hurried to find her coins and handed them to Elizabeth, watching as she slipped one into each shoe. "There," Elizabeth proclaimed. "I believe I'm nearly ready."
"Not until you have your breakfast." Malinda ushered her out of the bedroom. "The girls fixed something special."
"With all this attention, I am beginning to feel like a queen," Elizabeth said as she sat down at the table.
"You should be the queen on your wedding day. After that...well, I'm sure you'll be dethroned quickly enough." Malinda laughed.
After a breakfast of pancakes and syrup, it was time to proceed to the church. And to Elizabeth's relief, the dark clouds had not opened up yet. In fact, it almost looked as if they were rolling eastward. "Did JT hitch the team to the wagon?" Elizabeth asked Malinda.
"He certainly did." Malinda pointed toward the barn, where Elizabeth's team was just rounding the corner. Behind the shining Percherons was the wagon. Not only was it cleaned up, but someone had gone to the trouble of draping the sides with evergreen garlands and large bows made of white muslin.
"It's beautiful!" Elizabeth clapped her hands.
"All the children worked on it last night," Malinda told her.
As Elizabeth climbed up to drive, Malinda offered to take the reins, but Elizabeth just shook her head. "No thank you. I would prefer to drive."
Malinda just laughed, climbing into the seat beside her. Meanwhile the girls, all five of them, climbed into the back.
"Gid-up!" Elizabeth called out happily. "We have a wedding to attend."
Thankfully, the road had firmed up in the past couple of days. Still, Elizabeth was relieved to see that someone—probably Eli—had the foresight to leave a shovel and some planks in the back of the wagon.
"And now we shall sing you the wedding march song," Ruth proclaimed from behind Elizabeth.
"The wedding march song?" Elizabeth asked. "What is that?"
"Mrs. Taylor taught it to us," Ruth declared.
"It was written by a German composer named Richard Wagner," Amelia explained.
"And it was written in German," Belinda explained. "But Mrs. Taylor taught it to us in English."
"Because we don't speak very good German," Susannah told her.
"Mrs. Taylor said that the song is from an opera," Emily added.
"We just learned it last week," Ruth said.
"But we've been practicing it every day on our way to and from school," Amelia said.
"Well, I would love to hear it," Elizabeth told them.
"So would I," Malinda said.
After a bit of giggling and adjusting of their positions in the back of the wagon, the five girls began to sing.
Faithful and true, we lead thee forth
Where love, triumphant, shall crown ye with joy!
Star of renown, flow'r of the earth,
Blest be ye both, far from all life's annoy!
Champion victorious, go thou before!
Maid bright and glorious, go thou before!
Mirth's noisy revel ye have forsaken,
Tender delights for you now awaken;
Fragrant abode enshrine ye in bliss;
Splendor and state in joy ye dismiss!
"That is perfectly beautiful," Elizabeth told them when they finished. "Thank you so much, girls! In fact, that was so delightful, I would love to have you sing it in the church too."
"Can we, Mama?" Ruth asked hopefully. "Would it be all right?"
"I do not see why not. We always sing in church. Why not sing at a wedding?"
"We could sing it while you walk up the aisle," Belinda suggested.
"Yes. That would be lovely."
When they reached the church, both Matthew's wagon and her father's were parked in front. Elizabeth looked up at the cloudy sky, hoping now more than ever that the rain would hold off long enough so that all the wagons wouldn't be stuck in town.
They were just getting out of their own wagon and walking up to the church when Elizabeth spotted Mrs. Holmes rushing toward them with an urgent expression.
"Oh, Elizabeth," she said breathlessly. "I am just in time."
"What is wrong?" Elizabeth asked with concern.
Mrs. Holmes smiled. "Oh, nothing is wrong. But I was asked to bring you to my house, and I nearly missed you. You are to stay with me until they are ready to begin the wedding ceremony." She took Elizabeth by the arm. "Come along, dear."
Elizabeth didn't argue, and the girls all giggled as if they were in on this too.
"I have made us a nice pot of tea," Mrs. Holmes told her as they walked back to the parsonage. Inside the tiny house, Mrs. Holmes took Elizabeth's coat and hat and gloves. And soon they were seated by the fireplace, and Mrs. Holmes was handing Elizabeth a teacup.
"I want to assure you, dear, that I have been speaking to the reverend—morning, noon, and night—about how he needs to continue bringing God's goodness and love and mercy into his sermons. Just the way your father did when he preached that time."
"Oh, that is so good to hear. And how does the reverend respond to your encouragement?"
"At first he pretended not to listen to me. But then your father came by to speak to him. They have met several times now. And I do believe there is the making of a solid friendship between the two men."
"That is wonderful to hear." Elizabeth sipped her tea, trying to keep her eyes off the small mantle clock as it ticked steadily toward eleven o'clock. Was it truly possible that within the hour, she would become Mrs. Eli Kincaid? And furthermore, was she truly ready for it?
Chapter Twenty
It was a quarter past eleven, and Elizabeth was starting to feel a bit uneasy, when she and Mrs. Holmes both startled at a loud knock to the door.
"Hello?" called what sounded like her father's voice. "I have come to fetch the bride."
Elizabeth laughed as she stood. "I'm coming, Father."
He grinned at her. "Why, don't you look pretty."
"Thank you."
"Are you ready?" He peered into her eyes.
She nodded nervously. "I believe I am."
"The groom awaits." Asa took her arm and led her to the church with Mrs. Holmes following. He paused at the door, allowing Mrs. Holmes to slip in first. Then he looked at Elizabeth again. "You have chosen well with Eli. He is truly a good man. And I know he will be a fine husband and a good father. I have utmost confidence in him."
She blinked back tears. "That means so much to me, Father."
"It means everything to me, daughter." He leaned over and kissed her cheek. "Ready?"
She nodded with confidence. "Now I am."
Asa opened the door, leading her into the crowded church. She waited as he closed the door, taking in the faces that were eagerly looking back at them. And then, as Asa took her by the arm again, the girls began to sing their lovely wedding march song. As she and her father slowly walked down the aisle to the sweet lilting sound of the girls' voices, she noticed that someone had hung evergreen garlands clear down the aisle and then up over a graceful arch in front. White muslin ribbons were tied here and there. And a pair of candelabras glowed on cloth-covered tables on either side of the podium. She had never seen the little church look more beautiful than this.
But what captured her attention more than anything was the tall, handsome man standing beneath the evergreen arch. Eli's gaze was fixed on her, and although his expression was solemn, she could feel the warmth in his clear blue eyes as she stepped forward to take his hand. Standing next to Eli was Matthew. And standing next to Elizabeth was Malinda. Before her father sat down, Reverend Holmes invited him to say a prayer. And that's when Elizabeth knew the wedding ceremony would be perfect.
Just as Elizabeth had requested, it was a simple, straightforward wedding ceremony. But as she and Eli repeated their vows, she knew that it was heartfelt for both of them. And suddenly it was over and the reverend was proclaiming them man and wife, and after a shy kiss, the couple walked back down the aisle to the sweet singing of the wedding march song again. Perfect, Elizabeth thought as they went outside. Just perfect.
"You are the most beautiful bride in the world, Mrs. Kincaid," Eli whispered as he leaned over to kiss her with more passion than they'd shared in the church. "And you have made me the happiest man in the world."
Before she could respond as she would like, their friends and family began pouring out of the church. Gathering around the newlyweds, they extended their hearty congratulations. Many commented on the wedding ceremony and how much they enjoyed it. A few even mentioned their relief that the reverend didn't yell at anyone.
It wasn't long until Matthew and JT and Paddy McIntire were playing music, and the celebration grew even merrier as young couples started dancing outside and the women set up the food inside the church—just in case the weather changed.
"It hasn't rained," Ruth happily told Elizabeth. "God must have heard my prayer."
Elizabeth leaned over to kiss her daughter's cheek. "And why wouldn't he?"
Thanks to Clara's organization skills, people brought their own place settings, and many of the older people brought chairs to make themselves comfortable. And for the next couple hours, people were eating and dancing and merrymaking.
"We have plenty of food," Asa told Elizabeth with relief. "With so many people in attendance, I wasn't sure."
"Don't forget to take a plate home for Brady." Elizabeth sighed. "I wish he could have come."
Asa shook his head. "I'm afraid he made the right decision."
Elizabeth glanced around the crowded yard and knew her father was right. Everyone who lived within ten miles had come to this celebration, and they were all happy to eat of the wedding feast, but many of them would not approve of a Negro in attendance. As disturbing as that was, Elizabeth was determined not to let it spoil this day. Her reconciliation was that her family, including Brady, would celebrate with them next Sunday when she invited them all for dinner at her new house.
Eli and Elizabeth were just finishing up dessert when the first raindrop fell. She looked up at the gray sky and then laughed. "Well, like I kept telling Ruth, rain on your wedding day is good luck."
He chuckled. "Then it looks like we're about to get very lucky."
"Do you think we should try to leave?" she asked. "In order to get the wagon home?" She'd already told him of Malinda's idea to leave the wagon behind if it rained hard.
"I think we should honor our guests with one last dance," he told her.
She grinned. "Yes. I agree."
"And tell me, Mrs. Kincaid, what would you choose for your last wedding-day dance?"
She thought for a moment. "Would you mind if it was the Virginia reel?"
He laughed. "Would I mind? It's my favorite."
Now Eli went over to the musicians, who had been rotating off and on during the celebration in order to give everyone a chance to eat and dance. He requested a tune for the Virginia reel. Then he hollered to get everyone's attention as he rejoined Elizabeth, and the crowd got quiet.
"My wife and I wish to thank each and every one of you for helping us to celebrate today," he told them. Now he pointed to the sky. "As you can see, the weather is turning, and we do understand if some of you will need to make a run for home. But for the adventurous among us, we are going to enjoy one last dance. Who would like to join us for the Virginia reel?"
To Elizabeth's delight at least two dozen couples, including everyone in her family, came out to form several lines. Once they were ready, despite the raindrops that were falling more liberally, Eli waved to the musicians standing under the tarp awning that Asa and Matthew had constructed, and the lively song began.
Laughing and dancing in the rain, Elizabeth knew this was a day she would remember always. Seeing her parents and her children and everyone else enjoying themselves was the perfect way to end this celebration.
"Oh, my!" she exclaimed when the song ended. "Thank you all very much!" she called out. "Now we all better make a run for cover!"
Already, people were gathering chairs and dishes and belongings and making their hasty departures. Elizabeth and Eli offered to help her parents and family, but they shooed them away. "Not this time," Clara told her. "Off to your honeymoon." And so Elizabeth kissed her children goodbye, and she and Eli made their exit.
By the time they reached the wagon, it was raining hard enough that Elizabeth did not want to risk her team by taking them home. "Do you mind if we just ride horseback instead?" she asked Eli. "I've got my saddle."
"Not at all," he said as he started to unhitch the horses. "And I'm sure the wagon won't go anywhere parked right here in front of the church."
She laughed. "No, I think not."
Before long, they had saddled Bella, and Eli gave Elizabeth a hand getting up. Fortunately, her skirt was so full that she could easily straddle the horse without being indecent. Next he put the saddle blanket onto Beau, and using the wagon wheel hub for height, he mounted the horse. "Ready to go home?"
"Am I ever!" she declared.
People called out to them, wishing them well, and some teased as if they planned to follow the newlyweds, but thanks to the rain, no one seemed very serious. And before long, they were well on their way out of town. Although the road was getting muddy, it wasn't nearly as bad as on Thursday. And when they were about halfway home, the rain let up considerably. Even so, they were both fairly well soaked by the time they reached their property.
Elizabeth looked eagerly as they rounded the corner near the stand of fir trees and the cabin came into view. "Oh, Eli!" she cried out. "It is so beautiful."
He smiled. "Home sweet home."
She stared happily at the boxy cabin. The lower part of the walls consisted of solid logs, and the upper part was milled wood. The sloped roof was covered in cedar shingles, and she could see the top of a stone chimney in back. But what really got her attention was the big glass window on the left side of the front door. "It looks like a real house," she said with wonder.
"Wait until you see the rest." Eli slid off his horse and then helped her down from Bella. "You go on inside and get warmed up while I see to the horses."
She started to go but then stopped. "But you must carry me over the threshold," she declared stubbornly. "It's tradition."
He chuckled.
"What is so funny?" she demanded.
"You sounded just like Ruth when you said that."
She grinned at him. "Well then, it's about time you learned that every woman has a little girl inside of her."
"All right then. I'll carry you over the threshold. But would you like me to tend to the horses first?"
"Yes, please do."
"In the meantime, you could at least go up on the porch and stay out of the drizzle."
"I'll do that." As she walked up the stone-lined path leading to the porch, she spied what looked like an outhouse off to the right. Eli seemed to have thought of everything. On the covered porch there was a washing-up table with a bucket of water and a towel. Next to that was a bench, which looked like a handy place to remove muddy boots. And since she'd sent her carpetbag filled with clothes, including her moccasins, with Matthew the other day, she decided to remove her good shoes, which were now in need of a thorough cleaning.
She had just gotten her shoes unbuttoned and off when Eli came onto the porch. "Are you ready to be carried into your new home?"
She pointed out her stocking-clad feet. "I most certainly am." And before she could stand up he leaned over and scooped her up into his arms, nearly taking her breath away as he did so.
"Goodness!" she exclaimed as he opened the door.
"Right this way, my lady." He carried her into the warm room and closed the door with his foot. She was tempted to remind him that his boots were muddy too, but she stopped herself. Mud was not that difficult to clean. "What do you think?"
She stared in wonder at the big room. She wasn't sure if it was her imagination or wishful thinking, but it seemed even larger than Malinda's house to her. "Oh, Eli," she gushed. "It's the most beautiful house in the whole world."
"Beautiful?" He eased her down to the wooden floor. "You really think so?"
"I do," she said sincerely. "It is truly beautiful." As she removed her coat and hat, her eyes attempted to take it all in, which was impossible. Certainly this house was more rustic than her home in Kentucky, but the golden brown wood walls and floors seemed to glow with warmth. And the recently sawn lumber smelled clean and fresh. "I love every inch of it."
He grinned. "I'm glad to hear that. But maybe you'll want to see more of it before you make that claim."
"Of course, I want to see every bit of it." She hung her coat and hat on a peg on the door.
"So do you want me to show you around?" He lit a lantern and carried it over to the fireplace. "Or should I go out and call for Flax?"
"Flax?" She suddenly remembered. "Where is he anyway?"
"Probably off chasing a rabbit. He's not used to being left on his own too much. He and I have gotten quite close, you know."
"Yes, please, do go call him. And if you don't mind, I'll make a fire and look around. I want to admire your workmanship."
"This is your house, Elizabeth. Make yourself at home."
Chilled from the rain and feeling slightly damp, she went straight to the hearth, where sticks of kindling and wood shavings were stored neatly in a wooden crate. She stacked them carefully and found a box of matches on the mantle. She lit the shavings and blew a bit just to get it to catch. And then, with the light from the lantern, she admired the fir grain of the sturdy mantle. She ran her hand over a smooth round stone in the fireplace, one of the same stones she and the children had begun gathering from the creek after they'd first arrived. She thought they were for the foundation, but they were just perfect for the fireplace.
With the kindling crackling and snapping, she put on some large pieces of wood, and after warming her hands, she continued touring her new home. The rug she'd brought from Kentucky was just where she would have put it—in the sitting area near the fireplace. Her old rocker as well as a roughly hewn chair were in place as if waiting for the master and mistress of the house to hold court. Certainly, the area could use a woman's touch, but that would come. She went to the kitchen area, where the table and chairs she'd brought from her previous home were set into place. Directly across from the table was the large window. And hanging from the ceiling, directly over the table, was another kerosene lantern, which she lit. In the center of the table was a hand-carved wooden bowl filled with apples. On top of the apples was a note that appeared to have been written by Ruth. "To Mama and Eli. Love, JT and Ruth. PS. I picked the apples and JT made the bowl. Brady helped him."
Next it was the cookstove that captured her attention. New and black, it was spotless and ready to put to use. Resisting the temptation to light it, she examined the cupboards instead. Eli had given her the measurements for these, and she'd made some curtains she planned to hang on them to keep things tidier.
She was relieved to see that other than a few food items, some coffee and sugar and flour, the shelves and cupboards were still bare. The boxes of her household goods and the items she received at her bridal shower were stacked neatly against the wall. She could hardly wait to start putting everything into place.
Next to the stack of goods was a rustic but sturdy-looking dry sink. And on top of the dry sink was a little piece of paper. She immediately recognized her father's handwriting. "Lovingly made for Eli and Elizabeth by Brady." She ran her hand over the smooth surface. Unless she was mistaken, it was maple. Where had Brady found maple? She bent over to examine the cupboard space below. Nice and deep. She would hang a curtain over the front for now, and maybe someday it would have wooden doors.
She resisted the temptation to delve into the boxes containing her kitchen supplies and start putting the space into order. First she wanted to see everything in this sweet little house. She paused to look out the window, curious as to whether Eli had found Flax yet. Where could that dog have gone? As she peered out the shiny new window, she wondered why Eli had positioned it on this side of the house. She had assumed he would have placed it in the sitting area near the fireplace.
No matter, she told herself as she went to the rear of the house, where a rustic set of wooden steps, almost as steep as a ladder, was situated. Still in her stocking feet and holding her damp skirt up, she gingerly climbed the rungs. She knew this would be the children's sleeping quarters. She expected to emerge into a large open loft where her children would make beds on the floor out of blankets and quilts. But to her surprise, the loft space had been divided into two separate bedrooms, each complete with a small bed made from rough-hewn logs. Next to each bed was a bedside table, created from a wooden packing crate, with a small lantern on it. Along the dividing wall was a row of clothing pegs. Eli had thought of everything!
Curious as to how the beds had been constructed, she decided to investigate. Ropes were wrapped around the logs and then woven together to provide springy support for the mattress that rested on top. But where had Eli found mattresses? And what were they stuffed with? She gave the mattress a squeeze to discover that it was probably filled with straw. Perhaps it had been gathered from the dry grass that grew down by the river—back before the wet weather had set in. But how did Eli have time to do that?
On closer inspection, seeing the fine stitches on the mattress ticking and the colorful crazy quilts created from some familiar fabrics, Elizabeth realized it had to be her mother's doing. Of course—Clara would want to ensure that her grandchildren had comfortable places to sleep. The dear woman had even made them pillows. What more could anyone ask of a bedroom?
However, the most amazing and impressive part of these two sweet sleeping places was that each room had a small glass window above the bed and centered in the gable. It was a way to let light in, and thanks to the wooden hinges, which must have taken some time to carve, the windows could be opened to let in fresh air as well. Elizabeth was touched that Eli went to so much trouble for her children. And she couldn't wait until JT and Ruth saw everything.
Chapter Twenty-One
Elizabeth?"
"I'm up here," she called as she went over to the stair ladder.
"Taking a nap, are you?" he teased.
"Oh, Eli," she exclaimed as she came down the steps. "That loft is simply wonderful!"
He grinned as he reached for her hand. "You like it?"
"I never dreamed the children would have their very own rooms. Not at first anyway."
"I wanted to make the roof sturdier, and building a privacy wall seemed a good way to kill two birds with one stone." He helped her down to the floor. "Besides, we might have more children someday. The girls could be on one side and the boys on the other."
She laughed. "You have thought of everything."
"Oh, I'm sure there are some things I forgot."
"All the pegs for hanging clothes," she said. "And the bedside tables and lanterns. JT and Ruth will be over the moon when they see what you've done."
"I'm so glad you're pleased." He beamed at her.
"I'm beyond pleased," she proclaimed. "Everything is just perfect."
"And I found our dog down by the creek. He'd trapped some critter inside a hollow log and seemed determined to remain there until it came out." Eli chuckled as he pointed to where Flax was lying next to the fire. "I'm afraid he's plum worn out."
"Hello, old friend." She went over to pet Flax, but the tired dog didn't even bother to stand. However, his tail slapped happily against the floor. "We're finally home," she told him as she stood. "We'll all be back together soon."
"Did you see everything in the house?" Eli asked.
She smiled shyly at him. "Not everything...yet."
He pointed to her damp skirt. "Your personal things that Matthew brought over are in the bedroom. You should probably get into some dry clothes before you catch your death."
She looked over at the only interior door inside the house. It was just opposite the kitchen and she knew it must lead to their bedroom. "Yes," she said slowly. "Dry clothes would help keep the chill off."
"Meanwhile I'll stoke up this fire," he told her. "Get it nice and warm in here."
Elizabeth took in a deep breath as she slowly slid the wooden latch on the door that led to the bedroom—the room she would be sharing with Eli from now on. She had no idea what she was expecting to find, but when she saw the sweetly furnished room—with a bed similar to the ones upstairs except bigger—she felt a wave of relief wash over her.
Somehow, and to her amazement, this room felt very welcoming to her, as if she were truly home. On top of the bed was the beautiful quilt with the log cabin design. She felt the mattress to discover it was her own feather bed that she'd brought from home. And her feather pillows were here as well. All were covered with the new bed linens that Will's girls had given her at the shower. And beneath the feather mattress, resting on top of the rope supports, was yet another straw-filled mattress. They would sleep comfortably in this.
As wonderful as the bed looked, she was even more impressed that just like the bedrooms upstairs, this room had its own window. She went over to peer out, seeing that it looked toward the stand of fir trees behind the house. Hanging over the window was a lovely curtain that she felt sure had come from her mother's home in Kentucky. She fingered the crocheted lace and smiled. A very nice touch indeed.
Next she examined the washstand beneath the window. Like the dry sink in the kitchen, this piece was made of roughly hewn wood, which appeared to be maple. But on the front was a carved rose. Hanging on the towel bar were some fresh white linens, and positioned in the center of the cabinet was a creamy white pitcher and bowl—with water in the pitcher.
She picked up a bar of soap from a small china soap dish and sniffed it—lavender. Her mother had brought dried lavender from Kentucky with the plans of making soap once they got settled. She suspected this too was her mother's doing. Beneath the bar of soap was a slip of paper that said, "To Eli and Elizabeth with love from Father and Mother." They had obviously put the washstand and everything on it together. Bless them!
She looked back at the handsomely made bed. On one side sat a rustic chair—a good place to remove one's boots. On the other side was a small table covered in linen with lace trim. But on closer inspection, she discovered it was actually a crate. On this bedside table sat her small silver lantern as well as a little crystal vase that held a fragrant arrangement of evergreens and pretty dried grasses, giving it a feminine feel. Also there was a colorful rag rug on this side, perfect for keeping feet warm on a cold winter's night. Because of the more feminine touches, she suspected that this side of the room was meant for her. She had no complaints whatsoever.
At the foot of the bed was the trunk she'd brought in the wagon with her from Kentucky. As far as she knew, it was still packed with blankets and fabric and miscellaneous items of clothing for the children and her. But it looked nice at the foot of the bed. All in all, this room was unexpectedly sweet, albeit a little chilly.
And that reminded her that she'd come in here to get into a dry dress. Before long, her pretty blue wedding dress was hanging from a peg, and she had on her green calico and her moccasins. Putting her woolen shawl over her shoulders, she emerged to find Eli sitting by the fire.
"Are your clothes wet too?" she asked.
"My trousers are a little damp," he said as he stood.
"Well, you don't want to get a cold either."
"What did you think of the, uh...the bedroom?" He sounded slightly uneasy, which she found charming.
"It is beautiful," she told him. "And quite homey too. Although it is a bit chilly in there. Do you think we should leave the door open?"
"That's a good idea." He went to open it. "And the way the room is positioned, right across from the kitchen here, I expect that using the cookstove will help to warm it up some too."
"I'd love to try out the cookstove," she told him. "I'm not very hungry after our wonderful wedding feast, but perhaps I could make us some coffee or tea."
"Coffee would be good to warm up on. And I think your mother brought some over the other day, as well as some other staples she thought we might need. And she was very helpful in getting the bedding on the beds all put together. I couldn't have done it without her."
Elizabeth smiled. "I suspected she'd had a hand in some things."
"And that rug in our bedroom was made by Jessica. She just brought it by a few days ago."
"I wondered about that. It's so pretty."
"And I'm sure you saw that Brady made the dry sink and your father made the washstand. They found an old maple that had fallen a year or more ago. The wood had plenty of time to dry and was in perfect condition for making furniture."
"I wondered where they'd gotten the maple."
She held up the red coffeepot that Lavinia had given to her. "I'll put this pretty thing to good use now."
"And while you're doing that, I'll put on some dry trousers," he told her.
It wasn't long until Elizabeth had a little fire going in the cookstove. Then she filled the pot with water and measured out some coffee. Just as she was setting it on top of the stove, Eli emerged. He'd changed into a clean pair of work pants, but he still had on his good shirt. She blushed to think of how handsome he looked.
"You have no idea what a thrill it is to put a coffeepot on a real stove again."
"That's right, Malinda still cooks in the fireplace, doesn't she?" Eli sat down at the table, just watching her as she continued to explore her new kitchen.
She nodded. "I'm sure she'll be envious when she sees this one." She opened the iron door to poke at the fire before she slipped in some additional pieces of wood.
"I expect it'll just be a matter of time before Will buys her one, don't you think?"
"I'm sure you're right." She looked out the big window from where she was standing by the stove and sighed. "That window is just lovely. But I am curious as to why you put it over here instead of near the fireplace."
"Seemed to me you would enjoy it more over here. You can look out when you're cooking. And we can all look out when we're sitting here at the table and eating. But when we're sitting by the fire in the evening, it will be dark outside, and a window would be wasted.
She laughed. "You really did think of everything, didn't you?"
"I did my best."
She removed a parcel wrapped in brown paper from a cupboard and held it up. "Do you know what this is?"
"Something your mother left here yesterday."
She unwrapped the paper to find a loaf of bread, a crock of butter, and a jar of jam. "Looks like she thought we might want some supper tonight. Are you hungry?"
"I'm not very hungry, but I wouldn't mind some bread and jam with my coffee," he said.
She glanced around the homey-looking house, curious as to what time it was getting to be. Her best guess, based on the darkening sky, was that it was past five. Not too early for a light supper. Her livestock was still at Malinda's and her children were at her parents', so there wasn't much in the way of chores to be done this evening. She knew she should make the most of this unexpected bit of leisure.
"I have a clock packed somewhere," she absently said as she sliced some bread. "From Kentucky. If it still works, it might look nice on the mantle."
"I'm sure you'll have fun putting everything into place."
"I will. I am so eager to unpack my things and get settled. To make this our home."
"We are home, Elizabeth." He stood and came over to where she was buttering the bread. "Can you believe it?" He slipped his arms around her waist and pulled her close. "We are home at last."
"It feels a bit like a dream," she confessed. "A very sweet dream."
"Mr. and Mrs. Kincaid in their new home." He kissed her. "This is for real."
After a bit, she moved the coffeepot to the soapstone slab in the back of the stove, where it would stay warm without scorching. She suspected that they would have their light supper a little later.
######
On Sunday morning, Mr. and Mrs. Kincaid rode Beau and Bella back to the church, where they were warmly greeted by family and friends. Sitting with Eli and JT and Ruth, listening to Reverend Holmes' sermon, which was even more positive than the last one, Elizabeth felt very nearly perfectly happy. She probably would have been completely perfectly happy if she hadn't noticed something on the way to church.
Just as they were passing the brushy section near the creek, she'd observed the foliage moving. Eli had suggested it was the wind, although there didn't seem to be any breezes blowing. Then he said perhaps it was simply her imagination. And although she conceded that was possible, she felt fairly certain that wasn't it either. Somehow she knew that woman was still there. And the child. And somehow she was going to reach out to them. In the meantime, she would keep praying for them.
After church, while Eli was hitching the team to the wagon, Clara told Elizabeth that the children were welcome to stay with her and Asa for a few more days. "If you and Eli would like some more alone time."
"I don't think that's necessary," Elizabeth quietly told her. "Eli and I are both anxious to have the children settled in the home with us. I can't wait for them to see their rooms."
Clara smiled. "What did you think of everything?"
"It's all so wonderful and amazing. I could hardly believe it." Now Elizabeth began to gush, thanking her mother for all her sweet little touches. "One week from today, I want to have you all over for a thank-you dinner."
"That will be fun," Clara told her. "But I thought today you might want to join us for Sunday dinner. We had so much food left over from the wedding! We shared some with Matthew and Jess, but we want to send some of the food home with you. Also, the children's clothing and things are at our house. You'll need to pick them up."
"I'll check with Eli, but I'm sure we'd like to come," Elizabeth said. "Especially since I haven't got our kitchen very well set up yet. However, we might not linger much afterward. Eli hopes to get our livestock moved from Malinda's this afternoon." Elizabeth waved to her sister-in-law, who was approaching. "I know she'll appreciate it."
"Are you all settled in yet?" Malinda asked as she joined them.
"I haven't unpacked much," Elizabeth admitted. "But I feel completely at home. And we're taking the children home with us today."
"Already?" Malinda frowned. "I thought they were going to spend a few days at your parents'."
Elizabeth shrugged. "I'm eager to have my family all under one roof." Now she told Malinda about the loft bedrooms. "One for JT and one for Ruth."
"Oh, my. I look forward to seeing this house. When will I be invited over?"
"Give me a few days to get unpacked," Elizabeth told her. "How about coming over for tea? Sometime when the children are in school? I think I should have everything unpacked and put to order by midweek. How about Thursday?"
"I'll be there with bells on," Malinda promised.
"And Eli and JT will be by your place later this afternoon to pick up our livestock," Elizabeth promised. "And we've got the wagon, so we'll pick up the chickens and anything else we left behind on our way home from church."
It wasn't long until they were on their way home, with Elizabeth and Eli in front of the wagon and JT and Ruth in the back. Ruth was as chatty as a magpie, talking about yesterday's wedding and how many days until Christmas and a dozen other unrelated things. Meanwhile, JT seemed to be unusually quiet. Finally, worried that he might be coming down with something, Elizabeth decided to get to the bottom of his sullenness. "Are you feeling quite well, JT?"
"I feel fine, Ma."
"You seem rather quiet." She turned to peer at him, seeing that he had a slightly perplexed expression. "Is something wrong, son?"
"No. Nothing's wrong."
"It's about Eli," Ruth blurted out.
"Ruth!" JT glared at his little sister.
"I'm sorry," Ruth told him. "But you should tell Mama and Eli about what's troubling you."
Elizabeth glanced at Eli to see that he was looking a little uncomfortable. She wondered if JT could be having second thoughts about their marriage. However, it was a little late for that. Even so, perhaps this wasn't the best time to have a conversation like this. She wasn't sure what to do.
"Go ahead and tell them, JT," Ruth said. "Or else I will."
"Ruth!" JT sounded truly irritated.
"If you have something to say," Eli began slowly, "maybe you should just get it out in the open, JT. I'm sure it's something we can all sort out together. We're a family now."
"Come on," Ruth urged.
"All right," JT grumbled. "I was just wondering, now that you and Ma are married, what are we supposed to call you? Do we still call you Eli? Or do we call you Pa?"
Eli's countenance relaxed. "Well, I think that's up to you and Ruth," Eli said. "I'm comfortable with you calling me Eli if that's what you'd prefer. And I'd be rightly honored if someday you feel like calling me Pa. I realize I have some big shoes to fill, but I do want to be your father. When I married your mother, I married you children too. We are a family now."
"Well, I want to call you Pa," Ruth declared.
Eli turned to smile at her. "I am truly honored, Ruth."
"I reckon I should call you Pa too," JT said. "If you really don't mind."
"I do not mind in the least. Like I said, I am honored."
Elizabeth let out a relieved sigh. Well then, that was settled.
Chapter Twenty-Two
After dinner at Elizabeth's parents', they loaded up the wagon and continued on toward home. Elizabeth tried to act calm, but she was so excited about how JT and Ruth would react to the finished house. "Here we are," she said as Eli pulled the wagon in front.
"Oh, my!" Ruth exclaimed. "It's bigger than Grandma and Grandpa's cabin, isn't it?"
"Yes, but there are four of us and only two of them," Elizabeth pointed out as they climbed from the wagon. "And instead of unloading everything right now, do you want to go in and look around first?"
The children didn't argue, and she and Eli exchanged glances as they followed them into the house. Their reactions were similar to Elizabeth's, although Ruth was much more vocal and JT used his comments sparingly.
"And now you need to go up the stairs," Elizabeth told them. She and Eli waited down below, listening as the children exclaimed over the two separate bedrooms. A short argument erupted over who got which room, but eventually both JT and Ruth decided that the spaces were identical, and JT flipped a coin to determine who got what.
"I love my room," Ruth declared as she came down the stairs. "And the bed is really comfortable."
"I didn't know we'd have our own rooms," JT said happily as he rejoined them. "Thanks, Eli." He chuckled. "I mean, Pa."
"Yes! Thank you, Pa!" Ruth told Eli.
Although Elizabeth was glad the children accepted Eli enough to call him Pa, it was a bit unsettling too. The last man they'd called Pa had been James. And, the truth was, she wasn't sure she was ready for this. And yet she didn't want them to keep calling him Eli either. Really, she should be thankful.
"Let's get the wagon unloaded," Eli said to JT. "And then we'll change out of our church clothes and go fetch our livestock."
Elizabeth pointed at Ruth. "And let's get changed out of our Sunday clothes too. Then you can help me get those chickens all settled."
"Do you think they're glad to be home, Mama?"
"I'm sure they are."
"I think so too. Especially since some of Malinda's hens were bossy toward ours," Ruth said. "As if they thought they were better."
"Well, our hens were newcomers. Maybe they didn't like that."
"Or maybe they were jealous."
"Jealous?"
Ruth giggled. "Because of Reginald."
"The rooster?"
"Yes. Because Reginald probably liked our hens better than Aunt Malinda's."
Elizabeth laughed, thinking that could be beneficial to increasing their numbers. "Say, Ruth, now that we're home and settled, we won't collect the eggs for a while—maybe even a whole week."
"No eggs for a whole week?" She frowned. "What will we have for breakfast?"
"Oatmeal. Or biscuits and gravy."
"Like on the Oregon Trail?" she asked.
"A little like that. But don't worry, we won't starve. And it'll only be one week without eggs. Maybe not even that."
"But why don't you want to eat eggs, Mama?"
"Because I'd rather hatch some chicks."
"But it's not springtime."
"That's true. But the climate is mild here. I think we can raise some young chickens in the winter. And thanks to Reginald the rooster, we might have some fertile eggs in the next few days. And if we let the hens sit on the fertile eggs, we'll have baby chicks right around Christmas."
"Christmas chicks!" Ruth clapped her hands. "Yes! Yes! Yes!"
######
On Wednesday morning, after the hens had three undisturbed days of egg laying, Ruth carefully collected about a dozen eggs, and she and Elizabeth began candling them to see if any were fertile.
"See?" Elizabeth held the first egg up to the bright light of a kerosene lamp. "This one doesn't show any veins, so it's not fertile," she explained. "Put it in the bowl for later." Ruth handed her the next warm egg, and to their delight, it showed the faint traces of veins inside. "This one goes back in the basket," Elizabeth told her.
"Should we keep the basket near the stove to say warm?" Ruth asked. "Until we're done candling?"
"Yes. But not too close. We don't want to cook our chicks."
By the time they finished candling, they had discovered eight fertile eggs. Elizabeth marked each one with a spot of ink. "So when you see the spotted egg, you know not to bring it back in the house again," she explained to Ruth. "Now take these precious eggs back to the hens, and hopefully we'll get a few more before the week is out."
Elizabeth knew that it was unlikely that all the fertile eggs would safely hatch into chicks. But the more eggs they left with the hens, the greater the chance their chicken flock would grow. And at least one of the chicks was sure to be a cockerel. It was exciting to see that their farm could become productive. Besides the possibility of chicks, Elizabeth was hopeful that Goldie might be with calf, but they wouldn't be sure for a few more weeks.
She stood on the porch, watching as JT helped Ruth onto Molly's back. Then he climbed up behind her, taking the reins as they both called goodbye. "Have a good day at school," she called back.
Because they'd had several rainless days, Eli had the team out today, felling timber to use for fence rails. Elizabeth went back inside the house, which she'd spent the past two days putting to order. She smiled in satisfaction at how much homier it looked with curtains up, dishes on shelves, the clock on the mantle, and all the other housekeeping amenities that made a house a home.
She went to the dry sink, and removing the washtub from beneath, she filled it with hot water from the kettle on the stove and then happily went to work washing the breakfast dishes. Elizabeth had never disliked housework—not like some women she'd known—but she usually preferred being outdoors. However, she was so thankful to be in her own home, she relished every mundane little chore these days. Just the same, she had promised to join Eli by midmorning. She wanted to help him get as many fence rails into place as possible today. "It's like making hay while the sun's shining," she told him at breakfast. "I'll do whatever I can to help get our fences up. As long as I don't have to wade through the mud."
After she finished her housekeeping, she put on her split riding skirt and barn jacket and work boots. Then she wrapped a couple of buttered biscuits in a piece of linen and slipped it into her pocket for Eli. And grabbing her old felt hat and riding gloves, she headed out to where Eli was felling timber.
By the time she got there, he already had a load of wood ready to go. Because these logs were much smaller than what they'd used on the cabins, Eli had been using the wagon to transport them. "Looks like you've been busy," she said as she handed him the biscuits.
"Thanks." He pushed his hat back from his forehead, wiping his glistening brow as he took a bite. "Do you want to drive the wagon back and unload it? The poles aren't too heavy, and they slide out fairly easily."
"I'd be happy to."
He grinned at her. "I always liked seeing you in that outfit."
She looked down at her working clothes. "Truly?"
He nodded. "Oh, yes. You caught my eye many a time out on the Oregon Trail. I could spot you from almost a mile away, Elizabeth."
"That's rather hard to believe."
He chuckled. "Well, that might be an exaggeration. But believe me, Mrs. Kincaid, I kept my eye on you."
She reached over and touched his cheek. "And I'm so glad you did."
She got into the driver's seat of the wagon and drove the team over the hill and down toward the river, where they were constructing the fence. Eli's plan had been to use this section of river as part of the fence because the livestock wouldn't be likely to cross it and because they would have access to drinking water.
As the wagon slowly rolled along, she surveyed their land. There was so much potential here, so many possibilities. She wondered how long it would take to fully realize it. Perhaps in her children's lives...or her grandchildren's.
As she was gazing out over the meadow that led down to the river, she noticed a figure moving alongside a grove of trees—a woman. She squinted into the sunlight, thinking it could be Jess or her mother. But then she saw the smaller figure trailing behind the woman and realized it was the Indian woman. Without really thinking, Elizabeth pulled off her hat and waved, calling out what she hoped sounded like a friendly greeting.
But just as quickly, the woman and small child both ducked into the shadows of the trees. Elizabeth knew she'd probably frightened the poor woman. Yelling out like that must have sounded threatening. But didn't the woman know that Elizabeth was the one who'd taken her blankets and food? Even just yesterday, Elizabeth had walked a loaf of bread down to the place where she'd seen them before. She'd wrapped the loaf in one of her old dresses—a brown-and-black calico that she'd nearly worn out on the trail. It hadn't been easy to part with fabric because she had thought of using it for quilt pieces, but if it would keep the woman and her child warm, it was worth it. From the distance, it was impossible to see whether the woman was wearing the dress. And Elizabeth hadn't been back to the drop-off spot to check.
She pulled the team to a stop, set the brake, and climbed down to unload the poles. Eli had been right—they did slide fairly easily from the wagon. After she dropped a small pile, she moved the team farther on up, where she unloaded some more, spacing them out so they wouldn't need to be carried so far. She did this until the wagon was empty and then headed the team back to where Eli was working.
Using the crosscut two-man saw, they were able to cut down trees much more quickly. As they sawed back and forth, she told him about seeing the Indian woman and child again. "I called out to her," she admitted. "But I'm sure it frightened her."
"Did she have on your old dress?"
"I couldn't tell from the distance."
After they'd felled a number of trees, they used axes to chop off the limbs and make them into poles. Eli, much more adept than she, cleaned branches off four trees to her one. Still, she felt she was doing her part. Then they loaded them into the wagon together, and she took the second load. She decided to take this load over to the far side of what would be their livestock pasture—over by the trees where she'd spotted the woman and child.
Like before, she unloaded a few poles in one spot and then moved the wagon on up to the next, distributing the fencing materials evenly along the fence line. Every once in a while, she paused to catch her breath and glance around. She was curious if the woman was nearby because for some reason, Elizabeth felt she was being watched. After she dropped the last of the poles, she took off her hat and pushed the hair that had come loose from her bun back into place. And then she looked around again.
"Hello?" she called out in what she hoped was a cheerful sounding voice. "Is anyone there? Hello?" She waited quietly and then she heard a rustling sound back in the trees. "Hello?" she called again. "I want to be your friend. I want to help you if I can. Hello?" She stood still now, unsure of what to do. "Do you need my help?"
A small part of her questioned her behavior—and she could almost hear what Malinda would have to say. "What on earth are you doing standing out here like this calling out to an Indian?" And for all Elizabeth knew, there could be a whole bunch of them back in the woods. They could be very angry at the white men for taking over their land. For all Elizabeth knew, this land she considered her own might have truly belonged to the Indians. That thought alone made her feel slightly sickened. She closed her eyes and prayed silently. "Dear God, please help me to know what to do. Help me to help this woman...if she wants my help."
"Hello?" said a low voice.
Elizabeth opened her eyes to see the Indian woman standing at the edge of the woods. Her long black hair hung limply over her shoulders, and she was wearing Elizabeth's old dress, with her bare feet poking out from beneath the dirty hem. She peered curiously at Elizabeth with dark, somber eyes.
"Hello," Elizabeth said nervously.
"You need help?" the woman asked in a flat tone. "I hear you say help."
"No, I don't need help." Elizabeth made a stiff smile. She was tempted to approach this woman, but knew that might make her run. "You speak English?"
"Some."
Elizabeth pointed to herself. "I'm Elizabeth," she said slowly. "E-liz-a-beth."
"Elizabeth." The woman nodded as if she understood.
"You are?" Elizabeth pointed to her. "Charles' wife?"
The woman's eyes lit up ever so slightly. "You know Charles?"
"I know who Charles is. Are you his wife?"
"Yes. Charles' wife."
Elizabeth's smile grew bigger. "I'm Eli's wife."
"Eli?" The woman nodded toward where the sound of the ax was ringing.
"Yes." Now they stood just staring at each other. Elizabeth was trying to think of the right words when the woman glanced over her shoulder as if she were frightened. Was she about to run?
"I want to be your friend," Elizabeth said slowly. "Friend."
"Friend?" Her dark eyes looked suspicious as she folded her arms across her front.
"I think you need a friend," Elizabeth said gently. "You need help."
The woman pressed her lips together as if she understood the meaning, and yet she seemed somewhat confused. Probably she was unaccustomed to being befriended by white people. How had she met Charles?
"Charles is a good man?" Elizabeth asked.
"Yes." The woman nodded.
"Not all white men are good," Elizabeth said slowly. "Some white men hurt your people."
The woman nodded even more firmly. "Yes."
"Eli and I," she pointed at herself. "We want to be friends to Indians."
The woman narrowed her eyes, studying Elizabeth closely as if taking in every small detail.
"I want to help you," Elizabeth told her. "Do you need help?"
"You help me." She waved down at the hand-me-down dress. "You give food. You give blankets. You give clothes. You help."
"Yes. I gave those to you and your child. To help you. To be friends."
"Yes." Her features softened slightly. Almost as if she were trying to believe Elizabeth...to trust her.
"What is your name?" Elizabeth asked quietly.
"Marakeenakanaha."
Elizabeth attempted to repeat the name.
"Mara," the woman said.
"Mara." Elizabeth nodded.
"Your boy? Girl? What name?"
Elizabeth smiled. Mara had obviously been watching their whole family. "The boy is JT. The girl is Ruth. What is your child's name?"
"Rose."
"That is a pretty name. Rose."
Mara looked over her shoulder, calling out some words in her own language, and in the same moment a tiny barefoot girl emerged from the shadows.
"Hello, Rose," Elizabeth said warmly. The child looked to be about three years old. Her long hair was light brown and wavy, but she had her mother's eyes.
"Where do you live?" Elizabeth asked Mara.
Mara pressed her lips together and scowled.
"Do you have a home? A place to stay dry and warm?" Mara shook her head no.
"Where is Charles?"
Mara's eyes lit up again. "Charles go find gold."
"Gold mining?"
"Yes." She nodded eagerly. "Charles go find gold."
"Has he been gone for long?"
Mara looked uncertain.
"Did he leave in the summer? Or last year?"
Now Mara ran her hand over her midsection, which Elizabeth suddenly noticed was very well rounded despite Mara's overall thinness. Charles had obviously been here in the last nine months. But where was he now? Did he know his wife was with child? Did he know she was homeless? Did he even care?
Chapter Twenty-Three
Elizabeth looked up at the sky, realizing it was well past midday. Although she'd taken Eli those biscuits earlier, he was probably hungry for his dinner by now. For that matter, she was too. She pointed at Mara and Rose. "Are you hungry?"
"Yes," Mara muttered.
"Will you come with me?" She pointed to the wagon, wondering if she was expecting too much for them to ride with her, but to her surprise, Mara took Rose by the hand, and soon they were all loaded in the front seat. "I have food at my house," Elizabeth said simply as she drove the team toward home.
Elizabeth attempted to make small talk as she drove her impoverished passengers toward her cabin. Rose's dress was little more than a rag and couldn't possibly be enough to keep her warm. By the time they reached the house, Elizabeth had made a plan to alter one of Ruth's old dresses to fit the child.
Seeing Eli and Flax striding toward the house but peering curiously in her direction, she waved to let him know that everything was all right. "That's Eli," she told Mara and Rose. "And Flax is our dog."
"Dog," Rose pointed at Flax. "Dog."
"Yes," Elizabeth told her. "Dog. Flax."
"Flax." Rose repeated. "Flax. Dog."
"Rose is a smart girl," Elizabeth told Mara.
"Yes." She nodded.
"Hello," Eli said in a friendly tone as he approached the wagon.
"This is Mara and her daughter, Rose," Elizabeth said politely. "Mara and Rose, this is Eli, my husband."
Eli reached out to help them down from the wagon. First he gave Mara a hand, and then he simply lifted Rose, holding her high in the air until she giggled, and then he carefully set her on the ground. Then he helped Elizabeth down.
"Mara and Rose are joining us for dinner, Eli."
He nodded and smiled at Mara. "Good. Welcome to our home."
They went up to the porch, where they took turns washing up. The whole time, Elizabeth chattered cheerfully at her unexpected guests. She knew it was a strange situation, but she wanted them to feel at home. However, when it was time to go inside the house, Mara refused. "We stay here." She pointed to the porch.
"You are welcome to come in the house," Eli said.
"We stay here."
Elizabeth nodded. "I'll leave the door open," she told Mara. "If you want to come inside, please do."
While Elizabeth prepared food, Rose looked longingly into the house, but Mara spoke to her in their native language, and Rose did not step past the threshold. Instead, she played with Flax, who was happy to get this unexpected attention. Meanwhile, Mara went over to sit on the porch bench.
"They need help," Elizabeth quietly told Eli as he filled a coffee cup.
"I'm sure they do."
"They have no place to live, and it sounds like Charles has gone gold mining—probably down at the Rogue River. I hear that's where men go seeking their fortunes of late. Anyway, I'm not sure how long he's been gone, but Mara is with child, and I expect her baby is due in just a month or two."
"Oh..." He shook his head with a dismal sigh as he reached for the sugar bowl.
"But I have an idea."
"An idea?" He stirred some sugar into his coffee.
"Brady's cabin."
His eyes lit up. "Ah-ha."
"Is that all right with you?"
"It's perfectly fine with me. But you know you'll have to keep it a secret from everyone else."
"Even from the children?"
He shrugged. "I don't know, Elizabeth. Is that too much to expect of them? To keep a secret like that from their friends at school, their cousins, their teacher?"
"They know that they're not to speak of Brady in town."
"That's true."
"What is one more secret?" She smiled as she handed him a plate of smoked salmon, two slices of buttered bread, and one of the apple fritters she'd made last night.
"I'll leave that to your discretion."
She picked up the plate she'd made for Mara and Rose to share and carried it outside. "Here." She smiled as she handed it to Mara.
"Thank you," Mara muttered.
"You are most welcome." Elizabeth patted Rose on the head.
When she went back into the house, Eli handed her plate to her. "Why don't we go out and eat with them? It's a nice day today."
She grinned. "Good idea."
So it was that the four of them ate their picnic dinner out on the porch. And when Mara and Rose's plate was empty, Elizabeth went back inside and got them more food. After a while, Eli excused himself to return to work. "I'll take the wagon, and if you don't make it back out there to help me, I'll understand." He winked at her then tipped his hat to the porch. "Goodbye, ladies."
"Goodbye, Eli," Mara said politely.
"Goodbye, Eli," Rose said, imitating her mother.
Satisfied that Mara and Rose had eaten enough, Elizabeth took their plate, but once again she invited them to come inside her house. "You are my friends," she told Mara. "You can come in my house."
Mara looked uncertain. "Charles say no. I no talk to white man. I no go in white man house."
"I am a white woman," Elizabeth told her. "You can come into my house. Besides, I think I may have a dress for Rose."
"Dress for Rose?" Mara's eyes lit up.
"A dress that belonged to my little girl, Ruth." Elizabeth tipped her head toward the door. "Please, come inside."
"Yes." With a determined expression, Mara took Rose by the hand and cautiously followed Elizabeth into the house.
Elizabeth set the dirty dishes on the dry sink and then went to her bedroom and opened the trunk, where she had stored all the clothes that were too small or too big or only for special occasions. She dug until she found the blue-and-gray gingham dress that Ruth had nearly worn out on the Oregon Trail. It was patched and mended in places, but it was clean. She carried it out to where Mara and Rose were standing by the door waiting. She held the dress up to Rose. "It's too big now," she said to Mara, "but I can make it smaller so that it fits." She opened her sewing basket.
"I can sew," Mara declared.
"You can?"
Mara held up her hands in a helpless gesture. "No needle. No thread. No cloth."
Elizabeth took a spool of thread, slid a needle down into the side of it, and then wrapped it into the dress and handed it to Mara. "Yes. You can sew."
Mara held the dress to her chest. "Thank you."
"Now I want to get some food for you to take with you," Elizabeth told Mara. While Rose followed Flax to his favorite spot by the fireplace, Elizabeth went into the kitchen and began to fill a basket. Uncertain of Mara's cooking abilities, she put in smoked venison and fish and apples and bread. She also put in two drinking cups and a few other basic household items. Just enough to get them started. She had no idea whether Mara would even accept her offer. But she suspected by their soiled clothes that they'd been camping along the creek. She could imagine how miserable that would be in the rain.
She turned to Mara. "Let's go now."
Mara just nodded, still clutching Ruth's old dress to her chest. She called out to Rose to come—this time in broken English.
Elizabeth went outside with them, and still holding the basket of goods, she nodded in the direction of Brady's cabin. "I have a house for you to use," she told Mara.
"House?"
"Yes. You and Rose can stay there until Charles comes back for you."
"House?" Mara said again.
Elizabeth patted Mara on the back. "A very small house."
As they walked, Elizabeth tried to explain that her friend Brady was going to live in the small house but that he changed his mind.
"The dark man?" Mara asked.
"Yes. That is Brady. He lives with my mother and father now." She pointed in their direction.
Mara nodded as if she understood.
"Did Charles teach you English?" Elizabeth asked as they walked through the meadow.
"I learn some from Charles. I learn some at mission."
"What mission?"
"When I girl, many people die from white man sickness. My father die. Mission take village children. Girls learn to sew. Learn to cook white man food."
"Oh?"
"I go school. Learn English. Learn read and write."
"You can read and write?" Elizabeth tried not to sound too shocked.
"Some. Then school no more. Troubles start."
Although Elizabeth hadn't actually seen Brady's cabin, she knew exactly where it was because she was the one who had originally picked the spot as a good place for him to live out his final years. It was near the creek so that water would be handy, but not so low that it would flood. Set in a grove of fir trees, it was somewhat protected from the elements. Most of all, it was private. Walking up to the tiny cabin encased by tall evergreens, she was reminded of a fairy tale—as if she expected gnomes and fairies to emerge from the shelter.
She opened the door, which was barely her height, and peered inside. Like her house, it smelled of recently cut wood. But unlike her house, it had a packed dirt floor and no glass windows or new cookstove. It didn't even have a fireplace. Indeed, there would be no room for a fire in here. However, she had noticed the campfire area outside where she assumed Brady must have done his cooking during the short time he'd lived here. The cabin's interior, which was about the size of the bedroom she and Eli shared, was bare except for a shelf, a couple of clothes pegs, and a wooden bed that was attached to the wall.
"This is it," Elizabeth waved her hand. "If you want, you are welcome to make yourself at home."
"Home." Mara's dark eyes glistened as she looked around the tiny space.
"It is very small," Elizabeth said apologetically.
"Thank you," Mara said earnestly. "Thank you, thank you."
"Thank you," echoed Rose happily.
Elizabeth smiled in relief. "Do you still have the blankets I gave you?"
"Yes. I hang in sun. Dry."
"Good." Elizabeth looked around, wondering what more she could bring to make them more comfortable here. Perhaps another blanket. Maybe a pot to cook in. A bucket for fetching water. And yet she didn't want to interfere too much or overwhelm them with too many things. She suspected that Mara was accustomed to taking care of herself.
"We had house," Mara said quietly.
"You and Charles?"
"Yes. White men burned. Big fire."
"White men burned your house?"
Mara nodded sadly.
"I am so sorry." Elizabeth put her hand on her bony shoulder. "I think you are safe here. I will not tell anyone about you." Of course, this reminded her that her children would soon be home. And for the time being she had no intention of telling them either. Not only could it put Mara and Rose in danger, it might endanger her family as well. Hearing that white men had burned Mara's home was disturbing to say the least.
Elizabeth excused herself, promising to come back and visit in a day or two. Mara thanked her again, and then Elizabeth hurried away. As she walked back to her house, she experienced a mixture of conflicting emotions. On one hand, she was greatly relieved that Mara and Rose had accepted her offer of help and were now safely settled into Brady's little house. That in itself felt like a godsend. But on the other hand, she was greatly grieved to learn that white men had burned down Mara's house. How could supposedly civilized people be so thoughtless and selfish and cruel?
However, she wasn't only torn over Mara's situation. Now Elizabeth felt somewhat conflicted about returning to her own beloved home. As grateful as she was for her delightful cabin, it was unsettling to think that what she considered her land had probably belonged to Mara's people first. Suddenly home sweet home felt more like stolen treats.
Chapter Twenty-Four
On Thursday morning, Elizabeth wished she were more excited about Malinda's visit. Originally she had planned to make this visit into a memorable occasion. She'd planned to get out the tea set and silver spoons and linens and make Malinda feel like Queen Victoria. As it turned out, right after the children left for school, Elizabeth ran around the house gathering up items she felt would be useful to Mara and Rose. Loaded with a water pail, a cooking pan, two tin plates, some utensils, a tin bowl, another blanket, and more food, she was nearly out the door when she remembered Mara and Rose's bare feet.
Elizabeth knew that even the shoes Ruth had outgrown would be too large for Rose. But she could take her a pair of Ruth's stockings to keep her warm in the house. But what for Mara? That was when Elizabeth spied her moccasins by the fireplace. As much as she loved those, she knew that Mara needed them more.
Knowing that Malinda could very well arrive before she finished this errand, Elizabeth hurried as quickly as she could over to the little cabin. When she got there, she didn't see Mara or Rose around, so she knocked on the door. "It's me," she called out. "Elizabeth. I have some things for you. I'll just set them out here."
The door opened, and Mara smiled. "Come in my house," she told Elizabeth.
As much as Elizabeth wanted to make her excuses to get back home, she knew she needed to honor Mara by going inside. As her eyes adjusted to the dim light, she could see the blankets on the bed. She smiled. "I brought you some things." She set some of the supplies on a corner of the floor and then looked at Rose. She was wearing Ruth's old dress, and Mara, true to her word, had made some alterations. Certainly, they weren't the way that Elizabeth would have done it, but the child was clothed. "You look very pretty."
She pulled out the black stockings and handed them to Rose. "These are to keep you warm in the house." Rose stared at the stockings as if she didn't know what to do with them. "For your feet and legs." Elizabeth pulled up her skirt to show Rose her own stockings.
Now she handed the moccasins to Mara. "These are for you."
Mara stared down at the moccasins and then back up at Elizabeth. Her dark eyes grew misty. "Thank you," she said.
"I need to go back to my house," Elizabeth told them. "But I will come back again."
"Thank you," Mara said once more. "Thank you, friend."
"You are welcome, dear friend."
As Elizabeth hurried back to her house, she wondered what Malinda would think if she knew what her best friend and sister-in-law was up to this morning. Not that she had any intention of telling her. In fact, she had made up her mind that no one, besides her and Eli, would be in the know as far as Mara and Rose went. It just wasn't safe. However, she wouldn't mind having Will's legal counsel in this matter—as long as it could be done with confidentiality. She remembered how Will had offered her some advice in regard to Brady's emigration to Oregon. Of course, she hadn't appreciated all the answers or the narrowness of the new laws. But she did appreciate his legal knowledge.
To Elizabeth's relief, she managed to get her morning housekeeping chores finished and to set up what seemed a very respectable tea party before Malinda arrived. The table was covered in her best lace-trimmed tablecloth and set with her new china tea set and the silver teaspoons. A small plate of molasses cookies that Ruth had made yesterday graced the center. A cheerful fire glowed in the fireplace, and Elizabeth had just hung up her apron when she heard a knock at the door.
"Your little cabin is charming," Malinda said as she came into the house.
"Thank you." Elizabeth closed the door. "And welcome."
Malinda untied her bonnet, handing it to Elizabeth. "And I noticed you have quite a large glass window." She frowned. "But it seems oddly placed, over there by the kitchen."
"Eli thought we would enjoy it more there. As it turned out, he was right. I love looking out while I'm working in the kitchen. Or when we're eating."
"I'm surprised you have wood floors." Malinda tapped her toe on the solid floor. "Most newcomers must settle for packed dirt the first few years." She made a funny laugh as she removed her coat. "Perhaps that's why we call them settlers."
"Well, I don't feel that I've had to do much settling in that sense." Elizabeth took her coat, hanging it on a peg. "Eli thought of just about everything."
Elizabeth showed Malinda about the house, listening as her friend examined and commented on everything. But something about Malinda's tone made the comments feel more like complaints than compliments. It seemed that everything was either not good enough or too good. Elizabeth could not make heads or tails of her good friend.
"Glass windows in the loft?" Malinda questioned as they stood in Ruth's bedroom. "That seems awfully extravagant."
"Eli felt the children needed the light."
Malinda pointed to the bedside table, where Ruth had set her favorite bedtime storybook. "Lanterns provide good light."
"Yes, well, Eli thought the children might want fresh air in here too," Elizabeth explained.
"Do the windows open?"
"Oh, yes. Eli made wooden latches. See?"
"For a wagon train scout, he seems to know a fair amount about carpentry." Malinda gave Elizabeth a slightly suspicious look. "Did he learn that on the wagon train?"
"My father and Brady have helped him with some of the woodworking. Eli said he learned a lot while building the house."
"This ladder is a bit steep," Malinda said as they scaled down.
"Yes, I must agree. But the children don't mind it at all. They scurry up and down it like little mice. And it takes up so little space down here. I appreciate that." Now Elizabeth opened the door to their bedroom, and Malinda peered in.
"Another window?" Malinda's brow creased.
"It is awfully nice to have the sunlight coming into the room."
"Well, windows are lovely, but they don't put food on the table, Elizabeth. Don't forget, this is the frontier. Your priority is to feed your family—not impress people with your house."
"I'm not concerned for food. Eli is an excellent huntsman. He's providing us with plenty of venison and fish." Elizabeth almost reminded Malinda of how many times Eli and other members of her family had shared these sources with Malinda's household while Elizabeth and the children stayed with her. Instead, she simply smiled at Malinda. "Besides, as you and I know, God is our provider. We trust in him."
"Yes, yes, I know that as well as anyone. But I do want my best friend to be practical and wise and frugal. When hard times hit us, we need to be prepared to face them."
Elizabeth nodded as she removed the teakettle from the stove. "Yes, that's true..." As she poured a bit of hot water in the teapot, swishing it around to warm the china, she wondered why she was being lectured like this. And why was Malinda out of sorts? Surely it didn't have to do with Elizabeth's wedding. That was all over and done, wasn't it?
"When sickness runs through an isolated community such as ours, you realize how vital it is to be prepared for any sort of calamity."
"I feel that we are fairly well prepared." Elizabeth poured black tea leaves in the pot and added more hot water.
Malinda was staring at that cookstove now. "Does that little thing really work? It seems much too small to be useful."
So Elizabeth set the teapot aside and gave Malinda a quick lesson on how the cookstove worked. "The oven is large enough for two big loaves of bread or a large venison roast. And I'm sure it could hold a good-sized goose. Anything larger can always go in a roaster on top. It's actually quite convenient, and it helps to take the chill off the house. Sometimes we don't even use the fireplace. Also, the stove doesn't use very much firewood. Perhaps that's because it's smaller."
"Well, you don't have a large family to feed. Not like I do," Malinda smoothed her skirt. "I suppose when I get a cookstove, I will want a larger one."
"Yes, I can understand that." As Elizabeth moved to the table, she tried to stifle her irritation. Why was Malinda being so ungracious?
"This is an interesting piece." Malinda paused to examine the wet sink.
"Yes, it's very handy."
"Quite rustic. Did Eli make it too?"
Elizabeth almost mentioned that Brady had made it but stopped herself. She did not want Malinda to launch into a lecture about how Brady didn't really belong here and how he wouldn't be allowed to stay. "It was a wedding gift—from my family."
"Ah, yes. So much for your plans to have no wedding gifts. Honestly, I think you and Eli made off like bandits."
Elizabeth stared at Malinda. Why was she being so rude? "Would you like to sit now?" she asked in a stiff voice.
"Yes, of course." Malinda came over and took her seat at the table. "This looks very pretty, Elizabeth. Thank you." She made what seemed a strained smile.
"Malinda," Elizabeth began slowly as she filled Malinda's teacup. "Is something bothering you? Have I done something to offend you in some way?"
"No, no, of course not." Malinda reached for her teaspoon.
"Because you just seem a bit unhappy to me," Elizabeth said gently. "As if something is troubling you."
Other than the ticking clock and an occasional sound from the fire, the room grew very quiet. Elizabeth passed the plate of cookies to Malinda and then filled her own teacup with the hot amber liquid.
"I'm sorry," Malinda said contritely. "I suppose that I was being rather persnickety, wasn't I? Please, forgive my bad manners."
Elizabeth made an uneasy smile. "Certainly. But I'm concerned. You don't seem yourself today. Are you unwell?"
"No, my health is perfectly fine." Malinda made a small smile, but her eyes looked sad.
"I know something is troubling you." Elizabeth leaned forward. "You can tell me, Malinda. I'm your best friend."
Malinda set down her teacup with a loud clink then shook her head.
"Is it something with the children? No one is ill I hope..."
"The children are all fit as fiddles."
Elizabeth pondered the situation, wondering how hard she should push. "Does this have to do with Will?"
Malinda nodded sadly. "I'm afraid so."
"Is he ill?"
"No...but it's all coming apart."
"What is coming apart?"
"Everything."
"What do you mean?" Elizabeth peered at her friend with concern. She could see that Malinda was close to tears.
"Will wishes to postpone our wedding."
"Postpone it? Why?"
"Oh, Elizabeth!" She let out a sob. "I am devastated!"
"You poor dear." Elizabeth reached across the table and took her hand. "But I don't understand. Why postpone the wedding? I thought you were both eager to marry before Christmas. And you've made so many preparations."
Malinda retrieved a lace-trimmed hanky from her skirt pocket and dabbed her eyes. "Will is afraid we may have rushed into this without giving it sufficient consideration."
"Oh, dear." Elizabeth bit her lip. She had wondered about the same thing after they'd gotten engaged within days of meeting each other. It had seemed overly fast, not to mention that Malinda had not even been widowed for a year. Naturally, these were opinions Elizabeth kept to herself.
"I will admit that we don't know each other terribly well. But I do love him, Elizabeth. I know that I do. I felt it almost from the start. These are not things one imagines."
"I'm sure you do love him," she agreed. "And that's how I felt about Eli quite early on too. It just took a while for my heart to convince my head."
"So tell me, are you enjoying married life?"
"Very much so." Elizabeth smiled happily. Her first few days of marriage had been wonderful—far better than she'd even imagined. "But we are still newlyweds. I expect we'll have a disagreement before long. But I will not let it trouble me. Two people cannot always agree on everything."
There was another long silence, and Elizabeth regretted boasting about her marriage. Surely that was not what Malinda needed to hear just now. Elizabeth tried to think of a way to encourage her. "Remember how you encouraged me to consider a spring wedding? Perhaps that's what you will have. I can hardly wait to see how beautiful it will be here in the spring. I'm sure the wildflowers must be blooming profusely by—"
"There's more to this than postponement," Malinda blurted.
"What is it?" Elizabeth studied Malinda's troubled expression.
"Will feels he is not suited for pioneering."
"What do you mean not suited?"
"It seems that he's been a failure at cabin building. Even Jeremiah, who is only eighteen, has far better carpentry skills than Will. Lavinia confided that Hugh has been going out to help Jeremiah, and he comes home to tell her that poor Will is all thumbs."
"But that shouldn't matter to you," Elizabeth assured her. "Your house is already constructed. Your barn as well. You even have most of your fences up. Will won't have to build much of anything."
"I told him this very thing." She wiped her nose. "But Will pointed out that he is not a good hunter or fisherman or farmer. He feels that nothing in his life has prepared him for living in the frontier."
"And yet he was determined to come here." Elizabeth remembered when she'd first met the Bostonians—right from the beginning she'd been concerned that they would encounter problems. "I wondered whether the Prescotts and the Bramfords would even survive the Oregon Trail," she confessed. "And yet they did, whereas many did not. Will certainly proved himself a pioneer on the wagon train."
"Yes, but he had help."
"There are those who can help him here." She pointed to Malinda. "You yourself are a fine frontierswoman. You can certainly help him."
"I'm afraid there's more to it than that, Elizabeth." She sighed. "Will sees the Prescotts prospering with their store and their lovely home up above. Meanwhile he and Jeremiah have struggled so hard with their cabin—the cabin that Jeremiah and Mahala were supposed to live in after their wedding. But now Will even questions that."
"What does he question?"
"He's worried that Jeremiah is marrying beneath him."
"Mahala is a wonderful girl."
"Yes...but the Flanders...well, they are a bit rough around the edges."
"They are good people, Malinda. And if Jeremiah marries Mahala—and I hope he does—the Flanderses will be your in-laws too. Well, that is if you and Will marry." She frowned. "You say Will wants to postpone the wedding. For how long?"
"I don't know how long. Truly, Elizabeth, it feels as if he is questioning just about everything. He doubts himself and his ability to be a pioneer. He's worried that his choice to come out here has hurt his children. He frets that he has kept them from having a proper education. He is concerned that we made our decision to marry in haste." She threw up her hands. "Nothing...absolutely nothing seems to be working out for him—or for us."
"It's no wonder you were feeling a bit negative about my house this morning."
Malinda looked around the room and sighed. "Your home is perfectly lovely, Elizabeth. You are so very fortunate to have found a man like Mr. Eli Kincaid. I have no doubt that I am severely jealous. I wanted to find fault...just to make myself feel better. But in all honesty, there is nothing wrong with your dear home. The fault lies in me."
Despite her relief at hearing that her friend approved, Elizabeth felt sad. "What can be done for Will?"
Malinda sadly shook her head. "I don't know. We have talked and talked. It seems to get us nowhere. I suspect that if Will lasts here until springtime, he will be making a plan to move. He has mentioned San Francisco. He feels that attorneys would be welcome there."
"They will be welcome here too," Elizabeth declared. "In time." She almost divulged how she'd been wishing for his legal advice just yesterday, but she knew she couldn't.
"So now, besides feeling heartbroken, I have the additional humiliation of needing to inform everyone that there will be no wedding next week. Do you know how difficult that will be? And I feel so embarrassed when I consider how much trouble I created for you regarding your own wedding." She blew her nose. "And come to find out, my wedding was nothing more than a pipe dream all along. I am such a fool."
"Oh, my." Elizabeth didn't know what to say. "I am so sorry, Malinda."
"You made a lovely tea party for us," she said quietly. "A party that I have completely spoiled."
"You've not spoiled any—"
"Yes, I have! And I should probably just go home because I can't bear to—" She broke into fresh sobs.
Elizabeth went over to wrap her arms around Malinda. "You poor dear," she soothed. "If there's anything I can do to help, I will."
"Just keep being my friend," Malinda said softly. "That is most valuable."
"You know I will." She patted her back as Malinda stood. "Are you sure you don't want to stay and talk about this some more?"
"Will and I have talked it nearly to death." Malinda went for her coat and bonnet. "Mostly I want to just forget about it. Forget about the whole thing. Perhaps a year from now it will be just a distant memory."
"Would you like me to let people know about the postponement?"
Malinda brightened slightly. "Oh, Elizabeth, that would lighten my load considerably."
"I will get right to it. Perhaps I should ask Reverend Holmes to make an announcement on Sunday."
Malinda sighed. "Yes...please do that."
"And I won't give specific details when I tell people," she promised. "I'll just say that you realized you needed more time. After all, having a wedding right before Christmas was quite an undertaking. People should understand the sensibility of waiting."
"Yes. Hopefully they will. But Hugh and Lavinia are aware of the troubles we're having. No need to make up any excuses for Lavinia."
"Do Belinda and Amelia know?"
"I think they have their suspicions. But no, they are in the dark about the wedding getting called off. So are my children. Emily and Susannah will be so disappointed. They were looking forward to it. And they love their new stepsisters-to-be. I feel this is tearing our family apart." She sniffed. "A broken engagement is very much like a divorce."
"Perhaps it is only a postponement," Elizabeth reminded her.
"No...I fear that we are calling it a postponement simply because it feels less painful. But the truth is I do not think Will and I shall ever marry." As she went out the door she started crying all over again.
Elizabeth grabbed her coat and hat and insisted on walking Malinda partway home. She was unwilling to let her friend leave feeling so distraught. Linking arms with her, she was determined to remain with her until she appeared somewhat recovered. Eventually Malinda assured her that she would be fine.
"I just needed a shoulder to cry on," she confessed. "Thank you." She pointed to the sky, which was growing dark and foreboding. "You best get yourself home before it rains, dear one."
As Elizabeth hurried back toward home, she felt very thankful for Eli. Not only was he a dear husband and good father, he was a very capable frontiersman too. In fact, as she went into the house, she wondered if he might be able to help Will Bramford. Perhaps Eli could invite Will to go hunting and fishing with him. He could teach Will some valuable skills and possibly help the frustrated attorney to feel more confident about himself. Perhaps Eli could even ask Will for legal counsel in regard to their Indian friends.
Chapter Twenty-Five
On Saturday morning, Elizabeth and Ruth walked over to her parents' house. Leaving Ruth to visit with her mother, Elizabeth set out to find Brady. She wanted to inform him that his cabin was being inhabited. And she hoped that he wouldn't mind. She found Brady and Asa working on a very similar-looking cabin.
"Good morning," she called out.
They paused from chinking a log to greet her, and after some polite exchanges, Elizabeth asked Brady if she could speak privately with him. But then seeing her father's puzzled expression, she decided to include him. If she couldn't trust her father, she couldn't trust anyone. She quickly poured out the story of Mara and Rose.
"I thought I saw an Indian woman and child," Asa said. "A week or two ago. I mentioned it to Clara, but she thought I'd imagined it."
"You didn't imagine it, Father."
"Do they like my little cabin?" Brady asked.
She nodded. "Very much so."
Brady grinned. "Good. Musta been meant to be."
"Yes. That's exactly what I thought too."
"I never got a chance to make a table and chair for that cabin like I meant to do," Brady told her. "But maybe when I get this one all done, I can make Mara and Rose some furnishings too."
"I'm sure they would appreciate it. But in the meantime, I know they are just happy to be in out of the weather." She looked up at the gray sky. "Speaking of weather, I wanted to make it to town before it rains. I'll let you get back to your work."
"I'll walk back to the house with you," Asa said.
As she and Asa walked, she told him about Malinda and Will. Without going into much detail, she explained about the change in wedding plans.
"That's too bad," Asa said. "I know how much Malinda was looking forward to her wedding."
"Yes. She was fairly devastated." And now, since this was her father and she knew she could trust him, she confided some of the reasons for the "postponement." Her hope was that perhaps he would have some answers for the unhappy couple. "So Malinda feels that if Will felt more confident about hunting and fishing and farming, perhaps he would be more confident about their marriage as well."
Asa rubbed his chin. "I reckon that makes some sense. No man wants to feel as if he can't provide for his family and protect them."
"So I've spoken to Eli about it. And I thought perhaps you and Matthew could extend your friendship to him as well."
"I consider Will to be my friend. But I reckon I haven't reached out to him much. Everybody has been so busy."
"Eli wondered about inviting Will to go hunting with you men."
"That's right—we've been trying to plan our big hunting trip. In fact, Will and Malinda's wedding was complicating our plans. I reckon we don't need to be concerned about that now."
"Would you mind if Will went along too?"
"Not in the least." Asa paused a ways from the cabin. "Before we go inside, I want to talk to you about your Indian friends."
"Yes?"
"Well, I understand why you wanted to tell Brady. It would have been mighty awkward if he'd gone over there for something and found he had squatters."
"Yes. That was why I told him. And I suspected he'd be understanding."
"And I know we can trust him not to speak of it to anyone." Asa frowned. "I assume you don't plan on telling anyone else."
"No." She firmly shook her head. "Besides you and Brady and Eli and me, no one will know."
He put his hand on her shoulder. "That's wise. You do understand that keeping Indians on your land could be dangerous. And not just to the Indians."
She told him what Mara had said about white men burning her house down. "I felt so sorry for her. It was plain to see that she was terrified by the experience."
"She speaks English?"
She filled him in a bit more about the mission and how Charles had been teaching her. "She seems quite intelligent."
"Did she say when Charles is coming back?"
"She doesn't seem to know. But she is expecting another child."
"Oh, my." He shook his head.
She glanced over at their cabin now. "I wouldn't really mind if Mother knew about Mara and Rose. I know I can trust her."
"If you like, I'll explain the situation to her."
"Thank you."
"I expect she'll want to help them too."
Elizabeth nodded. "Well, they can certainly use it." She sighed. "Perhaps I should tell Matthew and Jess too. Just in case they've seen them."
"You know you can trust them not to speak of it. And I know Jess is a very compassionate young woman. She would probably want to help too."
"So feel free to tell them if you see them. And, please, tell them about the wedding postponement too. That is my task today. I'm going to town to let everyone know. I plan to ask the reverend to make an official announcement from the pulpit tomorrow."
"You are a good friend." He nodded toward the house. "I expect your ma will want to walk to town with you. She said she wanted to visit the mercantile."
"Then we had best be on our way." She glanced at the clouds. "Or we may have to stop and take shelter at Malinda's."
"And I'll get back to helping Brady."
"I thought you were going in the house."
He chuckled. "Sorry. I had to make an excuse to speak privately with you about my concerns for your Indian friends."
She nodded. "Well, they truly are friends, Father. They are both very dear."
######
Before long, Elizabeth and Clara and Ruth were on their way to town. As they walked, she told her mother about the postponed wedding.
"Oh, my—poor Malinda. She'd had her heart set on having the wedding before Christmas."
"But Mama said they might have a spring wedding now," Ruth said hopefully. "And there will be lots and lots of flowers then."
"That's true enough."
"So perhaps you can help me spread the word in town," she told her mother. "You don't need to say too much. Just that it's been postponed. That way it won't be so shocking when Reverend Holmes announces it at church tomorrow."
They walked at a fast pace, making it to the mercantile shortly after the rain began to fall. Shaking off moisture on the front porch, Elizabeth greeted Mrs. Taylor, who was just coming out. After exchanging pleasantries, Elizabeth told Mrs. Taylor about the change in wedding plans.
"Oh...well," Mrs. Taylor said. "You know the old saying."
"What's that?" Ruth asked.
"Thus grief still treads upon the heels of pleasure," she recited. "Married in haste, we may repent at leisure." She smiled. "William Congreve wrote that."
"Yes, well, perhaps waiting a bit is the wise route," Elizabeth told her.
As they shopped in the mercantile, the news of the postponement spread quickly. And although some expressed disappointment—ones who had been looking forward to the festivities—most were very kind and understanding.
Evelyn and her cousins Belinda and Amelia were helping to mind the shop, so Lavinia invited Elizabeth and Clara and Ruth to come upstairs. "Elizabeth has seen our home, but you two haven't," she told Clara and Ruth as she led them into the apartment.
"It's so beautiful," Ruth gushed.
"Oh, my." Clara clasped her hands. "I don't feel that I'm on the frontier anymore."
Lavinia laughed. "Yes. That's exactly my intention. Civilization in the midst of the wilderness. Would you ladies like some tea?"
Clara and Ruth both were happy to accept this unexpected invitation, but Elizabeth decided to use this opportunity. "I need to pay a visit to the Holmeses," she told Lavinia as she peered out the window. "And it appears the rain has let up. Perhaps Mother and Ruth could have tea with you while I run over to the parsonage."
"Mama's going to tell Reverend Holmes about Aunt Malinda," Ruth explained.
"Oh, yes," Lavinia said. "By all means go now, Elizabeth. We will be just fine here."
So Elizabeth hurried on over to the parsonage, and since the reverend was out, she explained the situation to Mrs. Holmes. "Malinda would appreciate it if the reverend could make an announcement."
Mrs. Holmes promised to relay the message, and then the two women visited for a while. For some reason Elizabeth felt that she could trust Mrs. Holmes, and although she didn't intend to tell her about the Indians, she wanted to question her about Mrs. Levine. "I know that you and the reverend are close friends with the Levines," she began. "I have only had a few limited conversations with Mrs. Levine, although I feel I have some commonality since I too have been a widow. Are you very close friends with her?"
"Oh, yes. Martha is my very best friend here. I don't know what I would do without her."
"I'm sure you must be a valued friend to her as well." Elizabeth weighed her words. "It must have been difficult for her when her Charles married an Indian woman. I'm sure she must have relied on your friendship then."
"Oh, they are not married, dear," Mrs. Holmes said quietly. "It's not legal, you know."
"Yes, I realize that. But I suspect they consider themselves married, don't you?"
She looked uncertain. "Well, I don't know. That all happened shortly before the reverend and I arrived here."
"So you don't know Charles then?"
"Certainly, I know him. Our families have been friends for quite some time."
"Were you shocked when you heard the news about him?"
"I was a bit surprised. But then I'd heard that happens sometimes. Especially in the frontier, where the white men often outnumber the white women and, well, you understand how it can be."
Elizabeth nodded. "Do you think that's how it was for Charles?"
"To be honest, I don't rightly know. Martha hasn't told me all the particulars. I suppose I've come to my own conclusions."
"What sort of man is Charles? I mean what sort of character?"
Her brow creased. "Oh, I always felt he was a fine young man. Charming and well-spoken and polite. Although Roland always felt that Charles was a bit willful. And he said that Charles had wild oats to sow."
Elizabeth nodded. "What are Mrs. Levine's feelings toward Charles now? Is their relationship broken beyond repair?"
"Oh, no, not at all. Martha loves Charles. Truth be told, I believe he was her favorite son. She calls him her prodigal, and I suspect if he asked to come home, she would welcome him with open arms."
"And his wife and child too?"
Mrs. Holmes put her hand over her lips.
"I take it that his wife and child would not be welcome in her home?"
"Oh, dear. I'm not sure what Martha would do. As a good Christian woman, she might wish to take in the woman and child, but she would be putting herself and her family in harm's way if she did."
"What would happen to her?"
"Goodness, I'm not altogether certain." Mrs. Holmes looked distressed.
"Are there people in our settlement who would make trouble for her?"
Mrs. Holmes simply nodded. "I'm afraid so."
"Would their actions be legal?"
"Legal?" She sighed. "In the frontier, there are laws...and then there are other laws, if you get my meaning."
"I think I do."
Mrs. Holmes peered curiously at her now. "Why are you asking these questions, dear? Why are you so interested in the Levines?"
Elizabeth had prepared an honest answer for this. "Being a newcomer here, I want a better understanding of my community," she explained. "I know that the Indians have been removed from these parts. And I'd heard bits and pieces about Charles Levine. I was simply curious as to the local sentiments."
"The local sentiments are divided. There are a few settlers who felt that removing the Indians was unfair and unkind, but they are not a majority. They believed that the Indians, for the most part, were peaceful—just like we believe that the settlers, for the most part, are peaceful. However, it only takes a handful of angry men to stir things up. And that is what happened."
"Would you and the reverend be in the minority of folks who hold no ill feelings toward the Indians?"
"We do believe that the Indians, like us, were created by God—in his image. Certainly, they are different from us. But they are not savages like some people claim."
"I'm relieved to know that, Mrs. Holmes."
"But be assured, this is not something that you will hear the reverend mention from the pulpit," she said firmly. "Don't expect him to."
"Why not?"
"He tried to voice his concerns. He'd seen and heard some things that were disturbing—some of it was related to Charles and his Indian woman. Roland preached on the subject shortly after our arrival here." She sighed. "Unfortunately, his sermon led to some severe disagreements, and the church nearly came to an end."
"I see."
"It has been a source of frustration for him."
"I can imagine."
"I hope that you will keep my confidence in these matters." Mrs. Holmes' brow creased. "I don't usually speak so freely about such things. And Roland might not approve."
"You can trust me to not say a word." She eyed the older woman. "I feel certain I can trust you too."
She nodded. "If you would like to become better acquainted with Mrs. Levine, I would be happy to arrange for us to meet. Perhaps you could come for tea."
"Or perhaps you and Mrs. Levine would like to come to my house for tea."
Her eyes lit up. "Oh, my. That would be delightful."
And so they set a date. The two women would come for tea on Tuesday morning at ten. "Unless it is raining," Elizabeth said. "In that case, I will understand."
"Then I shall pray that Tuesday breaks with no rain."
"And if it does rain on Tuesday, why don't we plan on Wednesday, whichever day is fair?"
"Yes." Mrs. Holmes smiled. "I like that plan."
"And speaking of rain, I see that the sun is out. I think I'll make a run for it and gather up my mother and daughter and hurry on home."
As Elizabeth went back to the mercantile, she wondered what she was getting herself into. Did she really intend to let Mrs. Levine know that her granddaughter was living nearby? No, Elizabeth decided, she could not betray Mara like that. First she would have to become thoroughly acquainted with Mrs. Levine. She would have to find out the woman's true character. Then she would try to do whatever was best for Mara and Rose.
Chapter Twenty-Six
The ladies were unable to come for tea until Wednesday. But thankfully, the day broke sunny and clear, and at just a bit past ten, Mrs. Holmes and Mrs. Levine showed up at her door.
"Welcome," Elizabeth said warmly. "Isn't it a fine day we're having?"
"It most certainly is," Mrs. Holmes said as she removed her bonnet.
"I hardly needed my shawl," Mrs. Levine told her.
"I cannot believe it's December," Elizabeth said. "It feels more like springtime to me." The three women made polite small talk as Elizabeth showed them through her home. Both of her visitors were suitably impressed. After Malinda's moody evaluations the previous week, Elizabeth appreciated her guests' kind and appreciative comments. Soon they were seated at the table, which was set just as grand as it had been for Malinda, and Elizabeth was pouring tea.
"It's such a pleasure getting to know you better, Mrs. Levine." Elizabeth handed her the plate of sugar cookies that she and Ruth had made the other day.
"Please, dear, if I am to keep calling you Elizabeth, you must call me Martha." She smiled. "After all, this is the frontier. We don't have to be overly prim and proper out here."
"And if you're going to call her Martha, it's about time you started calling me Georgia," Mrs. Holmes said.
"Thank you."
"I told Martha that you were curious about her son Charles," Georgia said a bit carefully.
"And naturally that aroused my curiosity," Martha admitted. "But Georgia assured me that you had no ill motives."
"Not at all," she told her. "In fact, you and I have some commonalities I believe. First I thought it was because we were widows, but when I learned more about Charles, well, I felt that I could take you into my confidence." She glanced at Georgia. "And I know I can trust you as well."
"Certainly." Georgia nodded.
Elizabeth had told Eli her plan several days ago, and he had given her his blessing to tell his story. "You see, my dear husband has had an experience similar to your son Charles," she said to Georgia. "And I felt it might comfort you to hear about it." And now she told them the story of how Eli had rescued the Indian woman and how her people had helped him and how Eli had fallen in love and married the woman.
"My goodness," Georgia exclaimed. "I had no idea."
"And they had a child," she told her spellbound guests. "A son." She described how happy the young family had been, how much at home Eli had felt with her people. And finally she explained how the mother and child had both died from a white man's disease. "Eli was brokenhearted over it."
"Oh, my." Martha wiped her damp eyes with her handkerchief.
"Such a sad tale." Georgia sniffed.
"Now I wouldn't share this story with just anyone," Elizabeth said to Martha. "But after hearing more about Charles and how much you love him, I felt that you especially needed to hear it."
Martha nodded. "I do believe I did need to hear it. Although it's a sad story, it does give this mother's heart hope. I admire your husband, Elizabeth. If he made a mistake like that but is living such a fine life now, well, it gives me hope for Charles."
Elizabeth bit her lip. This was not exactly the response she was looking for. "Eli did not consider his first marriage to be a mistake, Martha. He loved his first wife."
"Oh, well, yes...I suppose he thought he did. But don't you think his life would have been mighty complicated—wouldn't it have made things more difficult if his wife and child had survived?"
"Certainly, it would be more difficult for me." Elizabeth frowned. "But that's not my point. What I'm saying is that Eli is a good man. And I believe your Charles is a good man too."
"Yes, yes, he most certainly is," Martha assured her. "He was always a good boy. A little lively at times, but he had a good heart."
"And marrying an Indian woman doesn't make him less good, does it?"
"That all depends on who you're talking to." Martha shook her head. "Some folks round here would just as soon a man marry his horse rather than an Indian woman."
"Oh, Martha." Georgia shook her head with disapproval.
"It's true and you know it." Martha frowned. "And some folks have good reason to dislike Indians." She launched into a horrible story about how the Indians staged a brutal attack in the Rogue River War. "And that was just a couple of years ago."
"And wasn't that why the Indians got moved up north?" Elizabeth asked. "Although they didn't just move the warriors, but innocent women and children as well."
"That's true," Martha confirmed. "But I'm sure the women wanted to be with their men. But for all we know some of those same violent Indian warriors could still be lurking anywhere—even in these parts. I don't think the trouble with the Indians is over by any means."
Elizabeth studied Martha. She didn't seem to be a hateful woman, but she was apparently overly fearful of Indians. And yet how could she possibly be afraid of anyone as harmless as Mara or Rose? "Your son has a child, doesn't he?"
Martha made an uncertain nod. "Yes...I heard there was a baby, although I've never seen it. To be honest, I'm not sure I would want to see it."
"Your own grandchild?"
She twisted her handkerchief. "I just don't see what good could come of it."
Georgia patted Martha's hand. "You are caught between a rock and a hard place, my dear."
"I'm sorry to be so persistent about this," Elizabeth said as she refilled their teacups. "More than anything, I want you to realize that you are not alone. Eli and I have great compassion for you, Martha. And we want you to know that we are your friends. And if Charles should ever return from—from wherever he has gone—we would treat him as a friend as well."
Martha smiled. "That is a great comfort to me, Elizabeth. Thank you."
Feeling a bit guilty for being so intrusive, Elizabeth was determined that the duration of their tea party would remain cheerful and uplifting. They shared funny stories about their children and compared ailments and just laughed like girls together. And by the time the ladies left, Elizabeth felt certain that a strong friendship was being forged. However, she did not feel certain that Martha would welcome an Indian daughter-in-law and granddaughter.
######
The week before Christmas, the men planned to take off for a few days of hunting. But the night before they left, Eli shot a deer, leaving the best cuts of meat for Elizabeth and the children. "If you don't mind, I will take the rest of the meat as well as the skin to Mara and Rose."
"Thank you," she told him. "That will take a load off my mind."
The next morning, Eli went over to her parents' well before daybreak in order to help Asa pack the wagon and hitch up the oxen. Then, according to Clara, while it was still dark, the five hunters—Eli, Asa, Matthew, Brady, and Will—set out.
"I was so surprised when I heard that Will wanted to go hunting with your fellows," Malinda told Elizabeth. The hunting "widows" had gathered at Malinda's house to work on a quilt that was meant for Mahala and Julius' wedding. Thankfully, the young couple had dug in their heels, making it clear to everyone that their wedding was still scheduled for the first Saturday of February.
"Eli said they were grateful to have Will join them," Elizabeth told Malinda. "Last week, Eli scouted a nice herd of elk, and he hopes they'll bring home two cows. According to Eli, that's a lot of meat to pack back to the wagon. Will's muscles will come in handy."
"I appreciate that there are some young men along," Clara said. "Asa thinks he can keep up, but he's getting older. He needs to slow down."
"I feel like I'm slowing down," Jess admitted. "I've been so sleepy lately. Sometimes I feel like I can barely keep my eyes open."
"That's because of the baby," Elizabeth told her. "I had the same thing. Not when I was carrying Ruth so much, but with JT. Of course, maybe I had it with Ruth, but I was so busy chasing after JT that I didn't notice."
Now they all started comparing pregnancy and birthing stories. Malinda, having birthed four children, seemed to have the best ones. Or perhaps she just was a better storyteller. "Unfortunately, we don't have a doctor nearby," Malinda told them. "But I've helped with a number of births."
"Mother is a good midwife," Elizabeth said.
"And she's promised to help me when my time comes." Jess looked at Clara. "Right, Ma?"
Clara grinned. "You can count on it."
"I'm not as experienced as Mother—unless you're birthing animals." Elizabeth laughed. "But I'm happy to help too, Jess."
"And if by some weird stroke of bad luck they are both unavailable, you tell Matthew to come and fetch me," Malinda told her. "I'd be happy to deliver your baby."
Jess smiled. "I feel very well taken care of."
"Do you think Ruby and Doris will come in time for your birth?" Clara asked.
"I just got a letter from Ruby," Jess told her. "They don't expect to make it here until early fall next year. Her plan is to keep the business in Empire City going throughout the summer because so many settlers come through there. But in midsummer she expects to have enough profits to send out a carpenter to begin building an inn right next to the mercantile."
"Wonderful!" Malinda said happily. "Our town is growing by leaps and bounds."
Elizabeth was relieved to see Malinda in such good spirits today. She had been worried that her friend would be fretting over the wedding that never happened last Saturday. That was the very reason she'd suggested that they gather like this while the men were gone.
"I'll bet you all have plans for Christmas," Malinda said. "But I've been daydreaming about a little get-together I'd love to have here."
Elizabeth exchanged glances with her mother. Their plan was to have their family at Elizabeth and Eli's house. They hadn't invited Malinda because they weren't sure about the Bramfords and Prescotts, and Elizabeth knew her house wasn't big enough to contain all those people for that long.
"What sort of get-together?" Clara asked Malinda.
"I thought it would be lovely to have everyone here on Christmas Eve. I imagined a big bonfire, and we could have music and dancing in the barn. I suppose it's because I'd been dreaming of our December wedding," she confessed.
"I think that sounds like fun," Elizabeth said. Fortunately, their plan was to gather on Christmas Day. "I know the children would love it."
"So would Matthew," Jess told them. "He had been so looking forward to the...well..." She grimaced.
"The wedding," Malinda finished for her. "It's all right. You can say it. I'm resolved to my fate by now. You know what they say, the best laid plans of mice and men—and I suppose we could add overbearing brides to the list."
They all laughed.
"I'll promise you all this," Malinda continued. "If it turns out that Will and I do decide to get married—which is probably not going to happen, but if it does—I will try to follow Elizabeth and Eli's example."
"What example?" Elizabeth asked.
"Being more concerned about the marriage than the wedding."
"Oh." Elizabeth chuckled. "Well, a wedding is one day. A marriage, hopefully, is forever."
"Do you think the men will encounter any wild cats or wolves or bears where they are hunting?" Jess asked.
"If they do, you probably don't need to be worried," Elizabeth assured her. "Besides, I know Eli wouldn't mind bringing some skins or furs home. He's almost got that cougar skin ready to become a rug." Of course, as soon as she said this, she regretted it. The rug was meant to be a Christmas present for her parents, to go in their bedroom, to help cover their hard-packed dirt floor. So now she quickly changed the subject. "What can we bring to your Christmas Eve party?" she asked Malinda. And suddenly they were making plans for what was sure to be a fun evening for all.
######
Elizabeth tried not to feel worried when Eli didn't come home on the day she expected him. She assured JT and Ruth that it was probably something as mundane as a broken wagon wheel. "Or perhaps they were having such a good time, they decided to stay an extra day." And she couldn't blame them because the past two days had been exceptionally good weather. But when she finally blew out the lantern and went to bed, she felt a little concerned. And as usual, she prayed for the safety of all of the hunters.
"What if they don't get home in time for the Christmas party tomorrow?" JT asked as he picked up his guitar case the next morning.
"I'm sure they'll be home by today," she told him.
"In time to come to our Christmas program at school?" Ruth asked hopefully.
"I'm not sure about that." She tweaked a braid, adjusting the red bow that Ruth had insisted was necessary because they were doing their program. "But I am sure that Grandma, Aunt Jess, Aunt Malinda, and I will all be there. And I'm also sure that if you don't get going, you will be late for school."
After the children left and after she finished with chores as well as delivering some food to Mara and Rose, Elizabeth focused her attention on finishing some Christmas presents for the children and Eli. For Ruth, she had sewn a school dress from some of the green calico she'd brought from Kentucky. And then she'd made a matching dress for Ruth's doll. For Eli and JT she had sewn matching chambray shirts in a nice shade of blue. Just as she finished the last buttonhole on JT's shirt, it was time to go hitch up the team. She had promised to give her mother and Jess and Malinda a ride to school. She hoped that perhaps the men would be arriving at her parents' house by now, but to her dismay, there was no sign of them.
"You don't think anything went wrong with the hunters, do you?" Elizabeth asked Clara as she sat down on the seat next to her.
"I expect they've had good luck and it's just taking longer."
"Or they might have had a breakdown on the wagon."
"Your brother took tools. If they break down, he will fix it."
"Yes. I'm sure you're right." Elizabeth saw Jess waiting for them on the edge of her property. "And I won't act worried around Jess."
Before long, they were standing in the back of the school with the rest of the parents, watching as Mrs. Taylor led the children in a number of lively songs. Some, like JT, had instruments to play, and all in all, it was quite a nice production. But Elizabeth couldn't have been prouder than when Ruth sang "O Come, All Ye Faithful" accompanied by JT. It was so beautiful that tears filled her eyes. She wished Eli could hear it. Malinda's girls along with Will's older daughters sang a lovely rendition of "Hark! The Herald Angels Sing," and some of the older boys bellowed out "Joy to the World."
For the finale, all the children joined in to sing "Deck the Halls," and then the parents were treated to goodies that were provided by Mrs. Taylor, the Holmeses, and Prescotts' Mercantile. All in all, it was a very nice program. Considering they were just a little one-room frontier school, Elizabeth was impressed.
"We are so fortunate to have you," she told Mrs. Taylor afterward. "If you don't have plans for Christmas, we would love to have you join us."
"Thank you," Mrs. Taylor smiled happily. "But I promised to spend Christmas at the Levines. Reverend and Mrs. Holmes will be joining us."
Elizabeth squeezed her hand. "Merry Christmas!"
After visiting with the other parents and explaining more times than she cared to about why Eli and the other men were missing, Elizabeth suggested that it was time to go home. "Those clouds look like they could let loose before long," she told Malinda. "And we'll have a heavy load of people to get home."
The children insisted on singing Christmas songs in the back of the wagon. And for a while their merriment seemed to displace her concerns about the missing hunters.
"Susannah and Emily want to know if I can stay overnight at their house tonight," Ruth said when they reached Malinda's house. "There's no school in the morning."
Elizabeth glanced at Malinda. "Is that all right with you?"
"Sure. I'll put her to work helping prepare for tomorrow's party."
"What about JT?" Bart protested. "If Ruth gets to stay, can JT stay too? Please, Ma?"
"I'm fine with that," Malinda said as she climbed down.
"Ma?" JT asked hopefully. "Do you mind?"
Elizabeth made a tolerant smile. "Not at all. But both of you mind your manners and help Aunt Malinda."
They assured her they would, and suddenly it was just her and Clara and Jess. "Do you ladies want to keep singing?" Elizabeth asked tentatively as she drove the team on toward home.
"Not particularly," Jess answered solemnly.
"Yes, I think I've had enough music for now," Clara said. "Although it was a lovely program. And when JT and Ruth sang "O Come, All Ye Faithful"...oh, my. That was so beautiful. JT is such a musician and our dear Ruth has the sweetest voice."
"I just wish Eli and Father and Matthew could have heard it." Elizabeth sighed.
"Perhaps the children can do it again on Christmas Day," Clara suggested.
"Yes." Elizabeth nodded. "Of course."
"Do you think the men are all right?" Jess asked suddenly.
"Oh, sure," Clara told her. "I expect they're home by now. Probably dividing up the meat."
They speculated on this for a while, but when they reached Clara's there was no sign of Asa's wagon. "I'll walk from here," Jess told Elizabeth as she climbed down with Clara. "You better hurry and get that wagon home before the rain comes."
"Thanks," Elizabeth called out. "I don't want to get stuck in the mud again." She waved to them, and trying not to feel too dismayed over the missing men, she kept the team moving quickly, getting home just as the sky opened up.
By the time she'd unhitched and tended to the team, checked on the rest of the livestock, and gotten the hens safely into the henhouse, she was soaked to the skin. At least she had a safe and dry house to go to, albeit a house with no children and no Eli. But she did have a dog. "Hurry up," she told Flax as they were going inside. "Let's get ourselves dry."
She tried not to worry about the hunters as she busied herself making a fire in the cookstove and quickly changing out of her wet clothes. Since she was home alone, she simply put on her nightie and a shawl. But she missed having her moccasins. She brewed a pot of tea and then made herself some porridge. Then, to help take the chill off her and the house and to help her wet clothes to dry, she made a fire in the fireplace.
She tried to stay busy, but her eyes kept wandering to the window. But the darkness came quickly, and it wasn't long until all she could see outside was inky blackness. And now her imagination started to run away with her. As she pulled the curtains closed and returned to her chair by the fireside, she was gripped with an oppressive sense of fear. She tried to focus on the baby blanket she was knitting for Matthew and Jess' baby, but she kept dropping stitches and finally set it aside.
Something was wrong. She knew it. The men would not purposely stay away this long. Not unless something was wrong. She had already considered the various possible reasons that the men were late coming home. She wasn't all that concerned about wild animals. She knew that Eli was savvy about hunting in the wilderness and that he was a good shot. A wagon breakdown could explain a delay. And Eli had told her that because of the terrain, they could only take the wagon partway to the area he planned to hunt. But most breakdowns were repairable within hours, so that would only slow them by a day at most.
The next explanation that came to her was sickness. She knew this was only because of what her family had suffered from cholera back in 1853. She had not heard of any cholera outbreaks in this part of the country, but she knew that could change in a moment—and that cholera could come on quickly and without warning. But finally she had to assure herself that her family was unlikely to be ravaged by cholera again. Wasn't that a bit like lightning striking twice?
And so the peril that concerned her more than anything else seemed to be Indians. Fueled by Martha Levine's bloody tale about the Indian attacks on the Rogue and at Battle Rock, it was easy to imagine a band of displaced warriors hiding out from the army. And how outraged might they be, knowing their women and children had been forced off their land by the invasive white men? In all fairness, it would be a righteous anger too. But what would be the dire consequences if these incensed warriors encountered five white hunters invading their territory and killing their game? She couldn't bear to imagine. All she could do was pray.
Chapter Twenty-Seven
Elizabeth was still praying by the fireside when she heard the clunk of the door's latch. Jumping up and just inches from grabbing the old shotgun hanging above the fireplace, she watched as the door swung open—and Eli walked in.
"Eli!" she cried out, running to him. "You're safe!"
"Shhh," he hushed her as he held her in his arms. "The children."
"The children are at Malinda's," she said, choking back sobs of relief. "I was so worried."
"I'm sorry." He smoothed her hair. "Oh, Elizabeth, it's so good to be home."
"Welcome!" Regaining composure, she stepped back and looked at his face. He had the beginning of a beard and a happy grin. "Are you hungry?"
He nodded, unlacing his soggy buckskin shirt. "And wet. And cold."
She hurried over to the fire, tossing a couple more logs on. "Get undressed here by the fire, and I'll bring you something dry to put on."
She bustled about, happy to be tending to her husband, happy to have him home. Before long he was in dry clothes and sitting by the fire with two fat slices of bread and butter and jelly and a cup of coffee. "Your eggs are almost ready," she called out.
"Eggs?" He sounded disappointed.
"You don't want eggs?"
"What about the chicks?"
She laughed as she spooned the scrambled eggs onto a plate. "We already saved the fertile eggs—nineteen altogether. The hens are sitting on them. We won't get any more fertile eggs until we have our own rooster." She handed him his plate. "Now I want to hear the whole story, from beginning to end, and try not to leave anything out."
Between hungry bites, Eli explained why their hunting party was late. "It started out to be an uneventful expedition. We got the wagon up to the campsite just like I planned, and the weather was with us. On the first day your father shot a bull elk." He took a sip of coffee. "We ate like kings that night. The next day we were hunting for a cow because the roasts are so good, but no luck. So on the third day we split up. Brady seemed worn out, so Asa suggested he stay at camp and be our cook. Your pa went with me, and Matthew and Will set out together. I got a real nice cow early in the day, but it took Asa and me until midday to pack it back to camp. I left Asa and Brady to tend to the meat, and I went back to where I'd seen sign of turkeys. By the time I got back it was close to dark, but I got three turkeys." He paused to eat his last bite of eggs and then looked at her.
"Still hungry?" she asked.
"We haven't eaten since breakfast."
"You keep telling me your story and I'll make you some more eggs."
"So night came on, and it was just Asa and Brady and me at camp. None of us had seen hide nor hair of Matthew and Will. And we didn't let on, but I know we were all feeling a mite concerned."
"Oh, my." She cracked an egg. "Were they all right?"
"You said to tell the story from beginning to end," he reminded her. "So it was dark and late, but I decided to set out with a lantern to look for them. I knew which direction they'd taken, but I finally realized it was useless and returned to camp." He paused to refill his coffee cup.
"Oh, dear." She handed him another piece of bread with butter and jam.
"The next morning, the day we'd meant to go home, Will and Matthew weren't back. So Asa and I set out on horses to look for them at daybreak." He shook his head as he chewed. "We all knew that Will was the least experienced outdoorsman, and I couldn't help but think maybe he'd gotten lost or hurt or something."
"Yes." She nodded as she put the second portion of eggs on his plate. "I was just thinking that too."
"Your pa and I were calling and whistling, and I'm sure we were both praying, but it wasn't until midday that we found them."
"Oh, good." She sat down in the chair across from him.
"Your brother had fallen down a small ravine and broken his leg. Will was carrying him out."
"Oh, dear!" Her hand flew to her mouth. "Poor Matthew. Is he all right?"
"He's fine. Well, excepting for that broken leg. But thanks to Will, it looks like he will heal up just fine."
"Thanks to Will?"
"Turns out that Will knows a little about doctoring." Eli grinned. "Now, bear in mind that Will made us swear to secrecy, but I think I can trust you with this."
"Of course." She leaned forward with interest.
"Seems that Will started out to be a doctor. But when he was studying medicine, he discovered that he didn't have the stomach for working on cadavers."
She grimaced. "Can't fault him for that."
"So he switched from medicine to law. But apparently he had enough medical training to know how to set a bone and make a splint. Not only that, but Will shot a couple of rabbits to roast for their supper the previous night—otherwise they'd have gone hungry. Matthew swears that Will saved his life."
"Thank goodness for Will."
"So we loaded Matthew on a horse, but because he was in pain, it was slow going. By the time we reached camp it was too late in the day to head out. So this morning, we ate a hasty breakfast and broke camp first thing. I could tell the weather was about to turn, and I wanted to get back before it let loose. We were making good progress. But then just as we were crossing a creek, the axle broke."
"Oh, dear."
"Unfortunately our best mechanic was indisposed. So with Matthew's guidance I was doing my best to repair it, but I have to say that Will stepped right up, and if it hadn't been for his help, we'd probably still be out there."
"Did you tell Will that?" she asked eagerly.
"Sure did. We all did. In fact, by the time we made it back to Asa's, we all decided to adopt Will into the family." He gave her a sleepy smile. "You all right with that?"
She laughed. "That's fine with me. I just hope that it makes him want to marry Malinda now."
Eli let out a yawn, and she took him by the hand. "If there's any more story to tell, it will have to wait until morning," she said firmly. "It is time for bed, my dear."
"You won't get any argument from me."
######
The gray dawn light was already coming in the bedroom window by the time Elizabeth awoke, but Eli was already up. She dressed quickly, hurrying out to get a fire going and the breakfast started, but when she got out there she discovered the cookstove was hot and a pot of coffee was sitting on the soapstone. She set out some breakfast things, and then, remembering that Ruth was not home, she went out to tend to the chickens. It was still too early for eggs to hatch, but it wouldn't be long. According to her calculations, a few days after Christmas the first chicks could begin to hatch.
As she returned to the house, she spotted Eli and Flax coming toward the house. Eli was carrying something big and bulky. She paused to see better and realized it was an evergreen tree.
"Merry Christmas," he called out as he got closer.
"A Christmas tree!" she exclaimed as she went out to meet him.
"I thought we could surprise the children."
"Perfect." She peered at the tree. "And that is a beauty."
He leaned over to kiss her, and she stroked his smooth cheek. "You appear to have found your razor."
He chuckled. "Well, after all, it's Christmas Eve."
"Speaking of that..." As they walked to the house, she told him about Malinda's Christmas Eve party. "She was hoping you would bring your guitar tonight."
"I'm happy to."
"I expect poor Matthew won't be able to play fiddle now."
"It's his leg that's broken, Elizabeth, not his arm."
While Eli made a stand for the tree, Elizabeth fixed breakfast. Then, just as he carried the tree into the house, she set a stack of pancakes on the table. "I thought you could put the tree right here," she pointed to the center of the room. "That way we can admire it from every direction."
After breakfast and chores, Eli drove their wagon over to her parents' to help Asa with the meat as well as to bring some home. Most of it would be smoked, but some would be shared with friends in the community, and the best section of roast as well as a turkey would be used for their Christmas dinner tomorrow. While Eli was gone, Elizabeth looked at the Christmas tree. As pretty as it was, it seemed to be in need of some decorations, but all of hers had been left behind in Kentucky. She went to her trunk and her sewing basket, and before long she had crafted a few things. And she left some materials out for when Ruth got home because she knew she would want to help. She also took time to wrap the presents she'd recently finished, placing them under the tree.
JT and Ruth got home in time for dinner and, as expected, they were both delighted to see the Christmas tree. However, they were even happier to see Eli. After they all ate tender elk steaks and fried potatoes together, JT went out to help Eli with chores, and using fabric, paper, ribbon, sticks, and string, Ruth and Elizabeth proceeded to decorate the tree.
When it was time to get ready for the Christmas Eve party, Elizabeth decided she and Ruth would wear their wedding dresses. And why not—this was a time to celebrate. The men had made it safely home, and this was their first Christmas in Oregon. Why shouldn't they all kick up their heels tonight? She was only slightly disappointed when she realized that because of the slate-colored afternoon skies, they would not be taking the wagon and team to Malinda's. In all likelihood they would be walking in the rain.
"Perhaps we should wear our old dresses to walk in," Elizabeth said to Ruth as she watched the rain pelting against the big window. "Maybe we could change at Malinda's."
Just then Eli came into the house. "I forgot to tell you that your pa left the cover on their wagon, and he's double teaming just like we did on our hunting trip. He wants us to all ride together over to Malinda's."
Ruth clapped her hands. "And we will sing all the way!"
Within the hour, she and the children were all bundled into the back of Asa's wagon. Jess and Clara had brought blankets and quilts to keep everyone comfortable. And Matthew, with his leg wrapped in wooden splints and resting on a pillow, seemed in fairly good spirits. "Pa and Brady made me a set of crutches," he told them.
Up front, Eli was sitting with Asa, and Brady, bless his heart, had insisted on helping as well. The plan was to deliver the family members to the barn, and then Brady would park the wagon around back. After that Brady planned to tend to the teams and stay with the wagon.
"But you could play music with us," JT told him when it was time to unload.
"Yes," Ruth insisted. "Please, come to the party, Brady."
He smiled at them. "Truth be told, I'd druther stay out here and take a nap in the wagon. I'm plum wore out from hunting."
Asa slapped Brady on the back. "That sounds like a right smart plan. When I bring out your plate of food, I might just take a nap out here myself."
The barn was well lit and bustling with guests by the time they got inside. A few musicians were just starting to warm up, but the music became much more lively after JT and Eli pulled out their guitars. Then, after Matthew got situated on a chair with his leg resting on a crate and tuned up his fiddle, the music got into full swing.
"I'm so relieved to see the hunters are home," Malinda said as she joined Elizabeth.
Elizabeth stared at her friend in wonder. Dressed in the pale green gown that had been meant for her wedding, Malinda's dark hair was piled on her head, and her pearl drop earrings glistened in the lamplight. "Oh, Malinda, you look as beautiful as a portrait!"
"Really?" Malinda frowned down at her dress. "Just this morning, Ruth told me that green was not the best color for a wedding dress. She said green meant 'ashamed to be seen.'"
Elizabeth made an uneasy laugh. "Oh, well, that's a silly old rhyme."
"Maybe. But I realized that I truly had been ashamed to be seen. As you know, I didn't show my face in church last week. But this morning I thought, not only am I not ashamed to be seen, but I shall wear my wedding dress tonight. And I shall dance to my heart's content!"
"Good for you." Elizabeth patted her on the back. "And I suppose you've heard about Will's role in the hunting trip by now."
Malinda's eyebrows arched. "No...I haven't seen Will. Is he all right?"
"Yes. He's fine—"
"Come on, Ma." Bart grabbed his mother by the hand. "You promised to dance the first dance with me. Let's go."
Malinda laughed and waved as she and Bart hurried over to join in a circle dance that was just starting. Elizabeth went over to where her mother and Jess were standing on the sidelines, tapping their toes to the music and watching as merrymakers started the dance.
"Malinda hasn't seen Will yet," Elizabeth told them. "I wonder if he is even coming tonight."
"Maybe he wasn't invited," Jess said. "After all, Malinda made plans for this party while the men were out hunting."
"But wouldn't Jeremiah have told his father?"
Clara shook her head. "I expect we will have to wait to see."
"We don't have to wait long." Jess nodded toward the door.
Elizabeth glanced over her shoulder to see Will, dressed in party clothes, entering the barn behind the Holmeses and with Mrs. Taylor by his side. He looked around the barn, and then his eyes stopped as he spotted Malinda dancing with her son. And that was when he smiled. It was such a nice smile that it made Elizabeth giggle. Will still loved Malinda! She knew it.
When the dance ended, Will went over to greet Malinda, and the two of them disappeared. However, as Elizabeth took a turn dancing with Bart, she was not concerned. After a couple more dances, Will and Malinda reappeared in the barn, but now they went their separate ways. Malinda headed for the musicians, and Will, with a serious expression, returned to where the Holmeses and Mrs. Taylor were standing near the door. Was he going to leave and go home?
"Excuse me, folks," Eli said loudly from where he was standing with the other musicians. "We have an important announcement to make. If you could all gather round and give us your attention for a moment." Malinda was nowhere to be seen now. Had she gotten upset? Had she decided to put an end to this celebration? Surely not.
The partiers, caught off guard, grew quiet and looked at Eli. Meanwhile, Will was leading Reverend Holmes through the crowd. Once they got up front with the musicians, Will said something to Eli and then turned to face the guests.
"I have an announcement," Will spoke in a loud and clear voice. "As you all know, we had intended to hold a wedding here just one week ago. But that wedding was postponed. Now, if you would oblige us, we would like to hold that wedding tonight."
The surprised crowd began murmuring among themselves as Eli started to lead the musicians in playing a song—not one with a lively dancing tempo, but a quieter piece that seemed suitable for a wedding. At least for a barn wedding. And now Malinda, escorted by her older son, Todd, was coming forward and wearing the sweetest smile ever.
Reverend Holmes appeared to be prepared for this unexpected occasion, and opening his Bible, he proceeded to perform a simple ceremony. It wasn't long until the happy couple was repeating their wedding vows, and then the reverend pronounced them man and wife and the whole barn erupted into cheers and clapping.
Toasts were made and congratulations were shared, and the barn grew noisier and happier than ever. Young couples crowded around to dance while the older ones went to the house to fill their plates. All in all it was one of the merriest Christmas Eves that Elizabeth could remember.
Chapter Twenty-Eight
On Christmas Day, Elizabeth's family gathered at her house as planned. They opened presents first, and for a while the room was noisy and lively, but it slowly quieted down. And compared to the previous evening it was a calm celebration. Clearly everyone was a little worn out. Especially the men, still recovering from their hunting excursion. But Christmas dinner was tasty, and no one went hungry. Afterward, Ruth and JT performed "O Come, All Ye Faithful" for the family, and then they all sang some familiar Christmas carols together. Finally Asa opened the Bible—but then he handed it to Brady.
The room grew quiet as Brady cleared his throat. Haltingly but clearly, he began to read. "And it came to pass...in those days, that there went out...a decree from..." But he had to stop when he reached the name Caesar Augustus. Smiling shyly, he handed the Bible back to Asa while everyone clapped. And then Asa continued with the beloved story of the first Christmas.
######
Just a few days after Christmas, Ruth burst joyously into the house. "Mama, Mama, we have four baby chicks," she declared. "Four!"
Elizabeth paused from stirring biscuit batter. "That's wonderful, Ruthie. Do they appear to be in good health?"
"Yes. They are so sweet, Mama. Can I bring them in the house to show you?"
Elizabeth thought. "Perhaps not. But I will go out and see them right after breakfast."
When Elizabeth went out to survey their increasing flock, she could see that Ruth was right. All the chicks appeared to be sound and healthy. "Hopefully this is just the beginning," she told Ruth.
Seeing the baby chicks reminded Elizabeth that she needed to check on Mara and Rose. Between Christmas festivities and having the children on break from school, it had been difficult to slip away the last few days. But today JT was out helping Eli build the fence, and Elizabeth knew the perfect excuse to distract Ruth for an hour or so.
"We really have more eggs than we need," she told her. "And I don't plan on going to the mercantile until Friday. Why don't you take this basket over to share with Grandma and Grandpa? And perhaps you could take half of the eggs to Uncle Matthew and Aunt Jess. And of course, you will have to tell everyone about the new additions to our farm family."
Naturally Ruth was delighted to run this errand. She was barely out the door when Elizabeth began loading some food staples as well as the remainders of an elk roast into a basket. She hurried over to the cabin, but just as with each other time she'd gone, she felt uncertain as she went. What would happen if some of their Indian-hating neighbors found out? But as always, when she saw Mara's face as she opened the door, her fears evaporated.
"Come in," Mara insisted.
"Hello." Elizabeth stepped inside.
"Hello!" Rose declared loudly.
"Don't you look pretty!" She leaned down to stroke the little girl's head. Today her brown hair was smoothed into two tidy braids. Mara had obviously put the comb Elizabeth had given her to good use. She stood and began to unload the basket into Mara's arms and onto the crate that Mara was using as a small table. After the basket was empty, Elizabeth looked at Mara. "How are you?" she asked looking down at Mara's well-rounded tummy. "How is the baby?"
She rubbed her belly. "Good. We are good."
"Good. I'm glad to hear it."
Mara smiled faintly. "He is boy child."
Elizabeth blinked. "How do you know?"
"He kick like warrior."
"Oh." Elizabeth laughed. "It was my girl child that kicked most."
Now Mara laughed.
Elizabeth looked around the crowded space. "Do you need anything?"
Mara pressed her lips together then nodded. "Knife."
Elizabeth frowned. She had given her a knife already, though not a very good one. "I thought you had a knife."
"Yes." Mara nodded firmly. "I want sharp knife. Hunting knife."
Now Elizabeth felt slightly alarmed. Why would Mara need a sharp hunting knife? Surely she didn't plan to hurt anyone.
Now Mara went over to a corner of the room, picking up a skin of some sort. "I want to cut." She held it up. "To make moccasin for Rose."
"Oh." Elizabeth nodded eagerly. Now she understood. "Yes. That's a good idea. I will bring you a sharp knife."
"Thank you."
Elizabeth reached for the door. "I'll go get one and bring it right back," she promised. Mara thanked her again, and Elizabeth hurried back to the house. She wished she could send JT back on this errand, but she knew that was impossible. As it was, she wanted to get the knife to Mara before Ruth returned and got curious.
######
In the next few days, nine more baby chicks hatched. "Thirteen chicks," Ruth proclaimed proudly on the last day of December. "That's a baker's dozen."
"And I suspect that's all we can hope for," Elizabeth told her. "The other eggs should probably be thrown out by now."
"But maybe the chicks are still coming," Ruth said hopefully.
"You can let them sit a bit longer if you like, but remember what a rotten egg smells like when it gets accidentally broken," Elizabeth warned.
Ruth wrinkled her nose. "I'll get rid of them tomorrow," she promised.
On the first day of 1858, Ruth buried the six unhatched eggs back behind the chicken coop. Elizabeth was only mildly surprised to see that Ruth had planted a little wooden cross over the freshly dug grave. Then at around noon, the four of them traipsed over to her parents' for a New Year's Day dinner. Matthew was still on crutches but seemed to be in a little less pain than previously, and Jess' waist was just starting to thicken, which reminded Elizabeth about Mara. After dinner, seeing that Matthew and Brady were involved in checkers and the others had gone out for a stroll, Elizabeth remained behind to help her mother with the dishes.
"I think Mara's baby is not too far from arriving," she said quietly.
"Oh?" Clara looked at her. "Do you plan to help her when it comes?"
"I don't know exactly." Elizabeth glanced over to the checkers players, but they were oblivious. "Certainly, I'm willing. But she wouldn't have any way to send for me. Rose is too small."
"Yes." Clara nodded. "That is a problem."
"I'm not sure what to do."
"In all likelihood she won't need help." Clara frowned.
"Unless...she does need help." Elizabeth scrubbed the pot harder. "And what about poor Rose? She will be terrified if her mother is in great pain. And what can she do to help?"
"Oh, dear. I hadn't even thought about that."
"If they were in a village, other women would be on hand to help Mara and watch out for Rose."
"That's true."
"Well, she is in a village," Elizabeth declared. "You and I will have to be her village."
"What will we do?"
"I'm not sure." Elizabeth tried to think of a plan. "You are more knowledgeable than I am about birthing babies. Is there any way to know how far along she is?"
"Does she know?"
"She didn't seem to. But just looking at her and how much larger she's gotten these past few weeks convinces me that it can't be too far off now. I suppose it could be as much as a month...or as little as a few days." She set the pot down. "I don't know. And it's been difficult to spend much time with her when the children are out of school. And I really do not want them to know about her."
"Why don't you take me to visit Mara," Clara suggested. "The first day the children are back in school, you and I will go."
Elizabeth agreed. She just hoped that Mara didn't have her baby before that. In the meantime, she would be praying faithfully for her.
The day for school to be back in session broke rainy and gloomy, and Elizabeth almost didn't want to send the children out in it. But they were eager to be back with their teacher and friends. And JT put on his waxed coat, and Elizabeth insisted on wrapping her thick barn coat around Ruth before the two climbed up onto Molly. Sometimes she wondered if snow wasn't preferable to rain.
"Don't worry," Eli said as he joined her on the porch. "Those two are made of sturdy stuff. Besides, they have the heat from Molly to keep them warm. And once they get to school, Mrs. Taylor will have the stove cranked up and they will soon be toasty." He chuckled. "Or steamy."
She nodded. "I know you're right."
"Are you going to check on Mara this morning?"
"Yes." She peered in the direction of her parents' property, but thanks to the drizzling rain, she could barely see. "Although I won't blame Mother if she doesn't make it over."
"Come have a cup of coffee with me," he said. "Give her some time just in case she's coming."
It was pleasant to sit and share coffee with Eli. It had been some time since the two of them had been alone in the house. They were just finishing up when Clara arrived looking somewhat drenched. "Oh, Mother," Elizabeth exclaimed. "Let's get you in here and dried out." She peeled off the soggy coat and ushered her over to the chair by the fire, where she helped to remove her wet muddy boots. "At least your dress is dry."
"It really looks much worse than it is," Clara said as Eli handed her a cup of hot coffee. "But thank you."
After Clara was somewhat dried out and the rain shower had moved on, the two women bundled up and headed over to Mara's cabin. As they walked, Elizabeth confessed her fears to her mother. "I always wonder what would happen if someone in the settlement found out. Would they make trouble for Mara? Or burn down the cabin? Or come after Eli and the children and me?"
"Oh, goodness. I hope not."
"If you had seen the fear in Mara's eyes when she told me about how they burned down their cabin...you would understand."
"Poor dears."
Elizabeth knocked on the door, calling out to Mara. As soon as Mara opened the door, Elizabeth introduced her mother, explaining how Clara knew about babies and giving birth. At first Mara seemed uncertain, but as Clara began talking to her, Mara visibly relaxed.
"Can I take Rose outside with me?" Elizabeth asked Mara. "While you two talk?"
Mara looked unsure, but then nodded. "Put on your moccasins," she told Rose. Elizabeth watched as the little girl hurried to find her shoes.
"You made them," she said to Mara as she helped Rose tie the laces. "They are very nice." Soon Elizabeth and Rose were outside exploring and getting some fresh air. Rose seemed to appreciate this time, and Elizabeth was relieved that her mother was with Mara. She completely trusted her mother's wisdom and intuition when it came to things like health or childbirth.
After a while, Clara called them back inside, explaining to Elizabeth that Mara was close to giving birth. "It could be today or a week from today. But I don't think it will be much longer."
Elizabeth nodded, taking this in. "How does she feel about giving birth...alone? Or with only Rose here?"
Clara smiled at Mara. "Mara has agreed to come with me. She and Rose will stay with your father and me until her time comes."
Elizabeth was surprised. "Really? She wants to stay with you?"
Mara looked at Elizabeth with frightened eyes. "My mother...she die...when I birthed." Mara put a protective hand on Rose's head. "No good be alone."
Soon they packed up a few things and walked back toward Elizabeth's, and from there Clara, Mara, and Rose continued. Elizabeth prayed as she watched them walking through the soggy field. She prayed for all of them.
Then she went home and finished the baby blanket she'd been making for Jess' baby. However, she knew there was plenty of time to make another one before April. Mara would need it sooner. Elizabeth made a few other baby things as well. Nothing overly frilly or silly, just practical and warm.
Three days later, Mara gave birth to a small but healthy baby boy.
Chapter Twenty-Nine
January came with exceptionally mild weather that made the newest of the settlers imagine that it was spring. Mara and her baby, Charles Jr., continued to thrive, and after several days, the family of three moved back to their tiny cabin, where for the next couple of weeks, Elizabeth checked on them daily. But still there had been no sign of Charles Sr., and Elizabeth began to fear that harm may have come to the young man.
When February came, it brought enough rain and wind to down a number of trees and close the school for a few days. Not only that, it postponed Jeremiah and Mahala's wedding.
"The good news is that this delay has given Jeremiah and Will a chance to put some nice finishing touches on the cabin," Malinda told Elizabeth as they had tea at Malinda's house. It was the first time Elizabeth had been off of her property for nearly two weeks, and it was much needed.
"Have Will's carpentry skills improved?" Elizabeth asked curiously.
"Ever since that hunting trip, everything in Will's life seems to have improved," Malinda told her. "Truly, your father, brother, and husband were like a tonic for Will."
"I'm glad to hear it." This was the first time they'd been able to catch up since the surprise wedding on Christmas Eve. "I'm sure Will told you how he saved my brother's life."
Malinda waved her hand. "Oh, to hear Will talk, it was nothing special at all. But Jess took me aside after church a few weeks ago and thanked me. She told me what happened—in detail." She smiled proudly. "I felt as if I had married a real hero."
"You did."
"Well, he never would have had a chance to be a hero if the men in your family hadn't taken him in like they did. I am so thankful."
"So your marriage agrees with you?"
Malinda nodded. "Oh, there's some adjusting, I'll admit that. Will and I both have strong personalities. And as you know, I can be a bit stubborn at times. But I believe we are making progress." She seemed to be studying Elizabeth now. "How about you? How are things with you and Eli?"
Elizabeth smiled. "I have no complaints."
Malinda frowned. "And you are feeling well?"
Elizabeth looked down at her teacup.
"Elizabeth?"
She looked up and shrugged. "I've been a little under the weather. I blamed it on the weather. Wasn't it ghastly? Although Eli is using the fallen trees for our—"
"Don't change the subject, Elizabeth." Malinda peered at her. "You look pale and drawn to me. Are you sure that you are not unwell?"
Elizabeth took in a deep breath. "If you must know...and only if I can trust you."
"You can trust me, Elizabeth. I may be a bit willful and outspoken at times, but I am loyal to my best friend."
"All right." Elizabeth sighed. "I haven't said anything to Eli yet. And Mother doesn't even know. But I suspect I am with child."
"Oh, Elizabeth!" Malinda reached for her hand. "I'm so happy for you."
She made a weak smile. "Thank you. But I don't remember feeling so ill with JT or Ruth. My stomach seems to be upset every morning. In fact, that is what made me first suspect I was pregnant."
"I wasn't sick with my first three children, but oh, my, I couldn't keep anything down for the first three months with Susannah. John got quite worried."
"But you got better."
"Yes. By my fourth month I was hungry as a horse."
They visited for a while longer, and when Elizabeth left, she felt encouraged. Walking home in the sunshine, she tried to imagine what it would be like to have a baby in her arms next fall. She wondered what Eli would think. Or her children. Ruth would be thrilled. And Jess would be happy to find out that her baby would have a cousin close to the same age. Still, she was determined not to tell anyone until she'd reached her third month. If Malinda hadn't pressed her so, she never would have told her. As it was, she felt she could trust her.
On Saturday, everyone gathered at the church to celebrate the marriage of Mahala Flanders to Jeremiah Bramford. Elizabeth couldn't help but chuckle at the irony of this match. She remembered when she'd first met the Bostonians and how she'd felt they were stuffy and superior. And when she'd met the Flanders, she'd felt they were a bit loud and rough. Who would have guessed back then that the oldest Bramford son would wed the oldest Flanders girl? Yet here the two young people were, pledging their vows before God and their families and friends.
Afterward a lively potluck celebration was held at Will and Malinda's barn, and it seemed that everyone within a fifteen-mile radius was in attendance. Elizabeth danced with her husband and children and father, finally sitting out as the evening was drawing to an end. "How are you feeling?" Malinda asked with some concern. "You look a little pale."
"To be honest, my stomach is feeling a little sour," Elizabeth admitted. "Usually I get over this by midday."
"Just rest and take it easy," Malinda said. "I'll go get you some tea. That might soothe your belly."
Unfortunately, the tea didn't help. And by the time Eli was driving their wagon home, Elizabeth was feeling quite ill. To her dismay, Ruth seemed to be sick too. "Do you think you girls ate something bad?" Eli asked as he helped her into the house. JT was helping Ruth.
"I really don't know," she said weakly. "But please see to Ruth. I can take care of myself." With wobbly legs, she got a pot from the kitchen and went to her bed. If Ruth were not feeling ill, Elizabeth would assume this was related to her pregnancy. But as she continued to be violently ill, and knowing that Ruth was sick as well, she feared it was something more serious. The next hours passed in a painful blur.
By the time Elizabeth felt able to sit up in bed and sip some broth, it was late in the afternoon of the following day, and her mother was sitting with her. "How is Ruth?" Elizabeth asked in a hoarse voice.
"Ruth is just fine now. She's already eating solids," her mother assured her. "But I believe you both ate something tainted at the wedding potluck."
"Oh." Elizabeth let out a weak sigh.
"Unfortunately for you, it seems to have hit you harder than Ruth. Eli was so worried that he sent JT for me early this morning." She spooned more broth into her.
"Thank you for coming."
"I sent JT to check on Jess," Clara continued. "I was worried that in her condition she could be in trouble if she'd eaten something tainted. Thankfully, she is fine."
"Oh, I'm so glad to hear that."
Clara reached for Elizabeth's hand now. She had tears in her eyes. "But you, my darling, did not fare so well."
"What?" Elizabeth looked curiously into her mother's face.
"You lost your baby, Lizzie."
Elizabeth took in a jagged breath and then looked down at her lap as tears began to roll down her cheeks. They both just sat there in silence for a while.
"I'm so sorry, dear. I didn't even know. Nor did Eli."
Elizabeth looked back up. "Did you tell him?"
"I'm sorry if that was wrong. I just assumed he knew."
"No, I hadn't told him yet."
Clara squeezed her hand. "He is sad. But he was so worried about you...well, I think he's so relieved that you're all right." Now she started to cry, and they both just hugged and cried.
######
It took a full week before Elizabeth felt like leaving the house. Eli and the children were thoughtful and helpful, but a cloud seemed to hang over her head. Finally on a sunny Monday in March, she forced herself to go and check on Mara and her children. Clara had been checking on them, and Eli had been taking them food, but Elizabeth knew she was long overdue for a visit.
Mara greeted her warmly, inviting her in to her tiny cabin. She showed her baby Charles, and wrapped in the blanket Elizabeth had knit, he was growing nicely. Rose's hair was neatly braided. Despite their rather impoverished conditions, this little family seemed fairly content.
"I am sorry," Mara said quietly. "Clara told me."
Elizabeth just nodded.
"I have something." Mara turned now, going over to a corner of the room and then returning with something in her hand. "I make for you." She held out a pair of deerskin moccasins. "You wear. You get better."
Elizabeth hugged Mara. "Thank you. I will wear them. And I will get better."
As Elizabeth walked back to her house, she wondered at the strength Mara had to live in that cabin with her two young children, hoping that her husband was going to return to her. But the more Elizabeth considered the missing Charles Sr., the less she felt that he was likely to come back...or that he was even alive. And yet Mara was not giving up.
She looked at the moccasins Mara had painstakingly made for her. Mara was right—she would wear them, and she would get better. Like Mara, she would become strong again. And God willing, she and Eli would have another baby...someday. In the meantime, spring was coming on with lush meadow grass and bright green foliage and wildflowers blooming profusely along the river. Elizabeth's fruit tree seedlings had fared well over the mild winter and would be ready to transplant in the fall. Only about a fourth of them didn't survive, and that was better than she had expected. The other good news was that Goldie would calve by summer, and Molly would foal in the late fall. Besides that, their chicken flock was increasing regularly. Their farm was growing.
In mid-April, Jess delivered her first baby, a precious little girl whom they named Maude Ruby. Elizabeth and Ruth went over to visit the next day, taking a baby blanket and a few other things Elizabeth had made for the new addition to their family. Clara welcomed them into the house, holding her finger over her lips. "Jess is sleeping, but the baby is awake." Soon they were taking turns holding and admiring the newborn.
"She's so tiny," Ruth said as she held her new cousin, examining her fingers.
"So were you once," Elizabeth told her.
"And so was your mother," Clara said. "We all start out that small."
"Her hair is dark like Jess'," Ruth said.
"How is Jess?" Elizabeth asked her mother.
"Just fine. She did very well."
"How is Matthew?" she asked.
Clara chuckled. "I think he's recovering. He was mighty thankful to get outside this morning."
"What's wrong with Uncle Matthew?" Ruth asked.
"It's not easy being a father for the first time," Elizabeth explained. "But he'll get used to it."
"Will you have a baby, Mama?"
"God willing." Elizabeth nodded. "In his time." And as Elizabeth rocked her tiny niece, she did not feel in any hurry to have another baby. Really, there was so much to do with spring coming on. She would much prefer to put her energy into improving their farm.
######
On Sunday, they went to church as usual. But after the service, as they were visiting out in the sunshine, Elizabeth overheard some of the men talking. She could only hear bits of their conversation, but she was certain they were speaking about Indians. Excusing herself from the women she was chatting with, she went over to where Eli was talking with Asa.
"Excuse me," she said. "But the men over there, the ones gathered around Mr. Walters, seem to be talking about Indians. I couldn't catch all of it, but Mr. Walters seems to believe there are Indians nearby. And the way he was talking, it almost sounded like he wanted to round them up."
Eli's brow creased. "Mr. Walters always seems to be looking for a fight."
"Do you think he may have seen Mara? I know she takes the children out for sunshine now and then. She used to be afraid to go out in daylight, but she's gotten more comfortable lately."
"I suppose it's possible she's been seen," Eli conceded. "The Walters' farm is out past ours, and they go by our property on their way to town. But the way the cabin is placed, Walters would have to be well on our land in order to see Mara—unless she's wandered out a ways."
"I doubt she goes too far from the cabin," Elizabeth said. "And I'll warn her to be more careful."
"In the meantime, why don't we see if we can find out what Walters is talking about," Asa told Eli. "Maybe we can put a lid on it before it starts to smoke."
Elizabeth thanked them and casually walked over to where the older ladies were visiting. "Good day, Georgia and Martha," she said politely. "What do you women think of this glorious spring weather?"
"We were just saying how lovely it's been," Georgia told her. "I have daffodils and tulips blooming."
"Daffodils and tulips?" Elizabeth was surprised. "Where did you get bulbs?"
"I brought a few when we came out here. They have multiplied over the years. I'll be happy to share some bulbs with you in the fall."
"Oh, that would be delightful."
Elizabeth looked at Martha. "Have you had word from Charles?"
Martha sadly shook her head.
"I'm sure you must miss him. I want you to know that he's been in my prayers."
"Thank you," Martha told her. "I appreciate that more than you know."
"You are fortunate to be blessed with three fine sons," Georgia told her.
Martha brightened. "I s'pect you've heard that my Jacob and Belinda Bramford have set their wedding date for June," she said to Elizabeth.
"Yes, Malinda told me. That's wonderful news."
"June is such a nice time for a wedding," Georgia said.
"And now my youngest boy, Joseph, appears to be setting his cap for Belinda's younger sister, Amelia. So perhaps it won't be long until we have another wedding in the family."
"I still remember the day you ladies came to tea at my house," Elizabeth told them. "I hope you will come out again sometime."
"You just name the time and day, and I will be happy to come," Georgia said.
Martha nodded. "I would like that too." She lowered her voice. "Not everyone understands my...my suffering."
Elizabeth thought about Baby Charlie, as she'd taken to calling the infant. Such a sweet baby, contented and happy and getting cuter by the day. What would Martha think if she knew what she was missing out on?
"I just got to thinking, Elizabeth," Georgia said suddenly. "If you like flowers, you might want to come take a start from my violas."
"You have violas?"
"Oh, my, do I have violas!" She glanced at Martha.
"Yes, she certainly does. They grow in profusion behind the parsonage."
"They like the shade," Georgia explained.
"I would love to get some violas," Elizabeth told her.
"Why don't you get some today?" Georgia urged her. "No time like the present."
"And I should be getting home by now," Martha said. "Mrs. Taylor and I have invited the Prescotts for Sunday dinner."
Elizabeth told Ruth to let Eli know that she'd be right back, and she and Georgia hurried to the parsonage, where she was soon handed a trowel. After she dug up a good start of violas, Georgia gave her an old tin to contain them. "Thank you!" Elizabeth sniffed the fragrant blooms. "I know exactly where I'll plant these." Now she studied Georgia carefully. "Can I trust you with something that is extremely confidential?" she asked suddenly.
"I certainly hope so. I have not broken a confidence yet." She peered curiously at Elizabeth. "Is something wrong, dear?"
Elizabeth took a deep breath and said a silent prayer, but somehow she felt this was the right thing to do. "Georgia, I have been hiding Charles' Indian wife on my property."
Georgia blinked as her hand flew up to her mouth. "Oh, my word!"
"Her name is Mara, and she is a dear girl. She speaks English and was educated at a mission. I can understand why Charles fell in love with her."
"Oh, my...oh, my..." Georgia glanced around as if worried someone could be listening, but it was clear they were alone.
"Mara told me that Charles left to find gold, and then some of the settlers burned down the cabin that she and her daughter, Rose, were staying in. I found them in early December. They were living outside and suffering considerably. Mara had her second child—"
"She has two children?"
Elizabeth nodded solemnly. "Rose is about three or four. Baby Charlie will be four months soon. They are dear children. A sweet little family. But I am in constant fear for them." Now she explained what she'd overheard Mr. Walters saying.
Georgia scowled. "That man has always been a troublemaker."
"The reason I'm speaking to you of this is because I had hoped that I could somehow win Martha over by showing her these darling grandchildren. I thought perhaps she could be of some help to Mara." Elizabeth held up her hands. "I really don't know what else to do. It's not that Eli and I mind having Mara and the children on our property. But I worry they are not safe. I know Mara worries too. And hearing Mr. Walters just now..." She shook her head.
"That is certainly a conundrum. To think you've had them with you all these months, Elizabeth...and no one knew?"
"My parents know. And my brother and Jess know. But I haven't told my children. I feel that's too dangerous."
Georgia nodded.
"You know Martha much better than I do, Georgia. How do you think she would react if I presented her with her grandchildren?"
Georgia pursed her lips. "Martha has a good heart. I am certain she would love her grandchildren."
"But would she try to take them from Mara?" Elizabeth studied Georgia closely, looking intently into her gray eyes.
"I do not know for sure. I suspect it would depend..."
"Depend on what?"
"Her impressions of Mara. If she felt the girl was a savage or not fit to be a mother—"
"Mara is a wonderful mother. She loves her children dearly. I am certain she would die for them."
"Hopefully she will not have to."
"Do you think that if I could present Mara and her children to Martha in a way that would show Martha they are good people...would Martha consider helping them or taking them in? With two of her three grown sons living so nearby, perhaps Mara would be safer. And if they were out in the open, it wouldn't appear as it might now...as if we are hiding refugees." Elizabeth hated to think of what men like Walters might do if they were all worked up into an indignant rage—not only to Mara and her children, but to Elizabeth's family too.
Georgia grasped Elizabeth's hand. "I think your plan could work. But I will discuss it with Roland first. Do you mind?"
"I want to believe he can be trusted with this confidence, but I do not know him as well as I know you."
"The reverend can be a bit stern at times, but he is wholly trustworthy, Elizabeth. I promise you."
Elizabeth knew she could trust Georgia with this dilemma. But as she returned to rejoin her family, she silently prayed that God would help all of them to find a good solution for Mara and her children. Somehow that little family needed to carve a life for themselves in this hostile frontier—a land that had once belonged to their ancestors. However, Elizabeth knew that the odds were probably against this plan. The Indians had been removed, many were dead or dying, and still people like Mr. Walters continued to brim with hatred. It did not bode well for Mara and her children.
Chapter Thirty
On Monday morning, as soon as her children were on their way to school, Elizabeth paid a visit to Mara. After she shared a loaf of bread and some venison and played with the children, she turned to Mara with a serious expression. "We need to talk," she told her.
Mara made a worried frown and Elizabeth looked at Rose. "Can you stay here with Baby Charlie while your mother and I go outside to talk?"
Rose nodded solemnly.
Elizabeth smiled at her, patting her head. "You are a good big sister."
Now Rose smiled.
Outside, Elizabeth quietly told Mara about the conversation she'd overheard and how Eli and Asa had learned more. "It's possible that some of the men have seen you. Do you ever go far from here?"
"I gather. Along creek." She looked uncertain.
"You must be careful. These men are dangerous."
"I know."
Elizabeth sighed. "Of course you know." She put her hand on Mara's shoulder. "It's just that I worry about you."
"You good friend."
She nodded. "I have a plan, Mara. But you will have to help me."
"Me help you?"
Now Elizabeth explained how she wanted Mara and her children to look and act more like white people, but even as she said this she felt embarrassed. What right did she have to tell Mara how to live? But Mara's life could be in danger if she didn't. "I wish there was another way, Mara. But I'm afraid there is not."
Mara looked down at her hand-me-down dress. "I wear white-woman clothes."
"Yes. But we must do more. And we must work on your English. We have to do this to show the white man that you are a good mother."
"I am good mother."
"I know you are." Elizabeth stifled her frustration over the unfairness of this. "Will you do this with me?"
Mara looked confused. "I do not know how."
"I will help you. And my mother will help you too."
Mara agreed, and so for the next few days, Elizabeth and Clara took turns helping Mara and her children to fit in better with white society. They altered one of Elizabeth's dresses, a pretty blue calico, to fit Mara. They taught Mara how to comb and pin up her hair. They gave her shoes and a shawl and a new bonnet to wear. They altered one of Ruth's dresses as well—a red-and-white gingham that Ruth had outgrown. All the time they spent with Mara and her children, they insisted that Mara practice her English skills, speaking full sentences and practicing good grammar. Sometimes Mara complained, but all it took was one reminder that her efforts could very well save her children's lives, and she would comply.
By the end of the week, Mara had made good progress, and Elizabeth felt hopeful. After church on Sunday, she consulted briefly with Georgia. Encouraged that her plan was sound, Elizabeth then invited Georgia and Martha to come for tea on Thursday. She wished she could take more time, but seeing Mr. Walters and his cronies clustered together again warned Elizabeth that time might be of the essence.
Elizabeth and Clara had three more days to work on refining Mara. On Wednesday morning, they had a mock tea party. By now Mara was well aware that she would be meeting Charles' mother tomorrow. And she was trying her best to cooperate. By the time they finished, Mara not only poured tea and folded her napkin properly, she managed to converse a bit about the weather as well.
"I am so proud of you," Elizabeth told Mara as she walked her back to her cabin.
"Thank you," Mara said quietly.
"You will do just fine tomorrow," Elizabeth assured Mara before she left. "Just be sure to be ready before my mother comes to fetch you in the morning. Just like today."
"I will be ready," she promised.
The plan was to have Mara and the children at Elizabeth's before the ladies arrived. Mara would wait in the bedroom while Elizabeth and Clara had the children in the main room. As she walked back home, Elizabeth prayed that the plan would work.
######
"Welcome," Elizabeth said as she opened the door to let the ladies inside. "My mother and two young friends are joining us today." She led Georgia and Martha inside, and Clara greeted them from where she was sitting in the rocker, holding Baby Charlie.
"This is Rose," Elizabeth told Georgia and Martha. Rose looked very sweet in the red gingham dress. Her hair was in two neat braids, each tied with a red ribbon. On her feet were the moccasins Mara had made, but even that did not seem too unusual since many of the settlers' children went barefoot. And Ruth often wore her moccasins. "Rose, this is Mrs. Holmes and Mrs. Levine."
"Hello, Mrs. Holmes. Hello, Mrs. Levine." Rose repeated her lines just as she'd been instructed, even adding the curtsy Clara had taught her to do.
"Hello, Rose," Georgia said. "A pretty name for a pretty girl."
"Hello, young Rose," Martha said with a curious expression. "Where did you come—"
"Now you must come meet my little one," Clara called out. "Come and say hello."
"Oh, my," Georgia gushed as she went to see him. "What a beautiful baby!"
"Goodness," Martha exclaimed. "This cannot be your grandbaby, Clara. He is much too big."
"You're right, the young lad is not my grandchild," Clara confirmed. "But isn't he adorable?"
"May I hold him?" Georgia asked eagerly.
"Certainly." Clara stood and passed the cherub to her. "He is a most good-tempered baby."
"Where did these children come from?" Martha asked. "Are their parents new to the settlement? I haven't heard of any newcomers lately."
"They are not new to the settlement," Elizabeth told her.
"What's your name, little one?" Georgia cooed to the baby.
"This is Baby Charlie." Elizabeth watched Martha as she said this, but the woman did not seem to register any recognition at all. "He is four months old."
"He is a doll," Georgia gushed. "A sweet little doll."
"I'm confused," Martha told Elizabeth. "You say their parents aren't new. But I don't recall these children. They didn't come with the wagon train families last fall."
"Would you like to meet their mother?" Elizabeth asked, trying not to feel too nervous.
"Certainly." Martha nodded.
Elizabeth went to the bedroom door, slowly opening it and giving Mara an encouraging smile, but she could see a look of pure terror in her eyes. "Come and meet some friends." She linked her arm in Mara's, leading her out into the room. Once again, Martha didn't seem to understand.
"Georgia and Martha, I want to present Mara to you. Mara, this is Mrs. Holmes and Mrs. Levine."
Mara stepped forward just as she'd been taught, extending her hand to Georgia first. "I am pleased to make your acquaintance, Mrs. Holmes," she said in a shaky voice.
Georgia clasped her hand and smiled warmly. "I am pleased to meet you too, Mara. You have lovely children."
"Thank you." Now Mara extended her hand to her mother-in-law. "I am pleased to make your acquaintance, Mrs. Levine."
"Pleased to meet you too." Martha's brow creased. "But I did not catch your last name."
"I'm sorry," Elizabeth said. "I forgot to mention it. This is Mara Levine."
Martha's face visibly paled, and Clara hurried over to ease her into a chair by the table. "We did not wish to shock you," she said kindly. "But we felt it was time for you to become acquainted with your grandchildren and your daughter-in-law."
"I...I...don't know what to say." Martha's eyes were wide as she first stared at Mara and then Rose and finally the baby. "Charlie," she whispered. "For Charles." She looked back at Mara. "Where is Charles?"
"Charles went to find gold," Mara said simply.
"He left last summer," Elizabeth explained. "He wanted to get gold in order to care for his family. But Mara has not heard from him since then."
Martha looked at Charlie again. "The baby looks like his father."
"Mara and her children have been staying on my property," Elizabeth explained as she poured Martha a cup of tea. She glanced at Mara. "Perhaps you'd like to take Rose to see the chickens now." They had already worked this out. Elizabeth didn't want Rose to be exposed to all of their conversation. As soon as the two were out of earshot, Elizabeth told Martha about how settlers had burned Mara and Rose out of their home last fall.
"Naturally, Charles wouldn't know that his little family was homeless. I doubt that he even knew that Mara was with child. I'm sure if he did know, he would return to care properly for them."
"Yes," Martha said quickly. "I'm sure he would. Charles was always a good boy."
"I haven't minded having Mara here with me," Elizabeth said. "I consider her a friend, and her children are delightful."
"You have lovely grandchildren," Georgia told Martha. "Look at this little fellow."
"May I?" Martha asked.
Georgia handed over the baby, and soon Martha was talking and cooing to him. The other women exchanged hopeful glances.
"As you can see, Mara is a fine woman," Elizabeth told her. "A daughter-in-law that anyone could be proud of."
"I certainly would be," Clara said.
"She is an Indian," Martha said stubbornly.
"That's true," Elizabeth agreed. "And her people have been driven off their land by the white man. If men like Mr. Walters have their way, Mara and Rose and Charlie will be driven off as well. Or worse." She locked eyes with Martha. "It's possible that if they are discovered they will be burned out of their home again. And they might not survive this time."
"But that's wrong," Martha declared. "These children have Charles' blood in them. They cannot be treated like that. I won't allow it."
"Perhaps if we band together," Clara said. "If we could get enough settlers to stand by you and your grandchildren and daughter-in-law, perhaps we could stand up to the likes of Mr. Walters."
"Where is Charles in all this?" Martha asked in a slightly desperate tone. "He should be here to protect his family."
"I agree," Elizabeth said. "But perhaps he felt it was more important to provide for them. Mara said that he wanted to get enough money to get them away from here...to take them someplace safe."
With the baby still in her arms, Martha stood, looking at all of them with a defiant expression. "They will be safe with me."
"All of them?" Elizabeth asked. "Because I have promised Mara that I will let no one take her children from her. I want you to understand that."
"And I back her on it," Clara said.
"As do I," Georgia added. "And the reverend too."
Martha frowned but then looked back into her grandson's chubby face. "Yes. I will take all of them. I'll take them home today. My sons and I will keep them safe until Charles returns."
Elizabeth took in a sharp breath. She knew it was possible—maybe even likely—that Charles would never return. However, she did not intend to say this. Mostly she was relieved. Mara and her children would be cared for and protected—by their own kin.
"Georgia Holmes," Martha said a bit sharply. "You were in on this from the very beginning, weren't you?"
Georgia made a sheepish smile.
"That is why you insisted I drive the wagon out here today, wasn't it? You don't really have an ingrown toenail, do you?"
"I must confess...I told a falsehood. Please forgive me. It was for a good reason."
"Shall we load them up and go home now?" Martha asked.
"First we will have tea," Elizabeth told her. "Mara and Rose have been looking forward to it." She nodded to her mother. "Can you play hostess for a bit? I'll go ask Eli to get Mara's things loaded into Martha's wagon."
Elizabeth felt like dancing as she went out to where Eli was working to level a piece of land where their barn would eventually be located. "It worked!" she called out to him. "Martha is taking all three of them home with her."
Eli grinned. "I knew if anyone could pull that off it would be you, Elizabeth."
"Would you mind gathering up Mara's things from the cabin? She promised to have them packed up and ready to go."
"I would be happy to." He came over and hugged her. "Job well done, Elizabeth."
As she returned to the house, she knew that the job was not really done. Her friendship with Mara and her children would continue. She would continue to support this little family, helping them to fit into school and church and life in their town for as long as they remained here.
Chapter Thirty-One
In early June, shortly after school broke for the summer, Belinda Bramford and Jacob Levine were married. By the time of the wedding, everyone in the settlement was aware that Mara and her children were living at the Levines'. Reactions varied. Some settlers accepted them. Some seemed unsure. And a few, like Mr. Walters, were antagonistic. But Elizabeth was glad to see that Mara and the children attended the wedding in the church as well as the reception in Malinda's barn.
"I want you to meet my best friend, Malinda," Elizabeth told Mara as she linked arms with her and walked her over to where Malinda was standing with Will. Elizabeth made introductions, even introducing the children. And Mara, still practicing her English, graciously shook their hands.
"Mara and her children lived with us for a while," she said even though Malinda was well aware of this by now. "And I still miss her." She grinned at Mara. "We were neighbors, weren't we?"
"And friends," Mara said simply. "We miss you too."
"Well, now you will have my daughter for your neighbor," Will said in a friendly tone. "Belinda has spoken warmly of you."
"Belinda is a good girl." Mara smiled. "She is good with children too."
"She certainly loves your children," Malinda said. "She talks about them all the time."
Mara nodded down to Rose. "She made the dress for Rose."
"Very pretty," Malinda said.
Now Elizabeth changed the subject, telling Malinda that Goldie had calved several days ago. "A healthy heifer."
"I'm glad to hear it," Malinda told her. "I've had two heifers and a bull this spring."
"Do you have any interest in selling the bull?" Elizabeth asked.
"Now, now, ladies..." Will held his hands up. "Is this a wedding or a livestock auction?"
They both laughed, and soon they were joining in with the merriment, eating and dancing and congratulating the happy newlyweds. However, Elizabeth felt determined that somehow, between her father and brother and herself, her family would pool their resources to purchase that bull from Malinda.
######
As summer progressed, the town expanded. Will Bramford built a law office next to the mercantile, and because of his medical knowledge and the community's needs, it doubled as a pharmacy. On the other side of the mercantile, which now had a lean-to on one side for a post office, a large building was going up. This impressive structure would house a hotel and restaurant that would be run by Ruby and Doris before the year was out. Down the street a ways was Flanders' Blacksmith, and next to that was the new livery stable that Ezra Flanders was running with his new brother-in-law, Julius Prescott.
In just one year's time, their settlement had grown dramatically. And if the rumors were true, it would be twice as big by next year. As happy as Elizabeth was about these changes, she was even happier about a tiny change going on inside of her. She'd kept the news of her pregnancy within the confines of her family at first, but by early fall, her midsection had increased enough that people in town had begun to suspect her family would be growing too.
In mid-October two delightful surprises occurred. Actually, one was not a complete surprise, although it did arrive earlier than expected. Molly had a foal—a sweet little filly that Ruth named Pansy and claimed as her own. The second surprise was that Charles Levine came home. However, Charles was probably more surprised than anyone when he discovered that Mara was living with his mother.
"And when Charles saw his children," Martha excitedly told Elizabeth at the mercantile, "well, he was just beside himself. He had no idea he had a son." Then she happily boasted that Charlie, who wasn't even a year old yet, was trying to walk. "And Rose has been helping Mara with the garden, and I've never had such produce growing." She grasped Elizabeth's hands and squeezed them. "Thank you, dear."
"Thank you," Elizabeth told her.
By late October, the hotel was finished, and Doris and Ruby began to get settled in. "We made good money in Empire City," Ruby told everyone as they all enjoyed a happy reunion dinner at Eli and Elizabeth's house.
"But we never felt at home there," Doris finished for her.
"Doris had a suitor though," Ruby teased as she bounced Baby Maude on her knee. "A businessman named George Johnston. And I would not be surprised if we have a hardware store here by next summer."
"We told a lot of folks at Empire City about our plans to come help develop this town with our friends," Doris said. "But the problem was, this town does not have a name." She frowned. "Don't you locals think it's about time to name this place?"
"As a matter of fact, that's been a hot topic in these parts," Asa told her.
"Some families want to name it after themselves," Matthew said glumly.
"Like the Walters family," Ruth said.
"And they're not a real nice bunch," JT added.
"Then there's the Thompsons," Eli said. "They claim they were here first."
"Naturally, it's causing some arguments," Elizabeth explained.
"Some want to name it after the river, but seems there's a town east of here already going by that, although they're not incorporated."
"Not that we want to incorporate," Asa said.
"Some folks are trying to name it after plants or trees or fish, but that hasn't gone over too well," Clara explained.
"I suggested the name Riverside at the last meeting," Elizabeth said. "That's one thing most of us have in common...the river."
"I like that," Ruby said. "Riverside."
"Unfortunately, the Walters and Thompson families didn't like it," Eli said.
"And so the debate goes on."
"Why don't they put it to a vote?" Ruby asked.
"Good question," Asa told her. "That's what we keep encouraging them to do."
"Mr. Walters is afraid he'll lose," Ruth said.
"Godfrey Walters goes to our school," JT told her. "He's always bragging that his dad runs this town."
"Maybe someone should challenge him on that." Ruby pointed to Asa. "Like you. I think you'd make a good mayor." She looked around the crowded room. "Anyone else here willing to vote for Asa?"
Naturally, they all raised their hands.
Asa laughed. "Well, I don't think we need to be concerned about that. Our nameless town isn't big enough to need a mayor yet."
"Give it time and it will," Doris said. "From what we heard, Empire City has grown like a weed."
"That has to do with its location," Eli pointed out.
The energetic discussion about their nameless town and its uncertain future continued on into the afternoon until they finally realized it was getting late, and although it was Sunday, there were still chores to be done.
######
It wasn't until December, just a week after Eli and Elizabeth's first wedding anniversary, that a town meeting was called. All the men who lived within five miles were invited to attend, and after what Eli described as a rather loud and disagreeable evening, a vote was taken and the town's name was decided upon.
"What is it?" Elizabeth asked anxiously, keeping her voice low since the children were in bed. "Please tell me it's not Walters."
He grinned. "Or Thompson."
"What then?"
"Riverside."
"Riverside?" She grinned. "That was my idea!"
He nodded. "I know. It was your dad who suggested it. And he put forth a very convincing argument for it, saying some of the same things you said that day we were all talking about it right here."
Elizabeth felt torn. Of course, she liked the name and was relieved their town hadn't been named after a disagreeable family. But at the same time it seemed unfair that just because she was a woman, she had not been allowed to speak in behalf of the town's name. Would that ever change?
"After the vote had been taken and the town's name was secured, your pa let on that you were the one who thought of the name Riverside."
"He did?"
Eli chuckled. "Sure. He told everyone that he'd stolen the idea from you, but he was pretty sure you wouldn't mind."
She laughed. "Did they want to take another vote?"
"Oh, I'm sure that some of them did."
"Yes, I'm sure."
So it was that the little town of Riverside, Oregon, was born just before winter in 1858, a little more than a year after Elizabeth and her family settled there. To celebrate the town's name as well as the opening of its first hotel and restaurant, Ruby and Doris hosted a party at the Riverside Inn. Naturally, it was well attended and the event of the season.
The Dawson and Kincaid families had been blessed with four calves during the summer, and they'd gone in together to purchase Malinda's young bull. The calves were weaned and the grasslands were unlimited, so the cows were producing rich, creamy milk. And Asa's dream of Dawson's Dairy was beginning to be realized. Clara and Elizabeth were making two kinds of cheese, and Jessie was making butter. Not only that, but Elizabeth's ever-increasing flock of chickens was producing eggs by the dozens. All these wholesome food products were being sold and traded at the mercantile under the label of Dawson's Dairy. From what they could see, with more hard work and gumption, the future for their family and the town of Riverside was sunny and bright.
Chapter Thirty-Two
After a spell of wonderful springlike weather the first three weeks of January, the weather turned gray and grim and wet. Elizabeth, late in her pregnancy, felt clumsy and awkward trying to do farm chores, and after she took a bad spill, Eli put his foot down.
"I don't want to see you out here floundering around in the mud again," he said firmly. "Not until our baby is at least a month old. Understand?" He gently pulled her up from where she'd slipped just outside of the chicken yard. "Are you all right?"
"I'm sure my pride is a bit wounded." She made a lopsided smile.
He leaned down to kiss her. "I'm sorry. But I mean what I said. You are to stay in the house. The children and I will see to all the outdoor chores."
She shrugged. "Fine by me." Looking down at her muddy dress, she frowned. "But I still need to do laundry, and I normally do that outside."
"Yes, but you can do that on the porch. And in weather like this, you'll have to hang it in the house to dry anyway."
Elizabeth didn't really mind being housebound, but the gray weather was a bit dreary. However, she knew she could fill her time with knitting and sewing and getting ready for the new addition to their family. But in the next few days, she started to grow concerned, and on Saturday, she asked Ruth to take a note to her mother.
"Is it an invitation to a party?" Ruth asked excitedly.
"No, dear. It's just a note. I've missed her lately."
"Oh." Ruth nodded as she buttoned up her coat.
Elizabeth peered out the window at the darkening sky. "Don't dilly dally on the way, and maybe you will make it back before it rains."
Ruth hurried on out, and Elizabeth hoped that her mother wouldn't be too alarmed by the letter. Mostly she just wanted Clara to answer her questions. She'd described her symptoms in the note, reminding Clara that her baby was not due until early March. It wasn't even February yet. If the baby came too soon, it would not be good. Elizabeth had not had problems like these with JT or Ruth. But she had been considerably younger then. And though she didn't like to think of it, the two pregnancies after JT and Ruth had both ended sadly. To be fair, the first baby she'd lost was due to cholera. The second one...well, she didn't know for sure.
She sat down in the rocker, rubbing her rounded belly. "Be well, little one," she whispered. "And be patient. Don't come too soon." Then she picked up her knitting and attempted to distract herself with the pale blue blanket she was making for her unborn child. She hoped the baby would get to enjoy it.
"I brought Grandma with me," Ruth announced as they both came into the room.
"Oh?" Elizabeth looked up from her knitting. "That wasn't necessary." She started to push herself to her feet. "But I am happy to see you, Mother."
"Stay put," Clara demanded. "I've come to see you."
"Want to put on the teakettle, Ruth?" Elizabeth called out.
"How about if Ruth takes a snack out to the fellows," Clara suggested. "I said hello to them, and they looked hungry."
Elizabeth looked at the clock. "Oh, my. It is about dinnertime."
"Dinner can wait," Clara told her.
"Take them those doughnuts we made last night," Elizabeth told Ruth. "They can finish them off."
"Can I have one too, Mama?"
"Sure."
"Hurry now," Clara called "Before they come in here demanding to be fed."
Before long, it was just the two of them. "What is it, Mother?"
"What you wrote me." Clara sat down across from her frowning with concern. "You've got me worried, Lizzie." Now Clara began peppering her with questions. And she felt her forehead as well as her belly. "I brought you some red clover tea. That might help some. But as your mother and midwife, I am going to recommend bed rest for you until the problems you described to me stop. Or the baby is born."
"But the baby isn't due until early March."
"It's possible that your calculations are wrong."
"I do not think so, Mother."
"Well, that's even more good reason to get you to bed, my dear." Clara stood and then helped Elizabeth to her feet. "To bed with you. And do not argue with your mother."
"But I don't feel—"
"Elizabeth." Clara said sharply.
"Yes, Mother."
"I'll make you some tea. And then Ruth can help me fix dinner for your men. You just go get into your nightgown and climb into bed."
Elizabeth felt foolish going to bed in the middle of the day. Perhaps she'd been mistaken to call on her mother. And yet...she didn't want to put this baby at risk. She was just getting into bed when her mother appeared with a cup of red clover tea. "Drink this all up."
Determined to be a good patient, she drank the tea and remained in bed. She could hear them in the house, and she knew that Eli was probably concerned for her as well as the baby. After a bit, Ruth brought her a tray of food. "How are you feeling, Mama?" she asked with worried eyes.
"I feel just fine," she said. "I would like to get up and take care of—"
"Grandma says you have to stay in bed to keep the baby safe."
"Yes. I know."
"And I am to take care of you when I'm not at school. But while I'm at school, Grandma is going to be here."
"Oh, I don't think that's necessary."
"Grandma said you have to mind us, Mama." Ruth shook a finger at her. "We are all going to make you stay in bed. So you shouldn't argue about it."
Elizabeth laughed. "Well, I can see that your grandmother is ruling the roost now."
"Grandma and me."
######
Elizabeth had never been so bored in her life. It hadn't been too bad the first day or two, but after that, being confined to bed started to feel like a cruel punishment. But because her symptoms did not seem to be improving, she knew better than to question her mother's advice. And by now her whole household seemed to have turned against her. If she even made a move to get out of bed, they would scold her and show no mercy. Even the dog seemed to be keeping guard at the bedroom door.
On the third day, her father came to visit, bringing her some books to read. "Be patient," he told her. "Before you know it you'll be out on your feet and wondering why you didn't enjoy this respite a bit more."
"I don't think so, Father."
"You're a strong woman, Lizzie, but sometimes even a strong woman needs to let others help her." And then he asked if he could pray for her. Naturally, she didn't protest. By the time he left, she felt considerably better in spirit, but not well enough physically to be on her feet. Not if she wanted the baby to born healthy.
Word of her confinement began to spread in the community, and unless Elizabeth was mistaken, the women had made a schedule to ensure that she had a female visitor calling every afternoon. This provided a diversion in Elizabeth's boring days and allowed Clara to go home and see to her own household.
Her first visitors were Jess and Baby Maude. "See how big she's getting," Jess said as she held Maude up to stand on her little feet. "And she's only nine months old." She nodded to a small pile of baby clothes that she'd set on the washstand for Elizabeth. "She outgrew those a couple of months ago. And now she's crawling all over the place. Matthew says she'll be riding a horse in no time." Jess chattered cheerfully, telling Elizabeth the news from their farm and how Matthew was putting in a corral. "He's getting ready to start training our colt."
"I can't wait until I can be out there working with our filly."
"Don't be in a hurry." Jess nodded to the window. "There's mud everywhere right now. You might as well enjoy this break." She laughed. "Sometimes I think what I would give to take a nap." She looked at Maude. "This little girl only sleeps about an hour during the day now. But at least she sleeps soundly all night."
Jess continued chattering away, and Elizabeth couldn't help but remember the somber girl on the horse—the girl they all thought was a boy—and how she kept herself apart from everyone. And just look at her now. "Well, Jessica," Elizabeth said when they were finally getting ready to go. "You are a fine mother. And I am proud to have you for my sister."
Jess leaned down and kissed her cheek. "You take care of yourself. And mark my word, you'll be on your feet and chasing your little one in no time."
Malinda came the next day, presenting Elizabeth with a small quilt that she'd just finished for the baby. Like Jess, she chattered cheerfully almost nonstop. But Elizabeth sensed that underneath the happy chitchat, her friend seemed uneasy. It was clear that someone, probably her mother, had told these women that Elizabeth had been feeling a bit down in the dumps. And of course, that made Elizabeth feel a little guilty. "So how are you?" she asked Malinda finally. "Is it just my imagination, or do you seem a bit troubled?"
"Troubled?"
Elizabeth peered closely at her friend. She seemed to have shadows under her eyes. "And tired too. Malinda, are you all right?"
Malinda waved her hand. "I'm fine. It's you we're concerned for."
Elizabeth studied her. "Malinda Bramford—are you expecting?"
Malinda looked embarrassed.
"You are, aren't you!"
She shrugged. "To be honest, I'm not sure. I thought perhaps I was just getting old. You know I'll be thirty-five in June. I honestly felt I was too old to have another child."
"Mother had Matthew when she was in her early thirties."
"Yes, well...like I said, I'm not sure. I haven't said a word to anyone."
Elizabeth remembered the time she'd confided in Malinda...and how it turned out. "Don't worry," she told her. "Your secret is safe with me."
Belinda and Amelia came on the next day. They gave Elizabeth two sweet little nightshirts for the baby. And then they talked about how Belinda was helping Amelia to plan her wedding. "She's getting married in June, just like we did."
"Sisters marrying brothers," Elizabeth said. "One big happy family."
"It does make family gatherings simpler," Belinda admitted. Then Amelia asked Elizabeth's opinion on wedding dress colors, and Elizabeth told her to talk to Ruth about that.
On the following day Lavinia came, bringing a lace-trimmed christening gown for the baby. "Of course, I didn't sew it myself," Lavinia admitted. "But isn't it just fine?"
"It is beautiful. Thank you." They talked about new developments in town and how some of the businesses felt they should consider incorporating, but others, including the Prescotts, felt it was premature. "However, I do wish we had a mayor. Someone to help lead us in a good direction. I worry that a saloon will come to town. Can you imagine?"
Elizabeth grimaced. "That would be too bad."
"But Hugh says it's bound to happen."
They discussed the town's future at length until it was time for Lavinia to leave. Elizabeth sighed. She didn't like feeling sorry for herself, but it was strange feeling as if everyone and everything was moving right along and passing her by.
Mrs. Taylor came to visit on Saturday. She brought a pair of booties she'd made for the baby. She told about the latest goings on at school and how she hoped that Belinda and Amelia might start taking over for her more. "I still want to be involved, but to be honest, I am not getting any younger. I don't think I can keep up with the youngins the way I'd like to."
"You have done a wonderful job," Elizabeth assured her. "My children both love you. JT wouldn't be able to read music if you hadn't worked with him."
"Yes, and that is exactly what I've been wondering about. I think I could give music lessons to the children if the parents were able to pay. I would happily trade too. What do you think?"
"I think it's a great idea. I know my children would be happy to have lessons with you." They talked at length about how Mrs. Taylor could do this, and by the time she left, it seemed they had put together a good plan for next year.
On the women came throughout the next week, all of them bearing gifts for the baby. Doris and Ruby, Evelyn Prescott, Georgia Holmes, Martha Levine...but Elizabeth's favorite visitors were Mara and Rose. Dressed prettily in matching pink calico dresses that Mara and Martha had sewn, Mara and Rose looked happier than Elizabeth had ever seen them. "These are for your baby." Rose handed Elizabeth a tiny pair of deerskin moccasins with intricate beadwork.
"They are lovely," Elizabeth told her.
"Ma made them," Rose proclaimed proudly.
"Thank you."
"You're welcome," Rose said politely.
"And these are for you," Rose handed Elizabeth a beautiful pair of oyster white moccasins, also decorated with beadwork—delicate roses in shades of pink and red. They were very similar to the ones she'd given Mara.
"Oh, Mara, these are beautiful. Absolutely beautiful. Thank you so much."
Mara leaned over and grasped Elizabeth's hand. "Thank you, Elizabeth. I feel that you gave me a life."
######
On Monday, Elizabeth woke up with a backache and a sense of foreboding. But she tried to act cheerful as the children prepared to go to school.
"I have all my valentines finished," Ruth told her. "One for every person in school. Even the teachers."
"Good for you."
"I'm not giving valentines," JT said. "That's girl stuff."
Elizabeth smiled at him. "Well, someday you might have a sweetheart, and you might want to give her a valentine."
He wrinkled his nose.
"JT does have a sweetheart," Ruth teased. "He just doesn't want anyone to know."
He tossed his sister a dark look.
"If he doesn't want anyone to know, Ruth, why would you tell?"
"We better go or we'll be late," JT warned.
"Have a good day," Elizabeth called. But as soon as they were gone, she was in pain. "Mother," she called weakly.
Clara came quickly, tending to Elizabeth in every way she knew how, but it seemed clear that this baby was on its way. "I sent Eli for Malinda," Clara told Elizabeth as she urged her to drink some cool tea.
It wasn't long until both Malinda and Clara were with Elizabeth, doing what they could to make her comfortable and trying to cheer her. "This is only a couple of weeks early," Clara told her. "And it's possible you got the dates wrong."
"That's right," Malinda said as she wiped her forehead with a cold cloth. "Just look how big you are. That baby must be plenty big."
And when it came time for the baby to be born, Elizabeth had to agree with her friend. The baby was plenty big. And when it emerged into the world, screaming loudly, it sounded perfectly healthy too.
"You have a boy," Clara proclaimed. "A big, beautiful, baby boy."
As Clara worked to clean the infant off, Malinda announced that she was going to tell Eli the good news. It wasn't long until Clara handed the precious bundle to Elizabeth. The moment she looked into his wrinkled little red face and examined his fingers and toes, she burst into tears of happiness. "He looks just fine," she told her mother. "Just fine."
Soon Eli was in the room with them, inspecting his son with awe and relief. He leaned down to kiss her forehead. "You did well."
"She certainly did," Clara said.
"What are you going to name him?" Malinda asked.
Elizabeth looked up at Eli. "His name is Eli Dawson Kincaid."
Eli smiled.
"And he will grow up to be a good man in the Oregon frontier."
######
It wasn't until two weeks later, when Elizabeth was up and around and able to care for her household again, that some surprising news reached the small town of Riverside. It was a mild day, and Elizabeth had taken Eli Jr. outside with her, safely nestled in a basket, as she took down the dried wash. Meanwhile Eli was working nearby on the ever-expanding chicken yard. A scene of bucolic peace and contentment.
But suddenly their serene day was shattered when the children came running and shouting. "We have news!" they cried out together.
"Big news!" Ruth shrieked.
With his saw still in hand and a worried expression, Eli hurried over to see what was wrong.
"Let me tell it," JT said reaching them first.
"Wait for me!" Ruth cried as she hurried to join them.
"We have big news," JT said breathlessly. "Big, big news! It happened on February fourteenth—"
"That's Valentine's Day," Ruth exclaimed.
"Yes. And the same day our baby brother was born," JT added.
"But that's not the news," Ruth said.
"No." JT shook his head. "The news is that on that same day, the thirty-third state was added to the Union."
"Oh?" Elizabeth wasn't sure why that was such news.
"Our state!" Ruth told them.
"Oregon!" JT declared. "Oregon is the thirty-third state in the Union."
"We are a state!" Ruth shouted.
Eli and Elizabeth exchanged looks. "We are a state," she said with wonder.
"Imagine that," Eli said. "The state of Oregon was born on the same day Eli Jr. was born."
Elizabeth smiled down at her baby and then at her family. "May God bless them both—our new baby and our new state."
"And our family," Eli added. "God bless our family."
"And God bless our friends and neighbors," JT declared.
"And our animals," Ruth added as Flax came over, wagging his tail eagerly. "God bless them too."
OREGON STATEHOOD, FEBRUARY 14, 1859
Discussion Questions
1. While on the Oregon Trail, most of the pioneers couldn't wait for the arduous journey to end. Yet many hardships were still ahead. What do you think was the best thing about settling? What was the worst?
2. Imagine that you were a pioneer just arriving in the western frontier. What do you think you'd miss the most from your old life back east?
3. What was your initial reaction to Eli's disappearance with Elizabeth's wagon and beloved team? What did you imagine had happened to him?
4. Malinda is Elizabeth's best friend, and yet they sometimes seem at odds. Why do you think their relationship was so strained?
5. Imagine again that you're a new settler. What, besides food and shelter, would be your first priority in creating your homestead? How would you go about securing it?
6. Elizabeth's stay with Malinda was necessary but stressful. Fortunately it only lasted for a season. Describe an era in your life when you had to endure some discomforts for a spell. What helped you to get through that time?
7. Malinda seems driven to have the perfect wedding. Why do you think she was so obsessed with this?
8. What was your initial response to Reverend Holmes? To Mrs. Holmes? How did you feel about them later on, after you got to know them better?
9. When Malinda planned her quilting bee, she seemed to intentionally leave Elizabeth out. How did you react to her seemingly bad manners? How did you feel later on when you discovered she was actually making the quilt for Elizabeth?
10. Although the Indians were "relocated" from the area, Elizabeth manages to find one of the few still remaining. Considering the other settlers' opinions on these matters, as well as the welfare of her own children, do you think Elizabeth was wise to reach out this woman? Why or why not?
11. There are a number of strong women in this series. Elizabeth, Clara, Jessica, Malinda, Lavinia...to name a few. Describe the qualities you think best sustained women in the western frontier.
12. After Eli and Elizabeth wed, Elizabeth finally got to see the home Eli had built for her and her children. Naturally, she was ecstatic over every little detail. What did you like best about their rustic cabin?
13. Elizabeth knew Mara and her children would survive only if they integrated into White culture. How did you feel when Mara was getting her "makeover"? Describe your reaction to the tea party where Mara met her mother-in-law, Martha.
14. Confined to bed rest toward the end of her pregnancy, Elizabeth was understandably antsy and anxious. For a hands-on frontier woman, doing "nothing" was a challenge. Describe a time when you felt your hands were tied and all you could do was wait. What helped you get through that era?
15. Eli and Elizabeth's first son together was born on the same day Oregon became a state. What do you imagine Eli Jr. would have been doing 40 years later—in 1899?
16. Did anything in this series change the way you look at life? If so, describe.
About the Publisher
* * *
To learn more about Harvest House books and to read sample chapters, log on to our website:
www.harvesthousepublishers.com
HARVEST HOUSE PUBLISHERS
EUGENE, OREGON
* * *
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\section{Introduction}
\label{sec:introduction}
Cosmic voids represent a potentially powerful tool for measuring the cosmological parameters and probing the nature of dark energy \citep[e.g.][]{Li2011,Bos2012,Lavaux2012,Sutter2012,Cai2014a,Cai2014b,Hamaus2014c}. Most cosmological constraints are derived from the structure and dynamics of voids, which are a probe of modified gravity models \citep{Li2012,Clampitt2013,Cai2014,Barreira2015} as well as of the nature of dark matter \citep{Hellwing2010,Yang2014,Massara2015}. In the former case, void profiles are sensitive to the presence of a fifth force, which, while screened in higher density regions, can be large in voids. Such a force leads to emptier and larger voids, due to the faster evacuation of matter from low density regions \citep{Peebles2010,Clampitt2013}. In the latter case, replacing cold dark matter by warm dark matter or including massive neutrinos would lead to less evolved voids, and hence to shallower density profiles.
Up to now, the density and velocity structure of voids has been studied through the use of spherical profiles motivated by the fact that stacking many voids results into spherically symmetric structures \citep[e.g.][]{vandeWeygaert1993,Padilla2005,Ceccarelli2013,Ricciardelli2013,Hamaus2014b,Nadathur2015a}. But individual voids are distinctly non-spherical. While the simple picture of an expanding underdensity in a uniform background suggests that voids should become more spherical as they evolve \citep[][]{Icke1984}, in reality, voids are not isolated and this simplified picture does not hold. There are two major factors that affect the evolution of voids. Firstly, contrary to the case of collapsed structures, void evolution is strongly affected by the tidal field of the surrounding distribution of matter \citep{Platen2008,vandeWeygaert2011}. Secondly, as voids expand, they are squeezed by neighbouring voids. These effects lead to present-day voids that have highly complex shapes \citep{Platen2007,Platen2008,Neyrinck2008,Sutter2012a,Nadathur2014}.
The diversity of void shapes makes the traditional stacking procedure suboptimal for extracting cosmological information. Simply put, the cosmological constraints are derived by comparing the density inside voids with that at their boundaries. For example, in some modified theories of gravity the inner regions of voids are emptier than in the standard cosmological model, with the evacuated matter deposited at the void boundaries. Stacking randomly oriented voids of various shapes leads to an overlap of the voids inner regions and boundaries. This ``blurring'' decreases the density contrast between the inner and outer parts of voids, leading to a lower signal. \MCn{In addition, there is ambiguity in the definition of the void centre used for spherical stacking, with different choices resulting in different density profiles \citep[e.g.][]{Nadathur2015b}.}
In this work, we introduce a new method of both measuring void profiles and stacking voids by taking into account their shape. In contrast to the spherical method, we propose that void profiles should be measured with respect to the void boundary. This leads to a much sharper distinction between the inside, boundary and outside of voids, resulting in at least two major gains. Firstly, it leads to a better understanding of the structure and dynamics of cosmic voids enabling a closer comparison with analytical theories of void evolution. Secondly, it increases the stacked lensing signal of voids, which is the best probe for measuring void density profiles \citep{Higuchi2013,Krause2013}.
The paper is organized as follows. In \refsec{sec:whats_about} we outline the new method by applying it to a simplified void model; in \refsec{sec:data} we describe the cosmological simulation to which we apply the method as well as the void catalogues we construct from it; in \refsecs{sec:density_profile}{sec:weak_lensing} we present the density, velocity and weak lensing profiles obtained using the new \boundary{} stacking approach. We conclude with a short discussion and summary in \refsec{sec:conclusions}.
\section{The boundary profile of voids}
\label{sec:whats_about}
\begin{figure}
\centering
\includegraphics[width=0.66\linewidth,angle=0]{fig_pdf/ toy_example}
\caption{ Illustration of the new method for measuring void profiles. The top panel shows the void boundary, with the actual void shape selected randomly from voids found in an N-body simulation. For simplicity, the void is assigned a constant density, $1+\delta_{\rmn{inside}}=0.1$, inside its boundaries and is embedded in a uniform background with $1+\delta_{\rmn{outside}}=1$. The mass evacuated from inside the void is deposited at the void boundary, which has $1+\delta_{\rmn{boundary}}=30$. The center panel shows the spherical shells around the barycentre of the void that are used for computing the spherical profile. The bottom panel shows lines of equal distance from the void boundary (thick black curve) that are used for computing the \boundary{} profile proposed in this paper. }
\label{fig:toy_example}
\end{figure}
Here we give an overview of the proposed method for computing \boundary{} void profiles, which we illustrate using a simplified model of a void. We construct a void by randomly selecting a shape for it from a cosmological N-body simulation. A cross section through the boundary of the void is shown in the top panel of \reffig{fig:toy_example}. For simplicity, the inner region of the void is assigned constant density, $1+\delta_{\rmn{inside}}=0.1$, where, $\delta=\tfrac{\rho}{\overline{\rho}}-1$, denotes the density contrast. The void is embedded within a uniform background, $1+\delta_{\rmn{outside}}=1$, and the mass evacuated from within the void is deposited uniformly on the boundary, which is shown as a solid curve.
\begin{figure}
\centering
$\begin{array}{c}
\includegraphics[width=.93\linewidth,angle=0]{fig_pdf/ fig_toyProfile_11-sphericalProfile} \\
\includegraphics[width=.93\linewidth,angle=0]{fig_pdf/ fig_toyProfile_01-distanceProfile}
\end{array}$
\caption{ Void density profile. The top panel shows the spherical profile of the simple void model illustrated in \reffig{fig:toy_example}. The vertical grey line marks the effective radius of the void, $\reff$, defined in \eq{eq:effective_radius}. The bottom panel shows the density profile of the void as a function of the distance, $\dist$, from the void boundary. For clarity, we define $\dist$ as having negative values inside the void and positive outside. The vertical grey line marks to the boundary of the void. }
\label{fig:toy_example_profiles}
\end{figure}
Finding the \emph{spherically averaged} profile involves finding a void center, typically the volume-weighted barycentre,and growing concentric shells around it. This process is schematically illustrated in the centre panel of \reffig{fig:toy_example}, where, for clarity, we only show a few radial shells, but, in practice, we employ many more such shells. The spherical profile is given by the mean density of matter inside each shell. Applying this method to our model void provides the spherical density profile shown in the top panel of \reffig{fig:toy_example_profiles}. For small radial distances, which correspond to shells fully enclosed by the void, we recover the input density value, $1+\delta=0.1$. At larger radii, $r\ge9~h^{-1} {\rm Mpc}$, the shells intersect the void boundary giving rise to a ``noisy" profile. Due to the irregular shape of the void, different radial shells have varying degrees of overlap with the void boundary, giving rise to ``noisy" features\footnote{In contrast to our simplified model, in real voids the mass is not distributed uniformly along the void boundary, resulting in even larger ``noisy" features.}. These persist for as long as the shells intersect the boundary, corresponding to $r\le26~h^{-1} {\rm Mpc}$, while for even larger radii we recover the background density. This simple example illustrates that the spherical density profile is a complex convolution of the shape of the void and its actual density distribution.
To calculate the void profile with respect to the \emph{boundary} of the void we compute the boundary distance, $\dist{}$, that corresponds to the minimum distance from each point to the void boundary (see \eq{eq:void_distance} for a formal definition). The outcome is illustrated in the bottom panel of \reffig{fig:toy_example}, where each thin contour line corresponds to points that are at equal distance from the boundary of the model void. Now we can calculate the density profile as a function of $\dist{}$ by computing the mean density inside each shell of constant $\dist{}$ (in practice, we use many more shells than those shown in \reffig{fig:toy_example}). To distinguish between points inside and outside the void we adopt the convention that $\dist{}$ takes on negative values inside the void and positive outside, with $\dist{}=0$ at the void boundary. The resulting profile is plotted in the bottom panel of \reffig{fig:toy_example_profiles} and shows that we recover the actual input density distribution: $1+\delta=0.1$ inside the void, a large value of $1+\delta$ at the void boundary due to the mass evacuated from inside the void, and $1+\delta=1$ outside the void.
The new void profile has two major advantages. Firstly, it is independent of the shape of the void. For example, distorting the boundary of the void in \reffig{fig:toy_example}, while keeping the same density distribution inside and outside the void, would result in exactly the same density profile as a function of $\dist$. Secondly, on average, the mass displaced from inside the void is found at the void boundary, with the resulting density at the boundary being at least an order of magnitude higher than inside the void (\citealt{Sheth2004}, hereafter \hyperlink{labelHypertarget}{SvdW}{}, and \refsec{sec:density_profile}). Thus, while the spherical profile for radial shells that intersect the void boundary is dominated by the density at the boundary and not by the density inside the void, our proposed profile naturally differentiates between the boundary, the inside and the outside of the void.
\section{Void identification}
\label{sec:data}
We make use of the high-resolution Millennium cosmological N-body
simulation \citep[\textsc{MS}{};][]{Springel2005}. The \textsc{MS}{} follows the evolution
of cold dark matter (DM) using $2160^3$ particles, each of mass, $m_p=8.6\times10^8~h^{-1} \rm{M}_{\odot}$, to resolve
structure formation in a periodic cube $500~h^{-1} {\rm Mpc}$ on a side.
The \textsc{MS}{} assumes the WMAP-1 cosmogony \citep{Spergel2003} with the following cosmological
parameters: $\Omega_m=0.23$, $\Omega_\Lambda=0.75$, $h=0.73$, $n_s=1$
and $\sigma_8=0.9$.
We identify voids using mock catalogues constructed from the semi-analytic galaxy formation model of \citet{Guo2011_SAM}. For this, we select only galaxies with stellar masses, $M_\star\ge3.8\times10^{10}~h^{-1} \rm{M}_{\odot}$, such that the number density is $n=3.2\times10^{-3}~h^3 {\rm Mpc}^{-3}{}$, similar to that of typical redshift surveys \citep[e.g.][]{Zehavi2011}. These galaxies are used as input to the Delaunay Tessellation Field Estimator \citep[DTFE;][]{Schaap2000,Weygaert2009,Cautun2011}, which uses a Delaunay triangulation with the galaxies at its vertices to extrapolate a volume filling density field. The resulting density field is used as input to the void identification method. We also apply the DTFE method to the distribution of DM particles to obtain continuous density and velocity fields, which are used for computing the density, velocity and weak lensing profile of voids. Both the galaxy density field and the DM density and velocity fields are stored on a $1280^3$ regular grid with a grid cell size of $0.39 ~h^{-1} {\rm Mpc}$.
The voids are determined using the Watershed Void Finder \citep[WVF;][]{Platen2007}, which identifies voids as the watershed basins of the large scale density field, similar to the ZOBOV void finder \citep{Neyrinck2008}. Compared to other methods, the watershed void finders have the advantage of not imposing any a priori constrains on the size, shape and mean underdensity of the voids they identify \citep{Colberg2008}. The WVF proceeds by first smoothing the galaxy density field with a $2~h^{-1} {\rm Mpc}$ Gaussian filter, whose size corresponds to the typical width of the filaments and walls forming the void boundaries \citep[e.g.][]{Cautun2013,Cautun2014a}. This smoothing is applied in order to dilute any substructures present on the void boundaries \citep[e.g. see][]{Cautun2014a}, which could potentially give rise to artificial voids. The smoothed density field is segmented into watershed basins using the watershed transform implemented using the steepest descent method \citep[e.g.][]{Bieniek2000}. This process is equivalent to following the path of a rain drop along a landscape: each volume element, in our case the voxel of a regular grid, is connected to the neighbour with the lowest density (i.e. steepest descent), with the same process repeated for each neighbour until a minimum of the density field is reached. Finally, a watershed basin is composed of all the voxels whose path ends at the same density minimum.
\MCn{To overcome oversegmentation, the WVF joins the basins that share a boundary with a galaxy density, $\delta_{\rm g}\le-0.8$, since such low values typically separate subvoids embedded within larger voids. This threshold is motivated by the model of an expanding top-hat underdensity for which shell crossing takes place at $\delta=-0.8$ (\hyperlink{labelHypertarget}{SvdW}{}). This top-hat model has several shortcomings when compared to realistic voids, e.g. voids do not have initial top-hat profiles, their expansion is restricted by their environment and observations provide only the galaxy density, not the total matter density, which makes the extent to which $\delta_{\rm g}\le-0.8$ is a realistic threshold debatable.} \MCnn{This step leads to the merging of only $2\%$ of the watershed basins and hence it has no noticeable effect on the profiles of stacked voids.}
\begin{figure}
\centering
\includegraphics[width=\linewidth,angle=0]{fig_pdf/ fig_abundance_01-galaxy_voids}
\caption{ The abundance of galaxy voids in the \textsc{MS}{}. The shaded regions show the three ranges in effective void radius, $\reff{}$, for which we compute average density and velocity profiles. }
\label{fig:void_abundance}
\end{figure}
The distribution of the WVF voids is shown in \reffig{fig:void_abundance} where the voids are characterised by their effective radius, $\reff{}$. This corresponds to the equivalent radius of a sphere with the same volume as the void, i.e.
\begin{equation}
\reff{} = \left( \frac{3 V_{\rm void}}{4\pi} \right)^{\frac{1}{3}}
\label{eq:effective_radius} \;,
\end{equation}
where $V_{\rm void}$ denotes the volume of the void. The figure shows that we identify a wide range of void sizes, from $5$ to $50~h^{-1} {\rm Mpc}$, with the abundance peaking at $\reff\sim15~h^{-1} {\rm Mpc}$. For the rest of this work, we will calculate stacked profiles for voids in three intervals in void size corresponding to $\reff=8-12$, $18-22$ and $30-35~h^{-1} {\rm Mpc}$ (shown as dark shaded regions in \reffig{fig:void_abundance}), which contain 656, 643 and 100 voids, respectively.
These intervals were chosen to probe a variety of void sizes, while at the same time having enough voids to provide reliable statistics.
\MCn{The abundance of WVF voids is similar to that obtained using the ZOBOV void finder when applied to DM tracers with the same number density \citep[][Fig. 1]{Nadathur2015b}, but it is a factor of two higher than when applying ZOBOV to the galaxy distribution \citep[][Fig. 2]{Nadathur2015c}. The difference is likely due to the merging criteria employed by the two void finders (see \citealt{Nadathur2015c} who studied the dependence of the void abundance on merging criteria).} \MCnn{Regardless of these differences, the WVF voids have a similar minimum galaxy density to the ZOBOV voids \citep[see Fig. 1 in][]{Nadathur2015c}.}
\subsection{Spherical profiles}
\label{subsec:data_spherical_profiles}
The spherical profile of a void is computed as a function of the radial distance from the void centre, which we take as the volume-weighted barycentre. \MCn{While there are other potential choices of void centre (e.g. see \citealt{Nadathur2015b}) that result in slightly different spherical profiles, these differences are small when compared to the difference between the spherical and the boundary profile methods.} The void centre is given by $\Vector{x}_{\rm vc}=\sum_{i}\Vector{x}_i/N$, where the sum is over all the $N$ voxels that are part of the void and $\Vector{x}_i$ gives the position of each such voxel. The density of the void at radial distance, $r$, is then given as
\begin{equation}
\delta(r) = \frac{\sum_k w_k \delta_k}{\sum_k w_k}
\label{eq:spherical_profile} \;,
\end{equation}
where the sum is over all the voxels found at a radial distance, $r\pm\tfrac{1}{2}\Delta r$, with $\Delta r$ the radial bin width, and $\delta_k$ is the density of each voxel. The weights, $w_k$, give the overlapping volume between the voxel and the radial bin. This is calculated by generating 64 points regularly distributed inside each voxel; $w_k$ is then the fraction of those points that are found inside the radial bin. This method of calculating void profiles is very similar to the VTFE method of \citet{Nadathur2015a}, except that we use a Delaunay instead of a Voronoi triangulation. Besides the density profile, we also compute the profile of the velocity component along the radial direction, $\vrad$. This is computed similarly to \eq{eq:spherical_profile}, but with the density replaced by the radial component of the velocity, $\vrad = \Vector{v} \cdot \Vector{r} / r$.
\subsection{Boundary profiles}
\label{subsec:data_shape_profiles}
To calculate the shape independent profile we need to identify the boundary or border of each void, which is the density ridge delimiting the watershed basin corresponding to that void. In practice, we compute the void boundary as follows. We loop over all the neighbouring grid cells of each voxel that is part of a void. If one of the neighbours is not part of the same void, then the face connecting the two voxels is identified as the boundary of the void. To speed up the computation, each such face is stored as only one point corresponding to the centre of the face. Finally, the border of the void is given by the union of all those points, i.e. by all the centres of the faces connecting voxels that are not part of the same void. This procedure can be easily expanded to identify the boundary of ZOBOV voids too. In this case, the voids are composed of Voronoi cells, not the cells of a regular grid as for the WVF. The boundary of ZOBOV voids is given by the union of the faces of the Voronoi cells that connect two Voronoi cells that are not part of the same void.
The next step is to compute the distance of each point from the void boundary, which in computer science is referred to as the distance transform. This technique has been previously used to find the galaxies that are the farthest inside voids \citep{Kreckel2011} and to use voids for improving photometric redshift estimation \citep{Aragon-Calvo2014}. The minimum distance from the void boundary to a point with Cartesian coordinate, $\Vector{x}$, is given by
\begin{equation}
\dist =
\begin{cases}
+\; \min_i\left[ \; \left| \Vector{x} - \Vector{y}_i \right| \; \right] & \text{for } \Vector{x} \text{ outside the void}\\
-\; \min_i\left[ \; \left| \Vector{x} - \Vector{y}_i \right| \; \right] & \text{for } \Vector{x} \text{ inside the void}
\end{cases}
\label{eq:void_distance} \;,
\end{equation}
where $\{\Vector{y}_i\}$ denotes the set of points that give the void boundary and $|\,|$ denotes the magnitude of a vector. By convention, the boundary distance is negative for points inside the void and positive outside. One can further define a void boundary distance field, which at each point in space is a vector of magnitude, $|\dist{}|$, given by
\begin{equation}
\distVec = \dist \frac{\Vector{x} - \Vector{y}_j}{\left| \Vector{x} - \Vector{y}_j \right|}
\label{eq:void_distance_field} \;,
\end{equation}
where $j$ denotes the index of the point on the void boundary closest to $\Vector{x}$. The direction of $\distVec$ is perpendicular to the surfaces of constant $\dist$ (see bottom panel of \reffig{fig:toy_example}) and always points outwards. The void boundary distance field is computed separately for each void, using a kd-tree constructed from the set of points that gives the boundary of the void.
For each void, the boundary distance takes a minimum value, $\dist_{\rm min}$, that corresponds to the point inside the void farthest from the boundary. We always have $-\dist_{\rm min}\le\reff{}$ (note that $\dist_{\rm min}$ is negative), with equality only for spherical voids; on average, we find $\dist_{\rm min}/\reff{}=-(0.62\pm0.06)$ ($1\sigma$ standard deviation). For spherical voids, the boundary distance, $\dist$, is equivalent to the radial position, and, for this particular case, the spherical and boundary profiles are exactly the same.
The void density profile as a function of $\dist{}$ is computed similarly to \eq{eq:spherical_profile}:
\begin{equation}
\delta(\dist{}) = \frac{\sum_k w_k \delta_k}{\sum_k w_k}
\label{eq:distance_profile} \;,
\end{equation}
but now the sum is over voxels found at a distance $\dist{}\pm\tfrac{1}{2}\Delta\dist{}$ from the void boundary, with $\Delta\dist{}$ the width of the $\dist{}$ bin. The weight, $w_k$, is given by the fraction of the 64 uniformly distributed points inside each voxel that are within the required distance from the void boundary. As in \refsec{subsec:data_spherical_profiles}, we define the velocity component along the direction of $\distVec{}$ as $\vrad(\dist) = \Vector{v} \cdot \distVec / \dist$.
\subsection{Stacking}
\label{subsec:void_stacking}
The stacked profile is computed as an average over the individual profiles of voids in a narrow $\reff{}$ range. For spherical profiles, we average as a function of the rescaled radial distance, $r/\reff{}$. For \boundary{} profiles, we average over the individual voids at constant boundary distance, $\dist{}$. In this latter case, the minimum distance, $\dist_{\rm min}$, can differ even between voids of equal radii. As a result, at very low $\dist{}$ values only a subset of the voids contribute to the stacked \boundary{} profile. This effect, which our calculation accounts for, becomes important only for the lowest $\dist{}$ bins and can be easily spotted due to the large error bars associated with these points. The uncertainty for the two stacking procedures is given as the $1\sigma$ interval in the distribution of the mean value obtained using 1000 bootstrap samples.
\section{Void density profile}
\label{sec:density_profile}
We now apply the \boundary{} profile analysis to galaxy voids found in the \textsc{MS}{}. To demonstrate the power of this new method, we compare with the outcome of the conventional approach based on spherically averaged profiles.
\subsection{Individual voids}
\begin{figure}
\centering
$\begin{array}{c}
\includegraphics[width=0.95\linewidth,angle=0]{fig_pdf/ fig_individualProfile_01-dens_spherical} \\
\includegraphics[width=0.95\linewidth,angle=0]{fig_pdf/ fig_individualProfile_02-dens_distance}
\end{array}$
\caption{ The density, $1+\delta$, profile of six randomly selected voids that span a range of effective void sizes, $\reff{}$. The top panel shows the spherical profile as a function of the rescaled radial position, $r/\reff{}$. The bottom panel shows the \boundary{} profile as a function of the distance, $\dist{}$, from the boundary of the void. }
\label{fig:density_profiles_random_voids}
\end{figure}
\begin{figure}
\centering
$\begin{array}{c}
\includegraphics[width=0.95\linewidth,angle=0]{fig_pdf/ fig_stackedProfiles_01-dens_spherical} \\
\includegraphics[width=0.95\linewidth,angle=0]{fig_pdf/ fig_stackedProfiles_02-dens_distance}
\end{array}$
\caption{ The stacked density profile for voids in three ranges in effective radius, $\reff{}$.
The two panels show the spherical (top) and the \boundary{} (bottom) profiles. }
\label{fig:density_profiles_stacked}
\end{figure}
In \reffig{fig:density_profiles_random_voids} we show the density profile of six random voids selected to span a wide range of sizes. These profiles and the subsequent ones are computed using the full DM particle distribution and hence give the overdensity of matter. The \emph{spherical} profile is shown as a function of the rescaled radial distance, $r/\reff{}$, which is used for determining self-similar and universal void profiles \citep{Hamaus2014b,Nadathur2015a}. While there is a large variation between the different voids and between neighbouring radial bins, on average individual voids are underdense for $r/\reff{}\lsim0.5$ and show no consistent features at larger distances. Two of the voids have $\delta>0$ at their centres that can be explained either by the presence of a substructure inside the void \citep{Beygu2013,Rieder2013} or by the void centre being close to the boundary \citep{Nadathur2015b}.
Compared to the spherical profile, the \emph{\boundary{}} profile is very different and has more features, indicative of the fact that, since voids have highly complex shapes, taking a spherical average erases or damps many features. In addition, the \boundary{} profile shows a better qualitative agreement between the various voids: underdense for $\dist\le-3~h^{-1} {\rm Mpc}$; a sharp density peak at the void boundary, $\dist=0$; and close to mean background density for $\dist\ge5~h^{-1} {\rm Mpc}$. The density peak at the void boundary is expected since voids identified using watershed-based methods are delimited by a density ridge. The height of this ridge is largely given by the mass contained in the most massive haloes, which explains the variation in height between different voids. Massive haloes can also be found outside the void, resulting in sporadic peaks in the density profile, but only very rarely inside the void - no such example is present in \reffig{fig:density_profiles_random_voids}. The width of the density peak at the boundary is given by the typical size of the massive haloes as well as that of the filaments and walls that delimit the void \citep[e.g.][]{Cautun2013,Cautun2014a}.
\subsection{Stacked profiles}
In \reffig{fig:density_profiles_stacked} we present the mean density profiles of voids in three $\reff{}$ bins chosen to probe a variety of void sizes (see \reffig{fig:void_abundance}). The \emph{spherical} stacked profiles are underdense in the inner parts, with $\delta$ slowly rising to a maximum at $r\simeq\reff$, followed by a gradual transition towards the average background density \citep{Hamaus2014b,Nadathur2015a}.
The \emph{\boundary{}} profile paints a different picture of the structure of voids. In the inner most parts, $\dist\raisebox{-0.3ex}{\mbox{$\stackrel{<}{_\sim} \,$}}-4~h^{-1} {\rm Mpc}$, the density is very low, $-0.9\le\delta\le-0.5$, and nearly constant, with only a very small increase in $\delta$ with $\dist$. This is followed by a very steep rise of a density ridge at the boundary, which decreases nearly as fast at $\dist\ge0$. At even further distances, the density gradually reaches the background value.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth,angle=0]{fig_pdf/ void_neighbours}
\caption{ A simple model to understand the \boundary{} profile. The thick black curves show the boundary of the central void and that of its neighbours, which are coloured according to their density, with dark and light grey showing high and low density. The highest density regions correspond to the intersection points of two or more void boundaries, with the density decreasing farther away. The thin curves shows contours of constant distance, $\dist$, from the boundary of the central void, with two of those contours, $\dist=-5$ and $5~h^{-1} {\rm Mpc}$, highlighted in cyan. The outer contours intersect the boundary of neighbouring voids and hence correspond to a higher mean density than the inner contours. }
\label{fig:void_neighbours}
\end{figure}
The \boundary{} density profile can be understood within the multiscale picture of the cosmic web. Void interiors are not fully empty, but instead are criss-crossed by tenuous filaments and walls that become more densely packed as one approaches the massive structures that delimit the voids \citep{Cautun2014a}. Thus, the mean density is expected to increase close to the void boundary, in accord with the results shown in the bottom panel of \reffig{fig:density_profiles_stacked}. Close to the void boundary, the behaviour is dominated by the prominent filaments and sheets that delimit the void and that are substantially denser than the tenuous structures found inside the void \citep{Cautun2014a}. The picture outside the void is complicated by the presence of neighbouring voids and their own dense ridges, as illustrated in \reffig{fig:void_neighbours}. The density profile is not symmetric around $\dist{}=0$ since neighbouring voids can have different sizes, and hence different ridge thicknesses. In addition, the outer contours intersect the boundary of neighbouring voids. Due to clustering, the density varies along the void ridge, with higher density typically associated with the intersection points of two or more void boundaries. The density profile is sensitive to this clustering, which would explain why the slope, $\left|\tfrac{d\delta}{d\dist}\right|$, is shallower outside the void than inside the void.
Compared to the spherical profile, the average \boundary{} profile shows smaller differences between voids of different sizes and is close to a self-similar profile. Before discussing these differences, we proceed by fitting the \boundary{} profile with the empirical function:
\begin{equation}
\rho =
\begin{cases}
\rho_{\rm in} \left( 1 + \left(\frac{\rho_{\rm max}}{\rho_{\rm in}} -1\right) e^{-\frac{|\dist|}{t_{\rm in}}} \right) (1{-}\alpha|\dist|)
& \text{for } \dist<0\\
\rho_{\rm out} \left( 1 + \left(\frac{\rho_{\rm max}}{\rho_{\rm out}}-1\right) e^{-\frac{|\dist|}{t_{\rm out}}} \right)
& \text{for } \dist\ge0
\end{cases}
\label{eq:fit_function_density} \;,
\end{equation}
where \MCn{$\rho = {\bar\rho} (1+\delta)$ is the matter density and ${\bar\rho}$ is the mean background density}. The fit is a continuous function composed of two parts that describe the inner, $\dist<0$, and outer, $\dist\ge0$, mean density profiles, with $\rho(\dist{}=0)=\rho_{\rm max}$.
\begin{figure}
\centering
\includegraphics[width=\linewidth,angle=0]{fig_pdf/ fig_stackedProfiles_51-best_fit_errrors}
\caption{ The best-fitting function (Eq. \ref{eq:fit_function_density}) to the \boundary{} density profile of voids. The top panel shows the mean density for voids with $\reff=18-22~h^{-1} {\rm Mpc}$ (symbols with error bars) and the best-fit function (dashed line). The remaining panels show the ratio between the data and the best-fitting function for voids of different sizes. The fit was done using only data points with $\dist\le10~h^{-1} {\rm Mpc}$.}
\label{fig:density_profile_fit}
\end{figure}
The very interiors of the void are characterised by the density parameter, $\rho_{\rm in}$, and by the slope $\alpha$, the latter accounting for the fact that the density increases with $\dist{}$. The density ridge at $\dist\simeq0$ is well described by an exponential function that takes a maximum value, $\rho_{\rm max}$, at $\dist=0$. This ridge is not symmetric with respect to $\dist=0$ and so we have two parameters in the exponential, $t_{\rm in}$ and $t_{\rm out}$, that give the thickness of the inner and outer void boundary, respectively. Just outside the void boundary, the density has yet to converge to the background value, so there is an additional parameter, $\rho_{\rm out}$, to account for this effect. The $\dist{}\ge0$ part of the fitting function should include an additional component to account for the transition towards the background density at large $\dist$, but, for simplicity, we omit such a component. Our function is characterised by six parameters which is similar to other empirical fits to spherical void profiles: \citet{Hamaus2014b} proposed a four parameter fit that was latter extended by \citet{Barreira2015} to a five parameter fit to give a better description of voids identified in the DM density field. Compared to the \boundary{} profile, the spherical one smooths over many density features, so it is not surprising that the former requires more parameters.
\begin{figure}
\centering
\includegraphics[width=1.\linewidth,angle=0]{fig_pdf/ fig_stackedProfiles_53-best_fit_parameters_vs_t_in}
\caption{ The best-fitting parameters of \eq{eq:fit_function_density} obtained from stacked void density profiles. The top panel show the thickness of the inner void boundary, $\tin$, as a function of void radius. The remaining panels show the dependence of the other fit parameters: $\rho_{\rm in}$, $\alpha$, $\rho_{\rm max}$, $\rho_{\rm out}$ and $\tout$ as a function of $\tin{}$. The error bars give the $1\sigma$ uncertainty. The dashed lines show that the best-fitting parameters follow simple relations with $\reff{}$ (top panel) and $\tin{}$ (remaining panels). }
\label{fig:best-fitting_parameters}
\end{figure}
The upper panel of \reffig{fig:density_profile_fit} shows that the empirical function of \eq{eq:fit_function_density} describes, to very good approximation, the mean density profile. To better assess the fit quality, the lower panels of \reffig{fig:density_profile_fit} show the ratio between the measured profile and the best-fitting value for three void samples. The fit matches the data well, except for a few points around $\dist\simeq0$, which show a ${\sim}10\%$ difference, and for the $D\le-15~h^{-1} {\rm Mpc}$ region of the largest voids, which shows a systematic deviation from the best-fit.
\reffig{fig:best-fitting_parameters} shows the best-fitting parameters and their $1\sigma$ errors for voids of different size. These were computed using the Markov chain Monte Carlo method implemented in the emcee package \citep{Foreman-Mackey2013}. The figure shows $\tin{}$ as a function of $\reff{}$ and the remaining parameters as a function of $\tin{}$. The best-fitting parameters follow linear relations with $\tin{}$, which in turn can be parametrized as a quadratic function of $\reff{}$. This suggests that the parametrization of \eq{eq:fit_function_density} is overdetermined and that the number of free parameters is too large \citep[similar relations between the fit parameters have been reported by][]{Hamaus2014b}. \eq{eq:fit_function_density} can be rewritten by expressing $\rho_{\rm in}$, $\alpha$, $\rho_{\rm max}$, $\rho_{\rm out}$ and $\tout$ as a linear function of $\tin{}$ (two parameters in each case) and, in turn, by expressing $\tin{}$ as a quadratic function of $\reff{}$ (three parameters). This results in a 13 parameter function that fits in one step voids of various sizes. We repeated the fit using these parametrizations and obtained similarly good fits.
According to \reffig{fig:best-fitting_parameters}, void interiors are characterised by a nearly constant density, $\rho_{\rm in}$, but by different values of the density slope, $\alpha$, with larger voids having more slowly varying density profiles. The height of the density ridge, $\rho_{\rm max}$, is largest for small voids since these are typically embedded in overdense regions. This is illustrated also by the $\rho_{\rm out}/\bar\rho$ density parameter that is larger than $1$ for the smallest voids and that decreases with void size. The density ridge is asymmetric and is thinner inside the void, i.e. $t_{\rm in}<t_{\rm out}$ (see the discussion of \reffig{fig:void_neighbours}).
We also find that the smallest voids have lower $t_{\rm in}$ values and larger $t_{\rm out}$ values than the largest voids. The increase of $t_{\rm in}$ and decrease of $t_{\rm out}$ with void size can be a manifestation of the age characterising voids of different size. Just as low mass haloes, small voids are dynamically old, so the density ridge has been squeezed for a longer time. Larger voids, which originate from larger scale density fluctuations, have not had enough time to pile up mass at the ridge to the same extent as the small ones.
\subsection{The self-similarity of stacked profiles}
\label{subsec:self-similarity}
The \boundary{} density profile of voids of different size is very similar, but not exactly the same (see bottom panel of \reffig{fig:density_profiles_stacked}). Those differences are minimized, or even disappear entirely, when rescaling the inner profile by the thickness, $t_{\rm in}$, of the inner void boundary. The rescaled profiles are given in \reffig{fig:density_profiles_scaling} which clearly shows that all voids, independently of their size, have a self-similar profile. To better highlight this, in the bottom panel of the figure we take the ratio with respect to a weighted mean density. This weighted mean was obtained by averaging, at fixed $\dist/t_{\rm in}$ values, over voids of different sizes, with the contribution of each sample weighted by the inverse of its associated error. Small systematic differences with void size are seen only for $\dist\simeq0$, which probably arise because small void are embedded in overdense regions while large voids are found in predominantly underdense regions. For the rest, all the density profiles lie on the same curve with less than ${\sim}5\%$ scatter.
\begin{figure}
\centering
\includegraphics[width=\linewidth,angle=0]{fig_pdf/ fig_profileScalling_01-dens_distance}
\caption{ The self-similarity of voids. Top panel: the density profile, $1+\delta$, as a function of the rescaled void boundary distance, $\dist/\tin$, where $\tin{}$ is the thickness of the inner void boundary as determined by fitting \eq{eq:fit_function_density} to the density profile. The symbols correspond to voids of various effective radii, $\reff{}$. All voids have a self-similar density profile independent of $\reff{}$. Bottom panel: the ratio between the profiles and a weighted mean of the values in the various $\reff{}$ bins showing that there is less than $5\%$ scatter among voids of various sizes. }
\label{fig:density_profiles_scaling}
\end{figure}
The self-similar nature of \boundary{} profiles suggest that voids of different sizes have, on average, the same dynamical characteristics. In contrast, the same self-similarity is not seen for spherical profiles (see top panel of \reffig{fig:density_profiles_stacked}). This could be due to the limitations of spherical profiles, among which, most importantly, is the mixing and inability to separate between the inside, boundary and outside of voids, as we exemplified in \refsec{sec:whats_about}. This fits in with the results shown in the bottom panel of \reffig{fig:density_profiles_stacked} where the differences between voids of various sizes are most pronounced in the boundary and outside regions of the voids.
Self-similar profiles are obtained only after rescaling by the thickness of the inner void ridge, $t_{\rm in}$. This suggests that the void interior knows about the boundary or vice-versa, and that the two evolve together. The former possibility seems ruled out by the simple picture of an expanding spherical underdensity in which the evolution of a shell of matter of radius, $r$, depends only on the mass contained within $r$ \citep[][\hyperlink{labelHypertarget}{SvdW}{}, but see \citealt{Ruiz2015}]{Fillmore1984}.
Spherical void profiles have also been claimed to be self-similar \citep[e.g.]{Ricciardelli2014,Nadathur2015a}, but there are contradictory results in the literature \citep[e.g.][this work]{Hamaus2014b,Sutter2014b,Nadathur2015b}. The self-similarity of spherical profiles seems to be dependent on several factors: the void finder, the population of tracers used to identify the voids and the tracers used to measure the void profile. This could be the case for \boundary{} profiles too, though it is reassuring that self-similarity of \boundary{} profiles has been found for both voids identified using galaxies (this work) and for voids identified in the DM density field \citep{Cautun2015e}.
\subsection{Comparison to analytical predictions}
\begin{figure}
\centering
\includegraphics[width=\linewidth,angle=0]{fig_pdf/ fig_stackedProfiles_91-top-hat_voids}
\caption{ Comparison of analytical and measured density profiles of voids. \MCn{The solid line corresponds to an uncompensated top-hat spherical underdensity that gives rise to a void with mean density, $1+\delta=0.3$.} The dotted and dashed curves give the spherical and \boundary{} distance profiles of \textsc{MS}{} voids with $\reff=18-22~h^{-1} {\rm Mpc}$. The top-hat void shows a good qualitative agreement with the boundary distance profile of \textsc{MS}{} voids. }
\label{fig:density_profiles_top-hat}
\end{figure}
It is illustrative to compare with analytical predictions of void profiles, among which the isolated spherical underdensity model is the most popular (\citealt{Fillmore1984}, \hyperlink{labelHypertarget}{SvdW}{}; see \citealt{vandeWeygaert2011} for a more elaborate void evolution model that includes ellipsoidal collapse and that accounts for the effect of the external tidal field). For this purpose, we select a top-hat spherical underdensity that \MCn{gives rise to a void of radius, $\reff{}=20~h^{-1} {\rm Mpc}$, and density, $1+\delta=-0.3$, similar to the mean density of stacked \textsc{MS}{} voids with sizes, $\reff=18-22~h^{-1} {\rm Mpc}$.} While realistic voids do not have initial top-hat profiles, such a simple model captures most of the features of initial underdensities representative of cosmological environments (see Fig. 3 of \hyperlink{labelHypertarget}{SvdW}{}). \reffig{fig:density_profiles_top-hat} shows the density profile of the resulting void as a function of the rescaled radial distance, $r/\reff{}$. The figure also shows the spherical and \boundary{} profile of \textsc{MS}{} voids with sizes, $\reff=18-22~h^{-1} {\rm Mpc}$. To plot all three profiles on the same x-axis, we give the \boundary{} profile in terms of the rescaled coordinate, $(\dist+\reff)/\reff$, with $\reff{}=20~h^{-1} {\rm Mpc}$.
The top-hat profile shows large differences with respect to the spherical profile of \textsc{MS}{} voids, but is in approximate agreement with the \boundary{} profile of the same voids. In particular, the \boundary{} profile matches the main prediction of the analytical model, the formation of a density ridge at the edge of the void. Thus, this simple model offers a qualitative description of the density profiles of voids, but only after accounting for the fact that real voids are non-spherical.
\MCn{Note, however, that there are significant differences between the top-hat model and the \boundary{} density profile of realistic voids, which are driven by many factors. Our goal is not to test the accuracy of the analytical model, but rather to show that such a model performs better than one would naively expect from a comparison to spherical profiles. For example, the edge of \textsc{MS}{} voids contains more mass than the analytical prediction since the boundaries of realistic voids accrete matter also from outside the void (note the $1+\delta<1$ values of the \boundary{} profile for rescaled radial positions larger than $1.3$). Secondly, replacing the uniform top-hat underdensity with more realistic initial density profiles results in a more gradual increase of the density ridge (\hyperlink{labelHypertarget}{SvdW}{}), which is closer to the profile of \textsc{MS}{} voids.}
\reffig{fig:density_profiles_top-hat} is a first step towards testing one of the central assumptions of the \hyperlink{labelHypertarget}{SvdW}{} void abundance model, which is that void formation is well described by the evolution of \MCnnn{an isolated spherical underdensity}. We have shown that we can find \emph{a} top-hat model that qualitatively matches the \emph{mean} density of stacked voids. It remains to be seen if the parameters of this top-hat model are also the ones required to match the initial conditions of realistic voids. Furthermore, for the model to be realistic, the match should work not only for stacked samples, but also for individual voids.
\section{Void velocity profile}
\label{sec:velocity_profile}
The velocity field of voids is another property that can be better understood by analysing \boundary{} profiles. As for the density profile, we proceed by comparing the spherical and \boundary{} velocity profiles. We focus on the peculiar velocity component, $\vrad$, that gives the rate at which matter is evacuated in comoving coordinates through a surface of $r=\textrm{const}$ and $\dist=\textrm{const}$ for the spherical and \boundary{} profiles, respectively. Positive $\vrad$ values correspond to a net outflow of matter through the surface while negative values correspond to an inflow.
\MCn{For investigating void velocity profiles we use the same objects, both individual and stacked samples of voids, as we used when studying the density profiles in \refsec{sec:density_profile}. \reffigS{fig:velocity_profiles_random_voids}{fig:velocity_profiles_stacked} show the corresponding $\vrad$ profiles for individual and stacked voids. For brevity, we focus our discussion on the stacked velocity profiles, with individual voids showing similar trends, albeit with large individual variations.}
\begin{figure}
\centering
$\begin{array}{c}
\includegraphics[width=\linewidth,angle=0]{fig_pdf/ fig_individualProfile_11-velRad_spherical} \\
\includegraphics[width=\linewidth,angle=0]{fig_pdf/ fig_individualProfile_12-velRad_distance}
\end{array}$
\caption{ The peculiar velocity profile of the six randomly selected voids shown in \reffig{fig:density_profiles_random_voids}. It shows the velocity component, $\vrad$, along the direction of $\Vector{r}$ and $\distVec{}$, respectively. The two panels show the spherical (top) and \boundary{} (bottom) profiles of those voids. }
\label{fig:velocity_profiles_random_voids}
\end{figure}
\begin{figure}
\centering
$\begin{array}{c}
\includegraphics[width=\linewidth,angle=0]{fig_pdf/ fig_stackedProfiles_11-velRad_spherical} \\
\includegraphics[width=\linewidth,angle=0]{fig_pdf/ fig_stackedProfiles_12-velRad_distance}
\end{array}$
\caption{ The peculiar velocity profile as a function of radial distance, $r$, for spherical stacking (top) and as a function of void boundary distance, $\dist$, for \boundary{} stacking (bottom). It shows the velocity component, $\vrad$, along the direction of $\Vector{r}$ and $\distVec{}$, respectively. To guide the eye, the data points are connected with solid lines. The dotted lines show the linear theory prediction for $\vrad$ given the average density profiles of \reffig{fig:density_profiles_stacked}. }
\label{fig:velocity_profiles_stacked}
\end{figure}
The spherical velocity profiles show outflows from voids, which peak at ${\sim}0.6\reff$, and that are followed by regions with lower outflow velocities or even inflows. The nearly-linear increasing outflow for $r{\raisebox{-0.3ex}{\mbox{$\stackrel{<}{_\sim} \,$}}}0.5\reff$ indicates that void interiors expand faster than the average universe showing a so-called super-Hubble outflow \citep{Icke1984,vandeWeygaert2011,Aragon-Calvo2013}.
For the \emph{\boundary{}} profile, the velocity, $\vrad$, increases until near the void boundary, $\dist\raisebox{-0.3ex}{\mbox{$\stackrel{<}{_\sim} \,$}}-3~h^{-1} {\rm Mpc}$, and is then followed by a rapid switch from outflow to inflow. This behaviour at $\dist\simeq0$ is consistent with infall onto the void boundary, which, given its high density, is the main local driver of dynamics. At further distances from the void boundary, the velocity slowly converges towards $0$, as expected.
Given the density profiles shown in \reffig{fig:density_profiles_stacked}, we can use linear theory to predict the $\vrad$ values \citep[e.g. see][]{vandeWeygaert1993}, which are shown as dotted lines in \reffig{fig:velocity_profiles_stacked}. The linear predictions are given by
\begin{equation}
v_{\rm \parallel, lin} = -\frac{H f}{\overline{\rho}_{\rm m}} \frac{M(<x)}{S(x)}
\label{eq:velocity_linear_approximation} \;,
\end{equation}
with $H$ the Hubble factor, $f\simeq\Omega_{\rm m}^{0.55}$ the linear growth factor and $\overline{\rho}_{\rm m}$ the mean background density of matter. The symbol $x$ stands for the radial distance, $r$, for spherical profiles and for the distance, $\dist{}$, for \boundary{} profiles. The factor $M(<x)$ denotes the mass contrast inside $x$ and $S(x)$ denotes the area of a surface of constant $x$. See \refappendix{appendix:velocity_linear} for details and for a short derivation of the relation.
The linear theory prediction agrees with the data for the spherical profile, except for a few small systematic effects: the velocity of small voids is overpredicted while that of large voids is underpredicted. These discrepancies, seen also by \citet{Hamaus2014b}, have been attributed to the effect of surrounding structures on void interiors \citep{Ruiz2015}. In the case of \boundary{} profiles, the linear theory is in agreement only for the void interior, i.e. $\dist\lsim0$, and at large distances, $\dist\gsim10~h^{-1} {\rm Mpc}$. Large discrepancies are present at the void boundary and just outside the void where the linear predictions can be off by up to $100~\rm{km/s}$. Such differences are not surprising since linear theory is valid in the regime $|\delta|\ll1$. For spherical \MCn{stacking}, while the average $\delta$ is not very small, it is below unity at every point. In contrast, the \boundary{} \MCn{stacking} has very large values of $\delta$, as high as 3 at the void edge, which explains why large discrepancies are seen only at, and just outside, the void boundary. \MCn{Moreover, as shown in \reffig{fig:density_profiles_random_voids}, individual voids have density values above unity for spherical profiles as well, so linear theory would break down in such cases too. The difference is that for spherical profiles the position of the $\delta>1$ region varies from void to void, so departures from linear theory average out when stacking many such objects, whereas for the \boundary{} profile the departures are always at the same position, $\dist\simeq0$.}
\begin{figure}
\centering
\includegraphics[width=\linewidth,angle=0]{fig_pdf/ fig_stackedProfiles_13-vel_radial_distance}
\caption{ \MCn{The boundary profiles of the velocity component along the radial vector, $\Vector{r}$, (solid with symbols) and along the boundary distance vector, $\distVec$ (dotted). It shows that the outflow from voids is preferentially along the radial direction increasing until close to the void edge.} }
\label{fig:velocity_comparison}
\end{figure}
\MCn{We find that the $\vrad{}$ peak is highest for spherical profiles and that the same peak is up to $20\%$ lower for \boundary{} profiles, even though in the latter case the velocity increases until close to the void edge. To explain this, \reffig{fig:velocity_comparison} shows the \boundary{} profile for the velocity component along the radial direction. For comparison, the dashed lines show the profile of the velocity component along $\distVec$, which corresponds to the solid lines with symbols in the bottom panel of \reffig{fig:velocity_profiles_stacked}. For $\dist<0$, the radial velocity is larger than the velocity component along $\distVec$ which shows that the outflow from voids is preferentially directed radially. \reffig{fig:velocity_comparison} also shows that the radial velocity, when binned according to $\dist$, increases until close to the void boundary and it is then followed by a very steep decrease at the edge of the void. This contrasts with the spherical profile of the radial velocity (see top panel of \reffig{fig:velocity_profiles_stacked}), which shows a peak at $r\sim0.6\reff{}$ and not at the edge of the void, i.e. $r\sim\reff{}$.}
\begin{figure}
\centering
$\begin{array}{c}
\includegraphics[width=\linewidth,angle=0]{fig_pdf/ fig_stackedProfiles_81-boundary_velRad_vs_Reff} \\
\includegraphics[width=\linewidth,angle=0]{fig_pdf/ fig_stackedProfiles_82-boundary_velRad_PDF}
\end{array}$
\caption{ The velocity of the void boundary, $v_{\rm \parallel;\; boundary}$, as a function of void size. Negative values correspond to contracting voids and positive values to expanding voids. The top panel shows this velocity for voids stacked according to their size, $\reff{}$. It shows the velocity at $\dist=0$ (solid curve) and the mean velocity over the interval $|\dist|\le1~h^{-1} {\rm Mpc}$ (dashed curve), which is more robust. The bottom panel shows the probability distribution function (PDF) of the ridge velocity for individual voids of various sizes. The distribution is very broad with each sample having both expanding and contracting voids. }
\label{fig:velocity_boundary}
\end{figure}
The \boundary{} profile offers a natural way of discriminating between expanding and contracting voids. For example, expanding voids correspond to a positive $\vrad{}$ value at their boundary since the boundary is moving outwards. The top panel of \reffig{fig:velocity_boundary} shows the values of the velocity at the boundary, $\dist=0$, and also the $\vrad{}$ value averaged over the interval $|\dist{}|\le1~h^{-1} {\rm Mpc}$, with the latter being less prone to noise. The plot shows that, on average, small voids are contracting while large ones are expanding, with voids of $\reff\sim25~h^{-1} {\rm Mpc}$ being at the transition between the two behaviours. \MCnnn{Using the mean density of the large-scale region in which voids are embedded, previous studies have characterised the voids as under- or overcompensated \citep[e.g.][]{Ceccarelli2013,Hamaus2014b,Nadathur2015c}. For example, the smallest two void stacks in \reffig{fig:velocity_profiles_stacked} are overcompensated while the larger voids are slightly undercompensated. This distinction can be determined, for example, using the sign of the radial velocity at $r\gsim1.5\reff{}$ (see top panel of \reffig{fig:velocity_profiles_stacked}), with positive values corresponding to underdense regions and vice versa. Combining this with our analysis of the void boundary dynamics, we find that overcompensated voids are predominantly contracting while the undercompensated ones are predominantly expanding.}
Using the \boundary{} profile one can determine even for individual voids if they are expanding or contracting, as we show in the bottom panel of \reffig{fig:velocity_boundary}. For example, while most small voids are contracting, there is also a significant fraction that are expanding. Similarly for the largest voids: while most are expanding, there are large contracting voids too. \MCn{Thus, expanding and contracting voids cannot be differentiated using just their size, $\reff{}$, and additional void properties need to be considered \citep[see e.g.][]{Nadathur2015c}.}
\section{Weak lensing from voids}
\label{sec:weak_lensing}
We now address how \boundary{} stacking can be used to enhance the weak lensing signal of voids. Since it is a small effect, void lensing is difficult to measure \citep{Melchior2014}, although recently multiple detections of this signal have been reported \citep{Clampitt2014,Gruen2015}. Increasing the signal to noise of this measurement, by either having a larger sample of voids and/or by improving how voids are stacked, would result in a powerful cosmological probe, especially for tests of modified gravity theories \citep{Cai2014,Barreira2015}.
\begin{figure}
\centering
$\begin{array}{c}
\includegraphics[width=\linewidth,angle=0]{fig_pdf/ fig_weakLensing_01-spherical} \\
\includegraphics[width=\linewidth,angle=0]{fig_pdf/ fig_weakLensing_02-distance}
\end{array}$
\caption{ The stacked tangential shear, $\shear{}$, of voids in three ranges in effective radius, $\reff{}$. The top panel shows the spherically averaged result. The bottom panel shows the result when voids are stacked with respect to their boundary. The error bars show the 1$\sigma$ uncertainties due to object-to-object variation. }
\label{fig:lensing_tangential_shear}
\end{figure}
\begin{figure}
\centering
$\begin{array}{c}
\includegraphics[width=\linewidth,angle=0]{fig_pdf/ fig_weakLensing_11-spherical} \\
\includegraphics[width=\linewidth,angle=0]{fig_pdf/ fig_weakLensing_12-distance}
\end{array}$
\caption{ Same as \reffig{fig:lensing_tangential_shear}, but for the the stacked lensing convergence, $\conv{}$, of voids. }
\label{fig:lensing_convergence}
\end{figure}
\MCn{Within the thin lens and the Born approximation, the weak lensing signal is determined by the surface mass density,}
\begin{equation}
\Sigma(\bm{\xi}) = {\overline{\rho}_m} \int \delta(\bm{\xi},z) \mathrm{d}z
\label{eq:lensing_smd} \;,
\end{equation}
where $\bm{\xi}$ is the position vector in the plane of the lens and $z$ is the direction along the line-of-sight. We compute $\Sigma(\bm{\xi})$ for three lines-of-sight that correspond to the simulation principal axes. For each line-of-sight we obtain $\Sigma(\bm{\xi})$ on a $1280^2$ regular grid with grid spacing $0.39~h^{-1} {\rm Mpc}$. \MCn{We then proceed to compute the lensing potential, $\Psi$, via the relation
\begin{equation}
\nabla^2_{\xi}\Psi(\bm{\xi}) =2\frac{\Sigma(\bm{\xi}) }{\Sigma_{\rm c}}
\; ,
\end{equation}
with the Laplacian operator restricted to the plane of the lens.} The quantity, $ \Sigma_c = c^2D_{\rm S}/{(4\pi G D_{\rm L}D_{\rm LS})} $, is the critical surface mass density for lensing, where $D_{\rm S}$, $D_{\rm L}$ and $D_{\rm LS}$ denote the angular diameter distance between the observer and the source, the observer and the lens, and the lens and the source. The exact value of $\Sigma_c$, which depends on the characteristics of the lensing survey, is unimportant when comparing between the spherical and boundary stacking approaches.
\MCn{For each point, we compute the convergence, $\kappa$, and the shear, ${\bm\gamma}=(\gamma_1,\gamma_2)$, as
\begin{align}
\kappa(\bm{\xi}) & = \tfrac{1}{2} \left[\Psi_{11}(\bm{\xi}) + \Psi_{22}(\bm{\xi})\right] \equiv \Sigma(\bm{\xi}) / \Sigma_{\rm c}\\
\gamma_1(\bm{\xi}) & = \tfrac{1}{2} \left[\Psi_{11}(\bm{\xi}) - \Psi_{22}(\bm{\xi})\right] \\
\gamma_2(\bm{\xi}) & = \Psi_{12}(\bm{\xi}) \equiv \Psi_{21}(\bm{\xi})
\label{eq:lensing_shear} \;,
\end{align}
where the subscripts of $\Psi$ denote derivatives with respect to the two coordinate axes in the plane of the lens.}
\MCn{For \emph{spherical} stacking, the lensing signal is averaged as a function of the projected radial distance, $r_{\rm 2D}$, from the void centre. This results in the convergence, $\kappa(r_{\rm 2D})$, which is a mean value inside a spherical shell of radius $r_{\rm 2D}$. In the case of the shear, we are interested in the tangential component, $\gamma_{\rm t}$, given by
\begin{align}
\gamma_{\rm t} & = -\gamma_1 \cos(2\theta) + \gamma_2 \sin(2\theta)
\label{eq:lensing_tangential_shear} \;,
\end{align}
where $\theta$ is the angle between the first coordinate axis and the position of the point with respect to the void centre. After computing $\kappa(r_{\rm 2D})$ and $\gamma_{\rm t}(r_{\rm 2D})$ for each void, we stack the voids according to their effective radius and across the three different lines-of-sight for which we computed $\Sigma$. Since the projected matter distribution is different along those orthogonal lines-of-sight, averaging their lensing signal increases the signal-to-noise ratio.}
For \emph{\boundary{}} stacking, the procedure is slightly different, since we need to identify the boundary of the void in the lens plane. We do so by slicing the boundary of the void, which is a 2D surface, along the plane of the lens, with the slice centred at the point inside the void that is the farthest from the void boundary (this is the point corresponding to the minimum distance, $\dist_{\rm min}$). Following this, we obtain a closed curve in the lens plane that corresponds to one particular choice of the void boundary (see discussion below), which is then used to compute the distance in the plane of the lens, $\dist_{\rm 2D}$, of each surface element. Following this, for every void we compute the mean value of the convergence as a function of $\dist_{\rm 2D}$ resulting in the quantity $\kappa(\dist_{\rm 2D})$. \MCn{The tangential shear is computed using \eq{eq:lensing_tangential_shear} but with $\theta$ denoting the angle between the first coordinate axis and the 2D boundary distance vector, $\distVec{}_{\rm 2D}$, at that point. Finally, we stack all voids of similar size and across the three lines-of-sight.}
We note that this is just one possible choice for stacking with respect to the void \boundary{}, and may not be the optimal choice. For lensing studies, it is better to identify 2D voids in thin redshift slices, since this greatly enhances the lensing signal \citep{Clampitt2014}. The boundary of these 2D voids is a 1D curve in the plane of the sky. In such a case there is no ambiguity in choosing the 1D void boundary in the plane of the lens.
In \reffig{fig:lensing_tangential_shear} we show the void tangential shear obtained using the two stacking procedures. The spherically averaged $\shear$ shows the characteristic dip of void lensing at $r_{\rm 2D}\simeq\reff{}$, which is nearly the same for the three void samples. This depression is more pronounced when using \boundary{} stacking for which the signal is twice as large. Using \boundary{} stacking increases the convergence, $\conv$, also by a factor of about two, as can be inferred from \reffig{fig:lensing_convergence}. This doubling of the lensing signal is the result of a better separation between the void border, where most of the mass is, and the void interior, which is mostly empty. This factor of two represents only a lower limit to the potential improvements resulting from the use of \boundary{} stacking. Likely, the gain can be increased further by optimizing the selection of the void boundary in the plane of the sky.
\section{Discussion and conclusions}
\label{sec:conclusions}
We have proposed a new method for characterising voids that has several advantages over the conventional spherical approach, as demonstrated by our analysis of galaxy voids in the Millennium cosmological simulation. This approach, which we call the \emph{\boundary{} profile}, is based on describing the structure of voids as a function of the distance from their boundary, which allows for a natural segregation of the inner, boundary and outer regions of each void.
Voids are characterised by two defining features: they consist of large, fairly underdense volumes, with the evacuated matter found in a thin overdense region at the boundary, and they have very complex, non-spherical, shapes. The spherical averaging approach is inadequate for describing voids due to this very combination of features, as we exemplify for a simplified void model (\reffigS{fig:toy_example}{fig:toy_example_profiles}) and for realistic voids (\reffigS{fig:density_profiles_random_voids}{fig:density_profiles_stacked}). This is a consequence of the fact that taking a spherical average over an intrinsically non-spherical object leads to a complex juxtaposition of the inner, border and outer regions of that object, with each region having very different density. By contrast, the \boundary{} profile method differentiates, by construction, between those regions.
The \boundary{} profile analysis revealed that the interior of voids is characterised by low densities that increase slowly towards the void boundary. This is followed by a steep rise of a density ridge at the void boundary, which decreases nearly as fast outside the void. The peak of the density ridge corresponds to $1+\delta\simeq4$ while the interior of the void has $1+\delta\simeq0.2-0.4$. \MCn{We found a simple fitting function (Eq. \ref{eq:fit_function_density}) that describes fairly well the void density profiles and that can be parametrized in terms of a single quantity, the void effective radius (see \reffig{fig:best-fitting_parameters}). This parametrization provides a convenient way of describing the variation of density profiles with void size and allows for simple comparisons to theoretical models of void evolution, such as the spherical top-hat underdensity model (\hyperlink{labelHypertarget}{SvdW}{}).}
The \boundary{} density profile is self-similar, i.e. independent of void size, after rescaling the distance coordinate by the thickness of the void's inner density ridge (see \reffig{fig:density_profiles_scaling}). This suggests that the void interior knows about the void boundary or vice versa, and that the evolution of the two is coupled. This simple behaviour is reminiscent of the self-similar nature of dark matter haloes \citep{Navarro1996,Navarro1997} whose origin, while not well understood, must reflect the scale-free nature of gravity. \MCn{In contrast to haloes for which the characteristic scale is determined by the matter distribution in the innermost region, for voids the characteristic scale is determined by the matter distribution at the edge of the void.}
The \boundary{} profile of the peculiar velocity reveals outflows from voids, which peak at a few Megaparsecs from the edge of the void, and an external infall region onto the void boundary. \MCn{These outflows are preferentially directed along the radial direction, with the radial velocity being larger than the velocity component pointing towards the closest void edge (see \reffigS{fig:velocity_profiles_stacked}{fig:velocity_comparison}).} The \boundary{} profiles are especially suited for capturing the infall onto the void boundary, which is not seen for spherical profiles, and for determining if the voids are contracting or expanding (see \reffig{fig:velocity_boundary}).
The \boundary{} stacking method increases the weak lensing signal of voids by at least a factor of two when compared to the classical spherical stacking method. This gain can potentially be further increased by optimizing the selection of the void boundary on the plane of the sky (Cautun et al., in prep.). This gain in lensing signal boosts the utility of voids as cosmological probes, especially when applied to future large volume surveys like DESI \citep{Levi2013}, LSST \citep{LSST2009} and Euclid \citep{Laureijs2011}.
\section*{Acknowledgements}
We thank Rien van de Weygaert for insightful and valuable comments on an earlier draft of the paper and we are grateful to Alexandre Barreira, Shaun Cole, Jiaxin Han, Baojiu Li and Nuala McCullagh for helpful discussions.
We also thank the anonymous referee for their comments that have helped us improve the paper.
This work was supported in part by ERC Advanced Investigator grant COSMIWAY
[grant number GA 267291] and the Science and Technology Facilities Council
[grant number ST/F001166/1, ST/I00162X/1]. YC was supported by the Durham Junior Research Fellowship
This work used the DiRAC Data Centric system at Durham University,
operated by ICC on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk).
This equipment was funded by BIS National E-infrastructure capital
grant ST/K00042X/1, STFC capital grant ST/H008519/1, and STFC DiRAC
Operations grant ST/K003267/1 and Durham University. DiRAC is part
of the National E-Infrastructure. Data from the Millennium/Millennium-II
simulation is available on a relational database accessible from
http://galaxy-catalogue.dur.ac.uk:8080/Millennium .
\bibliographystyle{mnras}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,427 |
package org.apache.thrift.transport;
import java.nio.file.Files;
import java.nio.file.Path;
import java.util.Arrays;
import junit.framework.TestCase;
public class TestTSimpleFileTransport extends TestCase {
public void testFresh() throws Exception {
//Test write side
Path tempFilePathName = Files.createTempFile("TSimpleFileTransportTest", null);
Files.delete(tempFilePathName);
byte[] input_buf = new byte[]{1, 2, 3, 4, 5, 6, 7, 8, 9, 10};
TSimpleFileTransport trans_write = new TSimpleFileTransport(tempFilePathName.toString(),false, true, false);
assert (!trans_write.isOpen());
trans_write.open();
assert(trans_write.isOpen());
trans_write.write(input_buf);
trans_write.write(input_buf,2,2);
trans_write.flush();
trans_write.close();
//Test read side
TSimpleFileTransport trans = new TSimpleFileTransport(tempFilePathName.toString(),true, false);
assert(trans.isOpen());
//Simple file trans provides no buffer access
assert(0 == trans.getBufferPosition());
assert(null == trans.getBuffer());
assert(-1 == trans.getBytesRemainingInBuffer());
//Test file pointer operations
assert(0 == trans.getFilePointer());
assert(12 == trans.length());
final int BUFSIZ = 4;
byte[] buf1 = new byte[BUFSIZ];
trans.readAll(buf1, 0, BUFSIZ);
assert(BUFSIZ == trans.getFilePointer());
assert(Arrays.equals(new byte[]{1, 2, 3, 4}, buf1));
int bytesRead = trans.read(buf1, 0, BUFSIZ);
assert(bytesRead > 0);
for (int i = 0; i < bytesRead; ++i) {
assert(buf1[i] == i+5);
}
trans.seek(0);
assert(0 == trans.getFilePointer());
trans.readAll(buf1, 0, BUFSIZ);
assert(Arrays.equals(new byte[]{1, 2, 3, 4}, buf1));
assert(BUFSIZ == trans.getFilePointer());
trans.close();
Files.delete(tempFilePathName);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,232 |
Q: Probability random variable binomial distribution Every day, a lecture may be canceled due to inclement weather with probability 0.05. Class
cancelations on different days are independent.
(a) There are 15 classes left this semester. Compute the probability that at least 4 of
them get canceled.
(b) Compute the probability that the tenth class this semester is the third class that gets
canceled.
How to solve the part b of this question?
A: Hint:
For the tenth class this semester to be the third class that gets cancelled, you want
*
*exactly two of the first nine classes to be cancelled
*the tenth class to be cancelled
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,496 |
Keechi Creek is a stream in Palo Pinto County, Texas. It originates two miles east of Graford, and empties into the Brazos River northwest of Mineral Wells.
See also
List of rivers of Texas
References
USGS Geographic Names Information Service
USGS Hydrologic Unit Map - State of Texas (1974)
Rivers of Texas
Rivers of Palo Pinto County, Texas | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 450 |
\section{Introduction}
Quantum information protocols and quantum simulations with cold atomic
systems extensively utilize strong dipole-dipole interaction between
the laser-excited Rydberg states of atoms
\cite{Lukin2001, Saffman2010, Saffman2016, Petrosyan2017, Schauss1455, Labuhn2016, Bernien2017,Lienhard2017}.
Atoms are routinely trapped and manipulated near the surfaces
of superconducting atom chips \cite{Fortagh2007,Bernon2013},
and can couple to on-chip microwave planar resonators
\cite{Verdu2009,Hattermann2017}.
Atom chips are typically cooled by liquid helium to the temperature
of $T\simeq 4$~K.
At such temperatures, the black body radiation is sufficiently suppressed
and does not detrimentally affect the Rydberg state lifetime
\cite{Beterov2009,Saffman2016,Mack2015}. Rydberg qubits then exhibit
long coherence times on atom chips \cite{Avigliano2014,Teixeira2015}.
This should be contrasted with superconducting qubits that require
chips at $T\lesssim 100$~mK temperature to operate
\cite{Wallraff2004, Chiorescu2004, You2011}.
Huge electric dipole transitions between the Rydberg states of atoms
allow their strong coupling to microwave planar waveguides \cite{Hogan2012}
and resonators \cite{Petrosyan2009}. The cavity field can then serve
as a quantum bus to mediate long distance interactions and quantum gates
between the Rydberg qubits \cite{Petrosyan2008,Pritchard2014}.
At $T\simeq 4$~K, however, the microwave cavity or waveguide has large
population of thermal photons. This would preclude long-distance quantum
gates and coherent state transfer, unless inherently temperature-resistant
schemes are used \cite{Sarkany2015, Xiang2017, Vermersch2017}.
In this paper we consider a \textsc{swap} operation between a pair of Rydberg
qubits exchanging a virtual photon via a thermally populated microwave cavity.
We show that high-fidelity state or excitation transfer is only possible
if the coupling strengths of both atoms to the cavity mode is the same.
In the general case of unequal couplings, the photon number-dependent
Stark shifts of the atomic transitions precludes the state transfer.
In order to overcome this problem, we propose to continuously extract thermal
photons from the cavity mode using a separate, laser-driven ensemble of
cooling atoms. This breaks the symmetry between the cavity field population
from the thermal environment and depopulation due to the photon decay
and extraction, greatly reducing the effective mode temperature.
We find that despite the increased rate of photon decay from the cavity,
high-fidelity \textsc{swap} operation between the Rydberg qubits is
feasible with a realistic experimental setup.
The paper is organized as follows. In Sec.~\ref{sec:system} we describe
the system and show that the excitation transfer between two atoms
in non-equivalent positions of a thermal cavity is suppressed.
In Sec.~\ref{sec:cavcool} we present a method to reduce the number of
the thermal photons of the cavity by their continuous extraction with
an optically pumped atomic ensemble. In Sec.~\ref{sec:cavcooltrnans}
we show that this leads to increased transfer probability between the
two atoms, followed by conclusions in Sec.~\ref{sec:conclud}
\section{The compound system}
\label{sec:system}
\subsection{Mathematical formalism}
We consider a compound system consisting of two two-level (Rydberg)
atoms $i=1,2$ coupled to a common mode of a microwave resonator,
as sketched in Fig.~\ref{fig:ALscheme}(a). The system is thus
equivalent to the Jaynes--Cummings model for two atoms interacting
with a cavity field \cite{PLDP2007,Haroche2006}.
In the frame rotating with the frequency of the cavity mode $\omega$,
the Hamiltonian is given by
\begin{eqnarray}
\mathcal{H}/\hbar &=& \sum_{i=1,2}
\big[ \mbox{$\frac{1}{2}$} \Delta_i ( \hat{\sigma}_{bb}^{(i)} - \hat{\sigma}_{aa}^{(i)} )
+ g_i ( \hat{c} \hat{\sigma}_{ba}^{(i)} + \hat{\sigma}_{ab}^{(i)} \hat{c}^{\dag}) \big].
\end{eqnarray}
Here $\Delta_{i} \equiv \omega_{ba}^{(i)} - \omega$ is the detuning of atom
$i$ on the transition $\ket{a} \leftrightarrow \ket{b}$ from the cavity
mode frequency, $\hat{\sigma}_{\mu \nu}^{(i)} \equiv \ket{\mu}_i\bra{\nu}$
($\mu,\nu=a,b$) are atomic operators, and $g_i$ is the coupling strength
of atom $i$ to the cavity mode described by the photon creation
$\hat{c}^{\dag}$ and annihilation $\hat{c}$ operators. The coupling
strength $g_i = (\wp_{ab}/\hbar) \varepsilon_{\mathrm{c}} u(\mathbf{r}_i)$
is proportional to the dipole matrix element $\wp_{ab}$ of the atom
on the transition $\ket{a} \to \ket{b}$, field per photon
$\varepsilon_{\mathrm{c}}$ within the cavity volume, and the cavity mode
function $u(\mathbf{r}_i)$ at the position $\mathbf{r}_i$ of the atom.
Since atoms 1 and 2 may be at non-equivalent positions, $g_1$ and $g_2$
are in general different. We also assume that the detunings $\Delta_1$ and
$\Delta_2$ can be individually controlled by, e.g., spatially inhomogeneous
electric (dc Stark) or magnetic (Zeeman) fields or by a focused
non-resonant laser inducing an ac Stark shift of one of the Rydberg
states \cite{Leseleuc2017}.
\begin{figure}[t]
\centerline{\includegraphics[width=8.7cm]{aatsch.eps}}
\caption
Schematics of the system.
(a) Two atoms with levels $\ket{a},\ket{b}$ are coupled to the cavity field
$\hat{c}$ with the coupling strengths $g_{\mathrm{1}}$ and $g_{\mathrm{2}}$
and detunings $\Delta_{\mathrm{1}}$ and $\Delta_{\mathrm{2}}$, respectively.
(b) Transitions between states $\ket{ba,n}$ and $\ket{ab,n}$ containing
$n$ cavity photons can occur via two paths involving intermediate
non-resonant states $\ket{bb,n-1}$ and $\ket{aa,n+1}$. }
\label{fig:ALscheme}
\end{figure}
Several relaxation processes are affecting the system.
We assume that both Rydberg states $\ket{a}$ and $\ket{b}$
decay with approximately the same rate $\Gamma$ to some lower state(s)
$\ket{s}$ which are decoupled from the cavity field. These decay processes
are described by the Liouvillians \cite{PLDP2007,Haroche2006}
\begin{eqnarray}
\mathcal{L}_{\mathrm{a}_i} \hat{\rho} &=&
\mbox{$\frac{1}{2}$} \Gamma (2 \hat{\sigma}_{sa}^{(i)} \hat{\rho} \hat{\sigma}_{as}^{(i)} - \hat{\sigma}_{aa}^{(i)} \hat{\rho} - \hat{\rho} \hat{\sigma}_{aa}^{(i)})
\nonumber \\ & &
+ \mbox{$\frac{1}{2}$} \Gamma (2 \hat{\sigma}_{sb}^{(i)} \hat{\rho} \hat{\sigma}_{bs}^{(i)} - \hat{\sigma}_{bb}^{(i)} \hat{\rho} - \hat{\rho} \hat{\sigma}_{bb}^{(i)})
\end{eqnarray}
acting onto the density operator $\hat{\rho}$ of the total system.
The relaxation of the cavity field toward the thermal equilibrium
with rate $\kappa$ is described by
\begin{eqnarray}
\mathcal{L}_{\mathrm{c}} \hat{\rho} &=&
\mbox{$\frac{1}{2}$} \kappa (1+\bar{n}_{\mathrm{th}}) (2 \hat{c} \hat{\rho} \hat{c}^{\dag}
- \hat{c}^{\dag} \hat{c}\hat{\rho} - \hat{\rho} \hat{c}^{\dag} \hat{c} )
\nonumber \\ & &
+ \mbox{$\frac{1}{2}$} \kappa \bar{n}_{\mathrm{th}} (2 \hat{c}^{\dag} \hat{\rho} \hat{c} - \hat{c} \hat{c}^{\dag} \hat{\rho}
- \hat{\rho} \hat{c} \hat{c}^{\dag}), \label{eq:Lc}
\end{eqnarray}
where $\bar{n}_{\mathrm{th}} = (e^{\hbar \omega/k_{\mathrm{B}}T} -1)^{-1}$ is the mean number
of thermal photons in the cavity mode at temperature $T$
\cite{PLDP2007,Haroche2006}.
The density operator of the total system obeys the master equation
\begin{equation}
\partial_t \hat{\rho} = -\frac{i}{\hbar} [\mathcal{H},\hat{\rho}] +
\mathcal{L}_{\mathrm{a}_1}\hat{\rho} + \mathcal{L}_{\mathrm{a}_2}\hat{\rho}
+ \mathcal{L}_{\mathrm{c}}\hat{\rho}. \label{eq:MasterEq}
\end{equation}
We solve numerically the equations for the density matrix of the system
whose Hilbert space is a tensor product space of two three-state atoms,
$\{\ket{a},\ket{b}, \ket{s} \}_{i=1,2}$, and the cavity field with the photon
number states $\{\ket{n}\}$ truncated at sufficiently large $n \leq 100$.
\subsection{Adiabatic elimination of the cavity mode}
We are interested in the state or excitation transfer between
the atoms 1 and 2 using the cavity mode as a quantum bus.
Consider the states $\ket{ba,n}$ and $\ket{ab,n}$ with either
one or the other atom excited, while the cavity mode contains
$n$ photons, see Fig.~\ref{fig:ALscheme}(b).
There are two transition paths between these states via the
intermediate states $\ket{aa,n+1}$ and $\ket{bb,n-1}$ involving
a photon addition to or subtraction from the cavity mode.
In order to minimize the effects of relaxation and thermalization of
the cavity mode during the transfer, the atoms
should exchange virtual cavity photons. We therefore choose
the atomic detunings to be similar and large enough,
$\Delta_{\mathrm{1,2}} \gg |\Delta_1 - \Delta_2|, g_{\mathrm{1,2}} \sqrt{n_{\mathrm{max}}}$,
where $n_{\mathrm{max}}$ is the maximal number of photons that can be in the
cavity with appreciable probability; typically, $n_{\mathrm{max}}$ can be taken
as $10 \times \bar{n}_{\mathrm{th}}$, and we recall that at thermal equilibrium the
probability distribution of the cavity photon number is
\begin{equation}
P_n = \frac{\bar{n}_{\mathrm{th}}^n}{(1+\bar{n}_{\mathrm{th}})^{n+1}}.
\end{equation}
Using the perturbation theory, we adiabatically eliminate
the nonresonant intermediate states $\ket{aa,n+1}$ and $\ket{bb,n-1}$.
We then obtain that the energies of states $\ket{ba,n}$ and $\ket{ab,n}$
are Stark shifted by the cavity field as
\begin{subequations}
\begin{eqnarray}
E_{ba,n} & \simeq & \mbox{$\frac{1}{2}$} \delta + \frac{g_1^2(n+1)}{\Delta_1 +S(n+1)}
- \frac{g_2^2 n}{\Delta_2+Sn}, \\
E_{ab,n} & \simeq & - \mbox{$\frac{1}{2}$} \delta + \frac{g_2^2(n+1)}{\Delta_2 +S(n+1)}
- \frac{g_1^2 n}{\Delta_1 +Sn},
\end{eqnarray}
\end{subequations}
where $\delta \equiv \Delta_1 - \Delta_2$ and
$S \equiv \frac{g_1^2}{\Delta_1} + \frac{g_2^2}{\Delta_2}$.
Simultaneously, the transition amplitude
between the states $\ket{ba,n}$ and $\ket{ab,n}$ is given by
\begin{eqnarray}
G(n) &\simeq &\frac{g_1 g_2 (n+1)}{2}
\left[\frac{1}{\Delta_1 +S(n+1)} + \frac{1}{\Delta_2 +S(n+1)}\right]
\nonumber \\ & &
- \frac{g_1 g_2 n}{2} \left[\frac{1}{\Delta_1 +Sn} + \frac{1}{\Delta_2 +Sn}\right] ,
\end{eqnarray}
where the first and the second terms correspond to the amplitudes of
transitions via the states $\ket{aa,n+1}$ and $\ket{bb,n-1}$, respectively.
\subsection{Transfer probability}
States $\ket{ba,n}$ and $\ket{ab,n}$ have the energy difference
$\delta E(n) \equiv E_{ba,n} - E_{ab,n}$
and are coupled with rate $G(n)$.
Since we assumed $\Delta_1 \approx \Delta_2 = \Delta \gg \delta$, we can cast
the energy difference and transition rate as
\begin{eqnarray}
\delta E(n) & \simeq & \delta + \frac{g_1^2 - g_2^2}{\Delta} (2n+1)
, \label{eq:deEn} \\
G(n) &\simeq & \frac{g_1 g_2}{\Delta} \left( 1-\frac{\delta}{2\Delta}
- \frac{g_1^2 + g_2^2}{\Delta^2} (2n+1) \right), \label{eq:Geffn}
\end{eqnarray}
where we neglected terms of the order of $\delta \frac{g^2}{\Delta^2}$,
$\frac{g^4}{\Delta^3}$ and higher.
We thus see that both $\delta E(n)$ and $G(n)$ depend on the cavity
photon number $n$. If the coupling strengths $g_1$ and $g_2$ are
the same, the differential Stark shifts of levels $\ket{ba,n}$ and
$\ket{ab,n}$ becomes $n$ independent. Then, by choosing $\delta =0$,
we obtain that both atoms have the same transition frequency,
$\delta E(n) = 0 \, \forall \,n$. Simultaneously, due to interference
between the two transition paths, $G(n)$ only weakly depends on the
cavity photon number $n$, leading to resonant exchange of excitation
between the atoms. In Fig.~\ref{fig:oscil} upper panel, we show the
oscillations between the states $\ket{ba}$ and $\ket{ab}$ of the two
atoms with equal couplings $g_1=g_2$ to the cavity which has a large
mean thermal photon number $\bar{n}_{\mathrm{th}}$ corresponding to a broad photon
number distribution $P_n$.
\begin{figure}[t]
\centerline{\includegraphics[width=7.5cm]{oscil.eps}}
\caption
Oscillations between the states $\ket{ba}$ (red solid line) and
$\ket{ab}$ (blue solid line) in a cavity, as obtained from the
numerical simulations of the Master equation (\ref{eq:MasterEq}).
The parameters are $\Delta_1 = \Delta_2 = 30 g_2$,
$\kappa = 10^{-3} g_2$, $\bar{n}_{\mathrm{th}} = 5$,
$\Gamma = 3\times 10^{-4} g_2$,
while $g_1 = g_2$ (upper panel) and $g_1 = 1.4 g_2$ (lower panel).
The dashed black line results from the approximate expression
(\ref{eq:p0bosc}).}
\label{fig:oscil}
\end{figure}
When, however, the coupling strengths $g_1$ and $g_2$ are different,
the energy difference $\delta E(n)$ is photon number dependent. Then,
in a thermal cavity, $\bar{n}_{\mathrm{th}} > 0$, with a broad distribution $P_n$ of photon
numbers, $\delta E(n)$ cannot be made to vanish for all $n$. In fact
the difference between $\delta E(n)$ for various $n$ is of the order
of the transition rate $G$, which suppresses the amplitude of oscillations
between the states $\ket{ba}$ and $\ket{ab}$, as shown in
Fig.~\ref{fig:oscil} lower panel.
We can derive an approximate expression for the transfer probability
between the states $\ket{ba}$ and $\ket{ab}$ as follows.
Assume that we start in state $\ket{ba,n}$ at time $t=0$. Without relaxations,
the probability for the system to be in state $\ket{ab,n}$ would be given by
\begin{equation*}
p_{ab,n}(t) = \left| \frac{G(n)}{\bar{G}(n)} \right|^2
\sin^2 \left[\bar{G}(n) t \right],
\end{equation*}
where $\bar{G}(n) = \sqrt{|G(n)|^2 + \frac{1}{4} |\delta E(n)|^2}$ is the
generalized Rabi frequency for the oscillations between $\ket{ba,n}$
and $\ket{ab,n}$.
Relaxations will result in exponential damping of the transfer probability.
The decay of states $\ket{a}$ and $\ket{b}$ of each atom with rate $\Gamma$
leads to multiplication by $e^{- \Gamma t}$.
We neglect for now the cavity relaxation and will consider its effect later.
Thus the transfer probability from state $\ket{ba,n}$ to state $\ket{ab,n}$
is given by
\begin{equation}
p_{ab,n}(t) = e^{- 2 \Gamma t}
\left| \frac{G(n)}{\bar{G}(n)} \right|^2 \sin^2 \left[ \bar{G}(n) t \right],
\end{equation}
If initially there is an equilibrium photon number distribution $P_n$ in the
cavity, the total probability of transfer between $\ket{ba}$ and $\ket{ab}$
is given by
\begin{equation}
p_{ab}(t) = \sum_n P_n p_{ab,n}(t) , \label{eq:p0bosc}
\end{equation}
This expression approximates well the exact dynamics of the system
for small $\kappa \ll G$ as verified in Fig.~\ref{fig:oscil}.
\section{Reducing the number of thermal photons}
\label{sec:cavcool}
The thermal photons in the cavity thus preclude efficient state
transfer between the atoms when their couplings to the cavity mode have,
in general, different strength, $g_1 \neq g_2$.
We now outline a method to reduce the number of photons in the cavity,
which will significantly increase the transfer probability.
We will use an ensemble of stationary trapped atoms extracting photons
from the cavity by continuous optical pumping, attaining thereby an
equilibrium with a smaller mean number of photons.
We note a conceptually similar approach \cite{Raimond2001,Nogues1999,Bernu2008}
to extract thermal photons from a microwave cavity by sending across
it a sequence of atoms prepared in the lower Rydberg state, which is
typically done in the transient regime to achieve a nearly empty cavity
until it equilibrates with the thermal environment.
\subsection{Cavity cooling by photon extraction}
Our strategy to cool the cavity is to continuously extract the photons
from it with a rate $\gamma_{\mathrm{c}} \gg \kappa$. This process is described
by the Liouvillian
\begin{eqnarray}
\mathcal{L}_{\mathrm{c}}^{\prime} \hat{\rho} &=&
\mbox{$\frac{1}{2}$} \gamma_{\mathrm{c}} (2 \hat{c} \hat{\rho} \hat{c}^{\dag}
- \hat{c}^{\dag} \hat{c}\hat{\rho} - \hat{\rho} \hat{c}^{\dag} \hat{c} ) ,
\end{eqnarray}
which should be added to $\mathcal{L}_{\mathrm{c}} \hat{\rho}$ in Eq.~(\ref{eq:Lc}).
The photon extraction thus breaks the balance between the usual
photon decay to and addition from the thermal reservoir.
It follows from the Master equation for the cavity field that
the photon number probabilities obey the equation
\[
\partial_t P_n = d(n+1) P_{n+1} + a n P_{n-1} - d n P_n - a (n+1) P_n ,
\]
where $a= \kappa \bar{n}_{\mathrm{th}}$ and $d = \kappa (\bar{n}_{\mathrm{th}} + 1) + \gamma_{\mathrm{c}}$.
In the steady state, $\partial_t P_n = 0$, we have the detailed balance
$d (n+1) P_{n+1} = a (n+1) P_{n}$ [and $d n P_{n} = a n P_{n-1}$] for any
transition $n \leftrightarrow n \pm 1$. This leads to $P_n = (a/d)^n P_0$,
and upon normalization of the probability distribution $P_n$, we obtain
the usual expression
\begin{equation}
P_n = \frac{\bar{n}_{\mathrm{eff}}^n}{(1+\bar{n}_{\mathrm{eff}})^{n+1}} ,
\end{equation}
corresponding to the equilibrium with the effective thermal photon
number $\bar{n}_{\mathrm{eff}} = \frac{\bar{n}_{\mathrm{th}}}{1 + \gamma_{\mathrm{c}}/\kappa}$ reduced from $\bar{n}_{\mathrm{th}}$
by a factor of $(1 + \gamma_{\mathrm{c}}/\kappa)$.
In Fig.~\ref{fig:pncool} we show the equilibrium photon number distribution
in a cavity at a temperature corresponding to large $\bar{n}_{\mathrm{th}}$. Upon
continuous photon extraction with rate $\gamma_{\mathrm{c}} \gg \kappa$, we obtain
a thermal distribution, with a much smaller effective photon number $\bar{n}_{\mathrm{eff}}$,
which is highly peaked around $n=0$.
\begin{figure}[t]
\centerline{\includegraphics[width=7.5cm]{pncool.eps}}
\caption
Equilibrium photon number distribution $P_n$ in the cavity with
$\bar{n}_{\mathrm{th}} =5$ (blue narrow bars) and with continuous photon extraction with
rate $\gamma_{\mathrm{c}} = 9 \kappa$ (red thicker bars) leading to $\bar{n}_{\mathrm{eff}} = 0.5$.}
\label{fig:pncool}
\end{figure}
\subsection{Photon extraction by optical pumping}
To extract the photons from the thermal cavity, we envision a system
depicted in Fig.~\ref{fig:coolsch}. An ensemble of $N_\mathrm{c}$ ``cooling''
atoms in the ground state $\ket{g}$ are trapped near the cavity antinode.
A laser field acts on the transition from state $\ket{g}$ to a Rydberg
state $\ket{i}$ with the Rabi frequency $\Omega$ and large detuning
$\Delta_{\mathrm{c}} \gg \Omega$. Each atom is coupled to the cavity field $\hat{c}$
on the Rydberg transition $\ket{i}\to\ket{r}$ with strength
$g_{\mathrm{c}} \ll \Delta_{\mathrm{c}}$. Upon adiabatic elimination of the nonresonant
intermediate state $\ket{i}$, we obtain an effective Rabi frequency
$\Omega_{n}^{(2)} = \Omega g_{\mathrm{c}} \sqrt{n}/\Delta_{\mathrm{c}}$ for the two-photon
transition $\ket{g,n} \to \ket{r,n-1}$ which involves absorption
of a laser photon and a cavity photon.
We can write the equations for the amplitudes $A_{g,n}$ and $A_{r,n-1}$
of states $\ket{g,n}$ and $\ket{r,n-1}$ as
\begin{subequations}
\begin{eqnarray}
\partial_t A_{g,n} &=& -i \Omega_{n}^{(2)} A_{r,n-1} , \\
\partial_t A_{r,n-1} &=& - \mbox{$\frac{1}{2}$} \Gamma_r A_{r,n-1} - i \Omega_{n}^{(2)} A_{g,n} ,
\end{eqnarray}
\end{subequations}
where $\Gamma_r$ is the population decay rate of the Rydberg state $\ket{r}$
and we assume the two-photon resonance. Assuming $\Gamma_r \gg \Omega_{n}^{(2)}$
(see below), we can set $\partial_t A_{r,n-1}=0$ and obtain the incoherent
transition rate $\mbox{$\frac{1}{2}$} R_n = \frac{|\Omega_{n}^{(2)}|^2}{\Gamma_r/2}$ from
$\ket{g,n}$ to $\ket{r,n-1}$. With $N_\mathrm{c}$ cooling atoms, we can
then identify the photon extraction rate via $\gamma_{\mathrm{c}} n = N_\mathrm{c} R_n$,
leading to
\begin{equation}
\gamma_{\mathrm{c}} = N_\mathrm{c} \frac{4 \Omega^2 g_{\mathrm{c}}^2}{\Delta_{\mathrm{c}}^2 \Gamma_r} .
\end{equation}
Note that contributions of individual atoms add incoherently to the total
extraction rate $\gamma_{\mathrm{c}}$ and the possible variation of the coupling strength
$g_{\mathrm{c}}$ for different atoms can be absorbed into redefinition of the atom
number $N_\mathrm{c}$.
In the above analysis, we have also neglected the (Rydberg blockade)
interactions between the atoms in state $\ket{r}$. This approximation
is justified for moderate number of photons $n$ and large enough
$\Gamma_r \gg \Omega_{n}^{(2)}$, such that the probability of having
simultaneously two or more atoms in state $\ket{r}$ is small.
\begin{figure}[t]
\centerline{\includegraphics[width=2.5cm]{coolsch.eps}}
\caption
Schematics of the cooling setup.
Atoms in the ground state $\ket{g}$ are excited to the Rydberg state
$\ket{r}$ by a two-photon transition via intermediate nonresonant
state $\ket{i}$. The resonant two-photon transition involves a
photon absorption from the laser field acting on the transition
$\ket{g}\to\ket{i}$ with Rabi frequency $\Omega$ and detuning
$\Delta_{\mathrm{c}} \gg \Omega$, and a photon absorption from the cavity
field coupled with strength $g_{\mathrm{c}} \ll \Delta_{\mathrm{c}}$ to the Rydberg
transition $\ket{i}\to\ket{r}$ detuned by $-\Delta_{\mathrm{c}}$.
State $\ket{r}$ decays with rate $\Gamma_r$ either by ionization
or by an induced cascade to state $\ket{g}$ via intermediate state
$\ket{e}$ having large decay rate $\Gamma_s$.}
\label{fig:coolsch}
\end{figure}
Typically, Rydberg states have slow population decay rates. Larger decay
rate $\Gamma_r$ can be achieved by laser-induced ionization of state $\ket{r}$,
which, however, will result in continuous depletion of the number $N_\mathrm{c}$
of cooling atoms. A better alternative is to use an auxiliary laser
of Rabi frequency $\Omega_{r}$ to resonantly couple the Rydberg state
$\ket{r}$ to a lower excited state $\ket{e}$ having large decay rate
$\Gamma_e \gg \Omega_{r}$ back to the ground state $\ket{g}$,
as shown in Fig.~\ref{fig:coolsch}. This will induce a cascade
from $\ket{r}$ to $\ket{g}$ with sufficiently rapid rate
$\Gamma_r \simeq \frac{4 \Omega_{r}^2}{\Gamma_e}$. Thus, each cooling
cycle $\ket{g,n} \to \ket{r,n-1} \to \ket{g,n-1}$ will extract with
rate $\gamma_{\mathrm{c}}$ a cavity photon, while the number of cooling atoms will
be preserved. For $\gamma_{\mathrm{c}} \gg \kappa \bar{n}_{\mathrm{th}}$, we can then approach a
cavity vacuum by optically pumping out thermal photons.
\section{State transfer via cooled cavity}
\label{sec:cavcooltrnans}
In Fig.~\ref{fig:oscilcool} we demonstrate significant enhancement of
the amplitude of oscillations between the initial $\ket{ba}$ and the
target $\ket{ab}$ states in the presence of rapid extraction of thermal
photons from the cavity. Since the probability distribution of the cavity
photons is now highly peaked at $n=0$, we set the frequency mismatch
$\delta = \frac{g_2^2 -g_1^2}{\Delta}$ to satisfy the resonant
condition $\delta E(0) = 0$ in Eq.~(\ref{eq:deEn}).
\begin{figure}[t]
\centerline{\includegraphics[width=7.5cm]{oscilcoolN.eps}}
\caption
Time dependence of population $p_{ab}$ of state $\ket{ab}$ in the presence
of continuous photon extraction with different rates $\gamma_{\mathrm{c}}$, as obtained
from exact numerical simulations of Eq.~(\ref{eq:MasterEq}) (blue solid lines)
and from the approximate Eq.~(\ref{eq:p0bosc}) (black dashed lines),
for the initial state $\ket{ba}$. Dotted vertical line indicates
time $t_{\mathrm{tr}}=\frac{\pi}{2 G(0)}$ when the population
of $\ket{ab,0}$ is peaked.
The parameters are $g_1 = 1.4 g_2$, $\Delta = 30 g_2$,
$\delta = \frac{g_2^2 -g_1^2}{\Delta}$, $\Gamma = 3\times 10^{-4} g_2$,
$\kappa = 10^{-2} g_2$, $\bar{n}_{\mathrm{th}} = 5$, while $\gamma_{\mathrm{c}} = (0, 1.5, 4, 9, 49) \kappa$
corresponding to the mean photon numbers $\bar{n}_{\mathrm{eff}} = (5, 2, 1, 0.5, 0.1)$
for the graphs with progressively larger amplitudes. }
\label{fig:oscilcool}
\end{figure}
\subsection{Optimizing the transfer fidelity}
From Eq.~(\ref{eq:Geffn}) we have for the transition rate
$G(0) \simeq \frac{g_1 g_2}{\Delta} \left( 1-\frac{\delta}{2\Delta}
- \frac{g_1^2 + g_2^2}{\Delta^2} \right)$. Our aim is to transfer
the population of state $\ket{ba}$ to state $\ket{ab}$.
Using Eq.~(\ref{eq:p0bosc}), we can estimate the lower bound for
the transfer probability as being determined mainly by the $n=0$ term,
\begin{eqnarray}
p_{ab}(t) > P_0 p_{ab,0}(t) &=& \frac{1}{1+\bar{n}_{\mathrm{eff}}} e^{- 2 \Gamma t}
\nonumber \\ & &
\times \mbox{$\frac{1}{2}$} \{ 1 - \cos[2G(0)t] e^{-\kappa_{\mathrm{eff}} t} \} , \quad
\end{eqnarray}
where we included the effective damping rate
$\kappa_{\mathrm{eff}} = (\kappa+\gamma_{\mathrm{c}}) \frac{g_1^2 +g_2^2}{2\Delta^2}$
of the oscillation amplitude, which can be intuitively understood as follows:
During the transfer, states $\ket{ba,0}$ and $\ket{ab,0}$ have small admixture,
$\sim \frac{g_{\mathrm{1,2}}}{\Delta}$, of state $\ket{aa,1}$ containing
the additional exchange photon which is damped with rate $(\kappa + \gamma_{\mathrm{c}})$.
The transfer probability is peaked at time $t_{\mathrm{tr}}=\frac{\pi}{2 G(0)}$
when $\cos[2G(0)t_{\mathrm{tr}}] = -1$. Assuming $\bar{n}_{\mathrm{eff}} \ll 1$ and
$(\kappa_{\mathrm{eff}},\Gamma) t_{\mathrm{tr}} \ll 1$, the transfer
fidelity is then
\begin{eqnarray*}
F & \equiv & P_0 p_{ab,0}(t_{\mathrm{tr}})
\nonumber \\
& \gtrsim &
\left[ 1 - \frac{\kappa \bar{n}_{\mathrm{th}}}{\kappa + \gamma_{\mathrm{c}}} \right]
\left[ 1 - \pi \frac{\Gamma\Delta }{g_1 g_2} \right]
\left[ 1 - \pi \frac{\kappa + \gamma_{\mathrm{c}}}{4 \Delta} \frac{g^2}{g_1 g_2} \right] ,
\label{eq:Flb}
\end{eqnarray*}
where $g^2 = \mbox{$\frac{1}{2}$}(g_1^2 +g_2^2)$.
Although the right-hand side of this expression underestimates the maximal
fidelity, we can still use it to optimize the parameters of the system.
Thus, the transfer fidelity is reduced by three factors, and we therefore
require that each of them be small:
\begin{itemize}
\item[(i)] $\gamma_{\mathrm{c}} + \kappa \gg \kappa \bar{n}_{\mathrm{th}}$, i.e. the cooling rate $\gamma_{\mathrm{c}}$ should
be sufficiently large to have the mean photon number small, $\bar{n}_{\mathrm{eff}} \ll 1$;
\item[(ii)] $\frac{g_1 g_2}{\Delta} \gg \Gamma$, i.e., the transition
rate $G(0)$ should be large enough to have small probability of the
atomic decay during the transfer.
\item[(iii)] $\Delta \gg \gamma_{\mathrm{c}} + \kappa$, i.e., the cavity field should be
strongly detuned to have small photon population and therefore decay during
the transfer.
\end{itemize}
With $\gamma_{\mathrm{c}} \gg \kappa$, the total reduction of the fidelity, or infidelity,
can be estimated as
\begin{equation}
1-F \simeq \frac{\kappa \bar{n}_{\mathrm{th}}}{\gamma_{\mathrm{c}}} +
\pi \frac{\Gamma \Delta }{g_1 g_2} +
\pi \frac{\gamma_{\mathrm{c}}}{4\Delta} \frac{g^2}{g_1 g_2} .
\end{equation}
We can minimize this expression with respect to $\Delta$, with
the other parameters fixed, obtaining
\begin{equation}
1-F \simeq \frac{\kappa \bar{n}_{\mathrm{th}}}{\gamma_{\mathrm{c}}}
+ \frac{\pi g}{g_1 g_2} \sqrt{ \Gamma \gamma_{\mathrm{c}}} ,
\end{equation}
for $\Delta = g \sqrt{\frac{\gamma_{\mathrm{c}}}{4\Gamma} }$.
Next, we minimize the resulting infidelity with respect to
$\gamma_{\mathrm{c}}$, finally obtaining
\begin{equation}
\min(1-F) \leq 3 (\kappa \bar{n}_{\mathrm{th}})^{1/3}
\left( \frac{\pi g \sqrt{\Gamma}}{2 g_1 g_2} \right)^{2/3}
\end{equation}
for $\gamma_{\mathrm{c}} = \left( \frac{2 \kappa \bar{n}_{\mathrm{th}} g_1 g_2}{\pi g \sqrt{\Gamma}} \right)^{2/3}$.
\subsection{Experimental considerations}
We assume the following realistic parameters of the system:
The cavity mode resonant frequency is $\omega = 2 \pi \times 5\:$GHz,
and its quality factor is $Q = 10^5$ leading to the decay rate
$\kappa \simeq 300\:$kHz.
The Rydberg atom--cavity coupling rates are $g_1 = 2 \pi \times 5\:$MHz
and similar for $g_2$.
The decay rates of the atoms are $\Gamma = 10\:$kHz. With $\bar{n}_{\mathrm{th}} \sim 10$
at cryogenic environment,
we need to choose the cooling rate $\gamma_{\mathrm{c}} \gtrsim 15 \:$MHz,
which can be achieved with $N_{\mathrm{c}} \sim 1000$ cooling
atoms, with $\Omega \simeq g_{\mathrm{c}} = 2 \pi \times 0.1\:$MHz,
$\Delta_{\mathrm{c}} = 10 \Omega$ and $\Gamma_r = 1\:$MHz.
We then choose the detunings $\Delta_1 \simeq 30 g_1$ and
$\Delta_2 = \Delta_1 + \frac{g_1^2 - g_2^2}{\Delta}$ for the resonant
transfer of the excitation between the atoms.
The resulting fidelity is $F \gtrsim 0.95$ which we verified numerically.
To turn off the transfer, one of the atoms can be strongly detuned by
$|\Delta_1 - \Delta_2| \gg G(0) \simeq \frac{g_1 g_2}{\Delta}$, which can
be achieved by, e.g., Stark shifting the resonance with a focused laser
beam \cite{Leseleuc2017}.
\section{Conclusions}
\label{sec:conclud}
We have elaborated the conditions for coherent state transfer between two
two-level systems through a thermal microwave cavity. We have demonstrated
that by actively cooling a cavity mode by continuously removing photons
with a laser-driven ensemble of atoms, high-fidelity \textsc{swap}
operation between pairs of spatially separated Rydberg-atom qubits
is possible in state-of-the-art experimental systems
\cite{Hattermann2017,Avigliano2014,Teixeira2015,Hogan2012}.
The $\sqrt{\mbox{\textsc{swap}}}$ is a universal entangling quantum
gate \cite{PLDP2007}, which can also be realized by the present scheme.
Trapped ground-state atoms have good coherence properties and can serve
as reliable qubits. The atoms can be excited on demand by focused
lasers to the Rydberg states for realizing short distance quantum
communication and quantum logic gates. Our results will thus pave the way
for the realization of scalable quantum information processing with cold
atoms trapped on the integrated superconducting atom chips.
\begin{acknowledgments}
We acknowledge the financial support of the
DFG Schwerpunktsprogramm Giant interactions in Rydberg Systems (GiRyd SPP 1929).
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,527 |
Q: How can I check by a button click which ip to connect? On the MainActivity.java I have a method that connect with a server:
private byte[] Get(String urlIn)
{
URL url = null;
String urlStr = urlIn;
if (urlIn!=null)
urlStr=urlIn;
try
{
url = new URL(urlStr);
} catch (MalformedURLException e)
{
e.printStackTrace();
return null;
}
HttpURLConnection urlConnection = null;
try
{
urlConnection = (HttpURLConnection) url.openConnection();
InputStream in = new BufferedInputStream(urlConnection.getInputStream());
byte[] buf=new byte[10*1024];
int szRead = in.read(buf);
byte[] bufOut;
if (szRead==10*1024)
{
throw new AndroidRuntimeException("the returned data is bigger than 10*1024.. we don't handle it..");
}
else
{
bufOut = Arrays.copyOf(buf, szRead);
}
return bufOut;
}
catch (IOException e)
{
e.printStackTrace();
return null;
}
finally
{
if (urlConnection!=null)
urlConnection.disconnect();
}
}
I'm calling this method from onTouchEvent():
@Override
public boolean onTouchEvent(MotionEvent event)
{
float eventX = event.getX();
float eventY = event.getY();
float lastdownx = 0;
float lastdowny = 0;
switch (event.getAction())
{
case MotionEvent.ACTION_DOWN:
path.moveTo(eventX, eventY);
circlePath.addCircle(eventX, eventY, 50, Path.Direction.CW);
lastdownx = eventX;
lastdowny = eventY;
Thread t = new Thread(new Runnable()
{
@Override
public void run()
{
byte[] response = null;
if (is_start == true)
{
response = Get("http://10.0.0.2:8098/?cmd=start");
is_start = false;
}
else
{
response = Get("http://10.0.0.2:8098/?cmd=stop");
is_start = true;
}
if (response!=null)
{
String a = null;
try
{
a = new String(response,"UTF-8");
textforthespeacch = a;
MainActivity.currentActivity.initTTS();
} catch (UnsupportedEncodingException e)
{
e.printStackTrace();
}
Logger.getLogger("MainActivity(inside thread)").info(a);
}
}
});
t.start();
return true;
case MotionEvent.ACTION_MOVE:
path.lineTo(eventX, eventY);
break;
case MotionEvent.ACTION_UP:
circlePath.reset();
break;
default:
return false;
}
invalidate();
return true;
}
So now i'm connecting all the time to 10.0.0.2:8098
But that's when i connect my android device on my network on my pc room.
But if i move to the living room and connect to the network there a differenet network with another pc the pc ip is differenet in this case: 10.0.0.3:8099
So i added a button click event to the MainActivity.java:
public class MainActivity extends ActionBarActivity
{
private static final int MY_DATA_CHECK_CODE = 0;
public static MainActivity currentActivity;
TextToSpeech mTts;
private String targetURL;
private String urlParameters;
private Button btnClick;
private String clicking = "clicked";
@Override
protected void onCreate(Bundle savedInstanceState)
{
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_main);
addListenerOnButton();
currentActivity = this;
initTTS();
}
public void addListenerOnButton() {
btnClick = (Button) findViewById(R.id.checkipbutton);
btnClick.setOnClickListener(new OnClickListener()
{
@Override
public void onClick(View arg0)
{
}
});
}
Inside the button click event I want to check after connected to the network with a wifi if the pc ip is 10.0.0.3:8099 or 10.0.0.2:8098
I need that it will try to connect to this servers and if success then to set to a global variable global string the ip.
I added a global variable: string ipaddress
Now i use static address in my code but i need to check which ip address is correct and then to set this ip to the variable which i will use later in my code as the ip address.
How do I make the checking in the button click event ?
This is what i tried now:
At the top of my MainActivity i added:
private final String[] ipaddresses = new String[2];
private final Integer[] ipports = new Integer[2];
Socket socket = null;
Then in the onCreate:
ipaddresses[0] = "10.0.0.3";
ipaddresses[1] = "10.0.0.2";
ipports[0] = 8098;
ipports[1] = 8088;
addListenerOnButton();
new Thread(new ClientThread()).start();
Then
public void addListenerOnButton() {
btnClick = (Button) findViewById(R.id.checkipbutton);
btnClick.setOnClickListener(new OnClickListener()
{
@Override
public void onClick(View arg0)
{
try {
String str = btnClick.getText().toString();
PrintWriter out = new PrintWriter(new BufferedWriter(
new OutputStreamWriter(socket.getOutputStream())),
true);
out.println(str);
} catch (UnknownHostException e) {
e.printStackTrace();
} catch (IOException e) {
e.printStackTrace();
} catch (Exception e) {
e.printStackTrace();
}
}
});
}
And the ClientThread
class ClientThread implements Runnable {
@Override
public void run() {
for (int i=0; i<ipaddresses.length; i++)
{
try
{
InetAddress serverAddr = InetAddress.getByName(ipaddresses[i]);
socket = new Socket(serverAddr, ipports[i]);
} catch (UnknownHostException e1)
{
e1.printStackTrace();
} catch (IOException e1)
{
e1.printStackTrace();
}
}
}
}
This is a screenshot of the exception message i'm getting:
The exception is on the line:
new OutputStreamWriter(socket.getOutputStream())),
A: You must open sockets to check server connectivity. Here is an example on you send strings to server on click event:
public class Client extends Activity {
private Socket socket;
private static final int SERVERPORT = 8099;
private static final String SERVER_IP = "10.0.0.3";
@Override
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.main);
new Thread(new ClientThread()).start();
}
public void onClick(View view) {
try {
EditText et = (EditText) findViewById(R.id.EditText01);
String str = et.getText().toString();
PrintWriter out = new PrintWriter(new BufferedWriter(
new OutputStreamWriter(socket.getOutputStream())),
true);
out.println(str);
} catch (UnknownHostException e) {
e.printStackTrace();
} catch (IOException e) {
e.printStackTrace();
} catch (Exception e) {
e.printStackTrace();
}
}
class ClientThread implements Runnable {
@Override
public void run() {
try {
InetAddress serverAddr = InetAddress.getByName(SERVER_IP);
socket = new Socket(serverAddr, SERVERPORT);
} catch (UnknownHostException e1) {
e1.printStackTrace();
} catch (IOException e1) {
e1.printStackTrace();
}
}
}
}
So if you get an exception trying to connect to server, it means you haven't connectivity.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,388 |
← EA Releases Star Wars: Battlefront Reveal Trailer
Marvel's Daredevil Gets Second Season Along With New Showrunners →
Original Cinematographer for Gears of War Returns for Another Round With Franchise!
Well, Epic games lost a great worker as Greg M. Mitchell is leaving to work with Black Tusk Studios on a franchise he helped make so popular in Gears of War.
He broke the news over twitter which saw excitement from many fans. His work doesn't stop at Gears of War. His resumé also includes games like Infinity Blade and Shadow Complex. There's no official word on his exact role with the studio as he was pretty vague, but with his expertise in cinematography I think it's safe to assume that he'll be doing the same thing here.
Black Tusk had other ideas for another game that was in early production, but once they got the job of developing the next installment in the Gears series they were quickly stalled. No one was happier than Microsoft seeing as how they needed another developer to revive the franchise after purchasing it from Epic Games. Seems like the Vancouver-based company was more than happy to tackle the project.
Now It may not seem like a big deal at first, but in the past one of the calling cards of the Gears of War series was its original look and the familiar scenes that accompanied it. Being able to have one of the biggest cinematographers in the industry to help on a project that he is so familiar with and saw so much success with can do nothing but good. Couple that with one of the most refine gaming engines in the industry (Unreal 4), and you can expect superior visuals to enhance that brutal, yet fun gamer experience that so many fans of the series enjoyed so much. The only bad thing you can say about the news is that it makes the wait for E3 2015 that much harder, but really… it'll be worth it.
Tagged Black Tusk Studios, Gears of War, News, video games | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,758 |
Camaricus bipunctatus är en spindelart som beskrevs av Bastawade 2002. Camaricus bipunctatus ingår i släktet Camaricus och familjen krabbspindlar. Inga underarter finns listade.
Källor
Krabbspindlar
bipunctatus | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,428 |
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3 Ways to Start Doing Business in Canada
Business owners looking to massively scale their business should all be considering one thing;
Expanding to Canada.
Doing business in Canada can put you on the fast-track to multi-million-dollar revenue. From stellar tax breaks to additional write-offs, conducting business in Canada has numerous advantages for wealth builders.
It's crucial, however, to set things up right from the beginning. Canadian laws are different and it can be difficult to navigate the complexities of international insurance, taxes and trusts.
Your first step is to take advantage of a free Gap Analysis with a member of our team. Our cross-border specialists will help you create a plan that is individually tailored to your needs. Next, check out these three ways you can start doing business in Canada.
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Sole proprietorships are similar in Canada and the U.S. A main benefit is that they can give you write offs towards your taxable income.
For instance, Canadian sole proprietors can write off 10 to 20 percent of their houses if they're used for business. There are many similar write offs you can take advantage of on your T4 income – which is filed every year through the Canada Revenue Agency.
The many write offs make a sole proprietorship a great option for new businesses. Many business owners choose to go this route as they get established and have low profitability.
As their revenue generation grows, however, so do their tax burdens. If you start making seven figures, you can be taxed upwards of 50 percent. You will want to look at other entities if you expect to build wealth fast.
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Partnerships are an option for those looking to set up a business with others. In addition to combining financial resources, starting a Canadian partnership is an inexpensive way to establish a business.
While a partnership may be a viable option for those with business partners, they do have their disadvantages. Each partner is responsible for doing their own tax return. This means that any money you make on the partnership goes straight to your taxes.
You also need to be careful of your liability. Canadian partnerships have unlimited liability – so personal assets may be required to pay off debts.
3. Limited Companies
Setting up a limited company in Canada is a great way to have flexibility as a business. You can enjoy the benefits of more write off opportunities, more asset protection and more legal structure.
Canada also allows you to set up companies as C-Corps – and the structure is similar to C-Corps in the U.S. If you have C-Corps in both the U.S. and Canada, you can easily transfer money without paying lofty international fees.
For example, exchanges between U.S. and Canadian C-Corps have a 5 percent withholding tax. Bringing in money to Canada through an LLC? That has a much higher withholding tax at 15 percent. You can also avoid exchange rates by keeping your money in U.S. accounts. Using a U.S. credit card can help you keep avoid paying wiring fees and high Canadian taxes.
Starting a Canadian company is all about planning and having the correct team in place right from the beginning. By establishing your business the right way, you can get on the fast track to seven figure income. You can protect your assets, reduce your taxes and see much higher returns by establishing yourself as a Canadian business.
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[Insert download form codes here] | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,487 |
from app.core.models import *
from app.auth.models import User
import re
from unidecode import unidecode
class Category(Base):
name = ndb.StringProperty()
class Tag(Base):
name = ndb.StringProperty()
class Post(Base):
url = ndb.StringProperty()
title = ndb.StringProperty()
description = ndb.StringProperty()
content = ndb.TextProperty()
post_type = ndb.StringProperty()
post_status = ndb.StringProperty()
author = ndb.StructuredProperty(User)
category = ndb.StructuredProperty(Category)
tags = ndb.StructuredProperty(Tag, repeated=True)
def get_post_type(self):
if(self.post_type=='page'):
type = 'Page'
elif(self.post_type=='post'):
type = 'Post'
else:
type = ''
return type
def get_post_status(self):
if(self.post_status=='draft'):
status = 'Draft'
elif(self.post_status=='published'):
status = 'Published'
else:
status = ''
return status
def slugify(self, text, delim=u'-'):
"""Generates an ASCII-only slug."""
_punct_re = re.compile(r'[\t !"#$%&\'()*\-/<=>?@\[\\\]^_`{|},.]+')
result = []
for word in _punct_re.split(text.lower()):
result.extend(unidecode(word).split())
return unicode(delim.join(result))
class PostTag(Base):
post = ndb.StructuredProperty(Post)
tag = ndb.StructuredProperty(Tag)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,922 |
Q: How to convert Newtonsoft.Json.Linq.Array to an array of double? I have a Newtonsoft.Linq.Json.Array object which I obtained after running a camera and the string is received to my laptop via TCP/IP. The code which is used is shown below:
public async void ReadDataAsync(TcpClient mClient)
{
try
{
StreamReader clientStreamReader = new StreamReader(mClient.GetStream());
char [] buff = new char[1024];
int readByCount = 0;
while (true)
{
readByCount = await clientStreamReader.ReadAsync(buff, 0, buff.Length);
Directory.CreateDirectory("Camera o3D1");
if (readByCount <= 0)
{
Console.WriteLine("Disconnected from Server.");
mClient.Close();
break;
}
if (readByCount > 30)
{
var output = (new string(buff).TrimEnd('\u0000'));
Console.WriteLine(output);
Console.WriteLine(output.GetType());
var output1 = output.Split(new[] { ";;" }, StringSplitOptions.RemoveEmptyEntries).Select(s => s.Split(';')).ToArray();
JToken jsonParsed = JToken.FromObject(output1);
Console.WriteLine(jsonParsed);
Console.WriteLine(jsonParsed.GetType());
}
//File.WriteAllLines("Camera o3D1/file.json", jsonParsed);
//File.AppendAllText("Camera o3D1/file.json", type);
Array.Clear(buff, 0, buff.Length);
}
}
catch (Exception excp)
{
Console.WriteLine(excp.ToString());
}
}
The output I received is:
[
[
"17.623665",
"87.380157",
"51.530350",
"121.286850",
"+0.480"
],
[
"89.199364",
"47.660252",
"123.106056",
"81.566940",
"+0.521"
],
[
"\r\n"
]
]
Newtonsoft.Json.Linq.JArray
I want to convert it into an array of double so that if I used jsonparsed[0], I should get
[
"17.62",
"87.38",
"51.53",
"121.28",
"+0.48"
]
and jsonparsed[0][0], I should get 17.62 and everything in double.
Does anyone of you have any idea how to do this?
A: You can try something like this:
var input = $"[[\"17.623665\",\"87.380157\",\"51.530350\",\"121.286850\",\"+0.480\"],[\"89.199364\",\"47.660252\",\"123.106056\",\"81.566940\",\"+0.521\"]]";
var jarray = JsonConvert.DeserializeObject<JArray>(input);
foreach (var arrayItem in jarray)
{
var innerArray = arrayItem.ToObject<double[]>();
foreach (var item in innerArray)
{
Console.WriteLine(item);
}
}
You would have to filter out the last empty item first:
[
"\r\n"
]
"\r\n" does not deserialize to a double..
A: There are two problems you have to solve:
*
*The data is not a flat array. Instead it is an array of arrays of doubles.
*Some of you values are not valid double values and you have to decide what should happen with them.
So lets first solve the most inner problem, which would be the conversion of the values to doubles. Due to the fact, that you are using Newtonsoft you can write your own converter and provide that to the serializer:
public class SpecialDoubleConverter : JsonConverter<double>
{
public override double ReadJson(JsonReader reader, Type objectType, double existingValue, bool hasExistingValue, JsonSerializer serializer)
{
var rawValue = (string)reader.Value;
// Decide what should be returned if the value can not be read as double
// e.g. return some magic value like -17, double.NaN, 0, double.Epsilon, etc.
return double.TryParse(rawValue, out double result)
? result
: -17;
}
public override void WriteJson(JsonWriter writer, double value, JsonSerializer serializer)
{
// Writing a double into a json string is the same as the default.
writer.WriteValue(value.ToString());
}
}
If you use this serializer you could get a nested array by this call:
var jsonParsed = JsonConvert.DeserializeObject<List<List<double>>>(output1, new SpecialDoubleConverter());
Now we need to flatten this nested list, which can be achieved by using LINQ:
var doubleValues = jsonParsed.SelectMany(list => list).ToList();
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,883 |
La seconda stagione della serie televisiva Mary Kills People, composta da 6 episodi, è stata trasmessa in Canada su Global dal 3 gennaio al 7 febbraio 2018.
La storia riprende 8 mesi dopo la fine della prima stagione. Durante questo periodo di tempo Mary ha proseguito il suo business in autonomia rifornendosi del pentobarbital in Messico; Des ha scontato la sua condanna in galera e può uscire in anticipo per buona condotta. Non appena fuori vuole riprendere il business con Mary, ma l'incontro con Olivia Bloom li porterà ad affrontare una richiesta di morte inaspettata.
In Italia, la stagione è stata distribuita su TIMvision l'8 gennaio 2019. | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,747 |
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