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Q: calculating difference between array values php I have this small section of code.
<?php foreach($last_activity as $activity) : ?>
<tr>
<td><?= $activity['category'];?></td>
<td><?= $activity['activity'];?></td>
<td><?= $activity['datetime'];?></td>
<td><?= $activity[''];?></td>
</tr>
<?php endforeach; ?>
This runs a loop populating table rows with data from query.
What I want to be able to do providing possible this way...is in the final column in the row is calculate the difference between the datetime column in its present row with that of the row above.
is this possible (e.g by putting a snippet of php script in my final <td> column?
A: I suppose it can be done with something like this:
<?php $recent_datetime = null;
foreach($last_activity as $activity): ?>
<tr>
<td><?=$activity['category'];?></td>
<td><?=$activity['activity'];?></td>
<td><?=$activity['datetime'];?></td>
<td><?= ($recent_datetime
? $activity['datetime'] - $recent_datetime
: $activity['datetime'] );
?></td>
</tr>
<?php $recent_datetime = $activity['datetime']; endforeach; ?>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,071
|
Q: Some questions about __getattr__ and __getattribute__? The first demo:
class B:
def __init__(self):
self.name = '234'
# def __getattribute__(self, name):
# print('getattr')
def __getattr__(self, name):
print('get')
def __setattr__(self, name, value):
print('set')
def __delattr__(self, name):
print('del')
b = B()
print(b.__dict__)
b.name
b.__dict__ is {}, but the second demo:
class B:
def __init__(self):
self.name = '234'
def __getattribute__(self, name):
print('getattr')
def __getattr__(self, name):
print('get')
def __setattr__(self, name, value):
print('set')
def __delattr__(self, name):
print('del')
b = B()
print(b.__dict__)
b.name
b.__dict__ is None, why? And b.__dict__ invokes __getattribute__, but don't invoke __getattr__, does it mean __getattribute__ will prevent from invoking __getattr__?
A: The __getattribute__, __setattr__ and __delattr__ methods are called for all attribute access (getting, setting and deleting). __getattr__ on the other hand is only called for missing attributes; it is not normally already implemented, but if it is then __getattribute__ calls it if it could not otherwise locate the attribute, or if an AttributeError was raised by __getattribute__.
You replaced the standard implementations of the 3 main methods with methods that do nothing but print and return None (the default in the absence of an explicit return statement). __dict__ is just another attribute access, and your __getattribute__ method returns None, and never itself calls __getattr__ or raises an AttributeError.
From the Customizing attribute access documentation:
object.__getattr__(self, name)
Called when an attribute lookup has not found the attribute in the usual places (i.e. it is not an instance attribute nor is it found in the class tree for self).
and
object.__getattribute__(self, name)
Called unconditionally to implement attribute accesses for instances of the class. If the class also defines __getattr__(), the latter will not be called unless __getattribute__() either calls it explicitly or raises an AttributeError.
(Bold emphasis mine).
Either call the base implementation (via super().__getattribute__) or raise an AttributeError:
>>> class B:
... def __init__(self):
... self.name = '234'
... def __getattribute__(self, name):
... print('getattr')
... return super().__getattribute__(name)
... def __getattr__(self, name):
... print('get')
... def __setattr__(self, name, value):
... print('set')
... def __delattr__(self, name):
... print('del')
...
>>> b = B()
set
>>> b.__dict__
getattr
{}
>>> b.name
getattr
get
>>> class B:
... def __init__(self):
... self.name = '234'
... def __getattribute__(self, name):
... print('getattr')
... raise AttributeError(name)
... def __getattr__(self, name):
... print('get')
... def __setattr__(self, name, value):
... print('set')
... def __delattr__(self, name):
... print('del')
...
>>> b = B()
set
>>> b.__dict__
getattr
get
>>> b.name
getattr
get
Note that by calling super().__getattribute__ the actual __dict__ attribute is found. By raising an AttributeError instead, __getattr__ was called, which also returned None.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,859
|
Turkey: Military Makes Second Syrian Incursion
Published on 4 Settembre 2016 in Defence by Redazione
Turkey sent tanks over the Syrian border for the second time Sept. 3, crossing into al Rai, Kilis province, from Cobanbey, Dogan news reported.
The military also launched artillery strikes on the area, which has changed hands frequently between Islamic State and rebel forces. Turkey's first incursion occurred Aug. 24 at the city of Jarabulus 55 kilometers (34 miles) northeast. Also on Sept. 3, Turkish-backed Hamza Brigade and Failaq al-Sham rebels took control of the Syrian villages of Arab Ezra, Fursan, Lilawa, Kino and Najma.
These communities are all west of Jarabulus. After clearing a path for its troops to enter Syria, Turkey is now fully involved in a military campaign in northern Aleppo province. Turkey is attempting to move the Islamic State away from the border and, at the same time, prevent Kurdish militants in the Syrian Democratic Forces from establishing a foothold.
Source STRATFOR
Afghanistan, incontro su sicurezza delle provincie…
Missione in Kosovo: militari italiani partecipano…
Si e spento a 73 anni Pino Scaccia storico reporter Rai
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,698
|
The euro once again begins to look at the top. What is it: an attempt to turn and beginning of the wave (v) or another trap for buyers?
The answer to this question will soon become clear.
Given the fact for the long time that the euro either has been losing its positions or staying within any particular range as well as the wave picture, according to which formation of the wave (iv) in the form of the final diagonal triangle is over, we assume that growth is the most probable scenario.
Thus, it is worth trying to open a long position with Stop set at the Low, potential of the deal is very good. But so far, we have not received confirmation in the form of impulse upward movement so it is too early to speak about potential of this trade.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 274
|
# More Praise for _DODGERS_
**"Akin to William Gibson in its staccato blizzard of sentences, each as taut and tight as a drumhead, and reminiscent of James Ellroy in its sense of social injustice and of a life and upbringing etched caustically across the pages. A compelling debut."**
—Luke McCallin, _New York Times_ bestselling author of _The Man from Berlin_
**"** ** _Dodgers_** **is a wickedly good amalgamation of** **_Adventures of Huckleberry Finn_** **and** **_Clockers_** **that stands firmly on its own as a remarkable debut.** A harrowing road trip into the heart of America that will shock you, move you, and leave you marveling at its desolate poetry. A real accomplishment: a book that makes you see the familiar through new eyes. It will stick with me for a long, long time."
—Richard Lange, author of _Angel Baby_ and _This Wicked World_
**"Bill Beverly's gritty and propulsive debut novel,** **_Dodgers_** **, is more than a riveting read; it is a stunning literary achievement.** Our hero, East, a fifteen-year-old hit man, drives across America on a deadly mission, from the mean streets of Los Angeles to the heart of the heart of the country. **East is a character as memorable and as haunting as any I've met in contemporary fiction.** And he's not alone in that van, but there is room for one more. So hop in, but strap on your seat belt and hold on to your hat. The road's a little bumpy—and more than a little terrifying—up ahead."
—John Dufresne, author of _No Regrets, Coyote_
**"A terrific novel, urgent, thrilling, and dangerous from start to finish.** In East, Mr. Beverly has created a character who stays in the mind after the book is finished, an Odysseus straight out of Compton. His venture into the unknown lands of the American Midwest has a classic, mythic shape and scope. **And the writing throughout is lovely, economical, and exact. You could read this for the sentences alone."**
—Kevin Canty, author of _Into the Great Wide Open_
"I knew before I'd gotten very far into Bill Beverly's superb first novel that I was about to lose some sleep, since putting it down seemed to be beyond me. **To say it's a page-turner doesn't do it justice, though it certainly is. It's also much more. His characters are vivid and real and, yes, sometimes they'll break your heart. The world they inhabit—no matter where they may be at a given moment—all but leaps off the page. It's a winner. So is its author."**
—Steve Yarbrough, author of _The Realm of Last Chances_ and _Safe from the Neighbors_
"From the moment we encounter East, a mostly silent kid who 'didn't look like much,' we are initiated into his gaze on the malfunctioning world, a kind of concentrated, exquisite hypervigilance that is both his burden and his gift. It is this quality of attention that makes _Dodgers_ such an intense read—inescapable, inevitable, impossible to set aside. We can no more turn off East's vision—and the sense of urgency that comes with it—than he himself can, and we are along for the ride. **The truth-telling and pared-down purity of voice here are reminiscent of Denis Johnson, as if this novel were not written but channeled. This is a beautiful, extraordinary book."**
—Wendy Brenner, author of _Large Animals in Everyday Life_ and _Phone Calls from the Dead_
This is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, events, or locales is entirely coincidental.
Copyright © 2016 by Bill Beverly
All rights reserved.
Published in the United States by Crown Publishers, an imprint of the Crown Publishing Group, a division of Penguin Random House LLC, New York.
www.crownpublishing.com
CROWN is a registered trademark and the Crown colophon is a trademark of Penguin Random House LLC.
Library of Congress Cataloging-in-Publication Data is available upon request.
ISBN 9781101903735
eBook ISBN 9781101903742
_Cover design by Christopher Brand_
_Cover photograph by Gert Henning/EyeEm/Getty Images_
v4.1
ep
# Contents
Cover
Title Page
Copyright
Dedication
Epigraph
I: The Boxes
Chapter 1
Chapter 2
Chapter 3
II: The Van
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
III: Ohio
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Acknowledgments
About the Author
This book is for Olive, who gave me a new life
> And giving a last look at the aspect of the house, and at a few small children who were playing at the door, I sallied forth...my course lay through thick and heavy woods and back lands to town, where my brother lived.
>
> James W. C. Pennington, _The Fugitive Blacksmith_ (1849)
>
> Every cheap hood strikes a bargain with the world.
>
> The Clash, "Death or Glory"
# 1.
The Boxes was all the boys knew; it was the only place.
In the street one car moved, between the whole vehicles and skeletal remains, creeping over paper and glass.
The boys stood guard. They watched light fill between the black houses separated only barely, like a row of loose teeth. Half the night they had been there: Fin taught that you did not make a boy stand yard all night. Half was right. To change in the middle kept them on their toes, Fin said. It kept them awake. It made them like men.
—
The door of the house opened, and two U's stumbled out, shocked by the sun, ogling it like an old girl they hadn't seen lately. Some men left the house like this, better once they'd been in. Others walked easy going in but barely crawled their way out. The two ignored the boys at their watch. At the end of the walk, they descended the five steps to the sidewalk, steadied themselves on the low stone wall. One man slapped the other's palm loudly, the old way.
Again the door opened. A skeletal face, lip-curled, staring, hair rubbed away from his head. Sidney. He and Johnny ran the house, kept business, saw the goods in and the money out with teenage runners every half hour. Sidney looked this way and that like a rat sampling the air, then slid something onto the step. Cans of Coke and energy drink, cold in a cardboard box. One of the boys went and fetched the box around; each boy took a can or two. They popped the tops and stood drinking fizz in the shadows.
The morning was still chilly with a hint of damp. Light began to spill between the houses, keying the street in pink. Footsteps approached from the right, a worker man leaving for work, jacket and yellow tie, gold ear studs. The boys stared down over him; he didn't look up. These men, the black men who wore ties with metal pins, who made wages but somehow had not left The Boxes: you didn't talk to them. You didn't let them up in the house. These men, if they came up in the house and were lost, someone needed them, someone would come looking. So you did not admit them. That was another thing Fin taught.
—
Televisions came on and planes flashed like blades in the sky. Somewhere behind them a lawn sprinkler hissed— _fist, fist, fist_ —not loud but nothing else jamming up its frequency. A few U's came in together at seven and one more at about eight, crestfallen: he had that grievous look of a man who'd bought for a week but used up in a night. At ten the boys who had come on at two left. The lead outside boy, East, shared some money out to them as they went. It was Monday, payday, outside the house.
The new boys at ten were Dap, Antonio, Marsonius called Sony, and Needle. Needle took the north end, watching the street, and Dap the south. Antonio and Sony stayed at the house with East, whose twelve hours' work ended at noon. Antonio and Sony were good daytime boys. The night boys, you needed boys who knew how to stay quiet and stay awake. The day boys just needed to know how to look quiet.
East looked quiet and kept quiet. He didn't look hard. He didn't look like much. He blended in, didn't talk much, was the skinniest of the bunch. There wasn't much to him. But he watched and listened to people. What he heard he remembered.
The boys had their talk—names they gave themselves, ways they built up. East did not play along with them. They thought East hard and sour. Unlike the boys, who came from homes with mothers or from dens of other boys, East slept alone, somewhere no one knew. He had been at the old house before them, and he had seen things they had never seen. He had seen a reverend shot on the walk, a woman jump off a roof. He had seen a helicopter crash into trees and a man, out of his mind, pick up a downed power cable and stand, illuminated. He had seen the police come down, and still the house continued on.
He was no fun, and they respected him, for though he was young, he had none in him of what they most hated in themselves: their childishness. He had never been a child. Not that they had seen.
—
A fire truck boomed past sometime after ten, sirens and motors and the crushing of the tires on the asphalt. The firemen glared out at the boys.
They were lost. Streets in The Boxes were a maze: one piece didn't match up straight with the next. So you might look for a house on the next block, but the next one didn't follow up from this one. The street signs were twisted every which way or were gone.
The fire truck returned a minute later, going the other way. The boys waved. They were all in their teens, growing up, but everyone liked a fire truck.
"Over there," said Sony.
"What?" said Antonio.
"Somebody house on fire," said Sony.
The smoke rose, soft and gray, against the bright sky. "Probably a kitchen fire," East said. No ruckus, nobody burning up. You could hear the wailing a mile away when someone was burning up, even in The Boxes. But more fire engines kept rumbling in. The boys heard them on the other streets.
A helicopter wagged its tail overhead.
—
By eleven it was getting hot, and two men crashed out of the house. One was fine and left, but one lay down in the grass.
"Go on," Sony told him. "Get out of here."
"You shut the fuck up, young fellow," said the man, maybe forty years of age. He had a bee-stung nose, and under his half-open shirt East saw a bandage where the man had hurt himself.
"You go on," said East. "Go on in the backyard if you got to lie down. Or go home. Not here."
"This _my_ house, son," said the man, fighting to recline.
East nodded, grim and patient. "This _my_ lawn," he said. "Rules are rules. Go back in if you can't walk. Don't be here."
The man put his hand in his pocket, but East could see he didn't have anything in there, even keys.
"Man, you okay," East said. "Nobody messing with you. Just can't have people lying 'round the yard." He prodded the man's leg lightly. "You understand."
"I _own_ this house," said the man.
Whether this was true, East did not know. "Go on," he said. "Sleep in back if you want."
The man got up and went into the backyard. After a few minutes Sony checked on him and found him asleep, trembling, fighting something inside.
—
The fire's smoke seemed to thin, then came thicker. Trucks and pumps droned, and down the street some neighbor children were bouncing a ball off the front wall. East recognized two kids—from a neat house with green awnings, where sometimes a white Ford parked. These kids kept away. Someone told them, or perhaps they just knew. For the last two days there had been a third girl playing too, bigger. She could have grabbed every ball if she'd wanted, but she played nice.
East made himself stop watching them and studied the chopper instead where it dangled, breaking up the sky.
When he glanced back, the game had stopped and the girl was staring. Directly at him, and then she started to come. He glared at her, but she kept advancing, slowly, the two neighbor kids sticking behind her.
She was maybe ten.
East pushed off. Casually he loped down the yard. Sony was already bristling: "Get back up the street, girl." East flattened his hand over his lowest rib: _Easy._
The girl was stout, round-faced, dark-skinned, in a clean white shirt. She addressed them brightly: "This a crack house, ain't it?"
That's what Fin said: everyone still thought it was all crack.
"Naw." East glanced at Sony. "Where you come from?"
"I'm from Jackson, Mississippi. I go to New Hope Christian School in Jackson." She nodded back at the neighborhood kids. "Them's my cousins. My aunt's getting married in Santa Monica tomorrow."
"Girl, we don't give a fuck," said Antonio, up in the yard.
"Listen to these little gangsters," the girl sang. "Y'all even go to school?"
Probably from a good neighborhood, this girl. Probably had a mother who told her, _Keep away from them LA ghetto boys,_ so what was the first thing she did?
East clipped his voice short. "You don't want to be over here. You want to get on and play."
"You don't know nothing about what I want," boasted the girl. She waved at Antonio. "And this little boy here who looks like fourth grade. What are you? Nine?"
"Damn," Sony cheered her, chuckling.
Somewhere fire engines were gunning, moving again; East stepped back and listened. A woman and a daughter walked by arguing about candy. And the helicopter still chopping. It tensed East up. There were too many parts moving.
"Girl, back off," he said. "I don't need you mixed up."
" _You're_ mixed up," said the girl. She put one hand on the wall, immovable like little black girls got. A fighter.
"This kid," East snorted. The last thing you wanted by the house was a bunch of kids. Women had sense. Men could be warned. But kids, they were gonna see for themselves.
—
A screech careened up the flat face of the street, hard to say from where. Tires. East's talkie phone crackled on his hip. He scooped it up. It was Needle at the north lookout. But all East heard was panting, like someone running or being held down. "What is it?" East said. "What is it?" Nothing.
He scanned, backpedaling up the lawn.
Something was coming. Both directions, echoing, like a train.
He radioed inside. "Sidney. Something coming." The helicopter was dipping above them now.
Sidney, cranky: "Man, what?"
"Get out the back now," East said. "Go."
" _Now?_ " said Sidney incredulously.
" _Now._ " He turned. "You boys, get," he ordered Antonio and Sony. Knowing they knew how and where to go. Having taught them what to do. Everyone on East's crew knew the yards around, the ways you could go; he made sure of it.
The roar climbed the street—five cars flying from each end, big white cruisers. They raised the dust as they screeched in aslant. East thumbed his phone back on.
"Get out. Get out." Already he was sliding away from the house. _His_ house. Red Coke can on its side in the grass, foaming. No time to pick it up.
Sidney did not radio back.
How had this gotten past Dap and Needle? Without a warning? Unaccountable. Angry, he slipped down the wall to the sidewalk. The smell of engine heat and wasted tire rubber hung heavy. The other boys were gone. Now it was just him and the girl.
"I told you," he hissed. "Go on!"
Stubborn thing. She ignored him. Staring behind her at the herd of white cars and polished helmets and deep black ribbed vests: now, _this_ was something to see.
Four of the cops got low, split up, and gang-rushed the porch. Upstairs a window was thrown open, and in it, like a fish in rusty water, an ancient, ravaged face swam up. It looked over the scene for a moment, then poked out a gun barrel. East whirled then. The girl.
"Damn!" he yelled. "Get out of here!"
The girl, of course, did not budge. The _pop-pop_ began.
—
East hit the sidewalk, crouched below the low wall. Beneath the guns' sound the cops barked happily, ducking behind their cars like on TV. Everyone took shelter except the copter and the street dogs, howling merrily, and the Jackson girl.
East fit behind a parked Buick, rusted red. His breath fled him, speedy and light. The car was heat-blistered, and he tried not to touch it. Behind him the air was clouding over with bullets and fragments of the front of the house. Cop radios blared and spat inside the cruisers. The gun upstairs cracked past them, around them, off the street, into the cars, perforating a windshield, making a tire sigh.
The girl, stranded, peered up at the house. Then she faced where East had run, seeing he'd been right. She caught his eye.
With a hand he began a wave: _Come with me. Come here._
Then the bullet ripped into her.
East knew how shot people were, stumbling or crawling or trying to outrace the bullet, what it was doing inside them. The girl didn't. She flinched: East watched. Then she put her hands out, and gently she lay down. Uncertainly she looked at the sky, and for a moment he disbelieved it all—it couldn't have hit her, the bullet. This girl was just crazy. Just as unreal as the fire.
Then the blood began inside the white cotton shirt. Her eyes wandered and locked on him. Dying fast and gently.
The talkie whistled again.
"God damn you, boy," Sidney panted.
The police in the back saw their chance, and three of them aimed. The gun in the window fell, rattling down the roof. Just then the four cops on the porch kicked in the door.
"You supposed to warn us," crackled Sidney. "You supposed to do your job."
"I gave you all I got," East said.
Sidney didn't answer. East heard him wheezing.
He got off the phone. He knew how to go. One last look—windows blown out, cops scaling the lawn, one U stumbling out as if he were on fire. _His_ house. And the Jackson girl on the sidewalk, her blood on the crawl, a long finger pointing toward the gutter, finding its way. A cop bent over her, but she was staring after East. She watched East all the way down the street till he found a corner and turned away.
# 2.
The meet-up was a mile away in an underground garage beneath a tint-and-detail shop with no name. The garage had been shut down years ago—something about codes, earthquakes—but you could still get a car in, through a busted wall in the lot below some apartments next door. Nothing kept people away from a parking space for long.
East took the stairs with his shirt held over his nose. The air reeked of piss and powdered concrete. Three levels down he popped the door and let it close behind him before he breathed again. A few electric lights still hung whole and working from a forgotten power line. Something moved along a crack in the ceiling, surviving.
East wondered who'd be there. Fin had hundreds of people in The Boxes and beyond. After things went wrong, a meet like this might be strictly chain-of-command. Or it might be with somebody you didn't want to meet. Either way, you had to show up.
Down at the end he saw Sidney's car: a Magnum wagon, all black matte. Johnny reclined against it, doing his stretches. He squared his arms behind his head and curled his torso this way and that, muscles bolting up and receding. Then he bent and swept his elbows near the ground.
Sidney stood away in the darkness with his little snub gun eyeing East's head.
"Failing, third-rate, sorry motherfucker."
East went still. They said that down here people got killed sometimes, bodies dropped down the airshaft into the dark where nothing could smell them. He looked flatly past the gun.
Sidney was hot. "I don't like losing houses. Fin don't like losing houses."
"I ain't found out yet what happened," East said simply.
"Your boys ain't shit. Who was it?"
"Dap. Needle."
"Someone's stupid. Someone didn't care."
East objected, "They know their jobs. That was my house I had for two years."
" _I_ had the house, boy," Sidney spat. "You had the yard."
East nodded. "I was there a long time."
"Best house we had in The Boxes. Fin loves your skinny ass—you tell him it's gone."
It was not the first gun East had talked down. You did not fidget. You showed them that you were not scared. You waited.
Just then, Sidney's phone crackled. He uncocked his gun and stuck it away. Behind him, Johnny wagged his head and got off the car. Johnny was a strange go-along, dark black and slow-moving where Sidney was half Chinese, wound up all the time. Johnny was funny. He could be nice; he handled problems inside the house, kept the U's from fighting with each other. But you did not want to raise his temperature.
"Sidney don't relish the running," Johnny laughed. "In case you wasn't clear about that."
East breathed again. "Did everyone get out?"
"Barely. They got some U's. No money and no goods."
"Who was shooting out?"
"I don't know, man. Some old fool, shotgun in his _pants._ We was grabbing and getting. I guess you could say he was too."
Sidney put his phone away. He turned, fuming. "Someone _did_ get shot."
"I know it," said East. "Little girl." He could see the Jackson girl, the roundness of her face, like a plum, a little pink something tied in her hair.
"In the news it ain't gonna be no little girl," said Johnny. "Gonna be a very big girl. It's a little girl when _your_ ass gets shot."
This had been a bad time. Fin's man Marcus had been picked up three months ago. Marcus kept bank, never carried, never drove fast or packed a gun, quiet. He had a bad baby arm with seven fingers on his hand. He knew in his head where everything came from and where it went, where it was—no books, nothing to hide. Twenty-two years old, skilled, smart: Fin liked that. But they had him now, no bail. No bail meant the PD could just keep asking him questions till they ran out of questions. Since then, everything was getting tight. One lookout picked up just loitering—they kept him in for three days. Runners getting scooped off the street, just kids, police rounding them up in a boil of cars and lights, breaking them down.
Some judge wanted a war, so everything had gotten hard.
—
They rode the black wagon south unhappily. Sidney coughed wetly, like the running had made him lung-sick. He wiped his gargoyle face. "Don't look at the street signs," he snapped.
"Man, who cares? I know what street this is," said Johnny.
Something went _pumma, pumma, pum_ in the speaker box, and the AC prickled hard on East's face. He closed his eyes, like Sidney said, and didn't look out at the street.
Losing the house—it was going to be on him. He owned the daytime boys; he owned their failure. He'd run the yard for two years, and he'd taught the lookouts, and until today everyone said he'd done it well. His boys knew their jobs; they came on time, they didn't fight, didn't make noise. He could not see where it'd gone wrong. That girl—he shouldn't have talked to her for so long. Maybe she would have wandered off. He could have let Antonio muscle her a bit. She wound up dead anyhow.
What could he do? That many cops come to take a house, they're gonna take it.
A pair of dogs went wild as the car slowed, but East didn't open his eyes. Some of the neighbors' dogs likely were Fin's. Most people would keep a good dog if you gave them the food for it. And the cops looked where the dogs were. You didn't keep dogs where you stayed.
"Don't look at the house numbers."
"Man, how I'm not gonna see what house it is?" countered Johnny.
They parked down the street and walked. A little girl on a hollow tricycle scraped the sidewalk with her plastic wheels. The day had turned hot and windy. When Sidney said, _"Hyep,"_ they all turned and mounted two steps up toward a flat yellow house.
FOR SALE, said a sign. Someone had blacked out the real estate agent's name.
—
Answering the door was a short, stark-faced woman East had seen once before somewhere. On her hair she wore a jeweled black net. Her mouth was thin and colorless, slashed in. She showed them inside, then retreated, into a kitchen where something bubbled but gave no smell.
The room was empty, bare, brown wood floors. The drawn blinds muted the daylight into purples. A lonely nail on the walls here and there told of people who'd lived here once. There were two guns there too, Circo and Shawn. East had seen them before. It was never good, seeing them.
"Everyone get out your house when it happen?" asked Shawn. He was a tall kid, like Johnny.
"Little bitch didn't even give us a heads-up," flared Sidney.
East ignored it. It wasn't Sidney he had to answer to now. He wondered how much about it everyone knew.
Shawn wiped out the inside of his cheek with a finger and bit his lips unpleasantly.
"Gonna need me in Westwood tomorrow?"
"Depends. See what the day brings," said Sidney. "Miracles happen."
Shawn laughed once, more of a cough. He patted the bulk in the pocket of his jeans approvingly.
A security system beeped, and down the hall a door opened—just a click and a whisper of air. The woman exited the kitchen softly, on bare feet, and turned down the hall. She slipped inside the cracked door and shut it. A moment later it beeped open again. East watched the woman. She had a spell about her, like her time in this world was spent arranging things in another.
She pointed at East, Sidney, and Johnny. "You can come," she said calmly.
East had been in rooms like this before, where guns had talked vaguely amongst themselves. Until today, the day he'd lost the house, he'd found it exciting. Today he was glad to be summoned away. He caught a scent trailing from her body as he followed, and inhaled. Usually if he got this close to a woman, it was a U, heading in or out. Or one working the sidewalk, or stained from the fry grill. This woman was perfumed with something strange that didn't come out of a bottle. He held his breath.
The net in her hair glistened: tiny black pearls.
The system beeped again as she unsealed the door.
—
Fin's room: unlit except for two candles. He sat in a corner, barefoot and cross-legged atop a dark ottoman, his head bowed as if in prayer, a candle's gleam splitting his scalp in two. He was a big man, loose and large, and his shoulders loomed under his shirt.
This room had a dark, soft carpet. A second ottoman sat empty in the center of the room.
Fin raised his head. "Take off your shoes."
East bent and scuffled with his laces. In the doorway behind them appeared Circo, a boy of nineteen with a cop's belt, gun on one side and nightstick on the other. He stuck his nose in, looked around, and left. _Good._ The door beeped as the woman pulled it shut behind her.
Johnny took a cigarette out.
"Don't smoke in here, man," Fin said.
Johnny fumbled it back into the pack. "I'm sorry."
"House is for sale." Fin wiped the back of his head. "Purchase it if you like. Then you can do whatever you want."
The three boys arranged their shoes by the door.
Dust curled and floated above the candles. Fin sat waiting, like a schoolteacher. When he spoke, it was with an ominous softness.
"What happened?"
Sidney answered, grievous, wheezing. "No warning, man. Paying a whole crew of boys out there. When the time came, no one made a call. Didn't shout, didn't do shit."
"I did call you," East protested.
"When there was police already banging on the door."
So Sidney was here to saw him off.
"Why didn't they call?" Fin said it quietly, amused, almost as if he were asking himself.
Sidney jostled East forward unnecessarily. This meant him: this was his _why_.
"There was a lot going on," East began.
Fin, quizzical: "A lot?"
"Fire trucks. House fire," East said. "Lots of noise. The ends—Needle, Dap—maybe that's what they was thinking: police going to the fire. Maybe. I mean, I ain't spoken to them yet, so I can't say."
"I think your boys know to call when they see a police."
"Oh, yes, they know," East said. "Oh, yes."
"And why ain't you talked to them?"
"Something goes wrong, stay off the phone," East answered, "like you taught."
Fin looked from East to Sidney and back.
"Was there a fire for real?"
"I saw smoke. I saw trucks. I didn't walk and look."
"Maybe it don't matter," said Fin quietly, "but I might like to know." He gave East a hard look and then veiled his eyes. East felt a beating in his chest like a bird's wing.
A minute passed before Fin spoke again. "Close every house," he said. "Tell everybody. Submarine. I don't want to hear _anything_. I hate to say it, but people gonna have to look elsewhere a few days."
"I got it," said Sidney. "But what are we gonna do?"
"Nothing," Fin said. "Close my houses down."
"All right," said Sidney. "But how is it this little nigger fucked up, and _I'm_ not getting paid? Johnny neither?"
"I taught you to save for a rainy day," Fin said. "And, Sidney, I taught you not to say that word to me. You know better, so why don't you step out. You hear me?"
Sidney fell back and grimaced. "I apologize for that," he said, and he turned to pick up his shoes.
"You too, Johnny. You can go." Fin sighed. "East, you stay."
"You want us to wait for him?"
"No," Fin said. "Go on."
East stood still, not watching the two boys moving behind him. When the door beeped open and they left, the woman was there, outside, barefoot, waiting. She brought in a tray with two steaming clay cups. She stood mute, and something passed between her and Fin, no words, borne like an electrical charge. Then she placed the tray with the two cups on the empty ottoman.
Quietly she eyed East and then turned and left through the same door. _Beep._
—
"How you doing?" Fin said. "Shook up?"
East admitted it. He was aching where he stood. Tightened up more than ever. His knees felt unstrung. "Yeah."
"Sit."
East lowered himself to the second ottoman stiffly and sat beside the steaming tray in the dusky room. Fin spread his shoulders like a great bird. He moved slowly, top-heavy, as if his head were filled with something weightier than brains and bone.
Fin was East's father's brother—not that anyone had ever introduced East to his father. Others knew this; sometimes they resented East for it, the protective benevolence he moved under. But it shaped their world too, the special care that was given him, his house, his crew. When East was a child, Fin had been an occasional presence—not a family presence, like the grandmother whose house held a few Christmases, like the aunt who sometimes showed up in bright, baggy church clothes on Sunday afternoons with sandwiches and fruit in scarred plastic tubs. Fin was a visitor when East's mother was having hard times: to put a dishwasher in, to fetch East to a doctor when he had an ear infection or one of the crippling fevers he now remembered only dimly. Once Fin took East to the Lakers, good seats near the floor. But East didn't understand basketball, the spitting buzzers and the hostile rows of white people in chairs, and they'd left long before the game was decided.
But since East had grown, Fin was the quiet man in the background. East had never had to be a runner, a little kid dodging in and out of houses with a lunchbox full of goods or bills. He'd been a down-the-block lookout at ten, a junior on a house crew at twelve. He'd had his own yard for two years, directing and paying boys sometimes older and stronger than he was. Not often in that time had he laid eyes on Fin, but often he'd felt the quiet undertow of his uncle's blood carrying him deeper into the waves.
Did he want to do this? It didn't matter: it would provide. Did boys respect him because he could see a street and run a crew more tightly than anyone else, or because he was one of Fin's favorites? It didn't matter: either way he had his say, and the boys knew it. Was this a life that he'd be able to ride, or would he be drowned in it like other boys he'd kicked off his gang or seen bloody or dead in the street?
It didn't matter.
—
"Try it," Fin said.
East touched a cup, and his hand reared back. He was not used to hot drinks.
"Not ready yet?" Fin reached for his cup and drank soundlessly. The steam rose thick in the air. "Now tell me again why that girl got shot."
East saw her again, her face sideways on the street. Those stubborn eyes. He could still see them. "I tried," he said, and then his voice slipped away, and he had to swallow hard to get it back. He stared at the tea, the steam kicking up.
"I tried to make her go away," East said. "She been down the street all weekend playing ball and just came up that minute. Then the police. You couldn't tell her nothing."
"I see. Just her bad luck, then. Her bad timing."
"She was from Mississippi."
Fin sat and looked at East a long time.
"You know that girl hurts me more than the house," Fin said. "We got houses. We can move. Every time we move a house, we bring along the old and we pick up the new. It's that girl that costs us. It's that girl that goes down on my account."
"I know."
"She died."
East swallowed. "I know," he said.
Fin agitated his cup and stared down into it. "Get up and lock that door," he said. "I don't want nobody walking in on us, what happens next."
East stood. He stumbled in the carpet's pile. The lock was a push-button, nothing more. East pressed it gently.
Fin's dark eyes followed him back to the cushion.
"So you're free now. Had a house. Had a job. Lost the house. Lost the job."
East hung his head, but Fin waited on him to say something. "Yes, sir."
"You wondering what comes next?" Fin smacked his lips. "Because maybe nothing comes next. Maybe you should take some time."
_Take some time,_ East thought. What they said when they didn't want you anymore.
"There is something you might do for me," Fin said. "You can say yes or no. But it's quiet. We won't talk about it. Not now, next year, not ever. You keep it till you die."
East nodded. "I can keep quiet."
"I know. I know you can," Fin said. "So: I want you to go on a drive. At the end of that drive, I want you to do something." He curled a foot up and pulled on it. Fluid joints, a slow movement. "Murder a man."
East drew in his shoulder and carefully dried his mouth on it. A spark fired in his stomach; a snake curled.
"You can say yes or no. But once you do, you're in. Or out. So think."
"I'm in," East said automatically.
"I know you are," said Fin. And he drank down the rest of his tea, then shook his head twice, a long shudder that might have been a laugh or might have been something else entirely. East felt Fin's gaze then and swallowed the hard beating inside him.
"Be ready tomorrow, nine o'clock. You're gonna clear out straight. So bring clothes, shoes. That's all. No wallet. No weapons. We gonna take everything off you. Bring your phone, but you can't keep it. No phones on this trip. And no cards—we'll give you money. You hear?"
"All right."
"Keep your phone on, and Sidney will call."
"Sidney ain't too pleased with me right now."
"Sidney ain't got no choice," Fin said. "Okay? Be some other boys too going along. They a little older, more experienced. You might not feel you fit in. They might wonder too. Especially after today." He swabbed the moisture from the inside of his cup with his fingers. "But I think you got something they need."
This praise from Fin warmed him.
"Gone five, six days. You got a dog or a snake or something, find someone to feed it."
East shook his head.
"Good," said Fin. "Then we ain't talking about nothing in here. Just catching up. Stay a minute and drink your tea."
East picked up the heavy cup. He wet his tongue. The tea tasted old, like dust at first. Like something collected from the ground.
"You like this?"
East didn't, but he tried not to show. "What is it?"
"No name. It's good for you, though," said Fin. "That woman, she owned a tea shop. Then she fell into some things. I helped her. She knows business. She knows about bringing in off the docks. And she knows how to brew."
East nodded. "She's from China?"
"Half Thai. Half everything else," said Fin lazily. "How's your mother?"
East coughed once. "She's all right. She got a little sick, but she's better."
"House holding up?"
"Holding up," said East. "Hold up better if she cleaned a little."
"You the man of the family," said Fin. "You could up and clean it. Stop and see her before you go."
"Yes, sir. All right."
"All right," Fin said. "This is a big favor, man. This is not easy, what you're doing. I want you to know that it is important to me." Fin's hands clasped his feet and stretched them, twisted them. Like bones didn't matter, like they could be shaped any way you wanted. "I will remember it was you that did it," Fin said again. He put his cup down with East's. The two cups touching made a deep sound like the bell of a grandfather clock.
"Boy, go," Fin said. "Not a word. Nine o'clock. Sidney will see after your crew, take care of them. Don't worry. Stay low."
East stood. He felt childish in his white socks.
Fin brought out a thick fold of bills. He counted out twenties—five hundred dollars. He handed it over without looking at it.
"Some for your mother there."
"All right."
"One more thing you want to know. Your brother, he's in. He's part of the trip."
East nodded. But a little pearl of anger splattered inside his chest: his brother. Babysitting. Not that his brother was any baby.
"Maybe you ain't gonna like it. Figured I'd give you a night to get used to the idea." Fin rubbed down his feet, popped a toe. "You know why he's going."
East put the money away and laid his hand over his pocket. "Yeah, I know."
—
A bad street. Dogs bashed themselves against the fences. Televisions muttered house to house through caged doors and windows. East was the only person moving outside. He stepped up to a porch and unlocked the door.
In the living room, in a nest of dull air, his mother lay watching a game show. She looked older than her thirty-one: runny-nosed, fat and anemic at the same time. She drank from a plastic cup, a bottle of jug wine between her knees.
East approached from behind. She noticed, but late.
"Easton? What you doing here?"
Her fierceness, as always, was half surprise. She sat up.
"Hello, Mama," East said. He looked sideways at the game show.
"You come and sit down."
He sat beside her, and she smothered him in a hug that he received patiently, patting her arm. She did not turn down the TV: it made the windows hum. When she released him, her nose had grown wet again, and she was looking for somewhere to wipe it.
"I thought I might see you. I made eggs and bacon."
East stood up again. "I can't eat. I just came by to check."
"Let me take care of you," she reproached him.
East shrugged. The TV swerved into a commercial, even louder. It made him wince. He split off half the fold of bills Fin had given him, and she took it without resistance or thanks. The money curled unseen in her hand.
East said, "Nice day. You see it?"
"Huh?" his mother said, surprised again. "I didn't get outside today. Maybe. Where's Ty? You see him?"
"I ain't seen him. He's all right." He retreated to the kitchen, a little preserve behind a white counter littered with empty glasses. He could see her craning her neck, tracking him.
"He ain't been to see me."
"He's doing fine. He's busy."
"He my _ba_ by." Her voice rose frantically.
"Well, he's doing fine. He'll come around. I'll tell him."
"East," she commanded, "you eat some eggs. They're still in the pan."
_Let me take care of you._
When he flicked the switch, one of the two fluorescent tubes on the ceiling came to life. The kitchen was a wasteland. East bagged what could easily be thrown out. With a napkin from a burger bag he smashed ants. The eggs on the stove were revolting—cold and wet, visible pieces of shell. He turned away.
His mother had gotten up. She stood in the doorway.
"Easton," she breathed, "you gon stay here?"
Embarrassed, he said, "Mama, don't."
Proudly she said, "There's sheets on your bed."
"I can't tonight."
"I ain't seen either one of you," she sniffed.
Like every minute weighed a ton. "Mama, let me get this trash out."
"Whyn't you have some eggs?"
"Mama," he pleaded.
"Don't neither my boys love me," she announced to something on the opposite wall.
East dropped the bag of garbage. He found a fork in the congealed eggs, hacked out a mouthful, and shoveled it in. Sulfur. He tried to chew and swallow, eyes closed, and then turned to his mother. Eggs still milled around the sills of his teeth, horrible.
"You see." His mother beamed.
—
East's room was small but neat: twin bed with pillow, two photos on a shelf. A carpet he'd pulled up because he didn't like the pattern and laid back upside down. A little dust but no clutter. He shut the door, but the TV noise still buffeted him. He picked shirts, socks, and underwear out of the pressboard dresser and stuffed them into a pillowcase. He looked around for a moment before the door opened.
His mother, weary on her feet but still pursuing, stood in the doorway.
"Any of Ty's clothes here?" he asked.
She let out a sickly laugh. "Ty's clothes—he took them—I ain't seen—I don't know what Ty wear."
"Shirts? Anything?"
Two years younger, but Ty had left first. Even the room they'd shared for ten years—Ty barely ever seemed to live there. No toys, no animals, nothing taped to the wall. Like it was never his.
She zeroed in. "You going somewhere? You look like a tramp."
"Me and Ty need clothes for a few days."
She hummed, casual but knowing. "In trouble?"
"No."
"Suitcases in the closet. But they old."
"I don't need a suitcase," East said.
He stopped and waited stock-still till she retreated. After a moment he heard the squeak of the couch springs: she was down. He was alone. He checked the block of wood he'd mounted inside his bed frame, underneath: tight. He loosened it with the thumbscrew. He left his ATM cards there, then tightened it back down.
At the door he said, "I'll be back in a few days. Come see you then. Come and stay with you."
"I know you will. I know you gon come back," his mother cooed.
He took out the remaining money, peeled off three bills, and gave her the rest.
"I know you ain't in no trouble," she begged. "My boys ain't."
He tilted his face down, and she kissed him good-bye.
—
Down the street, freed from the shout of her TV, East heard the silence hiss like waves. He walked north until he entered an office park of sandy gray buildings nine stories high. Two of them stood in a sort of corner formation, and East walked around them. A faint hubbub of raucous people drinking came from somewhere in the darkness.
A narrow sidewalk led behind the air-conditioning island. The concrete pad full of AC units lent cover as East bent at the last building's foot. His fingers found the makeshift metal stay wedged between the panes of a basement window. The window fell inward, but he caught it before it made a sound. Quietly, twisting his body in one limb at a time, he crawled through.
The basement crawl space, dim behind dusty windows, was clean, its packed-dirt floor higher at the sides than in the middle. It was empty save for East's things and a faucet in one corner. It didn't turn on, but it wouldn't stop dripping either, and East had placed a wide stainless-steel bowl beneath it; there was always water, clear and cold. He tossed his bundle down and put his face over the bowl, watching his reflection swim in upside-down from the other side.
He drank. Then he washed his face, his hands, the caves of his armpits.
The spot where he slept was a pair of blankets, a pillow he'd bought at a roadside mattress store, and a large, heavy cardboard box the size of a washing machine. The air conditioners hummed all day, all night, washing out the hubbub and street noise. But that was not enough. East paused, stretched, then knelt on the floor beside the box. His hole. He tipped the cardboard up one side and straightened the blankets on the floor beneath it. He smacked the pillow straight and put his bundle of clothes down at the foot of the blanket. Then he slithered beneath and let the box drop over him. Like a reptile, a snake, calmest in the dark. Even the sound of the air conditioners vanished. Nothing. No one.
He breathed and waited.
# 3.
Near East's upturned box lay his pad of blankets and pillow. His shoes waited together in the shallow dirt, next to the old pillowcase full of clothes. Through the basement windows, early light crept in, the palest blue.
Sleeping through the night wasn't what he was used to. A long time, he'd been standing yard midnight to noon. He cleaned his teeth with the cold clear water brimming in the steel bowl. He cleaned his gums, fingers stretching his face to strange masks. He washed his arms again and his neck and face. He lowered his pants and washed his flanks and all around his balls, and shivered in the morning chill.
He checked his phone—Antonio, Dap, Needle, Sony. Nothing.
The house. He needed to go back and see. It was ten minutes' walk out of the office park to the house. East crossed the main street where the awnings were going up, taquerias and rim shops, the outer crust of the neighborhood. Then in, past the houses where everything was waking up, men hopping down their steps with cups and bags and keys, jumping into cars. Joggers and dog-walkers, old women smoking in their doorways.
A few blocks deeper, the cars turned older and were packed less tightly at the curb. The row houses sat blind behind plywood doors and windows—just a few at first, then more. Then two out of three. This was The Boxes.
He turned onto his street. This spot was where Dap kept one end's lookout. But he hadn't called. Needle watched the other end, five blocks down. He'd called too late. Sidney said back before cell phones was better: you knew who you could get in touch with and who you couldn't. You never sat and worried why somebody didn't answer his phone.
Two old gray women stood clucking on the shelf of their lawns. Most days when they rattled out to judge the morning, he'd been there for hours, eyes and skin already tuned to the movements of the day. But this morning they had the drop on him.
As he neared the house, he studied it sideways: brown face bullet-pocked, splintered, upper windows open like eyes. Smoke still seemed to hang. Now the door was just a sheet of plywood, bolted on. Yellow police tape stickied the yard side to side.
He ducked under to check. Power cords, recharging cables, all gone from the porch. So he had maybe an hour of battery left. Well, they'd be taking his phone. He gazed around the yard, but it told him nothing.
A patch on the sidewalk was still unmistakably bloody. He tried not to look. The Jackson girl's face was right near the top of his mind and he did not want to be seeing it, inside his head or anywhere else.
A man in a suit came walking past. Every day he walked by wordlessly, but today he nodded at East and thundered, "Good morning."
East nodded back.
"They caught you, didn't they, boy?" The man was jolly. "Shut you right down."
East ignored him, but the man kept on, made cocky by yesterday. "I see you got a pillowcase. Got your life in there?"
East shrugged and walked faster. He could have broken off a stick and beaten the man, bruised him, made him shut up and run. And for what?
He tried his phone in the open air, tried Dap, tried Needle. Neither of them answering. He left brusque messages: "Call me." But he'd trained his guys: if we run, stay off the phones. Now he was crossing that up.
It was ninety minutes before he needed to show up. He'd get a walk and eat breakfast. He picked his way south through The Boxes. Birds and small bugs stirred among the trees, buzzing like phones. Three little girls were out early, playing chalk on the sidewalk, colored _tizas_ as big as their wrists.
A cough from a porch was directed at him. "Hey, man. Hey," the voice came down.
East looked, then stopped. It was a man, maybe thirty-five, maybe forty, a U who'd come to his house some days. The man sat drinking out of a tall paper cup.
"What you need?" East said.
It was strange what he knew and didn't know. He knew this man, his secret hours. He remembered when he'd first showed up in the evening hours, still with a neatly made face, a thick gold band. Then he'd come more often. East remembered when he'd lost his job, and he'd marked the passage of drug time across the man's face, the thinness he'd taken on and the way his eyes now had that light to them, that ingenious, failing light.
The man's name was the one thing he didn't know.
"What you doing now?" the man said.
"Nothing."
"Where am I gon go?" asked the man indignantly.
East shrugged. He heard again what Fin had said. _Submarine_. _People gonna have to look elsewhere._ He knew a house a mile away, one that wasn't Fin's. But you didn't talk about other houses you knew. You didn't connect the dots.
The man coughed three times and spat out a large, silvery thing. "You don't _know_? Unacceptable, man."
East lowered his eyes and got walking again.
"Boy, don't deny me," the voice came, following him.
—
At eight, Sidney called him, told him where to go. A mile away—down off the south end of The Boxes. "Make sure you bring clothes for a couple days."
"Fin told me," East said.
"Course he did," sneered Sidney. Then East's phone died.
He bought a glazed doughnut to eat as he walked, then pulled down an orange off a low branch. He turned it in his hand as he walked, a small, heavy world. It was ripe, but he waited to eat it.
—
In the alleyway behind a line of stores, Michael Wilson was telling East about his car. Michael Wilson had the police interceptor and new glows down front and back and underneath. With a second battery Michael Wilson could run his system all night, loud as a club, and still crank it up and drive it away.
Michael Wilson was twenty—long body, long teeth, big brown eyes he liked to keep behind silver shades. Always laughing. Always telling a story. He had been a guy who came around The Boxes, sometimes keeping an eye, sometimes delivering a payday or food. He was an up-and-comer. He had gone away to college, UCLA, and East hadn't seen him since. He'd been a lot of noise to start with, and now he impressed himself even more.
Michael Wilson was shucking peanuts out of a blue plastic bag, tossing the nuts into his mouth, flipping the shells over his shoulder onto the pavement. Michael Wilson worried that the other guys might be stupid. Said he didn't have time to be riding with no one stupid, because stupid didn't stay cooped up. Stupid infected everyone. Little gangsters always thought they had a code, when really what they had was a case of stupid. East nodded, and he pretended to listen, because Michael Wilson was going to be a part. _But God damn,_ he thought.
For the longest time it was only the two of them. East checked his phone—dead. He shook his head. "What time you got?"
"Like nine," said Michael Wilson.
"What is _like_ _nine_?"
"It means almost nine. Approximately nine, motherfucker."
East sighed. "Let me see your watch." He grabbed Michael's wrist, eyed the watch's gold hands, the shining stones. "That's eight fifty-four. Not _like_ anything."
"It's like nine, old man," Michael said.
A couple of raggedy cars and a blue minivan shared the early light. Bulky air conditioners rusted on raised pads, muscular posts sunk in concrete to protect them from the drivers. Most of the stores were dark. One Chinese restaurant belched its fryer smells, and the women peered out at the boys and smoked in the safety of their doorway.
The next to arrive was a pumpkin-shaped boy in a green shirt. He waddled slowly, carefully; fat made his face young, his gait ancient. He breathed hard, excited, scraping his feet as he came. "Whew," he said. "Michael Wilson. What's up." They grabbed hands. Then Michael recognized him.
"I remember you. Walton? Wallace?"
"Walter."
Michael laughed at himself. "I remember you was up in those computers. A little science man."
"I remember you was going to college. You in charge of the laughing and lying, I guess."
"You in charge of the eating," said Michael Wilson. "Where's your bag at?"
Michael Wilson had things in a glossy contoured bag with a gym name on it. It looked like a new shoe. East had his pillowcase and his orange.
"I got no fuckin bag," Walter said. "I had no idea. I been all weekend at my uncle's in Bakersfield. They picked me up off the street fifteen fuckin minutes ago."
East eyed Walter. The _fuckin_. A soft boy sounding hard.
"I don't know what they told you. We gonna be gone for days, son," Michael Wilson said.
"I'll get some clothes on the road, I guess."
"If we can find a tent store," smirked Michael Wilson. He went to touch East's hand, but East looked the other way. _So._ There was a whole connection that came before. He leaned against the loading dock and studied the other two.
"You got the rundown? What the plan is?"
"No, man, they gonna tell us. They doing that here."
"And I heard you was at your leisure," the fat boy addressed East.
East looked up. "At what?"
"I said, I heard you was out of a job." Walter leaned against a post and addressed Michael Wilson. "This the boy whose house got shot up yesterday. They said there's three others coming," he explained, "so I asked who."
Michael Wilson cracked open a peanut and tossed the shell at East. "You lose your house? What you doing now?"
East swept his hand. "This."
"Moving up," said Michael Wilson. "What about you, Walt?"
"Everything," Walter said. "A couple days back they had me running a yard. Substitute teacher." He addressed East with a certain friendly contempt. "I used to work outside like you. Few years ago." He giggled.
East couldn't contain himself. "What you do now?"
"Projects," Walter said. "Research."
" _Re_ search?" said Michael Wilson. "How old are you, fat boy?"
"Seventeen."
"How about you, East?"
East looked away. "Fifteen."
The car arrived next. It was a burly black 300. Floating slow, the way cops sometimes did, all the way down the alley. At last the windows rolled down to reveal Sidney and Johnny.
"God damn, man," crowed Michael Wilson. "Could have walked here faster."
East saw that Michael laughed almost every time he talked. It wasn't that he thought everything was funny; it was like his sentence wasn't finished yet without it.
Sidney scowled at Michael Wilson and got out. He wore all white, a hot-day outfit. Johnny wore black jeans and no shirt.
"Where is the last one?" said Johnny.
"I don't know, shit," said Michael Wilson. "Number one is right here." Cackling.
"We ain't going over this twice," said Sidney. "What time is it?"
"Nine oh-five," said Michael Wilson.
"Fuck him then. He's late. Let's go on."
"I'll get him," Johnny said. "Fin said four boys, we gon have four boys." He fell back and started working his phone.
The fat boy scratched his face. "Who we waiting on?"
"My brother," said East calmly. There was a way to stick up without putting your neck out. Dealing with Ty— _Maybe you ain't gonna like it,_ Fin had said—would take plenty of neck.
"Oh. Ty," Sidney said. "That child cannot listen anyhow. So let's start. Just sit his ass in the back with a coloring book."
—
Sidney booted up a tablet on the back of the black car and swept his finger through a line of photos. A solid-looking black man, maybe sixty, a whitish beard cut thin. Broad, hammered-looking nose, a fighter's nose. Sharp eyes. In the pictures, he looked tired. His clothes cost good money: a black suit, a tie with some welt to it.
Sidney looked over their shoulders. "Judge Carver Thompson," he said. "When Fin's boy Marcus goes up on trial, he's the witness."
"Carver Thompson," said Michael Wilson. "If that ain't a name for a legal Negro, I don't know what is."
"Don't worry about his name. He used to be an asset to us. Now he ain't."
"That's why you going to kill him," said Johnny softly.
East looked around at the other boys. Michael Wilson nodded coolly. Looked like he knew. Walter didn't. Something falling out in the fat boy's throat, gagging him. East watched with satisfaction. _Little science man._ _Fuck you,_ he thought.
"Why this gonna take five days?" said Michael Wilson, quick on the pickup. "Why we ain't doing it already?"
Sidney put a road map down on top of the trunk. "Because here's where we at." He tapped Los Angeles. "And this man is way—over—here." He swept his hand across all the colors on the long stretch of land till he tapped on a yellow patch near a blue lake.
"Wisconsin?" said Michael Wilson.
Walter said, "What's a _black_ man doing in Wisconsin?"
"I guess a nigger likes to fish." Sidney shrugged. "Also likes to stay alive."
"How we gonna get there?" said Michael Wilson.
Now Walter's face turned cloudy. "Oh, shit. Oh, shit," he said, "I know what you're gonna say next. No flying, right? We about to drive all that?"
"Correct," said Sidney.
"You're tripping. That's a thousand miles," Michael Wilson said.
"Two thousand," said Walter despairingly. "That's why we ran them documents. Right? That's what you been setting up." He opened his hands, a little box, in front of Sidney.
"Crazy," said Michael Wilson. "We ain't gonna drive no two thousand miles. _And back_. Doesn't make sense."
"Michael Wilson," said Sidney softly. "You the oldest. You supposed to lead this crew. If you can't handle this trip, tell me, so I can shoot you and find someone that can."
Michael Wilson held his hands up, shifting gears smoothly. "Right, man," he sang. "Just running the numbers, man."
East breathed out and let his eyes adjust to the map, the thick red and black and blue cords inching state to state. Dense and jumpy. Every road had a number and joined up a hundred times with other roads. He saw how they would go. This was like the mazes they used to do in school while the teacher slept. What they said in school was: Don't worry. Keep looking at it. You can always get there.
—
After Sidney lectured them on the route and the job, Johnny handed each boy a wallet. East examined his. Inside, in a plastic window, was a California state driver's license with his face looking up at him. Dimly he remembered this, his picture taken in front of a blue cloth. Somebody he'd never seen, in some room the winter before. Some of the boys came in for it. Never asked why.
The work was good—the two photos, the watermarked top coat. Some kind of bar code on the back.
"Shit looks real," Michael Wilson said.
"It is real," Walter said.
" _Antoine Harris_. Sixteen years old," read East. "How you say it's _real_?"
"It ain't my name either," agreed Michael Wilson.
"Listen," said Sidney. "What is real? It's in the system. It's legit. Police pulls you over and looks you up, it's real. License like that cost a man on the street ten thousand dollars. So don't lose it. Read what it says and remember it in case some police asks you who you are."
"Kwame Harris," said Walter. "What, him and me supposed to be brothers?"
"You the one sat closer to the table," giggled Michael Wilson.
"Cousins," said Sidney. "Cousins. Know each other a little. Not too much."
"Here is one for you, Michael," Johnny said. "Give me yours. I'll keep it for you."
"If someone asks you where you're going," said Sidney, "you're going to a family reunion. If someone asks you where, you say Milwaukee, Wisconsin. If someone asks you where in Milwaukee, you don't know. That's three questions."
Johnny said, "Don't let nobody ask you more than three questions."
"We just some lying motherfuckers all across America," said Michael Wilson.
"You're getting it." Sidney snapped out a thick stack of money, and the air between the boys grew quiet and warm. All the eyes watched as he dealt twenties from his left hand into his right.
"Three hundred," he said to Walter. "Three hundred," he said to East. He passed the two piles. The rest, a bigger stack, went to Michael Wilson.
"Wait," Walter said. "I'm going out five days, _killing someone_ , and all I get is three hundred dollars?"
Johnny moved in. "Boy, this ain't compensation."
"This is expenses," bristled Sidney. "You pay cash. There is no credit card. There is no gas card. You dig? Ain't staying in motels. You go in and wash up in a rest stop, in a McDonald's. You ain't wasting time. You ain't making records of where you are. You the one supposed to _understand_ this."
"Walter, you're very smart," said Johnny. "But smarter people than you puzzled this out. So pretty please, shut the fuck up."
Walter nodded, swallowing it down.
"Michael Wilson, you got a thousand dollars. If there's a problem? You fix it. This money ain't for clothes or having a good time. This money ain't yours. The oldest one gets to hold. He gets to solve problems."
"All right," said Michael Wilson. He looked around at the younger boys significantly as he tucked the money away.
"So that's your ride today. Right there," said Johnny.
They all followed Johnny's eyes around the lot. The blue minivan was what he was looking at.
"What?" said Michael.
"Let me show you," Johnny said.
"Show me what? You chose the sorriest car you could find?"
Johnny took a handful of Michael Wilson and shoved him along ahead. "This a _job_ car, boys. This is my gift to you." He railed at them quietly. "Reliable. Invisible. Rebuilt. New six-cylinder, three-point-eight. New transmission. New suspension. New tires, brakes, battery. Doesn't look new, but it drives new. You can sleep in it. Most important, you ain't gonna look like ignorant gang boys, which is what, in fact, you _are_. Wisconsin plates. In this car you look like four mama's boys going to a family reunion, which is what you _want_ to look like. _Please don't give me a ticket, officer._ " He popped the back gate. Three cases of bottled water sat behind the rearmost seat. "You ain't got to love it. You ain't even got to bring it back. But this is the right car for the job."
"And what the fuck do we have here," Sidney said, looking up.
—
It was East's little brother. Shambling and grinning. He was small and two years younger than East. Lighter-skinned and already beginning to bald. But he had a sharp easiness. There was already something chiseled into him: Ty didn't care. He didn't want to be loved or trusted. He was capable and unafraid and undisturbed by anything he'd seen or done so far.
"Ty-monster, sneaking little thirty-six chambers motherfucker," said Johnny. They touched hands.
"Ty," said Sidney warily.
The other two boys stared. Ty ignored them, ignored East, completely. He sat down on the bumper of Johnny's black car and matter-of-factly drew a gun and reloaded the clip with bullets loose in the pocket of his blue T-shirt.
"This boy here," Johnny laughed.
Ty finished and put the gun straight down under his waistband. When he stood up from the bumper, the barrel stood out cock-straight in his pants.
"Which reminds me," Sidney said. "Give it up. Phones. Guns. Any ID you got. I need it right now."
" _Fuck_ that," Ty snorted.
"Whatever you got," said Sidney, unflinching. "Weapons. Knife or stick. Any digital other than a watch. If you got a bottle of something. Whatever you don't want the sheriff of White Town to find on you. Give it up right now."
East had come with just his phone, but the others all had something. Michael Wilson gave up a small bag and papers. Walter gave up a knife—a Korean type for street fights. So light and springy it would shiver inside you.
Sidney beckoned. "East? What else?"
"Nothing, man."
"Don't make me fuck you up."
He sniffed. "Fin told me, don't bring nothing."
"All right," said Johnny, and he and Sidney glanced at each other. They didn't ask Ty, just closed in. He twisted and swore while they held his hands. Johnny hung him up, and Sidney patted him down. They took just the one gun from his pants. Sidney examined it.
"Man, _fuck_ you," said Ty, wrenching his wrists free and rubbing them.
"Thank you for checking your weaponry at the door," Johnny said, taking it over.
"You best keep that for me."
"I'm keeping it already."
"Ridiculous," Ty snorted, shaking his clothes back right. "Sent to shoot a man with no guns."
East studied his brother. So content in his fury. Still little and raw but ready, happy to strike. So he had known about the job too. They hadn't had to tell him what to do—only the where and who.
"Get close to where you're headed, you'll get guns," Sidney said. "Until then, we need you clean. We need you to be angels." He wiped his mouth with his bare forearm—his tattoos glistened wet. "You think it's the same out there? But you don't know. It ain't. Them police don't budget on you. That's their country. They love a little Negro boy. They pat your ass down and you go to jail. You go to jail, the job ain't done. And if the job ain't done, Fin goes away." Suddenly, full-muscled, he lunged at Michael Wilson and bashed him back into the side of the van. East looked up, surprised.
"You listening to me, smiley-face motherfucker?" Sidney spat. He bared his teeth and raised his forehead to Michael's chin like a gouge.
"Sidney, man. We got it," East said. Straight bullying the lead man, he thought. Setting up the whole thing.
Sidney was wound way past tight. "The _job_. Do it the way we tell you. Got something funny to say now?"
"No," said Michael Wilson, clutching his sunglasses.
"Anyone?"
"No," said East. "We're good to go."
Johnny flexed the black rails of his arms. "See, we're being polite to you today, the better for your educational purpose. But do exactly what the fuck we say."
"We got it," said East again. _Patient,_ he reminded himself. A moment passed, the six of them wary and bareheaded in the sun.
"All right." Sidney wiped his mouth, then settled a little at last. "When you get into Iowa, look at the map. You gonna call for directions. Call this number." He opened the road atlas to the map of Iowa. A pink phone-sex street flyer was taped over its eastern half.
"This number here?" said Michael Wilson.
"This number here. When the operator asks what you want, you say, 'I want to talk to Abraham Lincoln.' "
"You what?" Michael Wilson broke up first.
Sidney waited bitterly. "Laugh all you want. But remember."
"Abraham Lincoln gonna say, 'Hi, Michael. Me so horny.' "
Even East bent over laughing.
Sidney waited, jaw hard like a fist. "Make that call," he said at last. "You'll get guns."
"Paid-for guns. Guns we selected for you," Johnny said. "Use them and lose them."
"The gun man is a white man. So be cool."
The van. East slipped away from the rest to examine it. Dingy outside—a few dents and scrapes untouched, dirty hubs, no polish for years. The upholstery showed wear. But the tires were brand-new, the tread still prickling. The windows were clean. Definitely submarine.
In his mind he was boiling it down: Drive the roads. Meet up for guns. _The job._ He tried to follow it in his mind, see where the problems were. But there was nothing to see. Only these boys. Kill a man? More like keep them from killing each other, these three boys, for two thousand miles in this ugly van. That was what they'd brought him in for. That was what he had to do to get back home.
—
Relieved of their things, armed with their new names and wallets full of twenties, they followed Johnny around the strip to the sporting goods store.
Above the clothes high banks of sick white lights spilled down.
"Dodgers cap. Dodgers shirts. Get you one," Sidney was repeating.
Walter squeezed between the triple-XL ends of the racks.
"Dodgers are faggots," said Ty.
"I don't disagree," sighed Johnny. "What can I say? White people love baseball. White people love the Dodgers."
"What I care what white people like?"
"Boy," Johnny said, "the world is made of white people. So you just pick out a nice hat."
All the clothes smelled of the chemicals that made them stiff and clean. The boys' hands sorted through the new and bright. East drifted back, found a rack marked CLEARANCE where the clothes didn't stink, grabbed two plain gray T-shirts with Dodgers script. Michael Wilson paid cash for it all at the register.
"Thank you for shopping with us today," the girl in her braids gushed. "Go Dodgers!"
"Thank _you_ ," Michael Wilson said over his sunglasses. "Okay, let's _vamos_ , kids."
Johnny reached for the receipt and crumpled it, then tore it to tiny bits.
Outside the damp, irrigated morning smell of Los Angeles flowers and fruit in the trees and small things rotting.
"Any problems? Any questions?" Sidney said. "Any last requests?"
East shrugged. Michael Wilson looked down into the white bag from the store.
"I don't think so," said Walter.
"Get going, then," Sidney said, already reaching for the door of Johnny's car. Like he couldn't be gone soon enough.
Michael Wilson had a key. East did too. Michael went for the driver's door. East ushered Walter up front. He tried the sliding door on the right side and popped it open.
In the dark of the van sat Fin, alone. Waiting in the middle seats, head bent low under the headliner, arms wrapped around his shoulders like pythons.
"Come on in," he said.
They exchanged glances and climbed in—Michael and Walter in the front, Ty sliding into the back. East sat on the middle bench next to Fin.
"You boys know how to lock a car?"
"Yeah," said Walter. "But we ain't gone anywhere yet."
"You got keys, though. Lock the doors. Or someone like me will be sitting in here when you boys return. Got it?"
They all nodded assent.
Fin's voice was deep, but his face was pinched, unhappy. "If I could," he said, "I'd do this myself. But I gotta trust you. You know the job?"
Four heads nodded yes.
"Michael Wilson, these the right boys?"
Michael Wilson found his voice, tried to steady it. "Yep."
"Anyone can't do it, walk away now."
Quietly they waited, reverent and impatient both.
From the center of the van came a black flash: Fin had drawn a fat pistol and cocked it, barrel at East's temple. East felt the cold metal burr scratching at his skin.
"You know East is my blood. Now I am sending you all out as my blood."
East's eyes kept a flat stare. Practiced. Like he didn't care.
"This gonna go all right?" said Fin.
"It's gonna go all right," Michael Wilson repeated.
Walter nodded, eyes large.
"I need you boys," said Fin. "And I don't like to need." Slowly he drew the gun back, then slipped it away somewhere. "Michael Wilson, you're in charge. You're the oldest. East and Walter, you keep him honest. Keep him straight. And, Ty, you make sure it gets done."
He slid the side door open.
"Any questions?"
Four boys shook their heads.
Fin lowered himself to the ground dryly, tenderly. "Don't make no friends."
Michael Wilson fired the engine. They watched Fin walk away down the row of loading docks, just a large, slow-moving man in the sun.
# 4.
A little compass on the ceiling glowed _E_. Only once did East turn to look back. Everything he recognized was already past: The Boxes. His gang. His mother. The abandoned, bullet-pecked house. The jimmied window and his crawl-space den. Fin and the gun he'd put to East's forehead _._ Now it was just these boys he had left, this van. Nothing.
Behind him lay his brother, a video game squeaking under his thumbs. Uninterested. East tried to remember the last time he'd seen Ty. Maybe late summer. A couple of months at least. He hadn't known where or how Ty was living. Not that Ty would tell. Talking to Ty, you ended up knowing less than you started with. He took a pleasure in sharing nothing, enjoying nothing, a scrawny boy who'd almost starved as a baby, didn't eat, didn't play— _failure to thrive,_ the relief doctor said. Smart but didn't like school, fast but didn't like running. Never cried as a baby, never asked questions. Never loved anything but guns.
Hadn't even said hi to East yet, his brother. Hadn't acknowledged that he'd noticed East there. East wasn't going to be the one to break the chill. He touched his forehead, the invisible scrape there, the gun's kiss that said, _This is important._ He stared away sideways through the curved, tinted glass: the traffic crawled and the streets changed, a scroll of places. East had never seen these streets or the ones beyond. Fifteen years old, he had never left Los Angeles before.
In the center seat he had an overview—he could watch the streets, watch these boys. Michael Wilson's head bobbed as he drove, talking, talking. Talking all the time, to everyone, even himself, a flow: he made music of it, he breathed through it. His sunglasses rode up top, and his head swung side to side, his white eyes dancing this way and that. So busy, thought East, working so hard. Walter, his head was lower down, bushier. He bulged off the seat into the middle and against the door. East had known some fat kids before, smart ones, worth something. But you couldn't work them in the yard. Not outside, a standing-up job.
But Walter was getting tight with Michael Wilson. Giggling at him. "Never thought you'd be driving a fuckin florist's van," he proposed.
Michael Wilson lifted his hands from the wheel. "It don't _smell_ like flowers."
East didn't mind the van. He liked the seat, the middle view, the drab shade. The carpet was blue. The seats were blue. The ceiling was a long faded grayish-blue, little pills of lint in the nap. Where he sat, the smoky windows were an arm's length away. They wouldn't roll down; they only popped out on a buckle hinge. That would do. Everything was an arm's length away.
He tucked his pillowcase under his seat and scouted in the knapsack hung on the back of Walter's. Pens, a notepad. Soaps and towels, toothbrushes and paste. The sort of kit somebody's mother would pack. And under the seat, a first-aid kit: gauze, ice packs, a flimsy red blanket that felt like wood splinters. CALL FOR HELP—AMBULANCE/POLICE was spelled out on one side. And four pairs of thin black nylon gloves.
Then the fat boy, Walter, cleared his throat and turned. He locked in on East. "Man, what you think when Fin held that barrel on you?"
"What'd I think?" East sniffed and looked off out the window. "Nothing." He didn't say: that is the second gun I've looked down in a day.
"Would stress me, man," Walter said. Still staring at East. Finally East looked back, and Walter gave a little nod and turned forward.
"Nigger, for blood, Fin must not like you all that much," said Michael Wilson.
"Fin says don't say _nigger_ ," East said flatly.
"Nigger, that's unrealistic." Silently Michael Wilson laughed at the windshield and bobbed his head to music, to a rocking horse in his mind. "In the man's personal presence, I defer. But be for real."
Walter said, "Michael. That word don't mean nothing to you?"
"Not like it means to Easy."
East shrugged. It was just a thing, a rule. Everyone broke it. But this trip was supposed to run tight. Rules were what they were. He supposed it would get wild at the end. That was later. Ty would see to that.
—
Out of habit, he studied them. Already bickering. "You got to get to I-15, man," Walter was saying. "You gotta hit Artesia Freeway before you get to 605."
"But you don't even say it right. Ar-tess-ya."
"It says it right here. Ar-tee-sia."
"You ever been there, man? Cause I had a girl that way, and she said Ar-tess-ya."
"You ain't never had no girl except off a corner in the ghetto," said Walter. "Just get us moving out this shit."
"Play some music, then, Walt," Michael Wilson ordered up.
For a minute, crouched over his stomach, Walter reached and fiddled with the radio. Nothing.
"Doesn't work."
"We supposed to have a genius," mourned Michael Wilson. "Can't even switch on a radio."
"It don't _work_."
Ty's game buzzed triumphantly. The screen for a moment turned his hands bright white.
"What you thinking back there, Easy?" said Michael Wilson, shimmying his head a few degrees over.
"He don't talk much," Walter opined. "He and his brother."
"No, he don't." Michael Wilson stopped at a light. "He don't like you."
Walter giggled. "He don't like _you._ "
East kept his mouth shut. Now they'd be teasing him. You just listened. You let it go, you came in when it was done, you kept your hands clean. Otherwise never built anything.
He didn't see how he'd ever get any sleep in this van.
"You know, we might want to get some food before we hit the road," Walter suggested.
"Oh, _we_ ," retorted Michael. "I didn't hear you taking no vote."
Walter too let it slide off, East noted. You didn't get to him by calling him fat boy. He'd heard it before. "Good chicken coming up on the right. What kind you want? They got both."
"Both what?"
"Both kinds. Black chicken and Spanish."
"If black chicken what made you fat, I'll take Spanish."
A noisy drive-through lane threaded between hedges and the little building. The loudspeaker squawked like a fire truck's horn. Michael Wilson paid and fetched out king-size drinks and a bucket. Walter passed the big drinks back, member VIP cups—you could refill them if you ever came back. Napkins. Hundreds of them.
They crawled onto the Artesia Freeway.
Ty finished a drumstick and twisted his fingers clean in a napkin before he picked up his game. The boy still barely ate. Barely moved sometimes. Unnatural.
They crawled past big yellow lights saying something was happening up ahead. East tried to see over the people in the other cars racked around the van. White man in an Acura to the right, phone pinned to the top of the wheel. Line of little Latin boys in the cab of a pickup, sullen and sleepy. Walter was fishing out a third piece of chicken. "Gonna be Christmas before we get out of LA," he mumbled.
East tried to look ahead at the brown mountains, to think about the way ahead. He tried not to watch. But it wasn't in him. A career of watching—a few years was a career in The Boxes—had made him the way he was. He spent his days watching, keeping things from happening. And sometimes he could see things coming. Sometimes he looked through a pair of eyes and saw what a U was deciding, what a neighbor feared.
Sometimes this watching wound East up. It was never finished. There would always be another thing, another thread, another pair of eyes. Days began sharp and tuned themselves sharper, until by the time it was over, the world made a quivering sound, like a black string humming. He could barely stand to be near things.
He slept in his box at night just to dampen the sound of the string. To blind him, deafen him, gratefully. Sometimes in the night beneath the box he woke up gasping.
Cars merged fitfully: they'd reached the obstruction. A vast brown Pontiac lay beached in the right lane, broken down and proud of it. Two women, bleached hair shining and loose, awaited the tow like parade queens. They sat up on the back, perched, putting lotion on. After them, the speed picked up.
—
The air conditioner was killing him. East weighed a hundred ten pounds, give or take. He reached into the first-aid kit and wrapped himself in the flimsy red blanket. The boys in the front seat laughed.
The traffic thinned out, and as the van climbed toward the mountains, they darkened, the solid brown breaking into purples and grays. All East's life the mountains had been a jagged base for the northern sky. It was the first time he'd been this far toward them, in them. He'd never seen them broken into what they were, single peaks dotted with plant scrub and rock litter, and the open distances between.
He couldn't stop looking—the slopes and tops and valleys that slowly revealed themselves. Like a new U moving in slowly, taking a look at the house, at the boys, maybe going away, maybe hanging around, deciding he'd just go up for a look, deciding he'd go inside. They all went inside sooner or later. He wanted those U's to decide, to go in or go away, come again another day, so he could stop watching them. In the same way he willed Michael Wilson to drive faster, to pell-mell it down the highway past the Pathfinders and Broncos and Subarus heading up into the hills for the afternoon. He wanted into the mountains. Almost wanted them over with.
Light poured between them, careened around them.
The purple and brown details opened, resolved themselves, then whipped away as they were passed. Shattered pieces of things. At length the van passed through a canyon where an entire hillside was on fire, white smoke fanning low in the wind.
The boys made no comment on this apparently normal disaster. East studied it, not breathing. Then it too was past.
—
Green signs flared by. Cajon Junction. Hesperia. Victorville. Thousands of windows punched into the valley, cars sliding on and off, streets littered with people. His eyes sought them out and registered. Hard to stop.
When he closed them, let his eyes rest, he felt, as he always did, that someone or something had turned its gaze on him.
"Go on and sleep if you like, Easy. I got it," Michael Wilson murmured. "Fat boy already checked out."
"All right," East said quietly. Walter had gone facedown on the tray of his breast.
Ty's video game behind him trumpeted happily. A new level.
In the afternoon darkness, they stopped for gas.
The ground had flattened—hills and fences, scrub and arroyo. But mostly empty. Sometimes East found his eyes had gone blurry, the miles spinning past like bus-window reflections. Normal afternoons, he'd sleep seven hours and get up for a while before heading to the house. Last night's sleep had been full but fruitless: his head was dried up, his lips stiff. The sun lay way over on the mountains, behind a shelf of cloud.
"Where are we?" said Walter, blinking too.
"This the desert, son," declared Michael Wilson.
Shimmering heat. The low sun wasn't sending it, but the pavement below them glowed white. East stepped quickly across.
The bathroom outside was locked, so he pissed in the back between two sun-cooked cars. Around him, the ground made a cracking sound, fine dry things touching in the breeze. Part of the moon was up, a white tab.
Inside the gas station, different oils, girl magazines. One ice cream cooler and one for drinks. Next to the counter where you paid for the gas was a grill.
"I can make you something, hon," said a woman, white, sun-scraped.
There was nothing in the place that he needed. He wandered outside. Michael Wilson stood squeegeeing the windshield.
"Boy, did you just piss out in the field?"
East glanced down. "Yeah. I just pissed out in the field."
The older boy laughed. "Country already."
East didn't answer. The jokes, the small laughs. He recognized Michael Wilson working, trying to do his job. The light talk of the leader. He dawdled past the van and stood on the apron of concrete and loose stones, looking around at the unbuilt land. Something moved in the distance, a ghost or tumbleweed. He'd seen a tumbleweed once, in a cartoon. This looked like that, a cartoon tumbleweed in cartoon land, hills and rubble, nothing with a name. Nothing real.
"Get in, Country," called Michael Wilson. "We got to find something for y'all to eat." And East turned back to the van, pale blue and dusty, like the sky.
# 5.
Only once did Michael Wilson try talking to Ty. They were riding out what was left of California. Or maybe it was Nevada. The land was dark, and sometimes the spread of headlights showed them low, brushy hills. Michael Wilson leaned away from the wheel.
"What's your story back there, young'un? Fighting them aliens still?"
A long, intent wait, and then the voice floated back, quavery, almost a little girl's: "Are you talking to me?"
"Yeah." Michael Wilson, for once, was not laughing.
"Tired of riding, man," Ty said. "Tired of bullshit. Why we ain't flying?" He switched back on the game with a musical flourish, as if the conversation were over.
This pushed Walter's button. "Oh, no. Really, man? Impossible. No way."
Michael Wilson: "We trying to keep low, young'un. You gotta use a credit card to buy a plane ticket. ID to buy, ID to get on. It don't matter if the ID ain't real; they still gonna track you."
Their song changed quick, thought East. Bitching about it themselves an hour ago.
"We'd be so fucked," said Walter. "They would know where we get off, where we catch a ride."
"So there's some trouble? Four little black boys done shot a man? A man who's a witness in LA next week? Look, I see where we got these four little Negroes just flown in from LA."
"I wonder when they flying back."
"Let's get a picture of them off the video."
"Let's call up the SWAT team."
Michael Wilson laughed. "Airborne Negro Detection System activated."
To all of this Ty snarled, "So what if they do?"
Michael Wilson and Walter glanced across at each other, holding their merriment.
"Fuck you," said Ty. "The both of you."
East sat with hands folded inside the red blanket. Pinning it shut. So this would be it. The funny faces in front tangling with Ty. For days.
"Oh, maybe you could fly," said Walter. "But you a wanted man then. This way, we're sneaking in, sneaking out. You remember that astronaut lady who wanted to kill her boyfriend's new girl? Drove all the way from Texas to Florida to get away with it? She had gas cans in the back, never stopped. She wore diapers, man, trying to keep off the cameras."
"You remember that _astronaut_ lady?" Ty sneered, and in the back he muttered softly to himself like an old injured dog. Again his thumbs worked the buttons of his game. And that was the end of Michael Wilson talking to Ty.
—
They were awake, but softly. Then Michael Wilson spoke. "You want to take a little drive into Vegas? See the sights?" The instrument panel lit the curve of his cheek like a moon.
"No," said East.
"Let's keep moving," said Walter.
But then the hollow dark of the desert was pierced, and colored light caught the undersides of the clouds. The boys stared ahead, rapt.
The van crept smoothly toward the emerging city's glow. It was like the orange flare of a cigarette—crawling, twinkling, growing bright. Buildings rose out of it like a great lighted forest. Vehicles of all descriptions visible on the road again in the spilled light. Then they were flashing through it, and everyone was watching, even Ty.
"Listen. We got to get off for gas anyway," Michael Wilson suggested, and _Yeah, yeah,_ they all agreed.
So much light. Crossing, cross firing, leaving nothing empty, no shadows. Barely had the van cleared the ramp before a pyramid appeared. A grand white pyramid, razor-edged, spotlit. Everything dressed up: even the parking ramp had a fairy on it, or a genie.
"City of sin, my niggers," drawled Michael Wilson, steering now with one hand.
He and Walter leaned out their windows, ogling, here and there letting out a holler. Evening, desert air. East sounded out the names as they passed: MGM. Aladdin. Bellagio. Flamingo. Treasure Island. Stardust. Riviera.
"Just like Disneyland, man," Michael Wilson added.
Couples. Women in pairs or threes. Men in vast teams. Families fitted out with prizes and bags. Wandering blind, their shadows spilling out in eight directions, walking like nobody walked in The Boxes.
" 'Biggest payouts on the strip,' " Michael Wilson read with a growl.
"But every sign's saying that," replied Walter.
Abruptly Michael Wilson sat up and swung the van into a parking lot. Wasn't any gas station here, East noted, but he let Michael snake the van back where low glaring lights caromed off tall buses and campers. A colonnaded canopy shone ahead, dancing in neon light. Michael aimed them that way.
"Michael," East prodded. "Just getting gas, right?"
"Sure," Michael Wilson said. They swooped in under the overhang to find one yellow rectangle outlined on the pavement just down from the door, a car just pulling out of it. Michael eased the van in there and parked it. Lights dancing in the wet of his eyes.
"You boys want to take a look?"
" _Hell_ yes," Walter said, already rolling out.
_"Hey,"_ East insisted.
"Don't worry, E," Michael Wilson said. "I know you're on tight. But you basic street Negroes don't get to Vegas every day. We got to _see_ this, man. Half a minute."
A tall lady in a silver cocktail dress and heels wafted by, shiny. Then it was Ty, brushing past without a word. East reached in vain, but his brother was already out the sliding door. _Damn._
So this is how Michael Wilson was going to do it. Sudden turns and promises.
To his left, a line of columns and potted palms. Lights moving on everything. They riled East, set his mind jumping. To the right, Ty wandered on the pavement, skinny, behind him a long row of golden doors. East clutched at his red blanket unhappily.
"Come on, Easy. It can't be that bad," Michael Wilson crooned. "If you scared, later I'll have fat boy read you a story."
—
The black carpet seemed limitless. Patterned neon curlicues forever, up steps, down ramps, no ceiling above, just the blinking lights on a thousand machines with their nonstop ringing jangle. The din was aggravating. East had seen a casino on TV—that gave no hint of how it would be, like a factory, a city, clanging, clanging, bells that weren't real, that couldn't be stilled, from distances that weren't real either. Clanging that didn't matter, signaled nothing, just made up the air of the place. Everything clanging.
Just the banks and banks of lighted boxes, and placed before each one, a person, rapturously lit.
Signs on the pillars warned, NO PERSONS UNDER THE AGE OF 18! But nobody was stepping after them: the doormen, the head-nodding security with ear coils, the waitresses with drinks in monogrammed glasses, the burly Mexican women wheelchairing old folks with oxygen tanks. Nobody accused Ty. No one watched with a purpose.
_These people looked drugged,_ East thought, _or lost._
He straightened his shirt and made to catch up. Michael Wilson was monologuing: _Yeah, yeah, I'm a show you what's tight._ Headed for something. Then at the end of a long, littered aisle, they came upon a clearing and a low, carpeted mesa—three shallow steps up. Glowing green tables in formation, ringed by white people. Michael Wilson took the steps at a trot, and the boys flocked along.
East hung back, parked himself on a column. Tried to see, not to be seen. He watched people eyeing the boys as Walter and Ty milled behind Michael's shoulder, peering down at the green felt.
Michael wedged himself in between two white women in dresses that noted the bones of their backs. "Deal me in, man," he demanded, fanning a handful of twenties. Sidney's money, East thought. Fin's money.
The dealer was the second black man at the table. Tall and prim with a silver clef on his tie. Neat. "Hey, brother," Michael Wilson addressed him, more directly. "Deal me in."
Now everyone looked up at this university Negro with his money hanging out.
The dealer pursed his lips; his politeness was contempt. "Please, sir. First you put value on a card. Then at the table you buy chips."
The money levitated in Michael Wilson's hand. His answer.
"This is not a cash game, sir."
"Oh. It isn't." Not a question, a challenge. No one else spoke. "Okay, my brother," Michael Wilson purred. "I see you in a minute." He broke away, pushing between Walter and Ty, and East caught his look: humiliated. An acted-out sweetness, packed with rage.
Now East fell in beside Michael, got up shoulder-close as they walked.
"Mike. We got to get. You said a half minute. We ain't supposed to be here."
"Ten minutes, E," Michael Wilson muttered, bulling high gear through the crowd. East glanced back at Walter and Ty, and they tried to keep up. Michael veered toward a spill of light jutting up: musical notes, blazing in turn, stepping up the wall into the dark. He found a service window, jailhouse bars over the counter, polished to a scream, and no one in line. Not a real window; like a window in a movie. Like _The Wizard of Oz._
East caught up just as Michael put his hands on the white marble counter, the stack of twenties flat under his left. "We ain't got time for this," he argued.
"Sir?" came the voice behind the bars.
The cashier was not a young woman, but her cheeks and eyes were dolled up with glitter. She eyed them each in turn: Michael, East, and then Ty and Walter as they jostled in.
Michael Wilson faced the woman and lit his face up just like hers.
"I want one hundred dollars' poker chips, ma'am," he announced.
"Sir." She inclined her head, as if reciting a rule in school. "You must be eighteen to enter, sir."
Michael's smile. "I'm twenty, ma'am."
"Yes, but, sir," the woman said. Patient, undeterred. "Are these gentlemen with you? Do they have ID?"
East watched Michael's eyes: one flash. Then his smile hooked itself back on. "They aren't _gambling_ ," he said. "So, what? They can't even watch? Can't see me?" He laughed. "How I'm gonna leave my babies in the car?"
The woman took a step back out of Michael's breathing room. She had decided. Michael saw it too.
Walter spoke first. "Mike. Let's step out, man. We don't want any trouble down here."
East caught a movement from the direction of the card tables. A big blue suit with a headset was bearing down. A security guard, the size of a football player. "Now look out," he warned.
At last something made Michael quit smiling. Now his strut became a hurry as he herded the three boys back. They skittered between the ringing machines, dodging players who careened, drugged, from stool to stool. But where had the door gone? Ty broke off ahead, scouting; East had lost his sense of direction entirely. Walter was lagging behind, and East waited up.
" _Go_ , man," he snapped.
"I'm going. I'm going."
Something made him cruel, made him jab at Walter. "This is your fault," he said. " _First_ one out the van."
"I said I'm going," Walter panted.
The players saw them coming now, and they got out of Walter's way, tokens rattling like chains in their plastic cups. East glanced back: the security man was cutting them room. But still he trailed, talking into a cupped hand.
A short whistle from ahead. They'd located the exits.
Past the first set of doors, they spilled out into the vestibule, piano music raining down. But now Michael Wilson had stopped, knelt to tie his shoe. East fished his keys out and slipped them to Walter: "Start the van." The doors sucked air as the seals broke, and East caught a slice of the night outside, the heat and sound of motors. The guard trailing them had stopped near the doors. They'd done what he wanted them to do.
Except Michael switched feet, began reknotting his second shoe.
East couldn't watch. "Quit stalling, man."
"Ain't the most family-friendly establishment you could ask for, is it, E?" Michael finished the knot and admired it before he stood. Cheery now.
"Mike, I'm gonna tell you something," East began.
"East, man."
The grin on him. It didn't matter what you said. It just came back.
East faced Michael Wilson up. "How long you gonna take in there? And how much money you gonna spend?"
_"East,"_ Michael Wilson purred. "Just a _taste._ " He sized his thumb and finger a half inch apart. Like a U in the yard—he wasn't even seeing East. He was staring back through the inner doors. "Slots, man—you put twenty dollars on a card, you can play it in a minute. Might even win. I'll let you play, man. You gonna like it."
"Fin's twenty dollars?"
"Fin ain't here. I'm in charge. Like Fin said."
"Then _be_ in charge."
"I am," grinned Michael Wilson. "All that's holding me up is one whiny little bitch."
The seal of the golden doors broke again as a pair of ancient women staggered in from outside, gasping, _"Oh, my goodness. Oh, my goodness!"_ Then the revolving lights outside found their way in too, announcing it, some new, bright kind of trouble.
—
Outside, under the canopy three cars wide, things were sudden and sharp. Every sound, every fidget of the lights was back in focus; every sound had a maker. An engine whined. A woman was shrieking. The palm leaves shivering in an invisible breeze.
The yellow light was spinning off a big white tow truck, and somewhere East heard Walter's hollering, a muffled squawk.
Michael Wilson: "Where's the _van_?"
East pointed; then he ran.
The tow truck was bulky, a wide silver bed tipped back like a scoop, and a steel cable ran taut down under the little van's nose, reeling it in. East's stomach slid. High on the wall he glimpsed the sign now: RESERVED PARKING/TOW ZONE. Of course.
East headed to Walter's window. Walter was pop-eyed and frantic in the driver's seat. Nobody was paying him any mind.
"What are you _doing_?"
"I'm in the car," Walter ranted. "They can't tow it. It's a rule. Tell him!"
"Tell who _what_?"
"Him!"
East saw then. Down low on the wrecker's left flank, a burly guy with a beard was working the levers, making the winch squeal and spool the thick line. But it was running the wrong way: he was turning them loose.
"He's letting us go?"
"Uh-huh," Walter hyperventilated.
"How come?"
A quiver passed down Walter's face. "Your brother."
Again East looked. Ty was poised high on the running board, staring down like a wildcat. Below him, the tow truck guy hurried, one-armed, shielding his head.
Michael Wilson stepped right up to the tow guy, bellowing: "Man, get my car the fuck off this thing."
Pushing a lever, the wrecker man stopped the winch. He stood and winced and spat something red on the pavement. "I _am_ ," he said, and East saw it: Something had made a mess of his mouth. Beard full of blood. Plainly afraid, the wrecker man nodded quickly at Michael Wilson and got away, rolling himself under the van's front bumper, out of sight.
Everything seemed to sizzle in the battling, shifting lights. Like they were caught in a camera flash that went on and on. Off to the left, by a concrete pillar, two security guys were watching everything.
East still could not comprehend. "Letting us go, right?" he asked Walter, and the fat boy said, "I _think._ "
"Fuck it, then." East stepped off and whistled, beckoned Michael and Ty. _Back in the van._ Because the security twins, they were getting ready. Bow ties, shiny patent leather shoes, but he could tell by the necks—all muscle. "Come _on_ ," East warned.
Michael Wilson cursed down at the wrecker man's legs as he scampered by. Ty hopped down off the tow truck. "Oh, God," said the shrieking woman, "look what you done!"
_Only a minute,_ East thought. A minute ago they were making time. Rolling. He knelt and watched the wrecker man work. He'd watched tows before, broken-down cars or repossessions. But never like this, peering up under the fender and counting seconds. The grips and chains came off the left wheel, and the guy shimmied over to work the right.
"Start it up," East barked to Walter.
"It _is_ started," Walter replied over the noise. A third security man arrived, triplet to the other two.
East tasted bile, spun around the back of the van, and climbed in shotgun. Michael and Ty huddled wide-eyed in the back. "You gotta wait for him," he instructed Walter, "but when he comes up, get us the fuck out."
They listened to the sounds, the wrestling going on below.
Then the tow driver's legs flailed out and spun, and he was lifting himself upright. He uttered something inaudible, his mouth wet again with blood. What was it? Did it matter? Walter was already crawling the van back. Two more bow ties came bursting out the golden doors. Walter was clear: he found Drive, and swung the van out around the big wrecker.
"Steady," East urged. The tow guy stood on the now-empty flatbed, cursing them. "Don't give them a reason."
"They _got_ a motherfuckin reason," moaned Walter. "They got one."
"Just be cool," East said. "Just get us out of here."
Walter muttered and steered. East glanced back at the security crew spreading out across the pavement where they'd been. "They're deciding do they want a piece of us."
"Got our plates. Got our pictures. Everything," mourned Walter.
"Drive, man," East said wearily. To nobody in particular he added, "Who was the girl?"
"What girl?"
"The girl that kept screaming."
"I don't know," Michael Wilson put in. "I didn't hear no girl."
"There was a girl," Walter sighed. "But that ain't had nothing to do with us."
Past the buses, toward the street, the white blaze where each light now seemed aimed straight at them. A sign by the curb read: PLAY IT AGAIN, SAM!
—
The shining monuments slid away, but all the boys were watching was the road behind them. Walter ran yellows to get them back on the interstate. Cars and trucks flashed by on the left, roaring; Walter was too tense to speak. After three miles he took them off at an exit, picked out a gas station, and stopped at the pump. He closed his eyes for a long minute.
At last East remarked politely, "I'm beginning to feel you. About the cameras."
"Got them here too," sighed Walter. "Why we need to keep from doing stupid shit."
East turned and shot a look at Michael Wilson. Michael saw East glaring and paused. "Mighty weird casino back there," he began.
"You shut the fuck up."
Walter bit his lips, looked sideways.
"Don't freak out, Easy," Michael Wilson said.
"Shit," East said. "Lucky we ain't facedown on a police car right now. You can't even _park_ without fucking up."
Meticulously Michael Wilson wiped something from his brow. "What do you want?" he said. "You want a little note, I'm sorry? I'm a get you flowers? _I'm sorry._ But don't say you didn't want to go in too."
"I didn't want to go in."
"But you went." Michael Wilson opened the door and climbed out. He dabbed at his hairline. "This is when I go pay for things. Who's pumping?"
East cursed. He climbed out and set the nozzle in, then stood waiting for the pump to click on. Just listening, the night sky starless, smeared pale by lights, by his pique. Unacceptable. He blamed Walter almost as much as Michael. But he was too mad to even begin with it.
At last the pump beeped and the orange numerals zeroed out. He began running the tank full of regular and banged on Walter's window. It rolled down.
"We got problems with that one."
Walter moaned, ghost-faced. "He's right, man. We all did go in."
"You first," East insisted. "None of us would have gone if you didn't."
"If I didn't," Walter said, "we'd still be there. Outside, waiting. Wondering was he gonna stop before all the money was gone. You think he was just gonna come back out in five minutes?"
East slid dead bug crisps around with his feet. "All right. What happened outside, then? The tow guy?"
Walter's face pinched shut. He shook his head.
"You best tell me. I need to know."
The pump kicked off, and East hung up the nozzle. Inside the bright station he saw Michael Wilson waiting in line, his head bobbing to a song inside it.
Walter squeezed himself out of the van. He glanced at East and stepped to the other side of the pump, furtively. East glanced back at the van where Ty was and followed Walter.
"It said _No Parking._ Right? We didn't see it. When I walked back out, the van was already hooked up. They probably keep that truck in the lot all the time. So. The law says you can't tow when somebody's in it."
"Don't give me law. This ain't California."
"It ain't just California."
"Stop with the truck," East sighed. "What happened with the guy?"
"So I'm yelling at the guy," Walter continued, "telling him stop. Then whoop, here comes your brother."
Walter swung his arm once.
"What'd he do? Hit him?" East scoffed. "Boy weighs a hundred pounds."
"Hit him with a gun," Walter whispered. "That's what I believe."
East frowned. "But Johnny searched him. He's clean. You saw."
"I know," said Walter. "Whatever it was, that guy changed his mind quick. And security, standing back watching like they did—explain that."
East looked up and tried to swallow the bad taste in his mouth. Above them, a big plastic dinosaur spun on a wire. Cars rushed by out on the highway, and East had to keep himself from staring down each one. Things moving. At first, the ride had felt like getting out, like being set free. Into nothing. But since Vegas, this felt like being stuck back in it. Like every headlight that rolled past was pointed at him.
"That boy is trouble," Walter said, looking away into nowhere.
"Which one?"
"Your brother."
East's back went up in spite of himself. "My brother is on the job. College boy is the problem."
"You talk like you're sure," said Walter, "but you best _be_ sure."
East was not sure. What East didn't know about his brother would fill the van. You heard stories. Things he'd done, scenes he'd been on, that he could get in anywhere, was too little to catch, too young for the police to bother with. Only stories, and nobody, least of all Ty, would say what was true.
Gloomily East glanced at the closed, smoky window where Ty lay listening to them talk. Or not. Then across the pavement came Michael Wilson, white shirt glowing, white teeth grinning, paid up and ready, his hands clean.
# 6.
Then it was late and dark, the scenery switched off, somewhere in the flat, empty Nevada that lay past Las Vegas. "So this the Wild West, huh?" Michael Wilson said. "Like, if the sun was up, they'd be riding horses and shit."
Walter rode beside Michael Wilson, who drove, and Ty slept, his video game switched off, across the back bench of the van. East sat tired and worried on his middle seat, crouched forward, hands uselessly figuring atop his knees, listening to the two boys in front telling lies.
"One time when I was at UCLA, man," Michael Wilson remarked, "we had a horse."
Walter said, "Horses don't like black people."
"Why don't horses like black people?"
"Why you think?" Walter said. "Who owns them?"
"But black dudes train horses. That one horse, what's his name, in the movie. Secretariat. Old nigger trained that horse."
"Train him to what?"
"He was a racehorse, man."
"Huh. He any good?"
"He won the Kentucky Derby."
" _Course_ he did," said Walter. "All right, that's one."
"Anyways," said Michael Wilson. "This horse liked me fine. He was a stolen horse."
"What do you mean, a stolen horse?"
"Some dude stole the horse," said Michael Wilson. "And he kept it on campus. The horse grazed the yard and shit on the sidewalk. Everybody giving it ice cream and pizza all the time."
"Horses don't eat ice cream."
"This one did," Michael Wilson said.
"Could you ride the horse?"
"Wasn't that kind of horse."
"What kind of horse was it?"
"I don't know what kind of horse it was, fat boy. Just stayed put and made a mess."
"What's interesting about that?"
Michael Wilson exploded. "You ain't supposed to have no horse in college, man. Simple."
"Why not?"
"Because you ain't. You got to follow the rules, or they kick you out."
"Why they kick you out?"
"They didn't," said Michael Wilson. "I left."
"You told Diamonds you got kicked out. I heard you."
"Oh, you were there for that?" Michael Wilson laughed. "Don't nobody tell Diamonds the truth, man."
"Who is Diamonds?" East put in.
"Eastside runner," singsonged Michael. "Sorry-ass Covina wannabe with one gun and a Nissan, trying to muscle in. Didn't know what he was doing. Fin involved him in a little business for about three weeks."
"Why they call him Diamonds?"
Walter said, "I think that's cause it's his name."
"Diamonds Wooten." Michael Wilson nodded. "Nice name. Back in Covina now."
"I don't like horses," Walter said. "They big and they bite and they mess you up. You like horses, East?"
"I never seen a horse," East said, "except with a cop on it."
"You a professional street nigger, East. I like you," Michael Wilson said. He laughed delightedly at himself.
Walter told a story about the U who'd walked into his house one day a few years ago with a Food 4 Less bag with two rattlesnakes in it, trying to scare his way into a fix. It might have worked, except the rattlesnakes went into a hole in the wall, and when the word spread, nobody wanted to use that house till Walter announced he'd gotten them out—though he never had. The snakes might still be in there.
Michael Wilson told about the research he'd done for Fin at UCLA. Michael Wilson said Fin wanted to know how much weed he could run at UCLA, thinking the college kids had to be underserved. What Michael Wilson found out was that there was more weed at UCLA than you could keep track of. More supply, more lines you never saw on the street, varieties, hybrids, designer weed, organic weed, heirloom weed, weed that was vanilla and weed that was chocolate, weed cut up this way and that, kindergarten weed all the way to cop grade. Selling for nothing. Giving it away. What happened rather than Fin trying to move in, said Michael Wilson, is that Fin started bringing weed out. UCLA was like Fin's docks. What UCLA did not have was cocaine. They didn't have it, didn't know how to get it, didn't know how much to pay for it. So that was a great couple of years, said Michael Wilson.
Walter said he thought Michael Wilson went to college one year. Michael Wilson said that for one year at college he studied; the second year was business.
Michael told about the day when he was sixteen and started working for Fin: his first job was secret shopper, just walking around buying drugs off everyone to see were they doing it right, were they straight, did they treat him right, what did they charge? Every hit he bought, he had to report. At the end of the day he had so much cocaine on him, he'd have gone inside, ten years mandatory, if he'd been hooked. His first day.
A motorcycle flashed by them in the left lane, doing ninety, a hundred, maybe more, a chainsaw roar in the dark. They watched the single red light shrink in the darkness.
"Never catch that," Michael Wilson said with conviction. "Those dudes got it made. Cops don't even try."
Then Michael Wilson asked East what it had been like, who was on his crew again, how long had he been on, when his house got policed. East stirred. Maybe he'd been dozing. His sleep was messed up.
"How was that, when it all came down?" asked Michael Wilson. And Walter, yawning, echoed, "Yeah, how was that?"
East didn't answer. It had been a weird day, all day a weird feeling, the fire trucks lost on the street, the old guy lying down to sleep in the backyard. For the first time he remembered that guy, the one who said he owned the house. For the first time since they'd pulled out of Los Angeles, he thought about his crew, thought about the police coming down.
He wished he'd gotten in touch with them, Dap and Needle, the ends who hadn't called in when the police cars had gone by them. He wished he'd found out why.
When he'd run, it had felt like he'd ditched everything. He'd been ready to accept whatever came—whatever Fin or one of his guns dished out. He was no different than the U's scrambling out, on a fix or wishing they were, trying not to get caught. From the castle of their getting some into the cold should-have-known.
In the darkness inside the van he remembered the whole yard—the porch, the walk, the beaten grass the color of dirt. The fingers of morning light spilling onto the street, the houses across. The helicopter and fire trucks.
And the girl who'd been shot there. He wasn't ready to think about that.
—
Walter snored once loudly and jerked. "Damn," he wheezed. "Fell asleep."
"You can move back," East offered. "I'll ride up front."
Michael Wilson stopped along the highway so Walter could maneuver out and climb back in. East tried to make out Walter's face as they traded places. But the night was unreasonably black.
He buckled in as Michael set the van rolling. It was just them and the white lines, one car fading away a mile ahead, a pair of red eyes.
" _'My crew is mad deep, I hope you niggas sleep,'_ " recited Michael Wilson.
"Oh, now you gonna rap for us?" East said. He lowered his seat belt and brought the seat forward a notch. "So, at the casino, man."
He watched in the dashboard lights as Michael Wilson reset his lips, then ducked low under the visor to read the road sign. "Make sure I get east on I-70, man."
East set his feet. Ignore the ignoring. "What was that, man? That mad-dog shit?"
Michael just rode his hands up on the wheel and bit his lips again. East stopped staring at him after a while, watched the reflectors pass instead. Little blots of light. A million of them already. His eyes were tired.
"Mad-dog shit?" Michael Wilson said at last. "You make that up last night standing yard in The Boxes?"
Now East stayed quiet. He'd cast his line.
"Listen," Michael Wilson came up with eventually. "You ain't gonna hear it. But when I stopped, I did it for you."
East snorted. "For me."
"For y'all. All of us. All right? It didn't work out. But I was trying, you know, to fire y'all up. I thought you'd _like_ it. I thought we might win or lose, man, but go in and look, play a bit. Come out as a team." Michael Wilson muttered as if the world were lined up against him. "Every good coach don't win every game."
"You ain't a coach is why."
"East, honey," Michael Wilson said. "You want to fuck with me? Do it straight. Not sideways."
"All right, then," said East. He took a pleasure in letting it out. Let everyone wake up. "You _ain't_ a coach. This ain't a team. This is a job. Keep on the job."
Michael Wilson nodded. "You done? Is that all?"
"If you can. If you can keep on the job," he taunted.
"You all, 'The boy stood on the burning deck.' "
"Well, I don't know what that is," East sniffed.
"I know you don't, my brother," said Michael Wilson. "I know. It's okay. Be a pal. Don't let me miss east I-70."
East let it rest. He didn't trust Michael Wilson.
"I went skiing up here one time," Michael said. "I went on this, like, black-diamond motherfucker. Dudes barreling past me on snowboards, I thought I was gonna die. I figured, okay, watch out for them two motherfuckers, then here comes another—"
"There it is." The sign. The wide green banner across the road. They were already under it. "Go that way, man."
Neither of them had been watching.
"You sure?"
"East I-70. Go right."
"Whoa, whoa, whoa, whoa!" Michael Wilson hollered, checking his flank and then banking the van right, sharp but smooth, across lines and the apron strewn with gravel bits. "Whoa. Thanks. See? We gonna help each other out!"
_If I'd never said,_ East wondered, _what would happen? Where would we be going to?_ Slowly the compass on the ceiling returned to _E_.
Michael went straight back to the story: "Anyway, these dudes on snowboards, they be zipping in there. And I think, and I'm, like, man, these dudes _gotta_ be getting high."
"You skiing?"
"Yeah. So I—"
"No," said East. "I mean, what? Skiing? You, man."
"Went up with my dad," said Michael insouciantly. "He had business."
"Huh." A dad. So that was another thing Michael Wilson had. "What he do?"
"Salesman. Pharmaceuticals. He sells medicine."
"Yeah? Why you do this, then? Why don't you do that?"
"He don't like his job."
"Yeah," East said.
"So I figure," Michael Wilson explained, "you could sell a lot up here. Same kind of deal like UCLA. But you got to find the right people. Local."
"So Fin lets you check things out," East said. "On the ground."
"Right," said Michael Wilson. "Market research, you call it. But the mountains—not for us, E. You can't be standing yard up here. It's different. You're alone. Black boys can't hide in the snow." He laughed at himself.
He rattled on. It didn't matter. Michael Wilson didn't hear him not answering. Right from the top East had known why Michael Wilson was along: to talk. To front them through. To bore any cop with his shiny record and UCLA smile.
But there was a problem. Michael Wilson was a fool. A rich boy and a gambler. Maybe that was all. You could bring worse problems. And if Fin had picked him, then Fin knew already what he was. What Michael _did_ , that was East's to handle. But carefully. Because Michael _was_ a pretty face, a storyteller; because Michael did pull Walter along. Even Ty. Even his brother, the real mad dog. He wasn't following East. He was following the tall boy with the year of college and the wad of cash out of the van.
East did not have those things, so he was pinned down. For now.
He breathed and watched the dark go past, the cold, relaxing nothing. White lines measuring it. There would be so much time just like this, waiting.
The road dipped, into a valley where no lights showed. Space like you never saw except in commercials. Strange dark land without people in it, miles of space between. For as long as he remembered, his business had been keeping people at arm's length. Keep the U's quiet and orderly, moving on. Keep his crew watchful, not too familiar. Keep people who didn't have business from even passing the yard. Keep his mother from worrying.
Standing yard.
Out here, everything kept at a distance. You could go an hour on the freeway without seeing a person walking, standing. One red ghost eye of a car a mile up. Maybe the same car. Maybe different. You'd never know.
He reached back onto the floor for the thin red first-aid blanket and doubled it up between his head and the window. After a minute he closed his eyes. But they would not obey; his lids would not soften. Every rise, every little tick in the inertia: they were hard. He checked Michael, checked the road ahead, the black nothing in the side mirror.
They were to trust each other, Fin had said. But East trusted nothing.
—
East stirred. Michael Wilson was slowing the van—a long parking lot, lights hung high, angled spaces. He pulled in and killed the engine.
"Gotta sleep," he murmured.
East sat up. Weird, sculpted land with no trees. Signs everywhere. "What is this place?"
"Rest stop, dummy." Michael Wilson breathed, his eyes shutting. "I miss my motherfucking phone."
East flexed his legs. He looked around outside. Nothing moving. Walter and Ty lay sleeping in back, and Michael was passing out against the window.
East took Michael's keys from the ignition and stepped out. The pavement seemed to clutch at him like pond mud. A whole day riding, his legs and ass had gone numb.
Half a dozen cars and trucks slumbered. A back lot over the rise showed the lights of big trucks, their box tops white under the high lights. At his feet, sprinklers fed a few planters, bright yellow flowers alert in the night. A pair of small buildings. Fiercely he looked around for whatever.
Restrooms. Cautiously he approached. THIS FACILITY BUILT AND MAINTAINED BY THE STATE OF UTAH. Utah, then. He pissed sleepily into a urinal in Utah. Green light. Moths stumbled drunkenly around the walls. The strange white soap gobbed automatically onto his hands when he reached. A man came in and looked at him with interest. He ducked his head and left.
The second building had Coke machines. East bought two cans. A dollar apiece. A clock said two-something in the morning.
Near the van, atop the weedy rise, sat a picnic table with a view of everything: front lot, back lot, all there was. East sat and drank. The van was dusty. He watched a truck, tipped with light like a fantastic ship, fly past.
There were no trees.
When he awoke with a start, the can was tipped over, dry to the touch. Grainy light. Flat clouds smudged the eastern horizon where the light was beginning. He cursed himself for falling asleep in the open.
Some of the parked cars had changed. A man combing his hair outside a Maxima was the same man who'd eyed him a few hours earlier. One of those. Dark birds hung like kites.
East descended the slope to the van, tapped Michael Wilson, and thumbed him over. Michael nodded, ashen, and squirmed across to the other seat. As soon as he'd fastened his belt, he was asleep again. East split open the warm can of Coke. He shortened the seat up and cranked the mirrors down.
Driving. He'd driven a few times for Fin, or dropped someone off, someone too gone to drive. He knew how.
On the other hand, he'd never been north of forty miles an hour.
For some miles he ran slowly, feeling the van track on the road, letting traffic funnel past. More cars now than in the evening. He could see them.
Then he picked it up to seventy-five.
The scene of the night before troubled East still: the flashing light under the canopy, the blood on the pavement, the chance that Ty was carrying a gun out here where they were supposed to run clean. And before that: the four of them bolting the van, leaving the job behind for—for what? _Just a taste._ It was all on camera—they'd made trouble. Maybe running down the road would let them leave it behind.
Only option he had.
The land changed—orange sky, light, the white of the flats giving rise to orange stone, crumpled and ridged, and dirt. The windshield filled with sunrise working its way up to blue.
He could glimpse the people in cars, the pickup-and-toolbox men, hidden behind wraparounds, heading to some job. The sleeping families, drivers with their coffee cups. The lone rangers, a man or a woman, sometimes intent on the road ahead, sometimes on the phone yammering. White people. Maybe some of them outrunning something too.
A half hour, an hour maybe, before everyone would wake up. That much alone, that much peace. The tires hummed, and he felt what Johnny had said about the van now: sorry-looking, yeah, but solid. He liked being up high, liked the firm seat. He could see the land, the flash movements in the brush, an animal, too fast to spot. Dog, maybe, or coyote. They had coyotes in LA, but they were skulking creatures, big rats. They ran down alleys and stayed in shadows, and before long somebody would shoot them dead. No law against doing it. Just another gunshot in the night.
When the other boys stirred and began cursing, East was sad. They rolled their limbs, spewed their night breath. They would be back with him soon. A chimneyed orange mountain loomed beside him, and East studied it closely as he passed it, its worn layers, saying good-bye. A secret. The last thing that was his alone.
—
Bright morning. East stopped at a shiny new gas station, TV screens humming at the pump. All the boys jumped out: East pumped the gas in the dry air. Los Angeles had dry air, but it smelled like something—always something. The air here smelled like nothing, or nothing that had a name.
Inside the store loomed hanging race cars and inflatable superheroes. A massive grill counter stretched across the back: no one there, yet people were taking food from a window. Walter studied it: you touched a screen till you had pointed out everything you wanted. Every item on every shelf had a price lit up in LEDs. Every little thing made a noise.
"This joint is fucking cool," said Ty.
Finished pumping, East headed for the bathroom and its cherry-cake smell. Some things never changed. A white boy pulled up at the next urinal. Hat on backwards.
"Sup, homes," he said.
East raised his eyebrows. This boy right up on him.
"Sup," he pronounced ironically.
The white boy finger-stabbed. "Manny Ramirez. My man."
East wasn't sure. He tightened his eyes, rushed his hands below a faucet, then rushed out. What was it with some people?
This was white boy's turf—he recognized that.
Outside the bathroom, Michael Wilson stood in line. Prepaid with twenties, so he had to wait for change every time—the price of doing business in cash. Blankly he stared, just a customer, and East watched him from behind. The cashier was an older lady with a huge amber stone caught in a fold of her throat.
"Seven dollars and thirty cents from sixty, sir," she said.
"Ma'am, you got scratch-offs?"
"There's no lottery in Utah, sir." The same note in her voice.
Michael Wilson bobbed his head agreeably, and East let him walk away. What was it? Buying lottery tickets. Stealing a little. How much?
What did Michael Wilson expect to do if he won?
Something hit him on the shoulder. It was the same bathroom white boy on his way back out. "Be cool, bro," he said, jabbing his finger. "Fly's open."
East looked down. Bro was right.
Just ten minutes here, but his morning calm was plucked. The dark string inside blurred and buzzed. He followed Michael to the van, every bit of air a puzzle, every person a future event. He climbed into the back and slid the door shut.
"Awesome station, man," said Walter, a fragrant family box of chicken biscuits steaming up his lap. "They all need to be like that."
East buckled his belt. "Who is Manny Ramirez?"
Both boys in the front let go a snort.
"Ninety-nine, Easy," said Michael Wilson.
"You ever check your shirt?"
East looked down. _Dodgers._ "What?"
"On the back. His name is on the back of your shirt, man."
"Yard boy don't get out much," Michael Wilson crowed.
After an hour, they crossed into Colorado. East felt the ground rising. He rode up front as Walter drove, Michael dozing, soft-eyed, in the middle. The hot food had upset East's stomach.
Mountains stood before them, above them, like in LA. But in The Boxes, the mountains were only a thing, like a wall or a tree: a sun-baked ridge above the valley full of everything. Here the ground was nearly empty of buildings and the mountains were like people, huddled figures, blue and gray and white, so high.
They were unmoving stone, but they tore East's eyes from the boys in the van and the unidentifiable people motoring up the same road. East gave up watching the people so much. They didn't stare back as much as they had in Utah. The people seemed younger, fitter. Some gazed at the mountains too. Some rode hollow-eyed. Families with kids drowned in their movie players; mountain boys with their racks of bikes and skis and packs; thin, straight-haired white women in their Subarus. They didn't stare back. Unsurprised by him.
Only one black man they saw, driving a moving truck.
Come noon they bought a tank of gas and two pizzas at an exit called Glenwood Springs. A bathroom stop, a round of sodas, little wooden buffalo roaming the counter near the gray cash register. Boxes so hot they singed East's fingers; they steamed on the van's floor till the windows ran wet.
They drove on until a sign announced a turnoff: SCENIC OVERLOOK. "Let's hit that," Walter said. "We can eat there."
"Oh, _we_?" laughed Michael Wilson, but Walter took it well.
The view, framed between two immense, square boulders, revealed just how far up they'd come. A gorge opened below, green, vertiginous. Two little kids from the gigantic white Navigator next to them hollered, "Wow! Wow!" East started, expecting them to be staring at him.
But they were just teetering on the edge, gaping down into the gorge below.
He slid out of the van. Again he had to find his legs, find his stance. Behind the handrail the ground was slippery pebbles. He approached the edge and looked down gingerly.
It took a moment for East's eyes to read the scene. He could see the valley's depth, feel the real wind dipping down it. But he could not convince himself that it was real. Space both vast and unattainable, opening up between the blue walls of stone. The air below was cold, he could feel it, a reservoir, and he could sense something about the chasm, all the time piled up there. Close to forever. More time than he had in a hundred lives like his.
Birds wheeled in midair, far below.
"Mommy, Daddy!" the kids cheered again. "Look! It's amazing!" East stood there too, the cold air streaming up his face, full of the smell of snow and stone.
—
For hours they worked in and out through the passes: town-size shadows sliding over the mountainsides, dark mossy valleys, clouds on the road that blinded their way. How blank it had looked on the map, this space, this state. How different to have to cross it. The road sank in and swelled out, like intestines. East asked for a turn at the wheel, but his stomach made him give it up right away. Sitting shotgun, next to the guardrail, was worse.
Their next stop was so that he could throw up. He was awake, dripping sweat; he had been dreaming of a terrible yellow goldfish. "Stop the van," he gasped.
Michael Wilson skidded off along the guardrail.
East fell out, a first taste like cement; then his backbone arched and his lunch rained over the rail.
Pizza, Coke, the rest. _Jesus_. He looked away, at the miles between him and the next solid ground. Same birds, flecks beneath him. The air smelled wet, like the rock.
He felt better—for a moment, he knew. But he breathed in wetly, the air of that moment.
"Who's next?" Michael Wilson said. Nobody in the van was even laughing.
"Never been up in the mountains before, huh, Easy?"
"I don't know," East grunted. "It's different than I thought."
"Never been nowhere, huh?"
It wasn't in him to argue.
—
Stickers covering the Jeeps and Subarus: THE EARTH DOES NOT BELONG TO US, WE BELONG TO THE EARTH. IT'S NOT A CHOICE, IT'S A LIFE. _CRISTO SALVA_. Bicycles on the back, in the bed, on the roof, wherever they could strap on. " _Crazy_ motherfuckers riding bicycles up here," said Walter. "You know there's no air? Go out and see if you can run a hundred yards."
" _You_ can't run no hundred yards," said Michael Wilson, " _any_ where."
Somewhere in the afternoon they topped out finally and started coasting downhill toward the city of Denver. East eyed the silver Colorado State Trooper cars, Chargers and Expeditions and the long, flat Fords. They scattered everywhere on the downhill, working the speeders like sharks tracking prey. Once a trooper dogged their back bumper for miles. "I'm going fifty-five, motherfucker," Michael Wilson protested. "Fifty-five minus two." The trooper hit the lights, jumped out from behind them, and bit on a Jeep. Everyone started breathing again.
_Ty's gun,_ thought East. _Ty's gun, Ty's gun, Ty's gun._
The van knifed past the city, the buildings low and shiny and suddenly too colorful below the cold blue sky long-grained with clouds. As they merged from one highway onto another, East turned back to look. The mountains stood in line behind them, still close but collapsed now, pressed together. No hint of what they were, what they held. Just another line, a little brighter and sharper than the brown line of home.
And they could see what was coming. Flatland, an endless sea of it.
"Someone else drive," said Michael Wilson. "I'm tired of seeing shit."
Walter took the wheel. Michael reclined in the shotgun seat, rubbing his face with a pair of fingers. East sat back on the middle bench and watched him fuss and prod. "You learned that where? Tokyo Spa?"
"East, baby, no," Michael said. "I learned this from your mother."
East smiled and watched the road, the eastbound trucks. After a while he shut his eyes too. Let himself fall off to sleep.
Except for the chirping. He peeked around at Ty. Ty did not look back. The muscles in his fingers twitched around the gray plastic tablet of his game. Something with aliens and bombs. Ty could lie around playing forever.
"Don't that thing run out of batteries?" East protested at last.
Ty's eyes zeroed in. "Run out all the time," he murmured. "But I don't."
"You go see your mother?"
"No," snorted Ty. "Did you?"
"Yes. I took her some money," said East. "Night before we left."
"Well," said Ty, more quietly, "ain't you nice."
East said, matching Ty's quiet, "Somebody said you might have a gun on you."
His brother's eyes ticked up and down, following something minuscule along an inch-long track. Then at last the game flashed in his face, and he relaxed.
"That's my business."
"You know you got no need to be holding," East pressed. "Fin said stay clean till we get the guns."
" _Fin_ said."
"I ain't trying to take it. But you should let me know."
Ty dialed madly with his thumbs, and his game trilled. "Shit be crazy, ain't it?" he murmured.
_Shit be crazy._ Between the two of them, it was a refrain, an old one. It meant nothing and everything at the same time, unreadable and obvious. Like a glance, like a wave in the street. It stood in lieu of ever being in their mother's house at the same time or knowing where the other slept. It stood in lieu of East having the slightest control over his little brother, or of his owning up to losing Ty. For Ty belonged to nobody now, an unknowable child, indolent as bees in autumn, until he rose up and moved in a spasm of energy and force.
Where Ty had come from, where Ty was now: these things East knew. What had made Ty what he'd become: that was the unseeable, the midair coil the whip made between handle and crack. That was anybody's guess.
Big brother taking the little, they called it babysitting. But it was not that. Nothing like it.
Deep in his game, Ty smiled. His thumbs drummed out a sprint. Then he relaxed his gaze. "You made me lose," he said.
# 7.
East liked driving here—the flat, unruffled fields with no one in sight, blind stubble mown down into splinters, maybe a tractor, maybe an irrigation rig like a long line of silver stitches across the fabric of earth. The flatness. There was more in the flatness than he'd expected. The van's shadow lay long, and the fields traded colors. The boys slept in intervals or complained. Riding in a car for more than a few hours, he thought, was like suspended animation—somewhere under the layers of frost, your heart beat. To the left, a thunderstorm hovered, prowling its own road.
They crossed under the front end of a line of storms, everything wet and alight in the slanting sun, and then they were out the other side but in the cloud's dark. The tank was low again, and East angled in for gas and stepped out. Little park of pumps under long white storm shelters and a steak-and-eggs place with a shop under a bright yellow plastic roof. Pickup trucks moved in the low, narrow roads on either side and climbed onto the highway, high and chromed or capped and rattling or stuffed with tools or crops or white bags of dirt. Men and women in their windows looked at him, eyed him with interest.
"You boys the only niggers they ever seen in real life," drawled Michael Wilson, "except Kobe."
"That was Colorado," said Walter. "We're in Nebraska now."
"Don't tell me Kobe ain't got some girls in Nebraska too."
East waited while Michael Wilson paid. Then he filled the tank and parked the van. Ty was sleeping, a reptile: East locked the doors around him and went in to sit in a bathroom stall. The farther east they got, the dirtier the toilets. Like every toilet in the country had been cleaned the moment they left LA and none of them since.
East shook his head. Sleepless. The person in the next stall wore his music through headphones and moaned along under his breath. Straining, suffering, only one word audible, at the end of the lines: _You. You._ He smelled like rotten eggs, like rot inside, and then he was gone. East grimaced and stopped breathing. Trying to press his gut out like a toothpaste tube. His thinking was frayed, sleepless: he had to think straight. They were close to getting there. He had to make sure everyone slept tonight. And walked around, cleaned out their heads.
He zipped up and left, no lighter.
Outside, the storm was about to catch them. It rose flat-faced, a gray curtain, sweeping loose trash along. Walter had taken the wheel and was idling at the curb. East swung himself up and in on the shotgun side. Then he noticed the smell. Like the mall, the kiosks where Arab girls tried to spray you: _Sample, sample? You like it._ That fruit-sweet smell.
The second thing he noticed was the shoe. A golden shoe, like a wedge of foil, with a girl's foot in it. It hovered brightly between the front seats.
The rest of the girl sat in the center of the van. Michael Wilson was beside her, all sideways and charming. In the back, Ty sat straight in his seat like an exclamation point. For once aroused but not sure what to do. Walter, steering the van away, was trying not to even look.
She was white. Sixteen, seventeen, red hair in curls and loop-the-loops. Bravely she looked at East, or curiously, as if she were nervous. But she was used to courage seeing her through.
No one else was saying anything, so East said it: "Girl, who the fuck are you?"
Michael Wilson made a crackling with his tongue. "E, this is Maggie. She just might ride for a while, over to Omaha. We can drop her off at the airport."
East said, "No. She ain't."
Michael let out a grin and a sigh.
"E," he began. "This girl needs help. She was just lost up in this rest stop." He had a hand snaked across the girl's belt, which, East saw, matched the golden shoes. "Wasn't nobody going her way. But we _are_ going her way. Right?"
The girl put her hand down on Michael Wilson's black track pants. Put it right on his dick.
A cold wave rolled up East's spine. The yellow-outlined parking space in Vegas. He made dead eyes at the girl.
"No she ain't. _Stop_ , man." He whacked Walter. "Drive back in there. Back where you were." Walter exhaled a shaky breath and swung the van back around the apron.
The girl kept her hand on Michael Wilson, and he rolled underneath it.
"E," Michael Wilson drawled. "Girl needs a ride. That's factual. May be something in it for all of us. Something in it for me, I _know_. So why don't we drive now so I don't have to fuck you up." The girl blushed uneasily and Michael laughed his little, trailing laugh. Something had happened to his face. His mouth crooked open as if dangling an invisible cigar. "Drive, Walt," Michael Wilson added. "Don't listen to this boy."
Walter rubbed his cheek, wasn't sure. "Right there," East insisted. He pointed out a space. "Right by the door."
"East, I'm gonna hurt you, man," Michael Wilson warned.
East dimmed his eyes, stared a cold hole through the girl. Her green eyes bright, but she stared back; she was used to sizing people up. For an instant he was looking at the black girl outside the house, the Jackson girl. The same: defiant. And curious.
"Get out, girl," East said. "It's nothing good for you here."
She did a little hitch with her lips, a smile. Then she leaned forward, her hair swinging like a fragrant bough. Her fingers climbed his left hip.
East uttered a strangled cry and slapped her hand, like a snake lunging. The girl drew her fingers back. He tried to go dead-faced again, but he was shaking.
"I'm sorry," said the girl, Maggie. Her voice was higher than East had imagined. "Maybe I shouldn't—"
"Aw, Maggie," Michael Wilson begged. "This boy, this little boy—Maggie, don't be listening."
He put his hands on her and she squirmed. "I better go."
East reset himself. He knew: the way she glanced back at the station. Like now it held things she'd forgotten: people, stuffed animals. She longed for it. Her nerve had fled.
"Maggie, aw," moaned Michael Wilson, sticky with desire.
Girls had sense. You could back them down. Girls saw bluster, knew its purpose. Boys, they just flew into the air over nothing, rose up with their dicks all hard, and then people got killed. Like at the house.
Michael Wilson was going to fly up now. Had to.
He pled with her first, grabbed at her. "But I _can't_ ," she said, and then the gold shoes were on the pavement. The glass door—WELCOME, THESE CARDS ACCEPTED HERE _—_ opened for her.
Michael Wilson made a little click in his mouth. "Damn," he said. "There she goes."
He clenched the hand he'd been grabbing at Maggie with and fired it at East's head.
East ducked and sheltered down. "Drive," he muttered, and Walter did. Michael rose up under the low blue ceiling, but he couldn't throw a right, not with East balled down in the shotgun seat. Between the seats, he came with two lefts, the second hard enough to light East's eyes with salt.
"Go!" East gasped. "Drive!"
The sliding door hung open, cool air rushing in. Again East ducked and the turn around the lot rocked Michael Wilson. He recovered and swung again, and East deflected it—another hard turn made Michael brace.
Walter sped the van up the ramp hard, as if he could stop this fight with gas. "Careful, Mike," he complained. "You gonna make me crash."
"Pull off, then," Michael steamed. "Because I'm gonna whup this bitch." He sat back, fists clenched. "Fuck you up," he promised East.
As soon as an EXIT 1 MILE sign came up, Walter hit the turn signal. East touched his face. His blood was in his ears and head now, the black string yawning, almost audible. He knew he had to play his cards now.
"What's up with you, Walt?" he appealed. "You just, 'Cool, we with this girl now'?"
Walter whined something indistinct. Hunkered down with his steering wheel. Behind him Michael Wilson sat and laughed, stretching his shoulders, limbering.
East turned on him. "Oh, now you're a muscleman. Just do your job."
In a high, public voice Michael Wilson declaimed: "Easy. I know where you're at. You just a little street faggot, ain't ever seen a girl. But I _have_. That girl was pussy for _everyone_."
Trying to line the boys up.
Walter braked the van down from eighty, seventy, fifty-five. The exit was a dead one, disused fields, one cracked concrete lot where someone had built once to make money. Across the highway, one lone gas station still hoisted its sign.
Nobody in sight. Here is where it was going to happen.
"Boy, you do not do to fuck with me," Michael Wilson boomed, "and now you will know."
Walter put the van in Park, and East just held on. _This is it,_ he thought. Didn't know when it would come, but he knew that it would.
If all there was was a fistfight, he was going to get beat. Maybe worse than beat. But if there was a vote, maybe he'd win it. East had Walter. Walter had wanted that girl, but he'd dropped her off too. Walter wanted what was right. Maybe he would help.
Ty? East didn't know what he had.
Michael Wilson was getting up. East hollered, "Everybody out!" and jumped out first. Sweating already.
He made two fists, weighed them. His arms had never seemed skinnier.
_This is it._
Then Michael Wilson was rushing across the cracked pavement.
They said that sometimes when you got your ass kicked, your mind sold out your body, stopped taking it personally, only crept back when the whupping was done. East's mind went nowhere. Calculating. Michael Wilson wasn't gonna kill him, not over this girl. But what did a fool do when he'd shown himself? He built up. He went pro on it. Became the hardest fool he could.
Michael Wilson stopped then, to strip off his meshy white Dodgers shirt and drape it on the van's side mirror. Gym muscles down his belly like puppies in a litter. The muscles were what broke the bottle of fear inside East.
"Listen, man," he pleaded. "I'll spell it out for you."
"Shut up," said Michael Wilson. "Should have done this a thousand miles ago." Some Chinese tattoo in the meat of his arm. He locked his fist and drove.
East ducked. But Michael was quick. He got an arm around and slugged East's kidneys: East felt that bitter spurt inside. "Come on, Easy," Michael grunted, wrapping East with his long arms. East grappled for footing. Stay up, you had to: the pavement was not your friend.
Michael Wilson tried harder. He wrenched sideways, lifted East around the ribs, trying to slam him. East spread his feet wide to catch himself. Michael sucked air, swore, and spun again. Again East got a foot down, fought to stay upright. Walter bounced by, shouting wildly. Michael's arms cinched, and East smelled his lotion. One fist peeled off and shot up, off his eye this time. At once he felt it throb and swell. With hard nails, Michael probed East's head; he took the ear and started to twist, to tear at it, until East let go a shriek. Then everything stopped.
Silent: the silence of hard, wet breathing. Something black and cold teased East's face, like a dog's nose. Ty had a small gun leveled at him, lazy and straight.
"Quit it," Ty said. Aiming the gun as if it didn't even hold his attention.
Michael Wilson cursed. He popped East free, right into Ty and the cold black barrel.
A big truck with a cartoon milkman on its side flew by.
"So, you got a gun," Michael Wilson said.
Ty didn't answer. East tried to clear his eyes, get his voice back. He'd bitten himself inside his mouth. But now was his best chance.
"Give Ty the money," he slurred, his mouth swelling around his teeth.
Ty kept the gun on _him_ , though.
"Fuck no," Michael Wilson said. He caressed one fist with the other. "Your boy gonna shoot? Don't look like he's decided who. So what you gonna do? Put me _out_?"
"Yes," said East. He'd thought it before. But now that Michael had said it, it was the only way.
Michael Wilson surged from his toes and hooked East once more, side of the gut. Sucker punch. It crumpled East, and he heaved with the pain. "See?" Michael Wilson smirked. "You ain't shit." He stepped and loaded up for another, when a hard crack like thunder hit them all, and East was untouched, backpedaling in the light.
Ty held the gun in the sky. Its hard gray pop echoed back from nothing.
Michael Wilson spat. "Oh, nigger, please."
Ty aimed the gun at Michael for the first time.
"I'll take that money now," he said.
Michael Wilson scowled a terrible scowl at Ty. From his hip pocket he threw a curve of twenties to the ground. They fluttered, and Ty put a foot on them.
"There you go."
Walter spoke up. "That ain't all the money."
Michael Wilson glanced across, measured the overpass to the station.
"Give up the rest of the money, Mike," said Walter.
"Let him keep the rest. He'll need it," East said. "Now you go."
Michael Wilson chuckled. "Just right out here on the farm?"
"That's right," East said. He grabbed the white mesh shirt off the mirror and tossed it at Michael Wilson. Michael shook it out and put it on. "Let me get my bag, then," he said. East nodded, and Michael fetched it out of the back.
"Pretty bag," East couldn't resist remarking.
"Let me tell you something," Michael Wilson announced. "I ain't sorry to leave you. I'm glad. I get home with one phone call. And you are lost. You can't get guns without me. You can't find the man without me. Don't none of you even look old enough to drive a car."
"We don't need you," East said.
"Ain't talking to you," Michael Wilson said. "I'm only talking to the youngster with the gun." He turned his back on East. "You a neighborhood boy. You ain't in no neighborhood now. There is plenty you don't know, gangster. You don't know you can't go back, because when you fail, there's no place for you. Johnny and Sidney will kill you just for knowing what you know. Or somebody will—it don't matter."
"Say what you got to say," said Ty.
"Just understand the picture." Michael Wilson chewed off the words. "You ain't even grown."
"I hear you," said Ty. "Good-bye."
With his immaculate sneakers, Michael Wilson tested the ground. Here, after the fight, in the middle of a cornfield, he looked as polished and bright as he always had: black track suit pants, glossy gym bag, white nylon shirt with his skin dark in the mesh. Stray raindrops blew at him and disappeared.
"Just remember," he said. "You will die. And fuck you."
Michael Wilson nodded—at the gun, not at East—and turned and took the first step away. Then he jogged. He ran, and Ty pocketed the gun. For a moment East couldn't believe it, that Ty had jumped in like that, and then he was letting Michael Wilson go. The Ty he'd expected, the Ty he wanted in the red, bruised part of his brain, would shoot Michael Wilson and leave him off the road for the birds. Not this. Not Michael Wilson on the country road, white shirt billowing under the roiling clouds, his necklace glinting. In a minute he had crossed the bridge and was descending. He did not look back.
# 8.
East's fingers knew more than the mirror did. His eye looked all right but felt fat and hot, liquid beneath. The napkin full of ice from Walter's drink just made him wet.
"How bad he get you?" Walter asked again. Checking the mirrors every second. As if Michael Wilson could some way be gaining on them.
Ty sat dully in the backseat, staring out.
East remembered the last time he'd been beaten up. He was eight or nine. Yes, nine—it was in third grade, a week before summer. Third graders were going up to the next school. But four who were being held back would catch a boy each day and whup him, just to say good-bye. The principal wouldn't suspend them—to be suspended was what they wanted. One of them was being held back for the third time. He was eleven already.
They'd bruised East's face and shoulders, blacked his eyes, loosened a tooth. His mother screamed how she'd go to the school and there'd be hell. But she'd never gone. That was worse. But this hurt more.
That was the year Fin started taking East under his wing. Started showing an interest, making sure East had what he needed. Ty, he didn't take much notice of. Ty was not his blood.
—
When East took the napkin off his eye, something was coming out of his skin. Walter took a look and bugged out.
"Telling you, man. Let's get to a pharmacy. Get you some ointment. You need medicine on that. And bandages."
East's voice came small and faraway. "How does it look?"
Walter stifled a giggle. "Like you got your _ass_ kicked."
East put the napkin back.
"You all right?"
East nodded. He didn't want to talk about it.
The high battling wall of cloud cut off the sun. Cars switched on headlights along the road. With stiff, trembling fingers, East opened his wallet and counted, one-eyed. He had two-sixty. He checked it again.
"How much money you got?" he asked Walter with his little hollow voice.
"Three hundred twenty-two dollars," Walter replied without looking.
"How you get three-twenty-two if we started with three hundred?"
"Man, I _had_ money. What, you don't carry any?"
"They said no wallet," East said. "What's two sixty and three-twenty?"
"Five-eighty. And whatever Ty has."
"Ty. How much money you got?"
They waited, Ty looking mutely out the window.
"Ty," East said again. "We trying to find out what we got."
Nothing.
"Here's a town," Walter announced. "Let's get off. I'll find you a store."
East surrendered. "All right. How's this gonna work, five hundred eighty dollars?"
"Minus gas," said Walter.
"Minus guns," said East.
Ty coughed. "They said you ain't have to pay for guns."
East said, "Oh? Did I hear a noise?"
"You heard me," said his brother.
Now East turned, showing his brother his swollen eye. It hurt, hot, like a wound that's poisoned, like a snakebite. "You want to tell me more?"
Ty stared mutely at the sunken median running by.
"You two, man." Walter shook his head. "I need to be getting combat pay."
—
East stayed in the van outside the drugstore. Ty didn't budge. They watched the doorway glowing blue and white, a plastic city. White people teemed in and out, carrying chips bags, cases of drinks. Everyone seemed to know each other, talking or at least waving.
"Ty," East called back. _Here goes nothing._ "What do you know about the guns?"
"This don't seem like a drugstore," Ty said.
"Says right there. _Drugs._ "
"Oh," Ty pronounced ironically. "Guess it is, then."
"You going to answer my question?"
"No," said Ty.
"Did you set it up? Do you know the people?"
Ty just snorted low, like an old man.
It was like that, talking to Ty. He'd been a willful baby, a stubborn child. Now he was a wall. Every conversation, he made East feel like the police. Sometimes he thought Ty must have been learned from being brought in once or twice, spending time in questioning, stone-facing it across a police desk. But you couldn't ask Ty. You would never find out.
Ty's inscrutability refused the mother's blood they shared. East could rally a gang of boys his age, shepherd junkies in and out safely. He could stare down a gun. But Ty had found a way to negate their childhood together, the two years of age East had on him. There was nothing East could do with him.
—
Walter brought antiseptic and a bandage as wide as a credit card. "I ain't wearing that," East said.
"You're welcome," Walter warned. "See how you feel by tonight. Might wish you had."
"We'll be at that gun house tonight?"
"You better hope Michael Wilson didn't tell that girl where we're headed."
East thought about it. "Too stupid. Even for Michael."
"Even to say Wisconsin, though. Even to say east. She _knew_ we were going east."
"He didn't say nothing." Here he was, defending Michael Wilson's good sense. "Man, everything's going to be all right."
"The reason we are out here is," Walter said, "everything is not going to be all right."
The ointment stung, but East made a slick of it, along his brow, under the eye. He rubbed with his fingertip as they regained the road. Now the van with only three in it seemed too long, voluminous, a dark burrow from front to back. East rolled the window half open, taking air and the afternoon light. The same storm was behind them now, piled up and coming. Across a fence East saw a new little neighborhood already policed by streetlights, rows of houses in a knot. Two white girls shot hoops in a lighted driveway.
His face had stopped hurting after the first hour. Now it was the drawn-out, scraped feeling inside, where Michael Wilson had slugged him in the back and then followed up, sucker-punched him, in the side. Hot mixed with cold. Like purple bruises going all the way, meeting in the middle: he pictured a rotten pear. His breath had a hitch, like a child trying not to cry. The pain angered him, and the anger made him quiet.
He took the road atlas up from between the seats, flipping through listlessly. State by state. Arizona. Arkansas. California. He stopped and studied the full-page city map of Los Angeles. He recognized names in the sprawl of towns. But could not find The Boxes on there. Nobody had ever taught him maps. It took faith in them, believing they were going the right way. Faith in the road, the book, the plan. That whatever they were following made it to somewhere.
He came to Iowa with its plastered pink flyer. Black woman with long tits like footballs. The phone number beamed out beneath her like a black seesaw.
Traced the number with his finger. "When you want to make this call?"
Walter sipped a drink. "When do you?"
"Next stop is good."
"All right. I'm a need to stop again anyway. That drugstore didn't have a bathroom."
"Piss out back."
" _You_ piss out back," said Walter. "With your ghetto ass. I was brought up with some dignity."
East let himself laugh.
"How you feeling?"
"Tell me how much money again."
"Five hundred eighty-two dollars. Minus five for your Band-Aid."
"All right," said East. "Don't ask me how I'm feeling no more."
—
"You gonna need quarters, E."
In back of a gas station, air hissing from the fill hoses, East and Walter huddled together at the phone, road atlas in East's hand.
"It ain't a cell phone. It don't dial free," Walter said.
"I know it," East said, but he gaped at the dial, the instructions. Resenting.
Walter read the number out: 213. Then 262. Then 8083. The buttons were sticky.
"Ask for Abraham Lincoln, then?"
"Abraham Lincoln." They kept straight faces.
"Shit. Get this done before it rains." East rechecked the pink flyer. A voice in his ear asked for the money then. "Give it up," he said spitefully, till Walter held out the silver coins, shining, warm.
Something in him was tired of Walter too, the chirpy voice, the can-do all the time. Something in him was not content yet.
At first came some slow jam and a woman's recorded voice: _Hi, baby. Glad you called me._ Half a minute. The live operator was quieter, just wanted payment information. East had to cough up his voice again. "Let me talk to Abraham Lincoln," he managed to say.
Walter giggled this time.
"I will connect you," agreed the woman.
Who was it? A cool voice, anonymous. The slit-mouthed woman at Fin's house? No, but he thought of her again, the net of shiny beads in her hair. Her hands, bringing the tea.
A man's voice came next.
"How you boys doing?"
Automatically East said, "All right." A new strangeness took the next moment to get over. For two days they'd been riding, mouths zipped, their mission buried deep. All the worries of the van. Now it was back on the table.
"Abraham Lincoln," he said warily.
"Right," said the man. Deep. Not Fin, but some of his gravity.
Walter hissed, "Who is it?"
"We're in Nebraska," East reported.
"Where at in Nebraska?"
"Gas station."
"Listen to me," the man said. "Where's the station at?"
"Oh. I don't know," East admitted.
"You don't know? You don't know what town you're in?"
"We been in Nebraska a while," East stammered.
"All right," said the voice, grudging. "You made good time. Good work. Call me back in an hour. Make sure you got a pen or pencil. I'll have directions for you."
"Got it," said East.
"I'm a confirm it," the man said. "Make sure you know where you're at. Exactly."
"Sorry."
"Call me."
The line went dead. East stared at the hot plastic receiver. He touched his face again where it still held the mouthpiece's warmth.
Walter was twitchy. "Who was it?"
"I don't know," East had to say.
"What's the deal?"
"Call back in an hour. Next time, know what town we're in."
Walter slapped the side of the phone box. "I know what town we're in."
"You should have said."
"You should have asked."
A little pout. _Fat boy missed his chance,_ East thought, _chance to ace the test._ He curled the atlas under his arm.
"I'm going to piss then," Walter said.
East went inside briefly, to buy a cup of lemonade, mostly ice. He took off his socks and shoes and shirt in the middle seat and bathed himself with melting ice. The cold bit clean through his tired skin, but the gray illness throughout his left side still dragged at him. He pulled fresh underwear out of his bag, and the second gray Dodgers shirt, and changed. He put his pants back on and iced his face.
It was beginning to be cold outside. Not night cold but winter cold. Big trucks pounded past both ways, their high exhaust pipes hammering.
Walter returned and they got rolling, the storm behind them again, looming high, boiling. East held the ripe, ripped Dodgers shirt outside his window and let it flap, tattered like a flag. The moment he let go his pinch, it was gone.
East dreamed interstate dreams, dreams he'd never had before, choppy, worrying. Running the wrong way, or driving against oncoming traffic, or impossible land: a highway emptying into a river, a bridge wobbling, a prairie breaking up. Or ahead of them in the east, LA, with its smog and brown mountain scrim, where it shouldn't be. All the land—people talked about America, someday you should see it, you should drive across it all. They didn't say how it got into your head.
He awoke. Walter was intent in the driver's seat, his lips mumbling something, repeating, like a song. He glanced at the rearview mirrors.
"How far we come?"
"I don't know. Fifteen hundred miles. Shh." Walter gestured back with two fingers.
East blinked, wriggled upright. "What?"
"Shh," Walter said again.
Out the back window he saw the dying light of the day below the shelf of the storm, a layer of smoky blue. One pair of headlights nosing behind them in the lane.
"That's state patrol," Walter said.
East looked again at the silhouette behind the headlights. "How you know?"
"Wasn't so dark out when he picked us up," Walter said. "He's on my ass this whole time."
"You speeding?"
"No. Right at sixty-five," Walter said. "I wish this van had cruise control."
"So maybe we're okay? If he wanted, he'd light you up."
"Maybe," Walter said. "Maybe. Maybe he's just taking his time? Maybe he's checking us out in the database."
"Database?" East yawned.
"Every plate," Walter said. "I forget what it's called now. Cops run a check. Find out are you wanted, are you missing, is the car stolen, do you owe money. Find out whatever they want."
"But Johnny said the van was straight."
"The van is registered in Johnny's family. Someone named Harris who can't be found. Insured. Licenses are straight. I know; I had them made. Everything was straight until Vegas."
"Stopping at that casino?"
"It wasn't the stopping," said Walter. "Wasn't even getting up on the tow. You can always pay the guy, give him a hundred dollars, the problem goes away. It was you two, jacking it up. Slugging the guy."
"That wasn't me."
"All right. Your brother did."
"That ain't on me," East insisted.
He glanced back. Nothing from Ty. No way to know if he was even alive.
"All right. Anyway. So what you gotta ask is: Did they call it in, those guys? Is there some bulletin out on us? Or did they keep it in house, make themselves a promise they'd kick our ass next time?" said Walter. "Wisconsin plates. So they knew there probably wasn't gonna be a next time."
East said, "How'd you learn all this?"
"That's what I do," Walter said. "Projects."
"Projects?"
"I know people," said Walter. "I got jobs. I watched a house for a bit like you. But I moved up."
"You made the licenses," said East. "What's that mean? That license ain't real, that's what it means."
"East. Fin has a whole setup. People in the DMV. People at the state. I got a part-time job there; they think I'm twenty-two. So we can float records. We can make people up. Some sit in the system for years, man, before we use them. When we need them, I take care of it. I make it happen."
"You make what happen?"
"I make a license," Walter said.
"You an artist? You print it?"
"No, state does that. I used to, but it wasn't good enough. Get you past a bouncer. But a cop would spot it. Now they're real."
"But they ain't real."
Walter said, "What is real? These got everything a California license has. There's backup in the statehouse says there is an Antoine Harris. State trooper calls it in, it checks out. That real enough for you?"
"I don't believe it," said East.
"Well," Walter sniffed. "Best hope you don't have to prove it. Antoine is clean, man. Keep him that way."
"I'm clean," said East. "I never been picked up. My name's good."
"Ain't you lucky," Walter said. "Shh, here comes the man."
The trooper had hit his light, blue and white pulsing off everything. Walter cut speed and shifted his path right.
But the cop sped on by.
"See?" East said. "Nothing to worry about."
Walter exhaled a long, tense breath. He smelled like fried food, sweat, and oil. "Oh, yeah," he said. "I should have woken you up earlier. Talking to you, it's sure to make me feel better."
—
Walter was floating before the headlights, big 4XL Dodgers jersey lit to boiling. Another dream East was having about the road—or so he thought. Then he saw the chain-link, that they were in a rest stop, not in a lane. Behind Walter's body was a pay phone.
Walter was making the second call. East bolted up in his seat. But Walter was already hanging up, the pink flyer in one hand, atlas in the other. He turned, spotted East behind the windshield, and nodded.
"Got it," he grunted, opening his door. "You want to drive, or should I?"
"You must have crept out smooth," East accused him.
"You were sleeping."
"You got no business making the call without me."
"I'll drive, then." Walter tugged the seat belt around himself uncomfortably. There was no good fit. "Yeah, I got business."
Bitterly East said, "I need to be on there when we call."
"East," Walter began, "I ain't your boy. I made the call. Like you did before. Do I feel better, since now I know? Yes. Who got directions? The one who can read did. Read a sign. Read a map."
"I can read," said East.
"Ain't you lucky," Walter said. "You ready?"
East snatched the flyer from Walter. Handwriting covered the back side now—terrible writing, scratches, loops slanting downhill.
"I can't read this," East said at last. "No one can read this."
"That's right, son," Walter said. "Nobody but me. I can read every word."
It was night now, thickening. A bread truck sat disabled two spaces down, the doors gaping. Like a foreclosed house.
"What they say about Michael Wilson?"
"We didn't talk about Michael Wilson. Why we want to talk about him?"
"Don't sneak around on me," East snapped.
"You're so angry," Walter said. "Put me out too. Go on. It was always gonna come down to you. You and your little brother. I knew it. You headstrong street Negroes. You were always gonna win. So do it now."
"Come on," East said, steaming.
"I mean it," Walter said. "Here the keys, then. Put me out."
Schoolgirl drama. But Walter had him beat.
"Come on," East groaned again.
Walter spat out the door and palmed the keys back. He started up the van and checked the mirrors. "About an hour," he said. "Then we get off the road."
Finally East relented. "What did they tell you? About the stop?"
"Easy. Grocery store about twenty miles off the road. We meet up with a truck. Follow them. Everything's paid for. They don't know us. We just get and go."
"What kind of people?"
Walter picked up speed. A strange field of lighted wire deer glowed in the distance. "They didn't tell me that."
# 9.
"What state are we in again?"
"Iowa."
Walter was still frosty at East. He'd dropped off the interstate without an announcement. When East had asked, after some driving, "You headed where the guns are?" Walter nodded once. Merely.
"This the way the directions said?"
"Uh-huh."
It was their first real stretch off the interstate. First time East had seen country like this. All East thought of _Iowa_ was a map and outlines of products: corn, a tractor, the smiling head of a dairy cow. That was Iowa to his mind.
This road had none of those. Houses thrown up like milk cartons in lonely space—dingy, flat, unpainted cinder-block foundations. Strips of siding hanging off the corners like bandages. In front of each waited a little collection of beat-up vehicles like a boy would arrange in a sandbox. Cheap LED Christmas lights glowed from behind the windows, and sometimes, behind the drainage ditches, a Baby Jesus was standing yard.
Their backyards stood empty, a darkness forever.
The boys rode quiet and tense, East feeling the shifting road through his seat. A little town arose with its dull light reflected on the bottoms of clouds. Signs, steeples, a little grocery store closed for the evening, red still glowing in the plastic letters along the roofline.
"This is it," Walter said. He coasted into the lot, splashed through the puddles. Out here it had already rained, or been hosed down, one or the other.
"This is it?"
"Ain't it enough for you?"
"Walt, man," East said. "Just, you don't run guns at the food store."
"They do." Walter pulled the van around once and then sat idling, lights on. "You feeling better?"
"Fine."
"You look a little less green than you did," said Walter. "Thought you'd be puking again."
East snapped, "It don't hurt so much if you don't talk about it."
A light drizzle speckled the windshield.
"How _you_ doing, Ty?" Walter called.
The reply came, "Fabulous."
"You psyched, man? Getting your hands on more guns?"
"Yeah. All I ever think about," said Ty drolly. "Guns. Guns. Guns."
Something, East thought. Something Walter knew about talking to his brother that he didn't. Even jiving him. As if they knew each other, when they didn't.
He and Ty, they _didn't_ know each other, when they did.
Then, from behind the grocery store, a small black pickup tiptoed out near them, not a glance from the driver, no signal, almost indifferent.
"Is that it?" said East.
Walter's eyes were buzzing. The truck's turn signal pulsed once. Toward the road.
Walter pulsed the light back, and the pickup revved and began to crawl out. "We're in business."
Walter crouched over the wheel, maintaining the distance between the van and the little black truck. Back down the highway the van unwound them, past the houses and signs and fields they'd already seen, then onto an eastbound route. Here, fewer houses and no maintenance—the pavement was lined by dropouts, potholes the size of dinner plates chipping off onto the shoulder. Both vehicles nosed along the center line.
"You set this up too, Walt?" said East.
"No. They did," said Walter. "I mean, there's a guy. A broker. Guy they call Frederick. He does it all by phone. He never handles a gun."
"Is he here? Or in LA?"
"This ain't my bailiwick."
"It ain't what?"
"I don't know where the man is. I didn't set this up," Walter said through gritted teeth. "Ask me a few more questions, why don't you?"
"All right," said East. "All right."
Another road, wider, ran straight between two fields dressed in stubble. The headlights touched the white, embarrassed carcass of a deer. Then the pickup slewed diagonally and stopped astride the center line. It surprised Walter: he chirped the tires stopping.
A passenger leapt out. Blue sweatshirt, hood knotted tight around his face. Just a nose, a white nose, a pair of eyes like coal-black holes. Walter grabbed the shift lever. But there was no time to do anything, nowhere to go.
"Be cool," East cautioned.
The passenger strode past them into the field, up the beginning of a beaten two-track. He popped a bolt on a metal gate. Then, turning, he beckoned.
"This freaks me out," said Walter. "They could lock us in."
"Well, that fence ain't much," East murmured.
Walter swung the van toward the track. The hooded passenger motioned to roll down the window.
"Grim reaper–looking motherfucker," Walter said under his breath, and cranked the window down.
The air pushed in, starry-cold. They saw the ball of the boy's head turning but not the face, heard his words but nothing in his voice. He _could_ have been the grim reaper. He could have been anyone. "You're going to go till you hit a barn that's got two Harvestores. Tall blue silos," the voice came. "It's about a mile up over the hill."
Courteously Walter said, "What hill?"
The passenger gave no sign Walter had spoken. "Follow that trail. On the other side, you will find it, down the hollow. You can't miss. Understand?"
Walter and East both nodded in a daze.
The boy's nod back was a single chop of his nose.
"You can get back out this way," the nose said. "Or there's a drive out the other side if you can find it." He pushed the long gate and it creaked open before them.
The boys sat stricken. Not sure of the etiquette. Like being little at Halloween, at the weird house, when somebody's dad answers the door in a costume and offers you a pull off his whiskey—what you do then.
"Go," the nose said. "It ain't no good out here waiting." He scented something up the road and tossed his head: _In._
Walter touched the accelerator and the van lurched through. In the taillights the boy swung the gate shut and departed. His small taillights moved away like tiny stamps.
Walter stopped, distracted, the van idling, working his chin with his fingertips.
"I don't know, man."
"What?"
"Do you like the feel of it?"
"The feel of it?" East sized up Walter. Got this far before he decided it was scary? "Like you said, it's set. It's all ready. No time to change our mind." More gently he said, "Go on, man."
Walter strapped his fingers around the wheel again.
But the van pitched this way and that, as if carried by hand. The track was a rough bargain between tires and ground—polished in places but muscled and bushy with weeds in others. The headlights danced ahead in the mangled fields. After a short time they made a shallow climb on a long, triangular bulge in the earth.
"This must be the hill," Walter said.
"Was that a mile?"
"I got no idea. I could use some coffee," he admitted.
On the other side, only dark fields.
Two more such ridges, and then they saw it: twin silos, strange and quiet, nearly invisible below their galvanized caps. A farmhouse two stories high, unlit, paintless, or left unpainted so long that its paint had darkened to match the wood. On the far side, a barn. They rattled their way past the farmhouse and descended to a large bald spot beaten flat by tires, not by treads or hooves. The barn was large, corrugated aluminum. One large low window suggested a light somewhere behind it. A little grass fared poorly.
Walter eased the van in, peering up at the alien silos.
Ty drew his breath loudly. "And here come the wolves," he announced.
East caught his door again. Two dogs came galloping—the footsteps sounded through Walter's window, and he rolled it up at once—loosed from somewhere, dull teeth flashing as they rounded and reared in the headlights, snarling. They knifed in and spurred back. Vicious and giddy, gang animals doing simple math, two of them, one truck, therefore with an edge. Was something wrong with them? East wondered. They reared and howled at the van, throats open, but made no sound—nothing but the scuffling of their paws on the beaten ground. East cracked his window: not a thing. Dogs without voices. Like in a movie.
"Bet y'all don't need no coffee now," cawed Ty from the back.
Then a whistle came from somewhere back in the shadow of the barn, and the dogs pointed and stopped. Their noses came down, ears spread out. Automatically. Someone hailed them again, and away they went.
"Jesus," Walter said. "That wigged me out."
"I hate dogs," Ty said.
"You do?" said East.
"Yes," Ty said. "Always making noise, drooling. Trying to be your friend."
East sat stunned and leery. He did not like dogs either. A dog changed the situation, always.
The bit of light their van threw had not given him any sense of the space around them. And nobody was coming out.
"Everything was right," Walter said. "Black truck. Short drive. Pickup at a farm. All this I got told by ol' Abe. All checks out."
"Let's wait then."
"You supposed to get out and go," Ty said.
For a flash of a moment, East felt everything he'd ever felt for his brother: righteousness and rage, exhaustion at the impudence.
"Take a look," Ty said. "That window. It's like a drive-through window off a bank."
East turned. The window was indeed the right shape, low and wide. A single metal drawer perched along the metal sill, a loudspeaker mounted there.
"I'll be damned," said Walter. "I never saw anything like that."
"They stole a drive-through window?"
"Likely bought it. At some auction for five dollars," Walter said. "Business ain't so good out here, if you hadn't noticed."
East said, "Maybe you can drive up?"
"They don't want you to," said Ty. "Ground isn't flat. Tip the van over."
"Shit." Walter clicked his belt open and unwrapped it.
East looked around again. "Ty? What you think?"
"Here's what we're gonna do," Ty said. "You two go out the front. Unlock the back gate for me, but don't open it. I'm a stay in here and watch."
"Oh?" said East. "You ain't gonna come?"
"I think that makes sense," Walter said. "Come out hard if we need you?"
"Right. Keep me a surprise."
East sighed. "All right," he said. "Stay in the van if you like."
"I like," Ty said.
East opened his door. The cold startled him—his breath became visible and lit in the stray light from the van. East headed around the back and unlatched the gate without opening it. Walter joined him in the exhaust and frosty red light.
"How cold is it, you think?" East asked.
"Not so bad," Walter said. "It's the wind that makes it feel cold."
"Not so bad?" said East. "It's cold as hell, son."
"You skinny boys," said Walter ruefully. "Well. Here goes nothin'."
They approached the window—loose scraps of concrete and chunks of sod made a pile at its foot, as if they did indeed mean to keep cars away. East looked for a way to step up to the little call button colored an unlikely red. A two-way speaker. It crackled now.
"You the boys?" said a voice from inside. "From out west?"
"That's us." East glanced at Walter.
The voice came high but quaky, an old man's voice. "First off, you boys is covered," it said. "So let's do like we said."
East wondered whether to believe this. Another gun sighted on him—how many was that? He kept himself from looking around.
"Where's the other two?"
"Other two what?"
The voice said, "The other two boys you got?"
"Oh," said Walter, taking over. "In the van."
The speaker crackled. "We got to see them. For safety's sake. Nothing funny. That's the deal we made."
"That's a problem," Walter said. "They're asleep."
"No matter about that," the voice said. "You come back when they wake up. Or you can even wake them up, can't you?"
"We don't want to wake them up."
"Seems strange, you'd drive out here," the voice said. "And then you aren't willing to wake them up."
"Well," Walter said. He bugged his eyes at East.
East had nothing.
"The deal is one, two, three, four. We done our part of the deal. I got a package here for you, exactly what you asked. And I'll be here when your boys wake up," the voice said.
"Hold on." East stepped back from the dark-tinted window and studied the barn. The wind-blasted house, the two anonymous silos. Every window and shadow too dark to read. There may have been half a dozen gunners covering them. Or no one at all.
The cold prickled on his bare arms.
Walter retreated with him and leaned in close.
"Like he's trying to trap us," said Walter. "Bank robbery: you put all the people together so you can cover them."
"Why'd you make it so we have to have four?" East hissed.
"I didn't make it, I told you," said Walter. "Be different if I did. I would have been happy doing the deal back there at the grocery store."
East rubbed his cold palms together and cursed.
"Old people," Walter said. "Country-ass religious people. Somebody told them four, now that's the scripture. I dealt with people like this before."
"You dealt with everyone."
"Tell you a story sometime. You want to fetch your brother out?"
East said, "He ain't gonna make four."
Shivering now, again they approached the window. The speaker shot a burst of static as they neared it, then cleared.
"Hello again."
Walter cleared his throat. A burst of mist rolling out. "We put one out on the road," he said. "So we're down to three. That's all we have."
"I seen you talking it over," came the voice. "I just do what I'm told."
"Come out and look. Ain't no fourth to see."
"I ain't stupid," said the voice. "And I cain't change on the plan. We agreed it was four. You show me four and I place your order in the drawer."
"The plan changed," Walter said. "Let me make a phone call."
The static came thick, like fry grease. "Go ahead."
Crestfallen, Walter said: "I mean, if we can use your phone."
"No phone," said the voice of the man.
East eyed the hard glass, the reflected blur of the van, the frost-lit world. His lips and skin were shrinking, emptying of blood. Black sky, taunting stars.
"Search us," Walter was saying. "We got three. That's all. Tell me what you need. But don't waste the whole day."
"I do what I'm told," the voice came back, unruffled.
"This is a whole organization we're here for," Walter tried. "You are stopping it up. I don't know how or why you got put in the way. But you got to understand that you are now a problem."
The old man coughed into the microphone. "That's why I sit in here, where it's safe."
"Can I make a new arrangement somehow?"
"Yes," said the voice. "You made an arrangement before. Follow it again."
"Can't you call your boss, whoever that is?"
The crackle insisted, without annoyance: "No phone."
In his shirt in the cold air, East had detached from the moment, detached from responsibility for it. Their strangeness in this wind-whipped Iowa farm field was plain—three black boys, deal gone bad, needing guns and knowing nobody. He watched Walter, watched the fat boy solving problems, inventing. Walter was an option man. He played around a thought, discarded it, played around another. That was what a furnace-size body afforded him, the time to try every key.
You could see what people liked about him.
East himself, he had been cold before. Never this cold. Back in the van there were sweaters. But he didn't want to break off from this old man. To credit the cold that the old man seemed the soldier of.
He wasn't sure he cared to stay. He wasn't sure he cared to win this. Guns, after all. Never had he cared for them. The noise, the mess. He'd held a gun before but never felt safer for it.
All the same, he was no fool. He knew guns made his world go round.
He shivered, and the air inside his mouth was no warmer than the outside. The bite inside his cheek stood out like a wound.
This black man, this judge they had come all these roads to shoot, the mission he had defended: he couldn't see it. Couldn't imagine. The bullets, the body. _Not_ shooting him, he saw that now. Not going on, not succeeding: that was real. _That_ was a bone in his freezing body now. There was no getting it out.
This old man saying no made that bone hurt. Made it harden.
But there was no sharing this with his brother, ever. No having that discussion. His brother, his blood, had different bones.
East breathed in the icy air. His eyes were caked with something, starting to freeze. Walter stood lit pink, making his arguments to the tinted window. East couldn't even hear beyond the blood churning in his ears. The problem was beyond discussing.
"Walter," East said. "Fuck it. I'm too cold."
Walter broke off and looked sideways. Surprised. Sometimes you could read him like a book. "All right," he said.
But as they turned away, a scraping sound at the window made them jump. An old, rusty machine sound. The drawer waggled open, like a silver tongue.
"Go to them boys," the amplified voice said.
Walter stepped back up the mound of rubble to the metal tray and pulled something out.
"Them boys will sell you what you need."
"Wait a minute," said Walter. "They'll set us up? How do you know that if you don't have a phone? If you didn't _call_ them?"
"I know. They'll sell to you," the old man said. "They'll sell to anyone."
East opened the back of the van, behind Ty's bench, and found the sweaters. There were four—woolen, all dark, the kinky weave, cold already, prickly on his skin. Two were small—one he left for Ty. One was a large—Michael Wilson's. He put it on over his own. The 4XL he handed across to Walter.
"What's this?"
His face was so cold, he couldn't make a word.
Walter looked amused, then sympathetic. "Easy," he said, "I ain't even cold yet."
East was half deranged with cold and lack of sleep. The dark of the night started flaking away. _Bugs,_ East's mind said, and then: _Something is wrong with me. Something wrong with my mind._ Then he saw it: the lightest snow. The most helpless bits, riding instead of falling on an imperceptible wind. Unseen, unstoppable, brushing past them like strangers.
By the time he had put words to it, it was gone.
Walter made the heater blow its hottest. East bent to it, but it did not warm him. He shook like a machine spinning off center, like a clothes dryer walking and breaking apart. He quaked. Slapped his arms, his palms, his sides, his thighs. "Ty, man," he began, and he lost it. Walter touched him: "East, man? East?"
Could not hold his jaw still on his face. Cold drool dripped.
"East? East?"
—
At last it lifted, the palsy, the shivering, and East's mind came back into his body, the touch came back to his fingers, he could hold his mouth closed. Embarrassed but surprised too, to feel himself together again.
With some effort he talked. "Ty, man. The gun you have. Is that enough? Will it do?"
Ty drew out the silence. As if, even after his brother's suffering, it cost him dignity to make an answer. At last he conceded: "Not this little gun. We gonna need more guns."
East nodded. At least he'd gotten an answer.
The soundless dogs poured forth to chase them out.
"Lord. Get us the fuck out of here," Walter said, though he was the one driving.
—
They regained the pavement, leaving the way they had come. East deciphered the old man's note. They made away north, to the same town with the glowing grocery sign: HY-VEE FOOD STORE _._ Three skateboarders in parkas traced the lot. One cop watched them from his Impala.
East was quiet. The cold had mortified him.
"I see three options," Walter said. "We can go on to this other house. I don't know. We could call ol' Abe and ask what he can do. Or we can drive around until we spot some black dude and ask if we can borrow him."
"You ain't finding nobody black out here," East said. "So let's call Abe."
A pair of phones waited in front of the Hy-Vee. But Walter didn't like the cop being there. So they searched until they hit a gas station: brushed steel box, quiet radius. Walter pulled the van up close.
"You want to do it this time?"
"You can," East said. But he got out and stood with Walter at the receiver—somehow the predawn neon buzz of the gas station made it seem less cold—and he made the call. Cold buttons, still sticky. The same quiet operator: "I will connect you." But then the phone went quiet.
Walter fished out more change, and East dialed again. The sexy girl blared, welcoming—East dreaded her now. She went on forever before the operator picked up.
"Abraham Lincoln, God damn it," East said. "It cut us off."
"No, sir. I tried to connect you, sir," protested the operator. "Please, sir. He isn't answering."
"All right," fumed East.
"Please, sir. I'll try again. It's a relay line: someone will always pick up."
East grunted. The operator was afraid of him. He put his elbow up to lean, but the steel box bit through the sweater, too cold. A truck splashed into the lot, and a woman with wet hair and a bright, glowing cigarette hurried in under the lights. One bicyclist wobbled up the road in the dark, dun jacket, gray hat, the reflectors on the bike the only concession to visibility.
"Look!" hollered Walter.
"Yeah."
"He's black."
"No he ain't," East answered automatically, and then the operator was in his ear.
"Sir? Sir? There is no one answering."
A dark buzz of alarm spread down East's back. "Did you try everybody? The relay, like you said?"
"I relayed to three numbers. Each number rang and rang. I tried each twice," the operator explained. "I'm sorry. It's late here. It's three in the morning. They're supposed to pick up, sir, but I can try again."
Her politeness infuriated East, moved him to fury. "Yes! Ring them again." He faced Walter, and Walter clutched at him.
"Let me go get that dude. He could be number four!"
"Let me finish."
"He's gonna get away!"
"On a kid's bike. On a highway," East said dubiously.
Walter said, "I'll circle back. Pick you up."
Like a ram East lowered his brow. "Not leaving me in the cold, man. Not for ten seconds."
The biker receded into the gloom.
The operator: "Sir. I rang them again. All three. No one is answering, sir...Sir?"
East thanked the operator and hung up the phone. "What do you think that means? Nobody answering the line?"
"No idea," Walter said. "Let's chase that damn bike while we can."
"Okay," said East. "We'll chase your damn bike."
—
The bicyclist was still weaving along the north-south road into town, advancing crazily. His knees chopped sideways like wings. Twice as large as his tiny bike. Walter backed into a driveway fifty yards ahead of the wobbling bike and rolled down his window.
"Hey, my man," he called. "Hey."
The black bicyclist stopped and stood astride his bike like a gray scarecrow. His gray hat was tied down over his cheeks with flaps. His coat was grime-streaked—this wasn't the first ride he'd had on the highway. In Los Angeles, East thought, this was a crazy man. Here and now, he envied the man's outerwear.
"Where you headed?" said Walter, friendly.
The man gave a minimal shrug, more a pinch, and pointed ahead. "Going down here, boy."
"Listen, man, we need somebody," said Walter. "We need somebody black. We got to pick something up, man, and we need another man for it."
"Somebody black?" the man said. "What you picking up? Like a sofa? At five in the morning?"
"We can pay you for your time," Walter said. He showed a split of twenties out the window.
The man's eyes dropped to the money, then came back up. "Good luck." Flat.
"It's nothing heavy. Just fifteen minutes. Take a ride with us."
"Oh, no," said the man on the bike. "No, no." He dropped his weight back onto the seat.
East leaned over and showed his face out Walter's window. "Hey. Check us out. We ain't bad boys. We'll give you a lift where you going."
"I'm just going down _here_ , son," the man said, and he put feet to the pedals. As he passed the van, he spat.
Ty laughed. "He's sure he's gonna die in this van."
Walter fished out the crumpled paper with the address. Now it was all they had.
—
Plain little bungalow the color of butter. East wanted no part of it. A street they'd never seen, a town they knew nothing about, a deal they didn't even know if they could make. But now this house seemed locked around their necks. They sidled the van down the street, squared around a few blocks, tried to feel things out. Regular. Chain-link sectioning off almost every yard. Small boats rusting on trailers, a few lawns with newspapers waiting in blue plastic bags. A few dogs out early, testing the air. Trees unlike the trees in LA: these rooted hard, grew up tall, muscular, their bare limbs grabbing all the air in the world.
Nothing moving. For East it was strange, this looking, this studying a neighborhood again. The way he had at the old house. The dogs, the doors, the windows. Scanning the surroundings for eyes.
Walter stopped down the block, and they watched the yellow house. The whitish farmhouse next door to it had security bars on every window.
_They'll sell to anyone._
Improvising now. What choice did they have?
He turned around. "Ty. You see what we're doing?"
"Do I see?" Ty said. "Am I stupid?"
Ever did anything like this again, he was getting his brother a secretary.
"Seems pretty straight," Ty said. Surprising East.
"Two go in, one stays out?"
"Right."
"So you got one gun," Walter said. "What do we need?"
"Walt. You know anything about guns?"
"A little."
"A better gun, then," said Ty. "I got this little popgun I can hide behind my dick. We get something real. Two guns. One of you can hold this. East can. Plus points."
Walter asked, "How much is that gonna cost?"
"Depends what the man charge," said Ty. "Begging your pardon, but we got a seller's market out here. Take all the money you got, try to bring some back."
"So who's going in?" Walter said.
East rubbed his eyes. Exhausted, he wanted no part of it. Fin had people who would work this, go in cool, shake hands like businessmen. Or there was Circo, who would go in with a gun in each hand and all the burners hot. He himself was a watchman. He could run a crew, keep them working all night. But walking in where people were ready to kill you was not his thing.
The feeling in his fingers had come back, and he rubbed them together, letting the skin heat.
"Well, Walt, my man," said Ty drolly. "Who you want on the outside, if you have a problem, coming to save your ass? Me or him?"
"It's you and me, man," Walter said across the front seats.
East nodded and closed his eyes. Plan was broke, gang was broke, Ty trying him at every chance. Might as well go.
Ty kept on. "Glock or a Tec. Glock or a Tec be nice. If they got real guns. If it ain't all duck-hunting shit. You got five hundred and some dollars. If you can't get two guns that work, fuck it, we driving back to LA." He laughed. "Now pull this up and park close. Right across the street. We doing business."
—
The cold air braced East. He decided not to waste words. The yellow house's door opened on a thick chain.
"We're here to buy some guns."
In the crack was a white face, beard, wire-rimmed glasses. "Show me money," it said.
Walter made a motion at his hip. Same handful of twenties he'd offered the black man on the bike.
"You packing?" said the bearded man. "Because if you are, hand me what you have now. Change and keys too. And Phillip will come around and wand you."
"We're clean," said East. "All right. But it's cold out here."
Around the corner of the house appeared a shamefaced man as skinny as a dog. He had a metal-detector paddle semiconcealed beneath his arm. It squelched as he jostled it. The man nodded uncomfortably before he climbed the stairs.
"You ready?" he said.
"That a metal detector?" said Walter.
"Yep."
"I know you ain't gonna scan us standing out here on your porch," said Walter.
"Yep."
They submitted to the nosing and grazing of the paddle. East wasn't sure the skinny man was using it right. Skinny enough that his shoulder joints bulged at the seams of his shirt; his knuckles stood out like knots. Hard to say how old he was.
"All right," the man Phillip said. "Now you may go in. But let me give you some advice. Okay? Don't be arguing. The nice man wants to sell you what you need. But everyone else in the house wants you dead."
Again the country manners, East thought. As if they'd come from a planet a million miles away.
"We just doing business," Walter soothed him.
Phillip stared with the same red-bitten face. "Remember what I told you." He nodded at the door, and they heard the chain come off. It opened, and Phillip led them in. Out of his collar curled the beginning of a tattoo, some ancient declaration.
They entered a parlor set with antique, dark furniture that was upholstered in pale peach. "Sit together there," Phillip said, indicating a sofa, and he slipped through a doorway at the back of the room. The bearded man with glasses split off and crept up the stairway running up behind the front door. To its railing was lashed a thick, transparent slab two inches thick—just roped on haphazardly with yellow nylon cord.
"Bulletproof glass, that is," Walter murmured.
East nodded.
Walter sat obediently, and East moved that way, but something over the sofa caught his eye. On the wall hung portraits, rectangular and oval, of tall, gaunt men with beards and white women with their hair in curls, or grandmothers with their last wisps. White faces, stolid expressions, their postures rigid in these antique frames. To East they were mesmerizing—the oldest pictures he'd ever seen. These strangers stood in poses that didn't fit them, that family lined up unhappily in front of a house. Pioneer faces, dead now, but their eyes still blazing, vigilant, even in sepia. He felt drawn, felt them watching.
Reluctantly he sat when Walter tugged at his arm.
Across a low, formal table sat the sofa's love-seat cousin, also peach. Not far away on the floorboards sat a bassinet in dull gray plastic. A large orange stuffed crab waited there—clutched in the hand, East saw now, of a damp, sleeping baby.
"Make yourself comfortable." Phillip's voice floated from the back.
Through the door frame East saw the antique dining room—long table and chairs, a tablecloth of lace, littered with plastic plates and mail. Cheerios had spilled across the floor.
Another creaking, and presently the doorway filled with a man in gray sweats. East almost whistled. The man was gigantic: Walter was 4XL fat, but this man made him seem a youngster. He was bald and he moved tenderly, shifting from foot to foot, sizing them up. His blond lashes made his blue eyes seem peculiar and dark.
"Hi," he said softly. "I'm Matt." He perched on the second sofa. "Pleased to meet you two. Maybe today is the day that the floor caves in."
Walter said, "You okay if I get right to it?"
"Be my guest," said Matt.
"We came to buy two pistols. One should be semiauto. Second one, anything that's straight. And bullets for both."
"Bullets for both," the man repeated dreamily. Almost a sigh. "How you boys this early morning? Come far?"
"A long way," reported Walter.
"You staying here? Or just passing through?"
"Passing through," Walter said concisely.
Walter was straight. It pleased East, reassured him—the straightest line between now and getting out. Made sense. He looked up at the other man, Matt, whose politeness felt effortlessly derogatory. Eyeing the two boys, Matt worked his mouth on something.
"Passing through. You sure? Ain't much here to steal."
"Leaving as soon," Walter said, "as we get what we need."
"Welp." Matt leaned slightly forward, a pivot somewhere in the base of his neck. "You seem like you're serious business. Let's see what I can show you. Phillip. Phillip. Bring out that bunch you'll find above the refrigerator."
The skinny man's footsteps scraped onto linoleum. East eyed the third man, the bearded one with wire glasses, lying in wait behind his barrier on the stairs. Was he armed? Figure he was. Figure Phillip too was just listening with the safety off. Small-town manners.
Phillip returned with a metal tray, tarnished, like it once was precious. The man named Matt accepted the tray with soft fingers. His blue eyes grew round, and he peered at the guns like a pawnbroker doubting jewelry. Then he placed the tray down before the boys.
Three guns. Two magazines.
"These are nice guns," sighed Matt as if the boys had given him heartburn.
East stared. Guns were not his thing. He'd carried a few, even fired a few to learn. But these guns were not for boasting or learning.
"This is yours to choose, man," he murmured to Walter.
Walter wiped his hands together and picked up the first gun. Checked the chamber first, then the action. Sighted it against a ceiling corner. _Click._ Quietly replaced it on the tray and tried the next.
"We good?" Matt said agreeably.
Walter said, "Not these."
"Oh?" Comically the fat man cleared his throat. His eyes went round again. "Not these? You want what, exactly?" Over his shoulder he said, "Phillip. He says not these."
"These guns are fine," said Walter, "but not these."
"These ain't the low-end Saturday-night specials city niggers use," said the man, licking his lips. "No offense."
There it was. East saw it fly out and watched it sink. Just a stone in the water.
Walter made a plain, thin line with his lips and let it go. "Why don't you show us what else you got."
"Tell me what road you came in on," said the fat man Matt.
"From the south," said Walter. "From the interstate."
"Didn't you go visit somewhere else first? Before you came here?"
Now Walter elbowed East. "This motherfucker," he scoffed. But East saw the cubes of windowpane light curving in Walter's eyes, the cock of his eyebrow: _Where_ _we going with this?_ Improvising was wearing thin.
They'd done some things right. But nobody would tell you how many things were left.
East intervened. "We went somewhere else. Come on. Ask us what you got to ask. Then let's see some more guns." His voice came harder than he meant it to. Maybe that was all right.
Matt chewed on something, encouraged. "Did you go to a barn? A big-ass barn in a field?"
"Maybe."
"Why didn't they sell you?"
"Who knows?" said East. "I guess they didn't like us."
"Did they not like you because a lot of little African Americans steal? Or because a lot of little African Americans actually turn out to be cops?"
"A lot of little African Americans trying to be polite here," East said. "Whyn't you bring another bunch of guns if you got any? Or we can just go buy them in the next town."
"Oh?" said Matt. "What town?"
Walter cut in: "Dubuque."
"Well, I don't know nobody who sells guns in Dubuque."
"We'll find someone," said Walter. "But we here doing business with you, or trying."
At this statement the baby stirred. It stretched one hand out, then emitted a loud complaint. Inside East, the black string sang murder. He hated the baby; he hated the men for using the baby, leaving the baby in the main room for show. Like the antique photographs, the upholstered furniture. As if it all meant respectability, as if you couldn't be touched.
Somewhere its mother was probably still asleep.
Matt moaned and shifted himself on the love seat. "Phillip," he crooned, "why don't you see what you can find in that drawer?"
"Which drawer, Matt?"
"Second one," said Matt. "Below the toaster."
"Second one below the toaster," Phillip repeated, retreating to the kitchen once again.
The big man Matt smiled, and in his sickly whine he said, "How about you, string bean? You know where you're headed next?"
East squinted. He'd been called String Bean sometimes as a kid. That Matt had put his finger right on it annoyed him, like someone had screamed his name inside a house. But he couldn't mean anything by it. He meant that East was long and skinny. That was all.
As much as East hated these men, he wanted to make the deal. He wanted it to be over. And he wanted to have done it.
"I'm going with him," he affirmed.
"Mmm," mused Matt. "That's good. Okay, here he comes."
The second tray: two guns, one extra clip. "Be my guest," Matt said again.
Walter picked up the first, a gray semi, jimmied the magazine out, checked it over. "Seventeen," he murmured tunefully.
"Good gun," said Matt. "Not Glock's best."
"Why ain't you bring this out first time?"
Matt smiled and said nothing.
"Do I get to fire it? You got room in the basement?"
"In the basement is my wife," Matt said. "Asleep, we both should hope. No, you don't get to fire it. If we go out in the fields, you can shoot it. But you're in my house, first thing in the morning. You're lucky we're even awake."
"You have a wife?" said Walter.
Again Matt smiled. "Big boys get it done, junior. You're on your way yourself."
"Not that big," Walter said. He pointed out the second gun. "Not much of a toss-in. Can't you sell me a better one?"
"That little Ruger in the last round," said Matt. "But it costs. Or you can have that Taurus."
"How much, these two?"
"Five-twenty-five."
"Four hundred."
"Oh, I'll say four-fifty," Matt said. "But I will also say: I came down. I come down one time only. Take it or leave it."
"Four-eighty," said Walter, "and you take back this cracker box and give me that Taurus."
"Five hundred and you can have all three."
"I don't want three," said Walter. "I don't want this leaky thing anywhere near me."
"A man who thinks he can spot shit," said Matt, "will still end up wondering why his shoes stink."
Walter said straight, "Four-fifty for the Glock and Taurus."
"Mostly now what I want, actually, is you to get out, actually," said Matt. "I care less and less if I get your money or not."
"Well," Walter said, "right now you get to decide."
East's stomach rolled. He watched Walter with a low, grudging admiration. Trading was all it was, maybe. But not everybody could trade.
"All right," said Matt with resignation. "Four-fifty for the Glock and Taurus."
East could not stop the little leap his hands made in his lap.
"Deal," said Walter. "And we stop outside town and see do they shoot. If they don't, we come back."
"You can _look_ at them and see they _shoot_. A child can see they _shoot_. The question is, can you aim?" Matt made to stand up but winced instead. "Phillip, get that little Taurus gun off the top of the fridge."
Walter counted out twenty-three twenties. "Got change?"
"Not if you want bullets. I got about a box and a half fits those both."
"Oh, shit," said Walter. "Yeah. Here's another twenty. Gimme it all."
Phillip opened a door in the dining room sideboard, near enough that East could watch. Red and black boxes were stacked beside vases. Phillip placed two in a bag that said Dollar General and walked them to the front door.
Walter said, "Where's he going?"
"Putting it outside for you," said Matt. "You think we invite boys into my house and hand them loaded guns?" He counted the twenties. "Four hundred eighty dollars. My handshake is my receipt."
They stood, but none of them shook hands.
—
Outside, behind the steering wheel, Ty waited, quiet-eyed. He slid back in the van as they approached. East peered down into the plastic bag. New bullets, a sealed box and a half one—nice.
"Those guys had guns in every drawer in the house," East said.
Walter snorted. "I know. A thousand guns. We could have been there all day."
"What you get?" said Ty the moment they opened the doors.
"Now we'll get schooled," said Walter. He unpocketed his gun and passed it back. East fished the Taurus out too. Walter set the van moving as Ty examined them.
"This Glock, nice," said Ty. "Other one, a piece of shit. You could smack somebody with it, I guess."
Walter smiled. "See?"
"What you pay?"
"Four-eighty for all of it and bullets."
Ty gaped. "Four-eighty? These guns? Four _hundred_ eighty?"
Walter turned a corner. "That a good price?"
"I get Glocks like this in The Boxes, two hundred," Ty said. "How many dudes were there?"
"Three," said Walter. "And a little baby."
Ty said, "Stop the truck."
"No!" said East. But Ty didn't wait. He threw back the side door and took the street at a leap. East popped his door too, but the seat belt caught, and then the van bounced as Walter pulled it over, and he cursed and fumbled with stinging fingers. Ty darted between houses and was gone.
Back on the seat he'd left two guns. He had the Glock. There was no chasing him.
East slammed his door. "Are you stupid? Never do what Ty says."
"What's he doing?"
"We'll find out," East said grimly. "Take us back. _Go_."
Lights were waking now in the kitchens, behind the porches with their hollow Christmas lights. Walter downed the windows: no dogs barking. Nothing. No sign of Ty. Silently they rolled toward the gun house.
"You want me to stop here?"
"Not right in front," East said. "Don't want them noticing us and wondering." He scanned the block, eyes burning.
"Do we go look for him?"
"No. Not yet."
Walter said, "What's he gonna do? Walk in and ask for a refund?"
"Walt," East steamed. "Nobody knows what Ty's gonna do. He don't plan things. It's always _pow_ with him."
"Ty saved our ass," Walter reasoned. "Got us out of Vegas all right—and when Michael was latched on to you? Ty improvises."
"Listen to you, man. _Last night_ you were saying he was trouble. He's an animal."
"Maybe he's lucky," said Walter. He pulled a three-point turn at a cross street. Jubilant accordion music spilled from a parked car there, all four doors open. A short Latin man was shining his dashboard in the cold.
They made one more flyby. East's stomach burned cold. He wished he'd marked the time. Walter pulled the truck up fifty yards from the house, and they quieted and watched.
Six-eleven on the clock. Six-thirteen. How long did they give? The pink of the morning clambered across the ceiling of the sky.
East knew this hour from years standing yard—in minutes, light would ooze down the treetops, color the chimneys, charge through the yards. The shabby street dawn tightened East in the way he had tightened for years—the standing there, regardless; the watching everything that moved. The not blinking.
The molding a group of boys you'd maybe met yesterday into the people your life depended on. And never to know whether you'd succeeded. Only to await the moments of test. Like this one.
Six-sixteen. A work truck with a big white Reading box mounted in the bed rumbled by slowly like a cart drawn by invisible horses.
Walter said, "How long you think we can wait?"
"I know," said East. "I _know_."
"Time is gonna come we have to decide."
East kept his eyes on the gun house.
"He's your brother."
"Not that I'd die for," said East curtly.
Six-eighteen.
He'd always been bigger than Ty, stronger. Always been older, and also the good son, such as it was. Always the one who tried to put a good face on it for their mother. Always the one she could count on, even if Ty was her baby. He remembered the first time he'd tried to celebrate his mother's birthday with a cake he'd bought with his own money. Brought it home and hid it, but Ty found out. He slipped out before dinner and stayed out, unaccountable, until three o'clock in the morning and her birthday was through. Just daring East to go ahead and serve it. Without him. And he didn't. That night he cried bitterly at his defeat. Nine years old and just trying to be the man of the family.
That unaccountability was the trick Ty had. The way he'd found of taking those two years East had on him and shattering them.
At nine he'd begun tomcatting out in the street for nights at a time. Had left the house entirely at eleven. His mother's baby.
"You decide," said Walter. "He's your brother. You want to gun up and go, I'm good. You want to drive away—either way. Who knows when one of these town ladies calls up the police—could be already. And then we're in some shit." He stopped and rolled his chin around on his knuckles. "Like, if he's still there at seven? At eight?"
"I should have followed him," East murmured.
"Let me ask you a question," Walter said. "We're supposed to have four guys. Without him, we got two. You and I, we're the same type. We're watchers. We manage. We ain't gunners, particularly. If we went, two of us, who does it? Who shoots?"
"You mean the guy?" East was annoyed, distracted, keeping his eyes peeled. "I can shoot."
"Yeah," said Walter, "but are you gonna shoot? You ain't that crazy about guns."
"I can shoot," said East blankly. "He'll be back."
"Not if he isn't."
Then dogs exploded, barking, and Walter started; East sat bolt upright. A dark slash between the houses. Ty was sprinting out, footing it down the street toward where he'd jumped out of the van. "Catch him," East said, and Walter put the van in Drive and veered it. Ty held the gun out plain as he ran: _Do not fuck with me._ Sprinting down the street as if he didn't even see them, didn't even care.
At the last moment he cut to the van and popped the door.
"What?" East demanded. "What did you do?"
Ty slid onto the middle seat. Panting and laughing both. "I told you, man, you paid too much."
Walter ran the stop sign heading them out to the county road. "What did you do?"
"I told you." Ty threw down the fold of twenties. "Four hundred eighty dollars. Now who's your daddy?" Regally he surveyed the world through the windows, the morning coming down.
# 10.
It was natural, Ty said, the way into the yellow house. A window up high was cracked open to the cold. He'd stolen a stepladder off a garage. If he could make the back-porch roof, he could get inside.
But he hadn't had to. He was crossing the backyard with the aluminum stepladder when someone came out of the house. "Crippled dude. Bad spine."
"Phillip," said Walter.
"Skinny. Look like he hurt to walk."
"Yeah. That's him."
"Phillip. First thing Phillip sees, black boy stealing a ladder. He came for me, man, gonna whup me with his car keys. You know, _crime stopper_? So I take my ladder and I knock Phillip on his ass, thinking, this will work: I'll just walk him back in with a gun in his ass. But guess what's in Phillip's hand with the keys?"
East said, "Four hundred eighty dollars." So crazy, he marveled, this charging on, with no idea what lay ahead.
"So beautiful," Walter cheered.
Ty grinned, angelic, contemptuous. "Thinks he got mugged by some kid from the block!"
—
Twenty miles east, they picked out a pancake house for the first sit-down meal in two days. Pancakes like no pancakes East had ever seen. Fluffy, meaty, thick as steaks.
"We'll make it today," Walter was saying, "just a few hours."
Euphoria had chased off the morning chill. It was easy to explain: new guns. Plenty of bullets. Money back. Ty smirking, all his dice landing sweet. That afternoon they could find a place, get some rest. But that wasn't everything on East's chest. The other thing that had warmed him that morning, even in that horrible house with the men and their pioneer ancestors standing guard together, and the baby on the gunpowder floor like a business card—even there, East had found himself hungry to make it work. To make the deal, straight, shake hands with the bastards. They had almost closed it, businesslike. Then Ty made his raid, and they all had that to hoot about, even if it was cheap and hard, even if it marked them, set them apart.
_So that's it,_ he thought, working on his stack of pancakes, which he never should have ordered; he could never eat all that. _So that's us. Just some thieving hoodlums, all across America._
—
Back outside, the cold was a jolt.
East drove. He nudged Walter: could they afford to stop somewhere, anywhere, get a room, shower, and a good sleep? "Don't want to do that," Walter replied. "Don't want to have to register. Not now, not this close to, you know. Where we're going."
So they would go until they got there, they decided. Arrive, circle, spot out the land. Make a plan and follow it.
Walter took the wheel back after a couple of hours. He exited onto a smaller highway, a Wisconsin state road, two-lane, rich black pavement, deep flood ditches dug on either side. The trees grew higher—and closer to the road. Pines, not thin and fire-hungry like California's, but tight-knit, impassable, winter-coated trees, their cones as thick as cats on the branches, green so deep it was blackish. Passing so close, they ripped East's eyes with their tiny, intimate spaces, tree to tree, branch to branch, too quick to see. They flashed by like the opposite of mountains, the grand spaces, the eons of time. Here, too many things to see and zero time to see anything. Around the back of every trunk, something could be hiding. East closed his eyes, but he didn't feel comfortable not watching either—Walter, the van, the narrow road. The deep, unforgiving ditches, the reaching trees. His eyes saw faces in them, every frightened bird an attacker, every mailbox a blaze of threatening color.
He was exhausted and could only watch. Walter was exhausted and could only drive. Like neither of them knew how to stop. And then they were there.
—
WILSON LAKE, read a tall green sign posted on redwood beams, surrounded by the emblems of clubs and lodges and churches of the town. Then another mile of jacketed pines.
Then a hill and a dip and the lake showed itself: just patches between the trees, a blur, the blue a murmur below the noon-white glare. The houses that appeared were not the bins of siding they'd seen for the last day, but triangles of stone and brown wood, frames of wood, walls of wood, jutting up like cabins, or in A shapes, from clearings in the pines. Names on signs out front by the driveways: WEE SLEEP, GREASY LAKE. And the mailboxes were fancy too: not just plain black U.S. Mail, but barns or jolly men with mail holes in their stomachs or monster animals whose heads hid the box.
"What the fuck is that one?" said East.
"It's a badger," said Walter.
Ty said, "A what?"
"A badger. It's the state animal. You ain't heard of badgers?"
They zigzagged quietly, getting the layout. There wasn't much. Big old houses. The lake was mostly round—half a mile across, maybe. Two beaches, three ramps, and a little strip of quiet stores. Three streets running parallel, a handful of connectors, and one road that looped around the far side of the lake. The address, Walter figured out off his page of scratchings, was 445 Lake Shore Drive. That turned out to be the road that circled the lake.
There on the far side, the houses were new, cabins and party houses, with skylights and roof decks thrust high like helipads, gas grills left out in their weather shrouds, flags East had never seen flying everywhere, in driveways, over doorways. He wondered at them as they drove by. People here were gonna know each other. Wedges of pines curtained the lots off, but next door could be a yard of noisy dogs, or a single nosy lady. It was a neighborhood. You never knew.
Some squirrel or small creature zipped in front, and Walter pumped the brakes: East looked up. Nobody watching them.
They were so tired.
"This is it," Walter announced.
The driveway forked. Two mailboxes at the foot, 435 and 445. Quietly the van crawled past: no other cars on the road right then. No one near, no one to mark the van cruising slowly. The house was an A-frame with bedrooms popped out on either side of the base. Two stories. Jagged corners with log-cabin beams, a grayish mortar holding them together.
Big windows cut either side of the door, and no flag.
"Big house," East said. One little sport truck out in front, black.
"These are vacation homes," said Walter. "Big and empty. By the way, shouldn't you wake your brother up? He might want to see it."
East peered back, tried to see Ty. "Naw. Let him sleep."
Two women approached, jogging down the road. In their fifties, wearing thin fleeces and mittens with reflectors. They raised hands at the crawling van, and Walter raised two fingers back. A natural.
No yards full of dogs. No high decks nearby with neighbors looking out over the trees. East's eyes ran a check automatically.
"Phone wires come in there," said Walter. "Pole behind the house, lines in on the back to both houses. We could take them out."
"Why? They got cells."
"But do they get a signal out here? Negative bars," Walter giggled.
East grunted, eyes on the woods. A tire-track path led back into the woods behind the row of houses that included 445. "You could park the van and walk up on that, get in from the back."
"Watch out for badgers," Walter said.
They looped back around the lake and headed into town. Found the police station, small, tucked behind the firehouse. Two black-and-whites in the lot and one unmarked, a little white SUV with good cop tires and a winch. Good to know.
"I got to sleep, man," said Walter. "I keep thinking I'm gonna throw up."
"All right," East said. His exhaustion had begun crashing down.
Walter put them back on the highway, the pines ever closer and closer around them, and cruised up until they found the next little village with its lake. It was smaller, this lake, the banks rough and muddy, the public lot an old reach of concrete leading down to some crumbling boat ramps. The homes along the shore had once been vacation homes, but the people living in them were no longer vacation people. Broken chairs and propane tanks in the yard, small sedans turning the color of dirt.
"We found the ghetto lake," East remarked.
"The _people's_ lake," insisted Walter. "You think it's safe?"
"We got guns."
Walter laughed and set the parking brake. The whole lot banked downward to the shallow, dark beach.
"Your brother," said Walter, "that boy can sleep through anything."
"I'm full awake, son," Ty spoke up.
"Better sleep," admonished East. "We gonna need to be awake and available later on."
"Oh, I will be."
They closed their eyes on the bright, final day.
# 11.
East slept like a drowned man. One time a pair of kids ran by with fishing lines, disturbing him. Their footsteps and yelling: his tongue in his mouth felt hard and lost. He remembered something people said: _Never eat a fish so sick you could catch it._
That was The Boxes. Who knew what kids caught out here?
Walter had moved back to the middle bench. East curled up in the shotgun seat. He closed his eyes again. Sleeping without cover wasn't as hard as he feared. Maybe there was something different here, out of the city.
Or maybe he'd just given up on peaceful sleep.
—
Later the sun crossed behind the pines, and now their jagged shadows lay on the icy water. Three metallic knocks sounded nearby.
East raised his head. It was a red-haired kid outside, seventeen or eighteen, maybe, his face as flat and empty as a dinner plate, a young moustache that looked just combed. And he was rapping on the window with a pistol.
"Open up." A cop? East's mouth was sour. He cranked down the window partway and said, "What?"
"Right now," the red-haired kid said hurriedly, "you're gonna need to give me your fucking money."
East squeezed his face and yawned. "Hello," he said. "You see we're _asleep_?"
"Wake up," ordered the red-haired kid. He might have been a year or two older than East. His moustache was brave, hairs ranging from orange to fishy white. He tapped the window with the gun barrel once more. Punctuation. Like a schoolteacher, East thought.
"Man," East yawned. "I don't want to make you sad. But everyone up in this van got a bigger gun than you, dig? The more people I wake up, the more people gonna be shooting at you."
From underneath the pale moustache came "Bullshit."
East considered this. A kid was gonna do what he wanted. It wasn't your job to change his mind.
"I'm gonna give you five dollars," he offered, "and you go away. Or else I'm gonna wake everyone in this van up, and then you in some shit."
" _Fuck_ that," the moustache said.
"Whatever you decide," said East. "You woke me, man. Only reason I'm not shooting you right now is I need you to go tell all your friends to let me rest."
The redheaded boy scratched his face with the barrel of his gun, then pointed it here and there, disheartened, as if picking out a substitute target.
"Here," said East. He rummaged around in his pockets. Nope: most of his roll was with the gun money Ty had retrieved. He had a ten and three ones left.
"You got change for this?" he said, holding the ten up behind the window.
The redhead squinted. "No, I ain't got change."
"Then you can have these," said East, offering the ones instead. "I ain't trying to scant you, man, but I need this money worse than you do, so."
This kid. East saw that bored excitement in his eyes. Every neighborhood had a few of these.
The redheaded boy put the gun away in the gut pocket of his sweatshirt. He reached up to take the ones. East held on tight to make a point.
"If I see you again," he warned, "I'm gonna shoot you in the stomach, man, first thing. Right in the stomach. Nine millimeter. You might live. It will hurt, though. It will change your life."
"Okay," said the boy, and East let the dollars go.
"Be cool, gunner," he said, and rolled up the window.
East watched the boy pocket the three dollars and go, walking off past a playground, where he gave each empty swing a frustrated shove. Soon the pines swallowed him. East closed his eyes.
But he couldn't go back to sleep. It wasn't smart, staying here. It wasn't The Boxes. No home-field advantage. Chances were that Gunner wasn't coming back, coming with ten or fifteen friends, all packing. Chance was that Gunner had his three dollars and was done for the day. But you could be wrong about that.
You could be wrong about anything.
He moved over to the driver's seat and ran the van slowly, gently, another mile up the road, finding a place to park it in the back of a Lutheran church. A group of boys and girls were holding a mad-scramble basketball game fifty yards away. Four adults—parents, maybe, or ministers, whatever, it didn't matter—stood with coffee and watched the battle. East parked the van in among their Fords, their Hondas. He would accept their company gratefully for an hour or two.
—
Thicker clouds, dull sun. Since late morning they'd slept. Six hours. Not a whole night. But it would focus them, East thought, let them work in the dark. Tonight's dark. On the middle bench, Walter was still knocked out, whistling and wheezing.
Then he realized with a punch: no head in the backseat, no knees or feet propped. Ty was gone.
He sat up, damp with sleep sweat. The gray lot, yellow lines. The kids had left their basketball game, and the lot of cars had emptied. Nearly six on the clock.
Then he located Ty, sitting at a picnic table across the grayish lawn, looking out across the yard into the trees. His sweater was dark green, like army leftovers. However cold it was, he wasn't bothered. Skinnier than East, even, but he stretched his head back, eyes closed in the feeble light. Cracked his neck.
East sat still, listening to the saw-blade whine of Walter snoring, watching his brother through the grimy windshield. He stretched his legs and arms, but they were leaden.
Ty. East had let him sleep as they'd found the house and checked it out. No—decided not to wake him, to let him sleep. But even that wasn't it. Postponed it.
Cursing silently, he took a breath and popped the door, then climbed out. Ty saw him coming and rose. "Wait a minute," East said, and Ty, with a sour look, loitered a few paces off.
"Get any sleep?" East inquired. "We rode past the guy's house. I didn't wake you up. Now thinking maybe I should have."
A shrug. "You thinking," Ty grumbled.
East swung his legs in and sat down at the table Ty had just abandoned.
"Cold," Ty said. "I'm about to get in."
"Talk a minute," East urged. "Walter's asleep. What do you need? Want to see the place while it's light?"
"Don't matter." Ty's voice was quiet, almost watery. "You two seen it."
"It looks pretty straight."
"Oh?" Ty said. "What did you learn?"
East fidgeted with his key on the pale, weather-brittle planks of the picnic table. Scarred with initials and names of the kids in this town: BEAU. RH AND JM. I LOVE SIGRID. The older marks swamped with the honey brown of the last coat of weather sealer. The new ones raw.
"You want to talk it over, how it's gonna go?"
Ty: "How?"
East blinked. "We're gonna do it, right? The way you want to do, man."
"You still want to do it?" Ty said softly.
"This is why we're here." A gust of wind, suddenly a note blowing in the gray air. East glared at it over the trees until it subsided. "But remember," he said, "be cool. We got a lot of getting away to do."
" 'Be cool,' he says," said Ty. "I _get_ away. The way I do it. Fact, if it was just me, it would be done, and I would _be_ away."
"Maybe," said East. No: he did not doubt. He imagined Ty flying in and out under his own name: luck and will and a supreme indifference to anything else. "But Fin sent us out like so. The four. So that is the way it has to go."
"You got all the answers," said Ty drily, "like always."
So Ty was making him call it. Making him, then scorning it. East let it go. He stared at his key scoring a line into the old table. "Tell me something, Ty. How'd you start doing this?" he asked.
"Huh?" said his brother, hands in pockets now, edgy. "Did you ask me something?"
"I asked you," said East, "what happened, man, that now you're a gunner?"
"Sure," said Ty. "What do you want to talk about? What I do? Or how I do it? Or you want to talk about why I left home?"
The constant difficulty. Like wrestling someone with three arms. East's key slipped, gouging a long splinter out of the table. A woody fiber. Exposing the light softwood below. With spit and his finger he patted it back.
"I don't know."
"Oh. So you don't _know_ what you wanna know." Balefully Ty eyed the basketball hoop. Its soft-laced net.
"I mean, like..." East said. Another day he'd have scratched his whole name in this table. In another life. "I don't see you, man. I don't know who you work for. Who taught you. Who you run with."
"Nigger, no one," Ty growled. "I'm here. I'm ready. I got nothing else to say."
East said, "You want to be that way, go ahead."
"I know what you think," said Ty. "I'm on the inside. Got a steady job, and when you lose it, you get another. That ain't me. I'm a contractor."
"You're thirteen years old, boy," East laughed. "You can't be no contractor."
"Tell Fin that," Ty said. "I live by my wits, man. Not like you." A thin, hot line of anger split his clear, high brow.
East stared at his brother for a long moment, then down at his hands. Digging the key along the grain of the wood again, doing nothing.
"Anyway," said Ty. "It's cold."
"So we'll go." East stood. "You want to plan it out, talk about it?"
"Ain't nothing to plan," Ty said, "and nothing to talk about."
—
Midway back, they found a drive-through: chicken sandwiches, milkshakes in the car. East wanted fruit, something natural. Somewhere in the van was the orange he'd picked up in LA. Couldn't find it now.
They made it back to the beach a quarter turn around the north side of Wilson Lake. The lot there was big and shaded, a couple of cars.
They idled the van while Ty checked guns and loaded. He took the Glock and handed the other to Walter. At East he flipped a glance.
"You want me to carry yours?"
An insult. East shrugged. "Give me that little one."
Ty handed over the little snub that he'd brought. "The lady's pistol."
Pushing it hard now that it was his time. East kept quiet. They spilled money out onto the seat and split it three ways—Ty put in the Michael Wilson money too. Most of three hundred dollars apiece. Then they pocketed it, because you didn't know.
"I don't have a key. So leave the van unlocked," Ty said.
"You don't think we're all coming back together?" said East. "You don't even drive."
Ty just said, "You don't know."
"It can be unlocked." Walter shrugged. "People out in the woods don't even care."
"They don't care. Ha-ha," Ty said. "All right, let's go for a walk."
A walk. East closed his door and stretched his arms inside the itchy sweater. The van's engine cooled, ticking. Walter bounced on his toes. Exercising. Ty went into motion without showing a thing. East tried to do the same. He didn't look at Walter. Walter would show back what East was feeling now, as surely as if they'd spoken it aloud. And as impossible to take back.
They walked the curve of Lake Shore Drive, the three of them in single file. But keeping close to Ty. Barely any light left in the day.
As the line of pine-rimmed houses drew near, they cut off on the track running through the trees and behind the cleared-out yards. The path led uphill from the lake. They found a loosening between trees and cut through to spot the houses.
"What's it? Fourth or fifth house?" said Walter. "No numbers on the back."
"Look for that black truck," said East.
A swish and hard squawk, and the pine straw beneath seemed to flip up and give forth a black ghost, a risen, screaming thing. East grabbed on to a tree, and Walter fell down. Ty nearly somersaulted to get away. It was a bird, a turkey or pheasant or something awakened in the pine straw, awakened from darkness. East could see nothing of it fleeing, but he heard the legs scrambling, the wings chop the air as the bird beat away, crying harshly.
"Damn," breathed Ty. "Could have had that."
"No shooting yet, junior," said Walter.
"No shooting. I could have _tackled_ that bitch."
East brushed off pine needles. They looked around. Lights burning on half the houses. An old white swing set like a gallows in the dark. No people around that they could see.
Three houses they'd passed. A couple more to get there. They moved together under the pine boughs in dark, scented air. East's eyes were opening up to the dark, but still he could not see all the branches, had no feel for space. There wasn't really space. He listened to Ty creeping ahead, Walter trying to stay on his feet. A snap of branch, a muffled curse.
He breathed it. He could sleep in here. The dark, the soft ground. Not even cold. But he too made his way. Nothing to carry, just the hard little spigot of the gun at his hip.
The ground kept climbing slightly. They passed a fourth house, lights on upstairs but quiet. A ceiling fan turning above the light. The fifth house was dark.
East was separated by fifteen, twenty yards. The fat boy had gotten himself snagged, had to unhook himself, fell behind. His brother likely was already there. That was it. That was the right house; he was certain. He picked a way under pines toward the dim light in the clearing.
Ty was already there, waiting just outside its edge.
"The house?"
"The house," Ty agreed.
Boxed in tight except for the drive—trees came to within ten or fifteen feet of the house. Not wide enough for a firebreak. The clearing was uncut field grasses, calf-high, still green.
Walter came creeping out, hands and knees. "Easier to crawl," he grunted. "Not so branchy."
One yellow light hung unlit over an empty deck, another over the back door.
For East, the house was stunning in its anonymity. They'd crossed all this land to an address: this was it. Just a brown house in the woods. Big _A_ on each end made of windows running light from front to back.
"Seems empty," Walter whispered.
"Nice to be sure," said Ty.
East looked up and gauged the sky. Seemed dark as they'd walked up, but now silver, strangely luminous, in the gap between the pines.
"Easy as pie," Ty whispered. "Angles on every inch of the place. Big windows on the bedrooms. No basement."
Walter said, "Where is the guy?"
"Can't see," said Ty. "Could be in bed in the dark. Could be out to dinner. Could be sitting right there on the sofa in the dark with a gun, waiting."
"You expect one or more than one?" East said.
Ty rolled his eyes. "I don't expect. We take what we get."
Walter said, "So what do you want to do?"
"How about we spread out a little and get some angles on this."
"All right," Walter said. "But stay back. It's no rush. Make sure we got the right guy."
"Did you call your George Washington?" needled Ty. "Is this the place?"
"It's the right house," East hushed. "Let's get the right guy."
"I don't see any guy," sniffed Ty. "Why don't you two collaborate on that. I'm gonna go see what I see." He began picking his way left along the seam of yard and woods, creeping along the flank of the house.
Walter stood breathing heavily beside East. They listened to the pine needles crackling under Ty's steps.
"Drive all that way thinking about it, man," Walter said. "And then here it is."
"I was thinking that," East said.
"Sure seems empty." Walter stood stock-still. "Seems nobody's home."
"You don't want to wake somebody up, and then you got a now-or-never in the dark," East reasoned.
"Yeah. Question is, how long you want to stand and wait?"
"I can wait awhile," said East.
Nothing sounded or moved. They'd lost track of Ty.
Walter said, "You gonna recognize him?"
"Who? The judge?"
East recalled the photos. The fierce, thick head on the man. The sides of gray. But it could have been sharper. The face swam with different faces in his mind: Fin's. Walter's. His own.
"I think so," he said.
"I'll know him."
"I will too," East said. But he wanted to get away from Walter.
"How many times your brother done this?"
"Ask him," East said. "Good luck," he added.
"How big was he when he started?"
"Who knows." He took a step away.
"He knows what he's doing," said Walter. "I mean, he goes right at it."
"He's got a reputation to protect," East said. "I'm a check the other side." He started tracing the seam around to the right.
"Fine. Stay invisible," Walter said uselessly.
The ground could be quiet if you slid your toe into each step. Like putting on slippers. East found a spot shaded by branches where he could see in the side windows and see the drive out front, the black truck and a glimpse of the road. He stood there, black face in a black hole. He could barely see his hands. He put one on his heart and tried to calm his blood down. Inside the black string was buzzing with irritation. Like he got sometimes at the crew at the house, when they ceased to watch, ceased to be in the moment. Became wild again. It annoyed East, made him bitter. And stubborn.
He made a smooth spot in the needles and tested it. Dry—but cold. He sat down anyway. Funny—a few days back, he was used to twelve-hour shifts on his feet, did six a week. Now he was eager to sit.
How many days had they been going again?
As if his mind was sand. The irregular sleep was one thing. But the road: as if he'd been brainwashed. As if he'd stared into a washing machine for days without closing his eyes. Even the lines and reflectors on the highway: like a code he couldn't read but couldn't stop, like a sound he'd wanted no longer to hear. His head felt out of shape, weak.
For years he'd guarded a place that mattered, looking out, seeing everything. Now he sat against a tree, staring in, seeing nothing at all. Nothing in the wooden house sounded or moved. It was wearing him out.
He thought he'd have time to think about it on the trip—killing a man. Or that in all the work of keeping things straight, the killing would become just another motion, another step. This was the address. They would find a man. He would be the man. Put it on the tab.
But he hadn't thought about it. What he hadn't seen was that in the rush across the country, the man would be forgotten. The face, the plan, inapparent. Only the miles and the goal remained. He'd declined the subject of the man until he was sitting outside his yard, waiting. Waiting on him to come home and die.
Darker. He heard something in the trees behind him—skitterings, like a bird. Something watching him. And then a heavier crunching. Walter coming like a freight train through the trees. East could not believe the noise he was making. He sat still until Walter had nearly blundered past him, then hissed, "Hey."
Walter stopped and peered around until he located East. He said, "Ty says it's empty. He's gonna check it out."
"What if somebody drives in?"
"Then we're loaded," said Walter. "We have the jump."
"He might drive in alone or not," East said. "We don't know. We're not even gonna know if it's him."
"You know what Ty said?"
"No," East exhaled. "I don't wanna know what Ty said. I know if there's anyone home on this road, they can hear your ass." His veins clouded with annoyance. He got up, but the cold stuck on his butt, an unpleasant circle.
He sized up the yard. "Tell you what. Go back. Take the back corner, left side. I'll get the front corner, right. We'll see all four sides. Then Ty can go close and look around."
"He's already doing it," Walter said.
"Still."
"All right." Walter turned and began again toward the back of the house. Making painful progress through the branches. Ty came out then. East watched his brother move. Casual, erect, no cat-stalking dramatics. He carried the gun in his hand but kept it shadowed. He crouched when he went against the house's frame. There he tested the ground and crept low along the window line. Popped up against a frame and peered inside. With one hand he tried the front doorknob. Didn't open.
East watched Ty working, recognized his careful pace. Taking time at each window, noting the rooms, the layout and angles. Shooters thought things through two ways. Where were people likely to be, before they knew about you? Then after, when they did, where were they likely to go? Where was shelter? Did they cover? Head for a closet where the guns were? If they shot back at you, from where? Or would they flush right out to the yard? A shooter understood a home just as well as the people who lived there. But to different ends.
Less cautious by the moment, Ty worked around the back. A car rolled by slowly on the stony road without slowing. East stood inside the clearing now, drawn to the house by impatience.
After four minutes Ty had made a full lap. No caution in how he stood now. Contempt for the house that had no people in it. Contempt for the time he'd spent working slowly. He spotted East along the face of the woods and approached, sticking the gun away with a swagger.
East felt almost apologetic. "Any minute he could be home."
"No." Ty shook his head. "No. No clothes, no suitcase. No dishes in the sink. No soap in the bathroom. Water's switched off. House is cold. Nobody's been here. Or if they were, they won't be back for a while."
"You want me to look?" said East.
Ty laughed. "Be my guest."
East made his own circuit. His eyes were hungry in the dark. He peered inside but nothing broke with Ty's account. Items to interest a thief—nice speakers, espresso maker, a flat TV up high. People with houses like this didn't skimp. Stealing wasn't his game but he knew enough from listening—most of the boys he'd led were thieves at some time. Once they stopped stealing, they stopped being quiet about it.
The back glass door was braced against sliding but not barred, like in the city. Probably an alarm, probably a glass-break sensor. No security badge on the windows, but he would have bet. It didn't matter much. If Ty saw his man, he'd be making noise.
Walter wandered up. "What you want to do? Stay here and stake out?"
"Wish we could bring the van up," said East. "It's cold."
"Scare him straight away too. Van full of black boys idling by the house?"
"I know."
Now the chill penetrated him. But the heat of the van seemed a hundred miles away.
"You want to go?" Walter said.
East slit his eyes. "No."
"I'm even getting cold myself," Walter said, undulating his bulk.
East turned away, played the judge's features in his head again. What he could remember.
"You want to go?" said Walter.
"Didn't you just say that?" said East viciously.
Ty came out around the corner, face pinched in, like he was chewing it up from the inside. "Shit. Forget it," he said. "I'm done with this."
East said, "Let's give it another hour."
"Oh?" said Ty. "You in charge? Fine, stay here. I know you like standing by a house. Me, I'm finished."
"I second that," said Walter.
East rolled his eyes at the sky. Fading but still silver above the grave-black square the trees made.
"We can call Abe back up. Might be a plan B," said Walter. "Come on, East. Sitting out here ain't gonna be nothing but cold."
"But what if they don't answer."
"Then they _don't._ Let's warm up at least. Get some food."
"I don't need to warm up."
Ty snorted. "Listen to him. Fucking cowboy, man. I seen you get cold last night."
Darkly East glared.
"I don't know," protested Walter. "I don't know if they'll answer. But maybe he's at a hotel or buying gas or at an airport. But I do know if he's somewhere else, we can find that out."
"How you gonna find that out?"
Walter pressed his lips together grimly. "Stuff you don't know about, man."
"You saying—"
"I'm saying I can't talk about it. But it's real."
"Fuck it," said Ty. "Fuck you both. I'll be in the van."
Walter glanced at Ty as he went. "I gotta agree," he said.
Quiet now. Even the birds stopped shifting in their trees.
"You coming?"
"Just trying to do my job," East said.
"Okay. I hear you. But I'm ready to get out of this icebox, man. Come on."
East hesitated, then followed Walter out to the road. Ty was a hundred yards ahead, out in plain view. At least it was dark. East hurried to catch up with Walter.
"Tell me one thing," he said. "The judge, could he have gone to LA already?"
"I guess," Walter said. "Possible. But I'm going to say no."
"Because of what?"
Walter winced. "Stuff you don't know about," he repeated.
"More stuff. Shit." East kicked a pinecone. "Burn these woods up, man."
—
Somehow they'd gotten low on gas. All tired. All angry. East took it personally. The empty house was another house lost. He had tried to keep it straight. But now everyone would kill one another at the first sideways look.
East fired the van through the little bubble of light that was the town and back into the dark, pine-jagged night. The big highway home lay to the south, so he took them north—toward the other lake, the ghetto lake, just by instinct. Just a glimpse of the big highway and it was over. The job would be over. If it weren't already.
All the miles, he thought. Nothing.
"Why we ain't got a cell?" grumbled Ty. "So much time wasted finding these phones."
"You know why," Walter said.
"You know there's a way to do it. You just too scary of everything."
"I'm Murder One scary," Walter replied. Staring out at nothing.
"All right. All right," East broke in. "Look. Wasn't there a pay phone up at Welfare Lake?"
"Yep."
"Do you know, while we were sleeping, some dude tried to rob us up there?"
Walter giggled. "What do you mean, tried to?"
"White dude with a little gun in his hand. I gave him three." Before it had seemed funny, a story he could save up and tell. Now it didn't have much in it.
"So he _did_ rob us," said Ty. "Whyn't you wake me?"
"Think about it," said Walter. "Think for a minute why he didn't wake you."
—
Yes: a phone booth, a fisherman's phone, at the second lake. It hung on a power pole beneath a light, the last few buzzing insects struggling through the cold air.
One woman, maybe thirty, forty, in dirty pink sandals, was on the line. Dolefully they watched her.
"What do you want to do?" asked East.
"Wait," said Walter. "What are we gonna do? Drive around? Go back and freeze? How long you think she can stand there in that housedress?"
"That housedress looks warm," Ty put in from the back. "I say she'll be there all night."
East parked the van two rows of spaces out and killed the lights. Left the van to idle. He broke open a water bottle, but it only made him colder.
"Wonder what she's talking about," Walter said, cracking his neck side to side.
Her feet were bare in the fuzzy sandals. She looked over her shoulder and winced at the sight of the van. Then turned back.
"When we get on there, talk if you want. But I got to ask a few questions," Walter said.
"I don't even care. You talk," East said. "What is _stuff I don't know about_? That's what I want to know."
"We see his credit cards. Okay?" Walter said. "Don't ask who, don't ask where. I'll tell you two things, then you forget them. One, we got people watching his credit cards. Two, if something like an airline ticket came through, Abraham Lincoln would have told us. Told us when, told us where. That's why I don't think so."
"But Abraham Lincoln ain't answering his phone."
"Today he ain't," said Walter gloomily. "I'll concede that."
The woman on the phone held out a hand and backhanded the air five, six times. Like she were reenacting the slapping of a child.
"Shit, I can't stand it," said East, and he swung the door open and jumped out. Colder here, now. And quiet.
The woman bared her teeth at him before he was ten feet away.
"You get away from me!" she raged. "You get back! I got here first!"
"Ma'am," East said. "Ma'am."
"You git back!" she hissed. "This is my call!" A yellow rubber bracelet on her wrist holding one key. Gripping the receiver jealously, ready to give up everything for it, her one treasure on Earth. "There's this _boy_ here," she shouted into the phone. "He wants the phone! Yes! And I told him, it's my call. I paid! I came first! And now he won't go away."
East pushed the air down with his empty palms down, trying to settle. "Ma'am," he cajoled. "Not hurrying you. Not hurrying you."
"Yes, you are. Yes, you are!"
He had Ty's little squirt gun in his pocket. It gave him an odd, lopsided feeling.
"How long you gonna be on, though? Ma'am!"
"How do I know?" the woman protested. "A minute, maybe? A few minutes? Damn!"
Blankly, East stared at the woman. Well, this sawed it. Ride for days, then crash up on this creature. Beggar-woman, they would have called her in The Boxes: steel-wool hair. Shoulder blades quivering under the housedress. His mother in white.
"A few minutes," he conceded bitterly. He turned and walked to the van, where they'd be teasing him, he knew. Yes. Pealing laughter as soon as he opened the door.
"God damn," wept Walter. "You should have seen her eyes flash."
East slid in with what he could salvage of his dignity. "A minute, she said."
"All night. I was right," Ty said. "Do what you wanna do. That was funny."
East tried to consider it funny.
"So, what did this robber look like, anyway?" Walter said. "The one who was gonna take us?"
East shrugged. _Stuff you don't know about._ "Big and white. Moustache. Red hair. Teenaged. Wore a green coat like the army."
"Probably _was_ the army," Ty said. "Oh! Look!"
The woman was scurrying off in her slippers. Wasn't taking a chance.
—
The operator was a new one. It took her forever to dial through. "Are you still on?" Walter asked twice. "Did we get disconnected?"
"I'm still here."
Then the silence of a new connection on the line. "Yeah," a male voice said.
East put his head in so close to Walter's that they were breathing each other's breath.
"Man," Walter said, "we're there. We reached it. You guys been offline. And the man isn't at the house. Do you understand me?"
"Yes," said the voice.
"You got any news for me? Anything I can use?"
"We got a different setup," reported the voice. "Police came down this morning. Arrested a few people."
The curve of Walter's ear just past the dark receiver, pinkish-brown.
"A few people? Who?"
"You know," the man said. "I don't really want to say."
"The big man?"
"Definitely got the big man."
_"Fuck,"_ said East. His body recoiled beneath him, wanted to smash something, to go running off. He bit down hard and listened.
Walter asked, "The two dudes who sent us?"
"Definitely arrested them. Definitely cleaned out that whole area."
"How many?"
"Maybe fifteen, twenty. Put it this way. Yesterday there was an organization. Tonight there ain't. We'll see what we can do tomorrow."
Walter put the phone to his chest. "You hear this?"
"Yeah," East whispered.
"Things are changed," the voice on the other end said. "I mean, it might be cool, you making that stop. More important now. There's more people he can talk about now, you dig?"
Walter said, "I dig."
"But I got no instructions. You do what you got to do," said the voice.
"You still got him flying out Sunday?"
"Flying Sunday," confirmed the voice. "Nonstop to LAX."
"Remind me," said Walter. "What day is it now?"
"Thursday night," the voice said tersely. "You boys strong? In your pocket?"
"Yeah, we're strong," said Walter.
"Okay," said the voice. "The next conversation should be like this conversation. Smart."
"So you're telling me to make up my own mind," Walter said.
"Yeah," returned the voice. "I guess you can do that."
Walter cupped the phone. "It's up to us," he whispered. "What you want to do?"
East looked long and hard at Walter's brown eyes. "We go," he said. "We do what Fin told us."
"All right," said Walter, and drew a long breath, the van full of guns idling behind them, blind.
# 12.
Once at the house in The Boxes a boy named Hosea was going to fight a boy they called Cancer. Everyone knew it would happen, and no one wanted it to, because Hosea was well liked while Cancer was not, and Cancer was going to whup Hosea's ass. None of that was a question. The boys knew, too, that it would happen, because Hosea had asked for it. He told Cancer that nobody liked him, then went on and explained it. He insulted Cancer and then insulted him again. Everyone knew Hosea was telling the truth. Hosea was a good kid.
But stronger than their feelings for Hosea or telling the truth was a principle: Know when you're fucking with someone. Know who you're fucking with. Know that things have their cost. The boys knew the fight was on, because Hosea could not say what he'd said and not pay.
And then the fight didn't happen. It was a windy, blast-furnace day. First Cancer was there, waiting for Hosea. When he left, Hosea showed up. Then they were both there, and ready, but Sidney called out from inside with some work: a U had stopped breathing on the kitchen floor. He had to be taken out. He was all quaky and blue. You did not have people dying in the house. It was best to save their lives, or have them die somewhere else. Either way, you took them out. Cancer and Hosea were both called into that, and they helped carry the U out and put him in a car. In the car, the U went ahead and died, so the event became less a drop-off and more a dropping-the-body. You could not just drop a body anywhere or at any time. Bodies were complicated. And the dead body dissipated the charge between Cancer and Hosea.
But not the other boys. That tension had not been released. And the rest of the day, East had to settle them, to separate them, to keep the electricity that had stabbed the air like a knife from cutting into them, from sending them honed at each other. No matter that Hosea and Cancer stood together and let it drop; no matter that a dead man lay in temporary storage, waiting to be cast aside when darkness came. A knife had been thrown up into the air, and the boys would not settle to their work until they saw it land.
There was a gas station. The lights in the cold made the cars gleam like licked suckers. East pumped and paid, and Walter tried calling the number again. Nothing. Nothing had changed. Nobody knew anything more.
They bought hot dogs out of a steamer and drove off.
It seemed they were reaching the end of the world of people. No towns on this road to speak of, only points where the trees peeled away, the road bent, and suddenly there'd be a house on the land, a single light atop a garage; it blazed, filled the yard as they passed, then the trees snapped shut like a curtain behind them. Roads as dark as rivers, absorbent of everything, only the reflectors in their measured rhythm, _here, here, here_ , on posts, and _here, here_ , down the center stripe.
Scuttling eyes disappeared along the edges of the road.
But Walter, having warned East not to ask, was now conversing on how you tracked a man through his credit cards. "Ain't hard to get someone's number," he rambled. "Any waiter can do it, any cashier at a store. If you don't mess with it, if you aren't trying to steal, then no one knows you're watching."
"So you doing this?" Ty put in. "On the judge?"
The judge. His name lay low in East's memory.
"I set the account up. I maintained it. Some people never check their shit online. Sometimes you gotta work at it. Sometimes you set it up once, and it works forever."
"How you learn?"
"Just learned," said Walter. "Kids at school."
East watched the inscrutable dark outside. Had to take this break, he reassured himself. It was necessary. Everyone needed to warm up. The night was even colder than the night before at the gun house, when it had snowed. This cold would freeze you. Everyone needed the heat. Everyone needed the food.
They ate the watery hot dogs and wadded the cartons up. No finding the trash bag anymore. East stuffed his into the crack of the seat.
Ty put his feet up on the back of Walter's chair. "So you're following the guy's cards. How come the dude don't know?"
"Know what?" said Walter.
"Somebody's watching him."
"How come he don't know? Everybody suspects. Nowadays everybody thinks somebody's on to their shit. But if you ain't losing money," he said, "if your money is still, you don't do anything about it. And we ain't taking his money."
Ty said, "If we got a computer, say at a library, could you see him?"
"No."
"I thought that was the point," said Ty acidly.
"I know," said Walter. "It's complicated, man. We were tracking more than one guy. It ain't like I had just one password. I don't know what his was. I ain't got these all memorized. And we was faking IP addresses, everything. We had a whole setup."
"What computer did you use?" said Ty.
"At school."
"No wonder you still in school," laughed Ty. "Tell me how come Fin ever had you standing yard. You a smart boy."
East opened his mouth, shut it again. It was more questions than he'd heard Ty ask in years.
Walter replied, "I stood yard so I'd know the job."
He coasted to a stop. The highway ended at a stop sign. Ahead, a maze of road signs peppered a luminous guardrail that kept cars from hurtling into the woods. Walter had been running squares, East knew from watching the roof compass switching _N, E, S, W_. Just going around the block, keeping the lake in the middle. And they'd been riding almost two hours.
"We can decide whatever we want," Walter said.
It was East they were waiting on.
He stirred. It had been easy to say _go_ in the light of the pay phone, tethered some way, however, to LA, to The Boxes. But here in the dark, the van filled with the things he didn't know.
Something about talking again seemed difficult. "Should we call them again and see what they know?"
Walter shook his head. "Maybe. But not till morning at best. School's closed. And I don't even know who's watching things. I don't know who's in jail and who isn't."
The turn signal pulsed in the ditch, shotgun side.
"If we went back," said Walter, "we could see if anything is going on."
"Back in them pine trees," East said hopelessly. "With nobody there."
"Maybe we'd find something," Walter said. "Maybe if we broke in. Something that would tell us."
Ty said, "I _did_ break in. At least, I popped a window up."
Walter turned in his seat. "Why didn't you say something? Was there any alarm?"
"If there was," said Ty, "I didn't hear it. If there was, cops already came and went."
"You think we should go?" Walter said over his shoulder. "Ty?"
Ty said, "Me? I seen it already. Doing my job. You two make up your minds." He lay down theatrically, retired.
"Yeah," East decided. "Circle back." He had been hugging his knees, and now that he let them go, his whole body hurt. Beaten up from inside. Heading back to the house did not make him happy. But Ty was right. You did your job.
—
Wilson Lake. Reflectors on every post, every driveway, eyeing them. The driveway of 445 Lake Shore empty but for the cold black truck.
Walter idled the van at the lake parking lot while once again they loaded up and got ready. They slipped on the thin, dark gloves. Once again they walked the back track off the road.
The grass had gone quiet, and the branches slapped back when Walter caught one in the dark. Ty led them through the field of pines to where they saw the yellow back porch light. A single moth clapped at it, ineffectual.
"There's the phone box," said Ty. A gray box below the kitchen windows.
"So?"
"If you got an alarm, it comes out in that."
"Some alarms work cellular," said Walter.
"Ain't no signal up here," Ty said.
"How do you know that?" Walter said. "How do you _know_ that?"
Ty had Walter spooked now.
The phone box was cobwebbed. Ty pried the lid up and pinched free the plug.
"If we're going in looking, what you want to find?"
Walter said, "Anything. Directions? Ticket receipt? A note? We'll just look. But we ain't going in there to break and fuck things up."
"Of course," said Ty.
"Take your shoes off," said Walter.
The window Ty had popped was on the middle of the back side, over the sink. There was a security lock on the frame, but it left enough room for Ty to squirm his head and shoulders in.
"Maybe they'll have a flashlight," Walter said. "You'd think we'd have a flashlight."
Ty kicked his shoes off. "Lift me up."
Walter clucked his tongue at East, and East started. Together they made stirrups of their hands. Ty climbed them and got an arm inside and went moving things—a bottle of soap off the sill, two glasses away down the counter. Then he began wriggling in. It was tight. He yelped at something. East passed Walter Ty's foot and went to help, felt at Ty, at the metal framing of the window. He found what was catching: his brother's ear. He touched it, little stub of cartilage, strangely warm, and for a flash he thought of their mother's face.
"Damn, that hurts," Ty groaned from inside. East pressed the ear, flattened it, helped pass it without abrasion through the hard, slotlike opening. And let Ty's head go. In. Ty's head was in.
East threaded his other arm around and began keying Ty's shoulders into the slot, feeding his body in, inch by inch, rib by rib. His waist and legs wriggled in midair.
"Fuck. Ouch. All right," Ty said from inside.
"You want us to stay back here?" East said.
"Yeah, let me look. Then I'll let you in."
"All right." East and Walter braced Ty's thighs as he wormed in in midair. His body seesawed, and his hips disappeared over the sill. Then the pair of headlights swung into the driveway.
# 13.
There would be a few more seconds where the car would be running, seconds where they might not be heard. East had Ty's legs in his hands. Ty was still pulling, wriggling in.
"We got to pull you out," East whispered fiercely through the window, past his brother's body.
"What? No!"
"They're here."
"I just got _in_ ," Ty cursed.
The pair of lights swung away as the car picked its way up the driveway. Then they stabbed again across the open _A_ of the house. The lights bounced off the countertops, the faucet. Then the car stopped—it was something new, little high-intensity bulbs—and the lights were doused.
"Now," said East, and he caught Ty's hand and hauled him back, up over the countertop and out the window. Ty's head bounced on the sill: " _Ow, ow, ow, ow_ ," he complained.
"Shh."
"The screen. The screen," said Walter.
"Forget the screen," East said. He slid the window down and scooped Ty up with him. They scuttled into the pines together. This little square of a yard, the house lights would fill entirely.
"Where my shoes at?" Ty whispered furiously.
"Shh," East replied.
The doors of the little car popped open. Two people got out: a full-grown man on the driver's side. Someone else on the other.
"What we got, a girlfriend?" said Walter.
Ty watched intently.
East picked up a pine bough, the needles dry and brittle but still arrayed. He held it over his face, peering through it. The man was coming to the house, thumbing through keys.
He stopped, the man, at the front door and caused an overhead light to come on. They could see him—large, a black man. East wondered: _Our_ _black man?_ The man opened the door and moved through the open space to the kitchen. On the counter he laid a satchel or briefcase. The lights he switched on there were furious, bright, paint-store white: yes. They threw a glare that filled the yard, that counted the trees. East flinched at it.
Were they covered? Deep enough? That last-second dilemma of hide-and-seek: Could you find a better place? Or did moving expose you now?
Slowly the man stepped back out of the kitchen, into one of the sides of the house, soundless—like watching TV with the mute on. Into the kitchen next came the girl. East watched her reach up. She took a cup down from a cabinet and a plastic jug of water from somewhere below and poured a drink.
Dark. Black dark. Hard to make out her features or her age.
"No water," Ty said. "They're just camping out here, man."
"Was that him?" Walter said. "East?"
East stared until his forehead hurt. "I'm not sure yet."
"Best make sure," said Ty. "If I shoot, he ain't coming back."
Madly East tried to flip through the pictures in the back of his mind. To see again what Johnny had shown them that morning. He could bring up the man's suit, his heft. Could remember nothing of his face.
Silently the man and girl moved around the lighted house, specimens in a box. So blindingly bright. East felt his stomach begin to knot.
"Just knock," Ty muttered. "If he answers, ask is he the judge we're here to shoot."
"What about the girl?" whispered Walter.
"Don't shoot her," said East.
"She ain't a target," Ty said, "but she better not fuck with me."
"That's cold," Walter said, "shooting the dude in front of her. Because he looks like, you know—her dad."
Ty didn't say anything. East held the fallen bough close. The dry needles prickled across his cheeks.
"You decided yet?" said Ty impatiently.
The neighbors' houses were distant, dull shapes. The stars wheeled above them, forgotten. East chewed a pine needle. Strange, bitter, sweet, like orange peel somehow. His eyes tightened on the blaring light.
The girl accelerated things. Two people there: one person you had to peel away. To ignore, to not shoot. But that's how it worked. You thought you had the rhythm, that your pace was the world's pace. Then someone busted a move. Someone drove up in a fleet of black-and-whites and disrupted. Someone opened a door. The world would have its way with you. You and your plan. There was only that lesson to learn.
You could pretend that it would not, that all your breathing and all your insulation would protect you. Ask Michael Jackson. Ask anyone in LA. Earthquakes rattled up out of nowhere. No radar, nobody yelling _Incoming_ , no warning text on your phone. Only the house jangling, things falling off the wall. East was fifteen. He'd never been through a big one.
_Stop. Stop it,_ he told himself.
He spat out the pine needle and began creeping, moving left along the line of woods around the house. The thrown light was as bright as a ball field: he had to stay well clear. Dark clothes, dark shoes, dark branch, dark skin. One night Sony had brought along his sister's astronomy book from school—she went to a special all-girl science school she had to ride an hour to get to—and they observed the stars they couldn't see in the LA sky, they studied the words that weren't used in The Boxes.
_Albedo._ Can a body throw back light. It might have been the last word he'd ever learned. East's albedo was near zero. Not much bounced back off him.
He'd been tracking toward the front of the house, but then the man appeared again at the back, in the kitchen. He lit a match, got the pilot going on the stove. He poured water from a bottle into a silver kettle and set it down. The light over his head drowned his face in shadows. Graying hair. Maybe fifty. Solid, thick shoulders. He washed and dried the girl's cup.
East stopped and sighted through the bedroom windows. Through one he could see the girl stepping into a bathroom on the other side. Light spilled into the hallway, then the door narrowed and snuffed it out.
_Give me a minute,_ he'd told Ty. Because from what he saw, he could tell nothing, could conclude nothing.
Then the man was moving again, stepping to the front door. East watched him flicker past one window, then the next. At the front door he rummaged in his pockets. Where was Ty now? Ty was waiting. Waiting on him.
The man couldn't find his keys. He stopped, retraced his path. Back to the kitchen. He reached, took the keys from the countertop.
The stripe of light at the bathroom opened again, and the girl stepped out. She went to the kitchen. Turned the tap, but the faucet gave no water. She reached again for the gallon jug, then the just-washed cup. As she poured the cup full again, she looked out the window, and East saw her eyes. Her face swam, seemed to look at him sideways. The face was the Jackson girl's face. The one he'd watched die. He caught his breath and looked away for a moment.
Yes. Just the girl, just the man. Nothing more.
Where was the man now? He'd come out the front. He was at the car! In the open air, away from her. Keys in his hand, even. East peered back along the trees' edge, but Ty and Walter weren't there where they'd been. Everyone was moving now: without anyone saying _Go_ , it was happening. East slipped left, farther toward the front, past the bedrooms, past the short brown stockade around the air conditioner. One lone pine stabbed up out of the earth, away from its pack. Then he was near the black truck and the car—a little Volvo wagon, snub-nosed, Illinois plates.
The man fobbed open the doors and lifted suitcases out of the back. He wrestled with them—big ones, not the little tote size, but monsters. He couldn't make it with both. He put one down on the beaten dirt and entered with the other. East watched him disappear inside.
The other suitcase sat beside the car, unattended.
Uncalculating, straight and quick, East rushed behind the truck, around the light that spilled out the front door, to the suitcase. Was there a tag? A name card? He reached for it, the pine bough still in his hand. Looked for a name card, something on the handle, down the side. Nothing.
Then he found the golden stitching on the highest flap. Faint in the house's glow: a monogram. CWT. For a moment his brain buzzed, pulling up the name of the man they were hunting. Then a shriek echoed inside the house: it all clicked in. Carver. Thompson. The right initials.
The right man.
The girl. "Daddy? Daddy? Someone—"
She came running. Not at him—not to shut the door. He saw her eyes. Never mind the initials—this suitcase was what she wanted.
He straightened, raised the pine mask idiotically to his face.
"Daddy!" She burst through the door, came outside.
East's heart hammered. Discovered now.
The father's first cry was faraway. Then he came barreling, yelling now: _Melanie! Melanie!_ East spun away—where was his gun, even? He kept his face hidden, for being spotted by her was not like being spotted by him. She was a witness; he was the target. He was the one who knew too much. East cleared his throat, but as he did, she reached her suitcase, and he heard the other pair of feet sliding to a stop on the piney ground: his brother. Arrived.
Ty said, "Here we go, E. Is it him?"
East nodded. "It's him."
The judge stopped at the door. East watched him stare and then smile. Half laughing, a curious voice: "Do I know you boys?"
Ty's arm came up and a growl rose from his throat, a reproach. Then he fired through the screen door. Two shots, three. East heard them punch the man, heard the long, failing gasp.
The girl tugged the suitcase and shut her eyes. Opened her mouth, but nothing came out.
Her father hit the floor. Walter arrived.
"He finished?"
Ty leveled the gun at the girl, who hadn't unsqueezed her eyes. She clutched the black handle.
"Not her," East said.
_Pock. Pock._ Twice his brother shot. The noise ripped the black yard. "No," East said, but already she was falling, the suitcase toppling over her.
Walter's face was pale, stretched. "Is he finished, I said."
"Three in the heart," Ty said. "That'll do. He's the right dude?"
East dropped the pine bough. "He's the one," he murmured.
And the girl. The girl with her face going still in the lamplight. In the dark, she had the Jackson girl's face. That same all-seeing look, into a world where nothing moves. East stood near her. "I told you not to," he said.
"My call. I did it. That was what we said," said Ty. "No time to talk about it."
"You want to lose that gun, man," Walter pointed out.
"No," said Ty. "Ain't the way it works. Right now I got to find my shoes."
He sprinted, dirty socks, back to where he'd mounted the window. East staggered away. The gunshots still echoed around the spaces of his brain. Seconds were passing.
Lights in a house deeply set behind pines. But they might have been on before.
Ty's feet scrambling in the needles behind the house. Every sound carrying sharply now, every breath leaving its shape like ghosts in the air.
"We got to go," Walter urged. "East. We got to leave."
"I know it." East's neck crawled. He did not look at the two dark piles, the man behind the door, and the other, under the black suitcase toppled over. He stared down the cleared channel beside the lit, empty house.
"Fuck!" Ty said in the useless dark.
Walter's eyes swiveled. "East?"
"Did you do something with his shoes?"
"I didn't do nothing," Walter protested. "He kicked them off. Maybe we should get a head start? East?" Backpedaling already, finding the road with his feet.
Then mad, flying footsteps, and Ty was coming, the stripes of light painting him as he sped beneath the windows. In one hand the shoes, in the other the gun. "Go!" he panted. East turned: already Walter was pumping down the road. No other noise, no movement, no response. The quiet banks of mailboxes marked them passing by.
Quickly they made their way back around the stony road, thudding heavy of foot as it swept downhill, the pines less dense on their left between them and the lake. Their breath came and fled in quick mouthfuls. Around the lakeside curve, the parking lot came into view, the lone light on its pole stained yellow, a glimpse of their blue van shining beneath the trees. Numb, East hastened his steps. _Get away, get away,_ his mind drummed. And also: _What happened?_
No talking, only the question the girl's face made.
Then he saw the other car, an old boat of a Chevy jacked up on fat wheels, parked near the van, black like the pines in the yellow light. And two kids swarming around the van. Trouble.
"Look," he said, pointing.
"Mother fuck," said Ty, taking the lead. "All right. Guns up, and spread out. I'm a handle it. But be ready."
"It's just neighborhood kids," said Walter.
Ty's scowl locked down. "Do what I say, Walt."
He fanned left, and East began a slow run across the lot, taking each yellow-lined parking space in two steps, Walter coming up behind. Ty's gun made a heavy _click_ , and East pawed at his pockets for the little gun. He found it clumsily, fumbling at it as he ran.
As yet the kids hadn't spotted them. One might have been the ghetto-lake thug from yesterday. Meaty shoulders, moustache.
"That one got a gun," East panted. Guessing. But just as much, telling Ty: _Careful._
Ty raised his hand and squeezed a shot into the trees. _Pock._ The white boys saw them now. They clutched at each other, then broke for their car. The rattly engine roared. Ty followed them left, and East went for the van, slapping his pockets for the keys. "Hey!" Ty was yelling. "Hey!"
The dark car burned rubber. It leapt toward a gap in the trees. Instantly its lights were gone, just its whine climbing the road away from the lake. East reached the van, panting, key ready in his fingers.
But it wasn't any use. The kids, they'd fucked it up. Walter came wheezing up behind, goggle-eyed. "Oh, shit," he groaned. "Oh, shit!"
East breathed and circled. One side window had been popped off its buckle. It hung askew on its hinges like a flap. That had gotten them in. They'd pulled out everything—the clothes, the food, the first-aid kit and blanket. The case of water bottles, scattered on the ground.
"They took my game, man," Ty swore from the back.
"We got the money, right?" said Walter. "Y'all still have the money?"
"They got my _game_."
"We have to leave," East said.
"Look, though," Walter said. East stepped back. The problem wasn't apparent. But down the hidden side of the van, the woods side, it said something in spray paint. It said, FUCK YOU NIGGERS _._
"Bashed out the taillights too," said Ty, shoes still dangling in his hand.
Walter chewed his lips. "We can't drive like this," he said. "Like, hello, help us out, every cop in the world. Right at the same time they're finding bodies."
"But right now we need to go," said East. "Get in."
—
He found a four-lane road westward and took it. Avoiding going back through town, where people would see them. Neighbors. The wind cut in through the broken side window, swirled through the van, searching. Ty was trying to rig a sling for it with the Ace bandage roll from the first-aid kit. East waited until his breath was steady before he turned around.
"The fuck was that, Ty?"
Ty, working on the window, was serene. "What? You hollering at me now?"
"What happened?"
"What happened?" Ty stopped and glanced forward for only an instant. "Mission accomplished, son."
"The girl. I told you, don't shoot the girl. Why don't you just kill everyone you see?"
Ty said, "Thinking on it."
East banged the steering wheel. "Just evil. Evil! And those dudes. Run right up where they can see you. Why don't you just tell them how many people you shot lately?"
Ty paused ironically. "Sure. Say. Maybe you're right. Maybe we should be subtle. Driving a van with _nigger_ wrote down the side."
"People can't see that in the dark," East snapped.
"People can see whatever they want."
It was what Ty did. Move East off the subject. Though East couldn't bear to move back. It was like a furnace behind them, something blinding hot.
Empty dark road before them. No lights or shapes. East fought his eyes—they kept searching the pavement rushing under them. Had to keep looking down the road.
For all the planning, they had nothing now.
"We got any help from here on out?" he asked Walter. "Or are we on our own?"
Walter opened his hands slowly and closed them.
"Ty?"
"You got a car," Ty said brutally. "You got a gun. Make your way."
"You got to change cars," Walter said. "We're just gonna have to get a new car."
"Could call Abe," said East. "See if he can hook us up."
Said Walter, "No. Listen to me. We're on our own now."
A little town was coming up, and East slowed for its speed limits. The van passed through the center of light—a gas station, bright as the vault of a refrigerator, and two late-night girls out pumping gas, brown hair fanned out over their fur hoods. They turned, watched the van roll past, painted words like a banner on the side they could see.
_Fuck you niggers._
If nobody knew who they were, still, everyone would remember seeing them. When it came to connecting the dots, everyone would draw lines.
They had most of a tank of gas and a few hours before daylight.
—
Two more towns and East found a gas station with a blue-lit phone on its outskirts. He pulled in. "I'm gonna call," he said.
"They won't know anything," Walter said. A beaten-down quiet in the middle of his voice. "Do what you want," he conceded.
Ty had gone silent in the back. Sunk back into his bench. Like a monster who rises out of the sea and then submerges.
East climbed out with a handful of quarters and gritted his teeth at the loud sexy recorded lady. Then the bloodless operator who tried to connect.
A red light blinked ceaselessly over an intersection.
He looked back at the van, battered now—they'd smashed the plastic grille trim with something, those kids. Taken out one of the front turn signals. _This,_ he thought. _This is what it's like._ It hadn't been his bullet, but he'd said, _It's him._ He'd risked his life to prove it. _Get him_ was going to be the next thing he said, if not for the girl standing there.
He didn't want to think about the girl right then.
—
"Y'all did it?" Abraham Lincoln said. Surprised. "I mean, the whole deal?" Like nobody had any idea.
Some sort of muttered consultation kept East waiting, looking sideways out of the wind. Finally Abraham Lincoln came back on the line. "Okay, thanks for calling."
"That's it?"
A long silence.
"If it makes sense, just go on home," the voice said.
—
They sat in the lot near the blue pay phone's glow, a bone-dry wind scraping under the van. It rocked them side to side like a cradle.
Walter sighed. "So we're here. Making it out is up to us."
East watched the red eye of the traffic light switch on and off.
"One idea," Walter continued. "Go on. Find a store, buy some paint and tape, fix the window and the taillight. Paint the van up. Another idea: we ditch the van and get home some other way. Plane. Bus. Whatever. We might not have ticket money for three. But two of us could hole up and one goes home and wires back."
If he'd brought his ATM cards, East thought. If he hadn't come clean the way they said but brought something extra. Like Ty did.
"Better idea," said Ty. "We carjack some bitch and get out of Dodge."
"I thought about that," said Walter. "That guarantees us attention, though. That gives them a car to look for, a car to hunt."
"Got to get rid of this van," Ty said. "For real. It ought to be burned. Soaked in gas and burned." He ejected his magazine and reloaded in the dark.
"Let's not waste time," East said. "Decide right here. Can you call someone, man, and fix us up some tickets?"
"Today?" said Walter. "Not _today_ , East. We got us a dead man. Federal witness. Three boys getting on a plane. Nobody wants their name on that purchase. And if they figure it out while we're still in the air, we're fucked. We're locked down. You like those chances?"
"We'll keep on, then," East said. "Do something about the van before daylight."
"Not soon enough," said Ty from the backseat.
"Ty," said Walter. "Don't you want to get rid of that gun?"
Ty said, "This gun's the one thing I got."
—
Westward they kept on, cutting a corner off Minnesota, then down into Iowa. Roads nobody hurried on, roads where they weren't expected. Soon the sun would come up. Walter laid his gun on the dashboard, but it kept rattling, sliding around. So East pocketed it.
_Fuck you niggers._ They carried it with them through a dozen little towns. First the tall, ghostly farmhouses, then the little shops with their new trades hand-painted over the old signs. AUTO BODY, BEEF JERKY, TAXIDERMY, _ASISTENCIA CON LOS TAXOS_.
The occasional truck in the oncoming lane, pushing its blast of air.
"It's getting light," said Walter. "If you see something that's open, let's get some paint."
They rolled by one large discount store in a shopping plaza, a thousand parking spaces deep. The letters were stripped off the concrete façade. Given up for dead.
"I gotta take a leak, man," said Ty. "Gas station right up here, man, please."
"You might want to park a way off," said Walter.
"We need gas anyway," East observed.
Maybe they'd been lucky. Maybe they should have expected to be stopped long ago. East walked into the little station shop and found a roll of red tape for broken taillights. FOR TEMPORARY USE, the package said. The cashier wore a red sweater the color of Christmas and a strange thing on her head. It appeared to be a pair of stuffed antlers. She was a big lady, big diamond ring, somebody's mom.
"You got any paint?"
The lady shook her head. "The new ordinance."
"What new ordinance?"
"When certain people," she said loudly, as if it were a rehearsed speech, "spray paint all over the middle school? They won't find it quite so easy next time."
East looked away then. He paid for the tape.
"Thank you anyway, ma'am," he said, his best manners.
The lady put the tape in a useless plastic bag. "You're welcome, hon."
Outside, the only person was Ty, standing in the cold in his socks. It was as if the cold and the van had driven something into him, because his eyes were big, and he quivered slightly, the way a cat does watching a yard mouse. The moment East saw him, Ty took the gun out. A car was pulling in, a low white sedan, Infiniti. Ty stepped right into its path and brought the gun up.
The tires chirped, and the sedan stopped short.
_What?_ thought East. Helplessly. For, he saw now, agreeing on a plan, he and Walter, meant nothing to Ty.
Nothing he said or did meant anything in Ty's mad story of the world.
He wanted to deny it, to return to the starting block and start over. No. In the station's noisy light, the Glock was a dark fact. The runners weren't stopping. Ty circled, taking aim on the driver, hollering, "Man! Get the fuck out the car!"
# 14.
The boy wearing dirty socks yanked the driver out of the car. He thrust the gun against the man's face until the man was up against the pumps, teetering.
"What are you doing, man?" the older boy demanded, one eye a bruise, the other wild.
The younger boy ducked inside the car to look. No wife. No babies. Just this early businessman in his buttons and tie. Unlucky. The boy popped the trunk release and stood up. He gestured with the gun arm. "Get in the trunk."
A gold tie. Gold with a pattern of bright blue pearls.
"Oh, no," the man said. Low and collected, even indignant. "I'm not going in there."
"Give me the keys," said the younger boy, "and get in." He leveled the gun again.
"No," gulped the man. He appealed to the taller boy, to the cashier inside, with his eyes, one raised hand.
The younger boy raised the gun up and shot one hole through the canopy. The echo bounced down, metallic.
"We can't do it like this," the older boy said. "It's crazy. You need to chill out. We can't come with you like this."
"Think I need you?" said the younger one. "You ain't fit for nothing but standing yard." To the man he said, "I'm a say it just once. To me you're just one more bullet."
The older boy thrust in then between the young one and his prey and shoved, made the gunner bounce off the car, nearly fall. The younger one found his footing and turned. The businessman pressed himself hard behind the gas pump.
"Oh," said the younger one. "So now we see."
He raised his gun, and the older boy, a gun in his hand though it hadn't been there until then, shot the younger in the chest. The younger boy uttered a short cry, and he fell. In a flash the older boy was on him, pinning the arm and taking the gun, rifling the pockets, taking a cache of bullets bunched carefully in a greasy wrapping. Opening the boy's pants and taking a fold of twenties out of the vent in the underwear. As if he'd known it would be there. He stood again and shut his eyes.
"Jesus," prayed the businessman. "Sweet Jesus."
"Shut up," the older boy said. He turned. Then something overtook him. He bent over the young boy's body, put something back inside the boy's pants. He buttoned them and stood again, looking down. The boy on the ground opened his mouth, rolled his head back. The muscles of his neck quivered in the night glare.
A fat boy in a van behind sat stricken at the wheel.
In the floodlights' glare, the older boy's face was a mask, angles of hardening bone, the eyes shadowy holes. He faced the man with the golden tie. "Run, God damn you," he said.
"Sweet Jesus," the businessman pleaded. In the station the woman with an antler headdress held her phone and stared, holding, holding.
# 15.
Walter sobbed. "I seen it," he blubbered. "I seen there wasn't nothing but this. We knew you would."
"We who?"
"Me and Michael," said Walter. "We knew it was gonna happen like this. Mike, he says, one of these brothers gonna kill the other. I thought he was joking. I said, that's bad news for Easy. And he said, naw, my money's on East."
"Bullshit," East said. "You making it up. Like everything."
"No," wept Walter. "It don't matter."
East chewed his lip.
"I'm sorry, man," Walter sobbed. "I saw it coming. I told myself, no, I couldn't do nothing. I didn't get out the van. I couldn't do nothing. I couldn't stop you."
"Stop me?" East said. "Ty's the one out of control."
"You the one who shot him. Who was cold to him the whole way."
"I'm not cold to him."
"You were," Walter insisted.
East raced through the dark, dread like two hard hands working his guts, reshaping them, remaking his body. He felt every poke, every lump of food inside him, every stone on the road coming up from the tires to his seat. His road-shocked mind could not even keep time.
He had run from police before: standing yard, after fights, after hurling stones at the windows of stores, hoping something would break. Flight, they called it. One part fear, one part the blindest excitement you'd ever known. It freed you from time, from who you were or the matter of what you'd done. You darted, like a fish away from a net, like a dog outrunning a dogcatcher.
No flashing lights behind them. But no time either to think of his brother, or the other two, knocked down along their trail. And Walter's grief was unending: his brother's stoniness, just taking other form.
"I'm sorry," he said once. But this just started Walter sobbing again.
None of us were perfect, East thought.
"Do you think I killed him?"
"You shot him in the chest, East."
"Yeah."
"Well," Walter said, "that's how you kill people."
—
Once, when Ty was about six, East had been sent to enforce his mother's word. Back then their mother's bedroom was sunlit and clean and the TV sat small and quiet in a corner—later, it took over the whole apartment. It was time for the TV to be off, their mother said, and Ty wouldn't turn it off— _SpongeBob_ or something. So East turned off the TV for him. Ty jumped up and hit him and turned it back on. East repeated their mother's warning and pulled the plug out of the wall. Ty threw the remote and hit East in the head. When East got back from checking his head in the mirror, was he cut, the TV was back on. Louder than ever. East picked up Ty and locked them both out of the bedroom. He caught hell for it—that old broken lock, if you locked yourself out, the only way back in was to take the door off the hinges.
He toted his brother like a laundry sack, one arm locked around his chest and the other around his hips. Ty caught the hand that had him at the collarbone and bit the fourth finger of East's left hand as hard as he could. First East screamed in anger and shock—it didn't really hurt yet. Screaming stopped things: startled them, embarrassed them, gave them satisfaction. With Ty it only stilled East so that his teeth could get a better grip. That was when it began to hurt. East had stopped screaming to begin to fight his brother then—put him down, free his arms up, begin wrestling himself out of a set of teeth that only clamped on tighter. That was when he began to imagine simply living as a sort of undying battle. Sometimes in the right kind of light, he could still see the indentations—the chain of little tooth marks circling—in the brown of his skin.
Different roads, different land, but East, half dreaming at the wheel, imagined the days in rewind: his brother, restored. The van, undamaged. The wooded house, undisturbed. The red moustache with his gun, bored and lonely near the swings. Back out the roads, without the guns, without all the food. Michael Wilson waiting for them by the foot of a bridge somewhere, welcoming. All forgiven. Like sometimes when he would sit up chilled and guilty from a horrible dream and stay there, the black string humming inside him, trying to breathe, to reassure himself that nothing had happened. But there would be no reassuring. Walter's weeping made sure of that. He looked down at his hands, still clad in the dark gloves they'd been issued.
He stopped the van just one time, to reach behind him, under the middle bench, and find the empty black shoes, sticky with pine sap and crumbly soil, a bud of pine needles caught in the laces. Maybe wet with something, maybe just cold, as the air was. He couldn't look. He rolled his window down, gunned into the oncoming lane, and let the shoes tumble into a ditch dark with reeds and litter, things that had grown there and things left behind.
—
Outside it would soon be the plain blue morning. The banks' signs gave different temperatures: 28, 31, 36. Air whistled in around the side window where Ty had managed to sling it in place. East scanned the roadsides for a store. An idea. Anything.
"We gonna paint this thing?" East proposed.
"Whatever you want," Walter replied, drained.
A small-town discount-store palace. Something about the big crumbly fort of the store, perched atop an old, sloping lot the size of a football field, promised to fix everything. There was a junction here, and around it had grown up a battlefield of paved spaces, a few of the buildings still doing business, some of the lots empty as if the sky had swiped them clean.
Walter sat straight. "Park up behind that Denny's there," he said. "Get between them trucks so we don't stick out."
The painted words along the blue side of the van were a dark, chalkboard green.
"What you want to do?" Walter asked, his white breath floating away.
East assessed the van. "Paint it over. Quick spray-paint job, good enough. And some duct tape to hold this window in. Tape the taillight red. And get going."
Even at sunrise the store was busy—early-morning women with hair tied back. Single, angry men lugging detergent or powdered milk. Everybody tired, even the people getting paid to be there. Everyone with eyeballs, noticing the black boys. The lady with chin-length, orange-dyed hair, bright sweater, staring in the candles aisle. East felt small, tried to stay small. Fluorescent tubes twenty feet up. Pine chips and needles still dotted his sweater, and a rank, anxious sweat ebbed from his skin.
He felt wired, sick without sleep.
"Back there," Walter said. "Paint and painting supply."
The aisle was dazzling. Most of the store was patchy, stock falling over, end caps picked clean. But the paint aisle was straight, tight-packed.
"Help you guys find something?" A twenty-year-old kid poked his head around the corner. Mexican goatee, tattoo on his neck.
"We're all right," Walter said.
East noted the kid. Gang stripes on him. Even here. He eyed them with a calmness that set East on edge.
"Right here if you need me," said the guy, and East could hear the _R_ kick and rumble like a motorcycle. Like the south side of Los Angeles. This guy was from home.
"Thanks," East said, nodding. As opaquely as he could.
Walter, studying the spray cans: "This will cover. Should we use primer?"
"No," said East.
"What would Johnny say?"
East said, "Johnny doesn't ever want to see this van again. Not now."
"Primer is cheaper," Walter said. "One can or two?"
"One. Jesus," said East.
Walter picked out a spray can of primer gray. Shook it once, the agitator bouncing around the hollow can.
The paint man passed suddenly behind them. _"Váyanse con Dios,"_ he whistled, making his accent plain. East watched him disappear around the end of the aisle.
They walked, Walter muttering to himself the whole way. They had paint, tape. Walter pulled a box of granola bars off a shelf in the aisle.
"We don't need that," said East moodily.
"Good price, man."
"Impossible, man, you out here trying to save a dollar."
"I like to eat, East," Walter said. "I like to live."
The cashier barely looked at them. East saw it now: the whole front end, the ceiling was dotted with cameras. Everywhere they went now would be like that.
"You think they had cameras going at that gas station?"
"Who knows?" Walter said. "I don't know if we should even get back in the van."
Outside, again, their breath rolled forth in sunrise plumes. East took the spray paint and shook it ready. "Where you wanna do this?"
"Anywhere. Not here," Walter said.
Then they saw the police cruisers. Behind the Denny's. Behind and around the van. The blue lights cut the air, and the red ones flashed high and stuck along the trucks, along the light poles. The cops milled, their uniforms and sticks on, all their cruisers churning smoke into the air.
"Man," Walter said.
Then, without a word, he turned the other way.
East froze for a moment, paint can in his hand, dumbfounded.
So this is how it went. Downhill. Maybe the police had just arrived. Maybe they hadn't thought to fan out yet, or were waiting on backup. Maybe they hadn't approached the van, were just boxing it in, not sure it was empty. If you only had a minute, you clung to that minute, you were thankful for it, you made an hour of it, did what you could. Downhill to the road, putting cars between them and the scene developing. Not toward anything, only away. The van after a week called out to him: _Defend me._ The way he'd guarded the house. A fool's voice. Gone.
They had the guns in their pockets and the money. Not the map, not the pink flyer, not the blanket, not the water, not their clothes, not the gloves, not Walter's directions in his inscrutable scratch. Not the heat.
No need for paint now. East ditched it in the back of a truck.
"Pick a direction," East said. "I'm with you."
The lot was studded with stones, protuberant like eyes, like once they'd been moored in concrete but the concrete had worn away. East and Walter left it and crossed a storm ditch, slick with frost. He stole a glance: now he couldn't see the van or cops. Just flashing. Just the store. Now no longer the store.
"Quit looking back," Walter said.
They reached the highway's shoulder. "Cross here?"
"Down there." Walter indicated the next intersection.
East checked the stoplights. "Got cameras on the lights down there."
Walter waved his hands, helpless. "All right, let's go here."
They sighted a hole between cars and took it. Across the street, the buildings were smaller, squashed between the highway and a running ten-foot fence. Hemmed in. Gas stations, doughnut shops: people and big windows everywhere.
Walter breathed his chugging breath. A little fire truck, a pumper with lights on, roared behind them. Going up the left-turn lane toward Denny's. _Ought to be burned. Soaked in gas and burned,_ Ty had said.
A stitch pierced East's side. Walter was panting already, his eyes worried like a dog's.
The first gas station. East evaluated the one truck parked at a pump: nondescript but old, tires knobby. Tough but slow. They hurried on. Next, a sort of post office. Closed as yet. Then a Laundromat. A beauty shop. Closed. Then a row of drive-throughs. East looked again to his left at the wire fence. Barbed on top, flecked with trash. Behind it, nothing.
There was nowhere to walk to, no hiding place. It was going to have to be here.
East closed his eyes and gave Walter a moment to catch up.
"We got to go gunning, right?"
"We could call," said Walter, wheezing.
"And say what?" said East. "Ask for some Superman shit?"
"You're right," said Walter. "Calling's no good."
"We got to go gunning."
"Yes," Walter agreed.
—
The time. In flight you used it. The space. Like a gunner checking a house. They examined the drive-throughs. The first was burgers. Hopping: two lanes in the drive-through, each one backed up. Nice, fast cars there: a sport Lexus. But any move you made, there'd be ten people watching.
The next was doughnuts. They studied the building, ugly and square, a little box of concrete with painted-on stripes. One asphalt snake around the back and up to the window. And a brawny green hedge five feet high all around.
"Let's look," East said.
They cut along the outside of the hedge until they stood across it from the black-eyed window. There was a gap a foot or two wide where the lane drained into a steel grate in the pavement.
"Come right through there," East said. "Wait till it thins out and a car comes. One of us blocks, one talks. Climb in and go. But time it right."
"Did you ever carjack before?"
"Never done it," East said. "I'm a yard man."
"And a proud one," said Walter. "How we keep the girl in the window from seeing?"
"Get it before the window, maybe," said East. "In the back."
"What about the driver?"
"What do you mean?"
"Take them or leave them? Ty was gonna put that dude in the trunk. What do you want to do?"
"I don't know," East said. "I guess I don't know."
Glumly they examined the hedge. "It ain't perfect," Walter decided.
"Keep looking?"
"I don't know," said Walter. "We ain't gonna find anything perfect. Not on this side. We can't go shooting no matter what. It's gotta be quiet."
"If we have to," East began.
"East. We ain't in the woods anymore. There's a hundred cops right over there. And the longer we stay here, the blacker we get." He gave East a worried look.
"All right," East said. "What? You think I'm trigger happy?"
"You shot your brother," Walter said bleakly.
"He was losing his shit."
"When we shot that judge, what, six hours ago, I didn't think we were going to jail. Now I do. For a while, everything that got in our way, we were on top of. But now it's a losing streak. And we ain't got your lucky brother to fall back on."
"He ain't lucky."
Walter said, "You can say that again."
—
They took up spots behind the hedge. Through the gap, they could watch the cars coming, look without being looked at.
Two cars came at once. First a van, tinted out, green, heavy, anonymous. Ideal, East thought, but impossible, for right behind it was a little Suzuki or Isuzu or something, a woman at the wheel, some sort of earring glinting near her ear. Right up smack behind the van, a witness. She missed seeing them entirely. But her child, a stout little thing with red on its chin, stared them through.
"I don't think this joint got a camera," Walter said. "I think we could stand right up in the back if we wanted."
"And what?" said East. "Just turn around and go out the way we went in?"
"That might work."
The next, a pickup, held two men and a boy. A full gun rack in the cab. "Not that one," Walter said.
"No."
The following car showed up so suddenly, it seemed as if East had been sleeping on his feet. Impossible, but maybe. Walter touched his arm, and he saw. Light brown Ford with after-market mud guards and golden seat covers. An old black lady, alone.
"She looks nice," said Walter.
"I ain't making that lady get in a trunk," East muttered.
The Ford pulled abreast and waited at the window. Suddenly Walter pushed through the hedge, the trimmed branches plowing at his clothes. He straightened his sweater and held his hand up in greeting. Like this was his great-aunt.
Crazy. East shivered. He had half a mind to run.
The old lady turned her head. Pouted, appraising him, through gold-trimmed glasses. Slowly her old power window whined down, and she said, in a high, chipped voice, painstakingly clear, "Do you two young men need some sort of _rod_ somewhere?"
"Ma'am?" said Walter.
"I said, are you young men waiting for a _ride_?"
Walter said, "Yes, ma'am. Thank you, ma'am."
The lady's sunny expression dimmed a bit as she peered around Walter at East, still hiding in the hedge.
"You with him?"
"He's my cousin, yes."
A silent _humph_. "Well, come on."
Walter pivoted, shaking his face at East. "Come _on_."
East wavered, then squeezed through the hedge. Squeezed his eyes shut like a kid jumping into water. The little branches tugged and ripped at him. Walter was twice as wide, but he'd come through easier.
"One black lady in the whole state," he whispered, "and you gonna steal her car."
"She _offered_ ," Walter hissed.
Across the roof of the car, the drive-through window folded open on a white girl with a round, pimpled face. "Good morning, Martha!" she hollered. "How are you?"
"Oh, I'm fine, fine," said the old woman. The cashier girl took the lady's money cheerfully and handed her down a dozen-doughnuts box with a gold seal on top. "Have a nice weekend—till next time!" she shouted. Sparing Walter and East one odd glance.
East stood at the back door, and the locks hammered up. The old lady set the box of doughnuts next to her on the seat, and she peered out at Walter again.
"Was you coming?"
"Yes, ma'am." Walter nodded vigorously. "Let me talk," he murmured at East.
"You better," East said.
Already Walter was taking the front seat, chirping his thanks. East lowered himself into the clean-swept interior. Extra rubber mats, one floral umbrella on the seat. The faint smell of lubricating oil. He reached for the belt; it clunked as he unrolled it. The old spring was soft and would barely click in.
In his pants, Ty's gun. A spur.
"What is your name?" the lady inquired, not going anywhere, not quite yet.
"Walter, ma'am."
Just like that, thought East, real name. Why not?
"A," said East. "For Andre."
"My name is Martha Jefferson," the lady said. "And where are you two headed today?"
"We'll just ride, ma'am," Walter said. "If it's all right."
The lady paused. She had that grandmotherly pout for thinking. "In my experience, a young person doesn't just ride. A young person has a good idea _where_ he's going."
"We've been going, but we're just stuck here," Walter said. "We need to get along."
"You in trouble?" said the woman, narrowing her eyes. "Or _are_ you trouble?"
"We hope we're not in trouble," Walter said.
She laughed. This pleased her. "Are you runaways?"
"No, ma'am," Walter said.
"Hoboes?" Her giggle a creak, like an old wooden chair.
"No." Walter giggled back artfully.
"Are you college students?"
"Yes."
"No you're not. Too young!" But at last, with this interrogation, she put the car in gear. Old but not simple, East observed. "Are you robbers?" she asked merrily.
"No, ma'am," said Walter again.
"Not here to rob me. Well, what, then?" Martha Jefferson said. Like sweethearts flirting, East thought.
"Together we could rob some other people if it helped you out."
The old lady creaked with mirth. "Aw, no," she said. "I be all right."
Now she'd put her sauce on.
"This morning," declared Martha Jefferson, "I am headed to the airport in Des Moines. I don't know if that helps you. But I can take you to the highway if you don't want the airport."
"We would ride all that way with you," said Walter. "With thanks."
Formally, Martha Jefferson agreed, "Then I will take you."
"Yes, ma'am."
East looked out. They were passing the Denny's. Police cars all up and down the lot. Then Martha Jefferson turned at the junction, and it all fell behind them: that van, that everything. East did not turn to look back.
"Chilly day for just a sweater, Mister Walter," the woman said.
The box of doughnuts wafted their smell into the car's box of air. Was she ever going to break into them? East thought rudely. Walter broke open his box of granola bars and offered one to Martha Jefferson first before passing one back. East unwrapped his and let a first, dry bite moisten in his mouth slowly, feeling it as much as tasting it, the chunks of it coming apart, turning to starch on his tongue. How something plain could open up like the whole world when you were starved. He was asleep before he could take a second bite.
—
He awoke. The little digital clock stuck to the dash said 9:20. No idea when they'd begun. Outside, the highway flowed by. They'd come through here before on this highway, heading east in the dark.
Walter was saying: "I'm going to study electrical engineering. Electronics. Wiring. Installing. You know. If I'm good, I can get a degree and begin to design. I could work up in Silicon Valley, you know. Not programming but helping on the engineering side. And if not, I can just install and repair. Computer. Cable. Alarm systems." On and on.
"You got it all figured out," said Martha Jefferson, admiringly.
"I'll come out okay either way," said Walter.
"It's that kind of world. You have to plan for either way."
Walter said, "Amen."
"My grandson lives in California too," she revealed.
The highway sang under them, and the two of them talked, like relatives, like people sure of their homes, till East felt himself falling away again. Walter was a good boy. East dreamed he was alone in the van, alone in a storm where he could see nothing ahead of him, knew there was nothing ahead of him. No one had seen Walter. Walter was a good boy. He slept until he felt the gentle lift of a ramp, and again he awakened, and a parking garage surrounded them now. Outside, the airport sky, cluttered, gray. A plane arced up behind a gray-windowed tower.
"Let me get it for you," Walter was saying. "Here's forty to cover it."
"No, no," said Martha Jefferson. "I'm only gone the two days. It ain't but five a day."
"Gas too, then," winked Walter. He tucked the twenties into her purse, and again she _humphed_ silently.
What was the act now? East had missed the whole conversation of lies. Keep his mouth shut.
"I knew," the lady said, "you was good people."
Walter. A good boy. East's stomach felt hard and scraped out. He felt as if he'd been punched in the chin with force. He wondered how he smelled. Shuddering, the old Ford pulled into a space near a ledge that looked down two stories. Walter leapt out, fetching the old lady's weekend bag from the trunk.
Then East was trudging behind him, the dumb cousin. The airport. Wasn't this what Walter had said to stay away from? The world was a gray smear. The patterned carpet running forever reminded him vaguely of the casino. Dog-tired on an invisible leash, he just followed.
People crossing his path were thin, wore black and gray, ears clipped to their phones. Long yellow and white lights blared above: they told him nothing.
"You're meeting at the gate?" Walter was saying.
"My sister? Yes, dear. But we couldn't get seats in the same row. Couldn't get that," the old lady said. "She's right in front of me, though. Oh. Oh, goodness." She stopped with a jolt. "We forgot the doughnuts."
Walter stopped too and grinned.
"She gonna have a fit!" he scolded Martha Jefferson.
Martha Jefferson laughed aloud. "She will!"
Walter turned on East in mock fury.
"Andre! Why didn't you bring the doughnuts!"
_Andre._ Right. Wearily East regarded Walter.
Either it was exhaustion or it was being on the outside, but he could see what they were playing at. As if a pair of sunglasses had been removed and now the light of day was bright and strange. Playing three black people, comic and noisy in an airport. Like a skit in school, the boys playing wolves, the girls playing lambs. Acting out what all these people expected them to be. Better than what they were.
He understood it, and yet did not know his lines.
Martha Jefferson said, "She'll be so _frustrated_ with me."
"I could—" began Walter.
The old lady, with searching eyes, asked, "Would you?"
Walter said, like a faithful nephew, "I will. I'll go right now. Andre, walk Mrs. Jefferson through to security now. I'm gonna be back before you go through the line. But you can't go through the line—you're not a passenger. Got it?"
East's stomach rolled. He nodded. Walter took Martha Jefferson's bag off his shoulder and traded it for her key ring.
"Andre. You can't go in the line," Walter said. "You hear me?"
"I won't go in the line," East murmured. _Jesus._ The lady was looking him up and down. Again East wondered how he smelled.
"I can carry your bag if you like," he offered.
"Very kind," said Martha Jefferson, but she kept it on her. He was the one she didn't like the looks of. All right.
Together they trudged off toward the security line.
A smattering of people funneling into line. They paused and shuffled, eyeing phones or conversing in low voices. Steering their luggage, casual but alert, along the switchbacks. East and Martha Jefferson came to a stop just outside the cordoned area.
A few of them watched the old, stern-faced Negro lady and her ragged boy. East was bilious. Acting sleepy was the best he could do.
"You can't go through, you know. Through the security. Not unless you have a ticket," Martha Jefferson reminded him.
"I know," he responded dully. "Walter said."
_How long was the fat boy going to be?_
He understood that he was standing by her now not to be quiet and kind, but to hide behind her. He was raw. He could barely stand. And Walter was dashing about, carrying things. Out in the parking lot now without him, a set of keys in his hand. Inventing things. He was standing by her to be the sort of boy who traveled with his great-aunt, who didn't have blood on his shoes. Late coming to the play, but happy to be in it.
As long as Walter came back.
A black couple inched forward in the line, one midsize son with his face in a video game and a little girl in braids, up on her father's shoulder. She eyed East with dread.
"I hitchhiked one time," Martha Jefferson reminisced. "I'll never forget. Course, it was in Louisiana, long ago."
East was going to say _Oh,_ but he hiccupped instead.
"You okay?" said Martha Jefferson.
"Fine."
"You don't look fine," said the lady. She stepped back, and East lacked the strength to argue. There wasn't anywhere to go. The little girl stared sideways. She closed her eyes and became the Jackson girl.
His stomach flipped, and instinctively he turned toward a trash can. He fought his jaw muscles, but they pried themselves apart, and he vomited into the cups and wrappers with a long, despairing cry.
"Oh, Lord," the lady was saying. "Look at you now."
Yellow heat in his mouth. He spat out the rest and felt in his pockets for something to wipe his face. Nothing. Just a granola bar wrapper, a key, a fold of twenties, and a gun. He cleaned himself with the back of his hand.
"I _told_ you," said the lady. Though she had not told him anything. But he saw that she was not speaking to him; she was speaking to the people in line, now gazing from their cordon at this small misfortune. He saw that she was not going to help him out like a grandmother might. She stood back, marking their distance.
At last, Walter. Bright fat idiot angel, he carried the box of doughnuts with its golden seal before him like a prize. "Here you are," he sang, flashing Martha Jefferson's key chain and helping her guide it back into her purse.
He looked around, seeing that something had happened. It was on everyone's faces. "What's wrong?"
"Your cousin is sick," announced Martha Jefferson.
Walter touched East's forehead, his hand there heavy and soft.
"I'm okay," East said.
A guard was coming up to see. The line slipped forward. The little sideways girl on her father's shoulder watched East again.
Walter said, "Well, maybe we should _go_ then, Andre."
Martha Jefferson agreed silently, stirring toward the line with her eyes. She was eager to be rid of them. Even Walter. She knew how to do it: she did it with such public sweetness. "You _have_ been good to meet. Such lovely young men."
She and Walter beamed brightly, falsely.
"So happy to have met you," Walter said.
She fluttered. "And I, you."
Walter handed the box of doughnuts over and then gasped at it. "Oh!" he exclaimed. "But, Mrs. Jefferson, are they going to let you take this on board?"
Mrs. Jefferson smiled, a final smile, not for them. "Well, it isn't allowed. But don't worry. They know me. Everyone in the sky knows me."
East, bleary though he was, saw different. Nobody knew this lady at all.
—
Walter: "What is it?"
East's stomach was tumbling to a halt. "Like food poisoning. Something."
"You're panicking, man. Your body's fooling with you."
East let this go. In the vast men's bathroom they washed with squirt soap and dried with brown paper. Morning businessmen made hurried passes under the seeing-eye faucets. East stood there a long time. In the mirror he was a different mess than he'd ever seen before: an eye still dark and swollen from Michael Wilson. He'd forgotten about it; he hadn't had time to hurt. Now, clean, the eye was fat and tender. No wonder Martha Jefferson had looked at him funny. His skin, even after he scrubbed it, was puffy, black, and greasy. The cold water brought his focus back a little.
"I was going to pass out," he grumbled.
"You did. You slept the whole way." Walter glanced around at the other men. "Can we go out? We got to talk."
East nodded. The bouncing daylight outside the bathroom braced East, brought him back into space and time. He and Walter made their way to the exit. _Iowa,_ he mused. Back in Iowa now. Behind them like beads on a string lay the other places: the van, his brother, the wooden house in Wisconsin. And The Boxes, the boarded-up house that was his. Strung out behind, not far, not long, but behind. Links in a chain. Behind, like his black eye, the bruise clouding over.
The noxious air of taxis idling. They found a bench, and Walter dug in his pocket.
"What is it?"
A single key. "Hers."
"Whose?"
"Miz Jefferson's. She had two. This is a valet key. It will start the car—I tried it. Just won't open the trunk."
East held the key in his fingers, examined it.
"And she won't be back for two days." Walter looked down the line of cabs. "Likely won't notice it's gone till then."
East whistled. "Smart," he said quietly.
Walter laid his legs out straight and crossed one over like an old cigar smoker. "I know," he said. "I impress myself."
"Walter," East said. "We gonna get caught?"
"I been thinking on that," Walter said. "Looks like police got the van. Question is, why? Because of Wisconsin? Or because of your brother?"
"What does it matter?"
"It does," Walter reasoned. "Two very different things."
"You gonna tell me they upset about Wisconsin," East guessed, "but they don't give a damn about my brother?"
"Well, there's that." Tightly Walter smiled. "Maybe you're right. Maybe they don't hunt the one at all. But they gonna hunt them differently. If it's just Ty they're looking for here, and they found the van, then we're just black boys who shot another. Probably didn't go far. Police are looking in that area. That Denny's, even. Not an airport. You follow?"
East nodded.
"But if it's Wisconsin, then they tie in Ty, maybe—there were witnesses, they saw us, man, they got a plate, probably an APB on us all night—then they find the van, that makes a direction. One, two, three. That points west. This way. The way home. Got me?"
"How you gonna know for sure which it is?"
Walter laughed. "Ha. I'm not. East, I'm guessing it was because of you and Ty. Because of the witnesses. I'm guessing it's your bullet they're following."
East sat back. The black string jangling inside.
"But the van isn't here. That's lucky—we drove it a few hundred miles last night and ditched it. We made a mystery jump. And we didn't dump it near an airport. And we didn't steal a car they can look for. That don't tell the cops we're looking to hop on a plane."
"But we're not," East said, "looking to hop on a plane."
"Well, I was thinking about it," Walter said.
"You what? You said it was dangerous, man. Even walking in there."
"Everything's dangerous now," Walter said. "Right? But we got cash. They can sell us walkup tickets. They gotta check our IDs, but they're clear, and we can drop them the minute we get to LA. Kill these names off and never look back."
Walter sat up out of his crouch and looked around, face wide open, as if they were waiting for a ride, as if he was unconcerned.
"What's it cost?" said East. "Do we have enough?"
"Don't know," said Walter, "but it sounds good to me. See the country from up top. Be home this afternoon." He traced the idea across his pants and stopped it with a dot. "Nobody knows where I've been all week," he confessed. "Probably worried."
"Oh, you got people?"
"Yeah," Walter replied. "Of course. I got people."
East watched the airport cop down the way, forty yards, directing people with suitcases.
"Quicker we drop these guns, the better, then," Walter said.
"If we're done shooting."
"How we gonna know if we're done shooting?"
"I'm done shooting," said East. He got up and strode over to the nearest trash can, poking around until he found a good fast-food bag, stiff white paper, a little greasy. He picked it out, straightened it, then palmed the little gun into the bag. He walked it back to Walter.
"See if you can put your trash in here. Be cool with it."
Walter emptied his pockets on the bench beside him: granola bars, van key, the money in a clip, paper napkins clean and used. He covered his pocket with a napkin and fished the gun out into it. Into the bag it went. East crumpled the bag and took it back, tucking it into a corner of the trash can, just so.
"Feel better now?" said Walter, up on his feet.
"No."
Walter frowned. Disappointment, maybe. What mattered to Walter, East saw, was solving problems. Inventing. Wasn't anything in East's stomach that Walter could solve.
A police car went by, white cop, black glasses, who knew what he was looking at. Airport security. Passed without slowing.
East's mind hurt and he could see only a part of it: the part that was made up. Nothing in him wanted a plane. He didn't trust it. Or maybe he didn't trust himself. Sick, tired, out of control. That person he'd always reined in, in himself or others—noisy, violent, fractious—he felt behind him, or just beside him, or attached, like a shadow. Outside him, maybe, but double. Visible.
He'd never been on a plane. But all he knew about them—the getting on, the staying put, the thousand people packed together—was not going to happen today.
"I'm gonna make it on the ground, man," East declared. "You take your plane if you want to. If you think it's safe."
Walter said, "Really?"
"Yeah."
"It's safe. I mean, this way," Walter said, and he could not help looking guilty. "For me. Nobody saw me, man. In Wisconsin, because I was running behind. Or with Ty, because I stayed in the van. They just saw you."
East toed at a spot on the pavement. "All right."
"I mean—" said Walter, then gave out.
"Come on," East said. He wanted to be done with it, finally alone. "Let's see how it goes."
—
He observed the line to the counter from a bench some distance away. What happened at the gate was out of his hands. He liked Walter. Walter was handy in ways he'd never imagined being. But there wasn't anything he could do for Walter. Across the country they'd come together—a team. A crew, right? Now scattered to the wind. They'd put bullets in the right guy—finished the job. But all East felt was beaten. That's what it cost. The week was a wound he hadn't even steeled himself to look at yet. Yet he felt it bleed.
He had a hundred dollars in his pocket. Walter had the rest. He was thinking about that thin hundred and the little clutch of ATM cards he hid under the block of wood in his bedroom, that he'd agreed not to bring, that he'd hidden away so he could follow Fin's rules. He wondered about the home Walter had, that steady place with a computer and a library, maybe a piano or a cat, and the people he had there. Who would be in LA when he got back, or when he didn't? Who might be wondering where he was right now and, if Walter got taken down at the counter, or at the gate, or pulled off the plane, would wish they were here where he was sitting right now, watching Walter make his mistake, watching him crawl up to the backlit ticket counter with his teacher's-pet grin and his pocketful of twenties? Overconfident. Or maybe just lucky.
_Rather be lucky than good,_ people said. East felt neither.
What East had—the house in The Boxes, a crew, the everyday job with Fin's gang, and Fin himself, maybe even his place under the office building, and Ty, whatever Ty had meant at the end of that invisible gravity that bound them unhappily to each other across the blocks and years—all that was gone. All defunct. The streets would be there, and the business, and he was skilled. But he was known too, one of Fin's. Even if he caught on with another outfit, he'd be secondhand, a refugee. Never a citizen. Nothing he'd ever done with Fin would make way for him.
He'd start again from the bottom, like a kid ten years old.
Watching the ticket counter in Des Moines, he thought of LA, the smell of the steady flowers mixed with the smell of sun and desert and cars and food frying. The people he knew, the ghost that was his mother, the guys who had scattered. None of it was anything he'd buy a plane ticket back to today.
That was the business, and the business was closed.
Walter's agent was a young man, thin, a thin moustache, a face without fat on it. East could see that the man didn't think much of Walter—fat, black, wrinkled clothes smelling of nights and days. Raggedly cheerful. All his brilliance invisible inside. The agent listened, his lips pursed on the edge of a grimace. Tapping his screen, asking, checking the license, asking again. Soundless from here. Eyebrows working like two small animals.
A moment passed when he was hoping the man would reject Walter. Would stop him dead at the counter, would take him down. So East could retrieve the guns. So East could blow holes right through the ticket agent, renounce himself. Throw everything down in an avenging storm.
He shook his head to clear it. A printed document came over the counter. Walter nodded again. And he stepped clear.
_Are we gonna get caught?_ he had asked Walter.
The fat boy hitched his pants on one side and sauntered back to East, incautious, a little proud of himself.
"How much it cost?" East said.
"Man." Walter whistled. "Three hundred and some. A lot. You buy at the last minute, they get you."
Grimly East remarked, "We got to plan ahead next time."
"There's one person," Walter said. "That girl at the doughnut shop. She saw us both. She saw us get into the car. She knew Martha Jefferson, knew she was going to the airport. If they asked her about us, man, she could trip us up."
"But you got your ticket," East said. Beyond considering possibilities. "You're gonna go anyway."
"I'm really gonna go," Walter said. "Let's find someplace to talk. I got to be at the gate in twenty minutes."
—
They sat apart in bathroom stalls, trying to purge, then huddled together at the sinks. Around them the businessmen cast down their eyes, wet their hands in the basins. Walter slipped East a wrinkled wad of bills. East counted it out. Seventy-one dollars.
"You take it," Walter said. "Give me back a twenty I can show. Get me on a bus or whatever. Get me home."
East fed him back a twenty. Now a hundred and fifty-one dollars was his stake in the world.
"Here," Walter said. He gave East a slip of paper from the airline counter. A suitcase tag, a 310 number on it and an elasticized string. "Give me till tonight, man. Then call me. I'll take care of you, man. I swear it."
East laughed hoarsely. "You gonna take care of me? Really?"
Walter stammered. "East. Why I'd wrong you? I mean, you're a bad man now, right?"
East lowered his head.
"You tell me where you're at. Town and address. I can buy you a ticket: plane, train, whatever you want. I can rent you a car. I can wire you money. I can probably find you a house to crash in."
"All right," East said. "I know it. We better ditch these van keys."
"Oh shit. You're right," Walter said. He grabbed a series of paper towels from the wall, dried his hands, and they wrapped the two keys up in them. Tossed it out.
"What happens to that trash?"
"They burn it. Some incinerator somewhere. They'll wind up in the ashes. But nobody ever looks at that shit," Walter said. "Whyn't we get out of this bathroom? I'm tired of the smell in here." Suddenly he was smiling, lighter. "E, man. We finished it."
They walked together into the spilling white carpeted light of the terminal. People flowing around them. East barely noticed them.
"This was terrible," Walter said. "I hated it. Hitting the dude. But I'm glad you were there. It wouldn't have gone on without you."
"I know," East said.
"We did it," Walter said. "I got to go."
"Be careful."
"I will." Walter slapped East's shoulder. "Love you, man."
He slapped Walter back and started walking. He burped; it came up raw, bitter. _Love you, man._ He didn't love Walter, and he didn't say shit like that. He made sure of the bills in his pocket and he looped the luggage tag with the number through the hole in Martha Jefferson's key. And he made his way out toward the garage. But before he left, he looked back. Walter was in the security line, watching him, his pants sagging already below his belly but the ticket clutched in his hand. East raised a hand and Walter smiled back. There. That much was enough.
He returned to the trash can out front and fetched the greasy white bag he'd stowed. He could feel its contents, the two loose weights inside. The popgun he'd shot Ty with and the other, the Glock that had killed Carver Thompson and his girl. The two guns with history. The third, the Taurus that no one had fired, the clean one, he'd left on his brother, tucked it into his pants at the end. He tightened the bag and clutched it to his hip.
"Can I help you get somewhere?" sounded a voice over his shoulder.
Maybe the airport cop had seen him. Maybe it was taking trash out of the can that had brought him hovering. Maybe it was just the way East looked. Or felt.
"Naw, man," he drawled without looking. "You can't help me." Then he was moving again, off into the garage. He was a bad man now.
—
First he drove south. South was away from the police and the van; it was away from Wisconsin; it was neither here nor there. Farms by the side of the road, naked highways, no trees. Sometimes he glimpsed ghostly barns lost on plains, herds of pigs thwarted behind wire.
His eyes felt caked, sticky, like they'd rolled around in the gutter before he'd popped them back into his face.
Brown signs pointed the way to a state park: picnic, camping, river access. Something in his mind started at the idea of a river—it promised crossing, crashing, the cold water dividing this from that. He turned off the highway down the old, tree-lined road.
A cruiser rolled by. STATE PARK RANGER. Cop of a different color is still a cop.
Then he neared a little farmhouse a quarter mile before the wooden park gate. The old house stood gray and alone on a big swath of forgotten land. Weathered past color, the way the gun barn had been. Weeds the height of a man had grown up all around. Two signs advertised it for sale, but sun had baked their red to pink. They'd been there awhile.
He nosed Mrs. Jefferson's car in. The drive circled around to the back. He could almost bury the car in these weeds. There was a space beneath a tree where small gray birds dove in and out. Birds so small they might have been babies. He hid the car in the shadows there, tucked the greasy bag under the seat, and watched the birds until he fell asleep.
# 16.
When he awoke, it was coming out of him like magma, like a volcano, punching a hole through the rock, spilling down his face. The black eye giving forth its flood. The back of the house, the abandoned oil tank, the meadow of weeds, the belly of the tree. A place where people had lived and left. He sobbed until it gave out. But he knew as it did that it had changed nothing. No words, no pictures, no thought. It was just a trick of the muscles, a release of the glands.
His skin grew dry and cold as the car cooled, and it was night when he woke up again.
—
Not afternoon dark but night dark. The stick-on digital clock on Martha Jefferson's dash read a quarter past eight. So he'd been asleep for six hours.
Not a night's worth, but a good sleep. He hadn't had a night's sleep since they'd left The Boxes on Tuesday.
If he was right, it was now Friday night.
He stood and pissed into the roots of the tree. A choir of small voices chirruped in the darkness. Something moved, and he startled. Wind. Weeds and wind. The back of the house, nobody's house.
He waited outside.
—
The car was old but well kept, well tuned. It rolled smoothly on the polished, rolling, back roads. Like the van, it was better than it looked. East filled the tank at a station where two empty roads crossed, each road a story lit by the station's glow. He wiped grime off Martha Jefferson's windshield and dried the runoff with a paper towel.
He was sorry to do the old lady wrong. But he wasn't going back to Des Moines, Iowa—to the airport or anywhere else. That lady had had a good look at him; she had heard him sleep. She had his measure. There was no guessing the things she knew about him, or what Walter had said while he snored. It was not impossible that after everything, Martha Jefferson would be the one who called him to account, for sleeping in her backseat, for stealing her car away. So he polished her windshield clean, and the cold biscuit sandwich he bought out of the cooler, he ate outside the car—no crumbs, no wrapper.
He switched on her radio, but it was all static. Loath to disturb the station knob where she had it set.
Each time a junction opened the four directions for him, he went away from home. He went east.
He passed signs for Chicago at four o'clock in the morning. For a moment he considered it. Chicago: everyone knew that was a town for gangsters, Michael Jordan's city, full of people and ways to make money. He had guns to travel by. But the gray-yellow light of the city under the clouds brought back Michael Wilson's crawling into Vegas. _Just a taste._
The billboards advertised everything, promising everything.
The long, steamy reach of the city flanked him for miles, gray factories lit by orange bulbs on his left, huge skeletons of iron that had held trains or loaded boats or moved the road over rivers or launched rockets into the sky.
In Indiana, on a larger road now, he had to stop with other cars at a booth and take a ticket out of a box, like at a parking ramp. He sat still and read it. It was going to cost him four dollars somewhere down the road. Someone behind him honked.
Booth, camera, stolen car. Keep doing what the others do.
Then the last signs of the city dropped away, and he was back in darkness, the highway narrowed down, the starry blocks of houses placed just so behind the trees.
At sunrise he'd reached the end of the state. A gauntlet of booths blocked the road; cameras everywhere now. He pulled in slowly where it said CASH and surrendered his ticket. "Four sixty-five," the woman said.
He'd barely heard humans speak since he'd left the airport the afternoon before. A sign advertised the last exit before a new toll pike. It was Ohio now. He took the little road south, Highway 49, then again turned east.
About eleven o'clock in the morning, in a small town with a sagging water tower, where the sun had quieted itself behind clouds, he had gone far enough in Mrs. Jefferson's car. His tank of gas was nearly gone. He had crossed three state lines.
On the main street, he located a little corner police office. Meters out front. He stopped in a space and made sure his money was with him. He took one gun out of the bag, rolled the bag shut on the other, and shared the guns out between his two front pockets. He left the key in the center ashtray and left the car unlocked.
For a moment he felt a panic. Was this an error, giving the car up? Walking out again into the winter?
One-hour parking. It would be found.
Ty would have pulled it behind a building and set it ablaze.
Quietly he dropped a quarter into the meter and walked away.
# 17.
Out of town he walked on tipped, sunken sidewalks, passing bacon-smelling restaurants and used-car lots laid out precisely, wheel facing wheel, headlights polished and aligned, everything eyeing everything else.
In a low gray midday, the windows of the buildings were mirrors, glass glaring back, insides invisible. The lots lay squared and similar, and seams and borders separated every this from every that. A fence, a rail, a line of grass or littered shrubs like the alley where they'd waited for Martha Jefferson. Or just a line of curb coated in tar. Each place marked the line between this and that, here and there. Each one he passed ticked off the passing from then to now.
He heard the motor first; then tires scratched the pavement behind and stopped at his heel. Little fat-tire police car with the windows rolled down. A moustache, bristling.
"How you doing this morning?"
East stopped. Helplessly he imagined the parking meter—it had to be still ticking. Not that. An age of dealing with cops had taught him what not to do.
"Fine," he mumbled.
"Going anywhere special?"
"Just a walk."
The cop looked skeptical and East squeezed his shoulders together, mute.
"Know anybody in this town?" the cop added. They asked a question, then stared into you like a fishing hole, like sooner or later the truth would swim up.
"Not really." East stood still, hands down in his pockets, where now there was only a small fold of bills and the bent California shapes of some guns.
The police radio burped and whistled. The cop glanced at it, then stilled it with a hand.
East gazed down the road and said, "You got what you need?"
The moustache stared through him. Like a gaze you could actually feel, could round on if you wanted to sleep tonight on a concrete floor.
"You take it easy," the cop said. Not a farewell. An instruction.
East volunteered nothing. Behind him the engine flared. The tires didn't crawl but cut a hard U-turn. So somebody had sent the cop out just to take a look at him.
The town smelled like corn cooked too long. Up the road, the two-lane broadened out as it sped up—flat shoulders spreading, cars at fifty or sixty, tossing up black grit and cinder tornadoes that bit his ears. He burrowed down in his sweater. No trees to hide behind, no woods line to trace—this would be straight country walking. Out where anyone could see.
A mile of level fields. Stalks sawed into splintered bits, scraps of wind-trash bright in the ruts. He heard Michael Wilson's laughing voice: _Country._
A sign said it was four miles to the next town.
—
At a junction gas station kept by a teenage girl, he bought a long sub sandwich and a stocking cap. The caps said BROWNS and came in two colors, orange and brown. He chose brown. He chose ham. Inside him, his stomach was a drifting ship. But he should eat. He knew that.
By the door, as the nervous girl prepared his sandwich, a map with a yellow tape arrow showed YOU ARE HERE. No Wisconsin, no Iowa, none of those places on a map that meant what they had done. Just this state, just Ohio, the town where the brown car slept somewhere to the west. He didn't remember the name of the town. He just wanted away. This was running. From the police and from the spinning inside his body, the yaw.
The girl patted his ingredients together through plastic-bag gloves. When he asked for more tomatoes, she took her phone out of her pocket and laid it on the chopping surface, near onion half-moons, within reach. Her movements sharpened: she pressed the sandwich and sliced it with the quick blade, and then she was done with him, everything but the making change.
He knew he must look horrible.
He took his bagged sandwich and used the little restroom. Everything looked like it had sat for years in rusty water. He would have liked to take a shit, but his insides were dry. The fountain, _Halsey Taylor_ , was just a husk.
When he stepped out, the girl was half-hidden in the back room, on her phone. She saw him and disappeared completely. He would have liked a cup of water.
Outside was a picnic table made of concrete with stones inset, under a hard steel umbrella. It was winter-cold, and the umbrella had once been white with a large red logo, but now the red had faded in the sun and was mostly rusted, and the rust had dripped down and begun to stick at the edges of the stones, like cuticles. It was the least comfortable table he'd ever seen, so he ate the sandwich as he walked, shedding bits of food, bouncing them off his trousers and onto the gritty shoulder of the highway.
In the bottom of the bag, a pawful of napkins, maybe twenty. A long slash of orange in the southern gray sky like a headache, like a crack.
The land changed, from flat and open to woods brooding over a series of prone, steplike hills. People here had hollowed out deep spaces—a long drive back from their armored mailboxes to their houses in the trees. Some were big and pointed, full of shapely windows; others were little boxes made of cinder-block. In front of or beside their houses the people here arrayed their cars and pickups and boats, and some kept dogs on leads or in stockades.
On the shoulder where he walked, East felt the eyes of every driver on the road. He would have kept to the woods, but the last thing he needed was some person's dog latching on.
Here and there he saw a horse or cow. Or raised boxes in a little cloud. Meat smoking, he thought—or something. Then he got a good look and saw one clouded by bees. Bees: one time back when he still went to school, they'd shown a movie about them. Little colonies, getting along, everyone doing for the queen. Some teacher tried to start a hive up on top of the school—got beekeeper suits and everything. But then a boy had been stung fifty times and had gone to the hospital, and that was the end of lessons with bees.
—
He kept walking, getting far from the little town, the little car. That car was the bone that connected him to Iowa. In Iowa, the van was the bone that connected him to Wisconsin. Somewhere the judge and his daughter lay, and somewhere his brother lay, unable to say his name. Bones. Walter and Michael Wilson, on the ground in LA or anywhere. They could be anywhere. Walter would keep it airtight. Michael Wilson: all East could hope was that he stayed out of trouble, stayed free of anything where telling what he knew was something he could bargain with.
He wondered who was watching. Not about the police—of course they were. Of course they looked funny at every step he took. He knew how to walk around the police. Even carrying the gun in his pocket, he didn't fear being stopped by them.
It wasn't just the police that worried him. It was the stop he wouldn't see coming. The little town centers had been swept and were quiet, older white men catching him in their eyes, holding him there a moment: _Single skinny black boy walking nowhere. Had a cap on._ Enough to remember him by. Enough to provide a description, or to find him again. They said "Hello there," and East nodded, walked by. Grateful for daylight.
It was on the unswept edges of town that East noticed the litter of the life—the little circles of stamped-down wrappers, abandoned bottles. He veered off his line on the shoulder of the road to look: yes. The vials and pinchies, the little knots of activity, torn-off matchbooks, plastic ice cream spoons scorched half through. _Firing up a plastic spoon,_ he thought, shaking his head. Over in the blacktop refuse and colorful trash that waited outside the border of the lot for nothing, splotches of vomit and the warm, sad smell of piss.
Right out on the edge of the main road they were doing it. It was these people he needed to avoid. He had had plenty of friends who'd used, who'd become deep U's before they were teenagers. He could resist them. But he had no friends out here. He wondered how young they would be.
—
The old lady's face returned to him, her pointed voice, and he heard her talking to Walter as he walked in the cold. He saw her looking at him. Or sometimes it was the face of the Jackson girl, leaking blood and going to sleep on the street, or the screaming mouth of the girl in Wisconsin. Her body, under the toppled suitcase. All of it inside him now, screaming to get out. Everything.
Another mile, he thought. Another mile.
As he passed a gas station, two men asked did he need a ride. He looked up, but then one was talking to the other, even though they were both looking at him. Two of them, one of him. He put his head down.
His sense of the direction was partial. But the winter sun kept slipping to the right. Soon he could walk under the cover of darkness.
A while after every intersection, the highway sign said EAST. So he kept on.
—
Inside he was used up, but his body pleased him, as if it were a specimen he were observing. Even after hours of walking, his body was light. When it rained lightly, the rain did not chill. He breathed it, absorbed it with his skin; it sated him, and the wind dried what it left in his sweater. The wind pushed him. His legs, so soft after driving, came back warm and good and hard. When he bent to retie a shoe, they quivered like engines idling.
How long he had walked he had no idea. It grew dark and colder, and he would have to decide. It was going to get a lot colder. He could see the plants shrinking, small plumes from chimneys. A regret hit him: couldn't he have driven the car one more day? Couldn't he have driven this all in an hour?
No. He had to keep telling himself. The car, he'd had to leave it behind.
By now it had a ticket. By now it was in the computer. Unless it was Sunday and the meters weren't enforced. This too he wondered about. He forgot what day it was.
—
Signs promised the next little town a mile out—churches, Neighborhood Watch, a billboard for a Chevy pickup built in America. The trees walled off the neighborhoods from the last ruined fields. The first liquor store, just outside the line. The streetlights began just beyond it, windowpane-yellow under the trees.
As he walked past houses again, his body came back into focus, took on color and shape. People crossed the street—walking together, older, twenty, dressed loosely or lightly, jackets open or sweaters and scarves. Dressed to walk outside but not for long, drinking from paper cups in the cold, disappearing into houses or coming out, holding hands, worrying, laughing. East stared at the big old houses with flags and open windows, even in the cold. At last a large stone sign the size of a bedroom wall announced it: it was a college. These were students, like Michael Wilson once was.
He couldn't stop looking. The way they walked—they met and bunched on the sidewalk, sat on the porches. They crossed the highway carelessly, and cars slowed for them. So sure of their world.
He broke from the road and followed a group of six that was cutting across a parking lot. Shiny sedans like he hadn't seen since they'd left LA: Volkswagens, Acuras, Hondas, Hondas, Hondas. Two students split off toward a building with a hundred long, lighted windows. East stuck behind the larger group. Three boys and a girl. They crossed between buildings and across a green field. Like kids rambling a neighborhood. At last they approached a large squarish building. East caught up and followed them in. A gymnasium: a desk said ALL VISITORS MUST SHOW ID, but the boy at the desk was on his phone. East ignored him as the others had, and the boy never looked up.
Inside, the four students turned left into a noisy arena: girls playing volleyball under blue-white lights. East followed the polished hallway. He knew what he wanted: MEN'S LOCKER ROOM. Inside, on a toilet, he sat and waited until he stirred inside, that ship, that load, until he could rid himself of it. Then he followed the sound to the showers—quiet lights, two older men cleaning off, pump tubs of soap and shampoo on the wall.
He opened an empty locker and stripped off. He hung the pants with the two guns gingerly. If someone were to find it now. Well, if someone were to find it ever.
But his money he drew out and took with him into the showers.
The tiles, square and divided by the lines of white grout, rectangles from here to forever, were gritty under his feet. That was his feet's grit. His body was caked with it, the grime of a week, all the way back to The Boxes. His body reddened in the scalding water and hurt the same inside and out, softening like meat under a hammer. He kept his head in the stream and tried to wash it clear.
—
In one locker, a forgotten pair of clean socks. A worn red T-shirt lay atop a locker. It had a dusting of sawdust, but that shook out. The breast read CHAMPION.
He wadded up the last bitter Dodgers T-shirt and stuffed it into the trash. Farewell to baseball. Farewell to all that white people's love.
His shoes, when he tied them again, seemed ragged on his feet. Dark drops on the toes and laces—he hadn't noticed them before. Spots. They could be anything. He padded the hallways, keeping quiet. No one stared, particularly; no one ejected him, yet. Weight rooms, a pool where the very air looked blue, the big gym where the game had ended. The crowd had gone, and he entered and drank from the fountain. A small man in a blue work shirt was motoring one set of stands back into a stack against the wall. A team of younger assistants removed the volleyball net with small, flashing tools, then carried off the chairs and official's tower. Efficiently the blue-shirted man swept the bleachers opposite, stopping here and there to polish with a towel anchored to his belt, then retracted them as well. The whole room changed. It was, to East, an interesting thing. He sat and watched against the wall, the two guns lumpy and hard against his thighs.
At last, the small man locked shut a storeroom, working off a large ring of keys, and then his assistants disappeared. Two black men came in with a basketball, taking the court at one end, dicing, lots of leaning and hand checks. _Damn,_ they laughed. _Damn._
One of the two left, and the second noticed East lingering by the stacked bleachers. "What's up, young man? You looking for a game?" he called over.
East shook his head. But the player stepped closer, scrutinizing. Something changed in his voice. "You all right? Need something to eat?"
East tried tucking his head away in his sweater. He couldn't begin to talk to the man.
"I can slide you into my dining hall. Get you some help if you need it."
East shook his head again, stood, crept away. The guy seemed okay. But nothing was enough to trust him. And no sense in just bringing the man his trouble.
—
He slept the night in the gymnasium. Collapsed, the stacked bleachers made a tower of thin shelves on their backside, each one twelve inches below the next. East slid himself onto one of these shelves, waist-high. The next plank was just a few inches above his nose. Like a tool in a drawer, he thought, like a body in a morgue. It had been more than twenty-four hours since he had slept, but even in his dreamless sleep he knew that he must stay quiet.
—
At five in the morning East was awake. The gym must be closed, he judged, but he lay in his safe slice of darkness, mind and body still, until the first sounds careened in the hallway: a mop bucket, the metallic knock of deadbolts. His fingers straightened the guns in his pocket, and he waited until he heard the first voices, early comers. Then he slipped out of the bleachers. He considered taking another shower. But it would only be luxury. He could not stay here in the gym forever.
He drank, used the bathroom, then saw three people on the way out of the building, enduring the briefest of greetings. Frosty air. Except for its watery lights, the campus was largely dark. He felt good after the long night's rest. Eight hours, nine?
Ninety dollars and change left after gas and tolls and the food of the first day. He bought a bagel with cream cheese at a walk-in place on one of the side streets. A bagel: three dollars and eighteen cents. His first college lesson.
—
Today the road rose and fell. The trees twisted low and misshapen, as if storms had combed them many times. A few dark apples still dangled. He hungered, but he dared not walk into the orchards. Once he found an apple on the side of the road and picked it up. Nearly perfect, like bait in a story for children. He put it in his pocket with the gun.
He noticed tags on the backs of the signs across the road: C+W. Some of the paint weathered, some still glossy fresh. On most of the signs. Then on all the signs. He turned around and looked behind the signs on his side. Tagged as well. He was walking through territory.
A figure came walking toward him on the shoulder gravel half a mile ahead. At a shorter distance he decided that the figure was a girl. He sank his head low into the collar of his sweater. Avoided even taking a good look. A mass of brown hair pinned back. She had a down jacket on, but she was shivering too. He felt relieved when she was by him.
One house caught his interest. He saw that it had burned from the center out, not long ago. The bricks were still smoked black. The roof's peak had given way to something like the mouth of a volcano. Heavy smudge still bloomed in the air. Across from the front yard, a school bus, still caution-yellow, but its long yellow sheet metal was marked in primer gray: CHRISTIAN WOLVES. Then the plus sign. Like a cross.
Like the van. He stamped it in his mind, then crossed to walk on the other side.
—
Slipping. A light rain had done little more than grease the roadside. His joints were webbed with red exhaustion. He had not reckoned on the cold. Toward midday he crossed a small junction, not even noticing it until a truck hurtled past.
He touched his pocket. The apple was gone. He could not remember eating it or throwing it away. The guns were still there in the loosening pockets, riding the bruises they were making on his thighs.
The next town was two miles off, and he thought that if it had a store, he would buy clothing. If it had a place to rest, he would rest. He imagined Walter walking with him, his voice, his hypotheses: what this closed factory had made, how those trees were planted, how this kind of church felt about black boys. But Walter could never have walked this far. Walter would be home, wondering why East hadn't called. And worrying about that old lady, Martha whatever. She'd be coming back on the plane, East recalled. Perhaps today. And Walter would be beside himself with guilt over it.
He wondered if he was far enough. He felt far. He felt lost. But if there were such a thing as far enough, it wasn't a place you could walk to.
The next town was what he settled on. Far enough was going to have to be here, at least for a while, even if he hadn't glimpsed it yet.
—
This town wasn't much. A couple of places by the highway, closed or indifferent, cigarette butts and harvest stubs blown clear up to the doors. Telephone books rotting in split plastic bags. East glanced up with effort at what remained of the signs: TIRES, VACUUM REPAIR. It didn't matter now. A nightclub that once had some style: a splashy sign, now a skeleton, and exterior walls studded up with white stones like a dinosaur's armor. Some sort of weird, fenced yard like an impound lot that said SLAUGHTERRANGE.COM. HELP WANTED in soaped letters in the window. A matronly farmhouse across the road disapproved of it all.
He turned down the side road and found a main street parallel to the highway. There, an old grocery, a post office full of cobwebbed shipping boxes, a pawnshop. Two stores said ANTIQUES but were falling apart themselves. The windowsills on the hardware store were rotting. A doughnut place seemed to hold the only people now, and one bar, blinking BUD, where they would be tonight. A Laundromat, machines scraped out and yawning.
And the little motel: Starlight. Two little wings of ten rooms each spread from an airy center office filled mostly with dust. Curls of neon tubing still clinging to the sign. It was the sort of motel you found on big north-south streets in The Boxes, but open there, sprawling, a clientele there to use or drink or hide, who sometimes just lived there for decades, disapproving of the others, fallen oranges rotting beneath their parked cars that never moved. But here in Ohio, the Starlight was empty, bleached sun-white and then dusted again, front door padlocked though the sign still said OPEN.
Town wasn't much. Clearly they'd run the highway just north of it at some point. But the highway had failed to keep it alive.
_Boy, this is why you get on the plane,_ he thought.
Outside the Starlight Motel, he climbed atop a concrete planter full of poisonous-looking dirt and surveyed. Relieved now of its relentless moving forward, his body cracked with want. A passing moment of sunlight lit the houses briefly. In the offing was a church, a shingled thing jumbled together like children's blocks, a big cross, gilt and dirtied.
Two days walking in the air had thinned out his stubbornness. But the stubbornness that remained was choosing. He had chosen. _This is where you said you'd stop,_ he reminded himself. Small, and no people out: everything he saw to hold against the town, to walk away from—to flee it, in fact—were reasons that, standing in the fading light, he steeled himself against.
A pickup truck with five kids in parkas in the back, sitting packed together, rumbled by slowly. All five kids turned their heads to look. Their mother gave him a single glance and blew a plume of smoke out her window, then flicked the lit cigarette butt after it.
He didn't even know the town's name.
—
After an hour, when his eyes had measured the town and his body had stiffened with cold, he climbed down from the planter awkwardly and walked to the grocery store. Closed. But oranges and cans on racks inside: at least that. At least the store was alive. He scrutinized the darkness inside, then backed off and read the sign. CLOSED ON SUNDAY. He had never heard of a grocery store being closed on Sunday.
Maybe, then, it was Sunday.
Next he walked to the doughnut shop. It was emptier than before, perhaps. But open. He paid his money for two large fried apple fritters, then added a cup of hot chocolate. He was beginning to treat the cold as a permanent adversary. The hot, doughy air of the shop was worth the people staring. He stood numb, breathing the steam off the scorching cup.
He used the bathroom for what he hoped was not too long. Hot water at the sink up his forearms, over his face, around the back of his neck. Carefully he dried himself. The clatter out by the main highway drew him back that way. The weird impound lot had, in the last two hours, filled with trucks and cars. The building there resembled a small barn backed up to a clumsy, bulldozed berm a story high, blocking off view from the road. A few dim lights burned cold on poles. The clatter back there was shooting. It had started just before sunset. He had heard the first shots from the doughnut shop as he stood there eating a fritter, dreaming on his feet. The first burst made him spill the rest of his hot drink on his fingers. Triggered, not automatic, four or five shots in all. He looked up, shaken: the locals in their booths had looked up too but were already back to their doughnuts.
Shots rang again as he stood at the mouth of the lot. The house. The wondering face of the girl. Involuntarily his body ducked. He looked around in the cold air—one split of dying orange in the dulling sky in the direction he'd come to know as south. A small car sat glowing its parking lights near the barn like a resting hog.
The shots weren't right—they didn't sound the same way. They lacked the knock a gunshot had. A _tump_ , a different bang—he didn't know how to describe it. No houses like in The Boxes to echo off. But the same rhythm, the exchanges of fire, he could hear that—the conversation. The old music of his streets.
A target range? But there were shouts from inside too, and scrambling. An occasional yelp.
Somebody _burning_ their ammo up, he told himself.
He slipped closer to the barn, finding a place to listen.
—
After he'd seen half a dozen men come in or out, singly or in pairs, carrying loads in heavy canvas bags like athletes used, or hunters, he got the nerve to open the door. _Slaughterrange._ An electronic beep announced him, but a noisy bouquet of Christmas bells rang too, duct-taped to the back of the door.
Long tube lights hung from the rafters hissed and flickered. Maybe the building had once been a garage: the floor was concrete and dipped slightly toward two long, steel-grated drains. The front half was given to two carpets, ratty and colorless, each of which anchored a sofa, a chair, a skid-marked coffee table, and a boxy TV. The back was a counter, antique with glass windows. Over the counter hung a range of weaponry, and a young man stood behind it.
"Hello," said the man, standing very still. He had a strong gaze and a weak nose, a nose that glistened and twitched like a rabbit's snout. It was a U's nose, East recognized.
East said nothing. He studied the guns over the man's head. Large and gripped out, sniper guns. He'd never seen guns like this in plastic bags before, like hairbrushes at the drugstore. They were not real guns, but he did not know what they were. Some of them looked like fantasy, outer-space guns, colored green and orange like children's toys.
"Can I help you?" said the man, looking East over. East recognized the small, secret fidgets of his face. He was, maybe, thirty. Behind the counter he would have a real gun.
"What sort of place is this?"
"Best paintball range north of the Ohio River," the man said automatically, as if it was a phrase he'd been paid to remember.
"Paintball?" said East.
The man reached and drew out a pearl of orange. He tossed it to East. "That's a stale one," he said. "That one will hurt when it hits."
East looked at the faintly luminescent nugget in his hand.
"What are you here for?" said the man. "The job?"
East shrugged.
"Perry's not here tonight. You might catch him in the morning. He's in charge; he'll see you about it."
"What is the job?"
"It's, like, assistant. Like a watchman."
East detected the accent, the harsh sound inside the words. Like a movie spy.
"I can do that," East said.
"This is a job for a grown man. You have to stay late."
East said, "I can do a man's job. I can stay late."
"What are you," the man said politely, "thirteen? Fourteen? In school?"
"I'm a man," East said. "I don't go to school no more."
"Ha. Good," said the man. He had a telltale wetness in his nose, a bubbling. Then two men came in with canvas bags, and East watched them pay and take the tubs of colored balls the man gave them and transfer them into their guns and their own containers with funnel-shaped loaders, perched on the edge of the sofas, mumbling profanely. At last they returned the tubs and moved up a stairway beside the counter and out a door that went to the back of the building, the berm side.
"Can I look?" East said.
"Not tonight," said the counterman. "Come back tomorrow." At first he'd seemed friendly, but now he'd seemed to have changed his mind.
In the parking lot East sat far apart from the other men's cars and trucks and listened to the shooting music fill the air. Dull night. No stars. He found himself starved for sleep. The strange hours.
He awoke with a gasp, leaning against the building. He had been dreaming of the van, someone terrible peppering it with bullets: Michael Wilson, somehow, but with Sidney's face, firing out the mouth of the cabin in Wisconsin where the daughter was the dead Jackson girl aiming at them from beneath the suitcase. There was no shaking this. There was no _far enough_ to settle his mind at sleep.
Leaned up against the wall was a roll of pink insulation. The wall type, fiberglass. He checked it, shook it—dry, no mice or vermin.
He found the largest truck in the yard, a jacked-up Ford with a work bed. He fed ten feet of insulation in between the huge back tires. Then he crawled in on top of it and fetched the rest of the roll around him, like a blanket, like a wrapper, sheltering under the huge round differential with its burnt-syrup smell. The cold and the lights muffled down, and he slept to the knocking of the shots in the yard beyond.
At one point in the night he woke and saw, through the hole in the end of his pink swaddling, the sky. The truck above him had gone. But he was still there.
# 18.
The morning light startled East, and he struggled to unbundle himself, unwrap, to find his feet and stumble free. Pink strands in his mouth, drying at his lips. Pink fleece in his hair. All his skin crawled.
Then his head cleared. He went back, picked up the insulation, and rolled it carefully, tightly. Leaned it back up where it had been.
He walked down the highway, furtive in the cold air, and revisited the doughnut shop, for its bathroom as much as the food. He washed in the sink with the palms of his hands. A spill of salt at the corners of his mouth, streaks of wet on his sweater. The black eye slowly reshaping itself. His muscles lank and dog-tired under his skin. He could not bear to look in the mirror at what was left.
Purchased two doughnuts but could not bring himself to sit in the warm little shop, among other people. He went instead down the street, to brood by himself outside the little closed motel.
—
When East opened the door again, there was one person sitting in the big white building of Slaughterrange: an old man, pink like a ham, larger than the bar stool he occupied. Sandpaper bristle of ginger hair. He was holding a small piece of machinery in his left hand and rendering it with the screwdriver in his right.
"You the one Shandor said was looking for a job?" he bellowed without looking up.
East sized the man up, and he stopped short. Six foot five, three hundred pounds, maybe. The sort of man who was used to moving things. He lay down the tool and the chunk of beaten metal, and he brushed his hands on his Carhartt overalls.
"Was that you lying in the yard last night, son?"
East did not lie. "I fell asleep."
"Cold," said the white man, "to be falling asleep, under Tim Crane's truck. Where you staying?"
East said nothing.
"You the one, then, that wants the job?"
East nodded. Kept his eyes on the old man's eyes. There was something wrong with them. Sticky. Not high, but lagging a bit.
"I got to ask you a few questions."
East said, "I know."
"Name?"
"Antoine."
"You had a job before?"
"I did security. Two years."
"How old does that make you?"
"Sixteen."
"You ain't sixteen and done two years security, son."
"I got a license," said East. "It says sixteen."
The big man shifted on his stool. He did not ask to see the license. He was not in that sort of pursuit. "You get high? You"—his voice became bitter, humorous—"tweak out?"
East shook his head.
"You in a gang?"
Shook his head again. It wasn't necessarily a lie.
"Christian Wolves? Any other gang, known or unknown? You got tattoos?"
East said, "No."
"Do you mind showing me?" the man said. "If you would lift your shirt up."
East held up his sweater and the red shirt beneath so that the old pink man could see him, ribs and the two black points on his chest.
The man was embarrassed too now. "Higher," he said. "I got to see your collarbones. That's where they put them."
"Who?"
"The Wolves. Their tattoos. I don't know," the man said.
East stripped his shirt off all the way and turned once, a dull outrage marking his face from inside. But the man, when he turned back, was looking away, with distaste. Maybe for East. Or maybe for having had to ask.
Maybe his willingness to be seen was all the man needed to know.
"Good," said the man. "I can show you how to work here. But I can't show you how to work. That, you got to know already."
"I know it," East said.
"I'm Perry Slaughter. I would be the owner. Excuse me." Now the big man seemed to be wilting in on himself. He turned away and bent down behind the long wooden counter with the thick glass windows at the front and top. He came back up with a thin rig of plastic tubing, which he fit over his ears and into his nostrils. For a moment he stood, taking hits of something through the tube.
"You ain't happy with me, I can move on," East suggested. "I don't need this job."
"Exactly why a person asks for a job," Perry Slaughter gasped below his tube, "because he don't need it. No, you're fine, for today at least."
He peeled off the tubing and stuffed it back in the drawer. He regarded East suspiciously over the bristled pink of his cheeks. "I take a little oxygen now and then," he admitted. "The good shit."
—
There was downstairs, and there was upstairs. Downstairs was the register and the counter and the cabinets full of paints, the front room, the bathroom. Upstairs, out the back door, was a covered deck with lockers and a four-man air station, chrome and shining and eager to hiss out air, and the sidewalk landing atop the plowed-up berm with its observation rail and lifeguard chair where someone could watch over the range.
The first two mornings East swept the range, raking up heavy litter and the clusters of paints he'd find, burst or spilled or trodden in. He'd board the stranded, wheelless school bus and the jeeps mired and moldering in the center of the range, picking up chunks of new-broken glass or metal. On foot, he pulled a light, knobby-tired tote behind him, emptying it into a Dumpster in the parking lot, which in turn was emptied by a black truck that stopped by and lifted the Dumpster above its head like a trophy, shaking it, the invisible driver jerking the hydraulic levers. At first Perry Slaughter had East taking directions from the other boy: upstairs or downstairs, what he should do. The other boy, Shandor, showed East the blaze-orange coat and helmet for entering the range when there were shooters, what the protocols were for sorting out problems or escorting the injured. Shandor showed him how the register worked, how to charge members, how to charge guests, how to sell time, paints, how to rent guns and headgear and check them back in, how to take a credit card, how to refuse one. Shandor showed East where the bathroom was and how to mop it out. The other boy seemed to prefer the outdoors, at least for the first string of warm afternoons, but then it no longer grew warm in the afternoon, and he left East out to look over the range.
East did either without complaint.
The men came every day, especially Sunday, new and curious, or regular, rumbling or limping, tires or boots crunching the frozen peaks of lot mud. They rented their guns or toted them in nylon bags with mighty, treaded zippers. They paid credit or they paid cash—money clips for the ones still working, pads of secret cash like squashed drink cups for the ones who were scraping by, who were no-good squandering, who were pilfering it out of a mattress or a mother or a wife. They came singly and in groups. There were posted hours, but they came before and they came after.
They paid entry and rentals, and they bought paintballs. Sometimes they bought gear—guns, helmets, pads, bags, military goggles. They bought drinks, crackers, beef jerky, chocolate bars. They lingered downstairs and stared at football on TV, from a skeptical distance or joining others on the sofas. Or they hurried up the stairs and out the back, to suit up and leave their wallets and phones and work shoes in the lockers there, pocketing the locker key or pinning it to a hip. Sweatpants, coveralls, track suits, dirty jeans. Then they went out to play, to shoot one another.
They left litter: the spent paints, the husks with their brilliant yolks. The blown bags and wrappers, the coffee cups, the tubes of liniment, the popper bottles, the bandages. The gum they chewed, the tobacco they dripped, the cigarettes they ground out, the gloves they lost. They left papers from the state and from their loans and from their joints and their wives. They left the _Plain Dealer_ and _Dispatch_ and _USA Today_. They left the penny-saver and the auto ads and their gun magazines and their computer-printed directions to here or somewhere else.
They scattered and held, playing their battles, dark shirts and light, red bandannas or blue, stalking one another with one paint or another. They scrambled through breaks in the land, burrowing themselves below the bulldozed hillocks and against the dead trailers and trees, the fortresses of fallen tree trunks, the one length of fieldstone wall older than anything. They hid in the school bus, or sometimes they swarmed in or out of it like a hive of bees. They revered the two surplus army jeeps painted army green mired near the trenches at the far end of the range, each with its white five-pointed star.
They scrambled and sighted, scrambled and sighted. Sometimes the men shouted at one another, coordinated, vague military directions, used phones, used walkie-talkies, working organizations in the brush. They formed teams, crews, alliances, factions. They turned on each other and then won each other back. Sometimes they ran singly, eyes blacked, sleeves peeled back in the cold, panting, waiting, shooting at anything or anyone, guarding their vantage points jealously, their long guns slung spanning their chests rigidly, extended skeletons.
They died, and they waited on the sidelines, rubbing bruises, watching the others. They died, and they came back to life.
—
In the beginning East got sixty dollars a day paid in cash—no discussion what a day was or how long it could last. It did not matter. Before the second week was out, Perry had made it a hundred. By then East knew the range. He knew the waiver forms and how to file them and how to talk back to someone who'd twisted an ankle or caught a paintball in the neck and now was angry, now threatened to sue. Soon it was East being asked the questions about how to clear a jammed barrel or punish an offender. Shandor had been there for four months, but Shandor did not work as hard as East or as much. And some days Perry's instructions were to tell Shandor he was not working that day.
Maybe Perry had begun to trust East, from seeing him, catching him working when he popped in for a few minutes now and then. East worked hard. Or maybe it was that Perry had never liked Shandor in the first place. Shandor was polite and handsome but evasive. He dabbed at his nose constantly. He could not remember, made things up. He had a thin, rabbity nose that was always wet with something.
At last one Monday, East asked where was Shandor, was he sick, knowing that wasn't the reason.
Perry looked away. His loud bray had, this morning, gone quiet and upset. "All right. I'm going to tell you. Shandor won't be back."
East raised his eyebrows.
"I put him in my truck and took him down to Columbus."
"Columbus? To the college?" He'd overheard the talk on the sofas, the games on TV.
"The university?" Perry said. "Nothing to do with that. It's what he wanted. He thought it would be a good place to start fresh. He had me turn him out on the street with his little suitcase and a handful of my money."
He shook his massive head and his body wagged along.
"If somebody were to ask after him," Perry concluded, "now you can tell them where to look. But I can't imagine who would, outside of Hungary or wherever."
"Why not?" East said.
"People don't connect with someone like that," Perry said, "who ain't from here."
"I ain't from here."
"Yeah, well," Perry said. He counted out twenties for the weekend and pushed them across the counter.
East unlocked the padlock that secured the back of the cabinet. Time to start the day.
"So, you gonna hire somebody new? Cause it's hard to watch the back and front the same time."
"I will. Right away," Perry said. "Never really took the HELP WANTED sign down. You need a day off?"
"I'm fine," said East. He wasn't sure of the calendar, but he thought he'd worked fifteen days in a row. "It's okay."
"You tell me if you need a day off," Perry said absently.
Perry didn't hire anybody new right away. But he began staying around. East figured he liked it: liked seeing the men and peppering them with his questions. Liked muttering in their presence and then holding forth.
Listening to Perry talk, East learned about the place. It was Perry who'd taken his wife's family's old field and plowed up the berm, Perry who'd backed the fences with sheeting to keep the noise and stray paintballs in, Perry who'd gutted the old barn and built the store inside. Plus the upstairs landing, where, through the long and dim afternoons until the lights came on at night, East oversaw the range.
Atop the berm, in his lifeguard's chair with its drop-hinged shield of spotty Plexiglas, East surveyed the men swarming like squirrels across the acres. Scrambled and sighted, scrambled and sighted. East admired some of the players, the small ones, the ones who shot less, who perched and waited, content to stand as rear gunners, hiding, conserving themselves. As Perry had told him, East ejected players whose behavior annoyed the others—cheaters, head shooters, overkillers in groups where overkill was unwelcome. Anyone with smuggled paint, with stale paint that did not break on contact, that bruised and bounced off. He protected the customers and protected the business.
Some of the men slighted East at first, or ignored him. But most came to accept him. They saw that he was always there. In the lifeguard's chair over the railing he was quiet and watched patiently, never hurrying them. They couldn't get much out of him. But he nodded once carelessly when a player asked if he was from out west, and that news got around, became the foundation of a dozen tales about who he was. He was no schoolboy. He was a runaway, an escapee. He was somebody Perry was sheltering, somebody's illegitimate son. He dealt with them directly and calmly, looked boys and men both in the eye, ended problems at once, kicked people out fairly and quietly if they had broken the rules, whether they were one-timers or regulars. He cut people off the way a bartender would. He seemed to have no fear and no body temperature: he sat out on the chair in a cotton shirt when everyone else was wearing a parka.
Some of the younger ones, kids buying paints out of their part-time jobs or the money their parents allowed them, idolized him. They called him Warlord, called him the Ancient, called him Gangsta.
The night the Buckeyes beat Michigan, a carload of Michigan guys had stopped at the range and rented out guns and snuck in a bunch of store-bought paintballs that were as stale as rocks and left bluish bruises where they hit. Then they lit a player up, a regular, mercilessly, four of them on one, fifty or sixty shots all around the back and shoulders, even when he was on the ground. East hadn't hollered or blown the air horn. He'd just switched out the lights. Then the regulars had all the advantages, knowing the range like a familiar block, like their own yard, and one of the Michigans had his teeth loosened with a gun butt. In the dark it was just an accident, and they got loud with East, and East faced up to all four, took their guns back and saw them, bleeding, out the door. That night East let the locals stay and play the TVs till five in the morning. More men arrived; they brought cold beer, they brought pizza, they watched the game again on a 1:00 AM rerun and sang through it.
They asked East to come and sit, to have a beer, but he mopped up and stayed away, keeping business.
Maybe he wasn't of age.
The shooters were white, all of them. Some of them had to forget what color he was before they spoke to him. But he had made his place. He was just Antoine, the new boy Perry had, and he was all right. Better than Shandor—they had not liked Shandor. He was something, Russian, Ukrainian, maybe, not from America. And Shandor was an addict. He was always looking past them, looking at something else. It made them uneasy, the men who came here, drank a few beers, went paintballing every day. Antoine, whoever he was, was American. Antoine looked them in the eye. He knew what he was doing, Antoine.
They respected him, stopped watching him all the time. But he never stopped watching them.
# 19.
The shooting, shooting, all the time. It filled his ears, was all he could hear. Then he didn't hear it anymore.
—
The range was just one of the things Perry ran—his deals, he called them. The range was a deal. He had another deal, plowing—city streets, in one truck, and driveways, which took another, and he got paid on state subcontract for plowing roads that the big trucks couldn't do. Another of Perry's deals was bulldozers. He could grade your yard or clear your lot or break your building up into a pile for hauling away. If you had a home and paid tax on it and you wanted to stop, wanted only to be bled for the land, Perry could start on a morning, and the place would be mud by night. He commanded Bobcats and bigger dozers and graders and a couple of backhoes, some of which he owned, some of which the state did, owned and maintained, though they stayed on his lot and bore his name in black letters on the side. BONDED. The truck that emptied the Dumpster every day, that was Perry's too.
Another deal: Perry was mayor of the town. Stone Cottage, Ohio, was what they called it, though they'd stopped quarrying stones, and there was no cottage anyone knew. He did not want to be mayor, but the mayor controlled zoning, and he wanted to control zoning, because no one wanted a paintball range a block from Main Street. Everyone had known that was why he'd run to be mayor. But he had bulldozed them too, one by one, and on voting day a little more than a year ago, he'd won. The range opened that month.
Maybe they'd known, Perry declared, maybe they knew all along how much he'd hate being mayor of their God damn town. Four years. Maybe that was their revenge on him, what they extracted in exchange.
"You could quit being mayor. Now that you got what you wanted," East said, head down, polishing the countertop glass. The counter was fourteen feet long, from an old candy store. It had glass in the top an inch thick. The glass got smudged under everyone's elbows, but East liked to keep it clear, keep the pans of paints visible underneath, glowing even without light on them.
"What I wanted _then_ , son," Perry said. "There's always a new thing to want. And I'll get it too, but they'll make it hell on me." He withdrew the oxygen rig from the drawer and slipped it on. A cannula, he called it. "It's only fair."
"Hell is forever," said East bravely. "Not just four years."
Perry coughed wetly. "If I make it four years," he said, "it will be the devil's last miracle."
—
It seemed a revelation to Perry that East didn't steal money. He kept the register straight, severe, didn't take IOUs or cut deals. But he couldn't see how Perry would know that. To slip out a little—the way Shandor apparently had—would be easy. There was a lot of money. Most of it passed quietly as cash. Sometimes loose bills were offered him to extend someone's range time, to pay for a gun someone'd broken, or for a handful of stale paints, duds, for laying a hurting on the birthday boy or the boss. He'd take it. But whatever came, he put in the register, included in the deposit. The deposit wasn't in a bank. It went through the front mail drop of the farmhouse across the highway, where Perry lived.
Perry trusted him. But maybe trust was a trick. Maybe trust was the act that not trusting put on when there was no better alternative.
Maybe trust was the trick that kept him working twelve, thirteen, fourteen hours a day. Mopping, raking, sitting counter on his ass.
"Son, I know you're gonna take me one of these days. I just don't know how," Perry hollered, laughing, as if his loudness could command this place forever.
The hundred per day still wasn't good money. Just the same, East got his handful every day, paid timely. And in this town he spent little—there was nothing to buy. An egg sandwich every morning at the shop, and fruit from the grocery twice a week. A used coat for outdoors at the resale, used jeans for a dollar or two, warm shirts. A good pillow still in its plastic wrapper. He bought gloves at the hardware—even the men with ragged coats and uncut hair wore good gloves. East studied, bought a pair new, thirty dollars. Warm, elastic, he could work all day in them without a complaint.
The old bank was built of stone with fat sandstone pillars in front but had the same name as East's bank in The Boxes. He acted trustworthy around banks, and banks so far had trusted him. Anyway, his wad of twenties was getting thick. He wasn't going to bury his money or squirrel it away, the way he'd buried the two guns behind the edge of the parking lot, plastic-bundled, in a wave of dirt Perry's bulldozer had left behind.
He asked that five hundred dollars be made into a cashier's check. He made a new account and deposited the rest. The woman seemed confused that he didn't want to order checks.
"Just the ATM."
"But you'll want the flexibility," she said. "Not everything can be paid online."
Patiently he listened to her explain the options, the fee-free checking, the online access. She looked intense, friendly but businesslike. Her dark black bob cut hung down. A stud in the side of her nose. She might have been twenty-three. He wondered if she walked or drove in to work, if she rued living here.
At last she concluded her pitch.
"Just the ATM," East repeated calmly.
"But you can. You could," she countered. Her hands paused, unsure. Then she lay them on the desk. "Well," she said. "We're happy to have you."
He got up, walked to the post office, bought a pre-stamped envelope, and sent the check to his mother, without a note.
—
The smell of the bleach and cleanser and water East liked to mix together in the mop bucket wasn't right; it rose and curled inside his nostrils, wasn't healthy. But it smelled clean. Twice each day, now that Shandor was gone, East made the bathroom clean. Since he'd started cleaning it and mopping cobwebs, wiping fixtures clean of the dust that got in somehow, the men had begun helping out, using the trash can, pissing the floor less. They spat or spattered somewhere else, wiped their blood off the sinktop, picked their bandages up and threw them away. They began to keep it right. They put their returns on the return box, their beer cans in the recycling.
The spiders had stopped coming down out of the ceiling and claiming corners every night.
The barn, East learned, had been the old garage for the farm trucks, before Perry Slaughter had it remade. All that work—the bathroom, the storeroom up top, the stairway out the back, the antique counter with its heavy glass—Perry had bartered.
"For paintballs," Perry said. "Some people will do anything for paintballs, God help them." Standing in the bathroom doorway as East mopped, he coughed and laughed both. "When I opened it, I thought it would be a weekend thing. But they wanted to come back. Then another range opened up five miles up the road and stayed open every day. Everyone I had went away. So I stayed open every day, and they all came back." He scratched his sandpaper chin with a middle finger. "The guys they fought up there, they brought them all back here. Every day."
"How these guys afford to come in here playing every day?" East said. He pressed the mop out.
Perry snorted. "They can't. Son, it's like jerking off. It's like meth. These boys can't stop, and they can't call it what it is."
"What do you mean?"
"This is what they lie about. Not having a girl. Not were they drinking. They come in here and then tell the world they didn't." He looked back over his shoulder: heavy tires rolled past on the road, whining. "It's the ugliest business I ever been in."
"If you don't like it," said East, "why you got it?"
"You got to sell what you got left." Vapor rose around Perry's boots. "You ever heard people talk about Peak America?"
"Peak?"
"Peak. Like a mountaintop, Antoine. Like high as you're ever gonna get."
East twisted a knot in the trash bag. "No."
"Well, that was Ohio, right here, fifty years ago," Perry said. "What a God damn country. But that's past now." He backed out and put himself down on the stool behind the counter, and East followed him out with the trash bag and the bucket.
"All these boys," the old man sighed. "Their daddies, fifty years ago, they worked foundry jobs, or machinists, or they were white-collar: sales, teacher, bank. Everything. Drove to Cleveland, drove to Youngstown, had a pension, second house. Ohio exported more steel than Japan. Than Germany. Than England or Spain. The whole countries. We were making babies then, believe me. Time you turned thirty, you'd have four or five. These days, most of these boys who come in here every day, secretly, the thing they want is for that girl to turn them out. She can keep the house. The sooner she gets another man, the sooner he is free. He can't fix nothing anyway. Gets an apartment that's tiny—the size of that bathroom. That's all he wants," Perry said. "Works a little when he can. Got his beer and his PlayStation. Can't look his dad in the eye. That's what I mean. We were up there, and we've come down to this."
East stood still. He thought of the road, the towns he'd walked through. "Everything out here is pretty chewed up," he volunteered.
"You have no idea." For a moment Perry eyed East, licking his thumb, like someone turning a page. "You think you want to be a dad, Antoine?"
East laughed.
Perry said, "Maybe you are already. Sometimes you _act_ like it," he added mysteriously.
East lugged the bag out the door into the cold parking lot and tossed it over the side of the Dumpster. He breathed the cold air, scanning the trees opposite, their damp black and bare white. One large bird perched, watching the road below.
His two guns lay in the wave of dirt just there. Every day he reminded himself of them. But rain kept packing the dirt down.
Back inside, East changed the subject. "What about you? You ever had any kids?"
"Early I did," said Perry. "With my first wife. Not much of a father. Sooner or later, every kid is gonna want to kick your ass." Wetly he coughed. "I gave them every opportunity back then."
East looked forward to these talks with Perry—he wasn't sure why. They went forever and everywhere, and the old man moved from mumbling to hollering about things as if they were East's fault: World War II. Miners who died. American steel and Japanese steel. All about lumber and what happened when the trees weren't old anymore. Perry had spent his life knowing what things were made of. He would talk about those things all day. Sometimes East caught himself thinking about it, wondering about things he hadn't even known he was listening to.
—
Perry was dying. It was not something he ever mentioned to East. It was something East slowly stopped denying to himself. Perry took a battalion of pills each day. All colors, all shapes, counting them out from a box he kept in his pocket. Under the counter were other pills he didn't want his wife knowing about. He'd count out a handful and chase them with a can of root beer, and he'd wince and clench his eyes.
No one took pills like that if they had a choice.
Perry's cough was a variable thing, like an engine that some mornings started and on others refused. Some of his teeth were coming loose. One day he pulled one out and lay it on the glass countertop. Then he was called away on his little silver flip phone, and he forgot about it. East didn't know what to do with the tooth. A tiny blackening spot of blood peered at him from between the roots. After a while he picked it up and put it in the register.
Perry hadn't hired anyone yet. East reminded Perry how he was stretched thin, running the place by himself. After that, Perry came and worked four full days in a row—from ten or eleven in the morning to helping East clean up at closing. It was fine with East this way. He didn't need some new kid to be the boss of. He'd had plenty of that.
—
One morning, before the range opened, Perry invited East over to the tall, yellowing farmhouse to eat. East did not want to go but did not know how to refuse. So he sat down to breakfast with Perry and his wife, sitting in a straight chair where the wicker curling around the back didn't make it any less uncomfortable.
Perry served out eggs and ham and potatoes and talked to various ends.
His wife, Marsha, sat mostly mute. She asked East a few polite questions about not much, then broke off, as if she'd prodded enough.
Perry talked about her as she sat, occasionally nodding assent. The land had been hers, her family living on it a hundred years: grapes once, and apples and cucumbers and pumpkins and squash and corn. Animals that fertilized the soil, soil that fed the animals. "That's done for," Perry said, and she got up then, cleared her plate, and sat back down with a glass of water into which she'd stirred something that sank and swirled.
Her sister had gone to California and was never heard of again. Two brothers were killed at war and the third driving drunk on the highway. When you could no longer hire men to work the fields, then not boys either, waves of Mexicans kept the farm for ten or fifteen years. But now the Mexicans were gone, and the farms up this road were farms no more: it cost more to run them than any crop could bring in. The windbreaks had grown out and filled the old furrows.
The paintball range, East guessed, was something Marsha had agreed to without knowing what it was, without knowing how high the bulldozers would cant the walls, or that a large red and white banner reading SLAUGHTERRANGE.COM would be her view out her front window thereafter, that her home would rock each afternoon with the sound of gunshots and idling trucks. She too had said yes to her husband with only a vague idea of what his bulldozers would do. And now, East could see, she was a woman whose business it was not to look out the window at how her money was made.
The walls of the dining room were lined with photographs of the white people who were Marsha's ancestors. Fierce faces in gray, the women with hair in elaborate plaits, the men always a little smaller than their suits. The children he could barely look at, wondering if they were the doomed ones.
East thought of the pictures in the gun house in Iowa. The same fierceness, white people with hard eyes, keeping still the faces that their hard lives had made.
After Perry picked up the dishes and washed them, still holding forth from the sink in the kitchen, East half bowed quietly and thanked Marsha before leaving her quiet dining room. He had barely heard a dozen sentences from her, but he knew he had wandered into a battle about what was left, about what could and could not be sold. He could see the fragile stacking it was not in his interest to upset.
—
His ATM card arrived. This new ATM, he could feed cash right into it, no deposit envelope. It counted it up. East was suspicious, but the machine was always right.
The little eye in the window above the keypad, observing him, the camera under the roofline—he didn't even hide his face anymore.
—
"Warmest day in thirty years of Decembers." Perry was drinking off a bottle of Old Crow. "There'll be a front down from Canada next," he added. "Then I'll have to be out in the plow. I'll get someone in here to help you."
"All right." To East the promise was already empty. To get Shandor any help, it had taken _him_ walking in half dead.
They sat together in canvas chairs at the end of the landing, away from the building, where the sky paced above them, great clouds, barely tinted from below, coiled like the entrails of some great creature.
"Another hour," Perry said, "these clouds will run out, and we'll have nothing but stars. If you can stay awake."
And truly, the shelf of cloud passed, unveiling a long, clear black road of sky. The number of stars was more than East had ever known. Like something scattered. He doubted his eyes.
Perry slugged from the bottle, sleepy and content.
"You ever do astronomy? The constellations and so forth?"
"What's a constellation?"
"You know, in the stars. Say, the Dipper. Say, the North Star." Perry turned his bull neck around with difficulty. The north was behind him. "You know the North Star? Underground Railroad stories and all that?" He pointed vaguely. "By the Dipper, there?"
East laughed. "What's a dipper?"
Perry's finger stopped tracing in midair. "A water scoop. A dipper. What do they call it? A ladle. There. Handle, handle, handle, then the box of four stars."
"Which box you mean?"
"Damn, son. That one. Then the last two point at the North Star." Perry took another drink and stopped before he said something else.
East wasn't sure. But did it even matter? So many stars. This is what old men did, sitting out in chairs, staring at these. In the morning, when East woke up slumped and startled in his chair, Perry's whiskey bottle sat empty on the ground. But he had gone.
—
The cold front rolled in as Perry had said. For two days the clouds gnarled and darkened; then it snowed, the sky trying to blot out the world.
East had seen flurries before—twice since he'd left The Boxes, and once when he was there, a strange cloud that came south off the mountains and glittered the air over The Boxes for five minutes one January day. But never anything like this. The road a foot deep, the trucks slipping, helpless, thunder roaring behind the farmhouse. No one came, and he was glad that they didn't. He didn't trust it to be safe, going out.
Perry came through about noon in the small plow, turned off the highway, pushed clear a rectangle of the buried lot. He jumped down then and left the truck idling outside.
"Jesus," Perry chuckled. "Not bad for a Saturday. I think you can take the rest of the day. I'll put a sign up, pay you anyway."
East could not contain his alarm. "It's supposed to be like this?"
In the disastrous cold, Perry seemed as young and happy as East had ever seen him. "Yeah. It's supposed to be just like this."
—
The men were just as pleased to go shooting in knee-deep snow. That Sunday after the Browns game there were twenty, thirty guys. Perry brought in a large red plastic drum of coffee, handing out cups. "Don't know what I did round here before this boy showed up," he said to the older ones who just came to lounge away the evening, talking not about paintball but things they'd done and why their knees and backs and hearts didn't work. Why they were retired but none of the younger ones would ever be able to do that. Why staying here in Ohio was what they'd do even if it was a bad idea. They'd be in it to the end or be damned. That was what they told each other.
The Browns game was replayed late, and the men stayed and talked and watched them lose again. Their season would end in a couple of weeks. But it was over a long time ago.
East swept the stamped-out snow from their boots out the door, mopped up the melt, repeated it once or twice an hour until at last they stopped coming. Menial work. Sometimes he tired of it, felt a bubble of resentment. But it was also true that Perry's praise gave him a soaring, stinging pride. With Fin, he supposed, it had been more or less the same way. It was the first time in a while he'd let himself think of Fin.
Sometimes when he was looking out over the range, watching the men hide and mass and surge and shoot, he thought of Ty, thought of The Boxes. But he no longer could find the phone number Walter had given him, and he didn't try to remember it. What was in The Boxes was safe without him.
—
At the top of the stairs, the back door onto the landing, the lockers, and the air-compressor station was to the right. To the left, latched and rarely used, was a little storeroom. A utility sink, a green skylight. All these weeks, East had been sleeping there. He could lay down a certain double sheet of cardboard on a pallet—it was comfortable, smooth, had a give to it. He had the pillow, a used blanket. And along the roadside he had found a box that a dishwasher had come in, still clean and dry. He could fold it flat, slide it behind the cabinet in the day. At night he opened it and slept underneath, the dark string humming quiet in his chest, in blackness, encased.
If Perry knew about this, he had not let on.
Sometimes in the night East dreamed of the Jackson girl. Or of the judge's daughter, screaming. Or of being here at the range with Walter and Michael Wilson, the three of them searching, hunting somebody. Or of nothing, just the yellow line broken on the road, a line of nothing, of questions. Sometimes in the day, watching the men stalk one another, he dreamed these things too.
—
But one day in December, when the players had left because of a steady rain, Perry came and called East to dinner. Refusing didn't seem to be an option. Perry counted the bills into a leather folder, then counted back change for tomorrow's register and hid it where they always did. East swept quickly and locked the back door. Then they hurried across the road, bent under their coats, Perry explaining. Marsha had a son. He couldn't make either holiday, Thanksgiving or Christmas. This was going to have to do. He had come in that day from Philadelphia, Pennsylvania. "Sorry," Perry concluded, "to spring this on you."
East took this to mean it was not Perry springing it at all.
The son was named Arthur. He was tall, an attorney—as he pointed out—and he sat to Marsha's left. They were already sitting when he and Perry came into the dining room. Perry brought the food from the kitchen and then sat down on East's side of the table. The room was dim, like for a celebratory dinner, but an overhead light shone down on the table, bright enough that someone might clearly read a document.
Marsha had made the butternut squash, green beans, and wild rice, Perry pointed out.
"But the turkey is from IGA," said Marsha soberly. "They do a nice job."
East nodded. He couldn't tell what this was about. Perry stood and carved the turkey with a long knife burnished black. The son said a tight little prayer, and meat was served onto the plates. It was warm and tender, the first real meat East had eaten in almost a month. He felt his stomach get confused about it, cramping dully.
Marsha waited until they had all refilled their plates before she spoke.
"You need to put Antoine in a decent place," she said. "Not on a sofa that others crouch on all day."
So this was the ambush he was in for. The lawyer son was Marsha's version of a gunner. Had she come in and looked at the range, at the storeroom? When he was out, getting breakfast, maybe? Did she guess that the sofa was his bed? Or had she found his nest, his box and bed?
"That is a garage, not a decent place for a human to live," Marsha said, "but you have one living there. I look out at night—he does not leave. I look out in the morning—he does not come back."
Perry, mayor of the town, looked down at his fork, then made parallel digs through his mashed potatoes. "All right. I didn't know where he was staying. I didn't know. Say the quarry hires a guy: they don't ask, where are you staying?" He coughed again and removed something from his cheek with cupped fingers. "Antoine. Did I know where you were staying?"
East stirred. "No, sir."
"They know if they want to know," said Marsha. "And if they ask. If they keep paper. If their records are at all legal, they know." Her son the lawyer gave a listening nod.
"There's paper enough," Perry said. "Precious little, but enough."
"If your luck holds out," Marsha said, spearing a green bean. "But you knew, Perry. You knew, and you didn't even take him anything to make it comfortable. A bed, a hot plate. You could have tried to make it nice. I mean, we must have a dozen toasters in the attic. Arthur gives me one every year."
"Not every year," interposed Arthur.
She looked at East then, sad half-moons under her eyes. A silent apology. Maybe she was sorry for watching. But she was watching.
Perry served himself two rolls from the basket. "We'll see what's possible."
Then she addressed East. "What brings this on is, we received the notice. The state: they'll inspect. Within a month. And if they find you staying there, they'll revoke the permit and close the business."
"Which would not make her unhappy," Perry said, chewing. "Which would not break her heart."
"Antoine." She spoke up—the loudest sound East had ever heard her make. Her eyes darkened. "He has to find you a place to live. I will make him. There are decent places."
"Not for what he's paying now," Perry grunted.
"If I have to remind you who owns the land and the building," said Marsha, "I will."
Perry wandered the subject around, mentioning an apartment building he knew, owned by a lady down near Chillicothe; down there they used to make truck axles, and they were once the capital of Ohio. Now they had a storytelling festival.
But Marsha cut him off. "They don't make truck axles now, do they?"
"No, Marsha, they don't."
"You haven't ever gone to the storytelling festival, have you?"
"No, dear. I have not."
"It's in September. Almost a _year_ away."
Perry coughed. The dinner had been an effort for him even without the conversation. He was sick, East had recognized from the beginning, but always florid, forceful somewhere back inside. Now he was tired inside.
"All right," he said under his breath. "I'll find you a place. I can help you pay for it." He stared at the meat piled before him, then at his wife. "Not tonight. Maybe tomorrow."
So the attorney son served himself more wild rice. He hadn't had to say a thing. East wondered if he considered it wasted time, coming out here. But his sitting by Marsha's side had given her courage.
Everyone finished in silence. East chewed his last green beans slowly, one at a time, crushing each little seed out and finding it with his tongue. His plate when he handed it over was as clean as if it had never held food. He stood and pushed his chair in square with the table, as if he'd never been there.
It had been difficult, sitting there. A conversation he could never really speak up into. But he would remember this meal—the good food of a family. Even this one, so far from his own, or what had once been his own.
—
It was the talk of the apartment that unsettled him—the moving, giving up space he knew for something different, threatening. He ruminated and could not sleep. The box was hot with his restlessness. He tipped it off him and listened to the wind creaking the skylight in its frame. He stretched his legs and drew them in again. Finally he sat up and found his shoes.
He locked the front door behind him and walked into town. In the night, the air had warmed, and the snow had thawed to gurgling mush. In the one bright window, a handful of people sat hunched at the doughnut counter. They all looked over at the one at the end, who was telling a story. It was three o'clock in the morning. The street smelled like the sweet, frying dough. The people didn't look up as he passed the little slurry of light.
Near the gas station hung two black pay phones. East picked up a cold handset and stared at the powdery buttons.
It had come back to him, the number off the flyer, seesawing out of that woman's body. He could see it now. He dialed.
"Abraham Lincoln, please."
"Oh, baby," the operator purred. "He ain't been to work lately."
"I got to talk to someone," he said. "What can you do for me?"
She could have been the operator from before. He couldn't tell. Some males paid a lot of attention to women, but he hadn't paid much.
"If you work there," he said, "you know about Abe Lincoln. You can get him, or you can't."
She said, "Hold on."
The music was high, quavering. Vampire hip-hop. A car slushed by and stopped at the doughnut shop; two men and a woman climbed out, all talking at once.
East heard the different, male breathing first. Impatient. Maybe just woken up.
"Yeah. Who is it?"
"Who's this?" replied East.
The male on the other end said flatly, "No. The first question is mine."
He remembered now the chaos of the last time he'd called this line. _They'd arrested them all._
"Did Walter get back?"
"Walter who?"
"Michael Wilson? Did Michael Wilson get back?"
"Michael Wilson _who_?" But the bite was fake. The voice knew this was real.
"From what I'm asking," East said, "you know who I am. Just tell me. Are they _back_?"
He stood waiting. The line hummed. The man was waiting him out.
"Just tell Walter for me," he said at last. "I was out here with him. I'm still out here. Get him on the line. I'll call you back, half hour."
"I ain't your secretary," the voice said. "Fuck you."
But this meant _yes_. That East was still being listened to, the phone still connected, meant _yes_. "Half an hour," he said again. He hung up the phone and turned around where he was standing.
Empty street. A big truck headed down the side road over the creek—they weren't supposed to. They ruined the little bridges that weren't load-rated, Perry said. But at night they came anyway. Somebody was driving them, and that somebody needed to go home.
—
He bought a chocolate doughnut and ate it at a corner booth. Rivers of glaze and grease on his fingers and the rich, fragrant cake. Dark crumbs on the waxed white bag. The half dozen customers and the night girl stole glances at him. So be it.
He listened. The three who had come in were discussing a card game, bowers and the dead hand. One of the men was the redheaded counter girl's brother. The girl with the two men had just given up smoking, East observed. It was in her restlessness. He could not smell the smoke on her, but he smelled the need. He knew. The counter girl did not like the ex-smoker who was there with her brother. She said nothing. Nobody knew the one at the end, the balding customer with the round, streamlined head. He tried to insinuate himself; he laughed pleasantly at everything. He wanted to be a part of this place.
East watched the clock, counting off thirty minutes. He zipped his coat and headed for the door. "Good night," ventured the counter girl from behind him with her plain, flat voice, clear above the murmuring and the bouquet of Christmas bells on the door.
"Good night," he answered.
Above, dim stars showed, like something unburied.
"I'm gonna put you through," the operator said. "Three-way call."
"I just want to speak to him," East said. "Don't want a three-way call."
"Shall I have him call you back at this number?"
"You got it, the number?"
"Yes, I have the number."
So now they would have an idea where he was.
"How you doing out there?" she added. Like someone had told her.
He grunted and hung up.
East scratched the dry skin above his jaw. For a moment he considered just walking away—abandoning the phone, leaving just this trace, this much of himself. But that assumed something, he knew. That someone remembered him. That somebody was still wondering.
—
Walter's voice was muffled. "Who is this?"
"How was your flight?"
"Damn." Walter whistled. "Man. Nobody knew what to figure about you. Was you dead or in some jail? Or did you get back here and hide out?"
"Is it okay to talk?" East said.
"Yeah," Walter said. "Yeah, I borrowed a phone, so be cool. But yeah."
East was about to spill a story. But he swallowed it. "So. Fin still inside?"
"Yeah. He's gonna be happy I heard from you, though."
"Who's running things?"
"Oh," moaned Walter. "It was gonna be Circo, you know? Everyone was gonna hate it. But then he got a DUI and had weed on him, and Fin, on the inside, said no. Fin hates that, you know, distractions. So it's kind of complicated, kind of in process. We're changing up. Getting out of the house business. Some dude actually bought thirty blocks, the houses, the U's, the kids, everything."
"He bought, like, the houses? That people live in?"
"He bought the rights," Walter said. "Any business going on in there, he gets to own that. Paid a lot of money. And—"
The strangeness of news from home. Like a message in a bottle. "Did Michael Wilson make it back?"
"I heard he did," Walter said. "I ain't seen him."
"Why is Fin still in?"
"Aw, man," said Walter. "He got, like, a billion dollars bail. At first it was a hundred thousand, and we had that easy, so the judge went sky high. They ain't letting him out. Ever."
East glanced around in the dark.
"Anybody talking about it?"
"No one saying _anything_ about that," Walter said.
East felt a twang of disappointment. In spite of himself.
Walter laughed. "You still out there, man. That's what mystifies me. What are you doing? You need money, I can send it. Or a ticket. You could fly home now, no problem, except there's a couple things I gotta fill you in on."
"No. I'm here now."
"I kept waiting on you to call."
"No, I'm here," East said. "So, man. What are you doing?"
"Me? I'm back in school," Walter chuckled. "Missed a week, nobody said shit. But I'm going to _private_ school in the spring. Last semester, college prep, no more Boxes. They sending me up, I'll live there. Do a little recon, you know."
"Like Michael Wilson at UCLA?"
"Maybe. Good school up in the canyons. Kids up there, they're either movie stars or geeks."
East tried to imagine it. He didn't know what to imagine.
"What are _you_ doing?"
"The old thing," said East. "Just watching."
"What? You found a house and crew?"
"Different."
"Different but the same?"
"Yeah," East said.
"That will hold you," Walter said, "but that's little boys' work, E—you know it. We did that when we were little boys."
East felt the sting. But it passed. "Walter. I need you to do something."
"What?"
"Send me something. Go over to my mother's house and tip my bed up. You'll find a wood block that don't belong, with a butterfly bolt. Get my ATM cards, and if you can, get my phone, and mail them to me."
He gave Walter his mother's address and the street address of the range.
"I saw that area code. Had to look it up," Walter said. "Ohio?"
"Ohio."
"Is it like Wisconsin, all cold?"
"Warm and mountains," East said, "just like LA."
"How I'm gonna get in there, your mom's house?"
"Tell her I need it. Give her fifty dollars, man. She'll let you in like anything." Grimly he added, "Probably let you in for five."
"You ain't coming back, are you?"
"I don't think. Don't tell anyone what I told you."
"Believe me, man. I'm keeping quiet about all this. You're not coming back?"
"Don't tell nobody I'm out here," East repeated.
"I won't say a word," Walter said. "But you're not coming back. I can't get my mind around it."
—
Perry put the mail on the counter and regarded it sideways. Express package, addressed to Antoine Harris. It took a day and a half.
"Last name Harris," he observed. "First time I knew you had a last name."
"I lose track myself," East said.
"I must engrave you a nameplate," Perry declared.
All day the package glowed in the cabinet, radioactive with his previous life.
That night he tore the brown paper and tape and unwrapped a shoebox that had come from under his bed at home, with a pair of his old, outgrown shoes, battered but still whole, still real. Why had Walter sent those? He almost chucked them into the trash. The stink of his socks, the bedroom at his mother's house. His salt.
Then he felt around. Down in one toe, a flat bundle was wrapped in brown paper with the scrawl he could barely make out: COULDN'T GET PHONE. WILL KEEP LOOKING. W. Inside the fold, a thousand dollars in twenties, wrapped around his ATM cards, and one more card: a license, State of California.
The name, his own. Strange to read it there, in that official type, beneath the watermark. The address, his mother's, the birthday, his own. A few days past now—he'd forgotten it. He was sixteen now. A licensed operator.
The photo he remembered taking in a drug house a year ago, a different shirt, a new haircut. Someone had been fighting in a bedroom upstairs as he had sat straight against the backdrop and looked the camera in the eye.
Maybe it hadn't been much work to make. But Walter had made it. He wasn't sure how to feel. Was it a reward? An invitation? Or was it a rope, tying him down to a spot on the ground? With Melanie and the Jackson girl clinging somewhere along the line?
He stared at his face for a while, till he felt sleepy. He concealed the license in a small crack behind the baseboard. He reached up and switched off the light.
At some hour in the dark, the thick, cloudy, pressing winter dark, he heard the front bolt scrape, the heavy door pop open. He tipped his box back. Soundlessly he rolled his body onto all fours and then uncoiled, balanced and erect.
With a quick twist he unscrewed a stout broomstick from the push broom and carried it before him, ready.
"Antoine?" came Marsha's voice.
—
She was too shaky to drive herself. He helped her into one of Perry's trucks and took the wheel. All that morning they kept vigil at the hospital, in a high room that looked down upon a snow-covered drainage pond. Perry lay sliced open, a network of tubes and lines crossing him like roads and wires on white countryside.
# 20.
His time at the range seemed to fall off the clock: the light came late, and even the short December day seemed to go on forever, across hours that hadn't been numbered yet. The regulars stopped in to ask what East knew. They weren't going to the hospital to see Perry, weren't sure they were even allowed. But they missed Perry; they wanted to talk. They leaned on the heavy glass of the candy-shop counter. Some of them told East about their fathers' heart attacks, some about their own. There was a little business going on in back but nothing that pulled East away from the counter. He rented time and guns and sold paints while the regulars loitered around him.
Some of these men had never said an extra word to East before. He hadn't been among them a month's time yet. What did it mean, that they came to him instead of Marsha, that they saw _him_ as Perry's friend, his confidant. They weren't telling East about Perry—with Perry gone, they were telling him about themselves. "Wish him well," they said. They left cards for East to take to the hospital, sometimes group-signed: GET WELL, they wrote in ballpoint pen, some neatly, some in childlike scrawl. SEE YOU SOON.
_Soon._ The time had crawled to this point. It stood still for him, poised like a cat at the top of its leap. He waited for it to begin to come down. He wondered at the future of Stone Cottage without Perry. He had come to think almost protectively of the town, though he had only just arrived.
On the third day she returned. She wore a brown sweater, corded and woolly: East imagined that once her hair had been brown like that. He asked how was Perry, and Marsha asked if he knew what a do-not-resuscitate order was.
"It means he'll get what he wants," she said dully. It meant they knew already that Perry would die tonight or tomorrow. Something seized in East's throat, and he reached the broom handle and leaned until he could speak again.
This could be decided by a piece of paper a man had signed.
"Does he know?"
"Does he know what?"
"Does he _know_ ," East said, betraying something like panic in his throat, "that he's going to die now?"
She lifted her hand and bit it. She turned, seeming very small, and he waited for her at some distance.
Again he went with her to the hospital. She drove this time, in a long old white car he had never laid eyes on before, a Plymouth.
Perry was still alive in his junction of tubing and wires, the monitors, the cannula, the intravenous lines, an undressed mountain. His eyes were turned up. They were cloudy like the eyes of the fish East would see in the Spanish markets just east of The Boxes, eyes that he never let himself look into, for what they had seen he knew he'd see one day. He let Marsha go to Perry's body; he stood waiting by the door. The nurses eyed him, this mysterious black boy: what did he mean? There were black nurses, young women he noticed in pale, flowery blue; there were black doctors and black men in hairnets pushing mop buckets along the floor. But this black boy with his bare head in this fat old white man's room, trying quietly to comfort the widow-to-be, what was that? He did not hide from their eyes.
He let Marsha have her time, and when she left his side and went to go sit at the window, where she cried with a little chuffing sound, he approached Perry's bedside.
"Have they given up on him?"
Marsha took a breath. "There isn't much to give up, Antoine. All that wiring on him, it's keeping him alive. When they take it off, then he dies. It's that simple."
"When they gonna do it?"
"Now," Marsha said. "I shouldn't have. You don't have to stay."
East looked down the pupils of Perry's eyes as if he was looking into a hole in the street, as if there were depths to the man, and in the depths there would be what the fish knew, what the fish saw: the end. The being swept up, the laying down. Sharply, as if she were here right now, he saw the Jackson girl in the street, her eyes: the seeing of the last thing and the fixing on it. He shuddered. Perry was a color part his wind-scorched red and part the opalescent white of paint, the white of the winter sky. East backed away. He stood again by the doorway, and when the two nurses returned, they had to push in past him, so much had he forgotten where he was.
"Mrs. Slaughter?" the younger of the two nurses said. The black one. The older nurse waited quietly, deferentially, much like a nun. "Are you ready?"
Again Marsha took her moment. "Yes," she said quietly.
"The doctor can be here in a few minutes."
Marsha stood and came to East in the doorway. Her body, the opposite of Perry's, female, small, her tiny birdlike bones visible in her wrists, the hands darkened with age. Her brown hair and dark eyes going gray.
"You don't have to stay, Antoine," she said. "I'll be all right."
Words in his throat curled under themselves. He shook his head.
It took less time than East would have guessed. The needles came out of Perry's body, and the breathing tube with its scarlet cloak of mucus scraped back up out of the mouth with only a little urging. Marsha had reviewed her signatures on the paper without very much looking, and she touched Perry and moved back to the window without very much looking at him either.
"It's what he wanted," the older nurse said by way of consolation, and Marsha laughed once, a hollow _pop._
Out the window was the highway. Winter cars rushed past in their coats of grime.
"Push this button if you need anyone," the nurse said.
It was as if Marsha had fallen into a trance, and East, after a minute, moved again toward the bedside. Was this it? Suddenly he was eager to know. He bent over Perry, watched though he had been unable to watch Ty, unable to stay where his brother and the ones before him lay and suffered.
Perry's breathing was soft and tinkly, glasslike, like a stone rattling in a bottle as it rolled. It tumbled and slowed, tumbled and slowed. _It isn't taking long,_ East thought. There isn't much more. Once Perry's breath nearly ended on the upstroke. Then he let the air out—another roll, another tumble. East put his hand near Perry's hand, and he leaned and looked again down the barrel of Perry's eyes. The last thing anyone saw. He supposed he was willing to be it. He put the fingers of his hand atop Perry's knuckles, and Perry let out a half cough, out of his chest, which was high and white and furred with hairs like bare winter trees on a mountain. The stone in the bottle rolled again. The eyes swam in their clouds, their baths of white. Then the bottle bumped up on something, rolled no farther, and the mountain knew it too, what the fish knew, that last thing of things.
# 21.
Then again it was windy and warm. Like summer warmth after the snow, like California warmth: another wave of southern air, men walking in shirtsleeves, cars with their windows open and music spilling. East kept the range closed. He was alone. No one pulled into the lot, now muddy, the tracks shimmering with melting snow under the blue sky. Melting water sounding everywhere. No one stopped. He supposed it was Marsha the regulars were greeting now, now that Perry was dead. A man's sickness you discussed with other men. When a man died, you spoke to his wife at last. But no cars at the house either. He did not cross the road. He was not sure how to approach the house or if, now that Perry was gone, she would receive him.
He worked at the building and the yard in the warmth that might not return, he knew. He worked like it was his own. He aired the building, cleaned the storeroom where the many jugs and trays of balls gave out their waxy smell. He put the stepladder up—sixteen feet, it frightened him to climb it—and cleaned the lights, replacing two old tubes that spat and flickered. These were the lights that would kill you—that's what Fin said once, dull your eyes, take the color out of paint. He dropped the old tubes into the bin and watched them break into curls of glass, puffs of white powder rising from them.
In the afternoon he put on a pair of hip boots Perry had left and wheeled a Dumpster around the range. In shadows lay thick slush like the kids drank with syrup, and it still lay heavy in the lee of the berm with its shading fence. But where the sun hit the ground, he could find the litter in the mud: chips bags, candy wrappers, sandwich papers, sweat towels, plastic flasks, cigarette packs, bloody socks, popper vials, zipper pulls, beer cans, stray gloves. Some of it windblown, most of it dropped. Each time he cleaned the range, he learned things he hadn't known: what they brought and dropped was what the range didn't sell. Cherry cigars, green tubs of chew. Wrappers from Chicken Lively—what was that? Glass pipes, someone getting high—he had ideas who. He pinched the pieces with the picker and raised them high for a look. The boots were enormous on his feet, and he tightened the laces until they bit rings around his ankles.
The streak of yellow-orange across the south. A rent in the clouds, or maybe their end.
He filled two bags with trash before dark. The bags weren't right. Some awkward, shifty plastic, thin and bulgy, not tough like the regular bags. Maybe they were for something else, not trash. But Perry wasn't here to ask about it. East lugged them out one at a time, opening the gate and scuffling down the shrubby bank to the bigger Dumpster. Then he stood outside eating a bagel dry and looking at the daylight dwindling down the road to his left. A thin, worried-looking dog came padding down the edge of the road from the east as if it were following the fading light. It looked at East and lowered its mutt head. East tossed the rest of the bagel, and it gave it a sniff, then picked it up and took it along. Silent thing, intent.
He left the boots outside the door. Inside, he put the roll of weak black plastic bags away in the storeroom and took his good gloves off.
He vacuumed the sofas and the two raggedy rugs. Swept out the corners.
The next day, just as warm. He allowed himself to sit for a few minutes at a booth in the doughnut shop. Ate his sandwich. A napkin holder of rust-speckled chrome. The morning light played over the salty lot like a single, insistent note.
He found the paint that Perry had used to paint the door and its frame white and, while the sun was bright, painted them again. He painted until the paint ran out; then he soaked the brush in thinner and cleaned the windows with a blade.
It wasn't clear what he was going to do next. It was clear to East that he was waiting for something, some sign, some sudden clearing that would allow him to glimpse his desire. Clear that he deserved to wait. Not clear that he deserved such a sign.
But the day was fresh, and the air moved through, dusty, hopeful. At dusk he sat outside again, watching the house, Perry's house. Marsha hadn't come or gone. Two days now without customers, without cold. He regarded the driveway curiously, the hump of the yard above the roadside ditch.
He tried not to feel left out.
After a little while, the dog passed by again. East saw it coming. Careful, stepping quietly along the roadside, skirting stones. It knew what it was doing.
He had saved a part of his lunch this time, egg and cheese with the bread. He called to the dog, not a word but a sort of yelp. The dog stopped, eyed him uneasily. He threw it a chunk of greasy bread, and after the dog wolfed this, it stood squarely, assessing him now. It came for the handful of food he held out. He did not touch the dog. He could see the worn path something had rubbed around the dog's neck, part scar, part dry riverbed. Like a landmark in the fur. He watched the dog eat, eyeing him shrewdly, and then move away.
He took a shovel from the locker atop the landing, and he dug his guns out of the piled dirt where they'd waited for him.
—
Long before midnight he was asleep, curled on his pallet, the single pillow. He slept the long and grateful sleep of men who work—a breathing sleep, a dreaming sleep of childhood, or flying, or pathways leading somewhere. He burped and shifted, and steadily inside him, like the ocean's tide, one great muscle drew air and pushed it out, drew it again.
It was late December, a Tuesday. It had been a busy day.
—
The back door was open, as if someone were minding the range, as if the building were still airing. East slept. Beneath his box he did not feel the air. And he had been in the air all day, this southern air, which did not feel or smell so different from the air that he'd grown up in.
Only the noise, the sustained clattering, woke him. Like something being dragged, being broken. His eyes opened, and he lay paralyzed, as happened sometimes in dreams. _The need to move._ He could not move. _It's the middle of the night._ Like a child's excuse.
The sounds echoed up. Now he smelled the air. The nighttime wind and smell of melt, the milky smell of rock being ground. He reached up under the sink. The tough plastic creased there, formed a crevice, and into it he'd pressed the two guns he'd dug up. He dislodged one and fit his hand around it, and silently he stood.
The open back door looked out across the range toward the shushed lights of town. Nobody there. Just the man-size furrows of the field.
He crept to the stairs. The noise persisted: was it Marsha? A clank and a sigh, a clank and a sigh. Like a giant lugging his tool kit, shifting it with every step. In his hand the gun was strange and cold. He took the corner, paused, and stepped down.
The blow came from beside, below. It smashed the gun out of his hand, sent it spinning over the ragged sofa. Again it came and smashed him off the step. He plunged down the half flight of stairs, hit the concrete, and rolled. He tried to come up— _the pavement is not your friend_ —but the blow struck a third time. Some sort of club, it bit into him. It smashed him down again, and now he heard the feet scrambling around, and he gave up getting away, just covered himself, his head shoved down by a kick. He locked it between his knees, rolled, took the ringing blows on his left side. They rained on him like whip strokes.
_Not the head, not the stomach, not the back,_ he pleaded silently. _Not the neck or the shoulder, the elbow or the arm._ Not the places he was being hit: he yearned to save them, as if the blows were making him miss every part of his body. He yearned to protect it. He cracked under two more strikes, rolled away, hid his face again, under his arm, the way a bird hides under a shivering wing. Whimpering. Dreading the blow that would break his head, that would send him to join the others.
The feet backed off then. He'd thought there were two pairs, but now he eyed the shoes circling, switching direction. The dark staff resting. Just one. The shoes rested, poised and small, as if picking out a target. Dark high-tops. He braced again and closed his eyes.
The voice came soft and exquisitely amused. "Damn, boy," it said. "You hold a gun like a girl."
Then he did not need to look.
"You ran from me, man. You _ran._ "
East opened his mouth, but his throat was stopped.
"I knew you'd run," Ty said. "I didn't think you'd shoot me. Didn't think of that."
He opened his eyes, but they flooded. The concrete floor blurred, faultlessly clean. He was a fool. Left Ty for dead. But _left him for dead_ was just something you told yourself _._ Dead had to be for real.
There was blood smudging his face; his arm was smacked open, raw. Painfully he moved it. "Ty," he coughed. "You gonna kill me?"
He dared to look up. Above the sofa he saw something swinging—a nylon tote, hung on its strap from a rafter. Something in there shifted, causing the racket.
A decoy.
"Found you," Ty said.
He closed his eyes.
"What are you doing here?"
"Found you," Ty said simply. "Stand up."
East was ready to die on the clean, hard floor. Like people did. Standing didn't mean anything now.
Ty bent over him. He pulled at East's shoulder, gave up. "I'll whip you till you get up, then," he said, almost helpfully. A foot helped nudge East onto all fours. His arm burned, and he sheltered it from Ty.
"Take a seat," Ty said. He pointed with the club. It was just a stick, the broomstick East had used, he saw now. Unscrewed.
"You gonna kill me?"
"I should," Ty said.
East eyed where the gun had ended up. Somewhere way past the other sofa. But Ty would be ready for that.
He sat.
Strange, long journey across the world. The world would have its way; you could not stop it. And his brother was the world.
Ty acknowledged it too. "Funny, seeing you again. Seeing where we left off."
In the dark, East nodded.
Lightly Ty spun the broomstick. "You ask what I'm doing here? Did you try to find out about me?"
"Find out what?"
"Find out _what_ ," Ty snorted. "Did I live? Did I live or die, nigger?"
"Where I'm gonna find that out?" East mumbled.
"Well, first, you got to care," snapped Ty. "Care enough to try. Internet, man. But you don't do that—I forget. Or call somebody. Call home. Let me tell you. They were gonna make me a state orphan. Give me to a farm lady. They asked me my name every day for a week. Then I walked out."
"Out of what?"
"Hospital."
"And then what did you do?"
"What you think?" Ty said. "I went _home_."
"How long you look for me?"
"For you?" Ty said. "Since yesterday."
East made a face.
"Didn't look. I waited till you popped up. I knew you'd be out here somewhere, man. Every cow you looked at, you fell in love."
"Did Walter snitch me?"
"Naw. Walter loves you, man. He thinks you the real thing."
"Then how?"
"You called Abraham Lincoln," said Ty. "That got back to me. 'Cause I'm on the inside now. Not scraping along begging for jobs. Fin changed his mind. I don't even see a gun most days. So I looked in the records, found out the number. Pay phone over there, right?" He cocked a finger over his shoulder, toward the town. "Flew out this afternoon, came into town, walked to your pay phone. I asked one person two questions, and I knew where to find you."
"You flew out here?" said East. "By yourself?"
"East. Don't insult me," Ty said. "Remember Bishop Street swimming pool? You had to keep your face out of the water?"
East remembered the old, grimy, city pool. Splashing, a war of noise. Kids drowned all the time. Neither of them had ever had a teacher. The teenage lifeguards were no better—they were just the ones who'd made it that far.
"You'd dogpaddle down to the deep end with your friends. I was, like, four or five, and I'd track your ass down? Splashing and gasping because I couldn't swim. But I found you."
"I remember."
"Well, nigger," Ty said quietly, "now I can swim."
East looked up at his brother. The light flick of his hands as the broomstick lashed this way and that, its alloy screw-tip flashing. Heels dug into the upholstery.
Almost impatiently, he said, "You gonna kill me?"
"No," Ty said. "I'm not."
"I don't believe you."
"Maybe you shouldn't."
The hanging bag had stopped swinging, now just twisted slowly. East reached down to mop his blood, wipe it elsewhere on his shirt. Tried not to drip it on the sofa. _Well, finally,_ he thought. _Family reunion._
—
"I'm hungry," Ty said.
In the dim light, Ty looked the same. Skinnier, though. If that was possible.
"How—" East began, then stopped, abashed. He didn't believe. A ghost. A ghost didn't fly across the country to trade vengeance in for a whupping.
"How are you alive?" he finished.
"You ask? How I didn't die?" Ty spun the stick and stopped it straight up, like a clock. " _Thank_ you. I woke up in an ambulance. Coughing up pink shit. All that day I chilled out on a ventilator. You know what that is? It's a machine that makes you keep breathing. Another four days with a tube going into my side. That bitch _hurt_. Hurt worse than the popgun did. That is how I didn't die."
"How you just walk out?" East breathed. "They didn't cuff you, suspect you? Didn't ask you about it?" He found the words. "The judge?"
Ty squinted. "Why would they?"
"I know they had police on that."
"Maybe," Ty said. "But we kept a low profile. Did it right. To them I'm just a victim, some cold, black-on-black shit. They _much_ more likely looking for you."
East hung his head. "So when you gonna kill me?"
"Was up to me, you'd already be getting cold," said his brother. "But I'm here on business. And I'm hungry. So let's eat."
—
Early morning. So there was just one place to go. East walked Ty out to buy a box of doughnuts before the light. Ty waited down the street.
Even before six, the place was warm, confusing, alive. East stole a glance back from the counter. But he saw nothing but the windows reflecting movies of the inside.
He could run. He was faster than Ty afoot, or used to be. He knew the yards and fields here. He knew where Perry kept a key to the old truck.
But there was nothing in it, running. He could open a gap between now and his old life. But only a gap.
The doughnuts waited in their bins, blessed and bright. The counter clerk today was a thin boy, hair brushed straight up. He folded the box together, and East picked out twelve and paid.
"You get one more. Thirteen for a dozen," the thin boy said.
East said, "No thanks."
"What," said the boy, "happened to your arm?"
East looked at it in the light for the first time. It was a whacked mess, the sleeve soaked and blackening. His stomach slunk downward. "Thanks," he said.
Five people. It didn't matter who was in there or what he might have spoken out loud: they couldn't change things. But what Ty had said was right. If he were here to kill East, it already would have happened.
Though he could still take a notion.
Ty waited in a doorway, reviewing the morning _Plain Dealer._ Seeing East, he rolled the paper again and bagged it. The angle of a pistol sprang in his pants.
He fell into step beside East. "So this your house? You standing yard still?"
"Kind of the same deal."
"Paintball? There's money in that?"
"There's money." It would only amuse Ty if he said how little.
"What you running besides?"
"Nothing."
"Straight up?" Ty laughed. "Huh." They walked back along the highway without a word.
The day was coming up, black thinning to silver. The two boys crossed the damp lot, and East unlocked the door. The holiday bells rang. They'd been on there since he arrived, but weeks had gone by without his really hearing them.
Ty stalked around the place, examining the counter and the merchandise, trying on a pair of goggles. East stood watching until he felt time again, stretching long. He went to unknot the rope that slung the bag from the rafters and lower it to the floor.
Ty finished his circuit. "Now you sit down," he said. "I brought a message."
Gingerly East sat on one of the sofas. Ty took his perch opposite.
"Ready to listen?"
"I guess," said East.
"Then—you're coming back. This ain't home. You don't belong here."
East shrugged.
"The organization changed. So I came to get you. Here on, it's business."
"Business," East repeated blankly.
"Maybe the fat boy told you. Somebody bought The Boxes."
"Walter told me," East said. "So, they sold Fin out?"
"Streets and houses. Those shit holes you stand by, man," Ty sneered. "Like this place. You remember how your place got taken down? That took five minutes. Police took two more while you were gone. It ain't even hard for them; they come before lunch. So, yeah, we sold them out."
East shook his head.
"Things change," Ty insisted. "They paid us like fools. Businessman from Mexico. In love with America, man. Paid us one and a half million dollars."
East whistled. "But what's left? What's the business now?"
Ty's face tightened. "Don't you ever pay attention, man? Houses got no future. Police like hitting them, mayors like hitting them, news likes hitting them." He wiped his mouth. "You the only one doesn't get it. Your boys, your crew? They back in school now. _Making_ something of themselves."
"What about us?"
"Us. We're making money. All that what Michael did at UCLA, we work other colleges now. Them schoolkids love weed. Smoke too much. Pay too much. They'll even go pick it up. Walter's back in school too. But he still works Saturdays at the DMV. They think he's, like, twenty-five. So far up in them computers now, they can't stop him." Ty smiled. "You know Walter just makes up people, man."
"He makes up licenses."
"No. He makes people. He made you, Antoine Harris. We _talked_ about this."
East's arm smarted. "So how you gonna make money on that?"
"Shit, boy. People pay. You know what a college kid will pay to be twenty-one, have a second name? What a Mexican dude will pay to be in the computer for years going back?" Ty wiped his mouth. "People make lives on that shit."
"Police gonna catch you on that too?"
"Walter is smart," Ty said. "And careful. You don't even know his name."
"Walter is his name," snapped East.
Ty laughed in his face. "You don't listen. You don't even know who you are."
"Your brother," East said.
"Half brother. Right," said Ty. "We got your mother in common. But since you always been Fin's boy, that organization gonna be yours. That's what Fin wants."
"Fin's boy?"
Ty's face filled then, no longer just a talkative skeleton. It filled and flexed with the old hatred.
"Nigger, you know," he said. "Half brother. But you are what I ain't."
People had always whispered at it. But there was nothing he could trust upon. A father wasn't anyone he'd ever known.
It wasn't anything he could use now.
"We left town for that reason," Ty said.
"What reason?"
"Protect the core."
"The core?"
"You think Fin gonna send the four of _us_ to kill a dude?" Ty said. "Makes no sense. Why not just two guns? Why not one?"
"To protect us," East said dubiously.
"Get you out of town. Walter and you. Brains and blood."
"But what about..." East said, and then it was as if he couldn't remember anyone's name. "What about Michael Wilson?"
"Michael Wilson was a babysitter. Bad one, we found out. He handled the polite situations. I handled the impolite ones."
"But people higher up," East said. "Sidney. Johnny."
"There is no Sidney or Johnny." Ty made a quick gesture that East didn't want to see. Something slipped, pulsed under East's ribs.
"But what about the dude?" he protested. "The judge? Why was that?"
"An excuse."
"An excuse?"
"Prosecution got a hundred witnesses, man. They didn't need Judge Carver Thompson."
"Why we _kill_ him, then?"
"You," said Ty.
"Me?"
"You were the only one took that seriously. It was you who kept on. Mission-focused—I hand it to you, man. Fin says something, you do it."
Ironically Ty bowed.
"No," East said. "That ain't how it was. Don't put it on me. We killed the man. What comes of that?"
"Nothing comes of it."
"Nothing? People are dead now, man."
Ty ran his fingers over the box and picked out another doughnut.
"Shit be crazy," he said.
—
All that time, flickering again inside his skull. The van, the hours of country. Rolling under their wheels like wave tips passing his brown calves at the beach. The same feeling, the same dull roar, tires, water, the same laying out of light on the sand and fence posts and all there is. Lightly flying. He still felt it.
He shook his head out, the way he would sometimes after waking from a dream.
"How is Fin?"
"Fin?" Ty chewed slowly. "Two days after we left town, he turned himself in."
"He did what?"
"He walked into a police station and sat down. Tired. Living house to house," Ty said. "Don't think they weren't surprised, though." He held up a finger while he worked something in his mouth. "Come back, man. It's what Fin wants. Fin _knows_ you shot me. But he don't have but one bastard son."
East rubbed his eyes. The light of day was finding the skylights.
"Terms of my employment," said Ty, "is, I have your back."
"I don't believe you," he said at last.
"If I was gonna kill you," said Ty, "this was my chance. No, come back. 'Cause Fin is giving us hell till you do."
East stood. He tested his shaky legs. Ty made no objection to it.
"These doughnuts." Ty was mumbling through a full mouth. "Good. I see why a girl like you would settle down."
"I never heard you talk so much," East said.
"Well," said Ty, "we all got to do things we don't want to do."
—
In The Boxes, when someone insulted you, you insulted them back, or if someone punched you, you punched them. But everything was subject to organization. If an insult came from inside, you threw it back. If it didn't, you found out first. Found out what you could get away with. It was possible you could get away with nothing. Possible you would need to swallow your pride.
If somebody really hurt you, bruised or beat or shot you, you didn't need to ask. Injury called for injury. No need for organization. These rules of living were inside out. These rules of living kept boys polite day to day, even if they had free rein to kill.
A brother putting a bullet into a brother was unacceptable. It had happened, no doubt. One must have had a reason. But there would be a response. East had known this without consultation. He knew it the way he knew walking, he knew language. He had come to Ohio expecting to be kicked, to be gutted, for the last bullet in the world to find him and spit in his face.
He did not expect to be fetched back. He did not expect doughnuts, still soft from the oven, or to be handed an air ticket reading FIRST CLASS LAX DATE OPEN with a name he'd never heard, clipped to a California state driver's license with that same name. Same name and a picture of his face, taken back one day when everything made sense.
—
Ty's car was a sleek gray Lincoln, parked a quarter mile down.
"How'd you get hold of this? You're thirteen."
Ty fiddled with knobs. "You forgot my birthday. I'm fourteen now."
_That's right, fourteen now,_ East thought. Both Sagittarius, born early December.
"Steal it?"
"No, man. Just a car service." Finally Ty exploded with disgust. "Fuck this motherfucker. How you get defrost and heat at the same time?"
East flipped through the knobs, chose something. Ty shook his head and gunned the engine. It was smooth, new and powerful. "Couldn't figure out the radio either," he confessed.
East had told a lie—that he had business to close up. Money hid and deposited, debts to be collected and paid. Otherwise, he feared Ty was just going to put him on a plane, today. It surprised him when Ty simply said okay.
As for Ty, he was flying home immediately.
"Don't make me come back," he warned. "This car, we got one week."
" _We_ got?"
"You got six days left. More than that, I have to make a call. I _don't_ want to make that call. So six days."
"All right," East said.
The wound was bandaged now. Ty had helped him disinfect it back at the range. The big first-aid kit had everything he needed. But picking the shirt out of the sticky, clotting blood hurt almost as much as the whupping. Ty bandaged the arm and taped it down. Squeezed it once, like a joke, and East screamed. "See?" Ty murmured. "See?"
Light rain fell. The temperature was dropping again.
"One thing I forgot to ask. How'd you _get_ here?" Ty said.
East considered. This too, this memory, seemed like a train on a different track than he'd been on. "Old lady rode us to an airport. Walter flew home. Then I stole her car."
"You stole a car? _You_ stole it? From a lady who gave you a ride? Cold," said Ty. "You get rid of it?"
"Left it two days back."
"Two days, what?"
"Two days of walking."
"You burned the car, right?"
"No. Left it by a police station."
"Crazy," Ty said. "So why you stop there? At your store, paint guns, whatever?"
East had to remember before Shandor, before Perry, when he was just a kid in the street. "I got cold, man. Cold and tired. Sign said HELP WANTED, so I went in."
"I know they liked you, didn't they? Had you pushing a mop?"
East shrugged. "Hundred dollars a day."
"White man stealing from you," Ty jeered. "I hope you stole a little back."
"I don't steal," East said.
"You stole that lady's car."
"Yeah." East's blood quickened. "That was different."
_"Oh."_ Ty's fingers tapped the wheel. "Tell you one thing different. Bet it wasn't no white lady's car you stole."
East shut his mouth and looked out at the dirty snow, stinging. Ty hummed. He drove fast, relaxed. Fourteen now, and he seemed to know driving by heart. He seemed to have the airport route in his head. He seemed just to pick things up like that.
East said, "Ty. Back at the gas pumps, when you held the gun on that dude. What were you thinking? What was the play?"
"You mean," Ty said, " _before_ you shot me?"
"Before I _had_ to. You were out of control, man."
"Maybe I was," said Ty. "But come off it. Who was saving your ass, every time? In Vegas, from Michael Wilson, from that hick-ass town? And who did the job?"
"What job?"
"The _judge_."
Then East remembered the judge: his name, his face. The dark shape of him moving around in the bright cabin like a rat in an experiment. "Ty. Did the judge know you? He looked at you."
"I bet he did."
"He _smiled_ at you," East said.
Ty just laughed.
"You ain't gonna tell me?"
"No," Ty said.
_This_ _shit,_ thought East. "But now you're saying I'm in charge."
"Play that," said Ty. "You always in charge. Fin had a hundred hungry niggers working, but you're the only one ever follows directions. Did you ever wonder how I had a gun after they took one off me?"
"You had a second one," East said, "they didn't find."
"Same one," said Ty. "Fin gave it back. You listening now?"
"Forget it," East seethed.
"East. You got your way. You understand that your way goes on top of my way. But your way don't stop me?"
"Just be quiet, man," said East. The rain was letting up, and Ty took the exit to the airport. But East was in his mother's apartment again, arguing. Arguing for his life against this impossible boy.
"You stick to business," said Ty. "But I am business, East."
They sat hot in their seats, hating each other like brothers, till they pulled up to the terminal, gray planes hanging in the sky.
# 22.
Along the departures curb, the gray Lincoln stopped. NO PARKING. DROP-OFF ONLY.
Ty said, "When you're ready to drop it off, call this number." A sticker on the dash. "Give them one hour. They meet you here. Or wherever you need."
"What do I need? Credit card? Fill the tank?"
"Bitch, this ain't Avis. Just give the car back." Ty handed East the keys. "Walter said you don't fly. So if you _got_ to drive this car back to LA, okay. But call and let them know."
East scanned the line of windows along the terminal. Airline porters. Police. Families pulling suitcases on small invisible wheels.
"Two more things," said Ty. He popped a compartment between the two seats. "There's your phone. A charger too."
East spun his phone in his fingers. The familiar weight and shape.
"Just be careful. Don't say much. Be smart. I called you. So my new phone is the last number on it."
East stared at the phone. "Thanks," he made himself say.
Ty reached down into the compartment again and came up with a roll of bills.
"Three thousand dollars, if something comes up," he said. "This is _my_ money, now. A loan to you from me. Understand? Say it."
"It's your money."
"All right. Take it."
East let the money sit in Ty's hand for a long time. A debt he didn't want. But there was nothing else now.
He put it away in his pocket.
"Don't lose it."
"I won't."
"Plus this." Ty fished a little silver gun out of the compartment, showed it, and replaced it. Then he took an envelope out and laid it on his thigh as he threaded the zipper on his jacket. FIRST CLASS, it said, just like East's. DATE OPEN. JOE WARNER.
"So, this is it," East said. "No luggage? Nothing?"
Ty shook his head.
"So. Six days. You got what you need. Any questions, you call me."
Ty picked up his ticket. East studied him for a long moment. The sharp easiness. His long little balding head. Kidney bean—that's what their mother had called him.
"I don't want to come back here. Winter and shit. If I do, it's the old rules. There will be consequences."
"I hear," said East.
A hard chuck on East's arm, right on the bandage—East kept from crying out. Ty unsealed the door with a rush of wind. He climbed out and straightened his colorless jacket. Ticket folded over once in his hand, he checked the traffic behind them and then mounted the sidewalk, passing people in parkas and colorful letter jackets. East watched Ty hurry toward the electric doors, which slid open for him, just another young man on the way somewhere else.
It took East a moment: now he was expected to drive the Lincoln off. DROP-OFF ONLY. Not to slide over: to get out and walk around the big car. He did so, his hands and chest tingling. Paused at the driver's door and checked the little chrome key ring: DODGERS. The brand of home.
A girl passed on the sidewalk, small and black, leading her parents, who were all burdened down with garment bags and ski poles. East didn't look: he knew she would be the Jackson girl, all big eyes and bravery. That face swimming atop her face.
Then she was gone.
He coaxed himself onward, opened the door, lowered himself into the driver's seat.
—
Barely any noise. Solid. He checked the mirrors, moved the seatback up. He wished he'd watched the route more on the way. Ty drove the roads as if he knew them. East knew only one town.
If he could find his way back to the long old highway, he'd be all right.
The wind moved the trees. The rain was stopping. But the pavement was already dry.
The steering wheel was thick and almost drowsy in its softness. He would need a little sleep as soon as he could get it.
—
Back at the range, he spent a few hours. He polished the countertop. He cleaned the storeroom. In the evening dark he dragged items out to the Dumpster—his bed of clean, flat cardboard. His blankets—he saved his new, still-fresh pillow. The box he'd fit under at night. He slept his last night on the sofa, comfortable without the heaters. The cold didn't bother him now. He wasn't as skinny as he used to be.
The last day. East took alcohol and rubbed down the register, the bathroom, the door handles, the cabinets—any place he'd touched, any place he'd made his. He went to the bank and cleaned out his Ohio account, took cash, more than a thousand dollars. Added it to Ty's money and Walter's.
At a table outside the little grocery where farmers sometimes came, he bought a handmade bouquet. Dried flowers, yellow and orange. He walked them across the highway to the leaning-forward yellow house where Perry had lived with Marsha. He stood for a moment on the porch but didn't knock. He left it on a rocking chair beside her door, with a note that said, _From Antoine, thanks. RIP._ An ambiguous good-bye. He didn't know if she'd ever get them. He doubted she'd ever step inside the range again, or her son would. It seemed to have been abandoned, except for him. It seemed to have been just the one man's dream, and when he died, it stopped. East had left it spotless. He hadn't stolen a thing.
He cleaned the two old Iowa guns inside and out. Reburied them, deep this time. He spread the three like licenses out on the counter—East, Antoine, and the new one, the person who matched the plane ticket—and studied his face: the three times, different expressions, different shirts. But each one of them him. Each one a different life.
It wasn't easy to decide.
The first he chopped to bits was Antoine. The van. The guns. The trail they'd made. He cut it to bits with Perry's wire snips. Then it was down to the new name and East.
He wondered what Walter had invented for him, what sort of life, what weakness. What sort of story, should he not come back. And he was never going back.
In the end, he and East said good-bye. Age sixteen, licensed driver, State of California. Snipped him too into tiny squares. Wrapped him in the shreds of the one-way ticket back to Los Angeles. And left him in the trash can outside the doughnut shop, with the wrappers and the half-crushed cups of coffee. Became the new name, and no one else. He would grow into it as he was growing into the body stirring beneath him, the strange and turning body, as uneasy and teenage as it was hard and loyal. He looked different anyway now, fuller, older. The wind, the food—something out here was giving him pimples.
—
For a few minutes he parked the car at the range. He put his clothes and toothbrush in a grocery bag. Threw his pillow into the trunk. Wiped down counter and doorknobs again before exiting the building for good. Leaving the Lincoln visible while he did so was maybe not smart, but it would be fine. The car would do to take him east, to the dense snarl on the map, that opposite coast with its tangled cities: Washington, Philadelphia, New York. He had the rest of the week.
The sun was going down. The dog, the colorless dog, followed the border of the road, weaving. He had no food to offer it this time, but he whistled, and it came. He touched its scarred and rippled neck, and it whined. Sharp, black, careful eyes.
He opened the back door of the Lincoln, and warily it looked the car over. "Get in," he said.
Before he left, he stopped and ate a doughnut. The place was nearly empty—the thin boy pouring coffee, two women from the grocery store, a lady truck driver in a fur collar eyeing her cab outside. He would miss this town, not that he'd ever liked it. It was a place he'd stopped and studied. But the way he was leaving felt like leaving home. He bought a second and third doughnut and had them packed in a paper bag. Take-away to anywhere. Outside, the strange dog lay asleep and breathing in the borrowed car.
He stood outside and took a last look around. It was growing night, but the town was not dark. Lights shone beside doors and over driveways, still air. Somewhere he sensed a clatter: he listened, then caught the voices, boys in a driveway shooting hoops, the echoes clashing. He could see the plumes of smoke, the exhalations of every chimney, rising and dispersing. Each house a quiet mystery.
Nobody watching. He fingered the keys his brother had handed him, and as he opened the door of the borrowed gray Lincoln, he caught himself, just a glimpse, in the curved window glass. Alone, the first few stars in the unswept sky behind him.
Then he was gone.
# ACKNOWLEDGMENTS
This book was largely drafted at the Virginia Center for the Creative Arts, and I thank its staff and its many Fellows for their listening and encouragement and kinship.
Thanks to many people who helped _Dodgers_ find its way: Dan Barden, Kevin Canty, my colleagues Wendy Bilen and Rewa Burnham. Wendy Brenner, my best-writer-friend from forever. Steve Yarbrough and other fellows and teachers at Sewanee Writers' Conference who helped me put the book in drive. My remarkable agent, Alia Hanna Habib, for her readings and advice, and everyone at McCormick Literary, especially Susan Hobson and Emma Borges-Scott. To Jon Cassir and his team at CAA.
At Crown: Nate Roberson, my editor, who has been unvaryingly patient and sharp-eyed and wise. To Barbara Sturman and Chris Brand for design. And to Danielle Crabtree, Dyana Messina, Rachel Rokicki, Lauren Kuhn, David Drake, and Molly Stern: I am deeply grateful for how you have received and imagined this story and what it might become.
And to my students at Trinity.
To my parents, for the world, and to Deborah Ager, for making it sweet.
# ABOUT THE AUTHOR
> Bill Beverly was born and raised in Kalamazoo, Michigan. He studied at Oberlin College and at the University of Florida, where he earned a PhD in American literature. He teaches at Trinity University in Washington, DC.
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1. Cover
2. Title Page
3. Copyright
4. Contents
5. Dedication
6. Epigraph
7. I: The Boxes
1. Chapter 1
2. Chapter 2
3. Chapter 3
8. II: The Van
1. Chapter 4
2. Chapter 5
3. Chapter 6
4. Chapter 7
5. Chapter 8
6. Chapter 9
7. Chapter 10
8. Chapter 11
9. Chapter 12
10. Chapter 13
11. Chapter 14
12. Chapter 15
9. III: Ohio
1. Chapter 16
2. Chapter 17
3. Chapter 18
4. Chapter 19
5. Chapter 20
6. Chapter 21
7. Chapter 22
10. Acknowledgments
11. About the Author
1. Cover
2. Cover
3. Title Page
4. Contents
5. Start
1. i
2. ii
3. v
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5. vii
6. ix
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|
{
"redpajama_set_name": "RedPajamaBook"
}
| 9,596
|
Die Scottish Football League Premier Division wurde 1995/96 zum 21. Mal ausgetragen. Es war zudem die 99. Austragung einer höchsten Fußball-Spielklasse innerhalb der Scottish Football League, in der der Schottische Meister ermittelt wurde. In der Saison 1995/96 traten 10 Vereine in insgesamt 36 Spieltagen gegeneinander an. Jedes Team spielte jeweils zweimal zu Hause und auswärts gegen jedes andere Team. Bei Punktgleichheit zählte die Tordifferenz.
Die Glasgow Rangers gewannen zum insgesamt 46. Mal in der Vereinsgeschichte die schottische Meisterschaft. Die Gers qualifizierten sich als Meister für die Champions League Saison-1996/97. Als unterlegener Pokalfinalist, qualifizierte sich Heart of Midlothian für den Europapokal der Pokalsieger. Der zweit- und drittplatzierte Celtic Glasgow und der FC Aberdeen qualifizierten sich für den UEFA-Pokal.
Partick Thistle und der FC Falkirk stiegen am Saisonende in die First Division ab. Mit 26 Treffern wurde Pierre van Hooijdonk von Celtic Glasgow Torschützenkönig.
Vereine
Statistik
Abschlusstabelle
Die Meistermannschaft der Glasgow Rangers
(Berücksichtigt wurden alle Spieler mit mindestens einem Einsatz)
Siehe auch
Edinburgh Derby
Old Firm
Liste der Torschützenkönige der Scottish Premier Division
Weblinks
Scottish Premier Division 1995/96 bei statto.com
Scottish Premier Division 1995/96 in der Datenbank der Rec.Sport.Soccer Statistics Foundation
Einzelnachweise
Fußballsaison 1995/96
1995 96
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,358
|
15,95 (2.57%)
1590,3 (-0.48%)
AngloGold Ashanti returning to Ghana mining town under a fairer deal, says president
Khulekani Magubane, Fin24
Zim policy changes show it is open for business - mining minister
Shape up SOEs if you want more investment, mining bosses tell government
We need a stronger Eskom if we want to save mining, says Mantashe
Gold Fields' Carolus: Mining companies can do better
Ghanaian President Nana Akufo-Addo is optimistic that the recently announced return of Anglo Gold Ashanti to one of Ghana's most lucrative mining towns will bring about economic benefits for ordinary Ghanaians, he told delegates on the second day of the 2019 Mining Indaba in Cape Town.
The town of Obuasi in Ghana is said to be the site of the richest Anglo Gold Ashanti gold mine in the world, he said, but in spite of this, the town looks nothing like a town where a multinational company has invested hundreds of millions.
The Mining Indaba is an international gathering of the mining industry, investors, government and labour, which is set to take place from Monday to Thursday.
"So why does Obuasi not look like a place from where hundreds of millions of dollars have been made? It should be the most beautiful city in Ghana, or one of the most beautiful cities in the world, but it is far from it.
For more Mining Indaba news read Fin24's Special Report
"After an absence of five years from the scene because of uncontrolled illegal mining, the company is back again. I had the pleasure of re-opening the mine two weeks ago under an agreement that balances, more fairly, the interest of the two sides," Akufo-Addo said to applause.
READ: Gold Fields' Carolus: Mining companies can do better
President Akufo-Addo was unapologetic about the negative impact certain mining companies have had on some African communities through displacement, environmental impact and investment deals that favoured companies at the expense of local communities.
Read more about: miningindaba2019 | nana addo dankwa akufo-addo | ghana | mining indaba
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 248
|
31-year-old man arrested for series of obscene acts in Yishun
Investigations revealed that the man was involved in at least four cases of obscene acts. PHOTO: ST FILE
Jun 17, 2016, 12:51 pm SGT
http://str.sg/4U2m
SINGAPORE - A 31-year-old man who is alleged to have committed a series of obscene acts over one month in Yishun was arrested on Thursday (June 16).
In a news release on Friday (June 17), police said it received reports of the acts involving an unknown man between May 6 and Tuesday (June 14).
The Straits Times understands that the man had exposed himself to his victims.
Officers from Yishun North and Yishun South Neighbourhood Police Centres conducted extensive ground enquiries to establish the man's identity, the news release added.
He was arrested along Yishun Ring Road.
Preliminary investigations revealed that he was involved in at least four cases.
If convicted, he may be fined and jailed up to three months.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,593
|
\section{Motivations}
Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with smooth boundary
$\partial\Omega$, and let $\nu = \nu(x)$ be the unit outward normal vector to
$\partial\Omega$ at $x\in \partial\Omega$.
We set
$$
-\mathcal{A}v(x) = \sum_{i,j=1}^d \partial_i(a_{ij}(x)\partial_jv(x))
+ \sum_{j=1}^d b_j(x)\partial_jv(x)
+ c(x)v(x), \quad x\in \Omega, \, 0<t<T, \eqno{(1.1)}
$$
where $a_{ij} = a_{ji}, b_j, c \in C^1(\overline{\Omega})$,
$1\le i,j\le d$ and we assume that there exists
a constant $\kappa > 0$ such that
$$
\sum_{i,j=1}^d a_{ij}(x)\xi_i\xi_j \ge \kappa \sum_{j=1}^d \vert \xi_j\vert^2,
\quad x\in \Omega, \, \xi_1, ..., \xi_d \in \mathbb{R}.
$$
Henceforth $\Gamma(\gamma)$ denotes the gamma function for $\gamma>0$:
$\Gamma(\gamma):= \int^{\infty}_0 e^{-t}t^{\gamma-1} dt$.
Our eventual purpose is to construct a theoretical framework
for treating initial boundary value problems for time
fractional diffusion equations with source term $F(x,t)$, which can be
described for the case of $0<\alpha<1$:
$$
\left\{ \begin{array}{rl}
& d_t^{\alpha} u(x,t) = -\mathcal{A}u(x,t) + F(x,t), \quad x\in \Omega, \, 0<t<T, \\
& u(x,0) = 0, \quad x\in \Omega,\\
& u(x,t) = 0, \quad x\in \partial\Omega, \, 0<t<T.
\end{array}\right.
\eqno{(1.2)}
$$
Here, for $0 < \alpha < 1$, we can formally define
the pointwise Caputo derivative:
$$
d_t^{\alpha} v(t) =
\frac{1}{\Gamma(1-\alpha)}\int^t_0 (t-s)^{-\alpha}
\frac{dv}{ds}(s) ds, \eqno{(1.3)}
$$
as long as the right-hand side exists.
As for classical treatments on fractional calculus and equations,
we can refer to monographs Gorenflo, Kilbas, Mainardi and Rogosin
\cite{GKMR}, Kilbas, Srivastava and Trujillo \cite{KST},
Podlubny \cite{Po} for example.
Henceforth let
$L^p(0,T):= \{ v;\, \int_0^T \vert v(t)\vert^p dt < \infty\}$ with $p\ge 1$
and $W^{1,1}(0,T):= \{ v\in L^1(0,T);\, \frac{dv}{dt} \in L^1(0,T)\}$.
We define the norm by
$$
\Vert v\Vert_{L^p(0,T)}:= \left(\int_0^T \vert v(t)\vert^p dt\right)
^{\frac{1}{p}}, \quad
\Vert v\Vert_{W^{1,1}(0,T)}:= \Vert v\Vert_{L^1(0,T)}
+ \left\Vert \frac{dv}{dt}\right\Vert_{L^1(0,T)}.
$$
In view of the Young inequality on the convolution,
we can directly verify that the classical Caputo derivative (1.3)
can be well-defined for
$v\in W^{1,1}(0,T)$ and $d_t^{\alpha} v\in L^1(0,T)$.
However, everything is not clear for $v\not \in W^{1,1}(0,T)$, and
it is not a feasible assumption that any solutions to (1.2) have
the $W^{1,1}(0,T)$-regularity in $t$.
The Young inequality is well-known and we can refer to e.g.,
Lemma A.1 in Appendix in Kubica, Ryszewska and Yamamoto \cite{KRY}.
\\
In (1.3), the pointwise Caputo derivative $d_t^{\alpha}$ requires the
first-order derivative $\frac{dv}{ds}$ in any sense.
Therefore, in order to discuss $d_t^{\alpha} v$ for a function $v$ which keeps
apparently reasonable regularity such as "$\alpha$-times"
differentiability, we should formulate $d_t^{\alpha} v$ in a suitable
distribution space.
Moreover, such a formulation is not automatically unique.
There have been several works for example,
Kubica, Ryszewska and Yamamoto \cite{KRY}, Kubica and Yamamoto
\cite{KY}, Zacher \cite{Za}. Here we restrict ourselves to
an extremely limited number of references.
In this article, we extend the approach in Kubica, Ryszewska and
Yamamoto \cite{KRY} to discuss fractional derivatives in
fractional Sobolev spaces of arbitrary real number orders and
construct a convenient theory for
initial value problems and initial boundary value problems for
time-fractional differential equations.
In \cite{KRY}, the orders $\alpha$ is restricted to $0<\alpha<1$, but
here we study fractional orders $\alpha\in \mathbb{R}$.
The main purpose of this article is to define such
fractional derivatives and establish the framework which enables us
for example, to uniformly consider weaker and stronger solutions with
exact specification of classes of solutions in terms of
fractional Sobolev spaces.
Thus we intend to construct a
comprehensive theory for time-fractional differential
equations within Sobolev spaces.
In (1.3), we take the first-order derivative and then $(1-\alpha)$-times
integral operator : \\
$\frac{1}{\Gamma(1-\alpha)}
\int^t_0 (t-s)^{-\alpha} \cdots ds$ to finally reach $\alpha$-times
derivative.
As is described in Section 2, our strategy for the definition of the
fractional derivative in $t$, is to consider $\alpha$-times detivative of $v$
as the inverse to the $\alpha$-times integral, not via $\frac{dv}{dt}$.
Moreover, for initial value problems for time-fractional
differential equations, we meet other complexity for how to pose initial
condition, as the following examples show.
\\
{\bf Example 1.1.}
\\
We consider an initial value problem for a simple time-fractional ordinary
differential equation:
$$
d_t^{\alpha} v(t) = f(t), \quad 0<t<T, \qquad v(0) = a. \eqno{(1.4)}
$$
We remark that (1.4) is not necessarily well-posed for all $\alpha \in (0,1)$,
$f \in L^2(0,T)$ and $a\in \mathbb{R}$,
because of the initial condition.
Especially the order $\alpha \in \left(0, \, \frac{1}{2}\right)$ causes
a difficulty: Choosing $\gamma > -1$ and
$$
f(t) = t^{\gamma} \in L^1(0,T),
$$
in (1.4), we consider
$$
d_t^{\alpha} v(t) = t^{\gamma}, \quad v(0) = a. \eqno{(1.5)}
$$
A solution formula is known:
$$
v(t) = a + \frac{1}{\Gamma(\alpha)}\int^t_0 (t-s)^{\alpha-1}s^{\gamma} ds
\eqno{(1.6)}
$$
(e.g., (3.1.34) (p.141) in Kilbas, Srivastava and Trujillo \cite{KST}).
For $0<\alpha<1$ and $\gamma > -1$, the right-hand side of (1.6) makes
sense and we can obtain
$$
v(t) = a + \frac{\Gamma(\gamma+1)}{\Gamma(\alpha+\gamma+1)}
t^{\alpha+\gamma}, \quad t>0. \eqno{(1.7)}
$$
However, the initial condition $v(0) = a$ is delicate:
\\
(i) Let $\alpha + \gamma > 0$. Then we can readily see that $v$ given by
(1.7) satisfies (1.5).
\\
(ii) Let $\alpha + \gamma = 0$. Then (1.7)
provides that $v(t) = a + \Gamma(1-\alpha)$.
However this $v$ does not satisfy $d_t^{\alpha} v = t^{-\alpha}$.
\\
(iii) Let $\alpha + \gamma < 0$. Then for $v(t)$ defined by (1.7), we see
that $\lim_{t\downarrow 0} v(t) = \infty$.
Therefore (1.7) cannot give a solution to (1.5), and moreover we are
not sure whether there exists a solution to (1.5).
\\
We have another classical fractional derivative called the
Riemann-Liouville derivative:
$$
D_t^{\alpha} v(t) = \frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int^t_0
(t-s)^{-\alpha} v(s) ds, \quad 0<t<T
\eqno{(1.8)}
$$
for $0 < \alpha < 1$, provided that the right-hand side exists.
Let $f\in L^2(0,T)$ be arbitrarily given.
In terms of $D_t^{\alpha}$, we can formulate an initial value problem:
$$
D_t^{\alpha} u(t) = f(t), \quad 0<t<T, \quad (J^{1-\alpha}u)(0) = a.
\eqno{(1.9)}
$$
The solution formula is
$$
u(t) = \frac{a}{\Gamma(\alpha)}t^{\alpha-1}
+ J^{\alpha}f(t), \quad 0<t<T \eqno{(1.10)}
$$
(e.g., Kilbas, Srivastava and Trujillo \cite{KST}, p.138).
Indeed, we can directly verify that $u$ given by (1.10) satisfies (1.9):
\begin{align*}
& D_t^{\alpha} J^{\alpha}f(t)
= \frac{1}{\Gamma(\alpha)\Gamma(1-\alpha)}\frac{d}{dt}
\int^t_0 (t-s)^{-\alpha} \left( \int^s_0 (s-\xi)^{\alpha-1}
f(\xi) d\xi \right) ds\\
=& \frac{1}{\Gamma(\alpha)\Gamma(1-\alpha)}\frac{d}{dt}
\int^t_0 f(\xi) \left( \int^t_{\xi} (t-s)^{-\alpha}(s-\xi)^{\alpha-1}
ds \right) d\xi\\
= & \frac{d}{dt}\int^t_0 f(\xi) d\xi = f(t) \quad \mbox{for
$f\in L^1(0,T)$}.
\end{align*}
and
$$
D_t^{\alpha} \left(\frac{a}{\Gamma(\alpha)}t^{\alpha-1}\right)
= \frac{1}{\Gamma(1-\alpha)}
\frac{d}{dt}\int^t_0 (t-s)^{-\alpha} s^{\alpha-1} ds = 0,
\quad 0<t<T,
$$
which means that $D_t^{\alpha} u(t) = f(t)$ for $0<t<T$. We can similarly verify
$$
J^{1-\alpha}\left( \frac{a}{\Gamma(\alpha)}t^{\alpha-1}
+ J^{\alpha}f(t)\right)(0) = a.
$$
By $0<\alpha<\frac{1}{2}$, the formula (1.10) makes sense in $L^1(0,T)$, but
$u\not \in L^2(0,T)$ for each $f \in L^2(0,T)$ if $a\ne 0$.
This suggests that the initial value problem (1.9) may not be well-posed
in $L^2(0,T)$.
Furthermore we can consider other formulation:
$$
D_t^{\alpha} u(t) = f(t), \quad 0<t<T, \quad u(0) = a. \eqno{(1.11)}
$$
We see that $u(t)$ given by (1.10) satisfies $D_t^{\alpha} u = f$ in $(0,T)$,
but it follows from $\alpha-1<0$ that
$\lim_{t\downarrow 0} \vert u(t)\vert = \infty$ by
$\lim_{t\downarrow 0} t^{\alpha-1} = \infty$ if $a\ne 0$, while
if $a=0$, then (1.10) can satisfy $u(0) = 0$, that is,
(1.11) by $f$ in some class such as
$f \in L^{\infty}(0,T)$.
\\
The delicacy in the above examples are more or less known, and
motivates us to construct a uniform framework for time-fractional
derivatives.
Moreover, we are concerned with the range space of
the corresponding solutions for a prescribed function space of $f$,
e.g., $L^2(0,T)$, not only calculations of the derivatives
of individually given function $u$.
Naturally, for the well-posedness of an initial value problem,
we are required to characterize the function space
of solutions corresponding to a space of $f$.
In other words, one of our main interests is to define a fractional
derivative, denoted by $\partial_t^{\alpha}$, and characterize the space of $u$
satisfying $\partial_t^{\alpha} u \in L^2(0,T)$.
Moreover, such $\partial_t^{\alpha}$ should be an extension of $d_t^{\alpha}$ and
$D_t^{\alpha}$ in a minimum sense in order that important properties of these
classical fractional derivatives should be inherited to $\partial_t^{\alpha}$.
Throughout this article, we treat fractional integrations and
fractional derivatives as operators from specified function spaces
to others, that is, we always attach them with their domains and
ranges.
Thus we will define a time fractional derivative denoted by $\partial_t^{\alpha}$ as
suitable extension of $d_t^{\alpha}$ satisfying the requirements:
\begin{itemize}
\item
Such extended derivative $\partial_t^{\alpha}$ admits usual rules of differentiation
as much as possible. For example, $\partial_t^{\alpha}\partial_t^{\beta}
= \partial_t^{\alpha+\beta}$ for all $\alpha, \beta \ge 0$.
\item
It admits a relevant formulation of initial condition even for
$\alpha \in \mathbb{R}$.
\item
In $L^2$-based Sobolev spaces,
there exists a unique solution to an initial value problem for
a time-fractional ordinary differential equation and
an initial boundary value problem for a time-fractional partial differential
equation for $\alpha>0$ and even $\alpha \in \mathbb{R}$.
\end{itemize}
In this article, we intend to outline foundations for a
comprehensive theory for time-fractional differential equations.
Some arguments are based on Gorenflo, Luchko and
Yamamoto \cite{GLY}, Kubica, Ryszewska and Yamamoto \cite{KRY}.
\\
This article is composed of eight sections and one appendix:
\begin{itemize}
\item
Section 2: Definition of the extended derivative $\partial_t^{\alpha}$:\\
We extend $d_t^{\alpha}$ as operartor in order that
it is well-defined as as isomorphism in relevant Sobolev spaces.
We emphasize that our fractional derivative coincides with
the classical Riemann-Liouville derivative and the
Caputo derivative in suitable spaces, and we never aim at
creating novel notions of fractional derivatives but we are
concerned with a formulation of a fractional derivative
allowing us convenient applications to time-fractional
differential equations, as Sections 5 and 6 discuss.
\item
Section 3: Basic properties in fractional calculus
\item
Sectoin 4: Fractional derivatives of the Mittag-Leffler functions
\item
Section 5: Initial value problem for fractional ordinary differential
equations
\item
Section 6: Initial boundary value problem for fractional partial differential
equations: selected topics
\item
Section 7: Applicaion to an inverse source problem:\\
For illustraing the feasibility of our approach, we conside one inverse
source problem of determining a time-varying function.
\item
Section 8: Concluding remarks.
\end{itemize}
\section{Definition of the extended derivative $\partial_t^{\alpha}$}
{\bf \S2.1. Introduction of function spaces and operators}
We set
$$
\left\{ \begin{array}{rl}
&(J^{\alpha}v)(t):= \frac{1}{\Gamma(\alpha)}\int^t_0
(t-s)^{\alpha-1} v(s) ds, \quad 0<t<T, \quad
\mathcal{D}(J^{\alpha}) = L^2(0,T),\\
& (J_{\alpha}v)(t):= \frac{1}{\Gamma(\alpha)}\int^T_t
(\xi-t)^{\alpha-1} v(\xi) d\xi, \quad 0<t<T, \quad
\mathcal{D}(J^{\alpha}) = L^2(0,T).
\end{array}\right.
\eqno{(2.1)}
$$
We can consider $J^{\alpha}, J_{\alpha}$ for $v\in L^1(0,T)$, but
for the moment, we consider them for $v \in L^2(0,T)$.
There are many works on the Riemann-Liouville time-fractional integral
operator $J^{\alpha}$, and we here refer only to
Gorenflo and Vessella \cite{GV} as monograph, and Gorenflo and
Yamamoto \cite{GY} for an operator theoretical approach.
We can directly prove
\\
{\bf Lemma 2.1.}
\\
{\it
Let $\alpha, \beta \ge 0$. Then
$$
J^{\alpha}J^{\beta}v = J^{\alpha+\beta}v, \quad
J_{\alpha}J_{\beta}v = J_{\alpha+\beta}v \quad \mbox{for $v\in L^1(0,T)$}.
$$
}
For $0<\alpha<1$, we define an operator $\tau: L^2(0,T)
\longrightarrow L^2(0,T)$ by
$$
(\tau v)(t):= v(T-t), \quad v\in L^2(0,T).
\eqno{(2.2)}
$$
Then it is readily seen that $\tau$ is an isomorphism between $L^2(0,T)$
and itself.
Throughout this article, we call $K$ an isomorphism between two
Banach spaces $X$ and $Y$ if the mapping $K$ is injective, and
$KX=Y$ and there exists a constant $C>0$ such that
$C^{-1}\Vert Kv\Vert_Y \le \Vert v\Vert_X \le C\Vert Kv\Vert_Y$
for all $v\in X$. Here $\Vert \cdot\Vert_X$ and $\Vert \cdot\Vert_Y$
denote the norms in $X$ and $Y$ respectively.
We set
\begin{align*}
& {_{0}{C^1[0,T]}}:= \{ v\in C^1[0,T];\, v(0) = 0\}, \\
& ^{0}{C^1[0,T]}:= \{ v\in C^1[0,T];\, v(T) = 0\} = \tau\,({_{0}{C^1[0,T]}}).
\end{align*}
By $H^{\alpha}(0,T)$ with $0<\alpha<1$, we denote the Sobolev-Slobodecki space
with the norm $\Vert \cdot\Vert_{H^{\alpha}
(0,T)}$ defined by
$$
\Vert v\Vert_{H^{\alpha}(0,T)}:=
\left( \Vert v\Vert^2_{L^2(0,T)}
+ \int^T_0\int^T_0 \frac{\vert v(t)-v(s)\vert^2}{\vert t-s\vert^{1+2\alpha}}
dtds \right)^{\frac{1}{2}}
$$
(e.g., Adams \cite{Ad}).
By $\overline{X}^{Y}$ we denote the closure of $X\subset Y$ in a normed space
$Y$.
We set
$$
H_{\alpha}(0,T):= \overline{{_{0}{C^1[0,T]}}}^{H^{\alpha}(0,T)},
\quad {^{\alpha}{H}}(0,T):= \overline{^{0}{C^1}[0,T]}^{H^{\alpha}(0,T)}.
\eqno{(2.3)}
$$
Henceforth we set $H_0(0,T):= L^2(0,T)$ and $^{0}{H}(0,T) := L^2(0,T)$.
Then
\\
{\bf Proposition 2.1.}
\\
{\it
Let $0<\alpha<1$.
\\
(i)
$$
H_{\alpha}(0,T) :=
\left\{ \begin{array}{rl}
&H^{\alpha}(0,T), \quad 0 < \alpha < \frac{1}{2}, \\
&\left\{ v\in H^{\frac{1}{2}}(0,T); \thinspace
\int^T_ 0 \frac{\vert v(t)\vert^2}{t} dt < \infty\right\},
\quad \alpha = \frac{1}{2}, \\
&\{v \in H^{\alpha}(0,T);\, v(0) = 0\}, \quad \frac{1}{2} < \alpha \le 1.
\end{array}\right.
$$
Moreover, the norm in $H_{\alpha}(0,T)$ is equivalent to
$$
\Vert v\Vert_{H_{\alpha}(0,T)} :=
\left\{ \begin{array}{rl}
\Vert v\Vert_{H^{\alpha}(0,T)}, \quad & \alpha\ne \frac{1}{2}, \\
\left( \Vert v\Vert^2_{H^{\frac{1}{2}}(0,T)} +
\int^T_0 \frac{\vert v(t)\vert^2}{t} dt \right)^{\frac{1}{2}},
\quad & \alpha = \frac{1}{2}.
\end{array}\right.
$$
(ii) Similarly we have
$$
{^{\alpha}{H}}(0,T) :=
\left\{ \begin{array}{rl}
&H^{\alpha}(0,T), \quad 0 < \alpha < \frac{1}{2}, \\
&\left\{ v\in H^{\frac{1}{2}}(0,T); \thinspace
\int^T_ 0 \frac{\vert v(t)\vert^2}{T-t} dt < \infty\right\},
\quad \alpha = \frac{1}{2}, \\
&\{v \in H^{\alpha}(0,T);\, v(T) = 0\}, \quad \frac{1}{2} < \alpha \le 1.
\end{array}\right.
$$
Moreover, the norm in $\, ^{\alpha}H(0,T)$ is equivalent to
$$
\Vert v\Vert_{{^{\alpha}{H}}(0,T)} :=
\left\{ \begin{array}{rl}
\Vert v\Vert_{H^{\alpha}(0,T)}, \quad & \alpha\ne \frac{1}{2}, \\
\left( \Vert v\Vert^2_{H^{\frac{1}{2}}(0,T)} +
\int^T_0 \frac{\vert v(t)\vert^2}{T-t} dt \right)^{\frac{1}{2}},
\quad & \alpha = \frac{1}{2}.
\end{array}\right.
$$
}
\\
We can refer to \cite{KRY} as for the proof of Proposition 2.1 (i).
Since $\tau: H_{\alpha}(0,T) \longrightarrow {^{\alpha}{H}}(0,T)$ is an isomorphism, we can
verify part (ii) of the proposition.
\\
\vspace{0.2cm}
\\
{\bf \S2.2. Extension of $d_t^{\alpha}$ to $H_{\alpha}(0,T)$: intermediate step}
We can prove (e.g., \cite{GLY}, \cite{KRY}):
\\
{\bf Proposition 2.2.}
\\
{\it
Let $0<\alpha<1$. Then
$$
J^{\alpha}: L^2(0,T) \longrightarrow H_{\alpha}(0,T)
$$
is an isomorphism. In particular, $H_{\alpha}(0,T) = J^{\alpha}L^2(0,T)$.
}
\\
By the proposition, we can easily verify that some functions belong to
$H_{\alpha}(0,T)$, although the verification by (2.3) is complicated.
\\
{\bf Example 2.1.}
\\
In view of Proposition 2.2, we will verify that
$$
t^{\beta} \in H_{\alpha}(0,T) \quad \mbox{if $\beta > \alpha - \frac{1}{2}$.}
$$
Indeed, setting $\gamma := \beta - \alpha$, we see
$\gamma > -\frac{1}{2}$, and so $t^{\gamma} \in L^2(0,T)$.
Moreover,
$$
J^{\alpha}t^{\gamma} = \frac{1}{\Gamma(\alpha)}
\int^t_0 (t-s)^{\alpha-1}s^{\gamma} ds
= \frac{\Gamma(\gamma+1)}{\Gamma(\alpha+\gamma+1)}t^{\alpha+\gamma},
$$
that is,
$$
t^{\beta} = t^{\alpha+\gamma}
= J^{\alpha}\left(
\frac{\Gamma(\alpha+\gamma+1)}{\Gamma(\gamma+1)}t^{\gamma}\right)
\in J^{\alpha}L^2(0,T).
$$
Thus we see that $t^{\beta} \in H_{\alpha}(0,T)$.
\\
In Proposition 2.4 in \cite{KRY}, the direct proof for $t^{\beta}
\in H_{\alpha}(0,T)$ is given for more restricted $\beta>0$ and
$0 < \alpha < \frac{1}{2}$, which is more complicated than the proof here.
We set
$$
\partial_t^{\alpha} := (J^{\alpha})^{-1} = J^{-\alpha} \quad \mbox{with}
\quad \mathcal{D}(\partial_t^{\alpha}) = H_{\alpha}(0,T)
= J^{\alpha}L^2(0,T). \eqno{(2.4)}
$$
The essence of this extension of $\partial_t^{\alpha}$ is that we define $\partial_t^{\alpha}$
as the inverse to the isomorphism
$J^{\alpha}$ on $L^2(0,T)$ onto $H_{\alpha}(0,T)$.
For estimating or treating $\partial_t^{\alpha} v = g$ later,
we will often consider through $J^{\alpha}g$, as is already calculated
in Example 2.1.
\\
From Proposition 2.2 we can directly derive
\\
{\bf Proposition 2.3.}\\
{\it
There exists a constant $C>0$ such that
$$
C^{-1}\Vert v\Vert_{H_{\alpha}(0,T)} \le \Vert \partial_t^{\alpha} v\Vert_{L^2(0,T)}
\le C\Vert v\Vert_{H_{\alpha}(0,T)}
$$
for all $v\in H_{\alpha}(0,T)$.
}
\\
{\bf Example 2.2.}
\\
We return to Example 2.1. Let $\beta > \alpha - \frac{1}{2}$.
Then $t^{\beta} \in H_{\alpha}(0,T)$ and
$$
J^{\alpha}t^{\beta-\alpha} = \frac{\Gamma(\beta-\alpha+1)}
{\Gamma(\beta+1)}t^{\beta}.
$$
Hence, by the definition of $\partial_t^{\alpha}$, we obtain
$$
t^{\beta-\alpha} = \frac{\Gamma(\beta-\alpha+1)}
{\Gamma(\beta+1)}\partial_t^{\alpha} t^{\beta},
$$
that is,
$$
\partial_t^{\alpha} t^{\beta} = \frac{(\beta+1)}
{\Gamma(\beta-\alpha +1)}t^{\beta-\alpha} \quad
\mbox{if $\beta > \alpha - \frac{1}{2}$ and $0<\alpha<1$.}
\eqno{(2.5)}
$$
If $0<\alpha<\frac{1}{2}$, then $\beta < 0$ is possible and for
$\beta < 0$ we have $t^{\beta} \not\in W^{1,1}(0,T)$. Therefore
$d_t^{\alpha} t^{\beta}$ cannot be defined directly.
\\
Next we will define $\partial_t^{\alpha}$ for general $\alpha > 0$.
Let $\alpha := m + \sigma$ with $m\in \mathbb{N} \cup \{0\}$ and
$0< \sigma \le 1$. Then we define
$$
\left\{ \begin{array}{rl}
& H_m(0,T) :=
\left\{ \begin{array}{rl}
& \left\{ v \in H^m(0,T);\,
v(0) = \cdots = \frac{d^{m-1}v}{dt^{m-1}}(0) = 0\right\},
\quad m \ge 1, \\
& L^2(0,T), \quad m =0, \\
\end{array}\right.
\\
& H_{m+\sigma}(0,T) := \left\{ v \in H_m(0,T);\,
\frac{d^mv}{dt^m}\in H_{\sigma}(0,T)\right\}
\end{array}\right.
\eqno{(2.6)}
$$
and
$$
\Vert v\Vert_{H_{m+\sigma}(0,T)}
:= \left( \left\Vert \frac{d^m}{dt^m}v\right\Vert^2_{L^2(0,T)}
+ \left\Vert \frac{d^m}{dt^m}v\right\Vert^2_{H_{\sigma}(0,T)}
\right)^{\frac{1}{2}}.
$$
We can easily verify that $H_{m+\sigma}(0,T)$ is a Banach space.
\\
We similarly define ${^{\alpha}{H}}(0,T)$ for each $\alpha>0$.
Then, in view of Proposition 2.2, we can prove
\\
{\bf Proposition 2.4.}
\\
{\it
Let $m \in \mathbb{N} \cup \{ 0\}$ and $0 < \sigma \le 1$.
Then $J^{m+\sigma}: L^2(0,T) \longrightarrow H_{m+\sigma}(0,T)$ is
an isomorphism.}
\\
{\bf Proof.}
\\
We can assume that $m\ge 1$.
By the definition, $v \in H_{m+\sigma}(0,T)$ if and only if
$$
\frac{d^m}{dt^m}v \in H_{\sigma}(0,T), \quad
v(0) = \cdots = \frac{d^{m-1}}{dt^{m-1}}v(0) = 0,
$$
which implies that $\frac{d^m}{dt^m}v = J^{\sigma}w$ with some
$w\in L^2(0,T)$. By $v(0) = \cdots = \frac{d^{m-1}}{dt^{m-1}}v(0) = 0$,
we see that
$$
v(t) = \frac{1}{(m-1)!}\int^t_0 (t-s)^{m-1}\frac{d^m}{dt^m}v(s)ds,
\quad 0<t<T.
$$
Therefore, exchanging the order of the integral, we have
\begin{align*}
& v(t) = \frac{1}{(m-1)!}\int^t_0 (t-s)^{m-1}\left(
\frac{1}{\Gamma(\sigma)}\int^s_0 (s-\xi)^{\sigma-1} w(\xi)d\xi
\right) ds\\
=& \frac{1}{(m-1)!\Gamma(\sigma)} \int^t_0 \left(
\int^t_{\xi} (t-s)^{m-1}(s-\xi)^{\sigma-1} ds\right) w(\xi)d\xi
= \frac{1}{\Gamma(m+\sigma)}\int^t_0 (t-\xi)^{m+\sigma-1}w(\xi) d\xi\\
=& J^{m+\sigma}w(t) \quad \mbox{for $0<t<T$.}
\end{align*}
Hence, $v \in J^{m+\sigma}L^2(0,T)$, which means that
$J^{m+\sigma}L^2(0,T) \supset H_{m+\sigma}(0,T)$. The converse inclusion is
direct, and we see that $J^{m+\sigma}L^2(0,T) = H_{m+\sigma}(0,T)$.
The norm equivalence between $\Vert v\Vert_{L^2(0,T)}$ and
$\Vert J^{m+\sigma}v\Vert_{H_{m+\sigma}(0,T)}$, readily follows from
the definition and Proposition 2.2.
$\blacksquare$
\\
For $\alpha = m + \sigma$ with $m\in \mathbb{N}$ and $0<\sigma \le 1$,
we now define $\partial_t^{\alpha}$ as extension of the Caputo derivative
$$
d_t^{\alpha} v(t) = \frac{1}{\Gamma(m+1-\alpha)} \int^t_0 (t-s)^{m-\alpha}
\frac{d^{m+1}}{ds^{m+1}}v(s) ds, \quad 0<t<T.
$$
We note that $d_t^{\alpha}$ requires $(m+1)$-times differentiablity of $v$.
By Proposition 2.4, the inverse to $J^{\alpha}$ exists for each
$\alpha \ge 0$, and by $J^{-\alpha}$ we denote the inverse:
$$
J^{-\alpha}:= (J^{\alpha})^{-1}.
$$
As the extension of such $d_t^{\alpha}$ to $H_{m+\sigma}(0,T)$, we define
$$
\partial_t^{m+\sigma} = J^{-m-\sigma} \quad \mbox{with} \quad
\mathcal{D}(\partial_t^{m+\sigma}) = H_{m+\sigma}(0,T). \eqno{(2.7)}
$$
Thus
\\
{\bf Proposition 2.5.}\\
{\it
Let $\alpha, \beta \ge 0$.
\\
(i) $$
\partial_t^{\alpha} : H_{\alpha+\beta}(0,T) \longrightarrow H_{\beta}(0,T)
$$
and
$$
J^{\alpha} : H_{\beta}(0,T) \longrightarrow H_{\alpha+\beta}(0,T)
$$
are isomorphisms.
\\
(ii) It holds
$$
J^{-\alpha}J^{\beta} = J^{-\alpha+\beta} \quad \mbox{on
$\mathcal{D}(J^{-\alpha+\beta})$}.
$$
Here we set
$$
\mathcal{D}(J^{-\alpha+\beta}) =
\left\{ \begin{array}{rl}
& L^2(0,T) \quad \mbox{if $-\alpha + \beta \ge 0$},\\
& H_{\alpha-\beta}(0,T) \quad \mbox{if $-\alpha + \beta < 0$}.
\end{array}\right.
$$
}
\\
{\bf Proof.}
\\
Part (i) is seen by (2.7) and Proposition 2.4.
Part (ii) can be proved as follows. Let $-\alpha+\beta \ge 0$.
If $\alpha = \beta$, then $J^{-\alpha}J^{\alpha} = I$: the identity
operator on $L^2(0,T)$ by (2.7) and the conclusion is trivial.
Let $\beta > \alpha$. Set $\gamma := \beta - \alpha > 0$.
Let $v\in L^2(0,T)$ be arbitrary. Then, by Lemma 2.1, we have
$$
J^{-\alpha}J^{\beta}v = J^{-\alpha}(J^{\alpha+\gamma}v)
= J^{-\alpha}(J^{\alpha}J^{\gamma})v = (J^{-\alpha}J^{\alpha})J^{\gamma}v
= J^{\gamma}v.
$$
Since $J^{\gamma}v = J^{-\alpha+\beta}v$,
we obtain $J^{-\alpha}J^{\beta}v = J^{-\alpha+\beta}v$ for
each $v \in L^2(0,T)$.
Next let $-\alpha + \beta < 0$. Given $v \in H_{\alpha-\beta}(0,T)$
arbitrarily,
by Proposition 2.4 we can find $w \in L^2(0,T)$ such that
$v = J^{\alpha-\beta}w$. Then
$$
J^{-\alpha}J^{\beta}v = J^{-\alpha}J^{\beta}(J^{\alpha-\beta}w)
= J^{-\alpha}(J^{\beta+(\alpha-\beta)}w) = J^{-\alpha}J^{\alpha}w = w
$$
by $\beta, \alpha-\beta > 0$ and Lemma 2.1. Therefore,
$J^{-\alpha}J^{\beta}v = w$. Since $v=J^{\alpha-\beta}w$
implies $w = (J^{\alpha-\beta})^{-1}v$, we have
$$
w = (J^{\alpha-\beta})^{-1}v
= J^{-(\alpha-\beta)}v = J^{-\alpha+\beta}v,
$$
that is, $J^{-\alpha}J^{\beta}v = J^{-\alpha+\beta}v$ for each
$v\in H_{\alpha-\beta}(0,T)$.
Thus the proof of Proposition 2.5 is complete.
$\blacksquare$
\\
Next, for $0<\alpha<1$, we characterize $\partial_t^{\alpha}$ as extension over
$H_{\alpha}(0,T)$ of
the operator $d_t^{\alpha}$ defined on ${_{0}{C^1[0,T]}}$.
Let $X$ and $Y$ be Banach spaces with the norms $\Vert \cdot\Vert_X$
and $\Vert \cdot\Vert_Y$ respectively, and let $K: X \longrightarrow Y$
be a densely defined linear operator. We call $K$ a closed operator
if $v_n \in \mathcal{D}(K)$, $\lim_{n\to\infty} v_n = v$ and $Kv_n$ converges to some
$w$ in $Y$, then $v \in \mathcal{D}(K)$ and $Kv = w$.
By $\overline{K}$ we denote the closure of an operator $K$ from $X$ to $Y$, that is,
$\overline{K}$ is the minimum closed extension of $K$ in the sense that if
$\widetilde{K}$ is a closed operator such that $\mathcal{D}(\widetilde{K}) \supset
\mathcal{D}(K)$, then $\mathcal{D}(\widetilde{K}) \supset \mathcal{D}(\overline{K})$.
It is trivial that $\overline{K} = K$ for a closed operator $K$.
Moreover we say that $K$ is closable if there exists $\overline{K}$.
Here we always consider the operator $d_t^{\alpha}$ with the domain
${_{0}{C^1[0,T]}}$: $d_t^{\alpha}: {_{0}{C^1[0,T]}} \subset H_{\alpha}(0,T) \, \longrightarrow\, L^2(0,T)$.
Then we can prove (e.g., \cite{KRY}):
\\
{\bf Proposition 2.6.}
\\
{\it The operator $d_t^{\alpha}$ with the domain ${_{0}{C^1[0,T]}}$ is closable and
$\overline{d_t^{\alpha}} = \partial_t^{\alpha}$.
}
\\
Proposition 2.6 is not used later but means that our derivative
$\partial_t^{\alpha}$ is reasonable as the minimum extension of the classical Caputo
derivative
$d_t^{\alpha}$ for $v\in {_{0}{C^1[0,T]}}$.
\\
\vspace{0.2cm}
\\
{\bf Interpretation of $\mathcal{D}(\partial_t^{\alpha}) = H_{\alpha}(0,T)$.}
\\
Let $\alpha > \frac{1}{2}$ and let $v \in H_{\alpha}(0,T)$. By the definition (2.3)
of $H_{\alpha}(0,T)$, we can choose an approximating sequence $v_n\in {_{0}{C^1[0,T]}}$,
$n\in \mathbb{N}$ such that $\lim_{n\to\infty} \Vert v_n-v\Vert
_{H_{\alpha}(0,T)} = 0$. By the Sobolev embedding (e.g., Adams \cite{Ad}),
we see that
$H_{\alpha}(0,T) \subset H^{\alpha}(0,T) \subset C[0,T]$ if $\alpha>\frac{1}{2}$,
so that $v \in C[0,T]$ and $\lim_{n\to\infty} \Vert v_n-v\Vert
_{C[0,T]} = 0$. Therefore,
$$
\lim_{n\to\infty} \vert v(0) - v_n(0)\vert
\le \lim_{n\to\infty} \Vert v_n-v\Vert_{C[0,T]} = 0,
$$
that is, $v(0) = \lim_{n\to\infty} v_n(0)$. Therefore,
$v\in H_{\alpha}(0,T)$ with $\alpha > \frac{1}{2}$, yields $v(0) = 0$.
In other words, if $\alpha > \frac{1}{2}$, then $v\in \mathcal{D}(\partial_t^{\alpha})$
means that $v=v(t)$ satisfies the zero initial condition, which is
useful for the formulation of the initial value problem in Sections 5 and 6.
We can similarly consider and see that
$v \in \mathcal{D}(\partial_t^{\alpha})$ with $\alpha > \frac{3}{2}$ yields
$v(0) = \partial_tv(0) = 0$.
\\
As is directly proved by the Young inequality on the convolution, we see that
$$
D_t^{\alpha} v \in L^1(0,T) \quad \mbox{if $v\in W^{1,1}(0,T)$}.
$$
However, we do not necessarily have $D_t^{\alpha} v \in L^2(0,T)$.
Indeed,
$$
D_t^{\alpha} t^{\beta} = \frac{\Gamma(1+\beta)}{\Gamma(1-\alpha+\beta)}
t^{\beta-\alpha} \quad \mbox{for $\beta > 0$.}
$$
If $\beta - \alpha < -\frac{1}{2}$ and $\beta > 0$,
then $D_t^{\alpha} t^{\beta} \not\in L^2(0,T)$,
in spite of $t^{\beta} \in W^{1,1}(0,T)$.
This means that if we want to keep
the range of $D_t^{\alpha}$ within $L^2(0,T)$, then even $W^{1,1}(0,T)$ of a space
of differentiable functions is not sufficient, although $D_t^{\alpha}$ are concerned
with $\alpha$-times differentiability.
\\
In general, we can readily prove
\\
{\bf Proposition 2.7.}
\\
{\it
We have
$$
\partial_t^{\alpha} v = D_t^{\alpha} v = d_t^{\alpha} v \quad \mbox{for $v \in {_{0}{C^1[0,T]}}$} \eqno{(2.8)}
$$
and
$$
\partial_t^{\alpha} v(t) = \frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}
\int^t_0 (t-s)^{-\alpha} v(s) ds = D_t^{\alpha} v(t) \quad \mbox{for
$v \in H_{\alpha}(0,T)$.} \eqno{(2.9)}
$$
}
\\
{\bf Proof.}
\\
Equation (2.8) is easily seen. We will prove (2.9) as follows.
For arbitrary $v \in H_{\alpha}(0,T)$, Proposition 2.2 yields that
$v = J^{\alpha}w$ with some $w \in L^2(0,T)$ and $w = \partial_t^{\alpha} v$.
On the other hand, Lemma 2.1 implies
$$
D_t^{\alpha} v = \frac{d}{dt}J^{1-\alpha}v = \frac{d}{dt}J^{1-\alpha}
(J^{\alpha}w)
= \frac{d}{dt}Jw = w.
$$
Therefore, $D_t^{\alpha} v = \partial_t^{\alpha} v$, and then the proof of Proposition 2.7 is
complete.
$\blacksquare$
\\
Equation (2.9) means that $\partial_t^{\alpha}$ coincides with the
Riemann-Liouville fractional derivative, provided that we consider
$H_{\alpha}(0,T)$ as the domain of $\partial_t^{\alpha}$ and $D_t^{\alpha}$.
This extension $\partial_t^{\alpha}$ of $d_t^{\alpha}\vert
_{{_{0}{C^1[0,T]}}}$ is not yet complete, and in the next subsection,
we will continue to extend.
\\
\vspace{0.2cm}
\\
{\bf \S2.3. Definition of $\partial_t^{\alpha}$: completion of the extension of
$d_t^{\alpha}$}
By the current extension of $\partial_t^{\alpha}$, we understand that
$\partial_t^{\alpha} 1 = 0$ for $0<\alpha<\frac{1}{2}$, but
$\partial_t^{\alpha} 1$ cannot be defined for $\frac{1}{2} \le \alpha < 1$:
$$
\left\{ \begin{array}{rl}
& 1 \in H_{\alpha}(0,T) = \mathcal{D}(\partial_t^{\alpha}) \quad \mbox{$0<\alpha<\frac{1}{2}$}, \\
& 1 \not\in \mathcal{D}(\partial_t^{\alpha}) \quad \mbox{$\frac{1}{2} \le \alpha<1$}.
\end{array}\right.
$$
Moreover, we note that $d_t^{\alpha} 1 = 0$ and
$D_t^{\alpha} 1 = \frac{t^{-\alpha}}{\Gamma(1-\alpha)}$ for all $\alpha \in (0,1)$.
Therefore, $\partial_t^{\alpha} 1$ is not consistent with neither the classical
fractional derivative $d_t^{\alpha}$ nor $D_t^{\alpha}$, which
suggests that our current extension from
$d_t^{\alpha}$ to $\partial_t^{\alpha}$ is not sufficient as fractional derivative.
Moreover, as is seen in Section 7, the current $\partial_t^{\alpha}$ is not convenient for
treating less regular source terms in fractional differential equations.
In order to define $\partial_t^{\alpha} v$ in more general spaces such as $L^2(0,T)$, we
should continue to extend $\partial_t^{\alpha}$.
We recall (2.1) and (2.3) for $\alpha>0$.
We can readily verify that $\tau: H_{\alpha}(0,T) \longrightarrow
{^{\alpha}{H}}(0,T)$ is an isomorphism with $\tau$ defined by (2.2).
Then
\\
{\bf Proposition 2.8.}
\\
{\it
Let $\alpha>0$ and $\beta \ge 0$.
Then $J_{\alpha}: \,^{\beta}{H}(0,T)\, \longrightarrow\,
^{\alpha+\beta}{H}(0,T)$
is an isomorphism}.
\\
We recall that $J^{\alpha}: H_{\beta}(0,T)\, \longrightarrow \,
H_{\alpha+\beta}(0,T)$ is an isomorphism by Proposition 2.5 (i).
Proposition 2.8 can be proved via
the mapping $\tau: L^2(0,T) \longrightarrow L^2(0,T)$ defined by (2.2).
\\
When $J^{\alpha}$ remains an operator defined over $L^2(0,T)$, the operator
$J^{\alpha}$ is not defined for $f \in L^1(0,T)$, but
the integral $\frac{1}{\Gamma(\alpha)}\int^t_0
(t-s)^{\alpha-1} f(s) ds$ itself exists as a function in $L^1(0,T)$ for
any $f\in L^1(0,T)$.
This is a substantial inconvenience, and we have to make
suitable extension of $J^{\alpha}\vert_{L^2(0,T)}$.
Our formulation is based on $L^2$-space and we cannot directly treat
$L^1(0,T)$-space. Thus we introduce the dual space of $H_{\alpha}(0,T)$ which
contains $L^1(0,T)$.
The key idea for the further extension of $\partial_t^{\alpha}$,
is the family $\{ H_{\alpha}(0,T)\}_{\alpha > 0}$ and
the isomorphism $\partial_t^{\alpha}: H^{\alpha+\beta}(0,T) \,
\longrightarrow \, H^{\beta}(0,T)$ with $\alpha,\beta>0$, which is called
a Hilbert scale. We can refer to Chapter V in Amann \cite{Am} as for
a general reference.
Let $X$ be a Hilbert space over $\mathbb{R}$ and let $V \subset X$ be a dense subspace
of $X$, and the embedding $V \longrightarrow X$ be continuous.
By the dual space $X'$ of $X$, we call the space of all the bounded linear
functionals defined on $X$.
Then, identifying the dual space $X'$ with itself, we can
conclude that $X$ is a dense subspace of the dual space $V'$ of $V$:
$$
V \subset X \subset V'.
$$
By $\,_{V'}<f, \, v>_V$, we denote the value of $f\in V'$ at $v \in V$.
We note that
$$
\, _{V'}<f, \, v>_V = (f,v)_X \quad \mbox{if $f\in X$},
$$
where $(f,v)_X$ is the scalar product in $X$.
We note that $H_{\alpha}(0,T)$ and ${^{\alpha}{H}}(0,T)$ are both dense in $L^2(0,T)$.
Henceforth, identifying the dual space $L^2(0,T)'$ with itself,
by the above manner, we can define $(H_{\alpha}(0,T))'$ and $({^{\alpha}{H}}(0,T))'$. Then
$$
\left\{ \begin{array}{rl}
& H_{\alpha}(0,T) \subset L^2(0,T) \subset (H_{\alpha}(0,T))' =: H_{-\alpha}(0,T),\\
& {^{\alpha}{H}}(0,T) \subset L^2(0,T) \subset ({^{\alpha}{H}}(0,T))'
=:\, {{^{-\alpha}{H}}(0,T)}, \quad \alpha>0,
\end{array}\right.
\eqno{(2.10)}
$$
where the above inclusions mean dense subsets.
We refer to e.g., Brezis \cite{Bre}, Yosida \cite{Yo}
as for general treatments for dual spaces.
Let $X, Y$ be Hilbert spaces and let $K: X \longrightarrow Y$ be a bounded
linear operator with $\mathcal{D}(K) = X$. Then we recall that the dual operator
$K'$ is the maximum operator among operators $\widehat{K}: Y' \longrightarrow X'$ with
$\mathcal{D}(\widehat{K}) \subset Y'$ such that
$_{X'}{<} \widehat{K}y, x>_{X} \,=\, {_{Y'}{<} y, Kx>_{Y}}$
for each $x\in X$ and
$y \in \mathcal{D}(\widehat{K}) \subset Y'$ (e.g., \cite{Bre}).
Henceforth, we consider the dual operator $(J^{\alpha})'$ of $J^{\alpha}:
L^2(0,T) \longrightarrow H_{\alpha}(0,T)$ and the dual $(J_{\alpha})'$ of
$J_{\alpha}: L^2(0,T) \longrightarrow \,
^{\alpha}H(0,T)$ by setting $X = L^2(0,T)$ and
$Y = H_{\alpha}(0,T)$ or $Y=\, ^{\alpha}H(0,T)$.
Then we can show
\\
{\bf Proposition 2.9.}
\\
{\it
Let $\alpha>0$ and $\beta \ge 0$. Then:
\\
(i) $(J^{\alpha})': H_{-\alpha-\beta}(0,T) \longrightarrow H_{-\beta}(0,T)$ is
an isomorphism.
\\
In particular, $(J^{\alpha})': H_{-\alpha}(0,T) \longrightarrow L^2(0,T)$ is
an isomorphism.
\\
(ii) $(J_{\alpha})' :\, {^{-\alpha-\beta}{H}(0,T)} \longrightarrow \,
^{-\beta}{H}(0,T)$ is an isomorphism.
\\
In particular,
$(J_{\alpha})' :\, {^{-\alpha}{H}(0,T)} \longrightarrow L^2(0,T)$ is
an isomorphism.
\\
(iii) It holds:
$$
J^{\alpha}v = (J_{\alpha})'v \quad \mbox{for $v \in L^2(0,T)$}.
$$
}
Henceforth we write $J_{\alpha}':= (J_{\alpha})'$.
We see
$$
J_{\alpha}'(J_{\alpha}')^{-1}w = w \quad \mbox{for $w \in L^2(0,T)$}
$$
and
$$
(J_{\alpha}')^{-1}J_{\alpha}'w = w \quad \mbox{for $w \in {^{-\alpha}{H}}(0,T)$}.
$$
\\
{\bf Proof.} From Proposition 2.8, in terms of the closed range
theorem (e.g., Section 7 of Chapter 2 in \cite{Bre}),
we can see (i) and (ii).
We now prove (iii).
We can directly verify $(J^{\alpha}v, \, w)_{L^2(0,T)}
= (v, \, J_{\alpha}w)_{L^2(0,T)}$ for each $v,w \in L^2(0,T)$.
Hence, by the maximality property of the dual operator $J_{\alpha}'$
of $J_{\alpha}$, we see that $J_{\alpha}' \supset J^{\alpha}$.
Thus the proof of Proposition 2.9 is complete.
$\blacksquare$
\\
Thanks to Propositions 2.5 and 2.8-2.9, for $\beta \ge 0$,
we can regard the operators $J_{\alpha}$ with $\mathcal{D}(J_{\alpha})
=\, {^{\beta}{H}}(0,T)$ and $J^{\alpha}$ with
$\mathcal{D}(J^{\alpha}) = H_{\beta}(0,T)$, and accordingly also
the operators $J_{\alpha}'$ with $\mathcal{D}(J_{\alpha}')
= \, ^{-\alpha-\beta}{H}(0,T)$ and
$(J^{\alpha})'$ with $\mathcal{D}((J^{\alpha})') = H_{-\alpha-\beta}(0,T)$.
We do not specify the domains if we need not emphasize them.
\\
We can define $J_{\alpha}'u$ for $u \in L^1(0,T)$ as follows.
We note that $J_{\alpha}'$ is the dual operator of
$J_{\alpha}: \, ^{\gamma}H(0,T)\, \longrightarrow\, ^{\alpha+\gamma}H(0,T)$,
where we choose $\gamma > 0$ such that $\mathcal{D}(J_{\alpha}')
\supset L^1(0,T)$.
To this end, choosing $\gamma > 0$ such that
$\alpha + \gamma > \frac{1}{2}$, we regard $J_{\alpha}'$ as an
operator :
${^{-\alpha-\gamma}{H}}(0,T)\, \longrightarrow \, {^{-\gamma}{H}}(0,T)$.
Then we can define $J_{\alpha}'u$ for $u\in L^1(0,T)$.
Indeed, the Sobolev embedding yields that
$$
{^{\alpha+\gamma}{H}}(0,T) \subset H^{\alpha+\gamma}(0,T)
\subset C[0,T]
$$
by $\alpha + \gamma > \frac{1}{2}$. Therefore, any $u \in L^1(0,T)$ can be
considered as an element in $(^{\alpha+\gamma}{H}(0,T))'$ by
$$
_{^{-\alpha-\gamma}H(0,T)}< u, \, \varphi >_{^{\alpha+\gamma}H(0,T)}
\, = \int^T_0 u(t)\varphi(t) dt \quad \mbox{for $\varphi \in \,
{^{\alpha+\gamma}{H}(0,T)}$},
$$
which means that $L^1(0,T) \subset \mathcal{D}(J_{\alpha}')
= \ ^{-\alpha-\gamma}H(0,T)$ and
$J_{\alpha}'u$ is well-defined for $u \in L^1(0,T)$.
Henceforth, we always make the above definition of $J_{\alpha}'u$ for
$u\in L^1(0,T)$, if not specified.
Then we can improve Proposition 2.9 (iii) as
\\
{\bf Proposition 2.10.}
\\
{\it
We have
$$
J_{\alpha}'u = \frac{1}{\Gamma(\alpha)}
\int^t_0 (t-s)^{\alpha-1} u(s) ds, \quad 0<t<T \quad
\mbox{for $u \in L^1(0,T)$}. \eqno{(2.11)}
$$
}
\\
We can write (2.11) as $J_{\alpha}'u = J^{\alpha}u$ for $u \in L^1(0,T)$,
when we do not specify the domain $\mathcal{D}(J^{\alpha})$.
\\
\vspace{0.1cm}
\\
{\bf Proof of Prposition 2.10.}
\\
Let $\gamma > \frac{1}{2}$. We consider $J_{\alpha}'$ as an operator
$J_{\alpha}':\, {^{-\alpha-\gamma}{H}}(0,T)\, \longrightarrow \,
{^{-\gamma}{H}}(0,T)$.
Let $u \in L^1(0,T)$ be arbitrary. Since $L^1(0,T)
\subset \, {^{-\gamma}{H}}(0,T)$ and $L^2(0,T)$ is dense in
$L^1(0,T)$, we can choose $u_n\in L^2(0,T)$, $n\in \mathbb{N}$ such that
$u_n \longrightarrow u$ in $L^1(0,T)$ as $n\to \infty$.
By Proposition 2.9 (iii), we have
$J_{\alpha}'u_n = J^{\alpha}u_n$ for $n\in \mathbb{N}$. By
Proposition 2.9 (ii), we obtain
$$
J_{\alpha}'u_n \longrightarrow J_{\alpha}'u \quad \mbox{in
${^{-\gamma}{H}}(0,T)$}. \eqno{(2.12)}
$$
On the other hand, by $u \in L^1(0,T)$, the Young inequality on the
convolution implies
$$
\Vert J^{\alpha}u_n - J^{\alpha}u\Vert_{L^1(0,T)}\le
\frac{1}{\Gamma(\alpha)}\Vert s^{\alpha-1}\Vert_{L^1(0,T)}
\Vert u_n - u\Vert_{L^1(0,T)},
$$
which yields $J^{\alpha}u_n = J_{\alpha}'u_n \, \longrightarrow\,
J^{\alpha}u$ in $L^1(0,T)$, that is,
$J^{\alpha}u_n \, \longrightarrow\,J^{\alpha}u$ in $^{-\gamma}{H}(0,T)$.
Therefore, with (2.12), we obtain $J_{\alpha}'u = J^{\alpha}u$.
Thus the proof of Proposition 2.10 is complete.
$\blacksquare$
\\
Now we complete the extension of $d_t^{\alpha}$ with the domain ${_{0}{C^1[0,T]}}$.
\\
{\bf Definition 2.1.}
\\
{\it
For $\beta \ge 0$, we define
$$
\partial_t^{\alpha} := (J_{\alpha}')^{-1} \quad \mbox{with} \quad
\mathcal{D}(\partial_t^{\alpha}) = \, H_{\beta}(0,T) \quad \mbox{or}\quad
\mathcal{D}(\partial_t^{\alpha}) = \,^{-\beta}H(0,T). \eqno{(2.13)}
$$
\\
}
Thus
\\
{\bf Theorem 2.1.}
{\it
Let $\alpha > 0$ and $\beta \ge 0$. Then
$\partial_t^{\alpha}: \, {^{-\beta}H}(0,T) \, \longrightarrow \,^{-\alpha-\beta}H(0,T)$ and \\
$\partial_t^{\alpha}: H_{\alpha+\beta}(0,T) \, \longrightarrow \, H_{\beta}(0,T)$
are both isomorphisms.
}
\\
The extension $\partial_t^{\alpha}: \, ^{-\beta}H(0,T)\, \longrightarrow\,
^{-\alpha-\beta}H(0,T)$ is not only a
theoretical interest, but also useful for studies of
fractional differential equations even if we consider
all the functions within $L^2(0,T)$, as we will do
in Sections 6 and 7.
We consider a special case $\beta = 0$. Then $(J_{\alpha}')^{-1}:
\mathcal{D}((J_{\alpha}')^{-1}) = L^2(0,T) \, \longrightarrow \,
{^{-\alpha}{H}}(0,T)$. Then, since Proposition 2.9 (iii) yields
$$
(J_{\alpha}')^{-1} \supset (J^{\alpha})^{-1}, \eqno{(2.14)}
$$
we see that this $(J_{\alpha}')^{-1} = \partial_t^{\alpha}$ defined on
$L^2(0,T)$ is an extension defined previously in
$H_{\alpha}(0,T)$. In particular, this extended $\partial_t^{\alpha}$ still satisfies
$$
(J_{\alpha}')^{-1}H_{\alpha}(0,T) = L^2(0,T). \eqno{(2.15)}
$$
Our completely extended derivative $\partial_t^{\alpha}$ of
$d_t^{\alpha}$ with $\mathcal{D}(d_t^{\alpha}) = {_{0}{C^1[0,T]}}$, operates similarly to the
Riemann-Liouville fractional derivative $D_t^{\alpha}$, and
is equivalent to $\frac{d}{dt}J^{1-\alpha}$ associated
with the domain $\, ^{-\beta}H(0,T)$ and the range
$\, ^{-\alpha-\beta}H(0,T)$ with $\alpha>0$ and $\beta \ge 0$.
Proposition 2.10 enables us to calculate $\partial_t^{\alpha} u = f$ provided that
$u, f \in L^1(0,T)$, and we show
\\
{\bf Example 2.3.}
\\
Let $0<\alpha < 1$. Choosing $\alpha+\gamma > \frac{1}{2}$ we consider
$\partial_t^{\alpha}$ with $\mathcal{D}(\partial_t^{\alpha}) = \, ^{-\gamma-\alpha}H(0,T) \supset
L^1(0,T)$.
Then $1 \in \mathcal{D}(\partial_t^{\alpha})$, and we have
$$
\partial_t^{\alpha} 1 = \frac{1}{\Gamma(1-\alpha)}t^{-\alpha}.
$$
Indeed, since $\frac{1}{\Gamma(1-\alpha)}t^{-\alpha}
\in L^1(0,T)$, Proposition 2.10 yields
$$
J_{\alpha}'\left( \frac{1}{\Gamma(1-\alpha)}t^{-\alpha}\right)
= \frac{1}{\Gamma(1-\alpha)}\left(\frac{1}{\Gamma(\alpha)}
\int^t_0 (t-s)^{\alpha-1}s^{-\alpha} ds\right) = 1.
$$
Therefore, the definition justifies $\partial_t^{\alpha} 1 =
\frac{1}{\Gamma(1-\alpha)}t^{-\alpha}$.
\\
Before proceeding to the next section, we will provide two propositions.
\\
{\bf Proposition 2.11.}
\\
{\it
Let $0<\alpha<1$. Then
$$
\partial_t^{\alpha} u(t) = \frac{d}{dt}\left( \frac{1}{\Gamma(1-\alpha)}
\int^t_0 (t-s)^{-\alpha} u(s) ds\right) \quad
\mbox{in $(C^{\infty}_0(0,T))'$}
$$
for $u\in L^1(0,T)$, where $\frac{d}{dt}$ is taken in $(C^{\infty}_0(0,T))'$.
}
In the proposition, we note $\frac{1}{\Gamma(1-\alpha)}
\int^t_0 (t-s)^{-\alpha} u(s) ds \in L^1(0,T) \subset
(C^{\infty}_0(0,T))'$.
We remark that we cannot take the pointwise differentiation
of $\int^t_0 (t-s)^{-\alpha} u(s) ds$ in general for $u \in L^1(0,T)$.
Proposition 2.11 enables us to calculate $\partial_t^{\alpha} u = f$ for
$u\in L^1(0,T)$ even if $\partial_t^{\alpha} u$ cannot be defined within
$L^1(0,T)$, as Example 2.4 (a) shows.
\\
\vspace{0.2cm}
\\
{\bf Example 2.4.}
\\
{\bf (a)}
Let $\frac{1}{2} < \alpha < 1$ and let
$$
h_{t_0}(t) =
\left\{ \begin{array}{rl}
&0, \quad t\le t_0, \\
&1, \quad t>t_0
\end{array}\right.
\eqno{(2.16)}
$$
with arbitrary $t_0\in (0,T)$. We can verify that
$h_{t_0} \in H_{1-\alpha}(0,T)$ if $\alpha > \frac{1}{2}$. Indeed
\begin{align*}
& \int^T_0 \int^T_0 \frac{\vert h_{t_0}(t) - h_{t_0}(s)\vert^2}
{\vert t-s\vert^{3-2\alpha}} dsdt
= 2\int^{t_0}_0 \left( \int^T_{t_0} \frac{1}
{\vert t-s\vert^{3-2\alpha}} dt\right) ds\\
=& \frac{1}{1-\alpha}\int^{t_0}_0 ((t_0-s)^{-2+2\alpha}
- (T-s)^{-2+2\alpha}) ds < \infty
\end{align*}
by $-2+2\alpha > -1$.
Therefore, by Proposition 2.2, we can choose $w\in L^2(0,T)$ such that
$J^{1-\alpha}w = h_{t_0}$. By Proposition 2.11, we can calculate
$\partial_t^{\alpha} w$ because $w \in L^2(0,T) \subset L^1(0,T)$:
$$
\partial_t^{\alpha} w = \frac{d}{dt}\left( \frac{1}{\Gamma(1-\alpha)}
\int^t_0 (t-s)^{-\alpha} w(s) ds\right)
= \frac{d}{dt}J^{1-\alpha}w = \frac{d}{dt}h_{t_0}
= \delta_{t_0},
$$
which is a Dirac delta function satisfying
$$
\, _{(C^{\infty}_0(0,T))'}<\delta_{t_0},\, \psi
>_{C^{\infty}_0(0,T)} \,= \psi(t_0) \quad \mbox{for all
$\psi \in C^{\infty}_0(0,T)$}.
$$
\\
\vspace{0.1cm}
\\
{\bf (b)}
We have
$$
\partial_t^{\alpha} t^{\beta} = \frac{\Gamma(1+\beta)}{\Gamma(1-\alpha+\beta)}
t^{\beta-\alpha} \quad 0<\alpha<1, \, \beta > -1
\eqno{(2.17)}
$$
in $^{-\alpha-\gamma}H(0,T)$ with some $\alpha+\gamma > \frac{1}{2}$.
See Example 2.3 as (2.17) with $\beta=0$.
We remark that $\beta-\alpha < -1$ is possible, and so
$t^{\beta - \alpha} \not\in L^1(0,T)$ may occur.
Indeed, by $\beta > -1$, we can see
$$
\frac{1}{\Gamma(1-\alpha)}\int^t_0 (t-s)^{-\alpha}s^{\beta} ds
= \frac{\Gamma(1+\beta)}{\Gamma(2-\alpha+\beta)}t^{1-\alpha+\beta}.
$$
Taking the derivative in the sense of $(C_0^{\infty}(0,T))'$ to see
\begin{align*}
& \frac{d}{dt}\left(
\frac{\Gamma(1+\beta)}{\Gamma(2-\alpha+\beta)}t^{1-\alpha+\beta}
\right)\\
= & \frac{\Gamma(1+\beta)}{\Gamma(2-\alpha+\beta)}
(1-\alpha+\beta)t^{\beta-\alpha}
= \frac{\Gamma(1+\beta)}{\Gamma(1-\alpha+\beta)}t^{\beta-\alpha}.
\end{align*}
Thus (2.17) is verified.
We cannot define $d_t^{\alpha} t^{\beta}$ in general for $-1 < \beta \le 0$,
but we can calculate $D_t^{\alpha} t^{\beta}$.
We remark that $\partial_t^{\alpha} t^{\beta}$ in the operator sense coincides with
the result calculated by $D_t^{\alpha} t^{\beta}$.
We here emphasize that our interest is not only computations of fracitional
derivatives, but also formulate $\partial_t^{\alpha}$ as an operator defined on
$^{-\alpha-\gamma}H(0,T)$ with the isomorphy.
\\
\vspace{0.1cm}
\\
{\bf (c)}
For any constant $t_0 \in (0,T)$, we consider a function of
the Heaviside type defined by (2.16).
By Proposition 2.10 or 2.11, we can see
$$
\partial_t^{\alpha} h_{t_0}(t) =
\left\{ \begin{array}{rl}
&0, \quad t \le t_0, \\
& \frac{(t-t_0)^{-\alpha}}{\Gamma(1-\alpha)},
\quad t>t_0.
\end{array}\right.
$$
We compare with the case $\alpha=1$:
$$
\frac{d}{dt}h_{t_0}(t) = \delta_{t_0}(t):\, \mbox{
Dirac delta function at $t_0$}
$$
in $(C^{\infty}_0(0,T))'$.
On the other hand, $\partial_t^{\alpha} h_{t_0}$ for $\alpha < 1$ does not generate
the singularity of a Dirac delta function.
Moreover,
$$
\partial_t^{\alpha} h_{t_0} \, \longrightarrow \, \delta_{t_0} \quad
\mbox{as $\alpha \uparrow 1$ in
$(C^{\infty}_0(0,T))'$},
$$
which is taken in the distribution sense.
More precisely,
$$
\lim_{\alpha\uparrow 1}
\, _{(C^{\infty}_0(0,T))'}<\partial_t^{\alpha} h_{t_0}, \, \psi
>_{C^{\infty}_0(0,T)} \,
= \, _{(C^{\infty}_0(0,T))'}<\delta_{t_0},\, \psi
>_{C^{\infty}_0(0,T)} = \psi(t_0)
$$
for each $\psi \in C^{\infty}_0(0,T)$.
\\
Indeed, by integration by parts, we obtain
\begin{align*}
& _{(C^{\infty}_0(0,T))'}<\partial_t^{\alpha} h_{t_0},\, \psi
>_{C^{\infty}_0(0,T)} \,
= \int^T_{t_0} \frac{(t-t_0)^{-\alpha}}{\Gamma(1-\alpha)} \psi(t) dt\\
=& \left[ \frac{(t-t_0)^{1-\alpha}}{(1-\alpha)\Gamma(1-\alpha)}\psi(t)
\right]^{t=T}_{t=t_0}
- \frac{1}{(1-\alpha)\Gamma(1-\alpha)} \int^T_{t_0}
(t-t_0)^{1-\alpha} \psi'(t) dt\\
= &\frac{1}{\Gamma(2-\alpha)} \int_T^{t_0}
(t-t_0)^{1-\alpha} \psi'(t) dt.
\end{align*}
Letting $\alpha \uparrow 1$ and applying the Lebesgue convergence
theorem, we know that the right-hand side tends to
$$
\frac{1}{\Gamma(1)} \int_T^{t_0} \psi'(t) dt = \psi(t_0).
$$
$\blacksquare$
\\
\vspace{0.2cm}
\\
{\bf Proof of Proposition 2.11.}
\\
We prove by approximating $u\in L^1(0,T)$ by $u_n \in {_{0}{C^1[0,T]}}$,
$n\in\mathbb{N}$ and using Proposition 2.7.
As before, we choose $\gamma > \frac{1}{2}-\alpha$, so that
$L^1(0,T) \subset \, {^{-\alpha-\gamma}H}(0,T)$ by the Sobolev embedding:
$\, ^{\alpha+\gamma}H(0,T) \subset C[0,T]$.
Since we can choose $u_n \in {_{0}{C^1[0,T]}}$, $n\in \mathbb{N}$ such that
$u_n \longrightarrow u$ in $L^1(0,T)$ as $n \to \infty$, we see that
$u_n \longrightarrow u$ in $\,^{-\alpha-\gamma}H(0,T)$.
Hence, Theorem 2.1 yields that
$\partial_t^{\alpha} u_n \longrightarrow \partial_t^{\alpha} u$ in $\, ^{-2\alpha-\gamma}H(0,T)$.
Since $C^{\infty}_0(0,T) \subset \, ^{2\alpha+\gamma}H(0,T)$ yields
$\, ^{-2\alpha-\gamma}H(0,T) \subset (C^{\infty}_0(0,T))'$, we see that
$\partial_t^{\alpha} u_n \longrightarrow \partial_t^{\alpha} u$ in $\, ^{-2\alpha-\gamma}H(0,T)$ implies
$\partial_t^{\alpha} u_n \longrightarrow \partial_t^{\alpha} u$ in $(C^{\infty}_0(0,T))'$, that is,
$$
\lim_{n\to \infty} {_{(C^{\infty}_0(0,T))'}<}\partial_t^{\alpha} u_n, \, \psi>_
{C^{\infty}_0(0,T)}
\, = \, {_{(C^{\infty}_0(0,T))'}<}\partial_t^{\alpha} u, \, \psi>_{C^{\infty}_0(0,T)}
$$
for all $\psi \in C^{\infty}_0(0,T)$.
On the other hand, by Proposition 2.7, we have
$\partial_t^{\alpha} u_n = D_t^{\alpha} u_n$, $n\in \mathbb{N}$.
Therefore,
$$
\lim_{n\to \infty} {_{(C^{\infty}_0(0,T))'}<}D_t^{\alpha} u_n, \, \psi>_
{C^{\infty}_0(0,T)}\,
= \, {_{(C^{\infty}_0(0,T))'}<}\partial_t^{\alpha} u, \, \psi>
_{C^{\infty}_0(0,T)} \eqno{(2.18)}
$$
for all $\psi \in C^{\infty}_0(0,T)$.
Then, for any $\psi \in
C^{\infty}_0(0,T)$, we obtain
\begin{align*}
& _{(C^{\infty}_0(0,T))'}< D_t^{\alpha} u_n, \, \psi>_
{C^{\infty}_0(0,T)}\,
=\, (D_t^{\alpha} u_n\, \psi)_{L^2(0,T)}
= \frac{1}{\Gamma(1-\alpha)}\left( \frac{d}{dt}
\int^t_0 (t-s)^{-\alpha} u_n(s) ds, \, \, \psi\right)_{L^2(0,T)}\\
= & -\frac{1}{\Gamma(1-\alpha)}\left(
\int^t_0 (t-s)^{-\alpha} u_n(s) ds, \, \, \frac{d\psi}{dt}\right)
_{L^2(0,T)}.
\end{align*}
Since the Young inequality on the convolution yields
$$
\left\Vert \int^t_0 (t-s)^{-\alpha} u_n(s) ds
- \int^t_0 (t-s)^{-\alpha} u(s) ds \right\Vert_{L^1(0,T)}
\le \Vert s^{-\alpha} \Vert_{L^1(0,T)}\Vert u_n-u\Vert_{L^1(0,T)}
\longrightarrow 0
$$
as $n \to \infty$, we have
$$
\lim_{n\to\infty}(D_t^{\alpha} u_n, \, \psi)_{L^2(0,T)}
= \left( -\frac{1}{\Gamma(1-\alpha)}\int^t_0 (t-s)^{-\alpha} u(s) ds, \, \,
\frac{d\psi}{dt}\right)_{L^2(0,T)}.
$$
Hence, with (2.18), we obtain
$$
{_{(C^{\infty}_0(0,T))'}<} \partial_t^{\alpha} u, \, \psi>_{C^{\infty}_0(0,T)}\,
= \lim_{n\to \infty} {_{(C^{\infty}_0(0,T))'}<}D_t^{\alpha} u_n, \, \psi>_
{C^{\infty}_0(0,T)}
$$
$$
= \left( -\frac{1}{\Gamma(1-\alpha)}\int^t_0 (t-s)^{-\alpha} u(s) ds, \, \,
\frac{d\psi}{dt}\right)_{L^2(0,T)} \eqno{(2.19)}
$$
for all $\psi\in C^{\infty}_0(0,T)$. Consequently, (2.19) means
$$
\partial_t^{\alpha} u(t) = \frac{d}{dt}\left( \frac{1}{\Gamma(1-\alpha)}\int^t_0
(t-s)^{-\alpha} u(s) ds\right)
$$
in the sense of the derivative of a distribution.
Thus the proof of Proposition 2.11 is complete.
$\blacksquare$
\\
\section{Basic properties in fractional calculus}
In this section, we present fundamental properties of $\partial_t^{\alpha}$ in the case
of $\mathcal{D}(\partial_t^{\alpha}) = H_{\alpha}(0,T)$. We can consider for
$\partial_t^{\alpha}$ with the domain $L^2(0,T)$ but we here omit.
We set $\partial_t^0 = I$: the identity operator on $L^2(0,T)$.
\\
{\bf Theorem 3.1.}
\\
{\it
Let $\alpha,\beta \ge 0$. Then
$$
\partial_t^{\alpha} (\partial_t^{\beta}v) = \partial_t^{\alpha+\beta}v \quad
\mbox{for all $v\in H_{\alpha+\beta}(0,T)$}.
$$
}
\\
This kind of sequential derivatives are more complicated for
$d_t^{\alpha}$ and $D_t^{\alpha}$ when we do not specify the domains.
For $\partial_t^{\alpha}$, the domain is
already installed in a convenient way.
\\
{\bf Proof of Theorem 3.1.}
\\
By (2.7) we have $\partial_t^{\alpha} = (J^{\alpha})^{-1}$ in
$\mathcal{D}(\partial_t^{\alpha}) = H_{\alpha}(0,T)$ with $\alpha \ge 0$.
Hence, it suffices to prove
$$
J^{-\alpha}(J^{-\beta}v) = J^{-(\alpha+\beta)}v, \quad v\in H_{\alpha+\beta}
(0,T).
$$
Setting $w = J^{-\beta}v$, we have $w\in H_{\alpha}(0,T)$ by Proposition 2.5 (i).
Then $J^{-\alpha}J^{-\beta}v = J^{-\alpha}w$ and
$$
J^{-(\alpha+\beta)}v = J^{-(\alpha+\beta)}J^{\beta}w
= J^{-(\alpha+\beta)+\beta}w = J^{-\alpha}w.
$$
For the second equality to last, we applied Proposition 2.5 (ii) in terms
of $w \in \mathcal{D}(J^{-\alpha})= H_{\alpha}(0,T)$. Hence, $J^{-\alpha}J^{-\beta}v
= J^{-(\alpha+\beta)}v$ for $v \in H_{\alpha+\beta}(0,T)$.
Thus the proof of Theorem 3.1 is complete.
$\blacksquare$
We define the Laplace transform by
$$
(Lv)(p) = \widehat{v}(p):= \lim_{T\to\infty} \int^T_0
v(t)e^{-pt} dt \eqno{(3.1)}
$$
provided that the limit exists.
\\
{\bf Theorem 3.2 (Laplace transform of $\partial_t^{\alpha} v$).}
\\
{\it
Let $u \in H_{\alpha}(0,T)$ with arbitrary $T>0$.
If $(\widehat{\vert \partial_t^{\alpha} u\vert})(p)$ exists for $p > p_0$ which is some positive
constant, then $\widehat{u}(p)$ exists for $p > p_0$ and
$$
\widehat{\partial_t^{\alpha} u}(p) = p^{\alpha}\widehat{u}(p), \quad p>p_0.
$$
}
For initial value problems for fractional ordinary differential equations
and initial boundary value problems for fractional partial differential
equations, it is known that the Laplace transform is useful if
one can verify the existence of the Laplace transform of solutions to
these problems. The existence of the Laplace transfom is concerned with
the asymptotic behavior of unknown solution $u$ as $t \to \infty$, which
may not be easy to be verified for solutions to be determined.
At least the method by Laplace transform is definitely helpful
in finding solutions heuristically.
\\
\vspace{0.2cm}
\\
{\bf Corollary 3.1.}\\
{\it
Let $u \in H_{\alpha}(0,T)$ with arbitrary $T>0$.
If $\partial_t^{\alpha} u \in L^r(0,\infty)$ with some $r\ge 1$, then
$$
\widehat{\partial_t^{\alpha} u}(p) = p^{\alpha}\widehat{u}(p), \quad p>p_0.
$$
}
\\
{\bf Proof of Theorem 3.2.}
\\
{\bf First Step.}
We first prove
\\
{\bf Lemma 3.1.}\\
{\it
Let $w \in L^2(0,T)$ with arbitrary $T>0$.
If $\widehat{\vert w\vert}(p)$ exists for $p>p_0$.
Then
$$
(\widehat{J^{\alpha}w})(p) = p^{-\alpha}\widehat{w}(p), \quad
p > p_0.
$$
}
{\bf Proof of Lemma 3.1.}
\\
We recall
$$
(J^{\alpha}w)(t) = \frac{1}{\Gamma(\alpha)}\int^t_0 (t-s)^{\alpha-1}w(s) ds,
\quad t>0 \quad \mbox{for $w\in L^2(0,T)$}.
$$
By $w\in L^2(0,T)$, we see that $J^{\alpha}w\in L^2(0,T)$ for
arbitrarily fixed $T>0$.
By the assumption on the existence of $\widehat{w}$, we obtain
$$
\lim_{T\to\infty} \int^T_0 e^{-pt}w(t) dt = \widehat{w}(p), \quad
p>p_0.
$$
Choose $T>0$ arbitrarily. Then $w\in L^2(0,T)$, and
$$
\int^T_0 e^{-pt} \left( \frac{1}{\Gamma(\alpha)}
\int^t_0 (t-s)^{\alpha-1} w(s) ds\right) dt
= \frac{1}{\Gamma(\alpha)}\int^T_0 w(s)
\left( \int^T_s e^{-pt} (t-s)^{\alpha-1} dt\right) ds.
$$
Changing the variables: $t \longrightarrow \xi$ by $\xi:= (t-s)p$, we
have
$$
\int^T_s e^{-pt} (t-s)^{\alpha-1} dt
= p^{-\alpha}e^{-ps} \int^{(T-s)p}_0 \xi^{\alpha-1}e^{-\xi} d\xi,
$$
and so
\begin{align*}
& \int^T_0 e^{-pt} \left( \frac{1}{\Gamma(\alpha)}
\int^t_0 (t-s)^{\alpha-1} w(s) ds\right) dt
= \frac{1}{\Gamma(\alpha)}p^{-\alpha}\int^T_0 e^{-ps}w(s)
\left( \int^{(T-s)p}_0 \xi^{\alpha-1}e^{-\xi} d\xi\right) ds\\
=& \frac{1}{\Gamma(\alpha)}p^{-\alpha}\int^T_{\frac{T}{2}} e^{-ps}w(s)
\left( \int^{(T-s)p}_0 \xi^{\alpha-1}e^{-\xi} d\xi\right) ds\\
+ &\frac{1}{\Gamma(\alpha)}p^{-\alpha}\int^{\frac{T}{2}}_0 e^{-ps}w(s)
\left( \int^{(T-s)p}_0 \xi^{\alpha-1}e^{-\xi} d\xi\right) ds
=: I_1 + I_2.
\end{align*}
We remark
$$
\int^{(T-s)p}_0 \xi^{\alpha-1}e^{-\xi} d\xi
\le \int^{\infty}_0 \xi^{\alpha-1}e^{-\xi} d\xi = \Gamma(\alpha).
$$
Then, since $\int^{\infty}_0 e^{-pt}\vert w(t)\vert dt$ exists,
we see
\begin{align*}
&\vert I_1\vert
= \left\vert \frac{1}{\Gamma(\alpha)}p^{-\alpha}\int^T_{\frac{T}{2}}
e^{-ps}w(s)
\left( \int^{(T-s)p}_0 \xi^{\alpha-1}e^{-\xi} d\xi\right) ds
\right\vert\\
\le& Cp^{-\alpha}\int^T_{\frac{T}{2}} e^{-ps}\vert w(s)\vert ds
\quad \longrightarrow 0 \quad \mbox{as $T\to \infty$}.
\end{align*}
Since $0 < s < \frac{T}{2}$ implies
$$
\frac{T}{2}p < (T-s)p < Tp,
$$
we obtain
$$
\int^{\frac{Tp}{2}}_0 \xi^{\alpha-1}e^{-\xi} d\xi
\le \int^{(T-s)p}_0 \xi^{\alpha-1}e^{-\xi} d\xi
\le \int^{Tp}_0 \xi^{\alpha-1}e^{-\xi} d\xi
\le \int^{\infty}_0 \xi^{\alpha-1}e^{-\xi} d\xi
= \Gamma(\alpha) \eqno{(3.2)}
$$
and so
$$
\lim_{T\to\infty} \int^{(T-s)p}_0 \xi^{\alpha-1}e^{-\xi} d\xi
= \int^{\infty}_0 \xi^{\alpha-1}e^{-\xi} d\xi = \Gamma(\alpha)
\quad \mbox{if $0<s<\frac{T}{2}$} \eqno{(3.3)}
$$
for any $p > p_0>0$.
Since
$$
\lim_{T\to\infty} \int^T_0 w(s)e^{-ps} ds
$$
exists for $p>p_0$ in view of (3.2) and (3.3), the Lebesgue
convergence theorem yields
$$
\lim_{T\to\infty} \int^{\frac{T}{2}}_0 e^{-ps} w(s)
\left( \int^{(T-s)p}_0 \xi^{\alpha-1}e^{-\xi} d\xi\right) ds
= \Gamma(\alpha) \int^{\infty}_0 e^{-ps}w(s) ds,
$$
and we reach
$$
\lim_{T\to\infty} I_2 = \frac{p^{-\alpha}}{\Gamma(\alpha)}
\int^{\infty}_0 w(s)e^{-ps} \Gamma(\alpha)ds
= p^{-\alpha}\widehat{w}(p), \quad p > p_0.
$$
Thus
$$
\lim_{T\to\infty} \int^T_0 e^{-pt}(J^{\alpha}w)(t) dt
= \lim_{T\to\infty} (I_1+I_2) = p^{-\alpha}\widehat{w}(p), \quad p > p_0.
$$
The proof of Lemma 3.1 is complete. $\blacksquare$
\\
{\bf Second Step.}
We will complete the proof of the theorem.
Since $u\in H_{\alpha}(0,T)$ for any $T>0$, we can find
$w_T \in L^2(0,T)$ such that $J^{\alpha}w_T = u$ in $(0,T)$.
For any $t \in (0,T)$, we can define $w$ satisfying $w(t) = w_T(t)$
for $0<t<T$. Therefore for all $t>0$, we can define
$w(t)$ and $J^{\alpha}w(t) = u(t)$ for any $t \in (0,T)$, that is,
for all $t>0$. Therefore $\partial_t^{\alpha} u(t) = w(t)$ for $t>0$ and
$w \in L^2(0,T)$ with arbitrary $T>0$.
Since $\partial_t^{\alpha} u(t) = w(t)$ for $t>0$ and
$\widehat{\vert \partial_t^{\alpha} u\vert}(p)$ exists for $p>p_0$,
we know that $\widehat{\vert w\vert}(p)$ exists for $p>p_0$.
Therefore Lemma 3.1 yields
$$
(\widehat{J^{\alpha}w})(p) = p^{-\alpha}\widehat{w}(p), \quad
p>p_0,
$$
that is,
$$
\widehat{u}(p) = p^{-\alpha}\widehat{\partial_t^{\alpha} u}(p), \quad p>p_0.
$$
Thus the proof of Theorem 3.2 is complete.
$\blacksquare$
\\
Moreover $\partial_t^{\alpha}$ with $\mathcal{D}(\partial_t^{\alpha}) = H_{\alpha}(0,T)$ is consistent
also with the convolution of two functions.
We set
$$
(u*g)(t) = \int^t_0 u(t-s)g(s) ds, \quad 0<t<T
\eqno{(3.4)}
$$
for $u\in L^2(0,T)$ and $g\in L^1(0,T)$. Then, the Young inequality
on the convolution yields
$$
\Vert u*g\Vert_{L^2(0,T)} \le \Vert u\Vert_{L^2(0,T)}\Vert g\Vert_{L^1(0,T)}.
\eqno{(3.5)}
$$
We prove
\\
{\bf Theorem 3.3.}
\\
{\it
Let $\alpha \ge 0$. Then:
\\
$$
J^{\alpha}(u*g) = (J^{\alpha}u)*g \quad \mbox{for $u\in L^1(0,T)$ and
$g\in L^1(0,T)$} \eqno{(3.6)}
$$
$$
\Vert u*g\Vert_{H_{\alpha}(0,T)} \le C\Vert u\Vert_{H_{\alpha}(0,T)}
\Vert g\Vert_{L^1(0,T)} \quad \mbox{for $u\in H_{\alpha}(0,T)$ and $g\in L^1(0,T)$.}
\eqno{(3.7)}
$$
$$
\partial_t^{\alpha} (u*g) = (\partial_t^{\alpha} u) *g \quad \mbox{for $u\in H_{\alpha}(0,T)$
and $g\in L^1(0,T)$.} \eqno{(3.8)}
$$
}
\\
{\bf Proof of Theorem 3.3.}
\\
{\bf Proof of (3.6).}
By exchange of the order of the integral and change of
the variables $s \to \eta:=s-\xi$, we can derive
\begin{align*}
& J^{\alpha}(u*g)(t)
= \frac{1}{\Gamma(\alpha)}\int^t_0 (t-s)^{\alpha-1}
\left( \int^s_0 u(s-\xi)g(\xi) d\xi \right) ds\\
=& \frac{1}{\Gamma(\alpha)}\int^t_0 g(\xi) \left(
\int^t_{\xi} (t-s)^{\alpha-1}u(s-\xi) ds \right) d\xi
= \frac{1}{\Gamma(\alpha)}\int^t_0 g(\xi) \left(
\int^{t-\xi}_0 (t-\xi-\eta)^{\alpha-1}u(\eta) d\eta \right) d\xi\\
=& \int^t_0 g(\xi)(J^{\alpha}u)(t-\xi) d\xi
= (g*J^{\alpha}u)(t), \quad 0<t<T,
\end{align*}
which completes the proof of (3.6).
\\
{\bf Proof of (3.7).}
Let $u \in H_{\alpha}(0,T)$ and $g \in L^1(0,T)$.
Then, by Proposition 2.5, there
exists $w \in L^2(0,T)$ such that $u = J^{\alpha}w$ and
$\Vert w\Vert_{L^2(0,T)} \le C\Vert u\Vert_{H_{\alpha}(0,T)}$.
By (3.6) we obtain
$u*g = J^{\alpha}w*g = J^{\alpha}(w*g)$.
Since $w\in L^2(0,T)$ and $g\in L^1(0,T)$ imply $w*g\in L^2(0,T)$,
by $u*g \in J^{\alpha}L^2(0,T)$ we see that
$u*g \in H_{\alpha}(0,T)$. Moreover, by Proposition 2.5 and
$u*g = J^{\alpha}(w*g)$, we have
$$
\Vert u*g\Vert_{H_{\alpha}(0,T)} \le C\Vert w*g\Vert_{L^2(0,T)}
\le C\Vert w\Vert_{L^2(0,T)}\Vert g\Vert_{L^1(0,T)}
\le C\Vert u\Vert_{H_{\alpha}(0,T)}\Vert g\Vert_{L^1(0,T)}.
$$
which completes the proof of (3.7).
\\
{\bf Proof of (3.8).}
For arbitrary $u\in H_{\alpha}(0,T)$, Proposition 2.5 yields that there exists
$w\in L^2(0,T)$ such that $u=J^{\alpha}w$. Applying (3.6), we have
$J^{\alpha}(w*g) = J^{\alpha}w * g$. Therefore,
$$
\partial_t^{\alpha} (J^{\alpha}w*g) = \partial_t^{\alpha} J^{\alpha}(w*g) = w*g.
$$
Since $u = J^{\alpha}w$ is equivalent to $w = \partial_t^{\alpha} u$, we reach
$\partial_t^{\alpha} (u*g) = (\partial_t^{\alpha} u) * g$. Thus the proof of Theorem 3.3 is
complete.
$\blacksquare$
\\
We will show a useful variant of Theorem 3.3.
\\
{\bf Theorem 3.4.}
\\
{\it
Let $\gamma > \frac{1}{2}$ and $\beta + \gamma > \frac{1}{2}$.
Then for $\partial^{\beta}_t: \, ^{-\gamma}H(0,T) \,\longrightarrow \,
^{-\beta-\gamma}H(0,T)$, we have
$$
\partial_t^{\beta}v\, *\, u = \partial_t^{\beta}(v\, *\, u)
$$
for all $u \in L^1(0,T)$ and $v \in L^1(0,T)$ satisfying
$\partial_t^{\beta} v \in L^1(0,T)$.}
\\
We note that $L^1(0,T) \subset \, ^{-\gamma}H(0,T)$ by the Sobolev
embedding and $\gamma > \frac{1}{2}$, and so if $v \in L^1(0,T)
\subset \mathcal{D}(\partial_t^{\beta}) = \,^{-\gamma}H(0,T)$, then
so $\partial_t^{\beta} v \in \, ^{-\beta-\gamma}H(0,t)$ is well-defined.
The assumption of the theorem further requires $\partial_t^{\beta} v \in L^1(0,T)$.
On the other hand, for $v, u \in L^1(0,T)$,
the Young inequailty implies $v\,*\, u \in L^1(0,T)$, and so
$v\, * \, u \in \, ^{-\gamma}H(0,t)$. Hence in the theorem,
$\partial_t^{\beta} (v\, *\, u) \in \, ^{-\beta -\gamma}H(0,t)$ is well-defined.
\\
{\bf Proof of Theorem 3.4.}
\\
The Young inequality yields $\partial_t^{\beta}v\, *\, u \in L^1(0,T)$.
Consequently, by Proposition 2.10, we see
$$
J_{\beta}'(\partial_t^{\beta}v\, *\, u) = J^{\beta}(\partial_t^{\beta}v\, *\, u).
$$
By (3.6) in Theorem 3.3, we have
$$
J^{\beta}(\partial_t^{\beta} v\, *\, u) = (J^{\beta}\partial_t^{\beta} v) \, *\, u.
$$
Since $\partial_t^{\beta} v \in L^1(0,T)$, by using the definition (2.13),
again Proposition 2.10 implies
$$
J^{\beta}\partial_t^{\beta} v = J_{\beta}'\partial_t^{\beta} v = J_{\beta}'(J_{\beta}')^{-1}v = v.
$$
Hence, $J_{\beta}'(\partial_t^{\beta} v\, *\, u) = v\, *\, u$.
Operating $(J_{\beta}')^{-1}$ to both sides and noting
$v\, *\, u \in L^1(0,T) \subset \, ^{-\gamma}H(0,T)$, by (2.13), we have
$$
(J_{\beta}')^{-1}(J_{\beta}'(\partial_t^{\beta} v\, *\, u))
= (J_{\beta}')^{-1}(v\, *\, u) = \partial_t^{\beta} (v\, *\, u),
$$
that is, $\partial_t^{\beta} v\, *\, u = \partial_t^{\beta}(v\, *\, u)$. Thus the proof of
Theorem 3.4 is complete.
$\blacksquare$
\vspace{0.2cm}
{\bf Theorem 3.5 (coercivity).}
\\
{\it
Let $0<\alpha<1$. Then
$$
\int^T_0 v(t)\partial_t^{\alpha} v(t) dt \ge \frac{1}{2\Gamma(1-\alpha)}
T^{-\alpha}\Vert v\Vert^2_{L^2(0,T)}
\quad \mbox{for $v\in H_{\alpha}(0,T)$}
$$
and
$$
\frac{1}{\Gamma(\alpha)} \int^t_0 (t-s)^{\alpha-1}
v(s)\partial_s^{\alpha} v(s) ds \ge \frac{1}{2}\vert v(t)\vert^2
\quad \mbox{for $v \in H_{\alpha}(0,T)$.}
$$
}
\\
The proof of Theorem 3.5 can be found in \cite{KRY}. Theorem 3.5 is not used
in this article, and is useful for proving the well-posedness for initial
boundary value problems for fractional partial differential equations
(see e.g., \cite{KRY}, \cite{KY}, \cite{Za}).
We can generalize Theorem 3.5 to arbitrary $v\in L^2(0,T)$, but
we omit the details for the conciseness.
\section{Fractional derivatives of the Mittag-Leffler functions}
Related to time fractional differential equations, we introduce
the Mittag-Leffler functions:
$$
E_{\alpha,\beta}(z) = \sum_{k=0}^{\infty} \frac{z^k}
{\Gamma(\alpha k+\beta)}, \quad \alpha, \beta > 0, \quad
z\in \mathbb{C}. \eqno{(4.1)}
$$
It is known (e.g., \cite{Po}) that $E_{\alpha,\beta}(z)$ is an entire
function in $z \in \mathbb{C}$.
Henceforth let $\lambda\in \mathbb{R}$ be a constant and, $\alpha>0$ and
$\alpha \not\in \mathbb{N}$.
We fix an arbitrary constant $\Lambda_0 > 0$ and we assume that
$\lambda > -\Lambda_0$.
First we prove
\\
{\bf Proposition 4.1.}
\\
{\it
Let $0<\alpha<2$.
We fix constants $T>0$ and $\lambda > -\Lambda_0$ arbitrarily. Then
\\
(i) We have
$E_{\alpha,1}(-\lambda t^{\alpha}) - 1\in H_{\alpha}(0,T)$,
$$
\partial_t^{\alpha} (E_{\alpha,1}(-\lambda t^{\alpha}) - 1)
= -\lambda E_{\alpha,1}(-\lambda t^{\alpha}), \quad t>0
$$
and there exist constants $C_1=C_1(\alpha,\Lambda_0,T) >0$ and
$C_2 = C_2(\alpha) > 0$ such that
$$
\vert E_{\alpha,1}(-\lambda t^{\alpha})\vert \le
\left\{ \begin{array}{rl}
& C_1 \quad \mbox{for $0\le t \le T$ if $-\Lambda_0 \le \lambda < 0$},\\
& \frac{C_2}{1+\lambda t^{\alpha}} \quad \mbox{for $t\ge 0$
if $\lambda \ge 0$}.
\end{array}\right.
\eqno{(4.2)}
$$
(ii) We have $tE_{\alpha,2}(-\lambda t^{\alpha}) - t \in H_{\alpha}(0,T)$,
$$
\partial_t^{\alpha} (tE_{\alpha,2}(-\lambda t^{\alpha}) - t)
= -\lambda tE_{\alpha,2}(-\lambda t^{\alpha}), \quad t>0.
$$
Furthermore we can find constants
$C_1=C_1(\alpha,\Lambda_0,T) >0$ and $C_2 = C_2(\alpha) > 0$ such that
$$
\vert E_{\alpha,2}(-\lambda t^{\alpha})\vert \le
\left\{ \begin{array}{rl}
& C_1 \quad \mbox{for $0\le t \le T$ if $-\Lambda_0 \le \lambda < 0$},\\
& \frac{C_2}{1+\lambda t^{\alpha}} \quad \mbox{for $t\ge 0$
if $\lambda \ge 0$}.
\end{array}\right.
\eqno{(4.3)}
$$
}
A direct proof for the inclusions in $H_{\alpha}(0,T)$ is complicated and
through the operator $J^{\alpha}$, we will provide simpler proofs.
\\
{\bf Proof of Proposition 4.1.}
\\
(i) Since
$$
E_{\alpha,1}(-\lambda t^{\alpha}) = \sum_{k=0}^{\infty}
\frac{(-\lambda)^kt^{\alpha k}}{\Gamma(\alpha k + 1)},
$$
where the series is uniformly convergent for $0\le t \le T$,
the termwise integration yields
\begin{align*}
& J^{\alpha}(E_{\alpha,1}(-\lambda t^{\alpha}))
= \sum_{k=0}^{\infty} \frac{(-\lambda)^k}{\Gamma(\alpha)}
\int^t_0 \frac{(t-s)^{\alpha-1}s^{\alpha k}}{\Gamma(\alpha k + 1)} ds\\
=& \sum_{k=0}^{\infty} \frac{(-\lambda)^k}{\Gamma(\alpha)}
\frac{\Gamma(\alpha)}{\Gamma(\alpha k + \alpha + 1)}t^{\alpha k + \alpha}
= -\frac{1}{\lambda} \sum_{j=1}^{\infty}
\frac{(-\lambda t^{\alpha})^j}{\Gamma(\alpha j + 1)}.
\end{align*}
Here we set $j=k+1$ to change the indices of the summation.
Therefore
$$
J^{\alpha}(E_{\alpha,1}(-\lambda t^{\alpha})) = -\frac{1}{\lambda}
(E_{\alpha,1}(-\lambda t^{\alpha}) - 1).
$$
Since $\partial_t^{\alpha} v = (J^{\alpha})^{-1}v$ for $v \in H_{\alpha}(0,T)$ and
$E_{\alpha,1}(-\lambda t^{\alpha}) \in L^2(0,T)$, we obtain
$$
-\frac{1}{\lambda}(E_{\alpha,1}(-\lambda t^{\alpha}) - 1) \in H_{\alpha}(0,T)
$$
and
$$
(J^{\alpha})^{-1}J^{\alpha}E_{\alpha,1}(-\lambda t^{\alpha})
= -\frac{1}{\lambda}\partial_t^{\alpha} (E_{\alpha,1}(-\lambda t^{\alpha}) - 1),
$$
that is, $\partial_t^{\alpha}(E_{\alpha,1}(-\lambda t^{\alpha}) - 1)
= -\lambda E_{\alpha,1}(-\lambda t^{\alpha})$.
On the other hand, for $0<\alpha<2$ and $\beta>0$,
by Theorems 1.5 and 1.6 (p.35) in
\cite{Po}, we can find a constant $C_0 = C_0(\alpha,\beta) > 0$ such that
$$
\vert E_{\alpha, \beta}(-\lambda t^{\alpha})\vert \le
\left\{ \begin{array}{rl}
& C_0(1+\vert \lambda\vert t^{\alpha})^{\frac{1-\beta}{\alpha}}
\exp(\vert \lambda\vert^{\frac{1}{\alpha}} t)
\quad \mbox{if $-\Lambda_0 \le \lambda < 0$},\\
& \frac{C_0}{1+\lambda t^{\alpha}} \quad \mbox{if $\lambda \ge 0$}
\end{array}\right.
\eqno{(4.4)}
$$
for all $t \ge 0$. Hence, we can choose constants
$C_3=C_3(\alpha,\beta, \Lambda_0,T) >0$ and $C_4 = C_4(\alpha,\beta) > 0$
such that
$$
\vert E_{\alpha, \beta}(-\lambda t^{\alpha})\vert \le
\left\{ \begin{array}{rl}
& C_3 \quad \mbox{for $0\le t \le T$ if $-\Lambda_0 \le \lambda < 0$},\\
& \frac{C_4}{1+\lambda t^{\alpha}} \quad \mbox{for $t \ge 0$
if $\lambda \ge 0$}.
\end{array}\right.
\eqno{(4.5)}
$$
In terms of (4.5), we can finish the proof of (4.2).
Thus the proof of (i) is complete.
\\
(ii) Since $tE_{\alpha,2}(-\lambda t^{\alpha}) - t \in L^2(0,T)$, noting
$\Gamma(2) = 1\times \Gamma(1) = 1$, we have
\begin{align*}
& J^{\alpha}(tE_{\alpha,2}(-\lambda t^{\alpha}))
= \frac{1}{\Gamma(\alpha)}
\int^t_0 (t-s)^{\alpha-1} \sum_{k=0}^{\infty} \frac{s(-\lambda)^ks^{\alpha k}}
{\Gamma(\alpha k + 2)} ds \\
=& t\sum_{k=0}^{\infty} \frac{(-\lambda)^k}{\Gamma(\alpha k + \alpha + 2)}
t^{\alpha k + \alpha}
= \frac{t}{-\lambda}\sum_{j=1}^{\infty}
\frac{(-\lambda)^j}{\Gamma(\alpha j + 2)}t^{\alpha j}\\
=& -\frac{t}{\lambda}\left( \sum_{j=0}^{\infty}
\frac{(-\lambda)^jt^{\alpha j}}{\Gamma(\alpha j + 2)}
- \frac{1}{\Gamma(2)}\right),
\end{align*}
that is,
$$
J^{\alpha}(tE_{\alpha,2}(-\lambda t^{\alpha}))
= -\frac{1}{\lambda}(tE_{\alpha,2}(-\lambda t^{\alpha}) - t), \quad
0<t<T.
$$
Therefore, $tE_{\alpha,2}(-\lambda t^{\alpha}) - t \in H_{\alpha}(0,T)$ and
$$
-\lambda t E_{\alpha,2}(-\lambda t^{\alpha}) = \partial_t^{\alpha} (tE_{\alpha,2}(-\lambda t^{\alpha})-t).
$$
In terms of (4.5), the proof of the estimate is similar to part (i).
Thus we can complete the proof of Proposition 4.1.
$\blacksquare$
We set
$$
(B_{\lambda}f)(t):= \int^t_0 (t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda (t-s)^{\alpha}) f(s) ds.
\eqno{(4.6)}
$$
Next we show
\\
{\bf Proposition 4.2.}
\\
{\it
Let $f \in L^2(0,T)$. Then
$$
B_{\lambda}f \in H_{\alpha}(0,T) \quad \mbox{and}\quad
\partial_t^{\alpha} (B_{\lambda}f)(t) = -\lambda (B_{\lambda}f)(t) + f(t), \quad 0<t<T.
\eqno{(4.7)}
$$
Moreover there exist constants $C_5 = C_5(\alpha,\Lambda_0,T)>0$ and
$C_6 = C_6(\alpha,T)>0$ such that
$$
\left\{ \begin{array}{rl}
& \Vert B_{\lambda}f\Vert_{H_{\alpha}(0,T)} \le C_5(\alpha,\Lambda_0,T)
\Vert f\Vert_{L^2(0,T)}\quad
\mbox{for all $f\in L^2(0,T)$ and $\lambda > -\Lambda_0$}, \\
& \Vert B_{\lambda}f\Vert_{H_{\alpha}(0,T)} \le C_6(\alpha,T)\Vert f\Vert_{L^2(0,T)}\quad
\mbox{for all $f\in L^2(0,T)$ and $\lambda \ge 0$}.
\end{array}\right.
\eqno{(4.8)}
$$
}
\\
The uniformity of estimate (4.8) on $\lambda \ge 0$ plays an important role in
Lemma 6.2 (ii) in Section 6. Such uniformity can be derived from the
complete monotonicity of $E_{\alpha,1}(-\lambda t^{\alpha})$
which is characteristic only for $0<\alpha<1$ and see (4.9) below.
Here we prove by
using $\partial_t^{\alpha} = (J^{\alpha})^{-1}$ in $H_{\alpha}(0,T)$, although other proof
by Theorem 3.4 is possible.
\\
{\bf Proof.}
\\
Since in view of (4.5), we can choose constants
$C_7=C_7(\alpha,\Lambda_0, T) > 0$ and $C_8=C_8(\alpha,T) > 0$ such that
$$
\vert E_{\alpha,\alpha}(-\lambda(t-s)^{\alpha})\vert \le
\left\{ \begin{array}{rl}
& C_7(\alpha,\Lambda_0,T) \quad \mbox{for $0\le t \le T$ and
$\lambda > - \Lambda_0$}, \\
& C_8(\alpha,T) \quad \mbox{for $0\le t \le T$ and
$\lambda \ge 0$},
\end{array}\right.
$$
we estimate
$$
\left\vert \int^t_0 (t-s)^{\alpha-1} E_{\alpha,\alpha}(-\lambda(t-s)^{\alpha})f(s) ds
\right\vert
\le C\int^t_0 (t-s)^{\alpha-1}\vert f(s)\vert ds, \quad 0<t<T.
$$
Here and henceforth $C>0$ denotes generic constants which depend on
$\alpha, \Lambda_0, T$ when we consider $\lambda > -\Lambda_0$ and
does not depend on $\Lambda_0$ if $\lambda \ge 0$.
Hence, the Young inequality yields that
$\int^t_0 (t-s)^{\alpha-1}\vert f(s)\vert ds
\in L^2(0,T)$ and so $B_{\lambda}f \in L^2(0,T)$.
Now
\begin{align*}
& J^{\alpha}(B_{\lambda}f)(t)
= \frac{1}{\Gamma(\alpha)}\int^t_0 (t-s)^{\alpha-1}(B_{\lambda}f)(s) ds\\
=& \frac{1}{\Gamma(\alpha)} \int^t_0 (t-s)^{\alpha-1}
\left( \int^s_0 (s-\xi)^{\alpha-1}E_{\alpha,\alpha}(-\lambda (s-\xi)^{\alpha})f(\xi) d\xi
\right) ds\\
=& \int^t_0 f(\xi) \left( \frac{1}{\Gamma(\alpha)}
\int^t_{\xi} (t-s)^{\alpha-1}(s-\xi)^{\alpha-1}E_{\alpha,\alpha}(-\lambda(s-\xi)^{\alpha})
ds \right) d\xi.
\end{align*}
Here
\begin{align*}
& \frac{1}{\Gamma(\alpha)}\int^t_{\xi} (t-s)^{\alpha-1}(s-\xi)^{\alpha-1}
E_{\alpha,\alpha}(-\lambda (s-\xi)^{\alpha}) ds\\
= &\int^t_{\xi} \frac{1}{\Gamma(\alpha)}(t-s)^{\alpha-1}(s-\xi)^{\alpha-1}
\sum_{k=0}^{\infty} \frac{(-\lambda)^k(s-\xi)^{\alpha k}}
{\Gamma(\alpha k + \alpha)} ds\\
=& \frac{1}{\Gamma(\alpha)}\sum_{k=0}^{\infty}
\frac{(-\lambda)^k}{\Gamma(\alpha k + \alpha)}
\int^t_{\xi} (t-s)^{\alpha-1}(s-\xi)^{\alpha k+\alpha -1} ds\\
= & \frac{1}{\Gamma(\alpha)}\sum_{k=0}^{\infty}
\frac{(-\lambda)^k}{\Gamma(\alpha k + \alpha)}
\frac{\Gamma(\alpha)\Gamma(\alpha k + \alpha)}{\Gamma(\alpha k+2\alpha)}
(t-\xi)^{\alpha k+2\alpha-1}\\
=& -\frac{1}{\lambda}(t-\xi)^{\alpha-1} \sum_{j=1}^{\infty}
\frac{(-\lambda (t-\xi)^{\alpha})^j}{\Gamma(\alpha j + \alpha)}
= -\frac{1}{\lambda}(t-\xi)^{\alpha-1}\left(E_{\alpha,\alpha}(-\lambda (t-\xi)^{\alpha})
- \frac{1}{\Gamma(\alpha)}\right).
\end{align*}
Therefore, we have
\begin{align*}
& J^{\alpha}(B_{\lambda}f)(t) = -\frac{1}{\lambda}(B_{\lambda}f)(t)
+ \frac{1}{\lambda}\int^t_0 (t-\xi)^{\alpha-1}\frac{1}{\Gamma(\alpha)}
f(\xi) d\xi\\
=& -\frac{1}{\lambda}(B_{\lambda}f)(t) + \frac{1}{\lambda}(J^{\alpha}f)(t),
\quad 0<t<T,
\end{align*}
that is,
$$
(B_{\lambda}f)(t) = -\lambda J^{\alpha}(B_{\lambda}f)(t) + (J^{\alpha}f)(t)
= J^{\alpha}(-\lambda B_{\lambda}f + f)(t), \quad 0<t<T.
$$
Hence, $B_{\lambda}f \in J^{\alpha}L^2(0,T) = H_{\alpha}(0,T)$ by $-\lambda B_{\lambda}f + f \in
L^2(0,T)$. Therefore, we apply $\partial_t^{\alpha} = (J^{\alpha})^{-1}$
to reach
$$
\partial_t^{\alpha} (B_{\lambda}f) = -\lambda B_{\lambda}f + f \quad \mbox{in $(0,T)$}.
$$
On the other hand, the termwise differentiation of the power series
of $E_{\alpha,1}(z)$ yields
$$
\frac{d}{dt}E_{\alpha,1}(-\lambda t^{\alpha})
= -\lambda t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha}).
$$
Furthermore we note the complete monotonicity:
$$
E_{\alpha,1}(-\lambda t^{\alpha}) > 0, \quad
\frac{d}{dt}E_{\alpha,1}(-\lambda_t^{\alpha}) \le 0, \quad t\ge 0 \eqno{(4.9)}
$$
(e.g., Gorenflo, Kilbas, Mainardi and Rogosin \cite{GKMR}). Therefore,
for $\lambda \ge 0$, we have
$$
\int^T_0 \vert \lambda t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha})\vert dt
= \int^T_0 \lambda t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha}) dt
$$
$$
= -\int^T_0 \frac{d}{dt}E_{\alpha,1}(-\lambda t^{\alpha})dt
= 1 - E_{\alpha,1}(-\lambda T^{\alpha}) \le 1. \eqno{(4.10)}
$$
For $-\Lambda_0 < \lambda < 0$, by (4.5) we have
$$
\int^T_0 \vert \lambda t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha})\vert dt
\le \vert \Lambda_0\vert C\int^T_0 t^{\alpha-1} dt,
$$
where the constant $C>0$ depends on $\alpha, \Lambda_0, T$.
Consequently,
$$
\lambda \Vert s^{\alpha-1}E_{\alpha,\alpha}(-\lambda s^{\alpha})\Vert_{L^1(0,T)}
\le C \quad \mbox{for $\lambda > -\Lambda_0$}.
$$
Applying the Young inequality in (4.6), we obtain
$$
\lambda\Vert B_{\lambda}f\Vert_{L^2(0,T)} \le \lambda\Vert s^{\alpha-1}
E_{\alpha,\alpha}(-\lambda s^{\alpha})\Vert_{L^1(0,T)}\Vert f\Vert_{L^2(0,T)}
\le C\Vert f\Vert_{L^2(0,T)}.
$$
Therefore,
\begin{align*}
& \Vert B_{\lambda}f\Vert_{H_{\alpha}(0,T)} \le C\Vert \partial_t^{\alpha}(B_{\lambda}f)\Vert_{L^2(0,T)}
= C\Vert -\lambda B_{\lambda}f + f\Vert_{L^2(0,T)}\\
\le & C(\Vert -\lambda B_{\lambda}f\Vert_{L^2(0,T)} + \Vert f\Vert_{L^2(0,T)})
\le C\Vert f\Vert_{L^2(0,T)} \quad \mbox{for all $f\in L^2(0,T)$}.
\end{align*}
Thus the proof of Proposition 4.2 is complete.
$\blacksquare$
\\
We close this section with a lemma which is used in Section 6.
\\
{\bf Lemma 4.1.}
\\
{\it
Let $0<\beta<\alpha$ and $\lambda > 0$.
Fixing $\gamma > \frac{1}{2}$, we consider
$\partial^{\beta}_t: \, ^{-\gamma}H(0,T) \,\longrightarrow \,
^{-\beta-\gamma}H(0,T)$. Then
\\
(i)
$$
t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha}) \in L^1(0,T) \subset
\, ^{-\gamma}H(0,T).
$$
\\
(ii)
$$
\partial_t^{\beta} (t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha}))
= t^{\alpha-\beta-1}E_{\alpha,\alpha-\beta}(-\lambda t^{\alpha})
\in L^1(0,T).
$$
\\
(iii)
$$
\partial_t^{\beta} (t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha})\, *\, v)
= (t^{\alpha-\beta-1}E_{\alpha,\alpha-\beta}(-\lambda t^{\alpha})\, *\, v)
\quad \mbox{for each $v \in L^2(0,T)$.}
$$
}
\\
{\bf Proof.}
\\
(i) By (4.4), we see $\vert t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha})\vert
\le Ct^{\alpha-1}$ for $t > 0$ and
$t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha}) \in L^1(0,T)$.
The embedding $L^1(0,T) \subset \, ^{-\gamma}H(0,T)$ is derived by the
Sobolev embedding.
\\
(ii) By $\alpha - \beta > 0$, we can similarly verify that
$t^{\alpha-\beta-1}E_{\alpha,\alpha-\beta}(-\lambda t^{\alpha})
\in L^1(0,T)$.
Therefore, Proposition 2.10 yields
$$
J_{\beta}'(t^{\alpha-\beta-1}E_{\alpha,\alpha-\beta}(-\lambda t^{\alpha}))(t)
= \frac{1}{\Gamma(\beta)}
\int^t_0 (t-s)^{\beta-1}s^{\alpha-\beta-1}
E_{\alpha,\alpha-\beta}(-\lambda s^{\alpha})ds.
$$
By $\beta > 0$ and $\alpha-\beta > 0$, we apply the formula (1.100) (p.25)
in \cite{Po}, which can be also directly verified by the expansion of
the power series of $E_{\alpha,\alpha-\beta}(-\lambda s^{\alpha})$,
so that
$$
\frac{1}{\Gamma(\beta)}
\int^t_0 (t-s)^{\beta-1}s^{\alpha-\beta-1}
E_{\alpha,\alpha-\beta}(-\lambda s^{\alpha})ds
= t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha}).
$$
Therefore,
$J_{\beta}'(t^{\alpha-\beta-1}E_{\alpha,\alpha-\beta}(-\lambda t^{\alpha}))(t)
= t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha})$.
By $\partial_t^{\beta} = (J_{\beta}')^{-1}$ in $L^1(0,T)$, we see part (ii).
Since $t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha})$,
$\partial_t^{\beta} (t^{\alpha-1}E_{\alpha,\alpha}(-\lambda t^{\alpha})$ are both in
$L^1(0,T)$, part (iii) follows from Theorem 3.4.
Thus the proof of Lemma 4.1 is complete.
$\blacksquare$
\section{Initial value problems for fractional ordinary differential
equations}
A relevant formulation of initial value problem for time-fractional
odinary differential equations is our main issue in this section,
in order to treat not so smooth data.
For keeping the compact descriptions, we are restricted to a simple
linear fractional ordinary differential equation.
The treatments are similar to Chapter 3 in \cite{KRY}.
We formulate an initial value problem as follows.
$$
\left\{ \begin{array}{rl}
& \partial_t^{\alpha} (u-a)(t) = -\lambda u(t) + f(t), \quad 0<t<T, \\
& u-a \in H_{\alpha}(0,T).
\end{array}\right. \eqno{(5.1)}
$$
As is mentioned in Section 2, if $\alpha > \frac{1}{2}$, then
$u-a \in H_{\alpha}(0,T) \subset {_{0}{C^1[0,T]}}$ and we know that $u(t) - a$ is continuous
and $u(0) = a$. Thus in the case of $\alpha > \frac{1}{2}$, if
$u$ satisfies (5.1), then a usual initial condition
$u(0) = a$ is satisfied.
Our formulation (5.1) coincides with a conventional formulation
$D_t^{\alpha} (u(t)-a) = f(t)$ of initial value problem, provided that we suitably
specify the regularity of $u$.
We emphasize that we always attach fractional derivative operators with
the domains such as $H_{\alpha}(0,T)$ or ${^{-\alpha}{H}}(0,T)$ with $\alpha \ge 0$,
which means that
our approach is a typical operator theoretic formulation, for example,
similarly to that one is prohibited to consider the
Laplacian $-\Delta$ in $\Omega \subset \mathbb{R}^d$ not associated with the domain.
In other words,
the operator $-\Delta$ with the domain $\{ u\in H^2(\Omega);\,
u\in H^1_0(\Omega)\}$, is different from
$-\Delta$ with the domain $\{ u\in H^2_0(\Omega);\, \nabla u \cdot \nu
= 0 \quad \mbox{on $\partial\Omega$}\}$.
Here $\nu = \nu(x)$ is the unit outward normal vector to $\partial\Omega$.
In particular, if we consider $\partial_t^{\alpha}$ with the domain $H_{\alpha}(0,T)$ and
both sides of (5.1) in $L^2(0,T)$,
we remark that the equality $\partial_t^{\alpha} (u-a)
= \partial_t^{\alpha} u - \partial_t^{\alpha} a$ does not make any sense for $\alpha > \frac{1}{2}$,
because a constant function $a$ is not in $H_{\alpha}(0,T)$.
On the other hand, if we consider $\partial_t^{\alpha}$ with the domain $L^2(0,T)$,
then we can justify
$$
\partial_t^{\alpha} (u-a) = \partial_t^{\alpha} u - \partial_t^{\alpha} a
= \partial_t^{\alpha} u - \frac{1}{\Gamma(1-\alpha)}t^{-\alpha}
$$
for any $u \in L^2(0,T)$.
\\
Now we can prove
\\
{\bf Theorem 5.1.}
\\
{\it
Let $f\in L^2(0,T)$. Then there exists a unique solution
$u = u(t)$ to initial value problem (5.1).
Moreover
$$
u(t) = aE_{\alpha,1}(-\lambda t^{\alpha})
+ \int^t_0 (t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda (t-s)^{\alpha}) f(s) ds,
\quad 0<t<T. \eqno{(5.2)}
$$
}
\\
Formula (5.2) itself is well-known (e.g., (3.1.34), p.141 in \cite{KST})
for $f$ in some classes. We can refer also to
Gorenflo and Mainardi \cite{GM},
Gorenflo, Mainardi and Srivastva \cite{GMS},
Gorenflo and Rutman \cite{GR}, Luchko and Gorenflo \cite{LG}.
On the other hand, we should understand that (5.2) holds in the
sense that both sides are in $H_{\alpha}(0,T)$ for each $f \in L^2(0,T)$.
\\
\vspace{0.2cm}
\\
{\bf Sketch of proof.}
\\
We see that (5.1) is equivalent to
$$
J^{-\alpha}(u-a) = -\lambda u + f \quad \mbox{in $L^2(0,T)$} \quad
\mbox{and}\quad u-a \in H_{\alpha}(0,T)
$$
and also to
$$
u=a-\lambda J^{\alpha}u + J^{\alpha}f \quad \mbox{in $H_{\alpha}(0,T)$}, \eqno{(5.3)}
$$
because $J^{\alpha}J^{-\alpha}(u-a) = J^{\alpha}(-\lambda u + f)$.
We conclude that
$J^{\alpha}: L^2(0,T) \, \longrightarrow \, L^2(0,T)$ is a compact operator
because the embedding $H_{\alpha}(0,T) \subset H^{\alpha}(0,T) \,\longrightarrow\, L^2(0,T)$
is compact (e.g., \cite{Ad}).
On the other hand, we apply the generalized Gronwall inequality
(e.g., Lemma A.2 in \cite{KRY}) to
$u = -\lambda J^{\alpha}u$ in $(0,T)$, that is,
$$
u(t) = \frac{-\lambda}{\Gamma(\alpha)}\int^t_0
(t-s)^{\alpha-1}u(s) ds, \quad 0<t<T.
$$
Then we obtain $u(t) = 0$ for $0<t<T$.
Therefore the Fredholm alternative
yields the unique existence of $u$ satifying (5.3).
Finally we define $\widetilde{u}(t)$ by
\begin{align*}
& \widetilde{u}(t) - a := a(E_{\alpha,1}(-\lambda t^{\alpha}) -1)
+ \int^t_0 (t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda (t-s)^{\alpha}) f(s) ds\\
=& a(E_{\alpha,1}(-\lambda t^{\alpha}) -1)+ (B_{\lambda}f)(t), \quad 0<t<T.
\end{align*}
Here $B_{\lambda}$ is defined by (4.6).
By Propositions 4.1 and 4.2, it follows that $\widetilde{u}(t)$ satisfies (5.1).
Thus the proof of Theorem 5.1 is complete.
$\blacksquare$
\\
We can discuss an initial value problem for a multi-term time-fractional
ordinary differential equation in the same way:
$$
\left\{ \begin{array}{rl}
& \partial_t^{\alpha}(u-a) + \sum_{k=1}^N c_k\partial_t^{\alpha_k}(u-a)
= -\lambda u + f(t), \\
& u-a \in H_{\alpha}(0,T),
\end{array}\right.
\eqno{(5.4)}
$$
where $c_1, ..., c_N \ne 0$, $0<\alpha_1 < \cdots < \alpha_N < \alpha \le 1$.
Theorem 2.1 implies that $J^{\alpha}\partial_t^{\alpha_k}
= J^{\alpha}J^{-\alpha_k} = J^{\alpha-\alpha_k}$.
By Proposition 2.2, we can obtain the equivalent equation:
$$
u-a = -\sum_{k=1}^N c_kJ^{\alpha-\alpha_k}(u-a) - \lambda J^{\alpha}u
+ J^{\alpha}f
$$
and we apply the Fredholm alternative to prove the unique existence
of solution, but we omit the details.
We further consider an initial value problem for
$f \in {^{-\alpha}{H}}(0,T)$:
$$
\left\{ \begin{array}{rl}
& \partial_t^{\alpha} (u-a)(t) = -\lambda u(t) + f(t), \quad 0<t<T, \\
& u-a \in L^2(0,T).
\end{array}\right.
\eqno{(5.5)}
$$
\\
Similarly to Theorem 5.1, we prove the well-posedness of (5.5) for
$f \in {^{-\alpha}{H}}(0,T)$.
\\
{\bf Theorem 5.2.}
\\
{\it
Let $0<\alpha < 1$ and $\alpha \not\in \mathbb{N}$. For $f \in {^{-\alpha}{H}}(0,T)$, there
exists a unique solution $u-a \in L^2(0,T)$ to (5.5).
Moreover we can choose a constant $C>0$ such that
$$
\Vert u-a\Vert_{L^2(0,T)} \le C(\vert a\vert + \Vert f\Vert_{{^{-\alpha}{H}}(0,T)})
$$
for all $a\in \mathbb{R}$ and $f \in {^{-\alpha}{H}}(0,T)$.
}
\\
{\bf Eaxmple 5.1.}
\\
We consider
$$
\partial_t^{\alpha} (u-a)(t) = f(t), \quad 0<t<T \eqno{(5.6)}
$$
with $f(t) = \frac{\Gamma(1+\beta)}{\Gamma(1-\alpha+\beta)}t^{\beta-\alpha}$,
where $\beta > -\frac{1}{2}$. Then, $u(t) = a + t^{\beta}$
satisfies (5.5) with $\lambda=0$ by (2.17).
However, if $-\frac{1}{2} < \beta < 0$ and $-\frac{1}{2} < \beta - \alpha$, then
$u, f \in L^2(0,T)$, but $u(t)$ does not satisfy
$\lim_{t\downarrow 0} u(t) = a$.
\\
\vspace{0.2cm}
\\
{\bf Proof of Theorem 5.2.}
\\
Setting $v:= u-a \in L^2(0,T)$, we rewrite (5.5) as
$$
\left\{ \begin{array}{rl}
& (J_{\alpha}')^{-1}v = -\lambda v - \lambda a + f, \\
& v \in L^2(0,T).
\end{array}\right.
$$
By the definition (2.13) of $\partial_t^{\alpha}$, equation (5.5) is equivalent to
$$
v = - \lambda J_{\alpha}'v + J_{\alpha}'(-\lambda a + f) \quad \mbox{in $L^2(0,T)$}.
\eqno{(5.7)}
$$
We set $g := J_{\alpha}'(f-\lambda a) \in L^2(0,T)$ and
$Pv := -\lambda J_{\alpha}'v$. Then we see that the solution $v=u-a$ is a fixed
point of $P$: $v=Pv + g$.
First the operator $P: L^2(0,T) \longrightarrow L^2(0,T)$ is a compact operator.
Indeed, Proposition 2.9 (iii) yields $J_{\alpha}'v = J^{\alpha}v$ for
$v\in L^2(0,T)$, and $J^{\alpha}: L^2(0,T) \longrightarrow H_{\alpha}(0,T)$ is
an isomorphism by Proposition 2.2.
Thus $J_{\alpha}'$ is a bounded operator from $L^2(0,T)$ to $H_{\alpha}(0,T)$.
Since the embedding $H_{\alpha}(0,T) \longrightarrow L^2(0,T)$ is compact, we see that
$J_{\alpha}': L^2(0,T) \longrightarrow L^2(0,T)$ is a compact operator
(e.g., \cite{Ad}).
Next we have to prove that $v=0$ in
$L^2(0,T)$ from assumption that $v=Pv$ in $(0,T)$.
Then, since
$$
J_{\alpha}'v(t) = J^{\alpha}v(t)
= \frac{1}{\Gamma(\alpha)}\int^t_0 (t-s)^{\alpha-1}v(s) ds
$$
by $v\in L^2(0,T)$, we obtain
$$
v(t) = -\frac{\lambda}{\Gamma(\alpha)}\int^t_0 (t-s)^{-\alpha}
v(s) ds, \quad 0<t<T.
$$
Hence,
$$
\vert v(t)\vert \le C\int^t_0 (t-s)^{\alpha-1}\vert v(s)\vert ds
\quad \mbox{for almost all $t \in (0,T)$.}
$$
The generalized Gronwall inequality (e.g., Lemma A.2 in
\cite{KRY}) implies $v(t) = 0$ for almost
all $t \in (0,T)$.
Therefore, the Fredholm alternative yields the unique existence of
a fixed point of $v=Pv + g$ in $L^2(0,T)$.
The estimate of $v$ follows from the application of the
generalized Gronwall inequality to (5.7) and Proposition 2.9 (ii):
$\Vert J_{\alpha}'f\Vert_{L^2(0,T)} \le C\Vert f\Vert_{{^{-\alpha}{H}}(0,T)}$.
Thus the proof of Theorem 5.2 is complete.
$\blacksquare$
\\
\vspace{0.2cm}
\\
{\bf Remark 5.1.}
\\
Now we compare formulation (5.1) with (5.5) for $f\in L^2(0,T)$.
\\
{\bf (a)}
For $f\in L^2(0,T)$, formulations (5.1) and (5.5) are equivalent.
Indeed, we immediately see that (5.1) implies (5.5).
Conversely, let $u$ satisfy (5.5). Then the second condition in
(5.5) yields $u \in L^2(0,T)$, and the first equation in (5.5)
concludes that $\partial_t^{\alpha} (u-a) \in L^2(0,T)$.
By Proposition 2.5, we see that $u-a \in H_{\alpha}(0,T)$, which means that
$u$ satisfies (5.1).
\\
{\bf (b)}
In formulation (5.1), as we remarked, we should not decompose
$\partial_t^{\alpha} (u-a) = \partial_t^{\alpha} u - \partial_t^{\alpha} a$, which is wrong for
$\alpha \ge \frac{1}{2}$ because $a \not\in H_{\alpha}(0,T)$.
We further consider this issue for $0 < \alpha < \frac{1}{2}$.
For the clearness, only here by $\partial_{t0}^{\alpha}$ we denote
$\partial_t^{\alpha}$ as operator with the domain $\mathcal{D}(\partial_{t0}^{\alpha})
= L^2(0,T)$, and we use the same notation $\partial_t^{\alpha}$ with the domain
$H_{\alpha}(0,T)$.
By means of (2.17) with $\beta = 0$, for $0<\alpha<\frac{1}{2}$, we see that
$1\in H_{\alpha}(0,T)$ and $\partial_{t0}^{\alpha}1 = \partial_t^{\alpha} 1
= \frac{1}{\Gamma(1-\alpha)}t^{-\alpha}$,
and also that $u-a \in H_{\alpha}(0,T)$ if and only if $u-a \in H^{\alpha}(0,T)$.
Therefore, we see:\\
If $0<\alpha<\frac{1}{2}$, then (5.1) is equivalent to
$$
\left\{ \begin{array}{rl}
& \partial_t^{\alpha} u = -\lambda u + \frac{a}{\Gamma(1-\alpha)}t^{-\alpha}
+ f(t) \quad \mbox{in $L^2(0,T)$},\\
& u \in H_{\alpha}(0,T).
\end{array}\right.
\eqno{(5.1)'}
$$
Noting $1 \in L^2(0,T)$ and using (2.17) with $\beta = 0$, for all
$\alpha \in (0,1)$, we can verify that (5.5) is equivalent to
$$
\left\{ \begin{array}{rl}
& \partial_{t0}^{\alpha} u = -\lambda u + \frac{a}{\Gamma(1-\alpha)}t^{-\alpha}
+ f(t) \quad \mbox{in ${^{-\alpha}{H}}(0,T)$},\\
& u \in L^2(0,T).
\end{array}\right.
\eqno{(5.5)'}
$$
Assming that $f\in L^2(0,T)$ and $0<\alpha<\frac{1}{2}$, we can conclude that
(5.1)' is equivalent to (5.5)'.
Indeed, $\frac{at^{-\alpha}}{\Gamma(1-\alpha)} \in L^2(0,T)$ and
so $\frac{at^{-\alpha}}{\Gamma(1-\alpha)} + f \in L^2(0,T)$.
Therefore, by $u \in L^2(0,T)$, the first equation in (5.5)' yields
$\partial_{t0}^{\alpha}u \in L^2(0,T)$, which means
that $u \in H_{\alpha}(0,T)$, and $\partial_{t0}^{\alpha}u = \partial_t^{\alpha} u$ in $(0,T)$.
Thus (5.1)' and (5.5)' are equivalent provided that $0 < \alpha < \frac{1}{2}$.
We emphasize that for $\frac{1}{2} \le \alpha < 1$, we cannot make
any reformulations of (5.1) similar to (5.1)' by decomposing
$u-a$ into $u$ and $-a$.
We can discuss similar reformulations also for
initial boundary value problems for fractional partial differential equations.
$\blacksquare$
\\
Now we take the widest domain of $\partial_t^{\alpha}$ according to classes in time
of functions under consideration, and we do not distinguish e.g.,
$\partial_{t0}^{\alpha}$ from $\partial_t^{\alpha}$, because there is no fear of
confusion.
\\
\vspace{0.2cm}
\\
{\bf Example 5.2.}
\\
Let $\alpha > \frac{1}{2}$ and let $f(t) := \delta_{t_0}(t)$:
the Dirac delta function at $t_0 \in (0,T)$.
In particular, we can prove that $\vert \varphi(t_0)\vert
\le C\Vert \varphi\Vert_{C[0,T]}$ and so $\delta_{t_0}\in
(C[0,T])'$:
$$
_{(C[0,T])'}<\delta_{t_0},\, \psi>_{C[0,T]} \, = \psi(t_0)
$$
for any $\psi \in C[0,T]$. Moreover, by the Sobolev embedding, we can see
that ${^{\alpha}{H}}(0,T) \subset H^{\alpha}(0,T) \subset C[0,T]$ by
$\alpha > \frac{1}{2}$. Therefore, $\delta_{t_0} \in {^{-\alpha}{H}}(0,T)$, and
$$
_{^{-\alpha}{H}(0,T)}< \delta_{t_0}\, \psi>_{^{\alpha}{H}(0,T)} \, = \psi(t_0)
$$
defines a bounded linear functional on ${^{\alpha}{H}}(0,T)$.
This $\delta_{t_0}$ describes an impulsive source term in fractional diffusion.
We will search for the representation of the solution to (5.5) with
$f = \delta_{t_0}$ and $a=0$:
$$
\left\{ \begin{array}{rl}
& \partial_t^{\alpha} u = -\lambda u + \delta_{t_0} \quad \mbox{in ${^{-\alpha}{H}}(0,T)$},\\
& u \in L^2(0,T).
\end{array}\right.
\eqno{(5.8)}
$$
Simulating a solution formula for
$$
d_t^{\alpha} u = -\lambda u + f(t), \quad u(0) = 0 \eqno{(5.9)}
$$
(e.g., \cite{KST}, p.141),
we can give a candidate for solution which is formally written by
$$
u(t) = \int^t_0 (t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda(t-s)^{\alpha}) \delta_{t_0}(s) ds,
\quad 0<t<T. \eqno{(5.10)}
$$
Our formal calculation suggests
$$
u(t) =
\left\{ \begin{array}{rl}
0, \quad & 0<t\le t_0, \\
(t-t_0)^{\alpha-1}E_{\alpha,\alpha}(-\lambda (t-t_0)^{\alpha}), \quad & t_0 < t \le T.
\end{array}\right.
\eqno{(5.11)}
$$
Now we will verify that $u(t)$ given by (5.11) is the solution to (5.8).
First it is clear that $u \in L^1(0,T)$.
Then we will verify $u = -\lambda J_{\alpha}'u + J_{\alpha}'\delta_{t_0}$
in $L^2(0,T)$.
Since $u \in L^2(0,T)$, we apply Proposition 2.10 to
have
$$
J_{\alpha}'u(t) = J^{\alpha}u(t) = \frac{1}{\Gamma(\alpha)}\int^t_0
(t-s)^{\alpha-1}u(s) ds.
$$
For $0<t<t_0$, by (5.11), we see $J^{\alpha}u(t) = 0$.
Next for $t_0\le t\le T$, we have
\begin{align*}
& -\lambda J^{\alpha}u(t) = \frac{-\lambda}{\Gamma(\alpha)}
\int^t_0 (t-s)^{\alpha-1} (s-t_0)^{\alpha-1} E_{\alpha,\alpha}(-\lambda (s-t_0)^{\alpha}) ds\\
=& \frac{1}{\Gamma(\alpha)}\int^t_{t_0} \sum_{k=0}^{\infty} (t-s)^{\alpha-1}
(s-t_0)^{\alpha-1}\frac{(-\lambda)^{k+1}(s-t_0)^{\alpha k}}
{\Gamma(\alpha k+\alpha)} ds\\
=& \frac{1}{\Gamma(\alpha)}\sum_{k=0}^{\infty} \left(\int^t_{t_0} (t-s)^{\alpha-1}
(s-t_0)^{\alpha k + \alpha -1} ds \right)
\frac{(-\lambda)^{k+1}}{\Gamma(\alpha k + \alpha)}\\
= & \frac{1}{\Gamma(\alpha)}\sum_{k=0}^{\infty} \left(\int^{t-t_0}_0 (t-t_0-\eta)^{\alpha-1}
\eta^{\alpha k + \alpha -1} d\eta \right)
\frac{(-\lambda)^{k+1}}{\Gamma(\alpha k + \alpha)}\\
=& \frac{1}{\Gamma(\alpha)}\sum_{k=0}^{\infty} \frac{\Gamma(\alpha)\Gamma(\alpha k + \alpha)}
{\Gamma(\alpha+\alpha k + \alpha)} (t-t_0)^{\alpha k + 2\alpha -1}
\frac{(-\lambda)^{k+1}}{\Gamma(\alpha k + \alpha)}
\end{align*}
$$
= (t-t_0)^{\alpha-1}\sum_{k=0}^{\infty} \frac{(-\lambda(t-t_0)^{\alpha})^{k+1}}
{\Gamma(\alpha(k+1) + \alpha)}
= (t-t_0)^{\alpha-1}\sum_{j=1}^{\infty} \frac{(-\lambda(t-t_0)^{\alpha})^j}
{\Gamma(\alpha j + \alpha)}. \eqno{(5.12)}
$$
For calculations of $J^{\alpha}u(t)$, we can apply
e.g., the formula (1.100) (p.25) in \cite{Po}, but we here
take a direct way.
On the other hand, by the definition of the dual operator $J_{\alpha}'$,
setting $v_0:= J_{\alpha}'\delta_{t_0}$, we have
$$
(v_0, \psi)_{L^2(0,T)} = \,\, _{^{-\alpha}{H}(0,T)}< \delta_{t_0},\, J_{\alpha}\psi>_{^{\alpha}{H}(0,T)}
\quad \mbox{for all $\psi \in L^2(0,T)$}.
$$
Since $J_{\alpha}\psi \in {^{\alpha}{H}}(0,T) \subset C[0,T]$ by $\alpha > \frac{1}{2}$
and Proposition 2.8, it follows that $v_0$ satisfies
$$
_{^{-\alpha}{H}(0,T)}< \delta_{t_0},\, J_{\alpha}\psi>_{^{\alpha}{H}(0,T)} \, = (J_{\alpha}\psi)(t_0)
= \frac{1}{\Gamma(\alpha)}\int^T_{t_0} (s-t_0)^{\alpha-1}\psi(s) ds.
$$
Therefore,
$$
\frac{1}{\Gamma(\alpha)}\int^T_{t_0} (s-t_0)^{\alpha-1}\psi(s) ds
= \, (v_0,\psi)_{L^2(0,T)}
= \int^{t_0}_0 v_0(s)\psi(s) ds
+ \int^T_{t_0} v_0(s)\psi(s) ds
$$
for all $\psi \in L^2(0,T)$. Choosing $\psi \in L^2(0,T)$ satisfying
$\psi = 0$ in $(t_0,T)$, we obtain
$$
\int^{t_0}_0 v_0(s)\psi(s) ds = 0,
$$
and so $v_0(s) = 0$ for $0 < s \le t_0$.
Hence,
$$
J_{\alpha}'\delta_{t_0}(s) = v_0(s) =
\left\{ \begin{array}{rl}
0, \quad & 0<s\le t_0,\\
\frac{(s-t_0)^{\alpha-1}}{\Gamma(\alpha)}, \quad & t_0<s<T.
\end{array}\right.
\eqno{(5.13)}
$$
In other words,
$$
\partial_t^{\alpha} v_0 = \delta_{t_0}, \eqno{(5.14)}
$$
where $v_0 \in L^2(0,T)$ is defined by (5.13).
Consequently, since
$$
\frac{(t-t_0)^{\alpha-1}}{\Gamma(\alpha)}
+ (t-t_0)^{\alpha-1}\sum_{k=1}^{\infty} \frac{(-\lambda(t-t_0)^{\alpha})^k}
{\Gamma(\alpha k + \alpha)}
= (t-t_0)^{\alpha-1}E_{\alpha,\alpha}(-\lambda(t-t_0)^{\alpha}),
$$
in terms of (5.12) and (5.13) we reach
$$
-\lambda J_{\alpha}'u(t) + J_{\alpha}'\delta_{t_0}(t) =
\left\{ \begin{array}{rl}
0, \quad & 0<t \le t_0, \\
(t-t_0)^{\alpha-1}E_{\alpha,\alpha}(-\lambda(t-t_0)^{\alpha}), \quad & t_0<t<T.
\end{array}\right.
$$
By (5.11) we verify
$$
u(t) = -\lambda J_{\alpha}'u(t) + J_{\alpha}'\delta_{t_0}(t).
$$
Thus we verified that $u(t)$ given by (5.11)
is the unique solution to (5.8).
$\blacksquare$
\\
We recall (4.6):
$$
(B_{\lambda}f)(t) := \int^t_0 (t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda (t-s)^{\alpha}) f(s) ds,
\quad 0<t<T, \quad \mbox{for $f\in L^2(0,T)$}.
$$
By Proposition 4.2, we know that $B_{\lambda}: L^2(0,T) \longrightarrow
H_{\alpha}(0,T)$ is a bounded operator.
\\
We close this section with
\\
{\bf Proposition 5.1 (Representation of solution to (5.5) with
$f \in {^{-\alpha}{H}}(0,T)$).}
\\
{\it
Let $\lambda > -\Lambda_0$ be fixed.
The operator $B_{\lambda}$ can be extended to
$S_{\lambda}: {^{-\alpha}{H}}(0,T) \longrightarrow L^2(0,T)$ as follows. For $f \in {^{-\alpha}{H}}(0,T)$, there
exists a sequence $f_n \in L^2(0,T)$, $n\in \mathbb{N}$ such that
$f_n \longrightarrow f$ in ${^{-\alpha}{H}}(0,T)$. Then
$$
\lim_{m,n\to \infty} \Vert B_{\lambda}f_n - B_{\lambda}f_m\Vert_{L^2(0,T)} = 0
\eqno{(5.15)}
$$
and $lim_{n\to \infty} B_{\lambda}f_n$ is unique in $L^2(0,T)$ independently of
choices of sequences $f_n$, $n\in \mathbb{N}$ such that $f_n \longrightarrow f$ in
${^{-\alpha}{H}}(0,T)$. Hence, setting
$$
S_{\lambda}f:= \lim_{n\to\infty} B_{\lambda}f_n \quad \mbox{in $L^2(0,T)$},
$$
we have
$$
\partial_t^{\alpha} (S_{\lambda}f) = -\lambda S_{\lambda}f + f \quad \mbox{in ${^{-\alpha}{H}}(0,T)$.} \eqno{(5.16)}
$$
}
\\
{\bf Proof.}
\\
First, since $L^2(0,T)$ is dense in
${^{-\alpha}{H}}(0,T)$, we can choose a sequence $f_n\in L^2(0,T)$, $n\in \mathbb{N}$
such that $\lim_{n\to\infty} f_n = f$ in $L^2(0,T)$.
\\
{\bf Verification of (5.15).}
\\
By Proposition 4.2, we see
$$
B_{\lambda} f = -\lambda J_{\alpha}'B_{\lambda}f + J_{\alpha}'f \eqno{(5.17)}
$$
and
$$
\Vert B_{\lambda}f\Vert_{L^2(0,T)} \le C\Vert J_{\alpha}'f\Vert_{L^2(0,T)}
\quad \mbox{for $f \in L^2(0,T)$}. \eqno{(5.18)}
$$
In view of (5.18), we obtain
$$
\Vert B_{\lambda}f_n - B_{\lambda}f_m \Vert_{L^2(0,T)}
\le C\Vert J_{\alpha}'f_n - J_{\alpha}'f_m\Vert_{L^2(0,T)}.
$$
Proposition 2.9 (ii) yields
$$
\Vert B_{\lambda}f_n - B_{\lambda}f_m \Vert_{L^2(0,T)}
\le C\Vert f_n - f_m\Vert_{{^{-\alpha}{H}}(0,T)}.
$$
Since $\lim_{m,n\to\infty} \Vert f_n-f_m\Vert_{{^{-\alpha}{H}}(0,T)} = 0$, we see
that $B_{\lambda}f_n$, $n\in \mathbb{N}$ converge in $L^2(0,T)$.
Similarly we can prove that $\lim_{n\to\infty} B_{\lambda}f_n$ is
determined independently of choices of sequences $f_n$, $n\in \mathbb{N}$ such that
$\lim_{n\to\infty} f_n = f$ in ${^{-\alpha}{H}}(0,T)$. Therefore, $S_{\lambda}f$
is well-defined for $f \in L^2(0,T)$.
\\
{\bf Verification of (5.16).}
\\
For an approximating sequence $f_n \in L^2(0,T)$, $n\in \mathbb{N}$ such that
$f_n \longrightarrow f$ in ${^{-\alpha}{H}}(0,T)$, we have
$$
B_{\lambda}f_n = -\lambda J_{\alpha}'B_{\lambda}f_n + J_{\alpha}'f_n \quad
\mbox{in $L^2(0,T)$ for $n\in \mathbb{N}$}.
$$
Since $\lim_{n\to\infty} B_{\lambda}f_n = S_{\lambda}f$ in $L^2(0,T)$,
letting $n\to \infty$, we obtain
$$
S_{\lambda}f = -\lambda \lim_{n\to \infty}J_{\alpha}'B_{\lambda}f_n
+ \lim_{n\to\infty}J_{\alpha}'f_n \quad \mbox{in $L^2(0,T)$}.
$$
We apply Proposition 2.9 (ii) to see $\lim_{n\to\infty} J_{\alpha}'f_n
= J_{\alpha}'f$ in $L^2(0,T)$ by $\lim_{n\to\infty} f_n = f$
in ${^{-\alpha}{H}}(0,T)$. Finally Proposition 2.9 (iii) implies that
$J_{\alpha}'\vert_{L^2(0,T)}: L^2(0,T) \longrightarrow L^2(0,T)$ is bounded, so that
$$
\lim_{n\to \infty} J_{\alpha}'B_{\lambda}f_n
= \lim_{n\to\infty} J^{\alpha}B_{\lambda}f_n
= J_{\alpha}'S_{\lambda}f \quad \mbox{in $L^2(0,T)$}.
$$
Therefore we reach
$$
S_{\lambda}f=-\lambda J_{\alpha}'S_{\lambda}f + J_{\alpha}'f \quad \mbox{in $L^2(0,T)$}.
$$
By the definition (2.13) of $\partial_t^{\alpha}$, this means
$$
\partial_t^{\alpha} (S_{\lambda}f) = -\lambda S_{\lambda}f + f \quad \mbox{in ${^{-\alpha}{H}}(0,T)$.}
$$
Thus the verification of (5.16) is complete, so that the proof of
Proposition 5.1 is finished.
$\blacksquare$
\\
We can discuss more about the reprensentation formula of solution to (5.5)
with $f \in {^{-\alpha}{H}}(0,T)$ in terms of convolution operators,
but we will postpone to a future work.
\section{Initial boundary value problem for fractional partial differential
equations: selected topics}
On the basis of $\partial_t^{\alpha}$ defined in Section 2, we construct a feasible
framework also for initial boundary value problems.
We recall that an elliptic operator $-\mathcal{A}$ is defined by (1.1),
and we assume all the conditions as described in Section 1
on the coefficients $a_{ij}$, $b_j$, $c \in C^1(\overline{\Omega})$.
Here we mostly consider the case $\alpha<1$, but cases $\alpha>1$ can be
formulated and studied similarly.
By $\nu = (\nu_1, ...., \nu_d)$ we denote
the outward unit normal vector to
$\partial\Omega$ at $x$ and set
$$
\partial_{\nu_A} v = \sum_{i,j=1}^d a_{ij}(\partial_jv) \nu_i \quad \mbox{on $\partial\Omega$}.
$$
We define an operator $-A$ in $L^2(\Omega)$ by
$$
\left\{ \begin{array}{rl}
& Av(x) = \mathcal{A}v(x), \quad x\in \Omega, \\
& \mathcal{D}(A) = \{ v\in H^2(\Omega);\, v\vert_{\partial\Omega} = 0\}
= H^2(\Omega) \cap H^1_0(\Omega).
\end{array}\right.
\eqno{(6.1)}
$$
Here $v\vert_{\partial\Omega}=0$ is understood as the sense of the trace (e.g.,
\cite{Ad}).
We can similarly discuss other boundary condition, for example,
$\mathcal{D}(A) = \{ v\in H^2(\Omega);\, \partial_{\nu_A} v + \sigma(x)v = 0 \,
\, \mbox{on $\partial\Omega$}\}$ with fixed function $\sigma(x)$, but we concentrate
on the homogeneous Dirichlet boundary condition $u\vert_{\partial\Omega} = 0$.
We formulate the initial boundary value problem by
$$
\partial_t^{\alpha} (u(x,t)-a(x)) + Au(x,t) = F(x,t) \quad \mbox{in $L^2(0,T;L^2(\Omega))$}
\eqno{(6.2)}
$$
and
$$
u-a \in H_{\alpha}(0,T;L^2(\Omega)). \eqno{(6.3)}
$$
We emphasize that we do not adopt formulation (1.2).
The formulation (6.2) - (6.3) corresponds to (5.1) for an initial value
problem for a time-fractional ordinary differential equation.
The term $Au$ in equation (6.2) means that $u(\cdot,t) \in \mathcal{D}(A)
= H^2(\Omega) \cap H^1_0(\Omega)$ for almost all $t\in (0,T)$, that is,
$$
u(\cdot,t)\vert_{\partial\Omega} = 0 \quad \mbox{for almost all $t\in (0,T)$.}
$$
In other words, the domain of $A$ describes the boundary condition which is
a conventional way in treating the classical partial differential equations.
Like fractional ordinary differential equations in Section 5, we understand
that (6.3) means the initial condition.
We first present a basic well-posedness result for (6.2) - (6.3):
\\
{\bf Theorem 6.1.}
\\
{\it
Let $0<\alpha<1$. Let $a\in H^1_0(\Omega)$ and $F \in L^2(0,T;L^2(\Omega))$.
Then there exists a unique solution $u=u(x,t)$ to (6.2) - (6.3) such that
$u-a \in H_{\alpha}(0,T;L^2(\Omega))$ and $u\in L^2(0,T;H^2(\Omega)\cap H^1_0(\Omega))$.
Moreover there exists a constant $C>0$ such that
$$
\Vert u-a\Vert_{H_{\alpha}(0,T;L^2(\Omega))}
+ \Vert u\Vert_{L^2(0,T;H^2(\Omega)\cap H^1_0(\Omega))}
\le C(\Vert a\Vert_{H^1_0(\Omega)} + \Vert F\Vert_{L^2(0,T;L^2(\Omega))})
$$
for all $a\in H^1_0(\Omega)$ and $F \in L^2(0,T;L^2(\Omega))$.
}
\\
Here we remark that
$$
\Vert v\Vert_{H_{\alpha}(0,T;L^2(\Omega))}
:= \Vert \partial_t^{\alpha} v \Vert_{L^2(0,T;L^2(\Omega))} \quad \mbox{for
$v \in H_{\alpha}(0,T;L^2(\Omega))$}.
$$
Unique existence results of solutions are known according to
several formulations of initial boundary value problems.
In Sakamoto and Yamamoto \cite{SY}, in the case of
symmetric $A$ where $b_j=0$ for $1\le j \le d$ in (1.1),
the unique existence is proved by means of the Fourier method, but the class of
solutions is not the same as here.
In the case where $b_j$ for $j=1,..., d$ are not necessarily zero,
assuming that the initial value $a$ is zero, Theorem 6.1 is proved in
Gorenflo, Luchko and Yamamoto \cite{GLY}. In both \cite{GLY} and
\cite{SY}, it is assumed that all the coefficients are independent of $t$ and
and $c=c(x) \le 0$ for $x\in \Omega$.
The work Kubica, Ryszewska and Yamamoto \cite{KRY} proved Theorem 6.1 in a
general case where $a_{ij}, b_j, c$ depends both on $x$ and $t$ without
extra assumpion $c\le 0$. In other words, Theorem 6.1 is a special case of
Theorem 4.2 in \cite{KRY}.
Furthermore, there have been other works on the well-posedness for
initial boundary value problems and we are
restricted to some of them: Bajlekova \cite{Ba}, Kubica and Yamamoto
\cite{KY}, Luchko \cite{Lu1}, \cite{Lu2}, Luchko and Yamamoto \cite{LuY1},
Zacher \cite{Za}. See also Pr{\"u}ss \cite{Pr} for a monograph on related
integral
equations, and a recent book Jin \cite{Ji} which mainly studies the symmetric
$A$. As for further references up to 2019, the
handbooks \cite{KoLu} edited by A.Kochubei and Y. Luchko
are helpful. Most of the above works discuss the case of
$F \in L^2(0,T;L^2(\Omega))$.
\\
Next we consider less regular $F$ and $a$ in $x$. To this end, we
introduce Sobolev spaces of negative orders in $x$.
Similarly to the triple $\, ^{\alpha}H(0,T) \subset L^2(\Omega) \subset
{^{-\alpha}{H}}(0,T)$
as is explained in Section 2, we introduce the dual space
$(H_1^0(\Omega))'$ of $H^1_0(\Omega)$ by identifying the dual space
$(L^2(0,T))'$ with $L^2(0,T)$:
$$
H^1_0(\Omega) \subset L^2(\Omega) \subset (H^1_0(\Omega))' = :H^{-1}(\Omega)
$$
(e.g., \cite{Bre}).
For less regular $F$ and $a$ in the $x$-variable, we know
\\
{\bf Theorem 6.2.}
\\
{\it
Let $0<\alpha<1$. For the coefficients of $A$, we assume the same
conditions as in Theorem 6.1.
Let $a\in L^2(\Omega)$ and $F \in L^2(0,T;H^{-1}(\Omega))$.
Then there exists a unique solution $u=u(x,t)$ to (6.2) - (6.3) such that
$u-a \in H_{\alpha}(0,T;H^{-1}(\Omega))$ and $u\in L^2(0,T;H^1_0(\Omega))$.
Moreover there exists a constant $C>0$ such that
$$
\Vert u-a\Vert_{H_{\alpha}(0,T;H^{-1}(\Omega))}
+ \Vert u\Vert_{L^2(0,T;H^1_0(\Omega))}
\le C(\Vert a\Vert_{L^2(\Omega)} + \Vert F\Vert_{L^2(0,T;H^{-1}(\Omega))})
$$
for all $a\in L^2(\Omega)$ and $F \in L^2(0,T;H^{-1}(\Omega))$.
}
\\
In Theorem 6.2, we consider both sides of (6.2) in $L^2(0,T;H^{-1}(\Omega))$.
The proof of Theorem 6.2 is found in \cite{KRY} and see also
\cite{KY}.
\\
\vspace{0.2cm}
\\
{\bf Remark 6.1.}
\\
As for general $\alpha>0$, by means of the space defined by (2.6), we can
formulate the initial boundary value problem as follows.
For $\alpha > 1$, we set $\alpha = m + \sigma$ with $m\in \mathbb{N}$ and
$0<\sigma\le 1$. Then for $\alpha > 1$ we formulate an initial boundary
value problem by
$$
\left\{ \begin{array}{rl}
& \partial_t^{\alpha} \left( u - \sum_{k=0}^m a_k\frac{t^k}{k!} \right)
= -Au + F(x,t), \quad x \in \Omega, \, 0<t<T,\\
& \left( u - \sum_{k=0}^m a_k\frac{t^k}{k!} \right)(x,\cdot)
\in H_{\alpha}(0,T) \quad \mbox{for almost all $x\in \Omega$}.
\end{array}\right.
$$
For $\sigma > \frac{1}{2}$, we can interpret the second condition as
usual initial conditions.
More precisely,
$$
\left( u - \sum_{k=0}^m a_k\frac{t^k}{k!} \right)(x,\cdot)
\in H_{\alpha}(0,T)
$$
if and only if
$$
\left\{ \begin{array}{rl}
\frac{\partial^ku}{\partial t^k}(\cdot,0) = a_k, \quad &k=0,1,..., m-1,\\
\frac{\partial^mu}{\partial t^m}(x,\cdot) - a_m\in H_{\sigma}(0,T).
\end{array}\right.
$$
\\
In this section, we pick up five topics and apply the results in
Sections 2 - 4. We postpone general and complete
descriptions to a future work.
Moreover, we are limited to the following $A$:
$$
-Av(x) = \sum_{j=1}^d \partial_j(a_{ij}(x)\partial_jv(x))
+ \sum_{j=1}^d b_j(x)\partial_jv + c(x)v, \quad x\in \Omega
\eqno{(6.4)}
$$
for $v \in \mathcal{D}(A):= H^2(\Omega) \cap H^1_0(\Omega)$
with
$$
a_{ij} = a_{ji} \in C^1(\overline{\Omega}), \quad
b_j\in C^1(\overline{\Omega})\quad \mbox{for $1\le i,j \le d$}, \quad
c\in C^1(\overline{\Omega}), \, \le 0 \quad \mbox{on $\Omega$}. \eqno{(6.5)}
$$
\\
{\bf \S6.1. Mild solution and strong solution}
We define an operator $L$ as a symmetric part of $A$ by
$$
-Lv(x) = \sum_{j=1}^d \partial_j(a_{ij}(x)\partial_jv(x))
+ c(x)v, \quad x\in \Omega, \quad
\mathcal{D}(L) = H^2(\Omega) \cap H^1_0(\Omega).
\eqno{(6.6)}
$$
Then there exist eigenvalues of $L$ and according to the multiplicities,
we can arrange all the eigenvalues as
$$
0 < \lambda_1 \le \lambda_2 \le \lambda_3 \le \cdots \, \longrightarrow \infty.
$$
Here by $c\le 0$ in $\Omega$, we can prove that $\lambda_1 > 0$.
Moreover we can choose eigenfunctions $\varphi_n$ for $\lambda_n$,
$n\in \mathbb{N}$ such that $\{ \varphi_n\}_{n\in \mathbb{N}}$ is an orthonormal
basis in $L^2(\Omega)$. Henceforth $(\cdot,\cdot)$ and
$\Vert \cdot\Vert_{L^2(\Omega)}$ denote the scalar product and the norm in
$L^2(\Omega)$ respectively and we write $(\cdot,\cdot)_{L^2(\Omega)}$
when we like to specify the space. Thus $L\varphi_n = \lambda_n\varphi_n$,
$\Vert \varphi_n\Vert_{L^2(\Omega)} = 1$ and $(\varphi_n\, \varphi_m) = 0$ if
$n\ne m$.
For $\gamma \in \mathbb{R}$, we can define a fractional power $L^{\gamma}$ of $L$ by
$$
\left\{ \begin{array}{rl}
& (L^{\gamma}v)(x) = \sum_{n=1}^{\infty} \lambda_n^{\gamma}(v,\varphi_n)\varphi_n(x),\\
& \mathcal{D}(L^{\gamma}) :=
\left\{\begin{array}{rl}
& L^2(\Omega) \quad \mbox{if $\gamma\le 0$}, \\
& \left\{ v\in L^2(\Omega);\, \sum_{n=1}^{\infty} \lambda_n^{2\gamma}
\vert (v,\varphi_n)\vert^2 < \infty\right\} \quad \mbox{if $\gamma > 0$}.
\end{array}\right.
\end{array}\right.
\eqno{(6.7)}
$$
We set
$$
\Vert v\Vert_{\mathcal{D}(L^{\gamma})}:= \left( \sum_{n=1}^{\infty} \lambda_n^{2\gamma}
\vert (v,\varphi_n)\vert^2 \right)^{\frac{1}{2}} \quad \mbox{if $\gamma > 0$}.
\eqno{(6.8)}
$$
Then it is known that
$$
\mathcal{D}(L^{\frac{1}{2}}) = H^1_0(\Omega), \quad
C^{-1}\Vert v\Vert_{H^1_0(\Omega)} \le \Vert L^{\frac{1}{2}}v\Vert_{L^2(\Omega)}
\le C\Vert v\Vert_{H^1_0(\Omega)}, \quad v\in H^1_0(\Omega).
$$
Here $C>0$ is independent of choices of $v\in H^1_0(\Omega)$.
We further define operator $S(t)$ and $K(t)$ from $L^2(\Omega)$ to
$L^2(\Omega)$ by
$$
S(t)a := \sum_{n=1}^{\infty} E_{\alpha,1}(-\lambda_nt^{\alpha})(a, \varphi_n)\varphi_n
$$
and
$$
K(t)a := \sum_{n=1}^{\infty} t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_nt^{\alpha})(a,\varphi_n)\varphi_n
$$
for all $a \in L^2(\Omega)$.
Then for $0\le \gamma \le 1$, we can find constants $C_1 > 0$ and
$C_2=C_2(\gamma) >0$ such that
$$
\left\{ \begin{array}{rl}
& \Vert S(t)a\Vert_{L^2(\Omega)} \le C_1\Vert a\Vert_{L^(\Omega)}, \\
& \Vert L^{\gamma}K(t)a\Vert_{L^2(\Omega)}
\le C_2t^{\alpha(1-\gamma)-1}\Vert a\Vert_{L^2(\Omega)} \quad
\mbox{for $t>0$ and $a\in L^2(\Omega)$.}
\end{array}\right.
\eqno{(6.9)}
$$
The proof of (6.9) is direct by the definition of
$S(t)$ and $K(t)$ and can be found e.g., in \cite{GLY}.
Here and henceforth we write $u(t):= u(\cdot,t)$ as a mapping from $(0,T)$
to $L^2(\Omega)$. The proofs of (6.8) - (6.10) can be found e.g.,
in \cite{GLY}.
We can show
\\
{\bf Proposition 6.1.}
\\
{\it
Let $a\in H^1_0(\Omega)$ and $F\in L^2(0,T;L^2(\Omega))$ and let (6.5) hold.
The following are equivalent:
\\
(i) $u \in L^2(0,T;H^2(\Omega) \cap H^1_0(\Omega))$ satisfies (6.2) and (6.3).
\\
(ii) $u \in L^2(0,T;L^2(\Omega))$ satisfies
$$
u(t) = S(t)a + \int^t_0 K(t-s)\sum_{j=1}^d b_j\partial_ju(\cdot,s) ds
+ \int^t_0 K(t-s)F(s) ds, \quad 0<t<T. \eqno{(6.10)}
$$
}
Equation (6.10) corresponds to formula (5.2) for an initial value problem
for fractional ordinary differential equation.
According to the parabolic equation (e.g., Pazy \cite{Pa}),
the solution guaranteed by Theorem 6.1 is called a strong solution
of the fractional partial differential equation, while the solution to
an integrated equation (6.10) is called a mild solution.
Proposition 6.1 asserts the equivalence between these two kinds of
solutions under assumption that $a\in H^1_0(\Omega)$ and
$F \in L^2(0,T;L^2(\Omega))$.
For the proof of Proposition 6.1, we show two lemmata.
Henceforth, for $v \in L^2(\Omega)$, we define $\partial_jv
\in H^{-1}(\Omega) := (H^1_0(\Omega))'$, $1\le j \le d$, by
$$
_{H^{-1}(\Omega)}< \partial_jv, \, \varphi>_{H^1_0(\Omega)}\, \, := -(v,\, \partial_j\varphi)_{L^2(\Omega)}
\quad \mbox{for all $\varphi \in H^1_0(\Omega)$}.
$$
Then by the definition of the norm of the linear functional, we have
$$
\Vert \partial_jv\Vert_{H^{-1}(\Omega)}
= \sup_{\Vert \varphi\Vert_{H^1_0(\Omega)}=1} \vert _{H^{-1}(\Omega)}< \partial_jv,\, \varphi>_{H^1_0(\Omega)}
\vert
$$
$$
\le \sup_{\Vert \varphi\Vert_{H^1_0(\Omega)}=1} \vert (v,\, \partial_j\varphi)_{L^2(\Omega)}
\vert \le \Vert v\Vert_{L^2(\Omega)}. \eqno{(6.11)}
$$
We recall that $L^{-\frac{1}{2}}$ is defined as an operator from
$L^2(\Omega)$ to $L^2(\Omega)$ by (6.7), while also
$L^{\frac{1}{2}}: H^1_0(\Omega)\, \longrightarrow \, L^2(\Omega)$ is defined by (6.7).
Then we can state the first lemma which involves the extension of
$L^{-\frac{1}{2}}: L^2(\Omega)\, \longrightarrow \, L^2(\Omega)$.
\\
{\bf Lemma 6.1.}
\\
{\it
(i) The operator $L^{-\frac{1}{2}}$ can be extended as a bounded operator from
$H^{-1}(\Omega)$ to $L^2(\Omega)$. Henceforth, by the same notation,
we denote the extension. There exists a constant $C>0$
such that
$$
C^{-1}\Vert v\Vert_{H^{-1}(\Omega)}
\le \Vert L^{-\frac{1}{2}}v\Vert_{L^2(\Omega)} \le C\Vert v\Vert_{H^{-1}(\Omega)}
\quad \mbox{for all $v \in H^{-1}(\Omega)$}.
$$
(ii) $K(t)v = L^{\frac{1}{2}}K(t)L^{-\frac{1}{2}}v$ for $t>0$ and
$v \in H^{-1}(\Omega)$.
\\
(iii)
$$
\Vert L^{-\frac{1}{2}}(b_j\partial_jv)\Vert_{L^2(\Omega)} \le C\Vert v\Vert_{L^2(\Omega)}
\quad \mbox{for all $v \in L^2(\Omega)$}.
$$
}
The second lemma is
\\
{\bf Lemma 6.2.}
\\
{\it
(i) For $a \in L^2(\Omega)$, we have $S(t)a \in \mathcal{D}(L)$ for $t>0$ and
$$
\partial_t^{\alpha} (S(t)a-a) + LS(t)a = 0, \quad 0<t<T.
$$
Moreover there exists a constant $C_0>0$ such that
$$
\Vert S(t)a - a\Vert_{H_{\alpha}(0,T;L^2(\Omega))}
\le C_0\Vert a\Vert_{H^1_0(\Omega)}
$$
for all $a\in H^1_0(\Omega)$.
Here $C_0 > 0$ is independent of $a \in H^1_0(\Omega)$ and $T>0$.
\\
(ii) We have
$$
\partial_t^{\alpha}\left( \int^t_0 K(t-s)F(s) ds\right)
+ L\left( \int^t_0 K(t-s)F(s) ds\right) = F(t), \quad 0<t<T
$$
for $F \in L^2(0,T;L^2(\Omega))$.
Moreover, there exists a constant $C_1 = C_1(T)>0$ such that
$$
\left\Vert \int^t_0 K(t-s)F(s) ds\right\Vert_{H_{\alpha}(0,T;L^2(\Omega))}
\le C_1\Vert F\Vert_{L^2(0,T;L^2(\Omega))}
$$
for each $F\in L^2(0,T;L^2(\Omega))$.
}
\\
We note that Lemma 6.2 corresponds to Propositions 4.1 (i) and 4.2.
\\
Let Lemmata 6.1 and 6.2 be proved.
Then, the proof of (i) $\longrightarrow$ (ii) of Proposition 6.1 can be done
similarly to \cite{SY}. The proof of (ii) $\longrightarrow$ (i) of the proposition
can be derived from Lemmata 6.1 and 6.2, and we omit the details.
The proofs of the lemmata are provided in Appendix.
\\
{\bf \S6.2. Continuity at $t=0$}
As is discussed in Sections 1,2 and 5, the continuity of solutions to
fractional differential equations at $t=0$ is delicate, which requires
careful treatments for initial conditions.
However, if we assume $F=0$ in (6.2), then we can prove the following
sufficient continuity.
\\
{\bf Theorem 6.3.}
\\
{\it
Under (6.4) and (6.5),
for $a \in L^2(\Omega)$, the solution $u$ to (6.2) - (6.3) satisfies
$$
u \in C([0,T];L^2(\Omega)).
$$
}
The theorem means that $\lim_{t\to 0} \Vert u(\cdot,t) - a\Vert_{L^2(\Omega)}
= 0$, but the continuity at $t=0$ breaks if
a non-homogeneous term $F \ne 0$, which is already shown
in Example 5.1 in Section 5 concerning a fractional ordinary differential
equations.
In the case of $b_j=0$, $1\le j \le d$, the same result is proved in
\cite{SY} and we can refer also to \cite{Ji}. However, it seems no proofs
for a non-symmetric elliptic operator $A$, although one naturally expect
the same continuity.
The proof is typical as arguments by the operator theory applied to
the classical
partial differential equations and we carry out similar arguments for
fractional differential equations within our framework.
\\
{\bf Proof.}
\\
{\bf First Step.}
\\
From Lemma 6.1, it follows that
the solution $u$ to (6.2) - (6.3) with $F=0$ is given by
$$
u(\cdot,t) = S(t)a + \int^t_0 K(t-s)\sum_{j=1}^d b_j\partial_ju(\cdot,s) ds
=: S(t)a + M(u)(t), \quad 0<t<T. \eqno{(6.12)}
$$
The proof of Theorem 6.3 is based on
an approximating sequence for the solution $u(t)$ constructed by
$$
u_1(t) := S(t)a, \quad
u_{k+1}(t) := S(t)a + (Mu_k)(t), \quad k\in \mathbb{N}, \, 0<t<T.
$$
First we can prove
$$
S(\cdot)a \in C([0,T];L^2(\Omega)). \eqno{(6.13)}
$$
\\
{\bf Verification of (6.13).}
For $\ell \le m$, using (4.4) with $\lambda_n>0$, for $0\le t \le T$
we estimate
$$
\left\Vert \sum_{n=\ell}^m (a,\varphi_n)E_{\alpha,1}(-\lambda_nt^{\alpha})\varphi_n
\right\Vert_{L^2(\Omega)}^2
\le \sum_{n=\ell}^m \vert (a,\varphi_n)\vert^2
\vert E_{\alpha,1}(-\lambda_nt^{\alpha})\vert^2
\le C\sum_{n=\ell}^m \vert (a,\varphi_n)\vert^2.
$$
Therefore,
$$
\lim_{\ell,m\to \infty}\left\Vert
\sum_{n=\ell}^m (a,\varphi_n)E_{\alpha,1}(-\lambda_nt^{\alpha})\varphi_n
\right\Vert_{C([0,T];L^2(\Omega))} = 0,
$$
which means that
$$
\sum_{n=1}^N (a,\varphi_n)E_{\alpha,1}(-\lambda_nt^{\alpha})\varphi_n \in C([0,T];L^2(\Omega))
$$
converges to $S(t)a$ in $C([0,T];L^2(\Omega))$. Thus the verification of
(6.13) is complete.
$\blacksquare$
\\
{\bf Second Step.}
We prove
$$
\int^t_0 L^{\frac{1}{2}} K(s)w(t-s) ds \in C([0,T];L^2(\Omega)) \quad
\mbox{for $w \in C([0,T];L^2(\Omega))$}. \eqno{(6.14)}
$$
\\
{\bf Verification of (6.14).}\\
Let $0<t<T$ be arbitrarily fixed. For small $h>0$, we have
\begin{align*}
& \int^{t+h}_0 L^{\frac{1}{2}} K(s)w(t+h-s) ds - \int^t_0 L^{\frac{1}{2}} K(s)w(t-s) ds\\
=& \int^{t+h}_t L^{\frac{1}{2}} K(s)w(t+h-s) ds
+ \int^t_0 L^{\frac{1}{2}} K(s)(w(t+h-s)-w(t-s)) ds\\
=: & I_1(t,h) + I_2(t,h).
\end{align*}
Then by (6.9) we can estimate
\begin{align*}
& \Vert I_1(t,h)\Vert\le C\left( \int^{t+h}_t s^{\frac{1}{2}\alpha-1} ds \right)
\Vert w\Vert_{C([0,T];L^2(\Omega))}\\
=& \frac{2C}{\alpha}((t+h)^{\frac{1}{2} \alpha} - t^{\frac{1}{2} \alpha})
\Vert w\Vert_{C([0,T];L^2(\Omega))} \longrightarrow 0 \quad \mbox{as $h \to 0$}.
\end{align*}
Next
$$
\Vert I_2(t,h)\Vert \le C\int^t_0 s^{\frac{1}{2}\alpha-1}
\Vert w(t+h-s) - w(t-s)\Vert ds.
$$
We have $\max_{0\le s\le T} \Vert w(t+h-s) - w(t-s)\Vert \longrightarrow 0$ as
$h\to 0$ with fixed $t$ by $w\in C([0,T];L^2(\Omega))$. Hence,
since $s^{\frac{1}{2}\alpha-1} \in L^1(0,T)$, the Lebesgue convergence theorem
yields that $\lim_{h\to 0} \Vert I_2(t,h)\Vert = 0$. We can argue similarly
also for $h<0$, $t=0$ and $t=T$, and so the verification of (6.14) is complete.
$\blacksquare$
Next we proceed to the proofs of (6.15) and (6.16):
$$
u_k \in C([0,T];L^2(\Omega)), \quad k\in \mathbb{N} \eqno{(6.15)}
$$
and
$$
\mbox{$u_k$ converges in $C([0,T];L^2(\Omega))$ as $k\to \infty$.}
\eqno{(6.16)}
$$
\\
{\bf Third Step: Proof of (6.15).}
We will prove by induction.
By (6.13), we see that $u_1(t) = S(t)a
\in C([0,T];L^2(\Omega))$. We assume that $u_k \in C([0,T];L^2(\Omega))$.
Then, by Lemma 6.1 (ii), we have
$$
M(u_k)(t) = \int^t_0 K(t-s)\left( \sum_{j=1}^d b_j\partial_ju_k(s)\right) ds
= \int^t_0 L^{\frac{1}{2}} K(t-s) L^{-\frac{1}{2}}\left( \sum_{j=1}^d
b_j\partial_ju_k(s) \right) ds.
$$
Lemma 6.1 (iii) yields
$$
\Vert L^{-\frac{1}{2}}(b_j\partial_ju_k(t)) - L^{-\frac{1}{2}}(b_j\partial_ju_k(t'))\Vert
_{L^2(\Omega)}
\le C\Vert u_k(t) - u_k(t')\Vert_{L^2(\Omega)}, \quad t,t' \in [0,T]
$$
for $1\le j \le d$, so that
$$
L^{-\frac{1}{2}}b_j\partial_ju_k \in C([0,T];L^2(\Omega)), \quad 1\le j \le d.
$$
Therefore, the application of (6.14) to $M(u_k)$ implies
$$
M(u_k) \in C([0,T];L^2(\Omega)).
$$
Consequently,
$$
u_{k+1} = S(\cdot)a + M(u_k) \in C([0,T];L^2(\Omega)).
$$
Thus the induction completes the proof of (6.15).
$\blacksquare$
\\
{\bf Fourth Step: Proof of (6.16).}
\\
We have
\begin{align*}
& (u_{k+2} - u_{k+1})(t) = M(u_{k+1} - u_k)(t)\\
=& \int^t_0 L^{\frac{1}{2}}K(t-s)\sum_{j=1}^d L^{-\frac{1}{2}}(b_j\partial_j
(u_{k+1}-u_k)(s)) ds, \quad k\in \mathbb{N}, \, 0<t<T.
\end{align*}
We set $v_k:= u_{k+1} - u_k$. Then Lemma 6.1 (iii) and (6.9) yield
$$
\Vert v_{k+1}(t)\Vert
\le \int^t_0 \left\Vert L^{\frac{1}{2}}K(t-s)\sum_{j=1}^d L^{-\frac{1}{2}}
(b_j\partial_jv_k (s))\right\Vert ds
$$
$$
\le C\int^t_0 (t-s)^{\frac{1}{2}\alpha-1} \Vert v_k(s)\Vert ds, \quad k\in \mathbb{N}.
\eqno{(6.17)}
$$
Setting $M_0:= \Vert v_1\Vert_{C([0,T];L^2(\Omega))}$, we apply (6.17) with
$k=1$ to obtain
$$
\Vert v_2(t)\Vert \le C\int^t_0 (t-s)^{\frac{1}{2}\alpha-1} ds M_0
= \frac{CM_0}{\Gamma\left( \frac{1}{2}\alpha\right)}t^{\frac{1}{2}\alpha}, \quad
0<t<T.
$$
Therefore, substituting this into (6.17) with $k=2$, we obtain
$$
\Vert v_3(t)\Vert \le \frac{C^2M_0}{\Gamma\left( \frac{1}{2}\alpha\right)}
\int^t_0 (t-s)^{\frac{1}{2}\alpha-1} s^{\frac{1}{2}\alpha} ds
= C^2M_0\frac{\Gamma\left( \frac{1}{2}\alpha+1\right)}{\Gamma(\alpha+1)}
t^{\alpha}, \quad 0<t<T.
$$
Continuing this estimate, we can prove
$$
\Vert v_{k+1}(t)\Vert
\le \frac{\frac{1}{2}\alpha M_0C^k\left( \Gamma\left(
\frac{1}{2}\alpha\right)\right)^{k-1}}
{\Gamma\left( \frac{1}{2} k\alpha+1\right)}t^{\frac{1}{2} k\alpha}, \quad 0<t<T
$$
for all $k\in \mathbb{N}$.
Consequently,
$$
\Vert v_{k+1}\Vert_{C([0,T];L^2(\Omega))}
\le \frac{C_1}{\Gamma\left( \frac{1}{2} k\alpha+1\right)}
\left( C\Gamma\left( \frac{1}{2} \alpha\right)T^{\frac{1}{2}\alpha}\right)^k,
\quad k\in \mathbb{N}.
$$
By the asymptotic behavior of the gamma function, we can verify
$$
\lim_{k\to \infty} \frac{
\left( C\Gamma\left( \frac{1}{2} \alpha\right)T^{\frac{1}{2}\alpha}\right)^k}
{ \Gamma\left( \frac{1}{2} k\alpha+1\right)}
= 0,
$$
so that $\sum_{k=0}^{\infty} \Vert v_k\Vert_{C([0,T];L^2(\Omega))}$ converges.
Therefore, $\lim_{k\to \infty} u_k$ exists in $C([0,T];L^2(\Omega))$.
Thus the proof of (6.16) is complete.
$\blacksquare$
From the uniqueness of solution to (6.16), we can derive that its
limit is $u$, the solution to (6.2) - (6.3) with $F=0$.
Since $u_k \in C([0,T];L^2(\Omega))$ for $k\in \mathbb{N}$, the limits is also in
$C([0,T];L^2(\Omega))$.
Thus the proof of Theorem 6.3 is complete.
$\blacksquare$
\\
{\bf \S6.3. Stronger regularity in time of solution}
Again we consider (6.2) - (6.3) with $F\ne 0$. Theorem 6.1 provides a basic
result on the unique existence of $u$ for $F \in L^2(0,T;L^2(\Omega))$ and $a\in H^1_0(\Omega)$,
while Theorem 6.2 is the well-posedness for $a$ and
$F$ which is less regular in $x$.
Here we consider stronger time-regular $F$ to improve the regularity of
solution $u$. Thanks to the framework of $\partial_t^{\alpha}$ deifned by (2.4),
the argument for improving the reglarity of solution is automatic.
\\
\vspace{0.1cm}
\\
{\bf Theorem 6.4.}
\\
{\it
Let $0<\alpha<1$ and $\beta > 0$. We assume
$a \in H^2(\Omega) \cap H^1_0(\Omega)$ and
$$
F - Aa \in H_{\beta}(0,T;L^2(\Omega)). \eqno{(6.18)}
$$
Then there exists a unique solution $u$ to (6.2) - (6.3) such that
$$
u-a \in H_{\beta}(0,T;H^2(\Omega) \cap H^1_0(\Omega)) \cap
H_{\alpha+\beta}(0,T;L^2(\Omega)), \eqno{(6.19)}
$$
and we can find a constant $C>0$ such that
$$
\Vert u-a \Vert_{H_{\beta}(0,T;H^2(\Omega))}
+ \Vert u-a\Vert_{H_{\alpha+\beta}(0,T;L^2(\Omega))} \le
C(\Vert F - Aa\Vert_{H_{\beta}(0,T;L^2(\Omega))} + \Vert a\Vert_{H^2(\Omega)}).
$$
}
If $0<\beta<\frac{1}{2}$, then (6.18) is equivalent to that
$F \in H_{\beta}(0,T;L^2(\Omega))$ under condition $a \in H^2(\Omega) \cap
H^1_0(\Omega)$. If $\frac{1}{2} < \beta \le 1$, then (6.18) is equivalent to
$$
F \in H_{\beta}(0,T;L^2(\Omega)), \quad Aa = F(\cdot,0) \quad
\mbox{in $\Omega$} \eqno{(6.20)}
$$
under condition $a \in H^2(\Omega) \cap H^1_0(\Omega)$.
Condition (6.18) is a compatibility condition
which is necessary for lifting up the regularity of the solution, and is
required also for the parabolic equation for instance (see Theorem 6 in
Chapter 7, Section 1 in Evans \cite{Ev}).
From Theorem 6.4, we can easily derive
\\
{\bf Corollary 6.1.}
\\
{\it
Let $a=0$ and $0<\alpha<1$.
We assume
$$
F \in H_{\alpha}(0,T;L^2(\Omega)) \cap L^2(0,T;H^2(\Omega)\cap H^1_0(\Omega)).
$$
Then
$$
A^2u, \, \partial_t^{2\alpha}u \in L^2(0,T;L^2(\Omega)).
$$
}
\\
{\bf Proof of Theorem 6.4.}
\\
We will gain the regularity of the solution $u$ by means of
an equation which can be expected to be satisfied by
$\partial_t^{\alpha} u$, although such an equation is not justified for the moment.
We consider
$$
\partial_t^{\alpha} v + Av = \partial_{\beta}(F-Aa), \quad v\in H_{\alpha}(0,T;L^2(\Omega)).
\eqno{(6.21)}
$$
Then, by $\partial_t^{\beta}(F-Aa) \in L^2(0,T;L^2(\Omega))$, Theorem 6.1 yields
that $v$ exists and
\\
$v\in H_{\alpha}(0,T;L^2(\Omega)) \cap
L^2(0,T;H^2(\Omega) \cap H^1_0(\Omega))$.
We set $\widetilde{u} := J^{\beta}v + a$. Then Proposition 2.5 (i) implies
$$
\widetilde{u} - a \in H_{\alpha+\beta}(0,T;L^2(\Omega))
\cap H_{\beta}(0,T;H^2(\Omega)\cap H^1_0(\Omega)). \eqno{(6.22)}
$$
In view of Proposition 2.5 (ii) and $v \in H_{\alpha}(0,T;L^2(\Omega))$, we have
$$
J^{-\alpha}(\widetilde{u}-a) = J^{-\alpha}J^{\beta}v
= J^{\beta}J^{-\alpha}v,
$$
and so $J^{-\alpha}(\widetilde{u}-a) = J^{\beta}(J^{-\alpha}v)$.
Moreover, by $v \in L^2(0,T;H^2(\Omega) \cap H^1_0(\Omega))$, we see
$$
A\widetilde{u} = A(J^{\beta}v+a) = J^{\beta}Av + Aa.
$$
Therefore,
$$
J^{-\alpha}(\widetilde{u}-a) + A\widetilde{u} = J^{\beta}(J^{-\alpha}v+Av)+Aa.
$$
In terms of (6.21), we obtain
$$
J^{-\alpha}(\widetilde{u}-a) + A\widetilde{u} = J^{\beta}(\partial^{\beta}_t(F-Aa)) + Aa
= F.
$$
Therefore, combining (6.22), we can verify that $\widetilde{u}-a$
satisfies (6.2) - (6.3), and
the uniqueness of solution yields $u=\widetilde{u}-a$.
Thus he proof of Theorem 6.4 is complete.
$\blacksquare$
\\
\vspace{0,2cm}
\\
{\bf Proof of Corollary 6.1.}
\\
Theorem 6.4 yields
$$
u \in H_{\alpha}(0,T;H^2(\Omega)\cap H^1_0(\Omega)) \cap H_{2\alpha}(0,T;L^2(\Omega)).
$$
Consequently, applying also Theorem 3.1, we deduce that
$\partial_t^{2\alpha}u \in L^2(0,T;L^2(\Omega))$.
Moreover, since $\partial_t^{\alpha}\pppa = \partial_t^{2\alpha}$ by Theorem 3.1,
operating $\partial_t^{\alpha}$ to (6.2), we obtain
$$
\partial_t^{\alpha} Au = -\partial_t^{2\alpha}u + \partial_t^{\alpha} F \in L^2(0,T;L^2(\Omega)).
\eqno{(6.23)}
$$
On the other hand, $A(\partial_t^{\alpha} u + Au - F) = 0$ in
$L^2(0,T;(C_0^{\infty}(\Omega))')$. Consequently, in terms of (6.23), we
reach
$$
A^2u = -A\partial_t^{\alpha} + AF \in L^2(0,T;L^2(\Omega)).
$$
Thus the proof of Corollary 6.1 is complete.
$\blacksquare$
\\
{\bf \S6.4. Weaker regularity in time of solution}
First we introduce a function space $\, {^{-\alpha}{H}}(0,T;X)$ for
a Banach space $X$.
Indeed, thanks to Proposition 2.9 or Theorem 2.1,
the operator $J_{\alpha}': \, {^{-\alpha}{H}}(0,T)\, \longrightarrow \,L^2(0,T)$ is
surjective and isomorhism for $\alpha > 0$, so that we define
$$
\left\{ \begin{array}{rl}
& {^{-\alpha}{H}}(0,T;X):= \{ v \in L^2(0,T;X);\, J_{\alpha}'v \in L^2(0,T;X) \}, \\
& \mbox{with the norm} \qquad
\Vert v\Vert_{{^{-\alpha}{H}}(0,T;X)}:= \Vert J_{\alpha}'v \Vert_{L^2(0,T;X)}.
\end{array}\right.
\eqno{(6.24)}
$$
By $\partial_t^{\alpha} := (J_{\alpha}')^{-1}: L^2(0,T)\, \longrightarrow\,
{^{-\alpha}{H}}(0,T)$, we see also that
$$
\mbox{$\lim_{n\to \infty} v_n = v$ in $L^2(0,T;L^2(\Omega))$ implies
$\lim_{n\to \infty} \partial_t^{\alpha} v_n = \partial_t^{\alpha} v$ in ${^{-\alpha}{H}}(0,T)$.}
\eqno{(6.25)}
$$
In this subsection, we consider
$$
\partial_t^{\alpha} (u-a) + Au = F \quad \mbox{in ${^{-\alpha}{H}}(0,T;L^2(\Omega))$}
\eqno{(6.26)}
$$
and
$$
u-a \in L^2(0,T;L^2(\Omega)), \quad u\in {^{-\alpha}{H}}(0,T;H^2(\Omega) \cap H^1_0(\Omega)).
\eqno{(6.27)}
$$
As is argued in Example 5.2, for $\alpha > \frac{1}{2}$, a source term
$$
F(x,t) = \delta_{t_0}(t) f(x) \in {^{-\alpha}{H}}(0,T;L^2(\Omega))
$$
with $f \in L^2(\Omega)$, describes an impulsive source at $t=t_0$, and it is
not only mathematically but also physically meaningful to treat
singular source $F \in {^{-\alpha}{H}}(0,T;L^2(\Omega))$ with $\alpha>0$.
We here state one result on the well-posedness for (6.26) - (6.27).
\\
{\bf Theorem 6.5.}
\\
{\it
Let $0<\alpha<1$. We assume
$$
F \in {^{-\alpha}{H}}(0,T;L^2(\Omega)), \quad a\in L^2(\Omega).
$$
Then there exists a unique solution $u\in L^2(0,T;L^2(\Omega)) \cap
{^{-\alpha}{H}}(0,T;H^2(\Omega) \cap H^1_0(\Omega))$ to (6.26) - (6.27) and
we can find a constant $C>0$ such that
$$
\Vert u\Vert_{L^2(0,T;L^2(\Omega))} + \Vert u\Vert_{{^{-\alpha}{H}}(0,T;H^2(\Omega)\cap H^1_0(\Omega))}
\le C(\Vert a\Vert_{L^2(\Omega)} + \Vert F\Vert_{{^{-\alpha}{H}}(0,T;L^2(\Omega))})
\eqno{(6.28)}
$$
for each $a\in L^2(\Omega)$ and $F \in {^{-\alpha}{H}}(0,T;L^2(\Omega))$.
}
\\
{\bf Proof.}
\\
We will prove by creating an equation which should be satisfied by
$J_{\alpha}'u$, and such an equation can be given by
$$
\partial_t^{\alpha} J_{\alpha}'(u-a) + AJ_{\alpha}'u = J_{\alpha}'F,
$$
where we have to make justification.
Thus we consider the solution $v$ to
$$
\partial_t^{\alpha} v + Av = J_{\alpha}'F + a, \quad v\in H_{\alpha}(0,T;L^2(\Omega)).
\eqno{(6.29)}
$$
Since $J_{\alpha}'F + a \in L^2(0,T;L^2(\Omega))$, Theorem 6.1 implies the
unique existence of solution $v \in H_{\alpha}(0,T;L^2(\Omega)) \cap
L^2(0,T;H^2(\Omega) \cap H^1_0(\Omega))$. Therefore, Proposition 2.10
yields $(J_{\alpha}')^{-1}v = J^{-\alpha}v = \partial_t^{\alpha} v$ by
$v\in H_{\alpha}(0,T;L^2(\Omega))$. Setting
$$
\widetilde{u}:= (J_{\alpha}')^{-1}v = J^{-\alpha}v, \eqno{(6.30)}
$$
we readily verify
$$
\widetilde{u} \in L^2(0,T;L^2(\Omega)) \cap {^{-\alpha}{H}}(0,T;H^2(\Omega) \cap H^1_0(\Omega))
$$
by Propositions 2.2 and 2.9 (ii).
Furthermore, since (6.30) implies $(J_{\alpha}')^{-1}v = \partial_t^{\alpha} v$,
and $v=J_{\alpha}'\widetilde{u}$, we obtain
\begin{align*}
& (J_{\alpha}')^{-1}(\widetilde{u} - a) + A\widetilde{u} - F
= (J_{\alpha}')^{-1}(\widetilde{u} - a + AJ_{\alpha}'\widetilde{u} - J_{\alpha}'F)\\
=& (J_{\alpha}')^{-1}((J_{\alpha}')^{-1}v - a + Av - J_{\alpha}'F)
= (J_{\alpha}')^{-1}(\partial_t^{\alpha} v + Av - J_{\alpha}'F - a) = 0
\end{align*}
in ${^{-\alpha}{H}}(0,T;L^2(\Omega))$.
Since $J_{\alpha}'$ is injective, the application of (6.29) implies
$\partial_t^{\alpha} (\widetilde{u}-a) + A\widetilde{u} = F$ and $\widetilde{u}-a\in L^2(0,T;L^2(\Omega))$.
Hence, $\widetilde{u}$ is a solution to (6.26) - (6.27), and (6.28) holds.
We finally prove the uniqueness of solution.
Let $w \in L^2(0,T;L^2(\Omega)) \cap \, {^{-\alpha}{H}}(0,T; H^2(\Omega) \cap H^1_0(\Omega))$ and
$\partial_t^{\alpha} w + Aw = 0$ hold in $\,{^{-\alpha}{H}}(0,T;L^2(\Omega))$. Then
$$
J_{\alpha}'\partial_t^{\alpha} w + J_{\alpha}'Aw = J_{\alpha}'\partial_t^{\alpha} w + AJ_{\alpha}'w
= 0 \quad \mbox{in $L^2(0,T;L^2(\Omega))$.} \eqno{(6.31)}
$$
By $w\in L^2(0,T;L^2(\Omega))$ and (2.13), we see
$$
J_{\alpha}'\partial_t^{\alpha} w = J_{\alpha}'(J_{\alpha}')^{-1}w = w.
$$
On the other hand,
$$
w = (J^{\alpha})^{-1}J^{\alpha}w = \partial_t^{\alpha} J^{\alpha}w
$$
by (2.4) and $J^{\alpha}w\in H_{\alpha}(0,T)$. Hence,
${J_{\alpha}}'\partial_t^{\alpha} w = \partial_t^{\alpha} J^{\alpha}w$.
Therefore, in terms of (6.31), we have $J^{\alpha}w
\in H_{\alpha}(0,T;L^2(\Omega)) \cap L^2(0,T;H^2(\Omega) \cap H^1_0(\Omega))$ and
$\partial_t^{\alpha} (J^{\alpha}w) + A(J^{\alpha}w) = 0$ in $L^2(0,T;L^2(\Omega))$.
Hence, the uniqueness asserted by Theorem 6.1 implies $J^{\alpha}w=0$
in $(0,T)$. Since $J^{\alpha}: L^2(0,T)\, \longrightarrow\, L^2(0,T)$ is injective,
we obtain $w=0$ in $(0,T)$.
Thus the proof of Theorem 6.5 is complete.
$\blacksquare$
\\
Finally in this subsection,
in order to describe the flexibility of our approach, we show
\\
{\bf Proposition 6.2.}
\\
{\it
Let $0<\alpha<1$. In (6.26) and (6.27), we assume
$$
a=0, \quad F \in {^{-\alpha}{H}}(0,T;L^2(\Omega)) \cap \, ^{-2\alpha}{H}(0,T;
H^2(\Omega) \cap H^1_0(\Omega)). \eqno{(6.32)}
$$
Then the solution $u$ satisfies
$$
u \in \, ^{-2\alpha}{H}(0,T;\mathcal{D}(A^2)) \subset
\, ^{-2\alpha}{H}(0,T;H^4(\Omega)).
$$
}
The proposition means that under (6.32), the solution $u$ can hold
the $H^4(\Omega)$-regulairty in $x$ at the expense of
weaker regulairty $^{-2\alpha}{H}(0,T)$ in $t$.
\\
{\bf Proof of Proposition 6.2.}
\\
By Theoren 6.5, the solution $u$ to (6.26) - (6.27) with $a=0$
exists uniquely. More precisely, we have
$(J_{\alpha}')^{-1}u + Au = F$ in ${^{-\alpha}{H}}(0,T;L^2(\Omega))$ and
$$
u \in {^{-\alpha}{H}}(0,T;H^2(\Omega) \cap H^1_0(\Omega)) \cap L^2(0,T;L^2(\Omega)).
$$
Therefore $J_{\alpha}'(J_{\alpha}')^{-1} = (J_{\alpha}')^{-1}
J_{\alpha}'$ in $L^2(0,T;L^2(\Omega))$ and $J_{\alpha}'Au = AJ_{\alpha}'u$, and so
$$
(J_{\alpha}')^{-1}J_{\alpha}'u + AJ_{\alpha}'u = J_{\alpha}'F
\in L^2(0,T;L^2(\Omega)).
$$
Hence, operating $J_{\alpha}'$ again, we obtain
$$
(J_{\alpha}')(J_{\alpha}')^{-1}J_{\alpha}'u + AJ_{\alpha}'J_{\alpha}'u
= J_{\alpha}'J_{\alpha}'F.
$$
Consequently, we obtain
$J_{\alpha}'(J_{\alpha}')^{-1}J_{\alpha}'u = (J_{\alpha}')^{-1}
(J_{\alpha}'J_{\alpha}'u)$. Setting $v:= J_{\alpha}'J_{\alpha}'u$, we have
$$
\partial_t^{\alpha} v + Av = J_{\alpha}'J_{\alpha}'F, \quad
v\in H_{\alpha}(0,T;L^2(\Omega)). \eqno{(6.33)}
$$
We note that Proposition 2.9 (iii) yields
$J_{\alpha}'(J_{\alpha}'u) = J^{\alpha}(J_{\alpha}'u)$ by
$J_{\alpha}'u \in L^2(0,T;L^2(\Omega))$ and that $J_{\alpha}'F
\in L^2(0,T;L^2(\Omega))$ by $F \in {^{-\alpha}{H}}(0,T;L^2(\Omega))$ and
Proposition 2.9 (ii). Moreover $J_{\alpha}'(J_{\alpha}'F)
\in H_{\alpha}(0,T;L^2(\Omega))$ by Propositions 2.9 (iii) and 2.2.
Applying Proposition 2.9 (ii) twice to $F \in \,
^{-2\alpha}H(0,T;H^2(\Omega) \cap H^1_0(\Omega))$, we obtain
$J_{\alpha}'F \in {^{-\alpha}{H}}(0,T;H^2(\Omega)\cap H^1_0(\Omega))$, so that
$J_{\alpha}'(J_{\alpha}'F) \in L^2(0,T;H^2(\Omega) \cap H^1_0(\Omega))$.
Therefore the application of Corollary 6.1 to (6.33), yields
$$
A^2v = A^2J_{\alpha}'J_{\alpha}'u \in L^2(0,T;L^2(\Omega)).
$$
By $u \in L^2(0,T;L^2(\Omega))$, we see that $J_{\alpha}'J_{\alpha}'u \in
L^2(0,T;\mathcal{D}(A^2)) \subset L^2(0,T; H^4(\Omega))$.
In terms of Proposition 2.9 (ii), it follows
$u \in \, ^{-2\alpha}{H}(0,T; \mathcal{D}(A^2))$.
Thus the proof of Proposition 6.2 is complete.
$\blacksquare$
\\
{\bf \S6.5. Initial boundary value problem for multi-term time-fractional
partial differential equations}
Let $N\in \mathbb{N}$, $0 < \alpha_1 < \cdots < \alpha_N < \alpha
< 1$, $q_k \in C(\overline{\Omega})$ for $1\le k \le N$. We recall that $A$ is
defined by (1.1) and (6.1). We discuss an initial boundary value problem
for a multi-term time-fractional partial differential equation:
$$
\partial_t^{\alpha} (u-a) + \sum_{k=1}^N q_k(x)\partial_t^{\alpha_k}(u-a) + Au = F
\eqno{(6.34)}
$$
and
$$
u-a \in H_{\alpha}(0,T;L^2(\Omega)). \eqno{(6.35)}
$$
Then we can will prove
\\
{\bf Theorem 6.6.}
\\
{\it
For each $a\in H^1_0(\Omega)$ and $F \in L^2(0,T;L^2(\Omega))$, there exists a unique solution
$u \in L^2(0,T;H^2(\Omega) \cap H^1_0(\Omega))$ to (6.34) - (6.35).
Moreover there exists a constant $C>0$ such that
$$
\Vert u\Vert_{L^2(0,T;H^2(\Omega) \cap H^1_0(\Omega))}
+ \Vert u-a\Vert_{H_{\alpha}(0,T;L^2(\Omega))}
\le C(\Vert a\Vert_{H^1_0(\Omega)} + \Vert F\Vert_{L^2(0,T;L^2(\Omega))})
\eqno{(6.36)}
$$
for each $a\in H^1_0(\Omega)$ and $F \in L^2(0,T;L^2(\Omega))$.
}
\\
Here the constant $C>0$ depends on $\Omega$, $T$, $A$, $\alpha_k$, $q_k$ for
$1\le k \le N$. \\
In the same way as in Subsections 6.3 and 6.4, we can
establish stronger and weaker regular solutions, but we omit the details.
The arguments are direct within our framework based on
$\partial_t^{\alpha}: \, ^{-\beta}H(0,T)\, \longrightarrow \, ^{-\alpha-\beta}H(0,T)$ and
$\partial_t^{\alpha}: \, H_{\alpha+\gamma}(0,T)\, \longrightarrow \, H_{\gamma}(0,T)$ with
$\gamma \ge 0$ and $\beta\ge 0$.
For fractional ordinary differential equations, we mentioned the
corresponding initial value problem as (5.4) in Section 5.
The multi-term time-fractional partial differential equations
are considered for example in Kian \cite{Ki},
Li, Imanuvilov and Yamamoto \cite{LIY}, Li, Liu and Yamamoto \cite{LLY}
for symmetric $A$ which makes arguments simple.
Non-symmetric $A$ contains an advection term $\sum_{j=1}^d b_j\partial_ju$,
and so it is physically more feasible model.
However, we do not know the existing results on the corresponding
well-posedness for non-symmetric $A$.
In fact, our method for initial boundary value problems is not
restricted to symmetric $A$.
Moreover we can treat weaker and stronger solutions in the same way as
in Subsections 6.3 and 6.4, and we omit the details.
We can easily extend our method to the case where the coefficients
$q_k$, $1\le k \le N$ and $b_j$, $1\le j \le d$ and $c$ depend on time.
\\
\vspace{0.2cm}
\\
{\bf Proof.}
\\
{\bf First Step.}
\\
We prove the theorem under the assumption that $b_j=0$ for $1\le j \le d$ and
$c\le 0$ on $\Omega$.
By Proposition 6.1 and Lemma 6.2, it suffices to prove that there exists a
unique $u-a \in H_\alpha(0,T;L^2(\Omega))$ satisfying
$$
u(t) - a = S(t)a - a + \int^t_0 K(t-s)F(s) ds
+ \sum_{k=1}^N \int^t_0 K(t-s)q_k\partial_t^{\alpha_k}(u(s)-a) ds,
\quad 0<t<T. \eqno{(6.37)}
$$
In this step, we prove
\\
{\bf Lemma 6.3.}
\\
{\it
Let $k=1,..., N$.
\\
(i)
$$
\Vert \partial_t^{\alpha_k}K(t)a\Vert_{L^2(\Omega)}
\le Ct^{\alpha-\alpha_k-1}\Vert a\Vert_{L^2(\Omega)}, \quad
0<t<T, \quad a\in L^2(\Omega).
$$
(ii)
For $G \in L^2(0,T;L^2(\Omega))$, we have $\int^t_0 K(t-s)G(s) ds
\in H_{\alpha_k}(0,T;L^2(\Omega))$ and
$$
\partial_t^{\alpha_k}\int^t_0 K(t-s)G(s) ds
= \int^t_0 \partial_t^{\alpha_k}K(t-s)G(s) ds, \quad 0<t<T.
$$
}
\\
{\bf Proof of Lemma 6.3.}
\\
(i) We set
$$
K_m(t)a:= \sum_{n=1}^m t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_nt^{\alpha})(a,\varphi_n)\varphi_n
\quad \mbox{for $m\in \mathbb{N}$ and $a\in L^2(\Omega)$.}
$$
Then, by Lemma 4.1 (ii), we see that
\begin{align*}
& \partial_t^{\alpha_k}K_m(t)a = \sum_{n=1}^m \partial_t^{\alpha_k}
(t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_nt^{\alpha}))(a,\varphi_n)\varphi_n\\
=& \sum_{n=1}^m t^{\alpha-\alpha_k-1}E_{\alpha,\alpha-\alpha_k}
(-\lambda_nt^{\alpha})(a,\varphi_n)\varphi_n.
\end{align*}
Therefore, applying (4.4), we obtain
$$
\Vert \partial_t^{\alpha_k}K_m(t)a\Vert_{L^2(\Omega)}^2
= \sum_{n=1}^m t^{2(\alpha-\alpha_k-1)}
\vert E_{\alpha,\alpha-\alpha_k}(-\lambda_nt^{\alpha})\vert^2
\vert (a,\varphi_n)\vert^2
$$
$$
\le Ct^{2(\alpha-\alpha_k-1)}\sum_{n=1}^m
\vert (a,\varphi_n)\vert^2. \eqno{(6.38)}
$$
Letting $m\to \infty$, we complete the proof of part (i).
\\
(ii) By Proposition 4.2, we can readily prove
$\int^t_0 K(t-s)G(s) ds \in H_{\alpha}(0,T;L^2(\Omega)) \subset
H_{\alpha_k}(0,T;L^2(\Omega))$.
Noting that $G(x,\cdot) \in L^2(0,T)$ for almost all $x\in \Omega$,
in terms of Lemma 4.1 (iii), we have
$$
\partial_t^{\alpha_k}\int^t_0 K_m(x,t-s)G(x,s) ds
= \int^t_0 \partial_t^{\alpha_k}K_m(x,t-s)G(x,s) ds, \quad x\in \Omega,
\, 0<t<T
$$
for $m\in \mathbb{N}$.
Similarly to (6.38), we esimate
\begin{align*}
& \left\Vert \int^t_0 (\partial_t^{\alpha_k}K_m - \partial_t^{\alpha_k}K)
(t-s)G(\cdot,s) ds\right\Vert_{L^2(\Omega)}
\le \int^t_0 \Vert (\partial_t^{\alpha_k}K_m - \partial_t^{\alpha_k}K)
(t-s)G(\cdot,s)\Vert_{L^2(\Omega)} ds \\
\le& C\int^t_0 (t-s)^{\alpha-\alpha_k-1}
\left( \sum_{n=m+1}^{\infty} \vert (G(s), \varphi_n)_{L^2(\Omega)} \vert^2
\right)^{\frac{1}{2}} ds.
\end{align*}
The Young inequality on the convolution yields
\begin{align*}
& \left\Vert \int^t_0 (\partial_t^{\alpha_k}K_m - \partial_t^{\alpha_k}K)
(t-s)G(\cdot,s) ds\right\Vert_{L^1(0,T;L^2(\Omega))}\\
\le & C\Vert s^{\alpha-\alpha_k-1}\Vert_{L^1(0,T)}
\left\Vert \left( \sum_{n=m+1}^{\infty} \vert (G(s), \varphi_n)_{L^2(\Omega)} \vert^2
\right)^{\frac{1}{2}} \right\Vert_{L^2(0,T)}\\
\le& CT^{\alpha-\alpha_k}
\left( \int^T_0 \sum_{n=m+1}^{\infty} \vert (G(s), \varphi_n)_{L^2(\Omega)} \vert^2
ds \right)^{\frac{1}{2}} \, \longrightarrow\, 0
\end{align*}
as $m\to \infty$ by $G \in L^2(0,T;L^2(\Omega))$. Therefore,
$$
\lim_{m\to\infty} \int^t_0 \partial_t^{\alpha_k}K_m(t-s)G(s) ds
= \int^t_0 \partial_t^{\alpha_k}K(t-s)G(s) ds \quad
\mbox{in $L^1(0,T;L^2(\Omega))$}. \eqno{(6.39)}
$$
Similarly, using (4.4), we can verify
$$
\lim_{m\to\infty} \int^t_0 K_m(t-s)G(s) ds
= \int^t_0 K(t-s)G(s) ds \quad \mbox{in $L^1(0,T;L^2(\Omega))$}.
\eqno{(6.40)}
$$
Choosing $\gamma > \frac{1}{2}$, we see that $L^1(0,T) \subset \, ^{-\gamma}H(0,T)$
and the series in (6.40) converges also in $\, ^{-\gamma}H(0,T;L^2(\Omega))$.
Since $\partial_t^{\alpha_k}: \, ^{-\gamma}H(0,T)\, \longrightarrow \,
^{-\gamma-\alpha_k}H(0,T)$ is an isomorphism by Theorem 2.1, we obtain
$$
\lim_{m\to\infty} \partial_t^{\alpha_k}\int^t_0 K_m(t-s)G(s) ds
= \partial_t^{\alpha_k}\int^t_0 K(t-s)G(s) ds \quad
\mbox{in $\, ^{-\gamma-\alpha_k}H(0,T;L^2(\Omega))$}.
$$
Combining this with (6.39), we finish the proof of Lemma 6.3.
$\blacksquare$
\\
\vspace{0.1cm}
\\
{\bf Second Step.}
\\
We will prove the unique existence of $u$ to (6.37) by the contraction
mapping theorem.
We set
$$
Qu(t):= S(t)a + \int^t_0 K(t-s)F(s) ds
+ \sum_{k=1}^N \int^t_0 K(t-s)q_k\partial_s^{\alpha_k} (u(s) - a) ds, \quad
0<t<T.
$$
By Lemma 6.2, we see
$$
S(t)a - a, \, \int^t_0 K(t-s)F(s) ds \in H_{\alpha}(0,T;L^2(\Omega))
\quad \mbox{for $a \in H^1_0(\Omega)$ and $F\in L^2(0,T;L^2(\Omega))$}.
\eqno{(6.41)}
$$
We estimate $\sum_{k=1}^N \Vert Qu - Qv\Vert_{H_{\alpha_k}(0,T:L^2(\Omega))}$
for $u-a,\, v-a \in H_{\alpha_k}(0,T;L^2(\Omega))$. To this end, we show
$$
\left\Vert \partial_t^{\alpha_j} \int^t_0 K(t-s)w(s) ds\right\Vert_{L^2(\Omega)}
\le C\int^t_0 (t-s)^{\alpha-\alpha_j-1}\Vert w(s)\Vert_{L^2(\Omega)} ds,
\quad 1\le j\le N, \, 0<t<T. \eqno{(6.42)}
$$
\\
{\bf Verification of (6.42).}
\\
The application of Lemma 6.3 (ii) and (i) yields
\begin{align*}
& \left\Vert \partial_t^{\alpha_j} \int^t_0 K(t-s)w(s) ds\right\Vert_{L^2(\Omega)}
= \left\Vert \int^t_0 \partial_t^{\alpha_j} K(t-s)w(s) ds\right\Vert_{L^2(\Omega)}\\
\le& \int^t_0 \Vert \partial_t^{\alpha_j} K(t-s)w(s) \Vert_{L^2(\Omega)} ds
\le C\int^t_0 (t-s)^{\alpha-\alpha_j-1} \vert w(s)\vert_{L^2(\Omega)} ds.
\end{align*}
Thus (6.42) follows directly.
$\blacksquare$
We note that if $u-a, v-a \in H_{\alpha_N}(0,T;L^2(\Omega)) \subset
H_{\alpha_k}(0,T;L^2(\Omega))$ with $1\le k \le N$, then
$(u-a) - (v-a) = u-v \in H_{\alpha_k}(0,T;L^2(\Omega))$ for $1\le k \le N$.
Therefore, using $q_k \in L^{\infty}(\Omega)$ for $1\le k \le N$, we have
\begin{align*}
& \Vert \partial_t^{\alpha_j} (Qu(t) - Qv(t))\Vert_{L^2(\Omega)}
= \left\Vert \sum_{k=1}^N \partial_t^{\alpha_j} \int^t_0 K(t-s) q_k
\partial_s^{\alpha_k}(u(s) - v(s)) ds
\right\Vert_{L^2(\Omega)}\\
\le & \sum_{k=1}^N \left\Vert \partial_t^{\alpha_j} \int^t_0 K(t-s)
q_k\partial_s^{\alpha_k}(u(s) - v(s)) ds\right\Vert_{L^2(\Omega)}\\
\le& C\sum_{k=1}^N \int^t_0 (t-s)^{\alpha-\alpha_j-1}
\Vert \partial_s^{\alpha_k}(u(s) - v(s))\Vert_{L^2(\Omega)}ds.
\end{align*}
Summing up over $j=1, ..., N$ and applying
$(t-s)^{\alpha-\alpha_j-1} \le C(t-s)^{\alpha-\alpha_N-1}$
for $1\le j \le N$, we reach
$$
\sum_{j=1}^N \Vert \partial_t^{\alpha_j} (Qu(t) - Qv(t))\Vert_{L^2(\Omega)}
$$
$$
\le C \int^t_0 (t-s)^{\alpha-\alpha_N-1}
\sum_{k=1}^N \Vert \partial_s^{\alpha_k} (u(s) - v(s))\Vert_{L^2(\Omega)}ds.
\eqno{(6.43)}
$$
Hence, applying (6.42) for estimating
$\sum_{j=1}^N \Vert \partial_t^{\alpha_j} (Q^2u(t) - Q^2v(t))\Vert_{L^2(\Omega)}$,
we obtain
\begin{align*}
& \sum_{j=1}^N \Vert \partial_t^{\alpha_j} (Q^2u(t) - Q^2v(t))\Vert_{L^2(\Omega)}\\
\le& C\int^t_0 (t-s)^{\alpha-\alpha_N-1} \sum_{k=1}^N
\Vert \partial_s^{\alpha_k} (Qu(s) - Qv(s))\Vert_{L^2(\Omega)}ds.
\end{align*}
Substituting (6.43) and exchanging the order of the integral, we
obtain
\begin{align*}
& \sum_{k=1}^N \Vert \partial_t^{\alpha_k} (Q^2u(t) - Q^2v(t))\Vert_{L^2(\Omega)}\\
\le& C^2\int^t_0 (t-s)^{\alpha-\alpha_N-1}
\left( \int^s_0 (t-\xi)^{\alpha-\alpha_N-1}
\sum_{k=1}^N \Vert \partial^{\alpha_k}_{\xi} (u-v)(\xi)\Vert_{L^2(\Omega)} d\xi
\right) ds\\
= & C^2\int^t_0 \left( \int^t_{\xi} (t-s)^{\alpha-\alpha_N-1}
(t-\xi)^{\alpha-\alpha_N-1} ds\right)
\sum_{k=1}^N \Vert \partial^{\alpha_k}_{\xi} (u-v)(\xi)\Vert_{L^2(\Omega)} d\xi\\
=& \frac{C^2\Gamma(\alpha-\alpha_N)^2}{\Gamma(2(\alpha-\alpha_N))}
\int^t_0 (t-\xi)^{2\alpha-2\alpha_N-1}
\sum_{k=1}^N \Vert \partial^{\alpha_k}_{\xi} (u-v)(\xi)\Vert_{L^2(\Omega)} d\xi.
\end{align*}
Continuing this estimation, we can reach
\begin{align*}
& \sum_{k=1}^N \Vert \partial_t^{\alpha_k} (Q^mu(t) - Q^mv(t))\Vert_{L^2(\Omega)}\\
\le & \frac{(C\Gamma(\alpha-\alpha_N))^m}{\Gamma(m(\alpha-\alpha_N))}
\int^t_0 (t-s)^{m(\alpha-\alpha_N)-1}
\sum_{k=1}^N \Vert \partial_s^{\alpha_k} (u-v)(s) \Vert_{L^2(\Omega)} ds, \quad
0<t<T, \, m\in \mathbb{N}.
\end{align*}
Consequently, the Young inequality yields
\begin{align*}
& \left( \int^T_0 \left(
\sum_{k=1}^N \Vert \partial_t^{\alpha_k} (Q^mu(t) - Q^mv(t))\Vert_{L^2(\Omega)}\right)^2
ds \right)^{\frac{1}{2}} \\
\le & \frac{(C\Gamma(\alpha-\alpha_N))^m}{\Gamma(m(\alpha-\alpha_N))}
\left\Vert s^{m(\alpha-\alpha_N)-1} \, *\,
\sum_{k=1}^N \Vert \partial_s^{\alpha_k} (u-v)(s) \Vert_{L^2(\Omega)}
\right\Vert_{L^2(0,T)}\\
\le& \frac{(C\Gamma(\alpha-\alpha_N))^m}{\Gamma(m(\alpha-\alpha_N))}
\frac{T^{m(\alpha-\alpha_N)}}{m(\alpha-\alpha_N)}
\left(\int^T_0 \left( \sum_{k=1}^N \Vert \partial_s^{\alpha_k} (u-v)(s) \Vert_{L^2(\Omega)} \right)^2
ds\right)^{\frac{1}{2}}.
\end{align*}
For estimating the right-hand side of the above inequality,
we find a constant $C_N > 0$ such that
$$
C_N^{-1}\sum_{k=1}^N \vert \xi_k\vert^2 \le \left( \sum_{k=1}^N \vert \xi_k\vert
\right)^2 \le C_N\sum_{k=1}^N \vert \xi_k\vert^2
$$
for all $\xi_1, ..., \xi_N \in \mathbb{R}$, we obtain
\begin{align*}
& \sum_{k=1}^N \Vert \partial_t^{\alpha_k} (Q^mu - Q^mv)\Vert_{L^2(0,T;L^2(\Omega))}\\
\le & \frac{(C\Gamma(\alpha-\alpha_N)T^{\alpha-\alpha_N})^m}
{\Gamma(m(\alpha-\alpha_N)+1)}
\sum_{k=1}^N \Vert \partial_t^{\alpha_k} (u-v)\Vert_{L^2(0,T;L^2(\Omega))},
\end{align*}
that is,
$$
\Vert Q^mu - Q^mv\Vert_{H_{\alpha_N}(0,T;L^2(\Omega))}
\le \frac{(C\Gamma(\alpha-\alpha_N)T^{\alpha-\alpha_N})^m}
{\Gamma(m(\alpha-\alpha_N)+1)} \Vert u-v \Vert
_{H_{\alpha_N}(0,T;L^2(\Omega))}
$$
for all $u-a,v-a \in H_{\alpha_N}(0,T;L^2(\Omega))$ and $m \in \mathbb{N}$.
By the asymptotic behavior of the gamma function, we know
$$
\lim_{m\to \infty}
\frac{(C\Gamma(\alpha-\alpha_N)T^{\alpha-\alpha_N})^m}
{\Gamma(m(\alpha-\alpha_N)+1)} = 0,
$$
and choosing $m\in N$ sufficiently large, we see that
\\
$Q^m:\, H_{\alpha_N}(0,T;L^2(\Omega)) \, \longrightarrow\,
H_{\alpha_N}(0,T;L^2(\Omega))$ is a contraction. Thus we have proved
the unique existence of a solution $u$ to (6.37) such that
$u-a \in H_{\alpha_N}(0,T;L^2(\Omega))$, which completes the proof of
Theorem 6.6, provided that $b_j=0$ for $1\le j \le d$ and $c\le 0$ in
$\Omega$.
For general $b_j, c$, setting
$$
-Lv(x) := \sum_{i,j=1}^d \partial_i(a_{ij}(x)\partial_jv)(x)
$$
in place of (6.6), we replace (6.37) by
$$
u(t) - a = S(t)a-a
+ \int^t_0 K(t-s)\sum_{j=1}^d b_j\partial_ju(s) ds
+ \int^t_0 K(t-s) cu(s) ds
$$
$$
+ \int^t_0 K(t-s)F(s) ds
+ \sum_{k=1}^N \int^t_0 K(t-s) \partial_s^{\alpha_k} (u(s) - a) ds, \quad 0<t<T
\eqno{(6.44)}
$$
and argue similarly.
For the estimation of the second term on the right-hand side of
(6.44), in view of Theorem 3.4, we see
\begin{align*}
& \partial_t^{\alpha_k} \int^t_0 K(t-s) b_j\partial_ju(s)
= \partial_t^{\alpha_k} \int^t_0 K(s) b_j\partial_ju(t-s) ds\\
= & \int^t_0 K(s) b_j \partial_t^{\alpha_k} \partial_ju(t-s) ds
= \int^t_0 L^{\frac{1}{2}}K(s) L^{-\frac{1}{2}}(b_j\partial_j\partial_t^{\alpha_k} u(t-s)) ds,
\end{align*}
and so we can apply Lemma 6.1 (iii) to argue similarly.
Here we omit the details. Thus the proof of Theorem 6.6 is complete.
$\blacksquare$
\section{Application to an inverse source problem: illustrating example}
We construct our framework for fractional derivatives.
In this article, within the framework we study
initial value problems for fractional ordiary differential equations and
initial boundary value problems for fractional partial differential equations
which are classified into so called "direct problems".
On the other hand, "inverse problems" are important where we are
required to determine some of initial values, boundary values, order
$\alpha$, and coefficients in the fractional differential equations
by observation data of solution $u$.
The inverse problem is indispensable for accurate modelling for analyzing
phenomena such as anomalous diffusion. After relevant studies of inverse
problems, we can identify coefficicients, etc. to determine equations
themselves and can proceed to
initial value problems and initial boundary value problems.
Thus the inverse problem is the premise for the study of the forward problem.
Actually, the main purpose of the current article is to not only establish
a theory for direct problems concerning time-fractional differential equations,
but also apply the theory to inverse problems.
The inverse problems are very various and here we discuss only one
inverse problem in order to illustrate how our framework for
fractional derivatives works.
Let $0<\alpha<1$ and $A$ be the same elliptic operator as the previous
sections, that is, defined by (6.1) and (6.4). We consider
$$
\left\{ \begin{array}{rl}
& \partial_t^{\alpha} u + Au = \mu(t)f(x) \quad \mbox{in ${^{-\alpha}{H}}(0,T;L^2(\Omega))$},\\
& u\in L^2(0,T;L^2(\Omega)),
\end{array}\right.
\eqno{(7.1)}
$$
where $\mu \in {^{-\alpha}{H}}(0,T)$, $\not\equiv 0$ and $f \in L^2(\Omega)$.
\\
{\bf Inverse source problem.}
\\
{\it Let $\theta \in L^2(\Omega)$, $\not\equiv 0$ be arbitrarily chosen, and
let $f \in L^2(\Omega)$ be given.
Then determine $\mu = \mu(t) \in {^{-\alpha}{H}}(0,T)$ by data
$$
\int_{\Omega} u(x,t)\theta(x) dx, \quad 0<t<T. \eqno{(7.2)}
$$
}
By Theorem 6.5, we know that the solution $u$ exists uniquely and
$u \in L^2(0,T;L^2(\Omega)) \cap {^{-\alpha}{H}}(0,T;H^2(\Omega)\cap H^1_0(\Omega))$.
Therefore the data (7.2) are well-defined in $L^2(0,T)$.
As other choice of data in the case where
the spatial dimensions $d \le 3$, we can choose:
$u(x_0,t)$ for $0<t<T$ with fixed point $x_0\in \Omega$.
By $1\le d\le 3$, the Sobolev embedding implies
$u\in {^{-\alpha}{H}}(0,T;H^2(\Omega)) \subset {^{-\alpha}{H}}(0,T;C(\overline{\Omega}))$ and so
$u(x_0,\cdot) \in {^{-\alpha}{H}}(0,T)$, which implies that
data are well-defined in ${^{-\alpha}{H}}(0,T)$.
However we are restricted to data (7.2).
Data (7.2) are spatial average values of $u(x,t)$ with the weight function
$\theta(x)$.
We can choose $\theta(x)$ in (7.2) such that
supp $\theta$ is concentrating near one fixed point $x_0\in \Omega$,
which means that data are mean values of $u(\cdot,t)$ in a neighborhood of
$x_0$.
If $\alpha > \frac{1}{2}$, then ${^{\alpha}{H}}(0,T) \subset C[0,T]$
by the Sobolev embedding, and so the space ${^{-\alpha}{H}}(0,T)$ can contain
a linear combination of Dirac delta functions:
$$
\sum_{k=1}^N q_k\delta_{t_k}(t),
$$
where $q_k\in \mathbb{R}, \ne 0$, $t_j \in (0,T)$ for $1\le k\le N$ and
$_{H^{-1}(\Omega)}< \delta_{t_k},\, \varphi>_{H^1_0(\Omega)} = \varphi(t_k)$ for all
$\varphi \in C^{\infty}_0(0,T)$. Then our inverse source problem is concerned
with determination of $N$, $q_k$, $t_k$ for $1\le k \le N$.
Now we are ready to state
\\
{\bf Theorem 7.1 (uniqueness).}
\\
{\it
We assume that $f$ satisfies
$$
\int_{\Omega} \theta(x)f(x) dx \ne 0. \eqno{(7.3)}
$$
If
$$
\int_{\Omega} \theta(x)u(x,t) dx = 0, \quad 0<t<T,
$$
then $\mu=0$ in ${^{-\alpha}{H}}(0,T)$.
}
\\
{\bf Proof.}
\\
The following uniqueness is known:
Let $v \in H_{\alpha}(0,T;L^2(\Omega)) \cap L^2(0,T;H^2(\Omega) \cap H^1_0(\Omega))$
satisfy
$$
\partial_t^{\alpha} v + Av = g(t)f(x), \quad 0<t<T, \eqno{(7.4)}
$$
where
$g \in L^2(0,T)$ and $f \in L^2(\Omega)$. Under assumption (7.3), condition
$$
\int_{\Omega} \theta(x)v(x,t) dx = 0, \quad 0<t<T, \eqno{(7.5)}
$$
yields $g=0$ in $L^2(0,T)$.
\\
This uniqueness for $g \in L^2(0,T)$ can be proved by combining
Duhamel's principle and
the uniqueness result for initial boundary value problem for fractional
partial differential equation with $f=0$.
We can refer also to pp. 411-464 by Li, Liu and Yamamoto
in a handbook \cite{KoLu},
Sakamoto and Yamamoto \cite{SY}. In particular,
Jiang, Li, Liu and Yamamoto \cite{JLLY}, which establishes the uniqueness
for the case of $f=0$ with non-symmetric $A$. Thus the uniqueness in the
inverse source problem of determining $g$ is classical within the caregory
of $L^2(0,T)$, and so we omit the proof.
\\
Now we can readily reduce the proof of Theorem 7.1 to the case of
$g\in L^2(0,T)$. We set $v:= J_{\alpha}'u$. Then, by Propositions 2.5
and 2.9 (iii),
we see that $v\in H_{\alpha}(0,T;L^2(\Omega)) \cap L^2(0,T;H^2(\Omega) \cap H^1_0(\Omega))$
satisfy
$$
\partial_t^{\alpha} v + Av = (J_{\alpha}'\mu)(t)f(x) \quad \mbox{in $L^2(0,T;L^2(\Omega))$},
$$
where $J_{\alpha}'\mu \in L^2(0,T)$.
Noting by Proposition 2.9 (ii) that $\int_{\Omega} \theta(x)u(x,\cdot) dx
\in L^2(0,T)$, by (7.2) we obtain
\begin{align*}
& 0 = J_{\alpha}'\left(\int_{\Omega} \theta(x)u(x,\cdot) dx\right)
= J^{\alpha}\left( \int_{\Omega} \theta(x)u(x,\cdot) dx\right)\\
= & \frac{1}{\Gamma(\alpha)}\int^t_0 (t-s)^{\alpha-1}
\left( \int_{\Omega} \theta(x)u(x,s) dx\right) ds
= \int_{\Omega} \theta(x)(J^{\alpha}u)(x,t) dx = 0, \quad 0<t<T.
\end{align*}
That is, we reach (7.5), which yields $J_{\alpha}'\mu = 0$ in
$(0,T)$ by the uniqueness in the inverse problem in the case of
$J_{\alpha}'\mu \in L^2(0,T)$.
Since $J_{\alpha}': {^{-\alpha}{H}}(0,T) \longrightarrow L^2(0,T)$ is
injective by Proposition 2.9 (ii), we obtain that
$\mu=0$ in ${^{-\alpha}{H}}(0,T)$. Thus the proof of Theorem 7.1 is complete.
$\blacksquare$
\section{Concluding remarks}
{\bf 8.1. Main messages}
\\
\vspace{0.1cm}
\\
{\bf (A)} We establish fractional derivatives as operators in
subspaces of the Sobolev-Slobodecki spaces, whose orders are not necessarily
integer nor positive, and consider fractional derivatives in
spaces of distribution.
Accordingly we extend classes of functions of which we take
fractional derivatives.
In such distribution spaces, we can justify
the fractional calculus, and we can argue
as if all the functions under consideration would be in $L^1(0,T)$.
Such a typical example is a fractional derivative of a Heaviside
function in Example 2.4.
\\
\vspace{0.1cm}
\\
{\bf (B)}
More importantly, with our framework of fractional calculus, we can
construct a fundamental theory for linear fractional differential
equations, which is easily adjusted to weak solutions and strong
solutions, and slso classical solutions.
\\
We intend this article to be introductory accounts aiming at
opertator theoretical treatments of time fractional partial differential
equations. Comprehensive descriptions should require more
works and so this artcile can provide the essence of foundations.
Next we give some summary and prospects.
\\
{\bf 8.2. What we have done}
\\
\vspace{0.1cm}
\\
{\bf (i)}
In Section 2, we introduced Sobolev spaces $H_{\alpha}(0,T)$
as subspace of the Sobolev-Slobodecki spaces $H^{\alpha}(0,T)$,
and we defined a fractional derivative $\partial_t^{\alpha}$ as
an isomorphism from $H_{\alpha}(0,T)$ to $L^2(0,T)$, and finally
as isomorphisms from
$H_{\alpha+\beta}(0,T)\, \longrightarrow\, H_{\beta}(0,T)$ and
$\, ^{-\beta}H(0,T)\, \longrightarrow\, ^{-\alpha-\beta}H(0,T)$ for
$\alpha > 0$ and $\beta \ge 0$, where $\, ^{-\beta}H(0,T)$ and
$\, ^{-\alpha-\beta}H(0,T)$ are defined by the duality and
are subspace of the distribution space.
The key for the definition of $\partial_t^{\alpha}$ is an operator theory, and we
always attach the fractional derivatives with their domains.
The extension procedure is governed by the Riemann-Liouville
fractional integral operator $J^{\alpha}$, and $\partial_t^{\alpha}$ is defined by
the inverse to $J^{\alpha}$ with suitable domain.
\\
\vspace{0.1cm}
\\
{\bf (ii)}
In Section 3, we established several basic properties of $\partial_t^{\alpha}$,
which are naturally expected to hold as formulae of derivatives.
We apply some of them to fractional differential equations.
\\
\vspace{0.1cm}
\\
{\bf (iii)}
In Sections 5 and 6, we formulate initial value problems for
linear fractional ordinary differential equations and
initial boundary value problems
for linear fractional partial differential equations
to prove the well-posedness and
the regularity of solutions within two categories: weak solution and
strong solutoin. Our defined $\partial_t^{\alpha}$ enables us to treat fractional
differential equations in a feasible manner.
\\
\vspace{0.1cm}
\\
{\bf (iv)}
Our researches are strongly motivated by inverse problems for
fractional differential equations. The studies of
inverse problems can be well based on the formulation proposed in this article.
Taking into consideration the great variety of inverse problems,
in Section 7, we are obliged to be limited to one simple inverse problem
in order to illustrate the effectiveness of our framework.
\\
{\bf 8.3. What we will do}
Naturally we had to skip many important and intersting issues.
Moreover many researches are going on and here we give up
comprehensive prospects for future research topics.
We write some of them, related to our framework.
\\
\vspace{0.1cm}
\\
{\bf (I)}
As fractional partial differential equations, we exclusively consider
evolution equations with
elliptic operators of second order for spatial part, which is
classified into fractional diffusion-wave equation.
The mathematical researches should not be limited to such evolution equations.
For exmple, the fractional transport equation is also significant:
$$
\partial_t^{\alpha} u(x,t) + H(x,t)\cdot \nabla u(x,t) = F(x,t).
$$
One can refer to Luchko, Suzuki and Yamamoto \cite{LuSY} for a
maximum principle, and Namba \cite{Na} for viscosity solution.
\\
\vspace{0.1cm}
\\
{\bf (II)}
In this article, we did not apply our framework to
nonlinear fractional differential equations, although necessary
steps are prepared. We can refer to Lucko and Yamamoto \cite{LuY2}
as one recent work.
\\
\vspace{0.1cm}
\\
{\bf (III)}
For fractional partial differetial
equations, we did not consider non-homogeneous boundary values in spite of
the necessity and the importance. We need more delicate treatments and
see e.g., Yamamoto \cite{Ya1}.
\\
\vspace{0.1cm}
\\
{\bf (IV)}
We should study variants of fractional derivatives according to
physical backgrounds. We mention a distributed derivative just as one example:
$$
\int^1_0 \partial_t^{\alpha} u(x,t) d\alpha.
$$
The work Yamamoto \cite{Ya2} studies similar treatments for a
generalized fractional derivative
$$
\frac{1}{\Gamma(1-\alpha)}\int^t_0 (t-s)^{-\alpha}g(t-s)\frac{dv}{ds}(s) ds
$$
with $0<\alpha<1$ and suitable function $g$, and discusses corresponding
initial value problems for time-fractional ordinary differential equations.
It is another future issue about how much our framework works for
such derivatives.
\\
\vspace{0.1cm}
\\
{\bf (V)}
Our approach is based essentially on the $L^2$-space, and our theory is
consistent within $L^2$-based Sobolev spaces of any orders
$\alpha \in \mathbb{R}$. Therefore, for example, for treating
$L^1$-functions in time as source term $F(x,t)$, we have to embed
such functions to an $L^2$-based Sobolev space of negative order.
This is not the best possible way for gaining $L^p$-regularity in time
with $p\ne 2$, and we should construct the corresponding $L^p$-theory
for $\partial_t^{\alpha}$. We can refer to Yamamoto \cite{Ya22} as for
a similar work discussing some fundamental properties in the $L^p$-case
with $1\le p < \infty$.
\\
\vspace{0.1cm}
\\
{\bf (VI)}
With the aid of our framework for the direct problem,
results on inverse problems can be
expected to be sharpened in view of required regularity for instance.
This should be main future topics after this article.
\section{Appendix: Sketches of proofs of Lemmata 6.1 and 6.2}
{\bf Proof of Lemma 6.1.}\\
{\bf (i)}
By (6.7) and (6.8), noting that
$$
L^{-\frac{1}{2}}v = \sum_{n=1}^{\infty} \lambda_n^{-\frac{1}{2}}(v, \varphi_n)_{L^2(\Omega)}
\varphi_n
$$
for $v \in L^2(\Omega)$, we obtain
$$
L^{\frac{1}{2}}L^{-\frac{1}{2}}v = v \quad \mbox{for $v\in L^2(\Omega)$}
\eqno{(9.1)}
$$
and
$$
(L^{\frac{1}{2}}y,\, z)_{L^2(\Omega)} = (y,\, L^{\frac{1}{2}}z)_{L^2(\Omega)}
\quad \mbox{for $y,z \in \mathcal{D}(L^{\frac{1}{2}}) = H^1_0(\Omega)$}.
\eqno{(9.2)}
$$
Next we will prove that there exists a constant $C>0$ such that
$$
C^{-1}\Vert v\Vert_{H^{-1}(\Omega)} \le \Vert L^{-\frac{1}{2}}v\Vert_{L^2(\Omega)}
\le C\Vert v\Vert_{H^{-1}(\Omega)} \quad \mbox{for $v \in L^2(\Omega)$}.
\eqno{(9.3)}
$$
\\
{\bf Verification of (9.3).}
\\
Let $v \in L^2(\Omega)$. Then we have $L^{-\frac{1}{2}}v \in H^1_0(\Omega)$ by (6.7)
and $L^{\frac{1}{2}}L^{-\frac{1}{2}}v=v$.
Hence, by applying (6.7) again, equalities (9.1) and (9.2) yield
\begin{align*}
& _{H^{-1}(\Omega)}< v,\, \psi>_{H^1_0(\Omega)} =\, _{H^{-1}(\Omega)}< L^{\frac{1}{2}}L^{-\frac{1}{2}}v,\, \psi>_{H^1_0(\Omega)}\\
=& (L^{\frac{1}{2}}L^{-\frac{1}{2}}v,\, \psi)_{L^2(\Omega)}
= (L^{-\frac{1}{2}}v\, , L^{\frac{1}{2}}\psi)_{L^2(\Omega)} \quad
\mbox{for $v \in L^2(\Omega)$ and $\psi \in H^1_0(\Omega)$}.
\end{align*}
Consequently, applying (6.8), we obtain
\begin{align*}
& \Vert v\Vert_{H^{-1}(\Omega)}
= \sup_{\Vert \psi\Vert_{H^1_0(\Omega)}=1}\,
\vert _{H^{-1}(\Omega)}< v,\, \psi>_{H^1_0(\Omega)} \vert\,
=\, \sup_{\Vert \psi\Vert_{H^1_0(\Omega)}=1}
\vert (L^{-\frac{1}{2}}v,\, L^{\frac{1}{2}}\psi)_{L^2(\Omega)} \vert \\
\le & \Vert L^{-\frac{1}{2}}v\Vert_{L^2(\Omega)}
\sup_{\Vert \psi\Vert_{H^1_0(\Omega)}=1} \Vert L^{\frac{1}{2}}\psi\Vert_{L^2(\Omega)}
\le C\Vert L^{-\frac{1}{2}}v\Vert_{L^2(\Omega)}.
\end{align*}
Therefore, the first inequality of (9.3) is proved.
Next we have
$$
\Vert L^{-\frac{1}{2}}v\Vert_{L^2(\Omega)}
= \sup_{\Vert \psi\Vert_{H^1_0(\Omega)}=1}
\vert (L^{-\frac{1}{2}}v,\, \psi)_{L^2(\Omega)}\vert
= \sup_{\Vert \psi\Vert_{H^1_0(\Omega)}=1}
\vert (v,\, L^{-\frac{1}{2}}\psi)_{L^2(\Omega)} \vert.
$$
Since
$$
\Vert L^{-\frac{1}{2}}\psi\Vert_{H^1_0(\Omega)}
\le C\Vert L^{\frac{1}{2}}(L^{-\frac{1}{2}}\psi)\Vert_{L^2(\Omega)}
= C\Vert \psi\Vert_{L^2(\Omega)},
$$
by (9.1) we obtain
\begin{align*}
& \sup_{\Vert \psi\Vert_{L^2(\Omega)} = 1}
\vert (v,\, L^{-\frac{1}{2}}\psi)_{L^2(\Omega)}\vert
\le \sup_{\Vert \widetilde{\psi}\Vert_{H^1_0(\Omega)} \le C}
\vert (v,\, \widetilde{\psi})_{L^2(\Omega)}\vert\\
\le& C\sup_{\Vert \mu\Vert_{H^1_0(\Omega)} = 1}
\vert _{H^{-1}(\Omega)}< v,\, \mu\, >_{H^1_0(\Omega)} \vert \le C\Vert v\Vert_{H^{-1}(\Omega)}.
\end{align*}
Hence, the second inequality of (9.3) is proved, and the verification of
(9.3) is complete.
$\blacksquare$
\\
Since $L^2(\Omega)$ is dense in $H^{-1}(\Omega)$, in view of (9.3),
we can extend $L^{-\frac{1}{2}}: L^2(\Omega) \, \longrightarrow\, L^2(\Omega)$ to
$L^{-\frac{1}{2}}: H^{-1}(\Omega) \, \longrightarrow\, L^2(\Omega)$.
More precisely, for any $v \in H^{-1}(\Omega)$, we choose $v_n \in L^2(\Omega)$,
$n\in \mathbb{N}$
such that $\lim_{n\to\infty} v_n = v$ in $H^{-1}(\Omega)$. Then (9.3) implies
$\lim_{m,n\to\infty} \Vert L^{-\frac{1}{2}}(v_m-v_n)\Vert_{L^2(\Omega)}
= 0$, that is, a sequence $\{ L^{-\frac{1}{2}}v_n\}_{n\in \mathbb{N}}
\subset L^2(\Omega)$ is a Cauchy sequence and we can define
$$
L^{-\frac{1}{2}}v:= \lim_{n\to\infty} L^{-\frac{1}{2}}v_n \quad
\mbox{in $L^2(\Omega)$}.
$$
Then, by (9.3), the limit is independent of choices of $\{v_n\}
_{n\in \mathbb{N}}$, and we see that (9.3) holds for $v\in H^{-1}(\Omega)$.
Thus the proof of part (i) of Lemma 6.1 is complete.
$\blacksquare$
\\
\vspace{0.1cm}
\\
{\bf (ii)}
By (6.7), we can directly verify
$$
K(t)v = L^{\frac{1}{2}}K(t)L^{-\frac{1}{2}} \quad \mbox{for $t>0$ and
$v \in L^2(\Omega)$}.
$$
Let $v\in H^{-1}(\Omega)$ be arbitrarily given. Since
$L^2(\Omega)$ is dense in $H^{-1}(\Omega)$, there exists a sequence
$v_n \in L^2(\Omega)$, $n\in \mathbb{N}$ such that
$v_n \longrightarrow v$ in $H^{-1}(\Omega)$ as $n\to \infty$.
By part (i) and the extended $L^{-\frac{1}{2}}$,
estimate (9.3) holds for $v \in H^{-1}(\Omega)$.
Hence, $L^{-\frac{1}{2}}v_n \,\longrightarrow \, L^{-\frac{1}{2}}v$ in $L^2(\Omega)$.
In terms of (6.9), we see that
$$
L^{\frac{1}{2}}K(t)L^{-\frac{1}{2}} v_n \,\longrightarrow \, L^{\frac{1}{2}}K(t)L^{-\frac{1}{2}}v
\quad \mbox{in $L^2(\Omega)$}
$$
as $n\to \infty$ for $t>0$.
Since $L^{\frac{1}{2}}K(t) L^{-\frac{1}{2}}v_n = v_n$ by $v_n \in L^2(\Omega)$,
we obtain
$v_n \,\longrightarrow \, L^{\frac{1}{2}}K(t)L^{-\frac{1}{2}}v$ in $L^2(\Omega)$
as $n\to \infty$ for $t>0$.
Using that $v_n \longrightarrow v$ in $H^{-1}(\Omega)$, we reach
$L^{\frac{1}{2}}K(t) L^{-\frac{1}{2}}v = v$ for $t>0$.
Thus the proof of part (ii) is complete.
$\blacksquare$
\\
\vspace{0.1cm}
\\
{\bf (iii)}
Let $b_j\in C^1(\overline{\Omega})$, $1\le j \le d$. Then
we will verify
$$
\Vert b_j\partial_ju\Vert_{H^{-1}(\Omega)} \le C\Vert u\Vert_{L^2(\Omega)},
\quad 1\le j \le d \quad \mbox{for all $u \in L^2(\Omega)$}.
\eqno{(9.4)}
$$
Here and henceforth constants $C>0$ depend on $b_j$.
If (9.4) is verified, then we can complete the proof of part (iii) of
the lemma as follows: by combining (9.4) with part (i) of the lemma,
we obtain
$$
\Vert L^{-\frac{1}{2}}(b_j\partial_ju)\Vert_{L^2(\Omega)}
\le C\Vert b_j\partial_ju\Vert_{H^{-1}(\Omega)} \le C\Vert u\Vert_{L^2(\Omega)}.
$$
$\blacksquare$
\\
{\bf Verification of (9.4).}
\\
By the definition of $b_j\partial_ju$ as element in $H^{-1}(\Omega)$,
in terms of $b_j \in C^1(\overline{\Omega})$, we have
\begin{align*}
& \Vert b_j\partial_ju\Vert_{H^{-1}(\Omega)}
= \sup_{\Vert \psi\Vert_{H^1_0(\Omega)}=1}
\left\vert _{H^{-1}(\Omega)}< b_j\partial_ju, \, \psi >_{H^1_0(\Omega)} \right\vert \\
=& \sup_{\Vert \psi\Vert_{H^1_0(\Omega)}=1}
\vert -(u,\, \partial_j(b_j\psi))_{L^2(\Omega)} \vert
\le \sup_{\Vert \psi\Vert_{H^1_0(\Omega)}=1} \Vert u\Vert_{L^2(\Omega)}
\Vert \partial_j(b_j\psi)\Vert_{L^2(\Omega)}
\le C\Vert u\Vert_{L^2(\Omega)}.
\end{align*}
Thus the verification of (9.4), and accordingly the proof of Lemma 6.1 are
complete.
$\blacksquare$
\\
{\bf Proof of Lemma 6.2.}
\\
{\bf (i)}
We set
$$
S_m(t)a:= \sum_{n=1}^m E_{\alpha,1}(-\lambda_nt^{\alpha})(a,\varphi_n)\varphi_n \quad
\mbox{for $m\in \mathbb{N}$.}
$$
Proposition 4.1 (i) yields
$$
\partial_t^{\alpha} (S_m(t)a-a)
= \sum_{n=1}^m -\lambda_nE_{\alpha,1}(-\lambda_nt^{\alpha})(a,\varphi_n)\varphi_n
= -L\left( \sum_{n=1}^m E_{\alpha,1}(-\lambda_nt^{\alpha})(a,\varphi_n)\varphi_n \right),
$$
that is,
$$
\partial_t^{\alpha} (S_m(t)a-a) + LS_m(t)a = 0, \quad 0<t<T, \, m\in \mathbb{N}.
\eqno{(9.5)}
$$
Since
$$
LS_m(t)a = t^{-\alpha}\sum_{n=1}^m -\lambda_n t^{\alpha}
E_{\alpha,1}(-\lambda_nt^{\alpha})(a,\varphi_n), \quad 0<t<T,
$$
by (4.4) we estimate
\begin{align*}
& \Vert LS_m(t)a\Vert^2_{L^2(\Omega)}
= t^{-2\alpha}\sum_{n=1}^m \vert -\lambda_n t^{\alpha}
E_{\alpha,1}(-\lambda_nt^{\alpha})\vert^2 \vert (a,\varphi_n)\vert^2\\
\le& Ct^{-2\alpha} \sum_{n=1}^m \vert (a,\varphi_n)\vert^2,
\quad 0<t<T.
\end{align*}
Therefore, $\lim_{m\to\infty} LS_m(t)a
= LS(t)a$ in $L^2(\Omega)$ for fixed $t>0$.
Hence, letting $m\to \infty$ in (9.5), we
obtain $\partial_t^{\alpha} (S(t)a-a) + LS(t)a = 0$ for $0<t<T$.
Next let $a \in H^1_0(\Omega)$. Since
$$
\lambda_n^{\frac{1}{2}}(a,\varphi_n) = (a,\, L^{\frac{1}{2}}\varphi_n)
= (L^{\frac{1}{2}}a,\,\varphi_n) \quad \mbox{for $a \in H^1_0(\Omega)$},
$$
we have
\begin{align*}
& LS(t)a = \sum_{n=1}^{\infty} E_{\alpha,1}(-\lambda_nt^{\alpha}) \lambda_n (a,\varphi_n)\varphi_n
= \sum_{n=1}^{\infty} \lambda_n^{\frac{1}{2}}E_{\alpha,1}(-\lambda_nt^{\alpha}) \lambda_n^{\frac{1}{2}}(a,\varphi_n)\varphi_n\\
=& t^{-\frac{\alpha}{2}}
\sum_{n=1}^{\infty} (\lambda_n t^{\alpha})^{\frac{1}{2}} E_{\alpha,1}(-\lambda_nt^{\alpha})
(L^{\frac{1}{2}}a,\,\varphi_n)\varphi_n,
\end{align*}
so that
\begin{align*}
& \Vert LS(t)a\Vert_{L^2(\Omega)}
\le Ct^{-\frac{\alpha}{2}}\left(
\sum_{n=1}^{\infty} \vert \lambda_n t^{\alpha}\vert\vert E_{\alpha,1}(-\lambda_n t^{\alpha})\vert^2
\vert (L^{\frac{1}{2}}a,\, \varphi_n)\vert^2 \right)^{\frac{1}{2}}\\
\le& Ct^{-\frac{\alpha}{2}}\Vert L^{\frac{1}{2}}a\Vert_{L^2(\Omega)}
\le Ct^{-\frac{\alpha}{2}}\Vert a\Vert_{H^1_0(\Omega)}
\end{align*}
by (4.4).
Therefore, this means that $LS(t)a \in L^2(0,T;L^2(\Omega))$ for $a \in H^1_0(\Omega)$ and
$$
\Vert LS(t)a\Vert_{L^2(0,T;L^2(\Omega))} \le C\Vert a\Vert_{H^1_0(\Omega)}.
\eqno{(9.6)}
$$
Since $\partial_t^{\alpha} (S(t)a-a) + LS(t)a = 0$ for $0<t<T$, by (9.6),
the proof of part (i) is complete.
$\blacksquare$
\\
{\bf (ii)}
We set
$$
\left\{ \begin{array}{rl}
& R(t):= \int^t_0 K(t-s)F(s) ds, \\
& R_m(t):= \int^t_0 K_m(t-s)F(s) ds\\
= &\sum_{n=1}^m \left( \int^t_0 (t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_n(t-s)^{\alpha})
(F(s),\, \varphi_n) ds\right) \varphi_n(x),\\
& F_m:= \sum_{n=1}^m (F,\,\varphi_n)\varphi_n
\end{array}\right.
$$
for $0<t<T$ and $m\in \mathbb{N}$.
Then by Proposition 4.2, we can readily verify
$$
\partial_t^{\alpha} R_m + LR_m = F_m \quad \mbox{in $L^2(0,T;L^2(\Omega))$ for all $m\in \mathbb{N}$.}
\eqno{(9.7)}
$$
Moreover
$$
\Vert LR_m(t)\Vert^2_{L^2(\Omega)}
= \sum_{n=1}^m
\left\vert \int^t_0 \lambda_n (t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_n(t-s)^{\alpha})
(F(s),\, \varphi_n) ds\right\vert^2.
$$
Hence,
\begin{align*}
& \Vert LR_m\Vert^2_{L^2(0,T;L^2(\Omega))}
= \sum_{n=1}^m \int^T_0
\left\vert \int^t_0 \lambda_n (t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_n(t-s)^{\alpha})
(F(s),\, \varphi_n) ds\right\vert^2 dt\\
=& \sum_{n=1}^m \Vert (\lambda_n s^{\alpha-1} E_{\alpha,\alpha}(-\lambda_ns^{\alpha})\,
*\, (F(s),\, \varphi_n) \Vert^2_{L^2(0,T)}\\
\le& \sum_{n=1}^m
\left( \int^T_0 \vert \lambda_n s^{\alpha-1} E_{\alpha,\alpha}(-\lambda_ns^{\alpha})\vert ds
\right)^2 \int^T_0 \vert (F(s),\, \varphi_n) \vert^2 ds\\
\le &\int^T_0 \sum_{n=1}^m
\vert (F(s),\, \varphi_n) \vert^2 ds
= \Vert F_m\Vert_{L^2(0,T;L^2(\Omega))}^2.
\end{align*}
For the last inequality and the second to last inequality,
we applied the Young inequality on the convolution and (4.10)
respectively.
Consequently,
$$
\Vert LR_m\Vert^2_{L^2(0,T;L^2(\Omega))} \le C\Vert F_m\Vert^2_{L^2(0,T;L^2(\Omega))}, \quad
m\in \mathbb{N}. \eqno{(9.8)}
$$
Since $F \in L^2(0,T;L^2(\Omega))$, we see that $LR_m \longrightarrow LR$ in $L^2(0,T;L^2(\Omega))$ as $m\to \infty$.
By (9.7) we see
$$
\lim_{m\to \infty} \partial_t^{\alpha} R_m = \lim_{m\to\infty} (F_m - LR_m)
= F - LR \quad \mbox{in $L^2(0,T;L^2(\Omega))$}. \eqno{(9.9)}
$$
Since $\lim_{m\to \infty} R_m = R$ in $L^2(0,T;L^2(\Omega))$, it follows from
(6.25) that $\lim_{m\to \infty} \partial_t^{\alpha} R_m = \partial_t^{\alpha} R$ in
${^{-\alpha}{H}}(0,T;L^2(\Omega))$. Hence (9.9) yields
$\partial_t^{\alpha} R = F-LR$ in ${^{-\alpha}{H}}(0,T;L^2(\Omega))$.
In view of (9.7) and (9.8), we reach
\begin{align*}
& \Vert \partial_t^{\alpha} R_m\Vert_{L^2(0,T;L^2(\Omega))} = \Vert -LR_m+F_m\Vert_{L^2(0,T;L^2(\Omega))}\\
\le &\Vert -LR_m\Vert_{L^2(0,T;L^2(\Omega))} + \Vert F_m\Vert_{L^2(0,T;L^2(\Omega))}
\le C\Vert F_m\Vert_{L^2(0,T;L^2(\Omega))} \quad \mbox{for each $m \in \mathbb{N}$.}
\end{align*}
Letting $m\to \infty$, we reach
$$
\Vert \partial_t^{\alpha} R\Vert_{L^2(0,T;L^2(\Omega))} \le C\Vert F\Vert_{L^2(0,T;L^2(\Omega))}.
$$
Thus the proof of Lemma 6.2 is complete.
$\blacksquare$
\\
{\bf Acknowledgements}.
\\
The author is supported by Grant-in-Aid
for Scientific Research (A) 20H00117, JSPS and by the National Natural
Science Foundation of China (Nos.\! 11771270, 91730303).
This paper has been supported by the RUDN University
Strategic Academic Leadership Program.
The author thanks Professor Bangti Jin
(University College London) for the careful reading and
comments.
|
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Stafford Bond is the Chief Digital Officer & Head of IoT and Analytics for Fujitsu Digital in North America. Stafford is responsible for the Internet of Things and Analytics businesses within Fujitsu North America as well as defining the digital transformation technology and partner eco-system that enable Fujitsu to take customers on their own digital transformation journey. As part of his role he is responsible for driving digital technology innovation and incubation within Fujitsu North America. He is passionate about delivering digital solutions to business problems driving tangible business value for Fujitsu's customers.
|
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The pantropical response of soil moisture to El Niño
Kurt C. Solander, Brent D. Newman, Alessandro Carioca De Araujo, Holly R. Barnard, Z. Carter Berry, Damien Bonal, Mario Bretfeld, Benoit Burban, Luiz Antonio Candido, Rolando Célleri, Jeffery Q. Chambers, Bradley O. Christoffersen, Matteo Detto, Wouter A. Dorigo, Brent E. Ewers, Savio José Filgueiras Ferreira, Alexander Knohl, L. Ruby Leung, Nate G. McDowell, Gretchen R. Miller
Show 7 others
Maria Terezinha Ferreira Monteiro, Georgianne W. Moore, Robinson Negron-Juarez, Scott R. Saleska, Christian Stiegler, Javier Tomasella, Chonggang Xu
The 2015-2016 El Niño event ranks as one of the most severe on record in terms of the magnitude and extent of sea surface temperature (SST) anomalies generated in the tropical Pacific Ocean. Corresponding global impacts on the climate were expected to rival, or even surpass, those of the 1997-1998 severe El Niño event, which had SST anomalies that were similar in size. However, the 2015-2016 event failed to meet expectations for hydrologic change in many areas, including those expected to receive well above normal precipitation. To better understand how climate anomalies during an El Niño event impact soil moisture, we investigate changes in soil moisture in the humid tropics (between <span classCombining double low line"inline-formula">±25</span><span classCombining double low line"inline-formula">ĝˆ </span>) during the three most recent super El Niño events of 1982-1983, 1997-1998 and 2015-2016, using data from the Global Land Data Assimilation System (GLDAS). First, we use in situ soil moisture observations obtained from 16 sites across five continents to validate and bias-correct estimates from GLDAS (<span classCombining double low line"inline-formula">r2Combining double low line0.54</span>). Next, we apply a <span classCombining double low line"inline-formula">k</span>-means cluster analysis to the soil moisture estimates during the El Niño mature phase, resulting in four groups of clustered data. The strongest and most consistent decreases in soil moisture occur in the Amazon basin and maritime southeastern Asia, while the most consistent increases occur over eastern Africa. In addition, we compare changes in soil moisture to both precipitation and evapotranspiration, which showed a lack of agreement in the direction of change between these variables and soil moisture most prominently in the southern Amazon basin, the Sahel and mainland southeastern Asia. Our results can be used to improve estimates of spatiotemporal differences in El Niño impacts on soil moisture in tropical hydrology and ecosystem models at multiple scales.
https://doi.org/10.5194/hess-24-2303-2020
10.5194/hess-24-2303-2020
Fingerprint Dive into the research topics of 'The pantropical response of soil moisture to El Niño'. Together they form a unique fingerprint.
soil moisture Earth & Environmental Sciences
temperature anomaly Earth & Environmental Sciences
data assimilation Earth & Environmental Sciences
sea surface temperature Earth & Environmental Sciences
Asia Earth & Environmental Sciences
humid tropics Earth & Environmental Sciences
land Earth & Environmental Sciences
Solander, K. C., Newman, B. D., Carioca De Araujo, A., Barnard, H. R., Berry, Z. C., Bonal, D., Bretfeld, M., Burban, B., Candido, L. A., Célleri, R., Chambers, J. Q., Christoffersen, B. O., Detto, M., Dorigo, W. A., Ewers, B. E., Ferreira, S. J. F., Knohl, A., Leung, L. R., McDowell, N. G., ... Xu, C. (2020). The pantropical response of soil moisture to El Niño. Hydrology and Earth System Sciences, 24(5), 2303-2322. https://doi.org/10.5194/hess-24-2303-2020
The pantropical response of soil moisture to El Niño. / Solander, Kurt C.; Newman, Brent D.; Carioca De Araujo, Alessandro; Barnard, Holly R.; Berry, Z. Carter; Bonal, Damien; Bretfeld, Mario; Burban, Benoit; Candido, Luiz Antonio; Célleri, Rolando; Chambers, Jeffery Q.; Christoffersen, Bradley O.; Detto, Matteo; Dorigo, Wouter A.; Ewers, Brent E.; Ferreira, Savio José Filgueiras; Knohl, Alexander; Leung, L. Ruby; McDowell, Nate G.; Miller, Gretchen R.; Monteiro, Maria Terezinha Ferreira; Moore, Georgianne W.; Negron-Juarez, Robinson; Saleska, Scott R.; Stiegler, Christian; Tomasella, Javier; Xu, Chonggang.
In: Hydrology and Earth System Sciences, Vol. 24, No. 5, 11.05.2020, p. 2303-2322.
Solander, KC, Newman, BD, Carioca De Araujo, A, Barnard, HR, Berry, ZC, Bonal, D, Bretfeld, M, Burban, B, Candido, LA, Célleri, R, Chambers, JQ, Christoffersen, BO, Detto, M, Dorigo, WA, Ewers, BE, Ferreira, SJF, Knohl, A, Leung, LR, McDowell, NG, Miller, GR, Monteiro, MTF, Moore, GW, Negron-Juarez, R, Saleska, SR, Stiegler, C, Tomasella, J & Xu, C 2020, 'The pantropical response of soil moisture to El Niño', Hydrology and Earth System Sciences, vol. 24, no. 5, pp. 2303-2322. https://doi.org/10.5194/hess-24-2303-2020
Solander KC, Newman BD, Carioca De Araujo A, Barnard HR, Berry ZC, Bonal D et al. The pantropical response of soil moisture to El Niño. Hydrology and Earth System Sciences. 2020 May 11;24(5):2303-2322. https://doi.org/10.5194/hess-24-2303-2020
Solander, Kurt C. ; Newman, Brent D. ; Carioca De Araujo, Alessandro ; Barnard, Holly R. ; Berry, Z. Carter ; Bonal, Damien ; Bretfeld, Mario ; Burban, Benoit ; Candido, Luiz Antonio ; Célleri, Rolando ; Chambers, Jeffery Q. ; Christoffersen, Bradley O. ; Detto, Matteo ; Dorigo, Wouter A. ; Ewers, Brent E. ; Ferreira, Savio José Filgueiras ; Knohl, Alexander ; Leung, L. Ruby ; McDowell, Nate G. ; Miller, Gretchen R. ; Monteiro, Maria Terezinha Ferreira ; Moore, Georgianne W. ; Negron-Juarez, Robinson ; Saleska, Scott R. ; Stiegler, Christian ; Tomasella, Javier ; Xu, Chonggang. / The pantropical response of soil moisture to El Niño. In: Hydrology and Earth System Sciences. 2020 ; Vol. 24, No. 5. pp. 2303-2322.
@article{d09ffb059ab54d7e93dd6870b4b1ff37,
title = "The pantropical response of soil moisture to El Ni{\~n}o",
abstract = "The 2015-2016 El Ni{\~n}o event ranks as one of the most severe on record in terms of the magnitude and extent of sea surface temperature (SST) anomalies generated in the tropical Pacific Ocean. Corresponding global impacts on the climate were expected to rival, or even surpass, those of the 1997-1998 severe El Ni{\~n}o event, which had SST anomalies that were similar in size. However, the 2015-2016 event failed to meet expectations for hydrologic change in many areas, including those expected to receive well above normal precipitation. To better understand how climate anomalies during an El Ni{\~n}o event impact soil moisture, we investigate changes in soil moisture in the humid tropics (between ±25ĝˆ ) during the three most recent super El Ni{\~n}o events of 1982-1983, 1997-1998 and 2015-2016, using data from the Global Land Data Assimilation System (GLDAS). First, we use in situ soil moisture observations obtained from 16 sites across five continents to validate and bias-correct estimates from GLDAS (r2Combining double low line0.54). Next, we apply a k-means cluster analysis to the soil moisture estimates during the El Ni{\~n}o mature phase, resulting in four groups of clustered data. The strongest and most consistent decreases in soil moisture occur in the Amazon basin and maritime southeastern Asia, while the most consistent increases occur over eastern Africa. In addition, we compare changes in soil moisture to both precipitation and evapotranspiration, which showed a lack of agreement in the direction of change between these variables and soil moisture most prominently in the southern Amazon basin, the Sahel and mainland southeastern Asia. Our results can be used to improve estimates of spatiotemporal differences in El Ni{\~n}o impacts on soil moisture in tropical hydrology and ecosystem models at multiple scales..",
author = "Solander, {Kurt C.} and Newman, {Brent D.} and {Carioca De Araujo}, Alessandro and Barnard, {Holly R.} and Berry, {Z. Carter} and Damien Bonal and Mario Bretfeld and Benoit Burban and Candido, {Luiz Antonio} and Rolando C{\'e}lleri and Chambers, {Jeffery Q.} and Christoffersen, {Bradley O.} and Matteo Detto and Dorigo, {Wouter A.} and Ewers, {Brent E.} and Ferreira, {Savio Jos{\'e} Filgueiras} and Alexander Knohl and Leung, {L. Ruby} and McDowell, {Nate G.} and Miller, {Gretchen R.} and Monteiro, {Maria Terezinha Ferreira} and Moore, {Georgianne W.} and Robinson Negron-Juarez and Saleska, {Scott R.} and Christian Stiegler and Javier Tomasella and Chonggang Xu",
note = "Funding Information: Acknowledgements. This project was supported as part of the Next-Generation Ecosystem Experiments – Tropics, funded by the United States Department of Energy Office of Science Office of Biological and Environmental Research through the Terrestrial Ecosystem Science program. Data obtained from French Guiana were recorded thanks to an "investissement d{\textquoteright}avenir" grant from the Agence Na-tionale de la Recherche (CEBA no. ANR-10-LABX-25-01; AR-BRE no. ANR-11-LABX-0002-01). Data obtained from one of the Panama sites were recorded thanks to an award from the National Science Foundation (NSF; no. 1360305). In situ data collected from Indonesia were made possible by the Deutsche Forschungsgemein- schaft (DFG; German Research Foundation; project no. 192626868 of SFB 990) and the Ministry of Research and Technology/National Research and Innovation Agency (Ministry of Research, Technology and Higher Education; Ristekdikti) in the framework of the collaborative German–Indonesian research project CRC 990. We would also like to thank Charu Varadharajan and Emily Robles for providing valuable assistance with database access and guidance on data storage during the course of this research. The Pacific Northwest National Laboratory is operated for the Department of Energy by Battelle Memorial Institute (contract no. DE-AC05-76RL01830). Funding Information: Financial support. This research has been supported by the U.S. Funding Information: Department of Energy, Office of Science (grant for Next-Generation Ecosystem Experiments – Tropics).",
doi = "10.5194/hess-24-2303-2020",
journal = "Hydrology and Earth System Sciences",
publisher = "European Geosciences Union",
T1 - The pantropical response of soil moisture to El Niño
AU - Solander, Kurt C.
AU - Newman, Brent D.
AU - Carioca De Araujo, Alessandro
AU - Barnard, Holly R.
AU - Berry, Z. Carter
AU - Bonal, Damien
AU - Bretfeld, Mario
AU - Burban, Benoit
AU - Candido, Luiz Antonio
AU - Célleri, Rolando
AU - Chambers, Jeffery Q.
AU - Christoffersen, Bradley O.
AU - Detto, Matteo
AU - Dorigo, Wouter A.
AU - Ewers, Brent E.
AU - Ferreira, Savio José Filgueiras
AU - Knohl, Alexander
AU - Leung, L. Ruby
AU - McDowell, Nate G.
AU - Miller, Gretchen R.
AU - Monteiro, Maria Terezinha Ferreira
AU - Moore, Georgianne W.
AU - Negron-Juarez, Robinson
AU - Saleska, Scott R.
AU - Stiegler, Christian
AU - Tomasella, Javier
AU - Xu, Chonggang
N1 - Funding Information: Acknowledgements. This project was supported as part of the Next-Generation Ecosystem Experiments – Tropics, funded by the United States Department of Energy Office of Science Office of Biological and Environmental Research through the Terrestrial Ecosystem Science program. Data obtained from French Guiana were recorded thanks to an "investissement d'avenir" grant from the Agence Na-tionale de la Recherche (CEBA no. ANR-10-LABX-25-01; AR-BRE no. ANR-11-LABX-0002-01). Data obtained from one of the Panama sites were recorded thanks to an award from the National Science Foundation (NSF; no. 1360305). In situ data collected from Indonesia were made possible by the Deutsche Forschungsgemein- schaft (DFG; German Research Foundation; project no. 192626868 of SFB 990) and the Ministry of Research and Technology/National Research and Innovation Agency (Ministry of Research, Technology and Higher Education; Ristekdikti) in the framework of the collaborative German–Indonesian research project CRC 990. We would also like to thank Charu Varadharajan and Emily Robles for providing valuable assistance with database access and guidance on data storage during the course of this research. The Pacific Northwest National Laboratory is operated for the Department of Energy by Battelle Memorial Institute (contract no. DE-AC05-76RL01830). Funding Information: Financial support. This research has been supported by the U.S. Funding Information: Department of Energy, Office of Science (grant for Next-Generation Ecosystem Experiments – Tropics).
N2 - The 2015-2016 El Niño event ranks as one of the most severe on record in terms of the magnitude and extent of sea surface temperature (SST) anomalies generated in the tropical Pacific Ocean. Corresponding global impacts on the climate were expected to rival, or even surpass, those of the 1997-1998 severe El Niño event, which had SST anomalies that were similar in size. However, the 2015-2016 event failed to meet expectations for hydrologic change in many areas, including those expected to receive well above normal precipitation. To better understand how climate anomalies during an El Niño event impact soil moisture, we investigate changes in soil moisture in the humid tropics (between ±25ĝˆ ) during the three most recent super El Niño events of 1982-1983, 1997-1998 and 2015-2016, using data from the Global Land Data Assimilation System (GLDAS). First, we use in situ soil moisture observations obtained from 16 sites across five continents to validate and bias-correct estimates from GLDAS (r2Combining double low line0.54). Next, we apply a k-means cluster analysis to the soil moisture estimates during the El Niño mature phase, resulting in four groups of clustered data. The strongest and most consistent decreases in soil moisture occur in the Amazon basin and maritime southeastern Asia, while the most consistent increases occur over eastern Africa. In addition, we compare changes in soil moisture to both precipitation and evapotranspiration, which showed a lack of agreement in the direction of change between these variables and soil moisture most prominently in the southern Amazon basin, the Sahel and mainland southeastern Asia. Our results can be used to improve estimates of spatiotemporal differences in El Niño impacts on soil moisture in tropical hydrology and ecosystem models at multiple scales..
AB - The 2015-2016 El Niño event ranks as one of the most severe on record in terms of the magnitude and extent of sea surface temperature (SST) anomalies generated in the tropical Pacific Ocean. Corresponding global impacts on the climate were expected to rival, or even surpass, those of the 1997-1998 severe El Niño event, which had SST anomalies that were similar in size. However, the 2015-2016 event failed to meet expectations for hydrologic change in many areas, including those expected to receive well above normal precipitation. To better understand how climate anomalies during an El Niño event impact soil moisture, we investigate changes in soil moisture in the humid tropics (between ±25ĝˆ ) during the three most recent super El Niño events of 1982-1983, 1997-1998 and 2015-2016, using data from the Global Land Data Assimilation System (GLDAS). First, we use in situ soil moisture observations obtained from 16 sites across five continents to validate and bias-correct estimates from GLDAS (r2Combining double low line0.54). Next, we apply a k-means cluster analysis to the soil moisture estimates during the El Niño mature phase, resulting in four groups of clustered data. The strongest and most consistent decreases in soil moisture occur in the Amazon basin and maritime southeastern Asia, while the most consistent increases occur over eastern Africa. In addition, we compare changes in soil moisture to both precipitation and evapotranspiration, which showed a lack of agreement in the direction of change between these variables and soil moisture most prominently in the southern Amazon basin, the Sahel and mainland southeastern Asia. Our results can be used to improve estimates of spatiotemporal differences in El Niño impacts on soil moisture in tropical hydrology and ecosystem models at multiple scales..
U2 - 10.5194/hess-24-2303-2020
DO - 10.5194/hess-24-2303-2020
JO - Hydrology and Earth System Sciences
JF - Hydrology and Earth System Sciences
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in: Pages containing cite templates with deprecated parameters, Stubs, Companies, Animation Studios
This article is a stub. You can help the Animanga Wiki by expanding it, or perhaps you could contribute to discussion on the topic.
feel (有限会社フィール, Yūgen-gaisha Fiiru)
Makoto Ryugasaki
Koganei, Tokyo, Japan
http://www.feel-ing.com/
Feel. Inc. (有限会社フィール, Yūgen-gaisha Fiiru), stylized as feel. is a Japanese animation studio founded in 2002.
2 Productions
2.1 Television series
2.2 Original video animations
2.3 Original net animation
Feel was established in Koganei, Tokyo on December 26, 2002 by ex-Studio Pierrot staff that specializes in the production of anime. To date, the studio have presented us various well-known works, including Kissxsis, Yosuga no Sora, Mayo Chiki!, Listen to Me, Girls. I Am Your Father!, So, I Can't Play H!, Outbreak Company, My Teen Romantic Comedy SNAFU TOO!, Dagashi Kashi, This Art Club Has a Problem!, and Tsuki ga Kirei.
Productions[]
Television series[]
First run
Da Capo Nagisa Miyazaki July 5, 2003 December 27, 2003 26 Based on an adult visual novel by Circus.
Collaboration with Zexcs. [1]
Jinki: Extend Masahiko Murata January 6, 2005 March 24, 2005 12 Adaptation of a manga series by Tunasima Sirou. [2]
Futakoi Alternative Takayuki Hirao April 7, 2005 June 27, 2005 13 Original work.
Collaboration with Studio Flag and ufotable. [3]
Da Capo: Second Season Munenori Nawa July 2, 2005 December 24, 2005 26 The second season of Da Capo. [4]
Otome wa Boku ni Koishiteru Munenori Nawa October 8, 2006 December 24, 2006 12 Based on an adult visual novel by Caramel Box. [5]
Nagasarete Airantou Hideki Okamoto April 5, 2007 September 27, 2007 26 Adaptation of a manga series by Takeshi Fujishiro. [6][7]
Da Capo II Hideki Okamoto October 1, 2007 December 24, 2007 13 Based on an adult visual novel by Circus.
The sequel to Da Capo.
Da Capo II: Second Season Hideki Okamoto April 5, 2008 June 28, 2008 13 The second season of Da Capo II. [8]
Corpse Princess: Aka Masahiko Murata October 2, 2008 December 25, 2008 13 Adaptation of a manga series by Yoshiichi Akahito. Collaboration with Gainax. [9]
Corpse Princess: Kuro Masahiko Murata January 1, 2009 March 26, 2009 12 The second season of Shikabane Hime.
Collaboration with Gainax. [10]
Kanamemo Shigehito Takayanagi July 5, 2009 September 27, 2009 13 Adaptation of a manga series by Shoko Iwami. [11]
Kissxsis Munenori Nawa April 4, 2010 June 21, 2010 12 Adaptation of a manga series written by Bow Ditama.
Yosuga no Sora Takeo Takahashi October 4, 2010 December 20, 2010 12 Based on an adult visual novel by Sphere. [12]
Fortune Arterial: Akai Yakusoku Munenori Nawa October 9, 2010 December 25, 2010 12 Based on an adult visual novel by August.
Collaboration with Zexcs. [13]
Mayo Chiki! Keiichiro Kawaguchi July 7, 2011 September 29, 2011 13 Based on a light novel series written by Hajime Asano. [14]
Listen to Me, Girls. I Am Your Father! Itsuro Kawasaki January 11, 2012 March 27, 2012 12 Adaptation of a light novel series written by Tomohiro Matsu. [15]
So, I Can't Play H! Takeo Takahashi July 6, 2012 September 25, 2012 12 Adaptation of a light novel series written by Pan Tachibana. [16][17]
Minami-ke: Tadaima Keiichiro Kawaguchi January 6, 2013 March 30, 2013 13 The fourth season of Minami-ke. [18][19]
Ketsuekigata-kun! Yoshihasa Ōyama April 7, 2013 June 30, 2013 12 An adaptation of a webtoon written by Real Crazy Man.
Collaboration with Assez Finand Fabric. [20]
Outbreak Company Kei Oikawa October 3, 2013 December 19, 2013 12 Adaptation of a light novel series written by Ichirō Sakaki. [21][22]
Locodol Munenori Nawa July 4, 2014 September 18, 2014 12 Adaptation of a manga series written by Kōtarō Kosugi. [23][24]
Jinsei Keiichiro Kawaguchi July 6, 2014 September 28, 2014 13 Adaptation of a light novel series written by Ougyo Kawagishi. [25][26]
In Search of the Lost Future Naoto Hosoda October 4, 2014 December 20, 2014 12 Based on an adult visual novel by Trumple. [27][28]
Ketsuekigata-kun! 2 Yoshihasa Ōyama January 8, 2015 March 26, 2015 12 The second season of Ketsuekigata-kun!.
My Teen Romantic Comedy SNAFU TOO! Kei Oikawa April 3, 2015 June 26, 2015 13 The second season of My Teen Romantic Comedy SNAFU. [29][30][31]
Bikini Warriors Naoyuki Kuzuya July 7, 2015 September 22, 2015 12 Animation of a Japanese Media franchise created by Hobby Japan. [32][33]
Suzakinishi the Animation Yoshihiro Hiramine July 8, 2015 September 23, 2015 12 Based on the anime radio program SuzakiNishi by Aya Suzaki and Asuka Nishi. [34][35]
Makura no Danshi N/A July 13, 2015 September 28, 2015 12 Original work.
Collaboration with Assez Finand Fabric.
Written by Yuniko Ayana and Ayumi Sakine. [36][37]
Ketsuekigata-kun! 3 Yoshihasa Ōyama October 13, 2015 December 29, 2015 12 The third season of Ketsuekigata-kun!.
Collaboration with Assez Finand Fabric and Zexcs. [20]
Dagashi Kashi Shigehito Takayanagi January 7, 2016 March 31, 2016 12 Adaptation of a manga series written by Kotoyama. [38][39][40]
Please Tell Me! Galko-chan Keiichiro Kawaguchi January 8, 2016 March 25, 2016 12 Adaptation of a manga series written by Kenya Suzuki. [41][42]
Ketsuekigata-kun! 4 Yoshihasa Ōyama January 11, 2016 March 29, 2016 12 The fourth season of Ketsuekigata-kun!.
This Art Club Has a Problem!' Kei Oikawa July 7, 2016 September 22, 2016 12 Adaptation of a manga series written by Imigimuru. [43][44]
Tsuki ga Kirei Seiji Kishi April 6, 2017 June 29, 2017 12 Original work.
Written by Yūko Kakihara. [45][46][47]
Hinamatsuri Kei Oikawa April 2018 TBA TBA Adaptation of a manga series written by Masao Ōtake. [48]
Island TBA 2018 TBA TBA Based on an all-ages visual novel by Frontwing.
YU-NO: A Girl Who Chants Love at the Bound of this World TBA 2018 TBA TBA Based on the adult visual novel remake of YU-NO by 5pb. Originally created by ELF Corporation.
Original video animations[]
Growlanser IV: Wayfarer of the Time Masahiko Murata March 10, 2005 1 A promotional OVA for Growlanser IV Returns.
Collaboration with Zexcs.
Jinki: Extend Masahiko Murata February 22, 2006 1 The 13th episode of Jinki: Extend.
Otoboku: Maidens Are Falling For Me! Munenori Nawa April 4, 2007 1 The 13th episode of Otoboku: Maidens Are Falling For Me!
Strait Jacket Shinji Ushiro November 26, 2007–
April 28, 2008 3 Adaptation of a light novel series written by Ichirō Sakaki. [49]
Kissxsis Munenori Nawa December 22, 2008–
April 6, 2015 12 Adaptation of a manga series written by Bow Ditama. [50][51]
Fortune Arterial: Akai Yakusoku - Tadoritsuita Basho Munenori Nawa February 23, 2011 1 OVA episode for Fortune Arterial: Akai Yakusoku
Papa no Iukoto wo Kikinasai! Itsuro Kawasaki July 11, 2012–
March 25, 2015 3 OVA episodes bundled with the 13th and 18th light novel volume.
Minami-ke: Omatase Keiichiro Kawaguchi October 5, 2012 1 OVA bundled with the 10th manga volume. [52]
Dakara Boku wa, H ga Dekinai Takeo Takahashi March 29, 2013 1 The 13th episode of Dakara Boku wa, H ga Dekinai
Minami-ke: Natsuyasumi Keiichiro Kawaguchi August 6, 2013 1 OVA bundled with the 11th manga volume.
Locodol Munenori Nawa September 24, 2014–
June 22, 2016 3 OVA episodes for Locodol.
In Search of the Lost Summer Break Naoto Hosoda August 29, 2015 1 The 13th episode of In Search of the Lost Future.
Makura no Danshi N/A November 27, 2015 1 The 13th episode of Makura no Danshi.
Bikini Warriors Naoyuki Kuzuya December 19, 2015–
December 7, 2016 3 OVA episodes for Bikini Warriors.
My Teen Romantic Comedy SNAFU TOO! Kei Oikawa October 27, 2016 1 Based on Volume 10.5 of My Teen Romantic Comedy SNAFU.
Please Tell Me! Galko-chan Keiichiro Kawaguchi January 23, 2017 1 OVA bundled with the fourth manga volume.
Original net animation[]
Augmented Reality Girls Trinary Itsuro Kawasaki April 12, 2017 November 29, 2017 34 Linkage with games. Consists of 6 Acts.
↑ "D.C.ダ・カーポ オフィシャルサイト - 初音島.com". http://www.hatsunejima.com/dc/staff/staff.html.
↑ "ADV Films Removes Titles from Website - Update". Anime News Network. http://www.animenewsnetwork.com/news/2008-01-30/adv-films-removes-titles-from-website-update.
↑ "フタコイ オルタナティブ". http://king-cr.jp/special/hutakoi/ufo/staff.html.
↑ "D.C.ダ・カーポ オフィシャルサイト - 初音島.com". http://www.hatsunejima.com/dc2ss/staff.html.
↑ "乙女はお姉さまに恋してる". http://king-cr.jp/special/otome/staff.html.
↑ "TV東京・あにてれ ながされて藍蘭島". http://www.tv-tokyo.co.jp/anime/airantou/.
↑ "ながされて藍蘭島". http://king-cr.jp/special/airantou/staff.html.
↑ "StarChild: D.C.II S.S.". http://king-cr.jp/special/dc2ss. Retrieved July 10, 2017.
↑ "Shikabane Hime Confirmed as Gainax's New TV Series (Updated)". Anime News Network. http://www.animenewsnetwork.com/news/2008-05-25/shikabane-hime-confirmed-as-gainax-next-tv-series.
↑ "Shikabane Hime: Kuro TV Anime Confirmed for January". Anime News Network. http://www.animenewsnetwork.com/news/2008-11-12/shikabane-hime/kuro-tv-anime-confirmed-for-january.
↑ "Kanamemo Four-Panel Manga Gets Anime Green-Lit". Anime News Network. http://www.animenewsnetwork.com/news/2009-01-17/kanamemo-four-panel-manga-gets-anime-green-lit.
↑ "ヨスガノソラ" (in ja). http://king-cr.jp/special/yosuganosora/staff/.
↑ "Crunchyroll Adds Heaven's Lost Property 2, Panty & Stocking, Fortune Arterial (Updated)". Anime News Network. http://www.animenewsnetwork.com/news/2010-09-15/crunchyroll-adds-heaven-lost-property-ii-panty-and-stocking-fortune-arterial.
↑ "まよチキ!|Staff&Cast" (in ja). http://king-cr.jp/special/mayochiki/staff_cast/.
↑ "Papa no Iu Koto o Kikinasai! Light Novels Get Anime". Anime News Network. http://www.animenewsnetwork.com/news/2011-08-01/papa-no-iu-koto-o-kikinasai-light-novels-gets-anime.
↑ "Dakara Boku wa, H ga Dekinai Light Novels Get TV Anime". Anime News Network. http://www.animenewsnetwork.com/news/2011-12-27/dakara-boku-wa-h-ga-dekinai-light-novels-get-anime.
↑ "Dakara Boku wa, H ga Dekinai Staff Listed". Anime News Network. http://www.animenewsnetwork.com/news/2012-03-30/dakara-boku-wa-h-ga-dekinai-staff-listed.
↑ "Minami-ke's New Anime Listed as '4th Season'". Anime News Network. http://www.animenewsnetwork.com/news/2012-03-21/minami-ke-new-anime-listed-as-4th-season.
↑ "みなみけ(4) ただいま - アニメ詳細データ - ◇テレビドラマデータベース◇" (in ja). http://www.tvdrama-db.com/drama_info/p/id-54611.
↑ 20.0 20.1 20.2 20.3 エンターテイメント, アース・スター. "TVアニメ「血液型くん!」公式サイト" (in ja). http://ketsuekigatakun.com/staff/.
↑ "Outbreak Company Anime's Promo Video Streamed". Anime News Network. September 26, 2013. http://www.animenewsnetwork.com/news/2013-09-06/outbreak-company-anime-promo-video-streamed. Retrieved September 8, 2013.
↑ TBS. "TBS" (in ja). http://www.tbs.co.jp/anime/obc/staffcast/.
↑ "Futsū no Joshikōsei ga Locodol Yattemita 4-Panel Manga Gets Anime". Anime News Network. http://www.animenewsnetwork.com/news/2014-01-18/futsu-no-joshikosei-ga-locodol-yattemita-4-panel-manga-gets-anime.
↑ TBS. "TBS" (in ja). http://www.tbs.co.jp/anime/locodol/staffcast/.
↑ "School Newspaper Anime Jinsei's Staff, Designs Unveiled". Anime News Network. http://www.animenewsnetwork.com/news/2014-05-19/school-newspaper-anime-jinsei-staff-designs-unveiled.
↑ "人生相談テレビアニメーション「人生」公式サイト" (in ja-JP). http://www.vap.co.jp/jinsei/cast/index.html.
↑ "Ushinawareta Mirai o Motomete Anime's 1st Ads Streamed". Anime News Network. http://www.animenewsnetwork.com/news/2014-09-20/ushinawareta-mirai-o-motomete-anime-1st-ads-streamed/.79012.
↑ "失われた未来を求めて|Staff/Cast" (in ja). http://ushinawareta-mirai.com/staff/.
↑ "My Teen Romantic Comedy SNAFU Anime Gets 2nd Season". Anime News Network. http://www.animenewsnetwork.com/news/2014-04-16/my-teen-romantic-comedy-snafu-anime-gets-2nd-season.
↑ "Outbreak Company's Oikawa to Direct My Teen Romantic Comedy SNAFU Season 2 at Studio feel.". Anime News Network. http://www.animenewsnetwork.com/news/2014-10-16/outbreak-company-oikawa-to-direct-my-teen-romantic-comedy-snafu-season-2-at-studio-feel/.80008.
↑ TBS. "TBS" (in ja). http://www.tbs.co.jp/anime/oregairu/staffcast/.
↑ "Bikini Warriors Fantasy Figure Series Gets TV Anime Project in July". Anime News Network. http://www.animenewsnetwork.com/news/2015-05-16/bikini-warriors-fantasy-figure-series-gets-tv-anime-project-in-july/.88235.
↑ 株式会社ホビージャパン「ビキニ・ウォリアーズ」. "TVアニメ『ビキニ・ウォリアーズ』公式サイト" (in ja). http://bikini-warriors.com/anime/.
↑ "Aya Suzaki, Asuka Nishi's Radio Program Gets TV Anime Adaptation". Anime News Network. http://www.animenewsnetwork.com/news/2015-05-24/aya-suzaki-asuka-nishi-radio-program-gets-tv-anime-adaptation/.88513.
↑ "スタッフ&キャスト -TVアニメ「洲崎西 THE ANIMATION」公式サイト-". http://suzakinishi.tv/staff/index.html.
↑ "Earth Star Produces Subjective Viewpoint TV Anime "Makura no Danshi" for July Premiere". Crunchyroll. http://www.crunchyroll.com/anime-news/2015/05/01/earth-star-produces-subjective-viewpoint-tv-anime-makura-no-danshi-for-july-premiere.
↑ "Natsuki Hanae Is 1 of 12 Pillow Boys in Makura no Danshi TV Anime". Anime News Network. http://www.animenewsnetwork.com/news/2015-05-01/natsuki-hanae-is-1-of-12-pillow-boys-in-makura-no-danshi-tv-anime/.87714.
↑ "Dagashi Kashi Manga Gets TV Anime Adaptation". Anime News Network. http://www.animenewsnetwork.com/news/2015-09-19/dagashi-kashi-manga-gets-tv-anime-adaptation/.93133.
↑ "Dagashi Kashi Anime's Staff, January Premiere Revealed". Anime News Network. http://www.animenewsnetwork.com/news/2015-10-14/dagashi-kashi-anime-staff-january-premiere-revealed/.94180.
↑ TBS. "TBS" (in ja). http://www.tbs.co.jp/anime/dagashi/staffcast/.
↑ "Web Manga "Oshiete! Galko-chan" Gets TV Anime in January 2016". Crunchyroll. http://www.crunchyroll.com/anime-news/2015/11/06/web-manga-oshiete-galko-chan-gets-tv-anime-in-january-2016.
↑ "Kenya Suzuki's Oshiete! Galko-chan Manga Gets TV Anime Adaptation in January". Anime News Network. http://www.animenewsnetwork.com/news/2015-11-06/kenya-suzuki-oshiete-galko-chan-manga-gets-tv-anime-adaptation-in-january/.95071.
↑ "Kono Bijutsu-bu niwa Mondai ga Aru! TV Anime's Staff, July Premiere, More Cast Revealed". Anime News Network. https://www.animenewsnetwork.com/news/2016-03-13/kono-bijutsu-bu-niwa-mondai-ga-aru-tv-anime-staff-july-premiere-more-cast-revealed/.99722.
↑ TBS. "この美術部には問題がある! 公式ホームページ|TBSテレビ" (in ja-JP). http://www.tbs.co.jp/anime/konobi/staffcast/.
↑ "Studio feel. Reveals Tsuki ga Kirei Original TV Anime Project for April". Anime News Network. http://www.animenewsnetwork.com/news/2017-01-29/studio-feel-reveals-tsuki-ga-kirei-original-tv-anime-project-for-april/.111612.
↑ "Studio feel.'s Tsuki ga Kirei TV Anime Premieres on April 6". Anime News Network. http://www.animenewsnetwork.com/daily-briefs/2017-03-08/studio-feel.s-tsuki-ga-kirei-tv-anime-premieres-on-april-6/.113143.
↑ "スタッフ&キャスト ☾ 月がきれい" (in Japanese). 月がきれい. http://tsukigakirei.jp/staffcast/.
↑ "Hinamatsuri TV Anime Reveals Visual, Staff" (in en). Anime News Network. https://www.animenewsnetwork.com/news/2017-11-17/hinamatsuri-tv-anime-reveals-visual-staff/.124160.
↑ "Anime Expo 2007 - New License Roundup". Anime News Network. http://www.animenewsnetwork.com/convention/2007/anime-expo/new-license-roundup.
↑ "Kiss×sis Manga from Mahoromatic's Ditama to be Animated". Anime News Network. http://www.animenewsnetwork.com/news/2008-06-20/kiss-sis-manga-from-mahoromatic-ditama-to-be-animated.
↑ "20th Century Boys 2 Film, Stitch!, Kissxsis Dated (Updated)". Anime News Network. http://www.animenewsnetwork.com/news/2008-08-06/20th-century-boys-2-film-stitch-kissxsis-dated.
↑ 桜場, コハル (October 5, 2012). "DVD付き みなみけ(10) 限定版". 東京: 講談社. ISBN 9784063584066. https://www.amazon.co.jp/dp/4063584062/.
Animation Studios
... more about "feel."
Makoto Ryugasaki +
Koganei, Tokyo, Japan +
Animation Studio +
More Animanga Wiki
1 Koroshi Ai
2 Boku no Pico
3 Life with an Ordinary Guy who Reincarnated into a Total Fantasy Knockout
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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In the entry above I erred,unusual I know,as it seems the re-brand may be a re-brand of the re-brand.
We may know more by Thursday.
It seems certain that Phil Carleton will have the services of Jon Ellis,currently Head Of Performance at/for Wales Rugby League and coach for the Under 18's.
The time has now come to announce the players who will be representing West Wales Raiders in 2018.
The club has elected to name them,one a day,in alphabetical order.
Quite taken by the brilliant Jamberoo Superoos,the name of the former club of the first named player,Mark Asquith.
Wonder if he is the middle brother?
Anyway,described as a half back - but in one of the videos I watched on that well known video sharing website,he was at full back.
The club have also signed the dual code Welsh winger Kristian Baller who was the club in it's earlier guise of South Wales Scorpions.
The 3rd player named to be in the 2018 squad is one of those,presumably,rare people amongst those playing our great sport,who has a medal from the recent Commonwealth Games.
Someone who has made his way through the age development system in Wales,Harry Boots,a prop forward.
'Tough as old boots' - one of my mother's sayings.
and has the name Karlin Claridge.
Karlin being the 2nd from Cardiff the local talent thus far is a fair indication of the club wanting to develop local talent.
Best wishes to Karlin.18 yrs of age,by the way.
The last signing,until after the weekend,to be announced is Matthew Cummins,who originates from the place where Max Boyce seemed quite happy about the front row.
Matthew plays at full back and was a member of the Dragonhearts side which lost to Germany,last October,by 4 points and lost the kicking from the tee competition despite the Germans having the advantage of Jimmy Keinhorst of Leeds Rhinos in their ranks.
The kicking side of things appears to have been won by St Helens player Brad Billsborough.
A good season ahead for Matthew,hopefully,and a victory by 4 points,or more,and success with the boot will be great.
for West Wales Raiders was announced.
in the League 1 side for 2018.
He had been playing with community club Torfaen Tigers.
A try scoring centre,so I certainly hope he continues.
The 7th signing to be announced is an international with Super League experience.
That Super League experience being with the Bridgend based Crusaders.
Welcome to the club - as the title of that album was -to Ashley Bateman.
With Ash having played on the wing and in the centres it will be interesting to see where he features in 2018.
The 8th signing warms the cockles of my heart - as the colder weather commences.
A bit of continuity from the back end of last season.
A prop with a lot of experience yet still only 25 yrs of age and with the best years ahead of him.
Delighted that Morgan Evans has signed.
More of the same will do for me.
The 9th signing now announced is the season-ticket-seller.
The opposition players will like him- not a lot!
Now that is what you call an early Christmas present.
He's played a bit of rugby union but no mention to suggest he will kick the same number of points,or get anywhere close,to the points scored by Neil Fox.
Still,I won't object if he makes a very good attempt at it in 2018.
Announcing the 3rd batch of players to be signed has begun tonight with signing number 11.
A former Dragonheart and GB Pioneers player,and a former Swansea university young man who was born in Norwich.
Louis Ford has also represented West Wales Raiders in the Conference.
Signing no.12 is Kurtis Haile,another centre and a former Dragonhearts and Torfaen Tigers player.
He was MoM in the Dragonhearts game v Germany,fairly recently,with the German side containing Leeds Rhino,Keinhorst,and he played,and scored for the losing side in the WRL Premier League Grand Final.
I think I'll just go listen to some Bob Marley...The family name isn't pronounced the same as....can't be.
I am pleased that todays announcement,of the signing of 18 yrs old Macauley Harris,is welcome news for the future.
My Haile Selassie quote for today is alternating between these 2 - Many discouraging hours will arise before the rainbow of accomplished goals will appear on the horizon.
Let us set our goals too high.Let us demand more of ourselves than we believe we possess.
who can play centre and possibly in the second row.
I thought about the pack and whether or not certain family names beginning with a low or higher alphabetical letter had more success.
The 114th player to be announced,tonight,is a prop.A young prop.Best wishes to Taine Hendy.
who came up with the findings about the dates of birth having a bearing.
Wrong date of birth and wrong family name for me - unfortunately.
I won't complain with vicarious happiness in 2018.
Very impressed by his performances last year and just wanting him to play with a consistent winger outside him.
He can also play in the 2nd row,but he is good both defensively and offensively.
It's been a good day with the announcement from Halifax of 3 south Wales youngsters gaining a place in the 'Fax first team squad.
Week 4 begins with the announcement of a young loose forward having signed for 2018.
Having played for Wales at Under 16 and Under 18 level,Rowland Kaye joins the club.
It has been a good weekend for Welsh folk showing faith in the youngsters.
A mixed bag of news today.
But there should be some positive news,shortly.
Craig Lewis is on board.
A second rower is another addition to a squad showing an interesting blend of youth and experience.
Dan Maiden has,unusually perhaps for a rugby league player,won a Commonwealth Games medal while playing the sport for his country.
I hope he goes onto further success with his rugby league in 2018.
It's back to experience and familiarity tonight.
A player signed by Lee Greenwood,at All Golds,at the end of June.Mentioned in despatches for their performance away to Toronto,before another club had an untimely demise,but the good news is the return of Shaun Owens.
He is signing no.19 and there seems some good battles for position(s) already established.
Well it's been such a long time it feels like it was a year ago when we last had a player announcement.
We've come crashing back,big style,with a player who last season played for University of Gloucester All Golds.
A strength and conditioning fella,a forward,one who has used his qualifications at Wigan.
Harrison Elliott,joins the Pack,ours not Toronto - though it seems he enjoyed playing in front of their supporters.
He's probably got more letters after his name than the Toronto/Leigh/Premier TV thread above has posts.
We have retained the services of Connor Parker,former Widnes Academy,and still just 20 yrs of age.
Connor is the 21st squad player to be named.
I'm beginning to become more than a little excited.
Pre-season game at Wrexham before this month is out - Kits to be disclosed - I hope I can keep the best till last and see the hallowed turf of Stebonheath Park at it's finest;despite the soccer season still underway,and the rainfall of late.
It is a brilliant pitch.Best groundsman in League 1 and further afield.
is experienced forward Barrie Phillips.
The only downside is all the away games on the bounce in April.
But,that pales into insignificance against the blend of youth and experience in the squad.
A Raider from the start - Dafydd Phillips will be another in a squad where there is some very interesting competition for places.
and not Connor Parker - but an eclectic array of skills is not to be ignored.
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{
"redpajama_set_name": "RedPajamaC4"
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Q: Issues with My Scanner I have Ubuntu 20.04.4 LTS and I have a Canon Pixma MG2522 Multi-function Printer I Reconnected the printer turned off the printer and the computer reinstalled the drivers from Canon's Website to no avail with the error "Fail to Scan error communicating with scanner" I am using the default Scan Document
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{
"redpajama_set_name": "RedPajamaStackExchange"
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Q: List all non unique columns of a dataframe I have a datafame that contains relationship between parent-child-origin-destination, it looks something like this.
parent_origin parent_destination child_origin child_destination
0 ABD NCL ABD ALM
1 ABD NCL ABD DHM
2 ABD YRK ABD ALM
3 ABD YRK ABD NTR
4 ABD KGX ABD SVG
I would like to group by on child_origin & child_destination to know if there are any child_origin, child_destination pairs that have 2 diffrent parent_origin & parent_destination and list out the result. I also want to print out the list of parent_origin & parent_destination that have the same child od pair.
For example, in the above dataframe i want the expected output to be like:
child_origin child_destination parent_origin parent_destination
1 ABD ALM ABD NCL
ABD YRK
What i have tried:
I can do a group by to get the values which have duplicate parents & the count of duplicates but i am not able to figure out how to diplay the actual parents values.
>>> grp = df.groupby(['child_origin','child_destination']).size().reset_index().rename(columns={0:'count'})
>>> grp[grp['count] > 1]
This gives me the count of all child_ods that have multiple parents but i want to knwo the value of parents as well.
PS: I am fairly new to pandas.
A: How about:
df.loc[df.groupby(['child_origin','child_destination'])['parent_origin'].transform("count") >1]
If you want the columns in order:
df.loc[df.groupby(['child_origin','child_destination'])['parent_origin'].transform("count") >1,
['child_origin', 'child_destination', 'parent_origin', 'parent_destination']]
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{
"redpajama_set_name": "RedPajamaStackExchange"
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Structural Engineering Blog: Adaptive Reuse
Adaptive Reuse: Benefits and Challenges and How We Should Properly Use Adaptive Reuse
Pratt Street Power Plant Source: http://www.sourcewatch.org/
Adaptive reuse construction is one of the latest trends in the structural engineering and urban revitalization industry. It is the method of reusing an existing structure or an old site that is deemed to have outlived its function, in order to recreate for new uses or functions (or the uses or functions of the changing needs of today's users) by implementing structural changes. This method puts emphasis on sustainable development since it conserves energy, land, and heritage. Construction of a poposed structure is not on new land, therefore reducing urban sprawl and increasing open space. There is no cost for demolition, so the cost of demolition machinery, energy, waste, and labor can be eliminated from project costs, as well as, the reduction of natural resources used in construction. Most importantly, adaptive reuse conserves culture and heritage by protecting structures with historical value and importance in the community. Our generation and our future generations will then have an opportunity to gain a sense of appreciation and understanding of history through the adapted historical buildings.
Often times, adaptive reuse is used onto old government buildings, commercial buildings, industrial structures, and churches. Examples of successful adaptive reuse include Home Depot's 23 Street store in Manhattan, New York, Pratt Street Power Plant in Baltimore, Maryland, and the Queen Victoria Museum and Art Gallery, Tasmania, Australia. All three of these structures were repurposed for new functions.
However, we should bear in mind that adaptive reuse is not just a glorified idea that glamorizes sustainable development and historical preservation. It can have some negative side effects too.
Hidden costs can be associated with adaptive reuse where there are unknown contaminants within the site, such as asbestos, which are harmful when inhaled. Additional ground investigative works and mitigation procedures might have to be used, in order to determine the levels of contaminants, adequate bearing capacity, and geological profiles of the site. These preliminary remediation tests and procedures might be costly.
Moreover, cost and time must be spent to determine the whole structural integrity of the building. There might be decay or erosion of the existing structure, therefore maintenance must be done. Also, there might be undiscovered structural flaws, such as the lack of reinforcement placement or concrete spalling in some areas of the structure. To determine the constructability of the structure, comprehensive investigative works need to be done on the ventilation systems, lighting flow, spatial flow, structural stability, settlement on foundation, etc. Initial maintenance must then be done, or else, additional alternations and additions will worsen the building.
Cost and time are also to be spent on adapting and modifying the existing structure to current building codes. For example, old industrial buildings cannot simply be adaptive to become a residential building. According to current building codes, old industrial buildings do not have sufficient ventilation flow, light flow, and fire requirements that can be adapted to become a residential building. There are also legal issues that might be involved, such as zoning requirements that need to be met and certain permits that need to be obtained. It sometimes takes immense creativity and problem solving skills to modify the structures to fit today's code.
Queen Victoria Museum and Art Gallery Source: http://www.gondwanastudios.com/
There is also the issue of satisfying the community, the client, and the government. The client's main concern is profitability and function of the building. The government's main concern is the existence of any historic preservation laws and development ordinances that may require strict review procedures. The community's concern is the biggest issue. Their questions will include whether the building is worth restoring or even saving, which buildings should first undergo adaptive reuse, how much of the building should be restored, etc. Nobody wants to have an old building with little or no historic significance anymore in the community. If that useless building undergoes adaptive reuse, the community will have to use that eyesore for another life cycle of the building.
It seems that I have listed more complexities and challenges of adaptive reuse than its benefits. But I believe that listing these challenges will help us learn to deal with these issues for future adaptive reuse projects. Although there are many challenges, adaptive reuse is often times a good investment and an amazing tool for long-term environmental sustainability and historic preservation. By reusing and modifying historical buildings for our current use, we can both respect the past and progress into the future. We just need to plan carefully amongst all parties and lay out the development's benefits and costs to determine whether adaptive reuse should be used. Preliminary assessment and planning should be done to make adaptive reuse successful. These include:
- Determining potential risks
- Identifying site conditions
- Researching whether development is viable for current regulations
- Determining historical significance
- Determining the level of support in the community and in the government
- Identifying structural integrity and limitations
- Identifying costs of alterations and additions
Working out these steps before even stepping into scheme design really helps with reducing cost and time of the project. Most importantly, it helps with finding solutions to the problems that are foreseen in this planning stage.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,691
|
\section{The main results}
The starting point of this paper is the results of Leary-Stancu in \cite{Leary-Stancu07} and Robinson in \cite{Robinson07}.
For any saturated fusion system $\F$ on a finite $p$-group $S$ they give recipes to construct groups $\pi_{LS}$ (for Leary-Stancu) and $\pi_R$ (for Robinson) which are generally infinite and which contain $S$ as a Sylow $p$-subgroup, see \S\ref{subsec:sfs-tansporter-and-linking-systems}, and realize $\F$ in the sense that $\F=\F_S(\pi)$.
Also, for primes $q=3,5 \mod (8)$, Aschbacher and Chermak constructed in \cite{Asch-Chermak} an infinite group $\pi_{AC}$ which contains a Sylow subgroup of $\Spin_7(q^n)$ as its Sylow $p$-subgroup and which realizes Solomon's fusion system $\F_{\Sol}(q^n)$, see \cite{LO1}.
The primary goal of this paper is to give a uniform and conceptual treatment for all these constructions.
Our approach is geometric rather than algebraic.
The main observation in this paper is Theorem \ref{thmA} below which we prove in the end of section \ref{sec:plfgs}.
Also, it was observed by Aschbacher that if a saturated fusion system $\F$ has the form $\F_S(\pi)$ where $\pi$ is possibly infinite, then a signaliser functor on $\pi$, see \S\ref{subsec:sfs-tansporter-and-linking-systems} for details, gives rise to an associated linking system $\L$.
A partial converse to this observation is point \ref{thmA:signaliser} of Theorem \ref{thmA}.
\begin{thm}\label{thmA}
Fix a $p$-local finite group $(S,\F,\L)$ and let $\pi$ be a group which contains $S$ as a Sylow $p$-subgroup.
Assume that $\F \subseteq \F_S(\pi)$ and that there exists a map $f \colon B\pi \to \pcomp{|\L|}$ whose restriction to $BS \subseteq B\pi$ is homotopic to the natural map $\theta \colon BS \to \pcomp{|\L|}$.
Then
\begin{enumerate}
\item
$\F=\F_S(\pi)$
\label{thmA:induce-fusion}
\item
There is a signaliser functor $\Theta$ on $\pi$ which induces $\L$, namely $\L$ is a quotient of the transporter system $\T^c_S(\pi)$.
\label{thmA:signaliser}
\item
$H^*(|\L|;\FF_p)$ is a retract of $H^*(B\pi;\FF_p)$ in the category of $\FF_p$-algebras.
It is equal to the image of $H^*(\pi;\FF_p) \xto{\res} H^*(S;\FF_p)$.
\label{thmA:retract}
\end{enumerate}
\end{thm}
The point of Propositions \ref{prop:thmA-conditions-R}, \ref{prop:ls-thmA-conditions} and \ref{prop:ac-thmA} below is that the groups $\pi_{LS}, \pi_R$ and $\pi_{AC}$ satisfy the hypotheses of this theorem, hence recovering the results of \cite{Leary-Stancu07, Robinson07} and \cite{Asch-Chermak}; See Remarks \ref{rem:we-do-as-good}--\ref{rem:cohomology-of-pi} below.
The conceptual reason that Theorem \ref{thmA} applies to these groups, which is the main message of this paper, is that all these groups are built from different homology decompositions which express $|\L|$ as the homotopy colimit of a diagram of classifying spaces of finite groups.
By taking homotopy colimits of carefully chosen subdiagrams we obtain classifying spaces $B\pi$ together with maps $B\pi \to \pcomp{|\L|}$ where $\pi$ induces the fusion system $\F$.
The groups $\pi_{LS}$ are built in \S\ref{subsec:Leary-Stancu} from the subgroup decomposition \cite[\S2]{BLO2}.
The groups $\pi_R$ are built in \S\ref{subsec:robinsons-construction} from the normalizer decomposition with respect to the collection of the $\F$-centric subgroups \cite{Lib06}.
The groups $\pi_{AC}$ are related to the normalizer decomposition with respect to the collection of the elementary abelian subgroup of $\F_{\Sol}(q^n)$ as we explain in \S\ref{subsec:Aschbacher-Chermak}.
An interesting conclusion of Theorem \ref{thmA}\ref{thmA:signaliser} is the following result whose proof is given in \S\S\ref{subsec:robinsons-construction}, \ref{subsec:Leary-Stancu}.
It settles a question which was open for some time.
\begin{cor}\label{corC}
Fix a saturated fusion system $\F$ on $S$ and let $\pi$ be either Robinson's group $\pi_R$ or Leary-Stancu's group $\pi_{LS}$ so that $\F=\F_S(\pi)$.
Then the following are equivalent.
\begin{enumerate}
\item
$\F$ has an associated linking system $\L$.
\label{corC:1}
\item
There exists a signaliser functor on $\pi$ which induces $\L$.
\label{corC:2}
\end{enumerate}
\end{cor}
\begin{remark}\label{rem:we-do-as-good}
We explain in Remark \ref{rem:R-avoid-linking-system} below, see also Proposition \ref{prop:stable-fusion}, that the structure of a $p$-local finite group is not essential to deduce point \ref{thmA:induce-fusion} of Theorem \ref{thmA} for arbitrary saturated fusion systems.
Thus, even though Theorem \ref{thmA} is not quite stated in this way, our results are as strong as those of \cite{Leary-Stancu07} and \cite{Robinson07} which apply to arbitrary saturated fusion systems.
\end{remark}
\begin{remark}
Our approach shows that Robinson's original construction isn't quite the natural one.
The group $\pi_R$ that we construct in \S\ref{subsec:robinsons-construction} is a factor group of Robinson's original construction \cite{Robinson07}.
See Remark \ref{rem:robinson-original-vs-pir} below.
Our costruction of $\pi_R$ is inspired by the results of the first author and Antonio Viruel in \cite{Lib-Vir}.
\end{remark}
\begin{remark}
The construction of $\pi_{AC}$ in \cite{Asch-Chermak} requires $q \equiv 3 \text{ or } 5 \mod (8)$.
This is particularly important for the construction of the signaliser functors.
The group $\pi_{AC}$ that we construct in \S\ref{subsec:Aschbacher-Chermak} is a variation of Aschbacher-Chermak's original construction.
Our group which is motivated by the geometric approach, requires no restriction on $q \mod (8)$, yet it admits a signaliser functor, see Remark \ref{rem:improved-ac}.
Thus, our construction improves Aschbacher-Chermak's albeit at the price of not being as explicit as their's.
\end{remark}
\begin{remark}\label{rem:cohomology-of-pi}
Finally, point \ref{thmA:retract} of Theorem \ref{thmA} applies to $\pi_{LS}, \pi_{R}$ and $\pi_{AC}$ and as far as we know, this cohomological information is new.
\end{remark}
From the cohomological point of view, the groups $\pi_R$ and $\pi_{AC}$ are very close to their ``target'' namely the ring $\H^*(\F)$ of the stable elements, see \cite[Theorem B]{BLO2}, in a sense made precise in the next result which is proven in \S\S\ref{subsec:robinsons-construction}, \ref{subsec:Aschbacher-Chermak}.
\begin{prop}\label{propB}
Let $\pi$ be either the group $\pi_R$ associated to a saturated fusion system $\F$ (see \S\ref{subsec:robinsons-construction}) or the group $\pi_{AC}$ if $\F=\F_{\Sol}(q^n)$ and $p=2$ (see \S\ref{subsec:Aschbacher-Chermak}).
Then
\begin{enumerate}
\item
$B\pi$ is $p$-good in the sense of \cite[I.\S 5]{Bous-Kan}.
\item
In the kernel of $H^*(B\pi;\FF_p) \xto{Bi^*} H^*(BS;\FF_p)$ the product of any two elements is zero.
\end{enumerate}
\end{prop}
\section{Saturated fusion systems and $p$-local finite groups}\label{sec:plfgs}
In this section we will \emph{very briefly} recall the notion of $p$-local finite groups and of transporter systems.
We will take the approach of Broto Levi and Oliver in \cite{BLO2} and Oliver-Ventura \cite[\S\S 2,3]{OV07}.
Also see \cite{BLO-survey}.
Fusion systems were first defined by Puig, e.g. \cite{Puig06}.
\subsection{Fusion systems}\label{subsec:sfs}
A \emph{fusion system} on a finite $p$-group $S$ is a small category $\F$ whose object set is the set of subgroups of $S$.
Morphism sets $\F(P,Q)$ are group monomorphisms which include the set $\Hom_S(P,Q)$ of conjugations by elements of $S$.
In addition, every morphism $P\to Q$ in $\F$ is the composition of an isomorphism $P \to P'$ in $\F$, which is necessarily a group isomorphism, followed by an inclusion $P' \leq Q$.
From the definitions, every endomorphism in $\F$ is an automorphism.
Subgroups $P, P' \leq S$ are called \emph{$\F$-conjugate} if they are isomorphic as objects of $\F$.
A subgroup $P \leq S$ is called \emph{fully centralized} if $P$ has the largest $S$-centralizer among all the subgroups in its $\F$-conjugacy class.
Similarly, $P$ is \emph{fully normalized} if its $S$-normalizer has the largest possible order among all the groups in its $\F$-conjugacy class.
We say that $\F$ is \emph{saturated} if the following two conditions hold.
First, every fully normalized subgroup $P \leq S$ is fully centralized and in this case $\Aut_S(P)=N_S(P)/C_S(P)$ is a Sylow $p$-subgroup of $\Aut_\F(P)$.
Second, if $\vp \in \F(P,S)$ is such that $\vp(P)$ is fully centralized then $\vp$ extends to a morphism $\tilde{\vp} \in \F(N_\vp,S)$ where $N_\vp \leq N_S(P)$ consists of the elements $g$ such that $\vp \circ c_g \circ \vp^{-1} \in \Aut_S(\vp(P))$ where $c_g$ is the inner automorphism $x \mapsto gxg^{-1}$ of $S$.
A subgroup $P \leq S$ is called \emph{$\F$-centric} if $P$ and all its $\F$-conjugates contain their $S$-centralizer.
We let $\F^c$ denote the full subcategory of $\F$ on the $\F$-centric objects.
A subgroup $P \leq S$ is called $\F$-radical if $\Inn(P)$ is the maximal normal $p$-subgroup of $\Aut_\F(P)$.
Alperin's fusion theorem \cite[Theorem A.10]{BLO2} says that any saturated fusion system $\F$ is generated by the $\F$-automorphism groups of its $\F$-centric $\F$-radical subgroups.
Namely, every morphism in $\F$ is obtained by composition of restrictions of $\F$-automorphisms of these groups.
An important source of saturated fusion systems are finite groups.
If $S$ is a Sylow $p$-subgroup of a finite group $G$, one forms the category $\F_S(G)$ whose objects are the subgroups of $S$ and whose morphisms are the group monomorphisms $c_g \colon P \to Q$ of the form $x \mapsto gxg^{-1}$ for some $g \in G$.
\subsection{Transporter systems, linking systems and signaliser functors}\label{subsec:sfs-tansporter-and-linking-systems}
A finite $p$-subgroup $S$ of a group $\pi$, possibly infinite, is a \emph{Sylow $p$-subgroup} if any finite $p$-subgroup $Q \leq \pi$ is conjugate to a subgroup of $S$.
Define $\F_S(\pi)$ where $S \leq \pi$ is a Sylow $p$-subgroup as we did above for finite groups; Up to isomorphism the choice of $S$ is immaterial.
The objects of the \emph{transporter system} $\T_S(\pi)$ are the subgroups of $S$; its morphism sets are
\[
\T_S(\pi)(P,Q)=N_\pi(P,Q) = \{ g \in \pi \colon gPg^{-1} \leq Q\}.
\]
There is an obvious surjective functor $\T_S(\pi) \to \F_S(\pi)$ which is the identity on objects and is the quotient by the action of $C_\pi(P)$ on morphism sets.
We denote by $\T^c_S(\pi)$ the full subcategory whose objects are the $\F$-centric subgroups of $S$ for some fusion system $\F$ on $S$ which will be understood from the context.
An associated \emph{centric linking system} $\L$ to a saturated fusion system $\F$ on $S$ is a category whose objects are the $\F$-centric subgroups.
It is equipped with a surjective functor $\pi \colon \L \to \F^c$ and an injective functor $\de \colon \T^c_S(S) \to \L$, both induce the identity on object sets.
In addition the following three conditions must be satisfied.
First, the image of $Z(P)$ under $\de \colon N_S(P) \to \Aut_\L(P)$ acts freely on $\L(P,Q)$, and $\F(P,Q)$ is isomorphic to the orbit set via $\pi$.
Second, for any $g \in N_S(P,Q)$ we have $\pi(\de(g))=c_g$.
Third, for any $f \in \L(P,Q)$ and any $g \in P \leq N_S(P)=\Aut_{\T^c_S(S)}(P)$ we have $\de(\pi(f)(g)) \circ f = f \circ \de(g)$ in $\L$.
A \emph{$p$-local finite group} is a triple $(S,\F,\L)$ of a saturated fusion system on $S$ together with an associated linking system.
It is not known if any saturated fusion system admits a centric linking system, and if so, if it is unique.
Now suppose that $\F=\F_S(\pi)$ for a group $\pi$ which contains $S$ as a Sylow $p$-subgroup.
A \emph{signaliser functor} on $\pi$ is an assignment $P \mapsto \Theta(P)$ for every $\F$-centric $P \leq S$ such that $\Theta(P)$ is a complement of $Z(P)$ in $C_\pi(P)$ and such that if $gPg^{-1} \leq Q$ then $\Theta(Q) \leq g\Theta(P)g^{-1}$.
Indeed, $\Theta$ is a contravariant functor from $\T^c_S(\pi)$ to the category of groups.
A signaliser functor $\Theta$, if it exists, gives rise to a centric linking system $\L=\L_{\Theta}$ where $\L(P,Q)=N_\pi(P,Q)/\Theta(P)$.
See \cite{Asch-Chermak} and also \cite[Section 3]{Lib-Vir} for more details.
\subsection{Homotopy theoretic constructions}\label{subsec:sfs-homotopy-theoretic-constructions}
Fix a finite $p$-group $S$ and a map $f \colon BS \to X$.
By \cite[Definition 7.1]{BLO2} $f$ gives rise to
a fusion system $\F_S(f)$ on $S$ whose objects are the subgroups of $S$ and a group monomorphism $\vp \colon P \to Q$ belongs to $\F_S(f)$ if and only if $BP \xto{B\vp} BS \xto{f} X$ is homotopic to $BP \xto{Bi} BS \xto{f} X$.
It follows directly from the definitions that any map $g \colon X \to Y$ induces an inclusion of fusion systems (not saturated in general)
\[
\F_S(f) \subseteq \F_S(g \circ f).
\]
The category $\L_S(f)$, see \cite[Definition 7.2]{BLO2}, has the same object set as $\F=\F_S(f)$.
The morphisms $P \to Q$ in $\L_S(f)$ are pairs $(\vp,[H])$ where $\vp \in \F(P,Q)$ and $[H]$ is the homotopy class of a path $H$ in $\map(BP,X)$ from $BP \xto{B\vp} BS \xto{f} X$ to $BP \xto{Bi} BS \xto{f} X$.
There is a surjective functor $\pi \colon \L_S(f) \to \F_S(f)$ which ``forgets'' $[H]$.
A map $g \colon X \to Y$ gives rise, by composition, to a functor
\[
g_* \colon \L_S(f) \to \L_S(g \circ f)
\]
over the projection functors to $\F_S(f) \subseteq \F_S(g \circ f)$, i.e. $\pi \circ g_* = \incl_{\F_S(f)}^{\F_S(gf)} \circ \pi$.
Define $\L^c_S(f)$ as the full subcategory of $\L_S(f)$ on the $\F_S(f)$-centric objects.
If $g \colon X \to Y$ is a map then $\F_S(f) \subseteq \F_S(g \circ f)$ so every $\F_S(g \circ f)$-centric subgroup of $S$ is also $\F_S(f)$-centric and we obtain a functor
\[
g_* \colon \L^c_S(f) \to \L_S^c(g \circ f)
\]
defined on the $\F_S(g \circ f)$-centric objects of $\L^c_S(f)$.
\subsection{Classifying spaces}\label{subsec:classifying-spaces}
The \emph{classifying space} of a $p$-local finite group is $\pcomp{|\L|}$.
The inclusion $\de \colon S \to \Aut_\L(S)$ thought of as a full subcategory of $\L$ gives rise to a map
\begin{equation}\label{eq:theta}
\theta \colon BS \to \pcomp{|\L|}
\end{equation}
which will play a fundamental role in this paper.
One of the key features of the map $\theta$ is \cite[Proposition 7.3]{BLO2}
\[
\F=\F_S(\theta) \qquad \text{ and } \qquad \L=\L^c_S(\theta).
\]
For any group $G$ let $\B G$ denote the category with one object $o_G$ and $G$ as its automorphism set.
A homomorphism $\vp \colon G' \to G$ induces a functor $\B\vp \colon \B G' \to \B G$.
It is well known that if $H \leq G$ then the functor $\B C_G(H) \times \B H \xto{(x,y) \mapsto xy} \B G$ gives rise, after realization and adjunction, to a homotopy equivalence $BC_G(H) \xto{\simeq} \map(BH,BH)_{Bi}$ where the space on the right is the path component of the inclusion $BH \xto{Bi} BG$.
\begin{prop}\label{prop:topological-linking-of-bpi}
Fix an inclusion $S \leq \pi$ of a Sylow $p$-subgroup and consider $Bi \colon BS \to B\pi$.
Then $\F_S(Bi)=\F_S(\pi)$ and $\L_S(Bi)=\T_S(\pi)$.
\end{prop}
\begin{proof}
The first isomorphism is clear because $[BP,B\pi]=\Hom(P,\pi)/\Inn(\pi)$.
Define a functor $\Phi \colon \T_S(\pi) \to \L_S(Bi)$ as follows.
It is the identity on objects and for a morphism $g \in N_\pi(P,Q)$ consider $c_g \colon P \to Q$ and the natural transformation $g \colon \B c_g \to \B i_P^\pi$ between the functors $c_g,i_P^S \colon \B P \to \B \pi$ given by $o_\pi \xto{g} o_\pi$.
This natural transformation gives rise to a homotopy $I \times BP \to B\pi$ from $Bc_g$ and $Bi_P^\pi$, whence to a path $H$ in $\map(BP,B\pi)$ from $Bc_g$ to $Bi_P^\pi$.
Define $\Phi(g)=(c_g,[H])$.
The functoriality of $\Phi$ is easy to check.
It is bijective because $\Phi$ maps the coset $C_\pi(P)g \leq N_\pi(P,Q)$ bijectively onto $\pi_1^\#(\map(BP,B\pi);Bc_g,Bi_P^\pi) \cong \pi_1BC_\pi(P)$ via $\B C_\pi(P) \times \B P \to \B \pi$.
\end{proof}
\subsection{The subgroup decomposition}\label{subsec:sfs-subgroup-decomposition}
Recall that the orbit category $\O=\O(\F)$ of a fusion system $\F$ has the same object set as $\F$ and $\O(P,Q)=\F(P,Q)/\Inn(Q)$.
See \cite[Definition 2.1]{BLO2}.
Let $\O(\F^c)$ denote the full subcategory on the $\F$-centric objects.
Fix a $p$-local finite group $(S,\F,\L)$.
By \cite[Proposition 2.2]{BLO2} there is a functor called the \emph{subgroup decomposition}
\[
\be \colon \O(\F^c) \to \Top
\]
which as a functor into $\hotop$ - the homotopy category of spaces - is isomorphic to the functor $P \mapsto BP$.
Moreover, there is a homotopy equivalence
\[
\hocolim_{\O(\F^c)} \be \xto{\simeq} |\L|
\]
whose restriction to $\be(S)\simeq BS$ is homotopic to the map $BS \to |\L|$ induced by the inclusion $S \leq \Aut_\L(S)$.
Hence $\pcomp{\theta}$ is homotopic to the map in display \eqref{eq:theta}.
\subsection{The normalizer decomposition}\label{subse:sfs-normaliser-decomposition}
Let $\R$ be a set of $\F$-centric subgroups of $S$ which is closed under $\F$-conjugation.
The subdivision category $\bar{S}(\R)$ is the following poset; see \cite{Lib06} for more details where it is denoted $\bar{s}d(\R)$.
The objects of $\bar{S}(\R)$ are the $\F$-conjugacy classes $[P_\bullet]$ of chains of inclusions $P_0 \lneq \dots \lneq P_k$ of subgroups in $\R$ called ``$k$-simplices'' ($k \geq 0$).
We say that $P_\bullet$ is $\F$-conjugate to $P_\bullet'=P_0' \lneq \dots \lneq P_k'$ if there is an isomorphism $P_k \to P_k'$ in $\F$ which carries $P_i$ isomorphically onto $P_i'$ for all $i$.
There is a unique morphism $[P_\bullet] \to [Q_\bullet]$ in $\bar{S}(\R)$ if $Q_\bullet$ is $\F$-conjugate to a sub-sequence (a ``subsimplex'') of $P_\bullet$.
Let $\Aut_\F(P_\bullet)$ be the subgroup of $\Aut_\F(P_k)$ which leaves $P_0,\dots,P_k$ invariant.
Let $\Aut_\L(P_\bullet)$ be its preimage in $\Aut_\L(P_k)$.
Suppose that $Q_\bullet=Q_0\lneq \dots \lneq Q_m$ is a sub-simplex of $P_\bullet$ where $Q_m=P_i$ for some $i$.
Consider $e \in N_\pi(P_i,P_k)$ and set $\hat{e}=\de(e)\in \L(P_i,P_k)$.
For every $\vp \in\Aut_\L(P_\bullet)$ we have $\pi(\vp) \in \Aut_\F(P_\bullet)$ and therefore $\pi(\vp)|_{P_i} \in \Aut_\F(Q_\bullet)$.
It follows from \cite[Lemma 1.10(a)]{BLO2} and the fact that $\hat{e}$ is both an epimorphism and a monomorphism in $\L$, see \cite[Corollary 3.10]{BCGLO} that there exists a unique $\vp' \in \Aut_\L(Q_\bullet)$ such that $\vp \circ \hat{e}=\hat{e} \circ \vp'$.
This sets up a canonical group monomorphism $\Aut_\L(P_\bullet) \hookrightarrow \Aut_\L(Q_\bullet)$.
See \cite{Lib06} for more details.
We obtain a functor $[P_\bullet] \mapsto B\Aut_\L(P_\bullet)$ into the homotopy category of spaces (here we use the fact that the self-map that an inner automorphism of a group induces on its classifying space is homotopic to the identity).
The \emph{normalizer decomposition} with respect to the collection $\R$ of $\F$-centric subgroups, see \cite[Theorem A]{Lib06}, is a functor
\[
\de \colon \bar{S}(\R) \to \Top
\]
which as a functor into $\hotop$ is isomorphic to the functor $[P_\bullet] \mapsto B\Aut_\L(P_\bullet)$.
In addition there is a map
\[
\hocolim_{\bar{S}(\R)} \de \xto{g} |\L|
\]
whose restriction to $\de([P_\bullet])$ is induced, upon taking nerves, from the inclusion of categories $\B\Aut_\L(P_\bullet) \subseteq \L$.
Hence, the restriction of $\hocolim_{\bar{S}(\R)} \de \simeq |\L| \to \pcomp{|\L|}$ to $BS\leq B\Aut_\L(S)$ is homotopic to $\theta$ in display \eqref{eq:theta}.
If $\R$ contains all the $\F$-centric $\F$-radical subgroups of $S$ then $g$ is a homotopy equivalence, \cite[Theorem A]{Lib06}.
\begin{proof}[Proof of Theorem \ref{thmA}]
\ref{thmA:induce-fusion}
Let $Bi \colon BS \to B\pi$ denote the inclusion.
By Proposition \ref{prop:topological-linking-of-bpi} the map $f$ induces an inclusion
\[
\F_S(\pi) = \F_S(Bi) \subseteq \F_S(f \circ Bi) = \F_S(\theta) = \F.
\]
By hypothesis $\F \subseteq \F_S(\pi)$ and therefore equality holds.
\ref{thmA:signaliser}
We deduce from Proposition \ref{prop:topological-linking-of-bpi} and from \S\ref{subsec:sfs-homotopy-theoretic-constructions} that $\L^c_S(Bi)=\T^c_S(\pi)$, that $\L_S^c(f \circ Bi)=\L$ because $f \circ Bi \simeq \theta$, and that $f$ induces a functor $\rho \colon \T^c_S(\pi) \to \L$ such that
\[
\xymatrix{
{\T^c_S(\pi)} \ar@{->>}[d]_{\pi} \ar[r]^\rho &
\L \ar@{->>}[d]^\pi
\\
{\F^c} \ar@{=}[r] &
\F^c
}
\]
commutes.
We will now prove, and this is the key observation, that $\rho$ is surjective.
Set $\T=\T^c_S(\pi)$.
Since $Z(P) \leq \Aut_\L(P)$ acts on $\L(P,Q)$ with orbit set $\F(P,Q)$ and since the functors $\pi$ in the square above are surjective, it suffices to prove that $Z(P)$ is in the image of $\rho_P \colon \Aut_\T(P) \to \Aut_\L(P)$ for any $\F$-centric $P \leq S$.
To do this consider the inclusion $Bi_P^S \colon BP \to BS$.
The composition
\[
\map(BP,BS)_{Bi_P^S} \xto{Bi_*} \map(BP,B\pi)_{Bi_P^\pi} \xto{f_*} \map(BP,\pcomp{|\L|})_{f \circ Bi_P^\pi}
\]
is a homotopy equivalence because $P$ is $\F$-centric so the first space is homotopy equivalent to $BC_S(P)=BZ(P)$ and because the third space is homotopy equivalent to $BZ(P)$ by \cite[Theorem 4.4(c)]{BLO2} since $Bi \circ f \simeq \theta$.
We see from the definition of $\L_S(-)$, see \S\ref{subsec:sfs-homotopy-theoretic-constructions}, that $\rho_P$ carries $Z(P) \leq N_\pi(P)$ isomorphically onto $Z(P) \leq \Aut_\L(P)$.
We will now define the signaliser functor $\Theta$.
For every $\F$-centric $P \leq S$ set
\[
\Theta(P) \overset{\text{def}}{=} \ker ( \Aut_\T(P) \to \Aut_\L(P) ).
\]
Now, $\Aut_\T(P)=N_\pi(P)$ so we obtain a commutative diagram with exact rows
\[
\xymatrix{
1 \ar[r] &
C_\pi(P) \ar[d] \ar[r] &
N_\pi(P) \ar@{->>}[d]^{\rho_P} \ar[r]^\pi &
\Aut_\F(P) \ar[r] \ar@{=}[d] &
1
\\
1 \ar[r] &
Z(P) \ar[r] &
\Aut_\L(P) \ar[r]_\pi &
\Aut_\F(P) \ar[r] &
1
}
\]
which implies that $\Theta(P) \leq C_\pi(P)$ and that it is a complement of $Z(P)$ because we have seen that $\rho_P(Z(P)=Z(P)$.
For $g \in \pi$ such that $gPg^{-1} \leq S$ we obtain a commutative diagram
\[
\xymatrix{
1 \ar[r] &
\Theta(P) \ar[r] \ar[d] &
N_\pi(P) \ar[r] \ar[d]^{x \mapsto gxg^{-1}}_\cong &
\Aut_\L(P) \ar[r] \ar[d]^{\vp \mapsto \rho(g)\vp\rho(g)^{-1}}_\cong &
1
\\
1 \ar[r] &
\Theta(gPg^{-1}) \ar[r] &
N_\pi(gPg^{-1}) \ar[r] &
\Aut_\L(gPg^{-1}) \ar[r] &
1
}
\]
which shows that $g\,\Theta(P)\, g^{-1} = \Theta(gPg^{-1})$.
Now suppose that $P \leq Q$ and set $N=N_\pi(P) \cap N_\pi(Q)$.
Set $\hat{e}=\de(e) \in \Aut_\L(P,Q)$ where $e \in N_S(P,Q)$ is the identity element.
Let $\Aut_\L(Q,\darrow_P)$ be the subgroup of $\Aut_\L(Q)$ of the automorphisms $\vp$ which restrict to an automorphism of $P$ in the sense that there exists $\vp' \in \Aut_\L(P)$ such that $\vp \circ \hat{e} = \hat{e} \circ \vp'$.
Note that $\vp'$ is unique because $\hat{e}$ is a monomorphism in $\L$ by \cite[Corollary 3.10]{BCGLO}.
Similarly, let $\Aut_\L(P,\uparrow^Q)$ denote the subgroup of $\Aut_\L(P)$ of the automorphisms $\vp'$ which extend to an automorphism of $Q$ in the sense that there exists $\vp \in \Aut_\L(Q)$ such that $\vp \circ \hat{e} = \hat{e} \circ \vp'$.
Note that $\vp$ is unique because $\hat{e}$ is an epimorphism in $\L$, \cite[Corollary 3.10]{BCGLO}.
Thus $\vp \mapsto \vp'$ sets up an isomorphism $\Aut_\L(Q,\darrow_P) \to \Aut_\L(P,\uparrow^Q)$.
Note that if $\vp = \rho_Q(g)$ for some $g \in N_\pi(Q)$ then $g \in N$ and $\vp'=\rho_P(g)$.
Clearly, the preimage under $\rho$ of $\Aut_\L(Q,\darrow_P)$ in $\Aut_\T(Q)=N_\pi(Q)$ is $N$ as is the preimage of $\Aut_\L(P,\uparrow^Q)$ in $\Aut_\T(P)=N_\pi(P)$.
Also, $C_\pi(Q) \leq N$.
We obtain
\[
\xymatrix{
1 \ar[r] &
\Theta(Q) \ar[r] \ar[d] &
N \ar[r]^(0.3){\rho_Q} \ar@{=}[d] &
\Aut_\L(Q,\darrow_P) \ar[d]_\cong^{\vp \mapsto \vp'} \ar[r] &
1
\\
1 \ar[r] &
\Theta(P) \cap N \ar[r] &
N \ar[r]_(0.3){\rho_P} &
\Aut_\L(P,\uparrow^Q)
\ar[r] &
1
}
\]
We deduce that $\Theta(Q) =\Theta(P) \cap N \leq \Theta(P)$, hence $\Theta$ is a signaliser functor.
\ref{thmA:retract}
Fix some $\vp \in \F(P,Q)$.
There is some $g \in \pi$ such that $\vp$ is the restriction of $c_g \in \Inn(\pi)$ to $P$ because $\F = \F_S(\pi)$.
Since $Bc_g$ is homotopic to the identity on $B\pi$, then following square commutes up to homotopy
\[
\xymatrix{
BP \ar[r]^{Bi_P^S} \ar[d]_{B\vp} &
BS \ar[r]^{Bi} &
B \pi \ar@{=}[d]^{Bc_g \simeq \id}
\\
BQ \ar[r]_{Bi_P^S} &
BS \ar[r]_{Bi} &
B\pi.
}
\]
Applying $H^*(-;\FF_p)$ to this diagram for every $\vp \in \F^c$ shows that that the image of $H^*(B\pi;\FF_p) \xto{Bi^*} H^*(BS;\FF_p)$ is contained in the ring of stable elements $H^*(\F^c;\FF_p)$, namely the subring of all the classes $\al \in H^*(S;\FF_p)$ such that $\vp^*(\al|_Q)=\al|_P$ for all $\vp \in \F^c(P,Q)$.
Now we consider $BS \xto{Bi} B\pi \xto{f} \pcomp{|\L|}$ which is homotopic to $\theta$.
By \cite[Theorem 5.8]{BLO2} $\theta^*$ is an isomorphism of $H^*(\pcomp{|\L|};\FF_p)$ onto $H^*(\F^c) \leq H^*(BS;\FF_p)$.
But $\theta^*=(Bi)^* \circ f^*$ so $Bi^*$ is left inverse of $f^*$, both are $\FF_p$-algebra homomorphisms.
\end{proof}
Ragnarsson defined in \cite{Ragnarsson} a classifying spectrum $\BB\F$ for any saturated fusion system $\F$.
It is equipped with a structure map $\si_\F \colon \BB S \to \BB\F$ where $\BB S=\Sigma^\infty BS$.
In the presence of a linking system, $\BB\F \simeq \Sigma^\infty \pcomp{|\L|}$ and $\si_\F=\Sigma^\infty \theta$, see display \eqref{eq:theta}.
He showed that the assignment of $\F \mapsto \BB\F$ is a functor from the category of fusions systems to the homotopy category of spectra.
In addition he showed that $\F$ can be recovered from $(\BB \F,\si_\F)$ in the sense that the morphisms in $\F$ are exactly the group monomorphisms $\vp \colon P \to Q$ such that $\BB P \xto{\BB \vp} \BB S \xto{\si_\F} \BB \F$ is homotopic to $\BB P \xto{\BB i} \BB S \xto{\si_\F} \BB \F$ where $\BB P = \Sigma^\infty BP$.
Compare this with the construction in \S\ref{subsec:sfs-homotopy-theoretic-constructions}.
Here is a variant of Theorem \ref{thmA}\ref{thmA:induce-fusion}.
Note that while linking systems are not known to exists, the classifying spectra of saturated fusion systems always exist.
\begin{prop}\label{prop:stable-fusion}
Suppose $\F$ is a saturated fusion system on $S$ and that $S$ is a Sylow $p$-subgroup of some group $\pi$.
Assume that there exists a map $f \colon B\pi \to \Omega^\infty \BB \F$ whose restriction to $BS$ is homotopic to the right adjoint $\si_\F^\# \colon BS \to \Omega^\infty \BB \F$ of $\si_\F \colon \BB S \to \BB\F$.
Then $\F_S(\pi) \subseteq \F$.
\end{prop}
\begin{proof}
By Proposition \ref{prop:topological-linking-of-bpi} and the constructions in \S\ref{subsec:sfs-homotopy-theoretic-constructions}, it only remains to prove that $\F=\F_S(\si_\F^\#)$.
This is straightforward from the definition of $\F_S(\si_\F^\#)$ and the adjunction $[ \BB P , \BB \F] \cong [\Sigma^\infty BP , \BB \F] \cong [BP,\Omega^\infty \BB \F]$.
\end{proof}
\section{Graphs and homotopy colimits}
\subsection{Categories of dimension $1$}\label{ss:cat-dim-one}
A small category $\C$ is called $1$-dimensional if its nerve is $1$-dimensional.
Thus, in any pair of composable morphisms in $\C$ at least one of them is an identity morphism.
Equivalently, there is a partition $\Obj(\C)=C_1 \coprod C_2$ such that if $\vp \colon x \to y$ is a non-identity morphism in $\C$ then $x \in C_1$ and $y \in C_2$.
Since there are no non-trivial compositions in $\C$, it is clear that if $\D$ is any category then any assignment $F \colon \Obj(\C) \to \Obj(\D)$ together with an assignment of a morphism $F(c_1) \to F(c_2)$ to any \emph{non-identity} morphism $c_1 \to c_2$ in $\C$, gives rise to a functor $F \colon \C \to \D$.
\begin{prop}\label{prop:1-diml-htpy-nat-maps}
Let $\C$ be a $1$-dimensional category and let $F,F' \colon \C \to \Top$ be functors whose values are CW-complexes.
Assume that there is an isomorphism $q \colon F \xto{\simeq} F'$ in the \emph{homotopy category of spaces $\hotop$}.
Then there is a homotopy equivalence $\bar{q} \colon \hocolim F \xto{\simeq} \hocolim F'$ with the property that the squares
\[
\xymatrix{
F(c) \ar[r] \ar[d]_{q_c}^{\simeq} &
\hocolim F \ar[d]^{\bar{q}}_{\simeq}
\\
F'(c) \ar[r] &
\hocolim F'
}
\]
commute up to homotopy for any $c \in \C$.
\end{prop}
\begin{proof}
Let $C_1 \coprod C_2$ be a partition of $\Obj(\C)$ as described above.
The isomorphism $q \colon F \to F'$ in $\hotop$ gives rise to the following maps which we will fix once and for all.
Homotopy equivalences $q_c \colon F(c) \xto{\simeq} F'(c)$ for every object $c \in \C$ and, for every non-identity morphism $\vp \colon c_1 \to c_2$ in $\C$, a homotopy $h_\vp \colon F(c_1) \times I \to F'(c_2)$ from $q_{c_2} \circ F(\vp)$ to $F'(\vp) \circ q_{c_1}$.
Define functors $H, H_0 , H_1 \colon \C \to \Top$ by describing their effect on objects of $C_1$ and $C_2$ and on non-identity morphisms as follows
\[
\begin{array}{lll}
H_0 \colon c_1 \mapsto F(c_1), &
H_0 \colon c_2 \mapsto F'(c_2), &
H_0(c_1 \xto{\vp} c_2) = q_{c_2} \circ F(\vp).
\\
H_1 \colon c_1 \mapsto F(c_1), &
H_1 \colon c_2 \mapsto F'(c_2), &
H_1(c_1 \xto{\vp} c_2) = F'(\vp) \circ q_{c_1}
\\
H \colon c_1 \mapsto F(c_1) \times I, &
H \colon c_2 \mapsto F'(c_2), &
H(c_1 \xto{\vp} c_2) = h_\vp.
\end{array}
\]
Now we will define natural transformations, all of which are homotopy equivalences:
\[
F \xrightarrow[\simeq]{(\id,q_2)} H_0 \xrightarrow[\simeq]{(i_0,\id)} H \xleftarrow[\simeq]{(i_1,\id)} H_1 \xrightarrow[\simeq]{(q_1,\id)} F'
\]
On objects of $C_1$ and $C_2$ the effect of these natural transformations is described by the rows of the diagram below and the columns show the effect of $F, H_0, H, H_1$ and $F'$ on a non-identity morphism $\vp$.
The commutativity of all the squares shows that these are indeed natural transformations.
\[
\xymatrix{
F(c_1) \ar[r]^{\id} \ar[d]_{F(\vp)} &
F(c_1) \ar[r]^{i_0} \ar[d]^{q_{c_2}\circ F(\vp)} &
F(c_1) \times I \ar[d]^{h_\vp} &
F(c_1) \ar[l]_{i_1} \ar[r]^{q_{c_1}} \ar[d]^{F'(\vp) \circ q_{c_1}} &
F'(c_1) \ar[d]^{F'(\vp)}
\\
F(c_2) \ar[r]^\simeq_{q_{c_2}} &
F'(c_2) \ar[r]_{\id} &
F'(c_2) &
F'(c_2) \ar[l]^{\id} \ar[r]_{\id} &
F'(c_2)
}
\]
We now obtain homotopy equivalences
\[
\hhocolim{\C}F \xto{\simeq} \hhocolim{\C} H_0 \xto{\simeq} \hhocolim{\C} H
\xleftarrow{\simeq} \hhocolim{\C} H_1 \xto{\simeq} \hhocolim{\C} F'.
\]
It is also clear from the commutative ladder above that the second diagram in the statement of this proposition is homotopy commutative for all $c \in \C$.
\end{proof}
\subsection{Graphs and their associated categories}
\label{subsec:graphs-associated-categories}
A special class of $1$-dimensional categories is associated to graphs.
Recall from Serre \cite{Serre:trees} that a graph $Y$ consists of the following data.
A set $V$ of vertices also denoted $\vertx(Y)$ and a set $E$ of (directed) edges denoted $\edge(Y)$.
There is a function $E \xto{y \mapsto (o(y),t(y)) } V \times V$ whose components are the \emph{origin} and \emph{terminus} of an edge $y$.
It is also equipped with an involution $E \xto{y \mapsto \bar{y}} E$ which has no fixed points and ``reverses the direction'' of an edge, namely $o(\bar{y})=t(y)$ and $t(\bar{y})=o(y)$.
The orbits of this involution $[y]$ are called ``geometric edges''.
The set of the geometric edges of $Y$ is denoted $[\edge(Y)]$.
A \emph{walk} in $Y$ from a vertex $u$ to a vertex $v$ is a sequence $y_1,\dots,y_n$ of edges such that $o(y_1)=u$ and $t(y_n)=v$ and $o(y_{i+1})=t(y_i)$ for all $i$.
A \emph{loop} is a walk from a vertex to itself.
A \emph{path} from $u$ to $v$ is a walk which contains no loops.
An \emph{orientation} of $Y$ is a choice $A\subseteq E$ of one representative from each geometric edge.
Given an orientation $A$ we will write $|y|$ for either the edge $y$ or $\bar{y}$ which belongs to $A$.
Any graph $Y$ gives rise to a small category $\C_Y$ whose object set is the disjoint union of $\vertx(Y)$ and $[\edge(Y)]$.
Apart from identity morphisms there is a unique morphism $[y] \xto{y} t(y)$ for any $y \in \edge(Y)$.
\subsection{Graphs of groups}
Let $Y$ be a graph.
Recall from \cite[Def. 8 in \S4.4]{Serre:trees} that a graph of groups $(G,Y)$ is an assignment of groups $G_v$ to every $v \in \vertx(Y)$ and $G_y$ to every $y \in \edge(Y)$ such that $G_y=G_{\bar{y}}$ and every edge is associated with a group monomorphism $G_y \to G_{t(y)}$ which is denoted $a \mapsto a^y$ (here $a \in G_y$ and $a^y$ is its image in $G_{t(y)}$, \cite[pp. 41--42]{Serre:trees}).
We obtain an associated functor $\G \colon \C_Y \to \{\Gps\}$ defined by
\[
\G(v)=G_v, \qquad \G([y])=G_{y}, \qquad \G([y] \xto{y} t(y)) = G_y \xto{a \mapsto a^y} G_{t(y)}.
\]
Clearly, $\G$ carries morphisms in $\C_Y$ to group monomorphisms.
Conversely, any functor $\G \colon \C_Y \to \{\Gps\}$ with this property defines a graph or groups $(G,Y)$.
\subsection{The fundamental group}\label{subsec:universal-cover}
Fix a graph of groups $(G,Y)$.
Let $F(\edge(Y))$ denote the free group whose generating set is $\edge(Y)$ and set
\[
\tilde{F}(G,Y) = F(\edge(Y)) * \big( \underset{v \in \vertx Y}{*} G_v \big).
\]
Define $F(G,Y)$ as the quotient group
\[
F(G,Y) = \tilde{F}(G,Y) \Big/ \langle \bar{y} = y^{-1}, ya^y y^{-1}=a^{\bar{y}} \rangle
\]
for all $y \in \edge(Y)$ and all $a \in G_y=G_{\bar{y}}$.
Let $T$ be a maximal tree in $Y$.
The \emph{fundamental group} of $(G,Y)$ is defined as the factor group, see \cite[p. 42]{Serre:trees}
\[
\pi_1(G,Y,T)=F(G,Y)/\langle y=1 \colon y \in T \rangle.
\]
The image of $y$ in this group is denoted $g_y$.
It is shown in \cite[p. 44]{Serre:trees} that different choices of $T$ yield isomorphic groups.
By \cite[Cor. 1, p. 45]{Serre:trees}, the inclusions $G_v \leq F(G,Y)$, for any $v \in \vertx(Y)$, induce injections
\[
G_v \hookrightarrow \pi_1(G,Y,T).
\]
It also follows from the definitions that for any $y \in \edge Y$ and for any $a \in G_y$
\[
g_y \cdot a^y \cdot g_y{}^{-1} = a^{\bar{y}}.
\]
Thus $g_y$ conjugates the image of $G_y \xto{a \mapsto a^y} G_{t(y)}$ onto the image of $G_y \xto{a \mapsto a^{\bar{y}}} G_{o(y)}$.
\subsection{Homotopy colimits and graphs of groups}\label{subsec:hocolim-groups}
By applying the classifying-space functor $B(-)$ to the functor $\G \colon \C_Y \to \{\Gps\}$ associated to $(G,Y)$ we obtain a functor $B\G \colon \C_Y \to \Top$.
The following result was observed by Dror-Farjoun in \cite[\S 4.1]{DF04}.
\begin{prop
Consider $\G\colon \C_Y \to \Top$ associated with $(G,Y)$.
Then
\[
\hocolim_{\C_Y} B\G \simeq B\pi
\]
where $\pi=\pi_1(G,Y,T)$ for some choice of a maximal tree $T$ and the maps $BG_v \to \hocolim_{\C_Y} B\G \simeq B\pi$ are homotopic to $B(\incl_{G_v}^\pi)$.
\end{prop}
\begin{prop}\label{prop:sylows}
Let $(G,Y)$ be a graph of groups and suppose that
\begin{itemize}
\item[(i)]
The groups $G_v$ and $G_y$ contain Sylow $p$-subgroups $P_v$ and $P_y$ for every $v \in \vertx Y$ and every $y \in \edge Y$.
\item[(ii)]
There exists a vertex $v_0$ such that for any other vertex $u$ of $Y$ there exists a path (directed, without loops) $y_1,\dots,y_n$ from $v_0$ to $u$ such that for any $i$ the map $G_{y_i} \xto{a \mapsto a^{y_i}} G_{t(y_i)}$ carries $P_{y_i}$ onto a Sylow $p$-subgroup of $G_{t(y_i)}$.
\end{itemize}
Set $\pi=\pi_1(G,Y,T)$
and set $S=P_{v_0}$.
Then $\hocolim_{\C_Y} B\G \simeq B\pi$ and
\begin{enumerate}
\item
$S$ is a Sylow $p$-subgroup of $\pi$.
\label{prop:sylows:exist}
\item
If the groups $G_v$ are finite and if $Y$ contains no loops then
\label{prop:sylows:finite}
\begin{enumerate}
\item
$B\pi$ is a $p$-good space (\cite[I.\S 5]{Bous-Kan}), and
\label{prop:sylows:finite:good}
\item
In the kernel $K$ of $H^*(B\pi;\FF_p) \xto{Bi^*} H^*(BS;\FF_p)$ the product of any two elements is zero.
\label{prop:sylows:finite:ker}
\end{enumerate}
\end{enumerate}
\end{prop}
\begin{proof}
\ref{prop:sylows:exist}.
By \cite[I.\S 5.3]{Serre:trees} the group $\pi$ acts without inversions on a tree $\tilde{X}$ such that the isotropy groups of the vertices of $\tX$ are conjugate to the subgroup $G_v$ of $\pi$ where $v \in \vertx Y$.
If $P$ is a finite $p$-subgroup of $\pi$ then by \cite[Prop. 19, \S I.4.3]{Serre:trees} it fixes a vertex of $\tX$ and it is therefore conjugate in $\pi$ to a subgroup of $G_v$ for some vertex $v$.
We therefore assume that $P \leq G_v$.
By hypothesis there is a a path $y_1,\dots,y_n$ from $v_0$ to $v$ with the properties listed in (ii).
We will use induction on $n$ to prove that $P$ is conjugate to a subgroup of $S$.
If $n=0$ then $P \leq G_{v_0}$ and the result holds because $S$ is a Sylow $p$-subgroup of $G_{v_0}$.
Suppose that $n >0$ and note that by hypothesis the image of $G_{y_n}$ in $G_{t(y_n)}=G_v$, which we denote by $G_{y_n}^{y_n}$, contains a Sylow $p$-subgroup.
Thus, by conjugating $P$ inside $G_v$ we may assume that $P \leq G_{y_n}^{y_n}$.
As we have seen in \S\ref{subsec:universal-cover} the element $g_{y_n} \in \pi$ conjugates $G_{y_n}^{y_n}$ onto $G_{y_n}^{\bar{y_n}} \leq G_{o(y_n)} = G_{t(y_{n-1})}$.
We deduce that $P$ is conjugate in $\pi$ to a subgroup of $G_{t(y_{n-1})}$ and we can now apply the induction hypothesis to the path $y_1,\dots,y_{n-1}$ and deduce that $P$ is conjugate in $\pi$ to a subgroup of $S$.
\ref{prop:sylows:finite}
Since $Y$ has no loops, it is a tree.
Orient the tree $Y$ in the only way so that there is a path from $v_0$ to any vertex of $Y$ which only uses edges from the orientation class $A$.
We now claim that by possibly replacing $(G,Y)$ with an isomorphic graph, the groups $P_v$ and $P_y$ can be chosen such that $(P,Y)$ is a sub-graph of $(G,Y)$, namely $G_y \xto{a \mapsto a^y} G_{t(y)}$ carries $P_y$ into $P_{t(y)}$.
In fact, by hypothesis (ii) $P_y \cong P_{t(y)}$ if $y \in A$.
To prove this, consider the poset $\T$ of the subtrees $Y'$ of $Y$ with root $v_0$ for which is it possible to replace $(G,Y)$ with an isomorphic tree of groups and to find Sylow $p$-subgroup $P_v \leq G_v$ and $P_y \leq G_y$ such that $(P,Y')$ is a subtree of $(G,Y')$.
First, $\T$ is not empty because clearly $\{v_0\} \in \T$.
Next, consider a maximal element $Y' \in \T$ and assume that $Y' \neq Y$.
Thus, we may assume, possibly by replacing $(G,Y)$ with an isomorphic tree, that $(P,Y')$ is a subtree of $(G,Y')$.
Choose an edge $y' \in A$ such that $v'=o(y') \in Y'$ and $v=t(y') \notin Y'$.
Let $G_{\bar{y'}}^{\bar{y'}}$ denote the image of $G_{y'}$ in $G_{o(y')}$ and let $P_{\bar{y'}}^{\bar{y'}}$ be the image of $P_{y'}$.
Then $g P_{\bar{y'}}^{\bar{y'}} g^{-1} \leq P_{o(y')}$ for some $g \in G_{o(y')}$ by the Sylow condition.
Let $c_g$ denote the inner automorphism $x \mapsto gxg^{-1}$.
By replacing $G_{y'}$ with $g G_{\bar{y'}}^{\bar{y'}} g^{-1}$ we obtain an isomorph $(\tilde{G},Y)$ of $(G,Y)$.
Now replace $P_{y'}$ with $g P_{\bar{y'}}^{\bar{y'}} g^{-1}$ and replace $P_{t(y')}$ with the image of $P_{y'}$ in $G_{t(y')}=G_v$ which is a Sylow $p$-subgroup by hypothesis (ii).
Note that $o(y') \neq t(y')$ so we have left the trees $(P,Y')$ and $(G,Y')$ unchanged.
By construction $(P,Y' \cup \{v\})$ is a subtree of $(\tilde{G},Y'\cup\{v\})$ which contradicts the maximality of $Y'$.
Thus, $Y \in \T$ which is what we wanted to prove.
\ref{prop:sylows:finite:good}.
Since $Y$ is a tree, $\pi$ is the amalgamated product of finite groups which is generated by the groups $G_v$.
Also notice that the images of $P_v$ in $\pi$ are subgroups of $S$ because $P_y \to P_t(y)$ are isomorphisms for $y \in A$ (consider all that rooted subtrees $Y' \subseteq Y$ for which $P_v \leq S$ and argue as above).
Let $K$ be the normal subgroup of $\pi$ generated by the elements of order prime to $p$.
Clearly $K$ contains $O^p(G_v)$ and therefore together with the groups $P_v$ it generates $\pi$.
Hence $\pi=KS$, so $\pi/K$ is a finite $p$-group.
Now $BK$ is $p$-good by \cite[Proposition VII.3.2]{Bous-Kan} since $K$ is $p$-perfect \cite[VII.3.2]{Bous-Kan} because its abelianisation must be a torsion group with no elements of order $p$.
The Bousfield-Kan spectral sequence $E^2_{ij}=\varinjlim{}_i H_j(B\G;\FF_p) \Rightarrow H_{i+j}(B\pi;\FF_p)$ where $\G$ is the diagram of groups associated to $(G,Y)$, clearly consists of finitely generated $\FF_p$-modules and therefore $H_*(B\pi;\FF_p)$ is finite in every dimension.
The action of $\pi/K$ on $H^*(BK;\FF_p)$ must be nilpotent by \cite[II.5.2]{Bous-Kan} so by the strong convergence of the Eilenberg-Moore spectral sequence for the fibration $BK \to B\pi \to B\pi/K$ proven by Dwyer in \cite{Dwyer73}, we deduce that $H_*(BK;\FF_p)$ is finite in every dimension.
By \cite[Chapter II, 5.1 and 5.2]{Bous-Kan} we obtain a nilpotent fibration $\pcomp{BK} \to \pcomp{B\pi} \to B\pi/K$ in which both the base and the fibre are $p$-complete, whence so is the total space.
This shows that $B\pi$ is $p$-good.
\ref{prop:sylows:finite:ker}.
Let $\G, \P \colon \C_Y \to \Gps$ be the associated functors for $(G,Y)$ and $(P,Y)$.
Observe that $\hocolim_{\C_Y} B\P = BS$ because $P_y \to P_{t(y)}$ are isomorphisms for all $y \in A$ (consider the poset $\T$ of rooted subtrees $Y' \subseteq Y$ for which $\hocolim B\P|_{\C_{Y'}} \simeq BS$ and argue as above).
We obtain a natural transformation of functors $B\P \to B\G$ which gives rise to a morphism of Bousfield-Kan spectral sequences $E^{*,*}_2(\G) \to E^{*,*}_2(\P)$ where
\begin{eqnarray*}
E^{i,j}_2(\P) &=& \varprojlim_{\C_Y}{}^i H^j(B\P;\FF_p) \Rightarrow H^{i+j}(BS;\FF_p), \\
E^{i,j}_2(\G) &=& \varprojlim_{\C_Y}{}^i H^j(B\G;\FF_p) \Rightarrow H^{i+j}(B\pi;\FF_p)).
\end{eqnarray*}
Since $P_v \leq G_v$ and $P_y \leq G_y$ are Sylow $p$-subgroups and $G_v$ are finite groups, the maps $H^*(G_v;\FF_p) \to H^*(P_v;\FF_p)$ and $H^*(G_y;\FF_p) \to H^*(P_y;\FF_p)$ are injective.
Since $\varprojlim{}^0$ is left exact, $E_2^{0,*}(\G) \to E_2^{0,*}(\P)$ is injective.
Also $E_2^{i,*}(\P)=E_2^{i,*}(\G)=0$ for all $i \geq 2$ because $\C_Y$ is $1$-dimensional.
Therefore $E_\infty^{**}=E_2^{**}$.
We obtain the following diagram with exact rows from which it is clear that $K=\ker(Bi^*)$ is contained in $E_\infty^{1,*}(\G)$.
\[
\xymatrix{
0 \ar[r] &
E^{1,*-1}_2(\G) \ar[r] \ar[d] &
H^*(B\pi) \ar[r] \ar[d]_{Bi^*} &
E_2^{0,*}(\G) \ar@{^(->}[d] \ar[r] &
0
\\
0 \ar[r] &
E_2^{0,*-1}(\P) \ar[r] &
H^*(BS)
\ar[r] &
E_2^{0,*}(\P) \ar[r] &
0
}
\]
It follows that $xy=0$ for any $x,y \in K$ because $xy \in E^{2,*}_\infty(\G)=0$.
\end{proof}
\section{Groups inducing fusion systems}\label{sec:examples}
\subsection{Robinson's construction - the normalizer decomposition}\label{subsec:robinsons-construction}
Fix a $p$-local finite group $(S,\F,\L)$ and a collection $\R$ of $\F$-centric subgroups which contains all the $\F$-centric $\F$-radical subgroups (e.g. all the $\F$-centric subgroups of $S$).
Consider the normalizer decomposition $\de \colon \bar{S}(\R) \to \Top$, see \S\ref{subsec:sfs-homotopy-theoretic-constructions}, and recall that in the homotopy category of spaces, $\de$ is isomorphic to the functor $[R_\bullet] \mapsto B\Aut_\L(R_\bullet)$.
Fix fully normalized representatives $R_0,R_1,\dots,R_n$ for the $\F$-conjugacy classes of $\R$ and choose for convenience $R_0=S$.
Let $\D$ be the subposet of $\bar{S}(\R)$ generated by the objects $[R_0],[R_1],\dots,[R_n]$ and $[R_1<S], \dots,[R_n<S]$.
It is easy to check that $\D=\C_Y$ where $Y$ is the tree of height $1$ with root $[S]$ and edges $[R_i<S]$ to the leaves $[R_i]$ ($i=1,\dots n$).
We obtain a tree of groups $(G,Y)$, see \S\ref{subse:sfs-normaliser-decomposition}
\begin{equation}\label{eq:robinson-type-amalgam}
\xymatrix{
&
\Aut_\L(S) \ar@{-}[dl]_{\Aut_\L(R_1<S)} \ar@{-}[d] \ar@{-}[dr]^{\Aut_\L(R_n<S)}
\\
{\Aut_\L(R_1)} \ar@{.}[r] & \Aut_\L(R_i) \ar@{.}[r] & \Aut_\L(R_n)
}
\end{equation}
This tree of groups satisfies the conditions of Proposition \ref{prop:sylows} because the groups $R_i$ are fully normalized so $N_S(R_i)$ is a Sylow $p$-subgroup of both $\Aut_\L(R_i)$ and $\Aut_\L(R_i<S) = N_{\Aut_\L(S)}(R_i)$.
Also note that $S$ is a Sylow $p$-subgroup of $\Aut_\L(S)$.
Therefore $\hocolim B\G \simeq B\pi_R$ for a group $\pi_R$ which is the amalgamated product of the tree \eqref{eq:robinson-type-amalgam} and which contains $S$ as a Sylow $p$-subgroup.
The crux, now, is that $B\G$ is the functor $[R_\bullet] \mapsto B\Aut_\L(R_\bullet)$ and therefore it is isomorphic to $\de|_\D$ in $\hotop$.
Proposition \ref{prop:1-diml-htpy-nat-maps} now yields a composite map $f$
\[
B\pi_R \simeq \hocolim_\D B\G \xto{\simeq} \hocolim_{\D} \de|_\D \xto{\incl} \hocolim_{\bar{S}(\R)} \de \xto{\simeq} |\L| \to \pcomp{|\L| }
\]
whose restriction to $BS$ is homotopic to $\theta$ in display \eqref{eq:theta}.
We now claim that $\F \subseteq \F_S(\pi_R)$.
\begin{proof}[Proof of claim]
First, $\F_S(\pi_R)$ contains all the groups $\Aut_\F(R_i)$ because $\pi_R$ contains $\Aut_\L(R_i)$ for all $i=0,\dots,n$.
Therefore $\F_S(\pi_R)$ contains the fusion system $\F'$ on $S$ generated by $\Aut_\F(R_i)$.
Clearly $\F' \subseteq \F$ and we next claim that, in fact $\F'=\F$.
To do this, we will prove by induction on $|S:P|$ that if $\vp \in \F(P,S)$ then $\vp \in \F'(P,S)$.
If $|S:P|=1$ then $P=S=R_0$ and therefore $\vp \in \Aut_\F(R_0) \subseteq \F'$.
Assume that $|S:P|>1$.
If $P$ is not $\F$-centric or not $\F$-radical then by Alperin's fusion theorem \cite[Theorem A.10]{BLO2} it is a composition of restriction of automorphisms of $\F$-centric $\F$-radical subgroups whose order must be strictly bigger than the order of $P$ and therefore, by the induction hypothesis, all these automorphisms must belong to $\F'$.
Hence $\vp \in \F$.
If $P$ is $\F$-centric $\F$-radical then it is $\F$-conjugate to $R_i$ for some $i$.
We now claim that $P$ is $\F'$-conjugate to $R_i$.
Indeed, fix some $\F$-isomorphism $\al \colon R_i \to P$.
By Alperin's fusion theorem $\al$ is the composition of restrictions of $\F$-automorphisms $\be_1,\dots,\be_k$ of some $\F$-centric subgroups $Q_1,\dots,Q_k$, thus $P=\be_k( \dots (\be_1(R_i)) \dots)$.
If $|Q_j|=|P|$ for some $j$ then $\be_j$ does not move $\be_{j-1}(\dots \be_1(R_i)\dots)$ and by omitting it we obtain another $\F$-isomorphism $R_i \to P$.
Thus, by possibly replacing $\al$ with another $\F$-isomorphism, we may assume that $|Q_1|, \dots,|Q_k| > |P|$, whence by induction hypothesis $\be_1,\dots,\be_k \in \F'$ and therefore $\al \in \F'(R_i,P)$.
Finally, consider $\vp \in \F(P,S)$ where $P$ is $\F$-centric $\F$-radical.
Note that $P$ and $\vp(P)$ are $\F$-conjugate to $R_i$ for some $i$.
Choose $\F'$-isomorphisms $\al' \in \F'(R_i,P)$ and $\be' \in \F'(\vp(P),R_i)$.
Then $\be' \circ \vp \circ \al' \in \Aut_\F(R_i) \subseteq \F'$ and therefore $\vp \in \F'$.
This completes the induction step.
\end{proof}
We summarize this in the following proposition.
\begin{prop}\label{prop:thmA-conditions-R}
Fix a $p$-local finite group $(S,\F,\L)$.
Then the group $\pi_R$ constructed above as the amalgamated product of \eqref{eq:robinson-type-amalgam} contains $S$ as a Sylow $p$ subgroup, it is equipped with a map $f \colon B\pi_R \to \pcomp{|\L|}$ whose restriction to $BS$ is homotopic to $\theta$ and furthermore, $\F \subseteq \F_S(\pi_R)$.
\end{prop}
We can now apply Theorem \ref{thmA}\ref{thmA:induce-fusion} to $\pi_R$ and deduce that $\F=\F_S(\pi_R)$.
Corollary \ref{corC} holds for $\pi_R$ because the implication \ref{corC:2}$\Rightarrow$\ref{corC:1} is a triviality and \ref{corC:1}$\Rightarrow$\ref{corC:2} follows from point \ref{thmA:signaliser} of Theorem \ref{thmA}.
Proposition \ref{propB} for $\pi_R$ follows from Proposition \ref{prop:sylows}\ref{prop:sylows:finite} because the graph of groups \eqref{eq:robinson-type-amalgam} has no loops.
\begin{remark}\label{rem:R-avoid-linking-system}
The groups which appear in the amalgam \eqref{eq:robinson-type-amalgam} can be constructed directly from $\F$ even without the existence of an associated linking system.
Thus, the group $\pi_R$ can be constructed for any saturated fusion system.
Indeed, since the groups $R_i$ are $\F$-centric the normalizer fusion systems $\F_i:=N_\F(R_i)$, see \cite[\S 6]{BLO2}, are constrained in the sense of \cite[Proposition C]{BCGLO}.
Therefore $\F_i$ admit unique linking systems $\L_i$ and there are canonically defined finite groups $L_i$ which contain $N_S(R_i)$ as a Sylow $p$-subgroup such that $\F_i=\F_{N_S(R_i)}(L_i)$.
In fact $L_i= \Aut_{\L_i}(R_i)$.
Similarly $\F_{i,0}:=N_\F(R_i<S)=N_{\F_0}(R_i)$ are constrained fusion sub-systems of $\F_0=N_\F(S)$ and therefore they have unique linking systems $\L_{i,0}$ and groups $L_{i,0}$ such that $\F_{i,0} = \F_{N_S(R_i)}(L_{i,0})$ and $L_{i,0}=\Aut_{\L_{i,0}}(R_i)$.
By construction, every morphism $P \to Q$ in $\F_0$ extends to an automorphism of $S$ and the arguments in \S\ref{subse:sfs-normaliser-decomposition} using the fact that $\hat{e} \in \L(R_i,S)$ is both a monomorphism and and epimorphism, show that there is a canonical isomorphism $\Aut_{\L_0}(R_i)=N_{L_0}(R_i)$.
Since $\F_{i,0}=N_{\F_0}(R_i)$ the uniqueness of the linking system for $\F_{i,0}$ and \cite[Definition 6.1]{BLO2} yield
\[
L_{i,0}= \Aut_{\L_{i,0}}(R_i) = \Aut_{N_{\L_0}}(R_i) = \Aut_{\L_0}(R_i) = N_{L_0}(R_i) \leq L_0.
\]
Observe that $\F_{i,0}$ is a subsystem of $\F_i$ with the same Sylow $N_S(R_i)$ which consists of the morphisms $P \to Q$ which extend to an $\F$-automorphism of $S$.
Let $\K$ denote the collection of all the subgroups of $N_S(R_i)$ which contain $R_i$.
Since both $\F_i$ and $\F_{i,0}$ contain $R_i$ as a normal centric subgroup, $\K$ contains all the $\F_i$-centric-radical subgroups and all the $\F_{i,0}$-centric-radical subgroups.
Define a sub-category $\L_i' \subseteq \L_i$ with object set $\K$ and such that $\L_i'(P,Q)$ is the preimage of $\F_{i,0}(P,Q)$ under the projection $\L_i(P,Q) \to \F_i(P,Q)$.
It is clear that $\L_i'$ is a ``partial'' linking system for $\F_{i,0}$ in the sense that it is only defined on the subgroups in $\K$, but satisfies the axioms for a linking system.
Our argument now follows \cite[Step 7 of Theorem 4.6]{BCGLO2}.
There is a functor $\Z \colon \O(\F_{i,0})^\op \to \Ab$ defined by $P \mapsto Z(P)$.
By \cite[Proposition 3.1]{BLO2}, elements of $\varprojlim{}^3 \Z|_\K$ classify, up to isomorphism, the partial linking systems associated to $\F_{i,0}$ defined on $\K$.
Since $\K$ is closed to over-groups and contains all the $\F_{i,0}$-centric-radicals, then \cite[Proposition 3.2]{BLO2} and well known properties of the $\Lambda$-functors imply that
\[
\varprojlim_{\O(\F_{i,0}^c)}{}^3 \Z \xto{\cong} \varprojlim_{\O(\F_{i,0}^\K)}{}^3 \Z|_{\K}
\]
is an isomorphism.
By \cite[Proposition 4.2]{BCGLO} the left hand side vanishes.
This shows that the partial linking system $\L_i'$ must coincide with $\L_{i,0}^\K$ and in particular
\[
L_{i,0}=\Aut_{\L_{i,0}}(R_i) =\Aut_{\L_i'}(R_i) \leq \Aut_{\L_i}(R_i)=L_i.
\]
We obtain the following tree of groups $(G,Y)$ which is defined purely by $\F$ without any reference to an associated linking system
\begin{equation}\label{eq:R-amalgam-L-not-needed}
\xymatrix{
&
L_0 \ar@{-}[dl]_{L_{1,0}} \ar@{-}[d] \ar@{-}[dr]^{L_{n,0}}
\\
L_1 \ar@{.}[r] & L_i \ar@{.}[r] & L_n.
}
\end{equation}
and define $\pi_{R}$ as its amalgamated product.
By Proposition \ref{prop:sylows} it contains $S \leq L_0$ as a Sylow $p$-subgroup because $N_S(R_i)$ are Sylow $p$-subgroups of $L_i$ and $L_{i,0}$.
In the presence of a linking system for $\L$, this tree of groups coincides with \eqref{eq:robinson-type-amalgam}.
We can now prove that $\F=\F_S(\pi_R)$ using our geometric methods.
The assignment $[R_i] \mapsto \F_i$ and $[R_i<S] \mapsto \F_{i,0}$ define a functor $\nu$ from $\C_Y$ to the category of fusion sub-systems of $\F$.
By applying Ragnarsson's functor $\BB$ we obtain a functor $\BB\nu \colon \C_Y \to \spectra$ together with a map $\BB \nu \to \BB\F$.
Since $\F_i$ and $\F_{i,0}$ are constrained, $\BB \nu \simeq \Sigma^\infty \pcomp{B\G}$ for the functor $\G$ associated to the tree \eqref{eq:R-amalgam-L-not-needed}.
By adjunction we obtain $B\G \to \Omega^\infty\BB \F$ in $\hotop$ and by Propositions \ref{prop:1-diml-htpy-nat-maps} and \ref{prop:sylows} we obtain $f \colon B\pi_R \to \Omega^\infty\BB \F$ whose restriction to $BS$ is homotopic to $\si_\F^\#$.
Proposition \ref{prop:stable-fusion} implies that $\F_S(\pi_R) \subseteq \F$ and the argument we used above exploiting Alperin's fusion theorem shows that $\F \subseteq \F_S(\pi_R)$.
\end{remark}
\begin{remark}\label{rem:robinson-original-vs-pir}
Robinson's originally constructed in \cite{Robinson07} a group $\Gamma$ as the amalgamated product \eqref{eq:R-amalgam-L-not-needed} but with the groups $L_{i,0}$ replaced with their Sylow $p$-subgroup $N_S(R_i)$.
The group $\pi_R$ that we construct is therefore a quotient of $\Gamma$.
Note however that if $\pi_R$ admits a signaliser functor, then $\F$ has an associated linking system and we can apply Theorem \ref{thmA} to $B\Gamma \to B\pi_R \to \pcomp{|\L|}$ in order to obtain a signaliser functor on $\Gamma$.
\end{remark}
\begin{remark}\label{rem:lib-vir-construction}
In \cite[Theorem 5.5]{Lib-Vir} it is shown that a $p$-local finite group with a ``compressible $\F$-centric'' collection $\R$ admits a group $\Gamma$ which contains $S$ as a Sylow $p$-subgroup with the property that $\F=\F_S(\Gamma)$ and that $\L$ is induced by some signaliser functor on $\Gamma$.
The group $\Gamma$ is simply the group $\pi_R$.
Our techniques are inspired by this paper.
\end{remark}
\subsection{Leary-Stancu's construction - the subgroup decomposition}\label{subsec:Leary-Stancu}
The following argument is given explicitly in \cite{Seeliger}.
Fix a $p$-local finite group $(S,\F,\L)$ and let $\Phi=\{\vp_1,\dots,\vp_n\}$ be a set of morphisms $\vp_i \colon P_i \to S$ in $\F^c$ which, together with $\Inn(S)$, generate $\F$ (recall Alperin's fusion theorem).
For example, $\Phi$ is the set of all the morphisms in $\F^c$.
Consider the oriented graph $Y$ with one vertex $v_0$ and oriented edges $y_1,\dots,y_n$.
Define a grapg of group $(G_,Y)$ where $G_{v_0}=S$, $G_{y_i}=G_{\bar{y_i}}=P_i$ and $y_i \mapsto P_i \xto{\vp_i} S$ and $\bar{y_i} \mapsto P_i \xto{\incl} S$.
Thus, the functor $\G \colon \C_Y \to \Gps$ is depicted by
\[
\xymatrix{
{P_1} \ar@/^0.3pc/[dr]^{\vp_1} \ar@/_0.3pc/[dr]_{\incl} & \cdots & {P_n} \ar@/^.3pc/[dl]^{\incl} \ar@/_0.3pc/[dl]_{\vp_n}
\\
& S &
}
\]
Define a functor $J \colon \C_Y \to \O(\F^c)$ by
\[
J(v_0)=S, \qquad
J([y_i])=P_i, \qquad
J(\bar{y_i})= P_i \xto{\incl} S, \qquad
J(y_i)= P_i \xto{\vp_i} S.
\]
Define $\pi_{LS}:=\pi_1(G,Y,T)$ where the maximal tree $T$ must be $\{ v_0\}$.
Observe that $J^*\be \cong B\G$ in $\hotop$ where $\be$ is the subgroup decomposition, see \S\ref{subsec:sfs-subgroup-decomposition}.
Now we obtain from Proposition \ref{prop:sylows}, whose conditions are satisfied trivially, and Proposition \ref{prop:1-diml-htpy-nat-maps} and the properties of $\be$ described in \S\ref{subse:sfs-normaliser-decomposition}, a sequence of maps
\[
B\pi_{LS} \simeq \hocolim_{\C_Y}B\G \xto{\simeq} \hocolim_{\C_Y} J^*\be
\xto{J_*} \hocolim_{\O(\F^c)} \be \xto{\simeq} |\L| \to \pcomp{|\L|}
\]
whose restriction to $BS$ is homotopic to $\theta$.
We also observe that $\F \subseteq \F_S(\pi_{LS})$ because $S \leq \pi_{LS}$ and by construction of $\pi_1(G,Y,T)$, see \S\ref{subsec:universal-cover}, every $\vp_i \colon P \to S$ in $\Phi$ is obtained by conjugation by the element $g_{y_i}$.
To summarize, we have shown:
\begin{prop}\label{prop:ls-thmA-conditions}
Fix a $p$-local finite group $(S,\F,\L)$.
Then the group $\pi_{LS}$ contains $S$ as a Sylow $p$-subgroup, it is equipped with a map $f \colon B\pi_{LS} \to \pcomp{|\L|}$ whose restriction to $BS$ is homotopic to $\theta$ and $\F \subseteq \F_S(\pi_{LS})$.
\end{prop}
Thus, we can apply Theorem \ref{thmA}\ref{thmA:induce-fusion} to recover the results of \cite{Leary-Stancu07}, namely $\F=\F_S(\pi_{LS})$.
Also Corollary \ref{corC} for $\pi_{LS}$ follows since the implication \ref{corC:2}$\Rightarrow$\ref{corC:1} is a triviality and \ref{corC:1}$\Rightarrow$\ref{corC:2} follows from point \ref{thmA:signaliser} of Theorem \ref{thmA}.
\begin{remark}
The construction of $\pi_{LS}$ appears in \cite{Leary-Stancu07}.
It is clear that it is defined purely in terms of $\F$.
The same ideas of Remark \ref{rem:R-avoid-linking-system} can be used to obtain $\F=\F_S(\pi_{LS})$ for any saturated fusion system without the assumption that a linking system exists.
\end{remark}
\subsection{Aschbacher-Chermak's construction - the normalizer decomposition}\label{subsec:Aschbacher-Chermak}
Fix an odd prime $q$ and some $n \geq 1$.
We now recall from \cite{LO1}, see also \cite{LO2}, the construction of Solomon's fusion system $\F_{\Sol}(q^n)$ at the prime $2$.
This fusion system is obtained from the fusion system of $\Spin_7(q^n)$ as follows.
Throughout we write $q^n$ for the finite field $\FF_{q^n}$ and $q^\infty$ for its algebraic closure.
There is a group homomorphism $\omega \colon \SL_2(q^\infty)^3 \to \Spin_7(q^\infty)$ whose kernel is the central subgroup of order $2$ generated by $(-I,-I,-I)$.
We will denote $\omega(X_1,X_2,X_3)=\trp[ X_1,X_2,X_3]$.
Let $H(q^\infty)$ be the image of $\omega$.
The centre of $\Spin_7(q^\infty)$ is a group of order $2$ generated by $z=\trp[ I,I,-I]$.
The set $\{ \trp[\pm I,\pm I, \pm I]\}$ forms an elementary abelian group $U$ of rank $2$.
By \cite[Lemma 2.3]{LO1} there is an element $\tau \in N_{\Spin_7(q)}(U)$ such that $c_\tau(\trp[ X_1,X_2,X_3])=\trp[X_2,X_1,X_3]$ where $c_\tau$ denotes conjugation $g \mapsto \tau g \tau^{-1}$.
Note that $\tau$ normalizes $H(q^{\infty})$.
Fix a copy of the quaternion group $Q_8$ in $\SL_2(q)$ and choose generators $A,B$ of order $4$.
Set (cf. \cite[\S 1]{LO2})
\begin{eqnarray*}
C(q^\infty) &=& \{ X \in C_{\SL_2(q^\infty)}(A) \colon X^{2^k}=I \text{ for some $k$}\} \cong \ZZ/2^\infty \\
Q(q^\infty) &=& \langle C(q^\infty),B \rangle \\
S_0(q^\infty) &=& \omega(Q(q^\infty)^3) \leq H(q^\infty) \\
S(q^\infty) &=& S_0(q^\infty) \cdot \langle \tau \rangle.
\end{eqnarray*}
We will write $\Theta(q^n)=\Theta(q^\infty) \cap \Spin_7(q^n)$ where $\Theta$ denotes one of $C(-),Q(-),S_0(-)$ or $S(-)$ etc.
By \cite[Lemma 2.7]{LO1}, $S(q^n)$ is a Sylow $2$-subgroup of $\Spin_7(q^n)$ hence also of $H(q^n) \cdot \langle \tau \rangle$.
Also $S_0(q^n)$ is a Sylow $2$-subgroup of $H(q^n)$.
There are automorphisms $\hat{\ga}$ and $\hat{\de}_u$ of $S_0(q^\infty)$ defined in \cite[Definition 1.6]{LO2} by
\[
\hat{\ga}(\trp[ X_1,X_2,X_3])=\trp[ X_3, X_1, X_2], \qquad
\hat{\de}_u(\trp[ X_1,X_2,A'B^j])=\trp[ X_1, X_2, (A')^uB^j ]
\]
where $A' \in C(q^\infty)$ and $X_1,X_2,X_3 \in Q(q^\infty)$ and $u$ is a carefully chosen $2$-adic unit.
Set
\[
\Gamma_n = \langle \Inn(S_0(q^n)) , c_\tau, \hat{\de}_u \circ \hat{\ga} \circ \hat{\de}_u^{-1} \rangle \leq \Aut(S_0(q^n)).
\]
The fusion system $\F_{\Sol}(q^n)$ on $S=S(q^n)$ is the fusion system generated by $\F_{S}(\Spin_7(q^n))$ and by $\F_{S_0(q^n)}(\Ga_n)$.
Set $S=S(q^n)$ and $S_0=S_0(q^n)$.
It easily follows from \cite[Proposition 2.5]{LO1} and the definitions that $C_S(U)=S_0$.
Also, $|S \colon S_0|=2$ because $|S(q^\infty) \colon S_0(q^\infty)|=2$ and therefore $S_0$ is fully normalized.
It is also $\F$-centric by the next proposition because by construction of $\F$, the $\F$-conjugates of $S_0$ are $\Spin_7(q^n)$-conjugates.
\begin{prop}\label{prop:spin-cent-s0}
$C_{\Spin_7(q^n)}(S_0(q^n)) \subseteq S_0(q^n)$.
\end{prop}
\begin{proof}
Since $U \subseteq S_0$ we deduce from \cite[Proposition 2.5]{LO1} that $C_{\Spin_7(q^n)}(S_0) \subseteq C_{\Spin_7(q^n)}(U)=H(q^n)$ which contains $S_0$ as a Sylow $2$-subgroup and therefore $C_{\Spin_7(q^n)}(S_0)=Z(S_0) \times C'$ where $C'$ has odd order.
We need to show that $C'=1$.
Choose $x \in C'$ of order $r$.
Since $\ker \omega$ is central of order $2$ then $\omega^{-1}(x)$ consists of an element $\tilde{x}_1$ of order $r$ and an element $\tilde{x_2}$ of order $2r$.
Now $(Q_8)^3 \subseteq \omega^{-1}(S_0)$ must leave $\{\tilde{x}_1,\tilde{x}_2\}$ invariant because $S_0$ centralizes $x$ and since the orders of $\tilde{x}_1,\tilde{x}_2$ are different $(Q_8)^3$ centralizes these elements.
By Lemma \ref{lem:centr-q8-in-sl2q} below $\tilde{x}_1 \in Q_8$ and therefore $\tilde{x}_1=1$ which implies that $x=1$.
\end{proof}
\begin{lem}\label{lem:centr-q8-in-sl2q}
Fix an odd prime $q$.
Then
\begin{enumerate}
\item
The centralizer of any element $X$ of order $4$ in $\SL_2(q^n)$ is a cyclic group.
\item
Any quaternion group $Q_8$ in $\SL_2(q^\infty)$ contains its centralizer.
\end{enumerate}
\end{lem}
\begin{proof}
(1)
The minimal polynomial of $X$ is $x^2+1$ which splits in $\FF_{q^{n+1}}$ which contains $\zeta=\sqrt{-1}$.
Thus, $X$ is conjugate in $\GL_2(q^{n+1})$ to the diagonal matrix $\diag(\zeta,-\zeta)$ whose centralizer is the subgroup of diagonal matrices and therefore its $\SL_2(q^{n+1})$ centralizer is the cyclic group $\FF_{q^{n+1}}^\times$.
Conjugating back we see that $C_{\SL_2(q^{n+1})}(X)$ is cyclic hence so is $C_{\SL_2(q^{n})}(X)$.
\noindent
(2)
Fix any $n$ such that $Q_8 \leq \SL_2(q^n)$ and consider $X$ in its centralizer.
Now $Q_8$ is generated by elements $A, B$ of order $4$, so $X$ belongs to the cyclic groups $C_{\SL_2(q^n)}(A)$ and $C_{\SL_2(q^n)}(B)$.
If $\langle A \rangle \subseteq \langle X \rangle$ and $\langle B \rangle \subseteq \langle X \rangle$ then $Q_8 \leq \langle X \rangle$ which is absurd.
So either $X \in \langle A \rangle$ or $X \in \langle B \rangle$.
\end{proof}
Set $\F=\F_{\Sol}(q^n)$ and $\L=\L_{\Sol}(q^n)$ and $Z$ is the subgroup generated by the element $z$ (this is the centre of $\Spin_7(q^n)$).
Set $\Aut_\L(S_0;Z)=C_{\Aut_\L(S_0)}(Z)$.
By \cite[Propositions 2.11, 3.3]{LO1} the linking system $\L_{\Sol}(q^n)$ contains $\L^c_S(\Spin_7(q^n))$ as a subcategory.
In fact, $\L^c_S(\Spin_7(q^n)) = C_\L(Z)$ is the centralizer linking system \cite[Definition 2.4]{BLO2} and by Proposition \ref{prop:spin-cent-s0} and \cite[remarks before Definiton 1.6]{BLO2}
\begin{multline*}
\Aut_\L(S_0;Z)=\Aut_{C_\L(Z)}(S_0) = N_{\Spin_7(q^n)}(S_0)/C'_{\Spin_7(q^n)}(S_0)
\\
= N_{\Spin_7(q^n)}(S_0) \leq \Spin_7(q^n).
\end{multline*}
We obtain inclusions of subcategories of $\L$
\[
\xymatrix{
\L_S(\Spin_7(q^n)) &
\B\Aut_\L(S_0;Z) \ar@{^(->}[r] \ar@{_(->}[l] &
\B\Aut_\L(S_0)
}
\]
It gives rise to the diagram of spaces below, all of whose squares commute, except the one on the top left which only commutes up to homotopy.
\[
\xymatrix{
B\Spin_7(q^n) \ar[d]_{\text{$\mod 2$ equivalence}} &
BN_{\Spin_7(q^n)}(S_0) \ar[r] \ar[l] \ar [d] &
B\Aut_\L(S_0) \ar[d]
\\
|\L_S(\Spin_7(q^n))|^{\wedge}_2 \ar[d] &
B\Aut_\L(S_0;Z)^{\wedge}_2 \ar[r] \ar[l] \ar [d] &
B\Aut_{\L}(S_0)^{\wedge}_2 \ar[d]
\\
|\L|^{\wedge}_2 &
|\L|^{\wedge}_2 \ar@{=}[r] \ar@{=}[l] &
|\L|^{\wedge}_2.
}
\]
We observe that $N_{\Spin_7(q^n)}(S_0)$ contains the Sylow subgroup $S$ because $|S \colon S_0|=2$.
By Proposition \ref{prop:sylows}, the homotopy colimit of the first row is homotopy equivalent to $B\pi_{AC}$ where $\pi_{AC}$ is the amalgam of the tree
\begin{equation}\label{eq:AC-amalgam}
\xymatrix@1{
{\Spin_7(q^n)} \ar@{-}[rrr]^{N_{\Spin_7(q^n)}(S_0)} & & & \Aut_\L(S_0)
}
\end{equation}
which contains $S$ as a Sylow $2$-subgroup.
By Proposition \ref{prop:1-diml-htpy-nat-maps} there is a map $f \colon B\pi_{AC} \to |\L|{}^\wedge_2$ whose restriction to $BS$ is homotopic to $\theta$.
Moreover, $\Spin_7(q^n)$ and $\Aut_\L(S_0)$ are subgroups of $\pi_{AC}$ and also $\Aut_\F(S_0)=\Aut_{\Aut_\L(S_0)}(S_0)$ contains $c_\tau$ and $\hat{\de}_u \circ \hat{\ga} \circ \hat{\de}_u^{-1}$ because these homomorphisms clearly normalize $S_0(q^n)$.
We therefore deduce that $\F \subseteq \F_S(\pi)$.
Thus we have proven:
\begin{prop}\label{prop:ac-thmA}
The group $\pi_{AC}$ satisfies the hypotheses of Theorem \ref{thmA}.
In particular $\F=\F_S(\pi_{AC})$ and $\pi_{AC}$ has a signaliser functor which induces $\L$.
It also satisfies the condition of Proposition \ref{propB} because \eqref{eq:AC-amalgam} has no loops.
\end{prop}
\begin{remark}\label{rem:improved-ac}
The group $\Gamma$ that Aschbacher and Chermak constructed in \cite{Asch-Chermak} is the amalgamated product
\[
\xymatrix@1{
{\Spin_7(q^n)} \ar@{-}[rrr]^{N_{\Spin_7(q^n)}(U)} & & & K
}
\]
where $|K \colon N_{\Spin_7(q^n)}(U)|=3$.
They need the hypothesis that $q \cong 3 \text{ or } 5 \mod 8$, however, which we don't.
Indeed, the groups $\Gamma$ and $\pi_{AC}$ are different.
We will show below that $U$ is the unique normal subgroup of $S_0=S_0(q^n)$ which is isomorphic to the Klein group $\ZZ/2\oplus \ZZ/2$ and therefore $N_{\Spin_7(q^n)}(S_0) \subseteq N_{\Spin_7(q^n)}(U)$.
The inclusion is proper because $H(q^n)=C_{\Spin_7(q^n)}(U)$ does not normalize $S_0(q^n)$.
In addition, $\Aut_{\Spin_7(q^n)}(U)=C_2$ and $\Aut_{\F_{\Sol}(q^n)}(U)=\Si_3$ and $\Aut_{\F_{\Sol}(q^n)}(S_0)$ contains $\hat{\de}_u\hat{\ga}\hat{\de}_u^{-1}$ from which we deduce that $|\Aut_\L(S_0) \colon N_{\Spin_7(q^n)}(S_0)|=3$.
Thus the groups we use in the amalgam \eqref{eq:AC-amalgam} are smaller than those used by Aschbacher and Chermak.
It is interesting to see how our approach interprets the signaliser functors geometrically in Theorem \ref{thmA} rather than algebraically.
\begin{claimx}
$U$ is the unique normal subgroup of $S_0(q^n)$ isomorphic to the Klein group.
\end{claimx}
\begin{proof}
Consider a Klein group $E \lhd S_0$ and set $\tilde{E}=\omega^{-1}(E)$.
Note that $\tilde{E}$ is normalized by $(Q_8)^3 \leq Q(q^\infty)^3$.
Fix an element $\tilde{X}=(X_1,X_2,X_3) \in \tilde{E}$ and note that $X_i^2=\pm I$ because all the elements of $E$ have order $2$ and $\ker \omega=\pm(I,I,I)$.
If $X_i$ has order $4$ then $X_1^2=X_2^2=X_3^2=-I$ so $X_1,X_2,X_3$ have order $4$ and Lemma \ref{lem:centr-q8-in-sl2q} implies that $\tilde{E}$ contains at least $2^3$ different $(Q_8)^3$ conjugates of $\tilde{X}$.
This is impossible, so the order of $\tilde{X}$ is at at most $2$.
If $E \neq U$ then we may assume that $X_1 \neq \pm I$ hence $X_1=WB^{\pm 1}$ for some $W \in C(q^\infty)$.
Now it follows that $\tilde{X} \cdot B\tilde{X} B^{-1} = (-W^2,I,I) \in \tilde{E}$ must also have order $2$.
This can only happen if $W \in \langle A \rangle \leq C(q^\infty)$ which implies that $X_1 \in Q_8$ whose only element of order $2$ is $-I$ and this is absurd.
\end{proof}
\end{remark}
\begin{remark}
The construction which we have presented here is motivated by the normalizer decomposition with respect to the collection of the elementary abelian subgroups $\E$, see \cite{Lib06}.
If $\D$ is the subposet of $\bar{S}(\E)$ of the form $[Z] \leftarrow [Z < U] \to [U]$ then the restriction of the normalizer decomposition to $\D$ has the form
\[
|C_\L(Z)| \longleftarrow |C_\L(U)|_{h \Aut_\F(Z<U)} \longrightarrow |\C_\L(U)|_{h\Aut_\F(U)}.
\]
Up to $2$-completion, the space on the left is $B\Spin_7(q^n)$.
A careful examination of the constructions in \cite[Theorem B]{Lib06} shows that these maps are induces by inclusion of categories which contain $\B\Aut_{C_\L(Z)}(S_0)$ and $\B\Aut_\L(S_0)$ as very natural full subcategories to look at.
We omit the details.
\end{remark}
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The 2008 World Wheelchair Curling Championship was held from February 2 to 9 at the Eishalle Sursee in Sursee, Switzerland.
Qualification
(Host country)
Top seven finishers from the 2007 World Wheelchair Curling Championship (not including host):
Top teams from qualifying event:
Qualification event
Two teams outside of the top finishers, Sweden and Italy, qualified from a qualifying event held in November 2007 in Inverness, Scotland.
Teams
Round-robin standings
Round-robin results
Draw 1
Saturday, February 2, 18:00
Draw 2
Monday, February 3, 10:00
Draw 3
Monday, February 3, 15:30
Draw 4
Tuesday, February 4, 10:00
Draw 5
Tuesday, February 4, 15:30
Draw 6
Wednesday, February 5, 10:00
Draw 7
Wednesday, February 5, 15:30
Draw 8
Thursday, February 6, 10:00
Draw 9
Thursday, February 6, 15:30
Tiebreaker
Thursday, February 7, 13:30
Playoffs
1 vs. 2 Game
Friday, February 8, 10:00
3 vs. 4 Game
Friday, February 8, 10:00
Semifinal
Friday, February 8, 15:30
Bronze medal game
Saturday, February 9, 10:00
Gold medal game
Saturday, February 9, 11:00
External links
Results at Curlit.com
World Wheelchair Curling Championship
2008 in curling
2008 in Swedish sport
International curling competitions hosted by Sweden
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"redpajama_set_name": "RedPajamaWikipedia"
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Q: Nutch 1.14 in windows giving exception java.lang.UnsatisfiedLinkError: org.apache.hadoop.io.nativeio.NativeIO$Windows.access0(Ljava/lang/String;I)Z While I am running Apache Nutch 1.14 I am getting following exception.
Injector: starting at 2018-07-08 10:15:56
Injector: crawlDb: crawl/crawldb
Injector: urlDir: urls
Injector: Converting injected urls to crawl db entries.
Exception in thread "main" java.lang.UnsatisfiedLinkError: org.apache.hadoop.io.nativeio.NativeIO$Windows.access0(Ljava/lang/String;I)Z
at org.apache.hadoop.io.nativeio.NativeIO$Windows.access0(Native Method)
at org.apache.hadoop.io.nativeio.NativeIO$Windows.access(NativeIO.java:609)
at org.apache.hadoop.fs.FileUtil.canRead(FileUtil.java:977)
at org.apache.hadoop.util.DiskChecker.checkAccessByFileMethods(DiskChecker.java:187)
I have installed Java and put hadoop native libraries(i.e, winutils.exe) in c:\winutil\bin and pointed to HADOOP_HOME.
Not to sure how to resolve it and cannot find any documentation how to run Nutch 1.14 in windows. If any one has solution please let me know.
A: The only way I have found to get around that error is from this thread:
Nutch on windows: ERROR crawl.Injector
The key is to comment out the following line:
JAVA_LIBRARY_PATH="`cygpath -p -w "$JAVA_LIBRARY_PATH"`"
Read the README.txt under $NUTCH_HOME/lib/native for more info regarding hadoop binaries which is what the JAVA_LIBRARY_PATH is referencing.
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"redpajama_set_name": "RedPajamaStackExchange"
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FAS | Military | DOD 101 | Systems | Ships ||||
Index | Search | Join FAS
AOE-1 Sacramento
Fast Combat Support Ship
The fast combat support ship (AOE) is the Navy's largest combat logistics ship. The AOE has the speed and armament to keep up with the carrier battle groups. It rapidly replenishes Navy task forces and can carry more than 177,000 barrels of oil, 2,150 tons of ammunition, 500 tons of dry stores and 250 tons of refrigerated stores. It receives petroleum products, ammunition and stores from shuttle ships and redistributes these items simultaneously to carrier battle group ships. This reduces the vulnerability of serviced ships by reducing alongside time.
The first two ships of the class, USS SACRAMENTO (AOE 1), USS CAMDEN (AOE 2), are assigned to the US Pacific Fleet. USS SEATTLE (AOE 3) and USS DETROIT are assigned to the US Atlantic Fleet.
The concept of creating a one station supply ship to perform the functions of three, to have the speed and be equipped with the weapons, sensors and communications equipment necessary to operate as an integral part of the battle group originated under the then Chief of Naval Operations, Admiral Arleigh Burke who embraced the concept as an answer to the logistics problems he had encountered during World War II. Previously battle group underway replenishments had to scheduled well in advance due to communications problems and these unreps were subject to change especially due to weather and tactical situation changes; The Underway Replenishment Groups which provided the critical supplies were slow, unwieldy and never seemed to be there when needed. To counter these problems the multi-product station ship concept originated and the Fast Combat Support ship (AOE) was designed and funded. When built Sacramento carried more fuel than the largest oiler and more ammunition than the largest ammunition ship in the US Navy as well as a significant portion of the supplies that a stores ships could carry.
The USS SACRAMENTO class ships combine the functions of three logistic support ships in one hull - fleet oiler (AO), ammunition ship (AE), and refrigerated stores ship (AF). As an oiler (AO) it carries (95% fully loaded) 5.2 million gallons of marine distillate fuel and 2.7 million gallons of aviation jet fuel. About 60-million gallons are pumped each year from this floating filling station to other ships. As an ammunition ship (AE), SACRAMENTO can replenish the entire ordnance requirements of an aircraft carrier in three to four hours. It also carries a complete assortment of missiles and ammunitions for cruisers and destroyers with a capacity of approximately six-thousand tons when fully loaded. In the capacity of a refrigeration ship (AF), SACRAMENTO carries over one-thousand tons of frozen, chilled, and dry food items. Mail is as important as food to sailors. During deployment the SACRAMENTO delivers approximately 100-thousand pounds of mail to forces at sea. Sacramento is capable of carrying over 300 tons of refrigerated provisions, 500 tons of dry provisions and 150 tons of other supplies. While not normally loaded, Sacramento carries 220 different items (dry and refrigerated) and 120 other items when deployed.
These multi-commodity, fast combat support ships have brought an increased capability to the Fleet for the underway replenishment of provisions, combat stores, ordnance and petroleum products. The advanced design of their replenishment facilities and sophisticated cargo-handling equipment effectively accomplishes the rapid transfer of combat consumables at sea by providing "one-stop shopping" for customer ships. In addition, because of their size, these ships also have facilities for limited ship repair and maintenance service, as well as, for other special assignments.
This capability for providing on-station combat logistic support contributes to the tactical effectiveness of Carrier Battle Groups or Amphibious Ready Groups operating in any theater of operation. As an example, one AOE can provide integral logistic support for one conventionally-fueled, large-decked aircraft carrier with a full air wing embarked, and three gas turbine powered guided missile escort cruisers or destroyers steaming 12,000 miles, flying 4,000 hours, and reloading a full magazine of ammunition for each.
SACRAMENTO is designed to operate with strike forces necessitating fast handling of cargo. A system of elevators and package conveyors permits ready access to ammunition and cargo holds. SACRAMENTO operates a fleet of fork lift trucks, allowing breakout and positioning of cargo and ammunition with greater efficiency.
Sophisticated automatic cargo equipment like the Standard Tensioned Replenishment Alongside Method (STREAM), provide rapid transfer of materials from SACRAMENTO to its customers. Fuel is delivered through four double and two single hose (STREAM) fuel lines which are fed by large turbine driven cargo fuel pumps capable of supplying fuel at 40 to 120 psi at a maximum flow rate of three-thousand gallons per minute per pump used. This process is termed FAS for Fueling At Sea.
Cargo, ammunition and supplies are moved vertically over seven decks (weather deck, cargo handling and staging deck and five levels of cargo holds) by seven different elevators. Elevators one through six service the four ammunition holds and number seven elevator and a package conveyor service the provisions hold (hold five). The supplies are moved horizontally by a fleet of 33 fork trucks of various types and sizes which allow breakout, positioning and stowage of cargo and ammunition with efficiency.
Cargo is delivered by four heavy and three standard lift STREAM cargo stations or by a detachment of two CH-46 helicopters which deploy with the Sacramento. The process of resupplying a ship alongside by means of a fuel or cargo station is termed CONREP for connected replenishment.
Vertical replenishment (VERTREP), adds another dimension to SACRAMENTO's ability to transfer freight, mail, personnel, provisions and ammunition. Two Boeing CH-46 jet turbine helicopters (Sacramento has hangar space for three), with replenishment rates of up to 6-thousand pounds per minute, allow delivery to customer ship flight decks. The VERTREP capability allows Sacramento to resupply ships in a dispersed formation, reduces alongside replenishment time, and allows transfer of cargo over greater distances. Additionally, SACRAMENTO can accept other helicopters, for VERTREP support.
To support the crew, the ship must be self sufficient in every respect. Personnel onboard have enough expertise in a variety of technical skills to maintain and operate all shipboard equipment. The ship is in essence a floating city and provides services to support and accommodate the crew.
The nearly 600 officers and crew are divided into seven departments with each performing a special type of work, coordinated to perform the mission of the ship. Those departments are: Deck, Engineering, Operations, Supply, Medical. Air and Administration. These trained professionals maintain, repair, and operate all shipboard systems. With medical and dental facilities, lounge, gymnasium and library, Sacramento provides for its crew who perform their duties day and night providing fuel, food, supplies, ammunition and mail to the striking forces of the fleet.
Sacramento is something of a bench-mark in west coast shipbuilding. It and two of its sister ships, Seattle and Detroit, are the largest ships ever built on the West Coast. Only the lowa class battleships and aircraft carriers have greater displacements than Sacramento. Sacramento's main engines came from the never completed battleship Kentucky and deliver in excess of 100-thousand shaft horsepower to the two 23-foot screws, the largest of any ship in the U.S. Navy.
It was originally armed with four, three-inch, 50 caliber guns. The two forward guns which were mounted forward of the bridge on the 01 level were replaced by the NATO Seapsparrow Missile System during the ships second regular overhaul in 1976 and the after gun mounts which were located aft of the stacks on the 04 level were replaced by two Vulcan/Phlanx mounts during the next overhaul in 1981. Sacramento's cargo handling methods and equipment have changed somewhat since 1964 as well. A new metal dunnage system was installed during its 1981 overhaul as were satellite communications and navigation systems and the Super Rapid Blooming Off-Board Chaff (SRBOC) system.
Sacramento was originally equipped with the first Fast Automated Shuttle Transfer (FAST) system to handle ammunition and stores. FAST relied on the use of cranes to move missiles and cargo containers on weather decks and hoists to move cargo from the 01 level to the main deck. In 1977 Sacramento's elevators were modified to lift cargo directly to the 01 level and the FAST cranes, transfer rails, and hoists were removed. the new system allowed cargo to be moved from hold to elevator to transfer station using standard electric fork truck and is far less prone to equipment malfunctions. In addition the modern Standard Tensioned Replenishment Alongside Method (STREAM) unrep system was installed.
The USS Detriot (AOE-4), a fast combat support ship of the Sacramento (AOE-1) class which is homeported near Philadelphia in Earle NJ, underwent a phased maintenance availability from May through October 1993. The repairs and alterations required 48,000 mandays and over $15 million. The work package included hull, mechanical and electrical work, but was much less intensive in combat systems and electronics than is the work on a combatant ship.
Propulsion plant 2 - turbines, producing 50,000 horsepower at 4,829 rpm
Boilers 4 - boilers - 600 psi at 856oF super-heated steam
Main Condenser cooling water flow rate is 50,000 gallons per minute
Main Reduction Gear Diameter 14 feet, 4 inches
FUEL CONSUMPTION 9,527 Gallons-Per-Hour with all boilers at 100%
Propeller two 23 foot diameter propellers with 6 blades
Length: Overall Length: 795 ft
Waterline Length: 770 ft
Beam: Extreme Beam: 107 ft
Waterline Beam: 107 ft
Draft Maximum Navigational Draft: 39 ft
Draft Limit: 41 ft
Displacement: 53,000 tons full load
Light Displacement: 20144 tons
Full Displacement: 51964 tons
Dead Weight: 31820 tons
Speed: 26 knots (30 miles, 48 km, per hour)
CARGO CAPACITY LIQUID CARGO CAPACITY 8,219,387 Gallons
ORDNANCE CAPACITY 3,000 Tons
PROVISIONS CAPACITY 675 Tons
Aircraft: 2 - CH-46E Sea Knight helicopters
BOATS 2 26' Motor Whale Boats
2 33' Personnel Boats
ANCHORS 2 12-1/2 tons (1,480 of Chain)
CRANES 1 15-Ton (Capacity)
Armament: NATO Sea Sparrow missiles
2 - Phalanx close-in weapons systems
Crew: 24 officers, 576 enlisted
Unit Cost: $458-568 million
Unit Operating Cost
Annual Average ~$37,000,000 [source: [FY1996 VAMOSC]
Name Number Builder Homeport Ordered Commissioned Decommissioned
Sacramento AOE-1 Puget Sound NSY Bremerton 08 Aug 1960 14 Mar 1964
Camden AOE-2 New York SB Bremerton 25 Apr 1963 01 Apr 1967
Seattle AOE-3 Puget Sound NSY Earle 29 Dec 1964 05 Apr 1969
Detroit AOE-4 Puget Sound NSY Earle 29 Dec 1965 28 Mar 1970
AOE-5 1968 cancelled 04 Nov 1968
FY1996 Ships Class Average Report Navy Visibility and Management of Operating and Support Costs (VAMOSC)
http://www.fas.org/man/dod-101/sys/ship/aoe-1.htm
Maintained by Robert Sherman
Originally created by John Pike
Updated Friday, March 05, 1999 5:24:28 PM
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Q: "cranial" vs "cerebral" Is there a substantial difference between the adjectives "cranial" and "cerebral"? Are these ones interchangeable in a not-so-medical context?
A: If you go by the meaning of their nominal cognates, there is a substantial difference: the adjective "cranial" derives from Latin "cranium" (skull) while the adjective "cerebral" is a derivate of the Latin "cerebrum" (brain).
But I am aware that this isn't your question. - What contexts are you thinking of?
I can think of at least one not-so-medical context where both of these adjectives are not interchangeable: a paleontologist speaks of "cranial fossils", he wouldn't speak of "cerebral fossils". But that's obvious.
Urban Dictionary provides definitions for colloquial uses of these adjectives and gives this interesting example sentence:
"He wanted to be cerebral but came across very cranial" (Art. "cranial")
According to the dictionary, a cranial person is stupid, while a cerebral person is an attentive and intelligent person.
But is this alleged colloquialism common anywhere or is it just an urban myth?
A: As @Ashwin states, in a medical context "cranial" refers to the skull, whereas "cerebral" refers to the brain. In other words they describe different bits of the body, and therefore are not interchangeable.
In more colloquial usage, "cerebral" tends to mean the brain as a source of thoughts (in other words, "the mind"), rather than the squishy physical organ. However, "cranial" still generally refers to the skull or the head.
So again, not interchangeable.
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\section{Introduction}
In the last years, Deep Learning methods have been receiving attention in the field of autonomous driving. The most common applications use cameras or laser scanners for tasks such as obstacle detection \cite{detection_camera}\cite{detection_lidar} and classification \cite{classification}. These tasks are mainly in the field of environment understanding and mapping. However, besides the need to understand the environment, the localization of the vehicle is still a challenging process and it has not yet been extensively explored by machine learning techniques.
To navigate unknown environments, methods known as Simultaneous Localization and Mapping (SLAM) allow to localize the vehicle while concurrently mapping the environment. Different types of sensors can be used for the mapping, such as cameras, laser scanners and radars. The use of 3D laser scanners has become very popular for autonomous driving, however its cost is still a major drawback for automobile manufacturers to maintain a reasonable price for the vehicles. Moreover, the amount of data provided by 3D laser scanners requires large computational resources. Considering this, we chose to focus on a solution for vehicles equipped with low cost 2D laser scanners. We propose a complete neural network method to localize a vehicle using only as input sequences of data acquired from 2D laser scanners.
\begin{figure}[H]
\centering
\includegraphics[width=1.0\linewidth]{intro.png}
\caption{Overview of the proposed system. The complete Recurrent Convolutional Neural Network takes a sequence of 2D laser scanner measurements as input, learns its features by a sequence of CNNs, which are used by the RNN to estimate the poses of vehicle. The output is a 2D pose of the vehicle composed by two values, one for translation and another one for rotation.}
\label{fig:System}
\end{figure}
Acquisition of a large amount of data is still one of the main challenges for the application of Deep Learning methods. Most of the datasets for autonomous driving with a significant number of sequences and good ground truth for localization are equipped with 3D laser scanners. For this reason, we use the KITTI \cite{kitti} odometry dataset and we simulate a 2D laser scanner by extracting one of the Velodyne layers. Although it is not a realistic position for the sensor, we can still use it as a proof of concept for estimating localization using this type of method.
The proposed network is in the format of a Recurrent Convolutional Neural Network, as shown in Figure \ref{fig:System}. The main idea is to use a sequence of CNNs in order to extract the features between two sequential laser scanners. Sequentially, these features are the input of a Recurrent Neural Network (RNN), more specifically a Long Short-Term Memory (LSTM) RNN, which learns how to estimate the pose of the vehicle.
The remainder of the paper is organized as follows. First, we present the related work in Section~\ref{relatedwork.sec}; the proposed method and the design of the network is presented in Section~\ref{method.sec}; experimental results are presented in Section~\ref{results.sec}; finally conclusion and perspectives are given in Section~\ref{conclusion.sec}.
\section{Related Work}\label{relatedwork.sec}
Deep learning research has presented notable results in different type of applications, among the most commons are object detection and classification using images. The success of these applications are mainly due to the fact that a large amount of datasets has become available in the last years. In the context of intelligent vehicles, interesting work has been developed in several different fields: trajectory prediction \cite{trajectory}\cite{trajectory2}, mapping \cite{mapping}, control \cite{control} and even end-to-end approaches \cite{endtoend}\cite{endtoend2}, where the car is controlled completely by a Deep Learning module.
At the same time, localization, which is still a very challenging problem for robotic systems, is not yet well explored by deep learning methods. Most of the proposed approaches are based on Visual Odometry (VO) \cite{VO}, which consists in using camera images for odometry estimation. The classic methods \cite{orb}\cite{svo} rely on finding geometry constraints from the images to estimate the camera's motion; while recent work tries to re-formulate this classic approach as a deep learning based VO, that potentially could deal with challenging environments and camera parameters difficulties. First, PoseNet \cite{posenet} proposed the use of CNNs to deal with the SLAM problem as a pose regression task by estimating the 6-DoF pose using only RGB images. Wang et al. \cite{deepvo} then introduced the DeepVO method, which uses RCNNs with the same goal. The same authors also presented the method UndeepVO \cite{undeepvo}, which proposes an unsupervised deep learning method to estimate the pose of a monocular camera. However, the classic VO methods still outperform deep learning based methods published to this date, considering the accuracy in the pose estimation.
The use of laser scanners is also popular for pose estimation in robotic systems. The most classic method consists in matching two point clouds to estimate the transformation between them by minimizing the matching error, this solution is known as Iterative Closest Point (ICP) \cite{icp}. More complex and robust methods to achieve the same goal have been proposed since then, like LOAM \cite{loam}, which runs two different algorithms in parallel to achieve real time processing. One algorithm can run faster to obtain a low fidelity odometry, and another one slower for a more precise matching. One of the main challenges for this type of localization methods is the error caused by moving obstacles or lack of information because of occlusions. For this reason, some localization solutions \cite{gridmatching}\cite{imls} are based on detecting what is static in the environment to only use this information for the scan matching process.
The application of deep learning techniques for this purpose using laser scanners is still considered as a new challenge and only few papers have addressed it. The use of this type of technique can eliminate the computational time problem found in most of the localization methods, and even be able to deal with the moving obstacles difficulties. A network that is properly trained could be able to distinguish what are the best matching features (static obstacles) from the moving obstacles that can cause pose estimation errors. Nicolai et al. \cite{laser1} were the first to propose to apply 3D laser scanner data in CNNs to estimate odometry. They presented an interesting approach that provided a reasonable estimation of odometry, however still not competitive with the efficiency of state-of-the-art scan matching methods. Later, Velas et al. \cite{laser3} presented another approach for using CNNs with 3D laser scanners for IMU assisted odometry. Their results were able to get high precision and close results compared to state-of-the-art methods, such as LOAM \cite{loam}, for translation, however the method is not able to estimate rotation with sufficient precision. Considering their results, the authors propose that their method could be used as a translation estimator and use together an Inertial Measurement Unit (IMU) to obtain the rotation. Another drawback is that, according to the KITTI benchmark, even using CNNs the method is slower than LOAM.
In our work we chose to use as the sensor only a 2D laser scanner, which could reduce considerably the price of future intelligent vehicles. At this moment, Li et al. \cite{laser2} were the only authors to use this kind of sensor to estimate odometry by the use of a deep learning approach. They propose a network to perform scan matching and loop closure using CNNs. Although the loop closure networks shows good accuracy, the scan matching results are still very inaccurate compared to classic methods.
Based on the idea presented in \cite{laser2}, we propose a new solution for 2D laser-based odometry estimation using deep learning networks. We explore the use of a RNN along with CNNs to learn temporal features in order to improve the odometry results. We also propose a new configuration for the CNNs where we were able to achieve better results. Finally, we explore this solution in outdoor environments, training and testing it with the KITTI \cite{kitti} dataset, which contains sequences different of type of scenarios.
\section{Method}\label{method.sec}
The proposed approach consists in finding the vehicle displacement by estimating the transformation between a sequence of 2D laser scanner acquisitions. From two consecutive observations, where each observation is a $\ang{360}$ set of points measured during one laser rotation, the network predicts the transformation $T = [\Delta{d}, \Delta{\theta}]$, which represents the travelling distance $\Delta{d}$ and the orientation $\Delta{\theta}$ between two consecutive laser scans $(s_{t-1}, s_{t})$ . We only consider the 2D displacement of the vehicle, since we are relying only on a 2D sensor. Therefore, the goal is to learn the optimal function $g(.)$, which maps $(s_{t-1}, s_{t})$ to $T$ at time $t$:
\begin{equation}
T_t = g(s_{t-1}, s_{t})
\label{learning_function}
\end{equation}
Once we learn these parameters, we can obtain the 2D pose $(x_t,y_t,\theta_t)$ of the vehicle in time $t$ as follow:
\begin{equation}
\begin{split}
x_t& = x_{t-1} + \Delta{d} \sin{(\Delta{\theta})} \\
y_t& = y_{t-1} + \Delta{d} \cos{(\Delta{\theta})} \\
\theta_t& = \theta_{t-1} + \Delta{\theta}
\end{split}
\end{equation}
In this way, we can accumulate the local poses of the vehicle and estimate the global position of the vehicle at any time $t$. Since the algorithm does not perform any sort of loop closure, drift can be also accumulated, thus reducing the accuracy of the vehicle's localization.
The following subsections will present in details the proposed method. First, we show the raw data of the laser scanner is encoded. Sequentially, we present the configuration of the network and the specifics of the training process.
\subsection{Data encoding}\label{sub.encoding}
We base our data encoding on the previous work \cite{laser2}, where the 2D laser scanner point set is encoded into a 1D vector. This can be done by first binning the raw scans into bins of resolution $\ang{0.1}$. Sequentially, since a group of points can fall into the same bin, we calculate the average depth value of this group. Finally, considering all the bins of a $\ang{360}$ rotation range, we store the depth values into a 3601 size vector, where each possible bin angle average depth is represented by the elements in the vector. This process is presented in Figure \ref{fig:encoding}.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth]{encoding.pdf}
\caption{Data enconding for the 2D laser scanner with a $\ang{360}$ rotation range. At each time step the raw data of the laser scanner is separated into $\ang{0.1}$ bins. The average depth of all the points in a specific bin is calculated and stored into a vector. The result is a 3601 size vector that stores the depth values for each bin.}
\label{fig:encoding}
\end{figure}
Once we have processed two 1D vectors from sequential scans, we concatenate them to use as input for the network. In this way, we create a form of image of size $2 x 3601$ that represents two acquisitions of the laser scanner. This format allows to use standard convolutional layers to extract the features detected by the sensor in the surrounding environment.
\subsection{Network Architecture}
\begin{figure*}
\centering
\includegraphics[width=0.92\linewidth]{architecture_deep.png}
\caption{Architecture of the proposed RCNN. Each block of the illustration presents the size of the tensors considering that the input is two sequences of laser scanners concatenated after being encoded as presented in subsection \ref{sub.encoding}. }
\label{fig:architecture}
\end{figure*}
The previous work on laser-based odometry estimation \cite{laser1}\cite{laser2}\cite{laser3}, for 2D and 3D sensors, only explored the use of CNNs without any deep temporal structure to estimate the local poses. We propose an architecture based on previous Visual Odometry (VO) methods, such as DeepVO \cite{deepvo}, that not only extracts the features of the input data but also estimate the possible connections among consecutive image inputs. The use of RNNs is convenient for this purpose because of its ability of modelling sequential dependencies. By adding this to our model, we aim to increase the accuracy of the pose estimation by using implicitly information that was detected by the laser scanner in previous frames. This process can be compared to graph-based approaches in classical SLAM methods \cite{graph}, where it takes a sequential of pose features and outputs a more precise estimation of these poses.
Figure \ref{fig:architecture} presents the architecture of the proposed network. Two pre-processed laser scanner acquisitions, represented as a 1 dimension vector of size 3601, are concatenated to create the input tensor of the network. Sequentially, the tensor is fed into the sequence of 1D convolutional and average pool layers to learn the features between the two acquisitions. Then, at each new data acquisition these features are received by the RNN to estimate new poses. We represent the 2D motion by two variables, the translation (travelling distance $\Delta{d}$) and rotation $(\Delta{\theta})$ . The main goal of this process is to use the neural network to learn the features of laser scanner data while simultaneously matching them using the proposed combination of CNN and RNN.
We inspire our network on the configuration used by DeepVO, which tries to achieve the same goal but using as input camera images. Since our data has a considerable smaller size we adapted the configuration for this purpose. This new configuration is presented in Table \ref{tab:table}. It has 6 1D convolutional layers, where each layer is followed by a rectified linear unit (ReLU) activation. Between each sequence of two convolutional layers, there is also one average pool layer. We added the pooling layers to reduce computation complexity by extracting the most important features, we tested both max and average pooling and we obtained better results applying the average pooling layer. Also, considering the size of the input and the size of features we can capture with this kind of sensor, we chose to use only kernels of size 3 after testing it with different configurations such as 5 and 7 size kernels.
\begin{table}[H]
\center
\begin{tabular}{cccc}
\hline
\textbf{Layer} & \textbf{Kernel Size} & \textbf{Stride} & \textbf{Number of Channels} \\ \hline
Conv1 & 3 & 1 & 32 \\
Conv2 & 3 & 2 & 32 \\
Conv3 & 3 & 1 & 64 \\
\multicolumn{1}{l}{Conv4} & 3 & 2 & 64 \\
\multicolumn{1}{l}{Conv5} & 3 & 1 & 128 \\
\multicolumn{1}{l}{Conv6} & 3 & 2 & 128 \\ \hline
\end{tabular}
\caption{Configuration of the convolutional layers in the proposed network.}
\label{tab:table}
\end{table}
After we learned the features, the output of $Conv6$ is passed to the RNN for sequential modelling. We use as our RNN, Long Short-Term Memory (LSTM) units, which are able to learn long-term dependencies \cite{lstm}. The use of stacked LSTM layers is common to learn high level representation and model complex dynamics. For this reason, we use the configuration of two LSTM layers with the hidden states of the first LSTM being used as input for the second one. The two layers are defined with 1024 hidden states. Finally, the last LSTM layer outputs two values, the rotation and translation at each time step.
\subsection{Training}
\label{sub:training}
The goal of using RNNs is to discover temporal correlations between the sequence of laser scans. While in principle the RNN is a simple and powerful model, in practice, it can be hard to train it properly to converge to precise results \cite{diff_rnn}. For this reason, the training of the network was performed in two steps. First, we trained the sequence of CNNs separately from the RNN. The objective in the first training is to pre-train the convolutional layers using no temporal information, only considering the information obtained from the two sequential laser scans input. Once we obtained the CNN part of the network pre-trained weights, we trained the complete network as presented in Fig. \ref{fig:architecture}.
\subsubsection{CNN Pre-Training}
To perform the pre-training, the output of the sequence of CNNs is fed to two different fully connected layers, one to estimate the rotation and another the translation. Sequentially, we train this convolutional network to learn the optimal function $g(.)$ presented in \autoref{learning_function}.
In \cite{laser3} the authors suggested that designing the network to regress the relative translation and rotation worked well only for translation, however the predicted rotation was still inaccurate. They were able to obtain better results reformulating the problem as a classification task for the rotation, and continuing as a regression one for the translation. This is possible because the range of possible rotations between consequent frames is quite reasonable. Considering this, we tested two configurations to pre-train our network, as a regression-only task, and as a regression and classification task.
For the regression-only task, we performed the training based on the Euclidean loss between the ground truth and the estimated translation and rotation values, defining the complete loss function as follow:]
\begin{equation}
\begin{split}
&\mathcal{L} = \mathcal{L}_{e}(\Delta{\hat{d}}, \Delta{d}) + \beta
\: \mathcal{L}_{e}({\Delta{\hat{\theta}},\Delta{\theta}})\\
&\text{where } \mathcal{L}_e(\hat{x}, x) = \norm{{\hat{x}} - {x}}_2
\end{split}
\label{loss_function}
\end{equation}
For the classification task, instead we used the Cross-entropy loss function to classify the angle, defining the new complete loss function for the training as:
\begin{equation}
\begin{split}
&\mathcal{L} = \mathcal{L}_{e}(\Delta{\hat{d}}, \Delta{d}) + \beta
\: \mathcal{L}_{c}({\Delta{\hat{\theta}},\Delta{\theta}})\\
&\text{where } \mathcal{L}_c(x,class) = - \log \Big( \frac{\exp({x[class]})} {\sum_j{\exp(x[j])}} \Big)
\end{split}
\label{loss_classification}
\end{equation}
In \eqref{loss_function} and \eqref{loss_classification}, $\Delta{{d}}$ and $\Delta{\theta}$ are relative ground-truth translation and rotation values, and $\Delta{\hat{d}}$ and $\Delta{\hat{\theta}}$ their output of the network counterparts. We use the parameter $\beta > 0$ to balance the scale difference between the rotation and translation loss values.
Considering all the possible variation of angles between two frames, we created classes for the interval $\pm\ang{5.6}$ with $\ang{0.1}$ resolution, resulting in 112 possible classes. As indicated in \cite{laser3}, the results as a classification task for rotation were better compared to the only regression, and for this reason we used the CNN network trained as a classification for angle estimation as input for the RNN.
\subsubsection{RCNN Training}
After initializing the weights of the CNN layers at the pre-training stage, we train the complete RCNN network. We define as the input of the RNN the output of the CNNs layers concatenated with the estimated result for rotation and translation, obtained from the pre-trained network, in a way that this could be used as a first estimation for the RCNN to refine the results.
\bgroup
\def1.5{1.5}
\begin{table*}[]
\begin{tabular}{c|cc
>{\columncolor[HTML]{EFEFEF}}c c|cc}
\hline
Sequence & CNN-Regression & CNN-Classification & RCNN-Regression & RCNN-Classification & DeepVO \cite{deepvo} & 3D LiDAR CNN \cite{laser3} \\ \hline
05 & 0.2888 & 0.2764 & \cellcolor[HTML]{EFEFEF}0.0293 & 0.2488 & 0.0262 & 0.0235 \\
07 & 0.1281 & 0.0756 & \cellcolor[HTML]{EFEFEF}0.0218 & 0.1251 & 0.0391 & 0.0177 \\ \hline
Mean & 0.2084 & 0.1760 & \cellcolor[HTML]{EFEFEF}0.0255 & 0.1869 & 0.0326 & 0.0206 \\ \hline
Time (s/frame) & 0.005 & 0.005 & 0.015 & 0.015 & 1.0 & 0.7
\end{tabular}
\caption{RMSE translation drift results for the two testing sequences along with the computation time per frame without GPU acceleration. We show the difference between the use of only CNNs and RCNN, the results for both of the RCNNs are using the pre-trained CNN-Classification network. In addition, we present the error for the different training configurations presented in Subsection \ref{sub:training}. We also compare the proposed approach to two other Deep Learning odometry estimation methods, one using as the sensor a monocamera \cite{deepvo} and the second using a 3D LiDAR \cite{laser3}. However, the result presented for the method \cite{laser3} is for a training dataset, since they chose different sequences for testing.}
\label{tab:error}
\end{table*}
For the RCNN we also tested the two different configurations (regression and regression with classification). Differently, we obtained more precise results treating the entire task (rotation and translation) as a regression problem, therefore using the loss function in \eqref{loss_function}. We presume that this occurs because the estimation of the rotation as a regression was easier once we had a first estimation of the class using the pre-trained CNNs, and with the possible information obtained by the RNN from previous frames.
\section{Results}\label{results.sec}
For validation we use the KITTI dataset \cite{kitti}, which provides several sequences in different conditions for outdoor environments. To obtain a 2D laser scanner dataset, we extracted one $\ang{360}$ layer from the Velodyne data. As mentioned before, the layer is extracted to simulate a low-cost 2D laser scanner, which could be an essential factor for the autonomous industry to maintain a reasonable price for future intelligent vehicles.
We use 10 sequences from the KITTI odometry dataset for the proposed method. In this dataset there are 11 sequences, however we eliminate the sequence 01 that consists of a trajectory mainly on a highway. During this sequence the 2D laser scanner detects almost no obstacle, making it impossible for the network to predict the odometry. Among the 10 sequences, we separate 8 for training and 2 for testing. We use for training the sequences 00, 02, 03, 04, 06, 08 and 09 and for testing the sequences 05 and 07. We chose these two sequences because they are not very long, leaving more data for the training, but they can still be challenging and present the potential of the proposed method. Additionally, during these two sequences we noticed that most of the time the simulated 2D laser scanner is able to detect obstacles, making it possible for the network to work as expected.
In order to validate our method and compare to other solutions, we calculated the drift according to the KITTI VO \cite{kitti} evaluation metrics, i.e., averaged Root Mean Square Errrors (RMSEs) of the translational error for all subsequences (100, 200,..., 800 meters). We adapt the proposed method to calculate the error only in 2D, since we only obtain 2D poses. The error score is then calculated by the mean of all subsequence errors.
Table \ref{tab:error} presents the error score for the testing sequences 05 and 07. The difference of scores between classification and regression shows why we chose to pre-train the convolutional layers as a classification task for the angle estimation, but to treat it as a regression task when we trained the entire RCNN. These results suggest that estimating the angle as a regression task was too hard for the network to learn; however, once we had a first estimation about the class in the training of the RCNN, we were able to refine the value and obtain a more precise angle.
\begin{figure}[H]
\centering
\begin{subfigure}[b]{1.0\linewidth}
\centering
{\includegraphics[width=0.9\linewidth]{seq_05.png} }
\caption{Sequence 05}
\end{subfigure}
\qquad
\begin{subfigure}[b]{1.0\linewidth}
\centering
{\includegraphics[width=0.9\linewidth]{seq_07.png} }
\caption{Sequence 07}
\end{subfigure}
\caption{Trajectories of the two test sequences (05 and 07) applying the proposed method. The blue lines represent the ground truth trajectory, while in red the predicted one.}%
\label{fig:trajectory}%
\end{figure}
\begin{figure*}
\centering
\begin{subfigure}{1.0\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{rot_trans_05.pdf}
\caption{Sequence 05.}
\label{fig:sub1}
\end{subfigure}%
\vspace{0.5cm}
\begin{subfigure}{1.0\textwidth}
\centering
\includegraphics[width=1.0\linewidth]{rot_trans_07.pdf}
\caption{Sequence 07}
\label{fig:sub2}
\end{subfigure}
\vspace{0.5cm}
\caption{Results of rotation and translation estimation for the two testing sequences. The ground truth values are presented in blue and in red the output of the network. }
\label{fig:rot_trans}
\end{figure*}
We also compare in \autoref{tab:error} the proposed network to other two Deep Learning approaches for odometry estimation: one using the mono-camera and another a 3D laser scanner. We chose these two approaches because they use the same dataset, allowing us to compare the drift results, and also their computational time is presented by the KITTI benchmark. Our approach has better results in comparison to the network using only mono-camera images, however slightly worse results as the 3D laser scanner method. The better result in \cite{laser3} is expected, since in our experiments we are extracting only one layer of the laser scanner, which provides a significant less amount of information. In addition, as the sensor is on top of the vehicle and we always extract the most parallel layer in relation to the vehicle, there are some frames where nothing or not much is detected, causing possible wrong estimations of translation and rotation. However, it is important to mention that the 3D laser scanner approach \cite{laser3} takes 0.23s to run with GPU acceleration, while our approach takes only 0.015s per frame without using GPU acceleration (2,6 GHz Intel Core i5, Intel Iris 1536 MB), and it can be as fast as 0.001s per frame with GPU acceleration (4,0 GHz Intel Core i7, GeForce GTX 1060); i.e. a 230-fold increase in speed (GPU configuration), while obtaining only a $0.49\%$ difference in drift score and using a much cheaper sensor. The faster processing is expected since we have a considerable smaller data input, and it allows to obtain odometry estimation in real-time with simple computational resources.
We can observe in Figure \ref{fig:trajectory} that even with the eventual errors that can occur, we can still obtain a trajectory close to the ground truth. However, since the proposed approach does not perform any sort of loop closure, one eventual large error can be accumulated over time, generating a large drift like the one we have by the end of sequence 07.
For this reason, a better way to understand the accuracy of the proposed approach is presented in \autoref{fig:rot_trans}. It shows the odometry estimation (rotation and translation) together with the ground truth for each frame of the testing sequences. Considering these two sequences, the average rotation absolute error is 0.05 degrees, while the average translation absolute error is 0.02 meters. However, we can encounter errors up to 0.4 degrees to rotation and 0.2 meters to translation in frames where it is harder to estimate the odometry. These values present how the network can most of the time estimate accurate odometry, however there are still some difficult cases that can result in inaccurate values.
The results show how promising is the proposed method and it could be used as a complement to traditional localization methods for intelligent vehicle or any mobile robot, when for example there are no wheel encoders or GPS signal. We can also expect that if the sensor was located in an ideal position, for example for an autonomous car as a set of 2D laser scanners around the vehicle in the level of the bumper, we could obtain even better results. It is also important to mention that we trained the network with a relatively small dataset compared to other deep learning applications, therefore the result could be improved using more sequences for training.
\section{Conclusion and Future Work}\label{conclusion.sec}
In this paper we presented a novel approach based on RCNNs to estimate the odometry using only the data of a 2D laser scanner. The combination of CNNs and RNNs allows us to achieve in real-time the extraction of scan features and learn their sequential model to obtain the localization of an intelligent vehicle. The proposed network presents that the use of 2D laser scanners can not only provide good accuracy with a low cost sensor, but also requires less computational resources to achieve real-time performance.
The results were evaluated using the KITTI odometry dataset making it possible to compare it with other Deep Learning approaches. Although the results were competitive to this type of approaches, we still do not expect that the deep learning methods could replace classic approaches at this moment, since they can still provide a better accuracy and a better understanding of the quality of the results. However, the proposed approach could be an interesting complement for classic localization estimation methods, since it can be run in real-time and could give relatively accurate values in systems where no wheel encoder data is provided or GPS signal is absent. Moreover, the proposed method shows a promising use of Neural Networks to understand the environment detected by a 2D laser scanner, since we can assume that the network learned what were the best features to match between different scans.
In the future work, we expect that better results could be obtained after training the network with a dataset that has real 2D laser scanners located at a better position at the vehicle. Additionally, the use of more than one sensor could be explored to increase the accuracy of the results. For example, the use of a mono-camera alone was not able to get precise results, but possibly by creating a network that could perform the fusion with a 2D laser scanner these results could be improved.
\addtolength{\textheight}{-12cm}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,055
|
<div id="month-viewer" data-final-month="<?php echo date('F-Y', end($articles)->getDate()); reset($articles); ?>">
<?php $current_date = date('F Y', $articles[0]->getDate()); ?>
<div class="row <?php if($category): echo $category->getCat(); else: echo 'felix_default'; endif; ?>">
<div class="small-12 columns">
<div class="bar-text"><?php echo date('F Y', $articles[0]->getDate()); ?></div>
</div>
</div>
<div class="row main-row top-row small-up-1 medium-up-2 large-up-3" id="<?php echo str_replace(' ', '-', $current_date); ?>">
<?php
for($i = 0; $i < count($articles); $i++):
$article = $articles[$i];
?>
<div class="date-article columns">
<?php
$theme->setHierarchy(array(
$articles[$i]->getCategory()->getCat() // category-{cat}.php
));
$theme->render('components/category/block_normal', array(
'article' => $articles[$i],
'equalizer' => str_replace(' ', '-', $current_date),
'show_category' => $show_category,
'headshot' => $headshot
));
?>
</div>
<?php
if(isset($articles[$i+1]) && date('F Y', $articles[$i+1]->getDate()) != $current_date) {
?>
</div>
<?php
$current_date = date('F Y', $articles[$i+1]->getDate());
if($category && $category->getCat()) {
$colour_key = $category->getCat();
} else {
$colour_key = 'felix_default';
}
?>
<div class="row <?php echo $colour_key ?>">
<div class="small-12 columns">
<div class="bar-text"><?php echo date('F Y', $articles[$i+1]->getDate()); ?></div>
</div>
</div>
<div class="row date-row top-row small-up-1 medium-up-2 large-up-3" id="<?php echo str_replace(' ', '-', $current_date); ?>">
<?php
}
?>
<?php
endfor;
?>
</div>
</div>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,853
|
require 'thread'
module Mutex_m
def Mutex_m.define_aliases(cl)
cl.module_eval %q{
alias locked? mu_locked?
alias lock mu_lock
alias unlock mu_unlock
alias try_lock mu_try_lock
alias synchronize mu_synchronize
}
end
def Mutex_m.append_features(cl)
super
define_aliases(cl) unless cl.instance_of?(Module)
end
def Mutex_m.extend_object(obj)
super
obj.mu_extended
end
def mu_extended
unless (defined? locked? and
defined? lock and
defined? unlock and
defined? try_lock and
defined? synchronize)
Mutex_m.define_aliases(singleton_class)
end
mu_initialize
end
# locking
def mu_synchronize(&block)
@_mutex.synchronize(&block)
end
def mu_locked?
@_mutex.locked?
end
def mu_try_lock
@_mutex.try_lock
end
def mu_lock
@_mutex.lock
end
def mu_unlock
@_mutex.unlock
end
private
def mu_initialize
@_mutex = Mutex.new
end
def initialize(*args)
mu_initialize
super
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,219
|
Palmarès
Europei
1 medaglia:
1 argento (Berlino 1930)
Morti per incendio
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,443
|
Linväxter (Linaceae) är en växtfamilj med omkring 275 arter, bland annat odlad lin. I Norden är familjen representerad med två släkten och fyra arter, förutom det odlade linet även vildlin och klipplin (vilka precis som det odlade linet hör till släktet Linum), och dvärglin (som hör till släktet Radiola).
Släkten i underfamilj Linoideae
Anisadenia
Cliococca
Hesperolinon
Linum
Radiola
Reinwardtia
Sclerolinon
Tirpitzia
Släkten i underfamilj Hugonioideae (Hugoniaceae)
Durandea
Hebepetalum
Hugonia
Indorouchera
Philbornea
Roucheria
Referenser
Lundevall, Carl-Fredrik & Björkman, Gebbe. Vilda växter i Norden, 2007, ICA bokförlag (s. 150).
Externa länkar
Den virtuella floran
Linväxter
Trikolpater
Li Växtindex
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,222
|
package org.jtwig.translate.message.source.cache.model;
import org.junit.Test;
import java.util.Locale;
import static org.hamcrest.core.Is.is;
import static org.junit.Assert.assertThat;
public class LocaleAndMessageTest {
@Test
public void equalsSameInstance() throws Exception {
LocaleAndMessage instance = new LocaleAndMessage(Locale.CHINA, "message");
assertThat(instance.equals(instance), is(true));
}
@Test
public void equalsNull() throws Exception {
LocaleAndMessage instance = new LocaleAndMessage(Locale.CHINA, "message");
assertThat(instance.equals(null), is(false));
}
@Test
public void equalsDistinctType() throws Exception {
LocaleAndMessage instance = new LocaleAndMessage(Locale.CHINA, "message");
assertThat(instance.equals("asd"), is(false));
}
@Test
public void equalsDistinctLocale() throws Exception {
Locale locale = Locale.CHINA;
String message = "message";
LocaleAndMessage instance = new LocaleAndMessage(locale, message);
assertThat(instance.equals(new LocaleAndMessage(Locale.CANADA, message)), is(false));
}
@Test
public void equalsDistinctMessage() throws Exception {
Locale locale = Locale.CHINA;
String message = "message";
LocaleAndMessage instance = new LocaleAndMessage(locale, message);
assertThat(instance.equals(new LocaleAndMessage(locale, "one")), is(false));
}
@Test
public void equalsHappy() throws Exception {
Locale locale = Locale.CHINA;
String message = "message";
LocaleAndMessage instance = new LocaleAndMessage(locale, message);
assertThat(instance.equals(new LocaleAndMessage(Locale.CHINA, "message")), is(true));
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,015
|
{"url":"https:\/\/www.centralbanking.com\/central-banking\/news\/1422618\/china-fx-reserves-reach-usd1-trln","text":"China 's FX reserves may reach $1 trln China's foreign exchange reserves will exceed 1 trillion US dollars eventually, economist and People's Bank of China adviser Yu Yongding said. Speaking to a business conference, Yu did not say when he expected foreign exchange reserves to reach that level. China's foreign exchange reserves stood at 769 bln US$ at the end of September this year.\n\nYu was also quoted in the China Securities Journal as saying that China should expand the floating range for trading the yuan against the dollar, and re\n\nLatest issue\n\nCentral Banking Journal\n\nRead the latest edition of the Central Banking journal\n\nYou need to sign in to use this feature. If you don\u2019t have a Central Banking account, please register for a trial.","date":"2018-06-19 12:11:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3003421127796173, \"perplexity\": 4368.830311524439}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-26\/segments\/1529267862929.10\/warc\/CC-MAIN-20180619115101-20180619135101-00107.warc.gz\"}"}
| null | null |
@interface VarToken : BasicToken
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,496
|
{"url":"https:\/\/www.msuperl.org\/wikis\/icsam\/doku.php?id=repository:solar_system_springs","text":"## Solar System Springs\n\n### Activity Information\n\n#### Learning Goals\n\n\u2022 Understand how a spring constant ($k$) affects simple harmonic motion\n\u2022 Understand effects of mass and gravity on simple harmonic motion\n\n#### Prior Knowledge Required\n\n\u2022 Free-body diagrams\n\u2022 Hooke's Law\n\u2022 $F=-kx$\n\n#### Code Manipulation\n\n\u2022 Creating\/modifying existing code\n\u2022 Translating physical concepts and equations into code\n\n\u2014-\n\n### Activity\n\n#### Handout\n\nSolar System Springs\n\nYou and your team have the opportunity to board the Sol Spacecraft to explore the solar system. Should you choose to accept the mission, NASA requires you to study the action of springs on each of the planets. Some planets do not have a solid surface, so your experiment will be conducted as you hover deep within the planet's atmosphere. Your experiments will examine the effect of each planet's gravitational field on the oscillation of a spring. You also have another container filled with many different masses.\n\nYour gravimeter gives you the following values for $g$ on each planet:\n\n(Dwarf*) Planet $g$ (m\/s\u00b2)\nMercury -3.6\nVenus -8.9\nEarth -9.8\nMars -3.8\nJupiter -26.0\nSaturn -11.1\nUranus -10.7\nNeptune -14.1\nPluto* -0.42\n##### Pre-coding Questions\n1. Complete a free body diagram for each of the 5 following cases:\n1. Case 1: The mass is at the equilibrium point\n2. Case 2: The mass is above the equilibrium point\n3. Case 3: The mass is below the equilibrium point\n4. Case 4: The mass is between equilibrium and the top\n5. Case 5: The mass is between the equilibrium and bottom\n2. Create a diagram showing the acceleration and velocity vectors for each of the aforementioned cases.\n3. Run the minimally working code (below). The code allows the user to input the value for the spring constant ($k$) and change the mass of the crate ($m$) using a slider. What problem(s) do you see with the simulation?\n\nFortunately, the code is not entirely complete; you have a fantastic opportunity to fix it! Modify the code to make the spring oscillate appropriately. Also, add some lines of code so that you can change the acceleration due to gravity and perform a check to make sure the value is in the \u201cdown\u201d direction (the \u201cup\u201d direction is set as positive). Make sure that the acceleration due to gravity is reported in the window that prints the mass and spring constant. Once you have verified that your code is running properly, complete the following questions.\n\n##### Post-coding Questions\n1. Sketch a graph of the position, velocity, and acceleration of the mass as functions of time. Use the graph produced by the program as a guide.\n2. When the mass is at the top of its oscillations, what is the velocity and acceleration? Are they a minimum, maximum, or intermediate value?\n3. When the mass is at the equilibrium point, what is the velocity and acceleration? Are they a minimum, maximum, or intermediate value?\n4. When the mass is at the bottom, what is the velocity and acceleration? Are they a minimum, maximum, or intermediate value?\n5. Run the program multiple times keeping the mass and acceleration due to gravity the same, but change the spring constant. Record your observations. What effect does changing the spring constant have on the amplitude for position, velocity, and acceleration? How does this affect the frequency and period of the oscillation?\n6. Now run the program multiple times keeping the spring constant and acceleration due to gravity the same, but change the mass. Record your observations. What effect does changing the mass have on the amplitude for position, velocity, and acceleration? How does this affect the frequency and period of the oscillation?\n7. Finally, run the program multiple times keeping the spring constant and mass the same, but change the acceleration due to gravity. Record your observations. What effect does changing the acceleration due to gravity have on the amplitude for position, velocity, and acceleration? How does this affect the frequency and period of the oscillation?\n\n#### Code\n\nGlowScript 2.7 VPython\u00a0running=True #Sets up condition for pause\/run button\u00a0totalTime=20 #in seconds\u00a0#Variables\/Objectsk=int(prompt(\"Spring constant?: \")) #prompt allows user to set spring constant#Graphs below measure position, velocity, and acceleration as a function of timef4 = graph(xmin=0,xmax=totalTime,xtitle='<i>t<\/i> in seconds')f5 = gcurve(color=color.green, label='position',fast=True)f6 = gcurve(type='scatter', color=color.blue, label='velocity',fast=True)f7 = gcurve(type='scatter', color=color.red, label='acceleration',fast=True)f1 = graph(xmin=0,xmax=totalTime,xtitle='<i>t<\/i> in seconds')f1 = gcurve(color=color.green, label='position',fast=True)f2 = graph(xmin=0,xmax=totalTime,xtitle='<i>t<\/i> in seconds')f2 = gcurve(type='scatter', color=color.blue, label='velocity',fast=True)f3 = graph(xmin=0,xmax=totalTime,xtitle='<i>t<\/i> in seconds')f3 = gcurve(type='scatter', color=color.red, label='acceleration',fast=True)\u00a0#The mass and spring hang from the ceilingceiling = box(pos=vec(0,11.5,0),size=vec(8,1,1), color=color.red)\u00a0Mass = box(pos=vector(0,-10,0),size=vec(2,2,2),velocity=vector(0,0,0),color=color.green,mass=1)\u00a0def setmass(s): return#Using a slider for the mass allows the user to change the mass, ranging from 1 to 5 kgmass = slider( min=1, max=5, value=1, align='right',step=0.5, length=220, bind=setmass)\u00a0pivot = vector(0,10,0)\u00a0spring = helix(pos=pivot,axis=Mass.pos-pivot,radius=0.8,coils=20,constant=k,thicnkess=0.1,color=color.green)#When you hang a mass from a spring, the spring stretches and the equilibrium point gets moved downeq = vector(0,Mass.mass*-9.81\/k,0)\u00a0#We are starting with time t=0, and changing every .05 secondst = 0dt = .05 #if time step is smaller than this, then completion time will not be the total time duration.print('mass = 1', 'spring constant = ', k)\u00a0#The code below allows the user to pause\/run the program by pushing a button at the topdef Run(b): global running running = not running if running: b.text = \"Pause\" running=True else: b.text = \"Run\" running=Falsebutton(text=\"Pause\", pos=scene.title_anchor, bind=Run) \u00a0#In the while loop below, the forces (gravity, spring, and net) are being updated#Also, the acceleration, velocity, and position of the box are being updated\u00a0while True:\u00a0\u00a0 if running==True: frameRate = 1\/dt rate(frameRate) Fg = vec(0,Mass.mass*-9.81,0) if eq.y < Mass.pos.y: Fs = vec(0, -k*mag(Mass.pos-eq),0) else: Fs = vec(0, k*mag(Mass.pos-eq),0) #something needs to happen here Fnet = Fg acc = Fnet\/(Mass.mass) \u00a0\u00a0 Mass.velocity = Mass.velocity+acc*dt Mass.pos = Mass.pos+Mass.velocity*dt spring.axis = Mass.pos-spring.pos\u00a0\u00a0 if mass.value != Mass.mass: print('mass = ', mass.value, 'spring constant = ', k)\u00a0 Mass.mass=mass.value\u00a0\u00a0 f1.plot(t,Mass.pos.y) f2.plot(t,Mass.velocity.y) f3.plot(t,acc.y) f5.plot(t,Mass.pos.y) f6.plot(t,Mass.velocity.y) f7.plot(t,acc.y)\u00a0\u00a0 t = t + dt if running==False:\u00a0 frameRate =1\/dt rate(frameRate) acc = (eq-Mass.pos)*(spring.constant\/Mass.mass)*0 Mass.velocity = Mass.velocity Mass.pos = Mass.pos spring.axis = spring.axis\n\n#### Handout\n\n##### Pre-coding Questions\n1. There are 2 major problems with the code:\n1. The program does not stop, despite there being code alluding to a 20 second run-time (line 5)\n2. The spring does not oscillate, but rather extends infinitely. This is due to the \u201cFnet\u201d in line 69 only including the force of gravity but not the force of the spring\n##### Post-coding Questions\n1. When the mass is at the top of its oscillation, the velocity is at a minimum and its acceleration is at a maximum (the force of gravity and the force of the compressed spring are pulling and pushing the mass down)\n2. When the mass is at the equilibrium point, its velocity is at a maximum, and its acceleration is at a minimum (the spring and the force of gravity counteract each other and the net force is 0)\n3. When the mass is at the bottom of its acceleration, the velocity is at a minimum and the acceleration is at a maximum value (although the force of gravity is still pulling the mass downwards, the force of the stretched spring pulling the mass upwards will result in a net force that causes the same magnitude acceleration as when the spring is at the top of its oscillation)\n4. Increasing the spring constant affects the stiffness of the spring. The higher the constant, the stiffer the spring is. A higher spring constant results in smaller amplitude for position, but higher amplitudes for velocity and acceleration. Additionally, the frequency of the oscillation increases, and therefore the period decreases\n\n#### Code\n\nGlowScript 2.7 VPython\u00a0running=True #Sets up condition for pause\/run button\u00a0totalTime=20 #in seconds\u00a0#Variables\/Objectsg=float(prompt(\"Acceleration due to Gravity?: \"))while g>0: g=0 g=float(prompt(\"ERROR: Acceleration due to Gravity is down: \"))k=int(prompt(\"Spring constant?: \"))#prompt allows user to set spring constant\u00a0#Graphs below measure position, velocity, and acceleration as a function of timef4 = graph(xmin=0,xmax=totalTime,xtitle='<i>t<\/i> in seconds')f5 = gcurve(color=color.green, label='position',fast=True)f6 = gcurve(type='scatter', color=color.blue, label='velocity',fast=True)f7 = gcurve(type='scatter', color=color.red, label='acceleration',fast=True)f1 = graph(xmin=0,xmax=totalTime,xtitle='<i>t<\/i> in seconds')f1 = gcurve(color=color.green, label='position',fast=True)f2 = graph(xmin=0,xmax=totalTime,xtitle='<i>t<\/i> in seconds')f2 = gcurve(type='scatter', color=color.blue, label='velocity',fast=True)f3 = graph(xmin=0,xmax=totalTime,xtitle='<i>t<\/i> in seconds')f3 = gcurve(type='scatter', color=color.red, label='acceleration',fast=True)\u00a0#The mass and spring hang from the ceilingceiling = box(pos=vec(0,11.5,0),size=vec(8,1,1), color=color.red)\u00a0Mass = box(pos=vector(0,-10,0),size=vec(2,2,2),velocity=vector(0,0,0),color=color.green,mass=1)\u00a0def setmass(s): return#Using a slider for the mass allows the user to change the mass, ranging from 1 to 5 kgmass = slider( min=1, max=5, value=1,label='Mass' ,align='right',step=0.5, length=220, bind=setmass)\u00a0pivot = vector(0,10,0)\u00a0spring = helix(pos=pivot,axis=Mass.pos-pivot,radius=0.8,coils=20,constant=k,thicnkess=0.1,color=color.green)#When you hang a mass from a spring, the spring stretches and the equilibrium point gets moved downeq = vector(0,Mass.mass*g\/k,0)\u00a0#We are starting with time t=0, and changing every .05 secondst = 0dt = .05 #if time step is smaller than this, then completion time will not be the total time duration.print('mass = 1 kg | ', 'spring constant = ', k,'N\/m', ' | gravity is:',g, 'm\/s\/s')\u00a0#The code below allows the user to pause\/run the program by pushing a button at the topdef Run(b): global running running = not running if running: b.text = \"Pause\" running=True else: b.text = \"Run\" running=Falsebutton(text=\"Pause\", pos=scene.title_anchor, bind=Run) \u00a0#In the while loop below, the forces (gravity, spring, and net) are being updated#Also, the acceleration, velocity, and position of the box are being updated\u00a0while True: if t > totalTime: running=False\u00a0 if running==True: frameRate = 1\/dt rate(frameRate) Fg = vec(0,Mass.mass*g,0) if eq.y < Mass.pos.y: Fs = vec(0, -k*mag(Mass.pos-eq),0) else: Fs = vec(0, k*mag(Mass.pos-eq),0) Fnet = Fg + Fs acc = Fnet\/(Mass.mass)\u00a0\u00a0 Mass.velocity = Mass.velocity+acc*dt Mass.pos = Mass.pos+Mass.velocity*dt spring.axis = Mass.pos-spring.pos\u00a0\u00a0 if mass.value != Mass.mass: print('mass = ', mass.value,'kg' ,'| spring constant = ', k,'N\/m', '| gravity is:', g, 'm\/s\/s')\u00a0 Mass.mass=mass.value\u00a0\u00a0 f1.plot(t,Mass.pos.y) f2.plot(t,Mass.velocity.y) f3.plot(t,acc.y) f5.plot(t,Mass.pos.y) f6.plot(t,Mass.velocity.y) f7.plot(t,acc.y)\u00a0\u00a0 t = t + dt if running==False:\u00a0 frameRate =1\/dt rate(frameRate) acc = (eq-Mass.pos)*(spring.constant\/Mass.mass)*0 Mass.velocity = Mass.velocity Mass.pos = Mass.pos spring.axis = spring.axis","date":"2022-05-25 01:19:19","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.39683783054351807, \"perplexity\": 2770.1047467151675}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662577757.82\/warc\/CC-MAIN-20220524233716-20220525023716-00426.warc.gz\"}"}
| null | null |
\section{Introduction}
As the data on single-particle distributions of identified hadrons produced in heavy-ion collisions become more abundant and precise \cite{sa, ja, ba, sa1, igb, iga, ba1}, more demands are put on theoretical models to reproduce them. It is generally recognized that in Au-Au collisions at $\sqrt{s_{NN}}= 200$ GeV at the Relativistic Heavy-Ion Collider (RHIC) the low transverse-momentum $(p_T)$ region $(p_T < 2$ GeV/c) is well described by hydrodynamics \cite{dat} and the high-$p_T$ region ($p_T > 6$ GeV/c) by perturbative QCD \cite {mv}, both subjects being reviewed recently in Ref.\ \cite{fn}. In the intermediate region ($2 < p_T < 6$ GeV/c) neither approaches work very well. What stands out in that region are the large baryon/meson ratio and quark-number scaling (QNS), which give empirical support to the recombination/coalescence models \cite{hy, gkl, fmnb, hwa, hfr, mv1}. The connection between the intermediate- and high-$p_T$ regions is smooth, since the dominance of shower-shower recombination is equivalent to parton fragmentation. The transition across the lower $p_T$ boundary at $p_T \sim 2$ GeV/c is not so smooth because of the difference in the continuum description in hydrodynamics and the parton description in hadronization. Our aim in this article is to extend our previous considerations \cite{chy, hz} to the lower-$p_T$ region and to describe in a self-consistent way both the $p_T$ and azimuthal $\phi$ behaviors of pions and protons without explicit reliance on hydrodynamics.
One specific point that motivates our study is related to the question of what happens to the initial system within 1 fm/c after collision. Semihard partons created within 1 fm from the surface will have already left the initial overlap region before thermalization is complete. There are many of them with parton transverse-momentum $k_T \sim $2-3 GeV/c even at RHIC, let alone at the Large Hadron Collider (LHC). They are minijets that can cause azimuthal anisotropy, not accounted for by conventional hydrodynamics. It is known that in events triggered by jets there is a ridge phenomenon in the structure of associated particles with narrow $\Delta\phi$ (azimuthal angle relative to that of the trigger) and extended $\Delta\eta$ (pseudorapidity relative to the trigger). Such a structure should be present in the inclusive distribution even if triggers are not used to select the jet events. When $k_T$ is low enough so that minijets are copiously produced, the corresponding effect on the $\phi$ anisotropy can become dominant, rendering the consideration of pressure gradients along different $\phi$ direction unreliable if semihard scatterings are ignored. In this paper we give specific attention to the ridge contribution to the single-particle distributions in the low-$p_T$ region. It is in this sense that we use the terminology: inclusive ridge distribution.
Another area of concern is the variation of the $p_T$ dependence as the focus is moved to the low-$p_T$ region, where pion and proton appear empirically to have different behaviors. In the parton recombination model the hadrons should have the same inverse slope as that of the coalescing quarks if the hadrons are formed by recombination of the thermal partons, but because of the difference in the meson and baryon wave functions, the net $p_T$ distributions turn out to be different. This line of analysis takes into account the quark degree of freedom just before hadronization, which is overlooked by the fluid description of the flow effect. The burden is to show that the data on $v_2(p_T)$ can be reproduced for both pion and proton at low $p_T$ without the hydro description of elliptic flow. That is indeed what we shall show for various centralities.
We confine our consideration in this paper to the physics at midrapidity. At larger $\eta$ there are other issues, such as large $p/\pi$ ratio \cite{iga} and large $\Delta\eta$ distribution of triggered ridge \cite{ba1}, which have been examined in Refs. \cite{hz1, ch}, and will not be further considered here.
\section{Single-particle distribution with ridge }
We begin with a recapitulation of our description of single-particle distribution \cite{hy, chy, hz}. At low $p_T$ we consider only the recombination of thermal partons, so the pion and proton spectra at $y=0$ are given by
\begin{eqnarray}
p^0 {dN^{\pi}\over dp_T} &=& \int \prod_{i=1}^2 \left[{dq_i\over q_i} {\cal T} (q_i)\right] {\cal R}^{\pi}(q_1,q_2,p_T), \label{1} \\
p^0 {dN^p\over dp_T} &=& \int \prod_{i=1}^3 \left[{dq_i\over q_i} {\cal T} (q_i)\right] {\cal R}^p(q_1,q_2,q_3,p_T), \label{2}
\end{eqnarray}
where ${\cal T} (q_i)$ is the thermal distribution of the quark (or antiquark) with momentum $q_i$, and ${\cal R}^h$ is the recombination function (RF) for $h=\pi$ or $p$. On the assumption that collinear quarks make the dominant contribution to the coalescence process (so that the integrals are one-dimensional for each quark along the direction of the hadron), the RFs are
\begin{eqnarray}
{\cal R}^{\pi}(q_1,q_2,p) &=& {q_1q_2\over p^2} \delta\left(\sum_{i=1}^2 {q_i\over p} - 1 \right), \label{3} \\
{\cal R}^p(q_1,q_2,q_3,p) &=& f\left({q_1\over p},{q_2\over p},{q_3\over p} \right) \delta\left(\sum_{i=1}^3 {q_i\over p} - 1 \right) \label{4}
\end{eqnarray}
where the details of $f(q_i/p)$ that depends on the proton wave function are given in \cite{hy}, and need not be repeated here. The main point to be made here is that if the thermal distribution ${\cal T}(q_i)$ has the canonical invariant form
\begin{eqnarray}
{\cal T}(q) = q{dN^q\over dq} = Cqe^{-q/T}, \label{5}
\end{eqnarray}
then the $\delta$-functions in the RFs require that $dN^h/p_Tdp_T$ has the common exponential factor, $\exp(-p_T/T)$, for both $h=\pi$ and $p$. The prefactors are different; we simply write down the results obtained previously
\begin{eqnarray}
{dN^{\pi}\over p_Tdp_T} &=& {\cal N}_{\pi}e^{-p_T/T}, \label{6} \\
{dN^p\over p_Tdp_T} &=& {\cal N}_p{p_T^2\over m_T}e^{-p_T/T}, \quad m_T = (p_T^2 + m_p^2)^{1/2}, \label{7}
\end{eqnarray}
where ${\cal N}_{\pi} \propto C^2$ and ${\cal N}_p \propto C^3$, and $C$ has the dimension (GeV)$^{-1}$. Note that the factor $p_T^2/m_T$ in the proton spectrum (that must be present for dimensional reason) causes the $p/\pi$ ratio to vanish as $p_T \rightarrow 0$ on the one hand, but to become large, as $p_T$ increases, on the other. When $p_T$ exceeds 2 GeV/c, shower partons become important and the above description must be supplemented by thermal-shower ({\bf TS}) recombination that limits the increase of the $p/\pi$ ratio to a maximum of about 1 \cite{hy}.
We restrict our consideration to $p_T < 2$ GeV/c, but now broaden it to include $\phi$ dependence. For non-central collisions the almond-shaped initial configuration leads to $\phi$ anisotropy. The conventional description in terms of hydrodynamics relates the momentum anisotropy to the variation of pressure gradient at early times upon equilibration \cite{kh}. The success in obtaining the large $v_2$ as observed gives credibility to the approach. We adopt an alternative approach and justify our point of view on the basis that we can also reproduce the empirical $v_2$, as we shall show. Furthermore, aside from offering a smooth connection with the intermediate $p_T$ region by the inclusion of {\bf TS} recombination, our approach describes also the effect of semihard scattering on the soft sector. The ridge phenomenon that we attribute to that effect can be with trigger \cite{ch, ch1, ch2} or without trigger \cite{rch, hwa, chy, hz}. Although data on the ridge structure must necessarily make use of triggers in order to distinguish it from background \cite{ja1, bia, ha, ba2}, inclusive distribution must include ridges along with background. Thus theoretically a single-particle distribution should have a ridge component in the soft sector due to undetected semihard or hard partons. That component has $\phi$ dependence that can be calculated from geometrical consideration \cite{hz}, and has been shown to be consistent with the dependence of the ridge yield in two-particle correlation on the trigger angle $\phi_s$ relative to the reaction plane \cite{ha}.
Let us use $\rho_1^h(p_T,\phi,b)$ to denote the single-particle distribution of hadron $h$ produced at mid-rapidity in heavy-ion collision at impact parameter $b$, i.e.
\begin{eqnarray}
\rho_1^h(p_T,\phi,b) = {dN^h\over p_Tdp_Td\phi}(N_{\rm part}), \label{8}
\end{eqnarray}
where $N_{\rm part}$ is the number of participants related to $b$ in a known way through Glauber description of nuclear collision \cite{mrss}. At low $p_T$ let $\rho_1^h$ be separated into two components
\begin{eqnarray}
\rho_1^h(p_T,\phi,b) = B^h(p_T,b) + R^h(p_T,\phi,b), \label{9}
\end{eqnarray}
where $B^h(p_T,b)$ is referred to as Base, not to be confused with the bulk that is usually determined in hydrodynamics; this is a change from earlier nomenclature \cite{hz}, where the use of ``bulk" did lead to some misunderstanding. Our emphasis here is that $B(p_T,b)$ is independent of $\phi$. In our approach we regard the semihard partons created near the surface, and directed outward, give rise to all the $\phi$ dependence of the medium before equilibrium is established; the recoil partons being directed inward are absorbed and randomized. The component expressed by $R^h(p_T,\phi,b)$ is referred to as ridge on the basis of its $\phi$ dependence discussed below. The $B^h(p_T,b)$ component consists of all the soft and semihard partons that are farther away from the surface and are unable to lead to hadrons with distinctive $\phi$ dependence. Thus the separation between $B^h(p_T,b)$ and $R^h(p_T,\phi,b)$ relies primarily on the $\phi$ dependence that the ridge component possesses.
In Ref.\ \cite{hz} we have given an extended derivation of what that $\phi$ dependence is. It is embodied in $S(\phi,b)$ that is the segment of the surface through which a semihard parton can be emitted to contribute to a ridge particle at $\phi$. From the geometry of the initial ellipse (with width $w$ and height $h$ that depend on $b$) and from the angular constraint between the semihard parton and ridge particle prescribed by a Gaussian width $\sigma$ determined earlier in treating the ridge formation for nuclear density not too low \cite{ch2}, it is found that
\begin{eqnarray}
S(\phi,b) = h[E(\theta_2,\alpha) - E(\theta_1,\alpha)], \label{10}
\end{eqnarray}
where $E(\theta_i,\alpha)$ is the elliptic integral of the second kind with $\alpha=1-w^2/h^2$ and
\begin{eqnarray}
\theta_i = \tan^{-1} \left({h\over w}\tan\phi_i \right), \quad \phi_1 = \phi - \sigma, \quad \phi_2 = \phi + \sigma, \label{11}
\end{eqnarray}
for $\phi_i \leq \pi/2$, and an analytic continuation of it for $\phi_2 > \pi/2$. Thus $S(\phi,b)$ is completely calculable for any given $b$, and $R^h(p_T,\phi,b)$ is proportional to it.
We can now rewrite Eq.\ (\ref{9}) unambiguously as
\begin{eqnarray}
\rho_1^h(p_T,\phi,b) = B^h(p_T,b) + {S(\phi,b)\over \bar S (b)} \bar R^h(p_T,b) \label{12}
\end{eqnarray}
where
\begin{eqnarray}
\bar S (b) = (2/\pi) \int_0^{\pi/2} d\phi S(\phi,b) \label{13}
\end{eqnarray}
and $\bar R^h(p_T,b)$ is a similar average of $R^h(p_T,\phi,b)$. According to Eqs.\ (\ref{6}) and (\ref{7}) the inclusive distributions $\bar \rho_1^h(p_T,b)$ should share the common exponential factor $\exp(-p_T/T)$, for $h = \pi$ and $p$, as for quarks. That does not take into consideration the enhancement of pions at very small $p_T$ due to resonance decay. We account for it by a phenomenological term $u(p_T,b)$, and write
{\begin{eqnarray}
\bar \rho_1^{\pi}(p_T,b) &=& {\cal N}_{\pi} (b)[1 + u(p_T,b)]e^{-p_T/T}, \label{14} \\
\bar \rho_1^p(p_T,b) &=& {\cal N}_p(b){p_T^2\over m_T}e^{-p_T/T}, \label{15}
\end{eqnarray}
where the resonance effect on the proton is neglected because of baryon-number conservation. These expressions are for the left-hand side of Eq.\ (\ref{12}) after $\phi$ averaging. The base term $B^h(p_T,b)$ on the right side is the soft component without the contribution from semihard scattering near the surface and should have the same common structure as in Eqs.\ (\ref{6}) and (\ref{7}) due to thermal parton recombination, except that the inverse slope is lower without the enhancement by the energy loss from the semihard partons. We can therefore write
\begin{eqnarray}
B^{\pi}(p_T,b) &=& {\cal N}_{\pi}(b)[1 + u(p_T,b)]e^{-p_T/T_B}, \label{16} \\
B^p(p_T,b) &=& {\cal N}_p(b){p_T^2\over m_T}e^{-p_T/T_B}, \label{17}
\end{eqnarray}
where $T_B < T$. It then follows that
\begin{eqnarray}
\bar R^{\pi}(p_T,b) &=& {\cal N}_{\pi}(b)[1+u(p_T,b)] \bar R_0(p_T) \label{18} \\
\bar R^p(p_T,b) &=& {\cal N}_p(b){p_T^2\over m_T} \bar R_0(p_T), \label{19}
\end{eqnarray}
where
\begin{eqnarray}
\bar R_0(p_T)=e^{-p_T/T}-e^{-p_T/T_B}=e^{-p_T/T_B}(e^{p_T/\tilde{T}}-1) \quad \label{19a} \\
{1\over \tilde{T}} = {1\over T_B} - {1\over T} = {\Delta T\over T_BT}, \qquad \Delta T = T - T_B. \qquad \label{20}
\end{eqnarray}
There are two undetermined inverse-slopes: $T_B$ and $T$, common for both $\pi$ and $p$. They are for single-particle inclusive distributions, so only $T$ is directly observable. We postpone phenomenology to a later section. In ridge analysis using triggered events for two-particle correlation the two corresponding inverse slopes are separately measured \cite{bia}. Here, however, we are dealing with single-particle distributions. The difference between $T_B$ and $T$ has to do with ridges and their effect on the $\phi$ distribution. Thus we expect $\Delta T$ to be related to azimuthal asymmetry, a topic we next turn to.
\section{Quadrupole Moments of $\phi$ Asymmetry}
This topic is usually referred to as elliptic flow, a terminology that is rooted in hydrodynamics. Since we have not used hydro in the previous section, it is more appropriate to use the unbiased language initiated in Ref.\ \cite{tk}, and call it azimuthal quadrupole. It is the familiar $v_2$ that is defined by
\begin{eqnarray}
v_2^h(p_T,b) = \langle \cos2\phi \rangle_{\rho_1}^h = {\int_0^{2\pi} d\phi \cos2\phi\rho_1^h(p_T,\phi,b)\over \int_0^{2\pi} d\phi\rho_1^h(p_T,\phi,b)}. \label{21}
\end{eqnarray}
Using Eqs.\ (\ref{12}) - (\ref{20}) yields
\begin{eqnarray}
v_2^h(p_T,b) &=& {[2\bar R(p_T,b)/\pi\bar S(p_T,b)]\int_0^{\pi/2} d\phi \cos2\phi S(\phi,b) \over B(p_T,b)+\bar R(p_T,b)} \nonumber \\
&=& {\langle \cos2\phi \rangle_S\over Z^{-1}(p_T) + 1}, \label{22}
\end{eqnarray}
where
\begin{eqnarray}
\langle \cos2\phi \rangle_S &=& {2/\pi\over \bar S (b)} \int_0^{\pi/2} d\phi \cos2\phi S(\phi,b), \label{23} \\
Z(p_T) &=& e^{p_T/\tilde T} - 1. \label{24}
\end{eqnarray}
These equations are remarkable in that the $b$ dependence resides entirely in Eq.\ (\ref{23}) and the $p_T$ dependence entirely in Eq.\ (\ref{24}); furthermore, there is no explicit dependence on the hadron type nor the resonance term represented by $u(p_T,b)$. As we have noted at the end of the preceding section, $T$ can be determined by the $p_T$ spectra, but $T_B$ is not directly observable. However, the quadrupole is measurable, so it can constrain $\tilde T$ and therefore $T_B$. In short, the two parameters $T$ and $T_B$ can be fixed by fitting the data on $\bar \rho_1^h(p_T,b)$ and $v_2^h(p_T,b)$.
Without using a model to describe the evolution of the dense medium, it is clear that we cannot predict the values of $T$ and $T_B$.
However, our aim is to discover how far one can go without using such a model.
Neither $T$ nor $T_B$ depend on $\phi$. Yet non-trivial $v_2^h(p_T,b)$ can be obtained because of the presence of the ridge term in Eq.\ (\ref{12}). If phenomenology turns out to support this interpretation of azimuthal asymmetry, as we shall do in the next section, then the ridges induced by undetected semihard partons play a more important role in giving rise to the $\phi$ dependence in inclusive single-particle distribution than hydro expansion that is based on assuming equilibration to be completely at a later time without semihard scattering.
From Eqs.\ (\ref{10}) and (\ref{23}) we can calculate $\left< \cos 2\phi\right>_S$ and obtain its dependence on $b$. For the initial elliptical configuration the width and height are
\begin{eqnarray}
w=1-b/2, \qquad h=(1-b^2/4)^{1/2} , \label{25}
\end{eqnarray}
where all lengths are in units of the nuclear radius $R_A$. Setting the Gaussian width $\sigma$ between the azimuthal angle $\phi_1$ of the semihard parton and $\phi_2$ of the ridge particle to be $\sigma=0.33$ \cite{ch2}, we determine $\left< \cos 2\phi\right>_S$ as shown in Fig.\ 1(a).
\begin{figure}[tbph]
\includegraphics[width=.4\textwidth]{fig1a.eps}
\includegraphics[width=.5\textwidth]{fig1b.eps}
\caption{(Color online) (a) Average of $\cos2\phi$ weighted by $S(\phi,b)$ vs impact parameter in units of $R_A$. (b) Common dependence of $v_2^h(p_T,b)$ on $N_{part}$ for various $p_T$, shifted vertically for comparison. The diamond and square points are horizontally shifted slightly from the points in circles to aid visualization. The solid line is from $\left<\cos2\phi\right>_S$ shown in (a), but rescaled and plotted in terms of $N_{part}$. The data are from Ref.\ \cite{ja}.}
\end{figure}
According to Eq.\ (\ref{22}) $\left< \cos2\phi\right>_S$ contains all the $b$ dependence of $v_2^h(p_T,b)$ for any $p_T$ in the soft region. To check how realistic that is phenomenologically, we show first in Fig.\ 1(b) the data on $v_2^h(p_T,N_{part})$ for three $p_T$ values from Ref.\ \cite{ja}, but shifted vertically so that they agree with the data for $p_T=0.975$ GeV/c for most of large $N_{part}$. The diamond and square points are slightly shifted horizontally to spread out the overlapping points for the sake of visual distinguishability. The fact that their dependencies on $N_{part}$ are so nearly identical is remarkable in itself. The solid line is a reproduction of the curve in Fig.\ 1(a) but plotted in terms of $N_{part}$, and reduced in normalization by a factor 0.25 to facilitate the comparison with the data points. For $N_{part}>100$ the line agrees with the data on $v_2$ very well, thus proving the factorizability of $p_T$ and $b$ dependencies of Eq.\ (\ref{22}). For $N_{part}<100$, corresponding to $b/R_A>1.3$ or centrality $>40$\%, there is disagreement which is expected because the density is too low in peripheral collisions to justify the simple formula in Eq.\ (\ref{22}). A density-dependent correction is considered in Ref.\ \cite{hz}, but will not be repeated here. Our focus in this paper is on the inclusive ridge, so we proceed to phenomenology on the basis that the formalism given above is valid for central and mid-central collisions at $N_{part}>100$. To have a compact analytic expression for $S(\phi,b)$ as given in Eqs.\ (\ref{10}) and (\ref{11}) to summarize the $\phi$ dependence is not only economical, but also provides a succinct feature to distinguish the ridge from the base components in Eq.\ (\ref{12}).
\section{Phenomenology}
We now determine the parameters in our model through phenomenology. A success in fitting all the relevant data can give support to our approach that emphasizes issues not considered in the standard model \cite{rv}.
Our first task is to determine the inverse slope $T$ that is shared by ${\cal T}(q), \bar\rho_1^\pi(p_T,b)$ and
$\bar\rho_1^p(p_T,b)$. Since the normalization factors in Eqs.\ (\ref{5}), (\ref{14}) and (\ref{15}) have not yet been specified, we consider first a particular centrality, 20-30\%, and fit the $p_T$ dependence of the proton spectrum for $p_T<2$ GeV/c, as shown in Fig.\ 2, and obtain
\begin{eqnarray}
T=0.283\ {\rm GeV}. \label{26}
\end{eqnarray}
Note that the one-parameter fit (apart from normalization) is very good compared to the data from Ref.\ \cite{sa}. It demonstrates that the proton is produced in that $p_T$ range by thermal partons and that the flattening of the spectrum at low $p_T$ is due to the prefactor $p_T^2/m_T$ arising from the proton wave function.
\begin{figure}[tbph]
\includegraphics[width=.45\textwidth]{fig2.eps}
\caption{Proton spectrum at $y\approx 0$ averaged over $\phi$ (hence, no $1/2\pi$ factor) at 20-30\% centrality. The solid line is a fit of the data by Eqs.\ (\ref{15}) and (\ref{26}) with free adjustment of normalization. The data are from Ref.\ \cite{sa}.}
\end{figure}
Having determined $T$, we next consider the pion spectrum $\bar\rho_1^\pi(p_T,b)$. According to Eq.\ (\ref{14}) it has the same exponential factor as does $\bar\rho_1^p(p_T,b)$, but has also an additional factor $[1+u(p_T,b)]$ due to resonance decay. We show in Fig.\ 3 the data from PHENIX \cite{sa} on the pion distribution\ for 20-30\% centrality; the $\exp(-p_T/T)$ factor is shown by the dashed line, the normalization being adjusted to fit (and to be discussed later). For $p_T>1$ GeV/c they agree very well, demonstrating the validity of the common $T$. For $p_T<1$ GeV/c there is resonance contribution to the pion spectrum which we cannot predict. Thus we fit the low-$p_T$ region by the addition of a term $\exp(-p_T/T_r)$, shown by the dash-dotted line, corresponding to $T_r=0.174$ GeV. The sum depicted by the solid line agrees with the data perfectly. The point of this exercise is mainly to show that the common $\exp(-p_T/T)$ behavior is valid for pion as for proton, but the reality of resonance contribution for $p_T<1$ GeV/c obscures that commonality. Converting the resonance term to the form given in Eq.\ (\ref{14}) we write
\begin{eqnarray}
u(p_T,b)=u_0(b) e^{-p_T/T_0} , \label{27}
\end{eqnarray}
where $T_0=0.45$ GeV and $u_0=3.416$ for 20-30\% centrality. We do not regard this $u$ term as a fundamental part of our model; we attach the factor $[1+u(p_T,b)]$ to all expressions of the pion distribution s, as in Eqs.\ (\ref{16}) and (\ref{18}). Of more significance is the role that $T$ has played in the phenomenology, and so far $T_B$ has played no role.
\begin{figure}[tbph]
\includegraphics[width=.45\textwidth]{fig3.eps}
\caption{Pion spectrum showing $e^{-p_T/T}$ by the dashed line, and the resonance contribution by the dash-dotted line. The sum is in solid line. The data are from Ref.\ \cite{sa}.}
\end{figure}
$T_B$ is not directly related to any observable spectrum, since it describes the $p_T$ dependence of the base $B^h(p_T,b)$ that lies under the ridge. The important concept we advance here is that it is $\phi$ independent, and that $R^h(p_T,\phi,b)$ carries all the $\phi$ dependence. Thus we turn to $v_2^h(p_T,b)$ in Eq.\ (\ref{22}) and examine its $p_T$ dependence for both $h=\pi$ and $p$. In order to emphasize the universality between $\pi$ and $p$, we consider $v_2^h$ versus the transverse kinetic energy $E_T$, for $E_T<0.8$ GeV, where
\begin{eqnarray}
E_T(p_T)=m_T(p_T)-m_h . \label {28}
\end{eqnarray}
We adopt the ansatz that $p_T$ is to be replaced by $E_T$ in Eq.\ (\ref{24}) so as to account for the mass effect, i.e.,
\begin{eqnarray}
Z(p_T)=e^{E_T(p_T)/\tilde T}-1 , \label{29}
\end{eqnarray}
where $\tilde T$ is as given in Eq.\ (\ref{20}). In Fig.\ 4 is shown the data from Ref.\ \cite{ja} when $v_2^h$ is plotted against $E_T$ for 20-30\% centrality. We fit the data points for both $h=\pi$ and $p$ by Eqs.\ (\ref{22}) and (\ref{29}) with the choice
\begin{eqnarray}
T_B=0.253\pm 0.003\ {\rm GeV} , \label{30}
\end{eqnarray}
\begin{figure}[tbph]
\vspace*{-.5cm}
\includegraphics[width=.45\textwidth]{fig4.eps}
\caption{(Color online) $v_2^h$ for $h=\pi$ and $p$. The shaded region corresponds to $T_B=0.253\pm0.003$ GeV. The data are from Ref.\ \cite{ja}.}
\end{figure}
\noindent which is represented by the shaded region in Fig.\ 4. The upper boundary of that region is for $T_B=0.25$ GeV that fits the pion $v_2$ almost perfectly, and the lower boundary is for $T_B=0.256$ GeV that fits well the proton $v_2$.
It is evident that $v_2$ is very sensitive to $T_B$ due to the exponential factor in Eq.\ (\ref{29}), yet the data support a common value for $T_B$ to within 1-2\% deviation for pion and proton production.
One cannot expect an accuracy better than that in the universality of $v_2^h$ for $h=\pi$ and $p$. We regard this result to be remarkable, since the normalization of $v_2^h$ is fixed by Eq.\ (\ref{22}) without freedom of adjustment. Note that we have not used any more parameters besides $T$ and $T_B$ to accomplish this, which is a fitting procedure not more elaborate than the hydro approach where the initial condition and viscosity are adjusted.
So far we have concentrated on 20-30\% centrality partly because we want to separate the $p_T$ and $\phi$ dependencies from the issue of centrality dependence, and partly because $v_2^h(p_T,b)$ is large at 20-30\% centrality for low $p_T$. To extend our consideration to other centralities, we fix $T$ and $T_B$ at the values obtained in Eqs.\ (\ref{26}) and (\ref{30}) so that $Z(p_T)$ is no longer adjustable. The centrality dependence of $v_2^h(\pt,b)$\ is then examined using Eq.\ (\ref{22}). Figure 5 shows the results for different centrality bins for both $h=\pi$ and $p$. The shaded regions due to the uncertainty in Eq.\ (\ref{30}) become narrower in more central collisions. The agreement with data from STAR \cite{ja} is evidently very good. Since there has been no more adjustment of free parameters to achieve that, we find substantial support from Fig.\ 5 for our view that the $\phi$ dependence arises entirely from the ridge component in the inclusive distribution. This raises serious question on whether viscous hydrodynamics is the only acceptable description of heavy-ion collisions, if the reproduction of $v_2^h(\pt,b)$\ is the primary criterion for the success of a model.
\begin{figure}[tbph]
\hspace*{-.5cm}
\includegraphics[width=.5\textwidth]{fig5.eps}
\caption{(Color online) Same as in Fig.\ 4 for four centrality bins.}
\end{figure}
It is possible to further improve the agreement between the values of $v_2^h$ for pion and proton in Fig.\ 5 if those figures are replotted in accordance to the idea of quark number scaling (QNS), i.e., $v_2^h/n_h$ vs $E_T/n_h$, where $n_h$ is the number of constituent quarks in hadron $h$ \cite{mv1,kcg}. As we have considered QNS and its breaking in the recombination model before \cite{chy}, we do not revisit that problem here, especially since our main goal to use $v_2$ to constrain $T_B$ has already been accomplished.
Having obtained the correct centrality dependence of $v_2^h(\pt,b)$\ that is calculable, we now consider the centrality dependence of the inclusive spectra $\bar\rho_1^h(p_T,b)$. We note that the unknown
normalization factors ${\cal N}_\pi(b)$ and ${\cal N}_p(b)$ in Eqs.\ (\ref{14}) and (\ref{15}) never enter into the calculation of $v_2^h(\pt,b)$\ because of cancellation, but for $\bar\rho_1^h(p_T,b)$ they must be reckoned with. As remarked after Eq.\ (\ref{7}), ${\cal N}_\pi(b)$ and ${\cal N}_p(b)$
are proportional to $C^2$ and $C^3$, respectively, due to $q\bar q$ and $qqq$ recombination. The magnitude $C$ of the thermal partons depends on $b$ in a way that cannot be reliably calculated. By phenomenology on the pion spectrum it was previously estimated for $p_T>1.2$ GeV/c \cite{hz}, but that is inadequate for our purpose here; moreover, ${\cal N}_\pi(b)$ and ${\cal N}_p(b)$ have different statistical factors that can depend on $b$ because of resonances. We give here direct parametrizations of the normalization factors in terms of $N_{part}$
\begin{eqnarray}
{\cal N}_\pi(N_{part})&=&0.516 N_{part}^{1.05} , \label{31} \\
{\cal N}_p(N_{part})&=&0.149 N_{part}^{1.18} , \label{32} \\
u_0(N_{part}) &=& 2.8+0.003 N_{part} . \label{33}
\end{eqnarray}
The parameters are determined by fitting the centrality dependence to be shown, but the essence of our prediction is in $p_T$ and $\phi$ dependencies that are not adjustable. Using the above in Eqs.\ (\ref{14}) and (\ref{15}) we obtain the curves in Fig.\ 6 (a) pion and (b) proton for three centrality bins. They agree with the data from PHENIX \cite{sa} very well over a wide range of low $p_T$. In all those curves $T$ is kept fixed at 0.283 GeV, thus reaffirming our point that both pions and protons are produced by the same set of thermal partons despite the apparent differences in the shapes of their $p_T$ dependencies.
\begin{figure}[tbph]
\includegraphics[width=.45\textwidth]{fig6a.eps}
\hspace{.5cm}
\includegraphics[width=.45\textwidth]{fig6b.eps}
\caption{Inclusive spectra at three centralities for (a) pion and (b) proton. The data are from Ref.\ \cite{sa}.}
\end{figure}
\section{Inclusive Ridge Distribution}
It is now opportune for us to revisit the two-component description of the single-particle distribution\ and focus on the ridge component, in particular. As stated explicitly in Eq.\ (\ref{12}), the $\phi$ dependence separates the $B^h(p_T,b)$ and $\bar R^h(p_T,b)$ components, the former being described by Eqs.\ (\ref{16}) and (\ref{17}), the latter by Eqs.\ (\ref{18}) and (\ref{19}). Upon averaging over $\phi$, we have
\begin{eqnarray}
\bar\rho_1^h(p_T,b)=B^h(p_T,b)+\bar R^h(p_T,b) . \label{34}
\end{eqnarray}
\begin{figure}[tbph]
\includegraphics[width=.45\textwidth]{fig7a.eps}
\hspace{.5cm}
\includegraphics[width=.45\textwidth]{fig7b.eps}
\caption{Inclusive distributions for pion showing the base (B) component by dashed line and ridge (R) component by dash-dotted line for (a) 0-5\% and (b) 20-30\%. The solid line is their sum. The data are from Ref.\ \cite{sa}.}
\end{figure}
\noindent Since the exponential factors are the same for $h=\pi$ and $p$, let us consider only the pion distribution\ specifically. In Fig.\ 7 we show $B$ and $R$ components by dashed and dash-dotted lines, respectively, for (a) 0-5\% and (b) 20-30\%. It is in those figures that we exhibit the basic difference between our description of inclusive spectra and those of others. Inclusive ridge represented by $R$ is always present in the single-particle distribution\ whether or not an experiment chooses to do correlation measurement to examine the ridge. Semihard scattering is unavoidable in any nuclear collisions at high energy. Its effect on soft partons is therefore also unavoidable. We quantify the effect by the $R$ component which is determined by the azimuthal anisotropy that is well reproduced in Fig.\ 5. Here in Fig.\ 7 we see it rising above the $\phi$-independent base $B$ component when $p_T$ is higher than 1.4 GeV/c. It is a consequence of the recombination of enhanced thermal partons. For $p_T>3$ GeV/c in addition to the inclusive ridge the jet component of the semihard partons themselves manifests in the spectra in the form of thermal-shower recombination that characterizes the intermediate-$p_T$ region. Thus we have a smooth transition from low- to intermediate-$p_T$ regions by recognizing the importance of the inclusive ridge component.
It is observed that the dash-dotted lines in Fig.\ 7 are not exactly straight because the ridge component is not exponential in $p_T$. However, for $p_T>1$ GeV/c, $\bar R^\pi(p_T,b)$ can be well approximated by pure exponential.
In Fig.\ 8 we show by the solid line the $p_T$ dependence of $\bar R_0(p_T)$, defined in Eq.\ (\ref{19a}); it is
the part of the ridge distribution s $\bar R^h(p_T,b)$ in Eqs.\ (\ref{18}) and (\ref{19}) that is common for $h=\pi$ and $p$ and is independent of $b$. From the values of $T$ and $T_B$ that we now know, we have $\tilde T=2.39$ GeV. The (red) dashed line is a straight-line approximation of the solid curve for $p_T>1$ GeV/c by
\begin{eqnarray}
\bar R_0(p_T)\approx R_0 e^{-p_T/T'}, \qquad T'=0.326\ {\rm GeV} . \label{36}
\end{eqnarray}
Thus the ridge distribution\ is harder than the inclusive distribution\ characterized by $T=0.283$ GeV. This is a property that is known from triggered ridges \cite{bia}, but now it is for untriggered inclusive ridge.
\begin{figure}[tbph]
\includegraphics[width=.45\textwidth]{fig8.eps}
\caption{(Color online) The $p_T$ dependence of $\bar R_0(p_T)$ defined in Eq.\ (\ref{19a}), represented by the solid line. The (red) dashed line is a straight-line approximation for $p_T>1$ GeV/c, expressed by Eq.\ (\ref{36}).}
\end{figure}
The enhancement of $T'$ over $T$ is an important point to note. Physically, it means that the ridge is a consequence of the passage of semihard partons through the medium, whose energy losses enhance the thermal partons in the vicinities of the trajectories. We know that the enhancement factor is $Z(p_T)$, which has the necessary $p_T$ dependence to render $v_2^h(p_T,b)$ to be in good agreement with the quadrupole data. In particular, the property that $Z(p_T)\to 0$ as $p_T\to 0$ is essential to guarantee that $v_2^h(p_T,b)\to 0$ in the same limit. The effect of $Z(p_T)$ at larger $p_T$ is to increase $T_B$ to $T'$. Although $Z(p_T)$ increases exponentially, its net effect on $\bar R_0(p_T)$ is suppressed by $e^{-p_T/T_B}$. The effective inverse slope $T'$ for $p_T>1$ GeV/c is larger than $T$ of the inclusive by 43 MeV, roughly the same as what Putschke reported on the first discovery of ridge, where the triggered ridge has $T'$ larger than that of the inclusive by 45 MeV \cite{jp}.
There are, however, subtle differences between triggered and untriggered ridges. Experimentally, it is necessary to do correlation measurements to learn about the properties of the ridge, which is extracted by a subtraction scheme. The inverse slope $T'$ can be compared to the inclusive $T$ of the background.
For single-particle distribution\ the only measurable inverse slope is $T$ for the inclusive spectrum.
Theoretically, we assert that ridges do not disappear just because triggers are not used.
The inverse slope $T'$ for $\bar R_0(p_T)$ cannot be measured directly. It is larger than both $T$ and $T_B$ because $\bar R_0(p_T)$ is the difference between the two exponentials for $\bar\rho_1$ and $B$,
represented by the middle term in Eq.\ (\ref{19a}), which vanishes as $p_T\to 0$.
A physically more sensible way to compare the various inverse slopes is to recognize that $T'$ is significantly larger than $T_B$ because of the enhancement effect due to semihard scattering, and that $T$ is the effective slope of the inclusive distribution, $B+\bar R$, that is measurable and is between $T_B$ and $T'$.
Although our concern in this paper has been restricted to the midrapidity region, the physics of inclusive ridge can be extended to non-vanishing pseudo-rapidity $\eta$. In Ref.\ \cite{ch} a phenomenological relationship is found between the triggered ridge distribution\ in $\Delta\eta$ and the inclusive distribution\ in $\eta$ with the implication that there is no long-range longitudinal correlation. However, there can be transverse correlation due to transverse broadening of forward (or backward) soft partons as they move through the conical vicinity of the semihard partons. The enhancement of the thermal partons due to energy loss is just as we have described in this paper. Indeed, the term representing the enhanced $p_T$ distribution\ in Ref.\ \cite{ch} is essentially identical to that expressed in Eq.\ (\ref{19a}). Similar consideration has also been used in the explanation of the ridge structure found at LHC \cite{cms,hy2}. In this paper we have presented the most detailed quantitative analysis of the RHIC data in the formalism of inclusive ridge that sets the foundation for the ridges at $|\Delta\eta|>0$.
\section{Conclusion}
Our study of inclusive ridge distribution s has consolidated earlier exploratory work with firm phenomenological support, and therefore succeeded in extending the hadronization formalism from intermediate-$p_T$ region to below 2 GeV/c, exposing thereby an aspect of physics that has not been included in other approaches. The effect of semihard scattering on soft partons is accounted for by the ridge component whose azimuthal behavior is totally characterized by $S(\phi,b)$; it is a calculable quantity bounding the surface segment through which semihard partons can contribute to the formation of a ridge particle at $\phi$. In an earlier paper \cite{hz} we showed the connection between $S(\phi,b)$ and the dependence of the triggered ridge yield on the $\phi_s$ of the trigger angle relative to the reaction plane. Now, we have exhibited the central role that $S(\phi,b)$ plays in determining the azimuthal quadrupole $v_2^h(p_T,b)$ of inclusive distribution s. Thus the inclusive ridge distribution\ that we have advanced in this series of work serves as a bridge between the single-particle distribution\ and the two-particle correlation. Since semihard partons are copiously produced before thermalization is complete, it is an aspect of physics that should not be ignored. The success in fitting $v_2^h(p_T,b)$ for all central and mid-central collisions and for both $\pi$ and $p$ by one parameter $T_B$ therefore leads to claim of relevance as much as viscous hydro does.
Another attribute of our approach is to unify the production of pions and protons in one hadronization scheme based on the recombination of enhanced thermal partons so that their spectra have the same inverse slope $T$ despite apparent differences in the low-$p_T$ data. That same scheme when extended to $p_T\sim 3$ GeV/c explains readily the observed large $p/\pi$ ratio. Since the ridge components in our formalism are the same for $\pi$ and $p$, we can predict that the $p/\pi$ ratio is also large in the triggered ridge as in the inclusive.
The property about ridge that most investigators are concerned about is the large $\Delta\eta$ range found in correlation experiments. That is an aspect of the problem that has been addressed in Ref.\ \cite{ch}. Our focus in this paper is on the hidden aspect of the ridge that is not easily detected, but is pervasive because it is in the inclusive distribution. Phenomenological success found in this paper puts the idea on solid footing. If the concept of inclusive ridge is important at RHIC, then its relevance at LHC will be predominant. Since the structure of $v_2^h(p_T,b)$ expressed in terms of $S(\phi,b)$ in Eq.\ (\ref{22}) is independent of initial density, viscosity, or even collision energy, except $\tilde T$, we would expect $v_2$ measured at LHC to be essentially similar to what is shown in Fig.\ 5 for both $\pi$ and $p$. A preliminary look at the data from ALICE \cite{aam} leads us to believe that such an expectation may not be unrealistic.
\section*{Acknowledgment}
This work was supported, in part, by the U.\ S.\ Department of Energy under Grant No. DE-FG02-96ER40972 and by the Scientific Research Foundation for Young Teachers, Sichuan University under No. 2010SCU11090 and Key Laboratory of Quark and Lepton Physics under Grant No. QLPL2009P01.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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Q: How do I create a Django superuser in ssh? I am deploying Django with PostgreSQL in Azure for a school project, per this tutorial. I am working with Django 2.1.2, and Python 3.7.5.
In the "Run Database Migrations" step of the tutorial, I am instructed to open an SSH session and run the following commands:
cd site/wwwroot
source /antenv/bin/activate
python manage.py migrate
python manage.py createsuperuser
When I run the 'createsuperuser' command, I expect a prompt for a username, email address, and password, but I am not prompted for any of these. Instead, the SSH session prompts me for another input, as below:
As such, I am unable to log in to my Django installation. When I attempt a login, I receive the following error message:
The traceback is at the end of this message.
I expect to be able to create a superuser in Django, and use that superuser account to configure the app. Instead, I receive the ProgrammingError message and am unable to proceed.
Any help would be gratefully received. I have Googled the various error message, searched StackOverflow, and searched YouTube for tutorials, and I have reached a roadblock.
Environment:
Request Method: POST
Request URL: http://nicholas-blog.azurewebsites.net/admin/login/?next=/admin/
Django Version: 2.1.2
Python Version: 3.7.5
Installed Applications:
['polls.apps.PollsConfig',
'django.contrib.admin',
'django.contrib.auth',
'django.contrib.contenttypes',
'django.contrib.sessions',
'django.contrib.messages',
'django.contrib.staticfiles']
Installed Middleware:
['django.middleware.security.SecurityMiddleware',
'whitenoise.middleware.WhiteNoiseMiddleware',
'django.contrib.sessions.middleware.SessionMiddleware',
'django.middleware.common.CommonMiddleware',
'django.middleware.csrf.CsrfViewMiddleware',
'django.contrib.auth.middleware.AuthenticationMiddleware',
'django.contrib.messages.middleware.MessageMiddleware',
'django.middleware.clickjacking.XFrameOptionsMiddleware']
Traceback:
File "/antenv/lib/python3.7/site-packages/django/db/backends/utils.py" in _execute
85. return self.cursor.execute(sql, params)
The above exception (relation "auth_user" does not exist
LINE 1: ...user"."is_active", "auth_user"."date_joined" FROM "auth_user...
^
) was the direct cause of the following exception:
File "/antenv/lib/python3.7/site-packages/django/core/handlers/exception.py" in inner
34. response = get_response(request)
File "/antenv/lib/python3.7/site-packages/django/core/handlers/base.py" in _get_response
126. response = self.process_exception_by_middleware(e, request)
File "/antenv/lib/python3.7/site-packages/django/core/handlers/base.py" in _get_response
124. response = wrapped_callback(request, *callback_args, **callback_kwargs)
File "/antenv/lib/python3.7/site-packages/django/views/decorators/cache.py" in _wrapped_view_func
44. response = view_func(request, *args, **kwargs)
File "/antenv/lib/python3.7/site-packages/django/contrib/admin/sites.py" in login
398. return LoginView.as_view(**defaults)(request)
File "/antenv/lib/python3.7/site-packages/django/views/generic/base.py" in view
68. return self.dispatch(request, *args, **kwargs)
File "/antenv/lib/python3.7/site-packages/django/utils/decorators.py" in _wrapper
45. return bound_method(*args, **kwargs)
File "/antenv/lib/python3.7/site-packages/django/views/decorators/debug.py" in sensitive_post_parameters_wrapper
76. return view(request, *args, **kwargs)
File "/antenv/lib/python3.7/site-packages/django/utils/decorators.py" in _wrapper
45. return bound_method(*args, **kwargs)
File "/antenv/lib/python3.7/site-packages/django/utils/decorators.py" in _wrapped_view
142. response = view_func(request, *args, **kwargs)
File "/antenv/lib/python3.7/site-packages/django/utils/decorators.py" in _wrapper
45. return bound_method(*args, **kwargs)
File "/antenv/lib/python3.7/site-packages/django/views/decorators/cache.py" in _wrapped_view_func
44. response = view_func(request, *args, **kwargs)
File "/antenv/lib/python3.7/site-packages/django/contrib/auth/views.py" in dispatch
61. return super().dispatch(request, *args, **kwargs)
File "/antenv/lib/python3.7/site-packages/django/views/generic/base.py" in dispatch
88. return handler(request, *args, **kwargs)
File "/antenv/lib/python3.7/site-packages/django/views/generic/edit.py" in post
141. if form.is_valid():
File "/antenv/lib/python3.7/site-packages/django/forms/forms.py" in is_valid
185. return self.is_bound and not self.errors
File "/antenv/lib/python3.7/site-packages/django/forms/forms.py" in errors
180. self.full_clean()
File "/antenv/lib/python3.7/site-packages/django/forms/forms.py" in full_clean
382. self._clean_form()
File "/antenv/lib/python3.7/site-packages/django/forms/forms.py" in _clean_form
409. cleaned_data = self.clean()
File "/antenv/lib/python3.7/site-packages/django/contrib/auth/forms.py" in clean
196. self.user_cache = authenticate(self.request, username=username, password=password)
File "/antenv/lib/python3.7/site-packages/django/contrib/auth/__init__.py" in authenticate
73. user = backend.authenticate(request, **credentials)
File "/antenv/lib/python3.7/site-packages/django/contrib/auth/backends.py" in authenticate
16. user = UserModel._default_manager.get_by_natural_key(username)
File "/antenv/lib/python3.7/site-packages/django/contrib/auth/base_user.py" in get_by_natural_key
44. return self.get(**{self.model.USERNAME_FIELD: username})
File "/antenv/lib/python3.7/site-packages/django/db/models/manager.py" in manager_method
82. return getattr(self.get_queryset(), name)(*args, **kwargs)
File "/antenv/lib/python3.7/site-packages/django/db/models/query.py" in get
393. num = len(clone)
File "/antenv/lib/python3.7/site-packages/django/db/models/query.py" in __len__
250. self._fetch_all()
File "/antenv/lib/python3.7/site-packages/django/db/models/query.py" in _fetch_all
1186. self._result_cache = list(self._iterable_class(self))
File "/antenv/lib/python3.7/site-packages/django/db/models/query.py" in __iter__
54. results = compiler.execute_sql(chunked_fetch=self.chunked_fetch, chunk_size=self.chunk_size)
File "/antenv/lib/python3.7/site-packages/django/db/models/sql/compiler.py" in execute_sql
1065. cursor.execute(sql, params)
File "/antenv/lib/python3.7/site-packages/django/db/backends/utils.py" in execute
100. return super().execute(sql, params)
File "/antenv/lib/python3.7/site-packages/django/db/backends/utils.py" in execute
68. return self._execute_with_wrappers(sql, params, many=False, executor=self._execute)
File "/antenv/lib/python3.7/site-packages/django/db/backends/utils.py" in _execute_with_wrappers
77. return executor(sql, params, many, context)
File "/antenv/lib/python3.7/site-packages/django/db/backends/utils.py" in _execute
85. return self.cursor.execute(sql, params)
File "/antenv/lib/python3.7/site-packages/django/db/utils.py" in __exit__
89. raise dj_exc_value.with_traceback(traceback) from exc_value
File "/antenv/lib/python3.7/site-packages/django/db/backends/utils.py" in _execute
85. return self.cursor.execute(sql, params)
Exception Type: ProgrammingError at /admin/login/
Exception Value: relation "auth_user" does not exist
LINE 1: ...user"."is_active", "auth_user"."date_joined" FROM "auth_user...
^
A: python manage.py makemigrations
python manage.py migrate
You might have forgotten to do makemigrations.
A: If it's not asking for username then it might be your shell is not allowing it. You can use --noinput for that
python manage.py createsuperuser --noinput --username testuser --email test@gmail.com
You cannot set the password though, it will be set automatically you'd have to change it from admin panel.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,574
|
PARTY HISTORY
Neil Morrison, Bruce Beetham, Gary Knapp
Social Credit MPs 1984
Fundamental Tenets
Party Objectives
H.M Rushworth
Vernon Cracknell
The first noteworthy proponent of Social Credit in New Zealand was H. M. Rushworth, Country Party MP for the Bay of Islands from 1928 to 1938. In the House he proposed major Social Credit arguments, namely, that credit and the monetary system should be controlled by the people through the Government; that a gap existed between purchasing power and production costs; that the gap could be bridged by the issue of credit; and that Parliamentary party control should not be unduly rigid.
At that time the New Zealand Social Credit Association was an educational body, prevented by its constitution from being a political party. In 1936 it was a major force in getting the first Labour government, who, under Michael Joseph Savage, campaigned for monetary reform, elected. Labour used social credit to good effect, using the newly nationalised Reserve Bank to fund the building of 40,000 state houses and to provide low interest overdrafts for the dairy board and other producer boards, but it didn't last.
NZ Social Credit Political League
In May 1953, the New Zealand Social Credit Political League was formed. Led by W.B. Owen it contested 79 seats in the 1954 Parliamentary elections, polling 11.31% of all votes cast, but did not win a seat. The election policies were similar to those put forward by Rushworth 20 years previously, but they also included pledges to reduce taxation progressively which would lead to falling costs of production and an increase in the purchasing power of incomes; to abolish social security tax without reducing benefits; to support private enterprise; and to enact legislation to eliminate unfair trade practices.
In 1957 the party gained 7.21% and 1960, under new leader P. H. Matthews, 8.62%. Additional policies included setting up a national monetary authority to ensure a balance between production and income; reducing bureaucratic controls; encouraging farming; reducing school classes sizes; improving health services; encouraging preventive medicine; introduction of a Bill of Rights limiting the powers of Government and safeguarding the rights of the individual; allowing pensioners to travel free on Government-owned services outside holiday periods. All those policies endure today.
The party gained its first MP in 1966 when a massive effort by supporters saw Kerikeri accountant Vernon Cracknell win the Hobson seat in Northland.
In 1981 under charismatic leader Bruce Beetham, who had won the Rangitikei seat in a by-election in 1977, the party gained its highest ever vote - 20.65% of all votes cast, but won only two seats under the "first past the post" electoral system. It has contested every election since 1954.
It has always advocated policies that were ahead of their time - proportional voting in 1972 (enacted 1993); an anti nuclear New Zealand in 1978 (enacted 1987).
It published "You and Your Environment", a 28 page document about protecting the environment, recycling, etc in 1973, and "Industry an People" a 16 page document proposing new concepts for industrial organisation, and ways to provide income for workers losing jobs due to the advancement of computers and robot technology.
Monetary reform is now being promoted in New Zealand and internationally by economists, professors of economics and economic commentators.
You can read more about that at www.tellmemore.org.nz
Vernon Cracknell (1966–1969)
Bruce Beetham (1977–1984)
Gary Knapp (1980–1987)
Neil Morrison (1984–1987)
Chris Leitch
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,730
|
_The sea-dragon_
_There it was, beyond the spires, rising up out of the stormy waters, bright as flame, with the sunlight itself looped around its neck._
_It was watching the only movement on the beach: Peri._
_She stopped, her mouth open. It lingered, massive and curious, its bright streamers swirling in the restless water. The delicate brow fins over its great eyes flicked up and down like eyebrows. It had a mustache of thin streamers above its mouth. It washed to and fro in the water, its eyes like twin red suns hovering above the sea. It seemed to wash closer with the tide. Peri stepped back nervously and bumped into something that snorted gently between her shoulder blades..._
FIREBIRD
WHERE FANTASY TAKES FLIGHT™
_The Beggar Queen_ Lloyd Alexander
_The Changeling Sea_ Patricia A. McKillip
_Crown Duel_ Sherwood Smith
_The Dreaming Place_ Charles de Lint
_The Ear, the Eye and the Arm_ Nancy Farmer
_Enchantress from the Stars_ Sylvia Engdahl
_Fire Bringer_ David Clement-Davies
_Growing Wings_ Laurel Winter
_The Hex Witch of Seldom_ Nancy Springer
_The Kestrel_ Lloyd Alexander
_I Am Mordred_ Nancy Springer
_I Am Morgan le Fay_ Nancy Springer
_Mariel of Redwall_ Brian Jacques
_Mattimeo_ Brian Jacques
_Mossflower_ Brian Jacques
_The Outlaws of Sherwood_ Robin McKinley
_Redwall_ Brian Jacques
_The Riddle of the Wren_ Charles de Lint
_Spindle's End_ Robin McKinley
_Treasure at the Heart of the Tanglewood_ Meredith Ann Pierce
_Westmark_ Lloyd Alexander
_The Winter Prince_ Elizabeth E. Wein
**_The
Changeling Sea_**
**PATRICIA A. McKILLIP**
FIREBIRD
Published by the Penguin Group
Penguin Putnam Inc., 345 Hudson Street, New York, New York 10014, U.S.A.
Penguin Books Ltd, 80 Strand, London WC2R ORL, England
Penguin Books Australia Ltd, 250 Camberwell Road, Camberwell, Victoria 3124, Australia
Penguin Books Canada Ltd, 10 Alcorn Avenue, Toronto, Ontario, Canada M4V 3B2
Penguin Books (N.Z.) Ltd, 182-190 Wairau Road, Auckland 10, New Zealand
Penguin Books Ltd, Registered Offices: Harmondsworth, Middlesex, England
First published in the United States of America by Atheneum,
Macmillan Publishing Company, 1988
Published by Firebird, an imprint of Penguin Putnam Inc., 2003
Copyright © Patricia A. McKillip, 1988
All rights reserved
ISBN: 978-1-101-65999-1
Except in the United States of America, this book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser.
FOR
JEAN KARL
# **Table of Contents**
One
Two
Three
Four
Five
Six
Seven
Eight
Nine
Ten
Eleven
Twelve
Thirteen
# One
NO ONE REALLY KNEW where Peri lived the year after the sea took her father and cast his boat, shrouded in a tangle of fishing net, like an empty shell back onto the beach. She came home when she chose to, sat at her mother's hearth without talking, brooding sullenly at the small, quiet house with the glass floats her father had found, colored bubbles of light, still lying on the dusty windowsill, and the same crazy quilt he had slept under still on the bed, and the door open on quiet evenings to the same view of the village and the harbor with the fishing boats homing in on the incoming tide. Sometimes her mother would rouse herself and cook; sometimes Peri would eat, sometimes she wouldn't. She hated the vague, lost expression on her mother's face, her weary movements. Her hair had begun to gray; she never smiled, she never sang. The sea, it seemed to Peri, had taken her mother as well as her father, and left some stranger wandering despairingly among her cooking pots.
Peri was fifteen that year. She worked at the inn beside the harbor, tending fires, scrubbing floors, cleaning rooms, and running up and down the kitchen stairs with meals for the guests. The village was small, poor, one of the many fishing villages tucked into the rocky folds of the island. The island itself was the largest of seven scattered across the blustery northern sea, ruled for four hundred years by the same family.
The king's rich, airy summer house stood on a high crest of land overlooking the village harbor. During the months when he was in residence, the wealthy people of the island came to stay at the inn, to conduct their business at the king's summer court, or sometimes just to catch a glimpse of him riding with his dark-haired son down the long, glistening beaches. In winter, the inn grew quiet; fishers came in the evening to tell fish stories over their beers before they went home to bed. But even then, the innkeeper, a burly, good-natured man, grew testy if he spotted a cobweb in a high corner or a sandy footprint on his flagstones. He kept his inn scoured and full of good smells.
He kept a weather eye on Peri, too, for she had a neglected look about her. She had grown tall without realizing it; her clothes were too loose in some places, too tight in others. Her hair, an awkward color somewhere between pale sand and silt, looked on most days, he thought, as if she had stood on her head and used it for a mop. He gave her things from the kitchen, sometimes, at the end of the day to take home with her: a warm loaf of bread, a dozen mussels, a couple of perch. But he never thought to ask her where she took them.
Occasionally her mother, who had simply stopped thinking and spent her days listening to the ebb and flow of the tide, stirred from her listening. She would trail a hand down Peri's tangled, dirty hair and murmur, "You come and go like a wild thing, child. Sometimes you're there when I look up, sometimes you're not. . . ." Peri would sit mute as a clam, and her mother's attention would stray again to the ceaseless calling of the sea.
Her mother was enchanted, Peri decided. Enchanted by the sea.
She knew the word because the old woman whose house she stayed in had told her tales of marvels and magic, and had taught her what to do with mirrors, and bowls of milk, bent willow twigs buried by moonlight, different kinds of knots, sea water sprinkled at the tide line into the path of the wind. The old woman's enchantments never seemed to work; neither did Peri's. But for some odd reason they fascinated Peri, as if by tying a knot in a piece of string she was binding one stray piece of life to another, bridging by magic the confusing distances between things.
The old woman had lived alone, a couple of miles from the village, in a small house built of driftwood. The house sat well back from the tide against a rocky cliff; it was shielded from the hard winter winds by the cliff and the thick green gorse that overflowed the fallow fields and spilled down around its walls. The old woman had made her living weaving. When Peri was younger, she would come to sit at the woman's side and watch the shuttle dart in and out of the loom. The old woman told stories then, strange, wonderful tales of a land beneath the sea where houses were built of pearls, and a constant, powdery shower of gold fell like light through the deep water from the sunken wrecks of mortals' ships. She was very old; her eyes and hair were the fragile silvery color of moonlit sand. One day, not long after Peri's father had died, the old woman disappeared.
She left a piece of work half-finished on her loom, her door open, and all her odd bits of things she called her "spellbindings" lying on the shelves. Peri went to her house evening after evening, waiting for her to return. She never did. The villagers looked for her a little, then stopped looking. "She was old," they said. "She wandered out of her house and forgot her way back."
"Age takes you that way, sometimes," the innkeeper told Peri. "My old granny went out of the house once to take her hoe to be mended. She came back sitting on a cart tail three days later. She never did tell us where she finally got to. But the hoe was mended."
Peri, used to waiting in the empty driftwood house, simply stayed.
She was fidgety and brusque around people then, anyway, and there was nothing in the house to remind her of her parents, both of them lost, in one way or another, to the sea. She could sit on the doorstep and listen to the tide and glower at the waves breaking against the great, jagged pillars of rock that stood like two doorposts just at the deep water. They were the only pieces of stone cliff left from some earlier time; the sea had nibbled and stormed and worn at the land, pushed it back relentlessly. It was not finished, Peri knew; it would wear at this beach, this cliff, until someday the old woman's house would be underwater. Nothing was safe. Sometimes she threw things into the sea that she had concocted from the old woman's spellbindings: things that might, she vaguely hoped, disturb its relentless workings.
"If you hate the sea so now," Mare asked in wonder one day, "why don't you leave?" Mare was a few years older than Peri, and very pretty. She came to work in the morning, with a private smile in her eyes. Down at the docks, Peri knew, was a young fisherman with the same smile coming and going on his face. Mare was tidy and energetic, unlike Carey, who dreamed that the king's son would come to the inn one day and fall in love with her green eyes and raven tresses. Carey was slow and prone to breaking things. Peri attacked her work grimly, as if she were going to war armed with a dust cloth and a coal scuttle.
"Leave?" she said blankly, knee-deep in suds. Mare was watching her, brows puckered.
"You haven't smiled in months. You barely talk. You scowl out the windows at the waves. You could go inland to the farming villages. Or even to the city. This may be an island, but there are places on it where you'd never hear the sea."
Peri's head twitched, as much away from Mare's reasonable voice as from the sound of the running tide. "No," she said shortly, not knowing why or why not.
Carey giggled. "Can you imagine Peri in the city?" she said. "With her short skirts and her hair like a pile of beached kelp?" Peri glowered at her between two untidy strands of hair.
"No," Mare sighed. "I can't. Peri, you really should—"
"Leave me alone."
"But, girl, you look like—"
"I know what I look like," Peri said, though she didn't.
"How will anyone ever fall in love with you looking like that?" Carey asked. Peri's glower turned into such an astonished stare that they both laughed. The innkeeper stuck his head into the room.
"Work on my time," he growled. "Laugh on your own."
They heard him shouting down the kitchen stairs a moment later. "Crab," Carey muttered.
"It's just," Mare said insistently, "you have such pretty eyes, Peri. But nobody can see them with your hair like—"
"I don't want anybody seeing them," Peri said crossly. "Leave me alone."
But later, after she had gone to the driftwood house and made something full of broken bits of glass and crockery and jagged edges of shell to throw into the great sea to give it indigestion, she looked curiously into the old, cracked mirror that the woman had left on her spellbinding shelf. Gray eyes flecked with gold gazed back at her from under a spiky nest of hair. She barely recognized her own face. Her nose was too big, her mouth was pinched. Some stranger was inhabiting her body, too.
"I don't care," she whispered, putting the mirror down. A moment later she picked it up again. Then she put it down, scowling. She went outside to a little cave of gorse where the old woman had found an underground stream wandering toward the sea, and had dug a hole to trap it. Peri knelt at the lip of the well and dunked her head in the water.
Shivering and sputtering, she threw more driftwood on the fire, and sat beside it for an hour, tugging and tugging at her hair with a brush until all the knots came out of it. By that time it was dry, but still she brushed it, tired and half-dreaming, until it rose crackling around her head in a streaky mass of light and dark. She remembered a long time past, when she was small and the old woman had brushed her hair for her, singing . . .
_"Come out of the sea and into my heart_
_My dark, my shining love._
_Promise we shall never part,_
_My dark, my singing love. . . ."_
Peri heard her own voice singing in the silence. She stopped abruptly, surprised, and heard then the little, silky sounds of the ebb tide washing against the shore. Her mouth clamped shut. She put the brush down and picked up a clay ball, prickled like a pincushion with bent nails and broken pieces of glass. She flung open the door; firelight ran out ahead of her, down the step onto the sand. But something on the beach kept her lingering in the doorway, puzzled.
There was an odd mass on the tide line. Her eyes, adjusting to moonlight, pieced it together slowly: a horse's head, black against the spangled waves, a long, dark cloak glittering here and there with silver thread, or steel, or pearl. . . . She could not find a face. Then the sea-watcher sensed her watching. A pale, blurred face turned suddenly away from the sea to her, where she stood in the warm light, with her feet bare and her hair streaming away from her face in a wild, fire-edged cloud down her back.
They stared at one another across the dark beach. A swift, high breaker made the horse shy. The rider swept the cloak back to free his arms; again came a moonlit spark of something rich, unfamiliar. He rode the dark horse out of the sea and Peri closed her door.
* * *
"The king came back to the summer house last night," Carey said breathlessly the next morning as the girls put on aprons and collected brooms and buckets in the back room. "I saw his ships in the harbor."
Peri, yawning as the apron strings tangled in her fingers, made a sour noise.
"It's early," Mare commented, surprised. "It's barely spring. The rainy season isn't over yet."
"Prince Kir is with him."
"How do you know?"
"I asked one of the sailors." Carey's eyes shone; she hugged her bucket, seeing visions. "Think of the clothes and the jewels and the horses and the men—"
"Think of the work," Mare sighed, "if they stay from now till summer's end."
"I don't care."
"Jewels?" Peri echoed suddenly. Something teased her brain, a glittering, moonlit darkness. . . .
"Girl, will you wake up?" Mare grabbed Peri's apron strings, tied them impatiently. "This place will be full by nightfall."
There were already strangers in the inn, tracking sand across the floors, demanding fires, spilling things. By the end of the day, the girls were almost too tired to talk. The innkeeper met Peri at the back door and gave her oysters to take home. He studied her, his brows raised.
"You washed your hair!"
It shouldn't have been all that surprising, Peri thought irritably, taking one of the cobbled streets through the village. A moment later she didn't care. She was climbing over a low stone wall to slip burrs into the back pockets of Marl Grey's fishing trousers, hanging on his mother's line. He had called her names a couple of days ago, laughing at her wild hair, her short skirt. "Let's see how funny you look," Peri muttered, "sitting down in a boat on those."
Then she went to see her mother.
She didn't decide to do that; she was just pulled, little by little, on a disjointed path through the village toward her mother's house. She didn't want to go: She hated the still house at the time of day when the boats were coming in. No matter how hard she looked, her father's small blue boat would not be among them. It would be idle, empty, moored to the dock as always. And yet she knew she would look. She opened the gate to her mother's yard. A hoe leaned against the wall among a few troubled clods of dirt. Already the thistles were beginning to sprout.
She went into the house, tumbled the oysters out of her skirt onto the table, and sat down silently beside the fire. Fish chowder simmered in a pot hung over the fire. Her mother sat at the window, gazing at the sunlit harbor. She turned her head vaguely as the shells hit the table, then her attention withdrew. They both sat a few moments without moving, without speaking. Then Peri's mother lifted one hand, let it fall back into her lap with a faint sigh. She got up to stir the soup.
"The king is back," Peri said abruptly, having an uncharacteristic urge to say something. She even, she discovered in surprise, wanted to hear her mother's voice.
"He's early," her mother said disinterestedly.
"Are you making a garden?"
Her mother shrugged the question away. The hoe had been standing up in the weeds for months. Her eyes went to the window; so did Peri's.
The sun was hovering above the horizon, setting the water ablaze. The first of the fishing boats had just entered the harbor; the rest of them were still caught in the lovely, silvery light. Peri's mother drew a soft breath. Her face changed, came gently alive, almost young again, almost the face Peri remembered.
"That's what I dreamed about. . . ."
"What?" Peri said, amazed.
"I dreamed I was watching the sun go down. The way it does just before it dips behind the fog bank, when it burns up the sea and the clouds, and the fishing boats coming home look like they're sailing on light . . . like they're coming from a land you could walk to, if you could step onto the surface of the sea and start walking. It's a country beneath the sea, but in my dream I saw the reflection of it, all pale and fiery in the sunlight. . . . And then the sun went down."
Peri's face was scarlet. "There is no country!" she shouted, and her mother's secret, dreaming face faded away, became the weary stranger's face once more. "There is no magic country in the sea! Stop watching for it!"
But her mother was already watching again. Peri ran out of the house, slamming the door so hard that a flock of sea gulls sunning on the roof wheeled into the air, crying. Her mother's face in the window was still as a sleeper's, hearing nothing in her dreams but the tide.
# Two
THE NEXT AFTERNOON, Peri climbed the cliff above the old woman's house. There was a moon-shaped patch of sand ringed with gorse at the top; on her days off she could sit in the sunlight and brood at the sea, yet feel protected from the world within the green circle. The gorse was beginning to bloom here and there, tiny golden flowers that made her sneeze. But so far her magic circle was ungilded.
She wrapped her arms around her knees and watched the white gulls wheel above the great weather-beaten spires of rock. Clouds scudded across the sea, making a mysterious weave of light and shadow on the water beyond the spires. Peri frowned at the mystery, chewing a thumbnail. What lay beneath the color and the shadow? Fish? Or some secret world within the kelp that sometimes floated too near the surface of the sea, disturbing those who dwelled on land? What would stop it from troubling her mother? She chewed a fingernail next, then took the finger out of her mouth and drew a spidery design in the sand.
She studied it critically, then drew another one. Hexes, the old woman had called them. She had bent soft willow branches into odd, angular shapes, and then wove webs of thread within them. Hung in doors and windows, they kept malicious goblins and irritating neighbors away. They protected cows from being milked at night by sprites. Perhaps, Peri thought, a few hexes floating across the sea might trap its strange magic underwater. She would make them out of tough dried kelp stalk, row out over the deep water to cast them. She would have to check her father's boat for leaks, get new oars, see if the rudder had been cracked. She had not looked closely at the _Sea Urchin_ since the fishers had cleaned the sand and seaweed out of it and moored it in the harbor. Someone had covered it, or it would have sunk under the weight of the heavy winter rains. It probably dragged a crust of barnacles on its bottom. . . .
She drew another hex, a crooked, crabbed design. The wind tossed a gull feather into the circle. She stuck it behind her ear, then broke off a couple of feet of a wild strawberry runner that was gliding across the sand, and wove that absently in and out of her hair. Her dress—her oldest one—barely covered her knees. It was loose around the waist and so tight in the shoulders the seams threatened to part. In the gorse circle, it didn't matter. She stretched out her legs, burrowed her feet under the warm sand, and devised another hex.
I wonder, she thought, if I have to say something over them to make them work. Then she stopped breathing. A feeling skittered up her backbone. She turned her head slowly, warily, to see who was watching her.
The dark horseman from the sea gazed up at her, mounted at the foot of the cliff. She caught her breath, chilled, as if the sea itself had crept noiselessly across the beach to spill into her circle. Then she blinked, recognizing him. It was only the young prince out for a ride in the bright afternoon. The dark horseman was Kir. Kir was the dark horseman. The phrases turned backward and forward in her mind as she stared at him. A wave boomed and broke behind him, flowing across half the beach, seeking, seeking, then dragged back slowly, powerfully, and, caught in the dark gaze of the rider, his eyes all the twilight colors of the sea, Peri felt as if the undertow had caught her.
Then his face changed again: the king's son, out for a ride. She blushed scarlet.
"Girl," he said, abrupt as one of the rich old lords who came to stay at the inn, though he was not even as old as Mare, "where is the old woman who lives in this house?"
Peri dragged her hair back out of her eyes; the strawberry runner dangled over one ear. "You know her?" she said, surprised.
"Where is she?"
"Gone."
"Where?"
Peri felt a sudden tightness in her throat; her brows pinched together. Too many people gone at once. . . . "She went away and never came back," she said, her sorrow making her cross. "So if you want a spell, you're too late."
"A spell," he repeated curiously. "Was she a witch? Who are you? Her familiar?"
Peri snorted. A waft of pollen from the gorse blooms caught up her nose and she sneezed wildly. The strawberry runner fell over one eye. "I clean rooms at the inn," she said stuffily. "Where do you work?"
He opened his mouth, then paused, his expression unfathomable. His horse shifted restively. There were pearl buttons on his shirt, Peri saw, under his black leather jacket. A ring on his forefinger held a stone that trembled with the same twilight shadows in his eyes. His brows were dark, slightly slanted over his eyes. The bones of his face made hollows and shadows that seemed, in spite of the hearty sunlight, as pale as pearl, as pale as foam.
"I sweep stables," he said at last. "My mother keeps sea horses."
Peri stared at him. A long, dark breaker swept endlessly toward the beach; it curled finally, turning a shade darker just before it crashed against the sand. The prince glanced back at the sound; his eyes, returning to Peri, seemed to carry, for a moment, a reflection of the sea.
"There is no land under the sea," she said uneasily. "There is no land."
His brows closed slightly; his eyes drew at her. "Why do you say that?" he asked abruptly. "Have you seen it?"
"No!" She bored holes in the sand with a twig, scowling at them. She added reluctantly, feeling his attention still pulling at her, "My mother has. In her dreams. So I am laying a hex on the sea."
"A hex!" He sounded too amazed to laugh. "On the entire sea? Why?"
"Because the sea stole my father out of his boat and it bewitched my mother so that all she does now is stare out at the water looking for the magic country under the sea."
"The land beneath the sea . . ." A yearning she knew too well had stolen into his eyes, his voice.
"There is no magic country," she said stubbornly, feeling her eyes prick with frustration.
"Then what does she see? And what are you making a hex against?"
Peri was silent. The warm wind bustled into her circle, tossed sand over her hexes, tugged her hair back over her shoulders. The prince's expression changed again, became suddenly peculiar.
"It was you then," he said.
"What was?"
"In the old woman's house, a night ago. You were standing in the doorway with the firelight in your hair, beneath your feet."
"Then it was you," she said, "watching the sea."
"For a moment I thought . . . I don't know what I thought. The light was moving in your hair like tide."
"For a moment I was afraid. I thought you rode out of the sea."
"How could I? There is no kingdom beneath the sea." He watched her a moment longer. Then, silently, he dismounted. He left the black horse flicking its tail at the sand flies, and found the trail through the gorse to the top of the cliff. When he broached Peri's circle, she shifted nervously, for her private sand patch seemed too small to hold such richness, such restlessness. He stood studying her hexes, still silent. Then he knelt in the sand across from her.
"What is your name?"
"Peri."
"What?"
"Peri—Periwinkle."
"Like the sea snail?"
She nodded. "When I was little, my father would spread his nets in the sand to dry, and I would walk on them and pick the periwinkles off."
"My name is Kir."
"I know."
He gave her another of his straight, unfathomable looks. She wondered if he ever smiled. Not, apparently, at barefoot girls who worked at the inn. He traced one of her designs lightly with his finger.
"What is this? Your hex?"
"Yes."
"This will terrify whatever watery kingdom lurks beneath the waves?"
"It's all I can think of," she said grumpily. "I'm trying to remember the old woman's spells. Is that what you wanted from her? A spell?"
"No." He was still gazing at the hex. His face seemed distant, now, aloof; she didn't think he would tell her. But he did, finally. "I wanted to ask her something. I met her one day long ago. I was standing out there watching the sun sinking down between those two stones, and the light on the water making a path from the stones to the sun. She came out to watch with me. She said things. Odd things. Stories, maybe. She seemed—she seemed to love the sea. She was so old I thought she must know everything. She—I came here to talk, I wanted to talk. To her."
His eyes had strayed to the sea. His ringed forefinger moved absently, tracing a private hex in the sand. Peri's eyes moved from the sandy scrawl to the stone on his hand, up to the black pearls on the cuff of his jacket, to the fine cream-colored cloth of his shirt, then, cautiously, to his face. It looked as remote, as expressionless, as the great spires weathering wind and sun and sea. His lashes were black as blackbirds' feathers against his pale skin.
She gave her skirt a sudden tug, trying to pull it over her callused knees. She closed her hands to hide the dry cracks on them. But nothing stayed hidden; she sat there with the king's son in full daylight, with workworn hands and red knees, in an old dress bleached so pale she'd forgotten what color it had ever been. She sighed, then wondered at herself. What did it matter, anyway? What was the matter with her?
The prince heard her sigh under the sigh of the tide; his head turned. He asked curiously, "How will you get these hexes out of the sand and into the sea?"
"I'll make them out of twigs and dry seaweed. I'll bend them and bind the ends, and weave the patterns inside with thread. Then I'll row out in my father's boat over deep water and throw them in."
"Will you—" He stopped, looked suddenly away from her. He began again, his hands closed tightly on his knees. "Will you give the sea a message for me? Will you bind it to one of the hexes?"
She nodded mutely, astonished. "What message?"
"I'll bring it here. When will you lay your hex on the sea?"
"On my next day off. In six days."
"I'll bring it when I can." He glanced at the sun, then over her shoulder at the summer house on its smooth green perch high above the sea. "I must go. I'll leave the message in the house if you're not here."
"I won't be," she said as he rose. "I mean, I'll be working."
He nodded. "But I'll come again," he said, "to see you. To find out what your hex did to the sea." He smiled then, a bittersweet smile that made her stare at him as he picked his way back down the cliff. Mounted, he glanced back at her once, then rode away: the dark horseman, the king's son, who was going to knock on Peri's door like any fisher's son, with a message for the sea.
* * *
She found his message on her table four days later among the hexes. The hexes, irregular circles and squares of sticks and seaweed, with jagged spiderwebs of black thread woven across them, carried, Peri thought, a nicely malevolent message. The prince's message was unexpected.
It was a small bundle of things tied up in a handkerchief so soft that its threads snagged on Peri's rough fingers. It was bordered with fine, heavy lace; one corner was embroidered with a pale crown and two letters: QV. Not Kir's initials. Puzzled, Peri untied the ribbon around it.
She sat fingering the small things within, one by one. A short black lock of hair. Kir's? A black pearl that was not round but elongated, irregular, tormented out of shape. Another lock of hair, black, streaked with gray. A ring of pure silver, with initials stamped into it. KUV. Kir? But who was Q? Then she dropped the ring as if it burned, and huddled on her stool as if the king himself had come into her house.
Q. K. Queen, king. King Ustav Var. Kir's father. That was his graying hair lying there on her table.
She tied everything back up, her fingers shaking, averting her eyes, as if she had caught the king in the middle of some small private act—counting the veins in his eyes or contemplating his naked feet to see how the years were aging them. She stuffed the handkerchief into an empty clay jar on the spellbinding shelf and slammed the lid down on it.
* * *
There was no way, she had to admit finally, that she could row out to sea in the _Sea Urchin_ by herself. Her back and arms were strong from carrying buckets of water and loads of wood, but it took more strength than she possessed to control heavy oars in open water with the sea roiling and frisking under her boat. Just getting out of the harbor with the hard waves feathering into the air above the breakwaters would be a nightmare. She'd lose the oars, she'd have to be rescued, teased and scolded by the fishers. Even the women who fished—Leih and Bel and Ami—were twice her size, with muscles like stones and hands hard as fence slats with rowing calluses.
But how could she get the hexes out so far that the sea would not simply spit them back at her?
She thought about the problem, her brows pinched tight as she worked. Carey was chattering about things she had seen unloaded from the king's ship: carved and gilded chests, milk-white horses, gray dogs as tall as ponies, with lean flanks and slender muzzles, and silver-gray eyes, looking as glazed and panicked as fine ladies from being tossed about on the sea.
"And their collars," Carey breathed, "studded with emeralds."
"Emeralds, my foot," Mare said witheringly. "Glass, girl, glass. This isn't such a wealthy land that the king would waste emeralds on a dog. Peri, your hair is in your bucket."
Peri twitched it out; a tangle landed soddenly on her shoulder. She wiped her nose with the back of her hand, thinking of the pearls on Kir's shirt, the silver ring.
"I want emeralds," Carey said dreamily. "And gowns of white lace and gold rings and—"
"You won't get them on your knees in the soapsuds."
"Yesterday when I brought clean towels to one of the rooms, a man in green velvet said I was beautiful and kissed me."
"Carey!" Mare said, shocked. "You watch yourself. Those fine men will migrate like geese in autumn, and you'll be stuck here with a belly full of trouble."
Carey scrubbed silently, sulking. Peri swam out of her thoughts, glanced up.
"Was it nice?" she asked curiously. For a moment Carey didn't answer. Then her mouth crooked wryly and she shrugged.
"His mustache smelled of beer."
"Green velvet," Mare muttered. "I hope a good wave douses him."
The tide was low that afternoon as Peri walked home, so low that even the great jagged spires stood naked in the glistening sand, and all the starfish and anemones and urchins that clung to their battered flanks were exposed. It was a rare tide. Beyond the spires the sea dreamed gently, a pale milky blue shot with sudden fires from the setting sun. Peri, her shoes slung over her shoulder, watched the bubbles from burrowing clams pop in the wet sand under her feet. The air was warm, silken, promising longer, lazy days, more light, promising all the soft, mysterious smells and colors of spring after the harsh gray winter. The sand itself was streaked with color from the sunset. Peri lifted her eyes, watched the distant sheen of light beneath the sun fall on water so still it seemed she could simply turn toward the tide and follow it. Her steps slowed, her lips parted; her eyes were full of light, spellbound. She could take the path of the sun to the sun, she could walk on the soft opal breast of the ocean as simply as she walked on the earth, until she found, there in the great glittering heart of light, the golden kingdom, the kingdom of—
She stopped, shaking her head free of thoughts like a dog shaking water off itself. Then she began to run.
She flung her shoes in a corner of the house, snatched the hexes from the table, Kir's message from the jar, ran back out, straight across the beach toward the spires and the sun illumining the false, tempting dream between them, as if they were some broken ancient doorway into the country beneath the waves, reflected in the light.
She stood between the spires at the edge of the idle tide, going no farther than that because the sand sloped sharply beyond the spires into deep water. She lifted the hexes, tied together and weighted with Kir's message, threw them with all her strength into the sea.
"I hex you," she shouted, searching for words as bitter as brine to cast back at the sea. "I hate you, I curse you, I lay a hex on you, Sea, so that all your spellbindings will unravel, and all your magic is confused, and so that you never again take anything or anyone who belongs to us, and you let go of whatever you have—"
She stopped, for the hexes, floating lightly along the crest of a wave, had suddenly disappeared. She waited, staring at the water, wanting nothing to happen, wanting something to happen. A bubble popped like a belch on the surface of the water a few yards away. She edged close to the wet starfish-dotted flank of one of the spires. Had she, she wondered uneasily, finally got the sea's attention?
The water beyond the spires heaved upward, flaming red. Peri shrieked. Still it lifted, blocking the sun: a wall of red, streaming waterfalls. Two huge pools of fire hung where the sun had been, so big she could have rowed the _Sea Urchin_ into either one of them. Long, long streamers of fire surfaced, eddied gracefully in the tide. And then gold struck her eyes, brighter than the sun.
She gasped, blinking, and the round pools of fire blinked back at her. A sigh, smelling of shrimp and seaweed, wafted over the water.
She edged backward, trying at the same time to cling to the rock like a barnacle. "Oh," she breathed, her throat so full and dry with terror she barely made a sound. "Oh."
In the deep waters beyond the stones, a great flaming sea-thing gazed back at her, big as a house or two, its mouth a strainer like the mouth of a baleen whale, its translucent, fiery streamers coiling and uncoiling languorously in the warm waters. The brow fins over its wide eyes gave it a surprised expression.
Around its neck, like a dog collar, was a massive chain of pure gold.
# Three
PERI STAYED WITH HER MOTHER that night. The sea, she decided, annoyed at her constant harassment, had sent some great monster out of its depths to eat her. Her mother didn't ask why she was there, but Peri's presence seemed to tug at her thoughts. She watched Peri. Sometimes a question trembled in her eyes; she seemed about to speak. But Peri would look away. Though sea-dragons garlanded with gold rode the waves in Peri's mind, she clamped the secret behind her lips, like an oyster locking away its pearl, rather than admit her mother might be right: There might be a land of light and shadow hidden among the slowly swaying kelp beneath the waves.
And Kir. His face rose clearly in her thoughts just before she fell asleep, pale and dark and restless against the vast wild blue of the sea. What message had he sent? she wondered in her dreams. And to whom?
The lovely promises of spring were nothing but dreams themselves by morning: The spring rains had started. Peri, walking soggily to the inn, felt all her fears of the sea-dragon dwindle under the dreary sky. Nothing alive in the world could have been as big as she remembered it. The gold around its neck could not have been real. And if it had wanted to eat her, it could have plucked her like an anemone away from the rock, the way she'd stood there frozen. Besides, she remembered, cheering up, it had no teeth. Only tiny shrimp and bits of kelp could pass through the strainer in its jaws; it was destined to a life of broth. It had simply been some great sea creature coming up for air as it passed along the island. It was probably scaring ships in the South Isles by now.
But who had put the chain around its neck?
She thought about that as she swept the stairs and made beds and carried buckets of ash to the bins behind the kitchen. Everyone was grumpy because of the rain. The guests tracked paths of water and mud on the floors and complained of smoking chimneys. Some of the fishers came in early off the heavy, swollen sea and trailed in more water, more sand. By the day's end, Peri felt as scoured as the flagstones and as damp. Carey burst into tears.
"My hands," she wailed. "I might as well be an old lobster."
"Never mind," Mare sighed. "Maybe you'll find some rich old lobster to love you."
"I won't! Ever! I'll never get out of this town. I'll never get out of this inn! I'll be scrubbing floors here when I'm ninety years old, and cleaning hearths and making beds to my dying day. The only pearls I'll ever see will be on someone else's fingers, I'll never wear velvet, I'll never sleep in lace, I'll never—"
"Oh, please, Carey. I've got a headache as it is."
"I'll never—"
"You'll never guess—" said Mare's lover, Enin, sticking his head into the doorway of the back room where they hung their aprons and stored their mops. Then he saw Carey's tears and ducked back nervously. "Oh."
"Enin!" Mare called. He was running water, too, from his rain cloak; his boat had just got in. Peri dumped her mop and her brush into her empty bucket and shoved it against the wall. She put her hands to her tailbone and bent backward, stretching. Enin's face reappeared. It was softly bearded, sunburned, and speckled with rain. His eyes, light blue, looked round as coins. Carey banged her brush into her bucket crossly, still sniffing. Enin's eyes went to her cautiously. Mare said, beginning to smile, since Enin's face was the most cheerful thing they had seen all day, "I'll never guess what?"
"You'll never guess what's out there in the sea."
"Mermaids in a coracle, I suppose."
"No." He shook his head, groping for words. "No. It's—"
"A sea monster?"
"Yes!" Peri's mop slipped as she stood staring at him. It rapped her on the head and Enin winced. "Are you all right there, girl? Mare, it's huge! Big as this inn! It came right up to our boats—mine and Tull Olney's—we went farthest out—and watched us fish!"
"Oh, Enin," Mare said, touching her forehead.
"Red as fire, even in the mist and rain, we could see that. And you'll never guess what else."
"It has a chain of gold around its neck," Peri said.
"It's wearing a chain of gold—pure gold—Mare, I swear! Stop laughing and listen!" Then he stopped talking, and Mare stopped laughing, and Carey stopped snuffling. They all stared at Peri.
"I saw it," she said, awkward in the sudden silence. "I saw it. Yesterday. Beyond the spires. The gold hurt my eyes."
Carey's long, slow breath sounded like the outgoing tide. "Gold."
"But what is it?" Mare said, bewildered. "Some great fish? A sea lion with a pattern around its neck?"
"No, no, bigger. Much bigger. More like a—a dragon, yes, that's what it's more like. And the gold is—Ah, Mare, you wouldn't believe—"
"You're right, I wouldn't." Mare sighed. "Probably some poor lost sea-something with a king's gilded anchor chain caught around its neck."
"No."
"No," Peri echoed him. "It's real. I saw the sun pouring off it like—like melting butter."
"Why didn't you tell us?" Carey demanded.
"Because it scared me," she said irritably. "All chained like that. Like someone's pet. I didn't want to think who might have made that chain. The links are so big I could have crawled through one."
There was a silence. Carey said suddenly to Enin, "Come in and shut the door."
Her voice was so high and sharp he did it. The noise from the inn faded.
"Why?" he said, puzzled.
"Because it's ours," she said fiercely. "Our gold. It belongs to us, to the village. Not to the king, not to the summer guests. To us. We have to find a way to get it."
Enin stared at her, breathing out of his mouth. Mare pushed her hands against her eyes.
"Oh, Carey."
"She's right, though," Enin said slowly. "She's right."
"It must be our secret," Carey insisted.
"Yes."
Mare turned abruptly, picked her cloak off a peg, and tossed it over her shoulders so fast it billowed like a sail. "I think," she said tautly, "you'd better take another look at this sea-dragon before you start counting your gold pieces. I think—"
"Mare—"
"I think you and Tull had too many beers for breakfast and you rowed right into the place where the sky touches the sea and you hear singing in the mist, and sea cows turn into mermaids, and old ships full of ghosts sail by without a sound. That's where you've been. Sea-dragons. Gold chains." She jerked the door open. "I have a headache and I'm famished. The only gold I want to see is in a cold glass of beer."
"But, Mare," Enin said, following her out. Carey gazed at Peri, her eyes suddenly wistful.
"Was that it? A sea-dream?"
Peri dragged a hand through her hair. "I didn't have any beer for breakfast when I saw it," she sighed. "I don't know what it is. But it's not our gold. I wouldn't like to be caught stealing from whoever made that chain."
Carey was silent; they both were, envisioning gold, so much gold that the damp air seemed to brighten around them. Carey reached for her cloak. "Nonsense," she said briskly. "Anyone who could waste that much gold on a chain for a sea-pet would never know it was missing."
* * *
Peri kept an eye out for the sea-pet when she walked back to the old woman's house. But there was no fire in the gray world, no gold, only the sea heaving sullenly between the spires. The rain clouds hiding the setting sun did not release a single thread of light. A false light dragged Peri's eyes from the sea: the gorse blazing gold above the old woman's house. And in front of her house: a black horse.
The rider, apparently, was inside. Peri's brows went up as high as they could go. As she trudged from the tide line across the stretch of beach toward the house, she saw that the top half of the door was open. Kir leaned against the bottom half, watching the breakers so intently that he did not notice Peri until she was nearly on the doorstep.
His head turned, his eyes still full of the sea. His thoughts tumbled over Peri like a wave, drenching her with a sharp, wild sense of restlessness and despair. She stopped short, one foot on the doorstep, staring at him. But he had already moved to open the door, while something shut itself away behind his face.
He didn't speak. Peri dumped scallops the innkeeper had given her out of her cloak and hung it up. He had started a fire, she noticed in surprise. He had picked up driftwood with his royal hands and piled it into the grate. But, it appeared, he did not know what a scallop was.
"What's this?" He was fingering a fan-shaped shell.
"My supper," Peri said, huddling close to the fire. He watched her shake water out of her hair. She added gruffly, recalling some manners, "You're welcome to stay."
He turned edgily away from the shells. "You cook, too."
"I have to eat," she said simply. He paced to the door and back again to the hearth, where she crouched, combing her hair with her fingers.
"Did you give my message to the sea?" he asked abruptly. She nodded, opening her mouth, but he had turned away again, speaking bitterly before she could answer. "I'm being stupid. It's just a child's game, your hexes, my message. They're probably lying out there now among the litter on the tide line. You can't talk to the sea by throwing things at it."
Peri, trying so hard to understand him that her forehead creased and her eyes were round as owls' eyes, asked bewilderedly, "Why did you want the sea to have your father's ring?"
"Why do you think?" he answered sharply.
"I don't know." She felt stupid herself. Something in her voice made him look at her again, as if he had never really seen her since she walked in damp and untidy from the rain, with red, chapped hands and tired eyes. His expression changed. Peri, recognizing his unhappiness if nothing else, said helplessly, "I don't know if the sea got your message. But after I threw it in, the biggest sea-thing I have ever seen in the world lifted its head out of the water to look at me. Around its neck there was a chain of gold—"
"What?"
"A chain. Gold. It—"
"Are you," he asked, his voice so thin and icy she shifted nearer to the fire, "making a fool of me?"
Peri shook her head, remembering the flame-colored wall rising out of the sea, blocking the sun, and the molten reflection of gold everywhere. "It was like a dragon. But with fins and long ribbons of streamers instead of wings. It was bigger than this house, and the gold chain ran down into the deep sea as if—as if it began there, at the bottom."
The prince's pale face seemed to glisten in the firelight like mother-of-pearl. He whirled; rain and wind blew across the threshold as he flung open the door. The waves fell in long, weary sighs against the sand. He stood silently, his eyes on the empty sea between the spires. Peri, her clothes still damp, began to shiver.
She moved finally to stop the shivering. She poured water from a bucket into a pot, dropped the scallops into it to steam them open. She hung the pot above the fire, then knelt to add more driftwood to the flames. Kir shut the door finally. He came to the hearth, stood close to Peri. Behind him he had left a trail of wet footprints.
Her eyes were drawn to them; her hands slowed. She heard Kir whisper, "All that gold to keep a sea-thing chained to the bottom of the sea."
"Why—" Her voice caught. "Why would— Who would—"
"There must be a way. There must be." He was still whispering. His hands were clenched. She stared up at him.
"To do what?"
"To get there."
"Where?" She rose, as he flung his wet cloak back over his shoulders. "Where are you going?"
"To the land beneath the sea."
"Now?"
"Not now," he said impatiently. "Now, I'm just going."
"Don't you want supper?"
He shook his head, his attention already ebbing away from her, caught in the evening tide. She scratched her head with a spoon, her face puckered anxiously. "Will you be back?" she asked suddenly. He looked at her from a long distance, farther than sleep, farther, it seemed, than from where the tide began.
"From where?"
She swallowed, feeling her face redden. "Here," she said gruffly. "Will you come back here?"
"Oh," he said, surprisedly. "Of course."
The door closed. She heard his horse nicker, then heard its hoofbeats, riding away from the village, down the long beach into the gathering night. She gazed at the door, envisioning Kir, dark and wet as the night he rode into, restless as the crying gulls and sea winds, with a hint of foam in the color of his skin. A frown crept into her eyes. Her foot tapped on the floor, on one of his footprints. He had left water everywhere, it seemed. Then her foot stilled; her breath stilled. She glimpsed something as elusive as a spangle of moonlight on the water. A lock of his father's hair thrown to the sea, a pearl . . . a message. . . .
She blinked, shaking her head until the odd thoughts and images jumbled senselessly, harmlessly. She grabbed the broom, swept at his watery footsteps across her threshold, at her hearth; they blurred and finally faded.
# Four
THE RAIN WITHDREW, crouched at the horizon; the fishers had a spell of blue sky to tempt them out, and then a wind and a rain-pocked sea to drive them back into the harbor. Out, in, out, in—it was like that for several days. Tales of the sea-dragon became as common as oysters. Then the teasing weather gave way to a full-blown storm that piled surf on the beach and tossed boats loose from the docks. The fishers could not get out past the swells raging at the harbor mouth. The sea-dragon rode out the storm alone. The fishers congregated at the inn to drink beer and stare moodily out at the weather. The guests, disdainful of the smells of wet wool and brine, withdrew to a private room, leaving the hearth and the flowing tap to the villagers. Peri, passing in the hall with her arms full of linen, or coming in to tend the fire, was aware, without really listening, of the thread of gold, glittering and magical, that wove in and out of their conversation.
"Links of gold. A link has to have an opening point, otherwise how would you make a chain? So we'll get a big lever of some kind, force a link apart—"
"And what's the monster going to be doing while you're standing on its neck and sticking a lever through its chain? Nibbling shrimp and watching gulls? It'll dive, man, and take you right down with it."
"Fire, then. Fire melts gold. We'll build a floating forge, row it out underneath the chain where it meets the sea. We'll distract the sea-monster with fish or whatever it eats—"
"Shrimp. Brine shrimp. How can you distract something as big as a barn with something you can hardly see yourself?"
"Then we'll sing to it. It likes singing."
"Sing!"
"Or Tull can play his fiddle. It'll be listening; we'll float the forge behind it, melt a link through, and then . . ."
"Gold," Mare sighed, mopping up the perpetual river of sand and water in the hall. "That's all they talk about these days. Even Ami and Bel. Enin's the worst. It's making them all loony."
"If it's out there," Carey said sharply, "they should get it. It's not doing the monster any good."
"Yes, but they're not thinking. None of them are. Nothing human could make a chain like that. That's what they should be considering first. Instead"—she gave her mop a worried, impatient shove—"they're going to do something stupid. I know it."
"The thing is," the fishers said, while the wind blustered and threatened at the closed windows, "there's still another problem. Even if we do pry open a link, what's the good of that? How could we possibly keep the chain from sliding back down to the bottom of the sea? It'd be like taking a whale into our boats. It'd crush us if we tried to hold it."
"Then we'll cut a link farther down. We'll kill the monster and let the sea itself float it to shore."
"Kill it! If we injure it at all, it'll just disappear on us. Or worse, come back and swamp our boats for us."
"Then how? How do we get the gold?"
Carey took to lingering in the doorway, listening. Peri was tempted to do the same. The fiery sea-dragon with its gold chain provided the only color in a world where everything—sand, sea, sky—had faded gray in the rain. It seemed a wonderful tale for a bleak, idle day, an elaborate fish story to tell over beer beside a warm hearth.
But Mare, bringing clean glasses into the bar, said crossly, "That's like you, all of you, to think of killing when you think of gold."
Enin said uncomfortably, "Now, Mare, we're just talking, let us be. It's the most we can do on a day like this."
"But you're not thinking!"
"Well," Ami said good-humoredly, "nobody pays us for that."
"Have you ever met a man or woman who could make a chain like that? Suppose whoever made that chain has something to say about your stealing that gold? Or freeing the sea-monster?"
"Oh, it's probably ancient, Mare, it's probably—"
"Oh, ho, then why is it so shiny it's left its reflection in all your greedy eyes? I haven't heard of so much as a barnacle on it, or a bit of moss. I think you should be a bit careful about who might treat a sea-monster that size like a pet. That's what I think, and there's no need to pay me for it."
They still talked, for the winds whipped up foam like the froth on cream on the surface of the sea, and the cold rain blew in sheets. But Mare had veered them away from the sea-dragon, Peri noticed. Now it was not "how?" but "who?" and no longer sea-monsters, but enchanted lands and witchery.
"Come to think of it," the fishers asked one another, "who did make that chain?"
Lands were invented on distant islands, at the bottom of the sea, or even floating upon the surface of the sea.
"Like the kelp islands, you see. Only they can skim the surface faster than a gull, and fade like light fades on the water, leaving no trace. Beautiful, rich, great floating islands of pearl and coral and gold. . . . The sea people keep the sea-monster the way a child keeps a pet. It's chained to the invisible island."
"It's not a pet. It's someone who did an evil deed, or crossed a wicked mage, and got chained to the sea bottom in punishment."
"It wants to be free then."
"It wants us to break the chain."
"Suppose we did. Will this wicked mage let us take his gold?"
"Ah, we should get the gold first and worry later."
Peri, working her mop desultorily, found herself daydreaming. Distant isles on the top of the world, past the glaciers and icebergs, past the winter lands, beyond winter itself, gleamed like summer light in her head. Magical isles, where fruit was forever ripe and sweet, and the warm air smelled of roses. Lands deep in the sea, where entire cities were made of pearls, and men and women wore garments of fish scales that floated about them in soft, silvery clouds. One of them had fashioned a chain of gold for a very special . . .
"Mare," she said abruptly.
"What?"
"Why do people do things?"
"Why? For as many reasons as there are fish in the sea."
"I mean, if you made a chain for a sea-dragon, would it be because you loved it and didn't want it freed? Or because you hated it and took away its freedom? Or because you were afraid of it?"
"Any one of those things. Why?"
"I was just wondering . . . was it love or hate or fear that made a chain like that?"
Mare looked surprised; Peri rarely used such complicated words. "I don't know. But the way they're talking in there, I think we'll soon find out."
Walking home wearily that afternoon, Peri searched the horizon for one hint of light in the monotonous gray of sea and sky. Rain flicked against her eyes; she pulled the hood of her cloak more closely about her face. Nothing but the slick, mute fish could possibly dwell in that sea, she decided. There were no wondrous deep-sea lands full of castles made of pearl and whalebone. No free-floating islands of perpetual summer. The sea-dragon's chain was nothing more than a ring of kelp that glowed with tiny, gold, phosphorescent sea animals. The sea-dragon itself had probably strayed from warm, distant waters where, in its own sea, it wasn't a monster. That was all. No mystery. Nothing strange, everything explained—
And there it was, beyond the spires, rising up out of the stormy waters, bright as flame, with the sunlight itself looped around its neck.
It was watching the only movement on the beach: Peri.
She stopped, her mouth open. It lingered, massive and curious, its bright streamers swirling in the restless water. The delicate brow fins over its great eyes flicked up and down like eyebrows. It had a mustache of thin streamers above its mouth. It washed to and fro in the water, its eyes like twin red suns hovering above the sea. It seemed to wash closer with the tide. Peri stepped back nervously and bumped into something that snorted gently between her shoulder blades.
She whirled, gasping. Kir's black mount whuffed at her again. Kir, never taking his eyes from the sea-dragon, held out his hand.
"Come up."
She stepped onto his boot, hoisted herself awkwardly behind him. He said nothing else, just sat there watching the sea-dragon, his eyes narrowed against the rain. It seemed to watch them as intently, all its fins and long streamers roiling to keep its balance in the storm.
And then it was gone, sliding fishlike back into its secret world.
Peri felt Kir draw a soundless breath. Then he lifted the reins, nudged his horse into a sudden gallop. Peri clutched wildly at him. He started under her touch, and slowed quickly.
"I'm sorry—I forgot you were there."
"I'll walk home," Peri suggested breathlessly.
"I'll take you." But he rode slowly in the hard rain, his face turned always toward the sea.
"What is it?" Peri asked again. "Where does the chain begin?"
For a long moment he didn't answer; she began to feel like a barnacle talking to the rock it was attached to. Then he answered her. "I think," he said, so softly she had to strain through the sound of the wind and the waves to hear him, "it begins in my father's heart."
Peri felt herself go brittle, like a dried starfish. Her mouth opened, but no words came out. Then she felt Kir shudder in her arms, and she could move again, the thoughts in her head as vague and elusive as shapes in deep water.
"There is a land under the sea."
"There must be," he whispered.
"That's what you look for, when you watch the sea."
"Yes."
"The way to get there. Where you—where you want to be."
"No one," he whispered. "No one knows this but you."
She swallowed drily, her voice gone again. Unguided, the horse had stopped; rain gusted over them. Kir's face lifted to the touch of water.
"It's why we came here early this year. I am not able, any longer, to be too far from the sea. My father—he thinks—I let him believe that I've lost my heart to some lord's daughter who lives near here. He doesn't suspect that I would give my heart to anyone who would show me the path to that secret country beneath the waves."
"But how—" Peri said huskily.
"Oh, Peri, do I have to spell it out to you? Are you that innocent?"
She thought a moment. Then she nodded, her face chilled as from inside, from a cold that had nothing to do with the rain. "I must be."
"My father took a lover out of the sea."
"A lover," she whispered.
"Yes."
"The king did." She thought of the lock of his gray hair, falling into the water. "That sounds cold."
"She bore me and gave me to him."
"Then he does know about you."
"No. He doesn't."
"I don't understand," she said numbly.
"I didn't, either, for a long time. . . . Even in the middle of this island, at the farthest point from the sea, I hear the tide, I know when it changes. I dream of the sea, I want to breathe it like air, I want to wear it like skin. But my father said my mother was a lady of the North Isles, with golden hair and a sweet voice, and that she bore me and died. . . . But how could she have had a son who wants to trade places with every fish he sees? I don't know where her son is. But I am not hers. My mother is tide, is pearl, is all the darkness and the shining in the sea. . . ."
A wave roared and broke; a lacework of foam unrolled across the sand almost to the horse's hooves. It withdrew before it touched them; the prince watched it recede. Peri felt herself shivering uncontrollably. Kir made a small sound, remembering her again. He gathered the reins; within minutes Peri was at her door.
She slid down; Kir's eyes were on her face for once, instead of the sea. "I'll come again," he told her, inarguable as tide, and she nodded, speechless but relieved that he was not going to leave her alone with this magical and frightening tale. She looked up at him finally, when he didn't ride away, and saw a sudden, strange relief in his own eyes. He left her then; she watched him until the sea mist swallowed him.
* * *
She was so quiet the next day at the inn that Mare said in amazement, "Peri, you look as if you're trying to swallow a thought or choke on it. Are you in love?"
Peri stared at her as if she were speaking a peculiar language. "I'm catching cold," she mumbled, for something to say. "All this rain. I stood in it yesterday watching the sea-dragon."
"You saw it?" Carey's voice squealed. "Why didn't you tell us?"
"I forgot."
"You forgot!" She hugged her wet scrub brush anxiously. "Is it true, then? About the gold? It wasn't a dream?"
"It's real."
"Is it dangerous?" Mare asked worriedly. "Would it attack the boats?"
Peri shook her head, shoving her bucket forward. "It seems friendly."
"Friendly!"
"Well, it seems to like watching people."
"That's what Enin said, that it likes listening to the fishers talk. He says it pokes its big head out of the water and listens, while the gulls land on it and pick at the little fish caught in its streamers."
"I bet it wouldn't miss that chain," Carey murmured.
"Yes, but how could they possibly get it off?" Mare said. "It sounds enormous. Whoever put it on meant it to stay."
"But they have to get it off!" Carey protested. "They have to! We'll all be rich! If it was put on, then it can be taken off."
Peri ducked over her work, thinking of Kir watching the cold, tantalizing waves, of the great chain disappearing down, down into a secret place. "Magic," she said, and was surprised at the sound of her voice.
Carey stared at her, openmouthed. "What's magic?"
"The chain. It must be."
"You mean wizards and spells, things like that?"
Mare straightened slowly, blinking at Peri. "You're right. You must be." A door slammed within the inn; she picked up the hearth brush again. "If magic put it on, then magic must take it off."
"It's not magic," the innkeeper said testily, poking his head into the room, "that does your work for you, as much as you may wish it." All their brooms and dust cloths moved again, to the beat of his footsteps down the hall. The front door opened; a wet wind gusted across the threshold. Carey groaned.
"I just mopped out there."
"Mare," Enin said, in the doorway. Mare gave him a halfhearted smile. "Working hard?"
"Go away."
"No, don't!" Carey cried, halting him as he turned. "Peri, tell him what you said. About the magic. Peri said the chain is magic, so Mare said you must take it off with magic."
"And where," Enin asked, "do we find magic? Among the codfish in our nets?"
"Find someone who does magic. There are people who can."
He rubbed his beard silently, blinking at Carey as she knelt among the suds. They had all stopped working, brushes suspended.
"A mage," he said. "A wizard."
"Yes!"
"We could offer gold. There's enough of it."
"Yes."
"With that kind of payment, we could get a good mage. The best. Someone who could break that chain and keep it from falling back down to the bottom of the sea."
"Someone who could keep you from killing yourselves over that gold," Mare said tartly.
His eyes moved to her. "Well," he admitted, "we have been a little carried away. But if you could see it, Mare, if you could—"
Peri shook her hair out of her eyes, suddenly uneasy. "Maybe you shouldn't," she said.
"Shouldn't what?"
"Disturb what's under the sea. Maybe the chain begins in a dangerous place."
They looked at her silently a moment, envisioning, amid the sea of suds, dangerous beginnings. Then Carey cried, "Oh, Peri, there's nothing to be afraid of!" The door opened again, slammed. Heavy boots stamped into the hall. Enin turned his head, cheerful again.
"Ami! We're going to get that gold!"
"How?"
"We're going to hire a magician!"
* * *
Magicians, Peri thought, huddling in her cloak as she walked down the beach. Kings. Sea-dragons. How, she wondered, had such words come to live in her head among plain familiar words like fish stew and scrub bucket? She wiped rain out of her eyes. Gulls sailed the wind over her head, crying mournfully. Her fingers were almost numb in the cold. She carried mussels the innkeeper had given her bundled in a corner of her cloak. The cloak was beginning to smell like old seaweed. As soon as the rains stopped, she would . . .
She heard hoofbeats and peered into the rain. A riderless horse galloped down the beach toward her. Against the dusky sky, she could not see its eyes, just its black, black head and body, like a piece of polished night. There was a long strand of kelp caught on one hoof. It passed her; the tide ran in and out of its path.
She made a wordless noise. And then she began to run, not knowing exactly why or where. The darkening world seemed of only two colors: the deep gray of sky and stone and water, and the misty white of foam and gulls' wings. The mussels scattered as she ran. The wind whipped her hood back, tugged her hair loose. She saw the old woman's house finally; there was no fire in its windows, the door was closed. She slowed, her eyes searching the beach. She saw a streak of black in the tide, half in, half out of the sea. She ran again.
It was Kir, face down in the sand. She dropped to her knees beside him, rolled him onto his back. His face looked ghostly; she couldn't hear him breathe. She gripped both his hands and rose, trying to tug him out of the encroaching grip of the tide. She pulled once, twice. His clothes were weighted with water and sand; she could barely move him. She shifted her grip to his wrists and gave a mighty tug. He pulled against her, coming alive. Sea water spilled out of his mouth. She let him go; he curled onto his side, his body heaving for air. The rasping, grating breaths he took turned suddenly into sobbing. She felt her body prick with shock; her own eyes grew wide with unshed tears.
"I don't know what to do," she heard him cry. "I don't know what to do. What must I do? I belong to the sea and it will not let me in, and I cannot bear this land and it will not let me go."
"Oh," Peri whispered; hot tears slid down her face. "Oh." She knelt beside him again, put her arms around his back and shoulders, held him tightly, awkwardly. She felt the grit of sand in his hair against her cheek, smelled the sea in his clothing. The tide boiled up around them, ebbed slowly. "There must be a way, there must be, we'll find it," she said, hardly listening to herself. "I'll help you find the path into the sea, I promise, I promise. . . ."
She felt him quiet against her. He turned slowly, shakily, on his knees to face her. He put his arms around her wearily, his hands twined in her hair, his chilled face against her face. He did not speak again; he held her until the tide roared around them, between them, forcing them to choose between land and sea, to go, or stay forever.
# Five
ENIN AND TULL WERE ABSENT from the sea and the inn for several days. The other fishers, whose hours on the water were intermittent and dangerous, told one another tales of other, wilder storms they had survived and of the strangest things they had ever pulled up from the bottom of the sea. Now and then, as Peri passed them, she heard a brief mention of magic, of wizardry, followed by a sudden silence, as if they were all envisioning, over their beers, the wondrous, powerful mage whom Enin and Tull were at that very moment enticing out of the city. There was no more talk of gold now, lest the word seep under the door into the curious, greedy ears of the visitors. The fishers, gold within their grasp, waited.
Then, for a while, the village was overrun with some very peculiar people. Carey counted fourteen jugglers, six fortune-tellers, nine would-be alchemists who, she said tartly, couldn't even make change for a gold coin, four inept witches, and any number of tattered, impoverished wizards who couldn't unlatch a door by magic, let alone unlink a chain. The fishers gave them a dour reception; they, in their turn, saw no sign of gold or dragon in the bitter, tossing sea, and jeered the fishers for drunken dreamers. They all trailed back to the city; the fishers slumped over their beers, mocking Enin and Tull for their thickheadedness and still seeing their fortunes glittering somewhere beyond the spindrift.
Peri, lost so deeply in her own thoughts, had barely noticed the motley crowd from the city, except when she dodged a juggler's ball or tripped over a witch's skinny familiar. On the morning the storm finally passed, she barely noticed the silence. She scrubbed at an unfamiliar patch of sunlight as if it were one more puddle to clean off the flagstones. She was trying to imagine the world beneath the sea that Kir yearned for so desperately. Where could he have got to, that night, if he had found entry to the sea's cold heart? What would he change into? A creature of water and pearl, son of the restless tide . . . She scowled, scrubbing away at memories: his hands in her hair, the chill kiss on her cheek, his need of her, someone human to hold.
On land, at least she could touch him.
"Peri," Mare said, and Peri, startled, came up out of layer upon layer of thought. "You've been working on the same spot for twenty minutes. Are you trying to scrub through to the other side of the world?"
"Oh." She pushed her bucket and herself forward automatically. Mare, her feet in the way, did not move.
"Are you all right, girl?"
"I'm all right."
"You're so quiet lately."
"I'm all right," Peri said. Mare still didn't move.
"Growing pains," she decided finally, and her feet walked out of eyesight. "You finish the hall, then help me upstairs when you're done."
Peri grunted, shoved her bucket farther down the hall. The frown crept back over her face. The wave of suds she sent across the floor turned into tide and foam.
There was a sudden crash. The inn door, with someone clinging to it, had blown open under a vigorous puff of spring wind. Peri looked up to see a stranger lose his balance on her tide. He danced upright a moment, and she noticed finally the blazing thunderheads and the bright blue sky beyond him. Then he tossed his arms and fell, slid down the hall to kick over her bucket before he washed to a halt under her astonished face.
They stared at one another, nose to nose. The stranger lay prone, panting slightly. Peri, wordless, sat back on her knees, her brush, suspended, dripping on the stranger's hair.
The stranger smiled after a moment. He was a small, dark-haired, wiry young man with skin the light polished brown of a hazelnut. His eyes were very odd: a vivid blue-green-gray, like stones glittering different colors under the sun. He turned on his side on the wet floor and cupped his chin in his palm.
"Who are you?"
"Peri." She was so surprised that her voice nearly jumped out of her.
"Periwinkle? Like the flower?" he asked.
"Is there a flower?" His eyes kept making her want to look at them, put a color to them. But they eluded definition.
"Oh, yes," the stranger said. "A lovely blue flower."
"I thought they were only snails."
"Why," the stranger asked gravely, "would you be named after a snail?"
"Because I didn't know there were flowers," Peri said fuzzily.
"I see." His voice was at once deep and light, with none of the lilt of the coastal towns in it. He regarded her curiously, oblivious to the water seeping into his clothes. His body looked thin but muscular, his hands lean and strong, oddly capable, as if they could as easily tie a mooring knot as a bow in a ribbon. He was dressed very simply, but not like a fisher, not like a farmer, not like one of the king's followers, either, for his leather was scuffed and the fine wool cloak that had threatened to sail away with him on the wind was threaded with grass stains. He popped a soap bubble with one forefinger and added, "I heard a rumor that someone here needs a magician."
She nodded wearily, remembering the tattered fortune-tellers, the alchemists in their colorful, bedraggled robes. Then she drew a sudden breath, gazing again into the stranger's eyes. That, she felt, must explain their changing, the suggestion in them that they had witnessed other countries, marvels. He looked back at her without blinking. As she bent closer, searching for the marvels, a door opened somewhere at the far side of the world.
"Peri!"
She jumped. The stranger sighed, got slowly to his feet. He stood dripping under the amazed stare of the innkeeper.
"Good morning," he said. "I'm—"
"You're all wet!"
"I'm all wet. Yes." He ran a hand down his damp clothes, and the dripping stopped. The flagstones were suddenly dry, too. So was the puddle outside the door. "My name is Lyo. I'm a—"
"Yes," the innkeeper said. He bustled forward, clutched the magician's arm as if he might vanish like Peri's scrub water. "Yes. Indeed you are. Come this way, sir. Peri, go down to the kitchen and bring the gentleman some breakfast."
"I'm not hungry," said the magician.
"A beer?"
"No," the magician said inflexibly. "Just Peri." He added, at the innkeeper's silence, "I'll see that her work gets done."
"That may well be," the innkeeper said with sudden grimness. "But she's a good, innocent girl, and we've promised to pay you in gold and not in Periwinkles."
Peri shut her eyes tightly, wishing a flagstone would rise under her feet and carry her away. Then she heard Lyo's laugh, and saw the flush that had risen under his brown skin.
He held out his hands to the innkeeper; his wrists were bound together by a chain of gold. "I only want her to take me to see the sea-dragon."
The innkeeper swallowed, staring at the gold. The chain became a gold coin in the magician's palm. "I'll need a room."
"Yes, your lordship. Anything else? Anything at all."
"A boat."
"There's the _Sea Urchin,"_ Peri said dazedly. "But it needs oars."
The odd eyes glinted at her again, smiling, curious. "Why would a _Sea Urchin_ not have oars?"
"It lost them when my father drowned."
He was silent a moment; he seemed to be listening to things she had not said. He touched her gently, led her outside. "Oars it shall have." She was still clutching her brush. He took it from her, turned it into a small blue flower. "This," he said, giving it back to her, "is a periwinkle."
The magician borrowed oars from no discernible place, stripped the barnacles from the _Sea Urchin_ 's bottom with his hand, put his ear to its side to listen for leaks, and pronounced it seaworthy. He rowed easily out of the harbor into open sea, his lank hair curling in the spray, his face burning darker in the sunlight. A pair of seals leaped in graceful arches in and out of the swells; seabirds the color of foam circled the blue above their heads. The magician hailed the seals cheerfully, whistled to the birds, and stopped rowing completely once to let a jellyfish drift past the bow. He seemed delighted by the sea life, as if he had seen little of it, yet he rowed fearlessly farther than Peri had ever gone, out to where the very surface of the world was fluid and dangerous, where the sea was the ruling kingdom they trespassed upon in their tiny, fragile boats, and the life and beauty in it lay far beneath them, in places forbidden to their eyes.
Peri's thoughts drifted to Kir, another sea secret. She had made him a promise: to help him find his way out of the world, away from her. But where was the bridge between land and water, between air and the undersea? She pulled her thoughts about her like a cloak, sat huddled among them, stirring out of them finally to see the magician's eyes, now as bottle-green as the water, on her face.
She shifted, disconcerted, as if he could pick thoughts like flowers out of her head. But he only asked worriedly, "What is it? Don't you like the sea?"
"No."
"Ah, I'm sorry. I shouldn't have asked you to come with me."
"It's not that," she said. "I don't mind being on it like this. I don't like what's under—" She stopped abruptly; he finished for her.
"What's under the sea." He sounded surprised. "What is under the sea besides kelp and whales and periwinkles?"
"Nothing," she mumbled, frightened suddenly at the thought of telling a sea tale that was as yet barely more than a secret in a king's heart.
"Then what are you—" He stopped himself, then, letting go of the oars to ruffle at his hair. The oars, upended, hung patiently in the air. "I see. It's a secret."
She nodded, staring at the oars. "Are you—are you rowing by magic?"
He looked affronted, while the _Sea Urchin_ 's bow skewed around toward land and the oars looked as if they were dipped in air. She laughed; the magician smiled again, pleased. He gripped the oars, brought them back down. "No, Periwinkle, I'm not rowing by magic, though my back and my shoulders and my hands are all shouting at me to get out and walk—"
"Can you do that?" she breathed. "Walk on the sea?"
"If I could, would I be blistering my hands with these oars? Besides, walking on the sea is very peculiar business, I've heard. You walk out of time, you walk out of the world, you find yourself in strange countries, where words and sentences grow solid, underwater, like the branches of coral, and you can read coral colonies the way, on land, you can read history." He laughed at her expression.
"Is it true?"
"I don't know. I've never been there."
"Where?"
"To the country beneath the sea." He was silent, then, watching her, his eyes oddly somber. "Why," he asked slowly, "do you want to know about that country?"
She almost told him, for if he knew it, he might know the path to it. But it was not her secret, it was Kir's. "I don't," she said brusquely. He only nodded, accepting that, if not, she sensed, believing it, and she had to resist the urge to tell him all over again.
"I can row awhile," she said instead. "I have strong arms." She changed places with him, took the oars. The _Sea Urchin_ skimmed easily over the first sluggish, heaving swell she dipped the oars into; surprised, she pulled at them again and felt she was rowing in a duck pond. Lyo had put a little magic into the oars, she realized, to help her, and he had not told her. And he had turned a snail into a flower.
He could break that chain.
"How did you get to be a magician?" she asked curiously. "Were you born that way? With your eyes already full of magic?"
He smiled, his eyes, facing the sun, full of light. "Magic is like night, when you first encounter it."
"Night?" she said doubtfully. She skipped a beat with one oar and the _Sea Urchin_ spun a half-circle.
"A vast black full of shapes . . ." He trailed his fingers overboard and the _Sea Urchin_ turned its bow toward the horizon again. "Slowly you learn to turn the dark into shapes, colors. . . . It's like a second dawn breaking over the world. You see something most people can't see and yet it seems clear as the nose on your face. That there's nothing in the world that doesn't possess its share of magic. Even an empty shell, a lump of lead, an old dead leaf—you look at them and learn to see, and then to use, and after a while you can't remember ever seeing the world any other way. Everything connects to something else. Like that gold chain connecting air and water. Where does it really begin? Above the sea? Below the sea? Who knows, at this point? When we find out, we'll never be able to look at the sea the same way again. Do you understand any of my babbling?"
Peri nodded. Then she shook her head. Then she flushed, thinking of the tipsy webs of black thread and twigs she had thrown into the sea. How could she have thought they held any power of magic? There was no more magic in her than in a broom. "Where?" she asked gruffly. "Where did you learn?" The magician's eyes were curious again, as if they were searching for those childish hexes in her head.
He opened his mouth to answer, and then didn't. His eyes had moved from her to a point over her shoulder; his hands moved, gripped the sides of the boat. His face had gone very still. She knew then what he was seeing. She pulled the oars out of the water into the boat, and turned.
The magician stood up. He balanced easily in the boat, under the great fiery stare of the sea-dragon. They were still a quarter of a mile from the fishing boats, but it had come to greet them, it seemed, lifting its bright barnacled head out of the water. Its body wavered beneath the surface of the water like a shifting flame.
Lyo whistled. The sea-dragon's brow fins twitched at the noise. _"Ignus Dracus,"_ the magician murmured. "The fire-dragon of the Southern Sea. It appears to be lost. No . . ." He was silent again, frowning. The fire-dragon watched him. The chain around its neck was blinding in the noon light.
The magician sat down again slowly. His face was an unusual color, probably, Peri thought, from the gold; everything around them was awash with gold. The fishers were hauling in their nets. They knew the _Sea Urchin,_ and they knew that only something momentous would send Peri that far out in it.
Peri felt a sudden surge of excitement at the imminence of magic. "Can you break that chain?"
He looked at her without seeing her. "The chain," he said finally. "Oh, yes. The chain is simple."
"Really?"
"It's just—" He waved a hand, oddly inarticulate. "There are just a couple of— Did anyone ever think to ask where this chain begins? Who made it and why?"
"Yes."
"Well?"
She shrugged, avoiding his eye. "They want the gold."
"The king? Did he see it?"
Peri stared at him. He was troubled; the colors in his eyes shifted darkly. Kir, she thought, and as she tried to hide the thought, his words came back to her: _The chain begins in my father's heart._ But his secret was not hers to give away. "No," she said briefly. His face stilled again; he watched her. How did he know? she wondered, rowing a little to keep the _Sea Urchin_ from drifting. How did he know to ask that?
"Peri," he said softly. "Sometimes two great kingdoms that should exist in different times, on different planes, become entangled with one another. Tales begin there. Songs are sung, names remembered. . . . This is not the first time."
She looked away from the magician again, letting the wind blow her hair across her face, hide her from his magic eyes. Kir, his blood trying to ebb with the ebb tide, the secret pain, the secret need in his eyes . . .
"Just break the chain," she pleaded.
"And then what?"
"I don't know. . . ." She added, "We'll pay you."
He studied her for a long time, while the fishing boats inched closer around them. "It's just the gold you want then."
She nodded, her eyes on a passing gull. "So we'll be rich. You'll be rich, too," she reminded him, and he made an odd noise in his throat.
"I'll do my best," he said gravely, "to make us all rich." He stood up again, then, and began to sing to the sea-dragon.
The boats gathered about the _Sea Urchin_ as he sang. The sea-dragon's big, round eyes never moved from him. Birds landed on its head, dove for the fish that followed it constantly; it never moved. Swells nearly heaved the magician overboard a couple of times; he only shifted his feet as if he had been born in a boat and never left it. His songs were in strange languages; the words and tunes wove eerily into the wind and waves and the cries of the birds. The fishers waited silently, plying oars now and then to keep from running into one another, for the bottom dropped away from the world where they floated and there was nothing but darkness to anchor in.
Gradually the songs became comprehensible. They were, Peri realized with astonishment, children's songs. Songs to teach sounds, letters, words, the noises of animals. The fishers glanced at one another dubiously. The sea-dragon drifted closer; its eyes loomed in front of the _Sea Urchin_ like round red doors.
The magician stopped finally, hoarse and sweating. He poured himself a beer out of nowhere, and the fishers grinned. This was no flea-bitten fortune-teller.
"My name," he said, "is Lyo. This, we'll assume for the moment, is a species of _Ignus Dracus,_ which originated in the warm, light-filled waters of the Southern Sea, where kelp and brine shrimp abound. This one apparently muddled into the vast, slow-moving maelstrom formed by the south current as it shifts upward to meet the cold northeast current, which circles downward in its turn to meet the warm south currents again on the other side of the sea." He paused to sip beer. The fishers listened respectfully. Some had even taken off their hats. "It got lost, in other words, which is, we'll also assume for the moment, what it's doing here. The chain, however is not a normal feature of _Ignus Dracus._ It is not a normal feature of anything I have ever laid eyes on in the sea. I will remove it for you. For a small fee, of course. A nominal fee. A quarter-weight of the salvaged chain."
They spoke then, jamming hats back on their heads. "A quarter of the gold! That's—that's—"
"At this point, a quarter of nothing."
"That's robbery!"
"No," Lyo said cheerfully. "Simple greed. I like gold. Take it or leave it. Remember: Three or four links alone will be enough to make the town rich. I can get you far more than that."
There was silence. "One link," someone growled. "One link yours. The rest ours."
Peri laid her arms along the side of the boat and rested her chin on them, watching the sea-dragon. It seemed to enjoy the bargaining: Its head turned ponderously at every new voice. What was it really? she wondered. What had the magician seen within its great, calm eyes? Something to do with Kir?
How had the magician known?
"All right," she heard him say finally. "The first link, and then one out of every five mine. Are we agreed?" He waited; a sea gull made a rude noise above his head. "Good. Now . . . just keep quiet for a couple of minutes, that's all I need. Just . . . silence. . . ."
The _Sea Urchin_ was inching toward the sea-dragon. The drift looked effortless, but it was against the tide, and Peri could see the drag of Lyo's magic at his back. He was frowning deeply; his face seemed blanched beneath its tan. The sea-dragon's eyes were dead ahead, twin portals of constant fire the _Sea Urchin_ seemed determined to enter. The magician's eyes burned like jewels in the bright reflection of the gold.
The boat bumped lightly against the chain. There was not a sound around them, not even from the gulls. Lyo leaned forward, laid a palm on the massive, glowing chain. His hand and face seemed transformed: He wore a gauntlet and a mask of gold.
And then the gold was gone. Peri blinked, and blinked again. It was as if the sun had vanished out of the sky. The sea-dragon made a sound, a quick, timbreless bellow. Then its head ducked down and it was gone.
All around the boats floated thousands upon thousands of periwinkles.
# Six
"OOPS," WAS ALL THE MAGICIAN SAID about it before he vanished. "Sorry." A fisher from one of the larger boats rowed Peri in; he was glumly silent all the way until he slewed the _Sea Urchin_ into its berth in the placid harbor. Then he spat into the water.
"There are some men born to be magicians. And some magicians born to be fish bait." He heaved himself out of the boat, headed toward the inn.
Peri tied up the _Sea Urchin,_ then stood a moment, feeling blank, a vast blue haze of periwinkles floating in her head. No more gold, the sea-dragon gone. . . .
"Periwinkles." Her own voice startled her. She walked down the dock toward the inn, then veered away from it. She had no desire to hear Carey's thoughts on the matter of gold turned into flowers. There would be days enough ahead for that. Months, likely. Where, she wondered, had the magician gone?
Where had the sea-dragon and the gold gone? Where all the moonlit paths to the country beneath the sea always went? To that elusive land called memory?
She sighed. In the bright, blustery afternoon, all the magic had fled, just when she had begun to believe it existed. And now, voices caught her ear from across the harbor, where the king's lovely, fleet ships were docked.
She stopped, staring. Sailors were scouring one of the ships, whistling; others loaded chests, white hens in cages. . . .
Someone else was leaving.
She felt her eyes ache suddenly. She twisted her hair in her hands, away from her face. "Well," she said to herself, her voice so swollen and deep it didn't seem to belong to her, "what did you expect?" A horse returning riderless, a prince coming home half-drowned in the sea—even an absentminded father would pay attention to that.
She dragged her eyes away from the ship, herself away from the harbor. She wandered through the village with its distant afternoon sounds of women chatting across walls as they worked in their vegetable gardens, children playing in the trees, calling to one another. Her rambling took her, as always, to her mother's gate. The hoe was still standing in the weeds. She stopped, glaring at it. Where were the furrows, the seeds for spring? Her mother had to eat.
She grabbed the hoe, ignoring the pale, listless face at the window, and attacked the rain-soaked ground.
Several hours later, she sat on the wall, dirty, sweating, aching, and surveyed her work. There were mud streaks even in her hair. A great pile of weeds lay to one side of the garden; the dark, turned path in the middle was ready for potatoes, cabbages, carrots, squash. The sun was lowering behind her, filling the yard with a mellow light. A sweet sea breeze cooled her face. She ignored the sea for a while, then gave up, turned to it. The boats were homing toward the harbor on a streak of silver fire.
She gazed at it, her heart aching again. She felt a touch on her shoulder; her mother stood beside her. They both watched silently. Peri's tangled head came to rest against her mother's shoulder. The sea, it seemed, had lured them both into its dreaming. Maybe there was no way out of the dream; they would be caught in it forever, yearning for a secret that was never quite real, never quite false. . . .
"Well." She stirred from the wall.
Her mother said softly, "Come in and eat, Peri."
She shook her head. "I'm not hungry. Get some seeds. I'll come and plant them."
"At least," her mother said in a more familiar tone, "wash before you go."
Peri drew water from the rain barrel, poured it over her hair and arms and feet until she was clean and half-drowned. She shook her soaked hair over her shoulders and drifted back out of the yard, through the village, to the beach. She followed the tide line, not looking at the sea except once: when she neared the spires and lifted her head to see the blinding light sinking down between them, showering its gold over empty waters.
She ducked her head again, trudged to the old woman's house. She opened the door, and found Kir, sitting on a stool with his feet on the window ledge, watching the sun go down.
He rose, went to her as she stopped in the doorway. He put his arms around her wordlessly; after a moment her hands rose shyly, touched his back. She closed her eyes against the light, felt him stroke her hair.
"You're wet," he commented, "this time."
"I was gardening."
"Oh." She felt him draw a long breath, loose it. Then he loosed her slowly, looking at her with his strange, clear, relentless gaze. "I'm leaving for a while."
"I know," she whispered. "I saw the ship."
"My father—" He stopped, a muscle working in his jaw. "My father is taking me to visit a lord and lady in the North Isles. They have a daughter."
"Oh."
"I'll be back."
"How do you know?"
His eyes left her, strayed toward the final, shivering path of light across the sea. "You know why," he whispered. "You know." His lips brushed hers, cold, yet she knew he gave her all the warmth he had. "If I could love, I would love you," he said softly. After a moment she smiled. "Why is that strange?"
"If you could love," she said simply, feeling as if she had taken an enormous step away from herself, and into the complex world, "you would not choose me to love."
He was silent. She moved away from him, sat down tiredly, and was instantly sorry that he was no longer close to her. He paced the room a little, looking out. Then he stopped behind her, put his arms around her again, held her tightly, his face burrowed into her hair. She took his hands in her hands, lifted them to her face. She said, "Promise me."
"What?"
"Stay safe, where you're going. Don't drown."
She felt his head shake quickly. "No. I wasn't trying to, that night. I was swimming beyond the spires, trying to follow the light. But the faster I swam, the farther it drew from me; I followed it until it was gone, and I was alone in the deep water, in the darkening sea. . . . I think—I think for the first time that night my father—the thought occurred to my father whose child I might be. I saw him look at me with changed eyes. Eyes that saw me for a moment. He does not want to believe it." He paused; she felt his heartbeat. "That night you pulled me out of the sea . . . before that night, I had never cried. Not even as a child. Not true tears. You made me remember I am half human." He moved to face her as she sat mute; he knelt, lifted her hands to his mouth. She drew a breath, felt herself pulled toward him helplessly, thoughtlessly. Something fiery brushed her closed eyes, her mouth; she lifted her face to it. But it was only the last finger of light from the setting sun.
Kir had moved to the window. She sat back, blinking, watching him watch the tide withdraw. Then, before she could feel anything—love, loss, sorrow—he looked down at her again. You know me, his eyes said. You know what I am. Nothing less. Nothing more.
She got up finally, took bread out of the cupboard, butter, a knife. "I'm lucky," she said, and heard her voice tremble.
"Why?"
She turned to look at him again, his hair black against the dusk, his eyes shades of blue darker than the dusk. She swallowed. "That you are only half like your mother," she whispered. "Because it would be very hard to say no to the sea."
His eyes changed, no longer the sea's eyes. He went to her, his head bent; he took the butter knife out of her hand, held her hand to his cheek. "Yes," he said huskily. "You are lucky. Because I would rise out of the tide bringing you coral and black pearls, and I would not rest until I had your heart, and that I would carry away with me back into the sea, and leave you, like me, standing on a barren shore, crying for what the sea possessed, and with no way but one to get it." He loosed her hand, kissed her cheek swiftly, not letting her see his eyes. "I must go. We're sailing on the outgoing tide. I'll be back."
He left her. The house seemed suddenly too still, empty. She sat down at the table, her eyes wide, her body still, feeling him, step by step, carrying away her heart.
* * *
She stood at the open door hours later, watching the moon wander through an indigo sky, watching the path it made across the sea constantly break apart and mend itself: the road into dreams, into the summer isles. She listened to the sea breathe, heard Kir's breathing in her memories. A tear ran down her cheek, surprising her.
"What have you done?" she asked herself aloud. "What have you done?" She answered herself a few moments later. "I've gone and fallen in love with the sea."
"I thought so," said a shadow beside her doorstep, and she felt her skin prickle like a sea urchin's.
"Lyo!"
He moved out of the shadow, or else ceased being one. The moonlight winked here and there on him in unexpected places; he smelled of sage and broom where he must have been lying earlier.
"Where did you go?" she demanded amazedly.
"Up the cliff."
"How? How did you get there from the _Sea Urchin_ on open sea?"
"As quickly as possible." She saw one side of his mouth curve upward in a thin, slanted smile. Then, as abruptly, he stopped smiling; his face was a pale mask in the moonlight, his eyes pools of shadow. "More easily than you left the sea."
She was silent. She gave up trying to see his eyes and sat down on the step, chewed on a thumbnail. Then she wound her hair into a knot at the back of her neck and let it fall again. "I thought I had more sense," she said finally. "Do you know what being in love is like?"
"Yes."
"It's like having a swarm of gnats inside you."
"Oh."
"They won't be still, and they won't go away. . . . What are you doing here? I thought you ran away."
He chuckled softly. "I'm waiting to be paid." He sat down beside her; she felt his fingers, light as moth wings, brush her cheek. "You were crying. It's a terrible thing, loving the sea."
"Yes," she whispered, her eyes straying to it. Waves gathered and broke invisibly in the dark, reaching toward her, pulling back. They were never silent, they never spoke. . . . Then she looked at the magician out of the corners of her eyes. "You know about Kir."
"I know."
"How? How could you know something like that?"
He reached down, picked up a glittering pebble beside his feet and flicked it absently seaward. "I listen," he said obscurely. "If you listen hard enough, you begin to hear things . . . the sorrow beneath the smile, the voice within the fire-dragon, the secret in the young floor-scrubber's voice, behind all the talk of gold. . . ."
"Gold," she said morosely, reminded. "Don't let the fishers see you still here."
"I won't."
"At least you tried. At least you showed them some magic."
"Perhaps," he said, chuckling again. "I won't expect to be overwhelmed by their thanks. But, not only can I turn gold into periwinkles, I can think as well. And what I think is this: There's someone missing."
"What do you mean?"
"There's Kir. There's his father, the king. There are two wives. Suppose this: Suppose they bore sons of the king at the same time. The son of the land-born queen was stolen away at birth and a changeling—a child of the sea—was slipped into his place. The queen died. But what happened to her true son? Kir's half-brother."
She was silent, trying to imagine a shadowy reflection of Kir. A shiver rippled through her. Somewhere in the night, a king's child wandered nameless, heir to the world Kir so desperately wished to leave. "Maybe he died."
"Perhaps. But I think he lived. I think he's living now, the only proof of the king's secret love. Does Kir suspect he might have a brother?"
She shook her head wearily. "He hasn't thought that far yet. He's barely guessed what he is, himself."
"Why did he tell you?" the magician asked curiously.
"I don't know. Because I was always thinking about the sea and so was he. Because . . ." Her voice trailed away; she put her face on her arms, swallowing drily. "He almost drowned one evening. I pulled him out of the surf. Another evening, he left wet footprints all over the house. Because he needed someone to tell, and I was here instead of the old woman. Because I work at the inn, and he can come and go in and out of my house, and no one would even think to look for him here. A few weeks ago all I did was scrub floors. I don't know how things got so complicated."
"They do sometimes when you're not paying attention. Will you let me help you both?"
"It's all right for now. He's going away to meet some lord's daughter."
"He'll be back."
"I almost wish he wouldn't. I almost wish he would sail away as far as he could sail, across all the days and nights there are to the end of the sea, and never come back." She saw Kir's face in the dark, dark sea, felt his touch, luring her out into deep water with cold kisses and promises of sea-flowers and pearls; she saw him again, crying in the surf, clinging to her on land, as she would have clung to him in the sea.
Her eyes had filled again with tears at the memory of his sorrow. Lyo said again, gently, "Will you let me help?"
"Yes," she whispered. "But be careful. You must always be careful of the sea."
* * *
She still watched, long after Lyo had left her. The moon was over the dunes now, queen of the fishes in a starry sea. The tide had quieted; the long, slow breakers whispered to her of magic hidden in that blackness: the great floating islands drifted past just beyond eyesight, the ivory towers on them spiraled and pointed like the narwhale's horn. Kir's world, the world his heart hungered for, elusive as moonlight, as water, yet constantly calling . . . She felt the touch of his lean, tense hands, saw his sea eyes, seeing her as nobody else did, heard his voice saying her name.
"Oh," she whispered, her throat aching, the stars blurring in her eyes. "I wish you were human." She blinked away tears after a moment. "No," she sighed, speaking to the waves since she had nothing else of Kir. "If you were human, you would never have given me a thought. A girl who works at the inn. You would never even have known my name. I wish—I just wish you were a little bit more human. So that you wouldn't always be turning away from me to the sea."
Something moved within the darkness. It flowed across her vision until it blotted out one star, and then another. Her skin prickled again; her eyes grew wide as she tried to separate dark from dark. Was it the sea country rising from the depths of the sea? Was it some dream island that shifted by night from place to place? Was it some vast, dark, high-riding wave? But the receding waves broke evenly, serenely against the shore. Still the darkness flowed until, by the outline of black against the stars and the foam, she realized what it was.
She stood up slowly. The sea-dragon was riding on the surf, closer to land than she had ever seen it. Lyo had freed it, yet it lingered, alone in the night, missing the fishers. Had it been drawn by her lamplight? She found herself moving quickly, impulsively across the sand, drawn toward it, trying to see it more clearly. It swallowed more stars. One of the spires disappeared behind its back, then the other. Her feet dragged abruptly in the sand, stopped.
It was coming out of the sea.
Her throat made a whimpering noise, but she was frozen, incapable of moving. She could see the great eyes reflecting moonlight, the mountain of its back, the enormous fins along its body pushing it through the shallow water into the tide. "Lyo," she said, but as in a dream, her voice had no sound. Was it coming out to die, she wondered, like the whales did sometimes? Or was it just coming out like the sea lions came out, to stretch its great body on the dry sand and sleep?
It was coming straight toward her; its fiery eyes saw her. She stepped back; it gave a mournful cry like a foghorn, and she stood still again. It can't eat me, she thought wildly. It could roll on me, but I can move faster. What does it want?
Its back fins heaved the bulk of it out of the surf. The tide played with its streamers awhile, rolled them up, stretched them out, delicate ribbons of smoke. Still it came, heave by heave, until only the long, tapering tail fin was left in the surf, and finally only its tail streamers.
And then it was all out, one final streamer tugged out of reach of the tide. And not six yards from her. Her hands were jammed over her mouth; she was poised to scream, to run all the way to the village if it decided to sit on her house. But it stopped heaving, stopped moving completely except for a whuff of a huge sigh out of its mouth.
Its eyes closed, red moons vanishing. And then all of it vanished.
A young man, naked as a fish, knelt on his hands and knees at the end of the path the sea-dragon had taken out of the sea.
# Seven
PERI MADE A NOISE. She still had her hands over her mouth, so it was a muffled noise. The sea-dragon's head lifted. He stared at her groggily, blinking, the salt water dripping from his hair into his eyes. He shook his head wildly, and stared at her again. She stood like one of the stone spires, like something to which barnacles and sea urchins could drift against and cling. But he did not confuse her with a stone. He turned his head to look at the stars, the waves, the sand, and finally his hands.
He touched his mouth with one hand. He said rapidly, experimentally, "One fish sat on a house, two fish ate a mouse, three fish—" His voice faltered. "Three fish . . . three fish . . ." He looked frightened, then desperate, as if he had forgotten some vital magic spell that was the lifeline to his new body. His eyes went back to Peri. She moved her hands after a moment; her bones felt brittle, like dried coral.
"Three fish rode a horse." Her voice seemed to come from somewhere else, out of the well under the gorse, perhaps. Her heart thumped raggedly. She felt as if she had stepped into some dream where anything might happen: Her head might float away and turn into the moon; starfish might walk upright onto the sand and dance a courtly dance.
The terror went out of the sea-dragon's face.
"Three fish rode a horse. Four fish swam in the gorse." It was one of the children's rhymes Lyo had chanted to the sea-dragon. "Five fish climbed a bee." The sea-dragon was beginning to shiver.
"A tree," Peri said numbly. "Five fish climbed a tree."
"Six fish caught a bee."
She took a step toward him, breaking out of the spell he was weaving about them both with his rhyming. "You're cold." He was silent at the unfamiliar word, watching her. In that moment, with his face still, uplifted to the moonlight, his hair dark with water, he looked eerily like Kir.
She closed her eyes, suddenly chilled herself. The king's missing son. He had crawled out of the sea, found his body and his voice and was kneeling stark naked under the stars counting fish in front of her.
"Seven fish dined with the king." His voice sounded strained again, uneasy at her silence. "Eight fish found a ring. Nine fish—"
Peri took another step toward him and he stopped speaking. She moved again and he stopped breathing. This time he was frozen in the sand, watching her come.
She reached him finally. He slid back on his knees to look up at her. His face, lit by the moon above the dunes, seemed unafraid. In his great, bulky underwater body, he would never have learned fear. Her hand moved of its own accord, another piece of the dream, and touched his shoulder.
At her touch, he began to breathe again. His skin was icy. He still watched her, his face curious, very calm. But when she lifted her hand, something flickered in his eyes; he put his own hand where hers had been.
She shivered again, glimpsing the complex, mysterious events that had hidden this king's son behind those huge, inhuman eyes, in that body with streamers for fingers, fins for feet, and the rest of him home for any passing barnacle. "Down in the sea," she whispered, "did anyone ever touch you? How could—how could anyone do such things to you and Kir? What makes people do such things?"
He was listening, as the sea-dragon had listened, alert for every pitch and shading in her voice. He picked up a word unexpectedly, hearing an overtone.
"Kir."
She swept at her hair with both hands, utterly perplexed. "I should take you to the king," she said, and was horrified at the thought. "I can't walk into his great house with you in the middle of the night and explain to him—me, Peri, who mops floors at the inn—that you are his human son and Kir is his sea-child and—I can't, anyway," she added with vast relief. "He left with Kir. Lyo. Lyo can tell me what to do with you." The sea-dragon was listening patiently, his teeth chattering. She put her arm around him, coaxing him to his feet. "At least I can find you a blanket. Can you walk? Not very well, but no wonder, you just got born."
She took him into the house, wrapped him in an old quilt and rekindled the fire under a pot of oyster soup. The light springing over his face made her eyes widen. His hair was as gold as the sea-dragon's chain; his eyes were sky blue under gold brows. Like Kir, he was tall, slender, broad-shouldered; like the sea-dragon, he was constantly in motion. He paced while he nibbled bread. He burned his fingers in the fire, poked himself with a needle, startled himself with the cracked mirror, tripped over the trailing quilt, and dropped everything he picked up, including his bread and a bowl of oyster soup. Peri made him sit down finally, curled his fingers around a spoon and taught him to feed himself. The first spoonful of hot milk, oysters, melted butter, salt, and pepper he swallowed amazed him. Peri laughed at his expression. An answering smile sprang to his face, mirroring hers. It was a smile startlingly unlike Kir's, sweet and free of bitterness. She gazed at him, silent again, forgetting to eat. He waited, alert to her silence, as curious and patient as the sea-dragon had been, balancing its lumbering body in the waves among the fishers' boats to listen to them talk.
"I wonder if you even have a name," she breathed finally. "I wonder what your mother called you, just before she died. You must look like her. I wonder if, down in the sea, they ever took off the chain, turned you back into this shape. . . . I wonder if they ever taught you anything at all, even 'yes' and 'no.'"
"Yes and no," the sea-dragon said obligingly, "and dark and light, sun and moon and day and night: They never speak, but to and fro together through the world they go."
"Or did they just keep you chained from the day they took you? Is this really the first time you have been human?"
"Human."
"Like me. Like the fishers."
He let the bowl sag forward in his hands as he listened, his eyes intent on her eyes, her mouth. His own eyes were heavy; the sea-dragon's struggle out of the sea had tired him. She righted the bowl before he spilled it, coaxed him to eat more soup before he nodded away and tumbled off his chair. She made him lie on a blanket beside the hearth; he was asleep, soundlessly, before she covered him.
She stood watching the firelight lay protective arms across him. Twice in one day, she thought; two princes have come into my house. One dark, one light, one day, one night. . . . And then she fell into bed without getting undressed.
She woke again in the dark to the roar of the high tide. A puddle of moonlight splashed through her open door. It creaked as she stared at it, then banged shut, making her jump.
The blankets lay scattered, empty beside the hearth. She was alone. She got out of bed, went to the window. The milky, moonlit sea dazzled her eyes; she leaned farther out, blinking, and saw it finally: the sea-dragon with all its streamers swirling about it taking the silver path of light between the spires, back into the sea.
* * *
She worked groggily at the inn that day. Carey, bewailing the loss of the gold unceasingly, it seemed, became to Peri's ears like a background noise of shrill, keening gulls' voices. Even Mare, with all her good sense and humor, was cross.
"How could he have been so stupid?" Carey demanded. "How could anyone with the power to turn gold into anything at all have turned it into a bunch of flowers? All that gold, Mare. . . . Peri, you were with him. Did you have any idea he would do anything so barnacle-brained as that?"
Peri shook her head and swallowed a yawn. Carey stood in front of her, wanting something, some word of explanation, of hope. When none came, she made a desperate noise and stared out a window.
"I'm going to run away."
"Oh, please," Mare sighed. "Stop caterwauling about the gold. It's gone. We've lived our lives so far without it, and if just living peacefully won't make you happy, I don't suppose a fortune will, either."
"You're sorry about it, too."
"All right, so I am. It would be fine not to have to scrub floors and listen to you complain every day; if you're going to go, then go for goodness sake, girl, and give us a rest."
"All right, then," Carey snapped. Peri lifted her head. Something in the tense, poised lines of Carey's body reminded Peri of Kir's desperation and helpless anger.
"Don't go," she said softly. Carey's miserable, furious gaze swung to her. "Maybe he'll be back. He has magic in him. He had the power."
The anger vanished from Carey's face. She went to Peri, grabbed her broom to keep it still. "Yes. You're right. If he could turn gold into flowers, why can't he turn the flowers back into gold? He could, couldn't he? If we could find him, if we could ask him."
"Those flowers are probably down in the south islands by now," Mare said. "And so, if he knows what's good for him, is the magician. You heard the fishers yesterday when they came in. If the magician had been anywhere within reach, they would have corked him into a beer barrel and tossed him back out to sea."
"But—" Carey said stubbornly.
"What?"
"The magic was real. It was real, Mare. Peri is right. He has the power."
Mare was silent, frowning uncertainly. Peri closed her eyes a moment, wishing she could curl up under a table and take a nap. An image of the sea-dragon exuberantly breasting the moonlit waves swam into her head. Yes and no and dark and light, his voice said, just as her head nodded forward against her broom and snapped back up again.
"Peri!" Mare exclaimed. "You're falling asleep!"
"Sorry."
"What have you been doing at night, girl? Meeting phantom lovers?"
"Yes," she said, yawning again. Unexpectedly, Carey laughed.
* * *
Peri's house was empty when she came home that afternoon. She ate bread and cheese beside the hearth, then crawled gratefully into bed before the sun went down. She slept deeply without dreaming; when she floated drowsily awake again, she wondered why it was still dark.
Someone was moving in the house. "Kir?" she said sleepily. "Lyo?" Then she came fully awake, for the door stood open and the bright moon hung like a lantern over her threshold.
A hand brushed her lightly. "Two fine ladies rode to town, in a carriage of seashells pulled by a prawn."
"You're back!"
"You're back," he echoed. He had already pulled a quilt over his shoulders; he was rubbing his dripping hair with a corner of it. "Peri," he added, and she started.
"Who taught you that?"
He burrowed more deeply into the quilt, then held out his hands to her cold hearth. Peri got out of bed. The fire she built chased away the fog in her head along with the darkness.
"Lyo," she exclaimed as the sea-dragon knelt to warm himself. "Has he been with you in the sea?"
He touched his mouth, feeling for words. Then he said in an arrogant, scholarly voice, "A species of _Ignus Dracus,_ which originated in the warm, light-filled waters of the Southern Sea—oops, sorry."
She smiled wonderingly. "Is that how you turned human? If so, his spell last night didn't work too well; maybe tonight you'll stay human. But—" A frown crept into her eyes. She stopped the curious sea-dragon from putting her hairbrush on the fire. "But all I can teach you are words," she said, groping with words herself to say what she meant. "How will you know what they mean? How can you say where you've been? What you want?"
"Want is empty, have is full," he said. "Want is hungry—" She turned; he stopped talking. But something in his eyes spoke, insisted, that when she taught him the language, he would have more to tell her than of kelp and shrimp and children's rhymes. He put his hands around his neck suddenly. "Chain," he told her. "Chain." And she saw the human feeling in his eyes.
She took his hands, held them toward the blaze. "This is fire," she said.
"Fire."
She tugged him up, led him to the open door. "Those are stars. That is the moon."
"Stars. Moon."
"Sand," she said, pointing to it. "Sea."
"Sand." He fell silent, gazing out at the restless tide. "Sea," he whispered, and as she led him in, she wondered if the sudden wash of fire across his eyes had hidden love or hate.
He roamed the house, touching everything, remembering many of the words she told him. Finally he turned to her with the same scrutiny, touched her tangled hair.
"Hair," she said.
"Hair." He bent slightly, peering into her eyes, then at her nose with such intensity that she laughed. He startled, ducking back as gracefully as a fish. Then he smiled.
"Nose."
"Nose."
"Eyes."
"Eyes." He stared into hers again; even his lashes were gold, she saw, against a milky skin that had never been touched by the sun. She drew a breath, pulled herself out of his summer blue gaze, and called his attention to the floor.
"Feet."
He grew drowsy soon; it taxed his strength, she guessed, to heave that sea-body onto dry land. But before he fell asleep beside the fire, he told her a story.
"Once upon a time," he said in Lyo's voice, "there was a king who had two sons: one by the young queen, his wife, and one by a woman out of the sea. The sons were born at the same time, and when the queen died in child-bed, her human son was stolen away, and the sea-born son left in his place. Why? No one truly knows, only the woman hidden in the sea, and the king. And perhaps the king does not even know. Why?
"Why is the wind, why is the sea, why is a long road between the world and me." He fell silent, watched the changing expressions on her face. He reached out, put his hand on her hand, and went to sleep.
When she woke in the morning, he was gone.
She went to work, puzzled. She searched for signs of the sea-dragon as she walked to the inn that morning, and again in the afternoon when she walked back. She cooked potatoes and sausages for supper, leaving her door open to catch the mild spring breeze. A shadow fell over her frying pan from the doorway; she whirled, and found the magician leaning against the doorpost.
"Lyo!"
He smiled. "I kept smelling something wonderful. I followed my nose." He looked thinner, she thought, and wondered what and where he ate. Certainly nowhere in the village. She took the pan off the flame and held it out to him. He took a smoking hot piece of potato, juggled it between his fingers, then bit into it and sighed.
"It's so good it must be magic."
"Lyo, where is the sea-dragon?"
"In the sea," he said with his mouth full. She gazed at him, perplexed; he took a sausage from the pan. She sucked at a tine of her stirring fork.
"Well, why?" she said. "Can't you make a spell work right for once?"
He raised his brows at her, speechless as he bit into the sausage. "What," he said when he could finally speak, "are you asking?"
"I'm asking why you can't change the sea-dragon into a prince for more than a couple of hours at a time."
"Why can't—"
"First you change gold into flowers, then you change the sea-dragon, only you—"
"I didn't."
"You didn't change him?"
He shook his head, reaching for the pan again. She set it down finally on the table; they sat down, nibbled out of it, he with his fingers, she with the oversized fork.
"Then who did?"
He shook his head, looking as curious and as baffled as she felt. "I have no idea. It doesn't make any sense. That's really what I came to ask you."
"Me?"
"If you knew why it—he—changed so suddenly. And at such an odd time. Did you see anyone? Hear anything?"
She shook her head. "I was there when it happened; I was still watching the sea. It just crawled out. There was no magic. It just changed. He. Lyo." She paused, groping while he waited. "He is so—so—"
"His mother," Lyo said, finding another sausage, "was said to be very beautiful."
"Then why did the king love a sea-woman? If he had a wife like that?"
"Well." He chewed a moment, thoughtfully. "As I have heard, they barely knew each other before they were married. The king knew the sea-woman longer than that, I suspect. I think she was not a passing fancy, but someone who came to love him. The king didn't realize he would come to love his own wife as well. He married and forgot about the sea-woman, but he saw her one last time just before he married. And that was one time too many. Nine months later the queen was dead, her child taken under the sea, and the changeling cried in the royal cradle instead."
"It is sad."
"It is."
"The sea-dragon doesn't even have a name." She poked holes in the potatoes with the fork, brooding, while Lyo watched her, his eyes sometimes smiling, sometimes secret. "I wish the king and Kir would come back. Then—oh." She put the fork down. "What will Kir say? He doesn't have a home on land or sea; the sea-dragon is only human in the middle of the night—"
"It's an odd pair of sons for a king to have."
"Lyo, you have to do something."
"I am." He bent over the pan again. "I'm going to finish your supper."
* * *
The magician taught the sea-dragon in the sea by day; by night it crawled out of the sea to tell Peri the words it had learned, and to learn more from her. Peri, keeping such odd hours, felt life begin to muddle like a dream. She found herself mumbling "scrub brush" and "soap" as she sloshed water across the floor at the inn, and stray bits of children's rhymes ran constantly through her head. She saw little of Lyo; she assumed he slept at night, along with everybody else who lived their lives oblivious of the double life of sea-dragons. The ship that had carried Kir away remained stubbornly away.
"I heard," Carey said, full of gossip one morning as they began to work, "that the king took Kir up to the North Isles to marry some lord's daughter."
A huge soap bubble, rainbows trembling in it, fascinated Peri. She stared at it and tried to imagine Kir married. Like a breath of dark wind, something of his own frustration and panic blew through her. He might marry, but he would never love, and then there would be yet another child trapped in one world, yearning for another. And another young woman cruelly betrayed by the sea. She sighed. The bubble popped. The story would go on and on. . . .
Mare tripped over her feet. "I'm sorry, Peri. Where did you hear that?" she asked Carey.
"From one of the girls who works in the kitchen. She was bringing supper to some of the guests and heard them talking. They said Kir was restless and unhappy. The king thought marriage would settle him."
"Poor Kir," Mare said, and Peri looked up from the hearth she was cleaning.
"Why?"
"There's no magic in marriage. If they become friends, though, that would be different. But royal folk rarely get to marry their friends. They have to marry power or wealth or land or—"
"Well," Carey said wistfully, "at least they have that."
"Oh, Carey!" Mare said, laughing. "You're impossible."
"I can't help it," Carey said stubbornly. "I want to be rich. I want that sea-dragon's gold. Then I'll be happy."
Peri cooked supper for herself, and crawled into bed as soon as the sun set. The sea-dragon woke her out of her dreams to the roar of the sea, the wind shaking her door, the little barque of the moon sailing among scudding clouds. "Saucepan," she taught him. "Wall, fork, bread, salt." When her house held no more new words, she taught him sentences. "I am hungry. I am thirsty. Where are you? I am here. What are you doing? I am stirring onions in a pan, I am combing my hair. . . ." As the nights passed and the sea-dragon consumed words like shrimp, they made the sentences into a game.
"What are you doing?" he asked as she drank water.
"I am drinking water. What are you doing?"
He moved to the door. "I am opening the door. What are you doing?"
"I am putting wood on the fire. What are you doing?"
"I am looking at your seashells. What are you doing?" he asked, with such a peculiar expression on his face that she laughed.
"I am jumping up and down. What are you doing?"
"I am walking to you."
"Toward you."
"I am walking toward you. What are you doing?"
"I'm still jumping. What are you doing?"
"I am walking closer toward you."
"To you."
"To you. And closer. What are you doing?"
She stopped jumping. "I am standing still," she said.
"And I am walking. Closer. Closer."
She stood very still, silent, watching him come, the sea-dragon in the prince's body, with the gold in his hair and the firelight sliding over his skin. "I am coming very close."
She swallowed. "Very close."
"I am touching you." His hands were on her shoulders. Then she saw the simple need in his eyes, and she put her arms around him.
"I am touching you."
"Yes," he said softly, and she felt the long sigh through his body. "You are touching me."
She watched him fall asleep in front of the fire. Her heart ached at his loneliness. Like Kir, he was bound to the sea, in body if not in heart, and loving him was no more possible than loving his brother, whose wild heart cried out to follow the tide.
"Lyo," she whispered, "what are we going to do?" But there was no answer from the sleeping magician.
# Eight
THEN THE SEA, missing its gold, perhaps, began to play tricks on the fishers. Enin told the first tale, coming in late on a spring evening with Tull Olney dragging behind him. Enin was soaking wet, his face pale, his eyes bloodshot from salt water. He stood at the bar, dripping on the floor, downing beer as if to wash salt out of his throat. Tull, as bedraggled as Enin, looked, Mare said later, as if he had been slapped silly by a dead cod. Peri, coming up from the kitchens with a warm loaf of bread wrapped in her skirt, stopped short on the top of the stairs when Tull said, "There is something going on in the sea."
"There's something going on in your head," Enin said brusquely. "I'll have another beer."
"You heard the singing!"
"I heard somebody blowing a conch. That's all." He turned to the fishers and the innkeeper, and Mare, who had slipped in at the sound of his voice. "Tull and I were fishing close by each other. He says he heard singing, I say a conch shell. It was near sundown, the sea was milky-blue under a sweet south wind. I heard a conch—"
"Singing," Tull muttered into his beer.
"It was that deep, foggy sound. A conch, like they use up in the north villages to call all the fishers together. I heard a splash, and there was Tull, leaping out of his boat to swim with his boots on after a seal!"
"It wasn't a seal!"
"I called out to him, he never answered, just swam on. Then the seal dove under, and Tull was left floundering in the water with his fishing boots filling up. So guess who got to leap in after him?" He downed half his second glass and glared at Tull. Peri, watching him with her mouth open, saw something frightened behind the glower. Tull banged his own glass onto the bar.
"It was singing! And it was a woman!"
"It was a seal! A white seal—"
"It was a white-haired woman, with—"
"With brown eyes."
"With brown eyes." Tull looked around the silent room, his own eyes round, stunned. "She sang. She was a small, pretty thing, white as shell, playing in the water as if she had been born in it. She flicked water at me, laughing, and then . . . there I was. Like Enin said. Jumping into the deep sea as careless as if I were a seal myself." He shuddered. "She vanished, left me hanging there in the empty ocean. Her singing . . . it was like singing out of a dream I wanted to find my way into. I started trying to drink the sea, then, and Enin pulled me out."
The fishers stared at him, lamplight washing over their still faces. Somebody snickered. Ami dropped her face in her arms, whimpering with laughter.
"A seal. You prawn-eyed loon, leaping into the deep sea to frolic with a seal!"
"It wasn't a seal!"
"Next it'll be the King of the Sea himself blowing his conch in your ear."
"I almost drowned," Tull said indignantly, but by then everyone was laughing too hard to listen to him.
Peri, hugging the warm loaf to her for comfort, left quickly, passing Mare in the doorway, staring at Enin and Tull without a glimmer of a smile on her face.
Next, it was Bel and Ami who came in late, quarreling bitterly over a lost net. Something, it transpired, had come up in their haul that Ami refused to pull in.
"It was an old dead hammerhead," Bel said disgustedly.
"It was a boy!" Ami wailed. "A luminous, shiny green-white mer-boy, caught in the net among all the fish. I thought he was dead, but he opened his eyes and smiled at me."
"She let go of the net. It was so heavy I couldn't hold it," Bel said. "Heavy as if someone were beneath, pulling it down. Ami was screeching in my face. I had to let go finally to shake her. A mer-boy, my left ear. It was nothing but a dead shark. Now we need a new net."
In the next week or so, half the fishers hauled in a tale along with their day's catch, and nobody was laughing anymore. One fisher had nearly rammed his boat against rocks, trying to join a pair of lovely sirens drying their long hair in the sun. Another, beckoned by a vague figure in a strange boat toward a wondrous school of fish, rowed dangerously far out to sea, only to watch the strange boat flounder and sink, as it had many years before. There were tales of hoary, tentacled, kelp-bedecked sea monsters, and of great ghostly ships from some forgotten past rising silently out of the water to sail right through the fishing boats like some icy, briny fog. The fishers' catch dwindled; the innkeeper had trouble keeping beer in stock. Worse, the summer visitors had caught wind of the tales and were passing them gleefully along to one another.
"We'll be the laughing stock of the island," Enin said glumly, leaning on a doorpost and watching the girls work. "Soon we'll be too frightened to stick our bare feet in the surf, let alone leave the harbor."
"They want their gold back," Mare said soberly.
"We haven't got it!"
"I know."
"That chowderhead mage turned it into flowers."
"I know."
"Well, what are we supposed to do?"
Mare stopped shoveling ashes out of the hearth. "I think you'd better find that magician before whatever is in the sea drives you onto land for good. But," she added, shoveling furiously again, "since you never paid any attention to me before—"
"Now, Mare," Enin said, coughing at the cloud she raised.
"It's not likely any of you will have enough sense to now."
"Where would he have got to, do you think?"
"You found him before, you can find him again."
Enin sighed. "We'll be a laughingstock."
"So? Who in this village is laughing anymore?"
Was it the gold, Peri wondered as she walked back that evening? Or was it the king's son the sea wanted back, chained again, not knowing any human language? He would come that night; he came every night. She was getting used to waking in the dark to his gentle voice saying unexpected things. She felt a chill down her back, though the air was balmy. The sea was troubling the fishers now. How long would it be before it found its way to her door? Lyo had freed the sea-dragon from the gold chain, but not from the sea. He could not live on land any more than his half-brother could live in water. Who could help them? Where was Lyo? Where were any answers for either of them? She stopped midstep in the sand, feeling too helpless and worried to think any longer; she could only shout hopelessly as loud as she could, in frustration, expecting no answer.
"Lyo!"
"What?" he said beside her. Her shout turned into a scream; she seemed to levitate before his eyes. He bent quickly to pick up the mussels she had dropped. She came back down to earth finally and glared at his shaking shoulders.
"Lyo, where have you been?"
"Here." His voice sounded constrained; he had to duck away from her again, while a sound like geese arguing came out of him.
"Well, why didn't you tell me?"
"Why didn't you call me before?"
"How should I know you would come?"
"I'm sorry—" He straightened finally, wiping his eyes. "You looked so—all your hair went straight for a moment, like a giant hedgehog. I've never seen anything like it."
"It did not." But she was smiling then, too, at the sound of his cheerful voice and at his secret, dancing eyes. She held out her skirt; he dropped the mussels back into it.
"Lyo, something is happening in the sea," she said.
"I know. I've been hearing the tales."
"Have you seen anything, when you're with the sea-dragon?"
"No."
"It's the sea-dragon the sea wants back."
"Do you think so?"
"What else could have upset it? The fishers think it's happening because they tried to steal the gold."
"So." His mouth curled up at one corner. "Now they want me to put the chain back on."
"But if you do that—"
"I'm not going to."
"But if you don't, the fishers will be frightened out of the sea. They have to make a living."
"I know."
"Then what will you do?"
He tugged his hair into spikes, smiling at her again. Then his eyes strayed to the sea idling between the spires. "Well. We know that paths exist between land and sea. Kir's mother found one. I have spent some time searching for a way for humans to reach that country beneath the sea."
"Walk there?" she breathed, appalled and fascinated at the same time.
"People do. Sometimes. But not easily, and sometimes at an extraordinary price. Time passes differently in the undersea; humans can lose years, memories, loves, other things they value. Getting back is even more difficult."
"Oh." She sighed slowly. "Then what—"
"The only thing I can think of that might help is to talk to Kir's mother."
Peri looked at him out of the corners of her eyes. "His mother."
"She stole the king's human son, she chained him, she bore the king's sea-child. Maybe she is responsible for the things happening to the fishers. Maybe she is trying to speak to the king that way, sending him a message, making him pay attention to the sea."
"Except that he's not here."
"But we are. We're listening."
"Would she want to talk to you?"
"Us."
"You. She has never even talked to Kir."
"Sometimes people get so angry they can't hear anything beyond their anger."
"Who is she angry at?"
"The king."
"Still? After all these years?"
"I suppose she still loves him."
"How can she love him and be so angry with him at the same time?" Peri asked, bewilderedly.
"It happens often," Lyo said. He stopped to pick up an agate in his lean, quick fingers and look through it at the sun. "Love and anger are like land and sea: They meet at many different places. The king has two sons. One he knows, the other he doesn't. It's about time he met his wife's true child."
"But he only looks human for a couple of hours every night. The rest of the time he looks like a sea-dragon. You can't row the king out to sea and introduce him to a sea-dragon."
"No."
"Well, then—" Her voice faltered. "Well, then, how can you—Lyo, no."
"It's the only way."
"No." She gripped his arm, pleading. "No. Please, no. You can't bring the king to my house."
"Peri, he has to know that he has another son. And if we don't do something soon, the fishers will be driven entirely out of the sea. Or else the sea-dragon will be chained again, so deeply that he will be lost forever. Were you thinking you could just teach him enough language so that he could find his own way to his father's house?"
Peri shook her head. "I don't know," she said numbly. "I wasn't thinking. But who knows when Kir and the king will be back?"
"Kir still doesn't know he has a brother?"
"He left just before the sea-dragon changed. He never guessed that part of the story."
Lyo grunted, thinking. "You tell him when he returns. I'll tell the king."
Peri stared at him. "You aren't afraid? You'll walk into his house and tell him he has a secret son in the shape of a sea-dragon living in the sea?"
Lyo shrugged imperturbably. "Someone has to. Three people know: you, me, and Kir's mother. That leaves me."
The next day and the next brought in more tales from the sea: of something coming alive in a net, wrapping caressing arms around a fisherman's neck and nearly pulling him under the water; of a strange cloud that swallowed two or three boats on a cloudless day. Fog-blind, lost, they drifted aimlessly for hours, hearing bells ring, occasional laughter, sometimes a sweet, astonishing harping, faint and light as a sudden patter of rain beyond the cloud. The boats came out of the cloud near nightfall, without a fish in their holds, and so far from the harbor it took them until midnight to get back. All the fishers were looking haunted; they sent messages traveling up and down the coast, pleading for the return of the magician.
"It's all the old stories out of the sea coming alive," Mare said wonderingly, as they put mops and brushes and dust cloths away at the end of the day. "I wonder who it was we offended, making all that gold disappear."
"And we never got so much as a coin out of it," Carey sighed. "It's not fair. The magician probably stole it. He probably picked all the flowers out of the water and turned them into gold again. He won't come back."
"Ah, don't say that. He's our only hope."
"Maybe. Maybe the king can do something when he comes back."
"What could he do? Even if he believed the fishers? I can't see him jumping out of his great ship with his boots on to swim after a seal. He watches the waves from his fine house; he sails from land to land on his ship; the only fish he sees are covered with sauce on his plate. What does he know about the sea?"
"Something," Peri muttered without thinking.
"What?"
She tugged at her hair until it fell over her eyes. "I said something. Maybe that's what he can do. Maybe."
There was a storm that night. Dark, swollen clouds gathered at the horizon at sunset, moved inland fast. Peri heard rain thump on the roof as she cooked supper. In the middle of the night, she woke to the crash of thunder, and she got up to watch the sea-dragon tumble in on the wild waves. The sea washed him out more quickly than usual; he was drenched with rain by the time he reached Peri's door.
She threw blankets over him; he stood in front of the fire, his teeth chattering. Then, as he drank the hot broth she gave him, gradually he began to speak. The storm had not upset him, she realized; to him it was just another form of water.
"I saw a boat," he said.
"A boat?" she repeated, horrified. "A fishing boat? In that storm?"
He shook his head, flicked water out of his bright hair with one hand. "Not boat. The word is too small. Bigger than a boat. After the sun went down. Far away. I swam so far the land was thin."
"A ship!"
"A ship," he agreed. "In the rain. I swam with it listening to the voices."
"It's a rough night for a ship to be out," Peri said. She was frowning, her arms folded tightly, protectively; she was nervous at what she wanted to say to him. He put his hand to her face suddenly, where her brows were trying to meet.
"What are you doing?"
"What? Oh—" Her brows jumped apart again. "I was frowning. That's a frown."
He tried it, his hand still touching her face. Then he laughed. He said, watching her closely when she didn't laugh, "Your face is talking. I can't hear it."
She held herself more tightly, drew a breath. "When you—when you swim in the sea, do you have a name?"
He was very still then; his hand dropped. His eyes left her face, went to the fire. He drew the blankets more closely about him. When she realized he would not, or could not, answer, she tried again.
"Who put the chain around you?"
Still he didn't answer. He kept his eyes on the fire, as if he were listening to its voice. She said softly, her brows puckered worriedly, "This is the world you belong in. Not the sea. You belong here, in this world of air and fire, you were born to walk on this land. All the words above the sea belong to you. Tell me. If you can. If you remember. Who chained you to the sea?"
He looked at her finally. Fire-streaked tears ran down his face; he made no sound. She swallowed, reaching out to him, touching him. He lifted one hand after a moment, brushed it across his cheek and stared down at it.
"What am I doing?"
"Crying," she whispered. "You are crying tears. Sea-children don't cry."
"Tears."
"You are sad." She put her hand on her heart. "Here. What made you cry?"
He looked down at the fire again, seeing in its drifting, eddying flames a land she could not imagine. "I don't have the words," he said softly. "You teach me."
"What—what words do you need?"
"All the words," he said, "under the sea."
Perplexed, she stopped on the beach the next evening and summoned the magician from whatever secret place he kept himself. She had caught him in the middle of a bite; he offered her a piece of his bread and cheese while he finished chewing.
"Lyo," she said, her mouth full.
"Yes."
"Where are you when you're not here?"
"Oh." He swallowed, waved a hand inland. "There's a bit of forest beyond the gorse. . . . What is it?"
"I need something."
"What do you need?"
"Something with words in it."
"A book?" he suggested. She frowned at him dubiously. He asked delicately, trying not to smile, "Can you read?"
"Of course I can read," she said witheringly. "Everyone can. It's just that after you learn how, it's not something you do."
"Oh."
"Not in this village, anyway. My mother has a book she presses flowers in. But it's not what I need."
"What—"
"I need something for the sea-dragon. Lyo, my house is too small; there are no more words in it. He wants to tell me something about the land under the sea, but he doesn't know the words, and I can't teach him, because I don't know what he's seeing."
"Ah," Lyo said, illuminated. Then his thoughts went away from her; his eyes grew blue-black, absent. "But," he said, coming back, "you must be very careful."
"Of what?"
"Of the book."
"What book?"
"Tut," he said. "Pay attention, Periwinkle. The spell book. Don't read it, just look at the pictures. They should help you. Promise you won't try to work the spells."
"I promise," she said, bewildered but entranced.
"I'm very serious. You'll make all your hair fall out, you'll turn yourself into something."
"A periwinkle?"
He laughed, then, forgetting his warnings. "Perhaps."
"Lyo. Did you turn that gold into periwinkles on purpose?"
His eyes grew light, dancing, making her smile. "Well. Your name was on my mind."
"Did you?"
"What a dull place the world would be if all the mysteries in it were solved. Wait here." He vanished, leaving, Peri's bemused eyes told her, his shadow on the sand. He was back in a moment, chewing again, with a huge black book under one arm. " _Elementary Dealings with the Sea,_ " he said, passing it to her; their hands seemed to blur a little into its darkness. Then Lyo murmured something, and the hazy lines of the book firmed. "It's open, now. It's a sort of primer for beginning mages."
"Oh."
"Don't worry," he said reassuringly, "it has lots of pictures." He paused, his voice on the edge of saying something more. Then he nudged at an expired jellyfish with his foot. "Call me again when you need me."
"How can you hear me out there in the forest?"
"It's easy. Your voice comes out of nowhere, catches me like a fishhook in my collar, and hauls me to where you are." She laughed, feeling a sudden color running up under her cheeks. He smiled his quick, slanting smile, then sobered. "Be careful," he said again; she nodded absently.
The book had wondrous pictures. She lay beside the hearth that evening, turning pages slowly and dropping crumbs between them as she nibbled her supper. Pictures accompanied each mysterious spell. At first glance they were simply paintings, but as she gazed at them longer, they began to move. Whitecaps swelled; wind picked up spindrift, flung it like rain across the surface of the sea: "How to Achieve a Minor Storm." Mermaids swam among languorous kelp forests: "How to Attract the Attention of Certain Inhabitants of the Sea." Between a glass-still sea and hot, windless blue sky, a ship's sails began to billow: "How to Inspire Breezes in a Dead Calm." A dark, beautiful horse rode out of the surf: "Recognizing Certain Dangerous Aspects of the Sea." Kir, she thought, recognizing him. The dark rider out of the sea . . . She fell asleep with her face against the dark horse and woke hours later, stiff and cramped beside the cold hearth, with the puzzled sea-dragon kneeling beside her, asking, "What are you doing?"
She built a fire quickly, showed him the pictures shifting under the trembling light. "Look," she said. "Lyo's book."
"Book."
"These are pictures. These are words." He looked dubiously at the faded writing on the pages, but the pictures fascinated him. Fish and sea-beasts swam through its pages. Sometimes he gave a chuckle of recognition and pointed for her to tell him a word.
"Sea-cow. Porpoise. Whale."
She turned a page that seemed nothing more, at first, than a painting of the bottom of the sea, full of giant kelp and coral colonies and clams and brightly colored snails strewn thickly across sand. Then the picture changed, as if water had rolled over it, altered it to reveal, behind the kelp, faint, luminous towers of shell and pearl. The sand turned into paths of pearl, the bright shells to gold and jewels scattered along the paths as if they might have fallen a long way from great ships wrecked and sunken and snagged by underwater cliffs on their cold journey down. Peri, her lips parted, peered closer. Was there a figure walking down a path? A woman, perhaps, clothed in pearl, her long hair drifting behind her, adorned with tiny starfish and sea anemones?
The sea-dragon made a sound. His face was very white; his open hand fell across the page as if to block it from his sight.
"Chain," he whispered. He looked at Peri, struggling to talk; the words were still trapped, in spite of everything Peri had done, behind his eyes. "Here." The woman took a slow step; the water shifted again, hid the magic kingdom. But the sea-dragon saw it, hidden within the dark kelp. "Here. It began."
# Nine
THE NEXT MORNING, Peri found a black pearl on her doorstep.
Her shriek startled Lyo out of his secret forest, brushing leaves out of his hair, his eyes so dark they looked black as the pearl. He took it from her silently, gazed at it as she babbled. It was the size of an acorn, perfectly round, with a sheen on it like dusky silk. He whistled.
"It's very beautiful."
"Lyo!"
"Well, imagine what great oyster fretted itself unknowingly, growing this in silence out of a grain of sand." He tossed it absently into the air, caught it, his eyes narrowed at the bright morning sea.
"Lyo, an oyster didn't roll across the sea and bring me this! She knows this is where the sea-dragon comes! She'll find him here, she'll chain him again—"
"No, she won't."
"But—"
"She sent you a message."
"Yes!"
"She said 'I know about you, you know about me.' If she wanted the sea-dragon back, she would take him, she wouldn't bother putting pearls on your doorstep. She permits the sea-dragon to come here. Although," he added, veering off into his own thoughts, "why for only a couple of hours in the dead of night is a mystery. Nothing about any of this makes much sense. The magic seems so confused. . . ."
"Well, what does she want?" Peri demanded, confused herself. "Lyo, what does she want? The sea-dragon recognized her last night—or someone like her. She walked through one of the paintings in your book. Someone saw her to be able to paint her like that, someone went down to the undersea, and came back up. So why can't Kir do it? Why can't you? Go down and ask her what she wants?"
"Have you ever seen a mermaid?"
"No."
"But you could draw one?"
"Yes."
"How? If you've never seen one?"
"I don't know. Everyone knows what a mermaid looks like. Now," she sighed, "they're even seeing them."
"But they knew the word before they saw the mermaid."
She nodded, perplexed. "People tell stories," she said finally.
"And words," Lyo said, "like treasures, get handed down through time. Very, very few people make a real journey to the undersea. It is a journey out of the world. But everyone who tells the tale of a sea-journey, or listens to it, travels there safely and comes back again. So don't assume the painter went down to view that world firsthand. Perhaps he painted his own sea-journey that he made through his mind when he first heard the tale."
"Yes, but," Peri said, "Lyo, the sea-dragon recognized her."
Lyo grunted. His fingers searched his hair, picked out a bit of twig. "Well," he admitted, "maybe you're right. Long ago, the painter went down and came back, bearing a sea-treasure of strange knowledge. . . . But neither you nor I are going to do that."
"Then how will you talk to Kir's mother?"
"We. We are going to do a little fishing with the fishers."
"Most of the fishers don't go out now," she said. "They say there's a storm at sea, and they'll wait it out like any other storm."
"Has anyone gotten hurt?"
"No. But—"
"Then let's go. Unless you'd rather wait and see what you find on your doorstep tomorrow morning."
"No," she whispered. "I wouldn't."
The few fishers braving the sea's tantrum had left the harbor by the time they reached the docks; no one saw the magician everyone was searching for except a half-dozen gulls sunning on the posts. Lyo charmed a few barnacles off the underside of the _Sea Urchin_ as he dipped oars into the water. This time, Peri knew, he used magic to row; they cleared the harbor and were into open sea faster than was decent for a small fishing boat. But instead of joining the vague dots on the horizon, he pursued his own course, rowing parallel to the land, heading toward the deep waters beyond the spires.
Peri, damp and numb, watched the tall rocks inch closer. She had never seen them from that angle. She had looked between them out to sea, but she had never seen them frame the land as if they were some giant broken doorway between the sea and the land. As Lyo rowed closer, the landscape between the spires changed: now empty, bright sea, now a wave tearing at a crumbling cliff, now white sand and a green wall of gorse, now the old woman's house between them, looking tiny and faded between the great, dark, sea-scoured stones, the way it might look to a sea-dragon or to someone swimming between them, carrying a black pearl like a message from the sea. . . . She blinked. Were the spires a doorway to the land or to the sea? Who was looking out? Who looked in? Which was the true country?
Then, as she blinked again, cloud fell over them, pearl white, chill, blinding. Lyo stopped rowing. They stared at one another, their hair beaded with mist. The sea, reflecting a cloudless sky moments before, had turned a satiny gray. Peri heard a light laugh, almost a sound water might have made lapping against the underside of a boat, but not quite. She slid to the floor of the boat, holding herself tightly, trembling with cold. Something shook the _Sea Urchin_ 's bow, a giant hand beneath the water playing with a toy boat. Peri tried to make herself smaller. Lyo, his face oddly milky in the strange mist, stood up and tossed the black pearl back into the sea.
A hand reached out of the water and caught it. A face looked up at them from beneath the cold, gentle water. Long hair coiled and uncoiled. Starfish clung to it, and sea-flowers, and long, long loops of many-colored pearls. The face was very pale; the heavy, almond-shaped eyes held all the darkest shades of mother-of-pearl . . . Kir's eyes.
She was very close to them, yet farther than a dream, just beneath the waves sliding softly over her face. She held the pearl underwater in her open palm, waiting, it seemed to Peri, for something to happen. Nothing did. Lyo looked transfixed by her. She watched them, swaying beneath the water, her eyes expressionless, or too strange to read. She said something finally; bubbles flowed upward. The words themselves, popping out of the water, sounded very distant. Lyo smiled. He picked periwinkles out of the mist and scattered them on the water. A few drifted down, clung to her hair. She smiled, then, a small, careful smile without much humor in it.
"What did she say?"
"She said," Lyo answered, "that I am very strong."
"That's a peculiar thing to say," Peri said morosely.
"Not really." His voice shook; she realized then that some of the mist beading his face was sweat. "We're having an argument at the moment about who is going to do what to the _Sea Urchin_ 's bow."
Peri closed her eyes. "I wish I were at work," she whispered. "I wish I were scrubbing floors, I wish I were—"
"Where's your sense of adventure?"
"I never had one. What happens if you lose?"
"I don't think I'd better."
A sudden thought cleared Peri's head. She opened her eyes, stared at the little pool of water that had splashed into the bottom of the boat. She was still shivering with cold, but her fear had gone. "Ask her," she said tightly, "if she tried to wreck this boat before. Ask her if she recognizes it."
Lyo rolled his eyes at her. A sea gull with blood-red eyes had come out of nowhere to sit on his shoulder. "You ask her," he said.
Peri leaned against the side of the boat, stared down at the woman with Kir's eyes who floated as easily as moonlight on the water. "Did you?" she whispered; in that close, strange mist it seemed even a tear falling into the water would echo. "Did you take my father out of this boat when he went out to fish? My mother thinks you did. She looks for the land beneath the sea. She thinks he's there now, that you took him there and sent his boat back to us empty."
The woman gazed back at her, her eyes secret, unblinking. She spoke again; her voice sounded like water trickling in some hidden place.
Peri looked at Lyo. "What did she say?"
"She said that no one from the world of air has come into the sea-kingdom for many years." The sea gull was nibbling at his ear; he shrugged a shoulder irritably and it flew off with a cackle.
"He went out to sea," Peri said, "and never came back."
"Many fishers do that," Lyo said gently. "They take such risks." The woman said something else, her hand closing and opening again on the pearl. Peri, listening closely, could not unravel her words from the murmur of the tide.
"If your father had cast his heart into the sea, his body might have wandered into her country," Lyo translated. "But his heart came with his boat into harbor every night. So his bones may be in this sea, but his heart remains where he kept it all his life."
Peri was silent. The woman, silent, too, studied her. The _Sea Urchin_ 's bow had not wavered from pointing at the same distant tangle of gorse; she still held it. Her face blurred slightly under a wash of foam; long, pale hair drifted. She spoke again.
"She says she has no quarrel with the fishers."
"Not for wanting her gold?"
"She says the gold falls out of the lost ships into her country like rain falls on land. It means little to her; it is the work of men, and belongs to the country above the waves."
"Then what does she want?"
The woman's other hand rose out of the water; she tossed something silver into the boat. It struck the wood at Peri's feet, and she jumped. Lyo picked it up.
"A ring," he said. His voice shook again, strained. "Letters on it—"
"U, V," Peri breathed. "It's the king's ring. I threw it into the sea."
"Of course. I should have guessed. Young floor-scrubbers are constantly throwing kings' rings into the sea."
"Kir brought it to me." Her eyes widened then; she added urgently, "Lyo, tell her about Kir—"
"She knows about Kir. She gave him to the king. What do you think this tempest is all about?"
"But, tell her—" She gripped the side of the boat and leaned far over, until it should have tilted, and said it herself. "Kir! Kir wants to come to you! Please let him in! Please—"
"Peri!" Lyo shouted. Pale hands came up out of the water, caught Peri's wrists, and pulled her down until the _Sea Urchin_ was almost on its side. Peri's own hair floated in the water; the sea washed over her face and she sputtered. The water was icy. She drew a breath to scream, and drew in another swell instead. Then the _Sea Urchin_ lurched; she tumbled down to the bottom of the boat, spitting out sea water, her eyes and nose running. The bow was free, drifting; the fog seemed to be thinning. Lyo was staring at her hands.
"What are you holding?"
She blinked salt water out of her eyes. Webs of pure moonlight attached to irregular circles and squares of twigs and dried seaweed . . . She sniffed, wiped her nose on her sleeve, still blinking. In the center of each web was a tiny crystal-white moon. She caught her breath.
"My hexes!"
"Your what?"
"My hexes. I made them to hex the sea. Lyo, look at them!"
"I am," Lyo said wonderingly.
"She turned my black thread into moonlight!"
"Wait—"
"I made them without moons and threw them into the sea with the king's ring. I thought the sea-dragon ate them!"
The fog had blown away; they were wallowing perilously close to the spires. Lyo plied the oars, fighting the tide. Gulls cried, circling the spires; a sea otter, on its back in the waves, cracking a shell against the stone on its belly, paused to give them a curious look.
"I think," Lyo said, with a neighborly nod at the sea otter, "you'd better begin at the beginning. Begin with the word hex. Who taught you to make one in that shape?"
"The old woman."
"What old woman?"
"The old woman who disappeared, whose house I've been staying in, she taught me. I made hexes to throw into the sea because it took my father—that's when I first saw the sea-dragon."
"When—"
"When I threw the hexes in. Kir was looking for the old woman, too, and he found me on the cliff drawing hexes in the sand."
Lyo pulled the oars up, leaned on them, looking at her. The _Sea Urchin_ continued along its course. "Kir knew the old woman, too?"
"He said she came out to watch the sea with him, once. He wanted to talk to her; he said she knew things. But she had already gone."
"Gone where?"
"Just gone. She went away and never came back." She sighed a little, her eyes on the tiny house tucked into the gorse.
Lyo said gently, "People left you, this past year."
She nodded. "My mother, too. She didn't leave—she—It's like you said. She went on a sea-journey, in her mind. She hasn't come back yet. Anyway, when Kir found me drawing hexes, he asked me to send the sea a message for him."
"And the message was?"
"Part of it was his father's ring."
The magician said, "Ah," very softly. His arms were still propped on the oars; his eyes, for some reason, were gray as gulls' wings. "And now the sea has returned the king's ring and your hexes."
"But she changed my hexes. They were ugly before, twisted and dark—that's how I meant them to be. Now they're full of magic."
Lyo touched one lightly, curiously. "So they are," he murmured, and shook birds off the upended oars.
"But why? Why did she give them back?"
"Why is the wind, why is the sea . . . ? She gave them back for a very good reason."
"What reason?"
"I haven't the slightest idea," the magician said, and dipped the oars back into the water. Then his eyes fixed on something beyond Peri's shoulder, narrowed, and changed again. "Look."
She turned and saw the king's ship on the horizon, all its white and gold sails unfurled to the wind, wending its way back to the village.
# Ten
KIR CAME WITH THE NIGHT TIDE. Peri, sleeping restlessly, listening beneath her dreams for a wash of pearls across her threshold, woke slowly to the full, hollow boom of the breakers. The moon, three-quarters full, hung in her window, distorted, veined with crystal; it reminded her of the tiny, mysterious moons the sea-woman had woven into the hexes. It fashioned magical shapes out of the night: pearly driftwood and tangled seaweed, a prince mounted on a dark horse at the edge of the tide.
Peri, tide-drawn, pulled a quilt over her shoulders and opened her door. The moon eyed her curiously. She walked across the cool, silvery sand, the ebb and flow of tide singing in her head. As she neared the dark prince, his face turned away from the sea to her. He said nothing, simply held out his hand for her to mount. A wave foamed around her feet, coaxing; Kir pulled her up out of the water. He put his arms around her, his cheek against her hair. They sat silently, watching the water arch and break against the stones.
"I missed you," Kir said finally. He sounded surprised. "I thought of you in the North Isles."
"I thought of you," Peri whispered. She stopped to swallow. "Kir—"
"I saw the sea-dragon last night—was it last night? In a storm. It followed us for a long time."
"Kir, I have something to tell you."
"Then tell me."
"I saw your mother."
Kir said nothing; the words, she realized, must have made no sense to him at first. Then she felt the shock of them through his entire body. "What did you say?"
"Kir, strange things have been happening to the fishers, they see mermaids, hear singing, they get lost in sudden fogs—"
"Peri," Kir said tightly.
"That's what happened to us—"
"Who? What are you—"
"Lyo. The magician. He rowed us out there, beyond the spires, to look for your mother, to speak to her for you. A cloud came down over us on a cloudless morning. And your mother stopped our boat." A wave glimmered around them, pulled at the dark horse, melted away. "She had your eyes. And your father's ring."
He gripped her almost painfully. "It reached her—"
"She got your message. And she got mine. She gave me back my hexes." She could hear his breathing now, shaken, unsteady, and she twisted anxiously to face him, breaking his hold. "Lyo says she is angry at your father. She threw his ring into the boat. Lyo says she still loves your father."
"Lyo? The magician who turned the gold chain into flowers?"
"Periwinkles."
"Peri—" He stopped suddenly, his tongue stumbling on her name. "Periwinkles . . . I thought that was an inept thing to do. Until now. Did my mother—did she—"
"She sent a kind of message to you. I tried to ask her about you, and she nearly pulled me overboard, giving me the hexes. I don't understand it."
"I don't understand," he said raggedly, "why it was you who saw her and not me. I have been waiting so long."
"I know." She pulled his arms around her again, feeling a chill that only he could give or take away. "It should have been you. But Lyo said we had to go out—"
"Why?"
"Because she left a black pearl on my doorstep."
His voice rose. "Why your doorstep? Why you?"
She swallowed, holding his hands tightly in hers. "Kir, there's something else. But you have to wait."
"I have waited," he said, his voice dangerously thin.
"I mean, just a few minutes. Then you'll see why she left the message at my door. Please." She pushed closer to him, feeling the cold again. "Please."
"Is she coming here?"
"No. I can't make her come and go. Just . . . wait. A few minutes. Tell me what you did while you were away."
He was silent; she sensed, like a gathering tidal wave, his anger, frustration, bewilderment. The sea roared around them, tugging at Peri's trailing quilt. The dark horse stirred restively, protesting. The reins flicked up; Kir guided it out of the water.
"I met a lady in the North Isles." His voice sounded haunted, weary. "She was the daughter of a lord there. She was very pretty. Her hair was not a tangled mess, nor did she walk barefoot in the sand by moonlight." Peri eased against him, her eyes wide; his hand touched her hair, smoothed it away from his mouth.
"Nor," she whispered, "did she scrub floors."
"No. She was sweet tempered and intelligent. We talked together, rode together. Sometimes we danced. My father was pleased. At night, after everyone had gone to sleep, I went down the cliffs over which her father's house was built, and I stood on the rocks and let the tide break over me as if I were another rock. I waited for it to pull me in. But it never did."
"Kir . . ."
"Nor did she pull me into her world. I wished she could have. . . . And then we came home. From the day we left the North Isles until this moment I have not spoken to my father. I can't. If I did, I would tell him he must let me go. But I have no world to go to, no place. So I cannot leave him."
"Did she love you? The lady from the North Isles?"
"I don't know. Perhaps, if I had been different, I might be there now, still dancing, watching her face in the moonlight. . . ." He touched Peri's cheek, turned her face; she felt the brush of his cold mouth. She put her arms around him, held him tightly, her eyes closed against the sea, as if by not seeing it, she could protect him. His hands slid through her hair, pulling her closer; she tasted the bitter salt on his lips. Then he pulled back, murmuring; she opened her eyes reluctantly. His face was turned as always toward the sea.
"What is that?" he breathed.
A shape, huge, dark, bulky, was rising out of the waves.
"It's the sea-dragon." Her voice shook. She felt her heartbeat, and a sudden chill that came from within her. The looming dark pushed closer to them through the surf; Kir rode farther up onto the sand.
"What is it doing?"
"It's coming out."
He was silent. The sea-dragon's eyes reflected moonlight, like two great, pale beacons. Its streamers tumbled in the tide, ribbons of light. Peri heard Kir swallow. "Why?" he asked abruptly. "Why does it come out?"
She shook her head slightly, too nervous to answer. The sea-dragon pulled relentlessly through the tide, up the gentle slope of wet sand, until it had coaxed all its fins and streamers out of the grasping tide. It was so close to them that its eyes seemed level with the moon. Kir's horse whuffed nervously at it; Kir held it still.
Then the extra moons vanished from the sky. While their eyes still searched bewilderedly for them, a young man rose from his knees on the sand, asked curiously, "What are you doing?"
Peri gathered breath. "He comes out," she said unsteadily, "to learn words."
Kir was still as a stone behind her. Then he moved, and she felt the cold at her back, all around her. Kir dismounted; the sea-dragon watched him calmly. The moonlight picked up strands of his gold hair. As Kir grew closer, the sea-dragon's expression changed; his brows twitched together. "What are you doing?" he whispered. He shivered suddenly, feeling the cold in his human form. "You are coming closer to me." Then his face smoothed again, with a look of wonder such as Peri had never seen on it before. He pulled a word out of nowhere like a mage: "Kir."
Kir stopped. Peri saw him trembling. Their faces, in profile against the bright waves, mirrored one another. Kir's hands moved; he unclasped the cloak at his throat, settled it over the sea-dragon's shoulders.
"What are you doing?" the sea-dragon asked again, pleading, Peri realized, for the sound of Kir's voice, a human voice answering his in the silence.
Kir spoke finally, his voice shaking. "I am looking at my brother."
Peri closed her eyes. She felt hands tugging at her, and she slid off the horse, hid her face against Kir, weak with relief. "You're not angry."
"How long—how long—"
"Since the night you left. It—he—came out of the sea then. The chain was gone. I never had a chance to tell you."
"No." She felt him still trembling. "I should have guessed. The chain—"
"Chain," the sea-dragon echoed. He hovered uncertainly at the tide's edge, watching them.
"What is his name?" Kir asked.
"I don't think anyone gave him a name. He can only stay out of the sea for a couple of hours in the night, then he must go back to the sea-dragon's shape."
"Does my father know?"
"Lyo is going to tell him."
He looked at her. "Lyo," he said flatly. "Lyo. Who exactly is this magician who likes periwinkles, and who isn't afraid to tell my father something like this?"
"I don't know," she said nervously. "Come to the house. I'll make a fire."
Sitting at her hearth, the two princes, one fair, one dark, looked startlingly alike. They were both of the same build, the same height. The sea-dragon, with Kir's dark cloak clasped with a link of pearls at his throat, studied Kir out of eyes a lighter blue than his own. That and their expressions differed. The sea-dragon, who had endured years of rolling winter storms, and who had been unthreatened by them, thwarted by nothing but a chain, seemed much calmer. Kir's face changed like the changing face of fire.
Peri opened Lyo's book, showed Kir the shifting, misty sea-gardens, the woman walking slowly away from them down the glittering path until the currents swirled through the kelp and the painting changed. The sea-dragon made a sound; Kir's eyes went to him.
"You know this place."
"When—when I was small, the chain was small," the sea-dragon said carefully. "The chain grew bigger. But it always began here."
Kir looked at the page, his eyes hidden again, but Peri saw the hunger in his face.
"It's like moonlight," he whispered, as the picture changed again. "You can see it, but you cannot hold it; it makes a path across the sea, but you cannot walk on it. I could look all my life and die before I found this place, and he is trying to escape from it." The sea-dragon was listening to him intently, trying to comprehend. Kir's eyes strayed to the writing beside the picture; the sea-dragon slid his hand over the words.
"What do you see?" he asked.
"A world I want." His face eased a little at the sea-dragon's expression. "You don't understand."
"I understand your words," the sea-dragon said. "I don't understand—" He made a little, helpless gesture. "Your eyes. You watch the sea. Even with Peri, you watch the sea."
Kir was silent, perhaps seeing himself on the shore through the sea-dragon's eyes. "Yes," he said softly. "I watch. I want to go there." He tapped the sea-world with his finger. The sea-dragon looked pained.
"You must not. You—" He shook his head, bewildered. Then things seemed to swirl together in his head into a picture; his eyes widened as he saw it. "Once there was a king who had two sons, one by the queen, his wife, and one by a woman out of the sea. . . . You," he said to Kir abruptly. "You." He touched Kir's face gently, near his eyes. Then he touched the woman with the starfish in her hair, whose heavy, blue-black eyes he had looked into beneath the sea. "You are the son out of the sea."
"Yes," Kir whispered. "Yes."
"I am not."
"No. You are not."
The sea-dragon looked bewildered again. "Then why am I in the sea?"
Kir's eyes rose, met Peri's. "Things happened," he said finally. "I don't understand all of them, either. I only know that you belong here on land, I belong in the country beneath the sea, with the woman who walks down those paths of pearl."
The sea-dragon was silent. His eyes shifted away to the fire; he gazed into it until Peri tugged at his arm, made him turn. He looked troubled, a new expression, one more movement into his human body.
"Kir," he said, his eyes on his brother's face. He paused, struggling for words; then he reached out, grasped Kir's shoulder. "I can see you. I can talk with you. To you. I come—I have come out of the sea to you. Stay. Here with Peri. In this world where I can see you."
"I can't stay," Kir said. His face looked white, stiff; Peri, watching anxiously, saw in amazement that he was close to tears. He moved after a moment, gripped the sea-dragon's wrist. "You can see me," he said huskily. "Peri can see me. No one else in the world can see what I really am. But I cannot stay with you here. I will die if I do not find my way into the sea."
"Die."
"Not live. Not see."
The sea-dragon loosed him reluctantly. "How?" he said, asking so many questions at once, it seemed, that Kir smiled.
"I don't know how," he said. "Perhaps the magician will find me a path. He seems adept at finding things." The sea shifted under his fingers then; he looked down at Lyo's book as if he had felt the changing. "There are ways," he said slowly, "written in here."
"Lyo said not to—"
"Lyo does not need to get into the sea."
"No," Peri said patiently, "but he said the spells are dangerous."
"Do you think I care?" he asked her as patiently, and she felt her hands grow cold.
"You are not a mage."
"I can read," he said inarguably, and did so, while the sea-dragon watched wonderingly and Peri got up and rattled pots and spoons, trying to distract Kir. She gave up finally, came to lean over his shoulder to see what he was reading.
"To find the path to the Undersea, find first the path of your desire," the spell book said mysteriously, next to a picture of a young woman standing in the surf, looking out to sea. Her hair was long, windblown; her feet were bare; a tear slid down her despairing face. Peri stared. Would she look like that when Kir finally left her? Her eyes went back to the script; her lips moved; she tried to memorize the spell in case she needed it.
"Call or be called," the spell said. Then: "Many paths go seaward. The path of the tide, the seal's path, the path of moonlight. The spiraled path of the nautilus shell may be imitated. Call or be called, be answered or answer. For those so called, this will be clear to their eyes. For others: You of a certain knowledge, a certain power, who wish for disinterested purposes to descend to the Undersea and return, it is imperative that a gift be taken. The gift must be of the value—or seem of the value—of the traveler's life. It may become necessary to make the exchange in order to return to time."
"I don't want to return," Kir murmured, frustrated.
"Wait," Peri said, fascinated. "'Possessing the gift, the traveler must then find the path of the full moon at full tide, at the point where the path of the moon meets land.' It's not a full moon."
"It's almost full."
"'There the traveler must reveal the gift to the sea and request, in fair and courteous voice, entry. Entry may be given by a dark horse appearing out of the foam, which the traveler will mount, a white seal, which the traveler will follow, by the sea queen herself, who will lead the traveler by moonlight to the country beneath the waves. The gift must be given at the time most appropriate for safe return. The journey is hazardous, not recommended unless all other courses are exhausted.'"
"A gift," Kir said heavily.
"You gave her a gift: your father's ring."
"It wasn't worth my life. And she gave it back."
"You tried to give her your life once, too," Peri said. Her eyes filled with sudden tears at the memory, at his hopelessness. He stared into the fire, his face sea-pale, bitter.
"Perhaps," he said, "she does not want me."
"I think she does. Lyo thinks—"
"How would you know?" he demanded. "How would either of you know?"
The sea-dragon, startled at his raised voice, said softly, "What are you doing?"
Kir's mouth clamped shut. Peri turned away, ruffling at her hair, wondering suddenly if she understood anything at all of Kir's mother, of the strange world she dwelled in. The hexes on the spellbinding shelf caught her eye. She grabbed them desperately, scattered them across the spell Kir was reading.
"Look. Your mother gave me these when I said your name. They must mean something. They must!"
Kir stared at the webs of pearl and crystal and moonlight strung on odd bits of bent kelp. He held one up; fire beaded on it like dew. The round crystal in the center glowed like the sea-dragon's eye. He breathed, "What are they?"
"My hexes."
"They're beautiful. How did you—"
"She did it. I made them with black thread, your mother put the magic into them—" Her voice faltered. "Oh, Kir, look!" All around them on the walls and ceiling, the reflection of the hex Kir held trembled like a great, shining web of fire.
The sea-dragon made a sound, entranced. Kir turned the hex slowly; the web revolved around them. His lips moved soundlessly. He lifted his other hand, traced a thread; the shadow of his fingers followed the fiery pattern on the wall.
"But what?" he whispered. "But how?"
There was a knock at the door. They stared at it, preoccupied, uncomprehending, as if it were a knock from another world. The door opened. The king walked into the firelight, into the web.
# Eleven
HE STOPPED SHORT at the sight of the silent faces turned toward him, spangled with fire from the hexes. He had come plainly dressed; his long, dark, wool cloak hid darker clothes, but nothing could disguise his height, the familiar uplift of his head. He had given Kir his dark hair and his winging brows, even his expression; his gray eyes, unlike Kir's, were fully human. They moved from Kir to Peri, and then were caught by the sea-dragon. Fire and shadow shifted over the gold hair, the light blue eyes; the king closed his eyes, looking suddenly haunted.
Lyo stepped in behind him. He gazed in rapt abstraction at the tangle of fire on the walls. Then he saw the open spell book, and his eyes went to Peri, wide, questioning. Kir dropped the hex he held then, and the web vanished.
He got to his feet; so did the sea-dragon. Peri, huddled beside the hearth, wanted to rifle through the book for a vanishing spell. Kir and his father seemed at a loss for words.
The king said finally, "The mage told me you would be here. That this is where you come."
"Sometimes I come here," Kir said. He stopped to swallow drily. "Sometimes I just watch the sea."
The king nodded, silent again. His eyes moved in wonder and disbelief to the sea-dragon. Kir's hands clenched; Peri saw the sudden pain in his face.
"He is your son," Kir said abruptly. "Your true son. Take him and give me back to the sea."
The king was wordless, motionless for another moment. Then he reached Kir in two steps, his big hands closing on Kir's shoulders.
"You are my son." His hold rocked Kir slightly, then loosened a little. "You are so much like your mother," he continued huskily. "So much. I tried not to see it all these years. I didn't understand how it could be so. You have her eyes. I kept finding her face in my mind when I looked at you. Yet how it could be so . . . ?" His gaze shifted beyond Kir again to the sea-dragon. "And this one, this son, wearing the face of the young queen, the woman I married, and was only beginning to know when she died."
The sea-dragon moved to Kir's side, uneasy, Peri guessed, in the sudden, bewildering tension. The king's eyes moved, incredulous, from face to face, one dark, one fair, both reflections of a confused past.
"What are you doing?" the sea-dragon asked tentatively, startling the king.
Lyo said gently, "He doesn't know many words yet. Peri has been teaching him at night when he takes his human shape."
"Why only at night?" the king demanded. "Why does she still keep him in that shape? Is there a price I pay to take him from the sea? Is that it?"
Lyo, crossing the room to the spell book, knelt next to Peri. "I don't know," he said simply. "I think you should ask her."
The king's shoulders sagged wearily; he seemed suddenly dazed, helpless. He looked at Peri again where she crouched beside the fire, trying to hide behind Lyo, and she felt all her untidinesses loom at once: her wild hair, her callused hands, the patched quilt she had wrapped around her faded nightgown.
"You are my son's friend."
Peri's face flooded with color. She could stand up, then, as if the king's word gave her and her frayed quilt a sudden dignity. The sea-dragon curiously echoed the word, "Friend."
The king picked Peri's cloak off a chair and sat down tiredly. "The mage brought me a ring," he said. "My own ring. He told me who had thrown it into the sea. And who returned it from the sea." He studied Kir as he sat, as if, once again, he saw long pale hair braided with pearls floating on the tide. His voice gentled. "I thought you had fallen in love with some fisher's girl."
"I did," Kir said tautly.
"I thought that was what troubled you. I hoped it was only that. The mage said, if I wanted to do something wise for once, I should ask you what you want. He said—"
"How did you know?" Kir asked Lyo. His voice was very tense; the sea-dragon stirred, disquieted. Lyo looked up from the hexes scattered on the spell book. He spoke calmly, but it seemed to Peri that he picked his words very carefully, as if he were devising a spell to avert a storm.
"Odd things draw my attention. Happiness, sorrow, they weave through the world like strangely colored threads that can be found in unexpected places. Even when they are hidden away, most secret, they leave signs, messages, because if something is not said in words, it will be said another way. In the city, I heard some fishers from a tiny coastal village wanting a great mage to remove a chain of gold from a sea-monster. Even before I saw the chain, I knew that the gold was the least important detail. What was important was the link someone had forged between water and air, between a mysterious place deep beneath the waves, and the place where humans dwell. And when I saw the sea-dragon, when I dipped behind its great eyes into its mind, I knew . . ."
"What did you know?" the king asked softly.
"Why it was drawn to the fishing boats, to every human voice. Why it rose above the waves to watch the land. And then I began to suspect why the king and his son came here so early this year, and why the prince was seen so often at odd hours of the day or night, riding that dark horse to the sea. . . . I didn't know then how much the king or the prince or the sea-dragon understood. I still don't know why it was finally permitted to be seen above the water. But I freed it. I turned the gold chain into flowers partly to disturb the sea, to send a message back to it. And partly because while gold will not float, periwinkles will. And then I tried to teach the sea-dragon a few things. It—he—found his own shape—I don't yet understand why or how. And he found Peri. Does he have a name?" he asked the king, who shook his head. The king's face was very pale.
"I named my son after my wife died. My changeling child. Kir. I don't know if I ever saw my wife's true child. I named the child I saw Kir, and I remember thinking how dark his eyes were—a twilight dark—and thinking they would change into his mother's summer eyes. But they never changed."
"And, in the sea, before she gave him to you, Kir's mother must have named him something. So Kir is twice-named—"
"Why would she have named me," Kir breathed, "to give me away? She must have hated us both—to chain him like that, to give me away—"
"She gave you to me," his father said sharply. "She knew I would love you. I loved her."
Kir was silent, his hands opening, closing.
The king rose slowly, stood in front of him. "Is it so terrible?" he asked painfully, "with me on land?"
"It is terrible," Kir said numbly. He lifted his face, so that the king could see his sea eyes. "I can't help it. I can't rest in this world. In the restless tide I can rest. I can't love in this world. Not even Peri."
"You have loved me," she said, her voice shaking.
"No."
"Yes. You have cared about me. You have thought of me."
He gazed at her, mute again; his face changed with a flicker of light. He reached out, touched his father lightly, pleadingly. "Please. You must let me go."
"How can you—" The king stopped, began again. "How can you be so sure that when you are in the sea, you will not long just as passionately for this land?" Firelight caught the glitter of unshed tears in his eyes. He swallowed, then added, the words coming with difficulty, "If you didn't want this so badly, I would never let you go."
"Please. Will you—will you talk to my mother?"
The king's eyes slipped away from him toward a memory. The harsh lines on his face eased, grew gentle, as if he might have been watching the soft blue sea on a summer's day. "Once," he whispered, "I could understand her strange underwater language."
The net of fire sprang around them once more. Lyo, toying with the hexes, had created such a tangle that they were cross-hatched with flaming threads. The sea-dragon murmured, "You are making the world into fire."
"It's not water, is it," Lyo said curiously. "Nothing that can exist in water . . . Strange, strange . . ."
"What are they?" the king asked. "Another message?"
"Yes. They're Peri's hexes. Kir's mother returned them like this, changed into moons and moon-paths, fire-paths."
"Why?"
"For us to use."
"How?"
Lyo shook his head, entranced, it seemed, by the weave of light. "I don't know," he whispered. "I don't know." Kir stood close to his father, watching. He seemed, Peri realized, finally becalmed; already he looked more like his mother, as if he were relinquishing his human experience. He found her looking at him wistfully; he gave her a sea-smile. She swallowed a briny taste of sadness in her throat. Already he was leaving her.
The sea-dragon stirred restlessly: The tide called him, luring him out of his shape. "Peri," he said, and she nodded. "I must go."
"What's to be done with him?" the king demanded of Lyo. The lines on his face deepened again. "Both my sons live in half-worlds. I will not lose them both to the sea."
The sea-dragon went to Kir, his fingers groping awkwardly at the clasp at his throat. Kir stopped him.
"Keep it," he said gently. "It's cold outside. I'll come with you to the tide's edge."
The sea-dragon shook his head. "No. Stay." They were all silent, hearing the tide as he listened to it. He smiled his untroubled smile, as if the rolling waves, the fish, and crying gulls were things he also loved, along with all the words he had learned, and Peri's human touch. Peri opened the door for him, put her arm around him in farewell. He started to take a step, then turned to look uncertainly at the king, as if struck by something—a web reflected around him—that he finally saw but barely understood.
"I want—" He struggled with the thought. "I must see you again."
The king's face eased with relief. "Oh, yes," he said. "Yes." Peri left the top half of the door open, leaned out to watch the vague, moonlit figure cross the sand. Unexpectedly, Kir came to her. He slid his arms around her, leaned his face against her hair, watching over her shoulder. The king stood behind them both. The sea-dragon reached the tide's edge. He dropped Kir's cloak and walked naked into the sea, a pale, moonlit figure that gathered bulk and darkness as it changed.
A twig snapped in the utter silence; they all started. The king said explosively to Lyo, "Do something."
Lyo nodded, looking determined but a little blank. "Yes."
"You need a full moon," Peri said, remembering, and Lyo looked at her reproachfully. Kir's arms dropped; he turned restively.
"It wouldn't work for me."
"It should work," Lyo said. "Spells are in spell books because they work. Which is why—" He closed the book, sent it back, Peri supposed, to whatever bush he kept it under. Kir's eyes clung to him.
"A gift—it says I need—"
"Ah," Lyo said, shaking his head. "That's for mages. You have your heart's desire; that should be your path. You are the gift."
"But she didn't—she won't—"
"I know. I don't understand." He slid his fingers through his hair, left it standing in peaks. "The hexes. She gave the hexes to us so that we could use them. They are vital, they are necessary."
"How—"
"I don't know," he sighed. "Yet. We can only try."
"When?" the king asked.
"Five nights from now. When the moon is full. Meet me near the spires."
Kir nodded wordlessly. The king dropped a hand on his shoulder. "Come home for now," he said wearily, "while I still have a few days left of you. Your heart may be eating itself up to get into the sea, but I had you for seventeen years and when you leave me, you'll take what I treasured most. If the sea needs a gift, I'll give it."
Kir's head bowed. He went to Peri wordlessly, kissed her cheek. Then he lifted her face in his hand, looked into her eyes. It won't be easy, his eyes said. It will not be easy to leave you.
"But I must," he said, and left her.
* * *
"The magician is back," Peri said absently, as she filled her bucket at the pump the next morning.
"Thank goodness," Mare breathed. "The fishers will be able to work again." Carey leaped a little with excitement, slopping water.
"Will he get us the gold?"
"I don't know about the gold," Peri said. "But I think he can stop the odd things happening in the sea."
"But what about the gold?"
"He didn't say about the gold."
"But why didn't—" She stopped, her eyes narrowing on Peri's face. "Why did he come to you? Where did you see him?"
Peri heaved her bucket aside to make room for Mare. "He rowed out with me in the _Sea Urchin_ yesterday. I think yesterday." It seemed suddenly a long time ago. She added to Mare, "You can tell Enin that he's back."
"I will." Mare's eyes were narrowed, too, contemplating Peri as if she were beginning to see the misty, magical fog Peri moved in, where sea-dragons turned into princes at her feet, and kings knocked at her door. "Why do I have the oddest idea that you know far more than you're saying about gold and mages and sea-dragons?"
Peri looked back at her mutely, clinging to her heavy bucket with work-reddened hands. Her shoes and the hem of her dress were already wet. Mare shook her head slightly, blinking.
"No," she said. "Never mind. Silly thought." She pumped water into her bucket. Peri gazed at the bright morning sea. She swallowed a lump of sorrow, thinking of Kir, and of life without Kir, without the sea-dragon. An endless succession of scrub-buckets . . . For the first time, she understood Carey. A path of gold glittered away from the inn, leading to . . . what? It was the goldless floor-scrubber the two princes came to; no gold in the world could have bought her that: the magical kiss of the sea.
"Wake up," Mare said. Peri sighed and hefted her bucket.
# Twelve
FIVE NIGHTS LATER, Peri sat at her window watching for the moon, waiting for Kir. Her face slid down onto her folded arms, she fell asleep and woke suddenly, hours later, drenched with light. A full moon hung above the spires; the breakers, slow and full, churned in its light to a milky silver before they broke.
She saw the rider beside the sea then, and her throat burned. Maybe, a tiny voice in her mind said, whatever Lyo does won't work, maybe he'll be forced to stay. . . . But even staying, he would always be someone found at the tide's edge, among the empty shells, looking seaward for his heart. Another horseman joined him: the king. They both looked seaward, down the dazzling path of light between the spires.
She opened her door, found Lyo on her doorstep. He, too, was watching the moonlight; his open hands were full of hexes.
He looked at her absently as she came out. "What do you think?" he asked. "One of them? All of them?"
"One of what?"
"The hexes."
"Are you going to hex the sea again?" she asked, confused, and he smiled.
"I hope not."
Her eyes went again to Kir; she sighed soundlessly, watching him, as he watched the sea. . . . Lyo was watching her. He gave her shoulder a quick, gentle pat.
"Come," he said, and she followed him across the sand. The beach between the house and the sea, between her and Kir, seemed to have stretched; the sand, strewn with driftwood and kelp, made her steps clumsy. She felt, as she reached the bubbling, fanning tide, that she had traveled a long way to the dark rider, whose face was still turned away from her. Then he turned, was looking down at her; he slid off his horse and came to her.
He held her wordlessly; she blinked hot, unshed tears out of her eyes. He loosed her, held her hands, put something into them.
"What is it?" Her voice sounded ragged, heavy, as if she had been crying for a long time.
"It's the black pearl," he said softly, "that I will never dare bring you when I am in the sea." He kissed her cheek, her mouth; he gathered her hair into his hands. She lifted her face to meet his dark, moonstruck eyes.
"Be happy now," she whispered, aware of all the shining waves behind him reaching toward him, withdrawing, beckoning again. She added, feeling the pain again in her throat, "When I'm old—older than the old woman who taught me to make the hexes—come for me then."
"I will."
"Promise me. That you will bring me black pearls and sing me into the sea when I am old."
"I promise."
She lifted her hands to touch his shoulders, his face. But already his thoughts were turning from her, receding with the tide. Her hands dropped, empty but for the black pearl. He kissed her softly, left her to the empty air.
She stepped out of the tide's reach, and bumped into Lyo. He steadied her. The king rode his horse past the tide line, up to dry sand, and dismounted.
"I don't know if she'll come," he said to Lyo.
"How did you call her before?"
"I didn't . . . at least, not knowingly. We called each other, I think. I would walk along the tide line wanting her, and soon I would see her drifting behind the breakers, with her long, pale hair flowing behind her in the moonlight." His eyes went to his son yearning at the tide's edge. "If she can't hear me now, it seems that she should hear him. That his longing would reach out to her."
"Yes," Lyo said gently. One of the hexes in his hands caught light; white fire blazed between his fingers. It pulled at the king's eyes.
"What will you do with those?"
"I'm not sure yet. . . . I'll think of something."
"You are young to be so adept."
"I pay attention to things," Lyo said. "That's all." His attention strayed to the sea; they all watched it. Something must happen, Peri thought, entranced by the glittering, weaving, breaking path of the moonlight across the water to . . . what? Something must happen.
Lyo gazed down at the moonlit weave in his hands. "Not fire," he whispered. "Here it is light. Moons and moonlight." He lifted a hex suddenly, threw it. Moonlight illumined it as it fell between the spires; an enormous, brilliant wheel of light cast its reflection across the water. Then the hex fell to the water, but did not sink. It floated, still shedding its reflection across the dancing waves. The angle of light changed. Peri's lips parted. Someone had caught it. The reflection no longer slid with the moving sea; it flung itself between the spires, a great web clinging from stone to stone just above the water, hiding the moonlit path across the sea from the watchers on the shore.
Lyo grunted in surprise. The king said tautly, "Are you doing that?"
"No."
Kir had moved toward the web; tide swirled around his knees. He seemed to have left them already. If the sea would not accept him, Peri thought, he would still be changed; even on land, the tides would roar, beckoning, louder than any human voice in his head. She hugged herself, chilled, marveling. Something moved through the fiery web between the spires, drifted beyond the breakers. . . .
The king made a soft sound. Waves rolled toward them, curled into long silver coils and broke, shuddering against the sand. Water frothed around Kir, twisting his cloak; he pulled it off, tossed it like a shadow into the tide and moved deeper into the sea. A pale, wet head appeared and disappeared in the surf. A glint of pearl, of bright fish scale . . . Lyo tossed another hex. This one hit the sand, made a shivering maze of light that the tide could not wash away. The figure in the surf moved toward it. Her shoulders appeared, and her long, heavy, tide-tossed hair. Her robe, carried for her by the currents, dragged down as she walked on land. The tide loosed her slowly.
The king moved to meet her. He stopped at the edge of the wide, burning web. A wave rolled over it; she stepped through the water, unerringly to the hex's bright center. Kir, still in the surf, had turned toward her. Lyo tossed him a hex; it grew under his next step as he turned back to wade out of the water to his mother. But instead of aiding him, the hex seemed to trap him, bind him, helpless and bewildered, in the heart of the maze. Lyo murmured something; Peri, one cold hand at her mouth, shook him with the other.
"Lyo!"
He muttered something else, exasperated, then quieted. "Shh," he said, both to himself and Peri. "Wait. The sea is working and unworking its own spellbindings."
The sea-woman's wet hair flowed to her feet; her shoulders were bowed under the weight of pearl. Her heavy-lidded, night-blue eyes seemed expressionless as she studied the king. Then she said something, and Peri heard Lyo's breath fall in relief.
"What did she say?"
"She said 'You've changed.'"
"It happens," the king said, "to humans."
She spoke again. Peri looked at Lyo, opening her mouth; he stooped suddenly and picked a shell out of the sand.
"Here."
"What should I—"
He tapped his ear patiently. "Listen."
She held it to her ear and heard the voice of the sea.
"Then," Kir's mother said, "I have been angry for a long time." Her voice was distant, dreamlike, passing from chamber to chamber within the shell.
"Yes."
"I did not realize how long it was until I felt my son's desire to come back to the sea. Is it long, by human time?"
"Yes," the king said softly. "Many years."
"Then many years ago, for many nights, I waited for you in the tide, and you did not come and you did not tell me why."
"You were like a dream to me. I had to turn away from you, return to my own world. I should have told you that."
"Yes."
"I should have told you that turning away from you was like turning away from wind and light. But I had to leave you. Can you forgive me?"
She lifted her hands slightly, opened them, as if letting something unseen fall. "I took your land-born child because I wanted you to have my child, our child. To love him as you could no longer love me. So you would look at him and remember me always."
"I did," he whispered.
"But I took your other son. I was angry, I made my anger into a chain, and changed your bright-haired son into something you would never see, never recognize. Can you forgive me for that?"
"How can I not, when I helped you forge that chain? All the twists and turns in it, your fault, my fault . . ."
"I kept him so long in that shape I nearly forgot what he was. Only the chain remembered my anger. Then one day the chain stretched beyond my magic and broke the surface of the sea. I could not hide your dragon-son any longer. He swam among the fishers, until their eyes turned to gold. And then even the gold vanished, my chain disappeared. . . ."
"So you let the sea-dragon take his shape on land?"
"No. I did nothing. The magic was out of my hands; it had become confused, unraveled. I had begun to hear my own son calling me, calling me, and I looked for him, but I could not reach him. All I could do was to disturb the fishers with small sea-spells, hoping they would go to you for help, and that you would find me." She sighed a little, a soft, distant breaking wave. "And you have finally come."
"To give you back our son. And to take mine out of the sea, bring him into the world where he belongs."
"I hope I have not kept him too long, that it is not too late for him in your world."
"I don't think so. But," he added, his voice low, weaving in and out of the sound of the breakers, "I have loved your restless sea-child, and taking him, you take another piece out of my heart. If there's a price to pay for his passage into the sea, that's all I have to give."
"There is no price." Her voice shook. "His desire is his path. But you must free him."
"I send him freely back. . . ." He paused, his eyes on her moonlit face, the pearls glowing here and there with a muted, silky light. "I was so young then, only a few years older than Kir, when I first saw you."
"I remember."
"It seems strange that, looking at your changeless face, I am not still that young man, walking beside the sea on a summer night, when all the stars seemed to have fallen into the water and you rose up out of the tide shaking stars out of your hair."
She smiled her delicate, careful smile; this time it had more warmth. "I remember. Your heart sang to the sea. I heard it, deep in my coral tower, and followed the singing. Humans say the sea sings to them and traps them, but sometimes it is the human song that traps the sea. Who knows where the land ends and the sea begins?"
"The land begins where time begins," the king said. "And it is time for Kir to leave me. Is it too late for him in your world?"
She turned her head, looked at Kir for the first time. Kir swayed a little, as if she or the undertow had pulled him off-balance. But still he could not step out of the web. She turned back to the king, her smile gone.
"I can hardly see his human shape, he is so much of the Undersea. His body is a shadow, his bones are fluid as water."
"Is it too late?"
"No. But he must leave your time now. No wonder he sang at me like the tide." Her shoulders were dragging wearily at the constant pull of the earth; even her hair seemed too heavy for her. "I must go now."
"Take him."
"I will. But you must free him." She lingered; waves covered the web under her feet, withdrew. It seemed to Peri that the king moved, or the sea-woman moved, or maybe the tide swirling about them made them only seem to move toward one another. For a moment, their faces looked peaceful. Then the woman said something too soft to carry past the web. She turned, stepped back into the sea, and melted into the foam.
Kir gave a cry of sorrow and despair that stopped Peri's heart. He turned, struggling against the web to follow the tide. But still he seemed trapped; he could only stand half in air, half in water, buffeted by waves that drenched him from head to foot but did not change him.
"Do something," Peri whispered. Tears slid down her face. "Lyo—"
"Do something," the king said, his voice sharp with anguish. "She said we must free him. Free him."
Lyo stared at the hexes in his hands. "They're so unpredictable," he murmured, baffled. "Peri, when you made them, did you say something over them? Or when you threw them into the sea?"
"I don't know, I don't know," she said distractedly. "I shouted at the sea—"
"What did you shout?"
"I don't know. Something—I was angry." Then she stopped. The world quieted around her, so hard was she thinking, suddenly. A lazy spring tide idled behind the spires . . . a malicious sea to be hexed. And like Kir's mother, she had woven her anger into a shape. . . . She felt the cold then, a chill of night, a chill of wonder. "I did it," she breathed. "Oh, Lyo, I did it."
"What did you do?" he and the king said together.
"I hexed the sea!" She drew wind into her lungs then, and shouted so hard it seemed there must be windows and doors slapping open all over the village, people putting their sleepy faces out. "I unhex you, Sea, I uncurse you! I take back everything I threw into you out of hate!" She stopped, wiping tears off her face, then remembered the rest of her spellbinding. "May your spellbindings bind again, and your magic be unconfused. Open the door again between the land and the sea, and take this one last thing that I love, that belongs to land and sea, to us and you!"
The last of the hexes whirled across the water. They struck the great web hanging between the spires, the doorposts of the sea. Strands sagged, tore, revealing stars, half a moon, ragged pieces of moon-path. A wave hit Kir, knocked him off his feet. It curled around him, drew back. When they finally saw him again, he had surfaced and was sputtering in the deep waters beyond the surf.
He did not look back. He dove deeply, heading toward the spires; when he surfaced again, it was with a seal's movement, sleek, balanced, graceful. He dove, stayed underwater far too long, so long that those watching him had stopped breathing, too. Another strand of the web loosened, fell like an old rotting net unknotting as it dried. The white fierce light of it was fading as the web burned itself out, thread by thread, in the sea.
They saw Kir at last, dangerously near the spires. He should have been flung against the rocks, battered in the merciless swells. But he slid from wave to wave, an otter or a fish, nothing human. He watched the web above his head, strung between him and the wide, dark sea. When another thread dropped toward the water, he reached up, caught it. He dove then, dragging the white, gleaming strand down with him. The great hex unraveled wildly between the stones, then fell, burning, into the burning path of the moon.
They watched for a long time, but they did not see Kir again. Of all the crystal lights only the moon remained, still weaving its own web between the spires.
The king turned finally. They had all tried to follow Kir into the sea, it seemed; they were standing in the surf. Peri found Lyo's arm around her, holding her closely. She was numb with cold, too numb for sorrow, and felt that she would never be warm again. They stepped out of the tide. The king took Peri's face between his hands, kissed her forehead.
"Thank you." He looked at Lyo. "Thank you both." There was no great happiness in his voice, just a blank weariness that Peri understood. Kir was gone, Kir was . . . Then a movement in the surf startled her.
It was the sea-dragon, coming out. "He's walking," Peri whispered. "He's walking out of the sea."
He pulled a human body out of the swells, as patiently as he had dragged the sea-dragon's great body out. Once he stopped to catch something in his hand: a bit of froth, an edge of moonlight. He reached them finally, shivering, his gold brows knit.
"Kir is gone," he said. The king took off his damp cloak, pulled it around his wet son.
"Yes."
"I watched him. Now, I am gone."
"No," Peri said, as the king looked at him puzzledly, "you have left the sea. You are here."
"I am here." He looked at his father, his expression hesitant, complex. "Your eyes want to see Kir."
"Kir wished to leave. He needed to leave."
"You are the king who had two sons."
"Yes."
The sea-dragon's shoulders moved slightly, as if feeling, one last time, the weight of the chain. "The sea did not want me. If you do not want me, maybe Peri will."
Peri nodded; Lyo shook his head. The king smiled a little, touched the sea-dragon's face. "You look so like your mother. Her gentle eyes and her smile . . . That will help, when I explain where Kir has gone, and why you are suddenly in his place."
"And why I have no name in the world," the sea-dragon said simply. He stood silently, then, looking at the sea, the cold, uncomplicated world he would never see again.
"Will you miss the sea?" the king asked abruptly. "Will you stand at the tide's edge, like Kir, wanting to change your shape, to return to it?"
The sea-dragon met his eyes again. Something fully human surfaced in his face: a strength, a hint of pain, a loneliness no one would ever share. "I have left the sea," he said. He held out his hand, showed them the hex he had rescued from the tide. The strange light had burned down to ragged black threads. But a tiny crystal moon still hung in the center, glowing faintly with an inner light.
Lyo took it from him, touched the moon; it kindled a moment, luminous, fire-white. He lifted his eyes from it to gaze at Peri.
"Do you realize what you did?" he asked. "You managed to unbind, confuse, and otherwise snarl up the most powerful magic in the sea."
Her face burned. "I'm sorry. I never thought it would work."
"You're sorry? When you threw the hexes into the water and confused the sea's magic, you caused the chain to stretch beyond its bounds, break the surface between land and sea, so that the sea-dragon could finally take a look at the world."
"But I trapped Kir on land, he couldn't get into the sea."
"Peri," Lyo said patiently. "You're not listening."
"I am, too," she said.
"You're not paying attention."
"Lyo, what are you—" She stopped suddenly, blinking at him. "I'm not paying attention," she whispered.
"You're swarming with magic like a beehive, Periwinkle."
"I must be. . . . I'd better watch what I hex."
"At the very least." His eyes narrowed slightly, glittering in the moonlight, fascinating her. "Now tell me this. The night the sea-dragon dragged itself out of the sea for the first time, with you watching, did you happen to say anything to make it do that?"
"No," she said, surprised.
"Think, Peri."
"Well, I was just watching the sky and the waves, thinking of Kir and wishing . . ."
"Wishing what?"
"Wishing that he could be . . ." Her voice faltered; she stared at the magician, not seeing him but the dark, star-flecked sea. "I said it. I said, 'I wish you were just a little more human.' But I meant Kir, not the sea-dragon!"
"So," the king murmured. "The sea-dragon, passing by at the moment, came out of the sea, a little bit human every night." He was smiling, a smile like his sea-son's, never quite free. "You have strange and wonderful gifts, Peri. You helped both my sons with your magic. Even more with your friendship." He sighed. "I wish you could have been powerful enough to keep Kir out of the sea, but in the world and under the sea, there is probably not enough magic for that. At least you brought this one out." He put a hand on the sea-dragon's shoulder; the sea-dragon started.
"You are touching me," he said wistfully. The king's face changed; he drew the sea-dragon into his arms.
"Yes," he said gruffly. "I am holding you. Humans touch. If they are foolish enough or wise enough. Come home with me now before you change your mind and follow the tide." He looked at Lyo. "I'll need your help with him. Can you stay?"
Lyo nodded, his mouth pulling upward into his private, slanting smile. "Oh, yes. I have some unfinished business involving periwinkles."
"Periwinkles," the sea-dragon echoed curiously.
"Small blue flowers," the magician said, and for the first time they heard both the king and the sea-dragon laugh.
# Thirteen
THE NEXT FEW DAYS, to Peri, seemed as colorless and dreary as the water she dumped out of her bucket at the day's end. The sky was a brilliant blue; the gorse, in full bloom, covered the cliffs with clouds of gold. The fishers went out every day; there were no more tales of singing sirens or ghostly ships. Peri, for the first time in weeks, could get a full night's sleep.
But she still woke late at night, listening for the sea-dragon; she still looked for Kir on the tide line; she still searched the sea between the spires, without thinking, watching for something unexpected, a message from the Undersea. She felt numb inside. All the magic was gone, nothing would ever happen to her again. Only the black pearl in her pocket told her that mystery had come into her life and gone, leaving her stranded at the tide's edge, yearning.
Mystery had stranded the villagers, too; they still longed, like Peri, for its return.
"I thought you said the mage was back," Enin said to Peri one afternoon, when she was putting her cleaning things away.
"He is," she said shortly.
"Then where is he?"
She shrugged, morose. "With the king, I guess."
Mare glanced at her oddly. "What's he doing there? We hired him."
"Helping with his son."
"What's the matter with Kir?"
"Nothing." She swallowed. "Nothing now. It's not Kir," she added, since everyone would know soon enough, anyway. "Kir went into the sea."
"He drowned?" Carey and Enin said incredulously.
"No." She took her apron off, bundled it up, hardly listening to what she was saying. "Kir went back to the sea. His mother is a sea-woman. The king's son by his true wife was the sea-dragon. That's why it was chained—the sea-woman was angry with the king. But she also loved him, which is why she gave him Kir. He came to my house at night in his human shape to learn words. The sea-dragon did. He's with the king now." They were staring at her, not moving, not speaking. She pulled her hair away from her eyes tiredly. "So that's where Lyo is, probably." She took the black pearl out of her pocket. "Kir gave me this before he left."
"Kir?" Carey's voice squeaked. The rest of her was immobile. Peri was silent, gazing at the pearl, remembering the full moon, Kir's hands in her hair, his promise to sing her into the sea. She lifted her head; faces blurred a moment, under tears she forced away.
"He used to come and talk to me. . . ." She slid the pearl back into her pocket and pulled her cloak off a hook.
Carey whispered, "What's he like? The new prince?"
"He has gold hair and blue eyes. Like his mother had. He can't talk very well yet, but he learns fast." She put her cloak over her arm and went to the door.
Mare said fiercely, "Girl, you take one more step, I will throw a bucket at you. You come upstairs with us and tell the story properly from one end to the other. You can't just go and leave us here with a jumble like that: sea-women, secret sons, princes wandering into your house at night giving you black pearls. . . ."
"I don't understand," Carey said plaintively, staring at Peri, "why it all happened to her. Look at her!"
They did, until she fidgeted. "I washed my hair yesterday," she said defensively. Mare groaned. Enin grinned.
* * *
She drifted to her mother's house the next afternoon. The days were growing longer; the air was full of delicate, elusive scents. Evening lay in dusky, silken colors over the sea. The sea-kingdom seemed very near the surface, just beneath the lingering shades of sunset. Peri found her mother leaning over the gate, watching the distant sea. Behind her, the garden was sprouting tidy rows of green shoots; there was a peculiar absence of weeds.
Peri's mother smiled as Peri came up the street. She opened the gate; they both leaned over it then, watching. Peri's eyes slid to her mother's hands. There was black dirt on her fingers, even a streak on her face.
"You've been gardening!"
"I thought I'd pull a few thistles. It seemed a nice day for it." Her voice sounded less weary than usual; the lines on her face had eased. Had the sea, Peri wondered suddenly, set her free, too?
They watched the fishing boats come into the harbor. When the last of them had slipped past the harbor-mouth, Peri's mother sighed, not in sadness, it seemed to Peri, but in relief that everyone was safely home. She said, "I miss you, Peri. The house seems empty suddenly. Do you think you might like to come back?"
Peri looked at her. The old woman's house felt that way, these days, too quiet, as empty as her heart. "Come back?"
"I never even asked where you've been living."
"Out at the old woman's house, near the stones. After she disappeared, I stayed there."
Her mother nodded. "I guessed, when I thought about it at all, that you might be there. I wonder where she got to, the old woman."
"Maybe," Peri said softly, "maybe into the sea. Maybe someone . . . someone special left a pearl on her doorstep and sang to her until she followed the singing."
"There is no land beneath the sea. You told me that."
"Well," Peri sighed, "I don't know everything, do I?"
"Do you think you'd like to come back?"
Peri turned to glance at the house. The door was open; a last thread of light pooled across the threshold. It might be nice, she thought, to have someone to talk to, now that her mother was talking again.
"Maybe," she said. "For a little while."
"You need some new clothes, child."
"I know. I forget things like that."
"You're growing again."
"I know." She picked at a splinter in the gate, her eyes straying to the sea. The last light faded; a thin band of blue stretched across the horizon, the shadow of night. She sighed. What did it matter where she lived? "All right," she said. "I'm tired of my cooking, anyway." She swallowed a sudden burning; her face ducked behind her hair. "What does it matter?" she whispered. She felt her mother's arm across her shoulders. The sea began to darken, the night-shadow widened, a deep, deep blue, the darkest shades of mother-of-pearl. . . .
They heard a whistling through the dusk. Peri jumped, for it had shifted abruptly from the street to her elbow.
"Lyo!"
"Goodness," her mother said, startled. Lyo gave her a deep bow, standing in her weed pile.
"This is Lyo," Peri explained. "He is the magician who turned the gold chain into flowers."
"What gold chain?" her mother said bemusedly. "What flowers?"
"Where did you get those clothes?" Peri asked. Lyo had put aside his scuffed and gorse-speckled leather and wool for a more familiar mage's robe of wheat and gold. It made him look taller somehow; even his hair had settled down.
"The king gave it to me. He said I was beginning to smell a bit briny."
"Oh. It looks very—very—"
He nodded imperturbably. "Thank you. It'll do for now. It'd be hard to row a boat in, though."
"Are you?"
"Am I what?"
"Going to sea to get the gold? The fishers keep asking me that."
"Oh," he said, chuckling.
"Well, are you?"
"Not exactly."
She looked at him, baffled. His eyes shifted colors mysteriously: the green of the seedlings, the brown of the earth; they pulled at her attention until she blinked herself free. "How is the sea-dragon?" she asked, since he wouldn't tell her about the gold.
"Aidon," Lyo said. "The king named him that."
"Is he learning to talk any better?"
"He's doing very well. I'm teaching him to read. Yesterday we added and subtracted periwinkles. That's what I—"
"A talking sea-dragon named Aidon," Peri's mother interrupted. "What are you talking about? A sea-dragon reading books?"
Lyo's brows rose. "You didn't tell her?"
"No."
"Tell me what? What sea-dragon? What gold chain?" She watched her daughter and the strange-eyed magician look at one another uncertainly. "Peri, what have you been doing while I haven't been paying attention?"
"Oh." She took a long breath. "It's a little hard to explain."
"Then you'd both better come in and have some supper and explain it to me," her mother said, sounding so much like her old self that Peri felt a sudden bubble of laughter inside her.
Lyo sat at the hearth, beginning in a calm and methodical fashion to explain while her mother chopped up carrots and onions for soup. Peri kept interrupting him; he gave up finally and let her tell the story for a while. Peri's mother sat down slowly in the middle of it, a paring knife in one hand and an onion in the other. The color came back into her face as she listened. She laughed and cried at different parts of the tale, and then, as Peri told her about the king and the sea-woman meeting each other under the moon, a stillness settled into her face, like the calm over water after a storm. She had finished her sea-journey, Peri realized; she had gone and come back to the familiar world, the one where she sang old sea chanteys and knew the names of all the shells on the beach.
She was silent for a long time when Lyo and Peri finished the story. Peri knew what she was seeing: the long, brilliant, fleeting path of sunlight between the spires. She saw the onion in her hand and got up finally. "Well," she said softly. "Well."
"That's partly why I came here," Lyo said. "The sea-dragon misses Peri."
"I can guess why. She's the first girl he ever saw."
"Yes." Lyo stopped a moment, his expression awry. "Yes. So the king wondered if Peri might consider coming to the summer house to teach Aidon again."
"You mean after work?" Peri asked, dazed.
"Peri, you can forget the brushes, the buckets. The king will pay you well for teaching. And the sea—Aidon will be happy to see you again. He likes being human, but he misses you. He had to give up his brother; he shouldn't have to lose you, too. Would you like to do that?"
"Teach the sea-dragon in the king's house?" She nodded vigorously, thinking of the prince's blue-eyed smile, his need of her. "Oh, yes. But doesn't the king want you to stay? I don't know very much beyond adding and subtracting."
"Oh, I'll stay awhile. Teach you a little magic," he added nonchalantly. "If you like. Just so you won't get into trouble. . . ." He paused again, staring so hard at a wooden nail in the floorboards that she thought it might rise out of the floor. Then he shook himself, ruffled his hair with both hands, and met her eyes. "Are you?" he inquired.
"What?"
"Planning to fall in love with any more princes?"
She thought about it, gazing back at him. Then she sighed deeply, her hand sliding into her pocket to touch the black pearl that held all her memories. "I don't think so. One prince is enough in one lifetime."
"Good," Lyo said with relief. He pulled beer out of the air then, and yellow daffodils, and a loaf of hot bread that looked as if it had come straight out of the innkeeper's kitchen.
"Lyo!" Her mother, face in the flowers, was laughing.
"It's all right, he'll get his payment tomorrow." He poured a basket of early strawberries into Peri's lap. "There will be a sea harvest of periwinkles coming in on the morning tide that this village will never forget."
Peri, her mouth falling open, saw periwinkles turning to gold all down the beach as the sea swept them tidily out of itself. "That will make Carey happy."
"Perhaps," Lyo said. "Perhaps not even that will make Carey happy. It's an odd thing, happiness. Some people take happiness from gold. Or black pearls. And some of us, far more fortunate, take their happiness from periwinkles." He leaned over Peri, impelled by some mysterious impulse, kissed her gently. "I've been wanting to do that for some time," he told her. "But you always had one king's son or another at hand."
Like him, she was flushed under her untidy hair. "Well," she said, "now I don't."
"Now you don't." He watched her, smiling but uncertain. Then, still uncertain, he sat down beside her mother to help her clean shrimp. Peri's eyes strayed to the window. But the magician's lean, nut-brown face, constantly hovering between magic and laughter, came between her and the darkening sea. After a while, watching him instead, she began to smile.
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 2,335
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Windows 10 For Phones May Launch On A High-End Smartphone
In case you are amid follower of the news that is being published about Microsoft in the last couple of months, you have started wondering about their plans for the high end segment. Rumours suggest that the company is planning to debut a high end phone with Windows 10.
The company at present is awaiting the official announcement of the Windows 10 before they can unveil its high end windows based phones. This information has been revealed by Ravi Kunwar, who is the Director - North, Nokia India Sales.
He had participated in a chat along with the company during the launch of Lumia 430. He added that their Company has been focusing on their budget segment, as they still have to work on the premium budget as well. Once the budget has been done, they will start working on the unveiling of the devices later in this year alongside Windows 10 launch.
From the perspective of the company, this move is appropriate, as it will allow them to continue and improvise on the computing power of all the features of the Windows 10 specially the Continuum aspect. This is the main reason why the highly rumored the high end Lumia 940 might debut later this year and somewhat the same time as that of Windows 10. What is the most important thing to remember is that the Windows 10 Continuum feature is similar to that of 'Continuity' feature of Apple and other similar features by Google as well.
What remains a surprise is that a company like Microsoft has been shying away from the 4G connectivity from a long time. The company has never tried to impose the 4G element in any of their devices apart from the Lumia 638. Compared to this the competitors of the company like Xiaomi, Samsung, and Motorola apart from others have already started incorporating 4G connectivity as critical part of many of their devices.
Kunwar has also confirmed that Microsoft is already 4G ready at the back end. This indicates that whether they are using the 4G or not, the current phones from the company are already capable of 4G connectivity since Microsoft has the capability of adding the 4G functionality by means of a software update. He further added that 3G is currently the most widespread method of internet connectivity in India. This is the main reason why even Microsoft is focusing on this area.
In the past, Microsoft has already unveiled many more devices like Lumia 540, Lumia 530, Lumia 430, 435, 532, 640XL and 640 among others. All these devices have been marketed in India despite the fact that they are operating only using the 3G connectivity.
They have never tried to market or sell any of their devices based on 4G element. From what it looks like all or atleast most of their devices are easily capable of working on 4G connectivity and the company is already working on enabling these features in India owing to increasing 4G connectivity.
Labels: smartphone, technology, technology news, Windows 10
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 624
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Getting letters and postcards in the mail is a joy! When I was a kid, I even loved getting junk mail! Years ago, a friend told me that only handwritten could suffice for thank you notes, never email. I try to follow her advice as often as I can.
How about you? Do you write and/or receive handwritten mail?
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{
"redpajama_set_name": "RedPajamaC4"
}
| 7,496
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Mellow – album di Sonny Stitt del 1975
Mellow – album di Donovan del 1997
Mellow – album di Maria Mena del 2004
Mellow Records – etichetta discografica italiana fondata nel 1991
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,476
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Q: PHP server sends json respond to swift client shows this error
PHP server sends json respond to swift client shows this error, knowing that the send is within a local network.
<?php
require("Conn.php");
require("MySQLDao.php");
$Id = htmlentities($_POST["Id"]);
$password = htmlentities($_POST["password"]);
$returnValue = array();
if(empty($Id) || empty($password))
{
$returnValue["status"] = "error";
$returnValue["message"] = "Missing required field";
echo json_encode($returnValue);
return;
}
//$secure_password = md5($password);
$dao = new MySQLDao();
$dao->openConnection();
$userDetails = $dao->getUserDetailsWithPassword($Id,$password);
header('Content-Type: application/json');
if(!empty($userDetails))
{
$returnValue["status"] = "Success";
$returnValue["message"] = "User is registered";
echo json_encode($userDetails);
} else {
$returnValue["status"] = "error";
$returnValue["message"] = "User is not found";
echo json_encode($returnValue);
}
$dao->closeConnection();
?>
Swift error msg:"UserInfo={NSDebugDescription=JSON text did not start with array or object and option to allow fragments not set.}"
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,543
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Albertville is een plaats (city) in de Amerikaanse staat Minnesota, en valt bestuurlijk gezien onder Wright County.
Demografie
Bij de volkstelling in 2000 werd het aantal inwoners vastgesteld op 3621.
In 2006 is het aantal inwoners door het United States Census Bureau geschat op 6001.
Geografie
Volgens het United States Census Bureau beslaat de plaats een oppervlakte van
12,1 km², waarvan 11,3 km² land en 0,8 km² water.
Externe link
Plaats in Minnesota
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,059
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HomeCorporate-galleryVIT-AP School of Law signs MoU with University of Birmingham University
VIT-AP School of Law signs MoU with University of Birmingham University
FPJ BureauUpdated: Friday, December 02, 2022, 01:16 AM IST
Signing ceremony of Memorandum of Understanding (MoU) between the VIT-AP School of Law (VSL), VIT-AP University and University of Birmingham, United Kingdom held on 1st December 2022.
Dr. S. V. Kota Reddy, Vice-Chancellor of VIT-AP University and Prof. Paul McConnel, Head of Global Engagement, University of Birmingham, United Kingdom signed the MoU and addressed the gathering. The Vice-Chancellor in his address highlighted the importance of MoU with University of Birmingham and informed the students that the agreement and collaboration will enhance visibility and opportunities to the students of VIT-AP School of Law and other departments in the university. Further he added that the University has committed to make VSL as a global law school and encourage students to be global legal professionals. He also highlighted that the university is committed to establish Centres of Excellence in various subjects of law, which includes International Law, Human Rights, Business and Commercial Laws, Intellectual Property Laws, New Emerging Technologies and Law, and other contemporary areas of law to encourage the clinical, research and practical application of law to the future legal professionals.
He further added that the centres of excellence would work holistically to provide suitable solutions to the society and to provide legal empowerment to the people. The academic collaboration with University of Birmingham would help students to equip with the opportunity to study one semester or one year at the University of Birmingham and enhance their skills with the help of expert professors and faculty members of the Birmingham University.
Further, VIT-AP students will get an opportunity to apply and study their master degree courses at University of Birmingham with concession of fee. This MoU will also facilitate research collaborations between faculty members of VIT-AP University and University of Birmingham. It will gives an opportunity to have knowledge partnership, organise guest lectures, workshops, symposiums, seminars, and International conferences in the current fields of law.
Prof. Paul McConnell after signing MoU said that VSL and University of Birmingham recognize the value of mutual cooperation and are desirous of entering into this Memorandum of Understanding (MoU) to facilitate academic collaboration between the two organizations to promote academic research, capacity building, academic collaborative activities and to develop the professional courses and leadership development programs and other academic and research collaborative activities in the filed of law. He also emphasised that the University of Birmingham is opened to have more collaboration with other disciplines of the VIT-AP University in future.
Dr. Jagadish Chandra Mudiganti Registrar, VIT-AP University and Dr. Benarji Chakka, Dean VIT-AP School of Law, students, faculty members and staff of the university witnessed the MoU signing ceremony.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,035
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\section{Introduction}\label{sect:Introduction}
In an increasingly connected world, our opinions on a phenomenon of interest or event are often not only influenced by our direct independent observations, but also by other people's public opinions on related events. In an online social network like Twitter or Facebook, users' opinions and postings are often influenced by the opinions of those they are connected to or are ``following'' in the social network \cite{AraWal:12,MitKilGot:13,Bakshy2011,LuoTayLen:J13}. For example, in viral marketing using social networks, marketing companies often target a few influential nodes in the network to help them push a product \cite{Leskovec2007,Zhang2010}. Similarly, widespread online access has made it easier for us to follow the opinions of experts like celebrities and industry insiders, who may have access to private information that we are unaware of. For example, we may be interested to determine the financial health of a publicly listed company. In addition to our own observations about the company through its annual financial reports and stock prices, we may also choose to incorporate the ``expert'' opinion provided by financial blogs like \cite{ZeroHedge} and \cite{DealBook}. In all these examples, inference about a phenomenon of interest is not only based on direct observations but also the opinions of other entities. This is known as \emph{social learning} \cite{AceDahLobOzd:11,KanTam:13}.
A further example is the Internet of Things (IoT) framework \cite{Kortuem2010,Ding2013}. Sensors each make their own private observations but collaborate by exchanging public information. This can be viewed as a ``physical social network'' of devices, cooperating to improve their situational awareness. Sensors originally designed for a specific purpose may collaborate with other sensors to perform inference on a phenomenon they were not specifically designed for. However, in order to ensure energy efficiency, each sensor needs to intelligently choose which other sensors to collaborate with since not all sensors may provide information relevant to it. For example, a sensor trying to estimate the temperature in a particular room of a building may choose to incorporate information from other temperature sensors or sensors tracking the number of occupants in the building, instead of information from a vibration monitoring sensor.
In this paper, we investigate the problem of multihypothesis social learning in which an agent can select an expert opinion from a group of experts, to incorporate into its final decision in order to minimize its expected loss.
Specifically, we consider an agent $0$ who wishes to choose from a set of $M$ hypotheses based on its own observations as well as the opinion of an expert chosen from a set of $K$ possible experts. Each expert has access to a set of private observations, which they use to form their own opinions about the true hypothesis, subject to their own local loss functions or biases, which may differ from that of agent $0$. By taking into account the experts' individual biases, our goal is to find an asymptotically optimal expert choice in order to minimize agent $0$'s expected loss.
\subsection{Related Work}
The problem of selecting the best expert opinion to follow is related to the problem of \emph{decentralized detection} or decentralized hypothesis testing, which has been extensively studied in \cite{Tsi:93a,CheVar:02,ChaVee:03,LinCheVar:05,WilSwaBlu:00,CheCheVar:12,TayTsiWin:J08b,TayTsiWin:J08a,TayTsiWin:J09a,ZhaPezMor:12,Tay:J12} and the references therein. In the decentralized detection problem, each agent has its own private observations, but cooperate with each other so that the whole network of agents reaches a decision regarding a common underlying phenomenon of interest. Here, which agent passes information to which other agent is determined by a known network topology like the parallel configuration \cite{Tsi:93}, tandem network \cite{PapAth:92,TayTsiWin:J08c,DraOzdTsi:13}, and tree architectures \cite{TayTsiWin:J08a,TayTsiWin:J08b,TayTsiWin:J09a,ZhaPezMor:12}. Therefore, all these works do not consider the problem of selecting which other agent's opinion to follow. Furthermore, in a decentralized detection problem, all agents are assumed to have the same hypotheses and loss functions for declaring the wrong hypothesis, or have the common goal of minimizing the loss at a last agent known as the fusion center.
We study a related but somewhat different problem from the decentralized detection problem. We consider the scenario where experts may make decisions according to their own biases, instead of minimizing the loss of a particular agent or fusion center. In \cite{AceDahLobOzd:11}, the authors consider binary hypothesis testing in a social network, in which agents sequentially observe the opinions of a stochastically generated neighborhood of agents in the network. Each agent has the same 0-1 loss function,\footnote{The 0-1 loss function assigns a loss of 0 if the agent declares the same hypothesis as the true underlying hypothesis, and a loss of 1 otherwise.} and conditions are derived for the Bayesian error probability of the $n$th agent to approach to zero as $n$ becomes large. The network model we consider in this paper is equivalent to a two-layer hierarchical tree network, which is much simpler compared to that studied in \cite{AceDahLobOzd:11}. This is because our goal is to analyze how an expert's opinion impacts that of a particular agent. In addition, we consider a multihypothesis testing problem in which each agent has different loss functions or even different number of hypotheses.
Since finding optimal decision rules for general decentralized hypothesis testing problems is NP-complete \cite{TsiAth:85}, it is difficult to find analytical characterizations for the best expert choice in networks of moderate size, and where agents have private observations that are not conditionally independent given the underlying hypothesis. Therefore, most of the literature in decentralized detection has focused on the case where observations are conditionally independent \cite{Tsi:93a,CheVar:02,ChaVee:03,LinCheVar:05,WilSwaBlu:00,CheCheVar:12,TayTsiWin:J08b,TayTsiWin:J08a,TayTsiWin:J09a,ZhaPezMor:12}. The same challenge applies to the social learning problem in this paper, and we therefore consider only the case where agents' private observations are conditionally independent. Although the problem becomes numerically tractable under this assumption, it remains hard to characterize the agents' optimal policies and decision rules analytically (see \cite{TayTsiWin:J09a} for a discussion). To overcome this difficulty, we consider the regime where each agent has access to an asymptotically large number of private observations (e.g., over a sufficiently long period of time) but each agent's observation sources and loss functions may differ. This allows us to adopt a large deviations perspective to the expert choice problem, and derive analytical characterizations of the optimal choice and agent strategies.
The reference \cite{NitAtiVee:13} considers a multihypothesis testing problem in a parallel configuration with a single agent, which is allowed to take actions that affect the observations it makes. The distribution of the observation at each time step depends on the action of the agent in the previous time step, and has bounded Kullback-Liebler divergences under any pair of hypotheses. The agent's aim is to minimize the error exponent corresponding to the 0-1 loss function. Our work can be viewed as a generalization of a result in \cite{NitAtiVee:13} (which also considers the sequential detection problem that we do not study here) to the case where one of the agent observations is the opinion of an expert, which itself has asymptotically many private observations leading to unbounded Kullback-Liebler divergences for its opinion. Furthermore, we consider general loss functions and the corresponding loss exponent, which is a generalization of the error exponent in \cite{NitAtiVee:13}.
The characterization of the error exponents corresponding to 0-1 loss functions in multihypothesis testing has also been studied in \cite{Tun:05,GriHar:10}, which derive the achievable error exponents region.
\subsection{Our Contributions}\label{subsection:contributions}
Our goal is to optimize the loss exponent of agent $0$, the absolute value of which is the rate of decay of the expected loss incurred by agent $0$ as the number of private observations grows large. We suppose that agent $0$ has a decision space consisting of $M$ hypotheses, and has a choice of $K$ expert opinions to follow.
\begin{enumerate}[(i)]
\item We consider a general $M$-ary multihypothesis social learning framework in which an agent $0$ and every expert have loss functions that depend on the number of private observations the agent or expert has access to. Each expert makes a decision that takes values from a decision space, the size of which may not be the same as $M$. We characterize the loss exponent of each expert, and provide an asymptotically optimal policy for an expert to achieve its optimal loss exponent (Theorem \ref{theorem:expert}).
\item We derive the optimal loss exponent for agent $0$ after it has incorporated the opinion of a particular expert, and provide an asymptotically optimal method for agent $0$ to choose the best expert to follow (Theorem \ref{theorem:expert_choice}). We also show that the worst loss exponent, up to asymptotic equivalence,\footnote{See Section \ref{subsection:asymptotic} for the definition of asymptotic equivalence.} for agent $0$ is achieved when agent $0$ adopts the 0-1 loss function (cf.\ Remark \ref{rem:worstloss} of Section \ref{subsect:choose}).
\item We introduce the concept of hypothesis-loss neutrality in Section \ref{subsection:special}, and show that if agent $0$ adopts a hypothesis-loss neutral policy, then the opinion of any expert who has a decision space with number of states strictly less than $M$, is ignored by agent $0$ (Proposition \ref{prop:expert_necessary}). In this case, additional information is useless, which is somewhat unexpected.
\item We show that if all experts have the same decision space as agent $0$, then it is not necessarily optimal for agent $0$ to choose the expert with the same loss function as itself. This is surprising as conventional wisdom seems to suggest otherwise. If agent $0$ adopts the 0-1 loss function, we derive sufficient conditions for when it is optimal to choose an expert who also utilizes the 0-1 loss function (Proposition \ref{prop:bestexpertloss}).
\end{enumerate}
In this work, we do not address the case where agent $0$'s opinion may be utilized by one of the experts, which result in much more complex opinion dynamics than that considered in this paper. This leads to the interesting question of whether there exists an equilibrium in the choice of loss functions in a network of agents who can incorporate opinions from each other. This is however out of the scope of the current work, and will be addressed in our future research.
The rest of this paper is organized as follows. In Section \ref{sect:Formulation}, we describe our system model, problem formulation and assumptions. We characterize the agents' loss exponents in Section \ref{sect:Optimal}, and provide asymptotically optimal policies to achieve the optimal loss exponents. In Section \ref{sect:OptimalChoice}, we discuss the optimal expert choice for agent $0$, and derive insights into this choice by making simplifying assumptions. We conclude in Section \ref{sect:Conclusion}. Appendix \ref{appendix:Mathematical} contains a brief review of some basic definitions of large deviations theory, and we defer all proofs to Appendix \ref{appendix:proofs}. Appendix \ref{appendix:Ak_properties} contains a characterization of the asymptotic decision regions, which are introduced in Section \ref{subsect:lossexp_experts}.
\section{Problem Formulation}\label{sect:Formulation}
In this section, we define our system model, assumptions and some notations. We consider an underlying measurable space $(\Omega,\mathcal{F})$ on which all random variables in this paper are defined. We adopt the following notations throughout this paper. Let $\mathbb{R}$ be the space of real numbers, and let $\ip{t}{z}$ be the inner product of the vectors $t$ and $z$ in $\mathbb{R}^{M-1}$. For $z \in \mathbb{R}^{M-1}$, let $z^0 = (0,z) \in \mathbb{R}^M$ be the vector augmented with a zero as the first element. The range of integers $a,a+1,\ldots,b$ is denoted as $[a,b]$. The notations $x[1:n]$ and $(x_i)_{i=1}^n$ are used to represent the sequences $x=(x[1],x[2],\ldots,x[n])$ and $(x_1, x_2, \ldots, x_n)$, respectively. We also make use of $x[i]$ to denote the $i$th element in the vector or sequence $x$.
\subsection{Learning From an Expert}\label{subsect:Learning_Network}
Suppose that an agent $0$ wishes to determine the underlying hypothesis $H$ associated with a phenomenon of interest, which from agent $0$'s frame of reference can be modeled by a set of probability measures $\{\P_m : m = 0,1,\ldots,M-1\}$ on the space $(\Omega,\mathcal{F})$. Let $\mathbb{E}_m$ be the mathematical expectation under $\P_m$, and let $H = m \in [0,M-1]$ if all agents' private observations have distributions derived from the probability measure $\P_m$. We suppose that $H=m$ has prior probability $\pi_m \in (0,1)$. The agent $0$ can choose to incorporate the opinion of an expert, chosen from a set of expert agents $\{1,2,\ldots,K\}$ (see Figure \ref{fig:two_layer}). If there is no need to distinguish between agent $0$ and the experts, we use the generic term ``agent'' to refer to either of them.
Each agent $k \in \{0,\ldots,K\}$ has access to a set of private observations $Y_k[1:n_k] = (Y_k[1],\ldots,\allowbreak Y_k[n_k])$. Conditioned on $H=m$, we assume that each $Y_k[l]$, for $l=1,\ldots,n_k$, is drawn independently from a conditional distribution belonging to the set $\{\P_m^\gamma: \gamma \in \Gamma_k\}$, where $\Gamma_k$ is a finite index set representing the information sources of agent $k$. The distribution set $\{\P_m^\gamma: \gamma \in \Gamma_k\}$ varies from agent to agent, and models the differences in quality of information that each agent may have access to.
We assume that for each $\gamma\in\Gamma_k$, the probability measures $\P_m^\gamma$ for $m=0,\ldots,M-1$, are absolutely continuous with respect to (w.r.t.) each other, and are known to all the agents.
We further assume that conditioned on $H$, all the random variables $(Y_k[1:n_k])_{k=0}^K$ are independent.\footnote{Although two experts may have the same information source, we assume their private observations from this source are independent due to their own noisy interpretations of the same piece of information.}
\begin{figure}[!tb]
\begin{center}
\includegraphics[scale=1]{Mary_network}
\caption{Learning network in which agent $0$ incorporates the decision of an agent $k \in\{1,2,\ldots,K\}$.
}\label{fig:two_layer}
\end{center}
\end{figure}
To motivate our setup, consider the example alluded to in Section \ref{sect:Introduction}, where an agent $0$ is interested to determine the financial health of a large publicly listed conglomerate. In addition to its own observations $Y_0[1:n_0]$ about the company through annual financial reports and other publicly available indicators like stock prices, the agent $0$ may also choose to incorporate the expert opinion of an investment analyst. Because of limited financial resources (since most analyst reports are not free), agent $0$ can only choose to subscribe to one analyst, and has to make an optimal choice of which analyst to use. In most of this paper, for simplicity, we restrict ourselves to the case where the opinion of a single expert is considered in the decision making of agent $0$, and show how to generalize our results to a finite set of experts later in Remark \ref{remark:multiexperts} of Section \ref{sect:OptimalChoice}. Note also that the number of information sources for each agent is also typically finite as in this example, which justifies our assumption that $|\Gamma_k| < \infty$ for all $k\geq 0$.
Let $x_k = (x_k[\gamma])_{\gamma\in\Gamma_k}$ be a vector of non-negative weights summing to one, where $\floor{x_k[\gamma]n_k}$ is the number of private observations of agent $k$ that have conditional distribution $\P_m^\gamma$ when the hypothesis $H=m$. We let the remaining $n_k - \sum_{\gamma\in\Gamma_k}\floor{x_k[\gamma]n_k}$ observations be drawn from an arbitrary distribution from the set $\{\P_m^\gamma: \gamma \in \Gamma_k\}$. Since we are concerned about asymptotics in this paper, the choice of the arbitrary distribution is immaterial to our analysis. We say that $x_k$ is a \emph{policy} for agent $k$. An agent chooses its policy in order to minimize its expected loss, as described below.
Let $\mathbb{S}(\Gamma_k)$ be the simplex consisting of all agent policies.
Based on its observations, each expert $k \geq1$ makes a decision $D_k = \gamma_k(Y_k[1:n_k]) \in [0,d_k-1]$ by minimizing a local loss criterion. We assume that every expert $k \geq 1$ chooses its policy $x_k$ and decision rule $\gamma_k$ so that
\begin{align}\label{Ck}
\EE{C_k(H, D_k, n_k)}
\end{align}
is minimized, where $C_k(m,d,n_k)$ is a non-negative finite loss incurred if the decision of expert $k$ is $d$ when the true hypothesis is $H=m$.
We can think of $C_k(\cdot,\cdot,\cdot)$ as encoding the ``bias'' of expert $k$. We call this the \emph{loss function} of expert $k$,
and
\begin{align}\label{exp_k}
\limsup_{n_k \to \infty} \ofrac{n_k} \log \EE{C_k(H, D_k, n_k)}
\end{align}
the \emph{loss exponent} of expert $k$. An example of a loss function is the 0-1 loss function: $C_k(m,m,n_k) = 0$, and $C_k(m,d,n_k)=1$ for all $n_k\geq 1$ if $d \ne m$, which results in \eqref{Ck} being the error probability considered in \cite{Tsi:88}. In this paper, we consider the regime where experts have asymptotically large number $n_k$ of private observations, which corresponds to the case where experts are very experienced in their respective fields (as determined by $\Gamma_k$).
By allowing the loss function to depend on the number of private observations, we can model various practical applications. For example, the expert $k$ may itself be part of a decentralized detection network like a tree configuration \cite{TayTsiWin:J08c,TayTsiWin:J09a}, in which case its goal is to minimize the Bayesian error probability at a fusion center. Then, its loss function decays exponentially fast in the number of private observations $n_k$. In the publicly listed conglomerate example, suppose that one of the hypotheses corresponds to the conglomerate being financially bankrupt. The loss associated with a missed detection of this hypothesis can be modeled to be exponentially larger than the loss associated with a missed detection of another more benign hypothesis.
In our model, expert $k$'s decision space consists of $d_k$ states, where $d_k$ may not equal to $M$, the number of hypotheses that agent $0$ is interested in. This allows us to model scenarios where different agents may have different models for the underlying state of the world. Using the conglomerate example described above, agent $0$ may be interested in the financial health of the whole conglomerate, while a particular analyst may only be interested in the real estate arm of the conglomerate. Nevertheless, the performance of the real estate arm has a bearing on the overall health of the conglomerate. In particular, expert $k$ may not distinguish between two hypotheses, say $H=M-2$ and $H=M-1$. This can be modeled by assuming that the expert declares the decision $d=m$ when $H=m < M-2$ and $d=M-2$ if $H=M-2$ or $M-1$. In this case, we let $C_k(M-2,d,n_k)=C_k(M-1,d,n_k)=1$ for all $d<M-2$ and $C_k(M-2,M-2,n_k)=C_k(M-1,M-2,n_k)=0$. This model is also general enough to model different decision spaces that may not map directly to any of the hypotheses of agent $0$.
On the other hand, we note that without loss of generality, there is no need to consider the case where an expert uses a model in which there are more than $M$ hypotheses, since agent $0$ can always choose a $M$ that is sufficiently large in its model. In addition, since our analysis is based on the frame of reference of agent $0$, it has no knowledge of any additional hypotheses, and in practical applications, it simply assumes that expert $k$ minimizes the expected loss given in \eqref{Ck}. We assume that based on the publicized expertise of each expert, agent $0$ knows its loss function decay rates, defined in Assumption \ref{assumpt:loss} below.
\subsection{Loss of Agent 0}
Let $D_0(k) = \gamma_0(Y_0[1:n_0],D_k) \in[0,M-1]$ be the decision made by agent $0$ after incorporating the opinion $D_k$ of agent $k$. The expected loss of agent $0$ is
\begin{align*}
\EE{C_0(H, D_0(k), n_0)},
\end{align*}
where $C_0(m, d, n_0)$ is the non-negative loss incurred by agent $0$ if it decides in favor of hypothesis $d$ when the true hypothesis is $H=m$, and the number of private observations is $n_0$.
We make the following assumptions regarding the loss function of each agent.
\begin{Assumption}\label{assumpt:loss}\
\begin{enumerate}[(i)]
\item\label{it:c_0} For all $n_0\geq1$ and $m \in [0,M-1]$, we have $C_0(m,m,n_0)=0$, and for every $d \in [0,M-1]$ such that $d \ne m$, we have $0< C_0(m,d,n_0) \leq 1$ with
\begin{align*}
\hspace{-8pt}-\hspace{-6pt}\lim_{n_0\to\infty} \hspace{-2pt}\ofrac{n_0}\log C_0(m,d,n_0) = c_0(m,d) \equiv c_0(m) \in [0,\infty).
\end{align*}
\item\label{it:c_k} For each agent $k \geq 1$, we have for every $m \in [0,M-1]$ and $d\in [0,d_k-1]$,
\begin{align*}
-\lim_{n_k\to\infty} \ofrac{n_k}\log C_k(m,d,n_k) = c_k(m,d) \geq 0.
\end{align*}
We assume that agent $0$ knows the loss function decay rates $c_k(\cdot,\cdot)$, for all $k=1,\ldots,K$, based on the publicized expertise of each expert.
\end{enumerate}
\end{Assumption}
Since the number of agents and decision states are finite, we can normalize $C_k(m,d,n_k)$ by $\displaystyle\max_{m,d,k} C_k(m,d,n_k)$ or $\displaystyle\sum_{d,k} C_k(m,d,n_k)$ so that there is no loss in generality in assuming that $C_k(m,d,n_k) \leq 1$ for all $m$, $d$, and $k$. In Assumption~\ref{assumpt:loss}\eqref{it:c_0}, we make the simplifying assumption that the loss function decay rate for agent $0$ w.r.t.\ a particular hypothesis is the same for all wrong decisions. This is because otherwise, easy examples can be constructed in which agent $0$ declares the wrong hypothesis with high probability for large $n_0$.
To see this, suppose that all hypotheses have the same prior probability, $H=0$ is the true hypothesis, and agent $0$'s observations are all drawn from the same conditional distribution. The loss of agent $0$ if it makes the decision $d$ is, with high probability, approximately proportional to
\begin{align*}
\sum_{m\ne d} e^{-n_0 (c_0(m,d) + K_{0m})},
\end{align*}
where $K_{0m}$ is the Kullback-Liebler divergence of the conditional distribution under $H=0$ versus that under $H=m$ \cite{CovTho:B05}. If $c_0(0,1) > c_0(1,0)+K_{01}$, while $c_0(m,1)>c_0(m,0)$ for all $m\ne 0,1$, then agent $0$ decides in favor of $H=1$ instead of $H=0$ when $n_0$ is large. This is clearly an undesirable model as $c_0(m,1) > c_0(m,0)$ for all $m \ne 0,1$ implies that the agent imposes an exponentially smaller loss when declaring $H=1$ versus $H=0$. In this case, an arguably more appropriate modeling approach is to merge the hypothesis $H=1$ into another hypothesis.
In Assumption~\ref{assumpt:loss}\eqref{it:c_k}, we assume that agent $0$ knows the loss function decay rates of the experts. This assumption may not hold in some practical applications, in which case we need to impose uncertainties on agent $0$'s knowledge about the experts, similar in spirit to \cite{HoTayQuek:C14}. This unfortunately makes the problem much more challenging, and is out of the scope of our current work. The results in this work serve as the basic foundation on which the more challenging problem in which agent $0$ has limited knowledge of the experts, can be addressed in future research.
\subsection{Asymptotic Equivalence and Optimal Loss Exponent}\label{subsection:asymptotic}
Consider two loss functions with loss decay rates $c_k(m,d)$ and $c_k'(m,d)$ respectively. If $c_k(m,d)$ and $c_k'(m,d)$ differ by the same constant for all $m$ and $d$, then their corresponding loss exponents \eqref{exp_k} are different but the optimal policies to minimize \eqref{exp_k} are the same. We therefore say that two loss functions are \emph{asymptotically equivalent} if their respective loss decay rates $c_k(m,d)$ and $c_k'(m,d)$ differ by the same constant for all $m$ and $d$. It can be shown that each loss function belongs to an equivalence class, with the equivalence relation being asymptotic equivalence. We call a loss function with $\min_{m,d} c_k(m,d) = 0$ a \emph{canonical loss function} of its equivalence class. For example, the 0-1 loss function is a canonical loss function of its equivalence class. In another example, consider the set of all loss functions $C_k(m,d,n_k)$ with $\sum_d C_k(m,d,n_k) = C$ for each $m$ and $n_k$, which imposes a total loss of $C$ for each hypothesis. Then, it can be shown that each of these loss functions is a canonical loss function.
We say that two loss exponents are equivalent if they have the same policy and their loss functions are asymptotically equivalent. For fair comparison of loss exponents, we will always assume canonical loss functions. We are interested to characterize the optimal loss exponent\footnote{Note that because of Assumption \ref{assumpt:loss}, the loss exponent is \emph{negative}, with a more negative loss exponent corresponding to a faster loss decay rate.}
\begin{align}\label{loss_exponent}
\min_{1\leq k\leq K}
\limsup_{n_0\to\infty} \ofrac{n_0}\log \EE{C_0(H, D_0(k), n_0)},
\end{align}
when the number of private observations of agent $0$ becomes large. Note that although agent $0$ is allowed an asymptotically large number of private observations, the quality of information available to agent $0$ is constrained by the set $\Gamma_0$. The experts' information sources $\{\Gamma_k:k=1,\ldots,K\}$ are different from $\Gamma_0$, and may thus improve the loss exponent of agent $0$. We assume that for all experts $k\geq1$, $q_k = \lim_{n_0\to\infty} n_k /n_0$ exists as a limit. If the number $n_k$ of private observations of expert $k$ is such that $q_k = 0$, agent $0$ will ignore the opinion of expert $k$ as its opinion becomes asymptotically negligible compared to agent $0$'s private observations. Therefore, without loss of generality, we assume that $q_k > 0$ for all $k \geq 1$.
\subsection{Technical Definitions}\label{subsect:Notations}
In this subsection, we define notations that will be commonly used throughout the paper. We rely heavily on the mechanisms of large deviations theory to characterize the agents' loss exponents \eqref{exp_k} and \eqref{loss_exponent}. We refer the reader to Appendix \ref{appendix:Mathematical} for a brief overview of some basic concepts, and to \cite{DemZei:98} for a detailed treatment of large deviations theory. As the theory of large deviations in hypothesis testing problems utilizes log moment generating functions of log likelihood ratios, and their Fenchel-Legendre transforms in order to characterize the loss exponents, we define these necessary quantities in the following.
For any given random variable $X$ with marginal distribution $\P_i^X$ under hypothesis $H=i$, we abuse notation by letting $\ell_{ij}(X)$ be the Radon-Nikodym derivative (or likelihood ratio) of $\P_i^X$ w.r.t.\ $\P_j^X$. Note that $\ell_{ij}(X)$ is a random variable that depends on the distributions and realization of $X$.
By convention, we let $\ell_{ii}(X) = 1$ for all values of $i$ and all realizations of $X$. In addition, for simplicity, we let $\ell_{ij}^\gamma = \mathrm{d}\P_i^\gamma/\mathrm{d}\P_j^\gamma$ be the Radon-Nikodym derivative of $\P_i^\gamma$ w.r.t.\ $\P_j^\gamma$, for each $\gamma\in\bigcup_k \Gamma_k$.
Let $Z^\gamma = (\log \ell_{m0}^\gamma)_{m=1}^{M-1}$ be a vector of log likelihood ratios, and the log moment generating function of $Z^\gamma$ under $H=m$ be
\begin{align*}
\xi_{m}(\gamma,t) = \log \Ec{m}{\exp(\ip{t}{Z^{\gamma}})},
\end{align*}
for all $t \in \mathbb{R}^{M-1}$. For a policy $x = (x[\gamma])_{\gamma\in\Gamma_k}$ of an agent $k$, the weighted log moment generating function is then given by
\begin{align}\label{logmomentgen}
\varphi_m(t,x) = \sum_{\gamma\in\Gamma_k} x[\gamma] \xi_m(\gamma,t),
\end{align}
and its Fenchel-Legendre transform \cite{DemZei:98} is
\begin{align}\label{Fenchel}
\Phi_m^*(z,x) = \sup_{t\in\mathbb{R}^{M-1}} \left\{ \ip{t}{z} - \varphi_m(t,x)\right\},
\end{align}
where $z \in \mathbb{R}^{M-1}$. The function $\Phi_m^*(z,x)$ will be shown in Theorem \ref{theorem:expert} to characterize the rates of decay of the probabilities $\P_m(D_k=d)$, similar to the rate functions in large deviations theory \cite{DemZei:98}.
When an expert $k$'s decision space has size $d_k=M$, the rates of decay of the probabilities $\P_m(D_k=d)$ become easier to characterize since we can now interpret a decision $d$ of the expert $k$ to be in favor of hypothesis $d$. Then, under the true hypothesis $H=m$, the log moment generating function of interest is the one involving the likelihood ratios of the observation distributions under hypothesis $d$ versus hypothesis $m$, given by
\begin{align}\label{def:Lambdaij}
\Lambda_{ij}(s,x) = \sum_{\gamma\in\Gamma_k} x[\gamma] \log \Ec{i}{(\ell_{ji}^\gamma)^s}.
\end{align}
The Fenchel-Legendre transform of $\Lambda_{ij}$ is then given by
\begin{align*}
\Lambda_{ij}^*(z,x) = \sup_{s\in\mathbb{R}} \left\{ sz - \Lambda_{ij}(s,x)\right\},
\end{align*}
where $z\in\mathbb{R}$.
We assume that the distributions $\{\P_m^\gamma$, $\gamma\in\bigcup_k\Gamma_k\}$ are well-behaved for all $m\in [0,M-1]$ in the following assumption, which holds for example in the case where all conditional distributions are from the exponential families. This assumption is required in all our proofs in order to show that an agent's loss exponent cannot be better than a certain achievable lower bound, which we characterize in order to derive the agent's optimal policy.
\begin{Assumption}\label{assumption:xi}
For all $m\in [0,M-1]$ and all $\gamma\in\bigcup_k\Gamma_k$, $\xi_m(\gamma,t) < \infty$ for all $t\in\mathbb{R}^{M-1}$.
\end{Assumption}
\subsection{Results Overview}\label{subsection:results}
If agent $0$ incorporates the opinion of expert $k$, we expect the loss exponent of expert $k$ to determine how useful its opinion is to agent $0$. Therefore, we first find the loss exponent of each expert $k$ in Theorem \ref{theorem:expert}, which also provides an asymptotically optimal policy for expert $k$. Then, we proceed to determine the loss exponent of agent $0$ if it incorporates the opinion of a particular expert $k$ in Theorem \ref{theorem:agent0}. To find the optimal expert, we simply optimize the loss exponent found in Theorem \ref{theorem:agent0} over the whole set of experts, which is essentially the content of Theorem \ref{theorem:expert_choice}. Based on the conclusions of Theorem \ref{theorem:expert_choice}, we provide detailed procedures in Remark \ref{rem:hardness} of Section \ref{subsect:choose} that allow agent $0$ to compute its asymptotically optimal expert choice and policy. Finally, we provide insights into the optimal expert choice under simplifying assumptions in Propositions \ref{prop:expert_necessary} and \ref{prop:bestexpertloss}.
\section{Optimal Loss Exponents}\label{sect:Optimal}
In this section, we first characterize the loss exponents of the experts $k=1,2,\ldots,K$, which leads to an asymptotically optimal policy for each expert. We then characterize the loss exponent of agent $0$, assuming that it is following the opinion of some expert $k$.
\subsection{Loss Exponents of Experts}\label{subsect:lossexp_experts}
Consider an expert $k$, where $k =1,\ldots,K$. By conditioning on the observations $Y_k[1:n_k]$, it can be shown (Proposition 2.3 of \cite{Tsi:93}) that the optimal decision rule for expert $k$ is given by
\begin{align}\label{optimal_D}
D_k = \arg \min_{0\leq d \leq d_k-1} \sum_{m=0}^{M-1} \pi_m C_k(m,d,n_k)\ell_{m0}(Y_k[1:n_k]),
\end{align}
where $\ell_{m0}(Y_k[1:n_k]) = \prod_{l=1}^{n_k} \ell_{m0}(Y_k[l])$. In general, the right hand side of \eqref{optimal_D} may have multiple minimizers.
In order to avoid having to consider the use of randomization to determine the final decision for agent $k$ (due to a mathematical technicality in the proof of Theorem \ref{theorem:expert} below), and for the ease of interpreting results, we will assume throughout this paper that the right hand side of \eqref{optimal_D} has a unique solution with probability one.
\begin{Assumption}\label{assumpt:unambiguous}
For every agent $k=1,\ldots,K$, and for any policy $x_k$, the minimization on the right hand side of \eqref{optimal_D} has a unique solution with probability one.
\end{Assumption}
Assumption \ref{assumpt:unambiguous} is satisfied if conditioned on any hypothesis $H \in \{0,\ldots,M-1\}$,
$(\ell_{m0}^\gamma)_{m\geq 1}$ are continuous random variables with a joint probability density function. To see this, it can be shown via an easy inductive argument that $(\ell_{m0}(Y_k[1:n_k]))_{m\geq 1}$ are in turn continuous random variables under any hypothesis $H$. Assumption \ref{assumpt:unambiguous} then follows by the same argument in Lemma 5.1 of \cite{Tsi:93a}, which we refer the reader to. This shows that Assumption \ref{assumpt:unambiguous} holds in most practical applications as private observations are usually modeled as noisy, with the noise typically having a probability density function like the Gaussian probability density \cite{You09, Viv96}.
If agent $0$ follows expert $k$, its loss exponent depends on the loss exponent \eqref{exp_k} of the expert $k$, which is in turn related to the probability exponents
\begin{align}\label{exp_k_md}
\limsup_{n_k \to \infty} \ofrac{n_k} \log \P_m(D_k = d),
\end{align}
where $m \in [0,M-1]$ and $d \in [0,d_k-1]$. In the following, our aim is to characterize the probability exponents \eqref{exp_k_md}. We use \eqref{optimal_D} to characterize \eqref{exp_k_md} by first letting
\begin{align*}
\tilde{g}_k(z, d, n_k) &= \ofrac{n_k} \log \sum_{m=0}^{M-1} \pi_m e^{\log C_k(m,d,n_k) + n_kz[m]},
\end{align*}
and
\begin{align*}
g_k(z,d,n_k) &= \tilde{g}_k(z,d,n_k) - \min_{d' \ne d} \tilde{g}_k(z,d',n_k),
\end{align*}
where $z = (z[0], z[1], \ldots, z[M-1]) \in \mathbb{R}^{M}$. Suppose that agent $k$ adopts the policy $x_{k,n_k}$ when it has access to $n_k$ private observations. Then from \eqref{optimal_D} and Assumption \ref{assumpt:unambiguous}, we obtain
\begin{align}\label{probD}
\P_m(D_k=d) = \P_m(g_k(\bar{Z}_{n_k}^0(x_{k,n_k}), d, n_k) < 0),
\end{align}
where
\begin{align}\label{barZ}
\bar{Z}_{n_k}(x_{k,n_k}) = \left( \ofrac{n_k} \log \ell_{m0}(Y_k[1:n_k]) \right)_{m=1}^{M-1}
\end{align}
is a vector in $\mathbb{R}^M$ of log likelihood ratios, and $\bar{Z}_{n_k}^0(x_{k,n_k}) = (0, \bar{Z}_{n_k}(x_{k,n_k}))$ is the vector of log likelihood ratios augmented with $0$ as the first entry (this corresponds to $1/n_k \cdot \log \ell_{00}(Y_k[1:n_k]) = 0$). In the following, we show that $g_k(z,d,n_k)$ converges uniformly in $z$ to
\begin{align}\label{fk}
f_k(z,d) &= \tilde{f}_k(z,d) - \min_{d' \ne d} \tilde{f}_k(z,d'),
\end{align}
where
\begin{align}\label{tfk}
\tilde{f}_k(z, d) &= \max_{0\leq m \leq M-1}\{z[m] - c_k(m,d) \}.
\end{align}
\begin{Lemma}\label{lemma:f_uniform}
Suppose that Assumption \ref{assumpt:loss} holds. For all $k=1,\ldots,K$, and all $d \in [0,d_k-1]$, $\tilde{g}_k(z, d, n) \to \tilde{f}_k(z, d)$ and $g_k(z,d,n) \to f_k(z,d)$ uniformly in $z \in \mathbb{R}^{M}$ as $n\to\infty$.
\end{Lemma}
\begin{IEEEproof}
See Appendix \ref{appendix:proofs}.
\end{IEEEproof}
We are now ready to present our first main result. Since we are working in the regime of large $n_k$, Lemma \ref{lemma:f_uniform} tells us that we can replace $g_k$ with $f_k$ in \eqref{probD}. For $k \geq 1$, let
\begin{align}\label{Ak}
A_k(d) = \left\{z \in \mathbb{R}^{M-1}: f_k(z^0, d) < 0,\textrm{ where } z^0 = (0, z)\right\},
\end{align}
The set $A_k(d)$ can be interpreted as the asymptotic decision region for expert $k$ to declare decision $d$ based on its sufficient statistics $\bar{Z}_{n_k}(x_{k,n_k})$; see Section \ref{subsect:ADR} for a discussion. Assumption \ref{assumpt:unambiguous} is required here to ensure that the sets $A_k(d), d\in[0,d_k-1]$ are disjoint.
\begin{Theorem}\label{theorem:expert}
Suppose that Assumptions \ref{assumpt:loss}, \ref{assumption:xi}, and \ref{assumpt:unambiguous} hold. Consider an agent $k \in \{1,\ldots, K\}$ who adopts a sequence of optimal policies $(x_{k,n_k})_{n_k \geq 1}$ that minimizes \eqref{Ck} for each $n_k$. Then, we have the following:
\begin{enumerate}[(i)]
\item\label{it:expert_opt} The loss exponent of agent $k$ is given by
\begin{align*}
\lim_{n_k\to\infty} \ofrac{n_k} \log \EE{C_k(H, D_k, n_k)} = - \max_{x \in \mathbb{S}(\Gamma_k)} I_k(x),
\end{align*}
where
\begin{align}\label{Ik}
&\hspace{-8pt}I_k(x) =\hspace{-8pt} \min_{\substack{0\leq m \leq M-1 \\ 0 \leq d \leq d_k-1}} \left\{\inf_{z \in A_k(d)} \Phi_m^*(z, x) + c_k(m,d)\right\}.
\end{align}
\item\label{it:policy_opt} There is no loss in optimality asymptotically, if we restrict the sequence of policies $(x_{k,n_k})_{n_k \geq 1}$ such that $\lim_{n_k \to\infty} x_{k,n_k} = x_k^*$, for some $x_k^* \in \arg\max_{x \in \mathbb{S}(\Gamma_k)} I_k(x)$.
\item\label{it:prob_exp} Let $x_k^* \in \arg\max_{x \in \mathbb{S}(\Gamma_k)} I_k(x)$. For every $m\in [0,M-1]$, and $d \in [0,d_k-1]$, we have
\begin{align*}
\lim_{n_k \to \infty} \ofrac{n_k}\log\P_m(D_k = d) = - \inf_{z \in A_k(d)} \Phi_m^*(z, x_k^*).
\end{align*}
\end{enumerate}
\end{Theorem}
\begin{IEEEproof}
See Appendix \ref{appendix:proofs}.
\end{IEEEproof}
Theorem \ref{theorem:expert} allows each expert to find an asymptotically optimal policy by maximizing \eqref{Ik} (see Remark \ref{rem:hardness} in Section \ref{subsect:choose}). In addition, we will see later that Theorem \ref{theorem:expert}\eqref{it:prob_exp} allows agent $0$ to characterize its own loss exponent and thus optimally choose the expert to follow.
\begin{Rem}\label{rem:Ik}
The quantity $I_k(x)$ in \eqref{Ik} can be interpreted as the rate of loss decay for agent $k$ adopting policy $x$, i.e., for $n_k$ sufficiently large, we have
\begin{align*}
\EE{C_k(H, D_k, n_k)} \approx g(n_k) e^{-I_k(x)},
\end{align*}
where $g(n_k)$ is a function that decays faster than exponentially in $n_k$. To achieve a smaller loss, agent $k$ then chooses a policy $x$ that maximizes $I_k(x)$. Observe that $I_k(x)$ is the minimization over all $m\in[0,M-1]$ and $d\in[0,d_k-1]$ of the terms $\inf_{z \in A_k(d)}\Phi_m^*(z, x) + c_k(m,d)$, where $\inf_{z \in A_k(d)}\Phi_m^*(z, x)$ is the absolute probability exponent in Theorem \ref{theorem:expert}\eqref{it:prob_exp}, while $c_k(m,d)$ is the loss function decay rate of declaring decision $d$ when the true hypothesis is $m$. Therefore, each term in the minimization of the right hand side of \eqref{Ik} is the absolute decay rate of $C_k(m,d,n_k)\P_m(D_k=d)$, and the overall loss exponent is dominated by the term that decays the slowest, which is what we expect from large deviations theory (cf.\ Lemma 1.2.15 of \cite{DemZei:98}). We next provide an intuitive interpretation for the regions $A_k(d)$ in the following subsection.
\end{Rem}
\subsection{Asymptotic Decision Regions}\label{subsect:ADR}
Suppose that agent $k\geq1$ adopts the sequence of policies $x_{k,n_k}\to x_k$ as $n_k\to\infty$. From \eqref{probD}, for large $n_k$, we have for each $m\in [0,M-1]$, and $d \in [0,d_k-1]$,
\begin{align*}
\ofrac{n_k}\log \P_m(D_k=d)
&\approx \ofrac{n_k}\log \P_m(f_k(\bar{Z}_{n_k}(x_{k,n_k}), d) < 0) \\
&\approx - \inf_{z \in A_k(d)} \Phi_m^*(z, x_k),
\end{align*}
where $A_k(d)$ is as defined in \eqref{Ak}, and $\Phi_m^*(z, x_k)$ can be interpreted as a rate function. In the same spirit as the G\"artner-Ellis Theorem \cite{DemZei:98}, we can interpret the sets $A_k(d)$ as the asymptotic decision region for deciding $D_k = d$ based on $\bar{Z}_{n_k}(x_{k,n_k})$ as $n_k\to\infty$, where the rate of decay of $\P_m(D_k=d)$ is dominated by the rate at a particular realization of $\bar{Z}_{n_k}(x_{k,n_k})$ in the region $A_k(d)$. We refer the reader to Appendix \ref{appendix:Ak_properties} for a characterization of $A_k(d)$ in terms of the union of intersections of multiple half-spaces.
In the following, we list some properties of the rate functions $\Phi_m^*(\cdot,\cdot)$ that will be useful in helping us to interpret our results.
\begin{Lemma}\label{lemma:Phi_properties}
For every $m\in[0,M-1]$, $\Phi_m^*(z,x)$ is non-negative, convex in $z$ and concave in $x$. Furthermore, for any policy $x$, $\displaystyle\min_{z\in\mathbb{R}^{M-1}} \Phi_m^*(z,x) = 0$, and the minimum is achieved at
\begin{align}\label{eqn:tildez}
\tilde{z}_m(x) = \sum_{\gamma} x[\gamma] \Ec{m}{Z^\gamma},
\end{align}
where $Z^\gamma = (\log \ell_{i0}^\gamma)_{i=1}^{M-1}$.
\end{Lemma}
\begin{IEEEproof}
See Appendix \ref{appendix:proofs}.
\end{IEEEproof}
In particular, if $d_k=M$, and the loss function of agent $k$ adopting the policy $x_k$ is such that $\tilde{z}_m(x_k) \in A_k(m)$ in \eqref{eqn:tildez} for every $m$, then we can interpret the decision $m$ of agent $k$ as in favor of hypothesis $H=m$ since $\P_m(D_k=m)$ is bounded away from 0 and has decay rate $\inf_{z \in A_k(m)} \Phi_m^*(z, x_k) = 0$.
\subsection{Loss Exponent of Agent 0}
In the following, we characterize the loss exponent of agent $0$ if it chooses expert $k$. Recall that $q_k = \lim_{n_0\to\infty} n_k /n_0$.
\begin{Theorem}\label{theorem:agent0}
Suppose that Assumptions \ref{assumpt:loss}, \ref{assumption:xi}, and \ref{assumpt:unambiguous} hold. Suppose that agent $0$ adopts the opinion of expert $k\geq 1$, which has the asymptotically optimal policy $x_k^*$. Then, the loss exponent of agent $0$ is
\begin{align*}
&\lim_{n_0\to\infty} \ofrac{n_0}\log \EE{C_0(H,D_0(k),n_0)} = - \max_{x_0\in\mathbb{S}(\Gamma_0)} \mathcal{E}_0(k,x_0),
\end{align*}
where \footnote{To avoid cluttered notations, we let $\min_{i\ne j}$ and $\max_{i\ne j}$ be the minimization or maximization over all unordered pairs $(i,j) \in [0,M-1]^2$ such that $i\ne j$, respectively.}
\begin{align}\label{C0_lowerbound}
&\mathcal{E}_0(k,x_0) \nonumber\\
&= \min_{\substack{i\ne j \\ 0\leq d \leq d_k-1}} \max_{s\in [0,1]} \bigg\{ (1-s)\left(q_k\inf_{z \in A_k(d)} \Phi_i^*(z, x_k^*) + c_0(i)\right) \nonumber\\
&\quad\quad + s\left(q_k\inf_{z \in A_k(d)} \Phi_j^*(z, x_k^*) + c_0(j)\right) - \Lambda_{ij}(s,x_0) \bigg\}.
\end{align}
\end{Theorem}
\begin{IEEEproof}
See Appendix \ref{appendix:proofs}.
\end{IEEEproof}
Following Remark \ref{rem:Ik}, we can again interpret each term in the minimization on the right hand side of \eqref{C0_lowerbound} as the absolute error exponent incurred when differentiating between hypotheses $i$ and $j$. From Theorem \ref{theorem:expert} and Theorem \ref{theorem:agent0}, we see that the choice of an expert $k$ affects agent $0$'s loss exponent through the rate of decay of the probabilities $\P_i(D_k=d)$ and $\P_j(D_k=d)$, which is to be expected. In particular, if the loss function of the expert $k$ is such that we can interpret the decision $d$ of the expert as in favor of hypothesis $d$, then $\P_i(D_k=i) > 0$ yields $\inf_{z \in A_k(d)} \Phi_i^*(z, x_k^*)=0$, and it does not contribute to the decay of the expected loss of agent 0.
\section{Optimal Choice of Expert}\label{sect:OptimalChoice}
We now address the question of how to choose an optimal expert for agent $0$. We first revisit Theorem \ref{theorem:agent0} to derive the optimal loss exponent for agent $0$, and then we make additional simplifying assumptions in order to provide more insights into our results. Finally, we discuss a numerical example to illustrate the use of our optimal loss exponent characterizations in finding the optimal policies and expert choice.
\subsection{Choosing an Expert}\label{subsect:choose}
If agent $0$ does not incorporate the opinion of any expert, we use the notation $\mathcal{E}_0(0,x_0)$ to denote its absolute loss exponent when using policy $x_0$. From Theorem \ref{theorem:agent0}, by setting $q_k=0$, we have
\begin{align}\label{E0}
\mathcal{E}_0(0,x_0) &= \min_{i\ne j} \max_{s\in [0,1]} \left\{
(1-s)c_0(i) + sc_0(j) - \Lambda_{ij}(s,x_0) \right\}.
\end{align}
In the case of minimizing the error probability at agent $0$ with 0-1 loss function, and without the help of any experts, we can further set $c_0(i)=c_0(j)=0$ to obtain the absolute error exponent
\begin{align}\label{EB}
\mathcal{E}_{0,B}(0,x_0) &= -\max_{i\ne j} \min_{s\in [0,1]} \Lambda_{ij}(s,x_0),
\end{align}
which recovers the result in \cite{Tsi:88}. Similarly, if agent $0$ chooses expert $k$, and adopts the 0-1 loss function, we let its absolute loss exponent be
\begin{dmath}[label={EkB}]
\mathcal{E}_{0,B}(k,x_0) = \min_{\substack{i\ne j \\ 0\leq d \leq d_k-1}} \max_{s\in [0,1]} \left\{ (1-s)q_k\inf_{z \in A_k(d)} \Phi_i^*(z, x_k^*) + sq_k\inf_{z \in A_k(d)} \Phi_j^*(z, x_k^*) - \Lambda_{ij}(s,x_0) \right\}.
\end{dmath}
The following proposition follows immediately from Theorem \ref{theorem:agent0}, and provides a method for agent $0$ to optimally choose which expert to follow.
\begin{Theorem}\label{theorem:expert_choice}
Suppose that Assumptions \ref{assumpt:loss}, \ref{assumption:xi}, and \ref{assumpt:unambiguous} hold. The optimal loss exponent of agent $0$ is
\begin{dmath*}
\min_{1\leq k\leq K}\lim_{n_0\to\infty} \ofrac{n_0}\log \EE{C_0(H,D_0(k),n_0)} = -\max_{1\leq k \leq K} \max_{x_0\in\mathbb{S}(\Gamma_0)} \mathcal{E}_0(k,x_0).
\end{dmath*}
Furthermore, for any $k\geq 1$ and policy $x_0$, we have
\begin{align}
&\mathcal{E}_0(k,x_0) \geq \mathcal{E}_{0,B}(k,x_0) \label{Eineq1}\\
&\mathcal{E}_0(k,x_0) \geq \mathcal{E}_0(0,x_0) \geq \mathcal{E}_{0,B}(0,x_0). \label{Eineq2}
\end{align}
\end{Theorem}
\begin{IEEEproof}
See Appendix \ref{appendix:proofs}.
\end{IEEEproof}
\begin{Rem}\label{rem:hardness}
Theorem \ref{theorem:expert_choice} shows that agent $0$ should choose an expert $k$ that maximizes $\max_{x_0} \mathcal{E}_0(k,x_0)$, which depends on $\inf_{z \in A_k(d)} \Phi_m^*(z, x_k^*)$, for $m\in[0,M-1]$ and $d\in[0,d_k-1]$. We now discuss procedures that can achieve this, depending on the amount of information that agent $0$ has about the experts:
\begin{enumerate}[(i)]
\item\label{it:probexp} If the experts' probability exponents $\{\inf_{z \in A_k(d)} \Phi_m^*(z, x_k^*): m\in[0,M-1],d\in[0,d_k-1]\}$ for all $k \geq 1$ are publicized, then agent $0$ simply needs to find its optimal policy $x_0$ that maximizes $\mathcal{E}_0(k,x_0)$ in Theorem \ref{theorem:expert_choice} for each $k$, and choose the expert that produces the largest $\max_{x_0} \mathcal{E}_0(k,x_0)$. To find the optimal policy $x_0$ corresponding to an expert $k$, we can use an iterative procedure as follows:
\begin{enumerate}[Step 1.]
\item An initial guess for the optimal policy $x_0^*(0)$ is made. Set $l=1$.
\item\label{step:s} For each pair $(i,j)\in[0,M-1]$, $i\ne j$, and $d\in[0,d_k-1]$, find $s_{ijd}(l)$ that is the maximizer of the optimization problem over $s\in[0,1]$ on the right hand side of \eqref{C0_lowerbound}. Note that this is equivalent to a convex minimization problem, which can be solved via standard convex optimization methods \cite{BoyVan:B04}.
\item\label{step:x0} Find $x_0^*(l)=(x[\gamma])_{\gamma\in\Gamma_0}$ by solving the linear program:
\begin{align*}
& \max_{r, (x[\gamma])_{\gamma\in\Gamma_0}} \quad r \\
& \textrm{subject to } \\
& r \leq (1-s_{ijd}(l))\left(q_k\inf_{z \in A_k(d)} \Phi_i^*(z, x_k^*) + c_0(i)\right) \\
&\quad\quad + s_{ijd}(l) \left(q_k\inf_{z \in A_k(d)} \Phi_j^*(z, x_k^*) + c_0(j)\right) \\
& \qquad - \sum_{\gamma\in\Gamma_0} x[\gamma] \log \Ec{i}{(\ell_{ji}^\gamma)^{s_{ijd}}} ,\\
& \qquad \forall i,j \in[0,M-1], i\ne j, \forall d\in[0,d_k-1], \\
& x[\gamma] \geq 0, \ \forall \gamma\in\Gamma_0,\\
& \sum_{\gamma\in\Gamma_0} x[\gamma] = 1.
\end{align*}
\item Set $l = l+1$ and repeat Steps 2-4 till $x_0^*(l)$ does not change significantly.
\end{enumerate}
\item\label{it:lossandpolicy} If agent $0$ knows only the experts' loss functions and optimal policies $x_k^*$ for all $k\geq 1$, then it can compute $\inf_{z \in A_k(d)} \Phi_m^*(z, x_k^*)$ for all $m\in[0,M-1]$ and $d\in[0,d_k-1]$ by searching for the minimum of $\Phi_m^*(z,x_k^*)$ on the boundaries of $A_k(d)$, if $\tilde{z}_m(x_k^*)$ in \eqref{eqn:tildez} is not in $A_k(d)$. This is because $\Phi_m^*(z,x_k^*)$ is convex in $z$ (cf. Lemma \ref{lemma:Phi_properties}), and the search can be performed using standard convex optimization methods \cite{BoyVan:B04}. The boundaries of $A_k(d)$ can be found through its characterization in Appendix \ref{appendix:Ak_properties} (see Section \ref{subsection:numerical} for an example). If $\tilde{z}_m(x_k^*)\in A_k(d)$, then from Lemma \ref{lemma:Phi_properties}, we obtain $\inf_{z \in A_k(d)} \Phi_m^*(z, x_k^*)=0$. Once the experts' probability exponents have been computed, the same procedure as in item \eqref{it:probexp} above can now be used to choose the optimal expert.
\item\label{it:lossonly} If agent $0$ knows only the experts' loss functions, it can utilize Theorem \ref{theorem:expert}\eqref{it:expert_opt} to determine the policy that each expert adopts. This however may not be an easy numerical procedure if $M$ and $d_k$ are large, even in the case where the expert has 0-1 loss function \cite{Tsi:88}. We propose the following alternating optimization approach for finding the optimal policy of expert $k$:
\begin{enumerate}[Step 1.]
\item An initial guess for the optimal policy $x_k^*(0)$ is made. Set $l=1$.
\item\label{step:t} For each $m\in[0,M-1]$ and $d\in[0,d_k-1]$, find $z_{m,d}(l) = \arg\inf_{z\in A_k(d)} \Phi_m^*(z,x_k^*(l-1))$. As noted in item \eqref{it:lossandpolicy} above, this can be obtained via standard convex optimization methods. Let
\begin{dmath*}
t_{m,d}(l)) \hiderel{=} \arg\max_{t\in\mathbb{R}^{M-1}} \left\{ \ip{t}{z_{m,d}(l-1)} - \varphi_m(t,x_k^*(l))\right\}.
\end{dmath*}
\item\label{step:x} Find $x_k^*(l)=(x[\gamma])_{\gamma\in\Gamma_k}$ by solving the linear program
\begin{align*}
& \max_{r, (x[\gamma])_{\gamma\in\Gamma_k}} \quad r \\
& \textrm{subject to } \\
& r \leq \ip{t_{m,d}(l)}{z_{m,d}(l)} \\
& \qquad - \sum_{\gamma\in\Gamma_k} x[\gamma] \xi_m(\gamma,t_{m,d}(l)) + c_k(m,d),\\
& \qquad \forall m\in[0,M-1], \forall d\in[0,d_k-1], \\
& x[\gamma] \geq 0, \ \forall \gamma\in\Gamma_k,\\
& \sum_{\gamma\in\Gamma_k} x[\gamma] = 1.
\end{align*}
\item Set $l = l+1$ and repeat Steps 2-4 till $x_k^*(l)$ does not change significantly.
\end{enumerate}
Unfortunately, there is no guarantee that the above procedure converges to the correct solution. However, in our numerical experiments, we were able to arrive at the correct optimal policy by using multiple initial guesses. A numerical example is presented in Section \ref{subsection:numerical}.
\end{enumerate}
\end{Rem}
\begin{Rem}\label{rem:worstloss}
The first inequality in \eqref{Eineq2} shows that there is no loss in optimality for agent $0$ to incorporate the opinion of any expert, verifying the adage that there is no harm in having more information. However, incorporating additional information does not necessarily improve its loss exponent; see Proposition \ref{prop:expert_necessary}.
The inequality in \eqref{Eineq1} shows that of all the loss functions satisfying Assumption \ref{assumpt:loss}\eqref{it:c_0}, the 0-1 loss is the worst canonical loss function for agent $0$. In particular, consider the set of loss functions with $C_0(m,d,n_0)=C_0(m,n_0)$ for all $m\in[0,M-1]$ and $d \ne m$, where the limit $c_0(m) = -\lim_{n_0\to\infty} (1/n_0)\allowbreak\log C_0(m,n_0)$ exists. Furthermore, each loss function has a total cost constraint,
\begin{align*}
\sum_{m=0}^{M-1} C_0(m,n_0) = M-1.
\end{align*}
It can be shown that all such loss functions satisfy Assumption \ref{assumpt:loss}\eqref{it:c_0}, and are canonical loss functions. The 0-1 loss function belongs to this set, and divides the total cost equally among all types of missed detections, which results in the worst loss exponent for agent $0$. This can be explained intuitively by observing that if there exists a $m$ such that $c_0(m) > 0$, then missed detection of $H=m$ incurs an exponentially decaying loss, so that the hypothesis $m$ can effectively be ignored. Agent $0$ then effectively has a smaller set of hypotheses, leading to a lower expected loss.
\end{Rem}
\begin{Rem}\label{remark:multiexperts}
It is easy to generalize Theorem \ref{theorem:agent0} or Theorem \ref{theorem:expert_choice} to the case where more than one agent can be chosen. In particular, if all $K$ experts' opinions are adopted by agent $0$, Theorem \ref{theorem:agent0} holds with $\mathcal{E}_0(k,x_0)$ replaced by
\begin{align*}
&\min_{\substack{i\ne j \\ (p_k)_{k=1}^K}}\hspace{-8pt} \max_{s\in [0,1]} \bigg\{ (1-s)\left(\sum_{k=1}^K q_k \inf_{z \in A_k(p_k)} \Phi_i^*(z, x_k^*) + c_0(i)\right) \nonumber\\
&\quad\quad + s\left(\sum_{k=1}^K q_k\inf_{z \in A_k(p_k)} \Phi_j^*(z, x_k^*) + c_0(j)\right) - \Lambda_{ij}(s,x_0) \bigg\}
\end{align*}
where the sequences $(p_k)_{k=1}^K \in \prod_{k=1}^K [0,d_k-1]$.
\end{Rem}
In the following subsection, we consider the special cases where the size of each expert's decision space is less than or equal to $M$ under additional simplifying assumptions.
\subsection{Special Cases}\label{subsection:special}
In this section, we assume that agent $0$'s policy has been fixed in advance. We first consider the case where an expert $k$ may have a decision space smaller than the number of hypotheses $M$ that agent $0$ has. To state our results, suppose that agent $0$ adopts policy $x_0$ and has loss decay rates given by the function $c_0$. We define the \emph{loss augmented} log moment generating function for $i,j\in[0,M-1]$ as
\begin{align}\label{eqn:oLambda}
{\overline{\Lambda}}_{ij}(s,x_0,c_0) = \Lambda_{ij}(s,x_0) - (1-s)c_0(i) - sc_0(j).
\end{align}
For the case where agent $0$ adopts the 0-1 loss function, we have $c_0(i)=0$ for all $i$. We observe that in this case, the Chernoff information $-\min_{s\in[0,1]}\Lambda_{ij}(s,x_0)$ tells us how well hypothesis $i$ can be differentiated from hypothesis $j$, where a larger Chernoff information corresponds to a faster error probability decay \cite{DemZei:98}. This leads us to the following general definition: For a given policy $x_0$, if there exists a constant $\alpha$ so that $\min_{s\in[0,1]}{\overline{\Lambda}}_{ij}(s,x_0,c_0) = \alpha$ for all $i\ne j$, we say that the policy $x_0$ is \emph{hypothesis-loss neutral} for agent $0$.
Intuitively, a policy is hypothesis-loss neutral if it can differentiate any pair of hypotheses equally well when adjusted for the agent's ``biases''. It may be argued that in practical scenarios, an agent would strive to be hypothesis-loss neutral, since otherwise there are hypotheses that are relatively less important than others, and can be dropped. In the following, we show that if agent $0$ is hypothesis-loss neutral, an expert's opinion is useless if it does not have a decision space as large as $M$.
\begin{Proposition}\label{prop:expert_necessary}
Suppose that Assumptions \ref{assumpt:loss}, \ref{assumption:xi}, and \ref{assumpt:unambiguous} hold. Suppose that agent $0$ adopts a policy $x_0$ that is hypothesis-loss neutral. Then, for any agent $k\geq 1$, if $d_k < M$, we have $\mathcal{E}_0(k,x_0) = \mathcal{E}_{0,B}(0,x_0)$, i.e., agent $0$ ignores the opinion of agent $k$.
\end{Proposition}
\begin{IEEEproof}
See Appendix \ref{appendix:proofs}.
\end{IEEEproof}
Proposition \ref{prop:expert_necessary} can be explained intuitively as follows in the case where agent 0 has 0-1 loss function. If an expert $k$ has $d_k < M$, then there exists two hypotheses $i$ and $j$ of agent $0$ that it does not discriminate between. The probability of making an error between these two hypotheses by the expert is therefore one, and the expert's opinion does not help agent $0$ to differentiate between hypotheses $i$ and $j$. The error probability of agent $0$ declaring $H=i$ when the true hypothesis is $H=j$ or vice versa, thus dominates when agent $0$ is hypothesis-loss neutral, and is the same as when agent $0$ ignores the expert's opinion.
We next consider the case where experts have the same decision space as agent $0$, and has zero loss if they decide on the true underlying hypothesis. We also make the following simplifying assumption.
\begin{Assumption}\label{assumpt:Bayesian2}
Every agent $k \geq1$ has $d_k = M$, with loss decay rates
\begin{align*}
c_k(m,d) = \left\{
\begin{array}{ll}
c_k(m) & \textrm{ if $d \ne m$,}\\
\infty & \textrm{ if $d = m$,},
\end{array}
\right.
\end{align*}
where $c_k(m) \geq 0$ for all $m\in[0,M-1]$.
\end{Assumption}
\begin{Proposition}\label{prop:bestexpertloss}
Suppose that Assumptions \ref{assumpt:loss}-\ref{assumpt:Bayesian2} hold. Suppose that agent $0$ adopts policy $x_0$, and follows the opinion of expert $k \geq 1$, who adopts the policy $x_k$. Then, the loss exponent of agent $0$ is
\begin{align}\label{sp_bestlossexp}
&\lim_{n_0\to\infty} \ofrac{n_0}\log \EE{C_0(H,D_0(k),n_0)} = - \tilde\mathcal{E}_0(k,x_0),
\end{align}
where \footnote{The notation $\min_{i,j: i\ne j}$ means minimization over all \emph{ordered} pairs $(i,j)$ with $i,j\in[0,M-1]$ and $i\ne j$.}
\begin{dmath*}
\tilde{\mathcal{E}}_{0}(k,x_0) = \min_{i,j: i\ne j}\max_{s\in [0,1]} \left\{ sq_k \Lambda^*_{ji}(c_k(i)-c_k(j),x_k) - {\overline{\Lambda}}_{ij}(s,x_0,c_0) \right\}.
\end{dmath*}
In addition, if for some $i\ne j$ such that $c_0(i)=c_0(j)$, and
\begin{align}\label{01condition}
\Lambda_{ij}(s,x_0) = \Lambda_{ji}(s,x_0),\quad \forall s\in [0,1],
\end{align}
then there is no loss in optimality for agent 0 to restrict to experts $k$ with canonical loss functions satisfying $c_k(i)=c_k(j)$.
\end{Proposition}
\begin{IEEEproof}
See Appendix \ref{appendix:proofs}.
\end{IEEEproof}
Surprisingly, Proposition \ref{prop:bestexpertloss} shows that it is not necessarily optimal for agent $0$ to choose an expert who utilizes the same canonical loss function as itself since agent $0$'s loss exponent depends only on the relative differences in agent $k$'s loss function at each hypothesis. We show a numerical example of this phenomenon in Section \ref{subsection:numerical} below. In particular, if agent $0$ adopts the 0-1 loss function, and \eqref{01condition} does not hold for some $i\ne j$, then an example can be constructed in which an expert with a loss function different from the 0-1 loss is optimal for agent $0$.
On the other hand, if agent $0$ is ``unbiased'' (in terms of loss) towards a pair of hypotheses $(i,j)$ and \eqref{01condition} holds, Proposition \ref{prop:bestexpertloss} tells us that it is optimal for agent $0$ to choose an expert, if any, who is also ``unbiased'' towards hypotheses $(i,j)$. This indicates that if agent $0$ has the same discriminatory power for a particular pair of hypotheses (which is the intuitive meaning of \eqref{01condition}), then it values information from an ``unbiased'' expert more than one who has the same ``bias'' as itself. Assuming that independent news agencies are in general ``unbiased,'' our result suggests that such news agencies are unlikely to be replaced by social news reporting or social blogs. The following corollary follows immediately from Proposition \ref{prop:bestexpertloss}.
\begin{Corollary}\label{Cor:01loss}
Suppose that Assumptions \ref{assumpt:loss}-\ref{assumpt:Bayesian2} hold. Suppose that agent $0$ adopts policy $x_0$, and has 0-1 loss function. If for every distinct pair of hypotheses $(i,j)$, \eqref{01condition} holds, then it is optimal for agent $0$ to choose an expert, if any, who also has the 0-1 loss function.
\end{Corollary}
\subsection{Numerical Example}\label{subsection:numerical}
In this section, we present a numerical example to illustrate our results. Suppose that $M=3$, there are 3 experts to choose from, and all agents have access to private observations from two information sources $\Gamma = \{\gamma_1, \gamma_2\}$, whose conditional distributions are shown in Table \ref{table:information_sources}. We use the notation $\N{\mu}{\sigma^2}$ to denote the normal distribution with mean $\mu$ and variance $\sigma^2$.\footnote{Normal distributions have been used to model social opinions and influences in the economics literature, including \cite{You09, Viv96}.} For example, the three hypotheses can correspond to the financial health of a company being ``bad'', ``neutral'', and ``good'' respectively. The information source $\gamma_1$ represents a view that tends to be optimistic when the true health of the company is ``neutral'', while $\gamma_2$ represents a view that tends to be pessimistic.
\begin{table}[!htb]
\centering
\caption{Conditional distributions of the information sources.}
\begin{tabular}{ccc}
\hline
Hypothesis $H$ & $\gamma_1$ & $\gamma_2$ \\
\hline
$0$ & $\N{-1}{\sigma^2}$ & $\N{-1}{\sigma^2}$ \\
\rowcolor{gray90} $1$ & $\N{\delta}{\sigma^2}$ & $\N{-\delta}{\sigma^2}$ \\
$2$ & $\N{1}{\sigma^2}$ & $\N{1}{\sigma^2}$ \\
\hline
\end{tabular}
\label{table:information_sources}
\end{table}
In this example, we take $\sigma=2$, $q_k=1$ for all $k\geq 1$, and assume Assumption \ref{assumpt:Bayesian2} for easier interpretation of our results. The asymptotic decision regions of each agent $k \geq 1$ can be shown, using the characterization in Appendix \ref{appendix:Ak_properties}, to be that in Figure \ref{Fig:Ak}. For each agent $k$, let $c_k = (c_k(0),c_k(1),c_k(2))$ be its vector of loss decay rates. Let $c_1=(0,0,0)$, $c_2=(0,0,0.2)$, and $c_3=(0,0.05,0)$. We use the alternating optimization procedure in Remark \ref{rem:hardness} item \eqref{it:lossonly} to find the optimal policies for each agent. An exhaustive search is done to determine the actual optimal policy for each expert, which is given in Table \ref{table:optimal policies}. Note that the experts' optimal policies are invariant of $\delta$.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.4\textwidth]{Akregions}
\caption{Asymptotic decision regions for agent $k$. The dotted lines denote the boundaries between the regions.}\label{Fig:Ak}
\end{figure}
\begin{table}[!htb]
\centering
\caption{Optimal policies for agents. The leftmost columns show the optimal policies for each expert. The three rightmost columns show the optimal policy and loss decay rate of an agent 0 with $c_0=c_3$, corresponding to each expert choice.}
\begin{tabular}{ccc|ccc}
\hline
Expert $k$ & $x_k^*[\gamma_1]$ & $x_k^*[\gamma_2]$ & $x_0^*[\gamma_1]$ & $x_0^*[\gamma_2]$ & $\mathcal{E}_0(k,x_0)$ \\
\hline
$1$ & 0.5 & 0.5 & 0.5 & 0.5 & 0.1099\\
\rowcolor{gray90} $2$ & 1 & 0 & 0.2117 & 0.7883 & 0.1158\\
$3$ & 0.5 & 0.5 & 0.5 & 0.5 & 0.1066\\
\hline
\end{tabular}
\label{table:optimal policies}
\end{table}
For experts 1 and 3, their optimal policies are both given by $x_1[\gamma_1]=x_1[\gamma_2]=0.5$, as is expected from the symmetry of the problem. For expert 2, its optimal policy is given by $x_2[\gamma_1]=1$ and $x_2[\gamma_2]=0$. This is because with a positive loss decay rate $c_2(2)=0.2$, expert 2 puts exponentially less loss on confusing hypothesis 2 for the others, therefore it chooses $\gamma_1$ as its sole information source. For the alternating optimization procedure, we started our initial guesses at $(x[\gamma_1] , x[\gamma_2]) = (0,1)$ and $(0.3,0.7)$.\footnote{Although a more natural initial guess is $(0.5,0.5)$, this was not selected in order to test the convergence for experts 1 and 3.} We repeated the procedure for different values of $\delta$. In all cases, we were able to find the correct optimal policies within 22 iterations (totaled over the two runs corresponding to the two initial guesses), as shown in Figure \ref{Fig:delta}. In some cases however, the procedure does not converge to the correct optimal policy if we restrict ourselves to the single initial guess $(x[\gamma_1] , x[\gamma_2]) = (0,1)$. This shows that the proposed procedure can get stuck at a suboptimal policy.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth]{iter_delta_1}
\caption{Total number of iterations required to find optimal policies for experts 1, 2, and 3.}\label{Fig:delta}
\end{figure}
We next suppose that $c_0 = c_3$, and use the procedure in Remark \ref{rem:hardness} item \eqref{it:probexp} to obtain the optimal policy for agent $0$ w.r.t.\ each expert $k=1,2,3$, when $\delta=0.9$. We found the optimal policies for agent $0$ within 4 iterations in each case, and that choosing expert 2 produces the best loss exponent for agent $0$, as shown in Table \ref{table:optimal policies}. In this case, it is not optimal for agent $0$ to choose expert 3, which has the same loss function as itself. Observe that even if agent $0$'s policy is fixed at $x_0=(0.5,0.5)$, it will still prefer expert 1 over expert 3. On the other hand, if $c_0=c_1$, we have $\mathcal{E}_0(1,x_0)=0.0884 > \mathcal{E}_0(3,x_0) = 0.0750 > \mathcal{E}_0(2,x_0) = 0.0566$, i.e., agent $0$'s optimal expert choice is expert 1. It can be shown that \eqref{01condition} holds for all pairs of hypotheses in this example, thus verifying Corollary \ref{Cor:01loss}.
\section{Conclusion}\label{sect:Conclusion}
We have studied a multihypothesis social learning problem in which an agent makes a decision with the help of a chosen expert. We have considered a general framework that allows the agent and experts to have different loss functions (biases), and different decision spaces. We have characterized the loss exponent of the agent in terms of the chosen expert's probability error exponent, which allows us to choose the asymptotically optimal expert as well as the agent's policy. We have shown that if the experts have the same decision space as the agent, then it is not necessarily optimal for the agent to choose an expert with the same loss function as itself. Moreover, if the agent is hypothesis-loss neutral, then it ignores any expert with a decision space smaller than the number of hypotheses.
The results in this paper are limited by our setup and assumptions, which however allows us to obtain analytical characterizations of the asymptotically optimal policies for the agent and experts, and an asymptotically optimal expert choice. In the regime of a finite number of private observations, these policies are in general sub-optimal. Finding the exact optimal policies in this regime is however analytically difficult, and the asymptotically optimal policies can be utilized when the numbers of private observations are large.
In this paper, we have assumed that the agent knows the experts' loss decay rates, which may not be valid in some practical scenarios. For example, expert opinions may very well depend on the mood of the expert at the time the opinion is publicized. Therefore, it is of interest to consider minimax loss exponents in which the expert's loss function or observation probability distributions are drawn from uncertainty classes. Minimax decentralized hypothesis testing has been studied in \cite{VeeBasPoo:94}, and in our recent work \cite{HoTayQuek:C14} in which we consider robust social learning in a tandem network. However, additional research in minimax decentralized hypothesis testing is required to address more complex network architectures and applications like that considered in this paper.
\appendices
\section{Mathematical Preliminaries}\label{appendix:Mathematical}
In this appendix, we briefly review some basic definitions and a result from large deviations theory. The reader is referred to \cite{DemZei:98,Str:93} for details. The notation $\trace{A}$ denotes the trace of the matrix $A$, $\nabla_t f(t)$ is the gradient of $f(t)$ w.r.t.\ the vector $t$, and $\nabla_t^2 f(t)$ is the Hessian w.r.t.\ $t$ of the function $f(t)$.
Let $\mathcal{X}$ be a Polish space. The function $\Phi : \mathcal{X} \mapsto [0,\infty]$ is said to be a good rate function if $\Phi$ is lower semicontinuous and has compact level sets $\{x \in\mathcal{X}: \Phi(x) \leq c\}$ for all $c \geq 0$. Let $(\P_n)_{n\geq1}$ be a sequence of probability measures on $\mathcal{X}$. This sequence of probability measures is said to satisfy a \emph{large deviation principle} (LDP) if for some good rate function $\Phi$, and for all closed sets $C \subset \mathcal{X}$, we have
\begin{align*}
\limsup_{n\to\infty} \ofrac{n}\log \P_n(C) \leq -\inf_{x\in C} \Phi(x),
\end{align*}
and for all open sets $O \subset\mathcal{X}$, we have
\begin{align*}
\liminf_{n\to\infty} \ofrac{n}\log \P_n(O) \geq -\inf_{x\in O} \Phi(x).
\end{align*}
A sequence of random variables $(Z_n)_{n\geq1}$ is said to satisfy a LDP if the sequence of marginal distributions $\P_n(\cdot) = \P(Z_n \in \cdot)$ satisfies a LDP. The celebrated G\"artner-Ellis Theorem \cite{DemZei:98} provides sufficient conditions for a sequence of random variables to satisfy a LDP. In the following, we present a partial version of the G\"artner-Ellis Theorem
\begin{Theorem_A}
Let $(Z_n)_{n\geq 1}$ be a sequence of $\mathbb{R}^M$-valued random variables, so that
\begin{align*}
\varphi(t) = \lim_{n\to\infty}\ofrac{n}\log \EE{\exp(\ip{t}{Z_n})}
\end{align*}
exists. Let
\begin{align*}
\Phi^*(z) = \sup_t \left\{ \ip{t}{z} - \varphi(t) \right\}
\end{align*}
be the Fenchel-Legendre transform of $\varphi$. Then, under the assumptions in Theorem 2.3.6 of \cite{DemZei:98}, the LDP holds with good rate function $\Phi^*$.
\end{Theorem_A}
When not all the conditions required by the G\"artner-Ellis Theorem hold (as is the case in some of our proofs), we instead make use of a \emph{uniform} lower bound given by the following lemma, which is a generalization of Theorem 1.3.13 of \cite{Str:93}. The proof is omitted here for brevity.
\begin{Lemma_A}\label{lemma:LDP_lowerbound}
Let $Z_1,Z_2,\ldots,Z_n$ be independent $\mathbb{R}^M$-valued random variables, with $x[\gamma]$ fraction of them having distribution $\P^\gamma$, for each $\gamma \in \Gamma$. Let $Z^\gamma$ have distribution $\P^\gamma$,
\begin{align*}
\varphi(t) = \sum_{\gamma\in\Gamma} x[\gamma] \log \EE{\exp(\ip{t}{Z^\gamma})},
\end{align*}
and
\begin{align*}
\Phi^*(z) = \sup_t \left\{ \ip{t}{z} - \varphi(t) \right\}.
\end{align*}
Suppose that $\EE{\exp(\ip{t}{Z^\gamma})} < \infty$ for all $t\in\mathbb{R}^M$, then for $t_z$ such that $\nabla_t\varphi(t_z) = z$, and any $\epsilon > 0$, we have
\begin{dmath}[label={A:lowerbound}]
\ofrac{n}\log \P\left( \ofrac{n} \sum_{l=1}^n Z_l \in B_\epsilon(z) \right)
\geq - \Phi^*(z) - \norm{t_z}\epsilon + \ofrac{n}\log\left(1 \hiderel{-} \ofrac{n\epsilon^2} \trace{\nabla_t^2 \varphi(t_z)} \right).
\end{dmath}
\end{Lemma_A}
In this paper, we apply Lemma \ref{lemma:LDP_lowerbound} in cases where $\varphi$ corresponds to $\varphi_m$ in \eqref{logmomentgen}. We will often need to further lower bound \eqref{A:lowerbound} by finding an upper bound for $\nabla_t^2 \varphi_m(t_z)$ that does not depend on $n$ or the particular policy adopted by an agent. Assumption \ref{assumption:xi} implies an upper bound for $\nabla_t^2 \varphi_m$, as shown in the following elementary lemma.
\begin{Lemma_A}\label{lemma:bounded_trace}
Suppose that Assumption \ref{assumption:xi} holds. Then there exists a non-decreasing function $G(r)$, finite for each $r\geq0$, such that for all $t \in B_r(0)$, and all $\gamma\in\bigcup_k \Gamma_k$, we have for all $m\geq0$,
\begin{align}\label{trace_bound}
0\leq \trace{\nabla_t^2 \xi_m(\gamma,t)} \leq G(r).
\end{align}
\end{Lemma_A}
\begin{IEEEproof}
The lower inequality in \eqref{trace_bound} holds because $\xi_m(\gamma,t)$ is convex in $t$ for each $\gamma$ (see Lemma 2.2.31 of \cite{DemZei:98}). Furthermore, we can define
\begin{align*}
G(r) = \max_{\substack{t\in B_r(0) \\ \gamma\in\Gamma}} \trace{\nabla_t^2 \xi_m(\gamma,t)},
\end{align*}
which is finite because of Assumption \ref{assumption:xi}, and the fact that $\trace{\nabla_t^2 \xi_m(\gamma,t)}$ is continuous on the compact set $B_r(0)$. The proof of the lemma is now complete.
\end{IEEEproof}
\section{Proofs of Main Results}\label{appendix:proofs}
\subsection{Proof of Lemma \ref{lemma:f_uniform}}
Let $\epsilon$ be a positive number. From Assumption \ref{assumpt:loss}\eqref{it:c_k}, for $n$ sufficiently large, we have $(\log M)/n \leq \epsilon /2$, $\min_m (\log \pi_m)/n \geq -\epsilon/2$, and
\begin{align*}
\left| \ofrac{n} \log C_k(m,d,n) + c_k(m,d) \right| \leq \frac{\epsilon}{2},
\end{align*}
for all $m \in [0,M-1]$. This implies that for all $z \in \mathbb{R}^M$, we have
\begin{align*}
\tilde{g}_k(z,d,n) &\leq \max_{m} \{\ofrac{n} \log C_k(m,d,n) + z[m]\} + \ofrac{n} \log M \\
&\leq \max_{m} \{-c_k(m,d) + z[m]\} + \epsilon \\
& = \tilde{f}_k(z,d) + \epsilon,
\end{align*}
and
\begin{align*}
\hspace{-2pt}\tilde{g}_k(z,d,n) &\geq \max_m \left\{\ofrac{n} \log C_k(m,d,n) + z[m] \right\} + \min_m \frac{\log \pi_m}{n} \\
&\geq \max_{m} \{-c_k(m,d) + z[m]\} - \epsilon\\
& = \tilde{f}_k(z,d) - \epsilon,
\end{align*}
which shows that $\tilde{g}_k(z, d, n) \to \tilde{f}_k(z, d)$ uniformly in $z$. Since the minimum of a finite set of uniformly convergent functions is also uniformly convergent, the lemma follows.
\subsection{Proof of Theorem \ref{theorem:expert}}
We prove claim \eqref{it:expert_opt} of Theorem \ref{theorem:expert} by first deriving a lower bound for the loss exponent of agent $k$, and showing that this bound is achievable. We have for any $\epsilon > 0$, and $n_k$ sufficiently large,
\begin{align}
&\ofrac{n_k} \log \EE{C_k(H, D_k, n_k)} \nonumber\\
&= \ofrac{n_k} \log \sum_{m=0}^{M-1} \sum_{d=0}^{d_k-1} \pi_m C_k(m,d,n_k) \P_m(D_k = d) \nonumber\\
& \geq \max_{\substack{0\leq m \leq M-1 \\ 0\leq d \leq d_k-1}} \left\{ \ofrac{n_k} \log C_k(m,d,n_k) + \ofrac{n_k} \log\P_m(D_k = d) \right\}\nonumber\\
&\qquad\qquad + \min_m\ofrac{n_k} \log \pi_m \nonumber\\
& \geq \max_{\substack{0\leq m \leq M-1 \\ 0\leq d \leq d_k-1}} \left\{ \ofrac{n_k} \log\P_m(D_k = d) -c_k(m,d)\right\} - \epsilon, \label{Ck_lowerbound}
\end{align}
which can be further lower bounded by lower bounds on the probability exponents. For each $z \in \mathbb{R}^{M-1}$, let $t_z$ be the solution to the equation
\begin{align*}
\nabla_t \varphi_i(t,x_{k,n_k}) = z,
\end{align*}
if the solution exists. For each $r > 0$, let $H_r = \{ z \in \mathbb{R}^{M-1} : t_z \textrm{ exists, } \norm{t_z} \leq r\}$, and $A_k(d,r,\epsilon) = \{z \in \mathbb{R}^{M-1}: f_k(z^0, d) < -\epsilon/2,\textrm{ where } z^0 = (0, z)\} \cap H_r$. From \eqref{probD} and \eqref{barZ}, we then have for any $u < -\epsilon$,
\begin{align}
&\ofrac{n_k} \log\P_m(D_k = d)\nonumber\\
& = \ofrac{n_k} \log\P_m(g_k(\bar{Z}_{n_k}^0(x_{k,n_k}), d, n_k) < 0) \label{eqn:Df}\\
& \geq \ofrac{n_k} \log\P_m(g_k(\bar{Z}_{n_k}^0(x_{k,n_k}), d, n_k) \in [u-\epsilon,u+\epsilon]) \nonumber\\
& \geq \ofrac{n_k} \log\P_m(f_k(\bar{Z}_{n_k}^0(x_{k,n_k}), d) \in [u-\frac{\epsilon}{2},u+\frac{\epsilon}{2}]) \nonumber
\end{align}
where the last inequality follows from Lemma \ref{lemma:f_uniform} for $n_k$ sufficiently large.\footnote{Note that Assumption \ref{assumpt:unambiguous} is required in \eqref{eqn:Df}, without which we need to replace $\P_m(g_k(\bar{Z}_{n_k}^0(x_{k,n_k}), d, n_k) < 0)$ with $\max\{\P_m(g_k(\bar{Z}_{n_k}^0(x_{k,n_k}), d, n_k) < 0), \P_m(g_k(\bar{Z}_{n_k}^0(x_{k,n_k}), d, n_k) = 0)\}$ if expert $k$ uses randomization to produce its final decision. The second term in the maximization unfortunately cannot be characterized using our existing approach.} Since $u < -\epsilon$ is arbitrary, we obtain for $n_k$ sufficiently large,
\begin{align}
&\ofrac{n_k} \log\P_m(D_k = d) \nonumber\\
& \geq \sup_{z : B_\epsilon(z) \subset A_k(d,r,\epsilon)} \ofrac{n_k} \log\P_m(\bar{Z}_{n_k}(x_{k,n_k}) \in B_{\epsilon}(z)),\label{lowerbound1}
\end{align}
where $B_\epsilon(z)$ is an open sphere of radius $\epsilon$ around $z$. From Lemma \ref{lemma:LDP_lowerbound}, we can further lower bound the right hand side of \eqref{lowerbound1} to obtain
\begin{align}
&\ofrac{n_k} \log\P_m(D_k = d) \nonumber\\
&\geq - \inf_{z : B_\epsilon(z) \subset A_k(d,r,\epsilon)} \bigg\{ \Phi_m^*(z, x_{k,n_k}) + \norm{t_z}\epsilon \nonumber\\
&\qquad - \ofrac{n_k}\log\left(1 - \frac{1}{n_k\epsilon^2} \trace{\nabla_t^2 \varphi_i(t_z,x_{k,n_k})}\right) \bigg\}\nonumber\\
&\geq - \inf_{z : B_\epsilon(z) \subset A_k(d,r,\epsilon)} \Phi_m^*(z, x_{k,n_k}) + r\epsilon \nonumber\\
&\qquad - \ofrac{n_k}\log\left(1 - \frac{1}{n_k\epsilon^2} G(r)\right),\label{lowerbound2}
\end{align}
where the last inequality follows from Lemma \ref{lemma:bounded_trace}. Combining \eqref{lowerbound2} with \eqref{Ck_lowerbound}, and letting $n_k\to\infty$, and then taking $r = 1/\sqrt{\epsilon}$ and $\epsilon \to 0$, we have
\begin{align}
\liminf_{n_k \to\infty} \ofrac{n_k} \log \EE{C_k(H, D_k, n_k)}
& \geq - \sup_{x \in \mathbb{S}(\Gamma_k)} I_k(x). \label{Ck_lowerbound2}
\end{align}
Since $\Phi_m^*(z,x)$ is continuous in $x$, $\inf_{z \in A_k(d)} \Phi_m^*(z, x)$ is upper semi-continuous in $x$, and $I_k(x)$ is upper semi-continuous in $x$. In addition, $\mathbb{S}(\Gamma_k)$ is compact, therefore the supremum over $x$ on the right hand side of \eqref{Ck_lowerbound2} is a maximization. Consider the policy $x_k^* = \arg\max_{x \in \mathbb{S}(\Gamma_k)} I_k(x)$. For each $n_k \geq 1$, let agent $k$ use the policy $x_{k,n_k}$ where $x_{k,n_k}[\gamma] = \floor{x_k^*[\gamma] n_k}/n_k$ for all $\gamma\in\Gamma_k$, and if $x_{k,n_k}[\gamma]$ do not sum to 1 over $\gamma\in\Gamma_k$, we simply choose the remaining private observations from an arbitrary distribution, and ignore them when making the decision for agent $k$. We have $x_{k,n_k} \to x_k^*$ as $n_k\to\infty$. We first show a simple lemma.
\begin{Lemma_A}\label{lemma:LDP_Z}
Suppose that Assumption \ref{assumption:xi} holds, agent $k$ adopts the policy $x_{k,n_k}=(x_{k,n_k}[\gamma])_{\gamma\in\Gamma_k}$ when it has access to $n_k$ private observations, and $x_{k,n_k} \to x_k$ as $n_k \to \infty$. Then, the sequence of random variables $(\bar{Z}_{n_k}(x_{k,n_k}))_{n_k \geq 1}$ defined in \eqref{barZ} satisfies a LDP under hypothesis $H=m$, for every $m\in [0,M-1]$, with good rate function $\Phi_m^*(\cdot,x_k)$.
\end{Lemma_A}
\begin{IEEEproof}
We apply the G\"artner-Ellis Theorem \cite{DemZei:98} to prove the lemma. Let $Z_i = (\log \ell_{m0}(Y_k[i]))_{m=1}^{M-1}$. We have $n_k \bar{Z}_{n_k}(x_{k,n_k}) = \sum_{i=1}^{n_k} Z_i$ since $Z_i, i=1\ldots,n_k$ are independent. For every $t\in\mathbb{R}^{M-1}$, we obtain
\begin{align*}
&\ofrac{n_k} \log \Ec{m}{\exp\left(\ip{n_k t}{\bar{Z}_{n_k}(x_{k,n_k})}\right)} \\
&=\ofrac{n_k} \log \Ec{m}{\exp\left(\Big\langle t, \sum_{i=1}^{n_k} Z_i \Big\rangle \right)} \\
&= \ofrac{n_k} \sum_{i=1}^{n_k} \log \Ec{m}{\exp\left(\ip{t}{Z_i}\right)} \\
&= \sum_{\gamma\in\Gamma_k} x_{k,n_k}[\gamma] \log \Ec{m}{\exp\left(\ip{t}{(\log\ell_{m0}^\gamma)_{m=1}^{M-1}}\right)} \\
& \to \varphi_m(t,x_{k}),
\end{align*}
as $n_k \to \infty$. From Assumption \ref{assumption:xi} and Lemma 2.3.9 of \cite{DemZei:98}, we have $\Phi_m^*(z,x_k)$ is a good rate function, and the lemma follows from the G\"artner-Ellis Theorem.
\end{IEEEproof}
From Lemma \ref{lemma:f_uniform}, we have for each $d \in [0,d_k-1]$,
\begin{align*}
\limsup_{n_k\to\infty} \sup_z |g_k(z,d,n_k) - f_k(z,d)| =0,
\end{align*}
and applying Theorem 4.2.23 of \cite{DemZei:98}, we obtain from \eqref{eqn:Df} and Lemma \ref{lemma:LDP_Z} that
\begin{align}\label{Dk_upperbound}
\limsup_{n_k\to\infty} \ofrac{n_k} \log \P_m(D_k = d) \leq - \inf_{z \in A_k(d)} \Phi_m^*(z,x_k^*).
\end{align}
We then have
\begin{align}
&\limsup_{n_k\to\infty} \ofrac{n_k} \log \EE{C_k(H, D_k, m_l)} \nonumber\\
&\leq \limsup_{n_k\to\infty} \max_{\substack{0\leq m \leq M-1 \\ 0\leq d \leq d_k-1}} \left\{\ofrac{n_k} \log\P_m(D_k = d) -c_k(m,d) \right\} \nonumber\\
&\leq -I_k(x_k^*), \label{Ck_upperbound}
\end{align}
where the last inequality follows from \eqref{Dk_upperbound}. Finally, \eqref{Ck_lowerbound2} together with \eqref{Ck_upperbound} gives us claim \eqref{it:expert_opt}.
To show claim \eqref{it:policy_opt}, fix any $x_k^* \in \arg\max_{x \in \mathbb{S}(\Gamma_k)} I_k(x)$. We note that if there exists a subsequence of policies $(x_{k,m_l})_{l\geq 1}$ with $\lim_{l\to\infty}x_{k,m_l} = x_k$ and $I_k(x_k) < I_k(x_k^*)$, then using the same arguments that lead to \eqref{Ck_lowerbound2}, we have
\begin{align*}
\liminf_{n_k \to\infty} \ofrac{n_k} \log \EE{C_k(H, D_k, n_k)}
& \geq - I_k(x_k) > - I_k(x_k^*),
\end{align*}
a contradiction to \eqref{Ck_upperbound}. Therefore, each policy subsequence converges to some $x_k$ with $I_k(x_k) = I_k(x_k^*)$, and there is no loss in optimality if we restrict the sequence of policies to converge to $x_k^*$.
Finally, to show claim \eqref{it:prob_exp}, we have from \eqref{lowerbound2} that
\begin{align}\label{Dk_lowerbound}
\liminf_{l\to\infty}\ofrac{n_k} \log\P_m(D_k = d) &\geq - \inf_{z \in A_k(d)} \Phi_m^*(z,x_k^*),
\end{align}
since $\Phi_m^*(z,x)$ is continuous in $x$. Together with \eqref{Dk_upperbound}, the claim now follows, and the theorem is proved.
\subsection{Proof of Lemma \ref{lemma:Phi_properties}}\label{appendix:lemma:Phi_properties}
The non-negativity and convexity of $\Phi_m(z,x)$ follows from Lemma 2.2.31 of \cite{DemZei:98}. From Jensen's inequality, for any $t\in\mathbb{R}^{M-1}$, we have
\begin{align*}
\varphi_m(t,x)
&\geq \sum_{\gamma} x[\gamma] \Ec{m}{\ip{t}{Z^\gamma}} = \ip{t}{\tilde{z}_m(x)},
\end{align*}
which implies that $\Phi_m(\tilde{z},x) = 0$, and the lemma is proved.
\subsection{Proof of Theorem \ref{theorem:agent0}}
We first present a generalization of Theorem 5 of \cite{ShaGalBer:67a} (see also \cite{Tay:J12} for a slightly more updated version). The proof steps are similar to that in \cite{ShaGalBer:67a} and \cite{Tay:J12}, and are provided below for completeness.
\begin{Proposition_A}\label{prop:D0_lowerbound}
Suppose that Assumptions \ref{assumpt:loss} and \ref{assumption:xi} hold, and agent $0$ adopts the opinion of agent $k \geq 1$ and policy $x_0$. Let
\begin{align*}
s_{ij}^*\hspace{-4pt} = \arg\hspace{-4pt} \max_{s\in [0,1]} \left\{\frac{s}{n_0} \log \frac{C_0(i,j,n_0)\P_i(D_k=d)}{C_0(j,i,n_0)\P_j(D_k=d)} - \Lambda_{ij}(s,x_0)\right\}
\end{align*}
where $\Lambda_{ij}(\cdot,\cdot)$ is as defined in \eqref{def:Lambdaij}. Then, for any $\epsilon > 0$, and any $d \in [0,d_k-1]$, there exists $n$ such that for all $n_0 \geq n$, we have for all $i\ne j$,
\begin{align}
& \ofrac{n_0}\log \left\{
\begin{array}{cc}
\min_{j':j'\ne i} C_0(i,j',n_0)\P_i(D_0(k) \ne i, D_k = d) \\
+ C_0(j,i,n_0)\P_j(D_0(k)=i, D_k=d)
\end{array}
\right\}\nonumber\\
& \geq (1-s_{ij}^*)\left(\ofrac{n_0}\log\P_i(D_k=d) - c_0(i)\right) \nonumber\\
& \qquad + s_{ij}^*\left(\ofrac{n_0}\log\P_j(D_k=d) - c_0(j) \right)
+ \Lambda_{ij}(s_{ij}^*, x_0) - \epsilon.\label{D0_lowerbound}
\end{align}
\end{Proposition_A}
\begin{IEEEproof}
From Theorem 5 of \cite{ShaGalBer:67a} (or Proposition A.2 of \cite{Tay:J12}), we have for $i,j \in [0,M-1]$, with $j \neq i$, and every $s \in [0,1]$, either
\begin{dmath}[compact,label={D0_lowerbound1}]
\P_i(D_0(k) \hiderel{\ne} i \hiderel{\mid} D_k \hiderel{=} d)
\geq \ofrac{4}\exp\left(n_0\Lambda_{ij}(s,x_0) \hiderel{-} s n_0\ddfrac{}{s}\Lambda_{ij}(s,x_0)
- s\sqrt{2n_0 \ddfrac{^2}{s^2}\Lambda_{ij}(s,x_0)}\ \right),
\end{dmath}
or
\begin{dmath}[compact,label={D0_lowerbound2}]
\P_j(D_0(k) \hiderel{=} i \hiderel{\mid} D_k \hiderel{=} d)
\geq \ofrac{4}\exp\left(n_0\Lambda_{ij}(s,x_0) \hiderel{+} (1 \hiderel{-} s)n_0\ddfrac{}{s}\Lambda_{ij}(s,x_0)
- (1-s)\sqrt{2n_0\ddfrac{^2}{s^2}\Lambda_{ij}(s,x_0)}\ \right).
\end{dmath}
If $s_{ij}^* \in (0,1)$, we have
\begin{align*}
\ddfrac{}{s}\Lambda_{ij}(s_{ij}^*,x_0) = \frac{1}{n_0} \log \frac{C_0(i,j,n_0)\P_i(D_k=d)}{C_0(j,i,n_0)\P_j(D_k=d)},
\end{align*}
and using Lemma \ref{lemma:bounded_trace}, \eqref{D0_lowerbound1} and \eqref{D0_lowerbound2}, we obtain
\begin{align*}
& \ofrac{n_0}\log \left\{
\begin{array}{cc}
\min_{j': j'\ne i} C_0(i,j',n_0)\P_i(D_0(k) \ne i, D_k = d) \\
+ C_0(j,i,n_0)\P_j(D_0(k)=i, D_k=d)
\end{array}
\right\}\\
& \geq \frac{1-s_{ij}^*}{n_0}\log (\min_{j': j'\ne i} C_0(i,j',n_0)\P_i(D_k=d))\\
& \quad\quad + \frac{s_{ij}^*}{n_0}\log (C_0(j,i,n_0)\P_j(D_k=d)) \\
& \quad\quad + \Lambda_{ij}(s_{ij}^*, x_0) - \ofrac{n_0}\log 2 - \sqrt{\frac{2G(2)}{n_0}}\\
& \geq (1-s_{ij}^*)\left(\ofrac{n_0}\log\P_i(D_k=d) - c_0(i)\right) \\
&\qquad + s_{ij}^*\left(\ofrac{n_0}\log\P_j(D_k=d) - c_0(j)\right) + \Lambda_{ij}(s_{ij}^*, x_0) - \epsilon,
\end{align*}
where the last inequality follows from Assumption \ref{assumpt:loss} for $n_0$ sufficiently large. On the other hand, if $s_{ij}^* =0$, we have
\begin{align*}
\ddfrac{}{s}\Lambda_{ij}(0,x_0)\geq \frac{1}{n_0} \log \frac{C_0(i,j,n_0)\P_i(D_k=d)}{C_0(j,i,n_0)\P_j(D_k=d)},
\end{align*}
since $\Lambda_{ij}(s,x_0)$ is convex $s$. The inequality \eqref{D0_lowerbound} then holds trivially. A similar argument holds for $s_{ij}^*=1$, and the proposition is proved.
\end{IEEEproof}
We next proceed to prove Theorem \ref{theorem:agent0}. Suppose that agent $0$ adopts the policy $x_0$. For any $\epsilon > 0$, and for $n_0$ sufficiently large, we have
\begin{align}
&\ofrac{n_0}\log \EE{C_0(H,D_0(k),n_0)} \nonumber\\
& = \ofrac{n_0}\log \sum_{d=0}^{d_k-1} \sum_{i\ne j} \pi_i C_0(i,j,n_0)\P_i(D_0(k)=j, D_k=d) \nonumber\\
& \geq\hspace{-8pt} \max_{\substack{i\ne j \\ 0\leq d \leq d_k-1}} \hspace{-8pt}\ofrac{n_0}\log\left\{
\begin{array}{cc}
\min_{j': j'\ne i} C_0(i,j',n_0) \\
\qquad \cdot \P_i(D_0(k) \ne i, D_k = d)\\
+ C_0(j,i,n_0)\P_j(D_0(k)=i, D_k=d)
\end{array}
\right\} \nonumber\\
&\qquad - \ofrac{n_0}\log 2 + \min_m \ofrac{n_0} \log \pi_m \nonumber\\
& \geq \max_{\substack{i\ne j \\ 0\leq d \leq d_k-1}} \min_{s\in [0,1]} \Big\{ (1-s)\left(\ofrac{n_0}\log\P_i(D_k=d) - c_0(i)\right) \nonumber\\
& + s\left(\ofrac{n_0}\log\P_j(D_k=d)- c_0(j)\right)
+ \Lambda_{ij}(s, x_0) \Big\} - \epsilon,\label{C0_lowerbound1}%
\end{align}%
where the last inequality follows from Proposition \ref{prop:D0_lowerbound}. By letting $n_0\to\infty$ and $\epsilon\to0$ in \eqref{C0_lowerbound1}, we obtain
\begin{align}
&\limsup_{n_0\to\infty} \ofrac{n_0}\log \EE{C_0(H,D_0(k),n_0)} \nonumber\\
& \geq \hspace{-10pt}\max_{\substack{i\ne j \\ 0\leq d \leq d_k-1}}\hspace{-10pt} \min_{s\in [0,1]} \Big\{ (1-s)\left(\lim_{n_0\to\infty}\ofrac{n_0}\log\P_i(D_k=d) - c_0(i)\right) \nonumber\\
& \quad + s\left(\lim_{n_0\to\infty}\ofrac{n_0}\log\P_j(D_k=d)- c_0(j)\right)
+ \Lambda_{ij}(s, x_0) \Big\}, \label{C0_lowerbound1_limit}
\end{align}
and the lower bound \eqref{C0_lowerbound} follows from Theorem \ref{theorem:expert}\eqref{it:prob_exp}.
We next show that there exists a decision rule for agent $0$ that achieves $\mathcal{E}_0(x_0)$ in \eqref{C0_lowerbound}. Given $D_k=d$, consider the following rule to differentiate between hypotheses $H=i$ and $H=j$ for $i\ne j$: declare $H=i$ iff $\ofrac{n_0}\log \ell_{ji}(Y_0[1:n_0]) \leq h_{ji}$,
where $h_{ji} = -q_k\inf_{z \in A_k(d)}\Phi_i^*(z, x_k^*) + q_k\inf_{z \in A_k(d)}\Phi_j^*(z, x_k^*) - c_0(i) + c_0(j)$.
By a simple generalization of Cram\'er's Theorem\footnote{Cram\'er's Theorem applies to independent and identically distributed (i.i.d.) observations. The private observations $Y_0[1:n_0]$ are not i.i.d., but are independent and can be divided into groups of i.i.d.\ observations.} \cite{DemZei:98}, and Theorem \ref{theorem:expert}\eqref{it:prob_exp}, we have for every $\epsilon >0$ and all $n_0$ sufficiently large,
\begin{dmath*}
\ofrac{n_0}\log (C_0(i,j,n_0)\P_i(D_0(k)\hiderel{=}j, D_k\hiderel{=}d))
= \ofrac{n_0}\log C_0(i,j,n_0)+ \ofrac{n_0}\log \P_i(D_k\hiderel{=}d)
+ \ofrac{n_0}\log \P_i(D_0(k)\hiderel{=}j\hiderel{\mid} D_k\hiderel{=}d)
\leq -c_0(i) - q_k\inf_{z \in A_k(d)}\Phi_i^*(z, x_k^*)
- \max_{s\in[0,1]} \left\{ sh_{ji} - \Lambda_{ij}(s,x_0)\right\} + \epsilon
\leq -\max_{s\in [0,1]} \bigg\{ (1-s)\left(q_k\inf_{z \in A_k(d)} \Phi_i^*(z, x_k^*) + c_0(i)\right)
+ s\left(q_k\inf_{z \in A_k(d)} \Phi_j^*(z, x_k^*)+c_0(j)\right) - \Lambda_{ij}(s,x_0) \bigg\} + \epsilon,
\end{dmath*}
from which we obtain
\begin{dmath*}
\ofrac{n_0}\log \EE{C_0(H,D_0(k),n_0)}
\leq \max_{\substack{i\ne j \\ 0\leq d \leq d_k-1}} \ofrac{n_0}\log\left(C_0(i,j,n_0)\P_i(D_0(k) \hiderel{=}j, D_k \hiderel{=} d)\right) + \ofrac{n_0} \log M
\leq -\mathcal{E}_0(k,x_0) + \epsilon.
\end{dmath*}
By taking $n_0\to\infty$ and $\epsilon\to 0$, we obtain the theorem by maximizing $\mathcal{E}_0(x_0)$ over all policies $x_0$. The proof is now complete.
\subsection{Proof of Theorem \ref{theorem:expert_choice}}
The first part of the theorem is a direct consequence of Theorem \ref{theorem:agent0}. From Lemma \ref{lemma:Phi_properties}, we have $\inf_{z \in A_k(d)} \Phi_i^*(z, x_k^*) \geq 0$ for any $i\in[0,M-1]$, $k\geq 1$, $d\in [0,d_k-1]$, and policy $x_k^*$. Furthermore, Assumption \ref{assumpt:loss} implies that $c_0(i) \geq 0$ for all $i\in[0,M-1]$. Therefore, the inequalities \eqref{Eineq1} and \eqref{Eineq2} follow from \eqref{C0_lowerbound}, \eqref{E0}, \eqref{EB}, and \eqref{EkB}, and the proof is complete.
\subsection{Proof of Proposition \ref{prop:expert_necessary}}
Suppose that agent $k$ adopts the policy $x_k$. From the pigeonhole principle, if $d_k < M$, there exists a region $A_k(d)$ in which both $\Phi_i(\cdot,x_k)$ and $\Phi_j(\cdot,x_k)$ achieve their minimum value of 0, for some $i\ne j$. Since for any $(i',j')$, we have
\begin{dmath*}
\max_{s\in [0,1]} \left\{ (1-s)\left(q_k\inf_{z \in A_k(d)} \Phi_{i'}^*(z, x_k^*) + c_0(i')\right)
+ s\left(q_k\inf_{z \in A_k(d)} \Phi_{j'}^*(z, x_k^*) \hiderel{+} c_0(j')\right) \hiderel{-} \Lambda_{i'j'}(s,x_0) \right\}
\geq \max_{s\in [0,1]} (-{\overline{\Lambda}}_{i'j'}(s,x_0,c_0))
= -\min_{s\in [0,1]} {\overline{\Lambda}}_{ij}(s,x_0,c_0),
\end{dmath*}
we obtain from \eqref{C0_lowerbound} and \eqref{E0},
\begin{align*}
&\mathcal{E}_0(k,x_0) = -\min_{s\in [0,1]} {\overline{\Lambda}}_{ij}(s,x_0,c_0) = \mathcal{E}_{0,B}(k,x_0),
\end{align*}
and the proposition is proved.
\subsection{Proof of Proposition \ref{prop:bestexpertloss}}
Since the proof is similar to that of Theorem \ref{theorem:agent0}, we provide only an outline here. From the proposition assumptions, we have for every $j \in [0,M-1]$, $\P_j(D_k=j)$ is bounded away from zero, i.e., $\lim_{n_k\to\infty}(1/n_k)\log \P_j(D_k=j) = 0$ because otherwise the expected loss of agent $k$ can be decreased. Therefore, we have $\inf_{z\in A_k(j)} \Phi_j^*(z,x_k) = 0$. By comparing \eqref{E0} (with $x_0$ replaced by $x_k$) and \eqref{Ik}, and using an inductive argument, we have $\inf_{z\in A_k(i)} \Phi_j^*(z,x_k) = \Lambda_{ji}^*(c_k(i)-c_k(j), x_k)$, and \eqref{sp_bestlossexp} follows.
We next show the second part of the proposition. Suppose that for some $i\ne j$, we have $c_0(i)=c_0(j)$ and \eqref{01condition} holds. Then from \eqref{eqn:oLambda}, we have ${\overline{\Lambda}}_{ij}(s,x_0,c_0)={\overline{\Lambda}}_{ji}(s,x_0,c_0)$ for all $s\in[0,1]$. Let $g_{ij}(\Delta) = \max_{s\in [0,1]} \{ sq_k \Lambda^*_{ji}(\Delta,x_k) - {\overline{\Lambda}}_{ij}(s,x_0,c_0) \}$. Note that $g_{ij}(\Delta)$ is non-decreasing in $\Delta$. Therefore, since ${\overline{\Lambda}}_{ij}(s,x_0,c_0)$ is symmetrical on $s\in[0,1]$, we have $\min(g_{ij}(\Delta), g_{ji}(-\Delta))$ is maximized if $\Lambda^*_{ji}(\Delta,x_k)=\Lambda^*_{ij}(-\Delta,x_k)$, which holds if $\Delta=0$. Take $\Delta = c_k(i)-c_k(j)$, and the proposition follows.
\section{Characterization of Asymptotic Decision Regions}\label{appendix:Ak_properties}
In this appendix, we give a characterization for the asymptotic decision region $A_k(d)$ for an agent $k$, and $d \in [0,d_k-1]$. For $i, j \in [0,M-1]$ and $p, q\in [0,d_k]$, define the halfspace
\begin{align}\label{half-space}
B_k(i,p,j,q) = \big\{ & z=(z[m])_{1\leq m\leq M-1}\in \mathbb{R}^{M-1}: \nonumber\\
& z[i] - z[j] \geq c_k(i,p) - c_k(j,q)\big\},
\end{align}
where $z[0] = 0$. For each $p \in [0,d_k-1]$, let $m_p \in [0,M-1]$ be a chosen corresponding index. From \eqref{tfk}, we have $z \in \cap_{i \ne m} B_k(m_p,p,i,p)$ iff $\tilde{f}_k(z,p) = z[m_p]-c_k(m_p,p)$.
Consider a $z\in\mathbb{R}^{M-1}$ such that $f(z,d) < 0$. Then, there exists a sequence $(m_p)_{0\leq p \leq d_k-1} \in [0,M-1]^{d_k}$ such that $\tilde{f}_k(z,p) = z[m_p]-c_k(m_p,p)$ for all $p\in[0,d_k-1]$ and $\tilde{f}_k(z,p) - \tilde{f}_k(z,d) = z[m_p] - z[m_d] - c_k(m_p,p) + c_k(m_d,d) > 0$ for all $p \ne d$, i.e.,
\begin{align*}
z \in & H_d((m_p)_{p=0}^{d_k-1}) \\
& \triangleq \bigcap_{p=0}^{d_k-1}\bigcap_{i \ne m_p} B_k(m_p,p,i,p)\bigcap_{p \ne d} B_k(m_p,p,m_{d}, d),
\end{align*}
where $H_d((m_p)_{p=0}^{d_k-1})$ is a polyhedron since it consists of intersections of halfspaces. On the other hand, if such a sequence $(m_p)_{p=0}^{d_k-1}$ exists, then $z \in A_k(d)$. Therefore, the set $A_k(d)$ is the union over all sequences $(m_p)_{p=0}^{d_k-1} \in [0,M-1]^{d_k}$ of the polyhedra $H_d((m_p)_{p=0}^{d_k-1})$.
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: can a derived class corrupt base class's implementation I am kind of new to c# and was studying about sealed class, when i came across
this
'A sealed class is mostly used for security reasons by preventing
unintended derivation by which the derived class may corrupt the implementation provided in the base class'
Is this really possible? can a derived class really corrupt base class's implementation? If so can someone please explain with an example.
A: Say you need some gate keepers:
public interface IGateKeeper
{
/// <summary>
/// Check if the given id is allowed to enter.
/// </summary>
/// <param name="id">id to check.</param>
/// <param name="age">age to check</param>
/// <returns>A value indicating whether the id is allowed to enter.</returns>
bool CanEnter(string id, int age);
... other deep needs ...
}
You may have a solid implementation of it to test the majority at the entrance of your bar:
public class MajorityGateKeeper : IGateKeeper
{
public virtual bool CanEnter(string id, int age)
{
return age >= 18;
}
... other deep implementation ...
}
And also have an implementation for the VIP room:
public class VipGateKeeper : MajorityGateKeeper
{
public override bool CanEnter(string id, int age)
{
// Do the majotity test and check if the id is VIP.
return base.CanEnter(id, age) && (id == "Chuck Norris");
}
}
And break it in a second:
public class DrunkGateKeeper : VipGateKeeper
{
public override bool CanEnter(string id, int age)
{
return true;
}
}
The DrunkGateKeeper is a VipGateKeeper so you can hide it's drunk (cast to VipGateKeeper). But it do a terrible job.
var gk = (VipGateKeeper) new DrunkGateKeeper();
var canEnter = gk.CanEnter("Miley Cyrus", 16); // true (sic)
If you make the VipGateKeeper sealed you are sure that it can't be drunk: an object of type VipGateKeeper is a VipGateKeeper nothing more.
A: You might corrupt base class implementations because of polymorphism.
If class A has a virtual method which can be overridable (i.e. polymorphic) and a class B overrides the whole method, and B doesn't output the same stuff as A's implementation, then, it seems like B has changed the actual behavior of A's implementation.
sealed in class level level (f.e. override sealed) is meant to avoid both inheritance and polymorphism on the entire class.
You might also prevent polymorphism (inheritance is still possible) on certain members overriding them in a derived class and marking them with sealed modifier:
public class A
{
public virtual void DoStuff() {}
}
public class B
{
public override sealed void DoStuff() {}
}
A: consider you defined and implemented a method in a class, in your program you need only this implementation. for example you return a message to user, whenever you call this method you expect that particular message be returned, consider you have a child class which overrides that method and shows another message, when you call that method you have another result which is not expected, if you mark a class as sealed, you prevent accidental inheritance.
sealed is obverse side of design coin from abstract. when you define an abstract class, you allow classes to derive from it and implement their own implementations for methods but sealed class does not allow others to derive it at all.
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{"url":"https:\/\/www.r-bloggers.com\/a-brief-primer-on-variational-inference\/","text":"A brief primer on Variational Inference\n\nOctober 30, 2019\nBy\n\n[This article was first published on Fabian Dablander, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)\nWant to share your content on R-bloggers? click here if you have a blog, or here if you don't.\n\nBayesian inference using Markov chain Monte Carlo methods can be notoriously slow. In this blog post, we reframe Bayesian inference as an optimization problem using variational inference, markedly speeding up computation. We derive the variational objective function, implement coordinate ascent mean-field variational inference for a simple linear regression example in R, and compare our results to results obtained via variational and exact inference using Stan. Sounds like word salad? Then let\u2019s start unpacking!\n\nPreliminaries\n\nBayes\u2019 rule states that\n\nwhere $\\mathbf{z}$ denotes latent parameters we want to infer and $\\mathbf{x}$ denotes data.1 Bayes\u2019 rule is, in general, difficult to apply because it requires dealing with a potentially high-dimensional integral \u2014 the marginal likelihood. Optimization, which involves taking derivatives instead of integrating, is much easier and generally faster than the latter, and so our goal will be to reframe this integration problem as one of optimization.\n\nVariational objective\n\nWe want to get at the posterior distribution, but instead of sampling we simply try to find a density $q^\\star(\\mathbf{z})$ from a family of densities $\\mathrm{Q}$ that best approximates the posterior distribution:\n\nwhere $\\text{KL}(. \\lvert \\lvert.)$ denotes the Kullback-Leibler divergence:\n\nWe cannot compute this Kullback-Leibler divergence because it still depends on the nasty integral $p(\\mathbf{x}) = \\int p(\\mathbf{x} \\mid \\mathbf{z}) \\, p(\\mathbf{z}) \\, \\mathrm{d}\\mathbf{z}$. To see this dependency, observe that:\n\nwhere we have expanded the expectation to more clearly behold our nemesis. In doing so, we have seen that $\\text{log } p(\\mathbf{x})$ is actually a constant with respect to $q(\\mathbf{z})$; this means that we can ignore it in our optimization problem. Moreover, minimizing a quantity means maximizing its negative, and so we maximize the following quantity:\n\nWe can expand the joint probability to get more insight into this equation:\n\nThis is cool. It says that maximizing the ELBO finds an approximate distribution $q(\\mathbf{z})$ for latent quantities $\\mathbf{z}$ that allows the data to be predicted well, i.e., leads to a high expected log likelihood, but that a penalty is incurred if $q(\\mathbf{z})$ strays far away from the prior $p(\\mathbf{z})$. This mirrors the usual balance in Bayesian inference between likelihood and prior (Blei, Kucukelbier, & McAuliffe, 2017).\n\nELBO stands for evidence lower bound. The marginal likelihood is sometimes called evidence, and we see that ELBO is indeed a lower bound for the evidence:\n\nsince the Kullback-Leibler divergence is non-negative. Heuristically, one might then use the ELBO as a way to select between models. For more on predictive model selection, see this and this blog post.\n\nWhy variational?\n\nOur optimization problem is about finding $q^\\star(\\mathbf{z})$ that best approximates the posterior distribution. This is in contrast to more familiar optimization problems such as maximum likelihood estimation where one wants to find, for example, the single best value that maximizes the log likelihood. For such a problem, one can use standard calculus (see for example this blog post). In our setting, we do not want to find a single best value but rather a single best function. To do this, we can use variational calculus from which variational inference derives its name (Bishop, 2006, p. 462).\n\nA function takes an input value and returns an output value. We can define a functional which takes a whole function and returns an output value. The entropy of a probability distribution is a widely used functional:\n\nwhich takes as input the probability distribution $p(x)$ and returns a single value, its entropy. In variational inference, we want to find the function that minimizes the ELBO, which is a functional.\n\nIn order to make this optimization problem more manageable, we need to constrain the functions in some way. One could, for example, assume that $q(\\mathbf{z})$ is a Gaussian distribution with parameter vector $\\omega$. The ELBO then becomes a function of $\\omega$, and we employ standard optimization methods to solve this problem. Instead of restricting the parametric form of the variational distribution $q(\\mathbf{z})$, in the next section we use an independence assumption to manage the inference problem.\n\nMean-field variational family\n\nA frequently used approximation is to assume that the latent variables $z_j$ for $j = \\{1, \\ldots, m\\}$ are mutually independent, each governed by their own variational density:\n\nNote that this mean-field variational family cannot model correlations in the posterior distribution; by construction, the latent parameters are mutually independent. Observe that we do not make any parametric assumption about the individual $q_j(z_j)$. Instead, their parametric form is derived for every particular inference problem.\n\nWe start from our definition of the ELBO and apply the mean-field assumption:\n\nIn the following, we optimize the ELBO with respect to a single variational density $q_j(z_j)$ and assume that all others are fixed:\n\nOne could use variational calculus to derive the optimal variational density $q_j^\\star(z_j)$; instead, we follow Bishop (2006, p. 465) and define the distribution\n\nwhere we need to make sure that it integrates to one by subtracting the (log) normalizing constant $\\mathcal{Z}$. With this in mind, observe that:\n\nThus, maximizing the ELBO with respect to $q_j(z_j)$ is minimizing the Kullback-leibler divergence between $q_j(z_j)$ and $\\tilde{p}(\\mathbf{x}, z_j)$; it is zero when the two distributions are equal. Therefore, under the mean-field assumption, the optimal variational density $q_j^\\star(z_j)$ is given by:\n\nsee also Bishop (2006, p. 466). This is not an explicit solution, however, since each optimal variational density depends on all others. This calls for an iterative solution in which we first initialize all factors $q_j(z_i)$ and then cycle through them, updating them conditional on the updates of the other. Such a procedure is known as Coordinate Ascent Variational Inference (CAVI). Further, note that\n\nwhich allows us to write the updates in terms of the conditional posterior distribution of $z_j$ given all other factors $\\mathbf{z}_{-j}$. This looks a lot like Gibbs sampling, which we discussed in detail in a previous blog post. In the next section, we implement CAVI for a simple linear regression problem.\n\nApplication: Linear regression\n\nIn a previous blog post, we traced the history of least squares and applied it to the most basic problem: fitting a straight line to a number of points. Here, we study the same problem but swap optimization procedure: instead of least squares or maximum likelihood, we use variational inference. Our linear regression setup is:\n\nwhere we assume that the population mean of $y$ is zero (i.e., $\\beta_0 = 0$); and we assign the error variance $\\sigma^2$ an improper Jeffreys\u2019 prior and $\\beta$ a Gaussian prior with variance $\\sigma^2\\tau^2$. We scale the prior of $\\beta$ by the error variance to reason in terms of a standardized effect size $\\beta \/ \\sigma$ since with this specification:\n\nAs a heads up, we have to do a surprising amount of calculations to implement variational inference even for this simple problem. In the next section, we start our journey by deriving the variational density for $\\sigma^2$.\n\nVariational density for $\\sigma^2$\n\nOur optimal variational density $q^\\star(\\sigma^2)$ is given by:\n\nTo get started, we need to derive the conditional posterior distribution $p(\\sigma^2 \\mid \\mathbf{y}, \\beta)$. We write:\n\nwhich is proportional to an inverse Gamma distribution. Moving on, we exploit the linearity of the expectation and write:\n\nThis, too, looks like an inverse Gamma distribution! Plugging in the normalizing constant, we arrive at:\n\nNote that this quantity depends on $\\beta$. In the next section, we derive the variational density for $\\beta$.\n\nVariational density for $\\beta$\n\nOur optimal variational density $q^\\star(\\beta)$ is given by:\n\nand so we again have to derive the conditional posterior distribution $p(\\beta \\mid \\mathbf{y}, \\sigma^2)$. We write:\n\nwhere we have \u201ccompleted the square\u201d (see also this blog post) and realized that the conditional posterior is Gaussian. We continue by taking expectations:\n\nwhich is again proportional to a Gaussian distribution! Plugging in the normalizing constant yields:\n\nNote that while the variance of this distribution, $\\sigma^2_\\beta$, depends on $q(\\sigma^2)$, its mean $\\mu_\\beta$ does not.\n\nTo recap, instead of assuming a parametric form for the variational densities, we have derived the optimal densities under the mean-field assumption, that is, under the assumption that the parameters are independent: $q(\\beta, \\sigma^2) = q(\\beta) \\, q(\\sigma^2)$. Assigning $\\beta$ a Gaussian distribution and $\\sigma^2$ a Jeffreys\u2019s prior, we have found that the variational density for $\\sigma^2$ is an inverse Gamma distribution and that the variational density for $\\beta$ a Gaussian distribution. We noted that these variational densities depend on each other. However, this is not the end of the manipulation of symbols; both distributions still feature an expectation we need to remove. In the next section, we expand the remaining expectations.\n\nRemoving expectations\n\nNow that we know the parametric form of both variational densities, we can expand the terms that involve an expectation. In particular, for the variational density $q^\\star(\\sigma^2)$ we write:\n\nNoting that $\\mathbb{E}_{q(\\beta)}[\\beta] = \\mu_{\\beta}$ and using the fact that:\n\nthe expectation becomes:\n\nFor the expectation which features in the variational distribution for $\\beta$, things are slightly less elaborate, although the result also looks unwieldy. Note that since $\\sigma^2$ follows an inverse Gamma distribution, $1 \/ \\sigma^2$ follows a Gamma distribution which has mean:\n\nMonitoring convergence\n\nThe algorithm works by first specifying initial values for the parameters of the variational densities and then iteratively updating them until the ELBO does not change anymore. This requires us to compute the ELBO, which we still need to derive, on each update. We write:\n\nLet\u2019s take a deep breath and tackle the second term first:\n\nNote that there are three expectations left. However, we really deserve a break, and so instead of analytically deriving the expectations we compute $\\mathbb{E}_{q(\\sigma^2)}\\left[\\text{log } \\sigma^2\\right]$ and $\\mathbb{E}_{p(\\sigma^2)}\\left[\\text{log } q(\\sigma^2)\\right]$ numerically using Gaussian quadrature. This fails for $\\mathbb{E}_{q(\\sigma^2)}\\left[\\text{log } q(\\sigma^2)\\right]$, which we compute using Monte carlo integration:\n\nWe are left with the expected log likelihood. Instead of filling this blog post with more equations, we again resort to numerical methods. However, we refactor the expression so that numerical integration is more efficient:\n\nSince we have solved a similar problem already above, we evaluate the expecation with respect to $q(\\beta)$ analytically:\n\nIn the next section, we implement the algorithm for our linear regression problem in R.\n\nImplementation in R\n\nNow that we have derived the optimal densities, we know how they are parameterized. Therefore, the ELBO is a function of these variational parameters and the parameters of the priors, which in our case is just $\\tau^2$. We write a function that computes the ELBO:\n\nThe function below implements coordinate ascent mean-field variational inference for our simple linear regression problem. Recall that the variational parameters are:\n\nThe following function implements the iterative updating of these variational parameters until the ELBO has converged.\n\nLet\u2019s run this on a simulated data set of size $n = 100$ with a true coefficient of $\\beta = 0.30$ and a true error variance of $\\sigma^2 = 1$. We assign $\\beta$ a Gaussian prior with variance $\\tau^2 = 0.25$ so that values for $\\lvert \\beta \\rvert$ larger than two standard deviations ($0.50$) receive about $0.68$ prior probability.\n\nFrom the output, we see that the ELBO and the variational parameters have converged. In the next section, we compare these results to results obtained with Stan.\n\nComparison with Stan\n\nWhenever one goes down a rabbit hole of calculations, it is good to sanity check one\u2019s results. Here, we use Stan\u2019s variational inference scheme to check whether our results are comparable. It assumes a Gaussian variational density for each parameter after transforming them to the real line and automates inference in a \u201cblack-box\u201d way so that no problem-specific calculations are required (see Kucukelbir, Ranganath, Gelman, & Blei, 2015). Subsequently, we compare our results to the exact posteriors arrived by Markov chain Monte carlo. The simple linear regression model in Stan is:\n\nWe use Stan\u2019s black-box variational inference scheme:\n\nThis gives similar estimates as ours:\n\nTheir recommendation is prudent. If you run the code with different seeds, you can get quite different results. For example, the posterior mean of $\\beta$ can range from $0.12$ to $0.45$, and the posterior standard deviation can be as low as $0.03$; in all these settings, Stan indicates that the ELBO has converged, but it seems that it has converged to a different local optimum for each run. (For seed = 3, Stan gives completely nonsensical results). Stan warns that the algorithm is experimental and may be unstable, and it is probably wise to not use it in production.\n\nAlthough the posterior distribution for $\\beta$ and $\\sigma^2$ is available in closed-form (see the Post Scriptum), we check our results against exact inference using Markov chain Monte carlo by visual inspection.\n\nThe Figure below overlays our closed-form results to the histogram of posterior samples obtained using Stan.\n\nNote that the posterior variance of $\\beta$ is slightly overestimated when using our variational scheme. This is in contrast to the fact that variational inference generally underestimates variances. Note also that Bayesian inference using Markov chain Monte Carlo is very fast on this simple problem. However, the comparative advantage of variational inference becomes clear by increasing the sample size: for sample sizes as large as $n = 100000$, our variational inference scheme takes less then a tenth of a second!\n\nConclusion\n\nIn this blog post, we have seen how to turn an integration problem into an optimization problem using variational inference. Assuming that the variational densities are independent, we have derived the optimal variational densities for a simple linear regression problem with one predictor. While using variational inference for this problem is unnecessary since everything is available in closed-form, I have focused on such a simple problem so as to not confound this introduction to variational inference by the complexity of the model. Still, the derivations were quite lengthy. They were also entirely specific to our particular problem, and thus generic \u201cblack-box\u201d algorithms which avoid problem-specific calculations hold great promise.\n\nWe also implemented coordinate ascent mean-field variational inference (CAVI) in R and compared our results to results obtained via variational and exact inference using Stan. We have found that one probably should not trust Stan\u2019s variational inference implementation, and that our results closely correspond to the exact procedure. For more on variational inference, I recommend the excellent review article by Blei, Kucukelbir, and McAuliffe (2017).\n\nI would like to thank Don van den Bergh for helpful comments on this blog post.\n\nPost Scriptum\n\nNormal-inverse-gamma Distribution\n\nThe posterior distribution is a Normal-inverse-gamma distribution:\n\nwhere\n\nNote that the marginal posterior distribution for $\\beta$ is actually a Student-t distribution, contrary to what we assume in our variational inference scheme.\n\nReferences\n\n\u2022 Blei, D. M., Kucukelbir, A., & McAuliffe, J. D. (2017). Variational inference: A review for statisticians. Journal of the American Statistical Association, 112(518), 859-877.\n\u2022 Kucukelbir, A., Ranganath, R., Gelman, A., & Blei, D. (2015). Automatic variational inference in Stan. In Advances in Neural Information Processing Systems (pp. 568-576).\n\u2022 Kucukelbir, A., Tran, D., Ranganath, R., Gelman, A., & Blei, D. M. (2017). Automatic differentiation variational inference. The Journal of Machine Learning Research, 18(1), 430-474.\n\nFootnotes\n\n1. The first part of this blog post draws heavily on the excellent review article by Blei, Kucukelbier, and McAuliffe (2017), and so I use their (machine learning) notation.\n\nR-bloggers.com offers daily e-mail updates about R news and tutorials about learning R and many other topics. Click here if you're looking to post or find an R\/data-science job.\nWant to share your content on R-bloggers? click here if you have a blog, or here if you don't.\n\nIf you got this far, why not subscribe for updates from the site? 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Josh Turner Details Forthcoming Album, 'I Serve A Savior'
The country star's faith-based collection to be released October 26th!
Lauren Hoffman
Josh Turner will release a brand new album, I Serve A Savior, on October 26. The country star will deliver a twelve track faith-based collection following his 2017 album, Deep South.
Related: Reba McEntire to Return as Host of 'CMA Country Christmas'
I Serve A Savior will feature hand-picked hymns and original songs. Turner has also recorded two of his previously released songs including "Me and God" and "Long Black Train" for the album as well.
"My faith is the most important thing to me," said Turner. "I want people to hear the sentiment and heart behind this record. I want them to feel the same inspiration that I felt. I want them to feel that and hopefully they'll make these songs a part of their life."
A gospel album is a dream come true for Turner especially since he had the opportunity to write "The River (of Happiness)" with his wife, Jennifer, and son, Hampton. The Turner's all had the chance to sing on it together in the studio which you'll hear when it's released next month.
While each new track reflects his personal manifesto, the project is something Turner's fans have been asking for since the start of his career. In addition the release of I Serve A Savior, the 40-year-old will deliver a 90-minute live performance DVD filmed at Gaither Studios in the coming months.
Josh Turner
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<p class="title">O Voo</p>
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★★★★★
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Wanderley Caloni, <a href="https://github.com/Caloni/cinetenisverde/commits/master/content/post/o-voo.md">February 10, 2013</a></p>
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<p>No início do drama intimista somos apresentados a Whip Whitaker, um piloto “old-timer” que pilota um avião comercial com 102 pessoas a bordo como quem dirige um carro. Só que seu próximo voo apresentaria um defeito mecânico que o deixaria na mão. Utilizando seu instinto e habilidade, decide girar o avião de cabeça para baixo e com isso ganha impulso o suficiente para um pouso forçado em campo aberto, sendo obstruído apenas — que ironia — por uma torre de igreja. A sequência é arrebatadora por sua dramaticidade e impacto visual de um acidente em tempo real acontecendo diante de nós. Não é um espetáculo visual, mas um drama visceral ocorrendo no ar. Quando tudo termina, o resultado frio e matemático é de 6 mortes.</p>
<p>Já as consequências do evento são apresentadas para Whitaker no hospital desde sua primeira conversa, e já de forma a dividir opiniões: enquanto o representante do sindicato diz que ele salvou 96 pessoas, os representantes da auditoria apontam que o voo sob a responsabilidade do piloto resultou em mortes. Os motivos da queda são irrelevantes desde o início, pois estão no fundo procurando por um culpado, e Whitaker pode ser um deles exatamente por ter ingerido álcool e cocaína antes do voo.</p>
<p>O tema mais interessante do novo filme de Robert Zemeckis (Forrest Gump, O Náufrago) é a inversão de expectativa que ele causa em nós: enquanto visto como um herói pelas pessoas e tendo de fato todo o controle na cabine do avião durante o acidente, sua vida parece carecer dessa mesma auto-confiança. Whip Whitaker é um alcoólico inveterado, além de extremamente desonesto, como podemos observar em sua primeira atitude ao sair do hospital: se isolar no sítio de seu avô e jogar fora todas as bebidas. (a princípio podemos pensar que fosse uma mudança de atitude, o que não se comprova quando ele descobre que os testes de álcool no sangue já haviam sido feitos no hospital).</p>
<p>O destino de um sujeito desses não seria difícil de imaginar caso ele não tivesse conhecido no hospital Nicole (Kelly Reilly), uma viciada em recuperação. Diferente de Whitaker, Nicole está ciente que sua vida depende que ela pare (e é interessante que o roteiro estabeleça essa relação entre drogas injetáveis e álcool, uma droga tão letal quanto as proibidas por lei). Whip a traz para sua vida, mas os efeitos positivos da companhia de Nicole nunca parecem penetrar na visão egoísta e descontrolada do piloto.</p>
<p>Robert Zemeckis parece não ter pressa em avançar a história, pois quer que conheçamos mais sobre o estado irrecuperável de Whip pela simples observação do seu dia-a-dia. Enquanto as investigações estão em andamento e o advogado prepara o terreno para um final feliz e indolor, adentramos cada vez mais na obscuridade e irrelevância que sua vida. Enquanto torcemos que a justiça seja feita, ao mesmo tempo não estamos tão certos que essa atitude seria a mais correta, pois outros voos podem não ter o mesmo final bem sucedido com um cara desses na ativa.</p>
<p>E é por isso que sua conclusão possui a força acumulada de nossas próprias conclusões a respeito. Não torcemos mais apenas por sua absolvição, mas por sua redenção. Ele é uma pessoa que tem tudo para dar certo, mas para isso precisa encarar a realidade à sua volta sem o efeito do álcool. Como não gostar de um filme desses, que abre questões tão relevantes sobre o caráter humano e possui coragem para seguir adiante?</p>
<p>Abaixo seção com spoiler e análise da decupagem de Zemeckis.</p>
<p>Um dos trabalhos do diretor é enquadrar as cenas de um filme de maneira que possam trazer um significado a mais apenas pela disposição dos elementos em cena. É conhecido na linguagem cinematográfica que os elementos postos à direita da tela possuem mais relevância e são uma posição dominante no imaginário do espectador. Já com o lado esquerdo ocorre o inverso.</p>
<p>No caso de O Voo, o piloto Whip Whitaker não possui controle sobre sua vida, e é constantemente dominado pela bebida. Zemeckis passa essa mensagem não apenas pelas inúmeras pilhas de garrafas de bebidas espalhadas por sua casa, mas o colocando quase que 100% do tempo do lado esquerdo da tela. Se você tiver oportunidade de ver/rever este filme preste atenção a esse detalhe (além de inúmeros outros, claro, mas continuemos o raciocínio apenas deste).</p>
<p>A partir do momento em que Whip Whitaker decide contar a verdade a respeito da bebida em sua vida e passa a ter responsabilidade sobre seus atos o diretor Robert Zemeckis faz uma belíssima transição girando a câmera em torno de Whip Whitaker e o colocando já na prisão, anos depois, falando aos presos. A câmera continua a enfocar o ex-piloto, só que nesse momento ele está no centro da tela. Daí ocorre algo que considero tão emocionante quanto brilhante: a câmera ao se aproximar do protagonista vai lentamente o colocando à direita da tela, em um claro reajuste da própria postura do sujeito a partir de então para com sua vida. Também no diálogo final com seu filho ele está pela primeira vez totalmente à direita da tela, quando fecha a última cena.</p>
<p>Repito a pergunta feita no texto sem spoilers: como não gostar de um filme desses?</p>
<a href="https://www.imdb.com/title/tt1907668/mediaviewer/" target="ctvimg">Imagens</a> e créditos no <a title="IMDB: Internet Movie DataBase" href="http://www.imdb.com/title/tt1907668">IMDB</a>.
</div>
<span class="entry-sidebar-stars">
★★★★★
</span>
O Voo ● O Voo. Flight (USA, 2012). Dirigido por Robert Zemeckis. Escrito por John Gatins. Com Nadine Velazquez, Denzel Washington, Carter Cabassa, Adam C. Edwards, Tamara Tunie, Brian Geraghty, Kelly Reilly, Conor O'Neill, Charlie E. Schmidt. ● Nota: 5/5. Categoria: movies. Publicado em 2013-02-10. Texto escrito por Wanderley Caloni.
<p><br>Quer <a href="https://twitter.com/search?q=@cinetenisverde O%20Voo">comentar</a>?<br></p>
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 623
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case publication
CASE Network Studies and Analyses
latest publications:
growth & trade, fiscal policy
demography, labor & social policy
The Migration of Belarusians to Poland and The...
Europe, ownership structure, Poland, Private sector development, innovation and knowledge-based economy, privatization, CASE Reports, CASE Network Studies and Analyses
Enterprise Performance and Ownership Changes in Polish Firms
The paper consists of two parts. In the first part, the authors briefly summarize the results of previous analyses devoted to such issues of relevance as the ownership structure of privatized companies in Poland and how it changed over the course of the 1990s, what factors seemed to have influenced those changes, the economic performance of these companies, and the composition of corporate governance organs such as supervisory and executive boards. In the second part, the authors present the results of econometric analysis of the relationship between performance and ownership structure evolution, focusing on concentration and the respective roles of three types of owners - managers, non-managerial employees, and strategic outside investors. In reference to the debate about whether ownership variables are exogenous or endogenous for performance, they test both hypotheses concerning the effect of ownership on performance and concerning the effect of performance on ownership change.
EuroPACE. Final Publishable Report
demography, labor & social policy, innovation, energy & climate
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,221
|
\section{Introduction}
\section{Shift retrieval problems}
Consider the square circulant matrices defined as:
\begin{equation}
\begin{aligned}
\mathbf{C} = \text{circ}(\mathbf{c}) \stackrel{\rm def}{=} & \begin{bmatrix}
c_1 & c_{n} & \dots & c_3 & c_2 \\
c_2 & c_1 & \dots & c_4 & c_3 \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
c_{n-1} & c_{n-2} & \dots & c_1 & c_n \\
c_n & c_{n-1} & \dots & c_2 & c_1
\end{bmatrix} \\
= & \begin{bmatrix} \mathbf{c} & \mathbf{P}\mathbf{c} & \mathbf{P}^2\mathbf{c} & \dots & \mathbf{P}^{n-1}\mathbf{c} \end{bmatrix} \in \mathbb{R}^{n \times n}.
\end{aligned}
\label{eq:circulant}
\end{equation}
The matrix $\mathbf{P} \in \mathbb{R}^{n \times n}$ denotes the orthonormal circulant matrix that circularly shifts a target vector $\mathbf{c}$ by one position, i.e., $\mathbf{P} = \text{circ}(\mathbf{e}_2)$ where $\mathbf{e}_2$ is the second vector of the standard basis of $\mathbb{R}^{n}$. Notice that $\mathbf{P}^{q-1} = \text{circ}(\mathbf{e}_q)$ is also orthonormal circulant and denotes a cyclic shift by $q-1$. The eigenvalue factorization of circulant matrices reads:
\begin{equation}
\mathbf{C} = \mathbf{F}^H \mathbf{\Sigma} \mathbf{F},\ \mathbf{\Sigma} = \text{diag}(\mathbf{\sigma}) \in \mathbb{C}^{n \times n},
\label{eq:cfactorizationr}
\end{equation}
where $\mathbf{F} \in \mathbb{C}^{n \times n}$ is the unitary Fourier matrix ($\mathbf{F}^H\mathbf{F} = \mathbf{F}\mathbf{F}^H = \mathbf{I}$) and the diagonal $\mathbf{\sigma} = \sqrt{n}\mathbf{Fc},\ \mathbf{\sigma} \in \mathbb{C}^n$.
The multiplication with $\mathbf{F}$ is equivalent to the application of the Fast Fourier Transform, i.e., $\mathbf{Fc} = \text{FFT}(\mathbf{c})$, while the multiplication with $\mathbf{F}^H$ is equivalent to the inverse Fourier transform, i.e., $\mathbf{F}^H\mathbf{c} = \text{IFFT}(\mathbf{c})$. Both transforms are applied in $\BigO{n \log n}$ time and memory.
Given two real-valued matrices $\mathbf{X}$ and $\mathbf{Y}$, both $n \times N$, an immediate application \cite{CDLA, OnLearningRusu} of the eigenvalue factorization with the Fourier matrix is to the solution of the problem:
\begin{equation}
\underset{\mathbf{\sigma}}{\text{minimize}} \quad \| \mathbf{Y} - \mathbf{CX} \|_F^2,
\label{eq:minimize}
\end{equation}
whose solution is given by
\begin{equation}
\sigma_1 = \frac{ \mathbf{\tilde{x}}_1^H \mathbf{\tilde{y}}_1 }{ \|\mathbf{\tilde{x}}_1\|_2^2 },\sigma_k = \frac{ \mathbf{\tilde{x}}_k^H \mathbf{\tilde{y}}_k }{ \|\mathbf{\tilde{x}}_k\|_2^2 },\ \sigma_{n-k+2} = \sigma_k^*,k=2,\dots,n,
\label{eq:optimalsolution}
\end{equation}
where $\mathbf{\tilde{y}}_k^T$ and $\mathbf{\tilde{x}}_k^T$ are the rows of $\mathbf{\tilde{Y}} = \mathbf{FY}$ and $\mathbf{\tilde{X}} = \mathbf{FX}$.
\subsection{Classic shift retrieval}
Given two signals $\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}$ assuming that $\mathbf{y}$ is a cyclic shift of $\mathbf{x}$ in order to find the shift amount we maximize the inner product:
\begin{equation}
\underset{q}{\arg \max} \quad \mathbf{x}^H \mathbf{P}^q \mathbf{y},
\label{eq:normalShift}
\end{equation}
where $\mathbf{P}^{q} \in \mathbb{R}^{n \times n}$ denotes a cyclic shift by $q$. The calculations above explicitly find all inner products between $\mathbf{x}$ and all possible circular shifts of $\mathbf{y}$, or vice-versa. Practically, to recover the shift we use the circular cross-correlation theorem:
\begin{equation}
\arg \max \quad \text{IFFT} ( \text{FFT}(\mathbf{x})^* \odot \text{FFT}(\mathbf{y})).
\label{eq:cross-correlation}
\end{equation}
The result follows directly from the factorization \eqref{eq:cfactorizationr} by computing correlations between all cyclic shifts of $\mathbf{x}$ and the vector $\mathbf{y}$ as $\mathbf{C}^H \mathbf{y} = \text{circ}(\mathbf{x})^H \mathbf{y} = \mathbf{F}^H \text{diag}(\mathbf{Fx})^H \mathbf{F} \mathbf{y}
= \text{IFFT}(\text{FFT}(\mathbf{x})^* \odot \text{FFT}(\mathbf{y}))$.
If the two signals are circularly shifted version of each other (by the amount $q-1$) then the result of this calculation is $\| \mathbf{x} \|_2^2 = \| \mathbf{y} \|_2^2$ at index $q$.
\subsection{A different perspective on the shift retrieval problem}
In this section, we provide a different view on the classic shift retrieval problem and give the following main result.
\noindent \textbf{Result 1.} We are given two signals $\mathbf{x}$ and $\mathbf{y}$ suh that there is a circular shift $q$ between them, i.e., $\mathbf{y} = \mathbf{P}^{q-1} \mathbf{x}$. Then:
\begin{equation}
\text{IFFT} (\text{FFT}(\mathbf{y}) \oslash \text{FFT} (\mathbf{x}) ) = \mathbf{e}_q.
\label{eq:theorem1}
\end{equation}
\noindent \textit{Proof.} Assuming that $\mathbf{y} = \mathbf{P}^{q-1} \mathbf{x}$, and then $\mathbf{y} = \mathbf{P}^{q-1-n}\mathbf{x}$, with $\mathbf{P} = \text{circ}(\mathbf{e}_2)$. We consider the problem
\begin{equation}
\underset{q}{\text{minimize}} \quad \| \mathbf{y} - \mathbf{P}^{q-1} \mathbf{x} \|_F^2.
\label{eq:minimizePq}
\end{equation}
Use $\mathbf{P}^{q-1} = \mathbf{F}^H \mathbf{\Sigma} \mathbf{F}$ and to develop $\| \mathbf{y} - \mathbf{P}^{q-1} \mathbf{x} \|_F^2
=\|\mathbf{y} - \mathbf{F}^H \mathbf{\Sigma} \mathbf{Fx} \|_F^2
= \| \mathbf{Fy} - \mathbf{\Sigma} \mathbf{Fx} \|_F^2
= \| \mathbf{\tilde{y}} - \mathbf{\Sigma} \mathbf{\tilde{x}} \|_F^2$,
where $\mathbf{\Sigma} = \text{diag}(\mathbf{\sigma}), \mathbf{\sigma} = \mathbf{F e}_q$ (the $q^\text{th}$ column of the Fourier matrix). If we relax the constraint and allow $\mathbf{P}^{q-1}$ to be any circulant matrix, to minimize the Frobenius norm, as the special case of \eqref{eq:minimize} for $N=1$, we have $\sigma_i = \tilde{y}_i / \tilde{x}_i, \ \tilde{x}_i \neq 0,$ and $\mathbf{Fe}_q = \mathbf{\tilde{y}} \oslash \mathbf{\tilde{x}}$.
The assumption that $\tilde{x}_i \neq 0$ seems restrictive (and is missing in \eqref{eq:cross-correlation}). We do not need to apply the inverse Fourier transform but instead compute only $\sigma_i$ where $\tilde{x}_i \neq 0$ and by inspection of all columns of $\mathbf{F}$ on the rows where this quantity was computed we find the shift $q$. Notice that $\sigma_1$ and $\sigma_{\frac{n}{2}+1}$ when $n$ is even are $\{\pm 1\}$ for all columns $j$ and thus they cannot provide an unambiguous answer, on the other hand, in the best case scenario, we would need to compute a single $\sigma_i$. This reduces the complexity of the shift retrieval problem to $\BigO{n}$, as also observed for the compressive shift retrieval result \cite{CSR} which is discussed in the next subsection. Generally, when $\mathbf{y} = \alpha \mathbf{P}^{q-1} \mathbf{x} + \beta \mathbf{1}$ where $\alpha, \beta \in \mathbb{R}$ then $\text{IFFT} (\mathbf{\tilde{y}} \oslash \mathbf{\tilde{x}}) \! = \! \alpha \mathbf{e}_q + \left( \beta / \sum_i x_i \right) \mathbf{1}$, $\mathbf{1} \in \mathbb{R}^n$ is the ones vector.$\hfill \blacksquare$
\noindent \textbf{Remark 1 (Connection to the classical circular cross-correlation theorem).} We can rewrite \eqref{eq:theorem1} as:
\begin{equation}
\begin{aligned}
\text{IFFT} ( \text{FFT}(\mathbf{y}) & \oslash \text{FFT}(\mathbf{x}) ) = \text{IFFT} ( \text{FFT}(\mathbf{y}) \oslash \text{FFT}(\mathbf{x}) ) & \\
= & \text{IFFT} ( \text{FFT}(\mathbf{x})^* \odot \text{FFT}(\mathbf{y}) \oslash | \text{FFT}( \mathbf{x}) |^{ 2} ),
\end{aligned}
\label{eq:similarity}
\end{equation}
where $|\text{FFT}(\mathbf{x})|^{2}$ computes the square absolute values of each element of the Fourier transform of $\mathbf{x}$. Notice that \eqref{eq:theorem1} represents a weighted variant of \eqref{eq:cross-correlation}. If the two signals $\mathbf{x}$ and $\mathbf{y}$ are shifted versions of each other then \eqref{eq:cross-correlation} and \eqref{eq:theorem1} provide the same answer. If this is not the case, or the signals are noisy, then \eqref{eq:theorem1} seems a weaker result in general since the minimizer $\mathbf{P}^{q-1}$ in \eqref{eq:minimizePq} might no longer have $\mathbf{P} = \text{circ}(\mathbf{e}_2)$, but some other circulant matrix that minimizes \eqref{eq:minimizePq}. Of course, in this high noise case we might not be able to interpret that the signals are shifted versions of each other. The circulant cross-correlation theorem does not have this feature as it will always provide the maximum correlation between the signal and is indifferent if there is any shift property between the two signals.$\hfill \blacksquare$
\noindent \textbf{Remark 2 (Calculation of the circular shift from one measurement).} Notice that \eqref{eq:theorem1} is equivalent to:
\begin{equation}
\text{FFT}(\mathbf{y}) \oslash \text{FFT} (\mathbf{x}) = \mathbf{f}_q,
\end{equation}
where $\mathbf{f}_q$ is the $q^\text{th}$ column of the Fourier matrix $\mathbf{F}$. This means that we can find the shift by computing a single entry $\tilde{y}_i/\tilde{x}_i$ and then inspecting the entries of only the $i^\text{th}$ column of the Fourier matrix. $\hfill \blacksquare$
\subsection{The compressive shift retrieval problem}
Recently, the compressive shift retrieval problem has been introduced \cite{CSR, CSR2}. Define the sensing matrix $\mathbf{A} \in \mathbb{C}^{m \times n}, m \leq n,$ and the compressed measurement signals $\mathbf{z} = \mathbf{Ay} \in \mathbb{C}^{m}$ and $\mathbf{v} = \mathbf{Ax} \in \mathbb{C}^{m}$. Assuming that $\mathbf{y}$ is a cyclic shift of $\mathbf{x}$, the goal is to determine the shift from $\mathbf{z}$ and $\mathbf{v}$. Similarly to \eqref{eq:normalShift}, consider the test (Corollary 2 in \cite{CSR2}):
\begin{equation}
\underset{q}{\text{argmax}}\quad \Re \{ \mathbf{z}^H \mathbf{\bar{P}}^q \mathbf{v} \},
\label{eq:argmaxq}
\end{equation}
where $\mathbf{\bar{P}}^q = \mathbf{A}\mathbf{P}^q\mathbf{A}^H$. It has been shown that when $\mathbf{A}$ is taken to be a partial Fourier matrix then (Corollary 4 in \cite{CSR2}):
\begin{equation}
\underset{q \in \{ 0, \dots, n-1\} }{\text{max}}\ \Re \left\{ \sum_{i=1}^m z_i^* v_i e^{\frac{-2 \pi j k_i q}{n}} \right\},
\label{eq:maxR}
\end{equation}
recovers the true shift if there exists $p \in \{1,\dots,m\}$ such that $\tilde{x}_{k_p} \neq 0$ (the $k_p^\text{th}$ coefficient of the Fourier transform of $\mathbf{x}$) and $\{1,\dots,n-1\}\frac{k_p}{n}$ contains no integers. The set $\mathcal{K} = \{ k_i \}_{i=1}^m$ contains the indices of the rows contained in the partial Fourier matrix $\mathbf{A}$. Following \cite[Theorem 1]{CSR2}, we assume that the sensing matrix $\mathbf{A}$ obeys: $\mathbf{A}^H\mathbf{AP}^{q-1} = \mathbf{P}^{q-1}\mathbf{A}^H\mathbf{A}$, $\exists \ \alpha \in \mathbb{R}$ such that $\alpha \mathbf{AA}^H = \mathbf{I}$ and all columns of $\mathbf{A}\text{circ}(\mathbf{x})$ are different so that there is no shift ambiguity in the measurements.
The compressive shift retrieval result is partly based on the fact that $\mathbf{A}^H \mathbf{AP}^{q-1} = \mathbf{P}^{q-1}\mathbf{A}^H\mathbf{A}$. Notice that $\mathbf{A}^H\mathbf{A} = \mathbf{F}^H \mathbf{\Sigma F}$ where the diagonal $\mathbf{\Sigma}$ contains $\{0, 1\}$ with ones on the positions where the rows of the Fourier matrix are selected ($\mathcal{K}$). Notice that $\mathbf{A}^H\mathbf{A}$ is a circulant and thus it commutes with $\mathbf{P}^{q-1}$ -- they have the same eigenspace. Also, given a set $\mathcal{K}$ of indices, we define the operation $(\mathbf{a})_\mathcal{K} = \mathbf{b}$ for vectors $\mathbf{a} \in \mathbb{C}^n, \mathbf{b} \in \mathbb{C}^m, m\leq n,$ as attributing values $\mathbf{b}$ in positions $\mathcal{K}$ of $\mathbf{a}$, leaving the rest $\left( \{1, 2, \dots,n \} \backslash \mathcal{K} \right)$ to zero.
\noindent\textbf{Result 2 (Circulant compressive shift retrieval with a proof based on circulant matrices).} Given $\mathbf{z} = \mathbf{Ay}$ and $\mathbf{v} = \mathbf{Ax}$ where $\mathbf{y} = \mathbf{P}^{q-1}\mathbf{x}$, assuming $v_i \neq 0, i=1,\dots,m$ then:
\begin{equation}
(\mathbf{Fe}_q)_\mathcal{K} = \mathbf{z} \oslash \mathbf{v}.
\label{eq:theorem3}
\end{equation}
\noindent\textit{Proof.} We start again from the least squares problem:
\begin{equation}
\underset{q}{\text{minimize}} \quad \| \mathbf{z} - \mathbf{A}\mathbf{P}^{q-1} \mathbf{A}^H \mathbf{v} \| _F^2.
\label{eq:minimizeCCompressive}
\end{equation}
With the assumption that $\mathbf{y} - \mathbf{P}^{q-1} \mathbf{x} = \mathbf{0}$ the objective reaches the zero minimum when $\mathbf{P}^{q-1} = \mathbf{F}^H \mathbf{\Sigma} \mathbf{F}, \mathbf{\Sigma} = \text{diag}(\mathbf{Fe}_q)$: $\mathbf{Ay} - \mathbf{AP}^{q-1} \mathbf{A}^H\mathbf{Ax} = \mathbf{A} (\mathbf{y} - \mathbf{P}^{q-1}\mathbf{x})$, where we used the commutativity of circulant matrices and that $\mathbf{A}\mathbf{A}^H = \mathbf{I}$. To develop \eqref{eq:minimizeCCompressive}, start again from \eqref{eq:cfactorizationr} and the expression of the matrix multiplication as $\text{vec}(\mathbf{A} \mathbf{F}^H \mathbf{\Sigma} \mathbf{F} \mathbf{A}^H \mathbf{v}) = \left( (\mathbf{FA}^H\mathbf{v})^T \otimes (\mathbf{AF}^H) \right) \text{vec}(\mathbf{\Sigma})$. We finally obtain:
\begin{equation*}
\begin{aligned}
\| & \mathbf{z} - \mathbf{A} \mathbf{P}^{q-1} \mathbf{A}^H \mathbf{v} \| _F^2 = \| \mathbf{z} - \mathbf{A} \mathbf{F}^H \mathbf{\Sigma} \mathbf{F} \mathbf{A}^H \mathbf{v} \| _F^2 \\
= & \| \text{vec}(\mathbf{z}) - \text{vec}(\mathbf{A} \mathbf{F}^H \mathbf{\Sigma} \mathbf{F} \mathbf{A}^H \mathbf{v}) \| _F^2 \\
= & \| \mathbf{z} - \left( (\mathbf{FA}^H\mathbf{v})^T \otimes (\mathbf{AF}^H) \right) \text{vec}(\mathbf{\Sigma}) \|_F^2 = \| \mathbf{z} - \mathbf{VFe}_q \|_F^2,
\end{aligned}
\end{equation*}
where the matrix $\mathbf{V} \in \mathbb{R}^{m \times n}$ contains only the columns of the Kronecker product that match the non-zero elements of the diagonal matrix $\mathbf{\Sigma}$. The matrix contains the elements of $\mathbf{v}$ in positions $(k_i, i)$. The second equality holds because the Frobenius norm is elementwise. It follows that $\mathbf{VFe}_q = \mathbf{z}$ and
\begin{equation*}
\begin{aligned}
(\mathbf{Fe}_q)_\mathcal{K} = & \mathbf{V}^H (\mathbf{V}\mathbf{V}^H)^{-1} \mathbf{z}
= \mathbf{V}^H (\mathbf{z} \oslash | \mathbf{v} |^{2}) \\
= & \mathbf{v}^* \odot \mathbf{z} \oslash | \mathbf{v} |^{2}
= \mathbf{z} \oslash \mathbf{v}.
\end{aligned}
\end{equation*}
The compressive shift retrieval is equivalent to \eqref{eq:theorem1}, the regular shift retrieval, on the set of Fourier components $\mathcal{K}$. This is a unified view of the shift retrieval solutions.$\hfill \blacksquare$
In relation to \eqref{eq:maxR}, we use the circulant structures to reach:
\begin{equation*}
\begin{aligned}
\mathbf{z}^H \mathbf{\bar{P}}^q \mathbf{v} = & \mathbf{z}^H \mathbf{A} \mathbf{F}^H \mathbf{\Sigma} \mathbf{F} \mathbf{A}^H \mathbf{v} = \text{vec}(\mathbf{z}^H \mathbf{A} \mathbf{F}^H \mathbf{\Sigma} \mathbf{F} \mathbf{A}^H \mathbf{v}) \\
= & ( (\mathbf{FA}^H \mathbf{v})^T \otimes (\mathbf{z}^H \mathbf{AF}^H) ) \text{vec}(\mathbf{\Sigma}) = \mathbf{r} \mathbf{Fe}_{q+1},
\end{aligned}
\end{equation*}
where we expressed the matrix multiplications as a linear transformation on $\mathbf{\Sigma} = \text{diag}(\mathbf{Fe}_{q+1})$, with $q \in \{{0, \dots, n-1}\}$ and $\mathbf{r} \in \mathbb{C}^n$ is the expression in the parenthesis with $(\mathbf{r})_\mathcal{K} = \mathbf{z}^* \odot \mathbf{v}.$ The matrix $\mathbf{FA}^H \in \mathbb{R}^{n \times m}$ is a partial permutation matrix -- only positions $(k_i, i)$ are non-zero. The products with $\mathbf{v}$ and $\mathbf{z}$ produce extended vectors $(\mathbf{v})_\mathcal{K}, (\mathbf{z})_\mathcal{K} \in \mathbb{C}^n$. Thus, maximizing $\mathbf{z}^H \mathbf{\bar{P}}^q \mathbf{v}$ reduces to the selection of $\mathbf{e}_{q+1}$.
Due to the natural appearance of the Fourier matrix $\mathbf{F}$ in the factorization of circulant matrices its rows are also the natural choice in the rows of the measurement matrix $\mathbf{A}$. Cancellations that occur because of this choice lead to the analytic results found. This shows a simple alternative, but equivalent, way to develop the result \eqref{eq:maxR} of \cite{CSR2}.
\section{Conclusion}
In this letter, we provide a new look at the shift retrieval problems and show how to find the circulant shift between two signals with a single measurement in frequency.
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,600
|
package cz.mzk.mapseries.tests.integration;
import cz.mzk.mapseries.dao.CurrentVersionDAO;
import cz.mzk.mapseries.dao.SerieDAO;
import cz.mzk.mapseries.dao.SheetDAO;
import cz.mzk.mapseries.managers.SeriesManager;
import org.junit.*;
import javax.persistence.EntityManager;
import javax.persistence.EntityTransaction;
import javax.persistence.Persistence;
public class PersistenceModelTestCase {
private EntityManager em;
private EntityTransaction tx;
@Before
public void setup() {
em = Persistence.createEntityManagerFactory("integration-test").createEntityManager();
tx = em.getTransaction();
}
@After
public void tearDown() {
em.clear();
em.close();
}
@Test
public void sheetsWithSameIdTest() {
CurrentVersionDAO currentVersion = new CurrentVersionDAO();
currentVersion.setValue(1);
SerieDAO serie1 = new SerieDAO();
serie1.setName("Serie 1");
serie1.setVersion(1);
SerieDAO serie2 = new SerieDAO();
serie2.setName("Serie 2");
serie2.setVersion(1);
SheetDAO sheet1 = new SheetDAO();
sheet1.setSerie(serie1.getName());
sheet1.setSheetId("1");
sheet1.setTitle("Sheet 1");
sheet1.setYear("2018");
sheet1.setThumbnailUrl("http://thumbnail");
sheet1.setDigitalLibraryUrl("http://digitalLibrary");
sheet1.setVufindUrl("http://vufind");
sheet1.setVersion(1);
SheetDAO sheet2 = new SheetDAO();
sheet2.setSerie(serie2.getName());
sheet2.setSheetId("1");
sheet2.setTitle("Sheet 2");
sheet2.setYear("2018");
sheet2.setThumbnailUrl("http://thumbnail");
sheet2.setDigitalLibraryUrl("http://digitalLibrary");
sheet2.setVufindUrl("http://vufind");
sheet2.setVersion(1);
tx.begin();
em.persist(currentVersion);
em.persist(serie1);
em.persist(serie2);
em.persist(sheet1);
em.persist(sheet2);
tx.commit();
SeriesManager seriesManager = new SeriesManager();
seriesManager.setEntityManager(em);
Assert.assertEquals("Serie 1 does not contain expected number of sheets", 1, seriesManager.getSheets("Serie 1", "1").size());
Assert.assertEquals("Serie 2 does not contain expected number of sheets", 1, seriesManager.getSheets("Serie 2", "1").size());
Assert.assertEquals("Serie 1 contains wrong sheet", "Sheet 1", seriesManager.getSheets("Serie 1", "1").get(0).getTitle());
Assert.assertEquals("Serie 1 contains wrong sheet", "Sheet 2", seriesManager.getSheets("Serie 2", "1").get(0).getTitle());
}
@Test
public void sheetsWithSameIdInOneSerieTest() {
CurrentVersionDAO currentVersion = new CurrentVersionDAO();
currentVersion.setValue(1);
SerieDAO serie = new SerieDAO();
serie.setName("Serie");
serie.setVersion(1);
SheetDAO sheet1 = new SheetDAO();
sheet1.setSerie(serie.getName());
sheet1.setSheetId("1");
sheet1.setTitle("Sheet 1");
sheet1.setYear("2018");
sheet1.setThumbnailUrl("http://thumbnail");
sheet1.setDigitalLibraryUrl("http://digitalLibrary");
sheet1.setVufindUrl("http://vufind");
sheet1.setVersion(1);
SheetDAO sheet2 = new SheetDAO();
sheet2.setSerie(serie.getName());
sheet2.setSheetId("1");
sheet2.setTitle("Sheet 2");
sheet2.setYear("2018");
sheet2.setThumbnailUrl("http://thumbnail");
sheet2.setDigitalLibraryUrl("http://digitalLibrary");
sheet2.setVufindUrl("http://vufind");
sheet2.setVersion(1);
tx.begin();
em.persist(currentVersion);
em.persist(serie);
em.persist(sheet1);
em.persist(sheet2);
tx.commit();
SeriesManager seriesManager = new SeriesManager();
seriesManager.setEntityManager(em);
Assert.assertEquals("Serie does not contain expected number of sheets", 2, seriesManager.getSheets("Serie", "1").size());
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,190
|
set -e
set -x
USERNAME='phpseclib'
PASSWORD='EePoov8po1aethu2kied1ne0'
sudo useradd --create-home --base-dir /home "$USERNAME"
echo "$USERNAME:$PASSWORD" | sudo chpasswd
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 245
|
Archived Seasons
# Mike Sharp - Profile
- Team - NORTH Lincoln Sioux City Sioux Falls St Paul SOUTH El Paso Fort Worth Gr Prairie Pensacola Shrev Boss
Hometown: Liberty, MO
Career Highlights: Second season of professional baseball and second with the Wingnuts…originally signed by the Wingnuts on 6/26/09…finished college career at Emporia State University in 2009, hitting .371 with 13 home runs and 56 RBI…threw out 17 potential base stealers…played at the University of Alabama in 2008, hitting .288 in 66 at bats…played first two seasons of college ball at Kansas City Community College…hit .385 in 2007, .401 in 2006…first team All-KJCCC selection in 2007…traded by Wingnuts to Frontier League's Traverse City Beach Bums during off-season…re-signed by Wingnuts on 5/22…also played at Kansas City Kansas Community College…named 1st Team Jayhawk Conference and All-Region First Team…KCKCC Male Athlete of the Year in 2007…enjoys fishing and video games
AA Reg Season 2010 Wichita Wingnuts 20 54 3 11 0 0 0 9 3 13 0 .204
Total 20 54 3 11 0 0 0 9 3 13 0 .204
Jul 31 @Pensacola W 6-4 1 0 0 0 0 0 0 0 0 0 0 0 0 0.237 0.204 0.204 0.441
Jul 25 St Paul W 8-0 5 1 1 0 0 0 0 0 0 0 0 0 0 0.241 0.208 0.208 0.449
Jul 18 @Sioux Falls W 9-3 4 0 0 0 0 0 0 0 2 0 0 0 0 0.245 0.208 0.208 0.453
Jul 17 @Sioux Falls L 8-9 4 0 1 0 0 0 1 0 1 0 0 0 0 0.265 0.227 0.227 0.492
Jul 16 @Sioux Falls L 5-11 3 0 0 0 0 0 1 1 0 0 0 0 0 0.267 0.225 0.225 0.492
Jul 11 Gr Prairie W 8-5 3 0 0 0 0 0 0 1 0 0 0 0 0 0.268 0.243 0.243 0.511
Jul 9 Gr Prairie W 8-7 1 0 1 0 0 0 1 0 0 0 0 0 0 0.270 0.265 0.265 0.535
Jul 4 @St Paul L 0-9 1 0 0 0 0 0 0 0 0 0 0 0 0 0.250 0.242 0.242 0.492
Jul 3 St Paul W 9-2 1 0 0 0 0 0 0 0 1 0 0 0 0 0.257 0.250 0.250 0.507
Jul 1 St Paul L 5-13 1 0 0 0 0 0 0 0 1 0 0 0 0 0.265 0.258 0.258 0.523
Jun 28 Sioux City W 5-2 2 0 0 0 0 0 1 1 1 0 1 0 0 0.273 0.267 0.267 0.540
Jun 27 @St Paul L 0-6 3 0 2 0 0 0 0 0 1 0 0 0 0 0.276 0.286 0.286 0.562
Jun 15 @Sioux Falls L 2-4 3 0 1 0 0 0 1 0 1 0 0 0 0 0.235 0.250 0.250 0.485
Jun 11 @Sioux City L 3-4 5 0 0 0 0 0 1 0 2 0 1 0 0 0.214 0.231 0.231 0.445
Jun 3 @Sioux Falls L 4-16 4 0 1 0 0 0 0 0 2 0 0 0 0 0.375 0.375 0.375 0.750
May 26 @Lincoln W 7-4 4 1 2 0 0 0 1 0 0 0 0 0 0 0.500 0.500 0.500 1.000
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,701
|
REVIVAL PRP® is an Advanced Separator Gel-based, easy to use, sterile system that is used to prepare PRP in a safe and effective manner to accelerate the body's own natural healing process.
PRP is used in various medical applications including tissue enhancement, tissue and cell regeneration, rejuvenation, reconstruction and for general healing and repair.
Supraspinatus tendinopathy (ST) is characterized by degeneration of the tendon fibers and occurs in young to middle-aged, large breed dogs with no gender predilection (Ho et al 2015) and is a common cause of forelimb lameness in dogs. The cause of ST is thought to be repeated strain and overuse from chronic repetitive activity, with a failure of adequate remodeling (Canapp Jr et al 2016). Extreme and rapid hyperflexion of the shoulder is likely the cause of tear or damage to the supraspinatus tendon.
Dogs are typically non-responsive to traditional treatments such as rest and NSAIDs, and additional reported treatment options such as rehabilitation therapy and surgery may not always be successful.
Following resolution of lameness, there is a lifelong increased susceptibility to re-injury due to the reduced strength of fibrotic scar tissue compared to the original tendon (Ho et al 2015).
Open wounds in dogs can occur from a number of different causes. Abrasions or scrapes can occur when the superficial skin layers are damaged. This will cause minor inflammation, some surface bleeding and may cause bruising. Abrasions can occur from an animal biting at their skin, jumping over or digging under fences, fighting or being dragged across a rough surface.
Lacerations are when an animal's skin has been cut or torn open. Some lacerations will have clean, smooth edges or they may have jagged edges. Some lacerations will affect multiple layers of tissue depending on what caused the actual laceration. Puncture wounds or bite wounds occur when an object or tooth pierces the skin and leaves a small hole in the surface.
The hole will most likely affect multiple layers of tissue and is most susceptible to bacterial infections. Puncture wounds can easily become abscessed, creating a bigger medical emergency for your client's pet.
The flexor tendons of horses' lower limbs are important weight-bearing structures at rest and during locomotion. The anatomic arrangement of tendons and joints in the lower limb provides an efficient transfer of muscular energy for rapid locomotion. Of the two flexor tendons, the superficial digital flexor tendon (SDFT) is more commonly injured than the deep digital flexor tendon (DDFT). The SDFT provides a spring-like store of energy in the galloping horse and is subject to strains close to its mechanical limits, making it susceptible to overstrain injuries. Tendon damage may occur as a result of either overstrain of the tendon or a traumatic penetrating injury (O'Sullivan 2007).
It appears that the SDFT matures early, after which time it has limited ability to adapt to stress and undergoes progressive degeneration (Dowling et al 2000). Tendinopathy of the superficial digital flexor tendon (SDFT) is a common injury in Thoroughbred racehorses and other horse breeds and is regarded as a career limiting disease. Clinical injury is mostly strain induced and characterized by chronic degeneration after repetitive microtrauma in sport horses.
Degenerative Joint Disease (DJD) refers to arthritis or osteoarthritis, which is the result of the gradual deterioration of the articular cartilage within one or more the joints. This smooth resilient cartilage degenerates, becoming brittle over time. With severe DJD, the degenerated cartilage may actually split away from the bone and become loose within the joint. DJD can follow a number of joint diseases, including infection, and may develop after bone or joint injury or surgery.
Obese animals are more likely to develop DJD, as a direct result of the mechanical stress that excess weight puts on the joints. Although DJD is not usually an inflammatory disease, mild inflammation plays a part in causing clinical signs. When the cartilage cells become damaged, they release substances that result in inflammation, causing pain and further damage to the cartilage. Thus, once DJD begins, it can become a vicious cycle. Most of the damage caused by DJD is irreversible.
Platelet Rich Plasma (PRP) is a portion of the animal's own blood plasma that is concentrated in platelets and growth factors. These are critical in triggering the body's regenerative mechanisms.
The most basic method to prepare PRP is through centrifugation. The animal's blood is drawn and centrifuged at a specific speed until it is separated into 3 layers: Platelet Poor Plasma (PPP), Buffy Coat/Platelet Rich Plasma (PRP) and red blood cells.
REVIVAL PRP's Advanced Separator Gel sits in between the platelet-rich plasma and the red blood cells to ensure that the physician obtains a plasma preparation that is free from any red blood cells. Red blood cell contamination may provoke an inflammatory effect which can affect healing.
Promotes angiogenesis, cartilage regeneration, fibrosis and platelet adhesion.
The purpose of this trial was to determine the efficacy of a single IA injection of an autologous platelet concentrate for treatment of osteoarthritis in dogs. Dogs were randomly assigned to a treatment or control group. Dogs in the treatment group were then sedated, and a blood sample (55 mL) was obtained. PRP was then injected intra-articularly within 30 minutes. Control dogs were sedated and given an intra-articular injection of saline (0.9% NaCl) solution. Assessments were repeated 12 weeks after injection of platelets or saline solution.
It was concluded that in dogs suffering from OA involving a single joint, administration of a single intra-articular injection of autologous platelets resulted in significant improvements 12 weeks later, as determined by subjective (ie, owner-assigned scores for severity of pain and lameness) and objective (ie, PVF) measures.
Using a placebo-controlled clinical trial design, 20 horses with naturally occurring tendinopathies of forelimb SDFTs were randomly assigned to the PRP-treated group (n = 10) or control group (n = 10) after clinical and ultrasonographic examination. The SDFTs received an intralesional treatment with autologous PRP or were injected with saline, respectively (day 0).
Compared to day 0, lameness decreased significantly by week 8 after treatment with PRP and by week 12 in the control group. Ultra-sonographically there was no difference in the summarized cross-sectional area between the groups at any time point.
Ultrasound tissue characterization showed that echo types representing disorganized matrix decreased significantly throughout the observation period in the PRP-treated group.
Echo type II, representing discontinuous fascicles, not yet aligned into lines of stress was significantly higher 24 weeks after PRP treatment. 80% of the PRP treated horses reached their previous or a higher level of performance after 12 months compared to 50 % in the control group.
A single intralesional treatment with PRP up to 8 weeks after onset of clinical signs of tendinopathy contributes to an earlier reduction of lameness compared to saline treatment and to an advanced organization of repair tissue as the fibrillar matrix is getting organized into fascicles while remodelling continues.
PRP treated wounds showed more granulation formation and angiogenesis, faster epithelialization and collagen deposition than control wounds.
Autologous PRP was separated from anticoagulant treated whole blood in three dogs. Cutaneous wounds were created and then treated by intralesional injection of PRP in the experimental group, while they were treated with saline in the control group on days 0, 2 and 4. The healing process was evaluated by gross examination throughout the experimental period and histologic examination on day 7, 14 and 21.
In PRP treated wounds, the mean diameter was smaller and the wound closure rate was higher than in the control. Compared with control wounds, total granulation indexes of PRP-treated wounds were increased on days 7 (p = 0.040) and 14 (p = 0.031).
The number of vessels was highest on day 7 in both the control and PRP-treated wounds and decreased gradually over the experimental period. More vessels were observed in the PRP-treated wounds than the control wounds on day 7 (p=0.0000). On day 7, PRP-treated wounds also displayed more collagen fibers than control wounds. The collagen fibers in all of the PRP-treated groups became denser and more abundant than those in the control group on day 14.
At day 21, most of the midportions and all of the upper portions of the dermis in PRP-treated wounds showed tightly packed collagen fibers running parallel to each other and the epidermis relative to control wounds.
If you have insurance for your pet, please contact the insurance company to see if PRP Therapy is covered.
For cranial cruciate ligament ruptures (CCLR) in small animals, surgery has typically been the only option. Click here to learn about how PRP therapy is a potentially beneficial adjunct treatment to surgery. This article talks about both the clinical benefits as well as limitations of PRP use in small animals with CCLR.
For many canine injuries, platelet-rich plasma has shown promising results in optimizing the healing environment after an injury. Click here to read more.
With recent advances made in the field of regenerative medicine, platelet-rich plasma therapy is becoming a more widespread treatment option offering vastly improved healing. Click here to read how this approach works to create real tendon tissue instead of scar tissue.
To learn more about Revival PRP®, please enter your name and email address below and click submit.
By clicking submit, you authorize Xediton Pharmaceuticals to send you information about Revival PRP for veterinary use and articles related to its application in veterinary medicine. You can unsubscribe at any time.
Platelet-rich Plasma (PRP) is a portion of the plasma fraction (fluid component of blood) of blood that has a platelet and growth factor concentration above baseline to stimulate and boost the body's natural healing process.
During PRP treatment, the animal's blood is drawn and centrifuged in order to separate and concentrate the platelets and growth factors that are essential for tissue healing.
PRP is then injected or applied directly into the injured area causing the platelets and growth factors to activate. There is also recruitment of other healing proteins and factors to the area resulting in the healing and regeneration of tissue.
PRP is used to accelerate healing and reduce pain by enhancing the body's innate ability to repair and regenerate in a significantly shorter period of time.
PRP has demonstrated efficacy and effectiveness in a number of different indications. Multiples studies have been published regarding the use of PRP with excellent results.
Yes, the use of PRP is safe. There is minimal risk involved because the treatment uses the animal's own blood plasma and platelets.
PRP is autologous and does not provoke an immune response and is therefore perceived to have a high margin of therapeutic safety (Textor 2014).
Since PRP is prepared from autologous blood, theoretically there are minimal risks for disease transmission or immunogenic reactions (Wang-Saegusa et al. 2011).
Adverse effects are rare, however, there is always a small risk of injection site morbidity, infection or injury to nerves or blood vessels. Scar tissue formation and calcification at the injection site have been reported as well (Dhillon et al. 2012).
It is also very rare that development of antibodies against clotting factors V and IX leading to life-threatening coagulopathies have been reported. However, to date there is no compelling evidence of any systemic effects of a local PRP injection (Dhillon et al. 2012).
Acting growth factors also attach to cell surfaces rather than to the nucleus, which minimizes the chance of tumor formation using a negative feedback control (Ko 2010).
PRP injections offer an alternative to surgery and are an ideal choice for pet owners who prefer a less invasive option or those who have pets that are unable to undergo a surgery. It also allows for a quick recovery and is less painful than surgery.
What are the advantages and benefits of PRP therapy?
Some advantages of PRP include an increase in collagen production, elimination of donor transmissible disease, its non-allergenic properties, and its ability to aid in tissue regeneration and rejuvenation.
Pet owners can see a significant improvement in symptoms and function in as little as two to three weeks. It reduces pain, increases mobility and increases flexibility as the treatment works to heal the injured or dysfunctional area. This may eliminate the need for more aggressive treatments such as long-term medication or surgery.
Since PRP treatment uses the animal's own blood, there are very few risks associated. While infection is possible, the risks are very remote. The procedure should always be performed by an experienced professional with extensive training to minimize potential risks.
Infection, bruising and pain in the injected area are possible side effects of PRP treatment. In some very rare cases, an allergic reaction or blood clot is possible (due to the risk of damaging a vein).
Animals with moderate injuries and/or have failed conservative treatments (i.e. medication, physical therapy, etc.) are potential candidates for PRP treatment. PRP is ideally used where faster healing, increased wound strength, pain relief, and reduced scarring is desired. Your veterinarian may recommend REVIVAL PRP® be used in tendon/ligament injuries, joint injuries, for wound healing, or for an injury where there is a need for increased blood flow.
Does PRP cure or just provide temporary relief?
PRP is perceived to heal the injured area by enhancing the natural regenerative process. Therefore, PRP cures and is not just a temporary solution to an injury.
Treatment time will depend on the condition treated, extent of the damage and each animal's unique profile, however, results from multiple studies show positive results as soon as the first week and up to 6 weeks post procedure.
There is a possibility of soreness after the injection due to the PRP induced inflammatory response, especially if a joint was injected. This soreness is a positive sign that a healing response has been set in motion. This effect can last for several days and gradually decreases as healing and tissue repair occurs. The animal can resume normal activity, simply following a suggested rehabilitation plan. Vigorous or intensive exercise during this period is not recommended.
How many rounds of therapy are usually needed?
It will depend on the extent of the injury and the type of procedure being performed. Typically animals may need 1-2 injections for optimal results.
How long will PRP treatment results last?
PRP treatment is intended to resolve pain and promote healing of damaged tissue, therefore results could prove to be long lasting. Initial improvements may be evident within the first few weeks following treatment and continue throughout the healing process. In some cases, it can last up to a year or more.
Can PRP be used in conjunction with other treatments?
PRP has been used in conjunction with other therapies such as stem cells and skin grafting with outstanding results.
Depending on the injury, the animal and the procedure being performed, approximately 8-10 mL is obtained through a simple blood draw. The blood is centrifuged to produce PRP and the platelets are then collected in a syringe leaving 3-5cc to be injected at the required site. The entire procedure should take about 15-30 minutes.
Equine PRP preparations will require more blood (about 20ml) to yield approximately 12-14ml of plasma.
How soon can my animal return to normal and athletic activities?
Return to normal and athletic activity depends on the type and site of injury. Most chronic tendon injuries that have failed to respond to any other type of treatment will generally take quite a few weeks to heal. Injections into joints and acute muscle injuries heal quicker. Wounds will also require a few weeks to completely heal and close. Your veterinarian will inform you, based on the specific injury, what steps you and your animal should take for an effective recovery process.
The whole procedure takes approximately 30-45 minutes (depending on the application), including preparation and recovery time.
What can I do for my pet after the procedure?
Once the numbing effects of anesthesia subside, your pet will experience some soreness at the site of the treatment. As with most pain, icing the injection area for 15 minutes 2-3 times a day for the first few days will help ease the pain and inflammation. Most pets experience noticeable improvement in their mobility within a week or two and can resume normal activity.
Is PRP right for my animal?
If your animal has a tendon or ligament injury, degenerated joints or a chronic wound that does not appear to be healing in a timely manner, and traditional methods have not provided lasting relief, then PRP may be the solution. It helps to heal tissue with minimal or no scarring and alleviate further degeneration. The procedure is non-invasive, less expensive than surgery, and may eliminate the need for long-term and potentially harmful medication use. Talk to your veterinarian about whether PRP is right for your animal's specific condition.
PRP is in fact bactericidal (a substance that kills bacteria). While platelets are increased during PRP processing, leukocyte (white cells that fight infection) concentration is increased 6 to 8 times. This is one reason why PRP is so effective in wound healing. Not only do the increased leukocyte levels enable the body to fight off infection, they also have a dual purpose. After platelets initially release their growth factors, they produce a secondary release of leukocytes or growth factors to promote healing.
PRP has no ability to induce tumor formation and has never done so (Marx, RE, 2001), (Schmitz et al., 2001). The growth factors immediately bind to the external surface of the cell membranes of cells in the graft, flap, or wound via transmembrane receptors. The importance of this is that the GWP growth factors never enter the cell or its nucleus, they are not mutagenic, and they act through the stimulation of normal healing, just at a much faster rate.
Is it necessary to have more than one treatment? How often does this treatment need to be repeated?
It will depend on the condition treated, extent of the damage and the animal's unique profile. While responses to treatment vary, most animals require 1 to 3 sets of injections; with majority of animals only needing 1 injection. If additional injections are necessary, the risks and side effects do not change.
Why is PRP such a great tool in wound care?
Platelets and inflammatory cells are the first to arrive at any site of injury and express and release growth factors that promote tissue repair and influence processes such as angiogenesis, inflammation and the immune response. Chronic wounds have been shown to stall in the inflammatory phase of wound healing and either do not heal or continue to deteriorate. By applying PRP which contains a higher than normal concentration of platelets to a site of injury, the healing process is accelerated and helps transition difficult to treat chronic wounds out of the inflammation phase and into the proliferative phase of healing.
How is REVIVAL PRP® different?
• REVIVAL PRP® has an advanced separator gel technology which allows for separation of the red and white blood cells so that only the plasma rich in healing platelets and proteins will be injected into the animal. The unique properties of the gel trap pro-inflammatory granulocytes below the gel. These granulocytes do not help regeneration and may contribute to a catabolic effect, so it is an ideal system when a veterinarian recommends it for your animal's joint injury.
• REVIVAL PRP® is a sterile and closed system, ensuring that your vet obtains a clean PRP preparation each time.
How can my vet order the product?
REVIVAL PRP® may be ordered online by your vet through our website, fax or by phone. Our Customer Service Department is always ready to help fill your vet's order, answer any questions and provide additional details regarding shipping and handling.
What should I do if my pet experiences an adverse reaction?
If your pet experiences an adverse reaction, please contact your veterinarian immediately.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 965
|
Visit of Enda O Gara and Vivienne O Halloran:
On the 24th of November, Vivienne O'Halloran and Enda O Gara visited 1st and 2nd class.Vivienne is an air hostess and Enda is a air traffic controller. Vivienne's job was to check that there was enough food on the plane, that the plane was clean and that the passengers and bags were in the correct place to balance the plane. Vivienne told us that 300 people can fit in a big plane,200 can fit in a medium plane and 42 can fit in a small plane.The biggest plane in the world is called the Antonov.Buster is a dog who travels to and from England with a blind man.Vivienne's longest flight yet was 11 hours.Enda works in the Irish Aviation Authority. They are in charge of all the aircraft in Ireland. Enda said the biggest problem with the gliders was the weather. The captain of the plane always sits on the left. All planes have a name. The Aer Lingus planes are named after Irish Saints.The bigger planes always fly higher than the smaller planes. They have to dump fuel if they are doing an emergency landing so that the plane is lighter and it will come down easier.Amelia Earhart was the first woman to fly across the Atlantic.Everything in the Oceanic Airspace has to get a clearance from Enda's company. We'd like to thank our teachers for organising it.
By Lucas Quinn,Lily Hegarty and Grace Prendeville.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,029
|
October Proclaimed Domestic Violence Month
Kim David Published: October 6, 2016
Gary Gershoff
Rochester, MN (KROC AM News) - An event was held at the Olmsted County Government Center Thursday to recognize domestic violence issues.
Local elected and other officials joined county employees at the noon hour event. Rochester City Council President Randy Staver read a statement proclaiming October as Domestic Violence Awareness Month.
Several displays dealing with domestic violence have been put up at various sites in the city and can be viewed through the end of the month.
"Clothesline Project" by Minnesota Coalition for Battered Women - DoubleTree Hotel Skyway
"Remember My Name" display - Olmsted County Community Services, 2117 Campus Dr SE
"Tombstone Honor" display - Rochester Community & Technical College, 851 30th Ave SE
"Clothesline Project/Children in Shelter" display - Rochester Downtown Skywalk
"From Victim to Survivor; Stories of Hope" - Women's Shelter, Inc. Facebook page
Here are statistics about the prevalence and seriousness of domestic violence in our community, state and nation are attached for your reference.
Domestic violence is the willful intimidation, physical assault, battery, sexual assault, and/or other abusive behavior as part of a systematic pattern of power and control perpetrated by one intimate partner against another. It includes physical violence, sexual violence, threats, and emotional abuse. The frequency and severity of domestic violence can vary dramatically.
Olmsted County Stats:
2015 Olmsted County Court System – Domestic assaults
511 New reports of DV
145 Domestic assaults are repeat offenders (# believed to be low/ or inaccurate)
284 Children were present at the time of the assault (# believed to be low /or inaccurate)
2014 Olmsted County Court system - Domestic assaults
52.5% of the domestic assaults are repeat offender
84% are male offenders 16% are female offenders
387 Detained arrests, 106 Citations, 19 NA
12 victims identified themselves as being in a LGBT relationship with the offender
311 Children were present at the time of the assault
Jan-Mar: 106
Apr-June: 86
Jul-Sept: 127
Oct-Dec: 94
Total: 413 New reports of DV
MN Stats:
DOMESTIC VIOLENCE IN MINNESOTA
• In 2011, Minnesota courts adjudicated 27,288 cases of domestic violence.
• Over 80% of domestic violence victims in 2002 did not report the violence to Minnesota law enforcement.
• 1 in 3 homeless women in Minnesota is homeless because of domestic violence.
• In 2013, at least 26 women, 7 men, and 6 family members/friends in Minnesota were murdered in domestic violence homicides.
• In 2014, 56% of Minnesota domestic violence homicides were committed with firearms.
• 1 in 3 women and 1 in 4 men in the United States have experienced some form of physical violence by an intimate partner.
• On a typical day, domestic violence hotlines receive approximately 21,000 calls, 15 calls per minute.
• Intimate partner violence accounts for 15% of all violent crime.
• The presence of a gun in the home during a domestic violence incident increases the risk of homicide by at least 500%.
• 72% of all murder-suicides involved an intimate partner; 94% of the victims of these crimes are female.
2013 - Minnesota
At least 25 women died from domestic violence
At least 6 family members/friends were murdered
At least 7 men died from domestic violence
2015- Minnesota
At least 34 Minnesotans were killed due to violence from a current or former intimate partner.
At least 22 women were murdered in cases where the suspected, alleged, or convicted perpetrator was a current or former husband, boyfriend, or male intimate partner.
At least 9 friends, family members or interveners were murdered in domestic violence-related situations.
At least 3 men were murdered in a case where the suspected, alleged, or convicted perpetrator was a current or former intimate partner.
At least 17 minor children were left motherless due to domestic violence murders.
Note: At the time this report was completed, MCBW was reviewing four additional cases. These deaths occurred in 2015 but MCBW is waiting for information on whether the deaths would be officially ruled homicides, and information on the nature of the relationship between the victim and suspected perpetrator.
National Stats:
DOMESTIC VIOLENCE AND ITS EFFECTS:
• Every 9 seconds in the US a woman is assaulted or beaten.
• In the United States, an average of 20 people are physically abused by intimate partners every minute. This equates to more than 10 million abuse victims annually.
• 1 in 3 women and 1 in 4 men have been physically abused by an intimate partner.
• 1 in 5 women and 1 in 7 men have been severely physically abused by an intimate partner.
• 1 in 7 women and 1 in 18 men have been stalked. Stalking causes the target to fear she/he or someone close to her/him will be harmed or killed.
• On a typical day, domestic violence hotlines nationwide receive approximately 20,800 calls.
• The presence of a gun in a domestic violence situation increases the risk of homicide by 500%.
• Domestic violence accounts for 15% of all violent crime.
• Domestic violence is most common among women between the ages of 18-24.
• 19% of domestic violence involves a weapon.
• Domestic victimization is correlated with a higher rate of depression and suicidal behavior.
• Only 34% of people who are injured by intimate partners receive medical care for their injuries.
ECONOMIC EFFECTS:
• Victims of domestic violence lose a total of 8 million days of paid work each year.
• The cost of domestic violence exceeds $8.3 billion annually.
• Between 21-60% of victims of domestic violence lose their jobs due to reasons stemming from the abuse.
• Between 2003 and 2008, 142 women were murdered in their workplace by former or current intimate partners. This amounts to 22% of workplace homicides among women.
WHY DO PEOPLE STAY IN ABUSIVE RELATIONSHIPS?
One of the most common questions people ask about victims of domestic violence is, "Why don't they just leave?" People stay in abusive relationships for a variety of reasons including:
• The victim fears the abuser's violent behavior will escalate if (s)he tries to leave.
• The abuser has threatened to kill the victim, the victim's family, friends, pets, children and/or himself/herself.
• The victim loves his/her abuser and believes (s)he will change.
• The victim believes abuse is a normal part of a relationship.
• The victim is financially dependent on the abuser.
• The abuser has threatened to take the victim's children away if (s)he leaves.
• The victim wants her/his children to have two parents.
• The victim's religious and/or cultural beliefs preclude him/her from leaving.
• The victim has low self-esteem and believes (s)he is to blame for the abuse.
• The victim is embarrassed to let others know (s)he has been abused.
• The victim has nowhere to go if (s)he leaves.
• The victim fears retribution from the abuser's friends and/or family.
Filed Under: crime, domestic violence, Olmsted County, police, Rochester, Rochester Police
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\section{Introduction \label{s1}}
It has been shown recently~\cite{b1} that the quantum Calogero
model~\cite{b2}
can be transformed by a similarity transformation into the set of
noninteracting harmonic oscillators. It is also known~\cite{b3}\cite{b4}
that
the Calogero--Moser model (i.e. the model with purely inverse square
interaction) is equivalent to the set of free particles, both on
classical
and quantum levels.
In the present paper we show that the ``inverted" classical Calogero
model,
i.e. the model obtained from the Calogero model by changing the sign of
the
oscillator term is equivalent to the set of inverted harmonic
oscillators.
The equivalence is proven by constructing, more or less explicitly, the
relevant time--independent canonical transformation.
Two canonical transformations doing the job are presented. First, we
construct
the classical counterpart of the similarity transformation proposed in
Ref.\cite{b1}. It is easy to check that for the inverted case this
transformation is unitary which, in classical case, corresponds to real
canonical transformation. In sec.\ref{s3} an alternative mapping is
presented
which is based on $sL(2,\mbox{\Bbb R})$\ dynamical algebra of rational Calogero
model. In order to show that they are really different transformation
we consider in some detail the $N=2$\ case. Both mappings are here
constructed explicitly which makes their inequivalence transparent.
\section{Equivalence of the models \label{s2}}
Let us start with ``inverted" Calogero model described by the
hamiltonian
\begin{eqnarray}
H_C&=&\sum\limits_i\left({p^2_i\over 2}-{\omega^2q^2_i\over 2}\r)+g/2
\sum\limits_{i\not=j}{1\over (q_i-q_j)^2}\label{w1}
\end{eqnarray}
In order to get rid of the last term on the right hand side we perform
three successive canonical transformations.\hfill{}\\
(i)\ \ First we put
\begin{eqnarray}
q_i&=&q_i'\label{w2}\\
p_i&=&p_i'+\omega q_i'\nonumber
\end{eqnarray}
This is a canonical transformation and transforms $H_C$\ into
\begin{eqnarray}
H_C&=&\sum\limits_i{p_i^{\prime 2}\over 2}+g/2\sum\limits_{i\not=j}
{1\over(q_i'-q_j')^2}+\omega\sum\limits_iq_i'p_i'\label{w3}
\end{eqnarray}
(ii)\ \ Next, consider the one--parameter family of canonical
tramsformations
of the form
\begin{eqnarray}
q_i'(\lambda)&=&e^{-\lambda H_{CM}(q'',p'')}*q_i''\equiv
q_i''+\lambda\{q_i'',
H_{CM}(q'',p'')\}+\ldots\label{w4}\\
p_i'(\lambda)&=&e^{-\lambda H_{CM}(q'',p'')}*p_i''\equiv
p_i''+\lambda\{p_i'',
H_{CM}(q'',p'')\}+\ldots\nonumber
\end{eqnarray}
where $H_{CM}$\ is Calogero--Moser hamiltonian
\begin{eqnarray}
H_{CM}(q,p,)&=&\sum\limits_i{p_i^2\over 2}+g/2\sum\limits_{i\not=j}
{1\over (q_i-q_j)^2}\label{w5}
\end{eqnarray}
Actually, eq.\mref{w4} describes the dynamics of Calogero--Moser model,
$\lambda$\ playing the role of time variable. It is easy to check that
the
chioce $\lambda=-{1\over 2\omega}$\ gives
\begin{eqnarray}
e^{{1\over
2\omega}H_{CM}(q'',p'')}*\left(H_{CM}(q'',p'')+\omega\sum\limits_i
q_i''p_i''\r)&=&\omega\sum\limits_iq_i''p_i''\label{w6}
\end{eqnarray}
Therefore, as a second canonical transformation we take
\begin{eqnarray}
q_i'&=&e^{{1\over 2\omega}H_{CM}(q'',p'')}*q_i''\label{w7}\\
p_i'&=&e^{{1\over 2\omega}H_{CM}(q'',p'')}*p_i''\nonumber
\end{eqnarray}
implying
\begin{eqnarray}
H_C&=&\omega\sum\limits_iq_i''p_i''\label{w8}
\end{eqnarray}
(iii)\ \ Finally, let
\begin{eqnarray}
q_i''&=&{1\over\sqrt{2}}(Q_i+{1\over\omega}P_i)\label{w9}\\
p_i''&=&{1\over\sqrt{2}}(P_i-\omega Q_i)\nonumber
\end{eqnarray}
so that
\begin{eqnarray}
\omega\sum\limits_iq_i''p_i''&=&\sum\limits_i\left({P^2_i\over 2}-
{\omega^2Q_i^2\over 2}\r)\label{w10}
\end{eqnarray}
The transformation $(q,p)\to(Q,P)$\ given by the composition of
(i)$\div$(iii)
reduces ``inverted" Calogero model to the inverted harmonic oscillator.
\section{An alternative formulation\label{s3}}
It is well known that the rational Calogero model posses dynamical
$sL(2,\mbox{\Bbb R})$\
symmetry. In fact, define
\begin{eqnarray}
T_+&\equiv&{1\over\omega}\left(\sum\limits_ip_i^2/2+g/2\sum\limits_{i\not=j}
{1\over(q_i-q_j)^2}\r)\nonumber\\
T_-&\equiv&\omega\sum\limits_iq_i^2/2\label{w11}\\
T_0&\equiv&\frac{1}{2}\sum\limits_iq_ip_i\nonumber
\end{eqnarray}
where $\omega$\ is a fixed, nonzero but otherwise arbitrary, frequency.
The
relevant Poisson bracket read
\begin{eqnarray}
\{T_0,T_\pm\}&=&\pm T_\pm\label{w12}\\
\{T_+,T_-\}&=&-2T_0.\nonumber
\end{eqnarray}
Define
\begin{eqnarray}
T_1&=&\frac{1}{2}(T_++T_-)\nonumber\\
T_2&=&\frac{1}{2}(T_+-T_-)\label{w13}\\
T_3&=&iT_0;\nonumber
\end{eqnarray}
the commutation rules~\mref{w12} take the form
\begin{eqnarray}
\{T_i,T_j\}&=&i\epsilon_{ijk}T_k\label{w14}
\end{eqnarray}
Consider now the canonical transformation
\begin{eqnarray}
q_i(\lambda)&=&e^{\lambda T_1(q',p')}*q_i'\label{w15}\\
p_i(\lambda)&=&e^{\lambda T_1(q',p')}*p_i';\nonumber
\end{eqnarray}
then
\begin{eqnarray}
T_2(q(\lambda),p(\lambda))&=&e^{\lambda
T_1(q',p')}*T_2(q',p')=\label{w16}\\
&=&T_2(q',p')\cos\lambda+T_3(q',p')\sin\lambda\nonumber
\end{eqnarray}
Therefore, puting
\begin{eqnarray}
q_i&=&e^{{\pi\over 2}T_1(q'',p'')}*q_i''\label{w17}\\
p_i&=&e^{{\pi\over 2}T_1(q'',p'')}*p_i''\nonumber
\end{eqnarray}
one obtains
\begin{eqnarray}
T_2(q,p)&=&T_3(q'',p'')\label{w18}
\end{eqnarray}
Let us define
\begin{eqnarray}
\tilde{T}_i&=&T_i(g=0)\label{w19}
\end{eqnarray}
Then $T_3=\tilde{T}_3$\ and the transformation
\begin{eqnarray}
q_i''&=&e^{-{\pi\over 2}\tilde{T}_1(Q,P)}*Q_i\label{w20}\\
p_i''&=&e^{-{\pi\over 2}\tilde{T}_1(Q,P)}*P_i\nonumber
\end{eqnarray}
converts $\tilde{T}_3=T_3$\ into $\tilde{T}_2$,
\begin{eqnarray}
\tilde{T}_2(P,Q)&=&\tilde{T}_3(q''.p'')\label{w21}
\end{eqnarray}
Composing the mappings~\mref{w17} and~\mref{w20} one gets finally
\begin{eqnarray}
H_C&=&\sum\limits_i\left({P_i^2\over 2}-{\omega^2Q_i^2\over
2}\r)\label{w22}
\end{eqnarray}
The canonical transformation~\mref{w20}, written out explicitly, reads
\begin{eqnarray}
q_i&=&{1\over\sqrt{2}}(Q_i-{1\over\omega}P_i)\label{w23}\\
p_i&=&{1\over\sqrt{2}}(P_i+\omega Q_i)\nonumber
\end{eqnarray}
and coincides with eq.\mref{w9}.
\section{The $N=2$\ case\label{s4}}
In secs.\ref{s2} and~\ref{s3} we defined two canonical transformations
reducing
the inverted Calogero hamiltonian to dilatation. However, we still do
not know
whether these are different transformations or two forms of the same
transformation. To show that they are really different it is sufficient
to
consider the $N=2$\ case. Separating the center of mass and relative
coordinates
\begin{eqnarray}
X=\frac{1}{2}(q_1+q_2)&,&\pi=p_1+p_2\label{w24}\\
q=q_1-q_2&,&p=\frac{1}{2}(p_1-p_2)\nonumber
\end{eqnarray}
and ignoring the former we get
\begin{eqnarray}
H_C&=&p^2+g/q^2-{\omega^2\over 4}q^2.\label{w25}
\end{eqnarray}
The transformations described in secs.\ref{s2} and~\ref{s3} can be given
explicitly. In particular, from eqs.\mref{w2} and~\mref{w7} one gets
\begin{eqnarray}
q&=&\mbox{sgn}(q'')\sqrt{g+(q''p''-E/\omega)^2\over E}\nonumber\\
p&=&\mbox{sgn}(q''){(q''p''-E/\omega)\over\sqrt{g+(q''p''-E/\omega)^2\over
E}}+
\mbox{sgn}(q''){\omega\over 2}\sqrt{g+(q''p''-E/\omega)^2\over E}\label{w26}\\
E&=&p^{\prime\prime 2}+g/q^{\prime\prime 2}\nonumber
\end{eqnarray}
We easily check that
\begin{eqnarray}
p^2+g/q^2-{\omega^2\over 4}q^2&=&\omega q''p''\label{w27}
\end{eqnarray}
as it should be.
On the other hand eq.\mref{w17} takes the form
\begin{eqnarray}
q&=&\mbox{sgn}(q'')\sqrt{{2E\over\omega^2}-{2q''p''\over\omega}}\nonumber\\
p&=&\mbox{sgn}(q''){(\omega q^{\prime\prime 2}/2-E/\omega)\over\sqrt{%
{2E\over\omega^2}-{2q''p''\over\omega}}}\label{w28}\\
E&=&p^{\prime\prime 2}+g/q^{\prime\prime 2}+{\omega^2\over
4}q^{\prime\prime 2}
\nonumber
\end{eqnarray}
Again one easily verifies that the eq.\mref{w27} holds. By comparying
eqs.\mref{w26} and~\mref{w28} we conclude that the transformations
described in secs.\ref{s2} and~\ref{s3} are really different.
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Theme of the ohrie blog
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Q: jQuery search/replace all body text problems I am doing a complex language translation in jQuery. I am copying some of the methods used in the jQuery/Google translate, but using our own XML files for getting original English and translated languages.
I am able to easily read English and other language into an array.
But my problem is that some of these text phrases, that we are paying someone to have translated, are going to be inside other sentences inside many different pages.
I have been blasting away, trying different approaches, and still no real luck in finding a useable solution.
In past efforts, I have used the $('body').nodesContainingText() from jQuery-translate, to parse through each text node, then search for that text inside the English array, grab that position, and use that to get the translated version in the other language array.
And that works fine where there are separate text nodes, but still not even close to consistently working on partial replacing.
Seems the more I try to fix the problem, the less it works.
So what I'd really like is some guidance about what I am doing wrong?
Here is the code I am using, maybe this will help.
http://crosenblum.pastebin.com/f6468aae8
A: It seems like you might want to try using <span> tags around the sub-section of the text. Something like this:
<!-- Header up to Here -->
<p>Lorem ipsum dolor sit amet, consectetur adipiscing elit. Quisque lacinia
vehicula interdum. Nam aliquet auctor mollis. Suspendisse non orci arcu, id
commodo augue. Aliquam quam enim, mollis in lacinia quis, pretium scelerisque
leo. Cras volutpat arcu sit amet purus malesuada non bibendum sem euismod.
Fusce ut augue in eros imperdiet <span class="translateMe">interdum. Mauris
eleifend rhoncus lacinia. Vivamus auctor scelerisque nulla</span>, sed
consectetur velit sodales porttitor. Nullam a neque eget mauris blandit facilisis.
Nunc bibendum enim vel est venenatis ac posuere ligula condimentum. Nullam tortor
neque, bibendum eget vestibulum at, interdum in diam.</p>
<!-- Rest of HTML after Here --> This way you can not only do the $('body').nodesContainingText(), but you can also do $('span.translateMe') to retrieve the nodes to translate.
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American Institute of Mathematical Sciences
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Quotient elastic metrics on the manifold of arc-length parameterized plane curves
June 2017, 9(2): 207-225. doi: 10.3934/jgm.2017009
The group of diffeomorphisms of the circle: Reproducing kernels and analogs of spherical functions
Yury Neretin 1,2,3,,
Math. Dept., University of Vienna, Oskar-Morgenstern-Platz 1,1090 Wien, Austria
Institute for Theoretical and Experimental Physics (Moscow), Russia
MechMath. Dept., Moscow State University, Institute for Information Transmission (Moscow), Russia
* Corresponding author: Yury Neretin
Received December 2015 Revised August 2016 Published May 2017
Fund Project: Supported by FWF grants P25142, P28421
Full Text(HTML)
Related Papers
The group $\text{Diff}\left( {{S}^{1}} \right)$ of diffeomorphisms of the circle is an infinite dimensional analog of the real semisimple Lie groups $\text{U}(p,q)$, $\text{Sp}(2n,\mathbb{R})$, $\text{SO}^*(2n)$; the space $Ξ$ of univalent functions is an analog of the corresponding classical complex Cartan domains. We present explicit formulas for realizations of highest weight representations of $\text{Diff}\left( {{S}^{1}} \right)$ in the space of holomorphic functionals on $Ξ$, reproducing kernels on $Ξ$ determining inner products, and expressions ('canonical cocycles') replacing spherical functions.
Keywords: Univalent functions, welding, reproducing kernel, Fock space, Virasoro algebra, highest weight representation, spherical function.
Mathematics Subject Classification: Primary: 81R10, 46E22; Secondary: 30C99, 43A90.
Citation: Yury Neretin. The group of diffeomorphisms of the circle: Reproducing kernels and analogs of spherical functions. Journal of Geometric Mechanics, 2017, 9 (2) : 207-225. doi: 10.3934/jgm.2017009
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Ha lavorato nel cinema e in televisione ed è forse più noto come direttore dell'orchestra Oregon Symphony Pops. Con Glenn Miller nel 1943 scrisse il tema I Sustain the Wings, che fu utilizzato come sigla dell'omonimo programma radiofonico trasmesso su CBS e NBC dal 1943 al 1945, durante la seconda guerra mondiale.
Biografia
Norman Leyden nacque a Springfield, nel Massachusetts, da James A. e Constance Leyden. Si laureò all'Università Yale nel 1938; nel 1961 frequentò il Domaine Musicale di Pierre Monteux a Hancock, nel Maine e conseguì un master (1965) e un dottorato (1968) presso la Columbia University (dove insegnò anche per diversi anni). Sposò Alice Curry Wells nel 1942 a Duval County, in Florida.
Carriera musicale
Iniziò la sua carriera musicale professionale suonando il clarinetto basso per la New Haven Symphony Orchestra mentre frequentava Yale. Leyden entrò nella Guardia Nazionale degli Stati Uniti nel 1940 per un anno di volontariato programmato e si arruolò come sergente di fanteria il 24 febbraio 1941 a New Haven, nel Connecticut. I suoi documenti di arruolamento riportano la sua altezza di 1,82 m. e il suo peso di 75 kg. e riportano come sua specialità musicista o bandleader. Durante la seconda guerra mondiale prestò servizio nell'esercito militare per cinque anni.
Mentre Leyden lavorava come sergente capo ad Atlantic City e provava musica, Glenn Miller lo sentì esibirsi. Miller gli disse: "Per un uomo di Yale, non suoni male il [clarinetto] tenore". Miller chiamò Leyden nel settembre 1943 quando dovette dirigere la spettacolare "Vittoria alata" della Moss Hart Army Air Force. Questo fu un grande spettacolo musicale nel teatro Shubert di Broadway con una band completa di tutto. Lo spettacolo iniziò nel novembre del 1943. Leyden richiese poi l'opportunità di arrangiare per Glenn Miller e fu accettato e prestò servizio come uno dei tre arrangiatori della Miller's Air Force Band. Il suo primo arrangiamento per la band fu "Now I Know". A volte Leyden avrebbe voluto scrivere gli spartiti in modo più complesso di quanto fosse desiderabile. Miller gli disse una volta "Hey Norm, è stato un bel tentativo, ma ricorda che non conta quello che scrivi, ma ciò che non scrivi". Nel 1943 Leyden compose la musica a tema per la serie radiofonica di guerra I Sustain the Wings con Glenn Miller, Chummy MacGregor e Bill Meyers. Il programma radiofonico fu trasmesso dal 1943 al 1944. Leyden arrangiò anche per la Glenn Miller Orchestra riorganizzata di Tex Beneke. Nell'agosto del 2000 diresse l'Air Force Falconaires della Air Force Band of the Rockies in uno speciale della televisione PBS, Glenn Miller's Last Flight.
Tra il 1956 e il 1959 fu direttore musicale del programma radiofonico di Arthur Godfrey. Lavorò anche come direttore musicale per The $ 64,000 Question (compresa anche la composizione del tema musicale) e come direttore musicale di The Jackie Gleason Show, originariamente chiamato You're in the Picture (1961). Organizzò anche la Westchester Youth Symphony a White Plains, New York, nel 1957 (un'organizzazione che diresse fino al 1968). Come arrangiatore personale presso la RCA Victor compose e arrangiò musica per Disney e altri musical tra cui Cenerentola, Alice nel paese delle meraviglie, Ventimila leghe sotto i mari, Winnie the Pooh, Peter Pan e Pinocchio. Leyden ha anche diretto e arrangiato molti artisti famosi, tra i quali Tony Bennett, Rosemary Clooney, Don Cornell, Vic Damone, Johnny Desmond, The Four Lads, Johnny Hartman, Gordon MacRae, Mitch Miller, Ezio Pinza, Frank Sinatra, Jeri Southern e Sarah Vaughan.
Leyden si trasferì a Portland, in Oregon, nel 1968 per rilevare la Portland Youth Philharmonic (poi Portland Junior Symphony) mentre il direttore di lunga data Jacob Avshalomov era andato in anno sabbatico. Entrò anche nel dipartimento di musica presso la Portland State University. Iniziò la sua lunga collaborazione con l'Oregon Symphony nel 1970 come direttore associato. Questo rapporto durò per 29 stagioni più 34 stagioni come direttore dell'Oregon Symphony Pops. Oltre un milione di persone parteciparono ai suoi concerti nell'Oregon Symphony Pops. Nel maggio del 2004 si ritirò e fu onorato con il titolo a vita di direttore associato laureato. Leyden ha anche lavorato come direttore musicale dei Seattle Symphony Pops per diciotto stagioni e come direttore dei Prairie Pops dell'Orchestra Sinfonica di Indianapolis per otto stagioni. Ha anche diretto l'Orchestra Chappaqua come secondo direttore musicale prima di trasferirsi sulla costa occidentale. Ha lavorato con la band di Portland Pink Martini e si può ascoltare un assolo di clarinetto sulla title track del secondo album della band, Hang On Little Tomato.
La libreria musicale personale di Leyden, ospitata in un arioso studio nel seminterrato, comprendeva oltre 1.200 arrangiamenti sinfonici e 300 lavori per big band. A 90 anni Leyden continuava a praticare il clarinetto tutti i giorni. Mercoledì 17 ottobre 2007 diresse un concerto per il 90º compleanno con la Norman Leyden Big Band di 17 elementi alla Arlene Schnitzer Concert Hall di Portland, intitolata Norman's Big Band Birthday Concert. È diventato uno dei soli due musicisti classici ad essere inserito nella Oregon Music Hall of Fame nel 2008. Si è esibito con Pink Martini a Seattle nell'agosto del 2012 e il 19 luglio 2013 ha debuttato all'Hollywood Bowl, sempre con Pink Martini.
Leyden è morto il 23 luglio 2014 per un motivo non specificato.
Il 28 agosto 2014 l'Oregon Symphony tenne un concerto commemorativo in onore di Leyden al Tom McCall Waterfront Park di Portland.
Premi
Oregon Governor's Arts Award, 1991
Oregon Music Hall of Fame, inducted 2008
Note
Bibliografia
"Air Force Band Brings Back Swing Sound of Glenn Miller", Regulatory Intelligence Data, September 29, 1999
Baker's Dictionary of Music and Musicians (2001)
Darroch, Lynn. "For Norman Leyden, 90, it's still all about music", The Oregonian, October 20, 2007.
Duval County Marriage Records 1942, certificate number 8657
Fifteenth Census of the United States: 1930 Population Schedule, New York State, Westchester County (gives parents and siblings names; also listed in Fourteenth Census of the United States 1920 Population Schedule, East Orange Ward 5, Essex, New Jersey)
Glenn Miller (2004) at Big Band Library.com
Leyden, Norman Fowler. A Study and Analysis of the Conducting Patterns of Arturo Toscanini as Demonstrated in Kinescope Films. Columbia University dissertation, 1968.
Salzman, Eric "Music: Six Conductors", New York Times, August 22, 1961, page 21.
Stabler, David "Farewell to that Pops Magic" The Oregonian, December 14, 2003, page F1.
Collegamenti esterni
Clarinettisti statunitensi
|
{
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We live imprisoned lives. Despite having our freedom we spend most or all of our waking hours inside our smartphone world. Play a little candy crush while waiting for the bus, watch a video while on the bus or listen to the latest songs on music apps, check out what our friends are up to on the Facebook app, indulge in some banking too through our banking app, in between of sending inane WhatsApp messages. We don't like to be interrupted by anyone, not even our smartphones.
When these devices our daily lives depend on, start draining out on battery, we start losing our mind. Never mind global warming or severe water crisis threatening our species, a long lasting battery is all we care for. 'Battery Low' is too dreaded a message to this new generation of consumers, and hence, Asus's recently-launched ZenFone Max smartphone.
The ASUS ZenFone Max is the first smartphone with a 5000 mAh battery to hit the Indian market. Its enormous battery enables it to deliver up to 38 days of standby time and 1.5 times longer talk time over a 3G network. Also, it can be used as a portable power bank to charge other smartphones and a host of personal electronic devices! All you need is an OTG cable that comes as a standard accessory with the smartphone.
With the aim to target people who consider battery life to be a deciding factor when buying a smartphone, Asus rolled out a hilarious vox-pop digital video. Conceptualized and executed by ASUS's digital agency BC Web Wise, the video begins with a survey of youngsters in Mumbai who talk about their worst 'zero battery' nightmares.
Some are believable, also relatable, some are outright outrageous nightmares – a teenager grounded by his upset mother, a worried couple who fear their daughter has been kidnapped, to an untimely battery issue leading to a break-up. They are then asked which smartphone brand and model they own and how long the battery lasts.
What you get are funny responses like 'not even Bandra to Churchgate," and can't survive more than an hour of gaming. They are then given a ZenFone Max and asked to guess its battery life. All their guesswork is wrong, but they are pleasantly surprised to know its battery power and that it costs only INR 10K.
ASUS goes on to ask the youngsters to connect their current smartphone with the ZenFone Max. As their gadgets start charging up, a universal antidote for their low battery nightmares makes itself obvious – 'Battery Low? Battery Lo…' Therefore the brand urges them to help a friend in need of power while also putting an end to their own low battery woes by getting the ZenFone Max.
The brand is also running a #BatteryLowBatteryLo sob stories contest on Facebook and Twitter.
Put your phone's low battery life into good use! Share your low battery nightmare story with us using #BatteryLowBatteryLo & win!
A funny digital video is a good way to reach out to the smartphone generation. Highlighting the phone's long battery life and its dual use as a power bank, via a vox-pop video is a good way to spread the word and also make it share-worthy with their friends and family. The video is a relatable one too; it catches the most common issue faced by a smartphone user today and also offers a solution to their woes. 'Battery Low? Battery Lo…' asks the question and provides the answer too. Moreover, the video is entertaining; despite a clever product association being slipped in, it does not feel like an advertisement.
Inviting youngsters to share their low battery stories through incentivized contests on social media is a further add-on to the video-driven campaign. "Battery Low? Battery Lo" could just become the mantra of the young smartphone generation!
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{
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## Applications
* Alexandria
* Football Scores
## Project description
Added enhancements to two apps in order to make them production ready. The work included ensuring errors were handled gracefully, building a widget for the home screen, support for screen readers, optimizations for localization, and barcode scanning via a library.
[Screenshots](Screenshots/)
## Libraries
* [Support Design libraries](https://developer.android.com/topic/libraries/support-library/features.html) - Material design library
* [Realm](https://github.com/realm/realm-java) - Realm is a mobile database: a replacement for SQLite & ORMs
* [Realm Recyclerview](https://github.com/thorbenprimke/realm-recyclerview) - A RecyclerView that is powered by Realm
* [Barcode Scanner](https://github.com/dm77/barcodescanner) - Barcode Scanner Libraries for Android
* [Butter Knife](http://jakewharton.github.io/butterknife/) - UI binding library
* [Timber](https://github.com/JakeWharton/timber) - Advanced logging library
* [Dagger 2](http://google.github.io/dagger/) - Dependency Injection library
* [Retrofit](http://square.github.io/retrofit/) - Web API library
* [LeakCanary](https://github.com/square/leakcanary) - A memory leak detection library
## Udacity Nanodegree SuperDuo Project
# License - Apache License, Version 2.0
```
# (C) Copyright 2016 by Marek Hakala <hakala.marek@gmail.com>
# 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.
```
|
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"redpajama_set_name": "RedPajamaGithub"
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Q: Need advice on what the following port could be? I've tried searching for different USB port types to identify what this port might be but to no avail. The mini USB cable fits in here however all the pins don't match up.
Any idea what it could be?
A: I believe that it is a:
MOLEX 47346-0001 USB Connector, Micro USB Type B, USB 2.0, Receptacle
As shown here
Or possibly a:
MOLEX 56579-0519 MINI USB TYPE AB CONNECTOR, RECEPTACLE
Shown here
|
{
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{"url":"https:\/\/blog.acolyer.org\/2019\/08\/23\/learning-to-prove-theorems-via-interacting-with-proof-assistants\/","text":"# Learning to prove theorems via interacting with proof assistants\n\nLearning to prove theorems via interacting with proof assistants Yang & Deng, ICML\u201919\n\nSomething a little different to end the week: deep learning meets theorem proving! It\u2019s been a while since we gave formal methods some love on The Morning Paper, and this paper piqued my interest.\n\nYou\u2019ve probably heard of Coq, a proof management system backed by about 30 years of research and developed out of INRIA. Here\u2019s how the Coq home page introduces it:\n\nCoq is a formal proof management system. It provides a formal language to write mathematical definitions, executable algorithms and theorems together with an environment for semi-interactive development of machine-checked proofs.\n\nCertified programs can then be extracted to languages like OCaml, Haskell, and Scheme.\n\nIn fully automated theorem proving (ATP), after providing a suitable formalism and a theorem, the goal is to be able to push a button and have the system prove that the theorem is true (or false!). The state-of-the-art in ATP is still some way behind human experts though it two key areas: the scale of systems it can tackle, and the interpretability of the generated proofs.\n\nWhat a typical theorem prover does\u2026 is to prove by resolution refutation: it converts the premises and the negation of the theorem into first-order clauses in conjunctive normal form. It then keeps generating new clauses by applying the resolution rule until an empty clause emerges, yielding a proof consisting of a long sequence of CNFs and resolutions.\n\nEnter interactive theorem proving (ITP). In ITP a human expert guides the structure of the proof, telling the system which proof tactics to use at each step:\n\nThe tactics capture high-level proof techniques such as induction, leaving low-level details to the software referred to as proof assistants.\n\n(Enlarge)\n\nIt would be nice if we could learn from all the ITP proofs that humans have generated, and use those learnings to improve the proof steps that a system can take automatically:\n\n\u2026 human experts have written a large amount of ITP code, which provides an opportunity to develop machine learning systems to imitate humans for interacting with the proof assistant.\n\nSuch a combination would get us closer to fully automated human-like proofs.\n\nTo pull all this together we need three things:\n\n1. A large ITP dataset we can learn from\n2. A learning environment for training and evaluating auto-ITP agents. Agents start with a set of premises and a theorem to prove, and then they interact with the proof assistant by issuing a sequence of tactics.\n3. A way to learn and generate new tactics which can be used in the construction of those proofs.\n\nThe first two requirements are satisfied by CoqGym, and ASTactic handles tactic generation.\n\n### CoqGym\n\nCoqGym (https:\/\/github.com\/princeton-vl\/CoqGym) includes a dataset of 71K human-written proofs gathered from 123 open-source Coq projects. Every proof is represented as a tree with goals (and their context) as nodes, and tactics as edges between nodes. On average proofs in the dataset have 8.7 intermediate goals and 9.1 steps. The projects are split into training, test, and validation sets resulting in 44K training proofs, 14K validation proofs, and 13K testing proofs.\n\nIn addition to the human-generated proofs, CoqGym also includes shorter proofs extracted from the intermediate goals inside long proofs. For every intermediate goal in a human-written proof, synthetic proofs of 1, 2, 3, and 4 steps are generated.\n\nLearning to theorem prove in Coq resembles a task-oriented dialog with an agent receiving current goals, local context, and global environment at each step, and issuing a tactic for the system to use. Agents can be trained using supervised or reinforcement learning.\n\n### ASTactic\n\nASTactic is a deep learning model for tactic generation that generations tactics as programs by composing ASTs in a predefined grammar using a subset of Coq\u2019s tactic language.\n\nCompared to prior work that chooses tactics from a fixed set, we generate tactics dynamically in the form of abstract syntax trees and synthesize arguments using the available premises during runtime. At test time, we sample multiple tactics from the model. They are treated as possible actions to take, and we search for a complete proof via depth-first search (DFS).\n\nASTatic is built using an encoder-decoder architecture. The encoder embeds a proof goal and premises expressed in ASTs, and the decoder generates a program-structured tactic by sequentially growing an AST.\n\n(Enlarge)\n\nThe production rule to apply is selected using a softmax over rule outputs. The argument(s) for the rule are then selected according to their type:\n\n\u2022 For rules that take a premise argument (e.g., H in apply H) a score is computed for each premise in either the environment or the local context, and then a premise is selected using a softmax over these scores.\n\u2022 For rules with an integer argument, most take an argument between 1 and 4, so a 4-way classifier is used to select one.\n\u2022 For quantified variables in the goal, a universally quantified variable in the goal is selected at random.\n\n### Can you prove it?\n\nWe evaluate ASTactic on the task of fully-automated theorem proving in Coq, using the 13,137 testing theorems in CoqGym. The agent perceives the current goals, their local context, and the environment. It interacts with Coq by executing commands, which include tactics, backtracking to the previous step (Undo), and any other valid Coq command. The goal for the agent is to \ufb01nd a complete proof using at most 300 tactics and within a wall time of 10 minutes.\n\nPerformance is compared against Coq\u2019s built-in automated tactics (trivial, auto, intuition, and easy), and against hammer. Hammer simultaneously invokes Z3, CVC4, Vampire, and EProver and returns a proof if any one of them succeeds. It sets a time limit of 20 seconds for all these external systems.\n\nThe automated tactics can also be combined with ASTactic, whereby at each step the agent first using an automatic tactic to see if it can solve the current goal. If it can\u2019t be solved automatically, the agent executes a tactic from ASTactic to decompose the goal into sub-goals.\n\nIn the best performing combination (ASTactic + hammer), 30% of theorems in the test set can be proved.\n\nThis demonstrates that our system can generate effective tactics and can be used to prove theorems previously not provable by automatic methods.\n\nAs an extra bonus, the generated proofs tend to be shorter than the ground truth proofs collected in CoqGym:\n\nPresumably you could take this system and use it as an even more powerful ITP, with human guidance still being provided where it struggles. After enough time, we could then learn from the human-assisted proofs that were generated, and go round the loop all over again!\n\nIf you liked this, you might also enjoy \u201cThe great theorem prover showdown.\u201d","date":"2021-02-27 15:52:51","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\": 0.5859065651893616, \"perplexity\": 1513.4560073531843}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178358976.37\/warc\/CC-MAIN-20210227144626-20210227174626-00234.warc.gz\"}"}
| null | null |
71st Annual Members' Exhibition
Opening Reception, Thursday, October 11, 5:30 - 7:30 PM
Awards Ceremony 6:15 PM
The most popular tradition in the history of the Quinlan, the 71st Annual Members' Exhibition showcases the work of the myriad of member-artists from Gainesville and surrounding North Georgia Communities, Atlanta and the Southeast. Each year of celebration marks yet another season of unparalleled excellence in visual arts programming and celebrates the artists that contribute to that level of excellence. The exhibition will be on view through November 24th and is a dazzling and eclectic array of art in all media including landscapes, abstracts, still life compositions, portraits, ceramics, wood turnings and much, much more.
The opening reception will be Thursday, October 11th from 5:30 p.m. until 7:30 p.m. Juror Christopher Dant, Assistant Professor of Photography at the University of North Georgia will award ribbons for Best of Show, First, Second, and Third places as well as several Honorable Mentions.
Chris Dant (b. Columbus, OH) is a visual artist who explores the ideas of empathy, shared experience, family history and loss through the use of the photographic medium. Chris received his Bachelor of Fine Arts degree from the University of North Texas in 2012 and his Master in Fine Arts degree from the University of Notre Dame in 2016. In 2015, he was commissioned by the South Bend Museum of Art to travel around the state of Indiana and photographically document the still existing projects constructed by the Works Progress Administration (W.P.A.) in the 1930s and 40s. This work was recently added to the South Bend Museum of Art's Permanent Collection. In addition, Dant has recently exhibited his work in New York City and received the Walter R. Beardsley Award for Best in Show, for his MFA Thesis Exhibition, Worthy Sons, from the Snite Museum of Art in Notre Dame, Indiana. After graduate school, Chris found his way back to the University of North Texas, where he held an adjunct position teaching the Advanced Darkroom course. In the Spring of 2017, he accepted the position of Assistant Professor of Photography at the University of North Georgia, in Dahlonega, Georgia, where he currently resides. You can find Chris hiking around the mountains of north Georgia, making photographs and looking for rocks to climb and a good watering hole to swim in.
The Roy C. Moore Gallery
The George and Anne Thomas Gallery
The Green Street Gallery
The Jonathan Mansfield Gallery
The Bailey Gallery
LISTING OF AWARD WINNERS
|
{
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| 7,108
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Primrose by Kate Halfpenny. Purchased and altered new this year. Snug size 10 fit. The straps claps together at the back and then need to be tied into the best bow your bridesmaids can manage! The dress has been cleaned. It has bust cups sewn in - great for an A/B cup. Don't think it's possible to wear a bra with the dress due to the low back.
Length was altered so that it just grazes the floor at the front for a 5 foot 3 bride (in low heels), leaving a nice train. Would rest higher for anyone taller.
Once you send a message, Kirsty can invite you to purchase the dress.
|
{
"redpajama_set_name": "RedPajamaC4"
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I have some exciting news to share with you! Tomorrow is my 11th anniversary as a Stampin' Up! Demonstrator. Holy cow, can you believe that…11 years? I received a very nice email from Rett Christensen who works at the home office; more than 6 years ago, he was assigned as my coach in the Demonstrator Business Development Program and he just about the sweetest person ever! It is always nice to get an email checking in once in a while form him.
Last week Sidney turned 10 years old and I made her some homemade treats. Not sure how that nose knew what I was doing, but she would not leave my side and even guarded the jar on the counter. She makes me so happy just by wagging her tail or snuggling close. I am always loving on her, I can't help it, she is my fur-baby and i love her so much!
She wasn't too interested in her new baby that we got her, it was just all about the food!
Here is the recipe that I made for her and I found it on Pinterest, gotta love that site for a ton of recipe ideas.
Soft treats will last no more than a few days on the counter, perhaps a week in the refrigerator. You can try freezing them for longer storage.
|
{
"redpajama_set_name": "RedPajamaC4"
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Q: opengrok Index database(s) not found I'm using OpenGrok and get an issue when I try to search on a project.
I get the error "Index database(s) not found..."
It's a bit weird, because there are other projects on this instance and they work perfectly. It's only with the last project in my list that I have this problem...
Does anyone have any idea?
A: You will see this error, when there is a problem with indexing of given repository. opengrok creates a "dirty" file in such cases under index folder eg. /home/opengrok/data/index/
Please follow the below steps to solve such errors.
*
*Identify the repository on which this error is being appeared eg.
test
*Login to opengrok server
*cd /home/opengrok/data/index/
eg. /home/opengrok/data/index/test
*Move the contents to a temp location
eg. mkdir /tmp/test; mv /home/opengrok/data/index/test/* /tmp/test
*Start indexing again.
A: Finally I found how to resolve it.
It was because the indexes where to big.
So I went in the OpenGrok directory and clear them.
Then I launched OpenGrok again and everything went fine.
Hope this can help someone else!
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{
"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"http:\/\/www.r-bloggers.com\/what-is-volatility\/","text":"# What is volatility?\n\nJanuary 19, 2014\nBy\n\n(This article was first published on Portfolio Probe \u00bb R language, and kindly contributed to R-bloggers)\n\nSome facts and some speculation.\n\n## Definition\n\nVolatility is the annualized standard deviation of returns \u2014 it is often expressed in percent.\n\nA volatility of 20 means that there is about a one-third probability that an asset\u2019s price a year from now will have fallen or risen by more than 20% from its present value.\n\nIn R the computation, given a series of daily prices, looks like:\n\nsqrt(252) * sd(diff(log(priceSeriesDaily))) * 100\n\nUsually \u2014 as here \u2013\u00a0log returns\u00a0are used (though it is unlikely to make much difference).\n\n## Historical estimation\n\nWhat frequency of returns should be used when estimating volatility? \u00a0The main choices are daily, weekly and monthly. \u00a0(In some circumstances the choices might be something like 1-minute, 5-minute, 10-minute, 15-minute.)\n\nThere is folklore that it is better to use monthly data than daily data because daily data is more noisy. \u00a0But noise is what we want to measure \u2014 daily data produces less variable estimates of volatility.\n\nHowever, this is finance, so things aren\u2019t that easy. \u00a0There is the phenomenon of variance compression \u2014 that is, it isn\u2019t clear that volatility from monthly and daily data are aiming at the same target.\n\nAnother complication is if there are assets from around the globe. \u00a0In that case asynchrony is a problem with daily data. \u00a0For example, the daily return of assets traded in Tokyo, Frankfurt and New York for January 21 are returns over three different time periods. \u00a0That matters (not much for volatility, but definitely for correlations). \u00a0Weekly returns as well as fancy estimation methods alleviate the problem.\n\n## Through time\n\nVolatility would be more boring if finance were like other fields where standard deviations never change.\n\nFinancial markets seem to universally experience volatility clustering. \u00a0Meaning that there are time periods of high volatility and periods of (relatively) low volatility.\n\nBut why?\n\nChanges in volatility would make sense if there were periods of greater uncertainty about the future. \u00a0But the future is always uncertain.\n\nThe day after Lehman Brothers failed there was a great deal more uncertainty about the markets than the week before. \u00a0However, that uncertainty was generated internally to the markets, it wasn\u2019t greater uncertainty about the physical world.\n\nAgent-based models of markets easily reproduce volatility clustering. \u00a0Rather than agents changing their minds about the best strategy at random times, mind-changing tends to cluster. \u00a0When lots of agents change strategies, then the market volatility goes up.\n\n## Across assets\n\nDifferent assets in the same market have different volatilities. \u00a0But why?\n\nSome assets are especially responsive to the behavior of the \u00a0rest of the market. \u00a0Some assets are inherently more uncertain.\n\nFactor models explicitly divide these two ideas into systematic and idiosyncratic risk.\n\n## Implied volatility\n\nImplied estimates come from using a model backwards. \u00a0Suppose you have a model that approximates X by using variables A, B and C. \u00a0If you know X, A and B, then you can get an implied estimate of C by plugging X, A and B into the model.\n\nAn example is implied alpha.\n\nImplied volatility comes from options pricing models. \u00a0The most famous of these is the Black-Scholes formula.\n\nBlack-Sholes uses some assumptions and derives a formula to say what the price of an option should be. \u00a0The items in the formula are all known except for the average volatility between the present and the expiration date of the option.\n\nWe can use known options prices and the Black-Scholes formula to figure out what volatility is implied. \u00a0You can think of this as the market prediction of volatility \u2014 assuming the model is correct (but it isn\u2019t).\n\n## Not risk\n\nSome people equate volatility with risk. \u00a0Volatility is not risk.\n\nVolatility is one measure of risk. \u00a0In some situations it is a pretty good measure of risk. \u00a0But always with risk there are some things that we can measure and some things that escape our measurement.\n\nPart of risk measurement should be trying to assess what is left outside.\n\n## Epilogue\n\nBefore I built a wall I\u2019d ask to know\nWhat I was walling in or walling out,\nAnd to whom I was like to give offense.\n\nfrom \u201cMending Wall\u201d by Robert Frost\n\n## Appendix R\n\nI made the statement that using simple returns rather than log returns is unlikely to make much difference. \u00a0We shouldn\u2019t take my word for it. \u00a0We can experiment.\n\nIn most blog posts I give only a sample of the R code. \u00a0Here, though, is the complete R code for this experiment.\n\n#### write a function\n\npp.dualvol <- function(x, frequency=252) {\nlv <- sd(x)\nsv <- sd(exp(x)) # same as sd(exp(x) - 1)\n100 * sqrt(frequency) *\nc(logReturn=lv, simpleReturn=sv)\n}\n\n#### get starting points of the samples\n\nstarts <- sample(length(spxret) - 253, 1000)\n\nThis statement presumes that the number sampled is smaller than the number of values being sampled since it is using the default of sampling without replacement.\n\n#### compute volatilities\n\nvolpairs <- array(NA, c(1000, 2),\nlist(NULL, names(pp.dualvol(tail(spxret, 10)))))\n\ntseq <- 0:62\n\nfor(i in seq_along(starts)) {\nvolpairs[i,] <- pp.dualvol(spxret[starts[i] + tseq])\n}\n\n#### write plot function\n\nP.dailyVolPair63 <-\nfunction (filename = \"dailyVolPair63.png\")\n{\nif(length(filename)) {\npng(file=filename, width=512)\npar(mar=c(5,4, 0, 2) + .1)\n}\nplot(volpairs, col=\"steelblue\")\nabline(0, 1, col=\"gold\")\nif(length(filename)) {\ndev.off()\n}\n}\n\n#### do the plot\n\nP.dailyVolPair63()\n\nFigure 1: Volatility estimates on a quarter of daily data \u2014 simple versus log returns.\n\nWe can see that almost all of the 1000 estimates are virtually the same when using either type of return. \u00a0There are a few large volatilities that are slightly different. \u00a0The differences will be due to large returns, which means that returns at lower frequencies will yield bigger differences between the volatility estimates.","date":"2014-11-28 19:19:43","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5928261280059814, \"perplexity\": 3306.7372204324956}, \"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-2014-49\/segments\/1416931010792.55\/warc\/CC-MAIN-20141125155650-00236-ip-10-235-23-156.ec2.internal.warc.gz\"}"}
| null | null |
import pyaudio
import wave
import os
defaultframes = 512
class textcolors:
if not os.name == 'nt':
blue = '\033[94m'
green = '\033[92m'
warning = '\033[93m'
fail = '\033[91m'
end = '\033[0m'
else:
blue = ''
green = ''
warning = ''
fail = ''
end = ''
recorded_frames = []
device_info = {}
useloopback = False
recordtime = 5
#Use module
p = pyaudio.PyAudio()
#Set default to first in list or ask Windows
try:
default_device_index = p.get_default_input_device_info()
except IOError:
default_device_index = -1
#Select Device
print (textcolors.blue + "Available devices:\n" + textcolors.end)
for i in range(0, p.get_device_count()):
info = p.get_device_info_by_index(i)
print (textcolors.green + str(info["index"]) + textcolors.end + ": \t %s \n \t %s \n" % (info["name"], p.get_host_api_info_by_index(info["hostApi"])["name"]))
if default_device_index == -1:
default_device_index = info["index"]
#Handle no devices available
if default_device_index == -1:
print (textcolors.fail + "No device available. Quitting." + textcolors.end)
exit()
#Get input or default
device_id = int(input("Choose device [" + textcolors.blue + str(default_device_index) + textcolors.end + "]: ") or default_device_index)
print ("")
#Get device info
try:
device_info = p.get_device_info_by_index(device_id)
except IOError:
device_info = p.get_device_info_by_index(default_device_index)
print (textcolors.warning + "Selection not available, using default." + textcolors.end)
#Choose between loopback or standard mode
is_input = device_info["maxInputChannels"] > 0
is_wasapi = (p.get_host_api_info_by_index(device_info["hostApi"])["name"]).find("WASAPI") != -1
if is_input:
print (textcolors.blue + "Selection is input using standard mode.\n" + textcolors.end)
else:
if is_wasapi:
useloopback = True;
print (textcolors.green + "Selection is output. Using loopback mode.\n" + textcolors.end)
else:
print (textcolors.fail + "Selection is input and does not support loopback mode. Quitting.\n" + textcolors.end)
exit()
recordtime = int(input("Record time in seconds [" + textcolors.blue + str(recordtime) + textcolors.end + "]: ") or recordtime)
#Open stream
channelcount = device_info["maxInputChannels"] if (device_info["maxOutputChannels"] < device_info["maxInputChannels"]) else device_info["maxOutputChannels"]
stream = p.open(format = pyaudio.paInt16,
channels = channelcount,
rate = int(device_info["defaultSampleRate"]),
input = True,
frames_per_buffer = defaultframes,
input_device_index = device_info["index"],
as_loopback = useloopback)
#Start Recording
print (textcolors.blue + "Starting..." + textcolors.end)
for i in range(0, int(int(device_info["defaultSampleRate"]) / defaultframes * recordtime)):
recorded_frames.append(stream.read(defaultframes))
print (".")
print (textcolors.blue + "End." + textcolors.end)
#Stop Recording
stream.stop_stream()
stream.close()
#Close module
p.terminate()
filename = input("Save as [" + textcolors.blue + "out.wav" + textcolors.end + "]: ") or "out.wav"
waveFile = wave.open(filename, 'wb')
waveFile.setnchannels(channelcount)
waveFile.setsampwidth(p.get_sample_size(pyaudio.paInt16))
waveFile.setframerate(int(device_info["defaultSampleRate"]))
waveFile.writeframes(b''.join(recorded_frames))
waveFile.close()
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,884
|
Q: postgres 'only' shows numbers with comma seperator, even if data is saved without my postgres shows me only comma separated number values like:
even when the original number comes without the separator.
lc_numeric shows me German_Germany.1252, which seems to be right.
So when the data is saved correctly, where can I change the format of the shown output?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,215
|
Territorial Army recruits from Windsor's 94 Signal Squadron have been 'on the front line' helping the Olympics go smoothly.
The armed forces have played a big role - the TA supplying much of the military's communications network.
The 94 Signal Squadron has been planning and rehearsing for a year, ready to link up the Olympic and security venues around London and Weymouth.
The TA recruits men and women aged 18 to 43.
Anyone interested in the TA at Windsor can ring 01753 860600 to find out more.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,378
|
O ácido tricarboxílico é um composto orgânico cujas moléculas contêm três grupamentos carboxila. O exemplo mais conhecido de um ácido tricarboxílico é o ácido cítrico.
Ver também
Ciclo do ácido cítrico
Referências
!3
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,515
|
Protocol
========
Pokémon Showdown's protocol is relatively simple.
Pokémon Showdown is implemented in SockJS. SockJS is a compatibility
layer over raw WebSocket, so you can actually connect to Pokémon
Showdown directly using WebSocket:
ws://sim.smogon.com:8000/showdown/websocket
or
wss://sim3.psim.us/showdown/websocket
Client implementations you might want to look at for reference include:
- Majeur's android client (Kotlin/Java) -
https://github.com/MajeurAndroid/Android-Unofficial-Showdown-Client
- pickdenis' chat bot (Ruby) -
https://github.com/pickdenis/ps-chatbot
- Quinella and TalkTakesTime's chat bot (Node.js) -
https://github.com/TalkTakesTime/Pokemon-Showdown-Bot
- Nixola's chat bot (Lua) -
https://github.com/Nixola/NixPSbot
- Morfent's chat bot (Perl 6) -
https://github.com/Kaiepi/p6-PSBot
- the official client (HTML5 + JavaScript) -
https://github.com/smogon/pokemon-showdown-client
The official client logs protocol messages in the JavaScript console,
so opening that (F12 in most browsers) can help tell you what's going
on.
Client-to-server messages
-------------------------
Messages from the user to the server are in the form:
ROOMID|TEXT
`ROOMID` can optionally be left blank if unneeded (commands like `/join lobby`
can be sent anywhere). Responses will be sent to a PM box with no username
(so `|/command` is equivalent to `|/pm &, /command`).
`TEXT` can contain newlines, in which case it'll be treated the same
way as if each line were sent to the room separately.
A partial list of available commands can be found with `/help`.
To log in, look at the documentation for the `|challstr|` server message.
Server-to-client messages
-------------------------
Messages from the server to the user are in the form:
>ROOMID
MESSAGE
MESSAGE
MESSAGE
...
`>ROOMID` and the newline after it can be omitted if the room is lobby
or global. `MESSAGE` cannot start with `>`, so it's unambiguous whether
or not `>ROOMID` has been omitted.
As for the payload: it's made of any number of blocks of data
separated by newlines; empty lines should be ignored. In particular,
it should be treated the same way whether or not it ends in a
newline, and if the payload is empty, the entire message should be ignored.
If `MESSAGE` doesn't start with `|`, it should be shown directly in the
room's log. Otherwise, it will be in the form:
|TYPE|DATA
For example:
|j| Some dude
|c|@Moderator|hi!
|c| Some dude|you suck and i hate you!
Some dude was banned by Moderator.
|l| Some dude
|b|battle-ou-12| Cool guy|@Moderator
This might be displayed as:
Some dude joined.
@Moderator: hi!
Some dude: you suck and i hate you!
Some dude was banned by Moderator.
Some dude left.
OU battle started between Cool guy and Moderator
For bandwidth reasons, five of the message types - `chat`, `join`, `leave`,
`name`, and `battle` - are sometimes abbreviated to `c`, `j`, `l`, `n`,
and `b` respectively. All other message types are written out in full so they
should be easy to understand.
Four of these can be uppercase: `J`, `L`, `N`, and `B`, which are
the equivalent of their lowercase versions, but are recommended not to be
displayed inline because they happen too often. For instance, the main server
gets around 5 joins/leaves a second, and showing that inline with chat would
make it near-impossible to chat.
Server-to-client message types
------------------------------
`USER` = a user, the first character being their rank (users with no rank are
represented by a space), and the rest of the string being their username.
### Room initialization
`|init|ROOMTYPE`
> The first message received from a room when you join it. `ROOMTYPE` is
> one of: `chat` or `battle`
`|title|TITLE`
> `TITLE` is the title of the room. The title is _not_ guaranteed to resemble
> the room ID; for instance, room `battle-gen7uu-779767714` could have title
> `Alice vs. Bob`.
`|users|USERLIST`
> `USERLIST` is a comma-separated list of `USER`s, sent from chat rooms when
> they're joined. Optionally, a `USER` can end in `@` followed by a user status message.
> A `STATUS` starting in `!` indicates the user is away.
### Room messages
`||MESSAGE` or `MESSAGE`
> We received a message `MESSAGE`, which should be displayed directly in
> the room's log.
`|html|HTML`
> We received an HTML message, which should be sanitized and displayed
> directly in the room's log.
`|uhtml|NAME|HTML`
> We recieved an HTML message (NAME) that can change what it's displaying,
> this is used in things like our Polls system, for example.
`|uhtmlchange|NAME|HTML`
> Changes the HTML display of the `|uhtml|` message named (NAME).
`|join|USER`, `|j|USER`, or `|J|USER`
> `USER` joined the room. Optionally, `USER` may be appended with `@!` to
> indicate that the user is away or busy.
`|leave|USER`, `|l|USER`, or `|L|USER`
> `USER` left the room.
`|name|USER|OLDID`, `|n|USER|OLDID`, or `|N|USER|OLDID`
> A user changed name to `USER`, and their previous userid was `OLDID`.
> Optionally, `USER` may be appended with `@!` to indicate that the user is
> away or busy.
`|chat|USER|MESSAGE` or `|c|USER|MESSAGE`
> `USER` said `MESSAGE`. Note that `MESSAGE` can contain `|` characters,
> so you can't just split by `|` and take the fourth string.
>
> If `MESSAGE` starts with `/`, it is a special message. For instance,
> `/me TEXT` or `/announce TEXT` or `/uhtml HTML`. A lot of these message
> types are abused to embed protocol messages in PMs (for instance, `/uhtml`
> is a stopgap before the client is rewritten to support `|uhtml|` etc in
> PMs).
>
> If the server wants clients to actually render a message starting with
> `/`, it will send a message starting with `//` (exactly like how users
> need to send those messages).
`|notify|TITLE|MESSAGE`
> Send a notification with `TITLE` and `MESSAGE` (usually, `TITLE` will be
> bold, and `MESSAGE` is optional).
`|notify|TITLE|MESSAGE|HIGHLIGHTTOKEN`
> Send a notification as above, but only if the user would be notified
> by a chat message containing `HIGHLIGHTTOKEN` (i.e. if `HIGHLIGHTTOKEN`
> contains words added to `/highlight`, or their username by default.)
`|:|TIMESTAMP`
`|c:|TIMESTAMP|USER|MESSAGE`
> `c:` is pretty much the same as `c`, but also comes with a UNIX timestamp;
> (the number of seconds since 1970). This is used for accurate timestamps
> in chat logs.
>
> `:` is the current time according to the server, so that times can be
> adjusted and reported in the local time in the case of a discrepancy.
>
> The exact fate of this command is uncertain - it may or may not be
> replaced with a more generalized way to transmit timestamps at some point.
`|battle|ROOMID|USER1|USER2` or `|b|ROOMID|USER1|USER2`
> A battle started between `USER1` and `USER2`, and the battle room has
> ID `ROOMID`.
### Global messages
`|popup|MESSAGE`
> Show the user a popup containing `MESSAGE`. `||` denotes a newline in
> the popup.
`|pm|SENDER|RECEIVER|MESSAGE`
> A PM was sent from `SENDER` to `RECEIVER` containing the message
> `MESSAGE`.
`|usercount|USERCOUNT`
> `USERCOUNT` is the number of users on the server.
`|nametaken|USERNAME|MESSAGE`
> You tried to change your username to `USERNAME` but it failed for the
> reason described in `MESSAGE`.
`|challstr|CHALLSTR`
> You just connected to the server, and we're giving you some information you'll need to log in.
>
> If you're already logged in and have session cookies, you can make an HTTP GET request to
> `https://play.pokemonshowdown.com/action.php?act=upkeep&challstr=CHALLSTR`
>
> Otherwise, you'll need to make an HTTP POST request to `https://play.pokemonshowdown.com/action.php`
> with the data `act=login&name=USERNAME&pass=PASSWORD&challstr=CHALLSTR`
>
> `USERNAME` is your username and `PASSWORD` is your password, and `CHALLSTR`
> is the value you got from `|challstr|`. Note that `CHALLSTR` contains `|`
> characters. (Also feel free to make the request to `https://` if your client
> supports it.)
>
> Either way, the response will start with `]` and be followed by a JSON
> object which we'll call `data`.
>
> Finish logging in (or renaming) by sending: `/trn USERNAME,0,ASSERTION`
> where `USERNAME` is your desired username and `ASSERTION` is `data.assertion`.
`|updateuser|USER|NAMED|AVATAR|SETTINGS`
> Your name, avatar or settings were successfully changed. Your rank and
> username are now `USER`. Optionally, `USER` may be appended with `@!` to
> indicate that you are away or busy.`NAMED` will be `0` if you are a guest
> or `1` otherwise. Your avatar is now `AVATAR`. `SETTINGS` is a JSON object
> representing the current state of various user settings.
`|formats|FORMATSLIST`
> This server supports the formats specified in `FORMATSLIST`. `FORMATSLIST`
> is a `|`-separated list of `FORMAT`s. `FORMAT` is a format name with one or
> more of these suffixes: `,#` if the format uses random teams, `,,` if the
> format is only available for searching, and `,` if the format is only
> available for challenging.
> Sections are separated by two vertical bars with the number of the column of
> that section prefixed by `,` in it. After that follows the name of the
> section and another vertical bar.
`|updatesearch|JSON`
> `JSON` is a JSON object representing the current state of what battles the
> user is currently searching for.
`|updatechallenges|JSON`
> `JSON` is a JSON object representing the current state of who the user
> is challenging and who is challenging the user.
`|queryresponse|QUERYTYPE|JSON`
> `JSON` is a JSON object representing containing the data that was requested
> with `/query QUERYTYPE` or `/query QUERYTYPE DETAILS`.
>
> Possible queries include `/query roomlist` and `/query userdetails USERNAME`.
### Tournament messages
`|tournament|create|FORMAT|GENERATOR|PLAYERCAP`
> `FORMAT` is the name of the format in which each battle will be played.
> `GENERATOR` is either `Elimination` or `Round Robin` and describes the
> type of bracket that will be used. `Elimination` includes a prefix
> that denotes the number of times a player can lose before being
> eliminated (`Single`, `Double`, etc.). `Round Robin` includes the
> prefix `Double` if every matchup will battle twice.
> `PLAYERCAP` is a number representing the maximum amount of players
> that can join the tournament or `0` if no cap was specified.
`|tournament|update|JSON`
> `JSON` is a JSON object representing the changes in the tournament
> since the last update you recieved or the start of the tournament.
> These include:
>
format: the tournament's custom name or the format being used
teambuilderFormat: the format being used; sent if a custom name was set
isStarted: whether or not the tournament has started
isJoined: whether or not you have joined the tournament
generator: the type of bracket being used by the tournament
playerCap: the player cap that was set or 0 if it was removed
bracketData: an object representing the current state of the bracket
challenges: a list of opponents that you can currently challenge
challengeBys: a list of opponents that can currently challenge you
challenged: the name of the opponent that has challenged you
challenging: the name of the opponent that you are challenging
`|tournament|updateEnd`
> Signals the end of an update period.
`|tournament|error|ERROR`
> An error of type `ERROR` occurred.
`|tournament|forceend`
> The tournament was forcibly ended.
`|tournament|join|USER`
> `USER` joined the tournament.
`|tournament|leave|USER`
> `USER` left the tournament.
`|tournament|replace|OLD|NEW`
> The player `OLD` has been replaced with `NEW`
`|tournament|start|NUMPLAYERS`
> The tournament started with `NUMPLAYERS` participants.
`|tournament|replace|USER1|USER2`
> `USER1` was replaced by `USER2`.
`|tournament|disqualify|USER`
> `USER` was disqualified from the tournament.
`|tournament|battlestart|USER1|USER2|ROOMID`
> A tournament battle started between `USER1` and `USER2`, and the battle room
> has ID `ROOMID`.
`|tournament|battleend|USER1|USER2|RESULT|SCORE|RECORDED|ROOMID`
> The tournament battle between `USER1` and `USER2` in the battle room `ROOMID` ended.
> `RESULT` describes the outcome of the battle from `USER1`'s perspective
> (`win`, `loss`, or `draw`). `SCORE` is an array of length 2 that denotes the
> number of Pokemon `USER1` had left and the number of Pokemon `USER2` had left.
> `RECORDED` will be `fail` if the battle ended in a draw and the bracket type does
> not support draws. Otherwise, it will be `success`.
`|tournament|end|JSON`
> The tournament ended. `JSON` is a JSON object containing:
>
results: the name(s) of the winner(s) of the tournament
format: the tournament's custom name or the format that was used
generator: the type of bracket that was used by the tournament
bracketData: an object representing the final state of the bracket
`|tournament|scouting|SETTING`
> Players are now either allowed or not allowed to join other tournament
> battles based on `SETTING` (`allow` or `disallow`).
`|tournament|autostart|on|TIMEOUT`
> A timer was set for the tournament to auto-start in `TIMEOUT` seconds.
`|tournament|autostart|off`
> The timer for the tournament to auto-start was turned off.
`|tournament|autodq|on|TIMEOUT`
> A timer was set for the tournament to auto-disqualify inactive players
> every `TIMEOUT` seconds.
`|tournament|autodq|off`
> The timer for the tournament to auto-disqualify inactive players was
> turned off.
`|tournament|autodq|target|TIME`
> You have `TIME` seconds to make or accept a challenge, or else you will be
> disqualified for inactivity.
Battles
-------
### Playing battles
Battle rooms will have a mix of room messages and battle messages. [Battle
messages are documented in `SIM-PROTOCOL.md`][sim-protocol].
[sim-protocol]: ./sim/SIM-PROTOCOL.md
To make decisions in battle, players should use the `/choose` command,
[also documented in `SIM-PROTOCOL.md`][sending-decisions].
[sending-decisions]: ./sim/SIM-PROTOCOL.md#sending-decisions
### Starting battles through challenges
`|updatechallenges|JSON`
> `JSON` is a JSON object representing the current state of who the user
> is challenging and who is challenging the user. You'll get this whenever
> challenges update (when you challenge someone, when you receive a challenge,
> when you or someone you challenged accepts/rejects/cancels a challenge).
`JSON.challengesFrom` will be a `{userid: format}` table of received
challenges.
`JSON.challengeTo` will be a challenge if you're challenging someone, or `null`
if you haven't.
If you are challenging someone, `challengeTo` will be in the format:
`{"to":"player1","format":"gen7randombattle"}`.
To challenge someone, send:
/utm TEAM
/challenge USERNAME, FORMAT
To cancel a challenge you made to someone, send:
/cancelchallenge USERNAME
To reject a challenge you received from someone, send:
/reject USERNAME
To accept a challenge you received from someone, send:
/utm TEAM
/accept USERNAME
Teams are in [packed format](./sim/TEAMS.md#packed-format). `TEAM` can also be
`null`, if the format doesn't require user-built teams, such as Random Battle.
Invalid teams will send a `|popup|` with validation errors, and the `/accept`
or `/challenge` command won't take effect.
If the challenge is accepted, you will receive a room initialization message.
### Starting battles through laddering
`|updatesearch|JSON`
> `JSON` is a JSON object representing the current state of what battles the
> user is currently searching for. You'll get this whenever searches update
> (when you search, cancel a search, or you start or end a battle)
`JSON.searching` will be an array of format IDs you're currently searching for
games in.
`JSON.games` will be a `{roomid: title}` table of games you're currently in.
Note that this includes ALL games, so `|updatesearch|` will be sent when you
start/end challenge battles, and even non-Pokémon games like Mafia.
To search for a battle against a random opponent, send:
/utm TEAM
/search FORMAT
Teams are in packed format (see "Team format" below). `TEAM` can also be
`null`, if the format doesn't require user-built teams, such as Random Battle.
To cancel searching, send:
/cancelsearch
### Team format
Pokémon Showdown always sends teams over the protocol in packed format. For
details about how to convert and read this format, see [sim/TEAMS.md](./sim/TEAMS.md).
If you're not using JavaScript and don't want to reimplement these conversions,
[Pokémon Showdown's command-line client](./COMMANDLINE.md) can convert between
packed teams and JSON using standard IO.
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,085
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,492
|
\section{INTRODUCTION}
The Gross-Pitaevskii (GP) equation
\cite{dalfovo99,pethick02,pitaevskii03} has been extremely
successful in describing a wide range of mean-field features for
experiments with Bose-Einstein condensates (BECs). In particular, the
Thomas-Fermi (TF) approximation
\cite{baym96,dalfovo96,lundh97}, where the kinetic
energy is neglected, has been very rewarding \cite{hau98}. This
approximation holds for repulsive condensates with positive scattering
length $a$ and large particle numbers.
In the regime of
validity of the TF approximation, the total energy is distributed between
interaction energy and potential energy from the confining trap, while
the kinetic energy becomes negligible.
Because of the non-linear nature of the GP equation, it is only solved
analytically in a few cases, e.g., vortices
and solitons in homogeneous condensates \cite{pethick02,pitaevskii03}.
The TF solution is also analytical, although it only holds in
the bulk of the condensate. At the surface the approximation breaks
down and is usually patched by including the kinetic energy at the surface
\cite{dalfovo96,lundh97}.
The interactions of the ordinary GP equation are based on the lowest
order zero-range potential, which is governed by the scattering length
alone. Although this approximation is usually very good, the
higher-order corrections to the scattering dynamics
\cite{roth01,fu03,collin07} can be crucial in certain cases, e.g., for
Rydberg molecules embedded in BECs \cite{collin07} and for narrow
Feshbach resonances \cite{zinner2009}. Inclusion of higher-order terms
is well known and applied in
Skyrme-Hartree-Fock calculations in nuclear physics \cite{brack85}. Here, they
often play a crucial role in order to get bulk nuclear properties right \cite{sky56,sie87}.
However, the effects of similar higher-order terms in the GP equation
have been less investigated.
In this paper, we solve the modified GP equation with higher-order
interactions analytically in the TF approximation. The paper is
organized as follows. In Sec.~\ref{MGPeq}, we introduce the
modified GP equation and its parameters and show how it is derived from
an appropriate energy density functional with careful treatment of
boundary terms. We present the analytical solution in the TF approximation
in Sec.~\ref{TFA} and discuss the condensate size and chemical potential
as function of the interaction parameters in Sec.~\ref{sizechem}. The
density profiles and energies are discussed in Sec.~\ref{densene}, and
in Sec.~\ref{cons}, we address the consistency of the TF approximation by
considering the kinetic energy of the solutions. We compare to some relevant
atomic systems in Sec.~\ref{compare} and finally
present our conclusions in Sec.~\ref{conc}.
\section{MODIFIED GP EQUATION}\label{MGPeq}
We assume that the condensate can be described by the GP equation.
Since we are interested in the ultracold regime, where the temperature
is much smaller than the critical temperature for condensation, we
adopt the \mbox{$T=0$} formalism. In order to include higher-order
effects in the two-body scattering dynamics, we use the modified GP
equation derived in \cite{collin07}, which in the stationary form reads
\begin{equation}
\mu\Psi
=\left[-\frac{\hbar^2}{2m}\nabla^2
+V(r)+U_0\left(|\Psi|^2+g_2\nabla^2|\Psi|^2\right)
\right]\Psi,
\label{GPE}
\end{equation}
where $m$ is the atomic mass, $V$ is the external trap,
$U_0=4\pi\hbar^2 a/m$, and $g_2=a^2/3-ar_e/2$, with $a$ and $r_e$
being, respectively, the $s$-wave scattering length and effective range
\cite{collin07}. We assume an isotropic trap, $V(r)=m\omega^2 r^2/2$, and introduce the
trap length $b=\sqrt{\hbar/m\omega}$. The single-particle density,
$\rho(r)=|\Psi(r)|^2$, is normalized to the particle number, $N=\int\mathrm{d}{\bm r}\rho(r)$,
and $\mu$ is the chemical potential.
As the boundary conditions are important for the TF
approximation applied below we now discuss the procedure for obtaining
the modified GP equation from the corresponding energy functional which is
\begin{equation}\label{efunc}
E(\Psi)=\int \mathrm{d}{\bm r} (
\epsilon_K +\epsilon_V +\epsilon_I +\epsilon_{I2} ),
\end{equation}
with kinetic, potential, and interaction energy densities
\begin{eqnarray}
&\epsilon_K=\frac{\hbar^2}{2m}|\nabla\Psi|^2,\qquad
&\epsilon_V=V(\bm r)|\Psi|^2,\\
&\epsilon_I=\frac{1}{2}U_0 |\Psi|^4,\qquad
&\epsilon_{I2}=\frac{1}{2}U_0 g_2|\Psi|^2 \nabla^2|\Psi|^2.
\end{eqnarray}
The corresponding integrated energy contributions are denoted $E_K$, $E_V$,
$E_I$, and $E_{I2}$, respectively.
To obtain Eq.~\eqref{GPE}, we vary Eq.~\eqref{efunc} with respect to $\Psi^*$ for fixed $\Psi$.
To first
order in $\delta\Psi^*$, we have
\begin{equation}\label{GPvar}
\begin{split}
\delta E
&=E[\Psi^*+\delta\Psi^*]-E[\Psi^*]\\
&=\int\mathrm{d}{\bm r} \bigg[ -\frac{\hbar^2}{2m}\nabla^2\Psi+ V(\bm r)\Psi\\
&\qquad\qquad+U_0\left( |\Psi|^2+g_2\nabla^2|\Psi|^2\right)\Psi \bigg] \delta\Psi^*\\
&+\int\mathrm{d}{\bm S}\cdot |\Psi|^2\nabla\left(\Psi\delta\Psi^*\right)
-\int\mathrm{d}{\bm S}\cdot \Psi\delta\Psi^*\nabla|\Psi|^2\\
&+\int\mathrm{d}{\bm S}\cdot \delta\Psi^*\nabla\Psi.
\end{split}
\end{equation}
Here, $\bm S$ is the outward-pointing surface normal. In the usual
analysis, one assumes that $\Psi$ and $\nabla\Psi$ vanishes at
infinity, drops the boundary terms, and Eq.~\eqref{GPE} is obtained by
varying $E-\mu N$. However, the existence of these surface terms is
essential for the inclusion of higher-order interactions as discussed below.
In the rest of this paper we use trap units, $\hbar\omega=b=1$, i.e.,
energies ($E$, $V$, $\mu$, etc.) are measured in units of $\hbar\omega$ and
lengths ($a$, $r_e$, $r$, etc.) in units of $b$. Note that
$g_2$ has dimension of length squared.
\section{THOMAS-FERMI APPROXIMATION}\label{TFA}
Let us briefly review the standard Thomas-Fermi approximation
\cite{baym96,dalfovo99,pethick02,pitaevskii03}. Neglecting the kinetic-energy
term, as compared to the trap and interaction energies, the GP
equation has the solution
\begin{equation}
\label{eq:density-TF}
\rho_{TF}=\frac{1}{4\pi a}(\mu_{TF}-\frac{1}{2}r^2),
\end{equation}
with chemical potential $\mu_{TF}$. This solution is used out to the
surface, $R_{TF}$, while outside $\rho_{TF}=0$. The
normalization and surface condition $\rho_{TF}(R_{TF})=0$ give
\begin{equation}
\label{eq:mu-rmax-TF}
\mu_{TF}=\frac{1}{2}R_{TF}^2.\qquad
R_{TF}=(15Na)^{1/5}.
\end{equation}
The total energy becomes
\begin{equation}
\frac{E_{TF}}{N}=\frac{5}{7}\frac{R_{TF}^2}{2}.
\end{equation}
The trap and interaction energies are $E_V=3E/5$ and $E_I=2E/5$,
respectively. Since $R_{TF}>0$ in Eq.~\eqref{eq:mu-rmax-TF}, these
results only hold for $a>0$.
The TF approximation is good for $Na\gg 1$, except at the surface
region where the kinetic-energy density diverges. Here, the solution
can be corrected as in \cite{dalfovo96,lundh97,pethick02,pitaevskii03}, essentially
giving a small exponential tail.
\subsection*{Inclusion of higher-order interactions}
We now consider the TF approximation with the
higher-order interaction term, $\epsilon_{I2}$. Ignoring the boundary
terms in Eq.~\eqref{GPvar}, the modified GP equation can then be
written in terms of the density $\rho(r)=|\Psi(r)|^2$ as
\begin{equation}
\label{eq:gpe-dimless}
{\mu} =\frac{1}{2} r^2+4\pi a \left(\rho+ g_2
\nabla^2 \rho\right).
\end{equation}
With scaled coordinate $x=r/\sqrt{g_2}$ (assuming $g_2>0$ for the moment) and density $f(r)=4\pi a x\rho(r)/g_2$, this becomes
\begin{equation}
\frac{\mathrm{d}^2 f}{\mathrm{d} x^2}+f= \frac{\mu}{g_2} x -\frac{1}{2} x^3,
\end{equation}
The inhomogeneous and homogeneous solutions with boundary condition
$f(0)=0$ are
\begin{equation}
\label{eq:f-solution}
f_i(x)=(\frac{\mu}{g_2}-\frac{1}{2}x^2 +3) x, \quad
f_h(x)=\frac{A}{g_2} \sin x,
\end{equation}
where $A$ is a constant (with dimensions of length squared) to be
determined later. The full solution is
\begin{equation}
\label{eq:density-solution}
\rho( x)=\frac{g_2}{4\pi a}\left[
\frac{\mu}{g_2} -\frac{1}{2} x^2 +3
+\frac{A}{g_2}\frac{\sin x}{x}
\right].
\end{equation}
For a given $A$, the chemical potential $\mu$ and the condensate radius $R$
are determined by the normalization and the surface condition,
\begin{equation}
\label{eq:norm-surf-cond}
\int_{0}^{x_0} 4\pi x^2 \rho(x)\mathrm{d} r=N
\quad\textrm{and}\quad
\rho(x_0)=0,
\end{equation}
where $x_0=R/\sqrt{g_2}$.
The solution $\rho$ should be positive for $x<x_0$ which must be
explicitly checked. Outside $x_0$, we use $\rho=0$.
We now consider the boundary terms in Eq.~\eqref{GPvar}. Above, we
assumed that $\rho(x_0)=0$ at some finite radius $x_0$ which we identify as the
condensate size. However, only the first two boundary terms in
Eq.~\eqref{GPvar} vanish on account of this
condition. For the last term in Eq.~\eqref{GPvar} to vanish we
need $\nabla_x\Psi(x_0)=0$, which implies that
\begin{equation}
\label{eq:surface-derivative}
\frac{d\rho}{dx}(x_0)=0.
\end{equation}
Notice that this latter derivative is in fact non-zero in the $g_2=0$ case, which is
the root of the divergence of the kinetic energy at the condensate surface as we discuss later.
Equation~(\ref{eq:surface-derivative}) gives a closed expression for the remaining free parameter $A$,
\begin{equation}
\label{eq:A-vs-z}
\frac{A}{g_2}=\frac{x_{0}^{3}}
{x_0\cos x_0-\sin x_0}.
\end{equation}
This additional requirement on the derivative at the edge of the
condensate implies that higher-order terms require a smoothing at the
surface of the cloud.
In addition, the
discussion of which kinetic operator structure to use
($|\nabla\Psi|^2$ or $\Psi^*\nabla^2\Psi$ \cite{lundh97}) is
obsolete in our treatment since the boundary term $\delta\Psi^*\nabla\Psi$
vanishes. In this sense the
inclusion of a higher-order term neatly removes some of the
difficulties of the traditional TF treatment.
The solutions with a finite boundary $R$ of the modified GP equation
only minimize the energy functional if
Eq.~\eqref{eq:surface-derivative} holds. We note that extremal states
of the energy functional always satisfy the virial theorem. Thus,
enforcing the virial theorem on the GP solutions is equivalent to
Eq.~\eqref{eq:surface-derivative}. We show in the Appendix that the virial
theorem approach also leads to Eq.~\eqref{eq:A-vs-z}.
\section{SIZE AND CHEMICAL POTENTIAL}\label{sizechem}
We now determine the condensate size $R$ and chemical potential $\mu$.
The normalization condition is
\begin{equation}
\label{eq:normalization-condition2}
\frac{Na}{g_2^{5/2}}=x_0^3\left(\frac{\mu}{3g_2}-\frac{x_0^2}{10}\right),
\end{equation}
while the surface condition reads
\begin{equation}
\label{eq:surface-condition2}
\frac{\mu}{g_2} -x_0^2/2+3+\frac{A}{g_2}\frac{\sin x_0}{x_0} =0.
\end{equation}
Combining eq.~\eqref{eq:A-vs-z}-\eqref{eq:surface-condition2} gives
\begin{equation}
\label{eq:rmax-equation}
\frac{Na}{g_2^{5/2}}=x_0^3\left(\frac{x_0^2}{15}-1+\frac{x_0^2/3}{1-x_0\cot x_0}\right),
\end{equation}
which determines $R$ for given $Na$ and $g_2$, and upon
back-substitution also $\mu$.
The $g_2<0$ case can be worked out analogously by replacing trigonometric
functions with hyperbolics and keeping track of signs. The two cases
can in fact be combined into one equation
\begin{equation}
\label{eq:rmax-equation2}
\frac{Na}{|g_2|^{5/2}}
=|x_0|x_0^2\left(\frac{x_0^2}{15}-1+\frac{x_0^2/3}{1-|x_0\cot x_0|}\right).
\end{equation}
This equation determines $x_0^2=R^2/g_2$ implicitly as function of $N
a/|g_2|^{5/2}$. The result is shown in Fig.~\ref{fig-rmax}. We notice that
in principle, $R$ becomes a multi-valued function. However, all the
higher solutions for $g_2>0$ [dotted in Fig.~\ref{fig-rmax}] are
spurious, since the density becomes negative on one or more intervals
inside $R$.
The non-spurious solutions [solid line in Fig.~\ref{fig-rmax}] define
$R$ as a single-valued function of $a$ and $g_2$, which was not
guaranteed a priori.
The four quadrants in Fig.~\ref{fig-rmax} correspond to the different
sign combinations of $a$ and $g_2$.
The sign of the extra interaction energy, $E_{I2}$,
is determined by $a
g_2\nabla^2\rho$. For a typical concave density, the Laplacian term will
be negative. We therefore see that for $ag_2>0$, the higher-order
interaction is attractive, whereas for $ag_2<0$, it is repulsive.
The TF solution only exists for $ag_2<0$. We discuss both cases separately below.
\begin{figure}[t!]
\epsfig{file=fig-rmax.eps,clip=true,scale=1.0}
\caption{(Color online)
Condensate size ($R$) as function of $Na$ and $g_2$ as
found in the modified TF approximation,
Eq.~\eqref{eq:rmax-equation2}. The solutions (a) and (b)
correspond to the sign combinations ($a>0$, $g_2<0$) and ($a<0$,
$g_2>0$), respectively. No solutions exist for $a g_2>0$. The
spurious solutions (dotted) have negative densities for one or
more intervals inside $R$. The branch (a) approaches the normal TF
result Eq.~\eqref{eq:mu-rmax-TF} when $Na\to+\infty$ or $g_2\to
-0$. Note that the convergence is only relative [see
eq.~\eqref{eq:rmax-equation2}] and the TF limit is better
represented in the logarithmic inset. Points indicate the data
from Tab.~\ref{tab:data}. All values are in trap units.}
\label{fig-rmax}
\end{figure}
\subsection{The attractive regime: $ag_2>0$}
For $a<0$, $g_2<0$ [third quadrant in Fig.~\ref{fig-rmax}] there are
no solutions, which is expected since the normal TF approximation has
no solutions for $a<0$ as the interaction energy $E_I$ is negative
and the kinetic energy that could prevent collapse is neglected.
The $g_2>0$, $a>0$ case in the first quadrant has only spurious
solutions. Here the $g_2$ term is attractive for the typical concave
density and a collapse towards a high-density state is possible in
complete analogy to the usual discussion of attractively interacting
condensates within the standard GP theory. Whereas there can be
metastable states at large values of $Na/g_{2}^{5/2}$, these are
stabilized by kinetic energy and thus are not present in our TF
approach. Thus, even when the total kinetic energy is small, it
is still needed to prevent the attractive higher-order term from
amplifying local-density variations.
This important point can also be established by considering the stability of the
homogeneous condensate through linearization of the GP equation.
By repeating the analysis of \cite{pethick02} with the higher-order term,
we find that for $g_2>0$
and $a>0$, the kinetic-energy term is crucial for the stability
of the excitation modes. In fact, exponentially growing modes
will always be present if the kinetic energy is neglected.
This will be discussed elsewhere in relation to the
numerical solution of the full GP equation \cite{thoger2009b}.
\subsection{The repulsive regime: $ag_2<0$}
For $g_2<0$, $a>0$ a single solution (a) exists. This was expected
since $E_{I2}>0$ gives extra stability. The solution approaches the
normal TF result in Eq.~\eqref{eq:mu-rmax-TF} when
$Na/|g_2|^{5/2}\to+\infty$, as can also be seen from
Eq.~\eqref{eq:rmax-equation2}. Of course in this limit $E_{I2}\ll E_I$.
However, the $-1$ term in Eq.~\eqref{eq:rmax-equation2} implies that
the convergence to the normal TF solution
is only on a relative scale and is better represented on a
logarithmic scale as in the
inset in Fig.~\ref{fig-rmax}.
For $g_2>0$,
$a<0$ there is a single solution (b) which connects smoothly to the
(a) solution. In the limit $Na/|g_2|^{5/2}\to -\infty$, which is
determined by $x_0\cot x_0=1$, we find $R^2/g_2=20.1907$. This solution
is possible when the $g_2$ term provides just enough repulsion to
cancel the usual $a<0$ collapse behavior.
\subsection{Chemical potential}
In Fig.~\ref{fig-mu} we show the chemical potential for the smoothly
connecting solutions (a) and (b). Again we see that (a) approaches the
normal TF limit for large $Na/|g_{2}|^{5/2}$. Here, it is
interesting to note how $\mu$ turns around near the origin
[amplified in the inset in Fig.~\ref{fig-mu}] and maintains a positive
value. This occurs in the
region where the lowest-order interaction gives a large
negative-energy contribution which the $g_2$ term is still able
to balance yielding a well-defined TF solution. This behavior is
analogous to the balancing of attraction by the kinetic term
in the usual $a<0,g_2=0$ case \cite{baym96,dalfovo99}. As $a$
becomes increasingly negative, so too does $\mu$ and collapse
is inevitable (and likewise when $g_2\rightarrow 0^+$).
\begin{figure}[t!]
\epsfig{file=fig-mu.eps,clip=true,scale=1.0}
\caption{(Color online) Chemical potential $\mu$ as function of $Na$
and $g_2$ as found in the modified TF approximation, using the
solutions (a) and (b) from Fig.~\ref{fig-rmax}. For branch (a)
and the upper part of branch (b) (see inset), we have $\mu>0$. The
lower part of (b) has $\mu<0$. Points indicate data from
Tab.~\ref{tab:data}. All values are in trap units.}
\label{fig-mu}
\end{figure}
\section{DENSITIES AND ENERGIES}\label{densene}
With $R$ and $\mu$ determined, we can find the density profile, energy
densities and integrated energy contributions. With
Eq.~\eqref{eq:density-solution}, the energy densities are given by
\begin{eqnarray}
&\epsilon_V=\frac{ x^2}{2} \rho,\qquad
\epsilon_I=2\pi a \rho^2,\\
&\epsilon_{I2} =-\frac{1}{2} \rho(3
+\frac{A}{ g_2 } \frac{\sin x}{x}).
\end{eqnarray}
Using Eq.~\eqref{eq:gpe-dimless}, the total energy density (without
$\epsilon_K$) becomes
\begin{equation}
\label{eq:energy-dens}
\epsilon \equiv \epsilon_V+\epsilon_I+\epsilon_{I2}
=\frac{1}{2} \rho(x) (V(x)+\frac{\mu}{g_2}).
\end{equation}
In Fig.~\ref{fig-dens-a}, we show the density profile of the (a) solutions
for $Na=10^4$ and selected $g_2<0$. We clearly see that the higher-order
term tends to expand the condensate through its repulsion. Importantly,
at the boundary, there is a smoothing caused by the
condition in Eq.~\eqref{eq:surface-derivative}
[see inset in Fig.~\ref{fig-dens-a}]. We will
discuss how this affects the estimated kinetic energy in the next section.
As $|g_2|$ grows, we see the condensate flatten and in the limit of
very large $|g_2|$, it becomes a constant density.
\begin{figure}[t!]
\epsfig{file=fig-dens-a.eps,clip=true,scale=1.0}
\caption{(Color online) Densities for branch (a) in
Fig.~\ref{fig-rmax} ($g_2<0$ and $Na=10^4$). The $g_2=-0.1$ curve
is on top of the normal TF result. Inset shows the smooth behavior at the surface
for $g_2<0$.
All values are in trap units.}
\label{fig-dens-a}
\end{figure}
Figure~\ref{fig-dens-b} displays the density profile for the (b) solutions with
$a<0$ for selected $g_2>0$. Here, we see the profile collapse toward the
expected delta-function with decreasing $g_2$.
It is interesting to follow the (a) solution through the origin in Fig.~\ref{fig-rmax} and
onto solution branch (b), passing from
$g_2=-\infty$ to $g_2=\infty$. On the (a) branch, the solution flattens as $g_2$ decreases
and eventually becomes effectively constant in space. This is also true for the (b) branch
at $g_2=\infty$, and as $g_2$ is decreased, the solution proceeds to shrink as the
$g_2$ term becomes unable to provide the repulsion needed to prevent the $a<0$ collapse
induced by the lowest-order term.
\begin{figure}[t!]
\epsfig{file=fig-dens-b.eps,clip=true,scale=1.0}
\caption{(Color online) Same as Fig.~\ref{fig-rmax} but for
solutions (b), i.e., opposite signs $g_2<0$ and $Na=-10^4$. }
\label{fig-dens-b}
\end{figure}
From the figures, we see that large $|g_2|$ induces large changes in cloud size. As the condensate
can be imaged with very good resolution \cite{hau98}, this should be measurable if the
regime of large $|g_2|$ can be accessed.
{\renewcommand{\tabcolsep}{0.3cm}
\begin{table*}[t!]
\centering
\begin{tabular}{cc|cccccccc}
\hline \hline
&$g_{2}$&$R$&$\mu$&$E_{V}/N$&$E_{I}/N$&$E_{I2}/N$&$E/N$&$E_R/N$&$E_{K}/|E|$\\
\hline
TF&---&10.8447&58.8040&25.2017&16.8011&---&42.0028&16.8012&3.135$\times10^{-3}$ \footnotemark[1] \\
\hline
&$-0.01$ &10.9447&58.8188&25.2164&16.7865&0.01465&42.0176&16.8012& 1.8$\times10^{-3}$\\
&$-0.1$ &11.1607&58.9481&25.3430&16.6635&0.13909&42.1456&16.8026& 1.4$\times10^{-3}$\\
(a)&$-1.0$ \footnotemark[2]&11.8364&60.1210&26.4309&15.6818&1.16330&43.2760&16.8451& 1.0$\times10^{-3}$\\
&$-10$ \footnotemark[2] &13.7835&68.4515&32.9856&11.3469&6.38609&50.7186&17.7330& 0.57$\times10^{-3}$\\
&$-50$ \footnotemark[2] &16.439 &87.8248&45.6836&6.99293&14.0777&66.7542&21.0706&0.30$\times10^{-3}$\\
\hline
&50 \footnotemark[2] &15.407 & 63.0102&38.9723 &$-8.92496$&20.9439&50.9912 &12.0189&0.43$\times10^{-3}$\\
&10 \footnotemark[2] &11.170& 15.9128&19.8375 &$-24.7434$&22.7810&17.8751 &$-1.9623$&2.3$\times10^{-3}$\\
(b)&5.14 \footnotemark[3]&9.1999 &$-13.1384$&13.1579 &$-46.0283$&32.8801&0.0097 &$-13.148$&6.098\\
&1.0 \footnotemark[2] &4.4801 &$-327.612$&3.04199 &$-416.359$&251.032&$-162.285$&$-165.33$&1.5$\times10^{-3}$\\
&0.1 &1.4204 &$-10456.4$&0.30571&$-13071.1$&7842.81&$-5227.98$ &$-5228.4$&0.47$\times10^{-3}$\\
\hline \hline
\end{tabular}
\footnotetext[1]{The kinetic energy estimated by surface corrections as in \cite{pethick02}.}
\footnotetext[2]{Values are indicated by points in Figs.~\ref{fig-rmax} and \ref{fig-mu}.}
\footnotetext[3]{The total energy $|E|$ is zero near $g_2=5.14$, hence the TF approximation is
invalid here.}
\caption{Chemical potential $\mu$ and condensate size $R$
for different $g_2$ and fixed
$N|a|=10^4$.
The integrated energies are trap ($E_V$), interaction ($E_I,E_{I2}$), total ($E=E_V+E_I+E_{I2}$), and
release energy ($E_R=E-E_V$).
The TF limit is approached for $g_2\to -0$. The ratio
of kinetic energy $E_K$ to total energy $E$ indicates
where the TF approximation is valid. The corresponding density
distributions are shown in Figs.~\ref{fig-dens-a} and
\ref{fig-dens-b}. All values are in trap units.}
\label{tab:data}
\end{table*}
}
We now discuss the energy contributions which are interesting since the
release energies are in fact measurable quantities \cite{pitaevskii03}. Since we neglect
the kinetic term in the TF approximation, the release energy is simply $E_R=E_I+E_{I2}=E-E_V$.
In Tab.~\ref{tab:data}, we give the integrated energy contributions for some relevant values of $g_2$
calculated for $N|a|=10^4$, whereas Fig.~\ref{fig-energy} gives the energies as function of $Na/|g_2|^{5/2}$.
We note that for smaller values of $N|a|$, the same overall behavior is found,
however, the kinetic term is more important and the TF approximation becomes worse.
\begin{figure}[t!]
\epsfig{file=fig-energy.eps,clip=true,scale=1.0}
\caption{(Color online) Different total-energy contributions. $N|a|=10^4$. All values are in trap units.}
\label{fig-energy}
\end{figure}
We observe that $E/N$ grows towards the $|g_2|=\infty$ point. This is due to the trap energy increasing
as the density flattens [$E_V$ diverges around the origin in Fig.~\ref{fig-energy}].
Furthermore, as $g_2\rightarrow 0^+$ the energy diverges toward $-\infty$
as the collapse sets in [$E_I$ diverges on the $g_2>0$ side in Fig.~\ref{fig-energy}].
The boundary where the energy vanishes is around $g_2=5.14$ for $N|a|=10^4$, but
this depends on the choice of $N|a|$. With respect to the release energy, we find that somewhere in
the region $10<g_2<50$, $E_R$ becomes negative. This is a result of the unavoidable collapse, and
also indicates that kinetic energy cannot be ignored at this point. Notice, however, that
the release energy changes considerably and could provide a way to measure the influence of the
$g_2$ term.
\section{CONSISTENCY OF THE THOMAS-FERMI APPROXIMATION}\label{cons}
We now address the validity of the TF approximation with the $g_2$ term included. In
order to do so, we must consider the contribution of the kinetic energy.
The kinetic energy density can be written as
\begin{equation}
\label{eq:eK}
\epsilon_K
=\frac{g_2}{8 \rho(4\pi a)^2}\left(
x+\frac{A}{g_2}
\frac{x
\sin x
-\cos x}{x}
\right)^2.
\end{equation}
Strictly speaking, this is not the true kinetic energy, since the
kinetic terms were neglected from the start. However
Eqs.~\eqref{eq:energy-dens} and \eqref{eq:eK} can be used to test
whether the TF approximation holds locally, i.e., $\epsilon_K \ll
\epsilon$ should hold for the solution $\rho$ to be consistent. In Tab.~\ref{tab:data},
we calculate the integrated contribution of the kinetic energy relative to
the total TF energy and we find that the contribution is small everywhere except
the point where $E=0$ on the $g_2>0$ side of Fig.~\ref{fig-energy}. Here, the kinetic energy is of course
the most important term and the TF approximation is poor.
In the standard TF, the kinetic energy causes trouble at the boundary of the cloud.
Here, $\nabla\Psi\propto \nabla\rho/\sqrt{\rho}$ and since the density vanishes
and the derivative is finite [see Eq.~\ref{eq:density-TF}], this diverges at $R_{TF}$.
When including the higher-order term we need to use the additional boundary
condition $\nabla\Psi=0$ at $R$, so the kinetic energy will be strictly zero
at $R$. However, as one approaches the boundary, the kinetic-energy density grows
rapidly before it descends towards zero within a very small interval
at $R$. The total energy density in Eq.~(\ref{eq:energy-dens}) goes to zero at this point
and we find that $\epsilon_K/\epsilon$ is very large near the boundary as in
the usual $g_2=0$ case.
We conclude that the inclusion of the higher-order term does not alleviate the
difficulties with kinetic energy at the boundary. The techniques for
addressing this problem described in \cite{dalfovo96,lundh97} should therefore
be generalized to include the higher-order interaction term in order to
improve the description at the boundary of the cloud.
\section{COMPARISON TO ATOMIC SYSTEMS}\label{compare}
The considerations above show that deviations from the usual TF approximation
can be strong when $g_2$ is large. In the following, we reintroduce explicit units
for comparison with real systems. We have to consider $g_2/b^2$.
Of course, the $b^2$ factor means
that this quantity is generally very small since $g_2$ is of order $a_{0}^{2}$
and $b$ is of order $10^4 a_0$.
We first consider some typical
background values for bosonic alkali atoms away from resonance. We estimate
the effective range to be the of order of the potential range and assuming a
van der Waals interaction, we have $r_e\sim 50-200a_0$.
For typical one-component gases we have
$-450a_0\lesssim a \lesssim 2500a_0$ \cite{chin2009}. Since $g_2=a^2/3-ar_e/2$,
we see that the $a^2$ term will dominate and in all cases
$0<g_2\lesssim 10^6a_0$. In trap units, this becomes
$g_2/b^2\lesssim 5\cdot10^{-3}(1\mu\text{m}/b)^2$.
In typical traps of $b\sim 1-10\mu\text{m}$,
the higher-order term is therefore very small.
These values also predominantly lie in the first quadrant of
Fig.~\ref{fig-rmax} and thus no TF solution exists.
Let us first consider Feshbach resonances in order to increase the influence of
the $g_2$ term.
We use a multi-channel Feshbach model \cite{bruun05},
which describes the full T matrix
as a function of resonance position
$B_0$, width $\Delta B$, magnetic-moment difference between the
channels $\Delta \mu$, and the background scattering length $a_{bg}$.
Performing an effective-range expansion \cite{zinner2009},
we have $a=a_{bg}[1-\Delta B/(B-B_0)]$ and
$r_e=r_{e0}/[1-(B-B_0)/\Delta B]^2$, where
$r_{e0}=-2\hbar^2/ma_{bg}\Delta \mu\Delta B<0$.
Combining these relations, we find $r_e=r_{e0}(1-a_{bg}/a)^2$ and
\begin{equation}
\label{eq:g2(a)}
g_2(a)=\frac{a^2}{3}-\frac{a r_{e0}}{2}(1-\frac{a_{bg}}{a})^2.
\end{equation}
Hence $g_2$ diverges when $a\to 0$ (referred to as zero-crossing) or
$a\to\infty$ (on resonance). Near zero crossing, the effective-range
expansion is, however, severely divergent and its validity is questionable.
Even so, the effective-range corrections near zero-crossing obtained
are in fact identical to those obtained from use of the full T-matrix \cite{zinner2009b}.
One finds $\lim_{a\rightarrow0}a g_2=|r_{e0}|a_{bg}^2/2$, where $r_{e0}<0$.
As a concrete example, we consider
the alkali isotope $^{39}$K where several Feshbach
resonances of vastly different widths were found recently \cite{errico07}.
First, we focus on zero-crossing and consider the very narrow resonance at $B_0=28.85$G with
$\Delta B=-0.47$G, $\Delta \mu=1.5\mu_B$, and $a_{bg}=-33a_0$.
We obtain
$r_{e0}=-5687a_0$ and $ag_2\rightarrow 93.8\cdot10^3a_{0}^{3}$ for $a\rightarrow 0$.
It is important to notice that $ag_2>0$ around $a=0$. This means that we are looking
for solutions in the first and third quadrants of Fig.~\ref{fig-rmax} and again we
have to conclude that no TF solutions can be found when higher-order terms are
taken into account.
Another case of interest is around resonance where $|a|=\infty$. Here, we have
$r_e\sim r_{e0}$ and $g_2\propto a^2>0$ on both sides of the resonance. Thus, the
$a>0$ side will be in the first and the $a<0$ in the second quadrant of Fig.~\ref{fig-rmax}.
This makes it difficult to imagine sweeping the resonance from either side to
probe the solutions on branch (b) in Fig.~\ref{fig-rmax}. One could imagine starting on the $a>0$
side with small $g_2>0$. The full GP equation will have perfectly sensible solution here, however,
when one approaches the resonance the $g_2$ term will diverge and induce collapse
already on the $a>0$ side. If we approach from the $a<0$ side, then we face the problem
that the critical number of particles decreases dramatically before $g_2$ grows
sufficiently and one therefore needs a very small condensate since
$Na/b\sim 0.5$ \cite{zinner2009}. At this point, the TF approximation is no longer valid.
The Feshbach resonance used to increase $g_2$ must be very narrow in order for $r_{e0}$
to be large. However, most experimentally known resonances are not narrow. For
broad or intermediate resonances, we have to consider the long-range van der Waals interaction
when calculating the effective-range corrections. Analytic formulas for this case have been
worked out in \cite{gao98}, and we note that the effective range diverges as $a^{-2}$ near zero crossing
exactly as
in the Feshbach model above. For very narrow resonances, we still have $\beta_6\ll r_{e0}$, where
$\beta_6$ is the characteristic length of the van der Waals interaction.
The model above should thus give the dominant contribution.
Using the van der Waals formulas we can estimate $ag_2$ at zero crossing. We find
\begin{align}
\frac{ag_2}{b^3}\rightarrow -\frac{1}{3x_e}\left(\frac{\beta_{6}}{b}\right)^3,
\end{align}
where $x_e=(\Gamma[1/4])^2/2\pi$, with $\Gamma$ the gamma function.
We have explicitly introduced the oscillator length which is the relevant length scale
of comparison. Importantly, we find that $ag_2<0$ for $a>0$ and we are thus in the fourth
quadrant where a TF solution exists.
For $a<0$, we pass to the second quadrant as $g_2$ becomes positive and
a single collapsed solution can be found.
We now estimate the parameters obtained from the van der Waals formulas.
With $b=1\mu$m and $\beta_{6}\sim 123.3a_0$ \cite{pethick02},
we have $ag_2/b^3\sim -10^{-8}$. We thus have $Na/|g_2|^{5/2}\propto 10^{8} (Na) a^{5/2}$.
For values of $a$ that are not extremely small, the solution is therefore
typically located far to the right in Fig.~\ref{fig-rmax} where it will look similar
to the $g_2=0$ case.
We can estimate how close to
zero one would have to tune $a$ in order to see deviations using the $a\rightarrow 0$ limit
of the van der Waals effective range. Let us aim for $g_2/b^2=- 10$ which should be observable
in the condensate profile according to Fig.~\ref{fig-dens-a}. With $b=1\mu$m, we need
$a\sim 10^{-6}\beta_6\sim 1.7\times 10^{-4}a_0$. Using broad resonances, one can tune
to zero at the level of $10^{-2}a_0$ in $^{39}$K \cite{fattori2008}. Observing the effect of the
$g_2$ term therefore seems out of reach at the moment, but might be possible in the near future.
Of course, we still have to maintain a large value of $Na$ for kinetic energy to be small,
and thus a larger condensate is needed close to zero crossing.
From the examples above, we see problems in accessing the TF solutions presented
above in current experiments with ultracold alkali gases.
In particular, we notice that
realistic systems which have been used for creation of BECs in alkali-metal gases for the last
decades have parameters that predominantly lie
in the first quadrant of Fig.~\ref{fig-rmax}. As we have discussed, there are no well-defined TF
solutions in that region. Therefore, we see that the kinetic energy plays
a decisive role and we are forced to consider it in principle, even if it is small
for all practical purposes. The physical reason is that for $a>0$ and $g_2>0$, the
higher-order interaction is effectively attractive and induces collapse which will have to
be balanced by a barrier from the kinetic term, similar to the $a<0$, $g_2=0$ case
\cite{dalfovo99}. Since we neglect the kinetic term in the TF approximation, we should not
expect to find solutions in the $ag_2>0$ case.
Only in the case of resonances
dominated by the long-range van der Waals interaction do the parameters allow
for TF solutions with non-zero $g_2$. However, here the length scale of the
trap makes the contribution very small and the TF solution becomes identical
to the $g_2$ case. One could in principle tune $a$ very close to zero-crossing
and obtain a significant contribution but the level of tuning required is
beyond current experimental reach.
\section{CONCLUSIONS}\label{conc}
We have considered the effect of higher-order interactions in Bose-Einstein
condensates within the Gross-Pitaevskii theory. We derived the GP
equation with effective-range corrections included and solved it
analytically in the Thomas-Fermi approximation.
Higher-order interaction terms act as
derivatives on the condensate wave function which means that the boundary conditions
on the solutions of the GP equation must be carefully considered.
We then discussed the
solutions for various parameters and presented the chemical potential, density profiles,
and the energy contributions.
We find that no TF solutions
are possible when the higher-order term is attractive. This conclusion holds
both in the
trapped system and in the homogeneous case \cite{thoger2009b}.
An estimate of the
relevant parameters for alkali atoms showed that away from resonances,
they typically lie in the region where the effective-range correction
is effectively attractive and likewise near very narrow Feshbach
resonances. We conclude that in those cases,
the kinetic energy, even if very small, is crucial in
order to stabilize collapse due to higher-order interaction terms.
For broader resonances where the long-range van der Waals
potential is dominant, we find that modified TF solutions exits. However,
for typical traps, the parameters are very small and tuning of the
scattering length near zero crossing at a level beyond current experimental
reach is necessary. This might of course become possible as experimental
control improves in the future.
\paragraph*{ACKNOWLEDGMENTS}
Discussions with D. V. Fedorov, N. Nygaard, and I. Zapata are highly
appreciated.
|
{
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}
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\section{Introduction}
\label{sec:introduction}
As one of the fundamental tasks in natural language processing, machine reading comprehension is fueled by avid neural modeling investigations in recent years. Given a certain textual content, the goal is to answer a series of questions based on semantic understanding. Many studies have focused on monological documents like Wikipedia \cite{rajpurkar2016squad_1} and news articles \cite{Hermann2015cnndata}, and some recent work are focusing on dialogue comprehension \cite{liu2019healthData,ma2018dialogueQA,2019Dream}. Different from passages, human-to-human dialogues are a dynamic and interactive flow of information exchange \cite{sacks1978simplest}, which are often informal, verbose, and repetitive, and this brings unique challenges for adopting document-oriented approaches on the conversational samples.
Recently, there is growing interest in automated extraction of relevant information from clinical dialogues \cite{du2019ext-sym,liu2019turn-by-turn,liu2019healthData}, in the form of dialogue comprehension. However, neural models via supervised learning usually require a certain amount of training data, and it is difficult to collect and construct large annotated resources for clinical-related tasks. Fine-tuning large-scale pre-trained language models has become a data-efficient learning paradigm, and achieves substantial improvement on various downstream tasks and applications \cite{devlin2019bert,liu2019roberta}. However, these general-purpose language backbones are trained on the online crawled data with universal objectives. Although this can provide feature-rich contextualized representations, it also limits their capability in specific domains. On the other hand, while some recent studies have proposed methods for pre-training on dialogue samples \cite{gao2020dialogueFeed,wu2020tod,zhang2020dialogpt,zhong2022-diaLM,zou2021-abs-pre}, they are focusing more on the generation of daily conversations (e.g., open-domain social chat). When adopting such generic language backbones on more targeted scenarios such as clinical-related tasks, their performance becomes sub-optimal due to the significant domain difference \cite{krishna2021soap}.
To address such challenges, in this work, we propose a domain-specific language pre-training, and adopt it to improve spoken dialogue comprehension performance on the clinical inquiry-answering conversations.
While the common methods for constructing pre-training samples (e.g., the token masking used in BERT \cite{devlin2019bert}, the infilling and deletion used in BART \cite{lewis2020bart}) proved effective for contextualized modeling, text manipulation designed for specific resources/tasks can bring further improvement \cite{gururangan2020dont-stop}. Therefore, considering the nature of human conversations \cite{sacks1978simplest} and the interactive flow of inquiry-answering dialogues, we propose a set of sample manipulations for conversational pre-training. At the token level, we introduce a masking strategy especially on speaker tokens, and we propose a permuting operation for better speaker role modeling. At the utterance level, we propose the utterance masking and intra-topic permutation scheme. More specifically, given a dialogue, we randomly select one utterance, and mask it or exchange it with another span in the same topic, and the language model is guided to reconstruct the coherent conversation flow based on the context. Moreover, we add an additional token at the beginning of each utterance to explicitly present the utterance boundary information.
Our experiments are conducted on a multi-topic symptom checking conversation dataset, where nurses inquire and discuss symptom information with patients. We empirically show that the proposed approach brings significant improvement on the dialogue comprehension task, especially in the low resource scenario.
\section{Domain-specific Language Pre-training}
\label{sec:methodology}
In this section, we elaborate on the six types of pre-training sample generation. We then describe the reconstruction-based learning process of language modeling to fuse domain-specific conversational features.
\subsection{Conversation-based Sample Construction}
Human-to-human spoken conversations are an interactive process of information exchange. Compared with the monological passages, a language backbone for multi-party dialogues is required to infuse the conversational linguistic features \cite{gu2020speaker}, such as speaker roles and utterance boundary information \cite{liu2019turn-by-turn}, as well as the underlying dialogue discourse structures \cite{sacks1978simplest}.
To guide language backbones to model the characteristics of conversations, we adopt the following six types of pre-training sample, as shown in Table \ref{table:pretrain-types}.\\
\textbf{Token Masking} Following the common pre-training scheme proposed in \cite{devlin2019bert}, for tokens in the sequence, 5\% of them are randomly sampled and replaced with a $<$mask$>$ token.\\
\textbf{Token Infilling} Following the denoising learning scheme proposed in \cite{lewis2020bart}, 5\% of input tokens are randomly sampled and replaced with random tokens extracted from the vocabulary.\\
\textbf{Speaker Masking} To encourage the model to grasp the speaker information \cite{gu2020speaker} in the interactive flow. We randomly sampled 10\% of the utterances, and replaced their speaker tokens with a $<$mask$>$ token.\\
\textbf{Speaker Permutation} Previous work shows that the text infilling scheme is helpful to tackle the training-testing inconsistency issue of masking-based methods \cite{lewis2020bart}. Thus, aside from the speaker masking, we randomly sampled 10\% of the utterances, and exchange its speaker with another one in the conversation.\\
\textbf{Utterance Masking} To encourage the model to grasp more contextual information at the utterance level, we further adopt a span masking scheme. More specifically, we randomly sampled 10\% of the utterances and mask the whole span, and the model will try to recover it based on the context understanding.\\
\textbf{Intra-Topic Utterance Permutation} While human conversations are often less structured than well-organized documents, they are inherently organized around the dialogue topics in a coarse-grained structure \cite{sacks1978simplest}. Therefore, we propose an intra-topic utterance permutation strategy, in which we randomly sampled 5\% of the utterances, and exchange them with another one in the same topic. This operation injects more noise into the conversation flow, and the model can only restore the original order by leveraging the underlying dialogue discourse information.\\
Moreover, we add a special $<$u$>$ token at the start position of each utterance, which can convey the utterance-level boundary information \cite{zhang2020dialogpt}, and is similar to the sentence-level $<$s$>$ token used in document-based approaches \cite{liu2019roberta}.
\begin{table*}[t!]
\caption{Pre-training data processing for dialogue samples. Here the raw utterances are extracted from the synthetic clinical inquiry-answering conversations. $<$mask$>$ denotes places replaced with the mask token. $<$random$>$ denotes replacing with a random token from a vocabulary. The permutation is to change the order of two randomly selected tokens/utterances.}
\begin{center}
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{p{6.5cm}p{6.5cm}}
\toprule
\textbf{Raw Utterances} & \textbf{Processed Utterances} \\
\midrule
\textbf{Token Masking} & \\
Nurse: Do you have any headache at night? & Nurse: Do you have any headache at \textbf{$<$mask$>$}? \\
Patient: No no headache, just a bit cough... & Patient: No no headache, just a \textbf{$<$mask$>$} cough.. \\
Nurse: Cough? you mean cough at every night? & Nurse: Cough? you \textbf{$<$mask$>$} cough at every night? \\
\midrule
\textbf{Token Infilling} & \\
Nurse: Do you have any headache at night? & Nurse: Do you have any \textbf{$<$random$>$} at night? \\
Patient: No no headache, just a bit cough. & Patient: No \textbf{$<$random$>$} headache, just a bit cough.. \\
Nurse: Cough? you mean cough at every night? & Nurse: Cough? you mean cough at every night? \\
\midrule
\textbf{Speaker Masking} & \\
Nurse: Do you have any headache at night? & Nurse: Do you have any headache at night? \\
Patient: No no headache, just a bit cough.. & \textbf{$<$mask$>$}: No no headache, just a bit cough.. \\
Nurse: Cough? you mean cough at every night? & \textbf{$<$mask$>$}: Cough? you mean cough at every night? \\
\midrule
\textbf{Speaker Permutation} & \\
Nurse: Do you have any headache at night? & Do you have any headache at night? \\
Patient: No no headache, just a bit cough.. & \textbf{Nurse}: No no headache, just a bit cough.. \\
Nurse: Cough? you mean cough at every night? & Nurse: Cough? you mean cough at every night? \\
\midrule
\textbf{Utterance Masking} & \\
Nurse: Do you have any headache at night? & Nurse: Do you have any headache at night? \\
Patient: No no headache, just a bit cough.. & Patient: \textbf{$<$mask$>$ $<$mask$>$ $<$mask$>$ $<$mask$>$}.. \\
Nurse: Cough? you mean cough at every night? & Nurse: Cough? you mean cough at every night? \\
\midrule
\textbf{Intra-Topic Utterance Permutation} & \\
Nurse: Do you have any headache at night? & Nurse: Do you have any headache at night? \\
Patient: No no headache, just a bit cough.. & \textbf{Nurse: Cough? you mean cough at every night?} \\
Nurse: Cough? you mean cough at every night? & \textbf{Patient: No no headache, just a bit cough..}\\
\bottomrule
\end{tabular}}
\end{center}
\label{table:pretrain-types}
\end{table*}
\subsection{Experiment Setup of Pre-training}
With the conversation samples built on the aforementioned strategies, we conduct the language backbone pre-training, and a Transformer-based neural architecture is used \cite{vaswani2017transformer}. To leverage the generic prior language knowledge, we select \textit{`RoBERTa-base'} model to initialize the Transformer model, and conduct the \textbf{reconstruction-based} learning process \cite{liu2019roberta}. More specifically, as shown in Figure \ref{fig-pretrain}, the input sequence is the dialogue content after text manipulation, and the target sequence consists of the tokens from the original utterances.
In our experiment, the pre-training data was the combination of the SAMSum corpus \cite{gliwa2019samsum}, a subset of OpenSubtitles \cite{lison2016opensubtitles2016}, and a set of synthetic clinical symptom checking conversations (40k samples). SAMSum is a social chat corpus consisting of 16k dialogues. OpenSubtitles is compiled from a large collection of TV and movie scripts across many languages, and we randomly selected 50k samples from the English part. The mixed conversational data contains a certain amount of dialogues with multiple participants, speaker role information, and conversational structures. During training, we fixed the max input length to 512 tokens. When constructing the pre-training samples, we first conducted text manipulations on tokens and speaker entities. Then the utterance-level operations were performed. We trained the language backbone with 5,000 warm-up steps. Batch size was set to 64 via applying gradient accumulation, and the initial learning rate was set at 1e-5. Cross-entropy was used as the loss function, and we selected the checkpoints at the knee point of loss decrease.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.75\textwidth]{images/PT_Pretrain.png}
\end{center}
\caption{Overview of the language model pre-training. Here we adopt the bi-directional mask-recovering training as in \cite{liu2019roberta}. Superscript values denote utterance ID.}
\label{fig-pretrain}
\end{figure}
\section{Dialogue Comprehension on Clinical Inquiry-Answering Conversations}
\subsection{Task Definition}
\label{ssec:task_definition}
One example of the dialogue comprehension task on clinical inquiry-answering conversations is shown in Table \ref{table:qa_example}. The input consists of a multi-turn symptom checking dialogue $D$ and a question $Q$ specifying a \textit{symptom} with one of its \textit{attributes}; the output is the extracted answer $A$ from the given dialogue. A training or test sample is defined as $S=\{D, Q, A\}$. Five attributes, specifying certain details of clinical significance, are defined to characterize the answer types of $A$: (1) \textit{time} the patient has been experiencing the symptom, (2) \textit{activities} that trigger the symptom (to occur or worsen), (3) \textit{extent} of seriousness, (4) \textit{frequency} occurrence of the symptom, and (5) \textit{location} of symptom. For each symptom/attribute, it can take on different linguistic expressions, defined as \textit{entities}.
\begin{table}[ht!]
\small
\caption{\label{table:qa_example} One example of the reading comprehension task on clinical inquiry-answering conversations. The synthetic dialogue is used for demonstration.}
\begin{center}
\begin{tabular}{p{9cm}}
\toprule
\textbf{Conversation Example (Truncated)} \\
\midrule
Nurse: Hi Mr.[Name], you were discharged on [date]. There are some questions I'd like to check with you.\\
Patient: Ok, Ok ... I think I feel better ...\\
Nurse: Is your left leg still {\color{blue}swollen}? You said so the last time I call you?\\
Patient: Yes, {\color{blue}only a bit} when I drink too much water ...\\
\midrule
\textbf{Question:} What is the extent of the swollen? \\
\midrule
\textbf{Reference Answer Span:} only a bit \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.67\textwidth]{images/PT_QA_model.png}
\end{center}
\caption{Overview of the machine comprehension model on a question-answering task. $Q$ presents the question sequence (in green), and it is concatenated with the conversation sequence as input. Following \cite{devlin2019bert,liu2019healthData}, the answer span is extracted from the conversation content by predicting the start and end positions (in purple).}
\label{fig-qa-model}
\end{figure}
\subsection{Clinical Dialogue Corpus}
\label{ssec:corpus}
The reading comprehension task is conducted on the data of nurse-to-patient symptom monitoring conversations.
The corpus was inspired by real dialogues in the clinical setting where nurses inquire about symptoms of patients \cite{liu2019healthData}. Linguistic structures at the semantic, syntactic, discourse, and pragmatic levels were abstracted from these conversations to construct templates for simulating multi-turn dialogues (40k samples in our settings). The informal styles of expressions, including incomplete sentences, incorrect grammar, and diffuse flow of topics were preserved.
A team of linguistically trained personnel refined, substantiated, and corrected the automatically simulated dialogues by enriching verbal expressions through different English speaking populations in Asia, Europe, and the U.S., validating logical correctness through checking if the conversations were natural, reasonable, and not disobeying common sense, and verifying the clinical content by consulting certified and registered nurses.
These conversations cover 9 topics/symptoms (e.g. headache, cough).
For each conversation, the average word number is 255 and the average turn number is 15.5. For the comprehension task, questions were raised to query different attributes of a specified symptom; e.g., \textit{How frequently did you experience headaches?} Answer spans in the dialogues were labeled with start and end indices, following the annotation scheme as in \cite{rajpurkar2016squad_1}. Note that if the queried symptom or attribute is not mentioned in the dialogue, the ground-truth output is \textit{``No Answer''}, as the same definition in \cite{liu2019healthData}.
\subsection{Baseline Models}
We further fine-tuned the Transformer-based model on the dialogue comprehension task, and compared it with several baselines, including \textbf{Pointer LSTM} \cite{wang2016matchLSTM}, Bi-Directional Attention Flow network (\textbf{Bi-DAF}) \cite{seo2016bidaf}, and \textbf{R-Net} \cite{wang2017rnet}.
To evaluate the effectiveness of our domain-specific language pre-training, we use the \textbf{Vanilla Transformer} and the original \textbf{RoBERTa-base} model as control, and our proposed model is
\textbf{RoBERTa-base w/ Domain-specific Pre-training}. As shown in Figure \ref{fig-qa-model}, we formulate the comprehension task as an answer extraction process. With the feature-rich contextualized representation, the answer span is generated by predicting its start/end position in the sequence, by adding a linear layer on the last layer hidden states from the language modeling \cite{devlin2019bert,liu2019roberta}.
\subsection{Training Configuration}
All models were implemented with Pytorch and Hugging Face Transformers \cite{paszke2019pytorch}.
For models without pre-trained language backbones (e.g. Bi-DAF, R-Net), Glove embedding \cite{pennington2014glove} was utilized, and out-of-vocabulary words were replaced with the \textit{$<$unk$>$} token. Hidden size and embedding dimension were 300, and those of Transformer-based models were 768. We used Adam \cite{kingma2014adam} with batch size 32, and gradient accumulation was applied. The initial learning rates were set at 2e-5, and dropout rate \cite{srivastava2014dropout} was set to 0.2. During training, the validation-based early stop strategy was applied. During prediction, we selected answer spans using the maximum product of $p_{start}$ and $p_{end}$.
\subsection{Evaluation: Comparison with Baselines}
\label{ssec:basic_result}
We conduct the evaluation on the synthetic clinical dialogue corpus, where the training, validation, and test size were 40k, 3k, and 3k, respectively. We adopted Exact Match (EM) and F1 score as metrics as the SQuAD benchmark \cite{rajpurkar2016squad_1}.
As shown in Table \ref{result-baseline}, the vanilla Transformer model obtains a slightly lower performance than the non-Transformer strong baselines (i.e. Bi-DAF and R-Net), and \textit{`RoBERTa-base'} is on par with them. This demonstrates that the prior knowledge from general language pre-training is beneficial for the downstream tasks. With the conversational pre-training, our proposed mode obtains substantial gains and achieve the best EM and F1 scores, showing that the domain-specific feature fusion is effective.
\begin{table}[t!]
\small
\caption{\label{result-baseline}
Evaluation result of the baseline models and our approach on the test set. \textit{Domain PT} denotes the proposed domain-specific pre-training.}
\begin{center}
\begin{tabular}{p{5.5cm}cc}
\toprule
\textbf{Model} & \textbf{\ \ EM Score\ \ } & \textbf{\ \ F1 Score\ \ } \\
\midrule
Pointer LSTM & 77.81 & 82.71 \\
Bi-Attention Flow (Bi-DAF) & 87.31 & 88.57 \\
R-Net (Our Implementation) & 88.22 & 90.13 \\
\midrule
Vanilla Transformer & 85.38 & 85.92 \\
RoBERTa-base w/o Domain PT & 88.37 & 90.15 \\
RoBERTa-base w/ Domain PT & \textbf{92.31} & \textbf{93.69} \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\subsection{Evaluation in Low-Resource Scenarios}
\label{ssec:low_res_result}
The limited amount of training data is a major pain point for clinical-related language tasks, as it is time-consuming and labor-intensive to collect and annotate the corpus at a large scale. Following the observation in previous work \cite{gururangan2020dont-stop}, we expect the domain-specific language modeling can result in more efficient learning on downstream tasks. To simulate the low-resource training scenario, we conducted experiments on a range of smaller training sizes (from 3k to 40k) with a fixed-size test set (3k samples). As shown in Figure \ref{fig-low-resource}, the proposed approach outperforms all other models significantly, especially when the training size is smaller than 20k.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.7\textwidth]{images/PT_lowRes.png}
\end{center}
\caption{\label{fig-low-resource} Experimental result on the low-resource training. X axis is the sample size, Y axis is the evaluation metrics, including exact match (EM) and F1 score.}
\end{figure}
\subsection{Evaluation: Pre-training Scheme Comparison}
\label{ssec:pretrain_type_exp}
To evaluate the effectiveness of the aforementioned strategies of pre-training sample construction. We conduct an experiment by adding different text manipulations to train the general-purpose language backbone. As shown in Table \ref{result-pt-table}, we observed that the token-level infilling can bring certain improvements, and introducing the conversation-related manipulations (i.e., speaker and utterance masking and permutation) are helpful for the final dialogue comprehension performance.
\begin{table}[t!]
\small
\begin{center}
\begin{tabular}{p{6.2cm}cc}
\toprule
\textbf{Model} & \textbf{ EM Score } & \textbf{ F1 Score } \\
\midrule
Pre-training on the \textit{RoBERTa-base} backbone \\
+ Token-level Masking & 88.79 & 90.30 \\
+ Token-level Infilling & 90.73 & 92.12 \\
+ Speaker Mask \& Permutation & 91.01 & 92.41 \\
+ Utterance Mask \& Permutation & \textbf{92.31} & \textbf{93.69} \\
\bottomrule
\end{tabular}
\end{center}
\caption{\label{result-pt-table}Performance comparison on pre-training schemes. The text manipulations are added from token level to utterance level.}
\end{table}
\section{Conclusions}
In this paper, we introduced a domain-specific language pre-training approach, and adopted it to improve performance on downstream tasks such as question answering. Based on the linguistic characteristics of spoken dialogues, we proposed a combination of six strategies to build samples for conversational language pre-training, and conducted reading comprehension experiments on a multi-topic inquiry-answering conversation data. The experimental results showed that the proposed approach can boost performance and achieve more efficient learning outcomes. Future work include extending conversational pre-training to other clinical tasks \cite{Kurisinkel2021report,liu2019conv_summ} and resources.
\begin{acknowledgement}
Research efforts were supported by funding and infrastructure from A*STAR, Singapore (Grant No. IAF H19/01/a0/023). We gratefully acknowledge valuable inputs from Angela Ng, Hong Choon Oh, Sharon Ong, Sheldon Lee, Weiliang Huang, and Ying Zi Oh at the Department of Cardiology, Health Management Unit, and Department of Health Services Research, Changi General Hospital, Singapore. We thank the anonymous reviewers for their precious feedback to help improve and extend this piece of work.
\end{acknowledgement}
\input{references}
\end{document}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: How to solve $\lim_{n\to\infty}P\left(\sum^n_{i=1}X_i\ge0\right)$? I am trying solve
$$\lim_{n\to\infty}P\left(\sum^n_{i=1}X_i\ge0\right)$$
where $X_1,X_2,...$ are centered random variables, i.i.d, and $0<EX^2_i<\infty$.
I don't know where to even start with this.
A: You should use the Central limit theorem, Classical CLT.
The classical CLT says that
$$ \frac{1}{\sqrt{n Var(X_1)}}\Big( \sum\limits_{i=1}^n X_i-n\mathbb{E}[X_1] \Big) \overset{\text{dist.}}{\to} Z $$
where $Z\sim N(0,1)$. In your case $\mathbb{E}[X_1]=0$, $Var(X_1)=\mathbb{E}[X_1^2]$, so
$$ \frac{1}{\sqrt{n \mathbb{E}[X_1]^2}} \sum\limits_{i=1}^n X_i \overset{\text{dist.}}{\to} Z. $$
In particular
$$ \underset{n\to \infty}{\lim} \mathbb{P} \Bigg( \frac{1}{\sqrt{n \mathbb{E}[X_1]^2}} \sum\limits_{i=1}^n X_i \geq 0 \Bigg) =\mathbb{P}\big( Z\geq0\Big). $$
You just need to connect the event $ \Big\{ \sum\limits_{i=1}^n X_i \geq 0 \Big\} $ to the event $ \Big\{ \frac{1}{\sqrt{n \mathbb{E}[X_1]^2}} \sum\limits_{i=1}^n X_i \geq 0 \Big\} $.
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"redpajama_set_name": "RedPajamaStackExchange"
}
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\section{The reconstruction conjecture and terminology}
A graph $G = [V,E]$ is a set of vertices $V$ and edges $E$. We restrict attention to simple graphs
(those which are undirected and with no self-loops or multiple edges).
The {\em degree sequence} of $G$
is the multiset of degrees of the vertices, in non-descending order. A vertex of degree 1 is called
a {\em pendant}.
The set of neighbours of a vertex $v$ will be denoted $N(v)$. The set $N(v) \cup \{v\}$
(called the closed neighbourhood of $v$) will be denoted $N[v]$.
For any graph $G$, a vertex-deleted subgraph $\Gv$ is called a {\em card}, and the multiset of
cards associated with all the vertices of $G$ is called the {\em reconstruction deck} of $G$,
or {\em deck} for short.
Two cards are deemed `the same' if they are isomorphic and distinct if they are not.
A property of $G$ which can be obtained from the deck, without
direct access to $G$, is said to be {\em recoverable}. For example, the number of vertices
$n = |V(G)|$
is recoverable since it is equal to the number of cards. The number of edges
$m = |E(G)|$ is recoverable since the number of edges appearing on all the cards is
$(n-2)m$ (since each edge appears on all but two of the cards). Further recoverable information
includes the degree sequence of $G$, and
for any given card $\Gv$, the degree sequence, in $G$,
of the neighbours of the associated vertex $v$.
A recoverable binary property, such as membership of a class, is said to be
{\em recognisable}. For example, the condition that the graph is regular is recognizable,
since it can be determined from a recoverable property, namely the degree sequence.
The question arises, whether a graph can be reconstructed in a unique way
from its deck. The graphs on two vertices cannot (they both have
the same deck). An automated (i.e. computer) calculation has confirmed that
graphs on 3 to 11 vertices can \cite{97McKay}, and for larger $n$ the question is open
apart from some special classes of graphs (see \cite{Bondy91,Babai95,14Bartha} for a review).
The {\em Reconstruction Conjecture} (Kelly and Ulam 1942) is the conjecture
that every graph having more than two vertices can be reconstructed in a unique
way (up to isomorphism) from its deck \cite{Kelly42,Kelly57,Ulam60}.
The conjecture is known to be true for most graphs
(that is, of all graphs of given order, most are reconstructible) \cite{Bollobas90,McMullen07}. However, despite
the wealth of information about a graph that is provided by its deck, no
one has yet found out how to do the detective work and show that
the evidence to be found in each deck points to one and only one `culprit' graph.
(It is not sufficient to find, for any given deck, a graph which has that deck.
Rather, one must show that no other graph has that same deck.)
A standard way to proceed in graph reconstruction problems is by a combination of
`recognise and recover'. One finds a reconstruction method which only applies to some
class of graphs, and one shows that membership of the class is recognisable.
\section{ds-reconstructibility}
It can happen that a reconstruction method may apply to all graphs whose degree
sequence has some given property. Such methods are useful because the degree
sequence is very easy to recover from the deck, or from any graph whose reconstructibility
is in question, and also because the degree sequence can be used as a starting-point for
generating sets of graphs. This connection can be used either to make
graph generation more efficient (see section \ref{s.generate}), or to avoid it
altogether when one is interested in the reconstruction problem.
In the latter case, if one wanted to use a computer to test for reconstructibility
all graphs of some given property, one might proceed by generating, for each degree sequence,
all graphs of that degree sequence and having the property, and testing each. If it is
already known that all
graphs of a given degree sequence are reconstructible, than one can avoid parts of
such a search.
Before approaching the issue in general, first let us note some well-known and simple cases:
\begin{lemma} \label{simple}
A graph with any of the following properties is reconstructible.\
\begin{enumerate}
\item There is an isolated vertex.
\item There is a full vertex (one of degree $d=n-1$).
\item No pair of degrees differs by 1.
\item There is a vertex of degree $d_v$ such that, after that degree is removed from
the degree sequence, the resulting sequence is one in which no pair of degrees differs by 1
(for example, the sequence
$2, 2, 2, 4, 4, 4, 5, 6, 6, 9$ has this property).
\end{enumerate}
\end{lemma}
\begin{proof}
These examples are straightforward. In each case, there exists a card
$\Gv$
for which all the neighbours of $v$ are recognisable by their degree (using the fact that the
degree sequence of the neighbours of $v$ is recoverable, and their degrees in $\Gv$ are
one less than their degrees in $G$).
In case 3, this is true of any card, and in case 4 it is true of the card associated
with the special vertex. \qed \end{proof}
{\bf Corollary}. Regular graphs are reconstructible; Euler graphs are reconstructible.
This follows immediately. The former was noted by Kelley 1957 and
Nash-Williams \cite{78NashWilliams}.
We would now like to generalize from these simple cases. That is, we want to define some
reasonably natural class of graphs that can be reconstructed by using degree information,
and then enquire how members of the class can be recognised.
To this end, let us introduce the following concept:
\begin{definition}
A card $\Gv$ is {\em ds-completable} if, for each degree $d$, vertices of degree $d$ in the card
are either all neighbours or all non-neighbours of $v$ in $G$.
We shall also say that a vertex $v$ is ds-completable in $G$
if the card $\Gv$ is ds-completable.
\end{definition}
For example, all the cards of a regular graph are ds-completable, and in the fourth case of lemma
\ref{simple} the card $\Gv$ is ds-completable, for the vertex $v$ defined there.
The idea of this definition is that, if a card has this property, then the set of neighbours of
the vertex which has been deleted
can be determined from the degree sequences $ds(G)$ and $ds(\Gv)$ without having to
bring in further information about the structure of the graph. For, we already know that the
degree sequence of the neighbours of $v$ is recoverable, so, for any given card $\Gv$, we know
what degrees the erstwhile neighbours of $v$ must have in that card (namely, one less than their
degrees in $G$). If the card has the property given in the definition, then there is no ambiguity in
determining, from the degree information alone, which vertices are the neighbours we want to find
(see the detailed example below).
Having identified them, the graph is reconstructed by introducing the missing vertex and attaching
it to those neighbours.
An automated method to identify the neighbours of $v$ in $\Gv$ is as follows \cite{14Bartha}.
First write out the degree sequence of the graph, and then, underneath it, write out the degree
sequence of $\Gv$, with a blank inserted at the first location where $d_v$ would appear.
For example:
\[
\begin{array}{lcccccccccccc}
G:&&2&2&3&3&3&5&5&6&7&7&9 \\
\Gv:&&1&1&2&2&3&4&4&5&-&7&9
\end{array}
\]
Next, insert vertical bars so as to gather the groups of given degree in the second line:
\[
\begin{array}{lccc|cc|c|cc|c|c|c|c}
G:&&2&2&3&3&3&5&5&6&7&7&9 \\
\Gv:&&1&1&2&2&3&4&4&5&-&7&9
\end{array}
\]
Then one inspects the groups. Since the degree sequence of the neighbours is recoverable, we
know which degrees in the lower line are candidates to belong to neighbours of $v$. If for a given
candidate group in the lower line, all the items in the top line for that group have the same
degree $d$, then all the vertices of degree $d-1$ in $\Gv$ are neighbours of $v$. If this
happens for all such groups, then no ambiguity arises, so the card can be completed.
The example shown above has this property. It is owing to the fact that,
whereas $v$ has neighbours of degree 3,6,7, which might in this example lead to ambiguity,
it is also adjacent to all the vertices of degree 2,5,6, which removes the ambiguity.
If, instead, the neighbours of $v$ had degrees $2,3,3,5,6,7,9$, then the procedure would give
this table:
\[
\begin{array}{lcc|ccc|c|c|cc|c|c|c}
G:&&2&2&3&3&3&5&5&6&7&7&9 \\
\Gv:&&1&2&2&2&3&4&5&5&6&-&8
\end{array}
\]
Now there are two groups (those for $d=2$ and $d=5$ in $\Gv$) where the items in the upper
line are not all the same, and therefore we have two sets of vertices in $\Gv$ where we can't tell,
from this information alone, which are neighbours and which non-neighbours of $v$.
With this concept in mind, we can define an interesting class of graphs:
\begin{definition}
A graph $G$ is {\em ds-reconstructible} iff its deck contains a ds-completable card.
\end{definition}
\begin{definition}
A degree sequence is {\em ds-reconstruction-forcing} (or {\em forcing} for short)
iff it is not possible to construct a non-ds-reconstructible graph realizing that degree
sequence.
\end{definition}
(Equivalently, a graphic sequence is forcing iff every graph realizing it is
ds-reconstructible, and non-graphic sequences are deemed to be forcing.)
A forcing degree sequence requires or forces the presence of a ds-completable card in any
associated graph. The use of the term `forcing' here is similar to, but not identical to, its use in \cite{Barrus08}.
In \cite{Barrus08} the authors use the term `degree-sequence-forcing' to describe a class of graphs.
A class is said to be `degree-sequence-forcing' if whenever any realization of a graphic
sequence $\pi$ is in the class, then every other realization of $\pi$ is also in the class. In the present
work we use the term `forcing' to describe a class of degree sequences. The set of
graphic sequences is divided into three distinct classes: those for which every realisation is
ds-reconstructible, those for which no realization is ds-reconstructible, and those for which some
realizations are ds-reconstructible and some are not. The first of these classes is under study here.
If we can show that a given degree sequence is ds-reconstruction forcing, then we
shall have proved the reconstructibility of all graphs realising that degree sequence.
We shall also have proved a stronger result, namely that for every graph $G$ realising the
forcing degree sequence $\pi$, there is a graph $G'$ on one fewer vertices,
such that $G$ is the only graph realising $\pi$ that can be obtained by adding a vertex
and edges to $G'$. This is relevant to graph generation, which we discuss in section \ref{s.generate}.
It is the main purpose of this paper to present methods to determine whether or not
a given degree sequence is ds-reconstruction-forcing.
Even if the reconstruction conjecture were to be proved by other methods, it would still remain
an interesting question, which degree sequences are ds-reconstruction-forcing, and whether
or not they can be easily identified. Connections to closely related ideas in graph theory are
explored briefly in the conclusion.
Our task is in some respects comparable to the task of determining whether or not a given
integer sequence is graphic (i.e., is the degree sequence of some graph). It is more closely
comparable to the task of determining whether a graphic sequence is
unigraphic.\cite{Johnson80,Tyshkevich00,Kleitman75,Koren76}
That question has been largely solved, in that there exists a small collection of conditions on
the degree sequence that can be evaluated in linear time and which suffice to establish
whether or not a sequence is unigraphic \cite{Kleitman75,Koren76}. Another method is
to construct a realization of a given sequence and apply switching operations to the
graph \cite{Johnson80,Tyshkevich00}.
The {\em switch}, also called {\em transfer}, is an operation which we shall write $\otimes^{ac}_{bd}$,
where $a,b,c,d$ are four distinct vertices of a graph $G$. If edges $ac$, $bd$ are present in the graph,
and $ad$, $bc$ are not, then $G$ is said to {\em admit} the switch $\otimes^{ac}_{bd}$. The image
of $G$ under $\otimes^{ac}_{bd}$ is obtained by replacing edges $ac$, $bd$ by $ad$, $bc$; i.e.
$\otimes^{ac}_{bd} G = G - ac - bd + ad + bc$. Such an operation preserves the degrees of all
the vertices, and furthermore it may be shown that one can move among all graphs of given degree
sequence via a finite sequence of switches \cite{Fulkerson65}.
\begin{lemma} \label{lemswtich}
If $\Gv$ is ds-completable, then $(\otimes^{ac}_{bd}G) - v$ is ds-completable when
$v \notin \{a,b,c,d\}$.
\end{lemma}
{\em Proof.} Such an operation does not change the neighbourhood of $v$, nor the degree of
any vertex, hence if $v$ is completable in $G$, it is also completable in $\otimes^{ac}_{bd}G$. \qed
One way to investigate whether a sequence $\pi$ is ds-reconstruction forcing is to
construct a graph $G$ realizing $\pi$, and then, if $G$ is ds-reconstructible, perform
a switch which changes the neighbourhood of the first ds-completable vertex found.
Such a switch may possibly result in a graph with fewer ds-completable vertices. One continues
switching until a graph is found having no completable vertex, or until the search is
abandoned. If a non-ds-reconstructible graph was found, or if all graphs realizing $\pi$ have
been considered, then one has discovered whether or not $\pi$ is forcing;
otherwise one has an inconclusive result.
When the number of realizations of a given sequence $\pi$
is small, one can move among them (and also be sure that all cases have been considered), by
exhaustive implementation of switching, or one can construct them all another way.
In this way the question, whether or not $\pi$ is forcing, can be decided.
However such an exhaustive method is
prohibitively slow for most sequences. It would be useful to find a method to choose a sequence
of switches
that efficiently arrives at a non-ds-completable $G$ if one exists. I have not discovered
such a method. Instead the present work is devoted to finding conditions on the degree sequence
that must be satisfied by forcing sequences. The following ideas will be useful.
\begin{definition}
A degree $d$ is {\em bad} if the graph contains a vertex of degree $d - 1$.
A degree $d$ is {\em dull} if the graph contains a vertex of degree $d + 1$.
A vertex is bad or dull (or both), according as its degree is bad or dull (or both).
A vertex which is not bad is good. A set of vertices is said to be
good, bad or dull iff all its members are good, bad or dull, respectively.
\end{definition}
Example: in the following degree sequence, the bad degrees are underlined and the dull
degrees are overlined:
$\overline{2,2},\underline{3,3,3}, 5,5,\overline{7,7}, \overline{\underline{8}}, \underline{9}$.
For every bad vertex, there is at least one dull vertex, and for every dull vertex,
there is at least one bad vertex. Vertices of least degree are never bad; vertices of highest
degree are never dull. If a vertex is bad in $G$ then it is dull in
the complement graph $\bar{G}$, and {\em vice versa}.
Bad neighbours make the job
of reconstruction harder. For, if a vertex $v$ has no bad neighbours, then, in the
associated card $\Gv$, the neighbours of $v$ are immediately recognisable since they have degrees
that do not occur anywhere in the degree sequence of $G$, and this is not true of
any other vertices in $\Gv$. Consequently the card is ds-completable.
If, on the other hand, a vertex has one or more bad neighbours then
this argument does not apply. This does not necessarily mean its card is not ds-completable, however:
\begin{theorem} \label{dsc} [Mulla 1978]
The card $\Gv$ is ds-completable iff in $G$, no non-neighbour of $v$ (excluding $v$ itself)
has a degree one less than a neighbour of $v$.
\end{theorem}
Another way of stating the theorem is as follows.
{\em For any given vertex $v$ in graph $G$, let $D_v$ be the set of vertices of degree one less than a
neighbour of $v$, and let $N[v]$ be the closed neighbourhood of $v$ in $G$. Then
the card $\Gv$ is ds-completable iff $D_v \subset N[v]$.}
\begin{proof}
This has been proved by Mulla \cite{78Mulla} (see also \cite{14Bartha})
but we present a proof suited to the concepts employed in this paper. The idea is to work along
the degree sequence of the card $\Gv$, determining which degrees must belong to neighbours of $v$.
The only neighbours which are not immediately identifiable are the bad neighbours,
and the condition of the theorem renders those identifiable too. This is because if
all vertices of some degree $d$ in $G$ are either absent from the card $\Gv$
or have had their degree reduced by one, then we know that vertices in the card of degree $d$
must be neighbours of $v$. Conversely, if this is not the case, then there remains an ambiguity
and the neighbours cannot be determined by the degree information alone. \qed \end{proof}
Note that we are not trying to capture cases of graphs which can be reconstructed
by methods that involve further considerations in addition to degree information.
For example, consider the path graph $P_4$. If one of the pendants is
deleted, the resulting card is {\em not} ds-completable (it does not satisfy the definition) because
there is an ambiguity: there are two vertices whose degree makes them candidates
to be the neighbour of the pendant which was deleted. The fact that in this example
one gets the same graph, irrespective of which of these two is picked, is not under
consideration in the definitions employed here, but we will return to this point at the end.
\begin{lemma} \label{lem.comp}
A graph $G$ is ds-reconstructible iff its complement $\bar{G}$ is
ds-reconstructible.
\end{lemma}
{\bf Corollary} A degree sequence is forcing iff the complement degree sequence is
forcing, where for any degree sequence $[ d_i ], \, i=1 \cdots n$, the complement degree
sequence is defined to be $[ n-1-d_n,\, n-1-d_{n-1},\, \cdots, n-1-d_1 ]$.
\begin{proof} By definition, a graph $G$ is ds-reconstructible iff it
contains a ds-completable vertex. Now, for any $G$,
a vertex which is bad in $G$ is dull in $\bar{G}$ and a vertex which is dull in $G$
is bad in $\bar{G}$. From lemma (\ref{dsc}) we have that a vertex $v$ is ds-completable
iff for any bad vertex $w$ adjacent to $v$, all the corresponding dull vertices
(i.e. those of degree $d_w-1$) are also either adjacent to $v$ or equal to $v$.
It follows that, in the complement
graph $\bar{G}$, $v$ has the property that for any dull vertex $w$ not adjacent to $v$, all the corresponding
bad vertices (i.e. those of degree $d_w+1$) are not adjacent to $v$ either. It follows that $v$
satisfies the condition of lemma (\ref{dsc}) in $\bar{G}$. \qed \end{proof}
To prove the corollary, it suffices to observe that for every graph with a given degree
sequence, the complement graph has the complement degree sequence. \qed
This lemma serves to show that if we wish to discover whether
a graph is ds-reconstructible,
then it is sufficient to determine the status of either one of $G$ or $\bar{G}$.
The same applies to the task of determining whether a given degree sequence
is ds-reconstruction-forcing.
\begin{definition}
A {\em stub} is half of an edge, such that each stub attaches to one vertex,
and each edge is the result of joining two stubs.
\end{definition}
We will also say that an edge or a stub is `bad' when it is attached to a bad vertex.
\begin{definition}
A vertex set is {\em neighbourly} if no vertex outside the set has a degree one
less than that of a vertex in the set. Thus, $K$ is neighbourly iff
$\{d_u : u \notin K\} \cap \{d_u-1 : u \in K\} = \emptyset$. A vertex set
is {\em $d$-neighbourly} if it contains a vertex $v$ of degree $d$ such that no vertex outside the
set has a degree one less than that of any vertex other than $v$ in the set.
Thus, $K$ is $d$-neighbourly iff
$\{d_u : u \notin K\} \cap \{d_w-1 : w \in K - v\} = \emptyset$ and $d_v = d$.
For example, a set of four vertices with degrees $[3,4,4,5]$ is 3-neighbourly if there is
no other vertex of degree 3 or 4, whether or not there is a vertex of degree 2.
\end{definition}
The significance of this concept is that if a vertex $v$ is adjacent to {\em all} the
members of a neighbourly set, then those neighbours can be deemed `good' as far as
$v$ is concerned. Indeed, it is obvious from the definitions that
a vertex $v$ is ds-completable iff $N[v]$ is the union of one or more neighbourly
or $d_v$-neighbourly sets.
Any set that contains all the dull vertices is neighbourly. Any set $j$ consisting of
all the vertices of degrees in some range $d_{\rm min}(j) \cdots d_{\rm max}(j)$ is
neighbourly iff $d_{\rm min}(j)$ is good.
\begin{theorem} \label{th.allv}
A graph has all of its vertices ds-completable if and only if no pair of degrees differ by 1.
\end{theorem}
\begin{proof}
({\em If}):
If no pair of degrees differ by 1, then all vertices are good, so no vertex has a bad neighbour,
therefore every vertex is ds-completable. \qed\\
({\em Only if}): We will show that if there is a bad vertex, then there is a non-completable vertex.
Let $v$ be a bad vertex which has degree $d$. Let $\alpha$ be the set of vertices of degree $d-1$.
Consider the edges of $v$. Suppose $a$ of these edges go to vertices in $\alpha$. Then $(d-a)$ of them
go to vertices not in $\alpha$.
In order that all vertices be completable, we require
in particular that each of the $(d-a)$ neighbours of $v$ that are not themselves in $\alpha$ must be adjacent to all of $\alpha$. Therefore edges between this group and $\alpha$ will use up
\be
(d-a) |\alpha| \label{alphastub}
\ee
of the stubs on vertices in $\alpha$. Next consider the $a$ vertices in $\alpha$ that are adjacent to $v$.
In order that they should be completable, we require that each is adjacent to all the other
vertices in $\alpha$. This requires $a(|\alpha|-a)$ edges between these $a$ and the
others in $\alpha$, plus a further $a(a-1)/2$ edges among the $a$ vertices. Hence the number of
stubs on vertices in $\alpha$ used up by the required edges among the vertices in $\alpha$ is
\be
2 a(|\alpha| - a) + a(a-1) .
\ee
Adding these to the number given by (\ref{alphastub}), and including also the $a$ edges
between $\alpha$ and $v$, we find that the number of stubs on vertices in $\alpha$ has to be
greater than or equal to
\be
(d-a) |\alpha| + 2 a(|\alpha| - a) + a(a-1) + a = |\alpha|(d+a) - a^2.
\ee
However, the total number of stubs on vertices in $\alpha$ is $(d-1)|\alpha|$. In order
that this should equal or exceed the number we just found to be required, one needs
$|\alpha| \le a^2/(a+1)$, i.e. $|\alpha| < a$, but this is clearly impossible. Hence
we have a contradiction, and it follows that at least one of the vertices is not completable. \qed
\end{proof}
The graphs having all good vertices form a type of error-correcting code, in the following sense.
If one party is going to send graphs to another, and it has been agreed beforehand that only
graphs of a given degree sequence will be sent, then if all degrees are good the receiver can determine
perfectly, from the received graph, which graph was sent, even when the communication channel
has a noise process whose effect is, with some probability, to delete one vertex
and its edges. (It is not my
intention to suggest that this is a physically realistic scenario, only to point out the mathematical
fact.) More generally, the degree sequence would not need to be agreed in full between the
communicating parties. If would suffice,
for example, that throughout the whole set of degree sequences which
are allowed, no degree differs from another by one.
The rest of the paper is concerned with trying to discover, by examining the degree
sequence $\pi$, whether or not it so constrains the graph that a completable vertex must
be present. For each $\pi$ this is like solving a logic puzzle, somewhat reminiscent of
the popular {\em Sudoku} puzzle. The degree sequence provides a `rule'
on how many edges each vertex has, and we have to fit these edges into the graph without
breaking the rule `no vertex is allowed to be completable'. If such a puzzle has
no solution, then $\pi$ is ds-reconstruction-forcing. One can enjoy this game
as a logical exercise in its own right, without regard to Kelly and Ulam's reconstruction
conjecture. The fact that we can write down such preliminary general observations as
lemmas \ref{lemswtich} and \ref{lem.comp} and theorems \ref{dsc} and \ref{th.allv} suggests
that we have a reasonably well-defined area of mathematics to explore.
However, it will emerge, in the subsequent sections, that the puzzle is a hard one!
\section{Easily recognised forcing sequences}
For each result to be presented, we shall give an example degree sequence
to which the result applies. For this purpose we adopt a notation
where we simply list the degrees, as in, for example, $[[1222]33\,56]$
(only single-digit examples will be needed), with square brackets
or bold font used to draw attention to vertices featuring in the argument.
\begin{lemma} \label{obv}
A degree sequence having one or more of the following properties is forcing:\\
\begin{enumerate}
\item There is no or only one bad vertex $($e.g. $[3333334])$.
\item There is no or only one dull vertex $($e.g. $[2333555])$.
\item There is a single vertex which is both bad and dull, and there are 2 bad degrees
$($e.g. $[1111233])$.
\item The sum of the degrees of bad vertices is less than $|V| + ( |B| \mbox{ \rm mod } 2)$,
where $B$ is the set of bad vertices.
$($e.g. $[1111[223]5])$.
\end{enumerate}
\end{lemma}
\begin{proof}
Parts 1--3 are merely ways of expressing examples of
case 4 of lemma \ref{simple}. Part 4
is owing to the fact that if there are too few bad edges then there must be a vertex with no bad
neighbour. Let us define $n = |V|$ and $\delta_B \equiv |B| \mbox{ mod } 2$.
The least number of stubs on bad vertices
required in order that every vertex can have at least one bad neighbour is
$n + \delta_B$ because we require at least $(n-|B|)$ edges
extending between $B$ and good vertices, and at least $\lceil |B|/2 \rceil$ edges
extending between bad vertices. The former use up one bad stub each, the latter use up two bad
stubs each. \qed \end{proof}
\begin{lemma} \label{BD}
If either of the following conditions hold:
\begin{enumerate}
\item there are exactly two bad degrees, and there is a dull vertex $v$ of unique degree, and a bad vertex $u$ of unique degree,
with $d_u \ne d_v + 1$,
$($e.g. $[{\bf 2}3333{\bf 4}])$
\item there are exactly three bad degrees, and there are two vertices $u$, $v$, each both bad and dull and of unique degree,
$($e.g. $[22{\bf 34}555])$
\end{enumerate}
then either $\Gv$ is ds-completable or $G\! - \! u$ is ds-completable.
\end{lemma}
\begin{proof}
Choose the labels $u,v$ such that $d_u \ne d_v + 1$.
If there is an edge $uv$ then $\Gv$ is completable.
If there is no edge $uv$ then $G-u$ is completable. (For further elaboration, see lemma \ref{unique}.)
\qed \end{proof}
\section{Application to graph generation} \label{s.generate}
As added motivation for the pursuing the subject further, we present in
this section an application.
The concept of `completing' a card by adding a vertex to it arises in the
area of graph generation. For example, one may wish to generate a set of
graphs of given degree sequence. A typical procedure is to generate smaller graphs
and extend them, and then eliminate duplicates by isomorphism checking. If one
first observes that the target degree sequence is forcing, then one knows that, for
each graph one is seeking, there exists a subgraph which is uniquely completable.
Therefore, in the graph generation algorithm, one may bypass those subgraphs
which are not uniquely completable (a property which is easy to check), in confidence
that a uniquely completable subgraph will eventually turn up, and then a graph
can be generated from that subgraph. In this way one obtains a
reduction in the amount of generation of isomorphic duplicates,
for only a modest computational effort.
Here is an example. Suppose we wish to generate all graphs having the degree
sequence $[2333555556]$ (there are 4930 such graphs). First we note, from lemma \ref{BD}
part 1, that the sequence is forcing. After deleting the vertex of degree 2,
there are 5 possibilities for the degree sequence of a graph on 9 vertices
which can be used as a template to generate the graphs we want. These are
$[223555556]$, $[233455556]$, $[333445556]$, $[233555555]$, $[33455555]$.
If we were to start by creating all graphs realising these degree sequences
and completing them by adding a vertex
in all ways that give the target degree sequence, then we would
generate $(119 + 1068 + 1810 + 6 \times 96 + 5 \times 351) = 5328$ graphs.
Isomorphism checking would then be required to reduce this set. If instead
we use the ds-reconstruction-forcing property, then we proceed as follows.
Let $u$ be the vertex of degree 2, and $v$ be the vertex of degree 6.
First, we note that it is the cases where $u$ and $v$ are
adjacent that give a non-uniquely-completable card after $u$
is deleted. Equally, it is precisely those cases where deletion of $v$
will yield a completable card. Therefore the set of degree sequences
to be considered is those where $u$ is removed if $u$ and $v$ are non-neighbours,
and where $v$ is removed otherwise. This yields the sequences
$[223555556]$, $[233455556]$, $[333445556]$ as before, and also the sequences
$[122244555]$, $[122344455]$, $[123344445]$, $[133344444]$. Now we consider
all graphs of order 9 that realise one of these sequences. In every case,
each such graph will yield one and only one graph of order 10 having the target
degree sequence, and no two of these will be isomorphic because they each have
a different degree neighbourhood of one or both of $u$, $v$.
The total number of graphs generated is now $(119+1068+1810+88+684+963+198)
= 4930$ and no isomorphism checking is required on the set of graphs of order
10 that has thus been generated.
This does not rule out that isomorphism
checking may be required to generate the graphs required on 9 vertices, but
we note that one of the order-9 sequences under
consideration is itself forcing.
\section{Recognising forcing sequences more generally}
Consider now the degree sequence [112223 5666].
In order to prove that this sequence
is forcing, we shall suppose that it is not, with a view to obtaining a contradiction.
Label the vertices by their degree, and suppose no vertex is completable.
Gather the vertices of degree 5 or 6 into a set called $j$, and the other vertices
into a set called $i$. Each vertex of degree 6 must have at least 3 edges to vertices
in $i$, and each vertex of degree 5 must have at least 2 edges to vertices in $i$,
so there are at least 11 edges from $j$ to $i$. But this is equal to the number of stubs
in $i$, so we deduce that all edges from vertices in $i$ go to $j$. The
pendants cannot be adjacent to 5, because 5 is good, so 5 is adjacent to at least one 2.
But the other edge from that 2 must go to a vertex in $j$, so it must go to a 6.
Therefore that 2 has a neighbourly neighbourhood, which contradicts the assumption
that no vertex is completable. It follows that there must be a completable vertex.
In order to formulate such arguments more generally,
divide the vertices into distinct sets, such that
all vertices of a given degree
are members of the same set, but it is allowed that a set may contain
vertices of more than one degree. The vertex sets will be labelled by indices
$i,j,k,c,g,h,p,q,s,t$. We shall choose these labels such that a set with label $i$ will
typically have vertices of low degree and a set with label $j$ or $h$
will typically have vertices of high degree. Throughout the rest of the paper,
the following conventions will be observed when choosing labels for sets of vertices:
\begin{enumerate}
\item $V$ is the set of all vertices in the graph,
$B$ is the set of bad vertices, $D$ is the set of dull vertices.
\item For any set $i$, define $\bar{i} \equiv V \setminus i$. Thus the set of good vertices is $\bar{B}$.
\item Sets labelled $i$ and $j$ are distinct.
\item $c = i \cap B,\; g = i \setminus c,\; p = \bar{i} \cap D,\; q = \bar{i} \setminus p.$
Thus $g,c,p,q$ are all distinct and together they account for all the vertices in $V$. Note that, by
definition, $g$ is good, $c$ is bad, $p$ is dull, and $q$ has no dull vertices.
\item The label $k$ is only applied to sets which are bad by definition;
the label $h$ is only applied to sets which by definition include the vertex or vertices
of highest degree.
\item $u,v,w$ refer to single vertices.
\item It will not be necessary to distinguish between a single vertex and the set whose
only member is that vertex.
\end{enumerate}
Let $\epsilon(j,k)$ be the number stubs in $j$ that are involved in edges between
$j$ and $k$. When $j$ and $k$ are distinct, this is equal to the number of edges between
$j$ and $k$. When $j=k$ it is equal to twice the number of edges within $j$, and more
generally $\epsilon(j,k) = \epsilon(j, k \setminus j) + \epsilon(j, k \cap j)$.
Define $\sigma(i,j)$ ($\tau(i,j)$) to be a lower (upper) bound on $\epsilon(i,j)$, i.e.
\be
\sigma(i,j) \le \epsilon(i,j) \le \tau(i,j) .
\ee
The bound $\sigma$ ($\tau$) will be raised (lowered) as we take into
account more and more information about the graph, based on
its degree sequence, under the assumption that it is a
graph and has no completable vertex. If eventually it is
found that $\tau(i,j) < \sigma(i,j)$ for some $i,j$ then
the conditions cannot be met: it amounts to a contradiction.
In this case we deduce that there must be a completable vertex
and therefore the degree sequence is forcing.
For any vertex-set $i$, define $n_i = |i|$, define
$\kappa(i,d)$ to be the number of members having degree $d$,
and $m_i$ the sum of the degrees:
\be
n_i &=& |i| \\
\kappa(i,d) &=& \left| i \cap \left\{v : d_v = d \right\} \right| \\
m_i &=& \sum_{v \in i} d_v
\ee
$m_i$ is the total number of stubs on vertices in $i$. It will also be useful to define
\be
\xi_i &=& n-1-n_i , \\
\chi_{ij} &=& m_j - (n-1-n_i) n_j .
\ee
For any vertex not in $i$, $\xi_i$ is the number of other vertices also not in $i$.
The following conditions follow from the definition of a graph;
they must hold for any graph whether or not there is
a completable card, for distinct $i,j,k$:
\be
\epsilon(i,\,i \cup \ibar) &=& m_i \label{epsitot} \\
\sigma(i,j) &\ge& 0 \label{sigpositive} \\
\tau(i,j) &\le& \min \{ n_i n_j,\; m_i,\; m_j \} \label{maxstubs} \\
\epsilon(i,j) &\ge& \sigma(j,i) \\
\epsilon(i,j) &\le& \tau(j,i) \\
\sigma(j,k) &\ge& \sigma(i \cup j, k) - \tau(i,k) \label{sigmtau} \\
\tau(j,k) &\le& \tau(i \cup j, k) - \sigma(i,k) \label{taumsig} \\
\sigma(i,j) &\ge & \chi_{ij} \label{minedge} \\
\sum_i \sigma(i,j) &\le& m_j \,\le\, \sum_i \tau(i,j) \label{sumeps}
\ee
where in (\ref{sumeps}) each sum indicates a sum over distinct sets that together contain all
the vertices in the graph. Most of the above are obvious. To prove (\ref{minedge}),
note that since a vertex of degree $d$ is non-adjacent to $n-1-d$ vertices, it can only be
non-adjacent to at most this number from any set of other vertices. Therefore
a vertex of degree $d$ is adjacent to at least $n_i-(n-1-d)$ members of any vertex set of
size $n_i$ (and this is true whether or not $n_i > n-1-d$).
The result follows by summing this over all members of $j$:
\[
\sum_{v \in j} \left( n_i + d_v + 1 - n \right) = m_j - n_j \left( n - n_i - 1 \right).
\]
The following also hold for any set of vertices in any graph:
\be
\epsilon(i,i) &\le& \tau(i,i) - (\tau(i,i) \mbox{ mod } 2) , \label{oddtau} \\
\epsilon(i,\bar{i}) &\ge& \sigma(i,\bar{i}) + ( (m_i - \sigma(i,\bar{i})) \mbox{ mod } 2) .
\label{oddsig}
\ee
These both follow from the fact that the number of stubs in a graph is always even. In
the first case, if we have established an upper bound $\tau(i,i)$ which is odd, then we know
the true number is at most one less than this. In the second case,
$(m_i - \sigma(i,\bar{i}))$ is an upper bound on the number of stubs
remaining in $i$ after all edges out of $i$ are accounted for. If this is odd
then the true value is at most one less than this, and therefore $\epsilon(i,\bar{i})$
is at least one more than $\sigma(i,\bar{i})$. These conditions are useful
for tightening constraints on $\epsilon$.
Define the following functions from vertex sets to binary values:
\be
\delta_i^j &:=& \left\{ \begin{array}{ll}
1 & \mbox{ if } i \cap j \ne \emptyset \\
0 & \mbox{ if } i \cap j = \emptyset \end{array} \right. \\
\nu_i &:=& \left\{ \begin{array}{ll}
1 & \mbox{ if $i$ is neighbourly} \\
0 & \mbox{ otherwise } \end{array} \right. \\
\mu_i &:=& \left\{ \begin{array}{ll}
1 & \mbox{ if $\ibar$ is neighbourly} \\
0 & \mbox{ otherwise } \end{array} \right.
\ee
\subsection{General conditions}
If a graph has no completable vertex, then
\be
\epsilon(i,j) &\le& m_i - (1 - \delta_j^B) n_i , \label{jgood} \\
\epsilon(i,j) &\le& m_i - \nu_j \kappa(i, n_j). \label{jneighbourly}
\ee
{\em Proof.} If $\delta_j^B = 1$ then (\ref{jgood}) follows from (\ref{maxstubs}); if $j$ is good (i.e.
$\delta_j^B = 0$) then no vertex in $i$ can have all its edges to $j$, or that vertex would
be completable, hence $\epsilon(i,j)$ must leave at least one stub unused on each vertex in $i$,
and (\ref{jgood}) follows. If $j$ is neighbourly, then after edges from $j$ to $i$ are accounted for
there must be an unused stub for each vertex in $i$ of degree $n_j$, which gives (\ref{jneighbourly}). \qed
Define the notation $\left[ x \right]^+$, where $x$ is a number, to mean
\be
\left[ x \right]^+ \equiv \max(0, x).
\ee
If there is no completable vertex then
\be
\lefteqn{\epsilon(i,j) \,\ge\, \chi_{ij}} \nonumber \\
&& + \left(1-\delta_i^D\right) \left[n_j + \left[ n_j n_s - \tau(s,j) \right]^+\right] \nonumber \\
&& + \delta_i^D \mu_i \left[ \kappa\left(j, \xi_i \right)
+ \left[ \kappa(j, n-n_i) - \tau(c,j) \right]^+ \right] \nonumber \\
\label{xminedge}
\ee
where $s = V \setminus (i \cup j \cup D)$ and $c = i \cap B$.
{\em Proof.}
This is eqn (\ref{minedge}), modified so as to include conditions that no vertex is completable.
First suppose $\delta_i^D = 0$. One has $\epsilon(i,j) = \chi_{ij}$ only on the assumption that each
vertex in $j$ has a closed neighbourhood containing all of $\bar{i}$, but if $\delta_i^D = 0$ then
$D \subset \bar{i}$ so this will result in all vertices in $j$ being ds-completable. To avoid this,
we must remove one edge from each vertex in $j$ to a dull vertex, and let that edge go to $i$ instead.
This accounts for the $n_j$ term in the first square bracket. We then note that we are still assuming
that every vertex in $j$ has an edge to every vertex in $s$ (hence $n_j n_s$ edges), but if this is more
than are possible (i.e. larger than $\tau(s,j)$) then we must correct for this, hence the second term
in the first square bracket.
Next suppose $\delta_i^D = 1$.
In this case we consider the vertices
in $j$ whose degree is such that their closed
neighbourhood must be $\bar{i}$ if they have no edge to $i$. But if $\bar{i}$ is neighbourly,
then this will not happen if no vertex is ds-completable, so such vertices must have at least
one edge to $i$. This accounts for the first term in the second square bracket in (\ref{xminedge}).
Next, consider the vertices in $j$ of degree $n-n_i$. The calculation using $\chi_{ij}$ allows that
each of these sends just one edge to $i$. But if $\bar{i}$ is neighbourly, then that edge must go
to a bad vertex in $i$, and if $\tau(c,j)$ is small (for example, because $m_c$ is small)
then this will not be possible for all the $\kappa(j,n-n_i)$ cases. The second term in the second
square bracket in (\ref{xminedge}) corrects for this.
\qed
The combination of
eqns (\ref{jgood})--(\ref{jneighbourly}) with (\ref{xminedge})
is sufficient to prove the forcing behaviour of many
degree sequences. For example, $[2222][4455]$ is forcing by
(\ref{jgood}), (\ref{xminedge}) and
$[1222]33[56]$ is forcing by (\ref{jneighbourly}), (\ref{xminedge}).
For $n=8$ there are 293 degree sequences having no isolated or full vertex,
and of these, 195 are forcing.
Eqns (\ref{jgood})--(\ref{xminedge}), together with lemma \ref{obv}, are sufficient
to prove the forcing condition in all but 29 of the 195 forcing cases.
In order to apply the constraints to any given degree sequence, one must make a judicious
choice of how the vertices are partitioned into sets. One wants to choose a case with
high $m_j$ and low $m_i$, and if $j$ is good or neighbourly, then so much the better, since
this will tighten the bound set by (\ref{jgood}) or (\ref{jneighbourly}) or both.
Using (\ref{jgood}) and
(\ref{xminedge}), we have $m_i + n_j (\xi_i - 1 + \delta_i^D) \ge m_j +
\delta_i^D \mu_i \kappa(j,\xi_i)$
for a non-forcing sequence. For a given set $j$, the quantity $(m_i +
\xi_i n_j)$ decreases by $n_j - d_v$ for each vertex of degree $d_v$
added to set $i$. Therefore,
to obtain a tight constraint, one should include in $i$ all vertices with
$d_v < n_j$, in the first instance, and also vertices of degree $d_v = n_j$ if this
makes $\delta_i^D \mu_i \kappa(j,\xi_i)$ increase. Further useful tests
can be obtained by considering the set $i \setminus D$ and/or $j \setminus B$.
For distinct sets $i,j$, if there is no completable vertex then
\be
\epsilon(i,j) \ge \chi_{ij} + \nu_j\left[ \kappa(s,n_j) + \kappa(j,n_j-1)\right]
\label{epsnuj}
\ee
where $s = V \setminus (i \cup j)$.
{\em Proof}. The argument is similar to that for eqn (\ref{xminedge}). $\chi_{ij}$ gives
the minimum number of edges extending from $j$ to $i$ on the assumption
that each vertex in $j$ is adjacent to all of $\bar{i}$ except itself, and
on this assumption, any vertex in $s$ of degree $n_j$ will have all its edges
to $j$. But if $j$ is neighbourly then this is not allowed, so for each such member of $s$ one of the vertices in $j$ is not a neighbour, which implies a further edge to $i$. This gives rise to the $\kappa(s,n_j)$ term in
(\ref{epsnuj}). The $\kappa(j,n_j-1)$ is obtained in the same way, applying
the argument to members of $j$. \qed
In order to detect the forcing condition more generally, one must
examine the degree sequence in more detail. The rest of the paper presents
a series of constraints that are sufficient to identify all forcing sequences for
graphs on up to 10 vertices. I have not been able to discover any single simply stated
condition sufficient to distinguish all the forcing from the non-forcing sequences.
Instead there are a range of cases to consider, but some broad
themes emerge. One theme is that to avoid being
forcing a sequence must not result in vertices that `greedily' consume all the bad stubs, so that
none are left for some other vertex. Another theme is that it is worth paying special attention
to the case where there is a vertex whose degree is not shared by any other vertex.
Each constraint is introduced by
an example degree sequence whose forcing nature can be proved by the
constraint under consideration, but not by most (or in some cases any) of the other constraints.
\subsection{Conditions relating to bad stubs}
[2 444555 [777]].
For any good vertex set $j$, if the sequence is not forcing then
\be
m_B \ge \sigma(B,j) + n - n_j + \delta_B . \label{mBbound}
\ee
{\em Proof}. The edges from vertices in $j$ use up at least $\sigma(B,j)$ of the bad stubs.
There are $n-n_j$ vertices not in $j$, and these require at least one bad stub each.
Therefore they require $n-n_j + \delta_B$ bad stubs. \qed
In order to apply this constraint, a good choice is to adopt for $j$ the set of good
vertices whose degree satisfies
\be
d > n - 1 - n_B - (1 - \delta_B^D).
\ee
Such vertices are liable to have more than one bad edge.
[222333 [666]7].
If the degree sequence is not forcing then, for any good set $j$,
\be
\sigma(B,j) \! &\ge& \sigma(g \cup B, j) - (m_g - n_g) , \label{epsBj} \\
\!\! \sigma(B,B) \! &\ge& 2 \sigma(c,k) + \left[ n_c - \sigma(c,k) \right]^+, \label{epsBB} \\
m_B &\! \ge& \! \sigma(B,j) + \sigma(B, B) + n \!- n_j\! - n_B , \label{mBbound2}
\ee
where $c = i \cap B$, $g = i \setminus c$, and $k = \bar{i} \cap B$
for some $i$ distinct from $j$.
{\em Proof}. (\ref{epsBj}) follows from (\ref{sigmtau}) and (\ref{jgood}).
(\ref{epsBB}) expresses the fact that each edge between $c$ and $k$ uses two bad stubs, and any
vertex in $c$ not adjacent to a vertex in $k$ also requires a bad edge.
(\ref{mBbound2}) is obtained by the same argument as for (\ref{mBbound}). \qed
In order to use
(\ref{epsBj})--(\ref{mBbound2}) one may use (\ref{xminedge}) or (\ref{epsnuj}) or another
method to obtain $\sigma(c,k)$ and $\sigma(g \cup B,j)$.
[11 334444 66]. For any degree sequence such that $D \cap B = \emptyset$, partition the vertices
into distinct sets $g,D,B,j$ where $m_g$ is small and $m_j$ is large. Then, if no
vertex is completable,
\be
\tau(D,j) &\le& \sum_{w \in j} \min(d_w, n_D-1) \le
n_j(n_D - 1), \nonumber \\
\tau(g,j) &\le& m_g - n_g, \nonumber \\
\tau(g,D) &\le& m_g - n_g, \nonumber \\
\sigma(B,j) &\ge& m_j - n_j(n_j-1) - \tau(D,j) - \tau(g,j) \nonumber \\
\sigma(D,B) &\ge& m_D - n_D(n_D-2) - \tau(D,j) - \tau(D,g), \nonumber \\
m_B &\ge& n_B+ \delta_B + \sigma(D,B) + \sigma(B,j) + n_g
\ee
{\em Proof}. The first constraint is the requirement that no vertex be adjacent to all the
dull vertices. The next two are examples of (\ref{jgood}), then we use (\ref{sigmtau})
twice (and require that no vertex in $D$ is adjacent to all the others in $D$)
and finally require that all vertices have at least one bad edge. \qed
\begin{lemma} \label{taukk_lem}
For any vertex set $i$ such that $\bar{i}$ is neighbourly, let
$p = \bar{i} \cap D$. If no vertex is completable, then
\be
\tau(p,p) \le n_p(n_p - 2) + \tau(i,p). \label{taukk}
\ee
\end{lemma}
\begin{proof}
Observe that since $\bar{i}$ is neighbourly and $p$ is all the dull vertices in $\bar{i}$,
a vertex in $p$ may only be adjacent to all other vertices in $p$ if
it has an edge to $i$. Hence there are at most $\tau(i,p)$ vertices in
$p$ adjacent to $n_p - 1$ fellow members of $p$, and the remaining $(n_p - \tau(i,p))$
are each adjacent to at most $n_p -2$ members of $p$. The
total for $\tau(p,p)$ is therefore at most $\tau(i,p)(n_p-1) +
(n_p - \tau(i,p))(n_p-2) = n_p(n_p-2) + \tau(i,p)$. \qed
\end{proof}
[12 5556666 8].
If there are 5 degrees, alternating between good and bad (i.e. $D \cap B = \bar{D} \cap \bar{B} = \emptyset$), then label the five sets of vertices $g,c,p,k,h$ (in increasing degree order) and let $i = g \cup c$.
These labels have been chosen in such a way that $p = \bar{i} \cap D$, so (\ref{taukk}) applies,
and $g,p,h$ are all good. If there is no completable vertex,
\be
\sigma(c,h) &\ge& \chi_{ih} - (m_g - n_g), \nonumber \\
\tau(i,p) &\le& (m_i - n_i) - \sigma(i,h), \nonumber \\
\tau(p,p) &\le& n_p(n_p-2) + \tau(i,p) - (\tau(p,p) \mbox{ mod } 2), \nonumber \\
\sigma(p,k) &\ge& m_p - (\tau(i,p)+\tau(p,p)+n_p n_h) ,\nonumber \\
\sigma(c,k) &\ge& \sigma(p,k) - n_k(n_p - 1), \nonumber \\
m_c &\ge& \sigma(c,h) + \sigma(c,k) .
\ee
{\em Proof}. The first inequality follows from (\ref{sigmtau}), (\ref{minedge}) and
(\ref{jgood}), noting that $h$ is good. The second inequality
follows from (\ref{taumsig}) and (\ref{jgood}), noting that $p$ and $h$ are both good.
The equation for $\tau(p,p)$ is (\ref{taukk}) adjusted using (\ref{oddtau}). Next, the constraint on
$\sigma(p,k)$ follows from (\ref{epsitot}) and (\ref{sigmtau}).
The constraint on $\sigma(c,k)$ follows from the fact that any vertex
in $k$ that is adjacent to all of $p$ must have an edge to $c$. Finally,
we employ (\ref{sumeps}) and (\ref{sigpositive}) to constrain $m_c$. \qed
If we have a good set $j$ and a distinct set $i$, such that $\bar{i}$ is neighbourly, then if
no vertex is completable,
\be
m_B \ge n +\delta_B + \left(n_j - \tau(i,j) + \chi_{ij} \right) n_k
\ee
where $k = \bar{i} \cap B$. {\em Proof.} $(\tau(i,j) - \chi_{ij})$ is the maximum number of
vertices in $j$ that are not adjacent to all of $\bar{i}$, therefore $n_j - (\tau(i,j)-\chi_{ij})$
is the minimum number of vertices in $j$ that are adjacent to all of $\bar{i}$. If $\bar{i}$
is neighbourly, these vertices require a bad edge to $i$ in addition to the $n_k$ bad edges they
each have to $\bar{i}$. The result follows. \qed
\subsection{Conditions relating to unique vertices}
\begin{definition}
A {\em unique degree} is one which appears only once in the degree sequence.
For a vertex $v$ of unique degree $d_v$, define
\be
\alpha_v &\equiv& \{ w : d_w = d_v -1 \}, \nonumber \\
\beta_v &\equiv& \{ w : d_w = d_v + 1 \}. \label{alphabeta}
\ee
\end{definition}
\begin{lemma} \label{unique}
{\bf Part A}.
If there is a vertex $v$ of unique degree $d$,
then if $v$ is not completable,
it must not be adjacent to all the dull vertices not in $\alpha_v$
and it must be adjacent to a bad vertex not in $\beta_v$.
Thus, if $s \equiv (D-v) \setminus \alpha_v$ and $t \equiv (B-v) \setminus \beta_v$
then $\epsilon(v,s) < n_s$ and $\epsilon(v,t) \ge 1$, where $\alpha_v$, $\beta_v$ are
defined in (\ref{alphabeta}).
{\bf Part B}.
If there is a vertex $v$ of unique degree $d$ and $\epsilon(i,v)=0$ for a neighbourly set $i$,
then if $v$ is not completable,
$\epsilon(v,\, s \cap \ibar) < |s \cap \ibar|$ and $\epsilon(v,\, t \cap \ibar) \ge 1$,
where $s$ and $t$ are defined as in part A.
\end{lemma}
{\bf Corollary 1.} If a vertex $v$ has unique degree and $B \setminus \beta_v \subset i$,
for some set $i$, then $\epsilon(v, \, i \cap B) \ge 1$.\\
{\bf Corollary 2.} If there is a $d$-neighbourly vertex set $j$
with $n_j = d+1$, and a distinct set $i$, then $\epsilon(i,j) \ge \chi_{ij} + 1$. \\
{\bf Corollary 3.} If a vertex $v$ has unique degree, then
for any distinct sets $i,j$ such that $s \subset j$, $v \notin i$,
and $d_{\rm min}(s) \ge n-n_i$, $\epsilon(i,j) \ge \chi_{ij} + 1$. \\
{\bf Corollary 4.} If there are exactly 3 bad degrees, and
there are two dull vertices $u,v$ with unique degrees $d_u, d_v \ne d_u+1$,
such that $u$ is good and $v$ is bad, then $\epsilon(u,\, \beta_v)
= \epsilon(v,\, \beta_u) = 1$ and $\epsilon(u,v)=0$ (see figure \ref{f.uv}).
\begin{figure}
\begin{picture}(200,30)
\put(0,0){$u$}
\put(6,0){\put(0,0){\line(0,1){2}}\put(0,0){\line(1,0){15}}\put(15,0){\line(0,1){2}}}
\put(33,0){\put(0,0){\line(0,1){2}}\put(0,0){\line(1,0){15}}\put(15,0){\line(0,1){2}}}
\put(50,0){$v$}
\put(56,0){\put(0,0){\line(0,1){2}}\put(0,0){\line(1,0){15}}\put(15,0){\line(0,1){2}}}
\put(52,6){\put(0,0){\line(0,1){4}}\put(0,4){\line(-1,0){40}}\put(-40,4){\vector(0,-1){7}}}
\put(2,-2){\put(0,0){\line(0,-1){5}}\put(0,-5){\line(1,0){60}}\put(60,-5){\vector(0,1){6}}}
\put(90,0){\mbox{or}}
\put(120,0){
\put(0,0){\put(0,0){\line(0,1){2}}\put(0,0){\line(1,0){15}}\put(15,0){\line(0,1){2}}}
\put(17,0){$v$}
\put(23,0){\put(0,0){\line(0,1){2}}\put(0,0){\line(1,0){15}}\put(15,0){\line(0,1){2}}}
\put(50,0){$u$}
\put(56,0){\put(0,0){\line(0,1){2}}\put(0,0){\line(1,0){15}}\put(15,0){\line(0,1){2}}}
\put(19,6){\put(0,0){\line(0,1){4}}\put(0,4){\line(1,0){45}}\put(45,4){\vector(0,-1){7}}}
\put(53,-2){\put(0,0){\line(0,-1){5}}\put(0,-5){\line(-1,0){25}}\put(-25,-5){\vector(0,1){6}}}
}
\end{picture}
\caption{Illustrating corollary 4 to lemma \protect\ref{unique}. $u$ and $v$ are vertices
of unique degree; the horizontal brackets represent groups of vertices arranged in degree order;
the arrows represent the presence of required edges in the associated graph if the sequence
is not forcing. There is no edge $uv$.}
\label{f.uv}
\end{figure}
\begin{proof}
The lemma follows immediately from the definitions;
the corollaries are simple examples. For the second and third corollaries, observe
that we cannot allow all the vertices in $j$ to have an edge
to the special vertex, because then that vertex would be completable.
For the fourth corollary (c.f. figure \ref{f.uv}),
$v$ must be adjacent to a member of $\beta_u$ since these
are the only bad vertices with degree not equal to $d_v$ or
$d_v+1$, and $v$ must not
be adjacent to $u$ because $u$ is the only dull vertex with degree not equal
to $d_v$ or $d_v-1$. It follows that $u$ must be adjacent to a vertex in $\beta_v$ since
these are the only bad vertices apart from $v$ whose degree is not $d_u+1$.
\qed
\end{proof}
Consider for example the sequences [1111{\bf 2}333 56] and [11111 3344{\bf 5}].
In the first case, let $v$ be the vertex of degree 2. Then the set $s$ consists
of one member, the vertex of degree 5.
This vertex is good, so it
cannot be adjacent to any of the pendants (if they are not completable), therefore it
has to be adjacent to $v$, so we have $\epsilon(v,s)=1=n_s$
which breaks the condition stated in the lemma,
so the sequence is forcing. In the second case, the vertices of degree 3 cannot be adjacent to
the pendants, and nor can either of them have a closed neighbourhood containing all of $D$.
Hence they
must both be adjacent to the vertex $v$ of degree 5, so we have $\epsilon(v,s)=2=n_s$ so
this is forcing also. This argument is generalised in eqn (\ref{epssvms}).
Corollary 1 of lemma \ref{unique}, when used in conjunction with (\ref{xminedge}),
suffices to prove that [1112 44{\bf 5}666] is forcing. The special vertex $v$ is the one
of degree 5. Let $j = \beta_v$, $i = \{ w : d_w < 3 \}$.
Since $v$ must be adjacent to the vertex of degree 2,
it uses up one of the stubs in $c = i \cap B$, with the result that
$\tau(c,j)=m_c-1=1$. Therefore (\ref{xminedge}) gives
$\epsilon(i,j) \ge 3 + (3-1) = 5 = m_i$. But $\epsilon(i,j) \le m_i -1$
owing to the edge from $i$ to $v$, so we have a contradiction, hence the
sequence is forcing.
Corollary 2 can be used to prove that [1112{\bf 3}455] is forcing, by taking
$j = \{ v : d_v \ge 3 \}$, $i =\bar{j}$.
One finds $\chi_{ij} = 5 = m_i$ so
$\epsilon(i,j) > m_i$ which contradicts (\ref{maxstubs}).
Corollary 3 can be used to prove that [1 333{\bf 4}55 778] is forcing, by taking
$d_v = 4$,
$j = \{ w : d_w > 6 \}$, $i = \{ w : d_w < 4 \}$.
We have $\chi_{ij} = 7$ so $\epsilon(i,j) \ge 8$ which contradicts (\ref{jneighbourly}).
[223334 {\bf 6}777]
If a good vertex $v$ has unique degree, then for any vertex set $i$,
\be
\epsilon(i,\, (\bar{i} \cap \bar{B}) \cup \beta_v + v) \le m_i - \sigma(i,v). \label{corollary4}
\ee
{\em Proof}.
Any vertex in $i$ that is adjacent to $v$ cannot
have all its other edges to members of $\bar{B} \cup \beta_v$, since with $v$
they form a $d_v$-neighbourly set. \qed
[1233 {\bf 5}6666 8].
If there is a set $j$ containing a vertex $v$ of unique degree $d=n_j-1$, and such that
all bad vertices in $j$ have degree $n_j$, and
$\chi_{ij} \ge m_i - 2$ where $i = \bar{j}$, then the degree sequence is forcing.
{\em Proof}. Using corollary 2 to lemma \ref{unique}, $\epsilon(i,j) \ge m_i - 1$,
and therefore, using (\ref{oddsig}), $\epsilon(i,j) = m_i$. To avoid being
completable, the vertex $v$ must have a neighbour in $i$. But then that
neighbour will be completable, because all its neighbours are in $j$ and
the conditions are such that any set consisting of $v$ plus some members of $j$
is neighbourly. \qed
[1222 {\bf 4}55555].
If there is a unique good vertex $v$, and $d_v+1$ is the only bad degree greater
than $d_v$, then let $i = B \setminus \beta_v$ and
$j = \{ d > d_v \}$. If $\sigma(i,j) \ge m_i - 2$ then the degree sequence is
forcing. {\em Proof}. If there is no completable vertex then $v$ must be
adjacent to a member of $i$. That member then requires an edge
to another member of $i$. Together these two edges require
3 stubs in $i$, but if $\sigma(i,j) \ge m_i - 2$ then there are
at most 2 stubs available after edges to $j$ have been accounted for. \qed
[11111 [33]44{\bf 5}].
If there is a non-dull vertex $v$ of unique degree, let $s = D \setminus \alpha_v$
and $i = (V-v) \setminus D$.
We have $\tau(s,D) \le n_s(n_D-2)$ since if there is no completable vertex, then no vertex can
have a closed neighbourhood containing
all of $D$. We have that $V$ consists of distinct sets $i,\, D, \, \{ v \}$,
so, using (\ref{sumeps}),
\be
m_s \le \tau(s,i) + \tau(s,D) + \tau(s,v)
\ee
which implies, using (\ref{jgood}) for $\tau(s,i)$,
\be
\!\! \epsilon(s,v) \ge m_s - (m_i - (1 - \delta_s^B) n_i) - n_s(n_D-2) . \label{epssvms}
\ee
If this number is $\ge n_s$ then, by lemma \ref{unique}, the degree sequence is forcing.
[1122 444 {\bf 6}77], [1122222 {\bf 5}67].
If there is a good dull vertex $v$ of unique degree $d \ge 3$, then let $j = \beta_v + v$
and $i =\{u : d_u < 3\}$.
Either of the following conditions imply that the degree sequence is forcing:
\be
m_B < m_j + 1 \\
d > n - (n_i + 3 + \mu_i)/2 \label{dnni}
\ee
{\em Proof.} In the first case, we have $m_B = m_k + (m_j-d)$ where $k = \bar{j} \cap B$.
If the sequence is not forcing, then every
neighbour of $v$ requires an edge to $k$, and so does $v$. This is not possible
if $m_k < d+1$, which gives the condition. In the second case, we have that
$v$ is adjacent to at least $d - \xi_i$ members of $i$, and these neighbours may not be pendants,
therefore they have degree 2. No vertex of degree $d+1$ may be adjacent to any of
these, or we would have a vertex of degree 2 with a neighbourly neighbourhood. Hence
if $d+1 > n-1-(d - \xi_i)$ then the sequence is forcing. Furthermore, if $d+1=n-1-(d-\xi_i)$
then a vertex has a neighbourhood which contains all of $\bar{i}$ and all the pendants,
and therefore all of $D$ when $\bar{i}$ is neighbourly, so in this case $d$ must be
larger still to avoid forcing. (\ref{dnni}) follows.
\qed
[333[4444] {\bf 7}[88]].
If there is a good dull vertex $v$ of unique degree $d$,
and no dull vertices of higher degree, then let
$h = \{ w : d_w > d \}$ and
$k = B \setminus h = B \setminus \beta_v$. If
\be
m_k < \sigma(k, h) + 2 \sigma(k,v) + (\sigma(k,v) \mbox{ mod }2) + x
\ee
where $x = n-(n_k+n_h)-2 + \delta_k^D - \kappa(V,1)$
then the degree sequence is forcing. {\em Proof}.
By lemma \ref{unique},
the conditions are such that if the sequence is not forcing, then any vertex adjacent to $v$
must be adjacent to a vertex in $k$. The right hand side of the inequality gives a lower
bound on the number of stubs in $k$ required. Each edge contributing to $\sigma(k,h)$ requires
one stub in $k$. Each edge contributing to $\sigma(k,v)$ requires one stub in $k$, and then another
so that the vertex also has a neighbour in $k$. The $(\sigma(k,v) \mbox{ mod }2)$ term accounts
for the extra stub required when vertices in $k$ that neighbour on $v$ can't be merely joined in pairs.
Finally, $x$ is the number of good vertices
that must be adjacent to $v$ on the
assumption that we are making $\epsilon(k,v)$ as small as possible.
For the number of vertices not in $k \cup h + v$ is $n-n_k-n_h - 1$, and
$v$ must be adjacent to all but $\kappa(V,1) + (1-\delta_k^D)$ of them when $\epsilon(k,v)$ is minimised.
\qed
[123 5555 77{\bf 8}].
Let $d_v$, $d_u$ be the highest and next highest bad degrees, respectively. If $d_v$ is unique
and $d_v-1$ is good, then let $i = \{ w : d_w \le d_u \}$ and $h = \{ w : d_w \ge d_v - 1 \}$.
If $i \cup h \ne V$ and $\chi_{ih} \ge m_i - 2$ then the degree sequence is forcing. {\em Proof}.
Let $v$ be the vertex having degree $d_v$.
We have $D \setminus \alpha_v \subset i$, therefore, from lemma \ref{unique}, $v$ cannot be adjacent to all
of $i$. Since $B - v \subset i$, this implies that at least one vertex in $i$ must have an edge to
another vertex in $i$. This edge uses up two of the stubs in $i$, so under the condition
$\chi_{ih} \ge m_i - 2$, every vertex in $j$ must be adjacent to all of $\bar{i}$ except itself.
It follows that each vertex not in $i$ or $j$ is adjacent to all of $\alpha_v$ and none of
$B - v$. Such vertices must be completable. \qed
If there is a bad unique vertex $v$, such that $\alpha_v$ is good, then let
$j = \alpha_v$. Then if there is no completable vertex,
\be
\tau(j,j) \le n_j(n_j-2) + m_B - d_v - 2. \label{taujjv}
\ee
{\em Proof}. This is an example of relation (\ref{taukk}).
Let $c = B - v$. By lemma \ref{unique}, if there is no completable
vertex then $v$ must have an edge to $c$, and $v$ must not be adjacent to all the
dull vertices whose degree is one less than a member of $c$.
Therefore there is an edge from $c$ to $v$, and there is an edge
between a vertex in $c$ and at least one other vertex not in $j$
(since every vertex requires a bad edge). Hence we have $\tau(c,j) \le m_c-2
= m_B - d_v - 2.$ One then obtains (\ref{taujjv})
by the same argument as for (\ref{taukk}). \qed
[3334 66666{\bf 7}].
In order to prove that this sequence is forcing, substitute
$\sigma(B,j) \ge m_j - (\tau(j,j) + m_g - n_g)$ into condition (\ref{mBbound}),
using (\ref{taujjv}) to provide a bound on $\tau(j,j)$, and
$g = \bar{B} \setminus \alpha_v$.
\subsection{Conditions relating to pendants}
If
\be
d_{\rm max}(\bar{B}) > n - 1 - \kappa(V,1) - x
\ee
where $x=1$ if $\kappa(V,2)=0$ and $x=0$ otherwise, then the sequence is forcing.
{\em Proof}. A good vertex having a degree above this bound is either adjacent to a pendant
(so the pendant is completable) or $x=1$ and it
is adjacent to all the vertices except the pendants and itself; in the latter case it is
completable since $x=1$ implies there are no vertices of degree 2. \qed
[11111 3344{\bf 5}].
If $\kappa(V,1) = n/2$ and
there is a vertex $v$ of unique degree $d_v = n - \kappa(V,1) = n/2$,
and $D = \{ w : d_v-2 \le d_w < d_v \}$, then the degree sequence is
forcing. {\em Proof}. This is an example of the situation treated by
eqn (\ref{epssvms}). In the case under consideration,
$s = D \setminus \alpha_v$ is good and $i$ consists of pendants, so
eqn (\ref{epssvms}) reads $\epsilon(s,v) \ge m_s - n_s(n_D-2)$. Also,
we have $m_s = n_s (d_v-2)$, so this becomes $\epsilon(s,v) \ge n_s(d_v-n_D)$.
Finally, $n_D < n - \kappa(V,1)$ since
pendants are never dull if there is no completable vertex, and by
assumption $v$ is also not dull. So if $d_v = n - \kappa(V,1)$ then
$d_v > n_D$ and we have $\epsilon(s,v) \ge n_s$,
which breaks the condition set by lemma \ref{unique}. \qed
[12 {\bf 45}66666 8].
If there are exactly 3 bad degrees, and these are $2, d+1, d+2$, with
$\kappa(V,2) = \kappa(V,d) = \kappa(V,d+1) = 1$, then if
\be
d > (n - 2 - \kappa(V,1)) / 2 \label{d1}
\ee
then the sequence is forcing. {\em Proof.}
Let $u$ be the vertex of degree $d$ and $v$ be the vertex of degree $d+1$.
If there is no completable vertex, then by lemma \ref{unique},
$v$ is adjacent to the vertex of degree 2, and not to the pendant.
It follows that $u$ may not be adjacent to the vertices of degree
1 or 2, and therefore it may not be adjacent to $v$ either (or its closed neighbourhood
will be $d$-neighbourly). Hence all neighbours of $u$ are in $h = \{ w : d_w > d+1\}$.
Equally, since there is no edge $uv$, vertex $v$ also has $d$ neighbours in $j=h$.
The number of vertices in $h$ adjacent to both $u$ and $v$ is therefore
greater than or equal to $2d - n_h$. At most one of these can have the vertex
of degree 2 as a bad neighbour, so if $2d - n_h > 1$ then there is a vertex
adjacent to $u$ and $v$ but not 2; such a vertex is completable. We have
$n_j = n-\kappa(V,1)-3$ and (\ref{d1}) follows. \qed
[1233$\ldots$ [good] $uv$].
If $\kappa(V,1) = \kappa(V,2) = \kappa(V,d) = \kappa(V,d+1) = 1$
for some $d$, and $\kappa(V,3) > 0$ and there are exactly three bad degrees
(namely $2,\, 3, \, d+1$), then if
\be
d \ge n-3
\ee
then the sequence is forcing. {\em Proof.}
Let $u,v$ be the vertices of degree $d,d+1$.
If there is no completable vertex, then the vertex of degree 2 must
be adjacent to $v$, and not to $u$. Therefore $v$ may not be adjacent
to the pendant (lemma \ref{unique}), and nor may $u$ (or the pendant
will be completable). Since the pendant requires a bad neighbour, it
is therefore adjacent to a vertex of degree 3. But if $d \ge n-3$ then
both $u$ and $v$ are adjacent to all the vertices of degree 3, so the
one with the pendant neighbour is completable. \qed
\subsection{Conditions relating to $\sigma = \tau$}
The theme of the conditions in this section is that when edges from a set $j$ of high degree
use up all the available stubs in a set $i$ of low degree, then the vertices of intermediate
degree must have all their edges to each other or to $j$.
\begin{lemma}
If $\sigma(i,\bar{i}) = m_i$ for a set $i$ with neighbourly $\bar{i}$,
and there is no completable vertex, then
\be
\tau(i,p) \le \sum_{v \in i} \min(d_v, n_p-1)
\ee
where $p = \bar{i} \cap D$.
\end{lemma}
{\em Proof}. When $\sigma(i,\bar{i}) = m_i$, all edges from $i$ go to vertices
in $\bar{i}$. In this case, when $\bar{i}$ is neighbourly no vertex in $i$ may be adjacent to all of $p$, or it will be completable. \qed
This lemma suffices to prove that [[122233] 566 8] is forcing. In this example,
$p$ consists of the vertex of degree 5 and we find $\tau(i,p) = 0$. However
$\sigma(i,p) \ge \chi_{ip} = 2$, so we have $\sigma(i,p) > \tau(i,p)$ which
is a contradiction. \qed
\begin{lemma} \label{lemgtoh}
If $\chi_{ij} \mu_i = m_i$ for distinct sets $i,j$ and there is no completable vertex, then
\be
\epsilon(g,s) = m_g \label{gtoh}
\ee
where $g = i \setminus B$ and $s = j \setminus \{ u : d_u = n-n_i \}$.
\end{lemma}
{\em Proof}. (c.f. figure \ref{f.gh}.) When $\chi_{ij} = m_i$, all the edges from $i$ go to $j$, and each
vertex in $j$ is adjacent to all the vertices in $\bar{i}$ except itself.
The vertices of degree $n-n_i$ then have exactly one edge each to $i$, and
when $\bar{i}$ is neighbourly, this edge must be to a bad vertex in $i$. Hence those
vertices have no edges to $g$. The result follows. \qed
\begin{figure}
\begin{picture}(200,30)
\put(0,0){\put(1,1){2233}\put(0,0){\line(0,1){2}}\put(0,0){\line(1,0){20}}\put(20,0){\line(0,1){2}}}
\put(25,0){\put(1,1){666778}\put(0,0){\line(0,1){2}}\put(0,0){\line(1,0){29}}\put(29,0){\line(0,1){2}}}
\put(0,9){\put(0,0){\line(0,-1){2}}\put(0,0){\line(1,0){10}}\put(10,0){\line(0,-1){2}}}
\put(39,9){\put(0,0){\line(0,-1){2}}\put(0,0){\line(1,0){15}}\put(15,0){\line(0,-1){2}}}
\put(5,10){\put(0,0){\line(0,1){4}}\put(0,4){\line(1,0){40}}\put(40,4){\vector(0,-1){6}}}
\put(8,-2){\put(0,0){\line(0,-1){5}}\put(0,-5){\line(1,0){30}}\put(30,-5){\vector(0,1){6}}}
\end{picture}
\caption{An illustration of lemma \protect\ref{lemgtoh}.
The arrows represent cases where all the edges from
the vertex set at the start of the arrow must go to the vertex set at the end of the arrow,
if no vertex is completable. The lower arrow represents the condition $\chi_{ij}=m_i$;
the upper arrow represents eqn (\protect\ref{gtoh}).}
\label{f.gh}
\end{figure}
\begin{lemma}
For any vertex set $i$ whose maximum degree is less than
$\xi_i$, let $h = \{ u : d_u > \xi_i \}$. If
there is no completable vertex and
\be
\left(\chi_{ih} + \kappa(\bar{i}, \xi_i) \right) \mu_i = m_i \label{ihcond}
\ee
then
\be
\epsilon(c, \, h) &\ge&
n_h + \sum_{w \in h} \left[ d_w - n+n_c+1-\delta_c^D\right]^+, \label{epsiBh} \\
\epsilon(i, \, s) &=& n_s , \label{epsink}
\ee
where $c = i \cap B,\; s = \{ u : d_u = \xi_i \}$.
\end{lemma}
{\em Proof}. If the condition holds then there is no room in $i$ for extra edges from $h$
in addition to the minimum number that go there on the assumption that every vertex in $h$
is adjacent to the whole of $\bar{i}$ except itself. When $\bar{i}$ is neighbourly, every
vertex in $h$ then requires a bad neighbour in $i$ to avoid being completable, which
gives the first term in (\ref{epsiBh}). The second term allows for the further edges
which are unavoidable when vertices in $h$ have high degree.
The condition also requires that each vertex in $s$ has one edge
to $i$, which gives (\ref{epsink}).
\qed
[122 44555], [1222 556667].
Suppose (\ref{ihcond}) holds, and therefore so do (\ref{epsiBh}) and (\ref{epsink}).
Using (\ref{epsink}), (\ref{sigmtau}) and (\ref{jgood}) we have
\be
\epsilon(c,s) \ge n_s - m_g + (1 - \delta_s^B)n_g \label{epsckg}
\ee
where $g = i \setminus c$. Hence if
\be
m_c < \sigma(c,h) + n_s - m_g + (1 - \delta_s^B)n_g \label{mchkg}
\ee
where $\sigma(c,h)$ is given by any applicable methods, such as (\ref{xminedge})
and (\ref{epsiBh}), then the degree sequence is forcing.
\begin{figure}
\begin{picture}(200,30)
\put(0,0){\put(1,1){123}\put(0,0){\line(0,1){2}}\put(0,0){\line(1,0){16}}\put(16,0){\line(0,1){2}}}
\put(20,0){\put(1,1){55}\put(0,0){\line(0,1){2}}\put(0,0){\line(1,0){10}}\put(10,0){\line(0,1){2}}}
\put(32,0){\put(1,1){6666 8}\put(0,0){\line(0,1){2}}\put(0,0){\line(1,0){28}}\put(28,0){\line(0,1){2}}}
\put(25,10){\put(0,0){\line(0,1){4}}\put(0,4){\line(1,0){18}}\put(18,4){\vector(0,-1){6}}}
\put(8,-2){\put(0,0){\line(0,-1){5}}\put(0,-5){\line(1,0){35}}\put(35,-5){\vector(0,1){6}}}
\end{picture}
\caption{An illustrative example. The arrows represent cases where all the edges from
the vertex set at the start of the arrow must go to the vertex set at the end of the arrow,
if no vertex is to be completable. The lower arrow is an example of (\protect\ref{ihcond});
the upper arrow is an example of (\protect\ref{taukk}) combined with (\protect\ref{taumsig})
and (\protect\ref{sumeps}).}
\label{f.hk}
\end{figure}
[123 [55][6666 8]].
For any vertex set $i$ such that $\bar{i}$ is neighbourly, let
$p = \bar{i} \cap D$ and $q = \bar{i} \setminus p$.
If the condition (\ref{ihcond}) holds, and all the vertices
in $p$ have degree $\xi_i - 1$,
and the only bad degree in $q$ is $\xi_i$,
then the degree sequence is forcing. {\em Proof}.
If we assume there is no completable vertex, then using
(\ref{taukk}), (\ref{sigmtau}) and $m_p = n_p( \xi_i - 1)$
we obtain $\sigma(p,q) = n_p n_q$. Hence all
of the vertices in $q$ are adjacent to all of $\bar{i} \cap D$,
(see figure \ref{f.hk} for an example). It follows that
when $\bar{i}$ is neighbourly, each vertex in $q$ requires an edge to a bad vertex in $i$.
Now, when (\ref{ihcond}) holds, the vertices of degree $\xi_i$ each have
one and only one edge to $i$, and furthermore no vertex in $i$ is
only adjacent to good vertices in $q$. It follows that
all the good vertices in $i$ are adjacent to vertices of degree
$\xi_i$ in $q$, and such vertices in $q$ have no other edge to $i$.
Consequently they are completable. \qed
[2233 [666[778]]]. (See figure \ref{f.gh})
If $\chi_{ij} \mu_i = m_i$ for distinct sets $i,j$
with $d_{\rm min}(j) \ge \xi_i$ and
$(i \cap B) \cap D = \emptyset$, then if
\be
m_g > n_s(n_g - 1) \label{mghg}
\ee
where $g = i \setminus B$ and $s = j \setminus \{ u : d_u = n-n_i \}$
then the degree sequence is forcing.
{\em Proof}.
Using (\ref{gtoh}) we have $\epsilon(g,s) = m_g$ if there is no completable vertex.
When $\chi_{ij} = m_i$ we also have that every vertex in $s$ is adjacent to all of
$\bar{i}$, from which it follows that no vertex in $s$ may be adjacent to all of
$i \cap D$. But the condition $(i \cap B) \cap D = \emptyset$ implies
$i \cap D \subset g$. So we require that no vertex in $s$ is adjacent to
all of $g$, and therefore $\epsilon(g,s) \le n_s(n_g - 1)$. This
condition cannot be realized when $\epsilon(g,s) = m_g$ and
(\ref{mghg}) holds. \qed
[112 [455][6666]].
For any vertex set $i$ such that $\bar{i}$ is neighbourly, let
$c = i \cap B$, $p = \bar{i} \cap D$, $q = \bar{i} \setminus p$.
If $n_q \ge m_c$
and $\sigma(i,q) = m_i$ and there is no completable vertex, then
\be
n_p(n_p+n_q-2) \ge m_p +n_q - m_c . \label{nknk}
\ee
{\em Proof}. Any vertex in $q$
that is adjacent to all the vertices in $p$ requires an edge to $c$, because otherwise
its neighbourhood would be neighbourly. Hence at most $m_c$ of the vertices
in $q$ can be adjacent to all of $p$, therefore $\tau(p,q) \le n_p m_c + (n_p-1)(n_q - m_c)$.
Using (\ref{sigmtau}) we have $\sigma(p,p) \ge m_p - (\tau(p,q) + \tau(p,i)) = m_p - \tau(p,q)$
since $\epsilon(i,p)=0$ when $\sigma(i,q)=m_i$. Finally, since $\epsilon(i,p)=0$ no
vertex in $p$ may be adjacent to all the other vertices in $p$, because when $\bar{i}$
is neighbourly, this set includes
all the vertices whose degree is one less than a vertex in $\bar{i}$. Therefore
$\tau(p,p) \le n_p(n_p-2)$. Eqn (\ref{nknk}) follows by asserting $\sigma(p,p) \le \tau(p,p)$. \qed
[11122[3] [55][66]].
If there is a good set $j$ and a distinct set $i$ such that
$\sigma(i,j) = m_i - n_i$ and $\chi_{ik} > n_i - \kappa(i,\, n_j+1)$, where
$k = \{w : d_w = d_{\rm max}(j)+1\}$,
then the degree sequence is forcing. {\em Proof}.
Let $s = \{ w : d_w = n_j + 1\}$ and suppose the sequence is not forcing.
The condition $\sigma(i,j)=m_i - n_i$ implies that
after edges from $j$ have been accounted for, there is one stub left
on each vertex in $i$.
Each vertex in $s$ is therefore adjacent to
all of $j$, and has one edge to a vertex not in $j$. That final neighbour
cannot be in $k$ or the vertex is $s$ is completable, so $\tau(s,k) = 0$.
Also, there is at most one edge from $i$ to $k$ for each vertex in $i$,
so $\tau(i \setminus s, k) \le n_i - n_s.$ Under these conditions,
eqn (\ref{sigmtau}) gives $\epsilon(s,k) \ge \chi_{ik} - (n_i - n_s)$.
The condition $\epsilon(s,k) \le \tau(s,k)$ then requires
$\chi_{ik} \le n_i - n_s = n_i - \kappa(i,\, n_j+1)$. \qed
[[12] 444455 [7]8].
If there is a good set $j$ and a distinct set $i$ such that
$\sigma(i,j) = m_i - n_i$, $\kappa(i,n_j) > 0$ and
$\sigma(s,k) > 0$, then the degree
sequence is forcing, where $s = \{ v : d_v = n_j \}$ and
$k = \{ v : d_v = d_{\rm max}(j) + 1 \}$. {\em Proof}.
After edges from $j$ are accounted for, each vertex in $i$ has
one stub remaining. We have $s \subset i$. If there is no
completable vertex, then the final edge from a vertex in $s$ cannot
go to $k$, but the condition $\sigma(s,k) > 0$ requires that it
does, which is a contradiction. \qed
\subsection{Conditions involving $\sigma=\tau$ and unique vertices}
[1233 [{\bf 5}6666 8]]. Suppose there is a good vertex $v$ of unique degree
$d = \xi_i$, and
$\sigma(i,j) = m_i$ where $j = \{w : d_w \ge d \}$, and $i$ is some vertex set
distinct from $j$. If $\bar{i}$ is neighbourly
and the only bad degree above $d$ is $d+1$, then
the sequence is forcing. {\em Proof}. The conditions are such that
(\ref{corollary4}) gives $\epsilon(i,j) \le m_i - 1$ which contradicts
$\sigma(i,j) = m_i$. \qed
[1122 4{\bf 5}[6667]]. We now extend eqn (\ref{mchkg}) in the case where
$n_s=1$, that is, there is a single vertex $v$ of degree $\xi_i$. If the condition
(\ref{ihcond}) holds, then there are no vertices of degree below $n_h$ in
$\bar{i}$, but there may be vertices of degree equal to $n_h$. Any such vertices
have all their edges to $h$ when the conditions holds, therefore they are not
adjacent to $v$. It follows that $v$, which must be adjacent to all
of $\bar{i}$ except two vertices (itself and one other),
has edges to all of $h$. Since $v$ is of unique degree $d$, it forms
a $d$-neighbourly set with $h$, so its edge to $i$ must go to a bad
vertex, and we have $\epsilon(c,s) = 1$ (whereas in this case
(\ref{epsckg}) merely gives $\epsilon(c,s) \ge 1 - m_g$ if $s$ is bad).
Hence under the assumed conditions, (\ref{mchkg}) can be replaced by
\be
m_c < \sigma(c,h) + 1.
\ee
[112 4445{\bf 6}[78]]. This case is similar to the previous one.
If condition (\ref{ihcond}) holds, and
$n_s = \kappa(\bar{i},\xi_i-1) = 1$,
$\epsilon(c,h) \ge m_c$,
and there are no bad degrees in $\bar{i}$ smaller than $\xi_i-1$, then the
degree sequence is forcing. {\em Proof}.
Let $u$ be the vertex of degree $\xi_i-1$ and $v$ be the vertex of degree $\xi_i$.
When condition (\ref{ihcond}) holds, both $u$ and $v$ are adjacent to all of $h$.
$h+v$ is $d_v$-neighbourly, so
to avoid being completable, $v$ requires an edge to a bad vertex not in
$h$. When $\epsilon(c,h) \ge m_c$ this edge cannot be to $c$ so it must be to $u$.
But this implies that $u$ is adjacent to all of $(p-u) \setminus \alpha_u$
and therefore, since $u$ can have no edges to $i$, it must be completable.
\qed
[122 {\bf 4}555 [77]8].
If $\chi_{ij} = m_i - n_i$ for a good vertex set $j$ and there
is a good unique vertex $v \in (\ibar \setminus j)$
such that $(D-v) \setminus i \subset j$
then the degree sequence is forcing. {\em Proof}. Under the conditions,
$v$ can have no edges to $i$, and each member of $j$ is adjacent to
all of $\ibar \setminus j$. It follows that $v$ does not satisfy
the condition stated in part B of lemma \ref{unique}. \qed
[{\bf 1}222 [44]555 [8]],
[{\bf 1}222 [444]56 [8]].
Suppose $d_j = n-2-n_i \ne d_{\rm min}$ is a good degree, where
$i = \{ w : d_w \le d_{\rm min}+1\}$ and $d_{\rm min}$ is the lowest
value in the degree sequence. Consider the case
where only one vertex $u$ has degree $d_{\rm min}$,
and either there are 2 bad degrees $[d_{\rm min}+1, \, d_j+1]$
or there are 3 bad degrees $[d_{\rm min}+1,\, d_j+1,\, d_j+2]$,
and in the second case
$d_j+1$ and $d_j+2$ are both unique.
Let $s = \{ w : d_w > d_j\} \setminus B$.
If $\sigma(i,s) = m_i - n_i$ then the sequence is forcing. {\em Proof}.
Under the condition on $\sigma(i,s)$
there can be no edges from $i$ to good vertices not in $s$.
Let $j = \{ w : d_w = d_j\}$. We have that $j$ is good, so
$\sigma(i,j) = 0$. Therefore all the edges from vertices in $j$ go to $\bar{i}$.
First suppose there are only 2 bad degrees. In this case
$j$ are the only dull vertices in $\bar{i}$, hence no vertex in $j$ may
be adjacent to all the other vertices in $j$. Hence, when the condition
$d_j = n-2-n_i$ holds, every vertex in $j$ is adjacent to every vertex in
$\bar{i} \setminus j$.
Now, by lemma \ref{unique}, $u$ must be adjacent to a bad vertex in $\bar{i}$.
That vertex is completable, since it is adjacent to all of $D$.
Next suppose we have 3 bad degrees, the highest two of which are
unique. Let $v$ be the vertex of degree $d+1$ and $w$ be
the vertex of degree $d+2$.
Any vertex in $j$ must be adjacent to $w$ because otherwise the
vertex in $j$ would have a closed neighbourhood
containing all of $\bar{i} \cap D$ and consequently be completable.
Also, noting that $v$ is both bad and dull, by using corollary 4 of
lemma \ref{unique} we have $\epsilon(u,w) = 1$.
It follows that $D \subset N(w)$ so $w$ is completable. \qed
[1111122 [4]{\bf 5}6].
If $\chi_{ij} = m_i - n_i$ for good $j$ with
$d_{\rm min}(c) > n_j$ and there is a vertex $v$ of unique degree
$d = d_{\rm max}(j) + 1$ and the only bad degrees in $\ibar$ are
$d,\, d+1$, then the sequence is forcing. {\em Proof}. The condition
$\chi_{ij} = m_i-n_i$ implies that all but one stub on each vertex in $i$
is used by edges to $j$. The condition $d_{\rm min}(c) > n_j$ implies
that, in order to fill all but one stub of any vertex in $c$, edges from
all the vertices of $j$ are required, so every vertex in $c$ is adjacent
to all of $j$. By lemma \ref{unique}, vertex $v$ requires
a neighbour in
$(B-v) \setminus \beta_v = c$. That neighbour is completable. \qed
[111 {\bf 4}5555{\bf 6}7].
If there are two unique vertices $u,v$ satisfying the conditions
of corollary 4 of lemma \ref{unique}, with $d_u + 2 = d_v = \xi_i$,
and $\chi_{ij} + n_j = m_i$ for a non-dull set
$i$ and the distinct set $j = \{ w : d_w \ge \xi_i\}$,
then the degree sequence is forcing. {\em Proof}.
The conditions are such that $D \subset \bar{i}$ and
the only bad degrees are $d_v,\,d_v \pm 1$. When $D \subset \bar{i}$,
to avoid being completable, no vertex
may be adjacent to all of $\bar{i} - w$ where $w$ is one of the vertices
having the highest bad degree (since such vertices cannot be dull).
The condition $\chi_{ij} + n_j = m_i$
implies that each vertex in $j$ is adjacent to
all but two of the vertices in $\bar{i}$ (itself and another),
and the same is true of each vertex in $s = \{ w : d_w = \xi_i - 1 \}$.
It follows that, if there is no completable vertex, then $v$ and
every vertex in $s$ is adjacent to every vertex in $\beta_v$.
Corollary 4 of lemma \ref{unique} also gives that $u$
is adjacent to one of the vertices in $\beta_v$. It follows
that that vertex must be completable, since its neighbourhood
includes all of $D$. \qed
\begin{figure}
\begin{picture}(200,30)
\put(0,0){$uv$}
\put(12,0){\put(0,0){\line(0,1){2}}\put(0,0){\line(1,0){15}}\put(15,0){\line(0,1){2}}}
\put(40,0){\put(0,0){\line(0,1){2}}\put(0,0){\line(1,0){15}}\put(15,0){\line(0,1){2}}}
\put(57,0){$w$}
\put(8,6){\put(0,0){\line(0,1){4}}\put(0,4){\line(1,0){52}}\put(52,4){\vector(0,-1){6}}}
\put(2,-2){\put(0,0){\line(0,-1){5}}\put(0,-5){\line(1,0){14}}\put(14,-5){\vector(0,1){6}}}
\end{picture}
\caption{Illustrating lemma \protect\ref{unique3}. $u,v,w$ are vertices
of unique degree;
the arrows represent the presence of required edges in the associated graph if the sequence is not forcing. There are no edges $uv,\,uw$. The illustration
shows the case $\beta_v \ne \alpha_w$; the lemma applies equally when
$\beta_v = \alpha_w$, and when there are further groups of good vertices.
}
\label{f.u3}
\end{figure}
\begin{lemma} \label{unique3}
If there are exactly 3 bad degrees, and there are three vertices $u,v,w$
of unique degree such that $d_v = d_u+1$, $u$ is good, $v$ is dull,
$w$ is bad and not dull, then if none of $u,v,w$ are completable then
$\epsilon(u,v) = \epsilon(u,w) = 0$,
$\epsilon(v,w)=1$, $\epsilon(u, \beta_v) \ge 1$.
(see figure \ref{f.u3}).
\end{lemma}
\begin{proof}
The edge $vw$ follows from lemma \ref{unique}. The same lemma then implies that
$w$ is not adjacent to $u$ because the conditions are such that $\{u,v \} = (D-w) \setminus \alpha_w$. It then follows that $u$ must be adjacent to $\beta_v$ and not to $v$. \qed
\end{proof}
[222 4{\bf 5}[66667]].
When the conditions of lemma \ref{unique3} apply, with $d_w > d_v$,
$u,v,w \in \ibar$, $d_v = \xi_i-1$, $j = \{ x : d_x \ge \xi_i \}$
and $\chi_{ij} + n_j = m_i$, the degree sequence is forcing. {\em Proof}.
The condition $\chi_{ij} + n_j = m_i$ implies that $v$ has no edge to $i$. It
follows that all its edges go to $\bar{i}$, and since $d_v = \xi_i-1$ it must
be adjacent to all of $\bar{i}$ except itself and one other vertex. By lemma 10,
there is no edge $uv$, so we deduce $N(v) = j$. The conditions are such that
$j+v$ is $d_v$-neighbourly, so then $v$ is completable. \qed
[{\bf 1}23333 [666]{\bf 7}].
If the conditions of lemma \ref{unique3} apply, such that $j = \alpha_w$
is good, then let $i$ be a distinct vertex set such that $(B - w) + u \subset i$.
If $\sigma(i,j) = m_i - n_i$ then the degree sequence is forcing. {\em Proof.}
After edges from $j$ have been accounted for, each vertex in $i$ has one stub
remaining, and that stub must be used to provide an edge to a bad vertex.
Hence this final edge cannot be to $u$, for each vertex in $i$. Hence the only
way $u$ can have a bad edge is if it is adjacent to $w$, but the edge $uw$
is ruled out by lemma \ref{unique3}. \qed
\subsection{Conditions involving $\sigma=\tau$, unique vertices and bad stubs}
[223 {\bf 5}[667777]]. Suppose we have distinct sets $i,\, j = \{ w : d_w \ge \xi_i \}$
such that $\bar{i}$ is neighbourly and (\ref{xminedge}) gives $\sigma(i,j) = m_i$.
If $\kappa(V, \, \xi_i - 1) = 1$, $\kappa(j, \xi_i) = 2$,
$\kappa(j, \, \xi_i+1) \ge m_c$ where $c = i \cap B$ and there are no dull degrees
in $\bar{i}$ except $\xi_i$ and $\xi_i-1$,
then the degree sequence is forcing. {\em Proof}. Let $v$ be the vertex of degree $\xi_i-1$,
let $s = j \cap D$, and suppose no vertex is completable.
$v$ is not in $j$ so it can have no edges to $i$. Therefore $v$
cannot be adjacent to both the vertices of $s$, or it will be completable.
It follows that the vertices of $s$ are adjacent, since whichever one is
not adjacent to $v$ must be adjacent to all other members of $j$ when (\ref{xminedge})
gives $\sigma(i,j) = m_i$. The degree of $v$ is such that it must be adjacent to at least
one of $s$, so we deduce that a vertex in $s$ has all of $\bar{i} \cap D$ in its closed
neighbourhood.
Such a vertex requires a bad edge to $i$ to avoid being completable,
but under the given conditions, all the bad stubs in $i$ have been used up by edges to
vertices of degree $\xi_i + 1$. It follows that there must be a completable vertex. \qed
[122 555{\bf 6}778], [112 4445{\bf 6}[78]].
Suppose there is a set $i$ such that condition (\ref{ihcond}) holds
and $\kappa(V,\xi_i)=1$ and $\epsilon(c,h) \ge m_c$.
Then if either there is no bad degree less than $\xi_i$ in $\ibar$,
or there is one and it is unique and equal to $\xi_i-1$,
then the degree sequence is forcing.
{\em Proof}. Let $u,v$ be the vertices of degree $\xi_i-1, \,\xi_i$.
When the conditions apply, all the vertices of degree above $d_v$ are
adjacent to $v$, and $v$ has no edge to $c$. It follows that, to avoid
being completable, $v$ requires
an edge to a bad vertex not in $(h + v) \cup c$,
because $h+v$ is $d_v$-neighbourly. If there is no such vertex,
then this cannot happen, so the sequence is forcing. If
there is a single such vertex ($u$) and its degree is $\xi_i-1$, then
$u$ has edges to all of $v$ and $h$ and none to $c$, therefore
its closed neighbourhood is $d_u$-neighbourly, so it is completable.
\qed
[22333 555{\bf 6} 8]. Let $j$ be the set of good vertices of degree above some value,
and $g$ be the set of all the other good vertices. Then, if there is no completable vertex,
and using (\ref{sigmtau}), (\ref{jgood}),
\be
\epsilon(B,j) \ge m_j - n_j(n_j-1) - (m_g - n_g) . \label{jgeps}
\ee
Now suppose we have that, when we use the right hand side of
this expression to give $\sigma(B,j)$ in (\ref{mBbound}),
we find that the bound (\ref{mBbound}) is saturated.
In this case, any argument which leads to the conclusion that $\epsilon(B,j)$ is in fact
larger than and not equal to $m_j - n_j(n_j-1) - (m_g - n_g)$
will lead to the condition (\ref{mBbound}) not being satisfied, so the degree sequence is forcing.
We consider the case where there are exactly two bad degrees, $d_k$ and $d_v$,
and only one vertex $v$ has the higher of these
two degrees, and $\alpha_v \subset j$.
Let $k = B - v$. If no vertex is completable,
then, on the assumption that (\ref{jgeps}) is an equality, we must have
\be
\epsilon(k, \bar{B}) \ge 1 + \sum_{w \in j} \max\left(1, \, d_w +1+n_B-n\right). \label{epscgsum}
\ee
For, in the conditions under consideration, each vertex in $j$ is adjacent to all the others in $j$,
and therefore requires an edge to $k$ in order that its closed neighbourhood is not neighbourly.
Each one of
those of degree higher than $n-n_k-2 = n-1-n_B$ requires $d_w-(n-1-n_B)$ edges to $k$ since
the closed neighbourhood may not contain all of $D$, and the conditions are such that $k$ is not dull.
This gives the sum in (\ref{epscgsum}). The additional $1$ is owing to the fact
that $v$ cannot be adjacent to all the vertices of degree $d_k-1$,
therefore at least one of these
requires an edge to $k$. If we now have $\epsilon(k,\bar{B}) > m_k - n_k$ then the condition
(\ref{jgood}) is not satisfied, so the degree sequence must be forcing.
[22{\bf 3}4 [66666]{\bf 7}].
If there are exactly three bad degrees, and there is a dull bad vertex $u$
of unique degree, and a bad vertex $v$ of unique degree $d_v \ge d_u + 4$,
and only one good degree greater than $d_u$, then if
\be
m_B &<& d_v + n_j + 2 + \delta_B , \label{mBkappa} \\
m_g - n_g &\le& d_v - n_j , \label{mgcond} \\
d_v &\ge& n_j + 2 \label{dvnjcond}
\ee
then the degree sequence is forcing, where
$j = \{ w : d_w = d_v - 1 \} = \alpha_v$ and
$g = \{ w : d_w < d_u\}$.
{\em Proof}. By lemma \ref{unique},
if there is no completable vertex then $u$
must be adjacent to $v$, and therefore $v$ may not be adjacent to all the vertices
of degree less than $d_u$. There are $n - n_j - n_B$ such vertices. It follows that
\be
\sigma(v,j) &\ge& d_v - (n-n_j-n_B-1) - (n_B-1) \nonumber \\
&=& d_v - n + n_j +2.
\ee
Let $t$ be the number of vertices in $j$ that are not
adjacent to all the other vertices in $j$. First consider the case $t=0$.
The conditions are such that the vertices in $j$ are
good, and any of them that are adjacent to both $v$ and all the others in $j$
require an edge to a vertex in $B - v$, so they have two bad edges. Therefore
when $t=0$ we have $\sigma(B,j) \ge 2 \sigma(v,j) + (n_j - \sigma(v,j))
= \sigma(v,j) + n_j$.
Substituting this into (\ref{mBbound}) gives (\ref{mBkappa}).
Next suppose $t>0$. $t$ cannot be equal to 1, so the smallest non-zero value is $t=2$.
If neither of these vertices are adjacent to $v$, then the count of bad edges is unchanged
and we obtain (\ref{mBkappa}) again. If one of them is adjacent to $v$, then the
only way it can avoid having another bad edge is if it has $d_v-2$
good neighbours. Therefore it has $(d_v-2)-(n_j-2) = d_v - n_j$ good neighbours
not in $j$. Under the condition (\ref{mgcond}) this is either not
possible, or else it is possible and no other vertex in $j$ can be adjacent to a vertex
in $g$. In the latter case, the second vertex in $j$ that is not adjacent to all others
in $j$ can have at most $n_j-2$ good neighbours, so it has $d_v - n_j + 1$ bad
neighbours. In this case we obtain
$\sigma(B,j) \ge 2(\sigma(v,j)-2) + (n_j-\sigma(v,j)) + 1 + (d_v - n_j + 1)
= \sigma(v,j) + d_v -2$. Using (\ref{dvnjcond}) we find
that $\sigma(B,j)$ is no smaller than before, and the condition
(\ref{mBkappa}) applies again. By similar arguments, $t > 2$ also does
not lead to a smaller $\sigma(B,j)$. \qed
\section{Statistics and extension of the concepts}
\begin{table}
\begin{tabular}{ccccccccc}
$n$ & seq & f-ds & graphs & ds-r & forced & weakly ds-r & good d & good G \\
\hline
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
\hline
3 & 4 & 4 & 4 & 4 & 4 & 4 & 2 & 2\\
4 & 11 & 10 & 11 & 10 & 10 & 11 & 6 & 6 \\
5 & 31 & 29 & 34 & 32 & 30 & 34 & 9 & 9 \\
6 & 102 & 88 & 156 & 128 & 106 & 152 & 30 & 34 \\
7 & 342 & 304 & 1044 & 768 & 616 & 930 & 54 & 76 \\
8 & 1213 & 1034 & 12346 & 7311 & 4385 & 9077 & 183 & 542 \\
9 & 4361 & 3697 & 274668 & 126152 & 60259 & 147812 & 379 & 2731 \\
10 & 16016 & 13158 & 12005168 & 4045030 & 1194438 & 4472390 & 1256 & 71984
\end{tabular}
\caption{Data for small graphs. The columns are: $n$=number of vertices; seq = number of graphic sequences;
f-ds = number of forcing degree sequences;
graphs=number of graphs; ds-r = number of ds-reconstructible graphs;
forced = number of graphs whose degree sequence is forcing; weakly dsr = number of graphs uniquely
specified by one card and the degree sequence; good d = number of sequences having all good degrees;
good G = number of graphs having all good vertices.
}
\label{t.data}
\end{table}
Table \ref{t.data} lists information about the number of ds-reconstructible graphs, the number
of ds-reconstruction-forcing degree sequences, and the number of graphs whose degree sequence is forcing
(the latter graphs are a subset of the ds-reconstructible graphs), for graphs on up to 10 vertices.
The numerical evidence is that the proportion of all graphs of given order $n$
that are ds-reconstructible is a decreasing function of $n$, and therefore the fraction of graphs
of large $n$ that are ds-reconstructible is small and may tend to zero as $n \rightarrow \infty$.
This does not rule out that these graphs may contribute a useful part of a proof
of the reconstruction conjecture (if one is possible) based on finding a collection of
reconstruction methods which together cover all graphs.
We have defined the notion of ds-reconstructibility so that it is simple and self-contained.
It is not defined in such a way as to capture as wide a class of graphs as possible, but
rather a readily-defined and -recognized class of graphs. For this reason it does not include
some graphs whose reconstruction is easy, such as the path graphs on more than three vertices.
There are several natural extensions to the idea of ds-reconstructibility that include path graphs
and some other graphs. For example, there is a class of graphs such that each graph in the class
contains a card $\Gv$, such that, up to isomorphism, there is only one graph having both the same
degree sequence as $G$ and having $\Gv$ as a subgraph.
Let us call such graphs `weakly ds-reconstructible'. This is a larger class than the
ds-reconstructible graphs (as we have defined them) because it includes cases where
degree information in $\Gv$ allows for more than one assignment of neighbours of $v$, but
all the assignments lead to the same graph, owing to automorphism of $\Gv$. The path
graphs are an example of this.
\begin{figure}
\basicgraph{0.4}
{{(0,0)/a}, {(1,0)/b}, {(2,0)/c}, {(3,0)/d}, {(1,1)/e}, {(2,1)/f}}
{a/b, b/c, b/e, c/d, c/f, e/f}
\hspace{5mm}
\basicgraph{0.4}
{{(0,0)/a}, {(1,0)/b}, {(2,0)/c}, {(0,1)/d}, {(1,1)/e}, {(2,1)/f}}
{a/b, a/d, b/c, b/e, c/f, d/e, e/f}
\caption{These two graphs, and their complements, are the smallest graphs that are not
weakly ds-reconstructible (i.e. reconstructible from one card and the degree sequence).}
\label{fig.6}
\end{figure}
The smallest graphs that are not weakly ds-reconstructible have six vertices; they are shown in figure
\ref{fig.6}.
Unigraphic degree sequences produce graphs that are obviously reconstructible from the degree sequence.
It can happen that, for some degree sequence,
(e.g. [11222]), all but one of the graphs of that degree sequence are ds-completable.
But in that case we can reconstruct all the graphs of that degree sequence, because the presence
of a completable card is recognisable. If such a card is present then it is used to reconstruct
the graph. If no such card is found then we know that the graph is the unique graph of the
given degree sequence that has no completable vertex.
Another extension is to ask, what further information would be required to make a card
uniquely completable, when degree information alone is not sufficient? One might for example
consider the class of graphs uniquely specified by a pair of cards and the degree sequence,
or by a single card and further information known to be recoverable from the deck.
The large number of equations in this paper is a symptom of the fact that I have not found
any single simply-stated criterion sufficient to fully distinguish the classes of forcing
and non-forcing degree sequences. A further avenue for investigation would be to
seek a way to manipulate a ds-reconstructible graph so as to obtain
a non-ds-reconstructible graph of the same degree sequence, if one exists. Another
avenue is to try to reduce the problem to a small number of canonical forms.
\bibliographystyle{plain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,402
|
Q: Why my personal hotmail account is Miscrosoft Exchange account type in Windows 7 Outlook 2016 I just found my personal Hotmail account's type is Microsoft Exchange. Is that because I have taken my laptop to my previous company? If that is the case, why it is still working? I have left that company and stayed at home for years. I also reinstalled and updated my Outlook application after left the company. Should I delete that account from the Win7 Outlook 2016 application and then add it back again with IMAP/SMTP type? Any difference between those two types?
A: Microsoft uses Exchange Server for hotmail.com addresses now. It will automatically create new mailboxes on their exchange servers if you create a new hotmail.com account. They still do allow setting up mail accounts through IMAP, but by default, autodiscover will find that it can connect through Exchange Server, which is always preferred over IMAP if available.
Outlook.com accounts are also on Exchange Servers now, so if you were to remove those accounts and add them again, they will also be added using the Exchange protocol.
As for IMAP vs Exchange, Always Exchange if possible. It is far more robust with many more features such as calendar, contacts, etc. Only if you really have a reason to use IMAP should you go back to this legacy format.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,977
|
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en-us" xml:lang="en-us">
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the License. You may obtain a copy of the License at
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Unless required by applicable law or agreed to in writing, software
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<meta name="DC.Title" content="SYSCS_UTIL.SYSCS_INVALIDATE_STORED_STATEMENTS system procedure" />
<meta name="abstract" content="The SYSCS_UTIL.SYSCS_INVALIDATE_STORED_STATEMENTS system procedure invalidates all stored prepared statements (that is, all statements in the SYSSTATEMENTS system table)." />
<meta name="description" content="The SYSCS_UTIL.SYSCS_INVALIDATE_STORED_STATEMENTS system procedure invalidates all stored prepared statements (that is, all statements in the SYSSTATEMENTS system table)." />
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<title>SYSCS_UTIL.SYSCS_INVALIDATE_STORED_STATEMENTS system procedure</title>
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<body id="rrefsyscsinvalidatestoredstmts"><a name="rrefsyscsinvalidatestoredstmts"><!-- --></a>
<h1 class="topictitle1">SYSCS_UTIL.SYSCS_INVALIDATE_STORED_STATEMENTS system procedure</h1>
<div><p>The <samp class="codeph">SYSCS_UTIL.SYSCS_INVALIDATE_STORED_STATEMENTS</samp>
system procedure invalidates all stored prepared statements (that is, all
statements in the SYSSTATEMENTS system table).</p>
<div class="section"><p>The next time one of the invalid stored prepared statements in the
<a href="rrefsistabs33768.html#rrefsistabs33768">SYSSTATEMENTS system table</a> is executed, it will
be recompiled, and a new plan will be generated for it.</p>
<p>Run <samp class="codeph">SYSCS_UTIL.SYSCS_INVALIDATE_STORED_STATEMENTS</samp> whenever
you think that your metadata queries or triggers are misbehaving -- for example,
if they throw <samp class="codeph">java.lang.NoSuchMethodError</samp> or
<samp class="codeph">java.lang.NoSuchMethodException</samp> on execution.
<span>Derby</span> stores plans for
triggers and metadata queries in the database. These should be invalidated
automatically on upgrade and at other necessary times. Should you encounter an
instance where they are not, you have found a bug that you should report, but
one that you can likely work around by running
<samp class="codeph">SYSCS_UTIL.SYSCS_INVALIDATE_STORED_STATEMENTS</samp>.</p>
</div>
<div class="section"><h2 class="sectiontitle">Syntax</h2>
<pre>SYSCS_UTIL.SYSCS_INVALIDATE_STORED_STATEMENTS()</pre>
<p>No result is returned by this procedure.</p>
</div>
<div class="section"><h2 class="sectiontitle">Execute privileges</h2>
<p>If authentication and SQL authorization are both enabled, only the
<a href="rrefattrib26867.html#rrefattrib26867">database owner</a> has
execute privileges on this procedure by default. See "Enabling user
authentication" and "Setting the SQL standard authorization mode" in the
<span><em>Derby Developer's Guide</em></span> for more information. The
database owner can grant access to other users.</p>
</div>
<div class="section"><h2 class="sectiontitle">JDBC example</h2>
<pre>CallableStatement cs = conn.prepareCall
("CALL SYSCS_UTIL.SYSCS_INVALIDATE_STORED_STATEMENTS()");
cs.execute();
cs.close();</pre>
</div>
<div class="section"><h2 class="sectiontitle">SQL Example</h2>
<pre>CALL SYSCS_UTIL.SYSCS_INVALIDATE_STORED_STATEMENTS();</pre>
</div>
</div>
<div>
<div class="familylinks">
<div class="parentlink"><strong>Parent topic:</strong> <a href="crefsqlbuiltinsystemprocedures.html" title="">Built-in system procedures</a></div>
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{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,876
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{"url":"http:\/\/xingyuzhou.org\/blog\/notes\/strong-convexity","text":"# Strong convexity\n\nStrong convexity is one of the most important concepts in optimization, especially for guaranteeing a linear convergence rate of many gradient decent based algorithms. In this post, which is mainly based on my course project report, I would like to present some useful results on strong convexity.\n\nLet\u2019s us first begin with the definition.\n\nA differentiable function $f$ is strongly convex if\n\nfor some $\\mu > 0$ and all $x, y$.\n\nNote: Strong convexity doesn\u2019t necessarily require the function to be differentiable, and the gradient is replaced by the sub-gradient when the function is non-smooth.\n\nIntuitively speaking, strong convexity means that there exists a quadratic lower bound on the growth of the function. This directly implies that a strong convex function is strictly convex since the quadratic lower bound growth is of course strictly grater than the linear growth.\n\nAlthough the definition in $\\eqref{eq:def}$ is commonly used, it would be quite useful for us to note that there are several equivalent conditions for strong convexity.\n\n### Equivalent Conditions of Strong Convexity\n\nThe following proposition gives equivalent conditions for strong convexity. The key insight behind this result and its proof is that we can relate a strongly-convex function ($e.g., f(x)$) to another convex function ($e.g., g(x)$), which enables us to apply the equivalent conditions for a convex function to obtain the result.\n\nProposition The following conditions are all equivalent to the condition that a differentiable function $f$ is strongly-convex with constant $\\mu > 0$. %\n\nProof:\n\n$(i) \\equiv (ii):$ It follows from the first-order condition for convexity of $g(x)$, i.e., $g(x)$ is convex if and only if $g(y)\\ge g(x) + \\nabla g(x)^T(y-x),~\\forall x,y.$\n\n$(ii) \\equiv (iii):$ It follows from the monotone gradient condition for convexity of $g(x)$, i.e., $g(x)$ is convex if and only if $(\\nabla g(x) - \\nabla g(y))^T(x-y) \\ge 0,~\\forall x,y.$\n\n$(ii) \\equiv (iv):$ It simply follows from the definition of convexity, i.e., $g(x)$ is convex if $g(\\alpha x+ (1-\\alpha) y) \\le \\alpha g(x) + (1-\\alpha) g(y), ~\\forall x,y, \\alpha\\in [0,1].$ $\\tag*{\\Box}$\n\nNote: The equivalence still holds for non-smooth function with gradient replaced by sub-gradient. For a proof, please refer to Lemma 1 of the note.\n\n### Implications of Strong Convexity\n\nThere are also some other conditions that are implied by strong convexity while are not equivalent to strong convexity, which means that these conditions are more general than strong convexity.\n\nProposition For a continuously differentiable function $f$, the following conditions are all implied by strong convexity (SC) condition. %\n\nNote: The condition (a) is often called Polyak-Lojasiewicz (PL) inequality which is quite useful for establishing linear convergence for a lot of algorithms.\n\nProof:\n\nSC $\\rightarrow$ (a): Taking minimization respect to $y$ on both sides of $\\eqref{eq:def}$ yields\n\nRe-arranging it we have the PL-inequality.\n\nSC $\\rightarrow$ (b): Using Cauchy-Schwartz on the equivalent condition (iii) in the last Proposition, we have\n\nand dividing both sides by $\\lVert x-y \\rVert$ (assuming $x \\neq y$) gives (a) (while (a) clearly holds if $x = y$).\n\nSC $\\rightarrow$ (c): Let us consider the function $\\phi_x(z) = f(z) - \\nabla f(x)^T z$. It is easy to see that $\\phi_x(z)$ is strongly-convex with the same $\\mu$ since\n\nwhere we again use the equivalent condition (iii) in the last Proposition for strong convexity.\n\nThen, the condition (c) can be directly obtained by applying the PL-inequality to the strongly-convex function $\\phi_x(z)$ with $z^* = x$\n\nSC $\\rightarrow$ (d): Interchanging $x$ and $y$ in condition (c) yields,\n\nCondition (d) is obtained by adding $\\eqref{eq:interchange}$ and condition (c). $\\tag*{\\Box}$\n\n### Application\n\nWe have already seen the application of condition (iii) in proving useful results in the last section. In this section, let us look at one simple application of condition (iv) for strong convexity.\n\nProposition Let $h = f + g$ where $f$ is strongly convex function and $g$ is a convex function, then $h$ is strongly convex function\n\nProof:\n\nwhere the first inequality follows from the fact that $f$ is strongly convex; the second inequality holds since $g$ is convex. $\\tag*{\\Box}$\n\nNote: Although the proof is simple, it has its own importance. For example, any L2-regularized problem of the form $h(x) = f(x)+ \\lambda \\lVert x \\rVert^2$, where $f$ is convex and $\\lambda >0$, is strongly convex.\n\n### Citation\n\nRecently, I have received a lot of emails from my dear readers that inquire about how to cite the content in my blog. I am quite surprised and also glad that my blog posts are more welcome than expected. Fortunately, I have an arXiv paper that summarizes all the results. Here is the citation form:\n\nZhou, Xingyu. \u201cOn the Fenchel Duality between Strong Convexity and Lipschitz Continuous Gradient.\u201d arXiv preprint arXiv:1803.06573 (2018).\n\nTHE END\n\nNow, it\u2019s time to take a break by appreciating the masterpiece of Monet.\n\nImpression Sunrise\n\ncourtesy of www.Claude-Monet.com","date":"2021-04-15 11:20:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 51, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 4, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8967349529266357, \"perplexity\": 250.07965144212434}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038084765.46\/warc\/CC-MAIN-20210415095505-20210415125505-00382.warc.gz\"}"}
| null | null |
Due to the digital nature of this item, no refunds will be given for PDF patterns.
Depending on the tail, not including beak.
You can reduce/enlarge the templates to make smaller hanging decorations or for soft toys. This can be done in any photocopying shop or using your home computer software.
Here's a few examples of what you can make; Key rings, brooch, magnets, Plush toys (patterns will need to be enlarged), Nursery mobile, Name banners, Appliqué: Cushion covers, quilts, notebook cover, bookmark, embroidery hoop art, fabric storage, bags, stitched canvas.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,021
|
//Apache2, 2017, WinterDev
//Apache2, 2009, griffm, FO.NET
namespace Fonet.Layout
{
internal class AreaClass
{
public static string UNASSIGNED = "unassigned";
public static string XSL_NORMAL = "xsl-normal";
public static string XSL_ABSOLUTE = "xsl-absolute";
public static string XSL_FOOTNOTE = "xsl-footnote";
public static string XSL_SIDE_FLOAT = "xsl-side-float";
public static string XSL_BEFORE_FLOAT = "xsl-before-float";
public static string setAreaClass(string areaClass)
{
if (areaClass.Equals(XSL_NORMAL) ||
areaClass.Equals(XSL_ABSOLUTE) ||
areaClass.Equals(XSL_FOOTNOTE) ||
areaClass.Equals(XSL_SIDE_FLOAT) ||
areaClass.Equals(XSL_BEFORE_FLOAT))
{
return areaClass;
}
else
{
throw new FonetException("Unknown area class '" + areaClass + "'");
}
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,833
|
FACT CHECK: 'If You're a Person in Power, Expert Fact-Checker Daniel Dale Is Watching Your Words'
Fact Check: CNN Fact-Checker Daniel Dale Is 'Fact-Checking the President'
The Tears of Van Jones
Blake Seitz
Former White House czar Van Jones is in touch with his sensitive side.
Jones, a CNN commentator, seasons his Democratic talking points with appeals to emotion.
When First Lady Michelle Obama delivered an address at the Democratic National Convention, Jones indulged a few tears.
'Van Jones, I've got to call you out. I saw you crying during this," CNN host Anderson Cooper said.
'If you weren't moved by that, go see the doctor," Jones responded.
'People are going to start getting emotions they didn't know they had," Jones said about Hillary Clinton's nomination for president. 'Sometimes on CNN we cover news. Today we're covering history."
Of course, the Democratic partisan can be selective in his use of pathos.
Jones said that a DNC appearance by the Mothers of the Movement, a group of mothers whose children were killed in encounters with police, was 'powerful, powerful, powerful." He felt differently about the RNC appearance by Patricia Smith, the mother of a U.S. foreign service officer killed in the 2012 terrorist attack in Benghazi, Libya.
'I worry about exploiting grief," Jones said about Smith's appearance.
In 2009, Jones resigned from his position as the Obama administration's 'green jobs czar" after his past political activism came to light. Jones helped form a communist group, called Standing Together to Organize a Revolutionary Movement (STORM), in 1994.
'I spent the next ten years of my life working with a lot of those people I met in jail, trying to be a revolutionary," Jones told the East Bay Express.
While Jones was still a member of STORM, the group advocated on behalf of convicted cop-killer Mumia Abu Jamal and participated in the 1999 WTO protests in Seattle. Jones claims he left the group for mainstream political activities in 2000, before STORM protested 'U.S. imperialism" in the wake of the September 11, 2001, terrorist attacks.
Jones' name appeared on a petition speculating that 'U.S. officials in the current administration" allowed 9/11 to occur 'as a pretext for war." Jones claims his name was added to the petition without his approval and that he was never a 9/11 Truther. His name was later removed at his request.
Published under: CNN , Van Jones
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 1,981
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Farrokhzad (moyen persan: Farrūkhzādag; nouveau persan: فرخزاد), également connu sous le nom de Pav, était un aristocrate iranien de la maison des Ispahbodhan et le fondateur de la dynastie Bavandide, gouvernant de 651 à 665. il a joué un rôle majeur dans les dernières années de l'empire sassanide et l'invasion arabe de l'iran.
Nom
Bien que son vrai nom soit «Farrokhzad» (qui signifie «le fils de Farrokh» ou «né avec chance et bonheur»), il est également connu dans d'autres sources sous divers autres noms, tels que Khorrazad, Zad Farrokh et Bav (ou Pav ).
Famille
Farrukhzad était le fils de Farrokh Hormizd, un aristocrate éminent de la famille des Ispahbodhan, qui a servi en tant que chef de l'armée (spahbod) des kusts d'Adorbadagan et de Khorasan et était l'un des généraux qui ont dirigé l'armée sassanide pendant la guerre byzantine-sasanienne de 602–628, mais en 626 avec son camarade Shahrbaraz se révolta contre le roi sassanide Khosro II (r. 590-628). Farrokhzad avait un frère du nom de Rostam Farrokhzad, celui qui commandera l'armée sassanide lors de la bataille d'al-Qadisiyya contre les musulmans arabes, qui résidaient à l'époque dans la province d'Adurbadagan.
Renversement de Khosro II
Farrokhzad est mentionné pour la première fois sous le règne de Khosro II, où il occupait de hautes fonctions et, selon le poète iranien Ferdovsi, il était "si proche de Khusro II que personne n'a osé l'approcher sans sa permission". En 626, Shahrbaraz avec le père et le frère de Farrokhzad se révoltent. En 627, Khosro a ensuite envoyé Farrokhzad pour négocier avec Shahbaraz, qui campait près de la capitale sassanide de Ctésiphon. Cependant, Farrokhzad s'est secrètement mutiné contre Khosro et a rejoint Shahrbaraz. Il a ensuite encouragé à rester indivis et à ne pas craindre la fureur de Khosro. En outre, il a également déclaré qu'aucun grand bozorgan ne l'avait soutenu.
Farrukhzad rassemblait plus de personnes qui s'opposaient à Khosro II afin de mettre en scène un coup d'État. En 628, Farrokhzad libéra Sheroe, le fils aîné de Khosro, ainsi que plusieurs familles féodales de l'empire sassanide, captura Ctésiphon et emprisonna Khosro. Sheroe a ensuite été couronné comme le nouveau roi et est devenu connu sous son nom dynastique de "Kavadh II".
Kavadh II a ensuite ordonné à son vizir (wuzurg framadar) Pirouz Khosro d'exécuter tous ses frères et demi-frères, y compris le fils préféré de Khosro II et héritier Mardanshah. Trois jours plus tard, Kavadh a ordonné à Mihr Hormozd d'exécuter son père. Avec l'accord des nobles de l'Empire sassanide, Kavadh fit alors la paix avec l'empereur byzantin Héracle. En outre, il a également pris toutes les propriétés de Farrokhzad et l'a mis en état d'arrestation à Istakhr.
La guerre civile sassanide de 628-632
Après la perte du territoire requis pour le traité de paix, l'aristocratie aigrie a commencé à former des États indépendants au sein de l'Empire sassanide, déclenchant ainsi la guerre civile sassanide de 628-632. Cela a divisé les ressources du pays. En outre, une peste dévastatrice a éclaté dans les provinces occidentales de l'Iran, tuant la moitié de la population avec Kavadh II, auquel son fils Ardachir III, 8 ans, a succédé. Pendant ce temps, Farrokh Hormizd a formé une faction dans le nord de l'Iran connue sous le nom de faction Pahlav (Parthe), une faction de Parthes de plusieurs familles qui s'étaient ralliées sous lui. Pendant la même période, cependant, Pirouz Khosro a également formé une faction dans le sud de l'Iran, connue sous le nom de faction Parsig (perse).
Le 27 avril 629, Ardachir a été renversé et tué par Shahrbaraz, qui a ensuite usurpé le trône. Quarante jours plus tard, cependant, Farrokh Hormizd l'a tué et a fait de la fille de Khosro, Boran, le nouveau monarque de l'Empire sassanide. Boran a ensuite nommé Farrokh Hormizd comme ministre de l'empire. Elle a cependant été rapidement destituée par le fils de Shahrbaraz, Shapur-i Shahrvaraz, qui après un bref règne a été remplacé par Azarmidokht, la sœur de Boran. Azarmidokht, sur les conseils des nobles sassanides, a rappelé Farrokhzad de son arrestation et l'a invité à servir à nouveau les Sasaniens. Farrokhzad, cependant, a décliné l'invitation et a refusé de servir sous une femme. Il s'est ensuite retiré dans un temple du feu à Istakhr.
En 631, Farrokh Hormizd, afin de prendre le pouvoir, a demandé à Azarmidokht de l'épouser. N'osant pas refuser, Azarmidokht le fit tuer avec l'aide de l'aristocrate mihranide Siavakhsh, petit-fils de Bahram Chobin, le célèbre spahbed et brièvement shahanshah. Elle a cependant été rapidement tuée par Rostam Farrokhzad, qui a ensuite remis Boran sur le trône. Plus tard en 632, Farrokhzad fut à nouveau invité à servir les Sasaniens, cette fois par le roi nouvellement couronné Yazdegerd III, qui était un roi fantoche du frère de Farrokhzad, Rostam Farrokhzad. Farrukhzad a accepté et toutes ses propriétés lui ont été restituées.
L'invasion arabe de la Perse
L'invasion de l'ouest de l'Iran
Cependant, au cours de la même année, la majorité des Arabes, unis sous la bannière de l'islam, ont envahi l'Empire sassanide. En 636, les Arabes se trouvaient à al-Qadisiyya, une ville proche de Ctésiphon, ce qui obligea Rostam Farrokhzad à agir lui-même. Cependant, Farrokhzad n'a pas pu participer, car il était le marzban de Balasagan, une province éloignée de Ctésiphon.
Alors que Rostam se préparait à affronter les Arabes, il a écrit une lettre à Farrokhzad, qui disait qu'il devait lever une armée et aller à Adorbadagan, tout en lui rappelant que Yazdegerd III était le seul héritage laissé aux Sasaniens. Rostam partit alors de Ctésiphon aux commandes d'une grande force sassanide pour affronter les Arabes. La bataille a duré trois jours, Rostam étant vaincu et tué au cours de la dernière journée.
Après la mort de son frère, Farrokhzad lui a succédé en tant que spahbed de Khorasan et Adorbadagan, et en tant que nouveau chef de la famille Ispahbodhan et de la faction Pahlav. Il a ensuite levé une armée à Adorbadagan et s'est rendu à Ctésiphon, où il a été nommé commandant par Yazdegerd III, qui s'est enfui à Holwan. Farrokhzad, cependant, s'est également enfui à Holwan après un petit affrontement décourageant avec les Arabes. En 637, Ctésiphon a été capturé par les Arabes. Pendant ce temps, Farrokhzad, avec Yazdegerd III, l'officier mihranide Mihran Razi, Pirouz Khosro et Hormouzan, ont quitté Holwan pour Adorbadagan, mais pendant qu'ils se déplaçaient vers l'endroit, ils ont été pris en embuscade par une armée arabe à Jalula, où ils ont été vaincus. Mihran Razi a été tué au cours de la bataille, tandis qu'Hormouzan s'était enfui au Khuzestan et Pirouz Khosro à Nahavand. En 642, les Arabes ont capturé Nahavand et Ispahan, tuant Pirouz Khosro, y compris d'autres officiers sassanides tels que Shahrvaraz Jadhouyih et Mardanshah. Au cours de la même année (ou en 643), Farrokhzad lève une autre armée avec son fils Isfandyadh et le général dailamite Mouta. Ils ont cependant été vaincus à Waj Roudh, un village de Hamadan. Pendant ce temps, Yazdegerd III s'est enfui dans le sud de l'Iran et y est resté jusqu'en 648. On ne sait pas si Farrokhzad était avec Yazdegerd III pendant son séjour dans le sud de l'Iran.
Fuite vers Khorasan, règne et mort
Vers 650, Yazdegerd III, avec Farrokhzad, est arrivé à Khorasan. Yazdegerd III a ensuite nommé Farrokhzad gouverneur de Merv et a ordonné à Baraz, le fils de Mahoe Souri, de lui donner le contrôle absolu de la ville. Mahoe, cependant, lui a désobéi. Farrokhzad a alors conseillé à Yazdegerd III de prendre le réfugié au Tabarestan. Yazdegerd III, cependant, n'a pas accepté ses conseils. Farrokhzad a ensuite fait une mutinerie contre Yazdegerd III et est parti pour Ray, pour venger son père contre Siavakhsh, qui était le chef de la ville. Sur le chemin de Ray, il a rencontré son allié Kanadbak, qui faisait partie du complot contre Khosro II et avait participé au choc de Rostam avec les Arabes, mais après que la défaite s'était enfuie à Tous, une ville qui faisait partie de ses domaines. Farrokhzad a ensuite poursuivi son voyage vers Ray, mais avant d'atteindre la ville a rencontré le général arabe Nouaïm près de Qazvin en 651, avec qui il a fait la paix.
Il a ensuite accepté d'aider les Arabes contre Siavakhsh. Le groupe combiné Ispahbodhan-Arab s'est ensuite engagé dans une bataille de nuit contre l'armée de Siavakhsh au pied de la montagne juste à l'extérieur de Ray. Farrokhzad a conduit une partie de la cavalerie de Nuoaïm par un chemin peu connu dans la ville, d'où ils ont attaqué l'arrière de l'armée mihranide, provoquant une grande effusion de sang. L'armée de Siavakhsh a finalement été vaincue et il a lui-même été tué. Pour donner l'exemple, les Arabes ont alors rasé le quartier aristocratique de Ray. Cependant, la ville a ensuite été reconstruite par Farrokhzad, qui est devenu le souverain de Ray.
Farrokhzad s'est ensuite rendu au Tabarestan, mais à son arrivée, il a appris la mort de Yazdegerd sous les ordres de Mahoe Souri, ce qui lui a fait se raser les cheveux et vivre comme un moine dans un temple du feu à Kusan. Lorsque les Arabes ont envahi le Tabarestan, Farrokhzad a été invité par les habitants à devenir leur roi, ce qu'il a accepté avec plaisir et qui a marqué la fondation de la dynastie bavandide. Il a ensuite rassemblé une armée, vaincu les Arabes et conclu un traité de paix avec eux. Farrokhzad gouvernerait le Tabarestan, y compris certaines parties d'Abarchahr pacifiquement en 14 ans, jusqu'à ce qu'il soit assassiné vers 665 par un noble karénide nommé Valash, dont la famille était en guerre avec la famille de Farrokhzad.
Après la mort de Farrukhzad, son fils Sohrab I s'est enfui dans un bastion Ispahbodhan/Bavandid nommé Kula. Plus tard en 673, Sohrab vengea son père en tuant Valash, puis reconquit le royaume de son père. Il s'est ensuite couronné ispahbodh (souverain) de la dynastie bavandide dans sa capitale à Périm.
Références
Sources
Histoire du Tabarestan
Personnalité militaire de l'Empire sassanide
Personnalité de l'Empire sassanide au VIIe siècle
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package net.paoding.rose.web.portal.impl;
import java.util.concurrent.ExecutorService;
import java.util.concurrent.ThreadPoolExecutor;
import org.springframework.beans.factory.FactoryBean;
import org.springframework.scheduling.concurrent.ThreadPoolTaskExecutor;
/**
* 此类用于配置在spring文件中,使获取Spring的 {@link ThreadPoolTaskExecutor} 的内部 {@link ExecutorService}对象
* @author 王志亮 [qieqie.wang@gmail.com]
*
*/
public class ThreadPoolExcutorServiceFactoryBean implements FactoryBean {
private ThreadPoolTaskExecutor threadPoolTaskExecutor;
public void setThreadPoolTaskExecutor(ThreadPoolTaskExecutor threadPoolTaskExecutor) {
this.threadPoolTaskExecutor = threadPoolTaskExecutor;
}
@Override
public Object getObject() throws Exception {
return threadPoolTaskExecutor.getThreadPoolExecutor();
}
@Override
@SuppressWarnings("unchecked")
public Class getObjectType() {
return ThreadPoolExecutor.class;
}
@Override
public boolean isSingleton() {
return true;
}
}
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{
"redpajama_set_name": "RedPajamaGithub"
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Plantations in Natchitoches Louisiana
Louisiana's plantations offer a fascinating look at lifestyles of the past and a crucial period in the history of the United States. Many of the state's amazing antebellum mansions remain intact, and are meticulously maintained and furnished with beautiful period pieces. Tour these majestic marvels and their manicured gardens along the Great River Road, and across south and central Louisiana, and learn more about life during an incredible era leading up to the Civil War. You can spend your day touring and even stay overnight at many of the properties. The experience is one you won't forget!
The City of Natchitoches is the heart of Natchitoches Parish. Founded in 1714 the site was established near a village of Natchitoches Indians. As the oldest permanent settlement in the Louisiana Purchase territory, its history is also a story of the development of our nation.
The City's Historic Landmark District, which fronts Cane River Lake and encompasses a 33-block area, includes many historic homes, churches and commercial structures. A mixture of Queen Anne and Victorian architecture, along with Creole style cottages can be seen throughout the district.
People settled in the Natchitoches area in large numbers after the Louisiana Purchase. Eventually, however, the river changed its course, bypassing Natchitoches. What was once known as the Red River is now Cane River Lake.
Nationally recognized as a Preserve America Community, a Distinctive Destination and a Great American Main Street, Natchitoches is a genuine heritage destination.
Famous Streets Across Louisiana
Get to know Louisiana a little better via its iconic streets in cities around the state.
Oak Alley's Shrimp Creole Recipe
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Explore the Louisiana Sports Hall of Fame and Northwest Louisiana History Museum
Every day is "game day" at the dazzling Louisiana Sports Hall of Fame and culture comes to life at the Northwest Louisiana History Museum in Natchitoches.
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There's no shortage of fall weekend fun in Louisiana, where autumn's arrival brings a variety of food, music and culture festivals.
Dancing Louisiana Style
Grab your dancin' shoes and boogie down at Louisiana's dance halls and clubs.
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To truly understand the appeal of Louisiana music, you've got to hear it live. Here's where to go to let the beat catch hold.
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Alexandria and Pineville lie in the heart of the state, situated in the Crossroads region of Louisiana.
Camping with Kids in Kisatchie National Forest
There is nothing like spending some quality time with the family in Louisiana's great outdoors.
Seasonal Souls: Making the Most of the Fall Season in Louisiana
Discover these fall things to do in Louisiana - Part 1!
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Ultimate List of Louisiana Products to Serve with Holiday Meals
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Locust Grove State Historic Site
Louisiana's plantation past comes alive at this small family cemetery.
10 Must-Visit Louisiana Festivals
Are you ready to party Louisiana style? Then punch your ticket for these 10 fabulous fests and experience what we do best: food, music and rollicking good times!
Holiday Open House – Tour Louisiana Homes, Gardens
History and holidays mix well throughout Louisiana, where you are invited to tour many beautiful, stately homes that date back a century or more.
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Rosedown Plantation State Historic Site
Tour the house and gardens of a luxurious 19th century cotton plantation.
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Louisiana Football Tailgating Tips
The boys of fall take to the field, and we couldn't be happier to cheer them on. Learn about a few of our local college teams—and find out where to celebrate their victories this season.
A Mix of Louisiana's History and Culture in Ascension Parish
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Louisiana's Historic Architecture
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Tour the River Road and Capital for Plantations, Dining and Outdoor Fun
Follow the Mississippi River as it flows through Baton Rouge and the heart of Plantation Country.
July 4th in Louisiana
Fireworks, food and fun are the main attractions at our many star-spangled Fourth of July celebrations.
Return to Grandeur at Nottoway Plantation
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Take the Scenic Route: Louisiana's Byways
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Things To Do With Kids: Central Louisiana
A mingling of cultures, a wealth of history and a beautiful landscape make the Crossroads region a hit with parents and children alike. Here are some of our top family-friendly picks in central Louisiana.
African-American Heritage Itinerary: Plantation Country
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Discovering Louisiana's Hidden Gems This Fall
Fall is perfect for discovering some of Louisiana's gorgeous attractions that shine in the autumn season.
A Visit to Frogmore Plantation, Where Cotton is Still King
Walk through high cotton and pick some yourself at this 19th-century plantation.
10 Things to Do in Natchitoches Parish
Explore Natchitoches Parish, the land of Cane River, teeming with history and beauty around every corner.
Discover the Plantation Parade
These four plantations will take you on a trip through time to discover the stories of Louisiana's past.
Cane River National Heritage Area
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This southern belle offers stately oaks, brick-lined streets, great oyster po-boys, Steel Magnolias and of course a meat pie!
Haunted At The Myrtles
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Fort St. Jean Baptiste State Historic Site
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Don't Miss Dishes - Red River Riches Trail
The Red River Riches trail will tempt your taste buds from Shreveport to Alexandria.
Making History at Frogmore Plantation
Learn the stories of Civil War soldiers, cotton kingpins, slaves and sharecroppers at a 19th-century plantation where cotton is still farmed today.
River Road Plantations
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Louisiana Culinary Dictionary
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Actress Faith Ford in Louisiana
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Kent Plantation House State Historic Site
Tour the grounds of an opulent 18th century sugar plantation, one of Louisiana's oldest.
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Historic Districts & Sites
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Melrose Plantation
City: Melrose
3533 Hwy 119 Melrose, LA 71452
We invite you to discover the two-hundred year history of the beautiful Melrose Plantation. A National Historic Landmark, Melrose contains nine historic buildings including the African House, Yucca...
Explore the church at the Whitney Plantation in Louisiana. ©Elsa Hahne
Explore the grounds of Whitney Plantation in Louisiana. ©Elsa Hahne
Melrose Plantation in Natchitoches, Louisiana
Fireworks displays are just part of the celebrations during Christmas in Natchitoches.
The Myrtles Plantation in St. Francisville
Take a tour of the beautiful Houmas House Plantation.
Louisiana Sports Hall of Fame Museum in Natchitoches, Louisiana
Nottoway Plantation in White Castle, Louisiana
The African House at Melrose Plantation is one of the nine buildings registered as historic landmarks.
Visit the unique San Francisco Plantation outside of New Orleans along River Road.
E. D. White Historic Site in Thibodaux
The African House at Melrose Plantation in Natchitoches has some of Clementine Hunter's famous murals.
Celebrate the holidays with a bang at the Annual Natchitoches Christmas Festival.
Houmas House Plantation sits along the famous River Road outside Baton Rouge, Louisiana
An old slave cabin at the Magnolia Plantation in the Cane River National Historical Park.
Explore the Sports Hall of Fame and Northwest Louisiana History Museum in Natchitoches.
Visit St. Joseph Plantation on the Great River Road.
Lasyone's Meat Pie Kitchen
Sports legends from across Louisiana are on display at the Sports Hall of Fame and Northwest Louisiana History Museum.
Oakland Plantation in the Natchitoches, Louisiana area.
Explore Laura Plantation, an 1805 sugarcane plantation with daily guided tours. Walk in the footsteps of four generations of one Créole family, both free and enslaved.
View a Clementine Hunter mural in the African House at Melrose Plantation.
Evergreen Plantation
Frogmore Cotton Plantation has modern gins that are a stark difference compared to technologies of the past.
Nathitoches Festival of lights
Nottoway Plantation
Saints Memorabilia ©Mark J. Sindler
LSU football has produced many sports legends. See the history at the Sports Hall of Fame and Northwest Louisiana History Museum.
Historic Front Street in Natchitoches
Stroll through the gardens at Houmas House and enjoy the fountains.
When standing on the Frogmore Cotton Plantation and Gin you can see the old church and quarters building amongst the cotton.
San Francisco Plantation recently reopened after more than $1 million in renovations.
Oak Alley Plantation, Restaurant and Inn, on River Road in Vacherie, Louisiana, boasts an Antebellum mansion and B&B accommodations in century-old cottages set in 25 acres of lush, oak-lined grounds.
As you wander through beautiful downtown Natchitoches, shop their unique stores.
Photo ©Natchitoches Tourism
Stroll along Front Street in Natchitoches which is the center of the historic district and full of shops and restaurants.
Clementine Hunter paintings on display at Melrose Plantation
Discover how cotton was baled in the past at the Frogmore Plantation and Gin where you can see the workings of a cotton plantation from field to bale, both historical and high-tech.
Natchitoches' historic buildings.
Visit Myrtles Plantation, one of Louisiana famously haunted locations, in St. Francisville
Pick cotton at the Frogmore Plantation, a modern-day working cotton plantation.
Rosedown Plantation in Louisiana
Festive holiday crowds gather to watch the grand fireworks display over Cane River during the Natchitoches Christmas Festival.
Visit the Oakland Plantation in Cane River National Heritage Area in Natchitoches.
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| 6,270
|
{"url":"http:\/\/tvim.info\/en\/node\/531","text":"# On Common Approach to the Construction of J-selfadjoint Dilation for a Linear Operator with a Nonempty Set of Regular Points\n\nAuthors:\nTretyakov D. V. On Common Approach to the Construction of J-selfadjoint Dilation for a Linear Operator with a Nonempty Set of Regular Points \/\/ Taurida Journal of Computer Science Theory and Mathematics, \u2013 2019. \u2013 T.18. \u2013 \u21164. \u2013 P. 92-106\nhttps:\/\/doi.org\/10.37279\/1729-3901-2019-18-4-92-106\n\nThe common approach to construction of J-selfadjoint dilation for linear operator with\nnonempty regular point set is considered in this article.\nLet $A$ \u2014 linear operator with nonempty regular point set $(\u2212i \u2208 \u03c1(A))$ and Clos dom(A) =$\\mathfrak{H}$,\nwhere $\\mathfrak{H}$ \u2014 Hilbert space,\n\n$B_+ := iR_{\u2212i} \u2212 iR^*_{\u2212i} \u2212 2R^\u2217_{\u2212i}R_{\u2212i}, B_\u2212 := iR_{\u2212i} \u2212 iR^\u2217_{\u2212i} \u2212 2R_{\u2212i}R^\u2217_{\u2212i}$,\n\n$Q_\\pm := \\sqrt{(|B_\\pm|)}, B_\\pm = \\mathfrak{J}_\\pm Q_\\pm$ - polar decompositions of $B_\\pm, \\mathfrak{Q}_\\pm = Clos(Q_\\pm\\mathfrak{H})$.\n\nLet $\\mathfrak{D}_\\pm$ \u2014 arbitrary Hilbert spaces and $F_\\pm : dom(F_\\pm) \u2192 \\mathfrak{D}_\\pm (dom(F_\\pm) \u2282 \\mathfrak{ D}_\\pm)$ \u2014 simple\nmaximal symmetric operators with defect numbers $(q_\u2212, 0)$ and $(0, q_+)$ respectively, moreover\ndim $\\mathfrak{Q}_\\pm=$ dim $\\mathfrak{N}_\\pm = q_\\pm, \\Phi_\\pm : \\mathfrak{N}_\\pm$ \u2192 $\\mathfrak{Q}_\\pm$ are isometries, $V_\\pm$ \u2014 Caley transformations of $F_\\pm$.\nLet $\\langle\\mathcal{H}_\\pm, \u0393_\\pm\\rangle$ are the spaces of boundary values of operators $F^\u2217_\\pm$, i.e.:\n\n1. \u2200 $f, g \\in$ dom$(F^\u2217_\\pm)$ $(F^\u2217_\\pm f, g)_{\\mathfrak{D}_\\pm} \u2212 (f, F^\u2217_\\pm g)_{\\mathfrak{D}_\\pm} = \\mp i(\u0393_\\pm f, \u0393_\\pm g)_{\\mathcal{H}_\\pm}$;\n2. the transformations dom$(F^\u2217_\\pm) \\in f \\mapsto \u0393_\\pm f \\in \\mathcal{H}_\\pm$ are surjective.\n\nConsider the Hilbert space $\\mathbb{H} = \\mathfrak{D}_\u2212 \u2295 \\mathfrak{H} \u2295 \\mathfrak{D}_+$. Define in this space indefinite metric\n$J = J_\u2212 \u2295 I \u2295 J_+$and operator $\\mathcal{S}:$\n\n\u2200 $h_\\pm = \\sum\\limits_{k=0}^\u221e V^k_\\pm n^\\pm_k \\in \\mathfrak{D}_\\pm, n^\\pm_k \\in \\mathfrak{N}_\\pm, J_\\pm (\\sum\\limits_{k=0}^\u221e V^k_\\pm n^\\pm_k \\in \\mathfrak{D}_\\pm) := \\sum\\limits_{k=0}^\u221e V^k_\\pm \\Phi^{\u22121}_\\pm \\mathfrak{J} _\\pm \\Phi_\\pm n^\\pm_k$.\n\nThe vector $h = (h_\u2212, h_0, h_+)^T \\in$ dom($\\mathcal{S]) if 1.$h_\\pm \\in$dom$(F^\u2217_\\pm)$; 2.$\\phi = h_0+Q_\u2212\\Phi_\u2212\u0393_\u2212h_\u2212 \\in$dom$(A)$3.$\\Phi_+ \u0393_+h_+ = T^*\\Phi_\u2212\u0393_\u2212h_\u2212 + i \\mathfrak{J}_+Q_+(A + i)_\\phi, where T^\u2217 = I + 2iR^\u2217_{\u2212i}$. . If this conditions are fulfil, that for all$h = (h_\u2212, h_0, h_+)^ T \\in$dom($\\mathcal{S}$)$\\mathcal{Sh} = \\mathcal{S}(h_\u2212, h_0, h_+)^T := (F^\u2217_\u2212h_\u2212, \u2212ih_0 + (A + i)\\phi, F^\u2217_+h_+)^T.$Theorem. Operator$\\mathcal{S}$is a$\\mathcal{J}$-sejfadjoint dilation of operator$A$. Different private cases of dilation$\\mathcal{S}$are considered too. Keywords:$ \\mathcal{J}\\$-selfadjoint dilation, maximal closed symmetric operator, defect operators.\n\nUDC:\n517.432","date":"2022-08-15 15:12:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9005327820777893, \"perplexity\": 4133.629885118044}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882572192.79\/warc\/CC-MAIN-20220815145459-20220815175459-00642.warc.gz\"}"}
| null | null |
Q: does "Ignoring APKINDEX" mean the image does not install apk search command tool? I am trying to use this docker image big-data-europe/docker-spark, based on "Alpine Linux v3.8"
when I try to search sbt
apk search sbt
I got
WARNING: Ignoring APKINDEX.adfa7ceb.tar.gz: No such file or directory
WARNING: Ignoring APKINDEX.efaa1f73.tar.gz: No such file or directory
does this mean the image does not install apk search command tool?
A: agree with @GracefulRestart
it seems that you forgot to run apk update first.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,703
|
{"url":"http:\/\/bootmath.com\/if-a2-i-then-a-is-diagonalizable-and-is-i-if-1-is-its-only-eigenvalue.html","text":"# If $A^2 = I$, then $A$ is diagonalizable, and is $I$ if $1$ is its only eigenvalue\n\nLet $A$ be a square matrix of order $n$ such that $A^2 = I$.\n\n1. Prove that if $1$ is the only eigenvalue of $A$, then $A = I$.\n2. Prove that $A$ is diagonalizable.\n\nFor (1), I know that there are two eigenvalues which are $1$ and $-1$, how do I go about proving what the question asks me?\n\n#### Solutions Collecting From Web of \"If $A^2 = I$, then $A$ is diagonalizable, and is $I$ if $1$ is its only eigenvalue\"\n\nIf $A \\neq I$, then there is a (non-zero) vector $v$ such that $v \\neq Av$, which is to say $v-Av \\neq 0$. But then we have\n$$A(v-Av) = Av \u2013 A^2v = Av \u2013 Iv = Av \u2013 v = -(v-Av)$$\nwhich means that $v-Av$ is a non-zero eigenvector of $A$ with eigenvalue $-1$. Thus we have proven\n$$A \\neq I \\implies A\\text{ has } -1\\text{ as an eigenvalue}$$\nwhich is the contraposition of what we were asked to prove, and we are therefore done.\n\nYou are right, eigenvalues are $1$ and $-1$.\nFor diagonalizable part:\n$$A^2=I$$ Therefore the characteristic polynomial is $( \\lambda-1)(\\lambda+1)=0$, hence characteristic polynomial splits. So $A$ is diagonalizable.\n\nHint for 2. Assume \u201coptimistically\u201d that every vector $v$ decomposes as a sum of eigenvectors of $A$:\n$$v = v_{1} + v_{-1}.$$\nApply $A$ to both sides to get a system of two equations in two unknowns, and solve formally for $v_{1}$ and $v_{-1}$.\n\nArmed with the expressions from the preceding paragraph, show that you have actually found eigenvectors of $A$, and therefore have proven that $\\mathbf{R}^{n}$ (or whatever field of scalars, so long as the characteristic isn\u2019t $2$) decomposes as a sum of eigenspaces of $A$.","date":"2018-06-18 18:59:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.9815692901611328, \"perplexity\": 101.73855008403609}, \"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-2018-26\/segments\/1529267860776.63\/warc\/CC-MAIN-20180618183714-20180618203714-00347.warc.gz\"}"}
| null | null |
Développée par le linguiste Morris Swadesh comme outil d'étude de l'évolution des langues, elle correspond à un vocabulaire de base censé se retrouver dans toutes les langues. Il en existe diverses versions, notamment :
une version complète de 207 mots, dont certains ne se retrouvent pas dans tous les environnements (elle contient par exemple serpent et neige),
une version réduite de 100 mots.
Il ne faut pas considérer cette liste de mots comme un lexique élémentaire permettant de communiquer avec les locuteurs de la langue considérée. Son seul but est de fournir une ouverture sur la langue, en en présentant des bases lexicales et si possible phonétiques.
Liste
Articles connexes
Grec ancien
Grammaire du grec ancien
Liste Swadesh du grec moderne
Liens externes
Henri Estienne, Thesaurus linguæ græcæ, 1572. C'est le plus complet des dictionnaires de la langue grecque. Il fut réédité en 2008 par l'éditeur italien La scula di Pitagora [archive] en 9 volumes.
Anatole Bailly (dir.), Dictionnaire grec-français, Paris, Hachette, 1895 (nombreuses rééditions, plusieurs refontes).
Liste Swadesh
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8
|
{"url":"http:\/\/mathonline.wikidot.com\/polynomial-functions-as-functions-of-bounded-variation","text":"Polynomial Functions as Functions of Bounded Variation\n\n# Polynomial Functions as Functions of Bounded Variation\n\nRecall from the Continuous Differentiable-Bounded Functions as Functions of Bounded Variation page that if $f$ is continuous on the interval $[a, b]$, $f'$ exists, and $f'$ is bounded on $(a, b)$ then $f$ is of bounded variation on $[a, b]$.\n\nWe will now apply this theorem to show that all polynomial functions are of bounded variation on any interval $[a, b]$.\n\n Theorem 1: Let $f$ be a polynomial function. Then $f$ is of bounded variation on any interval $[a, b]$.\n\u2022 Proof: Let $f$ be a polynomial function of degree $n$. Then for $a_0, a_1, ..., a_n \\in \\mathbb{R}$ with $a_n \\neq 0$ we have that:\n(1)\n\\begin{align} \\quad f(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n \\end{align}\n\u2022 Since $f$ is a polynomial, we have that $f$ is continuous on any interval $[a, b]$. Now consider the derivative of $f$ is therefore:\n(2)\n\\begin{align} \\quad f'(x) = a_1 + 2a_2x + ... + na_nx^{n-1} \\end{align}\n\u2022 Notice that $f'$ is itself a polynomial. Therefore $f'$ exists and is continuous on any interval $[a, b]$. Since $f'$ is continuous on the closed and bounded interval $[a, b]$ we must have by the Boundedness Theorem that $f'$ is bounded on $[a, b]$ and hence bounded on $(a, b)$.\n\u2022 Hence $f$ is continuous on $[a, b]$, $f'$ exists, and $f'$ is bounded on $(a, b)$, so by the theorem referenced earlier, we must have that $f$ is of bounded variation on $[a, b]$. $\\blacksquare$","date":"2017-10-18 05:50:32","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 2, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9848155379295349, \"perplexity\": 61.205025447181434}, \"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-2017-43\/segments\/1508187822747.22\/warc\/CC-MAIN-20171018051631-20171018071631-00620.warc.gz\"}"}
| null | null |
import warnings
from typing import Awaitable, Callable, Dict, Optional, Sequence, Tuple, Union
from google.api_core import gapic_v1
from google.api_core import grpc_helpers_async
from google.api_core import operations_v1
from google.auth import credentials as ga_credentials # type: ignore
from google.auth.transport.grpc import SslCredentials # type: ignore
import grpc # type: ignore
from grpc.experimental import aio # type: ignore
from google.cloud.asset_v1p4beta1.types import asset_service
from google.longrunning import operations_pb2 # type: ignore
from .base import AssetServiceTransport, DEFAULT_CLIENT_INFO
from .grpc import AssetServiceGrpcTransport
class AssetServiceGrpcAsyncIOTransport(AssetServiceTransport):
"""gRPC AsyncIO backend transport for AssetService.
Asset service definition.
This class defines the same methods as the primary client, so the
primary client can load the underlying transport implementation
and call it.
It sends protocol buffers over the wire using gRPC (which is built on
top of HTTP/2); the ``grpcio`` package must be installed.
"""
_grpc_channel: aio.Channel
_stubs: Dict[str, Callable] = {}
@classmethod
def create_channel(
cls,
host: str = "cloudasset.googleapis.com",
credentials: ga_credentials.Credentials = None,
credentials_file: Optional[str] = None,
scopes: Optional[Sequence[str]] = None,
quota_project_id: Optional[str] = None,
**kwargs,
) -> aio.Channel:
"""Create and return a gRPC AsyncIO channel object.
Args:
host (Optional[str]): The host for the channel to use.
credentials (Optional[~.Credentials]): The
authorization credentials to attach to requests. These
credentials identify this application to the service. If
none are specified, the client will attempt to ascertain
the credentials from the environment.
credentials_file (Optional[str]): A file with credentials that can
be loaded with :func:`google.auth.load_credentials_from_file`.
This argument is ignored if ``channel`` is provided.
scopes (Optional[Sequence[str]]): A optional list of scopes needed for this
service. These are only used when credentials are not specified and
are passed to :func:`google.auth.default`.
quota_project_id (Optional[str]): An optional project to use for billing
and quota.
kwargs (Optional[dict]): Keyword arguments, which are passed to the
channel creation.
Returns:
aio.Channel: A gRPC AsyncIO channel object.
"""
return grpc_helpers_async.create_channel(
host,
credentials=credentials,
credentials_file=credentials_file,
quota_project_id=quota_project_id,
default_scopes=cls.AUTH_SCOPES,
scopes=scopes,
default_host=cls.DEFAULT_HOST,
**kwargs,
)
def __init__(
self,
*,
host: str = "cloudasset.googleapis.com",
credentials: ga_credentials.Credentials = None,
credentials_file: Optional[str] = None,
scopes: Optional[Sequence[str]] = None,
channel: aio.Channel = None,
api_mtls_endpoint: str = None,
client_cert_source: Callable[[], Tuple[bytes, bytes]] = None,
ssl_channel_credentials: grpc.ChannelCredentials = None,
client_cert_source_for_mtls: Callable[[], Tuple[bytes, bytes]] = None,
quota_project_id=None,
client_info: gapic_v1.client_info.ClientInfo = DEFAULT_CLIENT_INFO,
always_use_jwt_access: Optional[bool] = False,
api_audience: Optional[str] = None,
) -> None:
"""Instantiate the transport.
Args:
host (Optional[str]):
The hostname to connect to.
credentials (Optional[google.auth.credentials.Credentials]): The
authorization credentials to attach to requests. These
credentials identify the application to the service; if none
are specified, the client will attempt to ascertain the
credentials from the environment.
This argument is ignored if ``channel`` is provided.
credentials_file (Optional[str]): A file with credentials that can
be loaded with :func:`google.auth.load_credentials_from_file`.
This argument is ignored if ``channel`` is provided.
scopes (Optional[Sequence[str]]): A optional list of scopes needed for this
service. These are only used when credentials are not specified and
are passed to :func:`google.auth.default`.
channel (Optional[aio.Channel]): A ``Channel`` instance through
which to make calls.
api_mtls_endpoint (Optional[str]): Deprecated. The mutual TLS endpoint.
If provided, it overrides the ``host`` argument and tries to create
a mutual TLS channel with client SSL credentials from
``client_cert_source`` or application default SSL credentials.
client_cert_source (Optional[Callable[[], Tuple[bytes, bytes]]]):
Deprecated. A callback to provide client SSL certificate bytes and
private key bytes, both in PEM format. It is ignored if
``api_mtls_endpoint`` is None.
ssl_channel_credentials (grpc.ChannelCredentials): SSL credentials
for the grpc channel. It is ignored if ``channel`` is provided.
client_cert_source_for_mtls (Optional[Callable[[], Tuple[bytes, bytes]]]):
A callback to provide client certificate bytes and private key bytes,
both in PEM format. It is used to configure a mutual TLS channel. It is
ignored if ``channel`` or ``ssl_channel_credentials`` is provided.
quota_project_id (Optional[str]): An optional project to use for billing
and quota.
client_info (google.api_core.gapic_v1.client_info.ClientInfo):
The client info used to send a user-agent string along with
API requests. If ``None``, then default info will be used.
Generally, you only need to set this if you're developing
your own client library.
always_use_jwt_access (Optional[bool]): Whether self signed JWT should
be used for service account credentials.
Raises:
google.auth.exceptions.MutualTlsChannelError: If mutual TLS transport
creation failed for any reason.
google.api_core.exceptions.DuplicateCredentialArgs: If both ``credentials``
and ``credentials_file`` are passed.
"""
self._grpc_channel = None
self._ssl_channel_credentials = ssl_channel_credentials
self._stubs: Dict[str, Callable] = {}
self._operations_client: Optional[operations_v1.OperationsAsyncClient] = None
if api_mtls_endpoint:
warnings.warn("api_mtls_endpoint is deprecated", DeprecationWarning)
if client_cert_source:
warnings.warn("client_cert_source is deprecated", DeprecationWarning)
if channel:
# Ignore credentials if a channel was passed.
credentials = False
# If a channel was explicitly provided, set it.
self._grpc_channel = channel
self._ssl_channel_credentials = None
else:
if api_mtls_endpoint:
host = api_mtls_endpoint
# Create SSL credentials with client_cert_source or application
# default SSL credentials.
if client_cert_source:
cert, key = client_cert_source()
self._ssl_channel_credentials = grpc.ssl_channel_credentials(
certificate_chain=cert, private_key=key
)
else:
self._ssl_channel_credentials = SslCredentials().ssl_credentials
else:
if client_cert_source_for_mtls and not ssl_channel_credentials:
cert, key = client_cert_source_for_mtls()
self._ssl_channel_credentials = grpc.ssl_channel_credentials(
certificate_chain=cert, private_key=key
)
# The base transport sets the host, credentials and scopes
super().__init__(
host=host,
credentials=credentials,
credentials_file=credentials_file,
scopes=scopes,
quota_project_id=quota_project_id,
client_info=client_info,
always_use_jwt_access=always_use_jwt_access,
api_audience=api_audience,
)
if not self._grpc_channel:
self._grpc_channel = type(self).create_channel(
self._host,
# use the credentials which are saved
credentials=self._credentials,
# Set ``credentials_file`` to ``None`` here as
# the credentials that we saved earlier should be used.
credentials_file=None,
scopes=self._scopes,
ssl_credentials=self._ssl_channel_credentials,
quota_project_id=quota_project_id,
options=[
("grpc.max_send_message_length", -1),
("grpc.max_receive_message_length", -1),
],
)
# Wrap messages. This must be done after self._grpc_channel exists
self._prep_wrapped_messages(client_info)
@property
def grpc_channel(self) -> aio.Channel:
"""Create the channel designed to connect to this service.
This property caches on the instance; repeated calls return
the same channel.
"""
# Return the channel from cache.
return self._grpc_channel
@property
def operations_client(self) -> operations_v1.OperationsAsyncClient:
"""Create the client designed to process long-running operations.
This property caches on the instance; repeated calls return the same
client.
"""
# Quick check: Only create a new client if we do not already have one.
if self._operations_client is None:
self._operations_client = operations_v1.OperationsAsyncClient(
self.grpc_channel
)
# Return the client from cache.
return self._operations_client
@property
def analyze_iam_policy(
self,
) -> Callable[
[asset_service.AnalyzeIamPolicyRequest],
Awaitable[asset_service.AnalyzeIamPolicyResponse],
]:
r"""Return a callable for the analyze iam policy method over gRPC.
Analyzes IAM policies based on the specified request. Returns a
list of
[IamPolicyAnalysisResult][google.cloud.asset.v1p4beta1.IamPolicyAnalysisResult]
matching the request.
Returns:
Callable[[~.AnalyzeIamPolicyRequest],
Awaitable[~.AnalyzeIamPolicyResponse]]:
A function that, when called, will call the underlying RPC
on the server.
"""
# Generate a "stub function" on-the-fly which will actually make
# the request.
# gRPC handles serialization and deserialization, so we just need
# to pass in the functions for each.
if "analyze_iam_policy" not in self._stubs:
self._stubs["analyze_iam_policy"] = self.grpc_channel.unary_unary(
"/google.cloud.asset.v1p4beta1.AssetService/AnalyzeIamPolicy",
request_serializer=asset_service.AnalyzeIamPolicyRequest.serialize,
response_deserializer=asset_service.AnalyzeIamPolicyResponse.deserialize,
)
return self._stubs["analyze_iam_policy"]
@property
def export_iam_policy_analysis(
self,
) -> Callable[
[asset_service.ExportIamPolicyAnalysisRequest],
Awaitable[operations_pb2.Operation],
]:
r"""Return a callable for the export iam policy analysis method over gRPC.
Exports IAM policy analysis based on the specified request. This
API implements the
[google.longrunning.Operation][google.longrunning.Operation] API
allowing you to keep track of the export. The metadata contains
the request to help callers to map responses to requests.
Returns:
Callable[[~.ExportIamPolicyAnalysisRequest],
Awaitable[~.Operation]]:
A function that, when called, will call the underlying RPC
on the server.
"""
# Generate a "stub function" on-the-fly which will actually make
# the request.
# gRPC handles serialization and deserialization, so we just need
# to pass in the functions for each.
if "export_iam_policy_analysis" not in self._stubs:
self._stubs["export_iam_policy_analysis"] = self.grpc_channel.unary_unary(
"/google.cloud.asset.v1p4beta1.AssetService/ExportIamPolicyAnalysis",
request_serializer=asset_service.ExportIamPolicyAnalysisRequest.serialize,
response_deserializer=operations_pb2.Operation.FromString,
)
return self._stubs["export_iam_policy_analysis"]
def close(self):
return self.grpc_channel.close()
__all__ = ("AssetServiceGrpcAsyncIOTransport",)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,034
|
Annual Rankings are for the best fighters in a given year. Tapology users submit their own lists which are combined into consensus rankings. Learn More.
As Tapology users create their own lists, the consensus rankings will update in real-time. For 2016 Fighter of the Year, the 150 most-recently updated member lists will be included.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,492
|
The first parts photos previewing the 2015 design of the Summit Super Series American Race Cars dragster have been released. Sources close to the mad mind of Jerron Settles and the Imagine That Customs crew tell us that they have been pulling some all-nighters on the American Race cars project and the JR Race Car junior project. The first parts are starting to ship out. No sleep till.....Memphis!
The Moser Engineering rear end will contain a complete Moser rear end assembly, axles and brakes when completed.
Abruzzi Transmissions featuring a Reid case will once again handle getting the power to the ground.
Trick Flow Specialties valve covers and intake manifold will top off the TFS cylinder heads that are the focal point of this year's engine build.
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"redpajama_set_name": "RedPajamaC4"
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Q: MAX682 with electrolytic capacitors In a new design I would like to use the MAX682. In the datasheet I found this typical operating circuit:
I would like to use it exactly like that except that I'm going to pull /SKIP to GND.
In table 2, I find the recommended values for the capacitors: CIN should be 2.2uF, CX should be 1uF and COUT should be 47uF (tantalum) or 10uF (ceramic).
First of all, why does the needed capacity depend on the type of capacitor? And secondly: from the circuit it looks like I can't use electrolytic capacitors for CIN and CX. Is that correct? Why is it so, because it seems clear that the polarity won't change. I would prefer to use electrolytic capacitors because I have them in stock.
I actually did a quick test myself and used electrolytic capacitors everywhere. In total, the circuit drew about 110mA, while the 5V circuit drew only 2mA. And besides that, the output has about 0.4V AC noise. However, it's not entirely sure that this is just because of the capacitors, because I for example didn't use a very short trace between GND and PGND (pin 1 is top left):
So it really isn't possible. The question remains why.
A: I see three things that are hurting your charge pump circuit.
The capacitors are aluminum electrolytic instead of ceramic, so they have very high dissipation factor and low self-resonant frequency. These charge pumps really do require ceramic capacitors.
Second, they are radial-leaded and axial-leaded capacitors instead of surface-mount. That adds some series inductance, which tends to defeat the capacitance.
Third, the whole thing is built on a white solderless breadboard, which adds even more inductance. You'd be better off using a PCB breakout board (like a Surfboard) and solder capacitors with trimmed short leads, or better yet, surface-mount capacitors.
Not all capacitors are created equal, and not even all ceramic capacitors are created equal. The X7R and X5R types give fairly large capacitance values stable over a good temperature range. The cheaper Y5V, Y5U, and Z5V types can lose a lot of their capacitance over temperature, so sometimes we have to compensate for that loss by using higher nominal starting capacitance. (The NP0 and C0G types are highly stable over temperature, but usually are only available in very low capacitance values.)
Back in the day, ceramic capacitors were only available up to about 10uF, but nowadays there are 100uF ceramic capacitors. So the need for using tantalum and aluminum electrolytics is somewhat diminished; now it's driven by a cost vs performance trade-off.
The self-resonant frequency (SRF) is important in switching circuits, because at frequencies higher than SRF, the capacitor stops behaving like a capacitor and starts behaving like an inductor. Surface-mount ceramic capacitors are usually higher SRF and thus closer to "ideal" capacitors.
Aluminum electrolytic capacitors are great for providing a large amount of capacitance and accepting large inrush current, these are typically seen on power supply inputs. But they don't work well for the "flying capacitor" of a charge pump.
Take a look at MAX619EVKIT http://datasheets.maximintegrated.com/en/ds/MAX619EVKIT.pdf for an example of a good charge pump layout, using surface-mount ceramic capacitors.
Disclosure: I am a Maxim employee and designed the MAX619EVKIT way back when great lizards roamed the earth.
A: The problem is likely to be because electrolytic caps have high relative ESR compared to ceramics.
At the bottom of page 8 of the data sheet it says: -
All capacitors must maintain a low (<100m Ω) equivalent series
resistance (ESR)
Such small values as typically used are not always found in electrolytic types.
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"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}
Cosmology is at the verge of a new era of spectroscopic and photometric data, with upcoming galaxy surveys such as Euclid, DESI, DES, JPAS, or LSST. These data will provide incredible thrust to better constrain the parameters of the cosmological model. At the same time, any deviation from it would indicate the emergence of new physics, thus gauging the room available for extending our models. Among the numerous, possible extensions to the simplest $\Lambda$CDM model, massive neutrinos occupy a privileged spot due to the strong evidence of their existence.
As a matter of fact, observations that neutrinos oscillate among their flavours set a lower bound to their mass: $M_{\nu} = \sum_i m_{\nu, i} \gtrsim 0.06 ~ \mathrm{eV}$ at 95\% confidence level \citep{Gonzalez-GarciaEtal2012, Gonzalez-GarciaMaltoniSchwetz2014, ForeroTortolaValle2014,EstebanEtal2017}. At the same time, the study of the endpoint energy of electrons produced in $\beta$-decay experiments places an upper bound to the neutrino mass, such as $m(\nu_e) < 2.2 ~ \mathrm{eV}$ at 95\% \citep{KrausEtal2005}. Future, planned experiments \citep[such as KATRIN and PTOLEMY,][]{BonnEtal2011,PTOLEMYEtal2019} are expected to increase the sensitivity at least by one order of magnitude. Complementing these efforts, cosmology is already able to produce competitive neutrino mass constraints by exploiting the modifications they induced on the large-scale structure (LSS) of the universe.
For example, from the sole analysis of the CMB temperature anisotropy power spectrum, the total neutrino mass can be constrained to be $M_\nu < 0.537$ eV, at 95\% credibility level, which decreases to $M_\nu < 0.257$ eV at 95\% when considering also CMB polarization signal \citep{PlanckResults2013,PlanckResults2015, PlanckResults2018}.
Combining this information and that coming from clustering measurements, one can tighten such constraints, for example obtaining $M_\nu < 0.28$ eV at 95\% level, from the joint analysis of the likelihoods of CMB, BAO peak position and smoothed clustering \citep{ZennaroEtal2018}, or even $M_\nu < 0.22$ eV combining CMB data and BOSS clustering using the so called double-probe approach \citep{Pellejero-IbanezEtal2016}.
Even more stringent constraints come from the analysis of the 1D power spectrum of distant quasars (of which we can see the Ly-$\alpha$ forest), giving $M_\nu < 0.12$ eV \citep{Palanque-DelabrouilleEtal2015}. A similar constraint of $M_\nu < 0.12$ eV at 95\% level has also been obtained by combining CMB and BAO data, including the high-$\ell$ polarization data of Planck \cite{VagnozziEtal2017}. Finally, there are even hints of detection from the combination of abundance and clustering of galaxy clusters \citep{EmamiEtal2017}.
Thanks to future surveys, we expect that the value of the total neutrino mass and, perhaps, their mass ordering, will be measured from LSS data. Therefore, making accurate predictions for the spatial distribution of matter and luminous tracers in the presence of massive neutrinos is, at this point, crucial.
By including neutrinos in the solution of Einstein-Boltzmann equations, it is possible to obtain predictions of the linear power spectra of matter in the presence of neutrinos, from codes such as \texttt{class} \citep{Lesgourgues2011} or \texttt{camb} \citep{LewisChallinorLasenby2000}. The linear density solutions, however, are not enough to describe the spatial distribution of matter at small scales and the clustering properties of collapsed objects and luminous tracers.
To this end, different approaches have been proposed and developed in the literature. On the one hand, one can extend perturbation-theory arguments to cosmologies that include massive neutrinos, as, for example, in \cite{BlasEtal2014,FuhrerWong2015}, or develop Effective Field Theories accounting for a hot massive component \citep{SenatoreZaldarriaga2017,deBelsunceSenatore2019}.
On the other hand, one can directly run full $N$-body simulations including a massive neutrino component. There are a number of possible implementations of neutrinos in cosmological simulations: neutrino perturbations can be solved linearly on a 3D grid, as in the \textit{grid-based} method proposed by \cite{BrandbygeHannestad2009}, or they can be traced by directly sampling the distribution function with additional $N$-body particles, as in the \textit{particle-based} method used by \cite{BrandbygeEtal2008, VielHaehneltSpringel2010, Villaescusa-NavarroEtal2013, CastorinaEtal2015, CarbonePetkovaDolag2016}, or even solved on a grid employing a linear response function that is sensitive to the nonlinearities of the cold perturbations, as in the method of \cite{AliHaimoudBird2013}.
Other methods are possible, in particular a hybrid method between the grid- and particle-based ones, proposed by \cite{BrandbygeEtal2010}, or a hybrid between the linear response and the particle-based method proposed by \cite{BirdEtal2018}. Finally, a method combining particle and fluid techniques has been used by \cite{BanerjeeDalal2016}.
Such large amount of attention drawn to neutrino cosmologies has made possible studying in detail many aspects of the matter distribution \citep{BirdViel2012}, halo and void clustering and abundance \cite{CastorinaEtal2014, CastorinaEtal2015, MassaraEtal2015, RaccanelliVerdeVillaescusa-Navarro2019, VagnozziEtal2018, SchusterEtal2019}, halo model \citep{MassaraVillaescusa-NavarroViel2014}, and redshift space distortions \citep{Villaescusa-NavarroEtal2018a, BelEtal2019}.
Nonetheless, comparing LSS observations to predictions remains a difficult task, especially when dealing with biased tracers and when accounting for degeneracies with other cosmological parameters, owing to the huge amount of computational resources needed to simulate the evolution of neutrino universes.
One approach is to develop emulators of the bias of dark matter tracers in the presence of neutrinos.
This approach has been presented in \cite{ValcinEtal2019}, who calibrated an emulator on $\Lambda$CDM simulations that provides predictions for the halo bias in cosmologies with arbitrary neutrino mass, provided that they share the same remaining cosmological parameters (most importantly the value of $\sigma_8$, the linear amplitude of matter fluctuations at scale $R=8~h^{-1}~\mathrm{Mpc}$).
In this work we propose an alternative method to make predictions for neutrino cosmologies, with emphasis on numerical efficiency and accuracy. Specifically, we extend the simulation rescaling algorithm presented in \cite{AnguloWhite2010} to cosmologies with massive neutrinos. This technique allows to quickly create 3D nonlinear mass density and velocity fields, including halos and their substructures, at an almost negligible computational cost.
The method has already been tested and applied in cases that did not include any non-cold component, finding that it works with very high accuracy \citep[e.g.][]{AnguloWhite2010, RuizEtal2011, MeadEtal2015, AnguloHilbert2015, RennebyHilbertAngulo2018}.
A distinctive advantage of our approach is that any computational effort used in the production of a given high-quality neutrinoless $\Lambda$CDM simulation is reused in the predictions for massive neutrino scenarios; explicitly, these predictions will inherit the quality and realism of the original simulation in terms of nonlinear growth of structures, internal properties of haloes, velocity bias or assembly bias. This enables a realistic and sophisticated modelling of the galaxy population and thus allowing an accurate way to forecast and interpret the wealth of upcoming LSS data.
This paper is organized as follows. In Sec. \ref{sec::neutrinos} we review the relevant aspects of neutrino cosmology, focusing on the quantities we seek to reproduce through the cosmology scaling. In Sec. \ref{sec::method}, after recalling the main aspects of the cosmology rescaling algorithm, we specify the modifications needed to account for massive neutrinos. We test such extended algorithm by quantifying departures in the linear variance and matter power spectrum in Sec. \ref{sec::theoryscale}. Finally, in Sec. \ref{sec::sims} we perform comparisons between neutrinoless $\Lambda$CDM simulations scaled to include massive neutrinos, and simulations actually run including massive neutrinos in their evolution. In Sec. \ref{sec::conclusion}, we present our conclusions.
\section{Massive neutrinos}\label{sec::neutrinos}
In this section, we review the main effects massive neutrinos induce on the evolution of the background and of the perturbations in the Universe. We do so both from an analytic standpoint, by adopting a fluid description of the cosmological neutrino distribution, and from a numerical perspective, by introducing a set of N-body simulations that incorporate a massive neutrino component. Please note that we focus on the aspects that are either useful for our implementation of neutrinos or that we want to reproduce in simulations. For a more general treatment of the effects of neutrinos in cosmology we refer the reader to the comprehensive work of \cite{LesgourguesPastor2006,LesgourguesPastor2012,LesgourguesPastor2014}.
We parametrize the total matter content of the Universe as the sum of a cold and hot component,
\begin{equation}
\Omega_{\rm m} = \Omega_{\rm cold} + \Omega_{\nu},
\end{equation}
\noindent where the cold component is given by the sum of the baryon and cold dark matter density parameters ($\Omega_{\rm b}$ and $\Omega_{\rm cdm}$, respectively) and the hot component is given by the neutrino density parameter. The remaining energy budget of the Universe is split among the contributions of radiation $\Omega_{\rm r}$, dark energy $\Omega_{\rm de}$ and curvature $\Omega_{\rm k}$. In this work we consider the case of three degenerate massive neutrinos, and we refer to the total neutrino mass as the sum of the masses of the single species, $M_\nu = \sum_i m_{\nu,i}$. Moreover, we will always work in the context of a flat universe, therefore assuming $\Omega_{\rm k} = 0$ in all cases.
Neutrinos become non-relativistic quite late in the thermal history of the Universe, their non-relativistic transition taking place at $1 + z_{\rm nr} \simeq 1890 ~ (m_\nu/1 ~ \mathrm{eV})$. This is much later than the time at which they decouple, i.e. when their interaction rate falls below the expansion rate of the Universe (roughly at $z_{\rm dec} \simeq 10^9$, when the background temperature is roughly $1 ~ \mathrm{MeV}$). For this reason, their momentum distribution remains frozen to that of a relativistic Fermi-Dirac species, even when they are already non-relativistic. The contribution of massive neutrinos to the Hubble function follows directly from the integration of their momentum distribution, which controls the time-dependent evolution of the neutrino density. Their density evolves as that of the radiation component of the Universe, proportionally to the expansion factor to the fourth power, until their non relativistic transition. Afterwards, it asymptotically tends to evolve as matter does, that is proportionally to the expansion factor to the third power.
We also note that the radiation content, in cosmologies without massive neutrinos, is usually given by the sum of the photon and massless neutrino contributions. In a more general context, this is given instead by
\begin{equation}
\Omega_{\rm r} = \Omega_{\gamma} \left[ 1 + \Delta N_{\rm eff} \dfrac{7}{8} \left( \dfrac{4}{11} \right)^{4/3} \right],
\label{eq::omega-rad}
\end{equation}
where $\Delta N_{\rm eff}$ is the corrected number of relativistic species, chosen to guarantee that the effective number of relativistic species in the early universe is $N_{\rm eff} = 3.046$, i.e. what is expected for three neutrinos undergoing non-instantaneous decoupling \citep{ManganoEtal2005}.
One of the consequences of the freezing of the neutrino momentum distribution is that they are characterized, throughout the thermal history of the Universe, by relatively large velocities compared to any other massive particle. Picturing neutrinos as a fluid \citep{ShojiKomatsu2010}, such velocities act as an effective pressure term and set a scale below which the growth of neutrino perturbations is strongly suppressed. Such scale is called the \textit{free-streaming scale} and, in fluid approximations, can be written as
\begin{equation}
k_{\rm fs} = \sqrt{\dfrac{3}{2} \Omega_{\rm m}(a)} \dfrac{a H(a)}{c_{\nu}(a)}.
\end{equation}
We have introduced here the speed of sound corresponding to the effective neutrino pressure \citep{BlasEtal2014},
\begin{equation}
c_{\nu}(a) = \dfrac{\delta P_\nu}{\delta \rho_\nu} \simeq \dfrac{123.423}{a ~ m_\nu} ~ \mathrm{km} ~ \mathrm{s}^{-1},
\label{eq::csound}
\end{equation}
where $m_\nu$ is expressed in electronvolts.
Not only are neutrino perturbations affected by such suppression, but, by solving the joint equations of growth of cold and hot matter, the neutrino pressure term induces a scale dependent suppression of the growth also for the cold components.
\subsubsection{Clustering}
In Fig. \ref{fig::neutrino-pk} we show the effect of massive neutrinos on the two point statistics of the mass field. We do so by studying a set of neutrino simulations whose technical aspects will be detailed in Sec. \ref{sec::sims}. We analyse the cold matter power spectrum computed in cosmologies with massive neutrinos, with increasing neutrino total mass $M_\nu = \{0.06, 0.1, 0.15, 0.2, 0.3\} ~ \mathrm{eV}$ and compare it to a $\Lambda$CDM case with $M_\nu = 0$ eV. At high wavelenghts, the neutrino induced suppression of clustering is $\Delta P/P \sim 8 \, \Omega_{\nu}/\Omega_{\rm m}$ -- spanning from $\sim 4\%$ at $k = 1 ~ h ~ \mathrm{Mpc}^{-1}$ for the lightest neutrino mass, to $\sim 25\%$ at the same scale in the case of the heaviest mass.
The evolution of the cold matter power spectrum in the nonlinear regime exhibits a well known, spoon like shape, \citep[see, for example,][]{BirdVielHaehnelt2012}. The reason resides in the combination of the suppression of the matter clustering induced by neutrinos, and the fact that neutrinos move the nonlinear scale $k_{\rm nl}$ to larger modes. Alongside our simulated results, we show nonlinear predictions obtained by applying the \texttt{halofit} algorithm \citep{TakahashiEtal2012} to the cold linear power spectrum predicted by \texttt{class}.
\begin{figure}
\includegraphics[width=0.48\textwidth]{{Plots/compare_pk}.pdf}
\caption{The cold matter (cdm + baryons) power spectrum in the presence of massive neutrinos. Points are measurements in the simulations, lines are predictions in linear theory (transparent colours) and applying \texttt{halofit} (non-transparent colours). The simulations employed have different neutrino masses, but same baseline cosmology, and share the same large scale normalization. Here results at redshift $z \simeq 0.5$ are shown. The upper panel shows the power spectra superimposed to their corresponding theoretical predictions; the grey, horizontal line marks the level of the shot-noise. The bottom panel shows the ratio between the measurements in the simulations and the predictions. The suppression due to neutrino free streaming is larger for larger neutrino masses. Moreover, neutrinos induce a peculiar spoon-like feature in the non linear matter power spectrum at relatively small scales ($k \simeq 1 ~ h ~ \mathrm{Mpc}^{-1}$).}
\label{fig::neutrino-pk}
\end{figure}
\subsubsection{Halo abundance}
We present in Fig. \ref{fig::neutrino-hmf} the halo mass function in the neutrino simulations, compared to the $\Lambda$CDM case. We compute the quantity $\mathrm{d} n / \mathrm{d} \ln M$, considering as halo mass the quantity $M_{200}$, that is the mass enclosed in a sphere whose average density is equal to 200 times the comoving critical density of the Universe.
To model the halo mass function in the presence of neutrinos, we use the fits obtained by \cite{AnguloEtal2012} without massive neutrinos, being careful to compute the variance of the matter distribution only in terms of the cold component, $\sigma_{\rm cold}$, \citep{CastorinaEtal2014,CastorinaEtal2015}. This is justified by the fact that the neutrino free-streaming scale is, at all times, significantly larger than the typical size of the most massive haloes. As a consequence, the dynamics of the collapse are dominated by the cold dark matter and baryons \citep[see, for example,][who showed that the collapse threshold changes by less than $0.1\%$ in the presence of neutrinos]{IchikiTakada2012}.
It comes therefore as no surprise that the main effect of neutrinos is to suppress the high mass end of the halo mass function. As a matter of fact, for the same $\Omega_{\rm m}$, haloes of a given mass are significantly rarer (since there is \textit{effectively} less matter collapsing). As shown in Fig. \ref{fig::neutrino-hmf}, this can result in finding from $\sim 20\%$ to $\sim 60\%$ less haloes in the high mass bin centered in $10^{15} ~ h^{-1} ~ \mathrm{M}_{\odot}$.
These effects are specific signatures of massive neutrinos and are dependent on the value of the neutrino total mass parameter. Therefore, they provide us with a way to constrain the neutrino total mass through cosmological observations. We aim at reproducing such effects in neutrino simulations that, instead of being run in actual neutrino cosmologies, can be obtained by rescaling a $\Lambda$CDM simulation, originally run without any hot massive component.
\begin{figure}
\includegraphics[width=0.48\textwidth]{{Plots/compare_dndlnM}.pdf}
\caption{The logarithmic halo mass function, $\phi(M) = \mathrm{d} n / \mathrm{d} \ln M$, multiplied by $M$ to reduce the dynamical range of the y axis. We show for each mass bin the associated Poisson error. The upper panel shows the two extreme cases of $M_\nu=0$ and $0.3$ eV. The lower panel shows, for six choices of neutrino mass spanning the above range, the ratio between the halo mass function in the presence of neutrinos and its corresponding $\Lambda$CDM case. Lines are theoretical predictions obtained as detailed in the text, while points are measurements in our simulations. In the presence of massive neutrinos, massive halos become rarer, when compared to halos of the same mass in pure $\Lambda$CDM cosmologies. As a consequence, the high mass end of the halo mass function is suppressed, the suppression being larger for larger neutrino masses.}
\label{fig::neutrino-hmf}
\end{figure}
It has been shown by \cite{AnguloWhite2010}, AW10 hereafter, that structure formation can be predicted by matching the linear density amplitude $\sigma(M)$. The fact that the neutrino induced effects can be captured by $\sigma_{\rm cold}$ leads us to postulate the Anstatz that neutrino simulations can be accurately reproduced by extending the ideas of AW10. We will develop and test this idea further in the next section.
\section{Method: extending cosmology rescaling to massive neutrino cosmologies}\label{sec::method}
In this section we will review the scaling method presented in \cite{AnguloWhite2010} and extend it to deal with the presence of massive neutrinos. Note that we will meticulously follow the philosophy of the original paper, stressing the modifications necessary to model massive neutrinos. The goal of the procedure is to find the transformation of times and lengths such that a simulation that was run assuming a given cosmology (that we will refer to as \textit{original cosmology} throughout this paper) resembles as closely as possible a simulation run in a different cosmology (that we will refer to as \textit{target cosmology}). Such resemblance can be quantified in terms of matter clustering, halo abundance and bias and galaxy clustering in redshift space.
In our case study, we will always consider the rescaling of a $\Lambda$CDM cosmology (original) so that it reproduces a neutrino cosmology (target). However, note that it is also possible to consider scaling a neutrino cosmology to a different value of $\Omega_{\nu}$.
The general idea of the scaling algorithm can be summarised in a few steps:
\begin{itemize}
\item determine the transformation of lengths ($s$) and time ($z^*$) that minimizes the difference between the linear amplitude of perturbations in the original and target cosmology;
\item select the simulation output in the original cosmology at the closest redshift to the one identified through the minimization, $z^*$;
\item apply the length transformation to the positions of particles/haloes/subhaloes in this snapshot, which corresponds now to a simulations with different volume $(sL_{\rm box})^3$ and particle mass ($s^3 \Omega_{\rm m}^\prime M_{\rm part} / \Omega_{\rm m}$, where primed refers to the target cosmology);
\item correct the large scales modes, to account for the different bulk motions between original and target cosmologies;
\item correct nonlinear velocities;
\item correct halo concentrations.
\end{itemize}
We will now go through these steps underlying where we need to account for the presence of massive neutrinos.
\subsection{Relevant modifications to include massive neutrinos}
In each cosmology, the typical level of inhomogeneity associated to a scale $R$ is given by the linear variance of the density field smoothed on that scale. In Press-Schechter theory, this uniquely determines the bias and halo mass function. For this reason, the first step of the procedure is to try to match, as closely as possible, the linear amplitude of the matter density field of the target cosmology.
More specifically, we fix the redshift at which we want to reproduce the target cosmology, $z_t$, and we assume we have run the original simulation in a cosmological box of comoving side $L_{\rm box}$. In order to mimic the target cosmology at redshift $z_t$ we will need to consider the original cosmology at a different redshift, $z_*$. Lengths will also be different and we will need to modify the simulation run in the original cosmology so that it has box size $L^\prime_{\rm box} = s L_{\rm box}$. The modified simulation will mimic a simulation run in the target cosmology with box size $L^\prime_{\rm box}$ at $z_t$. We find the scaled redshift and the length scale factor $s$ by minimizing the difference between the linear amplitude of fluctuations $\sigma(R)$ in the original and target cosmologies over a given interval of scales, $R \in [R_1, R_2]$,
\begin{equation}
\delta^2_{\rm rms}(s, z_*) \equiv \dfrac{1}{\ln(R_2/R_1)} \int _{R_1}^{R_2} \dfrac{\mathrm{d} R}{R} \left[ 1 - \dfrac{\sigma(s^{-1} R, z_*)}{\sigma^\prime(R, z_{\rm t})} \right]^2.
\label{eq::minimization}
\end{equation}
When including massive neutrinos, as discussed in Sec. \ref{sec::neutrinos}, the physical meaning of the linear amplitude of matter fluctuations is preserved provided that we substitute the total matter density field with the cold matter (cold dark matter + baryons) density field. Note that this is a direct consequence of neutrinos being characterized by a velocity dispersion that is, at all redshifts, comparable or larger than typical velocity dispersions in haloes, thus preventing the former from playing a significant role in the collapse process of the latter. We therefore modify $\sigma$ in Eq. \ref{eq::minimization} from its version in AW10, which now reads
\begin{equation}
\sigma^2(R,z) = \int_0^{\infty} \dfrac{k^2 \mathrm{d} k}{2 \pi^2} W^2(kR) D_{\rm cold}^2(k,z) P_0(k),
\end{equation}
where $W(x) \equiv 3 (\sin x - x \cos x) / x^3$ is the Fourier transform of the top-hat window function, $P_0$ is the total matter power spectrum at redshift zero and $D_{\rm cold}(k, z)$ is the growth factor (in Fourier space) for the cold matter component. Note that, in the presence of massive neutrinos, it depends on both scale and time, and, therefore, cannot be factorized out of the integral. The cold growth factor, in our convention, is always defined with respect to the total matter power spectrum, so that for any species considered we have $P_i(k,z) = D_i^2(k,z) P_0(k)$.
To find the scale dependent cold matter growth factor, we use the code \texttt{reps} \citep{ZennaroEtal2016}, which employs a three-fluid description, comprising a pressureless cold dark matter fluid, a presureless baryon fluid and a neutrino fluid characterized by the effective speed of sound presented in Eq. \ref{eq::csound}.
We therefore solve the set of linearised equations of growth of fluctuations for the three components. The pressure term introduced for neutrinos provides the correct redshift- and scale-dependent suppression for the cold matter growth factor. Note that neglecting the pressure term associated to the baryon fluid is in principle not justified.
However, in this work we are interested in characterizing neutrino effects, which have been shown to be separable from those of baryons \citep{MummeryEtal2017}.
In Fig. \ref{fig::nu-scaledep} we show an example of the scale dependent growth factor and growth rate, both as obtained using \texttt{reps} and \texttt{class}. The former provides purely Newtonian solutions, while the latter solves the Einstein-Boltzmann equations including non-gravitational coupling such as that of radiation and baryons on scales approaching the cosmological horizon.
Part of the discrepancy among the codes observable on large scales is due to such effect \citep[see, for example,][]{BrandbygeEtal2016}, while part is due to the fact that, while the results of \texttt{class} used here are in synchronous gauge, the results of \texttt{reps} are purely Newtonian (and could be interpreted in General Relativity assuming some \textit{ad hoc} gauge, different from the synchronous one).
Since the outputs of \texttt{reps} are designed to agree with the dynamics of a Newtonian simulation, we will use this code in order to make predictions for the (cold) matter power spectrum, growth factor and growth rate in the rest of this work. However, we will also present a method to reproduce, if desired, these large scale contributions to the power spectrum.
From Fig. \ref{fig::nu-scaledep} we can appreciate that the suppression of the clustering due to massive neutrinos evolves in time differently for different wavemodes. In particular, large scales, such as $k \simeq 10^{-3} ~ h ~ \mathrm{Mpc}^{-1}$, evolve very little, while small scales, such as $k \simeq 1 ~ h ~ \mathrm{Mpc}^{-1}$, go from showing a suppression with respect to the $\Lambda$CDM case of roughly 2\% at high redshift, to almost 6\% at low redshift.
Note that velocities in the scaled simulation need to be corrected to comply with the different growth history of the target cosmology. Following once again AW10 this can be achieved through
\begin{equation}
\boldsymbol{v}^{\prime} = \dfrac{a^{\prime} f^{\prime}(a^{\prime}) H^{\prime}(a^{\prime})}{a f(a) H(a)} ~ \boldsymbol{v},
\label{eq::vel-corr}
\end{equation}
where $f \equiv \mathrm{d} \ln D/ \mathrm{d} \ln a$ is the growth rate, primes quantities are in the scaled simulation and non-primed ones in the original cosmology. In the case of neutrino cosmologies the growth rate is scale dependent, $f = f(k, z)$. However, as shown in Fig. \ref{fig::nu-scaledep}, it exhibits two asymptotic regimes at large and small scales. We choose to use in Eq. \ref{eq::vel-corr} the value at small scales (obtained at the arbitrarily fixed wavemode $k=0.5 ~ h ~ \mathrm{Mpc}^{-1}$). This choice accurately corrects the velocity of the underlying bulk motions of overdensities on nonlinear scales, but is not accurate for those on larger scales. We will address in the next paragraphs residual corrections that further increase the accuracy of the rescaling procedure.
\begin{figure*}
\includegraphics[width=0.98\textwidth]{{Plots/theory_scaledep_growth}.pdf}
\caption{\textit{Left:} the ratio of the scale dependent square roots of the power spectra with and without neutrinos as computed using \texttt{reps} (solid lines) and \texttt{class} (dashed lines). The solutions for four wavemodes are shown as a function of the scale time. \textit{Right:} the ratio of the scale-dependent growth rates with and without neutrinos, computed with \texttt{reps} (solid lines) and \texttt{class} (dashed lines). The neutrino mass arbitrarily chosen for this example is $M_\nu=0.2$ eV.}
\label{fig::nu-scaledep}
\end{figure*}
First, even though the matching of the linear amplitude of fluctuations is tailored to provide a good agreement of the clustering of the scaled and target simulations on small scales, because it is dominated by the scales that contribute the most to the overall inhomogeneity, residual inaccuracies might affect the large scale clustering and bulk motion.
However, the evolution of the large scales is easily modelled in linear theory. The solution is, therefore, to correct the positions and velocities of the particles (or haloes, or subhaloes) of the scaled simulation.
This is done to greater accuracy in Lagrangian space. Once the corresponding Lagrangian position of each object is found, we apply a displacement field such that the large scale fluctuations of the original cosmology are subtracted. We then proceed with the rescaling procedure, and, at the end, we apply another displacement to add the contribution of the long waves of the target cosmology.
In terms of displacement fields, we can therefore move our particles so that
\begin{equation}
\boldsymbol{x} = \boldsymbol{q} - \boldsymbol{\psi}_{\rm o}(\boldsymbol{q}) + \boldsymbol{\psi}_{\rm t}(\boldsymbol{q}),
\label{eq::lss-correction}
\end{equation}
\noindent where $\boldsymbol{q}$ are the Lagrangian positions obtained iteratively with the method described in AW10, $\boldsymbol{\psi}_{\rm o}$ and $\boldsymbol{\psi}_{\rm t}$ are the displacement fields corresponding to the original and target cosmologies, respectively. Note that we only employ modes smaller than the nonlinear scale $k_{\rm nl} = 1 / R_{\rm nl}$, defined so that $\sigma(R_{\rm nl}) = 1$.
A second consideration regards the coupling, mentioned above, of radiation and baryon perturbations on large scales, an effect that is neglected in Newtonian N-body simulations. If the simulation is started (as in our case) with an initial power spectrum obtained scaling the $z=0$ spectrum back in time, then the matter power spectrum measured in the simulation (on large, linear scales) will match the solutions of the Einstein-Boltzmann equations only at low redshift.
At higher redshifts an unavoidable mismatch will appear on the scales approaching the cosmological horizon. If desired, this effect can be reintroduced, by incorporating it in the large scale correction presented in Eq. \ref{eq::lss-correction}. The target displacement field will therefore contain a double contribution
\begin{equation}
\boldsymbol{\psi}_{\rm t} = \boldsymbol{\psi}^{GR}_{\rm t} + \boldsymbol{\psi}^{NG}_{\rm t},
\end{equation}
where the term $\boldsymbol{\psi}^{GR}_{\rm t}$ corrects the difference in the large scale matter overdensities due to the different expansion and background gravitational potential between the original and the target cosmology (i.e. the gravitationally induced differences), while the term $\boldsymbol{\psi}^{NG}_{\rm t}$ accounts for these non-gravitational couplings.
Moreover, an additional correction can be applied to velocities of particles that reside in haloes, $\boldsymbol{v}_{\rm in-halo}$, to account for any residual nonlinear effect. We modify the correction presented in AW10, replacing the total matter density parameter $\Omega_{\rm m}$ with the cold matter one, $\Omega_{\rm cold}$, thus obtaining
\begin{equation}
\boldsymbol{v}^\prime_{\rm in-halo} = \sqrt{\dfrac{a ~ \Omega_{\rm cold}^\prime}{a^\prime ~ \Omega_{\rm cold}}} ~ \dfrac{h^\prime}{h} ~ s ~ \boldsymbol{v}_{\rm in-halo},
\end{equation}
where primed quantities are in the target cosmology and $s$ is the length scale factor and $h = H_0/100$ the Hubble parameter.
Finally, even better agreement of the clustering on scales dominated by the one-halo term can be achieved when accounting for the different concentrations haloes have in the two cosmologies. We expect halo concentrations to be different depending on the different halo formation time. We therefore define a displacement field accounting for the different density profile shapes expected at the time of the original simulation and at the time of the target simulation. We then proceed to apply it to all particles residing in a bound halo, on a halo-by-halo basis.
In summary, the fundamental considerations that we need to keep in mind to scale a $\Lambda$CDM cosmology to a neutrino cosmology can be outlined as:
\begin{itemize}
\item when finding the scale time and scale length factors, we need to minimize the difference in the amplitude of cold matter linear perturbations, instead of using the one referred to the total matter;
\item the growth factor, used to correct large scales, is scale dependent (and the scale dependence evolves with redshift);
\item consequently, also the growth rate, used to correct large scale velocities, is scale and redshift dependent;
\end{itemize}
\subsection{Examples}\label{sec::theoryscale}
\begin{figure*}
\includegraphics[width=\textwidth]{{Plots/theory_all}.pdf}
\caption{\textit{Top row}: The log-likelihood surface associated to the difference between the variance in the scaled and in the target cosmology. The white crosses mark the minima, i.e. the choices of length and time scaling factors that, applied to the original $\Lambda$CDM cosmology, allow it to best reproduce the variance in the target neutrino cosmology at the target redshift $z_{\rm t} = 0$. The diamonds mark the scaling time selected if we just match the value of $\sigma_8$.
\textit{Middle row}: The accuracy of the rescaled variance compared to the target one. The light grey and dark grey shaded areas mark the 5\% and 1\% levels, respectively. Blue solid, orange dashed and green dash-dotted lines show, respectively, the variance in the original cosmology at the target redshift, the variance obtained through the $\sigma_8$ matching and the variance of the scaled cosmology (this work), all divided by the variance computed in the target cosmology. The accuracy is at better than 1\% level for all of the smoothing scales considered.
\textit{Bottom row}: The same ratio, but for the linear power spectrum. Additionally, we show in red dotted lines the ratio for the case in which we do not apply the large scale (LS) correction described in the test. In the case involving the full scaling procedure, the ratio between the scaled and the target linear power spectrum results to be well within the 1\% accuracy level.}
\label{fig::theory}
\end{figure*}
In the present section we employ the method described up to here, to rescale the linear theory predictions of a cosmology that does not include massive neutrinos, to one with non-null neutrino total mass.
We consider three target cosmologies, with different neutrino masses, $M_\nu = \{0.1, 0.2, 0.3\}$ eV. All cosmologies share the same baseline cosmological parameters and have the same amount of total matter, the cold dark matter density parameter consequently being different, according to
\begin{equation}
\Omega_{\rm cdm} = \Omega_{\rm cdm}^{\Lambda\mathrm{CDM}} - \Omega_\nu,
\end{equation}
where the neutrino density parameter is linked to the neutrino total mass though $\Omega_\nu = M_\nu /(93.14~h^2)$.
We consider two cases: along with the full minimization of the linear density amplitude in terms of length and time scaling, we also analyse the result of just selecting the time at which there is a matching of the linear amplitude smoothed at a scale of $8 ~ h^{-1} ~ \mathrm{Mpc}$, $\sigma_8$.
An example of such minimization is presented in the top row of Fig. \ref{fig::theory}. In order to reproduce cosmologies with increasingly large neutrino masses, we need to consider the $\Lambda$CDM cosmology at increasingly earlier redshift. This is true both when we just match the value of $\sigma_8$ and when we perform the full minimization. However, the time selected in the two cases is slightly different. With the full minimization we also find that the matching is more accurate if we expand the lengths in the $\Lambda$CDM cosmology.
From the top row of Fig. \ref{fig::theory} we expect that both by matching $\sigma_8$ and by performing the full minimization the linear amplitude in the scaled cosmology and in the target one differ by less than $1\%$. This is confirmed in the middle row of the same figure, where we compare the linear variance of the matter field smoothed with spherical top-hat functions with different radii computed in the scaled $\Lambda$CDM cosmology and in the target neutrino cosmology. The filtering scales span the interval $R \in [0.1, 10] ~ h^{-1} ~ \mathrm{Mpc}$, which covers the range of typical Lagrangian sizes of haloes. We can see that the full minimization ensures an almost perfect agreement on all the scales considered, while, as expected, the $\sigma_8$ matching performs slightly worse the farther we are from $R=8 ~ h^{-1} ~ \mathrm{Mpc}$.
In the bottom row of Fig. \ref{fig::theory} we compare the linear matter power spectrum between the scaled $\Lambda$CDM cosmology and the neutrino cosmology. In this case it is even more apparent that by just matching $\sigma_8$ we introduce a (scale dependent) error going from 5\% to 15 \% for $M_\nu=0.1 ~ \mathrm{eV}$ and $M_\nu=0.3 ~ \mathrm{eV}$ respectively. We notice that even performing the full minimization, although the agreement largely improves on small scales, there is a mismatch at the 2-7\% level on large scales. However, we predict that, with the full minimization procedure and applying a correction of the large, linear scales, we can match the scaled power spectrum with sub-percent precision.
\section{Testing with simulations}\label{sec::sims}
In the present section we proceed to test our method using N-body simulations. In particular, we will scale a $\Lambda$CDM simulation in order to add massive neutrinos. Each time, we compare the scaled simulation to the corresponding simulation \textit{actually} run including neutrinos. We perform the comparison in terms of matter clustering in real space, halo abundance and galaxy clustering in redshift space.
We simulate the evolution of the matter field in six cosmologies, characterized by different neutrino masses, namely $M_\nu = \{0, 0.06, 0.1, 0.15, 0.2, 0.3\}$ eV. Our simulations are performed with a modified version of the N-body code \texttt{GADGET} \citep{Springel2005}, extensively based on its lean version, \texttt{L-GADGET-3}, presented in \cite{AnguloEtal2012}. To follow the evolution of neutrinos, we included the implementation presented by \cite{AliHaimoudBird2013}.
This method consists in updating the potential mesh with the contribution of neutrino densities, computed solving the Boltzmann equation in Fourier space. Even if the solution is found linearising for small neutrino overdensities, the total potential is the one computed in the simulation, and accounts for the nonlinear evolution of the cold components.
This approximation, albeit imperfect, works remarkably well because neutrinos are expected never to develop significant nonlinearities.\footnote{ It is important to note that, however small, the nonlinear evolution of neutrinos could in principle play a role in the nonlinear growth of cold matter, a case that this method would fail to describe. A more complete, though computationally expensive, version of the linear response method has been presented in \cite{BirdEtal2018}. However, discrepancies are mainly expected to be found at the neutrino power spectrum level, while \cite{BirdEtal2018} found that the cold matter power spectrum obtained either with the method utilized here, with the more accurate hybrid method, or with particle-based simulations differ by less than 0.1\%.}
Our baseline cosmology is described by a flat $\Lambda$CDM model, with total matter density parameter $\Omega_{\rm m} = 0.307112$, baryon density parameter $\Omega_{\rm b} = 0.048252$, Hubble parameter $H_0 = 67.77 ~ \mathrm{km} ~ \mathrm{s}^{-1} ~ \mathrm{Mpc}^{-1}$, optical depth at recombination $\tau = 0.0952$, spectral index of primordial fluctuations $n_s = 0.9611$ and amplitude of scalar perturbations at the pivotal physical scale $k_{\rm pivot} = 0.05 ~ \mathrm{Mpc}^{-1}$ equal to $A_s = 2 \times 10^{-9}$.
When adding massive neutrinos we keep the total matter density at present time fixed, thus modifying the cold dark matter fraction according to $\Omega_{\rm cdm} = \Omega_{\rm m} - \Omega_{\rm b} - \Omega_\nu$.
The photons have present day temperature $T_\gamma = 2.7255$ K \citep[in agreement with][]{Fixsen2009}, corresponding to $\Omega_\gamma = 5.3758 \times 10^{-5}$. The relativistic neutrino contribution to the radiation is obtained through Eq.(\ref{eq::omega-rad}).
All simulations follow the evolution of $1056^3$ particles of cold matter in a periodic comoving cube of side $L_{\rm box} \simeq 700 ~ h^{-1} ~ \mathrm{Mpc}$ and softening length set to 1/50th of the mean intraparticle distance, $\epsilon \simeq 13 ~ h^{-1} ~ \mathrm{kpc}$. They are initialized at $z_{\rm i} = 49$, using, for each cosmology, the present day power spectrum rescaled back in time at the initial redshift with the method described in \cite{ZennaroEtal2016}. To the end of reducing the importance of cosmic variance, for each cosmology we run two paired simulations, with the same initial fixed amplitudes and phase realizations shifted by $\pi$ radians \citep{AnguloPontzen2016}. We employ a slightly different volume for each simulation, in order to be able to compare them with the rescaled $\Lambda$CDM simulations (which, after applying the rescaling transformations, will have a effectively larger volume). We list in Table \ref{tab::sims} the different volumes used.
\begin{table}
\centering
\begin{tabular}{c c c c c}
$M_\nu$ & $\Omega_{\rm cold}$ & $L_{\rm box}$ & $m_{\rm p}$ & $\sigma_8$ \\
$[\mathrm{eV}]$ & & $[h^{-1} ~ \mathrm{Mpc}]$ & $[10^{10} \mathrm{M}_{\odot}]$ & \\
\hline
0.00 & 0.30711 & 700.0 & 2.482 & 0.800 \\
0.06 & 0.30571 & 705.2 & 2.527 & 0.790 \\
0.10 & 0.30477 & 707.9 & 2.548 & 0.783 \\
0.15 & 0.30361 & 711.6 & 2.577 & 0.773 \\
0.20 & 0.30244 & 715.3 & 2.608 & 0.763 \\
0.30 & 0.30010 & 723.6 & 2.677 & 0.742 \\
\hline
\end{tabular}
\caption{The specifications of the set of simulations considered.}
\label{tab::sims}
\end{table}
\begin{figure*}
\includegraphics[width=0.48\textwidth]{{Plots/n_scaling_test_lcdm_rad_N1056_L700.0_003_newconcludlow16_ratios_backward}.pdf}
\includegraphics[width=0.48\textwidth]{{Plots/n_scaling_test_lcdm_rad_N1056_L700.0_001_newconcludlow16_ratios_backward}.pdf}
\caption{The accuracy of the scaling method tested against simulations run in the target cosmology, in terms of real space cold matter power spectrum. The original simulation is characterized by a pure $\Lambda$CDM cosmology without massive neutrino. The target cosmologies, instead, span the mass interval $M_\nu \in [0.06, 0.3]$ eV. We present two target redshifts, namely $z_{\rm t} \simeq 0.5$ and $z_{\rm t} \simeq 2$. \textit{Upper panel}: the lines show the ratio of the matter power spectrum measured in the simulations run including massive neutrinos and in the neutrinoless case; points show the same ratio computed using the $\Lambda$CDM simulations scaled to include massive neutrinos, divided by the $\Lambda$CDM simulation at the same redshift. \textit{Lower panels}: the ratio between the cold matter power spectrum measured in the scaled $\Lambda$CDM simulation and in the simulations actually run with neutrinos. The gray shaded region marks the 1\% level. The black vertical line marks the scale at which the shot noise starts accounting for more than 80\% of the signal.}
\label{fig::sim-pk}
\end{figure*}
Structures in the simulations are identified employing a modified version of the \texttt{subfind} algorithm \citep{SpringelEtal2001} which we have inlined in \texttt{GADGET}. In order to define a Friend-of-Friends halo \citep{DavisEtal1985}, it must contain a minimum number of $N_{\rm min} = 20$ particles, with linking length $b = 0.2$ in units of the mean intraparticle distance. We also require that subhaloes contain at least $20$ particles.
\subsection{Matter power spectrum}
Fig. \ref{fig::sim-pk} shows the real-space matter power spectrum measured in the different neutrino simulations and in their corresponding $\Lambda$CDM scaled simulations. We consider the case in which we aim at reproducing neutrino cosmologies at $z \simeq 0.5$ and at $z \simeq 2$. Although not shown, we have checked that the results presented below also hold for redshift $z \simeq 1$.
All spectra presented here are measured on regular meshes characterised by a grid-size of $N_{\rm grid} = 512^3$ points. This corresponds to considering a range of wavemodes spanning from the fundamental model of the grid, $k_{\rm F} \simeq 9 \times 10^{-3} ~ h ~ \mathrm{Mpc}^{-1}$, to the Nyquist mode $k_{\rm N} \simeq 2.3 ~ h ~ \mathrm{Mpc}^{-1}$. We employ two interlaced grids in order to suppress the contribution of aliasing arising from Fast Fourier Transform methods and we assign mass particles to the grids following a Triangular Shape Cloud (TSC) algorithm \citep[see, for example,][for an exhaustive treatment of accurate power spectrum estimators and their sources of noise]{SefusattiEtal2016}.
We stress that at $z \simeq 0.5$ the shotnoise accounts for $\sim 30 \%$ of the power spectrum at $k = 2 ~ h ~ \mathrm{Mpc}^{-1}$, while at $z \simeq 2$, at the same scale, it is the dominant contribution, accounting for $\sim 90 \%$ of the spectrum itself.
In the scaled simulation, as detailed in Sec. \ref{sec::method}, besides applying our transformation of lengths and times, we have also corrected large scales according to the linear theory of the target cosmology, and the small scales according to the halo-concentration correction. In the case presented in this figure, we have used the concentration-mass relation of \cite{LudlowEtal2016} (in Appendix \ref{appendix:concentration}, we test the impact of choosing other descriptions).
The ratio between the matter power spectrum measured in the $\Lambda$CDM scaled simulations and in the target neutrino simulations is compatible with $1$, with sub-percent accuracy for all neutrino masses considered, at all scales $k<2 ~ h ~ \mathrm{Mpc}^{-1}$. We highlight that, even though the amplitude and location of the baryonic acoustic oscillations are expected to vary with cosmology and after our length transformation, the ratios show no evidence of any oscillatory feature, confirming the accuracy of our large-scale correction.
Although always within $1\%$, the accuracy of the scaling somehwat degrades with neutrino mass, especially on scales approaching $k\simeq 1 ~ h ~ \mathrm{Mpc}^{-1}$. This originates from neglecting the impact of neutrinos on the concentration-mass relation and on the outer profiles of dark matter halos, aspects which could be improved in the future. Nevertheless, we stress that this level of disagreement is smaller than that induced by numerical and algorithmic choices in $N$-body simulations or than that induced by baryonic physics. For instance, already in the pure $\Lambda$CDM context, different $N$-body codes exhibit a disagreement of the low-redshift matter power spectra at the 1\% level at $k = 1 ~ h ~ \mathrm{Mpc}^{-1}$ \citep{SchneiderEtal2016}. Additionally, baryonic processes such as feedback from massive black holes, gas cooling, and star formation induce modifications to the mass power spectrum at the $10\%$ level for $k > 0.5 h ~ \mathrm{Mpc}^{-1}$. As a consequence, our rescaling procedure appears to be sufficiently accurate for modelling the galaxy population, given the intrinsic uncertainties and approximations usually made in $N$-body simulations.
\subsection{Halo abundance}
Since we based our scaling algorithm on the matching of the linear amplitudes of our $\Lambda$CDM cosmology and neutrino cosmology, we expect to find a very good agreement between the halo mass functions of the scaled $\Lambda$CDM simulation and the simulation run with neutrinos. In Fig. \ref{fig::sim-hsmf} we show that we achieve such agreement with $1\%$ accuracy for halo masses $M_{200} \lesssim 10^{14} ~ h^{-1} ~ \mathrm{M}_\odot$. Larger masses exhibit more scatter, which is within $5\%$ up to $M_{200} \lesssim 10^{15} ~ h^{-1} ~ \mathrm{M}_\odot$, and reaches $10\%$ for $M_{200} \gtrsim 10^{15} ~ h^{-1} ~ \mathrm{M}_\odot$.
However, due to the sparseness of the sample, the behaviour of the high-mass bins is dominated by Poisson noise, which in this case makes up more than 60\% of the signal.
\begin{figure}
\includegraphics[width=0.48\textwidth]{{Plots/scaling_test_lcdm_rad_N1056_L700.0_040_hmf_ratios_backward}.pdf}
\caption{The accuracy of the scaling method for the halo abundance, tested against simulations actually run with massive neutrinos at the target redshift $z_{\rm t} \simeq 0.5$. \textit{Upper panel}: lines show the halo mass function measured in the simulations run in the five neutrino cosmologies, divided by the one measured in the $\Lambda$CDM simulation; points correspond to the ratio between the halo mass function measured in the scaled $\Lambda$CDM simulation and the one measured in the $\Lambda$CDM simulation at the same redshift. \textit{Lower panels}: the ratio between the halo mass function measured in the scaled $\Lambda$CDM simulation and the one measured in the corresponding neutrino simulation. The grey shaded region marks the 5\% level.}
\label{fig::sim-hsmf}
\end{figure}
As an additional test of our rescaling procedure we investigate whether we are able to reproduce the linear halo bias in the presence of neutrinos. We present in Fig. \ref{fig::sim-bias} the linear bias parameter obtained from the cross-power spectrum of the halo and matter fields,
\begin{equation}
b_1(M_{\rm h}) = \left\langle \left. \sqrt{\dfrac{P_{hc}}{P_{cc}}} \right|_{M_{\rm h}\in[M,M+\Delta M)} \right\rangle,
\end{equation}
where $\langle \cdot \rangle$ here denotes an average over the wavemode range $k_{\rm F} < k < 0.1 ~ h ~ \mathrm{Mpc}^{-1}$, $k_{\rm F}$ being the fundamental mode of the box. On this range, the linear bias is roughly constant, also in cosmologies including massive neutrinos (provided that it is defined with respect to the cold component, instead of the total matter field). In fact, slight deviations from scale independence have been observed \citep{ChiangEtal2018}, but are not considered here.
\begin{figure}
\includegraphics[width=0.48\textwidth]{{Plots/compare_hbiasmass_tinker10}.pdf}
\caption{The linear bias parameter, obtained from the cross-power spectrum of the halo and matter fields, $b_1 = \sqrt{P_{hc} / P_{cc}}$, dividing the halo population in different mass bins at redshift $z \simeq 0.5$. Different colours correspond to different neutrino masses, in the usual range $M_\nu = \{0,0.06,0.10,0.15,0.20,0.30\}$ eV. Lines show the predicted bias-mass relation in each cosmology. Open markers refer to the bias-mass relation measured in the target simulations, while filled markers to that measured in the scaled simulations. The shaded areas mark a 1\% accuracy level around the theoretical predictions.}
\label{fig::sim-bias}
\end{figure}
We find that our scaled simulations reproduce the bias mass relations of the simulations actually run including massive neutrinos with a 1\% accuracy in all mass bins. We also present in the same figure the predictions for the bias-mass relation obtained with the model presented in \cite{TinkerEtal2010}, applied, in the case with massive neutrinos, to the cold mass component. We notice that the accuracy of the rescaled simulations is actually higher than the agreement between the model and the measurements, especially in the high mass bins. This is competitive with the results of BE-HaPPY \citep{ValcinEtal2019}, the recently proposed bias emulator that also include a massive neutrino component. However, when the full minimization described in Sec. \ref{sec::method} is performed, our method guarantees the matching of the linear variance on a range of scales (i.e. the scales relevant to halo formation), thus avoiding the potential $\sigma_{8}$ mismatch discussed for this bias emulator.
\subsection{Galaxy clustering}
\begin{figure*}
\includegraphics[width=\textwidth]{{Plots/n_scaling_test_lcdm_rad_N1056_L700.0_003_mock5.00e-03_log_binning1_concdiemer18_ratios_backward}.pdf}
\caption{The accuracy of the scaling method for the monopole (left) and quadrupole (centre) and hexadecapole (right) of the galaxy 2-point correlation function in redshift space, tested against simulations run in the target cosmology. All cosmology share a fixed number density $n = 5 \times 10^{-3} ~ h^3~ \mathrm{Mpc}^{-3}$. \textit{Upper panel}: lines represent the 2-point correlation function multipoles measured in the simulations actually run including massive neutrinos; points correspond to the same quantities measured in the $\Lambda$CDM simulation, scaled to include neutrinos. \textit{Lower panels}: the ratio between the 2-point correlation function multipoles measured in the scaled $\Lambda$CDM simulations and the corresponding simulations actually run with neutrinos. The grey shaded region marks the 5\% level.}
\label{fig::sim-mock}
\end{figure*}
Finally, we investigate the accuracy of the rescaling procedure in terms of galaxy clustering in redshift space. To build our galaxy sample we apply the subhalo abundance matching (SHAM) technique, assuming all galaxies reside in subhaloes, ranked according to one meaningful physical property \citep{ValeOstriker2004,GuoEtal2010}. In our case we select the maximum circular velocity of a subhalo, $v_{\rm max}$, as the physical property tracing the depth of the potential well each subhalo lives in \citep{Trujillo-GomezEtal2011}, neglecting any possible scatter. Please note that ranking according to $v_{\rm max}$ is not, in general, the optimal choice to reproduce realistic stellar mass functions or galaxy correlation functions \cite[see][for a extensive treatment of this matter]{Chaves-MonteroEtal2016}. However, high fidelity to observed data not being the main scope of this work, we use this galaxy sample as realistic enough to test the rescaling procedure.
Once rank-ordered according to $v_{\rm max}$, we retain a number of subhaloes such to obtain two samples, one with fixed galaxy number density of $n_{\rm g} = 5 \times 10^{-3} ~ h^3 ~ \mathrm{Mpc}^{-3}$ and the other with $n_{\rm g} = 5 \times 10^{-4} ~ h^3 ~ \mathrm{Mpc}^{-3}$, irrespective of cosmology. These number densities roughly correspond to those expected in JPAS and SphereX, and in EUCLID and DESI, respectively.
Fig. \ref{fig::sim-mock} shows the monopole, quadrupole, and hexadecapole of the 2-point redshift-space correlation function of our $n_{\rm g} = 5 \times 10^{-4} ~ h^3 ~ \mathrm{Mpc}^{-3}$ sample. We computed these correlation functions with the \texttt{Corrfunc} code \citep{SinhaGarrison2017}.
The upper panels show the multipoles measured both in the scaled $\Lambda$CDM simulations and in the target neutrino simulations, whereas the ratio between them is shown in bottom panels. Once again we find our method to perform at a high accuracy, and we are unable to find any sizeable bias given the statistical uncertainties of our simulations.
Specifically, typical disagreements are a few $\%$ level, which implies that any potential error introduced by the rescaling procedure is well below the error associated with the SHAM technique itself \citep[which is expected to reproduce the galaxy 2-point correlation function with an accuracy of $\sim 10\%$ for scales $R > 2 ~h^{-1}~\mathrm{Mpc}$, as shown in][]{Chaves-MonteroEtal2016}. Although not shown here, we found the same level of agreement for our sparser galaxy sample.
Noticeably, such accuracy is confirmed also for the quadrupole and hexadecapole of the galaxy correlation function, indicating that, while rescaling, the velocity field is properly reconstructed, both in terms of large-scale bulk motion, and of small scale dispersion inside clusters. As a consequence, we conclude that galaxy catalogues can be created for simulations rescaled in arbitrary neutrino cosmologies retaining all the desirable properties linked to the SHAM technique, i.e. its ability to reproduce galaxy velocity bias and assembly bias, while retaining all the internal properties of haloes.
Finally, it is interesting to note the similarity among the clustering for different neutrino masses, this despite the underlying differences in the halo mass function, bias, and halo occupation numbers. We will explore these effects in the future, but here we highlight the importance of modelling the galaxy population to correctly and accurately predict the observable signatures of neutrinos in the Universe.
\subsection{Non-gravitational couplings}
As pointed out in Sec. \ref{sec::method}, if a simulation is started by rescaling a $z=0$ matter power spectrum back to the initial time with a gravity-only growth factor, the matter power spectrum measured at high redshifts will show a mismatch on large scales with respect to the linear solutions of the Einstein-Boltzmann equations. This mismatch is partly due to the lack of residual couplings between the radiation and baryon distributions, owing to a non-zero Compton scattering and to the fact that, if we consider the Boltzmann solutions in the synchronous gauge, we are actually comparing quantities in different gauges.
An interesting use of our scaling algorithm is that, if desired, one can correct for these effects by including them as an additional contribution to the displacement field used in the large scale correction. To test this idea we have run a neutrino simulation (although the same applies to a $\Lambda$CDM simulation) that shares all characteristics with the simulations of our main suite, but includes $N_{\rm part} = 512^3$ particles in a comoving cube of side $L_{\rm box} = 2000 ~ h^{-1} ~ \mathrm{Mpc}$. This choice has been motivated by a balance between the amplitude of the effect (few percent at $k\simeq 10^{-3} ~ h ~ \mathrm{Mpc}^{-1}$ at $z=49$), and the expected shot noise.
\begin{figure}
\includegraphics[width=0.48\textwidth]{{Plots/added_relativistic_corrections}.pdf}
\caption{The implementation of a large scale correction that includes non-gravitational couplings absent in Newtonian N-body codes. Lines show, for different redshifts, the mismatch expected due to neglecting the effect of radiation perturbations on large scales; this is computed as the ratio between the relativistic solution obtained with \texttt{class} and the Newtonian backscaling of a $z=0$ power spectrum performed with \texttt{reps}. Point show the ratio between a simulation rescaled to include these relativistic effects and the original simulation, which does not include them.}
\label{fig::relcorr}
\end{figure}
Fig. \ref{fig::relcorr} shows the ratio between the power spectrum obtained when adding the relativistic correction on large scales and the original one, corresponding to the purely Newtonian evolution of the initial conditions. We compare it with the prediction of the amplitude of this mismatch, computed taking the ratio of the relativistic solutions of \texttt{class} and the Newtonian rescaling of \texttt{reps}. We conclude that this method is able to reproduce the large scale clustering expected when solving the linearised Einstein-Boltzmann equations at high redshift. As expected, for $z<10$, this correction becomes superfluous, because the back-scaling method used to set-up the initial conditions of the simulations guarantees convergence at low redshift.
\subsection{Fitting formulae}
To the end of easing the computation of the length and time scaling parameters necessary for the scaling procedure, we present two empirical parametric fitting functions.
We find that the time and scale transformations mainly depend only on the total matter density parameter and on the neutrino density parameter, $\Omega_{\rm m,0}$ and $\Omega_{\nu,0}$ respectively. As in the rest of this work, we keep constant the total matter density parameter among the massive neutrino cosmology and the $\Lambda$CDM one.
As a result, in the case we want to add a neutrino component with $\Omega_{\nu,0} = M_\nu/(93.14 h^2)$ to a simulation run assuming a given $\Omega_{m,0}$ we will need to expand lengths by the factor given by
\begin{equation}
s = \alpha ~ \Omega_{\nu,0} + \beta ~ \Omega_{\rm m, 0} + \gamma ~ \Omega_{\rm m,0} \Omega_{\nu,0} + \delta,
\end{equation}
where $\alpha, \beta, \gamma$ and $\delta$ are free parameters, whose bestfitting values we report in Table \ref{tab::nu-recipe}.
This result does not depend on the target time we desire to reproduce in the massive neutrino cosmology of choice.
Moreover, we also have to consider a earlier snapshot of our $\Lambda$CDM simulation. We employ the same functional form as for the length rescaling, with different parameters (namely $\varepsilon, \zeta, \eta, \vartheta$). However, in this case, which previous output to consider depends on the target time we want to reproduce in the neutrino cosmology,
\begin{equation}
a^*(a_{\rm target}) = \varepsilon ~ \Omega_{\nu,0} + \zeta ~ \Omega_{\rm m, 0} + \eta ~ \Omega_{\rm m,0} \Omega_{\nu,0} + \vartheta,
\end{equation}
where $a^*$ and $a_{\rm target}$ are the expansion factor selected by the minimization procedure and the expansion factor at the target time respectively.
As a useful example, we present the fitting formula calibrated assuming four significant values of the target expansion factor, corresponding to the target redshifts $z_{\rm t} = \{0, 0.5, 1.0, 2.0\}$.
The results of the fitting procedures are presented in Table \ref{tab::nu-recipe}.
\begin{table}
\begin{tabular}{l c c c c}
\hline
& $\alpha$ & $\beta$ & $\gamma$ & $\delta$ \\
$z_{\rm t} = 0.0$ & $12.48094$ & $0.01508$ & $-0.24582$ & $0.99361$ \\
\hline
& $\varepsilon$ & $\zeta$ & $\eta$ & $\vartheta$ \\
$z_{\rm t} = 0.0$ & $-45.20044$ & $0.11578$ & $0.81595$ & $0.95602$ \\
$z_{\rm t} = 0.5$ & $-22.95795$ & $0.02504$ & $0.41368$ & $0.65882$ \\
$z_{\rm t} = 1.0$ & $-14.82997$ & $0.00763$ & $0.26286$ & $0.49853$ \\
$z_{\rm t} = 2.0$ & $-8.61413$ & $0.00118$ & $0.15002$ & $0.33387$ \\
\hline
\end{tabular}
\caption{Fitted values of the parameters of the empirical functions that allow to quickly reproduce the length scaling parameter and the expansion factor in the original $\Lambda$CDM cosmology corresponding to a target cosmology with arbitrary neutrino mass. The function does not depend on the choice of target redshift for the length scaling parameter (an arbitrary value of $z_{\rm t} = 0$ is reported here), while it does for the expansion factor.}
\label{tab::nu-recipe}
\end{table}
\begin{figure*}
\includegraphics[width=\textwidth]{{Plots/scale_mnu_recipe_atargets}.pdf}
\caption{The dependence of the scaling parameters on cosmology, for the case in which the only cosmological parameter changed is the total neutrino mass. The larger the neutrino mass, the larger the simulation box size must become, following the relation shown in the left panel. This result does not depend on redshift. On the contrary, the time scaling does depend on the target redshift at which we want to reconstruct the neutrino cosmology. In general, given a redshift of the target neutrino cosmology, the corresponding $\Lambda$CDM simulation output will always be at an earlier time. In the specific case presented here, the target redshift considered are $z_{\rm t}=\{ 0, 0.5, 1, 2 \}$.}
\label{fig::recipe}
\end{figure*}
\section{Conclusions}\label{sec::conclusion}
In this work, we have presented an extension of the cosmology scaling algorithm introduced in \cite{AnguloWhite2010} in order to include the effects of massive neutrinos. In particular, employing this method, it is possible to rescale a $\Lambda$CDM simulation, run without including massive neutrinos, to any neutrino cosmology.
The cosmology rescaling algorithm relies on the matching (on a range of scales) of the linear amplitude of fluctuations of the target cosmology at a given redshift with that of the original cosmology, at a redshift and with a length transformation to be determined. By matching the variance, in the Press-Schechter formalism, the halo abundance is automatically reconstructed as the one of the target cosmology. However, more care is needed to accurately reconstruct the clustering properties of the matter, halo and galaxy fields. In the general argument, in particular, one has to correct for different large-scale evolution (including the position of the BAO peak), for differences in the small-scale nonlinear velocities (due to the different growth history), and for different halo concentrations (due to differences in the formation time of haloes).
In the specific massive neutrino case, there are some additional aspects that need to be taken care of. First of all, the growth factor and growth rate of matter perturbations exhibit a peculiar scale dependence induced by the phenomenon of neutrino free streaming. Such scale dependence, moreover, evolves in amplitude and shape, being more pronounced at later redshifts. Secondly, the relevant quantity to be matched between $\Lambda$CDM and neutrino cosmology is no longer the total mass linear variance, but the cold matter linear variance. This is due to the neutrino free streaming scale being larger than typical halo sizes at all times, thus preventing neutrinos from playing any significant role in the context of halo collapse. Lastly, by correcting the large scale modes of the matter distribution, the correct neutrino scale dependence is recovered, with the expected, neutrino-mass dependent large-to-small scale ratio.
By comparing a $\Lambda$CDM simulation scaled to reproduce neutrino cosmologies with total masses $M_\nu = \{0.06,0.10,0.15,0.20,0.30\}$ eV to their corresponding simulations, run actually including massive neutrinos, we showed that we are able to reproduce the target matter power spectrum with 1\% accuracy on all scales $k<2~h~\mathrm{Mpc}^{-1}$, for all the choices of neutrino mass. Extending the comparison to halo abundance we verify that the halo mass function is reconstructed with better than 5\% accuracy.
Finally, also the redshift-space 2-point correlation function monopole and quadrupole of a galaxy population obtained with a SHAM technique are reconstructed in the target cosmology with a mismatch lower than 5\% with respect to the actual neutrino simulations.
This method is in principle applicable to any $\Lambda$CDM simulation, which is then converted into a neutrino simulation in a quick way and inheriting the, potentially high, accuracy.
As a consequence, the cosmology rescaling method opens up the possibility of performing analyses of cosmological datasets directly against simulations. As a matter of fact, considerable amount of information is encoded in the physical processes at play at the cluster or galaxy scale, where the evolution of the matter and galaxy fields is highly nonlinear. Besides improving models, another method is to consider the simulations themselves \textit{as the model}. Together with physically motivated galaxy formation models, this provides the most accurate predictions for LSS analyses of forthcoming surveys. Thanks to the work here presented, such analysis can also include (and constrain) the effect of massive neutrinos.
\section*{Acknowledgements}
We acknowledge the support of the Spanish MINECO and of the European Research Council through grant numbers \textit{AYA2015-66211-C2-2} and \textit{ERC-StG/716151}.
\bibliographystyle{mnras}
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{
"redpajama_set_name": "RedPajamaArXiv"
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Bailey and Barclay Halls at Urbana University are on the National Historic Registry and have an Ohio Historical Marker. The namesakes of the two historic buildings connect the University directly back to the American Revolution. This project was a combination addition and renovation/restoration of these important original buildings.
The university desired to bring both buildings back to original condition, while adapting them to the needs of the 21st Century University classrooms and offices. Additionally, Barclay Hall housed the Johnny Appleseed Museum, a tribute to John Chapman, a key leader in the founding of Urbana University. Both structures were also built in the Italianate style, and were designed by William Russell West, architect of the Ohio Statehouse.
Kapp Construction was presented with the Excellence in Construction Award for this project by the Associated Builders and Contractors, Ohio Valley Chapter, their highest recognition for a restorative project.
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"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}\label{Sec:Intro}
The Higgs sector in the Minimal Supersymmetric Standard Model (MSSM)
consists of two $SU(2)_L$ doublets, $H_d$ and $H_u$, with opposite
hypercharge. Five physical states remain after electroweak symmetry
breaking (EWSB). Assuming there are no CP-violating phases, these
physical states consist of two neutral CP-even states $h$ and $H$
with masses $m_h \le m_H$, one neutral CP-odd state $A$, and two
charged states $H^{\pm}$. The tree-level masses of $h$ and $H$ are
bounded, $m^{\textrm{\tiny{tree}}}_h \le m_Z \le
m^{\textrm{\tiny{tree}}}_H$, with $m_Z\simeq 91$ GeV.
Using the Large Electron Positron (LEP) collider, the LEP
collaboration searched for these Higgs bosons and published bounds
on their masses \cite{ALEPH:2006cr}. Their results have ruled out
substantial regions of the MSSM Higgs parameter space, and in much
of the remaining allowed regions it is clear that large radiative
corrections to the tree-level Higgs masses are required to satisfy
the LEP bounds. However, two very different scenarios are still
possible. One scenario is obtained in the Higgs decoupling limit in
which $h$ behaves like the Standard Model (SM) Higgs and all the
other Higgs bosons become heavy and decouple from the low energy
theory. Here the bound on $m_h$ coincides with the bound on the mass
of the SM Higgs, namely $m_h \ge 114.4$~GeV \cite{Barate:2003sz}.
The other scenario is obtained in the Higgs ``non-decoupling'' limit
in which $H$ behaves like the SM Higgs and the Higgs sector is
required to be light. It allows for 93 GeV $\lesssim m_h <$ 114.4
GeV, where the value of 93 GeV is the (somewhat model dependent)
lower bound that the LEP collaboration has obtained for $m_h$
assuming various decay scenarios for $h$ and a variety of different
``benchmark'' parameter choices for the MSSM parameters. If $m_h$ is
near 93 GeV, it seems to naively require much smaller radiative
corrections to the tree-level Higgs mass than when $m_h$ is near
114.4~GeV. Since the dominant radiative corrections to the
tree-level CP-even Higgs mass matrix, which determines $m_h$ and
$m_H$, come from loops involving the top quark and stop squarks, one
might naively suspect that $m_h$ near 93 GeV allows for much smaller
stop masses than $m_h$ near 114.4 GeV. Moreover, larger stop masses
would in general imply a more fine-tuned MSSM, and one might
therefore suspect that the MSSM is less fine-tuned for $m_h$ near 93
GeV. In this paper, we present lower bounds on the stop masses
consistent with the LEP Higgs bounds, both in the Higgs decoupling
region, with $m_h \ge 114.4$ GeV, as well as near the Higgs
non-decoupling region, with $m_h \simeq 93$ GeV. We compare the
constraints on the stop masses in these two regions of the Higgs
parameter space, and show that in certain regions of the MSSM
parameter space the lower bounds on the stop masses are not
significantly different from each other. Furthermore, although there
are regions in which the lower bounds are smaller, there are also
regions in which they are larger.
There are other constraints on new physics that may further tighten
bounds on the stop or Higgs sector. These additional constraints
include the electroweak $S$- and $T$-parameter, and the decays
$\bsg$ and $\Bmumu$. In this paper, we discuss the regions of the
MSSM parameter space in which these additional constraints are
important in restricting the stop and/or Higgs sector further.
The outline of this paper is as follows. In Section \ref{Sec:LEP
and MSSM Higgs sector}, we investigate the LEP constraints on the
neutral MSSM Higgs sector and its implication for the stop sector in
more detail. This will allow us to obtain a simple numerical
estimate of the lower bound on the stop masses in a particular limit
of the MSSM parameter space. This estimate is independent of the
size of $m_h$. Section \ref{Sec:Lower bounds on MSusy} contains the
main results of this paper. We give lower bounds on the stop masses
consistent with the LEP Higgs bounds. The analysis will include all
the important radiative corrections to the CP-even Higgs mass
matrix, and we discuss the importance of the top mass, the stop
mixing the gaugino masses and the supersymmetric Higgsino mass
parameter ($\mu$) on the lower bounds of the stop masses. In
addition, we present results on how the lower bounds on the soft
stop masses vary in the decoupling limit as a function of $m_h$. In
Section \ref{Sec:Constraints from new physics}, we discuss how other
constraints on new physics impact the results in Section
\ref{Sec:Lower bounds on MSusy}. In particular, we investigate the
effect of the electroweak $S$- and $T$-parameters, $\bsg$, and
$\Bmumu$. Section \ref{Sec:Implications for EWSB} contains a
discussion of the implications of our analysis for electroweak
symmetry breaking and the supersymmetric little hierarchy problem.
In Section \ref{Sec:Conclusion}, we summarize the results of this
paper. Appendix \ref{Sec:MSSM Higgs sector and LEP} gives the
relevant background to understand the LEP results for the MSSM Higgs
sector. In Appendix \ref{Sec:At fixed point}, we review the
quasi-fixed point for the stop soft trilinear coupling, $A_t$. The
trilinear coupling is the main ingredient in determining the amount
of stop mixing, and we use the quasi-fixed point value for $A_t$ in
some of the main results of this paper.
\section{LEP constraints on the Higgs sector and implications for the MSSM stop sector}\label{Sec:LEP and MSSM Higgs
sector}
In this section, we first review the LEP Higgs constraints, before
going on to discuss the implications of these constraints for the
stop sector.
\subsection{Constraints from LEP on the MSSM Higgs-sector}
The LEP collaboration searched for the production of Higgs bosons in
both the Higgsstrahlung ($e^+e^-$ $\rightarrow$ $Z$ $\rightarrow$
$Zh$ (or $ZH$)) and pair production ($e^+e^-$ $\rightarrow$ $Z$
$\rightarrow$ $Ah$ (or $AH$)) channels. The results from these
channels have been used to set upper bounds on the couplings $ZZh$
($ZZH$) and $ZAh$ ($ZAH$) as a function of the Higgs masses. These
couplings are proportional to either $\sin^2(\beta-\alpha)$ or
$\cos^2(\beta-\alpha)$. Here $\beta$ is determined from the ratio
of the two vacuum expectation values $v_u\equiv\langle{\rm Re}
(H_u^0)\rangle$ and $v_d\equiv\langle{\rm Re} (H_d^0)\rangle$ as
$\tan\beta = v_u/v_d$, and $\alpha$ is the neutral CP-even Higgs
mixing angle.
Within the MSSM, the results from the Higgsstrahlung channel give an
upper bound on $\sin^2(\beta-\alpha)$ and $\cossqbma$ as a function
of $m_h$ and $m_H$, respectively (see Fig.~2 in
\cite{ALEPH:2006cr}). The pair production channel, on the other
hand, gives an upper bound on $\cos^2(\beta-\alpha)$ and $\sinsqbma$
as a function of $m_h + m_A$ and $m_H + m_A$, respectively (see
Fig.~4 in \cite{ALEPH:2006cr}). Appendix \ref{Sec:MSSM Higgs sector
and LEP} contains a review on how these functions of $\alpha$ and
$\beta$ appear in the MSSM, and why LEP bounds them.
The LEP results from the Higgsstrahlung channel put several
interesting bounds on $m_h$, $m_H$ and $\sinsqbma$. In the
decoupling limit, $h$ behaves like the SM Higgs so that $\sinsqbma
\rightarrow 1$ and the bound on its mass is given by
\begin{equation}\label{Eqn:lower Region A} m_h \ge 114.4 {\textrm{
GeV}},~\sinsqbma \rightarrow 1. \end{equation}
If $m_h$ is less than 114.4 GeV, smaller values of $\sinsqbma$ are
required in order to suppress the production of $h$ in the
Higgsstrahlung channel and to allow it to have escaped detection.
For $m_h$ $\simeq$ 93 GeV, $\sinsqbma$ needs to be less than about
0.2, so that $\cossqbma \gtrsim 0.8$ (from Fig.~2 in
\cite{ALEPH:2006cr}). Larger values of $\cossqbma$, however,
increase the $HZZ$ coupling so that now $m_H$ needs to be large
enough to suppress the production of $H$ in the Higgsstrahlung
channel, and allow it, in turn, to have escaped detection. We find
that $m_H \gtrsim 114.0$ GeV (from Fig.~2 in \cite{ALEPH:2006cr}).
If $\sinsqbma$ is even smaller and approaches zero, i.e.~$\cossqbma
\rightarrow 1$, it is $H$ which behaves like the SM Higgs so that
the bound on its mass is given by $m_H$ $\ge$ 114.4 GeV (this will
be referred to as the Higgs ``non-decoupling'' limit).
In Section \ref{Sec:Lower bounds on MSusy}, we present lower bounds
on the stop soft masses for two regions in the Higgs parameter
space. These two regions are given by equation (\ref{Eqn:lower
Region A}) and by
\begin{equation} \label{Eqn:lower Region B}
m_h \simeq 93 {\textrm{ GeV}}, ~\cossqbma \ge 0.8, ~m_H \ge 114.4
\textrm{ GeV}.
\end{equation}
\noindent (We choose $m_H$ in equation (\ref{Eqn:lower Region B}) to
be at least above 114.4 GeV, in order
to allow the full range $0.8 \le \cossqbma \le 1$.)
We note that the bounds given in the previous paragraphs assume that
the MSSM Higgs boson $h$ decays like the SM Higgs boson (see
\cite{ALEPH:2006cr}). If we assume different Higgs decay branching
ratios, somewhat different bounds can be obtained. For example,
assuming $h$ decays completely into $\tau\bar{\tau}$ gives a
stricter bound on the $hZZ$ coupling for a wide range of $m_h$. The
LEP collaboration even considered the extreme case in which the
Higgs decays invisibly. In this case, the bound on the $hZZ$
coupling as a function of $m_h$ is in general not much worse than if
we assume that the Higgs decays like a SM Higgs. In fact, for some
range of $m_h$ the bound is even stricter if we assume that the
Higgs decays invisibly \cite{LHWG:2001xz}.
The lower bound on $m_h$ is also model dependent. For example, in
\cite{ALEPH:2006cr}, figures are presented that show excluded
regions in the MSSM parameter space for a variety of ``benchmark''
scenarios that consist of different choices for the MSSM parameters.
The LEP collaboration found that the lower bound on $m_h$ can be
slightly less than 93 GeV in certain cases. Moreover, the authors in
\cite{Belyaev:2006rf} claim that there are certain regions in
parameter space for which the $ZZh$ coupling and the $h/A\rightarrow
b\bar{b}$ branching ratios are both suppressed and that this allows
$m_h$ to be substantially less than 93 GeV.
\subsection{Implications for the MSSM stop sector}\label{Sec:Implication for MSSM stop sector}
Since the tree-level mass of the lighter neutral Higgs is bounded
above by $m_Z$, it is clear that substantial radiative corrections
are required to push the lighter Higgs mass above 114.4 GeV in the
Higgs decoupling limit.
We now discuss why substantial radiative corrections to the
tree-level Higgs masses are also required if the lighter Higgs mass
is near 93 GeV. This is most easily seen in the large $\tan\beta$
limit. In this limit, the CP-even Higgs mass squared matrix in the
($H_d$,$H_u$) basis is particularly simple if we only include the
tree-level pieces and the dominant radiative corrections. Since
$\tan\beta$ is large, the vacuum expectation value $v_d$ vanishes in
this limit, and the Higgs vacuum expectation value is thus
completely determined by $v_u$. In the absence of any radiative
corrections, one of the physical Higgs mass eigenstates lies
completely in the $H_u$ direction and thus behaves like the SM Higgs
(with a mass equal to $m_Z$), whereas the other mass eigenstate lies
completely in the $H_d$ direction (with a mass equal to the mass of
the CP-odd Higgs, $m_A$). This alignment of the two physical CP-even
Higgs mass eigenstates with the $H_u$ and $H_d$ direction,
respectively, remains unchanged when only the dominant radiative
correction is added. The reason for this is that, due to the large
top Yukawa coupling, the dominant radiative corrections to the Higgs
sector are to the up-type Higgs soft supersymmetry breaking
Lagrangian mass and come from loops involving the top quark and stop
squarks \cite{Okada:1990vk,Ellis:1990nz,Haber:1990aw}. This gives a
correction to the $H_u$-$H_u$ component of the CP-even Higgs mass
squared matrix. The matrix is thus particularly simple for large
$\tan\beta$, and is given by
\begin{equation}
\mathcal{M}^2 \approx \left( \begin{array}{cc} m_A^2 & 0 \\
0 & m_Z^2 + \delta \mathcal{M}^2_{uu}
\end{array} \right) ~~~~~~~({\textrm{for large}}~ \tan\beta),
\end{equation}
\noindent where $\delta \mathcal{M}^2_{uu}$ is the dominant top/stop
correction. Since the $H_u$-$H_u$ component for large $\tan\beta$
gives the mass of the physical Higgs that behaves like the SM Higgs,
its value is bounded below by 114.4 GeV, i.e.
\begin{equation}\label{Eqn:Size of delta M2uu}
m_Z^2 + \delta \mathcal{M}^2_{uu} \gtrsim (114.4 \textrm{ GeV})^2.
\end{equation}
\noindent The result in equation (\ref{Eqn:Size of delta M2uu}) is
independent of whether the lighter or the heavier Higgs lies in the
$H_u$ direction (this depends on the size of $m_A$). It also shows
that the lower bound on the size of the required radiative
corrections is fixed and independent of the mass of the lighter
Higgs, at least in the large $\tan\beta$ limit including only the
leading corrections. Moreover, it is $m_H$ which acquires the
dominant radiative corrections for $m_h\simeq$ 93 GeV.
A simple estimate of the lower bounds on the stop masses in the
large $\tan\beta$ limit may be obtained using equation
(\ref{Eqn:Size of delta M2uu}). For large $\tan\beta$, the dominant
radiative correction is given by
\begin{equation}\label{Eqn:delta_M_uu}
\delta\mathcal{M}^2_{uu} \simeq \frac{3 g^2 m_t^4}{8\pi^2 m_W^2}
\Bigg\{\ln\Bigg(\frac{\MSusy^2}{m_t^2}\Bigg) +
\frac{X_t^2}{\MSusy^2}\Bigg(1-\frac{X_t^2}{12
\MSusy^2}\Bigg)\Bigg\},
\end{equation}
\noindent where $m_t$ is the top mass, $g$ is the $SU(2)_L$ gauge
coupling, and $m_W$ is the mass of the $W$-bosons
\cite{Okada:1990vk,Ellis:1990nz,Haber:1990aw}. Furthermore, equation
(\ref{Eqn:delta_M_uu}) assumes that the stop soft masses are equal
to $\MSusy$, with $\MSusy$ $\gg$ $m_t$. The stop mixing parameter is
given by $X_t=A_t - \mu\cot\beta$ ($\simeq$ $A_t$ for large
$\tan\beta$), where $A_t$ denotes the stop soft trilinear coupling
and $\mu$ is the supersymmetric Higgsino mass parameter. The
dependence on the top mass to the fourth power is particularly
noteworthy. The first term in equation (\ref{Eqn:delta_M_uu}) comes
from renormalization group running of the Higgs quartic coupling
below the stop mass scale and vanishes in the limit of exact
supersymmetry. It grows logarithmically with the stop mass. The
second term is only present for non-zero stop mixing and comes from
a finite threshold correction to the Higgs quartic coupling at the
stop mass scale. It is independent of the stop mass for fixed
$X_t/\MSusy$, and grows linearly as $(X_t/\MSusy)^2$ for small
$X_t/\MSusy$.
It is apparent from equation (\ref{Eqn:delta_M_uu}) that the mixing
term is important for determining lower bounds on the stop masses.
Using equation (\ref{Eqn:delta_M_uu}) and assuming no mixing
($X_t=0$), we require $\MSusy \gtrsim 570$ GeV in order to satisfy
the LEP bound in (\ref{Eqn:Size of delta M2uu}). This value was
obtained using a running top mass of $m_t(m_t) \simeq 167$ GeV
\cite{Haber:1996fp}. The second (mixing) term in equation
(\ref{Eqn:delta_M_uu}), however, reaches a maximum of 3 for $X_t =
\sqrt{6}\MSusy$, called \emph{maximal-mixing}. In order for the
logarithm of the first term to be of the same order, $\MSusy$ needs
to be about 750 GeV. Thus the mixing term alone is more than enough
to give the required radiative corrections to satisfy the LEP bound.
Mixing in the stop sector therefore allows for much smaller stop
masses.
There are other radiative corrections to the Higgs masses which are
important, including negative radiative corrections that come from
charginos, for example. In Section \ref{Sec:Lower bounds on MSusy},
we include all the important radiative corrections to determine more
accurate lower bounds on the stop masses. For example, for no stop
mixing and $\tan\beta = 50$, a more accurate lower bound is given by
$\MSusy \gtrsim$ 980 GeV, assuming a physical top mass of 173 GeV,
$\mu = 200$ GeV, and a bino and wino mass of 100 GeV and 200 GeV,
respectively. This shows the importance of including higher order
corrections to the Higgs sector. Moreover, the lower bound is
approximately the same for $m_h\simeq 93$ GeV and for $m_h\ge 114.4$
GeV, as expected for large $\tan\beta$.
The above discussion assumes that $\tan\beta$ is large. In Section
\ref{Sec:Lower bounds on MSusy}, we obtain lower bounds on $\MSusy$
also for small and moderate values of $\tan\beta$, for which the
off-diagonal elements in the Higgs mass matrix become important. In
general, we find that the stop masses and/or mixing have to be
sizeable for all values of $\tan\beta$ and for both the Higgs
decoupling and non-decoupling regions. However, depending on the
size of the stop mixing, the lower bounds on the stop masses for
moderate values of $\tan\beta$ can be smaller for $m_h \simeq 93$
GeV than for $m_h \ge 114.4$ GeV (see also \cite{Kim:2006mb}).
Moreover, for small values of $\tan\beta$, the lower bounds on the
stop masses become larger for $m_h \simeq 93$~GeV than for $m_h \ge
114.4$~GeV.
\section{Lower bounds on the stop masses}\label{Sec:Lower bounds on MSusy}
In this section, we present lower bounds on the stop masses
consistent with the LEP Higgs bounds, and we discuss their
dependence on some of the other MSSM parameters. In particular, we
set lower bounds on the left-handed and right-handed stop soft mass,
$m_{\tilde{t}_L}$ and $m_{\tilde{t}_R}$, respectively, taking both
equal to a common value, which we denote by $m_{\tilde{t}}$. We
denote the lower bound on $\MSusy$ consistent with the LEP Higgs
bounds by $\MSusyMin$. We consider the two scenarios given in
equations (\ref{Eqn:lower Region A}) and (\ref{Eqn:lower Region B}),
namely the Higgs decoupling limit with $m_h \ge 114.4$ GeV (Section
\ref{Sec:m_h_greater_114_GeV}), and near the Higgs non-decoupling
limit with $m_h\simeq 93$ GeV, and the additional constraints
$\cossqbma\ge 0.8$ and $m_H \ge 114.4$ GeV (Section
\ref{Sec:m_h_simeq_93_GeV}). In addition, in Section
\ref{Sec:various mh}, we give lower bounds on the stop soft masses
as a function of the physical Higgs boson mass $m_h$ in the
decoupling limit. All the lower bounds on the stop masses that we
present are consistent with the 2$\sigma$ constraint on $\delta\rho$
(which is related to the electroweak $T$-parameter). In Section
\ref{Sec:Constraints from new physics}, we discuss the importance of
this parameter, as well as others, in constraining the stop masses.
\subsection{Lower bounds on the stop masses for $m_h \ge 114.4$ GeV} \label{Sec:m_h_greater_114_GeV}
For a given set of parameters, we minimize $\MSusy$ by starting it
at the lowest value that gives physical stop masses above 100 GeV
and increasing it until $m_h$ is above 114.4 GeV. We choose the
physical stop mass to be at least 100 GeV since this bound is
illustrative of the actual, slightly model dependent, lower bound
obtained from the Tevatron \cite{PDBook}. The Higgs masses were
calculated with version 2.2.7 of the program
\href{http://www.feynhiggs.de}{$\mathtt{FeynHiggs}$} which includes
all the important radiative corrections to the Higgs sector
\cite{Heinemeyer:1998yj,Heinemeyer:1998np,Degrassi:2002fi,Frank:2006yh}.
We set $m_A = 1000$ GeV to ensure that we are in the Higgs
decoupling limit.
In Fig.~\ref{Fig:M_S_vs_TB_114_mtop173_all_mixings}, we show
$\MSusyMin$ as a function of $\tan\beta$ for stop mixing $X_t/
m_{\tilde{t}}$ $=$ $0$, $\pm 1$, and $\pm 2$. All squark, slepton,
and gaugino soft masses are equal to $\MSusy$, $\mu=200$ GeV, $m_t =
173$ GeV, and all the soft trilinear couplings are equal to the stop
soft trilinear coupling, $A_t=X_t+\mu\cot\beta$. The lower solid
line shows the \emph{maximal-mixing} scenario,\footnote{The word
``maximal'' refers to the size of the radiative corrections, not to
the amount of mixing. Maximal mixing in $\mathtt{FeynHiggs}$ is
obtained by setting $X_t \simeq 2 \MSusy$, and not $X_t = \sqrt{6}
\MSusy$ as in Section \ref{Sec:Implication for MSSM stop sector}. In
the former case, $X_t$ is defined in the on-shell scheme used in the
diagrammatic two-loop results incorporated into
$\mathtt{FeynHiggs}$, whereas in the latter it is defined in the
$\overline{MS}$-scheme used in the RG approach. Moreover, $m_h$ is
not symmetric with respect to $X_t$ in the full two-loop
diagrammatic calculation in the on-shell scheme. For example, $m_h$
can be up to 5 GeV larger for $X_t=+2 \MSusy$ than for $X_t = -2
\MSusy$. The difference arises from non-logarithmic two-loop
contributions to $m_h$, see \cite{Carena:2000dp,Heinemeyer:2004ms}.}
$X_t = 2 \MSusy$, which approximately maximizes the radiative
corrections to the Higgs sector for a given set of parameters
\cite{Heinemeyer:1999be}. The dot-dashed line shows the
\emph{no-mixing} scenario, $X_t=0$, which approximately minimizes
the radiative corrections to the Higgs sector for a given set of
parameters. The lower dashed line shows the results for
$X_t=\MSusy$. An \emph{intermediate-mixing} scenario with
$X_t=-\MSusy$ is represented by the upper dashed line. We choose
this scenario since $A_t$ has a strongly attractive infrared
quasi-fixed point at $A_t$ = $-M_3$, see Appendix \ref{Sec:At fixed
point}. Thus, $A_t$ prefers to be negative due to renormalization
group evolution from the high scale down to the low scale (we choose
the convention in which $M_3$ is positive). In addition, we consider
a scenario which maximizes the Higgs mass for negative stop mixing,
and call it \emph{natural maximal mixing}. This scenario is given by
$X_t = -2 \MSusy$ and is represented by the upper solid-line in the
figure.
A feature that is common to all the curves is that $\MSusy$ becomes
very large for small $\tan\beta$. This is because the tree-level
contribution to the Higgs mass in the decoupling limit is given by
$m_h^{\textrm{\tiny{tree}}}$ $\simeq$ $|\!\cos 2\beta|m_Z$, and goes
to zero as $\tan\beta$ approaches 1. Larger radiative corrections,
and thus larger stop masses, are therefore required for smaller
$\tan\beta$ to push $m_h$ above 114.4 GeV.
Fig.~\ref{Fig:M_S_vs_TB_114_mtop173_all_mixings} clearly shows that
mixing in the stop sector has a large impact on the values of
$\MSusyMin$, with larger mixing allowing much smaller values of
$\MSusyMin$ (see also \cite{Kitano:2006gv,Dermisek:2007fi}). For
large $\tan\beta$, the difference in $\MSusyMin$ between no mixing
and maximal mixing is about 1000 GeV, with $\MSusyMin=1260$ GeV for
$\tan\beta=50$ in the no-mixing case.
A plot of the two \emph{physical} stop masses, $m_{\tilde{t}_1}$ and
$m_{\tilde{t}_2}$, versus $\tan\beta$ is given in
Fig.~\ref{Fig:stopmasses_vs_TB_mtop173_various_mixings} for no
mixing and for natural maximal mixing. For no mixing, there is no
discernible difference in the two stop masses since the only
difference that arises is from small $SU(2)_L$ and $U(1)_Y$ D-term
quartic interactions. For appreciable mixing, the two physical stop
masses are split by an amount that is on the order of $\sim\sqrt{m_t
X_t}$. For $X_t = -2 \MSusy$ and $\tan\beta \ge 7$, $\MSusyMin$ is
small enough that the lighter physical stop mass is all the way down
at its experimental lower bound of roughly 100 GeV. For this range
of $\tan\beta$, we find that $\MSusy$ is larger than that which is
required to get $m_h$ just above 114.4 GeV, and thus $m_h$ is
several GeV above 114.4 GeV here.
The current value of the top mass from the CDF and D0 experiments at
Fermilab is $m_t = 171.4 \pm 2.1$ GeV \cite{Group:2006xn}. The
values obtained for $\MSusyMin$ are, however, extremely sensitive to
slight variations in the value of the top mass (see also
\cite{Kitano:2006gv}). It is thus illustrative to plot $\MSusyMin$
as a function of $\tan\beta$ for various amounts of stop mixing and
for three choices of the top mass: 168 GeV, 173 GeV and 178 GeV. The
plots are shown in Fig.~\ref{Fig:M_S_vs_TB_Xt_0_all_mtops},
\ref{Fig:M_S_vs_TB_Xt_minus1_all_mtops} and
\ref{Fig:M_S_vs_TB_Xt_minus2_all_mtops} for $X_t/\MSusy$ = 0, -1,
and -2, respectively. These plots again assume that all squark,
slepton, and gaugino soft masses are equal to $\MSusy$, $\mu=200$
GeV, and all the soft trilinear couplings are equal to $A_t$.
All three figures show that $\MSusyMin$ is extremely sensitive to
small changes in $m_t$ for small $\tan\beta$. For intermediate and
vanishing stop mixing, this sensitivity persists for large
$\tan\beta$. For example, in the no-mixing case for $\tan\beta=50$,
we find $\MSusyMin\simeq$ 870 GeV, 1260 GeV, and 2570 GeV for $m_t =
$ 178 GeV, 173 GeV, and 168 GeV, respectively. The very large value
of $\MSusyMin$ for $m_t = 168$ GeV is particularly noteworthy,
especially if the central value of the measured top mass keeps
decreasing slightly as more data from the Tevatron becomes
available.
So far we assumed that the gaugino masses are all equal to $\MSusy$.
The bino and wino masses, $M_1$ and $M_2$, as well as $\mu$
contribute to the Higgs masses at one loop, whereas the gluino mass,
$M_3$, only appears at two loops (see e.g.
\cite{Haber:1996fp,Heinemeyer:1998np,Haber:1993an} and references
therein). Since large values of $M_1$, $M_2$ and $\mu$ can give
important \emph{negative} contributions to the Higgs masses
\cite{Carena:1999xa}, smaller values of $\MSusyMin$
are possible for smaller values of $M_1$, $M_2$ and $\mu$. For
example, setting $M_1$ = 100 GeV, $M_2$ = 200 GeV and $M_3$ = 800
GeV, we find in the no-mixing case for $\tan\beta=50$ that
$\MSusyMin\simeq$ 760 GeV, 980 GeV, and 1410 GeV for $m_t = $ 178 GeV,
173 GeV and 168 GeV, respectively. This may be compared with the values given in the
previous paragraph for the case where all the gaugino masses are
equal to $\MSusy$. Thus, setting the bino and wino masses to smaller values
decreases the size of $\MSusyMin$, especially if $m_t$ is
small. However, the large value of $\MSusyMin$ for $m_t =
168$ GeV is still noteworthy.
We show a further example of how a different choice for $M_1$ and
$M_2$ affects $\MSusyMin$ in
Fig.~\ref{Fig:M_S_vs_TB_different_gaug_M1M2M3_Xt_0_and_minus2} for
the no-mixing and natural-maximal-mixing scenario. For each
scenario, this figure shows a case for which $M_1$ and $M_2$ are
both large ($M_1 = M_2 = $ 800 GeV) or both small ($M_1$ = 100 GeV,
$M_2$ = 200 GeV). In both cases, $M_3$ is fixed to be 800 GeV, $\mu
= 200$ GeV, $m_t = 173$ GeV, all squark and slepton soft masses are
equal to the stop soft masses, and all the soft trilinear couplings
are equal to $A_t$. The plots show that $\MSusyMin$ is smaller for
smaller values of $M_1$ and $M_2$. For example, $\MSusyMin$ is
about 160 GeV smaller in the no-mixing case for $\tan\beta=50$ when
choosing the smaller set of values for $M_1$ and $M_2$, and the
difference in $\MSusyMin$ grows as $\tan\beta$ decreases. For
natural maximal mixing, no difference can be seen for most
$\tan\beta$ values, since here the condition $m_{\tilde{t}_1} \ge
100$ GeV again requires larger values of $\MSusyMin$ than the
condition $m_h \ge 114.4$ GeV. However, there is a difference in
$\MSusyMin$ for smaller $\tan\beta$, which again grows as
$\tan\beta$ decreases.
Fig.~\ref{Fig:M_S_vs_TB_different_gaug_M1M2M3_Xt_0_and_minus2} also
shows how a change in the gluino mass, $M_3$, affects $\MSusyMin$.
In general, $m_h$ tends to be maximized for $M_3$ $\simeq$ $0.8
\MSusy$ \cite{Heinemeyer:1998np}. In this figure, we compare
$\MSusyMin$ for two different gluino masses, namely $M_3 = 800$ GeV
and $M_3 = 1500$ GeV. The figure shows that the effect is not very
large for this choice of parameters. However, the gluino mass can
significantly affect the Higgs masses, and therefore $\MSusyMin$,
for large $\tan\beta$ and large and \emph{negative} $\mu$.
The variation of $m_h$ as a function of $\mu$ does not generally
exceed about 3 GeV \cite{Heinemeyer:1998np}. However, it can become
very large if one includes the all-order resummation of the
$\tan\beta$ enhanced terms of order $\mathcal{O}(\alpha_b (\alpha_s
\tan\beta)^n)$, where $\alpha_b = \lambda_b^2/4\pi$ and $\lambda_b$
is the bottom Yukawa coupling
\cite{Banks:1987iu,Hall:1993gn,Hempfling:1993kv,Carena:1994bv,Carena:1999py,Eberl:1999he,
Heinemeyer:2004xw}. This resummation is included in
$\mathtt{FeynHiggs}$. The origin of the enhancement is a change in
the bottom Yukawa coupling due to a loop containing, for example, a
gluino and a sbottom squark. The leading corrections to the bottom
Yukawa coupling can be incorporated into the one-loop result for the
Higgs masses by the use of an effective bottom mass,
$m_b^{\textrm{\tiny{eff}}}$. Large $|\mu|\tan\beta$ can
substantially change the effective bottom mass
$m_b^{\textrm{\tiny{eff}}}$ from its $\overline{\textrm{MS}}$ value.
\emph{Positive} $\mu$ can substantially \emph{decrease}
$m_b^{\textrm{\tiny{eff}}}$, making the sbottom/bottom sector
corrections to $m_h$ negligible. \emph{Negative} $\mu$ on the other
hand can substantially \emph{increase} $m_b^{\textrm{\tiny{eff}}}$,
making the sbottom/bottom sector corrections to $m_h$ important. The
bottom/sbottom corrections to $m_h$ are negative in the latter case.
Larger stop masses are then required for large and negative $\mu$ as
$\tan\beta$ increases to enhance the positive radiative corrections
from the stop/top.
This effect can be seen in
Fig.~\ref{Fig:M_S_vs_TB_different_mu_Xt_minus2} where we compare
$\mu = +200$ GeV and $\mu = \pm 500$ GeV for natural maximal stop
mixing. This figure again assumes that all squark, slepton, and
gaugino soft masses are equal to the stop soft masses, $m_t = 173$
GeV, and all the soft trilinear couplings are equal to $A_t$. For
large $\tan\beta$, slightly larger $\MSusyMin$ are required for $\mu
= -500$ GeV than when $\mu$ is positive (the effect would be
stronger for even larger negative $\mu$). Note that for small values
of $\tan\beta$ there is a region for which $\MSusy$ is larger for
both $\mu = -500$ GeV and $\mu = +500$ GeV than for $\mu = 200$ GeV.
As we discussed above, this is because larger chargino and
neutralino masses decrease the size of $m_h$.
Since the gluino mass also enters the equation that determines
$m_b^{\textrm{\tiny{eff}}}$, it can have a significant impact on
$m_h$ for large $\tan\beta$ and large negative values of $\mu$ as
demonstrated in \cite{Heinemeyer:2004xw}. Thus, some non-negligible
dependence of $\MSusyMin$ on the gluino mass is expected for
negative and large $\mu$.
\begin{figure}\begin{center}\includegraphics[scale=0.47]{M_S_vs_TB_mtop173_all_mixings_f.ps}
\caption{Minimum stop soft masses, $m_{\tilde{t}} \equiv
m_{\tilde{t}_L} = m_{\tilde{t}_R} $, for $m_h \geq 114.4$ GeV as a
function of $\tan \beta$ for stop mixing $X_t/ m_{\tilde{t}} = 0,\pm
1, \pm 2$. All squark, slepton, and gaugino soft masses are equal to
the stop soft masses, $\mu=200$ GeV, $m_A = 1000$ GeV, $m_t = 173$
GeV, and all soft trilinear couplings are equal to
$A_t=X_t+\mu\cot\beta$.
\label{Fig:M_S_vs_TB_114_mtop173_all_mixings}}
\end{center}\end{figure}
\begin{figure}\begin{center}\includegraphics[scale=0.47]{stopmasses_vs_TB_mtop173_various_mixings_f.ps}
\caption{Minimum physical stop masses, $m_{\tilde{t}_1}$ and
$m_{\tilde{t}_2}$, for $m_h \geq 114.4$ GeV as a function of $\tan
\beta$ for vanishing stop mixing ($X_t = 0$) and natural maximal
stop mixing ($X_t/ m_{\tilde{t}}= -2$). Other parameters are as
given in Fig.~\ref{Fig:M_S_vs_TB_114_mtop173_all_mixings}.
\label{Fig:stopmasses_vs_TB_mtop173_various_mixings}}
\end{center}\end{figure}
\begin{figure}\begin{center}\includegraphics[scale=0.47]{M_S_vs_TB_Xt_0_all_mtops_f.ps}
\caption{Minimum stop soft masses, $m_{\tilde{t}} \equiv
m_{\tilde{t}_L} = m_{\tilde{t}_R} $, for $m_h \geq 114.4$ GeV as a
function of $\tan \beta$ for vanishing stop mixing ($X_t = 0$) for a
top quark mass of $m_t = 168, 173, 178$ GeV. Other parameters are as
given in Fig.~\ref{Fig:M_S_vs_TB_114_mtop173_all_mixings}.
\label{Fig:M_S_vs_TB_Xt_0_all_mtops}}
\end{center}\end{figure}
\begin{figure}\begin{center}\includegraphics[scale=0.47]{M_S_vs_TB_Xt_minus1_all_mtops_f.ps}
\caption{Minimum stop soft masses, $m_{\tilde{t}} \equiv
m_{\tilde{t}_L} = m_{\tilde{t}_R} $, for $m_h \geq 114.4$ GeV as a
function of $\tan \beta$ for intermediate stop mixing ($X_t/
m_{\tilde{t}}= -1$) for a top quark mass of $m_t = 168, 173, 178$
GeV. Other parameters are as given in
Fig.~\ref{Fig:M_S_vs_TB_114_mtop173_all_mixings}.
\label{Fig:M_S_vs_TB_Xt_minus1_all_mtops}}
\end{center}\end{figure}
\begin{figure}\begin{center}\includegraphics[scale=0.47]{M_S_vs_TB_Xt_minus2_all_mtops_f.ps}
\caption{Minimum stop soft masses, $m_{\tilde{t}} \equiv
m_{\tilde{t}_L} = m_{\tilde{t}_R} $, for $m_h \geq 114.4$ GeV as a
function of $\tan \beta$ for natural maximal stop mixing ($X_t/
m_{\tilde{t}}= -2$) for a top quark mass of $m_t = 168, 173, 178$
GeV. Other parameters are as given in
Fig.~\ref{Fig:M_S_vs_TB_114_mtop173_all_mixings}.
\label{Fig:M_S_vs_TB_Xt_minus2_all_mtops}}
\end{center}\end{figure}
\begin{figure}\begin{center}\includegraphics[scale=0.47]{M_S_vs_TB_different_gaug_M1M2M3_Xt_0_and_minus2_f.ps}
\caption{Minimum stop soft masses, $m_{\tilde{t}} \equiv
m_{\tilde{t}_L} = m_{\tilde{t}_R} $, for $m_h \geq 114.4$ GeV as a
function of $\tan \beta$ for various values of the bino, wino, and
gluino soft masses, $M_1, M_2, M_3$. The upper three lines are for
vanishing stop mixing ($X_t=0$) and the lower three for natural
maximal stop mixing ($X_t/ m_{\tilde{t}}= -2$). Other parameters are
as given in Fig.~\ref{Fig:M_S_vs_TB_114_mtop173_all_mixings}.
\label{Fig:M_S_vs_TB_different_gaug_M1M2M3_Xt_0_and_minus2}}
\end{center}\end{figure}
\begin{figure}\begin{center}\includegraphics[scale=0.47]{M_S_vs_TB_different_mu_Xt_minus2_f.ps}
\caption{Minimum stop soft masses, $m_{\tilde{t}} \equiv
m_{\tilde{t}_L} = m_{\tilde{t}_R} $, for $m_h \geq 114.4$ GeV as a
function of $\tan \beta$ for natural maximal stop mixing ($X_t/
m_{\tilde{t}}= -2$) with $\mu=-500,200,500$ GeV and bino, wino, and
gluino soft masses of $M_1 = 100$ GeV, $M_2=200$ GeV, $M_3=800$ GeV.
Other parameters are as given in
Fig.~\ref{Fig:M_S_vs_TB_114_mtop173_all_mixings}.
\label{Fig:M_S_vs_TB_different_mu_Xt_minus2}}
\end{center}\end{figure}
\subsection{Lower bounds on the stop masses as function of the Higgs mass}\label{Sec:various mh}
In this section, we present lower bounds on the stop soft masses,
$m_{\tilde{t}} \equiv m_{\tilde{t}_L} = m_{\tilde{t}_R}$, as a
function of the Higgs mass, $m_h$. We assume the decoupling limit
($m_A = 1000$ GeV), and we set all squark and slepton soft masses
equal to the stop soft masses, $\mu=200$ GeV, $m_t = 173$ GeV, $M_1$
= 100 GeV, $M_2$ = 200 GeV, $M_3$ = 800 GeV, $\tan\beta = 30$, and
all the soft trilinear couplings equal to $A_t$. We allow $m_h$ to
range from 100 GeV upwards. This means that the values obtained
for $\MSusy$ in the range $m_h \in [100,114.4]$ GeV will be lower
than those consistent with the LEP results, since we set no
additional constraints on $\cossqbma$ and $m_H$. However, the main
point here is to show the dependence of $\MSusy$ on $m_h$ without
any other constraints. The lower bounds on $\MSusy$ are required to
give physical stop masses not less than 100~GeV.
We show the results for different amounts of stop mixing in
Fig.~\ref{Fig:M_S_vs_m_h_various_mixings}. This figure shows how an
increase in $m_h$ requires an exponential increase in $\MSusy$. In
addition to the no-mixing, intermediate-mixing and
natural-maximal-mixing cases, we also include the
$m_h^{\textrm{\tiny{max}}}$ benchmark scenario ($X_t = +2\MSusy$)
(but with $\mu = +200$ GeV, not $\mu = -200$ GeV)
\cite{Carena:1999xa}. This benchmark scenario is designed to
maximize the Higgs mass for a given set of parameters. Moreover, we
choose $M_3 = 800$ GeV for all cases, with the exception of the
latter one. In the latter benchmark scenario, we choose the
benchmark value $M_3 = 0.8 \MSusy$ instead, which gives slightly
higher values for $m_h$ \cite{Carena:1999xa}.
It is clear from the figure that there is some value of $m_h$ at
which a further small increase in $m_h$ would require an extremely
large increase in the stop masses. It is instructive to obtain the
values of $m_h$ from the figure if, for example, $\MSusy$ = 3000
GeV. We find for no stop mixing, $m_h \simeq 121$ GeV, for
intermediate stop mixing, $m_h \simeq 126$ GeV, for natural maximal
stop mixing, $m_h \simeq 131$ GeV, and for the
$m_h^{\textrm{\tiny{max}}}$ benchmark scenario, $m_h \simeq 134$ GeV
(see also \cite{Carena:2002es}, for example, and references
therein).
Since $A_t$ and $M_3$ most naturally have the opposite sign due to
renormalization group running and the presence of a strongly
attractive quasi-fixed point (see Appendix \ref{Sec:At fixed
point}), a negative value of $A_t$ is more natural. For negative
$A_t$, the upper bound of $m_h$ in the MSSM is around 131 GeV.
\begin{figure}\begin{center}\includegraphics[scale=0.50]{M_S_vs_m_h_various_mixings_f.ps}
\caption{Minimum stop soft masses, $m_{\tilde{t}} \equiv
m_{\tilde{t}_L} = m_{\tilde{t}_R} $, as a function of $m_h$. All
squark and slepton soft masses are equal to the stop soft masses,
$\mu=200$ GeV, $m_t = 173$ GeV, $\{M_1,M_2\}$ = $\{100, 200\}$ GeV,
$\tan\beta = 30$, and all the soft trilinear couplings are equal to
$A_t=X_t+\mu\cot\beta$. In the figure, the curved lines from left
to right are as follows: the dotted line is for no mixing
($X_t/\MSusy = 0$), the dash-dot line for intermediate mixing
($X_t/\MSusy = -1$), the dashed line for natural maximal mixing
($X_t/\MSusy = -2$), and the solid line for the
$m_h^{\textrm{\tiny{max}}}$ benchmark scenario ($X_t/\MSusy = +2$)
\cite{Carena:1999xa}. The gluino mass is set to be $M_3$ = 800 GeV
in all cases except in the $m_h^{\textrm{\tiny{max}}}$ benchmark
scenario, where $M_3 = 0.8\MSusy$. The vertical dotted line is at
$m_h = 114.4$ GeV, which is the lower bound set by LEP on $m_h$ in
the decoupling limit. \label{Fig:M_S_vs_m_h_various_mixings}}
\end{center}\end{figure}
\subsection{Lower bounds on the stop masses for $m_h \simeq 93$ GeV}\label{Sec:m_h_simeq_93_GeV}
In this section, we present results for the minimum stop soft
masses, $m_{\tilde{t}} \equiv m_{\tilde{t}_L} = m_{\tilde{t}_R}$, as
a function of $\tan\beta$, for various choices of the other MSSM
parameters, and consistent with the following set of constraints on
the Higgs sector obtained by LEP: $m_h\simeq 93$ GeV, $\cossqbma \ge
0.8$ and $m_H \ge 114.4$ GeV (see equation (\ref{Eqn:lower Region
B})).
The mass $m_A$ is allowed to be a free parameter, since $\MSusy$
needs to be minimized without enforcing the decoupling limit. We
vary $m_A$ between 93.5 GeV and 1000 GeV from the bottom up for a
given choice of $\MSusy$ and other MSSM parameters, until the
conditions 93 GeV $\le$ $m_h$ $\le$ 95 GeV, $\cossqbma \ge 0.8$, and
$m_H \ge 114.4$ GeV are satisfied. (The lower bound of 93.5 GeV for
$m_A$ is the approximate lower bound obtained within the same
benchmark scenarios as the bound on $m_h$; it turns out that the
actual values obtained for $m_A$ are slightly larger). If these
conditions cannot all be satisfied, we keep increasing $\MSusy$
until they are satisfied. Note that we require the lower bounds on
$\MSusy$ to give physical stop masses of at least 100 GeV. We again
denote the lower bounds on $\MSusy$ consistent with the LEP Higgs
bounds by $\MSusyMin$.
The Higgs masses, $m_h$ and $m_H$, are calculated with
$\mathtt{FeynHiggs}$, and $\cossqbma$ is calculated using the
$\mathtt{FeynHiggs}$ output of the radiatively corrected CP-even
Higgs mixing angle $\alpha$.
In Fig.~\ref{Fig:M_S_vs_TB_tongue_mtop173_all_mixings}, we show
$\MSusyMin$ as a function of $\tan\beta$ for stop mixing $X_t/
m_{\tilde{t}}$ $=$ $0$, $\pm 1$, and $\pm 2$. All squark, slepton,
and gaugino soft masses are equal to the stop soft masses, $\mu=200$
GeV, $m_t = 173$ GeV, and all the soft trilinear couplings are equal
to the stop soft trilinear coupling, $A_t$. This figure may be
compared with Fig.~\ref{Fig:M_S_vs_TB_114_mtop173_all_mixings} in
which we require $m_h \ge 114.4$ GeV in the Higgs decoupling limit.
Next, we show $\MSusyMin$ as a function of $\tan\beta$ for different
values of the top mass (168 GeV, 173 GeV and 178 GeV) and for
different amounts of mixing.
Fig.~\ref{Fig:M_S_vs_TB_tongue_Xt_0_all_mtops} is for no mixing,
Fig.~\ref{Fig:M_S_vs_TB_tongue_Xt_minus1_all_mtops} is for
intermediate mixing, and
Fig.~\ref{Fig:M_S_vs_TB_tongue_Xt_minus2_all_mtops} is for natural
maximal mixing. All squark, slepton, and gaugino soft masses are
again equal to the stop soft masses, $\mu=200$ GeV, and all the soft
tri-linear couplings are equal to $A_t$. These figures may be
compared with the figures in which we require $m_h \ge 114.4$ GeV in
the decoupling limit, namely Figs.
\ref{Fig:M_S_vs_TB_Xt_0_all_mtops},
\ref{Fig:M_S_vs_TB_Xt_minus1_all_mtops} and
\ref{Fig:M_S_vs_TB_Xt_minus2_all_mtops}, respectively.
We first compare $\MSusyMin$ in the two scenarios $m_h \simeq 93$
GeV and $m_h \ge 114.4$ GeV for large $\tan\beta$. Here, the
figures show that $\MSusyMin$ is the same in the case of maximal or
natural maximal mixing. For intermediate and vanishing stop mixing,
$\MSusyMin$ is only slightly smaller for $m_h \simeq 93$ GeV than
for $m_h \ge 114.4$ GeV. Assuming $m_t$ = 173 GeV and $\tan\beta =
50$, the difference is only about 15 GeV for $X_t$ = $-\MSusy$ and
70 GeV for $X_t$ = 0. We expected the values for $\MSusyMin$ to be
so similar from the discussion in Section \ref{Sec:Implication for
MSSM stop sector}.
For moderate $\tan\beta$, $\MSusyMin$ can be substantially smaller for
$m_h \simeq 93$ GeV than for $m_h \ge 114.4$ GeV. This is true in
particular for the no-mixing and intermediate-mixing cases, with
the difference being more pronounced for smaller values of $m_t$.
For example, the maximum difference between $\MSusyMin$ in the two
scenarios is about 600 GeV for $\tan\beta = 12.5$ if there is no
mixing and $m_t=173$ GeV.
As $\tan\beta$ decreases further, however, $\MSusyMin$ for $m_h \simeq
93$ GeV rises very steeply, and becomes larger than for $m_h \ge
114.4$ GeV.
Understanding this behavior of $\MSusyMin$ as a function of
$\tan\beta$ requires an understanding of the importance of the
constraints $\cossqbma\ge 0.8$ and $m_H\ge 114.4$ GeV. To this end,
we compare $\MSusyMin$ versus $\tan\beta$ for the case that the
constraint on $m_H$ is ignored, for the case that both constraints
are ignored, and for the case consistent with the LEP bounds that
includes both constraints. We again make the comparison for various
amounts of mixing in the stop sector.
Fig.~\ref{Fig:M_S_vs_TB_tongue_no114_93_Xt_0} shows the results for
no mixing, Fig.~\ref{Fig:M_S_vs_TB_tongue_no114_93_Xt_minus1} for
intermediate mixing, and
Fig.~\ref{Fig:M_S_vs_TB_tongue_no114_93_Xt_minus2} for natural
maximal mixing. Each of these figures has three lines. The solid
line shows the results which are consistent with the LEP bounds,
i.e. it includes the two constraints $\cos^2(\beta-\alpha) \ge 0.8$
and $m_H \ge 114.4$ GeV, in addition to requiring $m_h \simeq 93$
GeV. The dashed line, on the other hand, does not include the
constraint on $m_H$, but does require $\cos^2(\beta-\alpha)\ge 0.8$
and $m_h \simeq 93$ GeV. The dash-dot line only requires $m_h
\simeq 93$ GeV, and ignores the constraints on
$\cos^2(\beta-\alpha)$ and $m_H$.
As expected, both constraints from LEP in general increase
$\MSusyMin$. The constraint $\cossqbma \ge 0.8$ is more important
as $\tan\beta$ becomes smaller, but less important as $\tan\beta$
gets larger. The constraint $m_H \ge 114.4$ GeV, however, is more
important for larger $\tan\beta$ (if stop mixing is not too large),
but is less important as $\tan\beta$ becomes smaller. We now explain
these observations.
If the only condition is $m_h \simeq 93$
GeV, the theory tends to be in the Higgs decoupling limit where $\cossqbma$
$\rightarrow$ 0. The reason for this is that for a given set of parameters, including a
given value of $\MSusy$, $m_h$ is maximized in the
decoupling limit. (This is also the reason why ignoring both
constraints is in general equivalent to ignoring only the
constraint $\cos^2(\beta-\alpha)$ $\ge$ 0.8 but keeping $m_H \ge
114.4$ GeV as a constraint.)
The constraint $\cossqbma \ge 0.8$, however, forces all the MSSM
Higgs masses to be quite small. In particular, $m_A$ is forced to be
relatively small and degenerate with $m_h$, so that larger $\MSusy$
are required to obtain the same value for
$m_h$.\footnote{The results for $m_A$ for the case consistent with
the LEP results (which includes both constraints) are $m_A \in$
[96.1 GeV, 99.5 GeV] for natural maximal mixing, $m_A \in$ [94.3
GeV, 97.7 GeV] for intermediate mixing, and $m_A \in$ [95.1 GeV,
97.7 GeV] for no mixing. When the constraint on $m_H$ is ignored,
$m_A$ lies roughly in the same range. Note that from the
pair-production channel these values of $m_h + m_A$ give upper
bounds on $\cos^2(\beta-\alpha)$ consistent with
$\cos^2(\beta-\alpha)$ $\ge$ 0.8, depending on what one assumes for
the Higgs decay branching ratios, see \cite{ALEPH:2006cr}.}
Moreover, the maximum value reached by $\cossqbma$ decreases as
$\tan\beta$ decreases. Larger radiative corrections, in particular
larger values of $\MSusy$ or more stop mixing, can increase the
maximum value of $\cossqbma$. However, if $\tan\beta$ decreases too far,
exponentially larger values of $\MSusy$ are required to allow
$\cossqbma$ to be greater than 0.8.
For a given set of parameters, $m_h$ in general decreases as
$\tan\beta$ decreases. This is not the case for $m_H$, which in
general decreases as $\tan\beta$ increases. This explains why the
constraint on $m_H$ is more important for larger values of
$\tan\beta$. In the decoupling limit, $m_H$ is approximately
degenerate with $m_A$, and larger values of $\MSusy$ do not affect
$m_H$ much. In the non-decoupling limit, however, larger values of
$\MSusy$ can increase $m_H$. In fact, if we define
$m_h^{\textrm{\tiny{max}}}$ to be equal to $m_h$ in the decoupling
limit, then $m_H$ $\simeq$ $m_h^{\textrm{\tiny{max}}}$ for large $\tan\beta$ and
$\cossqbma \simeq 1$. This may be explained with the formula
\begin{equation} m_h^2 \sinsqbma + m_H^2 \cossqbma =
(m_h^{\textrm{\tiny{max}}})^2, \end{equation}
\noindent valid for large $\tan\beta$
\cite{Carena:2002es,Moreno:1996zm,Carena:1999bh}, and explains why
larger $\MSusy$ increases the value of $m_H$ in, or near, the
non-decoupling region (see also Section \ref{Sec:Implication for
MSSM stop sector}).
\vspace{2cm}
\begin{figure}[!bht]\begin{center}\includegraphics[scale=0.47]{M_S_vs_TB_tongue_mtop173_all_mixings_f.ps}
\caption{Minimum stop soft masses, $m_{\tilde{t}} \equiv
m_{\tilde{t}_L} = m_{\tilde{t}_R} $, for $m_h\simeq 93$ GeV, $m_H
\ge 114.4$ GeV, and $\cossqbma \ge 0.8$, as a function of
$\tan\beta$ for stop mixing $X_t/ m_{\tilde{t}} = 0,\pm 1, \pm 2$.
All squark, slepton, and gaugino soft mass parameters are equal to
the stop soft masses, $\mu=200$ GeV, $m_t = 173$ GeV, and all soft
trilinear couplings are equal to $A_t=X_t+\mu\cot\beta$. This
figure may be compared with
Fig.~\ref{Fig:M_S_vs_TB_114_mtop173_all_mixings}.
\label{Fig:M_S_vs_TB_tongue_mtop173_all_mixings}}
\end{center}\end{figure}
\begin{figure}\begin{center}\includegraphics[scale=0.47]{M_S_vs_TB_tongue_Xt_0_all_mtops_f.ps}
\caption{Minimum stop soft masses, $m_{\tilde{t}} \equiv
m_{\tilde{t}_L} = m_{\tilde{t}_R} $, for $m_h\simeq 93$ GeV, $m_H
\ge 114.4$ GeV, and $\cossqbma \ge 0.8$, as a function of
$\tan\beta$ for vanishing stop mixing ($X_t/ m_{\tilde{t}} = 0$) for
a top quark mass of $m_t = 168, 173, 178$ GeV. Other parameters are
given as in Fig.~\ref{Fig:M_S_vs_TB_tongue_mtop173_all_mixings}.
This figure may be compared with
Fig.~\ref{Fig:M_S_vs_TB_Xt_0_all_mtops}.
\label{Fig:M_S_vs_TB_tongue_Xt_0_all_mtops}}
\end{center}\end{figure}
\begin{figure}\begin{center}\includegraphics[scale=0.47]{M_S_vs_TB_tongue_Xt_minus1_all_mtops_f.ps}
\caption{Minimum stop soft masses, $m_{\tilde{t}} \equiv
m_{\tilde{t}_L} = m_{\tilde{t}_R} $, for $m_h\simeq 93$ GeV, $m_H
\ge 114.4$ GeV, and $\cossqbma \ge 0.8$, as a function of
$\tan\beta$ for intermediate stop mixing ($X_t/ m_{\tilde{t}} = -1$)
for a top quark mass of $m_t = 168, 173, 178$ GeV. Other parameters
are given as in Fig.~\ref{Fig:M_S_vs_TB_tongue_mtop173_all_mixings}.
This figure may be compared with
Fig.~\ref{Fig:M_S_vs_TB_Xt_minus1_all_mtops}.
\label{Fig:M_S_vs_TB_tongue_Xt_minus1_all_mtops}}
\end{center}\end{figure}
\clearpage
\begin{figure}\begin{center}\includegraphics[scale=0.4]{M_S_vs_TB_tongue_Xt_minus2_all_mtops_f.ps}
\caption{Minimum stop soft masses, $m_{\tilde{t}} \equiv
m_{\tilde{t}_L} = m_{\tilde{t}_R} $, for $m_h\simeq 93$ GeV, $m_H
\ge 114.4$ GeV, and $\cossqbma \ge 0.8$, as a function of
$\tan\beta$ for natural maximal stop mixing ($X_t/ m_{\tilde{t}} =
-2$) for a top quark mass of $m_t = 168, 173, 178$ GeV. Other
parameters are given as in
Fig.~\ref{Fig:M_S_vs_TB_tongue_mtop173_all_mixings}. This figure may
be compared with Fig.~\ref{Fig:M_S_vs_TB_Xt_minus2_all_mtops}.
\label{Fig:M_S_vs_TB_tongue_Xt_minus2_all_mtops}}
\end{center}\end{figure}
\begin{figure}\begin{center}\includegraphics[scale=0.4]{M_S_vs_TB_tongue_no114_93_Xt_0_f.ps}
\caption{Minimum stop soft masses, $m_{\tilde{t}} \equiv
m_{\tilde{t}_L} = m_{\tilde{t}_R} $, as a function of $\tan\beta$,
for no stop mixing ($X_t/ m_{\tilde{t}} = 0$). The solid line shows
the minimum stop soft masses for $m_h \simeq 93$ GeV, $m_H \ge
114.4$ GeV, and $\cos^2(\beta-\alpha) \ge 0.8$, consistent with the
LEP Higgs bounds. The dashed and dash-dot lines are not consistent
with the LEP bounds and used for comparison. The dashed line shows
the minimum soft masses for $m_h \simeq 93$ GeV and
$\cos^2(\beta-\alpha) \ge 0.8$ and without a constraint on $m_H$.
The dash-dot line shows the minimum soft masses for $m_h \simeq 93$
GeV and without constraints on $m_H$ and $\cos^2(\beta-\alpha)$.
Other parameters are given as in
Fig.~\ref{Fig:M_S_vs_TB_tongue_mtop173_all_mixings}.
\label{Fig:M_S_vs_TB_tongue_no114_93_Xt_0}}
\end{center}\end{figure}
\clearpage
\begin{figure}\begin{center}\includegraphics[scale=0.47]{M_S_vs_TB_tongue_no114_93_Xt_minus1_f.ps}
\caption{Minimum stop soft masses, $m_{\tilde{t}} \equiv
m_{\tilde{t}_L} = m_{\tilde{t}_R} $, as a function of $\tan\beta$,
for intermediate stop mixing ($X_t/ m_{\tilde{t}} = -1$). The other
parameters and the different lines are as for
Fig.~\ref{Fig:M_S_vs_TB_tongue_no114_93_Xt_0}.
\label{Fig:M_S_vs_TB_tongue_no114_93_Xt_minus1}}
\end{center}\end{figure}
\begin{figure}\begin{center}\includegraphics[scale=0.47]{M_S_vs_TB_tongue_no114_93_Xt_minus2_f.ps}
\caption{Minimum stop soft masses, $m_{\tilde{t}} \equiv
m_{\tilde{t}_L} = m_{\tilde{t}_R} $, as a function of $\tan\beta$,
for natural maximal stop mixing ($X_t/ m_{\tilde{t}} = -2$). The
other parameters and the different lines are as for
Fig.~\ref{Fig:M_S_vs_TB_tongue_no114_93_Xt_0}.
\label{Fig:M_S_vs_TB_tongue_no114_93_Xt_minus2}}
\end{center}\end{figure}
\clearpage
\section{Implications of new physics constraints for the
lower bounds on the stop masses}\label{Sec:Constraints from new
physics}
In Section \ref{Sec:Lower bounds on MSusy}, we presented lower
bounds on the stop soft masses that are consistent with the LEP
Higgs bounds (we again denote these bounds by $\MSusyMin$). In this
section, we consider additional constraints from the electroweak
$S$- and $T$-parameter and the decays $B \to X_s \gamma$ and $B_s
\to \mu^+ \mu^-$, which also constrain the Higgs and/or stop sector.
Some of these constraints may provide more stringent lower bounds on
the stop masses than those provided by the constraints from LEP on
the Higgs sector, or they might indirectly constrain the Higgs
sector more tightly than the LEP results.
\subsection{Constraints from Electroweak Precision Measurements: T- and
S-parameters}
The oblique parameters $T$ and $S$ parameterize new physics
contributions to electroweak vacuum-polarization diagrams. They
give a good parametrization if these diagrams are the dominant
corrections to electroweak precision observables \cite{Peskin:1991sw}.
Strong constraints on these parameters already exist
\cite{PDBook,Erler:2006vt}.
The MSSM includes new $SU(2)_L$ doublets that contribute to the $T$-
and $S$-parameter (which are defined to be zero from SM
contributions alone). The $T$-parameter is a measure of how strongly
the vector part of $SU(2)_L$ is broken, and is non-zero, for
example, for heavy, non-degenerate multiplets of fermions or
scalars. The $S$-parameter is a measure of how strongly the axial
part of $SU(2)_L$ is broken, and is non-zero, for example, for
heavy, degenerate multiplets of chiral fermions \cite{PDBook}.
The main contribution in the MSSM to the $T$-parameter in general
comes from the stop/sbottom doublet \cite{Heinemeyer:2004gx}. In
particular, large mixing in the stop and/or sbottom sectors can lead
to large differences amongst the two stop and two sbottom masses,
which gives a large contribution to the $T$-parameter. Moreover,
for a given set of parameters and fixed $X_t/\MSusy$, decreasing
$\MSusy$ tends to increase the value of the $T$-parameter. For these
reasons the $T$-parameter could provide more stringent lower bounds
on $\MSusy$ than those coming from the LEP Higgs bounds when the
mixing in the stop sector is large, since then the stop and sbottom
masses are split by large amounts and the LEP Higgs constraints
allow for small $\MSusy$.
We estimate the $T$-parameter with version 2.2.7 of
$\mathtt{FeynHiggs}$. This program calculates $\delta\rho$ which
measures the deviation of the electroweak $\rho$-parameter from
unity. The $T$-parameter and $\delta\rho$ are related by
$\delta\rho = \alpha T$, where $\alpha$ is the QED coupling. All
the results presented in this paper are consistent with the
2$\sigma$ constraint on the upper bound of $\delta\rho$, namely
$\delta\rho \le 0.0026$ \cite{PDBook}. The $T$- and $S$-parameters
are correlated, so that this bound corresponds to the 2$\sigma$
bound on $T$ for $S=0$.
We find that the 2$\sigma$ constraint on $\delta\rho$ does not provide
an additional constraint on the stop masses in essentially all the
analyses presented in this paper.
For (natural) maximal stop mixing with $\MSusy=\MSusyMin$, the value
of $\delta\rho$ is not consistent with its 1$\sigma$ bound, although
it is consistent with its 2$\sigma$ bound (for intermediate and less
mixing, it is consistent also with the 1$\sigma$ bound). For
example, $\MSusyMin$ = 283 GeV for large $\tan\beta$ and natural
maximal stop mixing ($X_t = -2\MSusy$) in order to obtain $m_h\ge
114.4$ GeV in the Higgs decoupling limit (this assumes all squark,
slepton, and gaugino soft mass parameters are equal to $\MSusy$,
$\mu=200$ GeV, $m_t = 173$ GeV, and all the soft trilinear couplings
are equal to $A_t$). This gives a value of $\delta\rho$ = 0.0014.
Increasing $m_{\tilde{t}}$ while keeping all other parameters fixed
decreases $\delta\rho$, and for $m_{\tilde{t}}$ = 420 GeV,
$\delta\rho$ is consistent with its 1$\sigma$ upper bound of 0.0009
found in the latest PDG review \cite{PDBook}. With $m_{\tilde{t}}$ =
530 GeV, $\delta\rho$ is consistent with its 1$\sigma$ upper bound
of 0.0006 found in the previous PDG review
\cite{Eidelman:2004wy}.\footnote{As
this paper was being completed, we noticed that version 2.5.1 of
$\mathtt{FeynHiggs}$ now uses
sbottom masses with the SM and MSSM QCD corrections added when
calculating $\delta\rho$. This
can give different values of $\delta\rho$, especially for small
sbottom masses, and it makes $\delta\rho$ more sensitive to $\mu$.
The results quoted in this paragraph change as follows. For $m_{\tilde{t}} = 283$ GeV,
$\delta\rho$ = 0.0011. Increasing $m_{\tilde{t}}$ to 310 GeV gives
$\delta\rho$ = 0.0009, and $m_{\tilde{t}}$ = 380 GeV gives
$\delta\rho$ = 0.0006.
Qualitatively the conclusions
presented in this section are unaffected. We thank
S. Heinemeyer for clarifying the difference between the older and
newer versions.}
The $S$-parameter in the MSSM is in general not very important
\cite{PDBook}. We estimated it using the formulae in
\cite{Inami:1992rb}. Including contributions from all squarks and
sleptons, the S-parameter does not reach a value higher than about
0.05 for $\MSusy=\MSusyMin$ in those cases that have large stop
mixing, with the main contribution coming from the stop/sbottom doublet. For
intermediate and vanishing mixing it is negligible. The constraint
on $S$ depends on $T$, but the 1$\sigma$ upper bound on
$S$ is about 0.07 for $T = 0$, whereas a positive value for $T$ allows for
larger values of $S$. Thus the $S$-parameter is a weaker constraint
on the stop masses than the LEP Higgs sector bounds.
\subsection{Constraints from $\bsg$}\label{Sec:bsg}
New physics can contribute at one loop to the decay $\bsg$, and can
therefore be just as important as the SM contribution mediated by a
$W$-boson and the top quark. This makes the decay $\bsg$ an
important tool in constraining new physics.
The SM contribution to the branching ratio $\mathcal{B}(\bsg)$ is
predicted to be
\begin{equation} \label{Eqn:SM bsg} \mathcal{B}(\bsg)_{\textrm{\tiny{SM}}}
\simeq (2.98 \pm 0.26) \times 10^{-4}, \end{equation}
\noindent see \cite{Becher:2006pu}, whereas the experimental bound
is given by
\begin{equation} \label{Eqn:Exp bsg} \mathcal{B}(\bsg)_{\textrm{\tiny{expt}}}
\simeq (3.55 \pm 0.26) \times 10^{-4}, \end{equation}
\noindent see \cite{:2006bi}. This allows, but does not require,
new physics contributions \cite{Becher:2006pu}.
There are several contributions to the decay $\bsg$ from the
additional particles in the MSSM, which we now discuss.
Within the Higgs sector, the charged Higgs ($H^{+}$) contributes at
one loop to the decay $\bsg$. The contribution is larger for smaller
$m_{H^{+}}$. If one only considers this contribution, as one would
in the two-Higgs-doublet model of type II (2HDM (II)), then this
sets a rather stringent lower bound on $m_{H^+}$. The bound of
course depends on the SM prediction and experimental measurement of
$\mathcal{B}(\bsg)$, and in the past used to be about $m_{H^+}
\gtrsim 350$ GeV, see \cite{Gambino:2001ew}, \cite{Hurth:2003vb} and
references therein. The latest results quoted in equations
(\ref{Eqn:SM bsg}) and (\ref{Eqn:Exp bsg}) are expected to change
this bound slightly, but we do not explore this in more detail
\cite{Becher:2006pu}. It is clear, however, that this bound is much
stronger than the bound coming from a direct search of $H^+$ at LEP
which is given by $m_{H^+} \gtrsim 78.6$ GeV \cite{LHWG:2001xy}.
Note that the charged Higgs contribution is mostly independent of
$\tan\beta$; only for very small values of $\tan\beta$ does it
increase substantially.
The charged Higgs, thus, does not contribute much to $\bsg$ in the
decoupling limit for large $m_A$, since here $m_{H^{+}}$ is large.
In the region $m_h \simeq 93$ GeV with $\cossqbma \ge 0.8$ and $m_H
\ge 114.4$ GeV, however, $m_{H^{+}}$ $\simeq$ 125 GeV. The
contribution from the charged Higgs to $\mathcal{B}(\bsg)$ is then
roughly $7.7 \times 10^{-4}$, more than a factor of two larger than
the SM contribution. We estimated this using version 2.5.1 of the
program $\mathtt{FeynHiggs}$,\footnote{Note that
$\mathtt{FeynHiggs}$ gives $\mathcal{B}(\bsg)_{\textrm{\tiny{SM}}}
\simeq 3.63 \times 10^{-4}$ which is larger than the latest value
quoted in equation (\ref{Eqn:SM bsg}). This is not of qualitative
importance here.} in the limit of large sparticle masses. Therefore,
the constraint on $\mathcal{B}(\bsg)$ rules out this region of the
Higgs parameter space if one only considers the charged Higgs
contribution.
There are, however, also chargino, neutralino and gluino contributions to
$\bsg$ within the MSSM with minimal flavor violation (MFV).\footnote{There are other
possibilities for flavor violation within the MSSM, and therefore additional
contributions to $\bsg$ are possible. The additional flavor
violation is small, however, assuming that the only source of flavor
violation comes from the mixing among the squarks and assuming that
this is of the same form as the mixing among the quarks,
i.e.~described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix. This
assumption is usually called minimal flavor violation (MFV).
The MSSM with general flavor violation allows for more contributions to the decay
$\bsg$, which can sometimes weaken constraints on parameters in the
MSSM with MFV \cite{Okumura:2004fh}.} NLO contributions can be very
important and need to be included in order to get an accurate
estimate of $\mathcal{B}(\bsg)$ \cite{Hurth:2003vb}. The
contribution from a chargino together with a stop in the loop is
often the most important one. The chargino-stop contribution can
become very large for small chargino and small stop masses, and it is
proportional to $\tan\beta$ in the amplitude. However, it vanishes
in the limit of large stop or chargino masses. From studying the
mSUGRA model, it is known that usually the chargino-stop contribution
to the branching ratio interferes constructively with the SM and the
charged Higgs contribution if the sign of $\mu A_t$ is
positive, whereas it interferes destructively if the sign of $\mu A_t$
is negative \cite{Hurth:2003vb}.
This means that the region $m_h \simeq 93$ GeV is not necessarily
ruled out, since a light stop and a light chargino could cancel the
charged Higgs contribution
\cite{Kim:2006mb,Drees:2005jg,Hooper:2005yg}. Using version 2.5.1 of
$\mathtt{FeynHiggs}$ to calculate the branching ratio of $\bsg$, we
verify this claim in the case of intermediate and larger stop
mixing, at least for $\tan\beta$ not too small. We find that the
contribution to $\bsg$ from the chargino-stop loop can easily be
large enough to interfere destructively with the charged Higgs
contribution and thus give an experimentally allowed value of
$\mathcal{B}(\bsg)$. Moreover, in some cases for sizeable stop
mixing, the chargino-stop contribution can be made much larger than
the SM and charged Higgs contribution. Thus, an experimentally
consistent value of $\mathcal{B}(\bsg)$ can also often be obtained
by finding a chargino mass that gives a chargino-stop amplitude
equal to the negative of the charged Higgs amplitude plus the
negative of twice the SM amplitude. We note that an experimentally
consistent value for $\mathcal{B}(\bsg)$ can always be found without
requiring the stop masses to be larger than $\MSusyMin$, but by
adjusting the chargino mass alone.
If $\tan\beta$ is small enough then $\MSusyMin$ becomes exponentially
large, and the constraint on
$\mathcal{B}(\bsg)$ rules out the $m_h\simeq 93$ GeV region since the chargino-stop
contribution cannot cancel the charged Higgs contribution.
In the case of vanishing stop mixing with \emph{degenerate} stop
soft masses, $\MSusyMin$ is so large that the chargino-stop
contribution to $\bsg$ is too small to cancel the charged Higgs
contribution. However, even in the no-mixing case one of the stops
can be chosen to be light by setting one of the stop soft masses to
a small value. In this case the other stop soft mass needs to be
very large in order for the radiative corrections to the Higgs
sector to be large enough to satisfy the LEP bounds. One light stop,
however, is able to give a sizeable chargino-stop contribution that
can cancel the charged Higgs contribution. For example, we find
$\MSusyMin \simeq 1100$ GeV for $m_h \simeq$ 93 GeV, $\cossqbma\ge
0.8$ and $m_H \ge 114.4$ GeV, with $\tan\beta=20$, $\mu = 200$ GeV,
$m_A\simeq 96$ GeV, and $m_t = 173$ GeV (this assumes that all
squark, slepton, and gaugino soft masses are equal to $\MSusy$, and
all the soft trilinear couplings are equal to $A_t$). Since the
charged Higgs then essentially provides the only contribution to
$\bsg$ beyond that of the SM itself, the branching ratio is again
about $7.7 \times 10^{-4}$. However, choosing, for example,
$m_{\tilde{t}_L} = 350$ GeV, $m_{\tilde{t}_R} = 2000$ GeV, all the
gaugino soft masses equal to $m_{\tilde{t}_L}$, and keeping all
other squark and slepton soft masses equal to 1100 GeV, gives a
consistent branching ratio of $3.6 \times 10^{-4}$.
In the Higgs decoupling limit, for which $m_h \ge 114.4$ GeV, the
charged Higgs contribution vanishes. Since $\MSusyMin$ is large for
very small $\tan\beta$ or vanishing stop mixing, the chargino-stop
contribution to $\mathcal{B}(\bsg)$ is small, and there is no
inconsistency with the experimental bound. On the other hand,
$\MSusyMin$ can be so low for appreciable amounts of mixing (and if
$\tan\beta$ is not too small) that the chargino-stop contribution
can easily be too large. In this case, however, we can find a
chargino mass that gives a branching ratio of $\bsg$ within the
experimentally allowed region, and again we find no further
constraint on $\MSusy$. We can achieve this by setting the chargino
mass to a very large value, in which case the chargino-stop
contribution becomes vanishingly small. For negative $\mu A_t$,
however, the chargino-stop loop interferes destructively with the SM
contribution so that we can also adjust the chargino mass until the
chargino-stop amplitude is equal to the negative of twice the SM
amplitude. This is what happens in the case depicted in
Fig.~\ref{Fig:BR_B_to_X_s_gamma_vs_mu_max_mix_TB20}, where we show
the branching ratio of $\bsg$ as a function of $\mu$. In this
figure, all squark, slepton, and gaugino soft masses are equal to
the stop soft masses, which are given by $\MSusyMin = 283$ GeV, $m_t
= 173$ GeV, $\tan\beta = 20$, $X_t= -2 m_{\tilde{t}}$, and all the
soft trilinear couplings are equal to the stop soft trilinear
coupling, $A_t$. We find an experimentally allowed value for
$\mathcal{B}(\bsg)$ in this case by choosing $\mu\simeq 330$ GeV. We
note that $\mu$ has to be chosen within about a 30 GeV window for
$\mathcal{B}(\bsg)$ to fall within the 3$\sigma$ allowed region.
\begin{figure}\begin{center}\includegraphics[scale=0.44]{BR_B_to_X_s_gamma_vs_mu_max_mix_TB20_f.ps}
\caption{$\mathcal{B}(\bsg)$ versus $\mu$ with stop soft masses
$m_{\tilde{t}} \equiv m_{\tilde{t}_L} = m_{\tilde{t}_R} = 283$ GeV,
and natural maximal stop mixing ($X_t/ \MSusy = -2$). All squark,
slepton, and gaugino soft mass parameters are equal to the stop soft
masses, $m_t = 173$ GeV, $m_A=1000$ GeV, $\tan\beta = 20$, and all
the soft trilinear couplings are equal to $A_t=X_t+\mu\cot\beta$.
\label{Fig:BR_B_to_X_s_gamma_vs_mu_max_mix_TB20}}
\end{center}\end{figure}
\subsection{Constraints from $\Bmumu$}
The decay $\Bmumu$ has not yet been observed. The SM contribution to
this decay is dominated by penguin diagrams involving the $Z$-boson
and box diagrams involving the $W$-bosons \cite{Inami:1980fz}. (The
SM Higgs does contribute to the decay $\Bmumu$ within the SM, but
relative to the dominant contribution it is suppressed by
$m_{\mu}m_{b,s}/m_W^2$, where $m_{\mu}$, $m_b$ and $m_s$ are the
masses of the muon, b-quark and s-quark, respectively, and $m_W$ is
the mass of the $W$-bosons \cite{Grzadkowski:1983yp}.) The SM
contribution to the branching ratio is quite small since it is fourth
order in the weak interactions. It is predicted to be
\begin{equation} \mathcal{B}(\Bmumu)_{\textrm{\tiny{SM}}} = (3.42 \pm 0.54)
\times 10^{-9} \end{equation}
\noindent (see \cite{Abulencia:2005pw} and references therein). This
is well below the current experimental bound from the CDF experiment
at the Tevatron given by
\begin{equation}\label{Eqn:Bmumu_CDF_bound}
\mathcal{B}(\Bmumu)_{\textrm{\tiny{expt}}} < 1.5\times 10^{-7} \end{equation}
\noindent at the 90$\%$ confidence level \cite{Abulencia:2005pw}.
There are several contributions to the decay $\Bmumu$ from the
additional particles in the MSSM, which we now discuss.
The contributions to the decay $\Bmumu$ coming only from the MSSM
Higgs sector are the same as those found in the 2HDM (II). They can
be enhanced by two powers of $\tan\beta$ in the amplitude, which can
compensate for the suppression by the muon mass. One can set an
approximate bound on $m_{H^+}$ assuming this is the \emph{only}
contribution within the MSSM. This bound depends on $\tan\beta$, but
for $\tan\beta=50$ one finds an experimentally allowed value for
$\mathcal{B}(\Bmumu)$ if $m_{H^+}\gtrsim 35$ GeV (see for example
\cite{Logan:2000iv,Chankowski:2000ng}). As we discussed in Section
\ref{Sec:bsg}, within the 2HDM (II) the constraint on
$\mathcal{B}(\bsg)$ alone forces $m_{H^+}$ to be larger than about
350 GeV. Such a large value for $m_{H^+}$ guarantees that
$\mathcal{B}(\Bmumu)$ is roughly of the same size as the SM result
even for quite large $\tan\beta$, so that it alone provides no
further constraint on the parameter space within the 2HDM (II)
\cite{Bobeth:2001sq}.
In the MSSM there are, however, further contributions to the decay
$\Bmumu$ coming from box and penguin diagrams that involve charginos
and up-type squarks
\cite{Chankowski:2000ng,Bobeth:2001sq,Babu:1999hn,Buras:2002vd,Dedes:2002er,Mizukoshi:2002gs,Dedes:2003kp}.
The penguin diagrams also contain the neutral Goldstone and Higgs
bosons. The self-energy MSSM Higgs penguin diagrams give the leading
contribution to $\mathcal{B}(\Bmumu)$ for non-negligible mixing in the
stop sector. (In an effective Lagrangian
approach these diagrams may be viewed as inducing a non-holomorphic
coupling between down-type quarks and the up-type Higgs field.) For
large $\tan\beta$, this leading contribution is roughly proportional
to $A_t^2 \tan^6\beta/m_A^4$, and can thus be significantly larger
than the contributions from the Higgs sector alone. Moreover, this
contribution becomes small for very small $\mu$. This contribution does not
vanish for degenerate squark masses, nor in the limit of large
sparticle masses. Thus, although the branching ratio depends on the
size of the stop masses, it is much more sensitive to the size of
the Higgs masses, $\tan\beta$ and the amount of stop mixing. A light
Higgs sector can give a branching ratio of $\Bmumu$ that
is more than three orders of magnitude above the SM prediction and
thus well ruled out, especially if the stop mixing and $\tan\beta$ are
large. Moreover, this is the case even for large sparticle masses.
Furthermore, such large values for $\mathcal{B}(\Bmumu)$ can be
reached within the MSSM without violating any other constraints,
including, for example, those on $\mathcal{B}(\bsg)$
\cite{Bobeth:2001sq,Bobeth:2002ch}.
There are further contributions to $\mathcal{B}(\Bmumu)$ which also
have a $\tan^6\beta/m^4_A$ behavior, even assuming that the CKM
matrix is the only source of flavor violation in the squark sector.
These appear if the left-handed up-type soft squark masses of the
three generations are not all equal, so that the left-handed
down-type soft squark mass-squared matrix has off-diagonal terms.
These lead to contributions from loops involving a neutralino or a
gluino and a down-type squark
\cite{Chankowski:2000ng,Babu:1999hn,Mizukoshi:2002gs,Dedes:2003kp,Bobeth:2002ch}.
Cancelations between the chargino and gluino contributions can occur
and the neutralino contribution, although usually smaller, can then
be important (see, for example, \cite{Bobeth:2002ch}).
We estimated the values for $\mathcal{B}(\Bmumu)$ with the program
\href{http://wwwlapp.in2p3.fr/lapth/micromegas/}{$\mathtt{MicrOMEGA}$}
1.3 \cite{Belanger:2001fz,Belanger:2004yn} and the subroutine
\href{http://www.hep.fsu.edu/~isajet/}{$\mathtt{IsaBMM}$} from
$\mathtt{IsaTools}$/$\mathtt{IsaJet}$
\cite{Mizukoshi:2002gs,Paige:2003mg}. We find that the branching
ratio of $\Bmumu$ is well within experimental limits for the region
$m_h$ $\simeq$ $93$ GeV, $m_H\ge114.4$ GeV and $\cossqbma \ge 0.8$
in the case of no (or very little) stop mixing and degenerate squark
soft masses. For intermediate mixing with degenerate squark masses
near $\MSusyMin$, $\mathcal{B}(\Bmumu)$ is consistent with
experimental limits for $\tan\beta \lesssim 20-25$. For natural
maximal mixing, $\mathcal{B}(\Bmumu)$ is consistent with
experimental limits for $\tan\beta \lesssim$ 15. For larger
$\tan\beta$, as well as for non-degenerate squark soft masses, a
scan over all relevant MSSM parameters is necessary in order to see
whether we can find an experimentally consistent value of
$\mathcal{B}(\Bmumu)$ for such a light Higgs sector. However, for
large stop mixing, it will become increasingly difficult to find a
parameter set that gives a branching ratio consistent with
experiments as $\tan\beta$ is increased. Of course, this assumes
that there are no fortuitous cancelations between the different
contributions, and also that there are no other flavor-violating
contributions such as from R-parity violating couplings. A scan over
the relevant MSSM parameters, even assuming MFV, is beyond the scope
of this paper. The reader is referred to the references found in the
previous two paragraphs, and especially \cite{Hooper:2005yg},
bearing in mind that the current CDF bound on $\mathcal{B}(\Bmumu)$,
equation (\ref{Eqn:Bmumu_CDF_bound}), is stronger than the one used
in these references.
In the Higgs decoupling limit, the dominant flavor-violating effects
involving loops of neutral Higgs bosons decouple, and these large
contributions to $\Bmumu$ become negligible. Using
$\mathtt{MicrOMEGA}$ 1.3, one may explicitly check that the decay
$\Bmumu$ does not provide stronger constraints on the stop masses
than do the LEP Higgs bounds in the decoupling limit found in
Section \ref{Sec:m_h_greater_114_GeV}.
\section{Implications for Electroweak Symmetry \\ Breaking} \label{Sec:Implications for
EWSB}
In Section \ref{Sec:Lower bounds on MSusy}, we presented lower
bounds on the stop masses consistent with the LEP Higgs bounds, and
in Section \ref{Sec:Constraints from new physics}, we discussed
whether the electroweak $S$- and $T$-parameter and the decays
$B\rightarrow X_s\gamma$ and $B_s\rightarrow \mu^+ \mu^-$ indirectly
put further constraints on the Higgs and/or stop sector. In this
section, we look at the implications for electroweak symmetry
breaking.
The mechanism of radiative electroweak symmetry breaking arises
rather naturally in supersymmetric extensions of the Standard Model
\cite{Ibanez:1982fr,Ellis:1982wr,Alvarez-Gaume:1983gj}. Because of
the large top Yukawa coupling, quantum fluctuations of the stop
squarks significantly modify the up-type Higgs potential, as studied
numerically for the physical Higgs boson mass in the previous
sections. The leading effect, however, is a tachyonic contribution
to the up-type Higgs soft supersymmetry breaking Lagrangian mass.
Over much of parameter space this tachyonic contribution is sufficient
to result in a stop squark quantum fluctuation-induced phase
transition for the Higgs fields, which is generally referred to as
radiative electroweak symmetry breaking.
The leading quantum contribution to the up-type Higgs soft mass
comes from renormalization group evolution below the supersymmetry
breaking messenger scale. The one-loop $\beta$-function for the
up-type Higgs soft mass-squared is, neglecting effects proportional to
gauge couplings,
\begin{equation} 16 \pi^2 \beta_{m^2_{H_u}} \simeq 6
\lambda_t^2 ( m^2_{H_u}+ m_{\tilde{t}_L}^2 + m_{\tilde{t}_R}^2
+ |A_t|^2 )
\label{betamhus} \end{equation} The light Higgs mass bounds require rather
large stop masses and/or stop mixing, where the stop soft trilinear
coupling is related to the mixing parameter by $A_t = X_t + \mu \cot
\beta$. This implies that the stop contributions to the
$\beta$-function in (\ref{betamhus}) proportional to the combination
$(m_{\tilde{t}_L}^2 + m_{\tilde{t}_R}^2 + |A_t|^2)$ are also
sizeable, at least at the low scale. Moreover, for generic
parameters this combination remains sizeable over the entire
renormalization group trajectory up to the messenger scale. For
generic messenger scale values of the up-type Higgs soft mass
squared, $m^2_{H_u}$, the large value of the combination
$(m_{\tilde{t}_L}^2 + m_{\tilde{t}_R}^2 + |A_t|^2)$, along with the
sizeable coefficient in the $\beta$-function (\ref{betamhus}), then
imply that $m^2_{H_u}$ evolves relatively rapidly under
renormalization group evolution.
This evolution is towards tachyonic values of $m^2_{H_u}$ which
reduce the magnitude of the $\beta$-function (\ref{betamhus}). For
running into the deep infrared, the up-type Higgs mass squared would
be driven to values near the zero of the $\beta$-function
(\ref{betamhus}) for which
\begin{equation} m^2_{H_u} \simeq - ( m_{\tilde{t}_L}^2 + m_{\tilde{t}_R}^2
+ |A_t|^2).
\label{zerobeta}
\end{equation} Although this relation is not strictly obtained with finite
running, the up-type Higgs mass squared can approach this value for
very high messenger scale. In Table \ref{Table:combination}, we show
the minimum allowed values of the combination $(m_{\tilde{t}_L}^2 +
m_{\tilde{t}_R}^2 + |A_t|^2)^{1/2}$ deduced from the results of
section \ref{Sec:m_h_greater_114_GeV} consistent with $m_h \geq
114.4$ GeV in the Higgs decoupling limit for large $\tan \beta$. The
minimum allowed values increase with decreasing $\tan \beta$.
\begin{table}\begin{center}\begin{tabular}{|c|c|c|c|}
\hline $X_t/m_{\tilde{t}}$ & $m_t = 168$ GeV & $m_t = 173 $ GeV &
$m_t=178$ GeV \\
\hline 0 & 3630 & 1780 & 1240 \\
\hline -1 & 1460 & 1000 & 770 \\
\hline -2 & 680 & 690 & 710 \\
\hline
\end{tabular}\caption{Minimum allowed values of the combination
$(m_{\tilde{t}_L}^2 + m_{\tilde{t}_R}^2 + |A_t|^2)^{1/2}$ consistent
with a physical Higgs boson mass of $m_h \geq 114.4$ GeV in the
Higgs decoupling limit for large $\tan \beta$, taking into account
only the LEP Higgs sector bounds. The minimum allowed values
increase with decreasing $\tan \beta$.
\label{Table:combination}}\end{center}\end{table}
The full Lagrangian mass squared for the up-type Higgs is a sum of
the soft mass squared and square of the superpotential Higgs mass,
$m^2 = m^2_{H_u} + | \mu|^2$. To leading order in $1/\tan^2 \beta$,
and ignoring the finite quantum corrections to the Higgs potential
which are not of qualitative importance for the present discussion,
this is equal to minus half the $Z$-boson mass squared in the ground
state with broken electroweak symmetry
\begin{equation} \frac{1}{2} m_Z^2 \simeq -( m^2_{H_u} + |\mu|^2). \end{equation}
\noindent For $m^2_{H_u}$ near the zero of its $\beta$-function
given by (\ref{zerobeta}), the bounds given in Table
\ref{Table:combination} imply that obtaining the observed value of
the $Z$-boson mass, $m_Z \simeq 91$ GeV, requires a rather sensitive
cancelation between the up-type Higgs soft mass and $\mu$-parameter.
The numerical magnitude of this tuning (which has come to be known
as the supersymmetric little hierarchy problem) is apparent in the
numerical data in Table \ref{Table:combination}, at least for
regions of parameter space which are driven under renormalization
group flow to near the zero of the $\beta$-function
(\ref{betamhus}).
The minimum allowed value of the combination $(m_{\tilde{t}_L}^2 +
m_{\tilde{t}_R}^2 + |A_t|^2)$ for a given lower limit on the Higgs
mass decreases with increasing stop mixing. This may be understood
from the leading expression for the quantum corrected Higgs mass
given in equation (\ref{Eqn:delta_M_uu}). For no stop mixing,
$X_t=0$, the leading correction to the Higgs mass squared comes only
from renormalization group running of the Higgs quartic coupling
below the stop mass scale, and is therefore proportional to $\ln(
m^2_{\tilde{t}} / m_t^2 )$. A linear increase in the Higgs mass
squared in this case requires an exponential increase in $\MSusy$.
However, the stop mixing correction to the Higgs mass squared with
$X_t \neq 0$ comes from a finite threshold correction to the Higgs
quartic coupling at the stop mass scale and is independent of
$\MSusy$ for fixed $X_t / m_{\tilde{t}}$. In this case a linear
increase in the Higgs mass squared only requires a linear increase
in $(X_t / m_{\tilde{t}})^2$. So increasing stop mixing allows
exponentially lighter stop masses in order to obtain a given Higgs
mass. While such a decrease clearly reduces the soft stop mass
contributions to $\beta_{m^2_{H_u}}$
\cite{Kitano:2006gv,Kane:2004tk} this is partially offset by an
increase in the mixing contribution from the stop trilinear
coupling. From the data in Table \ref{Table:combination}, it is
clear that large stop mixing can decrease the magnitude of
$\beta_{m^2_{H_u}}$ (\ref{betamhus}) by up to a factor of a few
depending on the top mass. However, the magnitude of the total stop
contribution including mixing is still quite sizeable for a Higgs
mass bound of $m_h \geq 114.4$ GeV. So large stop mixing alone
cannot appreciably ameliorate the tuning of supersymmetric
electroweak symmetry breaking or satisfactorily solve the
supersymmetric little hierarchy problem.\footnote{Although this
conclusion is valid for a generic choice of messenger scale values
for the sparticle masses, it is possible to reduce the amount of
tuning coming from the running of $m^2_{H_u}$ by a more judicious
choice. One example is to choose negative stop masses squared at
the high scale which allows the contribution to the tuning from the
running of $m^2_{H_u}$ to be arbitrarily small, as well allow for
the (natural) maximal mixing scenario to be radiatively generated at
the low scale \cite{Dermisek:2006ey}.}
This conclusion essentially remains unchanged for a physical Higgs
boson mass of $m_h \simeq 93$ GeV with $\cossqbma \ge 0.8$ and $m_H
\ge 114.4$ GeV, as seen from the numerical results in Section
\ref{Sec:m_h_simeq_93_GeV}. In general, one should bear in mind
that indirect constraints on new physics, especially from
$\mathcal{B}(\Bmumu)$, severely restrict the allowed MSSM parameter
space for $m_h\simeq 93$ GeV (see Section \ref{Sec:Constraints from
new physics}). However, for less than maximal stop mixing, the stop
masses can be somewhat smaller for moderate $\tan\beta$ near the
Higgs non-decoupling limit than in the Higgs decoupling limit (see
also \cite{Kim:2006mb}). The combination $(m_{\tilde{t}_L}^2 +
m_{\tilde{t}_R}^2 + |A_t|^2)^{1/2}$ is in fact the smallest in the
Higgs non-decoupling region near intermediate values for the stop
mixing and for $\tan\beta$ near 10. It reaches as low as about 650
for $m_t=178$, $\tan\beta = 10.5$, $X_t = -\MSusy$, and gaugino
masses equal to $\MSusy$. It can be decreased slightly further by
setting the bino and wino masses to smaller values. (For maximal
stop mixing, the combination is actually larger since here the
Tevatron bound on the lighter stop mass forces the stop soft masses
to be larger than required from the LEP Higgs bounds alone.) The
combination always remains sizeable though, and thus the tuning of
electroweak symmetry breaking cannot be ameliorated by much in the
$m_h\simeq 93$ GeV region.
\section{Conclusions}\label{Sec:Conclusion}
The dominant radiative corrections to the tree-level CP-even Higgs
mass matrix, which determines $m_h$ and $m_H$, come from loops
involving the top quark and stop squarks, with larger stop masses
implying larger radiative corrections. In this paper, we presented
lower bounds on the stop masses consistent with the LEP Higgs bounds
in two different regions in the MSSM Higgs parameter space. The one
region is the Higgs decoupling limit, in which the bound on the mass
of the lighter Higgs is equal to the bound on the SM Higgs, $m_h \ge
114.4$ GeV. The other region is near the Higgs ``non-decoupling''
limit with $m_h \simeq 93$ GeV in which the Higgs sector is required
to be light. In the latter region, there are two additional
constraints. One is on the mass of the heavier Higgs, which now
behaves like the SM Higgs, i.e.~$m_H\gtrsim 114.4$ GeV. The other
constraint is on size of the coupling of the lighter Higgs to two
$Z$ bosons which is controlled by the parameter $\sinsqbma$ and here
needs to be less than about 0.2 (i.e.~$\cossqbma \gtrsim 0.8$) for
the lighter Higgs to have escaped detection at LEP. We denote the
lower bounds on the stop masses consistent with the LEP Higgs bounds
by $\MSusyMin$.
We presented $\MSusyMin$ as a function of $\tan\beta$ in both these
regions in the Higgs parameter space for a variety of MSSM parameter
choices. In particular, we further elucidated the importance of the
top mass and stop mixing, and investigated numerically how larger
top masses and more stop mixing allow for substantially smaller
values of $\MSusyMin$. We also showed numerically how larger
gaugino masses and larger values of $\mu$ increase $\MSusyMin$.
Moreover, we saw how much $\MSusyMin$ increases if $\mu$ is negative
compared to $\mu$ positive if both $\tan\beta$ and the magnitude of
$\mu$ are large. In the non-decoupling region, we discussed how the
constraints on $\cossqbma$ and on $m_H$ lead to increased values for
$\MSusyMin$.
We also considered how $\MSusyMin$ changes as a function of $m_h$.
Since $A_t$ and $M_3$ most naturally have the opposite sign at low
scales due to renormalization group running, a negative value of
$A_t$ is more natural in a convention where $M_3$ is positive. For
negative $A_t$ and stop masses less than a few TeV, the upper bound
of $m_h$ in the MSSM is around 131 GeV.
We demonstrated that the two regions in the Higgs parameter space
have roughly the same $\MSusyMin$ if $\tan\beta$ is large. For
moderate values of $\tan\beta$ and non-maximal stop mixing,
$\MSusyMin$ is larger in the Higgs decoupling region than in the
Higgs non-decoupling region. As $\tan\beta$ decreases, however,
$\MSusyMin$ is larger in the Higgs non-decoupling region than in the
Higgs decoupling region.
We also considered additional constraints from the electroweak $S$-
and $T$-parameter and the decays $B \to X_s \gamma$ and $B_s \to
\mu^+ \mu^-$, which also constrain the Higgs and/or stop sector.
The main contribution to the $T$-parameter within the MSSM usually
comes from the stop/sbottom doublet and, for a given set of
parameters, is larger for larger stop (and sbottom) mixing as well
as for smaller stop and sbottom masses. We found that the value of
the $T$-parameter is well within its 2$\sigma$ bound for stop masses
equal to $\MSusyMin$. In fact, only for maximal stop mixing do we
find small enough values for $\MSusyMin$ that give a contribution to
the $T$-parameter that does not also fall within its 1$\sigma$
bound. For such large stop mixing one must then increase the stop
masses by a small amount above $\MSusyMin$ to also satisfy the
1$\sigma$ bound on the $T$-parameter.
We found that the contribution to the $S$-parameter is not
large, and that the $S$-parameter therefore does not provide an
additional constraint on the stop masses.
The indirect constraint on $\mathcal{B}(\bsg)$ in many cases does
not provide an additional constraint on the stop masses. In the
Higgs non-decoupling region for $m_h\simeq 93$ GeV, the Higgs sector
is required to be light, and the charged Higgs contribution to
$\bsg$ is large. The charged Higgs contribution can usually be
canceled by the chargino-stop contribution through a judicious
choice of the chargino mass. However, for vanishing stop mixing and
assuming degenerate stop soft masses, $\MSusyMin$ is so large that
the chargino-stop contribution is too small to cancel the charged
Higgs contribution. For vanishing stop mixing, we therefore require
non-degenerate stop soft masses with one light stop so that the
chargino-stop contribution can be large enough to give an
experimentally consistent value for $\mathcal{B}(\bsg)$ (the other
stop must then be very heavy so that the LEP Higgs constraints are
satisfied). In the Higgs decoupling limit, the charged Higgs
contribution vanishes. We find no further constraint on the stop
masses. Even for large stop mixing, for which $\MSusyMin$ can be
very small, one can always obtain an experimentally consistent value
for $\mathcal{B}(\bsg)$ by adjusting the chargino mass.
The main contributions to the flavor-violating decay $\Bmumu$ come
from flavor violating Higgs couplings, and these decouple in the
Higgs decoupling limit. Thus, the indirect constraint on
$\mathcal{B}(\Bmumu)$ is only important in the Higgs non-decoupling
region. In this region, however, it is able to severely restrict the
allowed parameter space, since the flavor violation does not
decouple in the limit of large sparticle masses. In fact, the region
for such a light Higgs sector is ruled out if stop mixing and
$\tan\beta$ are large, unless there are fortuitous cancelations
amongst the various contributions, or there are additional
flavor-violating contributions from, for example, R-parity violating
couplings that cancel these contributions.
We note that we did not consider the constraint on the anomalous
magnetic moment of the muon, $(g-2)_{\mu}$, since it decouples in
the limit of large sneutrino and smuon masses. It alone is thus
unable to directly provide a further constraint on the Higgs sector
or on the stop masses.
Lastly, we discussed the implications of our numerical analysis for
electroweak symmetry breaking. Large stop mixing generically
decreases the tuning of supersymmetric electroweak symmetry
breaking, but is unable to do so sufficiently to solve the
supersymmetric little hierarchy problem. Moreover, the tuning can be
ameliorated only slightly in the $m_h\simeq 93$ GeV region compared
to the $m_h \ge 114.4$ GeV region (for intermediate values of the
stop mixing and moderate values of $\tan\beta$), and thus the
supersymmetric little hierarchy problem cannot be satisfactorily
solved in either of the two regions.
\newpage
\noindent {{\bf \large Acknowledgements}}
\vspace{0.1cm}
I would like to thank my advisor S.~Thomas for suggesting this
project and for his support and guidance throughout. I would also
like to thank S.~Heinemeyer for helpful email exchanges about
$\mathtt{FeynHiggs}$. It is a pleasure to further acknowledge
useful conversations with J.F.~Fortin, D.E.~Kaplan, A.~Lath,
N.~Sehgal, and N.~Weiner, as well as R.~Derm{\'i}{\v s}ek who
pointed out two relevant references. I also want to thank J.~Shelton
for reading the manuscript and providing many helpful suggestions.
This research is supported by a Graduate Assistantship from the
Department of Physics and Astronomy at Rutgers University.
\vspace{0.5cm}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,886
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