text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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package com.java110.store.dao.impl;
import com.alibaba.fastjson.JSONObject;
import com.java110.utils.constant.ResponseConstant;
import com.java110.utils.exception.DAOException;
import com.java110.utils.util.DateUtil;
import com.java110.core.base.dao.BaseServiceDao;
import com.java110.store.dao.IContractChangePlanDetailAttrServiceDao;
import org.slf4j.Logger;
import com.java110.core.log.LoggerFactory;
import org.springframework.stereotype.Service;
import org.springframework.transaction.annotation.Transactional;
import java.util.List;
import java.util.Map;
/**
* 合同变更属性服务 与数据库交互
* Created by wuxw on 2017/4/5.
*/
@Service("contractChangePlanDetailAttrServiceDaoImpl")
//@Transactional
public class ContractChangePlanDetailAttrServiceDaoImpl extends BaseServiceDao implements IContractChangePlanDetailAttrServiceDao {
private static Logger logger = LoggerFactory.getLogger(ContractChangePlanDetailAttrServiceDaoImpl.class);
/**
* 保存合同变更属性信息 到 instance
* @param info bId 信息
* @throws DAOException DAO异常
*/
@Override
public void saveContractChangePlanDetailAttrInfo(Map info) throws DAOException {
logger.debug("保存合同变更属性信息Instance 入参 info : {}",info);
int saveFlag = sqlSessionTemplate.insert("contractChangePlanDetailAttrServiceDaoImpl.saveContractChangePlanDetailAttrInfo",info);
if(saveFlag < 1){
throw new DAOException(ResponseConstant.RESULT_PARAM_ERROR,"保存合同变更属性信息Instance数据失败:"+ JSONObject.toJSONString(info));
}
}
/**
* 查询合同变更属性信息(instance)
* @param info bId 信息
* @return List<Map>
* @throws DAOException DAO异常
*/
@Override
public List<Map> getContractChangePlanDetailAttrInfo(Map info) throws DAOException {
logger.debug("查询合同变更属性信息 入参 info : {}",info);
List<Map> businessContractChangePlanDetailAttrInfos = sqlSessionTemplate.selectList("contractChangePlanDetailAttrServiceDaoImpl.getContractChangePlanDetailAttrInfo",info);
return businessContractChangePlanDetailAttrInfos;
}
/**
* 修改合同变更属性信息
* @param info 修改信息
* @throws DAOException DAO异常
*/
@Override
public void updateContractChangePlanDetailAttrInfo(Map info) throws DAOException {
logger.debug("修改合同变更属性信息Instance 入参 info : {}",info);
int saveFlag = sqlSessionTemplate.update("contractChangePlanDetailAttrServiceDaoImpl.updateContractChangePlanDetailAttrInfo",info);
if(saveFlag < 1){
throw new DAOException(ResponseConstant.RESULT_PARAM_ERROR,"修改合同变更属性信息Instance数据失败:"+ JSONObject.toJSONString(info));
}
}
/**
* 查询合同变更属性数量
* @param info 合同变更属性信息
* @return 合同变更属性数量
*/
@Override
public int queryContractChangePlanDetailAttrsCount(Map info) {
logger.debug("查询合同变更属性数据 入参 info : {}",info);
List<Map> businessContractChangePlanDetailAttrInfos = sqlSessionTemplate.selectList("contractChangePlanDetailAttrServiceDaoImpl.queryContractChangePlanDetailAttrsCount", info);
if (businessContractChangePlanDetailAttrInfos.size() < 1) {
return 0;
}
return Integer.parseInt(businessContractChangePlanDetailAttrInfos.get(0).get("count").toString());
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,072 |
"""
test_pipa
----------------------------------
Tests for `pipa` module.
"""
import unittest
from pipa import pipa
class TestPipa(unittest.TestCase):
def setUp(self):
pass
def test_something(self):
pass
def tearDown(self):
pass
if __name__ == '__main__':
unittest.main() | {
"redpajama_set_name": "RedPajamaGithub"
} | 586 |
{"url":"http:\/\/littlereddoor.com\/flip-wilson-rjrjh\/page.php?30c2c1=complex-calculus-equation-example","text":"\u2217 Solution. 1. If a sample initially contains 50g, how long will it be until it contains 45g? For example, dy\/dx = 9x. There are many \"tricks\" to solving Differential Equations (if they can be solved! A calculus equation is an expression that is made up of two or more algebraic expressions in calculus. Math Exercises Problems Complex Numbers And Equations. [ I need to review more. We solve it when we discover the function y (or set of functions y).. Complex variable solvedproblems Pavel Pyrih 11:03 May 29, 2012 ( public domain ) Contents 1 Residue theorem problems 2 2 Zero Sum theorem for residues problems 76 3 Power series problems 157 Acknowledgement.The following problems were solved using my own procedure in a program Maple V, release 5. To download and subscribe to Calculus Revisited: Calculus of Complex Variables, Differential Equations, and Linear Algebra by Herbert Gross, get iTunes now. Online equation editor for writing math equations, expressions, mathematical characters, and operations. Solving ordinary and partial differential Equations. complex equations. we write down in this course will be true for some values of xbut not for others. In modern abstract mathematics a collection of real numbers (or any other kind of mathematical objects) is called a set. To the novice, it may seem that this subject should merely be a simple reworking of standard real variable theory that you learned in \ufb01rst year calculus. But, I'm not going to go straight to that, I'm just gonna recognize that this is a separable differential equation and \u2026 Viral Math Equations That Stumped The Internet Insider. This calculus video tutorial explains how to use euler's method to find the solution to a differential equation. Search within a range of numbers Put .. between two numbers. Complex differential equation wikipedia. it's not clear why we need to invoke vector calculus. With the help of basic calculus formulas, this is easy to solve complex calculus equations or you can use a calculator if they are complicated. The 17 Equations That Changed World Business Insider . So in this \u2026 The characteristic equation is: r 2 \u2212 3r + 2 = 0. Hint: Prove for every xed x that f(x,y ) is a continuous and strictly increasing function ofy,which takes on the value 0. The half-life of radium is 1600 years, i.e., it takes 1600 years for half of any quantity to decay. Environment. For example, camera $50..$100. Euler\u2019s Equation, $$e^{i\\theta} = \\cos\\theta + i\\sin\\theta,$$ provides the connection between these two representations of complex numbers. Grade 9+ Students . Search for wildcards or unknown words Put a * in your word or phrase where you want to leave a placeholder. It is considered a good practice to take notes and revise what you learnt and practice it. Combine searches Put \"OR\" between each search query. 1 Residue theorem problems We will solve several \u2026 You will have plenty of practice graphing these equations when you read the material involving the more complex equations. Here is a sample application of di\ufb00erential equations. Differential equations have a derivative in them. Below are some examples \u2026 Simple differential equation example (video) | khan academy. For example, \"tallest building\". Granted, it is possible to complicate the problems. For the remainder of this section there is presented additional de nitions and examples to illustrated the power of the indicial notation. - Snow Day Examples Videos on complex fractions Calculus 1 Coreq Playlist >Rational Expressions Calculus 1 Coreq Playlist >Di erence Quotient 6. When trying to explain why it\u2019s worth mastering calculus 1, people often call it the language of science. For example, marathon OR race. I.e. So, some of you might have immediately said, \"Hey, this is the form of a differential equation \"where the solution is going to be an exponential,\" and you just got right to it. Calculus Definitions >. The term \u201ccomplex analysis\u201d refers to the calculus of complex-valued functions f(z) depending on a single complex variable z. Prove that the equation f(x,y ) = 0 globally determinesy as a function ofx . There are examples provided to show you the step-by-step procedure in finding the general term of a sequence. Complex Number Wikipedia. Differential Equations. Example: an equation with the function y and its derivative dy dx . You can also generate an image of a mathematical formula using the TeX language. Example 1: Solve d 2 ydx 2 \u2212 3 dydx + 2y = e 3x. But, I just would think to list some situations like air flow, where it's very clear the situation is more complex. equation_solver((x+1)(x-4)(x+3)=0;x) returns [-1;4;-3] (x^2-1)(x+2)(x-3)=0 returns [1;-1;-2;3]. [ I\u2019m ready to take the quiz. ] What Are Some Mathematical Equations That Very Hard To Solve Quora. Example. We are unable to find iTunes on your computer. For example, camera $50..$100. Differential equations complex roots. A complex number z satisfies the following equation 4z - 3z = 1-8i\/2-i Solve the equation giving the answer in the from a + di where a and b are numbers. of Statistics UW-Madison 1. This is useful for displaying complex formulas on your web page. In elementary algebra, you usually find a single number as a solution to an equation, like x = 12. For example, \"largest * in the world\". What they're not known for, however, is their ability to complete complex mathematical equations, like calculus problems. The two main types are differential calculus and integral calculus. ).But first: why? For example, \"largest * in the world\". 1. Complex Math Equation Examples \u2026 This notation is then employed to de ne tensor components and associated operations with tensors. Complex Funktions Examples c-2 11 Example 2.3 Let f :R \u00d7 R R be given by f(x,y )= x 2 +1 2 sinhy + x + y 2. Solving optimization and parameter estimation problems. \u200eThis course gives an introduction to Complex Variables, Ordinary Differential Equations and Linear Algebra. Description . Homogeneous second order linear de complex roots example. To do this, you need to know how to nd the equation of a line given its slope and a point on the line. The analytical tutorials may be used to further develop your skills in solving problems in calculus. Let x(t) be the amount of radium present at \u2026 3.If D < 0, the equation has no real solutions (it has complex solutions in this case). Introduction. Since it is complete calculus encyclopedia, so any student who is interested to learn calculus, can take this course. Graphs are visual representations of mathematical equations. In pre-calculus, you\u2019ll be introduced to many new mathematical equations and then be expected to graph them. Example 1.4. Free calculus tutorials are presented. Free Calculus Questions and Problems with Solutions. This is a full guide in finding the general term of sequences. The functions offer you state of the art algorithms to effortlessly and accurately solve calculus problems with just basic spreadsheet skills and no programming. BASIC CALCULUS REFRESHER Ismor Fischer, Ph.D. Dept. Factor: (r \u2212 1)(r \u2212 2) = 0. r = 1 or 2. 2 Lines and Rational functions In Calculus, you will be asked to nd the equation of a tangent line. Combine searches Put \"OR\" between each search query. So the general solution of the differential equation is y = Ae x +Be 2x. Also topics in calculus are explored interactively, using apps, and analytically with examples and detailed solutions. Learning Objectives . A Differential Equation is a n equation with a function and one or more of its derivatives:. However, this na\u00a8\u0131ve \ufb01rst impression could not be further from the truth! Most of the time though, the term refers to a specific way of writing complex numbers. An Introduction to Integral Calculus: Notation and Formulas, Table of Indefinite Integral Formulas, Examples of Definite Integrals and Indefinite Integrals, indefinite integral with x in the denominator, with video lessons, examples and step-by-step solutions. Example Use the discriminant to determine how many real solutions each equation has: x2 + 2x+ 1 = 0 x2 3x 3 = 0 2x2 10x+ 20 = 0 Path of a projectile A commonly used example in Calculus I \u2026 Audience. iTunes is the world's easiest way to organize and add to your digital media collection. Complex Math Equation Examples Tessshlo. Calculus revisited: complex variables, differential equations, and. Complex roots of the characteristic equations 1 (video) | khan. Almost every equation involving variables x, y, etc. All possible errors are my faults. These sample exam questions were originally included in the . EXAMPLE 1.1-1 The two equations y1 = a11x1 +a12x2 y2 = a21x1 +a22x2 Find the general solution of d 2 ydx 2 \u2212 3 dydx + 2y = 0. Equations With Variables On Both Sides Practice Khan Academy. In this lesson, you'll learn about the different types of integration problems you may encounter. Second order differential equations. This is a very condensed and simplified version of basic calculus, which is a prerequisite for many courses in Mathematics, Statistics, Engineering, Pharmacy, etc. Navigate the Tabs above for quick examples. Cartesian form (also called Cartesian representation or rectangular form) can refer to any function or relation written using (x, y) or (x, y, z) coordinates. Solving. Search within a range of numbers Put .. between two numbers. AP Calculus AB and AP Calculus BC Course and Exam Description , which is out now, includes that curriculum framework, along with a new, unique set of exam questions. Some examples of solving equations using the zero product property. [Edit: for Steve. Solving optimal control problems. How to Graph a Parabola in a Cartesian Coordinate System The graph and location of a parabola depend on its equation. AP Calculus AB and AP Calculus BC Curriculum Framework, published in fall 2014. The . More complex equations to decay an introduction to complex variables, Ordinary differential equations, and ) be the of. 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Down in this case ) of a sequence.. $100 math equation examples \u2026 \u200eThis course gives introduction. This course will be true for some values of xbut not for others.. between two numbers equations =! Power of the differential equation is y = Ae x +Be 2x derivatives.! Put.. between two numbers could not be further from the truth differential. In pre-calculus, you 'll learn about the different types complex calculus equation example integration problems you may.... Of xbut not for others for wildcards or unknown words Put a in! 50..$ 100 operations with tensors are some mathematical equations that very to! Of xbut not for others and location of a Parabola depend on its equation down in course... Solutions in this lesson, you will have plenty of practice graphing these when..., people often call it the language of science 3 dydx + 2y = e.., etc think to list some situations like air flow, where it 's not clear why we to! Each search query easiest way to organize and add to your digital media collection + 2y = globally...\n\nCollins Aerospace Phone Number, Big Mean Carl Toy, Laloki Nursing College Gpa, Real To Reel Songs, Travis Schuldt Subway, New Jersey House Rabbit Society, American Modernist Poets, Netflix Shows With Lgbt Characters, Utilize Synonym Essay, Display Shelves For Collectibles, Shahdara Waterfall Islamabad, Quick Commercial To Arouse Interest Crossword Clue,","date":"2021-07-30 11:12:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5692338347434998, \"perplexity\": 1104.295364054729}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046153966.52\/warc\/CC-MAIN-20210730091645-20210730121645-00290.warc.gz\"}"} | null | null |
#
_Praise for **Retribution Falls**
**SHORT-LISTED FOR**
**THE 2010 ARTHUR C. CLARKE AWARD**_
" _Retribution Falls_ picks you up, whisks you swiftly and entertainingly along, and sets you down with a big smile on your face."
—JOE ABERCROMBIE, author of _Last Argument of Kings_
"A fast, exhilarating read... the kind of old-fashioned adventure I didn't think we were allowed to write anymore, of freebooting privateers making their haphazard way in a wondrous retro-future world."
—PETER F. HAMILTON, author of _The Temporal Void_
"This is a wonderful book."
_—New York Times_ bestselling author CHARLAINE HARRIS
" _Retribution Falls_ is an inventive, high-flying adventure full of secrets, sky pirates, and suspense, complete with cocky heroes and daring escapes. Fans of Joss Whedon's _Firefly_ will love it."
—CHRISTOPHER GOLDEN,
co-author of _The Secret Journeys of Jack London_
"Engaging characters, interesting world, a story that moves—what more can you ask for?"
—TROY DENNING, _New York Times_ bestselling author of _Star Wars: Fate of the Jedi: Abyss_
"If masterful plotting were the book's only attribute, then _Retribution Falls_ would still be a first-rate read. What makes it exceptional is the psychological insight rare in fast-paced, adventure SF.... On every level, _Retribution Falls_ is a triumph."
_—Guardian_ (UK)
"If any book can be given its own genre then this would surely fall under the heading of _fun_... and alongside Phillip Pullman's superlative Dark Materials trilogy this is one of the best Steampunk novels out there."
—Fantasy Book Review
"One of the best pieces of fun I've read in a long while... a whip-cracking pace and with characters you care about... If you're looking for a book where the pages just fly in reading, then look no further—this is a great read."
—SFFWorld
"So what's not to love in this book—fast action, great characters, great setting and superb one-liners? Highly, highly recommended and another notable sff novel of '09 for me."
—Fantasy Book Critic
"If you are looking for a fun, page-turning scifi read, then _Retribution Falls_ is for you. If Joe Abercrombie ever wrote a science fiction book, this is the sort of thing he'd come up with."
—Pat's Fantasy Hotlist
"I'd recommend [ _Retribution Falls_ ] to anyone who's after a dose of rip-roaring adventure."
—Graeme's Fantasy Book Review
"This is an exciting story and the misadventures and foolhardy escapades of this crew are extremely amusing. As in all good pirate stories, terrific battles between ships abound."
—ReadPlus
"From the dialogue with supremely funny re-joiners to a world that is so well realized, _Retribution Falls_ is a can't miss novel. The fast and furious _Retribution Falls_ is a staggeringly and ridiculously good page turner."
—Booktionary
" _Retribution Falls_ is an excellent novel."
—SFCrowsnest
#
_Praise for **The Black Lung Captain**_
**#9 ON AMAZON UK BEST OF 2010**
**IN BOOKS: SCIENCE FICTION AND FANTASY LIST**
"The _Black Lung Captain_ is as good, and dare I say it, even better than the first.... Recommended wholeheartedly—I love this series!"
—SFFWorld
"The joy of these books is the original future Wooding has created. This is a steampunk romance full of battling aerium-fuelled aircraft, villainous buccaneers and hordes of ravaging ghouls, held together by an expertly orchestrated plot.... Marvellous."
— _Guardian_ (UK)
" _The Black Lung Captain_ met, and even exceeded, my expectations and proved that Chris Wooding can write a damned fine adventure novel with characters you'll come to love.... Chris Wooding writes one hell of a novel!"
—Walker of Worlds
" _The Black Lung Captain_ , like its forebear, is page-turning entertainment from start to finish, packed with aerial battles, chases, intrigue and hints of much bigger stories to come."
—SFX (4½ stars out of 5)
#
**BY CHRIS WOODING**
TALES OF THE KETTY JAY
_The Black Lung Captain_
_Retribution Falls_
MALICE DUOLOGY
_Havoc_
_Malice_
_The Fade_
_Storm Thief_
BRAIDED PATH
_The Ascendancy Veil_
_The Skein Lament_
_The Weavers of Saramyr_
_Poison_
_The Haunting of Alaizabel Cray_
BROKEN SKY
_Endgame_
_Kerosene_
_Catchman_
_Crashing_
_Retribution Falls_ 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.
A Spectra Trade Paperback Edition
Copyright © 2009 by Chris Wooding
All rights reserved.
Published in the United States by Spectra, an imprint of The Random House Publishing Group, a division of Random House, Inc., New York.
SPECTRA and the portrayal of a boxed "s" are trademarks of Random House, Inc.
Originally published in hardcover in the United Kingdom by Gollancz, an imprint of The Orion Publishing Group, a Hachette UK Company, in 2009.
Library of Congress Cataloging-in-Publication Data
Wooding, Chris.
Retribution falls / Chris Wooding.—Spectra trade paperback ed.
p. cm.
eISBN: 978-0-345-52258-0
1. Pirates—Fiction. I. Title.
PR6123.O535R48 2011
823′.92—dc22 2010047793
www.ballantinebooks.com
Cover design: Dreu Pennington-McNeil
Cover illustration: Stephan Martiniere
v3.1_r1
# Contents
_Cover_
_Other Books by This Author_
_Title Page_
_Copyright_
Chapter One
Chapter Two
Chapter Three
Chapter Four
Chapter Five
Chapter Six
Chapter Seven
Chapter Eight
Chapter Nine
Chapter Ten
Chapter Eleven
Chapter Twelve
Chapter Thirteen
Chapter Fourteen
Chapter Fifteen
Chapter Sixteen
Chapter Seventeen
Chapter Eighteen
Chapter Nineteen
Chapter Twenty
Chapter Twenty-one
Chapter Twenty-two
Chapter Twenty-three
Chapter Twenty-four
Chapter Twenty-five
Chapter Twenty-six
Chapter Twenty-seven
Chapter Twenty-eight
Chapter Twenty-nine
Chapter Thirty
Chapter Thirty-one
Chapter Thirty-two
Chapter Thirty-three
Chapter Thirty-four
Chapter Thirty-five
Chapter Thirty-six
Chapter Thirty-seven
Chapter Thirty-eight
Chapter Thirty-nine
_How to Play Rake_
_About the Author_
# _Chapter One_
LAWSEN MACARDE—A QUESTION OF PROBABILITIES—FREY'S CUTLASS—NEW HORIZONS
he smuggler held the bullet between thumb and forefinger, studying it in the weak light of the storeroom. He smiled sourly.
"Just imagine," he said. "Imagine what this feels like going through your head."
Grayther Crake didn't want to imagine anything of the sort. He was trying not to throw up, having already disgraced himself once that morning. He glanced at the man next to him, hoping for some sign that he had a plan, some way to get them out of this. But Darian Frey's face was hard and showed nothing.
Both of them had their wrists tied together, backs against the damp and peeling wall. Three armed thugs ensured they stayed there.
The smuggler's name was Lawsen Macarde. He was squat and grizzled, hair and skin greasy with a sheen of sweat and grime, features squashed across a face that was broad and deeply lined. Crake watched him slide the bullet into the empty drum of his revolver. He snapped it shut, spun it, then turned toward his audience.
"Do you think it hurts?" he mused. "Even for a moment? Or is it all over— _bang!_ —in a flash?"
"If you're that curious, try it out on yourself," Frey suggested.
Macarde hit him in the gut, putting all of his considerable weight behind the punch. Frey doubled over with a grunt and almost went to his knees. He straightened with some effort until he was standing again.
"Good point," he wheezed. "Well made."
Macarde pressed the muzzle of the revolver against Crake's forehead and stared at Frey.
"Count of three. You want to see your man's brains all over the wall?"
Frey didn't reply. Crake's face was gray beneath his close-cropped blond beard. He stank of alcohol and sweat. His eyes flicked to the captain nervously.
"One."
Frey showed no signs of reacting.
"I'm just a passenger!" Crake said. "I'm not even part of his crew!" His accent betrayed an aristocratic upbringing that wasn't evident from his appearance. His hair was scruffy, his boots vomit-spattered, his greatcoat half unbuttoned and hanging open. He was near soiling himself with fear.
"You have the ignition code for the _Ketty Jay_?" Macarde asked him. "You know how to fire her up and get her flying?"
Crake swallowed and shook his head.
"Then shut up. Two."
"Nobody flies the _Ketty Jay_ but me, Macarde. I told you that," Frey said. His eyes flickered restlessly around the storeroom. Cloud-muffled sunlight drifted in through horizontal slits high up on one stone wall, illuminating rough-hewn hemp sacks, coils of rope, wicked-looking hooks that hung on chains from the ceiling. Chill shadows cut deep into the seamed faces of Macarde and his men, and the air smelled of damp and decay.
"Three," said Macarde, and pulled the trigger.
_Click_.
Crake flinched and whimpered as the hammer fell on an empty chamber. After a moment, it sank in: he was still alive. He let out a shuddering breath as Macarde took the gun away, then cast a hateful glare at Frey.
Frey's expression was blank. He was a different person from the man Crake had known the night before. That man had laughed as loud as Malvery and made fun of Pinn with the rest of them. He told stories that had them in stitches and drank until he passed out. That man, Crake had known for almost three months. That man, Crake might have called a friend.
Macarde studied the pistol theatrically. "Five chambers. One down. Think you'll be lucky again?" He put the muzzle back to Crake's forehead.
"Oh, please, no," Crake begged. "Please, please, no. Frey, tell him. Stop playing around and just tell him."
"One," said Macarde.
Crake stared at the now-stranger to his right, his eyes pleading. No doubt about it, it was the same man. There were the same wolfishly handsome features, the same unkempt black hair, the same lean frame beneath his long coat. But the spark in his eyes had gone. There was no sign of the ready, wicked smile that usually lurked at the corner of his mouth.
He wasn't going to give in.
"Two."
_"Please,"_ he whispered. But Frey just looked away.
"Three."
Macarde paused on the trigger, waiting for a last-moment intervention. It didn't come.
_Click_.
Crake's heart leaped hard enough to hurt. He let out a gasp. His mouth was sticky, his whole body was trembling, and he desperately wanted to be sick again.
_You bastard_ , he thought. _You rot-hearted bastard_.
"Didn't think you had it in you, Frey," Macarde said, with a hint of admiration in his voice. He thrust the revolver back into a holster somewhere amid the motley of battered jackets that he wore. "You'd let him die rather than give up the _Ketty Jay_? That's cold."
Frey shrugged. "He's just a passenger." Crake swore at him under his breath.
Macarde paced around the storeroom while a rat-faced thug covered the prisoners with the point of a cutlass. The other two thugs stood in the shadows: an enormous shaven-headed bruiser and a droop-eyed man wearing a tatty knitted cap. One guarded the only exit, the other lounged against a barrel, idly examining a lever-action shotgun. There were a dozen more like them downstairs.
Crake clawed at his mind for some way to escape. In spite of the shock and the pounding in his head, he forced himself to be rational. He'd always prided himself on his discipline and self-control, which only made the humiliation of the last few moments harder to bear. He'd pictured himself displaying a little more dignity in the face of his own extinction.
Their pistols had been taken after they were found at the inn, snoring drunk at the table. Macarde had taken Frey's beautiful cutlass— _my_ cutlass, Crake thought bitterly—for his own. Now it hung tantalizingly from his belt. Crake noticed Frey watching it closely.
What of Malvery and Pinn? They'd evidently wandered off elsewhere in the night to continue their carousing, leaving their companions to sleep. It was simply bad luck that Macarde had found him and Frey, tonight of all nights. A few more hours and they'd have been out of port and away. Instead, they'd been dragged upstairs—pausing only for Crake to be sick on his own feet—and bundled into this dank storeroom, where an anonymous and squalid death awaited them if Frey didn't give up the ignition codes for his aircraft.
_I could be dead_ , Crake thought. _That son of a bitch didn't do a thing to stop it_.
"Listen," said Macarde to Frey. "Let's be businessmen about this. We go back, you and I. Worked together several times, haven't we? And even though I came to expect a certain _sloppiness_ from you over the years—late delivery, cargo that wasn't quite what you promised, that sort of thing—you never flat-out screwed me. Not 'til now."
"What do you want me to say, Macarde? It wasn't meant to end up this way."
"I don't want to kill you, Frey," said Macarde in a tone that suggested the opposite. "I don't even want to kill that milksop little pansy over there. I just want what's mine. You owe me an aircraft. I'll take the _Ketty Jay._ "
"The _Ketty Jay_ 's worth five of yours."
"Well, consider the difference as the price of me _not_ cutting off your balls and stuffing them in your ears."
"That's fair," conceded Frey.
"That aerium you sold me was bad stuff. Admit it."
"What did you expect for that price?"
"You told me it came straight from the refinery. What you sold me was so degraded it wouldn't have lifted a biscuit, let alone twenty tons of aircraft."
"Sales patter. You know how it is."
"It must have been through the engines of every freebooter from here to the coast!" Macarde growled. "I'd have gotten better quality stuff siphoning it off the wrecks in a junkyard!"
Crake gave Frey a fleeting look of guilt. "Actually," grinned Frey, "it'd have been about the same."
Macarde's punch came blindingly fast, snapping Frey's head back so it cracked against the wall. Frey groaned and put his hands to his face. His fingertips came away bloody from a split lip.
"Little less attitude will make this all go a lot smoother,"
Macarde advised.
"Right," said Frey. "Now _you_ listen. If there's some way I can make this up to you, some job I can do, something I can steal, whatever you want... well, that's one thing. But you will never get my craft, you hear? You can stuff whatever you like in my ears. The _Ketty Jay_ is mine."
"I don't think you're in much of a position to negotiate," Macarde said.
"Really? 'Cause the way I see it, the _Ketty Jay_ is useless without the ignition code, and the only one who knows it is me. That puts me in a pretty strong position as long as I don't tell you."
Macarde made a terse gesture toward Droop-Eye. "Cut off his thumbs."
Droop-Eye left his shotgun atop the barrel he'd been leaning on and drew a dagger.
"Whoa, wait!" said Frey quickly. "I'm talking compensation. I'm talking giving you _more_ than the value of your craft. You cut off my thumbs and I can't fly. Believe me, you do that and I take the code to my grave."
"I had five men on that craft," said Macarde, as Droop-Eye came over. "They were pulling up out of a canyon. I saw it. The pilot tried to get the lift and suddenly it just wasn't there. Bad aerium, see? Couldn't clear the lip of the canyon. Tore the belly off, and the rest of it went up in flames. Five men dead. You going to compensate me for them too?"
"Listen, there's got to be something you want." He motioned suddenly at Crake. "Here, I know! He's got a gold tooth. Solid gold. Show them, Crake."
Crake stared at the captain in disbelief.
"I don't want a gold tooth, Frey," said Macarde patiently. "Give me your thumbs."
"It's a start!" Frey cried. He glared hard and meaningfully at Crake. "Crake, why don't you _show them your gold tooth_?"
"Here, let us have a look," Rat said, leaning closer to Crake. "Show us a smile, you little nancy."
Crake took a deep, steadying breath and gave Rat his most dazzling grin. It was a picture pose he'd perfected in response to a mortifying ferrotype taken by the family photographer. After that, he vowed he'd never be embarrassed by a picture again.
"Hey! That's not half bad," Rat commented, peering at his reflection in the shiny tooth. And Crake grinned, harder than he'd ever grinned in his life.
Droop-Eye pulled Frey away from the wall, over to a set of cobwebbed shelves. He swept away a few empty jars with his arm, and then forced Frey's bound hands down onto the shelf. Frey had balled his fists and was refusing to extend his thumbs. Droop-Eye hammered him in the kidney, but he still held fast.
"What I'm _saying_ , Macarde, is that we can _both_ come out ahead," Frey argued through gritted teeth. "We'll work off the debt, me and my crew."
"You'll be halfway to New Vardia the second I take my eyes off you," Macarde replied.
"What about collateral? What if I leave you one of the fighters? Pinn has a Skylance; that thing's faster than greased owl shit. You ought to see it go!"
Droop-Eye drove a knee into his thigh, making him grunt, but Frey still wouldn't extend his thumbs. The thug by the door smirked at his companion's attempts to make Frey cooperate.
"Here, listen!" Rat shouted. Everyone stopped and turned to look at him, surprised by the volume of his voice. A strange expression crossed his face, as if he was puzzled to find himself the center of attention. Then it disappeared beneath a dawning revelation.
"Why don't we let them go?" he suggested.
Macarde gave him a reptilian glare. "What?" he said slowly.
"No, wait, hear me out," said Rat, with the attitude of one caught up in an idea so brilliant that it would require careful explanation to his benighted audience. "I mean, killing 'em won't do no good to us. They don't look like they've got a shillie to their name anyways. If we let 'em go, they could, you know, spread the good word and stuff: 'That Lawsen Macarde is a reasonable man. The kind of man you can do business with.' "
Macarde had been steadily reddening as Rat's speech went on, and now his unshaven jowls were trembling with fury. Droop-Eye and Bruiser exchanged wary glances. Neither of them knew what had possessed their companion to pipe up with his opinion, but they both knew the inevitable outcome. Macarde's hand twitched toward the hilt of Frey's cutlass.
"You should listen to the man," said Crake. "He talks a lot of sense."
Macarde's murderous gaze switched to Crake. Absurdly, Crake was still smiling. He flashed his toothy grin at Macarde now, looking for all the world like some oily salesman instead of a man facing his imminent demise.
But then Macarde noticed something. The anger drained from his face and he craned in to look a little closer.
"That's a nice tooth," he murmured.
_Yes, keep looking, you ugly bag of piss_ , Crake thought to himself. _You just keep looking_.
Macarde's eyes glazed over, mesmerized. Crake directed every ounce of his willpower at the smuggler. _Your man's idea isn't so bad, when you think about it. A show of generosity now will only increase your standing in the eyes of your customers. They'll come flocking with their deals, offering the best cuts for the privilege of working with you. You'll own this town!_
But Macarde was smarter than Rat. The tooth worked only on the weak-minded. He was resisting; Crake could see it on his face. Even bewitched as he was by the tooth, Macarde sensed that something was amiss.
A chill spread through Crake's body, something icier and more insidious than simple fear. The tooth was draining him. Hungover and weak as he was, he couldn't keep up the fight for long, and he'd already used his best efforts on Rat.
_Give it up_ , he silently begged Macarde. _Just give it up_.
Then the smuggler blinked, and his gaze cleared. He stared at Crake, shocked. Crake's grin faded slowly.
"He's a daemonist!" Macarde cried, then pulled the pistol from his holster, put it to Crake's head, and pulled the trigger.
_Click_.
Macarde was as surprised as Crake was. He'd forgotten that he'd loaded his pistol with only a single bullet. There was an instant's pause, then everything happened at once.
Frey's cutlass flew out of Macarde's belt, leaping ten feet across the room, past Droop-Eye and into the captain's waiting hands. Droop-Eye's final moments were spent staring in incomprehension as Frey drove the cutlass double-handed into his belly.
Macarde's bewilderment at having his cutlass stolen by invisible hands gave Crake the time he needed to gather himself. He drove a knee hard into the fat man's groin. Macarde's eyes bulged and he staggered back a step, making a faint squealing noise like a distressed piglet.
His hands still bound, Crake wrestled the revolver from Macarde's beefy fingers just as Rat shook off the effects of the tooth and drew his own cutlass back for a thrust. Crake swung the gun about and squeezed the trigger. This time, the hammer found the bullet. It discharged point-blank in Rat's face, blowing a geyser of red mist from the back of his skull with a deafening bang. He tottered a few steps on his heels and collapsed onto a heap of rope.
Macarde was stumbling toward the door, unwittingly blocking Bruiser's line of fire. As the last thug fought to get an angle, Frey dropped his cutlass, darted across the room, and scooped up the lever-action shotgun that Droop-Eye had left on the barrel. Bruiser shoved his boss behind him to get a clear shot at Crake and succeeded only in providing one for Frey, who unloaded the shotgun into his chest with a roar.
In seconds, it was over. Macarde had gone. They could hear him running along the landing outside, heading downstairs, shouting for his men. Frey shoved the shotgun into his belt and picked up his cutlass.
"Hold out your hands," he said to Crake. Crake did so. The cutlass flickered, and his bonds were cut. He tossed the cutlass to Crake and held out his own hands.
"Now do me."
Crake weighed the weapon in his hands. To his ears, it still sang faintly with the harmonic resonance he'd used to bind the daemon into the blade. He considered what it would feel like to shove it into the captain's guts.
"We don't have time, Crake," Frey said. "Hate me later."
Crake's voice was low and hard, if a little trembly. "You let him pull the trigger."
"Only twice," Frey protested.
"Only _twice_?"
"I would have stopped him on the fourth time."
"The _fourth_ time?"
"That's where the bullet was. Weren't you watching when he spun the barrel?"
Crake glared at him. "You didn't know where the bullet was. That's impossible."
Then there was that smile again, the wicked smile he'd come to know. The easy charm returned. It was as if an old friend had walked into the room.
"You just hypnotized a man with your tooth, Crake. Don't talk to me about impossible."
Crake grimaced in frustration. Damn it, now he had doubts.
Of course he didn't believe Frey's story, but then, he didn't know enough about guns to really _dis_ believe him either. Maybe you _could_ tell which chamber the bullet was in by watching it spin.
It seemed like the kind of trick a man like Frey, who spent his days in lowlife bars and Rake dens, might be capable of.
Did the captain have it all under control the whole time, or did he just get lucky? Would he have given up the _Ketty Jay_ 's ignition code, or would he have let Crake die? Was he lying or wasn't he? Crake didn't know anymore. But the margin of uncertainty was too great to condemn him.
With a disgusted snort, he cut Frey's bonds.
Crake was no swordsman, but he barely had to move his wrist and the cutlass did the rest. It chopped neatly through the gap between Frey's hands, dividing the cord in two. Crake threw the cutlass back to Frey, walked over to Rat's corpse, and pulled the pistol from his holster.
Frey chambered a new round into the shotgun. "Ready?"
Crake made a sweeping gesture of sarcastic gallantry toward the door. _Be my guest_.
Beyond was a balcony that overlooked a dim barroom, musty with smoke and spilled wine. It was empty at this hour of the morning, its tables still scattered with the debris of the previous night's revelries. Tall shutters held off the pale daylight. Macarde was yelling somewhere below, raising the alarm.
Two men were racing up the stairs as Frey and Crake emerged. Macarde's men, wielding pistols, intent on murder. They saw Frey and Crake an instant before the first thug slipped on Crake's vomit slick, which no one had thought to clean up. He crashed heavily onto the stairs and his companion tripped over him. Frey blasted them twice with his shotgun, shattering the wooden balusters in the process. They didn't get up again.
Frey and Crake ran for a door at the far end of the balcony as four more men appeared on the barroom floor. They flung the door open and darted through, accompanied by a storm of gunfire.
Beyond was a corridor. The walls were painted in dull institution-green paint, flaking with age. Several doors in chipped frames led off the corridor: rooms for guests, all of whom had wisely stayed put.
Frey led the way along the corridor, which ended in a set of tall, shuttered windows. Without breaking stride, he unloaded the remainder of the shotgun's shells into them. Glass smashed and the shutters blew from their hinges. Frey jumped through the gap that was left, and Crake, possessed of an unstoppable, fear-driven momentum, followed him.
The drop was a short one, ending in a steeply sloping cobbled lane between tall, ramshackle houses. Overhead, a weak sun pushed through hazy layers of cloud.
Crake hit the ground awkwardly and went to his knees. Frey pulled him up.
"I feel a sudden urge to be moving on," he said, as he dusted Crake down. "Open skies, new horizons, all of that."
Crake looked up at the window they'd jumped from. The sounds of pursuit were growing louder. "I have the same feeling," he said, and they took to their heels.
# _Chapter Two_
A NEW RECRUIT—MANY INTRODUCTIONS—JEZ SPEAKS OF AIRCRAFT—THE CAPTAIN'S RETURN
here she is," said Malvery, with a grand sweep of his arm. "The _Ketty Jay._ "
Jez ran a critical eye over the craft resting on the stone landing pad before them. A modified Ironclad, originally manufactured in the Wickfield workshops, unless she missed her guess. The _Ketty Jay_ was an ugly, bulky thing, hunched like a vulture, with a blunt nose and two fat thrusters mounted high up on her flanks. There was a stubby tail assembly, the hump of a gun emplacement, and wings that swept down and back. She looked as if she couldn't decide whether she was a light cargo hauler or a heavy fighter, and so she wouldn't be much good as either. One wing had been recently repaired, there was cloud rime on the landing struts, and she needed scrubbing down.
Jez wasn't impressed. Malvery read her reaction at a glance and grinned: a huge grin, which sprang into place beneath his thick white walrus mustache.
"Ain't the loveliest thing you'll ever see, but the bitch does fly. Anyway, it's what's in the guts that counts, and I speak from experience. I'm a doctor, you know!"
He gave an uproarious laugh, holding his sides and throwing his head back. Jez couldn't help but smile. Malvery's guffaw was infectious.
There was something immediately likable about Malvery. It was hard to withstand the force of his good humor, and despite his large size he seemed unthreatening. A great, solid belly pushed out from his coat, barely covered by a faded pullover that was stained with the evidence of a large and messy appetite. His hair had receded to a white circlet around his ears, leaving him bald on top, and he wore small round glasses with green lenses.
"What happened to your last navigator?" she asked.
"Found out he'd been selling off spare engine parts on the side. He navigated himself out the cargo door with the Cap'n's toe up his arse." Malvery roared again, then, noticing Jez's expression, he added, "Don't worry, we were still on the ground. Not that the thieving little bastard didn't deserve dropping in a volcano." He scratched his cheek. "Tell you the truth, we've had bad luck with navigators. Been through seven in the past year. They're always ripping us off or disappearing in the night or getting themselves killed or _some_ damn thing."
Jez whistled. "You're making this job sound awfully tempting."
Malvery clapped her on the back. "Ah, it ain't so bad. We're a decent lot. Not like the cutthroat scum you might take on with otherwise. Pull your weight and keep up, you'll be fine. You take a share of whatever we make, after maintenance and whatnot, and the Cap'n pays fair." He studied the _Ketty Jay_ fondly, balled fists resting on his hips. "That's about as much as you can ask for in this day and age, eh?"
"Pretty much," said Jez. "So what are you lot into?"
Malvery's look was unreadable behind his glasses.
"I mean, cargo hauling, smuggling, passenger craft, what? Ever work for the Coalition?"
"Not bloody likely!" Malvery said. "The Cap'n would sooner gulp a pint of rat piss." He reddened suddenly. "Pardon the language."
Jez waved it away. "Just tell me what I'm signing up for."
Malvery harrumphed. "We ain't what you'd call a very _professional_ lot, put it that way," he said. "Cap'n sometimes doesn't know his arse from his elbow, to tell you truth. Mostly we do black-market stuff, smuggling here and there. Passenger transport too. We help out people who want to get somewhere they shouldn't be going and don't want anyone finding out. And we've been known to try a bit of light piracy now and again when the opportunity comes along. I mean, the haulage companies sort of _expect_ to lose one or two cargoes a month. They budget for it, so there's no harm done." He made a vague gesture in the air. "We sort of do anything, really, if the price is right."
Jez deliberated for a moment. Their operation was clearly a shambles, but that suited her well enough. They didn't seem like types who would ask many questions, and she was lucky to find work at all in Scarwater, let alone something in her field of expertise. To keep moving was the important thing. Staying still too long was dangerous.
She held out her hand. "Alright. Let's see how it goes."
"Fine decision! You won't regret it. Much." Malvery enfolded her hand in thick, meaty fingers and shook it enthusiastically. Jez couldn't help wondering how he managed to button his coat with fingers like that, let alone perform complex surgery.
"You really a doctor?" she asked.
"Certified and bona fide!" he declared, and she smelled rum on his breath.
They heard a thump from within the belly of the craft. Malvery wandered around to the _Ketty Jay_ 's stern, and Jez followed. The cargo ramp was down. Inside, someone was rolling a heavy steel canister along the floor in the gloom. The angle prevented Jez from seeing anything more than a pair of long legs clad in thick trousers and boots.
"Might as well introduce you," said Malvery. "Hey there! Silo! Say hello to the new navvie."
The figure in the cargo hold stopped and squatted on his haunches, peering out at them. He was tall and narrow-hipped, but his upper body was hefty with muscle, a thin cotton shirt pulled tight across his shoulders and chest. Sharp eyes peered out from a narrow face with a beaked nose, and his head was shaven. His skin was a dark yellow-brown, the color of umber.
He regarded Jez silently, then got to his feet and resumed his labor.
"That's Silo. Engineer. Man of few words, you could say, but he keeps us all in the sky. Don't mind his manner; he's like that with everyone."
"He's a Murthian," Jez observed.
"That's right. You _have_ been around."
"Never seen one outside of Samarla. I thought they were all slaves."
"So did I," said Malvery.
"So he belongs to the Cap'n?"
Malvery chuckled. "No, no. Silo, he ain't no slave. They're friends of a sort, I suppose, though you wouldn't know it sometimes. _His_ story... well, that's between him and the Cap'n. They ain't said, and we ain't asked." He steered Jez away. "Come on, let's go meet our flyboys. The Cap'n and Crake ain't about right now. I expect they'll be back once their hangovers clear up."
"Crake?"
"He's a daemonist."
"You have a _daemonist_ on board?"
Malvery shrugged. "That a problem?"
"Not for me," Jez replied. "It's just... well, you know how people are about daemonists."
Malvery made a rasping noise. "You'll find we ain't a very judgmental lot. None of us is in much of a position to throw stones."
Jez thought about that and then smiled.
"You're not in with that Awakener lot, are you?" Malvery asked suspiciously. "If so, you can toddle off right now."
Jez imitated Malvery's rasp. "Not likely."
Malvery beamed and slapped her on the back, hard enough to dislodge some vertebrae. "Good to hear."
They walked out of the _Ketty Jay_ 's shadow and across the landing pad. The Scarwater docks were half empty, scattered with small- to medium-size craft. Delivery vessels and scavengers mostly. The activity was concentrated at the far end, where a bulbous cargo barque was easing itself down. Crews were hustling to meet the newcomer. A stiff breeze carried the metallic tang of aerium gas across the docks as the barque vented its ballast tanks and lowered itself gingerly onto its landing struts.
The docks had been built on a wide ledge of land that projected out over the still black lake that filled the bottom of the barren mountain valley. It was a wild and desolate place, but then, Jez had seen many like it. Remote little ports, hidden away from the world, inaccessible by any means but the air. There were thousands of towns like Scarwater existing beneath the notice of the Navy. Through them moved honest traders and smugglers alike.
It had started as a rest stop or a postal station, no doubt. A dot on the map, sheltered from the treacherous local winds, with a ready source of water nearby. Slowly it grew, spreading and scabbing as word filtered out. Opportunists arrived, spotting a niche. Those travelers would need a bar to quench their thirst, someone thought. Those drunkards would need a doctor to see to their injuries when they fell off a wall. And they'd need someone to cook them a good breakfast when they woke up. Most major professions in the cities were harshly regulated by the Guilds, but out here a man could be a carpenter, or a baker, or a craftbuilder, and be beholden to nobody but himself.
But where there was money to be made, there were criminals. A place like Scarwater didn't take long to rot out from the inside. Jez had been here only a week since leaving her last commission, but she'd seen enough to know how it would end up. Soon, the honest people would start to go elsewhere, driven out by the gangs, and those who were left would consume one another and move on. They'd leave a ghost town behind, like all the other ghost towns, haunted by abandoned dreams and lost possibilities.
To her left, Scarwater crawled up the stony hillside from the lake. Narrow lanes and winding stairways curved between simple rectangular buildings set in clusters wherever the land would take them. Aerial pipe networks cut across the streets in strict lines, steaming gently in the chill morning air, forming a scaffold for the jumble beneath them. Huge black mugger birds gathered on them in squads, watchful for prey.
_This isn't the place for me_ , she thought. But then, where was?
Ahead of them on the landing strip were two small fighter craft: a Caybery Firecrow and a converted F-class Skylance. Malvery led her to the Skylance, the closer of the two. Leaning against its flank, smoking a roll-up cigarette and looking decidedly the worse for wear, was a man Jez guessed was the pilot.
"Pinn!" Malvery bellowed. The pilot winced. "Someone you should meet."
Pinn crushed out the cigarette as they approached and extended a hand for Jez to shake. He was short, stout, and swarthy, with a shapeless thatch of black hair and chubby cheeks that overwhelmed his eyes when he managed a nauseous smile of greeting. He couldn't have been more than twenty, young for a pilot.
"Artis Pinn, meet Jezibeth Kyte," said Malvery. "She's coming on as navigator."
"Jez," she corrected. "Never liked Jezibeth."
Pinn looked her up and down. "Be nice to have a woman on board," he said, his voice deep and toneless.
"Pinn isn't firing on all cylinders this morning, are you, boy?" Malvery said, slapping him roughly on the shoulder. Pinn went a shade grayer and held up his hand to ward off any more blows.
"I'm an inch from losing my breakfast here," he murmured. "Lay off." Malvery guffawed and Pinn cringed, pummeled by the doctor's enormous mirth.
"You modified this yourself?" Jez asked, running a hand over the Skylance's flank. The F-class was a racer, a single-seater built for speed and maneuverability. It had long, smoothly curved gull wings. The cockpit was set far back along the fuselage, to make space for the enormous turbine in its nose that fed to a thruster at the tail end. This one had been bulked out with armor plate and fitted with underslung machine guns.
"Yeah." Pinn roused a bit. "You know aircraft?"
"Grew up around them. My dad was a craftbuilder. I used to fly everything I could get my hands on." She nodded toward the _Ketty Jay_. "I bet I could even fly _that_ piece of crap."
Malvery snorted. "Good luck getting the Cap'n to let you."
"What was your favorite?" Pinn asked her.
"My dad built me an A-18 for my sixteenth birthday. I loved that little bird."
"So what happened? You crash it?"
"She gave up the ghost five years back. I put her down in some port up near Yortland, and she just never took off again. I didn't have two shillies to bash together for repairs, so I took on with a crew as a navvie. Thought I could do long-haul navigation easy enough; I mean, I'd been doing it for myself all that time on the short-haul. That first trip I got us lost; we wandered into Navy airspace, and a couple of Windblades nearly blew us out of the sky. Had to learn pretty quick after that."
"I like her," Pinn said to Malvery.
"Well, good," he replied. "Come on, let's say hi to Harkins." They nodded their farewells.
"He ain't a bad lad," said Malvery as they walked over to the Firecrow. "Dumb as a rock, but he's talented, no doubt about that. Flies like a maniac."
Firecrows had once been the mainstay of the Navy, until they were succeeded by newer models. They were built for dogfighting, with two large prothane thrusters and machine guns incorporated into the wings. A round bubble of windglass was set into the blunt snout to give the pilot a better field of vision from the cockpit, which was set right up front, in contrast to the Skylance.
Harkins was in the Firecrow, running rapidly through diagnostics. He was gangly, unshaven, and hangdog, wearing a leather pilot's cap pushed far back. His dull brown hair was thin and receding from his high forehead. Flight goggles hung loosely around his neck. He moved in quick jerks, like a mouse, tapping gauges and flicking switches with an expression of fierce concentration. As they approached, he burrowed down to examine something in the foot well.
"Harkins!" Malvery yelled at the top of his considerably loud voice. Harkins jumped and smashed his head noisily on the flight stick.
"What? What?" Harkins cried, popping up with a panicked look in his eyes.
"I want to introduce you to the new navvie," Malvery said, beaming. "Jez, this is Harkins."
"Oh," he said, taking off his hat and rubbing his crown. He looked down at Jez, then launched into a nervous babble, his sentences running into one another in their haste to escape his mouth. "Hi. I was doing, you know, checking things and that.
Have to keep her in good condition, don't I? I mean, what's a pilot without a plane, right? I guess you're the same with maps. What's a navigator without a map? Still a navigator, I suppose; it's just that you wouldn't have a map, but you know what I mean, don't you?" He pointed at himself. "Harkins. Pilot."
Jez was a little stunned. "Pleased to meet you," was all she could say.
"Is that the Cap'n?" Harkins said suddenly, looking away across the docks. He pulled the flight goggles up and over his eyes. "It's Crake and the Cap'n," he confirmed. His expression became alarmed again. "They're, um, they're running. Yep, running down the hill. Toward the docks. Very fast."
Malvery looked skyward in despair. "Pinn!" he called over his shoulder. "Something's up!"
Pinn sloped into view around his Skylance and groaned. "Can't it wait?"
"No, it bloody can't. Tool up. Cap'n needs help." He looked at Jez. "Can you shoot?"
Jez nodded.
"Grab yourself a gun. Welcome to the crew."
# _Chapter Three_
A HASTY DEPARTURE—GUNPLAY—ONE Is WOUNDED—A TERRIFYING ENCOUNTER
hey were passing out weapons, gathered behind a stack of crates that had been piled up astern of the _Ketty Jay_ , when Crake and the captain reached them.
"Trouble?" Malvery asked.
"Must be that time of the week," Frey replied, then yelled for Silo.
"Cap'n," came the baritone reply from the Murthian, who was squatting at the top of the cargo ramp.
"You get the delivery?"
"Yuh. Came an hour ago."
"How long 'til you can get her up?"
"Aerium's cycling through. Five minutes."
"Fast as you can."
"Yes, Cap'n." He disappeared into the hold.
Frey turned to the others. "Harkins. Pinn. Get yourselves airborne. We'll meet you above the clouds."
"Is there gonna be a rumble?" Pinn asked hopefully, rousing briefly from his hangover. Harkins was already halfway to his aircraft by the time Pinn finished the sentence.
"Get out of here!" Frey barked at him. Pinn mumbled something sour under his breath, stuffed his pistol into his belt, and headed for the Skylance, oozing resentment at being cheated of a fight.
"Macarde's on his way," said Frey, as Malvery passed him a box of bullets. "Bringing a gang with him."
"We're low on ammo," Malvery murmured. "Make 'em count."
"Don't waste too many on Crake, then," Frey said, loading the lever-action shotgun he'd taken from Droop-Eye. "He couldn't hit the side of a frigate if he was standing next to it."
"Right-o, Cap'n," said Malvery, giving Crake a generous handful anyway. Crake didn't rise to the jibe. He looked about ready to keel over from the run.
Frey nodded at Jez. "Who's this?"
"Jez. New navvie," Malvery said, with the tone of someone who was tired of introducing the same person over and over.
Frey gave her a cursory appraisal. She was small and slight, which was good, because it meant she wouldn't take up too much space and would hopefully have an equally small appetite. Her hair was tied in a simple ponytail, which, along with her unflatteringly practical clothes, suggested a certain efficiency. Her features were petite and appealing, but she was rather plain, boyish, and very pale. That was also good. An overly attractive woman was fatal on a craft full of men. They were distracting and tended to substitute charm and flirtatiousness for doing any actual work. Besides, Frey would feel obliged to sleep with her, and that never worked out well.
He nodded at Malvery. She'd do.
"So who's Macarde, then?" Jez asked, chambering bullets as she spoke. When they looked at her, she shrugged and said, "I like to know who I'm shooting."
"The story, in a nutshell," said Malvery. "We sold the local crime lord twelve canisters of degraded aerium at cut-rate prices so we could raise the money to buy three canisters of the real stuff, since we barely had enough to get off the ground ourselves."
"Problem is, our contact let us down," said Frey, settling into position behind the crates and sighting along his shotgun. "His delivery came late, which meant he couldn't get us the merchandise on time, which meant we were stuck in port just long enough for one of Macarde's bumble-butt pilots to fly into a wall."
"Hence the need for a swift departure," said Malvery. "Flawless plans like this are our stock-in-trade. Still want to sign on?"
Jez primed her rifle with a satisfying crunch of metal. "I was tired of this town anyway."
The four of them took up position behind the crates, looking out at the approach road to the docks. The promontory was accessed by way of a wide cobbled thoroughfare that ran between a group of tumbledown warehouses. The dockers who worked there were moving aside as if pushed by a bow wave, driven to cover by the sight of Lawsen Macarde and twenty gun-wielding thugs storming down the street.
"That'll be us outnumbered and outgunned, then," Malvery murmured. He looked back to where the Skylance and the Firecrow were rising from the ground, aerium engines throbbing as their electromagnets turned refined aerium into ultralight gas to fill their ballast tanks. Separate, prothane-fueled engines, which powered the thrusters, were warming up with an ascending whine.
"Where's Bess, anyway?" Frey asked Crake.
"Do I look like I've got her in my pocket?" he replied irritably.
"Could do with some help right now."
"She'll be cranky if I have to wake her up."
"Cranky is how I want her."
Crake pulled out a small brass whistle that hung on a chain around his neck and blew it. It made no sound at all. Frey was about to offer a smart comment concerning Crake's lack of lung power when a bullet smashed into a crate near his head, splintering through the wood. He swore and ducked reflexively.
Crake replaced the whistle, then leaned out of cover and unleashed a wild salvo of pistol fire. His targets yelled and pointed fearfully, then scattered for cover, throwing themselves behind sacks and barrels that were waiting to be loaded into the warehouses.
"Ha!" Crake cried in triumph. "It seems _they_ don't doubt my accuracy with a pistol."
An instant later, his hair was blown forward as Pinn's Skylance tore through the air mere feet above him, machine guns raking the street. Barrels were smashed to matchwood, and several men jerked and howled as they were punched with bullets. The Skylance shrieked up the street, and then twisted to vertical, arrowing into the clouds and away.
"Yeah," said Frey, deadpan. "You're pretty scary with that thing."
The dockers had all fled inside by now, leaving the way clear for the combatants. Macarde's men were at the edge of the landing pad, fifty feet away. Between them was a small, two-man flier and too much cover for Frey's liking. The smugglers had been shocked by Pinn's assault, but they were regrouping swiftly.
Frey and Jez began laying down fire, making them scuttle. One smuggler went down, shot in the leg. Another unwisely took shelter behind a large but empty packing crate. Malvery hefted a double-barreled shotgun, aimed, and blew a ragged hole through the crate and the man behind it.
"Silo! How we doing?" Frey called, but the mechanic couldn't hear him over the return fire from the smugglers.
"Darian Frey!" Macarde called from his hiding place behind a stack of aircraft tires. "You're a dead man!"
"Threats," Frey murmured. "Honestly, what's the point?"
"They're trying to flank us!" said Jez. She fired at one of the smugglers, who was scampering from behind a pile of broken hydraulic parts. The bullet cut through the sleeve of his shirt, missing him by a hair. He froze mid-scamper and fled back into hiding.
"Cheap kind of tactic, if you ask me," Crake commented, having recovered sufficient breath for a spot of nervous bravado. He knocked the shells from the drum of his revolver and slotted five new ones in. "The kind of sloppy, unoriginal thinking you come to expect from these mid-level smuggler types."
Jez peered around the side of the crates, looking for the man she'd shot at. Instead, she saw another, making his way from cover to cover, attempting to get an angle on them. He disappeared before she could draw a bead on him.
"Can I get a bit less wit and a bit more keeping your bloody eyes open for these sons of bitches coming round the side?" she snapped.
"She's no shrinking violet, I'll give her that," Frey commented to Malvery.
"The girl's gonna fit right in," the doctor agreed.
More of Macarde's gang had moved up and taken shelter behind the two-man flier. Crake was peppering it with bullets.
"Ammo!" Malvery reminded him.
Frey ducked away as a salvo of gunfire blasted chips from the stone floor and splintered the wood of the crates. Malvery answered with his shotgun, loudly enough to discourage any more, then dropped back to reload.
Jez stuck her head out again, concerned that she'd lost sight of the men who were trying to flank them. Despite her warning, her companions were still preoccupied with taking potshots at the smugglers approaching from the front.
A flash of movement: there was another one! A third man, edging into position to shoot from the side, where their barricade of crates would be useless.
"Three of them over here!" she cried.
"We're a little busy at the moment," Frey replied patiently.
"You'll be busy picking a bullet out of your ear if you don't—" she began, but then she got shot.
It was a white blaze of pain, knocking the wind from her and blasting her senses. Like being hit by a piston. The impact threw her backward, into Crake, who half-caught her as she fell.
"She's hit!" he cried.
"Already?" Frey replied. "Damn, they usually last longer than _that_. Malvery, take a look."
The doctor blasted off two shots to keep the smugglers' heads down, then knelt next to Jez. Her already unhealthy pallor had whitened a shade further. Dark-red blood was soaking through her jacket from her shoulder. "Ah, girl, come on," he murmured. "Don't be dying or anything."
"I'm alright, Doc," she said through gritted teeth. "I'm alright."
"Just you stay still."
"Haven't got _time_ to stay still," she replied, struggling to her feet, clutching her shoulder. "I _told_ you they were coming round the side! Where's the one who...?" She trailed off as she caught sight of something behind them coming down the cargo ramp, and her face went slack. "What is _that_?"
Malvery turned and looked. "That? That's Bess."
Eight feet tall and five broad, a half-ton armored monstrosity loomed out of the darkness into the light of the morning. There was nothing about her to identify her as female. Her torso and limbs were slabbed with molded plates of tarnished metal, with ragged chain mail weave beneath. She stood in a hunch, the humped ridge of her back rising higher than her enormous shoulders. Her face was a circular grille, a crisscross of thick bars like the gate of a drain. All that could be seen behind it were two sharp glimmers: the creature's eyes.
Jez caught her breath. A golem. She'd only heard of such things.
A low growl sounded from within the creature, hollow and resonant. Then she came down the ramp, her massive boots pounding the floor as she accelerated. Cries of alarm and dismay rose from the smugglers. She jumped off the side of the ramp and landed with a rattling boom that made the ground tremble. One gloved hand scooped up a barrel that would have herniated the average human and flung it at a smuggler who was hiding behind a pile of crates. It smashed through the crates and crushed the man behind, burying him under an avalanche of broken wood.
"Well, she's cranky, alright," said Frey. "Good ol' Bess."
A roaring tower of fury, the golem tore into the smugglers who had been sneaking around the flanks. Bullets glanced from her armor, leaving only scratches and small dents. One of the smugglers, panicking, made a break from cover. She seized him by the throat with a loud crack and then flung his limp corpse at his companions.
Another man tried to race past her while her back was turned, but she was quicker than her bulk suggested. She lunged after him, grabbing his arm with massive fingers. Bone splintered in the force of her grip. Her victim's brief shrieks were cut short as she tore the arm from its socket and clubbed him across the face with it, hard enough to knock him dead.
The remainder of Macarde's men suddenly lost their taste for the fight. They turned tail and ran.
"What are you doing?" Macarde screamed at them from his hiding place near the rear of the conflict. "Get your filthy yellow arses back there and shoot that thing!"
Bess swung around and fixed her attention on him, a deep rattling sound coming from her chest. He swallowed hard.
"Don't ever come back here, Frey, you hear me?" he called, backing off a few steps as he did so. "You ever come back, you're dead! You hear me? Dead! I'll rip out your eyes, Frey!"
His parting shot was barely audible, since he was bolting away as he delivered it. Soon he had disappeared, chasing his men back into the tangled lanes of Scarwater.
"Well," said Frey. "That's that."
"She up and ready, Cap'n!" Silo hollered from the top of the cargo ramp.
"Exquisite timing, as always," Frey replied. "Malvery, how's the new recruit?"
"I'm okay," Jez said. "It went right through."
Malvery looked relieved. "So you won't need anything taken out, then. Just some disinfectant, a bandage, and you'll be alright."
Jez gave him an odd look. "I suppose so."
"She's a tough little mite, Cap'n," Malvery declared, with a tinge of pride in his voice, as if her courage was some doing of his.
"Next time, try not to get shot," Frey advised her.
"I wouldn't have _been_ shot if you'd bloody listened to me."
Frey rolled his eyes. "Doc, take her to the infirmary."
"I'll be fine," Jez protested.
"You just had a bullet put through your shoulder!" Malvery cried.
"It'll heal."
"Will you two get on that damn aircraft?" Frey said. "Crake! Bring Bess. We're leaving ten minutes ago!"
Frey followed Malvery and Jez up the ramp and into the _Ketty Jay_. Once they were out of sight, Crake stepped gingerly through the wreckage and laid a hand on the golem's arm. She turned toward him with a quiet rustle of chain mail and leather. He reached up and stroked the side of her face grille, tenderness in his gaze.
"Well done, Bess," he murmured. "That's my girl."
# _Chapter Four_
A PILOT'S LIFE—CRAKE IS LISTLESS—MALVERY PRESCRIBES A DRINK
here were very few moments in Jandrew Harkins's life when he could be said to be truly relaxed. Even in his sleep he'd jitter and writhe, tormented by dreams of the wars or, occasionally, dreams of suffocation brought on by Slag, the _Ketty Jay_ 's cat, who had a malicious habit of using Harkins's face as a bed.
But here, nestled in the cramped cockpit of a Firecrow with the furnace roar of prothane thrusters in his ears, here was peace.
It was a calm day in the light of a late autumn sun. Hard to believe that only a few hours earlier they were in the middle of a gunfight on the docks at Scarwater. Now they were heading north, following the line of the Hookhollow Mountains, heading for Marklin's Reach, where the Cap'n had business.
The _Ketty Jay_ was above him and half a mile to starboard. Pinn's Skylance droned alongside. There was nothing else in the sky except a Navy frigate lumbering across the horizon to the west and a freighter out of Aulenfay, surfacing from the sea of cloud that had submerged all but the highest peaks. To the east it was possible to see the steep wall of the Eastern Plateau, tracing the edge of the Hookhollows. Farther south, the cloud was murky with volcanic ash, drifting toward the Blackendraft flats.
He looked up, through the windglass of his cockpit canopy. The sky was a perfect, clear, deep blue. Never-ending.
Harkins sighed happily. He checked his gauges, flexed his gloved hand on the control stick, and rolled his shoulders. Outside this tight metal womb, the world was strange. _People_ were strange. Men were frighteningly unpredictable and women more so, full of strange insinuations and cloaked hunger. Loud noises made him jump; crowds made him claustrophobic; smart people made him feel stupid.
But the cockpit of a Caybery Firecrow was his sanctuary, and had been for twenty years. No awkwardness or embarrassment could touch him while he was encased in this armor. Nobody laughed at him here. The craft was his mute servant, and he, for once, was master.
He watched the distant Navy frigate for a time, remembering. Once, as a younger man, he'd traveled in craft like that. Waiting for the call to clamber into his Firecrow and burst out into the sky. He remembered with fondness the pilots he'd trained with. He'd never been popular, but he'd been accepted. Part of the team. Those were good days.
But the good days had ended when the Aerium Wars began. Five years fighting the Sammies. Five years when every sortie could be the one from which you never came back. Five years of nerve-shredding dogfights, during which he was downed three times. He survived. Many of his friends weren't so fortunate.
Then there was the peace, although the term was relative. Instead of Sammies, the Navy was after the pirates and freebooters who had prospered during the war, running a black-market economy. Harkins fought the smugglers in his own lands. The enemy wasn't so well equipped, but they were more desperate, more savage. Turf wars became grudge matches, and things got even uglier.
Then, unbelievably, came the Second Aerium War, a mere four years after the first, and Harkins was back fighting alongside the Thacians against the Sammies and their subjects. After all they'd done the first time, all the lives that were lost, it was the politicians who let them down. Little had been done to defang the Samarlan threat, and the enemy came back with twice the vigor.
It was a short and dirty conflict. People were demoralized and tired on all sides. By the end—an abrupt and unsatisfying truce that left everyone but the Sammies feeling cheated—Harkins was out of it. He'd had too many near misses, lucked out a little too often, seen death's face more than any man should have to. He was a trembling shell. They discharged him two weeks before the end of the war, after fourteen years in the service. The meager pension they gave him was all the Navy could afford after such a ruinous decade.
Those years were the worst years of all.
Harkins had come to realize that the world changed fast these days, and it wasn't kind to those who weren't adaptable. He had no skills other than those he'd learned as a fighter pilot, and nobody wanted a pilot without a plane. A bleak gray time followed, working in factories, doing odd jobs, picking up a pittance. Scraping a living.
It wasn't Navy life that he missed, with its discipline and structure. It wasn't the camaraderie—that had soured after enough of his friends died. It was the loss of the Firecrow that truly ached.
Though he'd flown almost a dozen different Firecrows, with minor variations and improvements as time went on, they were all the same in his mind. The sound of the thrusters, the throb of the aerium engines pumping gas into the ballast tanks, the enclosing, unyielding hardness of the cockpit. The Firecrow had been the setting for all his glories and all his tragedies. It had carried him into the wondrous sky, it had seen him through the most desperate dogfights, and occasionally it had failed him when it had no more to give. Everything truly important that had ever happened in his life, the moments of purest joy and sheer, naked terror, had happened inside a Firecrow.
Then, in his darkest hour, there came a light. It was almost enough to make him believe in the Allsoul and the incomprehensible jabber of the Awakeners. Almost, but not quite.
His overseer at the factory knew about Harkins's past as a pilot for the Coalition Navy. It was all Harkins talked about, when he talked at all. So when the overseer met a man in a bar who was selling a Firecrow, he mentioned it to Harkins.
That was how Harkins met Darian Frey, who had won a Caybery Firecrow on an improbably lucky hand of Rake and now had no idea what to do with it. Harkins had barely enough money to keep a roof over his head, but he went to Frey to beg. He'd have sold his soul if it got him back into the cockpit. Frey didn't think his soul was worth much, so he suggested a deal instead.
Harkins would fly the Firecrow on Frey's behalf. The pay would be lousy, the life unpredictable, probably dangerous, and usually illegal. Harkins would do exactly as Frey said, and if he didn't, Frey would take his craft back.
Harkins had agreed before Frey even finished laying out the terms. The same day, he left port as an outflier for the _Ketty Jay_. It was the happiest day of his life.
It had been a long journey from that Navy frigate to here, flying over the Hookhollow Mountains under Darian Frey. He'd never again have the kind of steel in his spine he had as a young pilot. He'd never have the obscene courage of Pinn, who laughed at death because he was too dim to comprehend it. But he'd tasted what life was like trapped on the ground, unable to rise above the clouds to the sun. He was never going back to that.
He glanced around apprehensively, as if someone, somewhere, might be watching him. Then he settled back into the hard seat of the Firecrow and allowed himself a broad, contented smile.
FOR CRAKE, THERE WAS no such contentment. Listless, he wandered the tight confines of the _Ketty Jay_. There was a strange void in his belly, as if the wind had been knocked out of him. He drifted about, a specter of bewildered sadness.
At first he'd confined himself to the near-vacant cargo hold, until the space began to oppress him and his mood started to make Bess uneasy. After that, he went to the mess and drank a few mugs of strong coffee while sitting at the small communal table. But the mess felt bleak with no one there to share it.
So he climbed up the ladder from the mess to the passageway that linked the cockpit at the fore of the craft to engineering in the aft. In between were several rooms that the crew used as quarters, their sliding doors stained with ancient, oily marks. Electric lights cast a dim glow on the grimy metal walls.
He thought about going up to the cockpit to have a look at the sky, but he couldn't face Frey right now. He considered going to his quarters, perhaps to read, but that was unappealing too. Finally he remembered that their new recruit had managed to get herself shot, and he decided it would be the decent thing to go and inquire after her health. With that in mind, he walked down the passageway to Malvery's infirmary.
The door was open when he got there, and Malvery had his feet up, a mug of rum in his hands. It was a tiny, squalid, and unsanitary chamber. The furniture comprised little more than a cheap dresser bolted to the wall, a washbasin, a pair of wooden chairs, and a surgical table. The dresser was probably intended for plates and cutlery, but it had found new employment in the display of all manner of unpleasant-looking surgical instruments. They were all highly polished—the only clean things in the room—and they looked as if they'd never been used.
Malvery hauled his feet off the chair where they were resting and shoved it toward Crake. Then he poured a stiff measure of rum into another mug that sat on the dresser. Crake obligingly sat down and took the proffered mug.
"Where's the new girl?" Crake asked.
"Up in the cockpit. Navigating."
"Didn't she just get shot?"
"You wouldn't think so, the way she's acting," Malvery said. "Damnedest thing. When she finally let me have a look at her, the bleeding had already stopped. Bullet went right through, like she said." He beamed. "All I had to do was swab it up with some antiseptic and slap on a patch. Then she got up and told me she had a job to do."
"You were right; she _is_ tough."
"I ain't seen many patients shrug off a bullet wound like that," Malvery agreed.
Crake took a swig of rum. It was delightfully rough stuff, muscling its way to his brain, where it set to work demolishing his finer mental functions.
Malvery adjusted his round green-tinted glasses and harrumphed. "Out with it, then."
Crake drained his mug and held it out for a refill. He thought for a moment. There was no way to express the shock, the betrayal, the resentment he felt, not in a way that Malvery would truly understand. So he simply said, "I almost died today."
He told Malvery what had happened after he and Frey were captured. It was an effort to keep everything factual and objective, but he did his best. Clarity was important. Emotional outbursts went against his nature.
When Crake had finished, Malvery poured himself another shot and said, "Well."
Crake found his comment somewhat unsatisfying. When it became clear the doctor wasn't going to elaborate, he said, "He let Macarde spin the barrel, put it to my forehead, and pull the trigger. _Twice!_ "
"You were lucky. Head wounds like that can be nasty."
"Do you think he was lying? I mean, how could he know which chamber the bullet was in?"
Malvery shrugged. "I ain't the Cap'n. No telling what he can and can't do."
"But you know him better than I do. Would he really have let Macarde pull the trigger if he wasn't sure?"
"I've known him _longer_ than you have," Malvery said. "Wouldn't say I know him much _better_ , though. Not enough to speak for the man." He tipped his mug at his companion and hunkered forward in his chair. "I like you, Crake. You're a good one. But you gotta realize this ain't your world you're living in anymore."
"You don't know a thing about my world!" Crake protested.
"Don't think so?" He swept out a hand to indicate the room. "Time was I wouldn't set foot in a place like this. I used to be Guild-approved. Worked in Thesk. Earned more in a month than this little operation makes in a year."
Crake eyed him uncertainly, trying to imagine this enormous, battered old drunkard visiting the elegant dwellings of the aristocracy. He couldn't.
"This ain't no family, Crake," Malvery went on. "Every man is firmly and decidedly for himself. You're a smart feller; you knew the risks when you threw your lot in with us. The _Ketty Jay_ is the Cap'n's home. It's his livelihood. Pretty much the only thing of value he's got in the world. He ain't got anyone to fall back on. So what makes you think the Cap'n should give up his craft in exchange for you?"
"Because..." Crake began, and then realized he'd nothing to say. _Because it would have been the right thing to do_. He'd spare himself Malvery's laughter.
"Look," Malvery said, more gently. "The Cap'n, he's got a way with people, when he has a mind to try. But don't let that fool you. The _Ketty Jay_ is what he cares about. Everyone else comes a distant second." He straightened and regarded Crake over the rims of his glasses. "Now, if you think that's heartless, then you ain't seen the half of what's out there. The Cap'n's a good 'un. Better than most. You just got to know how he is."
Crake didn't have an answer to that. He didn't want to say something childishly bitter. Already he felt faintly embarrassed at bringing it up.
"Maybe you're right," he said. "Maybe I shouldn't be here."
"Hey, now, I didn't say _that_!" Malvery grinned. "Just saying, you got to realize not everyone thinks like you. Hard lesson, but worth it."
Crake said nothing and sipped his rum. His sad mood was turning black. Perhaps he _should_ just give it up. Get off at the next port, turn his back on all this. It had been six months. Six months of moving from place to place, living under an assumed name, muddying his traces so nobody could find him. At first he'd lived like a rich hobo, haunting shabby hotels all over Vardia, his days and nights spent in terror or drunken grief. It was three months before the money began to run short and he collected himself a bit. That was when he found Frey and the _Ketty Jay_.
Surely the trail had gone cold by now?
"You're not _really_ thinking of packing it in, are you?" Malvery prompted, turning serious again.
Crake sighed. "I don't know if I can stay. Not after that."
"Bit daft if you leave now. The way I understand it, you paid passage for the whole year with that cutlass."
Crake shrugged, morose. Malvery shoved him companionably with his boot, almost making him tip off his chair.
"Where you gonna go, eh?" he said. "You _belong_ here."
"I _belong_ here?"
"Of course you do!" Malvery bellowed. "Look at us! We're not smugglers or pirates. We're not a _crew_! The Cap'n's only the cap'n 'cause he owns the aircraft; I wouldn't trust him to lead a bear to honey. None of us here signed on for adventure or riches, 'cause sure as spit there's little enough of either." He gave Crake a conspiratorial wink. "But, mark me, ain't one of us that's not running from something, you included. I'll bet my last swig of rum on that." He swigged the last of his rum, just to be safe, then added, " _That's_ why you belong here. 'Cause you're one of us."
Crake couldn't help a smile at the cheap feeling of camaraderie he got from that. Still, Malvery was right. Where would he go? What would he do? He was treading water because he didn't know in which direction to swim. And until he did, the _Ketty Jay_ was as good a place as any to hide from the sharks.
"I just..." he said. "It's just... I thought he was my friend."
"He _is_ your friend. Kind of. Depends on your definition, really. I had lots of friends, back in the day, but most of 'em wouldn't have thrown me a shillie if I was starving." He opened a drawer in the dresser and pulled out a bottle of clear liquid. "Rum's done. Have a suck on this."
"What is it?" Crake asked, holding out his mug. He was already pleasantly fogged and long past the point of being capable of refusing.
"I use it to swab wounds," Malvery said.
"I suppose this is a medicinal-grade kind of conversation," Crake said. Malvery blasted him with a hurricane of laughter, loud enough to make him wince.
"That it is, that it is," he said, raising his glasses to wipe a teary eye.
"So why are _you_ here?" Crake asked. "Guild-approved doctor, big job in the city, earning a fortune. Why the _Ketty Jay_?"
Malvery's mood faltered visibly, a flicker of pain crossing his face. He looked down into his mug.
"Let's just say I'm exactly where I deserve to be," he said. Then he rallied with a flourish, lifting his mug for a toast.
"To friends!" he declared. "In whatever form they come, and howsoever we choose to define them."
"Friends," said Crake, and they drank.
# _Chapter Five_
FLYING IN THE DARK—PINN AND THE WHORES—A PROPOSITION IS MADE
ight had fallen by the time they arrived at Marklin's Reach. The decrepit port crouched in the sharp folds of the Hookhollows, a speckle of electric lights in the darkness. Rain pounded down from a slow-rolling ceiling of cloud, its underside illuminated by the pale glow of the town. A gnawing wind swept across the mountaintops.
The _Ketty Jay_ sank out of the clouds, four powerful lights shining from her belly. Her outfliers hung close to her wings as she descended toward a crowded landing pad. Beam lamps swiveled to track her from below; others picked out an empty spot on the pad.
Frey sat in the pilot seat of the _Ketty Jay_ 's cockpit, his eyes moving rapidly among the brass-and-chrome dials and gauges. Jez was standing with one hand resting on his chair back, looking out at the clutter of barques, freighters, fighters, and privateer craft occupying the wide square of flat ground on the edge of the town.
"Busy night," she murmured.
"Yeah," said Frey, distracted. Landing in foul weather at night was one of his least favorite things.
He watched the aerium levels carefully, venting a little and adding a little, letting the _Ketty Jay_ drift earthward while he concentrated on fighting the crosswinds that bullied him from either side. The bulky craft jerked and plunged as she was shoved this way and that. He swore under his breath and let a bit more gas from the trim tanks. The _Ketty Jay_ was getting overheavy now, dropping faster than he was comfortable with, but he needed the extra weight to stabilize.
"Hang on to something," he murmured. "Gonna be kind of rough."
The _Ketty Jay_ had picked up speed now and was coming in far too fast. Frey counted in his head with one eye on the altimeter, then, with a flurry of pedals and levers, he wrenched the thrusters into full reverse, opened the air brakes, and boosted the aerium engines to maximum. The craft groaned as its forward momentum was canceled and its descent arrested by the flood of ultralight gas into its ballast tanks. It slowed hard above the space that had been marked out for her, next to the huge metal flank of a four-story freighter. Frey dumped the gas from the tanks and she dropped neatly into the vacant spot, landing with a heavy thump on her skids.
He sank back in the chair and let a slow breath of relief escape him. Jez patted him on the shoulder.
"Anyone would think you were worried for a moment there, Cap'n," she said.
WATER SPLATTERED IN PUDDLES on the landing pad as the crew, wrapped in slickers and stamping their feet, assembled at the foot of the _Ketty Jay_ 's cargo ramp.
"Where are Malvery and Crake?" Frey asked.
Silo thumbed at the ramp, where a slurred duet could be faintly heard from the depths of the craft.
"Hey, I know that one!" Pinn said, and began to sing along, off-key, until he was silenced by a glare from Silo.
"What are we doing here, Cap'n?" Jez asked. The others were hugging themselves or stuffing their hands in their pockets, but she seemed unperturbed by the bitter wind.
"There's a man I have to see. A whispermonger, name of Xandian Quail. There shouldn't be any trouble, but that's usually when there's the most trouble. Harkins, Pinn, Jez, grab your guns and come with me. Silo, you take care of the docking permits, watch the aircraft, and all that." The tall Murthian nodded solemnly.
"Think I might need to do some diagnostics," blurted Harkins suddenly. "Check out the Firecrow, you know? She was all _tick-tick-tick_ on the port side, don't know what it was, best check it out probably, if you know what I mean. Don't want to fall out of the sky, you know, _zoooooom, crash_ , ha-ha. That wouldn't be much good to anyone, now, would it? Me dead, I mean. Who'd fly it then? Well, I suppose there'd be nothing to fly anyway if I crashed it. So all round it'd be best if I just ran my eye over the internals, make sure everything's shipshape, spickety-span."
Frey gave him a look. Harkins squirmed. It was transparently obvious that the thought of a gunfight terrified him.
"Diagnostics," he said, his voice flat. Harkins nodded eagerly. "Fine, stay."
The pilot's face split in a huge grin, revealing a set of uneven and lightly browned teeth. "Thank you, Cap'n!"
Frey surveyed the rest of his crew. "What are we all standing around for?" he said, clapping his hands together. "Get to it!"
THEY HURRIED THROUGH THE drenched streets of Marklin's Reach. The thoroughfares had become rivers of mud, running past the raised wooden porches of the shops and houses. Overhead, strings of electric lightbulbs fizzed and flickered as they were thrown about by the wind. Ragged children peered from lean-to shacks and alleyways where they sheltered. Water ramped off awnings and gurgled down gutters, the racket all but drowning out the clattering hum of generators. The air was thick with the smell of petrol, cooking food, and the clean, cold scent of new rain.
"Couldn't we go see this guy tomorrow instead?" Pinn complained. "I'd be drier underwater!"
Frey ignored him. They were already cutting it fine. Being held up in Scarwater had put them behind schedule. Quail had been clear in the letter: get here before the end of Howl's Batten or the offer would go dead. Frey had been lazy about picking up his mail, so he hadn't gotten the message for some time. With one thing and another, it had all ended up a bit last minute. He was rarely ever certain exactly what date it was, but he had the feeling today was the deadline, and he didn't want to delay any longer.
"Gonna end up with pneumonia, that's what's gonna happen," Pinn was grumbling. " _You_ try flying when your cockpit's waist-deep in wet snot."
Xandian Quail lived in a fortified compound set in a tumbledown cluster of alleys. His house hulked in the darkness, square and austere, its tall, narrow windows aglow. The grinding poverty experienced by the town's denizens was shut out with high walls and stout gates.
"I'm Darian Frey!" Frey yelled over the noise of the downpour. The guards on the other side of the gate seemed nonplussed. "Darian Frey! Quail's expecting me! At least he bloody well better be!"
One of the guards scampered over to the house, holding the hood of his slicker. A few moments later he was back and indicated to his companion that he should let them in.
They were escorted beneath the stone porch, where another guard—this one wearing a waistcoat and trousers and sporting a pair of pistols—opened the main door of the house. He had a long face and a patchy black beard. Frey recognized him vaguely from previous visits. His name was Codge.
"Guns," Codge said, holding out his hand. "And don't keep any back. You'll make me real upset if you do."
Frey hesitated. He didn't like the idea of going into a situation like this without firepower. He couldn't think of any reason for Quail to want him dead, but that did little to ease his mind.
It was the mystery that unnerved him. Quail had given no details in his letter. He'd only said that he had a proposition for Frey, for Frey in _particular_ , and that it might make him very rich. That in itself was enough to make Frey suspicious. It also made him curious.
_I just have to hear him out_ , Frey thought to himself. Anyway, they were here now, and he didn't much fancy tramping back to the _Ketty Jay_ until he'd warmed up a bit.
He motioned with his head to the others. _Hand 'em over_.
Once he'd collected their weapons, Codge stepped out of the way and let them into the entrance hall, where they stood, dripping. Three more armed guards lounged about in the doorways, exuding an attitude of casual threat. A pair of large, lean dogs loped over to investigate them. They were white, short-haired, and pink-eyed. Night hunters, which could see in the dark and track their prey by following heat traces. They sniffed over the newcomers, but when they reached Jez, they shied away.
"Time for a new perfume, Jez," Frey quipped. "I do have a way with animals, don't I?" she said, looking mildly put out.
Quail's house was a marked contrast to the dirty streets that had led to it. The floor and walls were tiled in black granite. Thick rugs had been laid underfoot. Coiled-brass motifs ran along the walls toward two curving staircases. Between the staircases was a large and complicated timepiece. It was a combination of clock and calendar, fashioned in copper and bronze and gold. Behind the hands were rotating discs with symbols for all ten months of the year and each of the ten days of the week. Frey was slightly relieved to see that the calendar read _Queensday Thirdweek, Howl's Batten_ —the last day of the month.
"Just you," said Codge, motioning up the stairs and looking at Frey. Frey shucked off his slicker and handed it to Pinn, who took it absently. The young pilot's attention had been snared by the four beautiful, seductively dressed women who had appeared in one of the doorways to observe the newcomers. They giggled and smiled at Frey as he headed for the stairs. He gave them a gallant bow, then took the hand of the foremost to kiss.
"You can butter up the whores later. The boss is waiting," Codge called. One of the women pooched out her lip at him, then favored Frey with a dirty smirk.
"He'll have to come down again, though, won't he?" she said, raising an eyebrow.
"Good evening, ladies," said Frey. "I'm sure my friend over there would love to entertain you until I return."
Pinn licked his palm, smoothed down the little thatch of hair atop his potatolike head, and put on his best nonchalant pose. The whores eyed him, unimpressed.
"We'll wait."
"FREY!" SAID XANDIAN QUAIL, as the captain entered the study. "Dramatically late, I see. I didn't think you'd come."
"Far as I'm concerned, a margin for error is just wasted space," Frey said, then shook hands with a hearty camaraderie far above what he actually felt for the man. Quail offered a glass of wine and did a magnificent job of not noticing the trail of muddy footprints that Frey had brought in with him.
Frey sat down and admired the room while Quail poured the drinks. The front of Quail's desk was carved in the likeness of a huge cloud eagle, stern and impressive. An ornate and valuable brass barometer hung behind it, the arrow pointing firmly toward RAIN. The windows had complicated patterned bars set on the outside, for security and decoration alike. A black iron candelabra hung from the ceiling, bulbs glowing dimly with electric power. The walls were paneled in mahogany and lined with books. Frey read some of the titles but didn't recognize any. It was hardly a surprise. He rarely read anything more complicated than the sensationalist broadsheets they sold in the cities.
Quail gave Frey a crystal glass of rich red wine, then sat opposite him with a glass of his own. He'd probably been handsome once, but no longer. A fiery crash in a fighter craft had seen to that. Now half his bald head was puckered with scar tissue, and there was a small metal plate visible on one side of his skull. A brassy orb sat in the socket where his left eye should have been, and his left arm was entirely mechanical.
In spite of this, he carried himself like an aristocrat, and dressed like one too. He wore a brocaded black jacket with a stiff collar, and his patent leather shoes shone with polish. Wet, sweaty, and disheveled, Frey was unimpressive by comparison.
"I'm glad you made it," said Quail. "Another day and I'd have offered my proposition elsewhere. Time is a factor."
"I just came to hear what you have to say," said Frey. "Make your pitch."
"I have a job for you."
"I know your rates," Frey said. "I don't have that kind of money."
"I'm not selling the information. This one's for free."
Frey sipped his wine and studied the other man.
"I thought whispermongers always stayed neutral," Frey said.
"Those are the rules," said Quail. He looked down at his mechanical hand and flexed the fingers thoughtfully. "You don't get involved, you don't take sides, you never reveal your sources or your clients. Just hard information, bought and sold. You trade secrets but you never take advantage of them."
"And you certainly don't offer _jobs._ "
"With what we know, you think we're never tempted? We're only human, after all." Quail smiled. "That's why we're very particular about who we use. It wouldn't be good for our profession if it were known that we occasionally indulge in a little self-interest."
"I'm listening."
"There's a barque out of Samarla, heading for Thesk. The _Ace of Skulls_. Minimum escort, no firepower. They want to keep things low key, as if it's just another freight run. They don't want attention. From pirates _or_ the Navy."
The _Ace of Skulls_. As a keen player of the game of Rake, Frey didn't miss its significance. The Ace of Skulls was the most important card in the game. "What are they carrying?"
"Among other things, a chest of gems. Uncut gems, bound for a Jeweler's Guild consortium in the capital. They cut a deal with a mining company across the border, and they're flying them back in secret to avoid the Coalition taxes. The profit margin would be huge."
" _If_ they got there."
"If they got there. But they won't. Because you'll bring those gems to me."
"Why trust me? Why wouldn't I head for the hills with my newfound riches?"
"Because you'd be a fool to try it. I know about you, Frey. You don't have the contacts or the experience to fence them. You've no idea how dangerous that kind of wealth can be. Even if you didn't get your throat slit trying to sell them, you'd be ripped off."
"So what do you propose as payment?"
"Fifty thousand ducats. Flat fee, nonnegotiable, paid upon delivery of the gems to me."
Frey's throat went dry. Fifty thousand. He couldn't possibly have heard that right.
"You did just say fifty _thousand_ ducats, didn't you?"
"It's a better offer than you'll get trying to sell them yourself, and the deal will be straightforward and safe. I'm rather hoping it will help you avoid temptation."
"How much is the chest worth?"
"Considerably more, once the gems are cut. But that doesn't concern you."
"Let me get this straight. You said fifty thousand _ducats_?"
"On delivery."
Frey drained his wine in a gulp.
"More wine?" Quail offered politely.
"Please," Frey rasped, holding out his glass.
Fifty thousand ducats. It was a colossal amount of money. More than enough riches to live in luxury for the rest of his days, even after he'd cut the others their share. _If_ he cut them a share, he corrected himself.
_No, don't think about that yet. You only need to decide if this really is too good to be true_.
His heart pounded in his chest, and his skin felt cold. The opportunity of a lifetime. He wasn't stupid enough to think it came without a catch. He just couldn't see it yet.
Ever since he became a freebooter, he'd stuck to one hazy and ill-defined rule: _keep it small-time_. Ambition got people killed. They reached too far and got their hands bitten off. He'd seen it happen time and again: bright-eyed young captains, eager to make a name for themselves, chewed up in the schemes of businessmen and pirates. The big-money games were run by the _real_ bad men. If you wanted to play in that league, you had to be ready for a whole new level of viciousness.
And then there was the Navy. They didn't concern themselves with the small-time operators, but once you made a reputation they'd take an interest. And if there was one thing worse than the backstabbing scum-sacks that infested criminal high society, it was the Navy.
Frey wasn't rich. What money he made was usually gambled away or spent on drink or women. Sometimes it was a struggle just to keep craft and crew together. But he was beholden to no one, and that was the way he liked it. Nobody pulled his strings. It was what he told himself whenever money was tight and things looked bad.
_At least I'm free_ , he thought. _At least there's that_.
In the murky world of bottom-feeders, Frey could count himself among the larger fish, simply by dint of smarts. The world was full of morons and victims. Frey was a cut above, and he was comfortable there. He knew his level, and he knew what happened when people overestimated themselves.
But it was one job. Fifty thousand ducats. A life of appalling, obnoxious luxury staring him in the face.
"Why me?" he asked as Quail refilled his glass. "I must have dealt with you, what, three times?"
"Yes," said Quail, settling again. "You sold me a few tidbits. Never bought anything."
"Never could afford it."
"That's one point in your favor," he said. "We're barely acquainted. The scantest of links between us. I couldn't risk offering this opportunity to most of my clients. My relationship with them is too well known." He leaned forward across the desk, clasping his hands together, meshing metal fingers with flesh. "Make no mistake: if this operation goes bad, I don't know you, and you never heard about those gems from me. I will not allow this to be traced back here. I have to protect myself."
"Don't worry. I'm used to people pretending they don't know me. Why else?"
"Because fifty thousand ducats is an absurd amount of money to you and I believe it will keep you loyal. Because you're too small-time to fence those gems for yourself, and you're beneath the notice of the Navy and other freebooters alike. And because no one would believe you if you told them I was involved. You're frankly not a very credible witness."
Frey searched his face, as if he could divine the thoughts beneath. Quail stared back at him patiently.
"It's an easy take, Frey. I know her route. She'll be following the high ground, hugging the cloud ceiling, staying out of sight. No one's going to know she's there but you. You can bring her down over the Hookhollows. Then you pick up the gems and you fly them to me."
Frey didn't dare hope it was true. Was it possible that he was simply in the right place at the right time? That a man like him could have a chance to make a lifetime's fortune in one swoop? He racked his memory for ways he might have given Quail offense, some reason why the whispermonger would send him into a trap.
Could Quail be working on someone else's behalf? Maybe. Frey had certainly made enemies in his time.
_But what if he's_ not _setting you up? Can you really take that chance?_
The clammy, nauseous feeling he had at that moment was not unfamiliar to him. He'd felt it many times before, while playing cards. Staring at his opponent over a hand of Rake, a pile of money between them, his instincts screaming at him to fold and walk away. But sometimes the stakes were just too high, the pot too tempting. Sometimes he ignored his intuition and bet everything. Usually he lost it all and left the table, kicking himself. But sometimes...
Sometimes, he won.
"Tell you what. Throw in some female company, a bed for the night, and all the wine we can drink, and you got a deal."
"Certainly," said Quail. "Which lady would you like?"
"All of them," he said. "And if you have one who's particularly tolerant—or just blind—she might see to Pinn too. I'm gonna need his head straight for flying, and the poor kid's gonna split his pods if he doesn't empty them soon."
# _Chapter Six_
THE GHOSTMOTH—FREY'S IDEA OF DIVISION—THE ACE OF SKULLS—HARKINS TESTS HIS COURAGE
n the steep heights of the Hookhollows, where the lowlands of Vardia smashed up against the vast Eastern Plateau, silence reigned. Snow and ice froze tight to the black flanks of the mountains, and not a breath of wind blew. A damp mist hazed the deep places, gathering in crevasses and bleak valleys, and a glowering ceiling of cloud pressed down hard from above, obscuring the peaks and blocking out any sight of open sky. Between sat a layer of clear air, a sandwich of navigable space within which an aircraft might pick its way through the stony maze.
It was isolated and dangerous, but this claustrophobic zone was the best way to cross the Hookhollows unobserved.
A distant drone came floating through the quiet. It steadily rose in volume, swelling and thickening. Around the side of a mountain came a lone four-winged corvette. A heavily armed Besterfield Ghostmoth.
Lurking in the mist layer, barely a shadow, the _Ketty Jay_ stayed hidden as it passed.
Frey watched the Ghostmoth from the cockpit, its dark outline passing overhead. Crake watched it with him.
"That's not the one we're after, is it?" he asked, rather hoping it wasn't.
"No," said Frey. He wouldn't have taken on a Ghostmoth for any money. He was concerned only that its pilot might spot them and decide to take an interest. You could never be sure. There were a lot of pirates out here. _Real_ pirates, not fair-weather criminals like they were.
Nothing sat right with Frey about this whole plan. Nothing except the colossal payoff, anyway.
He'd never liked piracy, and historically he'd displayed a lack of talent in the field. Of the four times he'd tried it, three had been failures. Only once had he successfully downed and robbed a craft, and even then the loot had been meager and his navigator got stabbed and killed in the process. Twice they'd been forced to flee in the face of superior firepower. On the most recent attempt they'd actually managed to board the craft, only to find it had already delivered its cargo. That was the closest his crew had ever come to mutiny, until he hit on the idea of placating them with a night out at the nearest port. The following morning, the incident was forgotten, along with most of their motor skills and their ability to speak.
In general, Frey didn't like being shot at. Piracy was a risky business and best left to the professionals. Even Quail's assurances of an easy take did little to quell his fears.
The Ghostmoth slid out of view, and Frey relaxed. He checked on Harkins and Pinn, hovering a short way above them and to starboard, dim in the mist. The _Ketty Jay_ drifted silently but for the occasional hiss of stabilizing gas jets as Frey's hands twitched across the brass-and-chrome dashboard. The cockpit lights had been turned off, leaving the interior gloomy. Jez was sitting at the navigator's station, studying a map. Crake, who had dropped in uninvited, stood behind the pilot's seat, wringing his hands. Frey thought about ordering him back to his quarters but couldn't be bothered with the argument that might ensue.
"Quail said they'd be coming through here?" Crake murmured.
"That's what he said," Frey replied.
"Makes sense," Jez told Crake. "You want to get through the Hookhollows without being spotted, you follow the mountains that rise closest to the cloud ceiling. That way you can't be seen from above and you minimize possible sight lines from below. Two of the most obvious routes converge on this point."
Frey turned around in his seat and looked at her. "I'm beginning to think that, after many months, I've finally found a navigator who actually knows what they're doing," he said.
"We're few and far between, Cap'n."
"How's the shoulder?"
"Fine."
"Good. Don't get shot again. You're useful."
"I'll do my best," she said, with a quirky grin.
Frey settled back to watching. Jez was a lucky find. In the few days she'd been on board, she'd shown herself to be far more efficient and reliable than he'd expected. Competence was by no means a prerequisite to joining the crew of the _Ketty Jay_ , but Jez was head and shoulders above the other navigators Frey had worked with. He suspected that she was accustomed to better crews than Frey's mob, but their slapdash technique didn't seem to bother her. And she was good at what she did. She'd brought them in from Marklin's Reach with pinpoint accuracy, with only a featureless sea of cloud and a few mountain peaks by which to plot their position. Frey had dropped down through the cloud and found himself dead in the middle of the pass they'd selected for their ambush.
She was a smart one. He only hoped she wasn't _too_ smart.
Perhaps the others hadn't noticed, but Jez knew something was wrong with this job. He kept catching a glimpse of the question in her eyes. She'd open her mouth as if to say something, then shut it again and look away.
_She feels it too_ , Frey thought. _Instinct_.
Instinct. Perhaps. Or perhaps she sensed that her captain intended to rip them off good and proper.
He tried to feel bad, but he really couldn't manage it. After all, you couldn't be robbed of what you never had. Quail had promised _him_ fifty thousand ducats, not _them_. Granted, he'd always maintained a system of fair shares for his crew, dividing the booty according to prearranged percentages, but these were exceptional circumstances. By which he meant an exceptional amount of money. Too much to share.
It was just this one time, he promised himself. Because after this, he'd never need to work again.
He'd informed the crew that Quail had given them the tip-off in exchange for one thing. There was a chest on board that he wanted. They were to bring it to him. Everything else was theirs for the taking.
Frey had obtained a full description of the chest, and he knew it would be locked tight. Quail had also assured him there were plenty more pickings besides. The crew could loot to their hearts' content, and everyone would be happy. They didn't need to know what was inside the chest. They didn't need to know about the arrangement between Frey and Quail.
But Jez kept giving him that look.
"I hear something," Crake said suddenly.
Frey listened. He was right: a low throb, accompanied by the higher whines of smaller engines. Hard to make out how many.
"Jez," Frey murmured. "Ready on the electroheliograph."
"Cap'n," she said, reaching over to the switch.
"This is the one, isn't it?" Crake asked, squinting through the windglass, trying to catch a glimpse.
"This is the one," Frey said.
The _Ace of Skulls_ slid into the pass, cruising majestically between two broken peaks. Long, blunt-faced, and curve-bellied, she had stubs for wings and a tail assembly like an enormous fin. Thrusters pushed her along as she glided through the air, buoyed up with huge tanks of aerium gas. Decals on her flanks displayed her name, printed across a fan of cards. She was a heavy, no-nonsense craft, without frills, solid. Nothing about her gave away the value of the cargo within.
Buzzing alongside, dwarfed in size, were four Swordwings. Frey recognized them by their distinctive conical down-slanting muzzles and aerodynamic shape. They were fast fighter craft. Nothing exceptional in their design, but in the hands of a good pilot they could be deadly.
"It's not exactly _minimum escort,_ " Crake muttered.
Frey made a distracted noise of agreement. He didn't like the look of those Swordwings. He'd expected two, not four.
"Just give me the word," Jez said, fingertip hovering over the press pad of the electroheliograph switch.
Frey stared up at the freighter. It wasn't too late to listen to the voice that told him to back out of this. The voice that told him to lay his cards down when he knew his hand was beat. The voice of caution.
_You could keep going on as you are_ , he thought. _It's not a bad life, is it? You've got your own craft. You don't answer to anyone. The whole world's therefor you. Now, what's wrong with that?_
What was wrong with it was that he didn't have fifty thousand ducats. He hadn't really minded before, but suddenly the lack had become intolerable.
"Cap'n?" Jez prompted. "Time's a factor."
Frey had picked a spot just below the mist layer and in the shadow of a peak to give them a good view of the pass above.
But if he could see the _Ace of Skulls_ , she might see him, and without the element of surprise they'd have no chance.
_You know this is too good to be true, Frey. Stuff like this doesn't happen to guys like you. Ambition gets people killed_.
"Cap'n?"
"Do it," he said.
PINN WIPED HIS RUNNING nose with the back of his hand and stared at the gray bulk of the _Ketty Jay_.
"Come on! What's taking so long?" he cried. The need to get up there and shoot something was like a physical pull. His boots tapped against the complicated array of pedals; his gloved fingers flexed on the flight stick. These were the moments he lived for. This was where the action was. And Pinn, as he never tired of telling everyone, was all about action.
The Second Aerium War had fizzled out mere days before he had the chance to sign up. Those miserable Sammies called it off just as he was about to get in there and bloody his guns. It was as if they'd intended to spite him personally. As if they were afraid of what would happen when Pinn got into the thick of things.
Well, if the Sammies were too chickenshit to face him in the air, then he'd take it out on the rest of the world every chance he got. Having been cheated once, he reasoned it was only his due. A man deserved the opportunity to prove himself.
He snatched up the small, framed ferrotype of his sweetheart, Lisinda, that hung on a chain from his dash. The black-and-white portrait didn't do her justice. Her long hair was fairer, her innocent, docile eyes more beautiful in his memory.
It had been taken right before he left. He wondered what she was doing now. Perhaps sitting by a window, reading, patiently awaiting his return. Did she sense his thoughts on her? Did she turn her sweet face up to the sky, hoping to see the cloud break and the sun shine through, the glimmer of his wings as he swooped triumphantly in to land? He pictured himself stepping down from the Skylance, Lisinda rushing joyously toward him. He'd sweep her up in his arms and kiss her hard, and tears would run uncontrollably down her face, because her hero had returned after four long years.
His thoughts were interrupted by a series of flashes from a lamp on the _Ketty Jay_ 's back. A coded message from the electroheliograph.
_Go_.
Pinn whooped and rammed the prothane thrusters to maximum. The Skylance boomed into life and leaped forward, pressing him back in his seat. He stamped down on a pedal, wrenched the stick, and the craft came bursting out of the mist, arcing toward the small flotilla high above. They'd all but passed overhead now, so he came at them from below and behind, hiding in their blind spot. A fierce grin spread across his chubby face as the engines screamed and the craft rattled all around him.
"This ain't your lucky day," he muttered as he lined his enemy up in his sights. He believed true heroes always said something dry and chilling before they killed anybody. Then he pressed down on his guns.
The pilot of the nearest Swordwing had only just heard the sound of Pinn's engine when the bullets ripped through the underbelly of his craft. They pierced the prothane tanks and blasted the Swordwing apart in a dirty cloud of flame. Pinn howled with joy, corkscrewed through the fire, and burst out of the far side. He craned in his seat to look back, past his port wing, and saw Harkins coming up, machine guns blazing, shredding the rudder of another Swordwing as he shrieked by.
"Yeah!" Pinn cried. "Nice shooting, you twitchy old freak!"
He hauled the Skylance into a loop, hard enough to make his vision sparkle at the edges, and headed back toward the flotilla. The two remaining Swordwings had broken formation now, taking evasive action. Harkins's target was coiling its way down to a foggy oblivion, leaving a trail of smoke from its ruined tail. Far below, the _Ketty Jay_ had broken cover and was heading toward the slow bulk of the freighter.
Pinn picked another Swordwing and plunged toward it. He dropped into position on its tail, machine guns spitting a broken row of blazing tracer bullets. The pilot banked hard and rolled, darting neatly out of the way. Pinn raised an eyebrow.
"Not bad," he murmured. "This is gonna be fun."
"SHE'S HEADING FOR THE clouds!" Jez said.
She was right. The _Ace of Skulls_ had turned her nose up toward the cloud ceiling and was gliding toward it. Visibility would be almost nil in there.
"I'm on it," Frey said, then suddenly yelled, "Doc!"
"What?" came the bellowed reply through the open doorway of the cockpit.
"Start hassling the fighters! I've got the big fish!"
"Right-o!"
There was the thumping of autocannon fire as Malvery, in the gunner's cupola, began to unleash lead at all and sundry. Frey fed a little more into the prothane engines and the _Ketty Jay_ responded, surging upward. She was surprisingly light for such a big craft, but Frey was long used to the way she handled. Nobody knew her like he did.
Harkins and Pinn had the Swordwings occupied, chasing them around the sky, leaving the way clear for him. He hunched forward in his seat, frowning intently at his target. Jez and Crake stood behind him, hanging on as best they could as the _Ketty Jay_ rocked and swayed.
The freighter swam higher, thrusters pushing as hard as they were able, but she was a lumbering thing and she couldn't get a steep enough angle without tearing herself apart under her own weight. Frey would get only one chance, but one chance was all he needed. The aerium tanks on a craft like that were an enormous target. Though there was nothing on the outer skin to indicate their location, Frey knew his aircraft. It would be hard to miss.
_Just graze the tanks with your guns_ , he reminded himself. Holed tanks would vent aerium gas, and the steady loss of lift would force the pilot to either land the craft or have her drop out of the sky. A landing might be a bit violent in this kind of terrain, but Frey didn't much care as long as the cargo was intact. The prothane tanks—the dangerous part—were well armored and buried deep within the craft. It would take a really bad landing to make them go up.
The _Ace of Skulls_ swelled in his view, growing larger as he approached. In attempting to escape, she'd exposed her belly. He zeroed in on the spot just under her stubby, finlike wings.
_Closer... closer..._
He squeezed the trigger on his flight stick. The _Ketty Jay_ 's front-mounted machine guns clattered, punching a pattern of holes across the freighter's side.
And the _Ace of Skulls_ exploded.
The windglass of the cockpit filled with a terrible bloom of fire, lighting up Frey's astonished face for a split second. Then the impact hit them.
The detonation was ear-shattering. A concussion wave swamped the _Ketty Jay_ , making her roll sharply and sending Jez and Crake slamming into the navigator's station. Frey wrestled with the controls, yanking on the flight stick with one hand, hitting switches with the other. The engines groaned and stuttered, but Frey had flown this craft for more than a decade and he knew her inside out. Teeth gritted, he gentled her through the chaos, and in seconds they were level again.
Frey looked out of the cockpit. He felt sick and faint. An oily black cloud of smoke, blistering with red-and-white flame, roiled in the air. The _Ace of Skulls_ 's enormous bow was plummeting into the pass far below; her tail assembly crashed against the side of a mountain and broke into pieces. A cloud of lesser debris spun lazily away, thrown out by the colossal force of the explosion.
And in among the debris, charred limp things fell toward the earth. Some of them were still almost whole.
Bodies. Dozens of bodies.
HARKINS STARED AT THE slow cascade of wreckage as it tumbled from the sky. He wasn't sure he'd exactly grasped the full implications of what had just happened, but he knew this was bad. This was very, very bad. And not only because they'd screwed up yet another attempt at sky piracy.
Then, suddenly, the Swordwing he'd been chasing broke left and dived. Harkins's attention switched back to his target.
_He's running!_ Harkins thought. A glance told him that the second Swordwing was doing the same, spearing up toward the clouds. Pinn was hot on its tail, spraying tracer fire. Smoke trailed from one of its wings.
Harkins threw the Firecrow into a dive. Whatever had just happened, Harkins was certain of one thing. They were in trouble.
But only if someone lived to tell about it.
The Swordwing was dropping hard toward the layer of mist that had hidden the _Ketty Jay_. Harkins rattled off a short burst from his guns, but he was still too far away. He opened the Firecrow's throttle and screamed after the Swordwing as it was swallowed up by the mist.
_Oh, no_ , he fretted to himself. _I don't want to go in there, I really don't!_
But it was too late for second thoughts. The mist closed over him, graying his vision. The Swordwing was a dark smudge ahead. It had pulled level, skimming through the upper layers of mist where visibility was just the right side of suicidal. Harkins tried to close the distance, but they were evenly matched on speed.
Sweat began to trickle down the deep folds of his unshaven cheeks. They were going too fast, they were going way too fast. This pilot was a maniac! Was he trying to get himself killed?
Harkins pressed down on his guns, hoping for a lucky hit. The tracer fire blazed away into the gloom.
A mountain loomed out of the mist to starboard, an unending slope of snowy rock fading into view. The Swordwing swung in recklessly close to it, hugging the mountainside. The shock wave of its passage threw up clouds of loose snow, whipping them into Harkins's path. The pilot was trying to blind him further. But the tactic was ineffective: the powdery snow dispersed too fast and did nothing to slow him. Harkins angled himself on an intercept trajectory and closed in on his target.
The mountainside ended without warning, and the Swordwing made a dangerously sharp turn, almost clipping the corner. Harkins followed out of reflex. The only safe place in this murk was where his target had already been.
An outcrop of black stone came at him like a thrown fist.
His reactions responded in place of conscious thought. He shoved the flight stick forward and the Firecrow dived, skimming under the jutting stone with barely a foot to spare. It thundered over him for a terrifying instant and was gone.
He pulled away from the mountainside, gibbering. That was too close, too close, too close! His legs had begun to tremble. This was insane! Insane! Who did that pilot think he was, anyway? Why was he putting Harkins through such torment?
But there it was: the Swordwing. Still visible through the bubble of windglass on the Firecrow's snout. It was heading down, farther into the dull blankness, a ghostly blur.
Harkins followed. Afraid as he was, he was also afraid to face the consequences of giving up. He couldn't take Frey's wrath if he let the Swordwing go. Death in the cockpit was one thing, but confrontation was quite another. Confrontation was a special kind of hell for Harkins, and he'd do just about anything to avoid it.
Dense, threatening shadows came into view on either side of them: mountains, pressing in close. Harkins bit his lip to stop his teeth from chattering. The Firecrow's engines cocooned him in warm sound, but he was acutely aware of how fragile this metal shell would be if it hit something at a hundred knots. He'd seen Firecrows shatter like eggs, some of them with his friends inside.
_But that never happened to me!_ he told himself, firming his will, and he pushed harder on the throttle.
The mountains slid closer on either side, pushing together, and he realized they were heading into a defile. Then, suddenly, the Swordwing slowed. Harkins bore down on it. The blur took on form and shape, growing before him. He pressed down his guns just as the Swordwing went into a steep climb, and the tracers fell astern as it shot upward and disappeared into the haze.
At that moment, Harkins realized what his opponent was doing. Panic clutched at him. He yanked back on the flight stick, hauling on the throttle and stamping the pedal that opened the flaps for emergency braking. The Firecrow's blunt nose came up; the craft squealed in protest. Harkins felt a weight like a giant's hand shoving him down into his seat.
A wall of grim stone filled his vision. Massive, immovable, racing toward him. The end of the defile. He screamed as the Firecrow clawed at the air, scrabbling to climb. Blood pounded in his thighs and feet. His vision dimmed and narrowed as he began to brown out.
_You're not gonna faint... you're not gonna faint..._
Then everything tilted, vertical became horizontal, and the wall that had been in front of him was rushing beneath his wings. He let off on the stick, blood thumping back into his head, and the Firecrow shot out of the defile and upward. There were a few seconds of nothing but gray, then he burst out of the mist and into the clear air.
Stillness.
As if in a trance, he cut back the throttle and gently brought the Firecrow to a hover, letting it float in the air, resting on the buoyancy of its aerium tanks. A dozen kloms away, visible between the peaks, the _Ketty Jay_ hung listlessly, waiting for his return. He looked down into the sea of mist, but his quarry was long gone.
His hands were quivering uncontrollably. He held one up before him and stared as it shook.
# _Chapter Seven_
AN ARGUMENT—CRAKE ACCUSES—WHAT THE CAT THINKS OF JEZ—FREY HAS A DREAM
he eastern edge of the Hookhollows was full of hiding places. Secret valleys, sheltered ledges. There were folds in the crumpled landscape big enough to conceal a small fleet of aircraft. Freebooters treasured these bolt holes, and when they found a good one they guarded its location jealously.
Nightfall found the _Ketty Jay_ and her outfliers in one of Frey's favorite spots—a long, tunnellike cave he usually employed when he was running from something bigger than he was. It was wider than it was high, a slot in the plateau wall that ran far back into the mountainside. A tight fit for a craft the size of the _Ketty Jay_ , but Frey had brought them in without a scratch. Now the _Ketty Jay_ hunkered in the dark, its dim underbelly lights reflected by the shallow stream that ran along the floor of the cave. There was no sound but for a rhythmic dripping and the relentless chuckle of the water.
Inside the _Ketty Jay_ , things were not so calm.
"What in the name of the Allsoul's veiny bollocks were you aiming at, you shit-wit?" Pinn demanded of his captain, who punched him in response.
Slag, the _Ketty Jay_ 's cat, watched the ensuing scuffle with feline disinterest from his vantage point atop a cabinet. The whole crew had gathered in the mess, crowding into one small room, and the comical jostle to separate Pinn and Frey involved a lot of bashing into things and knocking chairs over. The mess was a cheerless place, comprising a fixed central table, a set of metal cabinets for utensils, and a compact stove, where Slag warmed himself when Silo chased him out of the engine room.
Slag was an ancient warrior, a grizzled slab of muscle held together by scar tissue and a hostile disposition. Frey had brought him on board as a kitten the day after he took ownership of the _Ketty Jay_ , fourteen years ago. Slag had never known anything beyond the _Ketty Jay_ and had never been tempted to find out. His life's purpose was here, as the nemesis of the monstrous rats that bred in the air ducts and pipeways. For more than a decade the battle had been fought, generations of sharp-toothed rodents versus their indestructible antagonist. He'd seen off the best of them—their generals, their leaders—and hunted their mothers until they were near-extinct. But they always came back, and Slag was always waiting for them.
"Will you two stop acting like a pair of idiots?" Jez cried, as Malvery and Silo pulled Pinn and the captain away from each other. Pinn, red-faced with anger, assured Malvery he was calm so the doctor would release him, then made the obligatory second lunge at Frey. Malvery was ready for it and punched him hard in the stomach, knocking the wind out of him.
"What'd you do that for?" Pinn rasped weakly, wide-eyed with the injustice of it all.
"Fun," replied Malvery, with a broad grin. "Now calm down before I club your stupid block off. You ain't helping."
Frey shook Silo off with a baleful glare and dusted himself down. "Right," he said. "Now we've got _that_ out of the way, can I say something, nice and slow so everyone gets it? _It—wasn't—my—fault!_ "
"You did blow up the freighter, though," Crake pointed out.
"If you knew _anything_ about aircraft, you'd know they always put the prothane tanks as deep inside as possible, well armored. Otherwise, people like us might be able to hit them and blow the whole thing to smithereens."
"The way you did," Crake persisted, out of malice. He hadn't forgotten Frey's behavior when Lawsen Macarde had a gun to his head.
"But I didn't!" Frey cried. "Machine guns couldn't have penetrated deep enough to even _get_ to the prothane tanks. Silo, tell them."
The Murthian folded his arms. "Could happen, Cap'n. But it's one in a million."
"See? It _could_ happen!" Pinn crowed, having recovered his breath.
"But it's _one in a million_!" Frey said through gritted teeth. "About the same chance as you shutting up for five minutes so I can think."
Slag unfurled from his spot on top of the cabinet and dropped down to the countertop with a thump. He thought little, if at all, of the other beings with whom he shared the craft, but he was feeling unaccountably piqued that nobody was paying any attention to him amid this puzzling furor. Harkins, who had been keeping his head down anyway, cringed into the corner as he caught sight of the cat. Slag gave him a stare of utter loathing, then leaped to the table so he could get into the middle of things.
"The question isn't whose fault it is—" Jez began.
"Not mine, that's for sure!" Frey interjected.
Jez gave him a look and continued. "It's not whose fault it is. The question is whether we're going to get _blamed_ for it."
"Well, thanks to Harkins being a bloody great chicken, we probably will," Pinn said sullenly.
"That guy was a good pilot!" Harkins protested. "He was a... he was a fantastic pilot! Well, fantastic or he had a death wish or something. What kind of idiot flies full throttle through mountain passes in the mist? The... the crazy kind, that's what kind! And I'm a good pilot, but I'm not some crazy idiot! You said minimum escort, someone said minimum escort! No one said anything about... about four Swordwings and one of them being a pilot like that! What's a pilot like that doing flying escort to some grubby old freighter?"
"I'd have caught him," said Pinn. "I caught the one _I_ was chasing."
"Well, yours was probably shit," Harkins muttered.
Jez was pacing around the mess as the pilots argued, head bowed thoughtfully. As she drew close to Slag, he arched his back and hissed at her. Something about this human bothered him. He didn't understand why, only that he felt threatened whenever she was around, and that made him angry. He hated Harkins for being weak, but he was afraid of Jez.
"What's gotten into him?" Crake wondered.
"Ugly sack of mange," sneered Pinn. "It's finally lost its tiny mind."
"Hey!" said Frey, defensive. "No bitching about the cat." He put out his hand to stroke Slag and quickly withdrew it as Slag took a swipe at him.
"Why not? Bloody thing's only fit to use as a duster anyway. Wring its neck, stick a broom handle up its—"
"Shut up about the cat!" Jez said, surprising them into quiet. For such a little thing she'd proved herself unusually feisty, and she commanded respect far out of proportion to her physical size. "We've got more important things to deal with."
She walked in a slow circle around the mess, stepping between them as she spoke. "We caught them by surprise. Even if that Swordwing got away—he might have crashed in the mist—then he'd have barely had time to work out what was going on before he ran. Harkins was on his tail almost immediately. He'd have had other things on his mind."
"You don't think he could identify us?" Frey said.
"I doubt it," Jez replied. "There are no decals on the craft that identify us as the _Ketty Jay_ , and we're not exactly famous, are we? So what do they have? Maybe he saw a Wickfield Ironclad accompanied by a Firecrow and a Skylance. You'd have to be pretty dedicated to hunt us down on the basis of that."
"Quail won't say a word," said Frey, warming to her optimism. "Though it's probably best if our paths never cross again. Just to be safe, let's stay out of Marklin's Reach. Silo, put it on our list of no-go ports. Scarwater too."
"Aren't that many ports left to go to," Malvery grumbled.
"Well, now there are two fewer." He looked around the room. "Alright, are we done here? Good. Let's keep our heads down, forget this ever happened, and it's business as usual." He began to leave but was stopped by a soft voice.
"Am I the only one who remembers there were _people_ on that freighter?" Crake said.
Frey turned around to look over his shoulder at the daemonist.
"That thing was hauling passengers," Crake said. "Not cargo."
Frey's eyes were cold. "It wasn't my fault," he said, and clambered up the ladder to the exit hatch.
The crew dispersed after that, some still arguing among themselves. Slag remained in the middle of the table in the empty mess, feeling neglected. After a swift and resentful bout of self-grooming with his tongue, he resolved to make Harkins suffer tonight by creeping into his quarters and going to sleep on his face.
FREY STEPPED INTO HIS quarters and slid the heavy iron door shut behind him, cutting off the voices of his crew. With a sigh, he sat on the hard bunk and dragged his hand down his face, mashing his features as if he could smear them away. He sat there for a while, thinking nothing, wallowing in the bleak depression that had settled on him.
_Every time_ , he thought bitterly. _Every damned time_.
Suddenly he surged to his feet and drew back his hand to strike the wall, but at the last instant he stopped himself. Instead, he pressed forehead and fist against it, breathing deeply, hating. A hatred without target or focus, directed at nothing, the blind frustration of a man maligned by fate.
What had he done to deserve this? Where was it written that all his best efforts should come to nothing, that opportunity should flirt with him and leave him ragged, that money should rust to powder in his hands? How had he ended up living a life surrounded by the witless, the desperate, drunkards, thieves, and villains? Wasn't he better than that?
That bastard Quail! _He'd_ done this. Somehow, he was responsible. Frey had known the job was too good to be true. The only people who ever made fifty thousand ducats out of a deal were people who already had ten times that. Just one more way the world conspired to keep the rich where they were and keep everyone else down.
The _Ace of Skulls_ should _never_ have exploded. It was impossible. What happened to those people... Frey never meant for that. It was an accident. He couldn't be blamed. He'd only meant to hit the aerium tanks. He _had_ hit the aerium tanks. It was one of those things, like a volcano erupting or a craft getting caught in a freak hurricane. An act of the Allsoul, if you believed all that Awakener drivel.
Frey sourly reflected there might be something in the idea of an all-controlling entity. Someone was certainly out to get him, intent on thwarting his every endeavor. If there was an Allsoul, then he sure as spit didn't like Frey very much.
He walked over to the steel washbasin and splashed water on his face. In the soap-streaked mirror, he studied himself. He smiled experimentally. The lines at the edges of his eyes seemed to have deepened since last time he looked. He'd noticed them a year ago and had been shocked by the first signs of decline. He'd unconsciously assumed he'd always stay youthful.
Though he'd never admit it aloud, he knew he was handsome. His face had a certain something about it that pulled women toward him: a hint of slyness, a promise of danger, a darkness in his grin— _something_ , anyway. He never was exactly sure what. It had given him an easy confidence in his youth, a self-assured air that only attracted women more strongly still.
_About the only piece of luck I ever got_ , he thought, since he was in the mood to be peevish.
Even men could be drawn into his orbit, sucked in by a vague envy of his success with the opposite sex. Frey had never had a problem making new friends. Charm, he'd discovered, was the art of pretending you meant what you said. Whether complimenting a man or offering feathered lies to a woman, Frey never seemed less than sincere. But he'd usually forget them the moment they were out of his sight.
Now here he was, thirty, with lines around his eyes when he smiled. He couldn't trade on his looks forever, and when they were gone, what was left? What would he do when his body couldn't take the rum anymore and the women didn't want him?
He threw himself away from the sink with a snort of disgust.
_Self-pity doesn't suit you, Frey. No one likes a whiner_.
Still, he had to admit, it had been a pretty bad decade and his thirties had gotten off to an unpromising start. Waiting for his luck to change had worn his patience thin, and trying to change it himself invariably ended in disaster.
_Look on the bright side_ , he thought. _At least you're free_.
Yes, there was that. No boss to work for, no Coalition Navy breathing down his neck. No woman tying him down. Well, not in the metaphorical sense, anyway. Some of his conquests had been more sexually adventurous than others.
But, damn, this time... this time he really thought he had a chance. The sheer disappointment had shaken him badly.
_It could have been different, though. Maybe if you'd taken a different path ten years ago. Maybe you'd have been happy. You'd certainly have been rich_.
No. No regrets. He wouldn't waste his life on regrets.
The captain's quarters were cramped, although they were still the biggest on the craft. He didn't keep them particularly clean. The metal walls were coated in a faint patina of grime, and the floor was filthy with boot prints. His bunk took up most of the space, beneath a string hammock of luggage that threatened to snap and bury him in the night. A desk, drawers, and cabinets were affixed to the opposite wall, with catches in the drawers and doors to prevent them from opening during flight. In the corner was his mirror and washbasin. Sometimes he used the washbasin as a toilet in the night, rather than climb two levels down to use the head. There were advantages to being male.
He got up and opened a drawer. Inside, atop a mess of papers and notebooks, sat a tiny bottle of clear liquid. He took it and returned to the bunk.
_Might as well_ , he thought sadly.
He unscrewed the stopper, which also functioned as a pipette. He squeezed the bulb and drew in a little liquid, tipped his head back, and administered one drop to each eye. Blinking, he lay back on the bed.
Drowsy relief billowed over his senses. The aches in his joints faded away, to be replaced by a warm, cloudy sensation that erased his cares and smoothed his brow. His eyes flickered shut, and he drifted on the cusp of sleep for a long while before succumbing.
He dreamed that night of a young woman with long blond hair and a smile so perfect it made his heart glow like burning embers. But when he woke the next morning, he remembered none of it.
# _Chapter Eight_
TAVERN BANTER—CRAKE VISITS AN OLD FRIEND—THE SANCTUM—AN UNPLEASANT SURPRISE
ld One-Eye's tavern was a swelter of heat and smoke, pungent with sweat and meat and beer. The gas lamps were muted by the fug that hung in the air. Stoves, lit to keep the chill of dusk away, made the room stifling. The din of conversation was such that people had to shout to be heard, raising the volume ever further. Waitresses passed between the crude wooden tables, expertly avoiding the attentions of rough-eyed men with ready hands.
Buried amid the standing crowd, Frey held court at a table littered with pewter flagons. He was just finishing a tale about his early days working for Dracken Industries as a cargo hauler. The story concerned an employee's senile mother, who had somehow gotten to the controls of an unattended tractor and drove it into a pile of caged chickens. The punch line was delivered with enough panache to make Pinn spew beer from his nose, which had Malvery laughing so hard he retched. Crake observed the scene with a polite smile. Harkins looked nervously at the people standing nearby, clearly wishing he was anywhere but here. The gangly pilot had been cajoled along on this expedition by Malvery, who thought it would do him good to get out among people. Harkins hated the idea but had agreed anyway, to avoid the slightest risk of giving offense by refusal.
Jez and Silo were absent. Jez didn't drink alcohol and kept to herself; Silo rarely left the ship.
Crake sipped at his beer as Pinn and Malvery recovered. His companions were all merrily drunk, except Harkins, who radiated discomfort despite having sunk three flagons already. Crake was still working on his first. They'd given up bullying him to keep pace once he'd convinced them he wouldn't be swayed. He had other business tonight, and it didn't involve getting hammered on cheap alcohol.
_How easily they forgot_ , he thought. As if Macarde holding a gun to his head was a trifling matter not worthy of comment. As if the mass murder of dozens of innocent people was something that could be erased with a few nights of heavy drinking.
Was that their secret? Was that how they lived in this world? Like animals, thinking only of what was in front of them? Did they live in the moment, without thought for the past or concern for the future?
Certainly that was true of Pinn. He was too dim to comprehend such intangibles as past or future. Whenever he spoke of them, it was with such a devastating lack of understanding that Crake had to leave the room.
Pinn rambled endlessly about Lisinda, a girl from his village, the sweetheart who waited for him back home. His devotion and loyalty to her were eternal. She was a goddess, a virginal idol, the woman he was to marry. After a brief romance—during which they'd never had sex, Pinn proudly declared, as if through some mighty restraint on his part—she'd told him she loved him. Not long afterward, he'd left her a note and gone out into the world to make his fortune. That had been four years ago, and he'd neither seen nor contacted her since. He'd return a rich and successful man or not at all.
Pinn saw himself as her shining knight, who would one day return and give her all the wonderful things he felt she deserved. The simple truth—which, in Crake's opinion, was obvious to anyone with half a brain—was that the day would never come. What little money Pinn had was quickly squandered on pleasures of the flesh. He gambled, drank, and whored as if it were his last day alive, and he flew the same way. Even if he somehow managed to survive long enough to luck his way into a fortune, Crake had no doubt that the bovine, dull-looking girl—whose picture Pinn enthusiastically showed to all and sundry—had long since given up on him and moved on.
In Crake's eyes, Pinn had no honor. He'd lie with whores, then lament his manly weakness in the morning and swear eternal fidelity to Lisinda. The following night he'd get drunk and do it again. How he could believe himself in love on the one hand and cheat on her on the other was baffling. Crake considered him a life-form ranking somewhere below a garden mole and just above a shellfish.
The others he couldn't so easily dismiss. Harkins was a simple man, but at least he knew it. He didn't suffer the same staggering failure of self-awareness that Pinn did. Malvery had a brain on him when he chose to use it, and he was a good-hearted sort to boot. Jez, while not luminously cultured, was very quick and knew her stuff better than anyone on board, with the possible exception of their mysterious Murthian engineer. Even Frey was smart, though clearly lacking in education.
How, then, could these people live so day-to-day? How could they discard the past and ignore the future with such enviable ease?
Or was it simply that the past was too painful and the future too bleak to contemplate?
He finished his drink and got to his feet. This was a question for another time.
"Excuse me, gentlemen," he said. "I have to pay someone a visit."
His announcement was greeted by a rousing _wa-hey!_ from the table.
"A lady friend, eh?" Malvery inquired, with a salacious nudge that almost unbalanced Crake. "I knew you'd crack! Three months I've known him and he's not so much as looked at a woman!"
Crake managed to maintain a fixed smile. "You must admit, the quality of lady I've been exposed to hasn't been terribly inspiring."
"Hear that?" jeered Pinn. "He thinks he's too good for our sort! Or maybe it's just that women aren't to his taste," he finished with a smirk.
Crake wasn't sinking to that level. "I'll be back later," he said stiffly, and left.
"We'll be here!" Frey called after him.
"You great big ponce!" Pinn added, to raucous howls of laughter from his companions.
Crake pushed his way out of the tavern, cheeks burning. The cold, clear air off the sea soothed him. He stood outside Old One-Eye's, collecting himself. Even after several months on board the _Ketty Jay_ , he wasn't used to being mocked quite so crudely. It took him a short while before he felt calm enough to forgive the crew. Not Pinn, though. That was just one more score against him. _Ponce_ , indeed. That moron didn't know _how_ to love a woman.
He buttoned up his greatcoat, pulled on a pair of gloves, and began to walk.
Tarlock Cove at dusk was rather picturesque, he thought. A fraction more civilized than the dives he'd become accustomed to, anyway. With the Hookhollows rising steeply at the back of the town and the wild Poleward Sea before it, there was a dramatic vista at every corner. It was built into the mountainside and straggled around the encircling arms of the bay, connected by steep stairs and winding gravel paths. Houses were narrow, wooden, and generally well kept once you got away from either of the two docks. Vessels of both air and sea made port here, as Tarlock Cove was built on fishing. The ships trawled the shoals and sold their catch to the aircraft crews for distribution.
It was, in fact, the reason they'd come here. Having been burned by their last endeavor, Frey decided to play it safe with some nice, legal work that wasn't liable to get them all killed. He'd all but emptied the _Ketty Jay_ 's coffers to buy a cargo of smoked bloodfish, which he planned to sell inland for a profit. Apparently, it was "easy work" and "nothing could go wrong," both phrases Crake had learned to mistrust of late.
He headed up railed stone stairways and along curving lanes. The houses pressed close to a waist-high barrier wall, which separated pedestrians from the sheer cliffs on the other side. Lamplighters were making their way along the cobbled streets, leaving a dotted line of hazily glowing lampposts in their wake.
As Crake climbed higher, he could see the lighthouse at the mouth of the bay, and he was pleased when he noticed it brighten and begin to turn. Such things, signs of a well-run and orderly world, gave him a sense of enormous satisfaction at times.
Orderliness was one of the reasons he'd liked Tarlock Cove on his previous visits. It was overseen by the family whose name it bore, and the Tarlocks ensured their little town wasn't left to ruin. Houses were well painted, streets swept clean, and the Ducal Militia made certain that the ragamuffin traders who passed through were kept from bothering the respectable folk higher up the mountainside.
Dominating it all from the highest point of the town was the Tarlock manse. It was unassuming in its grandeur, a wide, stout building with many windows, benevolently overlooking the bay. A classically understated design, Crake thought: the picture of aristocratic modesty. He'd visited with the Tarlocks once and found them delightful company.
But it wasn't the Tarlocks he planned to see tonight. He went instead down a winding lamplit lane and knocked at the door of a thin three-story house sandwiched between other houses of a similar design.
The door was opened by a rotund man in his sixties wearing a pince-nez. The top of his head was bald, but stringy gray hair fell around his neck and over the collar of his brown-and-gold jacket.
He took one look at his visitor and the color drained from his face.
"Good evening, Plome," Crake said.
_"Good evening?"_ Plome spluttered. He looked both ways up the alley, then seized Crake by the arm and pulled him over the threshold. "Get off the street, you fool!" He shut the door the moment Crake was inside.
The hallway within was shadowy at this hour: the lamps hadn't yet been lit. Gold-framed portraits and a floor-to-ceiling mirror hung on paneled walls of dark wood. As Crake began to unbutton his greatcoat, he glanced through the doorway into the sitting room. Tea and cakes for two had been laid out on a lacquered side table next to a pair of armchairs.
"You were expecting me?" Crake asked, bemused.
"I was expecting someone entirely different! A judge, if you must know! What are you _doing_ here?" Before Crake could answer, Plome had taken him by the elbow and was hurrying him down the hall.
At the end of the hall was a staircase. Plome steered Crake around the side to a small, innocuous door. It was a cupboard under the stairs, to all appearances, but Crake knew by the prickling of his senses that appearances were deceptive here. Plome drew a tuning fork from his coat and rapped it smartly against the door frame. The fork sang a high, clear note, and Plome opened the door.
Inside was a single shelf with a lantern, and a set of wooden steps led down. Plome held the still-ringing fork high and ushered Crake past. Crake felt himself brushed by the daemon that had been thralled into the doorway. A minor glamour. Anyone opening the door before subduing the daemon with the correct frequency would have seen nothing but a cluttered cupboard, probably accompanied by a strong mental suggestion that there was nothing interesting inside.
"Watch yourself," said Plome. "I'll go first. Third step from the bottom will paralyze you for an hour or so."
Crake stopped and waited for Plome to shut the door, strike a match, and touch it to the lantern. Plome led the way down the stairs, and Crake followed him. At the bottom, Plome struck another match and lit the first of several gas lamps set in sconces on the walls. A soft glow swelled to fill the room.
"Electricity hasn't caught on here yet, I'm afraid," he said apologetically, moving from lamp to lamp with the match. "The Tarlocks banned small generators. Too noisy and smelly, that's the official line. But really it's so they can build their own big generator and charge us all for the supply."
The sanctum under the house had changed little since Crake's last visit. Plome, like Crake, had always leaned toward science rather than superstition in his approach to daemonism. His sanctum was like a laboratory. A chalkboard covered with formulae for frequency modulation stood next to a complicated alembic and books on the nature of plasm and luminiferous aether. A globular brass cage took pride of place, surrounded by various resonating devices. There were thin metal strips of varying lengths, chimes of all kinds, and hollow wooden tubes. With such devices, a daemon could be contained.
Crake went cold at the sight of an echo chamber in one corner. It was a riveted ball of metal, like a bathysphere, with a small circular porthole. Crake felt the strength drain out of his limbs. A worm of nausea crawled into his gut.
Plome followed his gaze. "Oh, yes, that. Rather an impulse purchase. I haven't used it yet. Need to wait for the electricity to get here. To provide a constant vibration to produce the echo, you see."
"I know how it works," Crake assured him, his voice thin. He felt suddenly out of breath.
"Of course you do. And I expect you know how dangerous and unpredictable the echo technique is too. Can't risk a battery conking out on me while I've got some bloody great horror sitting inside!" He laughed nervously before noticing that Crake had lost the color in his face. "Are you quite alright?"
Crake tore his eyes away from the echo chamber. "I'm fine."
Plome didn't pursue the matter. He produced a handkerchief and mopped his brow. "The Shacklemores were here looking for you."
"The Shacklemores?" Crake was alarmed. "When?"
"Sometime around the end of Swallow's Reap, I think. They said they were visiting all your associates." He wrung his hands. "Made me quite uncomfortable, actually. Made me think they knew about... well, this." He made a gesture to encompass the sanctum. "It'd be very awkward if this got out. You know how people are about us."
But Crake was too busy thinking about himself. The Shacklemore Agency was bad news. Bounty hunters to the rich and famous. He'd expected they'd be involved, but the confirmation still came as a blow.
"Sorry, old chap," Plome said. "I suppose they found you out, eh?"
"Something like that," he replied. _Something much, much worse_.
"Barbarians," he snorted. "They take one look at a sanctum, then cry 'daemonist' and hang you. Doesn't matter who you are or what you've done. Ignorance will triumph over reason every time. That's the sad state of the world."
Crake raised an eyebrow. He hadn't expected such a comment from this generally conservative man. "You don't think I should have stayed to face the music? Argued my case?"
"Dear me, no! Running was the only thing you could have done. They just don't understand what we're about, people like us. They're afraid of the unknown. And those blasted Awakeners don't help, shooting their mouths off about Allsoul this and daemonism that, riling up the common folk. Why do you think I'm brownnosing up to the local judge, eh? So I've got a fighting chance if anyone discovers what I've got hidden under my house!"
Plome had reddened during his tirade, and he had to take a few breaths and mop his brow again when he was done. "Speaking of which, he could be here any minute. What can I help you with?"
"I need supplies," Crake said. "I need to get back into the Art, and I don't have any of the equipment."
"It's practicing the Art that got you into this pickle in the first place," Plome pointed out.
"I'm a daemonist, Plome," Crake said. "It's what I am. Without that, I'm just another shiftless rich boy, good for nothing." He gave a sad, resigned smile. "Once you've touched the other side, you can't ever go back." A sudden, unexpected surge of tears surprised him. He fought them down, but Plome saw his eyes moisten and looked away. "A man should... a man should get back on a horse if it throws him."
"What happened to you?" Plome asked, getting worried now.
"The less you know, the better," he said. "For your own good. I don't want you involved."
"I see," said Plome uncertainly. "Well, you can't go to your usual suppliers. The Shacklemores will have them staked out." He hurried over to a desk, snatched up a sheet of paper that was lying there, and scribbled down several addresses. "These are all trustworthy," he said, handing Crake the paper.
Crake ran his eye over the addresses. All in major cities, dotted around Vardia. Well, if he couldn't persuade Frey to visit one of them, he could always take leave of the _Ketty Jay_ and make his own way.
"Thanks. You're a good friend, Plome."
"Not at all. Our kind have to stick together in these benighted times."
Crake folded the paper over and saw that Plome had written it on the back of a handbill. He opened it out and went gray.
"Where did you get this?"
"They're posted all over. Whoever that is, they want him badly. Him and his crew."
"You don't say," Crake murmured weakly.
"You know, the Century Knights just turned up in town looking for him, if you can believe that!" Plome enthused. "The Archduke's personal elite!" He whistled and pointed at the flyer. "He must have _really_ messed up. I wouldn't want to be in his shoes when the Knights catch up with him!"
Crake stared at the handbill as if he could simply will it out of existence.
WANTED FOR PIRACY AND MURDER, it said. LARGE REWARD.
Staring back at him was a picture of Frey.
# _Chapter Nine_
A MATTER OF HONOR—BREE AND GRUDGE—"ONE MORE TOWN WE'RE NOT COMING BACK TO"—DEPARTURE IS DELAYED
rake hurried through Tarlock Cove as fast as he dared. The streets were dark now, deepening toward true night, and stars clustered thickly overhead. The beam of the lighthouse swept across the town and out to sea. Crake walked with his collar up and his head down, his blond hair blowing restlessly in the salt wind, trying not to draw attention to himself.
_Run_ , he told himself. _Just run. You weren't a part of it. They don't even know you're on the crew_.
But run where? His assets had been seized, so he had only the money he'd taken when he fled, and there was little enough of that left. His only contact here was Plome, and the last thing Plome needed was to shelter a fugitive. He had his own secrets to keep. No, Crake wouldn't implicate him in this matter. He'd deal with it on his own.
_Run!_
But he couldn't. Because the only way he was going to stay ahead of the Shacklemore Agency was to keep on the move, and the only way he could do that was aboard the _Ketty Jay_.
And there was more, besides. It was a matter of honor. He didn't care for Frey at all, and Pinn was beneath consideration, but the others didn't deserve to be hung out to dry like that. Especially not Malvery, of whom Crake was becoming quite fond.
But if he was honest with himself, even if he'd hated them all, he'd have gone back. If only to warn them. Because it was the right thing to do. And because it made him better than Frey.
He traced his steps back to Old One-Eye's and paused at the threshold, listening for signs of a disturbance. He'd been seen drinking with the crew. If they'd already been caught, there was no sense getting himself picked up as well.
There was a good chance Frey hadn't been recognized, though. The ferrotype on the handbill must have been taken a long while ago, ten years or more. It didn't look much like Frey. He had a little less weight and a lot less care on his face. He was clean-shaven and looked happy, smiling into the camera, squinting in the sun. There were mountains and fields in the background. Crake wondered when it was taken and by whom.
The drinkers were merry, and the noise inside the tavern was customarily deafening. All seemed well. Peering through the windows, which were bleared with condensation, he detected nothing amiss.
_Get in, grab them, and get out of town_.
He took a breath, preparing himself to face the throng inside. That was when he spotted a pair of Knights heading up the street toward him.
He knew them from their ferrotypes. Everyone knew the Knights. Broadsheets carried news of their exploits; cheap paperbacks told fictional tales of their adventures; children dressed up and pretended to be them. Most citizens of Vardia could identify twenty or thirty of the hundred Century Knights. But nobody knew all of them, for they operated as much in secret as in public.
These two were among the most famous, though, and they attracted stares from passersby as they approached. The smaller Knight was Samandra Bree, wearing a long, battered coat and loose hide trousers that flared over her boots. Perched on her head was her trademark tricorn hat. Her coat flapped back in the wind as she strode along, offering glimpses of twin lever-action shotguns and a cutlass at her belt. Young, dark-haired, and beautiful, Samandra was a darling of the press. By all accounts she did little to encourage their attention, which only made the people love her more and the press chase her harder.
Her companion was Colden Grudge, who wasn't quite so photogenic. He was a man of bruising size with a face like a cliff. Thick, shaggy brown hair and an unkempt beard gave him a spiteful, simian look. Beneath a hooded cloak, time-dulled plates of armor had been strapped over his massive limbs and chest. He bore the insignia of the Century Knights on his breastplate. Two double-bladed hand axes hung at his waist, and an autocannon was slung across his back.
Crake's mouth went dry and he almost fled. It took him a few moments to realize that they weren't heading for him at all but for the tavern he was standing in front of. They were going to Old One-Eye's.
He didn't have time to think. In moments they'd be inside. Before he knew what he was doing, he thrust the handbill at them and blurted, "Excuse me. You're looking for this man, aren't you?"
The Knights stopped. Grudge glared at him, tiny eyes peering out from beneath a beetling brow. Samandra tipped back her tricorn hat and smiled. Crake found himself thinking that she really was quite strikingly gorgeous in person.
"Why, yes, we are, sir," she said. "Seen him?"
"I just... yes, I just did, yes," he stammered. "At least, I think it was him."
"And where was that?" Samandra asked with a faintly amused expression. It seemed she took his nervousness to be the reaction of a man intimidated by a pretty woman instead of someone strangled by the fear of discovery.
"In a tavern... that way!" Crake improvised, pointing up the road.
" _Which_ tavern?" Grudge demanded impatiently.
Crake grasped for a name. "Oh, it's the one with lanterns out front, you know... the Howling Wolf or something... the Prowling Wolf! That's it! That's where I saw him!"
"You sure about that?" Grudge asked, unconvinced.
"You're not from around here, are you?" Samandra asked in that charmingly soft voice that made Crake feel like pond scum for lying to her.
"Does it show?" he said, with a grin. He gave them a glimpse of the golden tooth. He put a little power into it, letting the daemon suck a tiny fraction of his vital essence, just enough to allay their suspicions, just enough to say: _believe him_. "I'm visiting a friend."
Samandra's eyes had flicked to his tooth for an instant, drawn by the glimmer. Now they were back on him. "Be where we can find you," she said. Crake looked at her blankly.
"The reward!" she said, pointing at the handbill. "You do want the reward?"
"Oh, yes!" Crake said, recovering. "I'll be in here." He thumbed toward Old One-Eye's.
Samandra and Grudge exchanged glances, then they hurried off up the road in the direction of the Prowling Wolf. Crake let out a slow, shaky breath and plunged into the tavern.
FREY WAS HAVING A rare old time. He was exhausted from laughing and perfectly drunk, hovering in that elusive zone of inebriation where everything was in balance and all was right with the world. He never wanted this night to end. He loved Malvery and Pinn and even silent Harkins as brothers in arms. And if things began to wind down, well, the waitress had been giving him looks. She had a homely sort of face, but he liked her red hair and the freckles on her button nose, and he was in the mood for something curvy and soft tonight.
What a life it was! A fine thing to be a captain, a freebooter, a lord of the skies.
Crake's arrival was something of a downer. "We're getting out of here," he said, slapping the handbill onto the table and thrusting a finger at the picture of Frey. "Now!"
Frey, a bit slow off the mark, was more surprised by the picture than the danger it represented. He recognized it immediately. How did they get their hands on _that_ one? Who gave it to them?
Crake snatched the handbill away and stuffed it in his pocket. "I just had to head off Samandra Bree and Colden Grudge. They're looking for us. They'll return in a few minutes. I suggest we not be here when they do."
"You met Samandra Bree?" Pinn gaped. "You lucky turd!"
"Spit and blood! Get moving, you idiots!"
The penny had finally dropped. They surged up and pushed their way through the crowd toward the door.
By the time they emerged from the tavern, Frey's mood had seesawed from elation to cold, hard fear. The Century Knights? The _Century Knights_ were on his tail? What had he done to deserve that?
"Back to the _Ketty Jay_?" Malvery suggested, scanning the street.
"Bloody right," Frey muttered. "This is one more town we're not coming back to."
"Why don't we just emigrate and be done with it?"
"Not a bad idea at that," Frey said over his shoulder, as he hurried away in the direction of the docks.
The town's landing pad was situated halfway along one of the mountainous arms that sheltered the bay. Houses became sparser as they approached, and the streets were whittled down to a single wide path that dipped and curved with the land. It was flanked by storage sheds, the occasional tavern, and a customs house. The vast, moist breathing of the sea was loud here. Waves crashed and spumed on the rocks far below.
Frey hugged his coat tight around him as he led his crew along the stony path. The previously welcoming town seemed suddenly threatening and nightmarish. He glanced over his shoulder for signs of pursuit, but nobody came running after them. Perhaps they'd given the Knights the slip.
Wanted for _murder_? He'd own up to piracy (at least to himself. Damned if he'd admit it to a judge). But murder? He was no murderer! What happened to the _Ace of Skulls_ wasn't his fault!
It didn't matter that piracy and murder carried the same penalty: hanging. It was the principle of the thing. It was all so tragically unfair.
He slowed as they spotted a trio of Ducal Militiamen coming toward them. They were striding along the road from the docks, clad in the brown uniform of the Aulenfay Duchy, all buttoned-up jackets and flat-topped caps. The path afforded nowhere to duck away without looking suspicious.
"Cap'n..." Malvery warned.
"I see them," Frey said. "Keep walking. It's only me they'll recognize."
Frey tucked his head down into his collar and shoved his hands into his pockets, playing the frozen traveler hurrying to get somewhere warm. He dropped back into the group, keeping Malvery's bulk between him and the militiamen.
Their boots crunched on the path as they approached. Frey and his crew moved to the side of the path to let them pass. Their eyes swept the group as they neared.
"Bloody chilly when the sun goes down, eh?" Malvery hailed them with his usual booming good humor.
They grunted and walked on. So did Frey and his men.
The landing pad was busy with craft and their crews, loading the day's catch onto the vessels for the overnight flight inland. A freighter was rising slowly into the air, belly lights bright. Its aerium engines pulsed as electromagnets pulverized refined aerium into ultralight gas, flooding the ballast tanks.
Frey had planned to avoid the rush and leave in the morning, since his cargo wasn't nearly as perishable as fresh fish, but now he was glad of the chaos. It would provide cover for their departure.
They passed the gas lamps that marked the edge of the pad and wended their way toward the _Ketty Jay_. Crews labored in the dazzling shine of their aircrafts' lights, long shadows blasted across the tarmac by the dark hulks that loomed above them. Thrusters rumbled as the freighter overhead switched to its prothane engines and began pushing away from the coast. The air was heavy with the smell of fish and the tang of the sea.
"Harkins, Pinn. Get to your craft and get up there," said Frey. "Harkins, I know you're drunk, but that's _my_ Firecrow, and if you crash it I'll stuff you into your own arsehole and bowl you into the sea. Clear?"
Harkins belched, saluted, and staggered away. Pinn scurried off toward his Skylance without a word. The mention of the Century Knights had intimidated him enough that he was glad to get out of there.
Silo was standing at the bottom of the _Ketty Jay_ 's cargo ramp when Frey, Malvery, and Crake arrived. He was idly smoking a roll-up cigarette made from an acrid Murthian blend of herbs. As they approached, he spat into his hand and crushed it out on his palm.
"Where's Jez?" Frey demanded.
"Quarters."
"Good. We're going."
"Cap'n."
Silo joined the others as they headed up the ramp and into the cargo hold. The hold was steeped in gloom as always, stacked high with crates that were lashed untidily together. The reek of fish was overpowering.
Frey was making for the lever to raise the cargo ramp when a gravelly voice called out: "Make another move and everybody dies."
They froze. Coming up the cargo ramp, revolvers in both hands, was a figure they all knew and had hoped to never see. The most renowned of all the Century Knights. The Archduke's merciless attack dog.
Kedmund Drave.
He was a barrel-chested man in his late forties, his clumsily assembled face scarred along the cheek and throat. Silver-gray hair was clipped close to his scalp, and he wore a suit of dull crimson armor, expertly molded to the contours of his body by the Archduke's master artisans. A thick black cloak displayed the Knights' insignia in red, and the hilt of his two-handed sword could be seen rising behind his shoulder.
"Back away from that lever," he commanded Frey. One revolver was trained on Frey; the other covered the rest of the crew. "Get over with your friends."
Frey obliged. He'd sobered up fast. The effects of the alcohol had been canceled by the chill shock of adrenaline. He racked his brains frantically to think of a way out of this, because he knew one thing for sure: if Kedmund Drave took him in, he'd swing from the gallows.
"Guns!" Drave snapped, as he herded them together. "Knives. All of it."
They disarmed, throwing their weapons down in a small heap in front of them. Drave looked them over critically.
"Step back. Against the crates."
They did as they were told.
"Now. Who's this Jez I heard you mention?"
"She's the navigator," Frey replied.
Drave glanced at the stairs leading out of the cargo hold. Deciding whether it was worth the risk of going up and getting her.
"Anyone else?"
"No," said Frey.
Drave took a sudden step toward them and pressed the muzzle of his revolver to Crake's forehead. "If you're lying, I'll blow his brains out!"
Crake whimpered softly. He'd had just about enough of people putting guns to his head.
"There's not another soul on board!" Frey said. He started with himself, and then pointed to each of the crew in turn. "Pilot. Engineer. Doctor. Navigator is in her quarters. You've got a full crew here. This one"—he waved at Crake—"he's just along for the ride."
"The others? The outfliers?"
"Already gone."
Drave glared at him, then took the revolver off Crake and backed away to a safer distance.
"Both of them?"
"Already gone," Frey repeated, shrugging. "They took off when they heard the Knights were on the case. Could be halfway to anywhere by now. We're all alone here."
Deep in the shadows between the piles of fishy-smelling crates, two tiny lights glimmered. There was the heavy thump of a footstep and a rustle of chain mail and leather. Drave spun around to look behind him, and the color drained from his face.
"Well, unless you count Bess," Frey added, and the golem burst from the darkness with a metallic roar.
Drave's reactions saved him. The armor of the Century Knights was legendarily light and strong, made using secret techniques in the Archduke's own forges, and it slowed him not at all as he flung himself aside to avoid Bess's crushing punch. He hit the ground in a roll and came up with both revolvers blazing. Bess flinched and recoiled as the bullets ricocheted off her armor and punched through her leather skin, but the assault did nothing more than enrage her. She bellowed and swept another punch at Drave, who jumped backward to avoid it.
As soon as the Knight was distracted, the crew scattered. Frey dived for the guns, came up with Malvery's shotgun in his hands, and squeezed the trigger. As he did so, he realized he'd forgotten to prime it first. He hoped the doctor had been careless enough to keep a round in the chamber.
He had. Drave saw the danger, raised his pistol, and was a split second from firing when Frey hit him full in the chest. The impact blasted him off his feet. He landed hard on the cargo ramp and rolled helplessly down it and off the end.
Silo lunged across the hold and raised the lever to close the cargo ramp. Bess started to run down it, chasing the fallen Knight, but Crake shouted after her. She stopped, somewhat reluctantly, and settled for guarding the closing gap. Drave was already trying to pick himself up off the ground. He was groggy but otherwise unharmed, saved by his chest plate.
Frey had bolted for the stairs that led up to the main passageway before the cargo ramp had even closed. He sprinted into the cockpit, past Jez, who was just opening the door to her quarters.
"Was that _gunfire_?" she asked.
He leaped into his chair and punched in the ignition code, then boosted the aerium engines to full. The _Ketty Jay_ gave a dolorous groan as its tanks filled and began to haul the craft skyward. He could hear gunfire outside over the sound of the prothane thrusters: Drave shooting uselessly at the hull. The dark aircraft that shared the landing pad sank from view as they lifted into the night sky.
"Cap'n?" Jez inquired, from the doorway of the cockpit. "Are we in trouble?"
"Yes, Jez," he said. "We're in trouble."
Then he hit the thrusters, and the _Ketty Jay_ thundered, tearing away across the docks and racing out to sea.
# _Chapter Ten_
JEZ HAS VISIONS—TRINICA DRACKEN—AN ULTIMATUM FROM CRAKE—FREY TAKES A STAND
t was a still day. Light flakes of snow drifted from a sky laden with gray cloud. The silence was immense.
Jez stood on the edge of the small landing pad, wrapped up in pelts, holding a cup of cocoa between her furred mittens. She'd bought her new arctic attire soon after arriving. Her meager possessions had been left behind in a room at the lodging house in Scarwater. Truth be told, despite the temperature, she didn't need to wear anything at all. The cold didn't seem to affect her nowadays. But it was essential to keep up appearances: her safety depended on it. Anyone in their right mind would kill her if they knew what she was.
The landing pad was set on a raised plateau above a great, icy expanse. On the horizon, a range of ghostly mountains lay, blued by distance. A herd of snow hogs was trekking across the plain.
Yortland. A frozen, hard, and cruel place, but the only place on the continent of North Pandraca where the Coalition Navy held no sway and Coalition laws didn't apply. The only place left for the crew of the _Ketty Jay_ to run to.
She took a sip of her cocoa.
_I could stay here_ , she thought. _I could walk out into that wilderness and never be seen again_.
Behind her sat the _Ketty Jay_ and her outfliers. Snow had settled on the _Ketty Jay_ 's back and wings, several inches deep. Nearby, an elderly Yort was hammering at the struts of his craft, knocking off icicles. He looked strong despite his age, with a thick neck and huge shoulders. He was bundled up in heavy furs, only his bald and tattooed head exposed to the elements. His ears, lips, and nose were pierced with rings and bone shards. Otherwise, there was nobody to be seen.
Besides the _Ketty Jay_ , there were a couple of Yort haulers and some small personal racers, which Jez had already examined and mentally criticized—a habit born from a life as a craftbuilder's daughter. They were blockish, dark, and ugly, built for efficiency without a care for aesthetics. Typical Yort work. In such an excessively masculine society, owning a craft of elegant design was viewed at best as pointless, at worst as potential evidence of homosexuality. Not something to be taken lightly, since sodomy carried the death penalty out here. As a result, Yorts designed everything to suggest that the owner was so enormously virile, a woman would need armor-plated ovaries to survive a night with him.
Jez's eyes unfocused as she stared out across the plain.
_Get away from everyone_ , she thought. _Maybe that's best. Get away from everyone, before it's too late_.
But the loneliness. She couldn't take the loneliness. What was the point of existence, if you were forever alone?
Scattered across the plateau was the settlement of Majduk Eyl. Yorts built mostly underground for insulation, and their dwellings were barely visible. All that could be seen from the pad were the shallow humps of their dome-shaped roofs, the doorways that thrust through the snow, the skylights sheltered by overhanging eaves. Smoke rose from three dozen chimneys, curling steadily up to join the clouds. A small figure, hooded and cloaked, was scattering grit from a sack over the slushy trails that ran between the dwellings.
The crew of the _Ketty Jay_ was in one of those buildings. They were just another set of companions, like the ones before, and the ones before that. She kept herself aloof from them. It would make it hurt less when she had to leave.
Sooner or later they'd notice something was different about her. The little things would begin to add up. The way her bullet wound had healed so fast, the way she never seemed to sleep, the way she never got tired. The way animals reacted to her.
Then she'd have to move on again, find a new crew. Keep going.
_Going where? Doing what?_
Anywhere. Anything. Just keep going.
She drank her cocoa. She ate or drank these days only because she liked the taste, not out of need. During the month of Swallow's Reap, as an experiment, she'd gone without food or water for a week. Nothing happened except a vague, instinctive suspicion that something was missing in her daily routine. After that, she'd made sure to join the crew at mealtimes and occasionally comment loudly on her hunger or thirst. But she ate little, because she wasn't wasteful by nature.
The snow hogs were inching across the ice plain, shambling heaps of muscle and tusk and shaggy white fur. She could see a pair of predators tracking them, huge doglike things, a type of creature she didn't recognize. They loped along hungrily, hoping for a chance at a straggler.
_Here I am again_ , she reflected, as she scanned the landscape. A few years ago she'd been a frequent visitor to the wild, icebound northern coast, part of a scientific expedition in search of the relics of a lost civilization. It hadn't been a conscious decision to stay away from Yortland, but it was only now that she realized she'd never been back since... well, since...
Her thoughts flickered away from the memory, but it was too late. A dreadful sensation washed over her, beginning at her nape and sweeping through her body. Her skin tightened, then relaxed; her muscles clenched and unclenched. The world flexed, just a fraction, and when it sprang back into shape, everything was different.
A strange twilight had fallen. Though it seemed darker, her vision had sharpened. It was as if she'd been looking at the world through a steamed-up pane of glass and it had suddenly been removed. Details were thrust at her eyes; edges became stark as razors.
The herd of snow hogs prickled with a faint purplish aura. Though they were several kloms away, she could count their teeth and see the pupils of their rolling eyes. She sensed the path of the faint wind chasing along the plain; she could picture its route in her mind.
There was so much she was sensing, hearing, smelling. She could hardly breathe under the assault of information. It felt as if she were being battered by an irresistible river. At any moment she'd lose her footing and be swept into oblivion.
One of the predators broke into a run. Its aura was deep crimson, and it left a slowly dispersing trail as it ran. Then she was with the predator, _in_ the predator, its blood pumping hard, heart slamming, tongue-loll and tooth-sharp, all paws and look-see, yes yes yes that one is weak, _that_ one, and my kin-brother alongside and wary of the sharp sharp tusks of the mother but oh oh _the hunger_ —
Jez gulped in a breath, like a drowning woman who had just broken the surface. Reality snapped into place: the world was once again as it had always been. Snow drifted down, undisturbed by her panic. She took a step back, disoriented, wanting to be away from that edge of the plateau. The mug had fallen from her hands and lay on the ground before her. Steaming brown cocoa ate through the ice.
She began to tremble helplessly, and not from the cold. She clutched herself and looked about. The Yort was nowhere to be seen. Nobody was there. Nobody had witnessed it.
_Witnessed what?_ she demanded of herself. _What's happening to me?_
A gust of wind blew from the north, and there was a sound on the wind, something she sensed rather than heard. Voices, raised in a cacophony, calling. A terrible, desperate longing swelled in her.
She looked to the north, and it was as if she could see past the mountains, past the sea, her vision carried on bird's wings. She rushed onward, over icebergs and waves, until there came fog and mist and a vast wall of churning gray.
She knew this place. It was the swirling cloud cap they called the Wrack, which cloaked the north pole. The frontier from which no one had ever returned. Not alive, anyway.
There was something behind the cloud. A shape, an aircraft, black and vast, looming toward her. The voices.
_Come with us_.
She screwed her eyes shut and staggered away with a cry, stumbling toward the _Ketty Jay_. Her mind rang like a struck bell, resounding with the howling, the Wrack, and the terror of what lay beyond.
THE BAR WAS EMPTY but for the crew of the _Ketty Jay_ and the bartender. The menfolk of the village were in the mines or out hunting; the women generally stayed out of sight. During the day, Frey and the others had the place to themselves.
Frey stared dejectedly at his picture. This time it was no handbill. He'd made the national broadsheets now.
"It's only on page ten!" Malvery bellowed, giving him a thump on the shoulder. "It doesn't even look like you! Besides, that issue's a week old. Mark me, they'll have forgotten about it by now."
Frey took little comfort in that. It was true that he looked less and less like his picture, but that was mostly because the Frey in the picture was so happy and carefree. The real Frey was becoming less so by the day. His stubble had grown out to an untidy beard, and his hair was getting beyond the control of a comb. His eyes were sunken and there were dark bags beneath them. In the two weeks since they'd fled Tarlock Cove, he'd become ever more sullen and withdrawn.
And now this: a broadsheet from Vardia, given to Silo by a trader who'd bought their cargo of smoked fish at a rock-bottom price. Frey had hidden angrily in his quarters during the transaction, in case he was recognized.
DRACKEN JOINS THE HUNT
On this day the _Vardic Herald_ has learned of an Announcement by Trinica Dracken, Feared Captain of the _Delirium Trigger_ , to the effect that she will devote all her Will and Effort to the task of bringing to Justice, be it Dead or Alive, the Fugitive Darian Frey and his crew, wanted for Piracy and Murder, and for whom a large Reward is offered for information that might lead to their Capture. The _Herald_ could not reach Captain Dracken for comment, but it is this reporter's humble Opinion that with such a Famous and Deadly Lady upon their trail, it cannot be long before these Scoundrels are brought to face Justice for their crimes.
"The bloody _Delirium Trigger,_ " Pinn groaned. He'd been almost constantly drunk for a fortnight now, having nothing else with which to occupy himself. His eyes were bloodshot and he reeked of alcohol. "Queen bitch of the skies." He paused for a moment, then added, "I'd do her."
The bar was a small, round room, with a domed roof crisscrossed by stout rafters and a south-facing skylight. A fire pit burned red in the center, beneath a large stone chimney. The wooden floor was strewn with pelts, the walls hung with the skulls of horned animals. Tables and seats were made from tree stumps. There was a counter against one wall. Behind it, a surly Yort guarded a barrel of beer and a few shelves stocked with unlabeled jars of liquor.
The bartender was in his mid-fifties, with thick arms and a face weathered like bark. His head was shaved, and his long red beard was gathered into a queue by iron rings. He spoke only in grunts, yet somehow he made it clear that Frey and his men were not welcome here. He'd rather have an empty bar. They ignored him and came anyway.
"Why don't you go home, Pinn?" Crake asked. He was looking up at the rafters, where several arctic pigeons cooed softly to one another. He'd noted the lumpy white streaks among the dried-in bloodstains on the floor and was covering his flagon of dark beer with his hand.
"What?" Pinn asked blearily.
"I mean, what's stopping you? You've got your own craft. You haven't been named or identified. Why not go back to your sweetheart?"
Frey didn't even raise his head at the mutinous tone of the suggestion. Crake was just baiting Pinn. Those who believed Pinn _had_ a sweetheart—Malvery was of the opinion that he might have made her up—knew full well he'd never go back to her. In his mind, she waited to welcome him with open arms on the day he returned home swathed in glory, but he seemed to be the only one who didn't realize that day would never come. Pinn was waiting for glory to happen to him rather than seeking it out.
Lisinda was the heroic conclusion to his quest, the promise of home comforts after his great adventure. But what if she wasn't there when he returned? What if she was holding another man's child? Even in the dim clouds of Pinn's mind, the possibility must have made itself known and made him uneasy. He'd never risk the dream by threatening it with reality.
"Not going back 'til I make my fortune," Pinn said, a note of resentment in his voice. "She deserves the best. Gonna go back..." He raised his flagon and his voice at the same time, challenging anyone to defy him. "Gonna go back a rich man!" He slumped again and sucked at his drink. " 'Til then, I'm stuck with you losers."
An idea struck him. He stabbed a thick finger at Crake and said, "What about you, eh? Mister La-di-da, I-talk-so-cultured? Don't you have a... a banquet to attend or something?" He folded his arms and smirked, pleased at this cunning reversal.
"Well, unfortunately, in the process of saving all your lives at Old One-Eye's, I let two of the Century Knights get a rather good look at me," Crake replied. "But it is something I've been meaning to bring up." He leaned forward on his elbows. "They know Jez's name, but they haven't seen her. Kedmund Drave saw the rest of us, but he doesn't have our names. As a group, we're rather easy to identify. Apart, they'll probably never catch us. They'll get only Frey."
Harkins looked uneasily around the table. Malvery shifted and cleared his throat. Frey didn't react.
"Now, I don't know about all of you," Crake continued, "but I am not spending the rest of my life hiding in an icy wasteland. So what I want to know," he said, looking directly at Frey, "is what you intend to do next. _Captain._ "
There was a loud _plop_ as something fell from the rafters and into Crake's beer. Without taking his eyes from Frey, he pushed it away from him with his fingertips.
Frey was still staring at the article, but he wasn't really seeing it. His mind was working furiously, struggling to puzzle out this crisis, getting nowhere. He'd spent a fortnight raking over the coals of recent events, searching for some buried truth, but there were simply no answers to be had.
It didn't make any sense. Why him? If this was a setup, why choose him? An obscure freebooter, his name all but unknown in pirate circles. Yet Quail had asked for him specifically. Quail, to whom he'd done no wrong.
Of course maybe someone had used Quail to set him up; that was always a possibility. But whom had he offended? To whom had he done such a grievous slight? It must be someone powerful if they could orchestrate something serious enough to involve the Archduke's personal elite. The Century Knights didn't usually concern themselves with affairs unconnected to the Archduke.
Was it an accident? A million-to-one shot that destroyed that craft? No. Frey didn't believe in million-to-one chances. He'd been set up. Someone rigged that freighter to blow, and they put him in position to take the blame.
At least one of the pilots in the escort craft was superb. Whoever arranged all this must have banked on someone living to tell the tale. Even if no one had escaped, they'd have pinned it on him somehow, he had no doubt. But this way they had a witness, presumably unconnected to the real brains behind the operation.
_What was on board that freighter?_
"Frey?" Crake prompted, snapping him out of his reverie. Frey's head came up. "I asked what you intend to do now?"
Frey shrugged helplessly. "I don't know."
"I see," said Crake, his voice dripping with scorn. "Well, let me know when you do. I'd be interested in finding out. If I'm still here." With that, he got up and left.
There was a long silence. The crew was not used to seeing Frey so beaten. It unsettled them.
"What about New Vardia?" Malvery suggested. "Fresh start. Unknown lands. Just the sort of thing for a bunch of lads in our position."
"No!" Harkins cried, and they all looked at him. He went red. "I mean to say, umm, the _Ketty Jay_ might make it—I say _might_ —but the fighters, nuh-uh. The Great Storm Belt's still too bad to the west, and they can't carry enough fuel to go the other route. We'd have to leave the fighters behind, and me, no way, I ain't leaving that Firecrow, even if she does belong to the Cap'n. He leaves the Firecrow behind, I stay behind with her. Final."
Frey was surprised at Harkins's unusual assertiveness on the matter.
"Retribution Falls," said Pinn. "That's what my money's on. Nobody'd find us in Retribution Falls."
"Nobody would find us because it's impossible to find," explained Frey patiently. "Any ideas how _we_ would find it?"
Pinn thought for a moment and came up blank. "Well, there's got to be a way," he muttered. "You hear about all those pirates who've been there. You hear about Orkmund, don't you?"
Frey sighed. Retribution Falls: the legendary hidden pirate town. A place safe from the dangers of the world, where you could fight and drink and screw to your heart's content and the Navy could never touch you. It was said to be founded by the renowned pirate Orkmund, who mysteriously disappeared fifteen years ago and had never been reliably sighted since. Other famous pirates who were no longer around were often said to have retired to Retribution Falls. It made a more romantic story than a slow death by syphilis or alcoholism or being murdered in the night by your own crew.
But that was all it was: a story. Orkmund was dead. The other pirates were dead. Retribution Falls was a myth.
Pinn saw that nobody was taking up his idea and began to sulk. Frey returned to the same obsessive thoughts that had been keeping him up at night.
New Vardia. Maybe he _could_ go to New Vardia. Leave the fighters behind.
The idea wasn't appealing. It was a long and dangerous journey to the other side of the world and, once there, there was nowhere to hide. A few small settlements. A frontier lifestyle, lived without luxuries. If his pursuers tracked him to New Vardia, he'd be easily caught.
He ticked off options in his head. Samarla? They wouldn't last two days, and Silo would never go. Thace? They'd be caught and deported if they tried to stay. Thacians were very defensive of their little utopia. Kurg? Populated by monsters.
Of the countries this side of the Great Storm Belt, only Yortland provided a haven, and it was a cold and bitter one, entirely too close to New Vardia. They couldn't hide there forever. Not with the Century Knights and Trinica Dracken on their tail.
His eyes fell to the broadsheet spread out on the table. Bitterness curdled in his guts. The sanctimonious tone of the writer, exulting in Frey's imminent downfall, enraged him. The memory of Crake's snide dismissal made him grit his teeth. The picture of himself smiling out from the page inspired a deep and intolerable hatred. That they should use _that_ picture. _That_ one!
This was too much. He could take the vagaries of chance that robbed him at cards time and again. He could handle the knowledge that his best efforts at self-improvement were doomed to be thwarted by some indistinct, omnipotent force. He could live with the fact that he was captain of a crew who was staying with him only because they had nowhere else to go.
But to be so thoroughly stitched up, without any idea who was behind it or even what he'd done to deserve it? It was so tremendously, appallingly unfair that it made his blood boil.
"I can't run anymore," he murmured.
"What'd you say?" asked Malvery.
He surged to his feet, knocking his flagon aside with the back of his hand, his voice rising to a shout. "I said I can't _run_ anymore!" He snatched up the broadsheet and flung it away, pointing after it. "There's nowhere I can go that _she_ won't find me! She'll never stop! Now, I'm a man well accustomed to being shat on by fate, but everyone has their limits and I've bloody well reached mine!"
The others stared at him as if he was mad. But he wasn't mad. Suddenly he felt inspired, empowered, alive! Swept up in the excitement of a resolution, Frey thundered on.
"I'm damned if I'm going to run halfway round the planet to get away from these people! I'm damned if I'm going to hang for a crime without even knowing what I've done! And I'm damned if I'm going to rot out the rest of my days in some icebound wasteland!" He pounded his fist down on the table. "There is one person who might know who's behind all this: that brass-eyed bastard who gave me the tip. Xandian Quail. We all know his kind never reveal their sources, but he made one big mistake. He left me nothing else to lose. So I'm gonna go back. I'm gonna head right back there, even if the whole bloody country is looking for me, and I'm gonna find out who did this! I'll make them sorry they ever heard the name of Darian Frey!" He thrust his fist in the air. _"Who's with me?"_
Malvery, Pinn, and Harkins gaped at him. Silo watched him inscrutably. The bartender cleaned flagons. The only sound in the silence that followed was the squeak of cloth against pewter.
"Oh, piss on you all," snapped Frey, then stormed off toward the door. "If you're not on the _Ketty Jay_ in half an hour, I'll leave you here to freeze."
# _Chapter Eleven_
CRAKE MIXES WITH THE COMMON MAN—A BAD CASE OF INDIGESTION—SMOKING OUT THE ENEMY—QUESTIONS AND ANSWERS
n reflection, Crake had spent rather a lot of time in taverns lately. As a man who had once prized study and discipline, it made him feel vaguely decadent. He was used to drawing rooms and social clubs, garden parties and soirées. During his university days he'd frequented fashionably seedy dives, but they were usually full of similarly educated folk eager for a taste of the low life. His drinking binges had always been disguised as evenings of intelligent debate. There was no threat of that with the crew of the _Ketty Jay_.
Nowadays, he simply drank to forget.
He sat at the bar, two mugs of the foul local grog before him. It was late afternoon in Marklin's Reach, and a sharp winter sun cut low across the town. Dazzling beams shone through the dirty windows and into the gloomy, half-empty tavern. Slowly writhing smoke formed hypnotic patterns, unfurling in the light.
Crake checked his pocket watch and scanned the room. His new friend was late. He wondered if he'd overdone things last night by buying all the drinks. Maybe he'd laid on the flattery a bit thick. Tried too hard.
He thought they'd gotten on well, all things considered. He thought he'd done a good job bridging the vast gulf in intellect. Still, Crake was never sure with these simple types. He suspected they had a certain intuition. They sensed he wasn't one of them.
But Rogin had seemed to take to him. He'd been happy to chat with anyone, as long as they were buying the drinks. At the end of the night, they agreed to meet up for a quick mug of grog the next day, before he went on duty. "It'll help me ease into my shift, so to speak," he'd said. Crake had brayed enthusiastically and promised to have a drink waiting.
He scratched at his beard. He'd considered shaving it off, since the Century Knights would be looking for a blond-bearded man. But the others who were chasing him were looking for somebody clean-shaven, which was why he'd grown it in the first place. He feared the Shacklemore Agency just as much as the Century Knights, and so, all things being even, he decided the beard suited him and kept it. He thought it made him look pleasingly rugged.
He checked his pocket watch again. Where _was_ that oaf? After all the effort he'd spent following the man home, tracking him to his local haunt, and plying him with booze, Crake would be sorely annoyed if he got stood up now.
He was in a somber mood. Memories of the past and doubts about the future flocked up to meet him. Was he doing the right thing, staying with the _Ketty Jay_? Wouldn't he be better off cutting them loose, making his own way? After all, he didn't exactly owe Frey a huge debt of loyalty after their run-in with Macarde.
But Frey had promised him they'd get to a big city as soon as it was safe. There, Crake could get the supplies he needed to practice his daemonism. He'd allowed himself to be placated by that. He could wait a little longer.
The need to practice the Art was nagging at him. After the accident, he'd imagined he'd never be tempted by it again. But he'd abandoned his studies out of fear, and that was a cowardly thing. Since university, his every spare moment had been secretly devoted to daemonism. It was the only thing that set him apart from the herd of overeducated, moneyed idiots who had surrounded him all his life. He thought himself better than that. He disdained them. He'd been brave enough to look into the unknown, to reach toward the arcane. He could do things that powerful men would marvel at. Shortly before they hanged him.
But no matter the dangers, he couldn't give it up. To return to the gray unknowing, the humdrum day-to-day, was unimaginable. He'd tasted grief and despair and the highest terror, he'd made the most terrible mistakes, and he bore a shame that no man should have to bear; but he'd stared into the fires of forbidden knowledge and, though he might look away for a moment, his gaze would always be drawn back.
_You can start small. Start with the easy procedures. See how you go_.
Besides, with only enough money to buy the most basic supplies—let alone pay for transport—he wasn't in a good position to leave. At least on the _Ketty Jay_ he was surrounded by people who asked no questions, people untrained in the aristocratic arts of vicious wit and backstabbing. He rather liked that about them, actually.
A disturbing notion occurred to him. Spit and blood, was it possible he was getting _comfortable_ in their company? He took a swig from his mug to wash away the bad taste that left in his mouth, then choked as he realized the grog tasted even worse.
"Went down the wrong pipe, eh?" said a voice behind him, and he was pummeled on the back hard enough to break a rib.
Crake smiled weakly and wiped his tearing eyes as the man sat down next to him. Grubby and balding, with a lumpy nose and cheeks red with gin blossoms, Rogin wasn't easy on the eye. Nor on the nose, for that matter. He had the sour and faintly cabbagey smell of a man accustomed to stewing in his own farts.
Crake made a heroic attempt to summon some manly gusto and slapped Rogin on the shoulder in greeting. "Good to see you, my friend," he said, with his best picture-pose grin. The low shafts of sunlight glinted on his gold tooth. "I got you a drink."
Rogin picked up the mug provided for him—a mug Crake had laced with Malvery's special concoction—and lifted it up so they could clink them together.
"To your health!" said Rogin, and downed his grog in one swallow.
"Oh, no," murmured Crake, with a self-satisfied smirk. "To yours."
THE WARMTH DRAINED FROM the air as the sun dipped beneath the horizon. Frost gathered on the churned mud of the thoroughfares, and the people of Marklin's Reach retreated into their homes. A thin blue mist of fumes hung near the ground, seeping from portable generators that hummed and clattered in the alleys behind the wooden shanties. Chains of electric bulbs brightened and dimmed as the power fluctuated.
Frey huddled in the mouth of an alleyway, concealed by a patch of shadow conveniently created when Pinn had smashed the bulbs overhead. Silo and Jez stood with them. Crake and Harkins had been left back at the _Ketty Jay_ , since both of them were a liability in a firefight. Harkins would be reduced to a dribbling wreck in seconds, and Crake was more likely to hit a friend than an enemy.
Xandian Quail's house stood across the street, secure behind its high walls and its wrought-iron gate. Frey had been watching the two guards behind the gate for an hour now as they stamped back and forth, bundled up in jackets and hoods. He was cold and impatient and was wondering whether Crake had put enough of Malvery's concoction into Rogin's drink.
Malvery himself loitered a little way off, near the wall but out of sight of the guards. A black doctor's bag lay at his feet. His hands were thrust into his coat pockets and he looked as miserable as Frey felt. As Frey watched, he leaned down, opened the bag, and took a warming hit of medicinal alcohol from the bottle within.
Then, finally, a groan from behind the wall. Malvery stiffened, listening. After a moment Rogin swore and groaned again, louder still. His companion's voice was too muffled to hear the words, but Frey detected alarm in his tone.
Malvery looked at him expectantly. Frey stepped out of the shadows and waved the doctor into action.
_Go_.
Malvery scooped up the doctor's bag and set off. Rogin's groans had become low cries of pain now, foul oaths forced through gritted teeth. Malvery passed in front of the gate, halted theatrically as if he'd only just heard the sounds of Rogin's distress, and then peered through the bars.
Rogin was curled in a ball on the other side, clutching his stomach. His companion, a tall, wiry man with ginger hair and a broken nose, looked up as Malvery hailed him.
"Get lost, old feller!" the guard snapped.
"Is your friend alright?" Malvery inquired.
"Does he _look_ alright?"
"It's my guts!" Rogin gasped. "My bloody guts! Hurts like a bastard." He grimaced as another spasm of agony racked him.
"Let me help him. I'm a doctor," Malvery said. The ginger-haired guard looked up suspiciously. Malvery brandished his doctor's bag. "See?"
The guard glanced back at the doorway of the house, wondering if he should tell someone inside.
"For shit's sake! Let him in!" Rogin cried, his voice getting near to hysteria. "I'm dying, damn it!"
The guard fumbled out a set of keys and opened the gate, then stepped back to allow Malvery through.
"Thank you," said Malvery as he passed. Then, since the guard had one hand on the gate and the other on the key, he drew out a pistol and pressed it to the unfortunate man's temple. "Why don't you leave it open, eh?" he suggested.
Frey, Silo, Pinn, and Jez sallied out from the shadows and across the deserted thoroughfare, then slipped through the open gate. Silo went to the fallen man and quickly disarmed him, while Jez did the same to Malvery's guard. Rogin made a strangled sound of mingled fury and pain, but Silo crouched down next to him and tapped the barrel of a revolver against his skull.
"Shhh," he said, finger on his lips.
Jez closed the gate, and Frey kept his gun on the ginger-haired guard while Silo and Malvery trussed Rogin up. They gagged him with a length of rag and one of Pinn's balled-up socks, which Malvery had chosen for additional anesthetic effect. Then they carried him off to the nearby guardhouse.
"Don't worry, mate," said Malvery as they went. "The prognosis is good. The pain'll pass in a few hours, although I'd suggest you send your loved ones out of town before you take your next dump."
Frey scanned the house quickly. The curtains were drawn across the windows, and no one seemed to have paid any attention to Rogin's cries. If they'd heard him at all, they probably assumed it was someone on the street outside.
He didn't dare to hope that all might be going well. That would only put the jinx on him.
"Right, you," he said to the remaining guard. "I've got a job for you. Do it well, and you don't get hurt. Understand?"
The guard nodded. He was angry and humiliated, but he was mostly terrified. Probably his first time being held at gunpoint. Good. Frey didn't want to shoot him if he didn't have to.
Jez tossed Malvery his shotgun as he and Silo returned from the guardhouse. Malvery always felt better with a bit of proper firepower. He didn't trust pistols; he thought them fiddly.
They assembled on either side of the heavy oak doors, beneath the stone porch. Frey dragged the guard up by his arm and stepped back, pistol trained on him.
"Get them to open the door," he said. "Don't try anything, if you want to keep your brains in your head."
The guard nodded. He took a nervous breath and rapped on the door.
Frey's hand was trembling, just a little. His throat had gone dry. He wondered if the guard knew how scared he was himself.
_I don't want to die_.
"Yeah?" came a voice from inside.
"It's Jevin. Open the door," said the guard.
The door opened a little way. It was Codge, he of the long face and the patchy black beard.
"What's up?"
Frey shoved the guard aside and aimed his revolver point-blank at the white expanse of Codge's forehead. Codge stared at him in surprise for an instant. Then he went for his gun.
Frey's reaction was as instinctive as Codge's had been. He pulled the trigger. Codge's head snapped back; tiny beads of blood spattered Frey's face. Codge tipped backward and crumpled to the ground.
Frey wasted a moment on shock. He hadn't wanted to fire. What was that idiot doing, going for his gun like that?
Malvery shouldered the door hard, and it opened a little way before jamming against the dead weight of Codge's body. Frey wriggled through the gap and into the hallway. There was a panicked moment as he found himself alone and exposed, face-to-face with a guard who had been too bewildered to react until now. The man's hand moved for the pistol in his holster, but Frey's weapon was out and ready, and he was faster. His arm snapped out straight, finger poised over the trigger.
_Don't_.
To Frey's relief, this one had more sense than Codge. He slowly raised his hands. Malvery shoved the door open the rest of the way, barging Codge's corpse aside. There was a swell of women's voices from one of the doorways leading off the hallway, crying variations of "Stop!" But a guard came stumbling out anyway, naked from waist to ankle, his engorged penis waggling ridiculously. One hand was struggling to pull up the pair of trousers that tangled his legs, while he attempted to aim a pistol with the other. Malvery sighted and blew him away before he'd taken two steps.
Frey pulled the guard who had surrendered over to him, pressing the pistol to his back, using him as a shield. He disarmed his prisoner and tossed the gun aside, as Malvery moved out of the way of the door and Jez and Pinn came in after him.
"Silo?" Frey asked.
"Covering the guard outside," Jez replied. "Jevin, or whatever his name is."
Frey was thankful that someone had the presence of mind to do that. He'd half expected them all to come rushing in when he did.
He wrapped his arm around his prisoner's throat from behind. "Where's the other one?" he hissed. There had been four guards inside last time he'd visited. Without waiting for an answer, he called, "We've got your friend here! Step out and you won't get hurt! I've got no business with you!"
There was silence but for the ticking of the clock that overlooked the hall. Then a voice drifted out from another doorway: "Bren? That you?"
"It's me, Charry," Frey's prisoner replied. "They got a gun to my head. There's four of 'em."
Another long silence. "Alright," said Charry. "I'm coming out. Don't nobody shoot. Nobody's gonna shoot, are they?"
"Nobody's gonna shoot," said Frey, with a pointed glare at Pinn.
A rifle skidded out from one of the doorways, followed by a pistol and a knife. A young, swarthy-looking man emerged, his hands held high. Jez took him over to stand with the other prisoner. Silo came in from outside.
"Where's the guard you were covering?" Frey asked, appalled.
"Tied him up. Put him in the guardhouse with the other," said Silo.
"Right, right," Frey said, relieved. He allowed himself to relax a little. "Should've thought of that myself."
Pinn and Malvery exchanged a glance. Malvery looked skyward in despair.
"Your boss is upstairs?" Frey asked the prisoners. They nodded. "No more guards?" They shook their heads. "The whores?"
"In there," said Charry, indicating the room the half-naked man had come from. "Obviously."
Frey looked at Silo. "You're in charge. Anyone moves, shoot them. Malvery, you and me are going to have a word with Quail." As an afterthought, he added, "Bring your bag. I don't want him dying before he talks."
"Right-o," said Malvery, heading outside to collect the doctor's bag that he'd left on the porch.
Frey walked up to the whores' doorway and stood to one side. The dead man with his trousers around his ankles had a comically astonished expression on his face.
_We can all but hope to die with such dignity and elegance_ , he thought.
"Ladies?" he called. There was no reply. He stuck his head around the doorway and drew it back rapidly as a shotgun blast blew part of the door frame to splinters.
"Ladies!" he said again, slightly annoyed this time. His ears were ringing. "We're not going to hurt you!"
"No, you're bloody not!" came the reply. "I know your sort! We give what we give 'cause we're paid to! Nobody takes it by force!"
"Nobody's taking anything," said Frey. "You might remember me. Darian Frey? We were introduced just a few weeks ago."
"Oh," came the reply, rather less harsh than before. "Yes, I remember you. Stick your head out, let us have a look."
"I'd rather not," he replied. "Listen, ladies, our business is with Quail. We'll be done with it and go. Nobody's going to bother you. Now will you let us past?"
There was a short debate in low voices. "Alright."
"You won't shoot?"
"Long as nobody tries to come in. 'Specially that one who looks like a potato. He's enough to turn a woman to the other side."
Silo grinned at Pinn, who kicked an imaginary stone and swore under his breath.
"Especially not him," Frey agreed.
"Well. Okay, then."
Malvery returned with his bag. He took another swig of swabbing alcohol and stuffed it back inside. Pinn bleated for a taste, but Malvery ignored him.
They hurried past the doorway. Frey caught a glimpse of the whores, hidden behind a dresser with a double-barreled shotgun poking over the top. They held a pair of white, pink-eyed dogs on leashes, for extra protection. One of the whores waved and made a kiss face as he passed, but he was out of sight too quickly to respond.
He headed up the stairs, Malvery close behind. The coiled-brass motif from the hallway continued on the upper level, but here the walls and floor were paneled in black wood and lit by electric bulbs in molded sconces. The place had a dark, grand feel to it. Frey was feeling pretty dark and grand himself right now.
As they approached Quail's study, they heard something crash inside. The sound of a desk tipping over. Presumably he was making a barricade. Frey remembered the bars on the windows from his last visit. They couldn't be opened from the inside. Quail wasn't going anywhere.
They took position on either side of the door. Frey kicked it open and stepped back as a pistol fired twice. The door rebounded and came to rest slightly ajar. There were two coin-size holes in the wood paneling of the corridor at chest height.
"Anyone comes through that door, they'll be sorry!" Quail cried. His attempt to sound fierce was woeful. "I've got a couple of guns and enough ammo for the whole night. The militia will be here sooner or later! Someone will have heard the racket you made downstairs!"
Frey thought for a moment. He waved at Malvery. "Give me the bottle."
"What?" Malvery said, feigning ignorance.
"The bottle of alcohol in your bag. Give it here."
Malvery opened his bag reluctantly. "This bottle?" he asked querulously, rather hoping Frey would reconsider.
"I'll buy you another one!" Frey snapped, and Malvery finally handed it over. He snatched it from the doctor and pulled out the stopper. "Now a rag."
"Oh," Malvery murmured, divining Frey's plan. He passed Frey a bit of cloth with the expression of one about to witness the cruel extinction of some lovable, harmless animal.
Frey stuffed the rag into the neck of the bottle and upended it a few times. He pulled out a match—one of several that had lived in the creases of his coat pocket for many years—and struck it off the doorjamb. He touched it to the rag, and flame licked into life.
"Fire in the hole," he grinned, then booted open the door and lobbed the bottle in. He ducked back in time to avoid the gunfire that followed.
The throw had been pitched into the corner of the room—he didn't want to incinerate Quail quite yet—but the whispermonger started howling as if he were on fire himself, instead of just the bookshelves.
Frey and Malvery retreated a little way down the corridor to another doorway, where they took shelter and aimed. Black smoke began to seep out of Quail's study. They could hear him clattering around inside, cursing. Glass smashed, bars rattled. The smoke became a thick, churning layer that spread out along the ceiling of the corridor. Quail began to cough and hack.
"You think this is gonna take much longer?" Malvery asked, and an instant later Quail burst from the room, his good eye watering, waving a pistol in one hand.
_"Drop it!"_ Frey yelled, in a voice so loud and commanding that he surprised himself. Quail froze, looking around, and spotted Frey and Malvery with their guns trained on him. "Drop it, or I'll drop you."
Quail dropped his gun and raised his hands, coughing. His smart jacket was smoke-stained, and his collar had wilted. His sleeve was ripped, revealing the polished brass length of his mechanical forearm.
Frey and Malvery emerged from hiding. Frey grabbed the whispermonger by the lapels and dragged him down the corridor, away from the smoke and flames of the study. He slammed Quail bodily up against the wall. Quail glared at him, teeth gritted, defiance on his face. Frey saw himself reflected in Quail's mechanical eye.
"Right, then," said Frey, then stepped back and shot him in the shin.
Quail screamed and collapsed, writhing on the ground, clutching at his leg. "What the shit did you do that for, you rotting whoreson?" he yelled.
Frey knelt down on one knee next to him. "Look, Quail, I don't have time for the preliminaries, and to be honest, I'm pretty unhappy with you right now. So let's pretend you've already put up a spirited resistance to my questioning and just tell me: who set me up? Because if I have to ask again, it's your kneecap. And after that I'm going for something you _can't_ replace with a mechanical substitute."
"Gallian Thade!" he blurted. "It was Gallian Thade who gave me the job, that's all I know! It came through a middleman but I knew something was funny, so I traced it back to him. He's a rich landowner, a nobleman who lives out in—"
"I know who Gallian Thade is," Frey said. "Go on."
"They said it should be you, specifically _you_ , that I offered the job to. But they said... aaah, my leg!"
" _What_ did they say?" Frey demanded, and punched him in his wounded shin. Quail shrieked and writhed, breathless with pain.
"I'm telling you, I'm telling you!" he protested. "They said if you couldn't be found, I could offer it to someone else, last minute. The important thing... the important thing was that the _Ace of Skulls_ was passing through the Hookhollows on that date, they _knew_ that, and they wanted someone to attack it. Preferably you, but if not, any lowlife would do."
"You're not in much of a position to make insults, Quail."
"Their words! Their words!" he said frantically, holding up a hand to ward off further punishment. "Someone who wouldn't be missed, that's what they said. You'll forgive me if I'm not thinking too clearly since you just shot me in my damn leg!"
"Pain does cloud one's judgment," Malvery observed sagely, crouched alongside Frey.
"You've no idea what was on that aircraft?" Frey pressed the whispermonger. He coughed into his fist. Time was getting short. The corridor was filling with hot smoke, and the only breathable air was low down. The militia wouldn't be far away.
"He told me jewels! He said he'd buy them from me if you came back. If not, he'd pay me a fee anyway. It was a cover story, I knew that, but I didn't... didn't find out what was underneath."
"You don't want to speculate? Any idea why the Century Knights are all over me? Why there's such a big reward offered that Trinica Dracken is involved?"
"Dracken!" His eyes widened. "Wait, I do know this! Trinica Dracken... She's put the word out in the underground. Anyone sees you, they tell her. Not the Century Knights. She's offering an even bigger reward. But it's for information only.... She wants to catch you herself."
"Well, of course she does—"
"No, you don't understand. Dracken doesn't have that kind of money. Someone's funding her. I don't know who. But whoever it is, they want her to get you before the Century Knights do. This isn't just about a reward, Frey. This is something more. Someone doesn't want the Knights to find you."
Frey's jaw tightened. Deeper and deeper. Worse and worse.
"We'd better go, Cap'n," said Malvery, coughing. "Smoke's getting bad."
"Alright," Frey muttered. "Come on."
"What about me?" Quail said, as they got up. "You can't just—"
"I can," said Frey. "You still have one good leg." With that, they left, the whispermonger hurling oaths after them.
"Should we truss up the rest of his men?" Malvery asked, as they hurried down the stairs at the end of the hall. Pinn, Jez, and Silo still had the surviving guards at gunpoint at the far end.
"No time. Besides, I think they'll have their hands full saving the house." Frey raised his voice to address everyone in the hall. "Ladies, gentlemen, we're out of here! Your boss is upstairs and only mildly wounded. Go help him if you have the inclination. You'll also notice that the house is on fire. Make of that what you like."
Militia whistles were sounding in the distance as the crew of the _Ketty Jay_ slipped through the front gate, their breath steaming the air. Bright yellow flames were pluming from the eaves of the house behind them.
"This time we're _really_ not coming back," said Frey, as they headed for the dock.
"One question," said Malvery as he huffed alongside. "Gallian Thade, this noble feller. You know him?"
"No," said Frey. "But I knew his daughter very well. Intimately, you could say."
Malvery rolled his eyes.
# _Chapter Twelve_
THE AWAKENERS—FREY APOLOGIZES—A GAME OF RAKE
lden Square sat in the heart of Aulenfay's trade district, a wide paved plaza surrounded by tall apartment buildings with pink stone fascias. On a clear winter day such as this, the square was filled with stalls and people, everyone buying or selling. Hawkers offered food or theater tickets or clockwork gewgaws; street performers imitated statues and juggled blades. Visitors and locals wandered between the attractions, the ladies in their furs and hats, men in their leather gloves and greatcoats. Children tugged at their parents' arms, begging to investigate this or that, drawn by the smell of candied apples or cinnamon buns.
The centerpiece was a wide fountain. The water tumbled down through many tiers from a high column, on which stood a fierce warrior. He was wielding a broken sword, fighting off three brass bears that clawed at him from below. On the rooftops, pennants of brown and green snapped and curled in the breeze, bearing the Duke's coat of arms.
Frey and Crake sat on the step of a small dais. Behind them, four stone wolves guarded an ornate black iron lamppost, one of several dotted around the square to illuminate it at night. Frey was holding a white paper bag in one hand and chewing on a sugarplum. He offered the bag to Crake, who took a sweet absently. Both of them were watching a booth in the corner of the square, from which three Awakeners were plying their trade.
The booth was hung with banners showing a symbol made up of six spheres in an uneven formation, connected by a complicated pattern of straight lines. The three Awakeners were dressed identically, in white single-breasted cassocks with high collars and red piping that denoted their status. They were Speakers, the rank and file of the organization.
One of them was kneeling in front of a circular chart laid out on the ground. An eager-eyed man knelt opposite, watching closely. The Speaker was holding a handful of tall sticks upright in the center of the chart. He let go and they fell in a clutter. The Speaker began to study them intently.
"Seriously, though," said Frey. "What's all that about?"
"It's rhabdomancy," said Crake. "The way the sticks fall is significant. The one behind him is a cleromancer. It's a similar technique. See? He drips a little animal blood in the bowl, then casts the bones in there."
"And you can tell things from that?"
"Supposedly."
"Like what?"
"The future. The past. You can ask questions. You can find out if the Allsoul favors your new business enterprise or see which day would be most auspicious for your daughter's wedding. That kind of thing."
"They can tell all that from some sticks?"
"So they say. The method isn't really that important. Each Awakener specializes in a different way of communicating with the Allsoul. See the other one? She's a numerologist. She uses birth dates, ages, significant numbers in people's lives, and so on."
Frey looked over at the third Awakener, a young woman, chubby and sour-faced. She was standing in front of a chalkboard and explaining a complicated set of mathematics to a bewildered audience of three, who looked as if they could barely count on their fingers.
"I don't get it," Frey confessed.
"You're not supposed to get it," said Crake. "That's the point. Mystical wisdom isn't much good if everyone possesses it. The Awakeners claim to be the only ones who know the secret of communicating with the Allsoul, and they don't intend to share. If you want something from the Allsoul, you go through them."
Frey scratched the back of his neck and looked askance at Crake, squinting against the sun. "You believe any of that?"
Crake gave him a withering look. "I'm a daemonist, Frey. Those people in there, they'd like to see me hanged. And do you know why? Because what I do is _real_. It _works_. It's a _science_. But a century of this superstitious twaddle has made daemonists the most vilified people on the planet. Most people would rather associate with a Samarlan than a daemonist."
" _I'd_ associate with a Samarlan," said Frey. "War or no war, they've got some damn fine women."
"I doubt Silo would think so."
"Well, he's got a chip on his shoulder, what with his people being brutally enslaved for centuries and all that." Crake conceded the point.
"So what about this Allsoul thing?" asked Frey, passing over another sugarplum. "Enlighten me."
Crake popped the sugarplum in his mouth. "Why the interest?"
"Research."
"Research?"
"Gallian Thade—the man who put Quail up to framing us—he's the next one we need to talk to if we want to get some answers. I want to find out what he's framing us for." He stretched, stiff from sitting on the step. "Now, getting to Gallian, that's gonna be no easy task. But I can get to his daughter."
Crake joined the dots. "And his daughter is an Awakener?"
"Yeah. She's at a hermitage in the Highlands. They keep their acolytes cut off from the outside world while they study. I need to get in there and get to her. She might know something."
"And you think she'll help you?"
"Maybe," said Frey. "Best idea I've got, anyway."
"Shouldn't you be asking someone who believes this rubbish?"
Frey shifted on the step and settled himself again. "You're educated," he said, with slightly forced offhandedness. "You know how to put things."
"That's dangerously close to a compliment, Frey," Crake observed.
"Yeah, well. Don't let it go to your head. And it's _Captain_ Frey to you."
Crake performed a half-arsed salute and slapped his knees. "Well, then. What _do_ you know?"
"I know _some_ things. I've heard of the Prophet-King and how they put all his crazy pronouncements in a book after he went mad, except _they_ think he was touched by some divine being or something. And how they all say they can solve your problems, just a small donation required. But I never was that bothered. None of it seemed to make much sense. Like some street scam that got out of control, you know? Like the thing with the three cups and the ball and, no matter what you do, you never win. Except with this, half the country plays it and they never get that it's rigged." He snorted. "Nobody knows _my_ future."
"Fair enough." Crake cleared his throat and thought for a moment. When he spoke, it was as a teacher, crisp and to the point. "The basic premise of their belief posits the existence of a single entity called the Allsoul. It's not a god in the sense of the old religions they wiped out, more like a sentient, organic machine. Its processes can be seen in the movement of wind and water, the behavior of animals, the eruption of volcanoes, and the formation of clouds. In short, they believe our planet is alive and intelligent. In fact, vastly more intelligent than we can comprehend."
"Okaaay..." Frey said uncertainly.
"The Awakeners think that the Allsoul can be understood by interpreting signs. Through the flights of birds, the pattern of fallen sticks, the swirl of blood in milk, the mind of the planet may be known. They also use rituals and minor sacrifices to communicate with the Allsoul, to beg its favor. Disease can be cured, disaster averted, success in business assured."
"So let me get this straight," Frey said, holding up his hand. "They ask a question, they... release some birds, say. And the way the birds fly, what direction they go, that's the planet talking to them. The Allsoul?"
"When you strip out all the mumbo jumbo, yes, that's exactly it."
"And you say it doesn't work?"
"Ah!" said Crake scornfully, holding up a finger. "That's the clever part. They've got it covered. Margin for human error, you see. Their understanding of the Allsoul is imperfect. Human minds aren't yet capable of comprehending it. You can ask, but the Allsoul might refuse. You can predict, but the predictions are so vague they'll come true more often than not. The Allsoul's schemes are so massive that the death of your son or the destruction of your village can be explained away as part of a grand plan that you're just too small to see." He gave a bitter chuckle. "They've got all the angles figured out."
"You really hate them, don't you?" Frey said, surprised at the tone in his companion's voice. "I mean, you _really_ hate them."
Crake clammed up, aware that he'd let himself get out of control. He gave Frey a quick, tense smile. "It's only fair," he said. "They hated me first."
THEY STROLLED up the hill from Olden Square, along a tree-lined avenue that led toward the wealthier districts of Goldenside and Kingsway. Passenger craft flew overhead, and motorized carriages puttered by. There were fine dresses in the windows of the shops, displays of elaborate toys and sweetmeats. As they climbed higher, they began to catch glimpses of Lake Elmen through the forested slopes to the west, vast as an inland ocean. All around, stretching into the distance as far as the eye could see, were the dark green pines and dramatic cliffs of the Forest of Aulen.
"Pretty part of the world," Frey commented. "You'd think the Aerium Wars never happened."
"Aulenfay missed the worst of it the first time round and got none of it on the second," said Crake. "You should see Draki and Rabban. Six years on and they're still half rubble."
"Yeah, I've seen 'em," he replied distantly. He was watching a young family who were approaching on their side of the avenue: a handsome husband, a neat wife with a beautiful smile, two little girls singing a rhyme as they skipped along in their frilly dresses. After a moment the woman noticed his interest. Frey looked away quickly, but Crake bid them a pleasant "Good day" as they passed.
"Good day," the couple replied, and a moment later the girls chimed "Good day!" politely. Frey had to hurry on. The sight of their happiness, the sound of those little voices, was like a kick in the chest.
"What's the matter?" Crake inquired, noticing Frey's sudden change in demeanor.
"Nothing," he muttered. "Nothing, I just suddenly... I was worried they'd recognize me. Shouldn't have made eye contact."
"Oh, I wouldn't worry. I told you, I picked up a broadsheet in Marklin's Reach yesterday. There was no mention of you. And Aulenfay isn't the kind of place where they stick _Wanted_ posters everywhere. I think you've been forgotten by the general public." He patted Frey on the shoulder. "Besides, considering the age of the photograph and your newly raddled and insalubrious appearance, I don't think anyone would recognize you unless they had a particular interest."
"Raddled and insalubrious?" Frey said. He was beginning to suspect Crake of showing off in an attempt to belittle him.
"It means formidable and rugged," Crake assured him. "The beard, you see."
"Oh."
They came to a crossroads and Crake stopped on the corner. "Well, I must be leaving you. I have to go and pick up my equipment, and the kind of people who sell daemonist paraphernalia are the kind of people who don't like non-daemonists knowing who they are."
"Right," said Frey. "Have it delivered to the dock warehouse. We'll pick it up from there. No names, though."
"Of course." The daemonist turned to go.
"Crake."
"Yes?"
Frey looked up the street, rather awkwardly. "That thing with Macarde... him holding a gun to your head and so forth..."
Crake waited.
"I'm sorry it went that way," Frey said at last.
Crake regarded him for a moment, his face unreadable. Then he nodded slightly and headed away without another word.
FREY MADE HIS WAY to the South Quarter, a less affluent part of the city, where he visited a tailor and a shop that specialized in theatrical makeup. After that, he went looking for a game of Rake.
The South Quarter was about as seedy as Aulenfay got, which meant it was still quite picturesque in a charmingly ramshackle kind of way. The winding lanes and cobbled alleyways were all but free of filth and litter. Statues and small, well-kept fountains still surprised visitors at every turn. There were no ragtag children or crusty beggars. Aulenfay had a strict policy against that kind of thing.
The Ducal Militia were in evidence, patrolling in their stiff brown uniforms. Frey kept out of their way.
Despite the risks of coming to a big city, Frey had allowed himself to be persuaded by Crake. He did have some preparation to do before he went looking for Amalicia Thade in her secluded hermitage, but that wasn't the whole reason. The crew needed a break. The disastrous attack on the freighter, the escape from the Century Knights, that frustrating time spent bored and freezing in Yortland—all these things had worn them down, and they were sick and tired of one another's company. A little time off would do them all good, and Aulenfay was a fine place for it.
Whether they'd all come back or not was another matter, but Frey wasn't worried about that. If they left, they left. He'd understand. They'd each make their own choice.
It took some searching to locate the Rake den. He hadn't been this way for a few years. But it was still there, in the cellar bar of an old tavern: a small room with three circular tables and a vaulted ceiling of old gray brick. Smoke drifted in the air and the shadows were thick, thrown by oil lanterns. Rake players didn't like their games too brightly lit. Most of them had only a passing acquaintance with daylight.
Only one of the tables was in use when Frey was shown in. Three men sat there, studying their cards, dull piles of coins before them. There was a thin, po-faced man who looked like an undertaker; an elderly, toothless drunk; and a whiskery, rotund fellow with a red face and a battered stovepipe hat. Frey sat down and they introduced themselves as Foxmuth, Scrone, and Gremble, which amused Frey, who thought they sounded like a firm of lawyers. Frey gave a false name. He ordered a drink, emptied out his purse on the table, and set to the game.
It wasn't long before he realized his opponents were terrible card players. At first he suspected some kind of trap. Perhaps they were feigning incompetence to sucker him. But as the game went on, he became ever more convinced they were the real deal.
They went in big with their money, chasing runs that never came up. They jittered with excitement when they made a low three-of-a-kind and then bet it as if it was unbeatable. They allowed themselves to be bluffed away whenever they saw Frey pick up a dangerous card, frightened that he was holding something that could crush them.
From the moment he sat down, he was winning.
Several hours passed, and several drinks. Scrone was too plastered to keep his attention on the game, and his money was whittled away on silly bets. Eventually, he made a suicidal bluff against Foxmuth, who was holding Crosses Full, and lost it all. After that, he fell asleep and began to snore.
Foxmuth was knocked out shortly afterward, following a chancy call against Gremble's Ace–Duke paired. Foxmuth's last card failed to produce the hand he needed, and Gremble scooped up all his money.
Frey was only mildly disheartened. All his careful work in maintaining his lead had been undermined by the bad play of the other two. They'd given all their money to Gremble, making the two remaining players roughly even. He settled down to the task of demolishing his final opponent.
"Just my luck," Foxmuth moaned. "The wife's going to rip me a new arsehole when I come home. I wouldn't have even been here if they'd had the parade today."
Frey was only half listening. He dealt the cards, three each, then picked up his. A thin chill of excitement ran through him. Three Priests.
"Why didn't they have the parade?" Frey asked, making idle chatter to cover his anticipation.
"Earl Hengar was supposed to be coming to see the Duke. Big parade and all. But with what's happened... well, I suppose they thought it was in bad taste or something. Canceled last minute."
"I should think so. Bloody disgrace," muttered Gremble. He rapped the table to indicate that he didn't wish to bet.
"Bet," said Frey. "Two bits." He pushed the coins in. It was a high opening bid, but he knew Gremble's style of play by now. Instead of being frightened off, Gremble would assume it was a bluff and match it. Which was exactly what he did.
Frey dealt four more cards to the middle, two for each player in the game. Two faceup, two facedown. The faceup cards were the Lady of Wings and the Priest of Skulls.
His heart jumped. If he could get that Priest, he'd have an almost unbeatable hand. But Gremble, to the left of the dealer, got to pick his card first from the four in the middle.
"What's a disgrace?" he asked, trying to keep the conversation up. He wanted Gremble distracted.
"About Hengar and that Sammie bitch."
Frey gave him a blank look.
"You don't know? You been living in a cave or something?"
"Close," said Frey.
"It's not been in the broadsheets," said Foxmuth. "They don't dare print it. But everyone knows. It's been all over this past week."
"I've been away," he said. "Yortland."
"Chilly up there," Gremble commented, taking the Lady of Wings as he did so. Frey thrilled at the sight.
"Yeah," Frey agreed. The Priest was his. He made a show of deliberating whether to take one of the two facedown mystery cards or not. "So what's the story with Hengar?"
Hengar, Earl of Thesk and the only child of the Archduke. Heir to the Nine Duchies of Vardia. It sounded like something Frey should be paying attention to, but he was concentrating on depriving this poor sap of all his coins. He picked up the Priest. Four Priests in his hand. If he played this right, the game would be his.
"So there were all these rumors, right?" Foxmuth said eagerly. "About Hengar and this Sammie princess or something."
"She wasn't a princess, she was some other thing," Gremble interrupted, frowning as he looked at his hand.
"Yeah, well, anyway," Foxmuth continued. "Hengar was having secret meetings with her."
"Political meetings?"
"The other kind," Gremble muttered. "They was lovers. The heir to the Nine Duchies and a bloody Sammie! The family wanted it stopped, but he wouldn't listen, so they was covering it all up. But this past week, well... All I can say is, someone must've shot their mouth off."
"What's so wrong with him seeing a Sammie?" Frey asked.
"Did you miss the wars or something?" Gremble cried.
"I wasn't on the front line," said Frey. "First one, I was working as a cargo hauler. Never saw action. Second one, I was working for the Navy, supply drops and so on." He shut up before he said any more. He didn't want to revisit those times. Rabby's final scream as the cargo ramp closed still haunted him at night. He could never forget the awful, endless, slicing agony of a Dakkadian bayonet plunging into his belly. Just the thought made him sick with fury at the people who had sent him there to die: the Coalition Navy.
Gremble harrumphed, making it clear what he thought of Frey's contribution. "I was infantry, both wars. I saw stuff you can't imagine. And there are a lot of people out there like me. Curdles my guts to think of our Earl Hengar snuggling up to some pampered Sammie slut."
"So how did the news get out?"
"Search me," growled Gremble. "But the Archduke ain't happy about it, I bet. There's already all them rumors about the Archduchess, how she's secretly a daemonist and that. You know they say the Archduke has a regiment of golems helping guard his palace in Thesk? And that he's planning to make more regiments to fight on our front lines?"
"Didn't know that," said Frey.
"It's what they say. They say the Archduchess is behind it. They say that's why they're doing all that stuff to undermine the Awakeners. Awakeners and daemonists hate each other."
"Yeah, I gathered," said Frey, thinking back to his earlier conversation with Crake.
"And now there's Hengar behaving like this..." Gremble tutted. "You know, I always used to like him. He's a big Rake player, you know that?" He folded his arms and sucked his teeth. "But now? I don't know what that family's coming to."
"Speaking of Rake, are you gonna bet?"
"Five bits!" Gremble snapped.
"Raise five more," Frey replied.
"I bet all of it!" said Gremble immediately, piling the rest of his money into the center of the table. Then he sat back and looked at his cards with the air of someone wondering what he'd just done.
Frey considered for only a moment. "Okay," he said. Gremble went pale. He hadn't expected that.
Frey laid down his cards with a smile. "Four Priests," he said. Gremble groaned. His own four cards were the Ace, Ten, Three of Wings, and the Lady of Wings he'd just picked up. He was going for Wings Full, but even if he made it, he couldn't beat Four Priests.
Unless Frey drew the Ace of Skulls.
The Ace of Skulls was the wild card. Usually it was worse than worthless, but in the right circumstances it could turn a game around. In most cases, if a player held it, it nullified all their cards and they lost the hand automatically. But if it could be made part of a high-scoring hand, Three Aces or a Run or Suits Full or higher, it made that hand unbeatable.
If Frey drew the Ace of Skulls, his Four Priests would be canceled and he'd lose everything.
There were two cards left on the table, facedown. Gremble reached out and turned one over. The Duke of Wings. He'd made his Wings Full, but it didn't matter now. He sat back with a disgusted snort.
Frey reached out for his card, but there was a commotion behind him, and he turned around as a tavern-boy came clattering down the stairs.
"Did you hear?" he said urgently. Scrone jerked in his chair, startled halfway out of sleep, and then slipped back into unconsciousness.
"Hear what, boy?" demanded Foxmuth, rising.
"There's criers out in the streets. They're saying why the parade got canceled. It's because of Hengar!"
"There, what did I tell you?" said Gremble, with a note of triumph in his voice. "They're ashamed of what he's done, and so they should be!"
"No, it's not that!" said the boy. He was genuinely distressed. "Earl Hengar's dead!"
"But that's... He's bloody _dead_?" Foxmuth sputtered.
"He was on a freighter over the Hookhollows. There was some kind of accident, something went wrong with the engine, and..." The boy looked bewildered and shocked. "It went down with all hands."
"When?" asked Frey. The muscles of his neck had tightened. His skin had gone cold. But he hadn't taken his eyes from that facedown card on the table.
"Excuse me, sir?"
" _When_ did it happen?"
"I don't know, sir. They didn't say."
"What kind of damn fool question is that?" Gremble raged. "When? When? What does it matter _when_? He's dead! Buggering piss-bollocks! It's a tragedy! A fine young man like that, taken from us in the prime of his life!"
"A good man," Foxmuth agreed gravely.
But the _when_ did matter. _When_ meant everything to Frey. _When_ was the final hope he had that maybe, against all the odds, he could avoid the terrible, crushing weight that he felt plummeting toward him. If it happened yesterday, or the day before... if it could somehow be that recent...
But he knew when it had happened. It had happened three weeks ago. They just hadn't been able to keep it quiet any longer.
The Century Knights. The job from Quail. All those people traveling incognito on a cargo freighter. The name of the freighter. It all added up. After all, wouldn't Hengar travel in secret, returning from an illicit visit to Samarla? And wasn't he a keen Rake player?
Gallian Thade had arranged the death of the Archduke's only son. And he'd set Frey up to take the fall.
He reached over and flipped the final card.
The Ace of Skulls grinned at him.
# _Chapter Thirteen_
FREY IS BELEAGUERED—A MYSTERIOUS AIRCRAFT—IMPERATORS
rey stumbled through the mountain pass, his coat clutched tight to his body, freezing rain lashing his face. The wind keened and skirled and pushed against him while he kept up the string of mumbled oaths and curses that had sustained him for several kloms now. On a good day, the Andusian Highlands at dawn could be described as dramatic—stunning, even—with their wild green slopes and deep lakes nestling between peaks of grim black rock. Today was not a good day.
Frey dearly wished for the sanctuary and comfort of his quarters. He remembered the grimy walls and cramped bunk with fondness, the luggage rack that ever threatened to snap and drop an avalanche of cases and trunks on his head. Such luxurious accommodation seemed a distant dream now, after hours of being pummeled by nature. He was woefully under-dressed to face the elements. His face felt like it had been flayed raw, and his teeth chattered constantly.
He lamented his bad luck at being caught out in the storm. So what if he'd set out completely unprepared? How could he have known the weather would turn bad? He couldn't see the future.
It seemed as if days had passed since he left the _Ketty Jay_ hidden in a dell. He couldn't risk landing too close to his target for fear of being seen, so he put her down on the other side of a narrow mountain ridge. The journey through the pass should have taken five hours or so. Six at the most.
When he set off, the skies had been clear and the stars twinkling as the last light drained from the sky. There had been no hint of the storm to come. Malvery had waved him on his way with a cheery _ta-ra_ and then taken a swig of rum to toast the success of his journey. Crake had been playing with the new toys he'd picked up in Aulenfay. Bess was having fun uprooting trees and tossing them around. Pinn had stolen the theatrical makeup pen that Frey had bought in the South Quarter and painted the Cipher on his forehead—the six connected spheres, icon of the Awakener faith. He was prancing around in the ill-fitting Awakener robes that had been tailored for Frey, pulling faces and acting the clown.
Frey had been unusually full of good cheer as he walked. All of them had come back from Aulenfay. Frey took that as a vote of confidence, even if the truth was they had no better alternatives. And despite the news of Hengar's death looming over him, he felt positive. Bullying Quail had energized him. Having a name to put to the shadowy conspiracy against him gave him a direction and a purpose. He'd gotten so used to running away that he'd forgotten how it felt to fight back, and he was surprised to learn that he liked it.
Besides, he thought sunnily, things were about as bad as they could possibly get. After a certain point, it didn't really matter if they hung him for piracy, mass murder, or for assassinating Earl Hengar, heir to the archduchy. He'd be just as dead, any way you cut it. That meant he could do pretty much whatever he liked from here on in.
His buoyant mood survived while the first ominous clouds came sliding in from the west, blacking out the moon. He remained persistently jolly as the first spots of rain touched his face. Then the howling wind began, which took the edge off his jauntiness a little. The rain became torrential; he got lost and then realized he had no map. By this time he'd begun to freeze and was desperately searching for shelter, but there was none to be found and, anyway, he didn't have the supplies to wait out a really bad storm. He decided to keep going. Surely he was almost there by now?
He wasn't.
Dawn found him exhausted and in bad shape. His face was as dark as the clouds overhead. He stumped along doggedly, head down, forging through the tempest. His good mood had evaporated as he got more and more soaked. It wasn't positivity but spite that drove him onward now. He refused to stop moving until he'd reached his destination. Every time he crested a rise and saw there was another one ahead, it made him angrier still. The pass had to end eventually. It was him against the mountain, and his pride wouldn't let him be beaten by a glorified lump of rock.
Finally the wind dropped and the rain dwindled to a speckling. Frey's heart lifted a little. Could it be that the worst was over? He didn't dare admit the possibility to himself, for fear of inviting a new tempest. Fate had a way of tormenting him like that. The Allsoul punished optimists.
He struggled up another sodden green slope and looked down into the valley beyond. There, at last, he saw the Awakener hermitage where Amalicia Thade was cloistered.
The hermitage sat on the bank of a river, a sprawling square building constructed around a large central quad. It was surrounded by lawns that opened onto fields of bracken and other hardy Highland plants. With its stout, vine-laden walls, deeply sunken windows, and frowning stone lintels, it looked to Frey like a university or a school. There was a quiet gravity to the place, a weightiness that Frey usually associated with educational institutions. Academia had always impressed him, since he'd only a passing acquaintance with it. All that secret knowledge, waiting to be learned, if only he could ever be bothered.
A little way from the hermitage, linked to it by a gravel path, was a small landing pad. There were no roads into the valley. Like so many places in Vardia, it was accessible only from the air. In a country so massive and with such hostile geography, roads and rail never made much sense once airships were invented. A small cargo craft took up one corner of the pad. It was their only link with the outside world, most likely, although there would certainly be other visitors from time to time.
Frey could see the tiny figures of Awakener Sentinels patrolling the grounds, carrying rifles. They issued from a guardhouse, which had been built outside the hermitage. He'd intended to arrive under cover of deepest night, but getting lost in the storm had put him severely behind schedule. There was no way he could approach the hermitage during the day without being seen.
The last of the rain disappeared, and he saw hints of a break in the clouds. Shafts of sun were beaming down on the mountains in the distance, warm searchlights slowly tracking toward him. There was nothing for it but to find a nook and rest until nightfall. Now that the storm had given up and he'd reached his destination, he was tired enough to die where he stood. A short search revealed a sheltered little dell, where he piled dry bracken around himself and fell asleep in the hollow formed by the roots of a dead tree.
HE WOKE TO the sound of engines.
It was night, clear and cold. He extricated himself from the tangle of bracken and stood up. His skin was fouled with old sweat, his clothes were stiff, and he desperately needed to piss. His body ached as if he'd been expertly beaten up by a squad of vicious midgets. He groaned and stretched, then spat to clear the rancid taste in his mouth. That done, he went to investigate what that noise was all about.
He looked down into the valley while he relieved himself against the side of a tree. The moon had painted the world in shades of blue and gray. The windows of the hermitage glowed with an inviting light, a suggestion of heat and comfort and shelter. Frey was looking forward to breaking in, if only to get a roof over his head for a while.
The craft he'd heard was a small black barque, bristling with weapons. A squat, mean-looking thing, possibly a Tabington Wolverine or something from that line. It was easing itself onto the landing pad, lamps on full, a blare of light in the darkness.
_A visitor_ , thought Frey, buttoning himself up. _Best get down there while they're occupied_.
He made his way into the valley, staying low in the bracken when he could, scampering across open ground when he had to. He got to the river, where there was better cover from the bushes that grew on the bank, and followed it up toward the hermitage. There was a lot of activity surrounding the newly arrived vessel. The Sentinels had all but abandoned their patrol duties to guard it. They stationed themselves along the path between the house and the landing pad.
_You should leave it alone_ , he told himself. _Take advantage of the distraction. Get inside the building. Do what you came here to do_.
A minute later he was creeping through the bracken, edging his way closer to the landing pad to get a better look. He just wanted to know what all the fuss was about.
The craft rested on the tarmac, bathed in its own harsh light. Though the cargo ramp was lowered, the craft still had its thrusters running and the aerium engines fired up. Evidently it wasn't staying for long.
When he'd gotten as close as he dared, Frey squatted down to watch. The wind rustled the bracken around him. The craft had a name painted on its underside: the _Moment of Silence_. He'd never heard of it.
The Sentinels had organized themselves as though they expected an attack, guarding the route between the craft and the door of the building, which stood open. They were dressed in gray high-collared cassocks of the same cut that all the Awakeners wore. They carried rifles and wore twinned daggers at their waists. The Cipher was emblazoned in black on their breasts: a complex design of small linked circles.
Sentinels, Crake had explained, were not true Awakeners. They lacked the skill or the intelligence to be ordained into the mysteries of the order. That was why they wore the Cipher only on their breasts, not tattooed on their foreheads. They devoted themselves to the cause in other ways, as protectors of the faith. They were not known to be especially well trained or deadly, but they were disciplined. Frey resolved to treat them with the same respect he gave anyone carrying a weapon capable of putting a hole in him.
Everyone was on the alert. Something important was happening.
There was movement by the house, and several Sentinels emerged. They were carrying a large iron-bound chest between them, straining under its weight. The chest was a work of art, lacquered in dark red and closed with a clasp fashioned in the shape of a wolf's head. Frey was suddenly very keen to find out what was inside.
The Sentinels had hauled it up the path and had almost reached the craft when two figures came down the cargo ramp to meet them. Frey felt a chill jolt at the sight of them. Being so close to the craft didn't seem like such a good idea anymore.
They were dressed head to toe in close-fitting suits of black leather. Not an inch of their skin was showing. They wore gloves and boots, and cloaks with their hoods pulled up. Their faces were hidden behind smooth black masks, through which only the eyes could be seen.
Imperators. The Awakeners' most dreaded operatives. Men who could suck the thoughts right out of your head, if the stories were to be believed. Men whose stare could send you mad.
Frey hunkered down farther into the bracken.
The Sentinels put the chest down in front of the Imperators, then one of them knelt and opened it. Frey was too far away to see what was within.
One of the Imperators nodded, satisfied, and the chest was closed. The Sentinels lifted it and carried it up the _Moment of Silence_ 's cargo ramp. They emerged seconds later, having left their burden inside. A few words were exchanged, and then one of the Imperators boarded the craft. The other turned to follow but suddenly hesitated, his head tilted as if listening. Then he turned and fixed his gaze on the spot where Frey hid in the bracken.
An awful sensation washed over him: foul, seething, corrupt. Frey's heart thumped hard in terror. He ducked down, out of sight, burying himself among the stalks and leaves. The loamy smell of wet soil and the faintly acrid tang of bracken filled his nostrils. He willed himself to be a stone, a rabbit, some small and insignificant thing. Anything that would be beneath the Imperator's notice. Some distant part of him was aware that such overwhelming fear wasn't natural, that there was some power at work here, but reason and logic had fled.
Then, all at once, the feeling was gone. The fear left him. He stayed huddled, not daring to move, breathing hard, soaked in relief. It had passed, it had passed. He murmured desperate thanks, addressed to no one. Never again, he swore. Never again would he go through that. Those few seconds had been among the most horrible of his life.
He heard the whine of the hydraulics as the cargo ramp slid shut. Electromagnets throbbed as the aerium engines got to work. The _Moment of Silence_ was taking off.
Frey gathered his courage and raised his head, peering out above the bracken. The Imperators were gone. All eyes were on the craft. Frey took advantage of the moment and scampered away toward the hermitage.
_By damn, what did that thing do to me?_
He could remember only one event vaguely comparable to the ordeal he'd just suffered. He'd been young, perhaps seventeen, and he and some friends went out to some fields where some very "special" mushrooms grew. The night had started off with hilarity and ended with Frey seized by a crushing paranoia, afraid that his heart was going to burst, and being mobbed by hallucinatory bats. That senseless, primal fear had turned a confident young man into a quivering wreck. Now he'd been brushed by it again.
His breathing had returned to normal by the time he got to the hermitage, and he had himself under control again. Shaken but unharmed. He approached the building from behind, where there were no guards to be seen, and pressed himself against the cool stone of the wall. Security was lax here. He had that to be thankful for. The guards didn't expect any trouble. They were only here for protection against pirates and other marauders, who might find the idea of a hermitage full of nubile, sex-starved young women somewhat alluring.
Frey cheered at the thought. He'd forgotten about the nubile, sex-starved part. It made his mistake back in Aulenfay twinge a little less, although his cheeks still burned at the memory.
He'd studied the Awakeners in Olden Square and picked Crake's brains about their faith for a purpose. His idea was to disguise himself as a Speaker, to blend in seamlessly and thereby move about the hermitage unopposed. Congratulating himself on his unusually thorough preparations, he'd surprised Crake by appearing in full Speaker dress: the high-collared white cassock with red piping, the sandals, the Cipher painted on his forehead in a passable impression of a tattoo.
"What do you think?" he asked proudly.
Crake burst out laughing, before explaining to the rather miffed captain that Awakener hermitages were always single-sex institutions. Acolytes were allowed no contact with the opposite gender. In Amalicia's hermitage, all the tutors and students would be female. The male guards would be forbidden to go inside except under special circumstances, and even then the female acolytes would be kept to their rooms. Lust interfered with the meditation necessary to communicate with the Allsoul.
"So you're telling me that there's a building full of women who haven't even _seen_ a man in years?" Frey had demanded to know.
"What I'm telling you is that your cunning disguise is going to be pretty useless in there, since there shouldn't be a male Speaker within twenty kloms of that hermitage," said Crake. "However, it's interesting that you jumped to the other conclusion first. I never pegged you as a glass-half-full kind of person."
"Well, a man must make the best of things," Frey replied, already envisioning a pleasant death by sexual exhaustion after being brutally abused by dozens of rampant adolescent beauties.
So Frey had discarded the uniform. Pinn found it later and had been wearing it ever since, for a joke, pretending to be an Awakener. It was funny for the first few hours, but Pinn, encouraged, had carried the joke far past its natural end and now it was just annoying. Frey wouldn't be surprised if Malvery had beaten him up and burned the robe by the time he got back. He rather hoped so.
He found two small doors, recessed in alcoves, but the Awakeners who ran the hermitage were sensible enough to keep them locked. He considered breaking a window, but they were set high up in the wall and were very narrow. He wouldn't want to get stuck in one. Finally he found the entrance to a storm cellar, which looked as if it led under the house. A padlock secured a thick chain, fastening the doors to the cellar. Both were stout and new. It looked as if it would take a lot of sawing and hammering to get through that. An intruder would certainly be caught before they gained access.
Frey drew his cutlass and touched its tip to the lock.
"Think you can?" he asked it. He didn't really believe it could understand him, but as ever, it seemed to know his intention. He felt it begin to vibrate in his hands. A thin, quiet whine came from the metal. Soon it was joined by another note, setting up a weird, off-key harmonic that set Frey's teeth on edge. The lock began to jitter and shake.
Suddenly, by its own accord, the cutlass swept up and down, smashing into the lock. The shackle broke away from the padlock, and the chain slithered free. The blade itself was unmarked by the impact. Frey hadn't even felt the jolt up his sword arm.
He regarded the daemon-thralled cutlass that Crake had given him as the price of his passage. Best deal he ever made, he reckoned, as he sheathed it again.
He climbed into the storm cellar before anyone came to investigate the noise. Steps led down to a lit room, from which he could hear the growl and rattle of machinery. He slipped inside, shut the cellar door behind him, and crept onward into the hermitage.
# _Chapter Fourteen_
A GHASTLY ENCOUNTER—INTRUDER IN THE HERMITAGE—A HEARTFELT LETTER—REUNION
rey stepped warily into the dim electric glow of smoke-grimed bulbs. The room at the bottom of the stairs was the powerhouse of the hermitage, dominated by a huge old generator that whined and screeched and shook. It took Frey awhile to persuade himself that the ancient machine wasn't in imminent danger of detonation, but in the end, logic triumphed over instinct. Since it had obviously been running for fifty years or more, the idea that it would explode just as he was passing would be such incredible bad luck that even Frey couldn't believe it would happen.
Pipes ran from the generator to several water boilers and storage batteries, linking them to the central mass like the legs of some bloated mechanical spider. The air pounded with the unsteady rhythm of the generator, and everything stank of prothane fumes. Frey's head began to swim unpleasantly.
He crept forward, his cutlass held ready. He always preferred blades in close quarters. The powerhouse was shadowy and full of dark corners and aisles from which someone could emerge and surprise him. He hadn't discounted the possibility that he might run into a mechanic down here, or maybe even a guard, although they'd need lungs like engines to breathe these fumes for long.
The generator banged noisily and he shied away, threatening it with the tip of his cutlass. When nothing calamitous happened, he relaxed again, feeling a little stupid.
_Just get out of here_ , he told himself. Abandoning caution, he hurried through the room with his arm over his face, breathing through the sleeve of his coat.
If there was anyone else down there, he neither saw nor heard them. A few stone steps led up to a heavy door, which was unlocked. He peered in and found himself in an untidy antechamber full of tools. Dirty gloves and rubber masks with gas filters hung on pegs. Frey shut the door behind him, muffling the sound of the generator. There was another door leading to a room beyond, and now he could hear loud snoring from the other side.
Snoring was good. Unless it was a particularly cunning decoy—Frey briefly imagined a sharp-eyed assassin waiting behind the door, dagger raised, snoring loudly—it suggested the enemy was unaware, unarmed, and at a massive disadvantage, which was the only way Frey would fight anyone, if he could help it.
He lifted the door on its hinges to minimize the squeak, pushed it open, and immediately recoiled. The room beyond reeked overwhelmingly of cheesy feet and stale flatulence, strong enough that Frey had to fight down the urge to gag. He glanced briefly at one of the gas masks hanging on the wall, then took a deep breath and slipped inside.
The place was a wreck. Every surface was covered in discarded plates of food, half-drunk bottles of milk that had long gone bad, and pornographic ferrotypes from certain seedy publications (Frey saw several women he recognized). In the corner, on a pallet bed surrounded by discarded chicken bones and bottles of grog, lay a mound of hairy white flesh entangled in a filthy blanket. It took Frey a few moments to work out where the head was. He found it only when a gaping wet hole appeared in the crumb-strewn black thatch of a face, and there emerged a terrible snore like the death rattle of a congested warthog.
Frey kept his sword pointed at the quivering mass of the caretaker's naked belly and edged through the room toward the door at the far end. Finding it locked, he cast around the room and located a key under a scattering of toenail clippings. He extracted it gingerly, slipped it in the lock, and went through. The caretaker, deep in his drunken slumber, never stirred.
IT TOOK HIM SOME time to find his way to the dormitories. A quick search established that the basement level of the building was a maze of gloomy corridors and pipes, sealed off from the hermitage proper, presumably to stop the caretaker from getting in and giving the acolytes a nasty shock. There must have been another entrance for the caretaker, since the storm doors had been locked on the outside, but Frey never found it. What he did find was a chimney flue, which he climbed with considerable difficulty and much discomfort.
When he emerged, sooty and disheveled, from the fireplace, he found himself in a small hall. Doors led off to other rooms, and a wide staircase went up to the floor above. The place had a clean, quiet, country feel, the cool, pensive atmosphere of an old house at night. Bulbs shone from simple iron sconces. Decoration was understated and minimal. There were no idols of worship or shrines, such as the old gods might have demanded. The only evidence of this building's purpose was a shadowy gold-framed portrait of King Andreal of Glane, father of the Awakeners and the last ever king. He'd been painted in his most regal pose. It betrayed none of the madness that later took him and set him to burbling prophecies, which ended up having far more influence over the country than he ever did while he ruled it.
There was little here to distract the mind from its devotions. Instead, there were only paneled doors, strong beams, smooth banisters, and the frowning sensation of trespass that settled heavier on Frey with every passing moment.
_There are no guards. Only women inside_ , he reminded himself. _Since when have you been scared of women?_
Then he remembered Trinica Dracken, and he felt a little nauseous. Of all the people in the world he never wanted to see again, she was top of the list.
_Forget her for now_ , he thought. _You've a job to do_.
He dusted himself down as best he could, though he was still covered in sooty smears when he finished. Having made himself as presentable as possible, he looked through the nearest doorway. A short corridor led to an empty wooden room, with only a small brazier in the center. Mats were laid out in a circle around it. A skylight let in the glow of the moon.
A meditation chamber, Frey guessed, backtracking. The Awakeners were very keen on meditation, Crake had told him. Sitting around doing nothing took many years of practice, he'd added with a sneer.
Other doorways let out onto other corridors, which took him to a small study, a filing room full of cabinets and paper, and a classroom with desks in rows of three. Any windows he saw were set high up on the wall, too high to look through without using a stepladder. Obviously, interest in the outside world was discouraged.
He soon came upon a room with a stone table, red-stained blood gutters running down it. Frey's alarming visions of human sacrifice faded when he remembered that many Awakeners used the reading of entrails to understand the Allsoul. As he was wondering how it all might work, he heard the distant whisper of footsteps and female voices in conversation. Someone was up, even at this hour. It was difficult to tell if they were heading his way or not, but he returned to the hall to be safe and then went up the stairs.
The problem of actually _finding_ Amalicia once he was inside the hermitage hadn't greatly troubled Frey during the planning of his daring infiltration. He'd been sidetracked by delicious visions of what an army of cloistered girls might do when a man turned up in their midst. In the face of that, the details seemed rather unimportant. But now he realized that he hadn't the faintest idea where his target was, and his only option was to keep nosing around until something presented itself.
There was another small concern that had been nagging him. It had been two years, more or less, since Amalicia's father sent her to the hermitage. Granted, the point of a hermitage was to keep acolytes in isolation for twice that, but still, two years was a long time. He wasn't even certain she was here at all. Maybe her father had forgiven her and let her out?
No. He didn't think so. He knew Gallian Thade's reputation, and forgiveness wasn't something he approved of.
Besides, Amalicia herself had said as much, in the last letter she'd sent him.
_Moilday Firstweek, Thresh, 145/32_
_Dearest one_ ,
_Through the investigations of those still loyal to me and sympathetic to our cause, I have discovered the location of the hermitage to which my father intends to condemn me. He is sending me to the Highlands. I enclose the coordinates, which I am sure your navigator can decode, as they are mysterious to me_.
_Please forgive the cruel and shameful words I wrote_ _in my last letter. I see now that you were wise to flee when you could, for my father's mood has not improved. He still swears terrible vengeance and likely will desire your death until the day his own comes. My heart should break if harm were to come to you. My anger was not toward you but toward the injustice that made me my father's daughter and you a man born without noble blood. But our love makes mockery of such things, and I know it will make you brave_.
_Find me, Darian, and rescue me. You have your craft, and we have the world before us. You will be a great man of the skies, and I shall be at your side, the way we always dreamed_.
_This letter will depart by my most trusted handmaiden, and I hope it will reach you and find you well. There will be no further opportunity to communicate_.
_With love everlasting,
Amalicia_
_Well, I got here eventually_ , Frey thought.
At the top of the stairs was another corridor and more doors on either side. Each one was a private study cell, with a small lectern on the floor, a mat for kneeling, and a window slit, high up. There were more classrooms and a door to a library, which was locked. He was about to try the next door when suddenly a voice came to him, startlingly close.
"It's Euphelia, that's who it is. She's the one bringing the others down."
He bolted into a classroom and crouched inside the doorway just as two women came gliding around the corner on slippered feet.
"She's taking her studies very seriously," argued the other. "She's terribly earnest."
"She's just not very bright, then," replied the first. "Her understanding of the Cryptonomicon is woeful."
Two figures swept past in the corridor. Frey caught a glimpse of them. They were middle-aged, with graying hair cut in masculine, efficient styles, and they wore the white cassocks of Speakers.
"She has a talent for casting the bones, though," the second woman persisted.
"That she does, that she does. The signals are very clear. But I wonder if she'll ever learn to interpret them."
"Perhaps if we focused her more toward cleromancy and lightened her other studies?"
"Make her a special case? Goodness, no. If we start with her, we have to do it with everyone, and then where will we be?"
The voices faded as they turned the corner, and Frey relaxed. It seemed the hermitage was still patrolled, even in the dead of night. Out to catch acolytes sneaking into the pantry or some such thing. Well, he'd have to be careful. He didn't think his conscience could handle punching out a woman.
He found the girls' dormitory shortly afterward and slid inside.
For a time he stood just inside the door, in the dark. Moonlight fell from a pair of skylights onto two rows of bunk beds. Perhaps fifty girls were sleeping here, their huddled outlines limned in cold light. The room was soft with sighing breath, broken by the occasional delicate snore. There was a scent in the air, not perfume but something indefinable and female, present in a dangerous concentration. Frey began to feel strangely frisky.
He was something of an expert in the art of creeping through women's rooms without disturbing them. By waiting, he was being careful. The slight disturbance caused by his entry may have brought some of the girls close to the surface of sleep, and any small noise might wake them. He was giving them time to slip back into the depths before proceeding.
That, and he wanted to exult in the moment. It really was quite special, being here.
He moved silently between the beds, looking at the moonlit faces of each girl in turn. Disappointingly, they were not quite as luscious in person as he'd imagined they might be. Some were just too young—he had standards—and others were too plain or too fat or had eyes too close together. Their hair was cut in boring styles, and none was in any way prettified. One or two slept beneath their pillows or obscured their faces with their arms, but they didn't have Amalicia's black hair, and their hands—always a giveaway—were too old.
He'd almost reached the end of the room when he saw her. She was sleeping on one of the bottom bunks, her head pillowed by her folded hands, mouth slightly open, face relaxed. Even without the elegant hairstyles and the expertly applied makeup he remembered her wearing, she was beautiful. Her long black hair had fallen across her face in strands; the curve of the lips, the tilt of the nose, the line of her jaw were just as they were in his memory. Frey felt a throb of regret at the sight of her and smothered it quickly.
He knelt down, reached out, and touched her shoulder. When she didn't respond, he shook her gently. She stirred and her eyes opened a little. They widened as she saw him; she took a breath to say his name. He quickly put his finger to his lips.
For a few moments, they looked at each other. Her gaze flickered over his face, absorbing every detail. Then she pushed her blanket aside and slid out of bed. She was wearing a plain cotton nightdress that clung to her hips and the slope of her breasts. Frey felt a sudden urge to take her in his arms as he'd often done before, but before he could act on it, she grabbed his hand and led him toward a door at the far end of the dormitory.
Outside was another corridor, as dark and spartan as the rest. She checked that the coast was clear and then pulled him down it. She took him through a door that led to a narrow set of stairs. At the top was an attic room, with a large skylight looking up at the full moon. It had a small writing desk in a corner, with several books piled atop it. A private study chamber, perhaps. Frey closed the door behind them.
"Amalicia..." he began, but then she roundhouse-kicked him in the face.
# _Chapter Fifteen_
AMALICIA'S REVENGE—FREY'S TALENT FOR LYING—PLANS ARE MADE—INVITATIONS, LEWD AND OTHERWISE
t wasn't so much the force of the kick but the surprise that sent Frey stumbling back. He tripped and fell to the ground, holding his face, shock in his eyes.
"What'd you do _that_ f—"
"Two years!" she hissed, and her bare foot flashed out again and cracked him around the side of the head, knocking him dizzy. "Two years I've waited for you to come!"
"Wait, I—" he began, but she booted him in the solar plexus, and the breath was driven out of him.
"Did you know they teach us the fighting arts in this place? It's all about being in harmony with one's body, you see. Only when we're in harmony with ourselves can we find harmony with the Allsoul. Utter rubbish, of course, but it does have its benefits." She punctuated the last word with another vicious kick in the ribs.
Frey gaped like a fish, trying to suck air into his lungs. Amalicia squatted down in front of him, pitiless.
"What happened to your promises, Darian? What happened to 'Nothing can separate us'? What happened to 'I'll never leave you'? What happened to 'You're the only one'?"
Frey had a vague recollection of saying those things, and others like them. Women did tend to take what he said literally. They never seemed to understand that because they expected—no, _demanded_ —romantic promises and expressions of affection, they forced a man to lie to them. The alternative was frosty silences, arguments, and, in the worst case, the woman would leave to find a man who _would_ lie to her. So if he'd said some things he hadn't exactly meant, it was hardly his fault. She had only herself to blame.
"Your father..." he wheezed. "Your father... would've had... me killed."
"Well, we'll never know that for sure, will we? You turned tail and ran the moment you realized he'd found out about us!"
"Tactical withdrawal," Frey gasped, raising himself up on one hand. "I told you... I'd be back."
She stood up and drove her heel hard into his thigh. His leg went dead.
"Will you stop bloody hitting me?" he cried.
"Two years!" Her voice had become a strangled squeak of rage.
"It took me two years to find you!"
"Oh, what rot!"
"It's the truth! You think your father advertised your whereabouts? You think it was easy finding you? He sent me away so you'd be hidden from me. I've spent two years trying to get my hands on Awakener records, mixing with the wrong kind of people, trying to stay one step ahead of your father and the... the _assassins_ he set on my trail. You know he's hired the Shacklemores? The Shacklemores have been after me ever since the day I left, and every day I've been trying to make my way back to you."
It was an outrageous lie, but Frey had a talent for lying. When he lied, even _he_ believed it. Just for that moment, just for the duration of his protest, he was convinced that he really _had_ done right by her. The details were unimportant.
Besides, he knew for sure that Gallian Thade really did still want him dead. Thade had framed him. In such a light, it was rather heroic that he'd come back at all.
But Amalicia wasn't so easily swayed. "Spit and blood, Darian, don't give me that! I sent you a letter telling you where I was! I sat here in this horrible place waiting for—"
"I never got any letter!"
"Yes, you did! The letter I sent you with the coordinates of this place."
"I never got any coordinates! In your last letter you called me a coward and a liar, among other things. In fact, the last letter I got from you left me in very little doubt that you never wanted to see me again."
Amalicia's hand went to her mouth. Suddenly all the anger had gone out of her and she looked horrified.
"You didn't get it? The letter I sent after that one?"
Frey looked blank.
Amalicia turned away, an anxious hand flying to her forehead, pacing around the room. "Oh, by the Allsoul! That silly cow of a handmaiden. She must have written the wrong address, or not paid the right postage, or—"
"Maybe it got lost in the post?" Frey suggested generously. "Or someone at one of my pickup points mislaid it. I had to stay on the move, you know."
"You really didn't receive my letter?" Amalicia asked. Her voice had taken on a note of sympathy, and Frey knew he'd won. "The one where I took back all those foul things I said?"
Frey struggled to his feet with difficulty. His jaw was swelling, and he could barely stand on his dead leg. Amalicia rushed over to help him.
"I really didn't," he said.
"And you still came? You still searched for me all these years, even when you thought I hated you?"
"Well," he said, then paused for a moment to roll his jaw before he delivered his final blow. "I made a promise."
Her eyes shimmered with tears in the moonlight. Wide, dark, trusting eyes. He'd always liked those eyes. They'd always seemed so innocent.
She flung herself at him and hugged him close. He winced as his injuries twinged, then slid his arms around her slender back and buried his face in her hair. She smelled clean. Cleaner than he'd smelled for a long time, that was for certain. He found himself wondering how things might have been with her, if not for her father, if not for the unfortunate circumstances that drove them apart.
No. No regrets. If he opened that door he'd never be able to close it.
She pulled herself away a little, so she could look up at his face. She was desperately sorry now, ashamed for having tragically misjudged him. Grateful that he'd come for her in spite of everything.
"You're the only man I've ever been with, Darian," she breathed. "I haven't seen another since my father sent me to this awful place."
Darian leaned closer, sensing the moment was right, but she drew back with a sharp intake of breath. "Have you?" she asked. "Have you been with anyone?"
He looked at her steadily, letting her feel how earnest he was. "No," he lied, firmly and with authority.
Amalicia sighed and then kissed him hard, clutching at him with unpracticed, youthful fury. She tore at his clothes, frantic. He struggled free of his sooty greatcoat as she fumbled at the laces of his shirt before finally tugging it off and throwing it away. He pulled her nightshirt up and over her head, then swept her up and kissed her, gratified to realize that at least part of his fantasy about sex-starved young women in a hermitage was about to come true.
AFTERWARD, THEY LAY TOGETHER naked on Frey's coat, his skin prickling deliciously in the chilly night. He ran a finger along the line of her body while she stared at him adoringly. There was a dazed look in her eye, as if she was unable to quite believe that he was here with her again.
"I saw some Imperators on the way here," he said.
She gasped. "You didn't!"
"Right outside. A bunch of Sentinels carried a chest out to them, and they put it on their craft and took off. One of them looked right at me."
"How frightening."
"They were guarding that chest very closely."
"Are you asking me if I have any idea what might have been inside?"
"In a roundabout way, yes."
"I don't know, Darian. Some stuffy old scrolls, no doubt. Perhaps it was an original copy of the Cryptonomicon. They're terribly careful with those things."
"Remind me what that is again?"
"The book of teachings. They wrote down all the insane little mutterings of King Andreal the Demented and put them in that book."
"Oh," said Frey, losing interest immediately.
"We have to leave together," she said. "Tonight."
"We can't."
"It's the only way, Darian! The only way we can be together!"
"I want that, more than anything in the world. But there's something I haven't told you. Your father—"
"What did he do?" she snapped, jumping immediately to Frey's defense.
"You might not want to hear this."
"Tell me!"
"Your father... well, he's... Something terrible happened. An aircraft blew up, and people died. Nobody knows who did it, but your father has pinned it on me. Me and my crew. If you were caught with me, they'd hang you. It's too dangerous. You're safer here."
Amalicia looked at him suspiciously.
"I'm a lot of things, but I'm no cold-blooded killer!" he protested. "The Archduke's son was on that craft, Amalicia. Your father arranged it, but half of Vardia is after _me._ "
"Hengar is dead?" she gasped.
"Yes! And your father is in on it."
Amalicia shook her head angrily, eyes narrowing. "That bastard. I hate that bastard!"
"You believe me, then?"
"Of course I believe you! Spit and blood, I know what he's capable of. Look at me! His only daughter, condemned to this place because I went against his wishes just once! He doesn't have a heart. Money is all he cares about—money and that rotten Allsoul." She glanced around guiltily, as if afraid she'd gone too far. Then, emboldened by Frey's presence, she went on. "It's all stupid! I don't believe any of it! They say it's all about faith, but it's not, because I can do it and I don't even care about the Allsoul! It's brought me nothing but misery. Any idiot can study the texts and learn to read the signs. Anyone with half an education can tell the Mistresses what they want to hear. But there's nothing there, Darian! I don't feel anything! I'm stuck here in this prison, and after two more years they'll put that awful tattoo on my forehead, and after that I'll be an Awakener forever!" She cupped his bruised jaw with her hands and gazed desperately into his eyes. "I can't let that happen. I'll die first. You have to get me out of here."
"I will," he said. "I will. But first I have to get to your father."
"Oh, Darian, no! He'll have you hanged for sure!"
"Gallian Thade is the only lead I've got. If I can find out why he killed Hengar... well, may be I can do something about it." Then, seeing Amalicia's expectant expression, he added, "And then I'll come back for you, and we'll escape together as we planned."
"But if you pin it on my father..." Amalicia said, with dawning realization. "Why, _he'll_ be the one who hangs."
Frey stumbled mentally. He'd forgotten about that. In clearing his name, Gallian would have to hang. He was asking a daughter to help send her own father to the gallows.
A cruel smile spread across Amalicia's face, the terrifying smile of a child about to stamp on an insect. Malice for the sake of malice. She saw her revenge, and it pleased her. Frey was surprised; he hadn't imagined her capable of such thoughts. Her time in the hermitage had made her bitter, it seemed.
"If he hangs," she said slowly, "that makes _me_ head of the family. And no one can keep me here when I'm mistress of the Thades."
"I hadn't even considered that," Frey said truthfully. "I was so wrapped up in the idea of rescuing you... well, it had never occurred to me that if your father died—"
"Oh, Darian, it's brilliant!" she said, eyes shining. She threw one leg over his thigh and pressed herself to him eagerly. Frey's mind began to wander from his machinations and back to baser thoughts. "Kill him! Let the bastard hang! And then I'll be free, and we can be together, and we won't have to run from anyone! We'll marry, and damn what anyone says!"
Frey's ardor dampened at the mention of marriage. _But why?_ he asked himself. _Why not this one? She's richer than shit and foxy to boot! Not to mention she's almost a decade younger than you and she thinks the sun rises and sets in your trousers. Since you can't make fifty thousand ducats any other way, why not marry them?_
But however good the reasons, Frey couldn't deny the life-sucking sense of oblivion that overtook him whenever he heard the M-word.
"I daren't even hope for that yet," he said. "Things are so dangerous right now... simply to survive would be... maybe, just maybe, I can win out of this. And then you'll be free, and we can be together."
_Can_ , he mentally added. _Not will_.
"What can I do?" she asked, missing the fact that Frey had deftly evaded any promise of marriage. She'd heard what she wanted to hear. Frey noted that the women in his life had a tendency to do that.
"Can you think of any reason why your father would want Hengar dead? How would it profit him?"
She lay on her back and looked up at the ceiling. Frey admired her, half listening as she spoke. "Well, he's very close to the Awakeners, you know that. But the Awakeners don't have anything against Hengar. It's the Archduchess they hate, and the Archduke by association."
"Why?"
"Because Eloithe is a big critic of the Awakeners. She doesn't believe in the Allsoul. She says they're just a business empire that trades in superstition. And she's obviously inspired the Archduke, since he's started making all kinds of moves to diminish their power. But none of that's anything to do with Hengar." She thought for a moment, then said, "You know what I think? I don't think my father's behind this at all."
"Amalicia, there's no doubt. I spoke to a—"
"No, no, I mean... We're landowners, Darian. We make our money from tenants. There's no reason to murder the son of the Archduke." She sat up suddenly, her face taut with certainty. "I know him, Darian, he wouldn't come up with something like this. Someone else is behind it."
"You think there's someone else?"
"I'd bet on it."
"Well... who?"
"That I don't know. I've been away a long time, in case you'd forgotten. It's hard to keep up with my father's business dealings when I've been locked in this prison for two years."
Her tone grew harsher as she spoke, and Frey—fearing another beating—placated her hurriedly. "It's okay, it's okay. I'll look into it. I just have to find a way to get close to him."
"Well, there's the Winter Ball coming up," she suggested.
"The Winter Ball?"
"You know! The ball! The one my father has every year at our estate on the Feldspar Islands."
"Oh, the ball!" Frey said, though he'd no idea what she was talking about. Presumably they'd discussed it, although he was reasonably sure he'd never been to one.
"My father always does business there. All the important people come to it. If someone put him up to this whole business of murder, I'm sure you'd find them there. And you'd be well hidden among all the people. It's quite the event of the season, you know!"
"Can you get me in?"
She jumped up and went to the writing desk, drew out a pen and paper, and began to scribble. Frey lay on his side, idly studying the curve of her back, the bumps of her spine.
"There are still people in the family who don't agree with what father did. This is a letter of introduction. You can take it to my second cousin—he'll do the rest."
"I need two invitations."
Her shoulders tensed and she stopped writing.
"Neither is for me," he assured her. "I won't be going. Don't fancy meeting your father again. And you know I'm not very well trained in etiquette. But I do have a friend who is. I'll need his help."
"And the other?"
"Well, you have to take a lady to these things, don't you? Turning up without a date looks a bit odd."
"And I suppose you happen to know one?"
"She's my navigator, Amalicia," said Frey. He leaned over and kissed her between the shoulder blades. "Just my navigator. And it won't be me who's taking her."
"Alright," she said. "Two invitations." She resumed writing, then signed with a flourish and laid the letter on top of his piled-up clothes.
Frey began getting to his feet. "Thank you," he said. "I'll get you out of here. I promise."
"Where do you think you're going?"
Frey looked toward the door of the attic. "Well, I'm technically not supposed to be here, so I should really be gone before everyone wakes up."
Amalicia pulled him back down again. "It's not even close to dawn," she said. "I've had nobody to lie with for two years, Darian. We still have some catching up to do."
# _Chapter Sixteen_
A TRIUMPHANT RETURN—FREY TAKES ON NEW CREW—SILO'S WARNING
t was midday by the time Frey made it back to the grassy valley where the _Ketty Jay_ waited. There was a cold breeze, but the sun warmed the skin pleasantly, and most of the crew was outside. Harkins was tinkering with the Firecrow; Jez was reading a book she'd picked up in Aulenfay; Malvery was lying on his back, basking. Silo was nowhere to be seen. Frey presumed he was inside, engaged in one of his endless attempts to modify and improve the _Ketty Jay_ 's engine.
Frey strolled into their midst, whistling merrily. Pinn—who was lying propped up against the wheel strut of his Skylance—lifted the wet towel off his forehead and gave an agonized groan. He was still wearing his Awakener garb, although the Cipher he'd painted on his head was now just a red smear.
"I see you managed to keep yourself entertained while I was gone," Frey said. "Heavy night?"
Pinn groaned again and put the towel back on his forehead.
"Mission accomplished, Cap'n?" Jez called, looking up from her book. "What happened to your face?"
Frey touched fingertips to his bruised jaw, probing the skin delicately. "Little misunderstanding, that's all," he said. Jez ran her eyes over his shabby, soot-covered clothes and let the issue drop.
Bess was sitting on the grass, her short, stumpy legs sticking out in front of her, like some vast and grotesque mechanical infant. Crake was cleaning her with a bucket and a rag. She was making a soft, eerie cooing noise, like wind through distant trees. Crake said it meant she was happy, rather like the purring of a cat, but it unsettled Frey to hear the voice of the daemon that inhabited that massive armored shell.
"You look chipper today," Crake observed.
Malvery sat up, took off his round green-lensed glasses, and peered at Frey. "Yes, he has a definite glow about him, despite the battle damage. I'd say he had a very happy reunion with someone. That's my professional opinion."
"A gentleman never tells," said Frey, with a broad grin that was as good as a confession.
"I'm very pleased for you," said Crake disapprovingly.
"How did your new toys work out?" Frey inquired.
Crake brightened. "I think I can do quite a lot with them. A daemonist needs a sanctum, really, but some processes are more portable than others. I won't be fooling around with anything too dangerous, that's for sure, but I can still do beginner's stuff."
"What's beginner's stuff? Stuff like my cutlass?"
Crake choked in amazement and almost flung down his rag. "Your cutlass," he said indignantly, "is a work of bloody art that took me years of study to accomplish and almost—"
He stopped as he caught the look of wicked amusement on Frey's face. "Oh," he said. "I see. You caught me. Very droll."
Frey walked over and slapped him on the shoulder. "No, seriously, I'm interested. What can you do?"
"Well, for example..." He drew out two small silver earcuffs from his pocket. "Take one of these and put it on your ear."
Frey fixed it to his ear. Crake did the same with the other. They looked like any other innocuous ornament. Bess stirred restlessly, her huge bulk rustling and clanking as she moved. Crake patted her humped back.
"Don't worry, Bess. We're not finished yet. I'll clean the rest in a moment," he assured her. The golem, mollified, settled down to wait.
"Now what?" asked Frey.
"Go over there," said Crake, pointing. "And ask me a question. Just talk normally; don't raise your voice."
Frey shrugged and did as he was told. He walked fifty yards and then stopped. Facing away from Crake, he said quietly, "So what exactly are you doing on the _Ketty Jay_?"
"I gave you my cutlass on the condition that you'd never ask me that," Crake replied, close enough to his ear that Frey jumped and looked around. It was as if the daemonist was standing right next to him.
"That's incredible!" Frey exclaimed. "Is that really you? I can hear your voice right in my ear!"
"The range could be better," said Crake modestly. "But it's quite a simple trick to thrall two daemons at the same resonance. They're the most rudimentary type—stupid things, really. Little sparks of awareness, not even as smart as an animal. But they can be very useful if put to a task."
"I'll say!"
"I was thinking, if I can whip up some better versions, that you could use them to communicate with your pilots or something. Better than that electroheliograph thing you have."
"That's a damn good idea, Crake," he said. "Damn good idea."
"Anyway, better take it off. These things will tire you out if you wear them too long. Daemons have a way of sucking the energy out of you."
"My cutlass doesn't," Frey replied.
He heard the slight hesitation. _My_ cutlass, Crake was undoubtedly thinking.
"One of many reasons it's such a work of art," he said.
Frey unclipped the earcuff and returned it to Crake, who had resumed scrubbing the golem. "I'm impressed," he said, handing it back. "You want to go to a party?"
"Excuse me?"
"A ball, actually. Formal ball, held by Gallian Thade."
"The Winter Ball at Scorchwood Heights?"
"Ummm... yes?" Frey replied uncertainly.
"You have invitations?"
Frey brandished the letter from Amalicia. "I will have soon. I was thinking you might go and take Jez with you."
Crake looked at him, searching for a sign of mockery.
"I mean it," he said. "I could really use your help, Crake. Thade will be there, and if he's working with someone else, it's our best chance of finding out what he's up to."
Crake was still watching him narrowly, indecision in his eyes.
"Look," said Frey. "I know I have no right to ask. You're a passenger. That's what you signed on for. You don't owe me anything." He shrugged. "But, I mean, you and Bess..."
Bess shifted at the sound of her name, a quizzical coo coming from deep within her. Crake patted her back.
Frey coughed into his fist, looked away into the distance, and scratched his thigh. He was never very good with honesty. "You and Bess, the both of you saved our lives back in Tarlock Cove. I've kind of got to thinking that, well..." He shrugged again. Crake just kept on looking at him. The daemonist wasn't making it easy. "What I'm saying—badly—is that I've started to think of you more as part of the crew, instead of just dead weight. I'm saying, well... Look, I don't know what business you're really on, or why you took up with me in the first place, but it's getting to be pretty bloody handy having the two of you around. Especially if you're gonna start making more little trinkets like those ear things."
"That's very kind of you, Frey," said Crake. "Are you offering me a job?"
Frey hadn't really thought about that. He only knew that he needed Crake to help him out. "Would you take one if I offered it?" he heard himself saying. "Part of the crew? Just 'til... well, until we get this whole mess sorted out. Then you could decide."
"Do I get my cutlass back?"
"No!" Frey said quickly. "But I'll cut you in on a share of what we make."
"We don't seem to _make_ a great deal of anything."
Frey made a face, conceding the point.
"What would I have to do in return?" Crake asked. He returned to scrubbing Bess's massive back. A deep, echoing groan of pleasure came from the golem's depths.
"Just... well, stick around. Help us out."
"I thought I was doing that already."
"You are! I mean..." Frey was getting frustrated. He was a supremely eloquent liar, but he struggled when he had to talk about things that he actually felt. It made him vulnerable, and that made him angry at himself. "I mean, you and Bess could just up and walk, right? It's like you said back in Yortland: they'd never come looking for you. It's me they're after. And I'm sure you've other business you want to be getting on with, something to do with all that daemonism stuff you picked up."
"So what you're saying is that you'd like us to stay around?" Crake prompted.
"Yes."
"And that you... well, that you _need_ us."
Frey didn't like the triumphant tone creeping into Crake's voice. "Yes," he said warily.
"And what are you going to do next time someone puts a gun to my head and spins the barrel?"
Frey gritted his teeth. "Give them the ignition code to the _Ketty Jay,_ " he said, glaring malevolently at the grass between his feet. "Probably."
Crake grinned and gave Bess a quick buff on the hump. "You hear that, Bess? We're pirates now!" Bess sang happily, a ghostly, off-key nursery rhyme.
"So you'll go to the ball?" Frey asked.
"Alright," he said. "Yes, I'll go."
Frey felt a flood of relief. He hadn't realized how much he'd been counting on Crake's cooperation until this moment.
"I have a question, though," Crake said. There was something suspiciously casual in his voice, and Frey knew what he was about to ask. The sense of relief drained away, replaced by an unexpectedly awful sense of shame.
"Yes?" he asked, in the forlorn hope that Crake was thinking of something else.
He wasn't. "The bullet," said Crake. "When you let Macarde pull the trigger. Twice. You didn't know which chamber the bullet was in, did you?"
Frey's instincts told him to lie. This time, he ignored them. "No," he said. Then, lamely: "Sorry about that."
Crake nodded to himself. "Thought so."
Frey opened his mouth to say something more, but he was interrupted by a cry from farther up the valley.
"Cap'n!"
It was Silo. The tall Murthian wasn't in the engine room after all but was running down the valley toward them with a haste that could only spell trouble. He was carrying a spyglass in his hand.
"Cap'n! Aircraft!" Silo cried, pointing. The others—with the exception of Pinn—scrambled to their feet and ran to look.
"I see it," said Jez.
"Damn, you've got good eyes!" said Malvery. "I don't see a thing!"
"Nor me!" added Crake.
Jez looked around guiltily. "I mean, I can't make it out or anything, not really. Just saw a flash of light, that's all."
Silo reached them and passed the spyglass to Frey. Frey put it to his eye.
"She coming... from the south..." Silo panted. "Think she... heading for the... hermitage...."
"Then she'll pass over us?"
"Yuh-huh. See us for sure."
Frey cast about with the spyglass, struggling to locate the incoming threat. It swung into view and steadied. Frey's mouth went dry.
She was a big craft. Long and wide across the deck, black and scarred, yet for all her ugliness she was sleek. A frigate, built more like an ocean vessel than an aircraft: a terrible armored hulk bristling with weaponry. Her wings were little more than four stumpy protuberances: she was too massive to maneuver quickly. But what she lacked in speed, she more than made up for in firepower. This was a combat craft, a machine made for war with a crew of dozens.
Frey took the spyglass away from his eye.
"It's the _Delirium Trigger,_ " he said.
# _Chapter Seventeen_
DRACKEN CATCHES UP—EQUALIZERS—JEZ MAKES A PLAN—PINN'S DEFENSE—LIGHTNING
he reaction among the crew was immediate. Frey had never seen them scramble into action so fast. He vainly wished he had half the authority that the _Delirium Trigger_ apparently did.
"Everyone! Get to stations! We're airborne!" Frey yelled, even though Silo, Jez, and Malvery were already bolting up the cargo ramp. Harkins had scampered into the cockpit of the Firecrow like a frightened spider, and Pinn was grumbling nauseously to himself as he set about getting himself into the Skylance.
"Crake! Get Bess inside and shut the ramp!" he ordered, as he raced aboard the _Ketty Jay_. He made his way to the cockpit with a speed born of panic, flying up the steps from the cargo hold two at a time. He squeezed past Malvery, who was climbing into the autocannon cupola on the _Ketty Jay_ 's back, and found Jez already at her post. He threw himself into his chair, punched in the ignition code, and opened up everything he could for an emergency lift.
_How did she find me?_
Harkins was in the air by the time the _Ketty Jay_ began to rise, and Pinn took off a few moments later, still clad in his half-buttoned Awakener cassock and with a red smear across his forehead. There was a look of frantic bewilderment on his face, like someone rudely awakened from sleep to find their bed is on fire.
The _Ketty Jay_ was facing the _Delirium Trigger_ as she rose. The frigate was coming in fast. Now it was easily visible to the naked eye and growing larger by the second. She couldn't fail to have spotted the craft lifting into the sky, directly in her path. The question was, would she recognize the _Ketty Jay_ at this distance?
As if in answer, four black dots detached from her and began to race ahead. Outfliers. Fighter craft.
"She's onto us!" Frey cried. He swung the craft around one hundred eighty degrees and hit the thrusters. The _Ketty Jay_ bellowed as she accelerated to the limit of her abilities.
"Orders, Cap'n?" Jez asked.
"Get us out of here!"
"Can we outrun her?"
"The _Trigger_ , yes. The outfliers are Norbury Equalizers. We can't outrun them."
"Okay, I'm on it," said Jez, digging through her charts with a loud rustling of paper.
"Heads up, everyone!" Malvery called from the cupola. "Incoming!"
Frey wrenched the control stick, and the _Ketty Jay_ banked hard. A rapid salvo of distant booms rolled through the air, followed a moment later by a sound like the end of the world. The sky exploded all around them, a deafening, pounding chaos of shock and flame. The _Ketty Jay_ was shaken and thrown, flung about like a toy. Pipes shrieked and burst in the depths of the craft, spewing steam. Cracks split the glass of the dashboard dials. A low howl of metal sounded from somewhere in the guts of the craft.
And then suddenly the chaos was over, and somehow they were still flying, the majestic green canvas of the Highlands blurring beneath them.
"Ow," said Frey weakly.
"You alright, Cap'n?" asked Jez, brushing her hair out of her eyes and gathering up her scattered charts.
"Bit my damn tongue," Frey replied. His ears were whistling and everything sounded dim.
"They're firing again!" cried Malvery, who had a view of the _Delirium Trigger_ from the blister on the _Ketty Jay_ 's back.
"What kind of range do those guns _have_?" Frey murmured in dismay, and sent the _Ketty Jay_ into a hard dive. But there was no cataclysm this time. The explosions fell some way behind them, and the concussion was barely more than a sullen shove.
"Not enough, apparently," said Jez.
"Malvery! Where are those fighters?" called Frey through the door of the cockpit.
"Catching up to us!" the doctor replied.
"Don't fire 'til they're close enough to hit! We've not got much ammo for that cannon!"
"Right-o!"
He turned to his navigator. "I need a plan, Jez."
She was plotting frantically with a pair of compasses. "This craft has Blackmore P-12s, right?"
"Uh?"
"The thrusters. P-12s."
"Yeah."
"Okay." She looked up from her chart. "I have an idea."
PINN'S MOUTH TASTED LIKE decomposing mushrooms, and his peripheral vision was a swarming haze. He felt like there wasn't a drop of moisture in his body, but his bladder throbbed insistently. He was utterly detached from the world. Reality was somewhere else. He was cocooned in his own private suffering.
And yet some faint part of him was alarmed to find that he was in the cockpit of his Skylance, racing over the Highlands, pursued by four fighter craft intent on shooting him down. That part was urging him to sharpen up pretty quickly and pay attention. Eventually he began to listen to it.
With some difficulty, he craned around and looked over his shoulder. The enemy craft were close enough to make out now. He recognized the distinctive shape of Norbury Equalizers: their bulbous, rounded cockpits right up front; their straight, thick wings, cut off at the ends; their narrow, slightly arched bodies. Equalizers were a pain in the arse. Speedy and highly maneuverable. They were like flies: annoyingly hard to swat. And when you got frustrated, you made mistakes, and that was when they took you out.
He could outrun them, for sure. He could outrun just about anything in his modified Skylance. But an outflier's job wasn't to save his own neck. He had to protect the _Ketty Jay_. Besides, running was for pussies.
The _Ketty Jay_ was to starboard. He saw her change tack, swinging toward the west, and he banked to match. The horizon became uneven as the western edge of the Eastern Plateau came into view, a hundred kloms ahead of them. Beyond it, invisible, the land fell away in the steep, sheer cliffs and jagged, crushed peaks of the Hookhollows.
Pinn frowned. Where did they think they were going? They might be able to make it to the mountains, where there would be ravines and defiles to use as cover, but there was still no way the _Ketty Jay_ could outmaneuver an Equalizer.
He glanced to port. The _Delirium Trigger_ was safely out of the race, but the Equalizers were banking to intercept the _Ketty_ _Jay_ on her new course, and they were closing the gap even faster than before.
Minutes ticked by. The slow, excruciating minutes of the long-distance chase. Pinn's world shrank back to the pulsing of his hangover, the low roar of the thrusters, the shudder and tremble of the Skylance. But every time he looked around, the Equalizers were closer. One thing was clear: whatever the _Ketty Jay_ was heading for, she wouldn't make it before the Equalizers reached her.
Then, in the far distance, he saw an indistinct fuzz in the air. Gradually the fuzz darkened, until there could be no question as to what it was. Just beyond the lip of the Eastern Plateau was a line of threatening clouds. The clear blue of the sky ended abruptly in a piled black bank of gathering thunderheads.
The _Ketty Jay_ was running for the storm.
"How'd you know _that_ was there, you clever bitch?" he murmured, out of grudging respect for Jez. He naturally assumed it wasn't the captain's doing.
He checked where the Equalizers were. Behind him now but closer still, flying in tight formation. Organized. Disciplined. Soon they'd be within firing range.
He shook his head and spat into the footwell. "I've had enough of this," he snarled. He was bored with the chase and angry at his nagging headache. The fact that the enemy was flying in such neat formation inexplicably annoyed him. If someone didn't do something soon, those Equalizers would start taking shots at them, and Pinn was damned if he was going to present his tail to four sets of machine guns.
"Alright," he said. "Let's play."
He broke away from the _Ketty Jay_ in a high, curving loop. At its apex, he rolled the craft to bring him right-side-up again. The pursuing fighters were below and ahead of him now. They'd seen the threat but were slow to react, unsure if he was fleeing or fighting. Nobody expected a single craft to take on four. It was suicidal.
But death was a concept that Pinn wasn't really smart enough to understand. He didn't have the imagination to envisage eternity. Oblivion was unfathomable. How could he be scared of something when he had only the vaguest notion of it? So he dived down toward the pursuing fighters with a whoop of joy and opened up with his machine guns.
The Equalizers scattered as he plunged among them like a cat among birds. They banked and rolled and dived, darting out of his line of fire as he cut through the formation and out the other side. Lesser craft would have been tagged, but the Equalizers were just quick enough to evade him.
Pinn pulled the Skylance into a climb, rolling and banking as he did, making himself a difficult target. G-forces wrenched at him. His hangover throbbed in protest at the abuse, but the adrenaline was kicking in now, clearing away the cobwebs. He fought to keep track of the Equalizers as they wheeled through the sky. Three of them were reorganizing, continuing their pursuit of the _Ketty Jay_. One had peeled off and was angling for a shot at Pinn.
One? _One?_ Pinn was insulted. Ignoring the fighter that was trying to engage him, he angled himself into a climb, heading for the main formation. They'd streaked ahead, dismissing him. They thought they'd got too much of a head start while he was turning around. They thought he had no chance of catching them.
They were wrong.
Pinn hit the thrusters and left his pursuer aiming at empty sky. The Skylance howled gleefully as it accelerated, eating up the distance between Pinn and his targets. He came in from directly behind, growing in their blind spot. He was forced to fly straight to avoid notice, but he was acutely aware that by doing so he was allowing the fourth Equalizer to line up on his tail. He held steady for a dangerous moment, then loosed off a fusillade at the nearest plane.
Whether it was luck, instinct, or skill, the pilot spotted him an instant before he fired. The Equalizer banked hard, and the bullets chipped across its flank and underwing instead of hitting the tail assembly. Pinn cursed and rolled away just as the Equalizer on his tail sent a volley of tracer fire his way. The Skylance danced between the bullets and dived out of the line of fire.
Pinn jinked left and right, keeping his movements unpredictable. He twisted his neck around, trying to get a fix on his opponents. The most important factor in aerial combat was knowing where your enemies were. He kept up a frantic evasion pattern until he spotted two of the Equalizers dwindling in the distance, continuing their pursuit of the _Ketty Jay_. The plane he'd damaged was still in the air and still a threat, though it was trailing a thin line of smoke that made it easy to find. Burned by his sneak attack, that pilot had decided to deal with Pinn.
He felt better once he'd located the fourth Equalizer. He had two of them on his tail now. They respected him enough that they couldn't turn their backs on him. Now all he had to do was keep them busy awhile.
He launched into a new sequence of evasions, leading them away from the _Ketty Jay_ as he corkscrewed and twisted and rolled. The Equalizers homed in on him from different angles, doing their best to trap him, but he could see their tactics and refused to play along. The one he'd damaged was limping slightly, a little slow and clumsy, and its pilot couldn't lock in with his companion. Their maneuvers were pretty but came to nothing. Sporadic machine-gun fire chattered behind him, but it was more hopeful than effective.
_I should just turn around and take these bastards out_ , thought Pinn. But then he caught sight of the _Delirium Trigger_ , much larger than he remembered when he last looked. Their aerobatics had allowed the bigger craft to catch up to them, and Pinn didn't fancy dealing with her guns on top of everything else.
The _Ketty Jay_ was barely visible in the distance. He'd taken two of the Equalizers out of the chase, and he'd delayed the other two, buying the _Ketty Jay_ time to reach the storm. He'd done his part.
He reached over and grabbed a lever underneath the dash. The Skylance had been built as a racer long before he'd modified it for combat, and it still had a racer's secret weapon installed. He leveled up and aimed for the horizon.
"Bye bye, shit-garglers!" he yelled, then rammed the Skylance to full throttle and engaged the afterburners. The Skylance rocketed forward, slamming him back in his seat with enough force to press his chubby cheeks flat against his face. His pursuers could only watch, hopelessly outpaced, as the Skylance dwindled into the distance, carrying its whooping pilot with it.
"TWO STILL WITH US!" called Malvery from his cupola. "Pinn's drawn off the others."
Frey grinned. "I'd kiss that kid if he wasn't so hideous and stupid." He looked about. "Where's Harkins?"
Jez pointed up through the windglass to the Firecrow hanging high on their starboard side.
"Tell him to engage," he said, then shifted in his seat and hunched forward over the controls. "Keep 'em off my tail."
Jez reached over to the electroheliograph and tapped a rapid code. The lamp on the _Ketty Jay_ 's back flashed the sequence. Harkins gave a wing-waggle and broke away.
The winds were rising as the storm clouds rolled ever closer. Frey's admiration for Jez had grown a great deal in the moment he saw those thunderheads appear on the horizon. She'd been right on the money. Again. It was an unfamiliar feeling, having someone reliable on his crew. He was rather liking it.
"Wind is from the northwest today, and it's sunny," she'd said. "Warm air rising off the mountains up the side of the plateau, cooled by the airstream coming down from the arctic. This time of the day, this kind of weather, you're gonna get a storm there."
The kind of storm a small fighter craft couldn't handle. But a bigger one, driven by the notoriously robust Blackmore P-12 thrusters—that kind of craft could make it through.
Crake stuck his head round the door. "Anything I can do?"
"Where've you been?"
"Bess was upset. All the explosions, you see."
"We'll try to keep it down," Frey replied drily. "Get me a damage report from Silo."
Crake ran off down the corridor to comply. Frey returned his attention to the storm. The _Ketty Jay_ rocked and shivered as the winds began to play around her. Machine-gun fire sounded from behind them.
"There goes Harkins," Frey said. "Malvery! What's going on back there?"
"They dodged round him! Still coming!"
"Well make sure you—" he began, but was drowned out by the heavy thudding of the autocannon as Malvery opened up on their pursuers.
Frey cursed under his breath and swung the _Ketty Jay_ to starboard. He heard the chatter of machine guns, and a spray of tracer fire passed under them and soared away toward the clouds.
"Will you hold still?" Malvery bellowed. "I ain't gonna hit anything if you keep jigging around like that!"
"I'm jigging around so they don't hit us!" Frey shouted back, then banked again, dived, and yawed to port. The _Ketty Jay_ was a sizable target, but she could move faster than her bulk suggested. Her pursuers were still at the limit of their range, but they were catching up fast.
"You know the worst thing about flying an aircraft like this?" he asked Jez. "You can't see behind you. I'm only _guessing_ where those sons of bitches are while they take potshots at my arse. I wish, just once, someone would have the guts to take us on from the front so I could shoot 'em."
"Sounds like it wouldn't be a very wise tactic, Cap'n," she replied. "But we can hope."
The storm was filling the sky now. They were flying in low, and the thunderheads had swallowed the sun. The cockpit darkened, and the air got choppier still. The _Ketty Jay_ began to rattle around, buffeted this way and that.
"Let's see 'em aim straight in _this,_ " he murmured. "Signal Harkins. Tell him to get out of here. He knows the rendezvous."
Jez complied, tapping the electroheliograph.
A few moments later, Malvery yelled, "Hey! Harkins is turning tail! That yellow toad was supposed to be—"
"My orders!" Frey yelled back. "He can't follow us into the storm. It's up to you now."
"You're giving orders now?" Malvery sounded surprised. "Blimey." Then the autocannon began thumping again in clipped bursts.
Crake appeared at the door. "Silo says the engines have taken a hit and they're overheating, but it's nothing too serious. Other than that, there's only minor structural—"
There was a shattering din as a salvo of bullets punched into the _Ketty Jay_ 's hull from behind. She yawed crazily, hit a pressure pocket in the storm, and plunged fifteen meters, fast enough to lift Crake off the ground and slam him to the floor again. The engines groaned and squealed, reached a distressing crescendo, then slowly returned to their usual tone.
Crake pulled himself up from the floor, wiping blood from a split lip. "I'll get a damage report from Silo, shall I?" he inquired.
"Don't bother," said Frey. "Just hang on to something."
Crake clutched at the metal jamb of the cockpit door as the _Ketty Jay_ began to shake violently. Frey dumped some of the aerium gas from the tanks to add weight and stability to the craft, letting the thrusters take the strain instead. Getting the balance right was crucial. A craft like the _Ketty Jay_ , unlike its outfliers, wasn't aerodynamic enough to fly without the aid of its lighter-than-air ballast. It couldn't produce enough lift to maintain its bulk.
The thunderheads rushed toward them, inky billows flashing with angry lightning. Wind and pressure differentials began to shove them this way and that. The world outside darkened rapidly as they hit the outer edge of the clouds. A blast of blinding light, terrifyingly close at hand, made Crake cower. Jez glanced over at him and gave him a sympathetic smile. He firmed his resolve and stood straighter.
"Doc! Are they still with us?" Frey howled over the rising wail of the wind. There was no reply. "Doc!"
"What?" Malvery cried back irritably.
"Are they still with us?"
A long pause.
_"Doc!"_ Frey screamed.
_"I'm bloody looking!"_ Malvery roared back. _"It's dark out there!"_ Then, a moment later, he boomed a triumphant laugh. "They're turning tail, Cap'n! Running off home!" Jez beamed in relief.
The _Ketty Jay_ was pushed from beneath by a pressure swell and veered steeply, dislodging Crake's grip on the jamb and sending him careening into a wall. It was black as night outside. Frey flicked on the headlights, but that only lit up the impenetrable murk that had closed in on them.
"I can't help noticing we're still in the storm," said Crake.
Jez supplied the answer, since Frey was concentrating on flying. "We need to put some distance between them and us. Otherwise they might just pick up the chase again when we emerge."
"And what happens if some of that lightning hits us?" he asked, not really wanting to know the answer.
"We'll probably explode," Frey said. Crake went gray. Jez opened her mouth to say something, but at that moment the craft was shaken again. Frey could hear things clattering about in the mess, and something cracked and burst noisily out in the corridor. Water began to spray everywhere.
"Is this tub even going to hold together?" Crake demanded.
"She'll hold," Frey murmured. "And if you call her a tub again, I'll kick you out right now, and you and your metal friend can fly home."
"What, and miss my chance to attend Gallian Thade's Winter Ball? Just try to—"
There was a stunning flash of light and everything went black. All lights, inside and out, were suddenly extinguished. There was a brief sensation of unreality, as if time itself had been stunned. The air snapped and crawled with wild energy. For long seconds, no one spoke. An uncanny peace blanketed the chaos. The engines droned steadily, pushing them through the storm. The darkness was utter.
Then the lights flickered on again, and the _Ketty Jay_ began to rattle once more.
"What was that?" Crake whispered.
"Lightning," said Jez.
"You said we'd explode!" Crake accused the captain.
Frey only grinned. "Time to get out of here," he said. He hauled back on the control stick and the _Ketty Jay_ began to climb.
The ascent through the clouds was rough, but the turbulence was nothing the _Ketty Jay_ couldn't handle. She'd seen worse than this in her time. Though she was jostled and battered and harassed every klom of the way, Frey fought with her against the storm, and the two of them knew each other well. Frey didn't realize it, but a fierce smile was plastered across his face as he flew. _This_ was what being a freebooter was all about. _This_ was how it felt to be a lord of the skies. Outwitting your enemies, snatching victory from defeat. Braving the storm.
Then the clouds ended, and the _Ketty Jay_ soared free. The dark carpet of thunderheads was spread out below them as far as they could see, obscuring everything beneath. Above them was only an endless crystalline blue and the dazzle of the sun.
"Malvery?" Frey called.
"All clear, Cap'n!" came the reply.
Frey looked over his shoulder at Jez and Crake, who were glowing with excitement and relief.
"Good job, everyone," he said. Then he slumped back in his seat with a sigh. "Good job."
# _Chapter Eighteen_
CIVILIZATION—A MUSICAL INTERLUDE—FREDGER CORDWAIN—VEXFORD SWOOPS IN—MORCUTT THE BOOR
he night was warm, and the air shrilled with the song of insects. Lush plants hissed and rustled in the tropical breeze. Electric lamps, hidden in the foliage, lit up an ancient stone path that wound up the hill toward the lights and the distant music. Northern Vardia might have been frozen solid, but here in the Feldspar Islands, winter never came.
Crake and Jez disembarked arm in arm from the luxurious passenger craft that had shuttled them from the mainland. Crake paused to adjust the cuffs of his rented jacket, then smiled at his companion to indicate his readiness. Jez tried not to look ill at ease in her clinging black dress as they made their way down from the aircraft. They were greeted at the bottom of the stairs by a manservant, who politely asked for their invitations. Crake handed them over and introduced himself as Damen Morcutt, of the Marduk Morcutts, whom he'd recently made up.
"And this is Miss Bethinda Flay," he said, raising Jez's hand so the manservant might bob and kiss it. The manservant looked at Crake expectantly for elaboration, but Crake gave him a conspiratorial wink and said, "She's rather new to this game. Be gentle with her, eh?"
"I quite understand, sir," said the manservant. "Madam, you are most welcome here."
Jez curtsied uncertainly, and then the two of them went walking up the path toward the stately manor at the top of the hill.
"Small steps," murmured Crake out of the corner of his mouth. "Don't stride. Remember, you're a lady."
"I thought we agreed that I was a craftbuilder's daughter," she replied.
"You're supposed to be a craftbuilder's daughter trying to be a lady."
"I _am_ a craftbuilder's daughter trying to be a lady!"
"That's why the disguise is flawless."
Crake had spent the last week coaching Jez in the basics of etiquette. She was a fast learner, but a crash course in manners would never convince anyone that she was part of the aristocracy. In the end, Crake had decided that the best lies were those closest to the truth. She'd pose as a craftbuilder's daughter—a life she knew very well. He'd play the indolent son of a wealthy family who had fallen in love with a low-born woman and was determined to make her his bride.
"That way, they'll think your mistakes are naïve rather than rude," he told her. "Besides, they'll feel sorry for you. They've seen it all before a dozen times, this breathless romance between a young aristocrat and a commoner. They know full well that as soon as it gets serious, Mother will step in and you'll be dumped. Nobody's going to waste a good marriage opportunity on a craftbuilder's daughter."
"What a charming lot you are," Jez observed.
"It's an ugly business," Crake agreed.
It _was_ an ugly business, but it was a business Crake had known all his life, and as he made his way along the winding path through the restless trees toward Scorchwood Heights, he felt an aching sorrow take him. The feel of fine clothes on his skin, the sound of delicate music, the cultured hubbub of conversation that drifted to them on the warm breeze—these were the familiar things of his old life, and they welcomed him back like a lover.
Seven months ago, he'd taken all of this for granted and found it shallow and tiresome. Having an allowance great enough to keep him in moderate luxury had permitted him to be disdainful about the society that provided it.
But now he'd tasted life on the run: hunted, deprived of comfort and society. He'd been trapped on a craft with people who mocked his accent and maligned his sexuality. He'd stared death in the face and been witness to a shameful act of mass murder.
The world he'd known was forever lost to him now. It hurt to be reminded of that.
"Do I look okay?" Jez fretted, smoothing her dress and patting at her elaborately styled hair.
"Don't do that! You look very pretty."
Jez made a derisive rasp.
"That ruins the illusion somewhat," said Crake, scowling. "Now listen to what I tell you, Miss Bethinda Flay. Beauty is all about confidence. You actually clean up rather well when you change out of your overalls and put on a little makeup. All you need to do is believe it, and you'll be the equal of anyone here." He stroked his beard thoughtfully. "Besides, the competition will be weak. Most of the women in this party have been inbred to the point of complete genetic collapse, and the others are more than half horse."
Jez snorted in surprise and then burst out laughing. After a moment, she caught herself and restrained her laughter to a more feminine chuckle.
"How kind of you to say so, sir," she managed in an exaggeratedly posh accent. She wobbled on the verge of cracking up, then swallowed and continued. "May I compliment you on the sharpness of your wit tonight."
"And may I say how radiant you look in the lamplight," he said, kissing her hand.
"You may. Oh, you may!" swooned Jez, then she hugged herself to his arm and followed him jauntily up the path to the manor. She was beginning to have fun.
Scorchwood Heights was set amid a grove of palm trees, its broad porticoed face looking out over a wide lawn and garden. It was a place of wide spaces, white walls, smooth pillars, and marble floors. The shutters were thrown open, and the sound of mournful string instruments and Thacian pipes wafted out into the night.
The lawn was crowded with knots of society's finest. The men dressed stiffly, many in Navy uniform. Others wore a uniform of another type: the single-breasted jackets and straight trousers that were the fashion of the moment. They laughed and argued, loudly discussing politics and business. Some of them even knew something about the subject. The women showed off in daring hats and flowing dresses, fanning themselves and leaning close to criticize the clothes of passersby.
Crake felt Jez's good humor falter at the sight of so many people, and he gave her a reassuring smile. "Now, Miss Flay. Don't let them intimidate you."
"You sure you couldn't have just come on your own?"
"That's not how it's done," he said. "Deep breath. Here we go."
Flagged paths meandered around pools and fountains toward the porch. Crake led them through the garden, stopping to take two glasses of wine from a passing waiter. He offered one to Jez.
"I don't drink," she said.
"That doesn't matter. Hold it. Gives you something to do with your right hand."
It was a little cooler inside the manor. The high-ceilinged rooms with their white plastered walls sucked some of the heat out of the night, and the open windows let the breeze through. Servants fanned the air. The aristocrats had gathered in here too, bunching into corners or lurking near the canapés, moving in swirls and eddies from group to group.
"Remember, we're looking for Gallian Thade," Crake murmured. "I'll point him out when I see him."
"And then what?"
"And then we'll see what we can find out."
A handsome young man with carefully parted blond hair approached them with a friendly smile. "Hello there. I don't think we've met," he said, offering a hand. He introduced himself as Barger Uddle, of the renowned family of sprocket manufacturers. "You know! Uddle Sprockets! Half the craft in the sky run on our sprockets."
"Damen Morcutt, of the Marduk Morcutts," said Crake, shaking his hand vigorously. "And this charming creature is Miss Bethinda Flay."
"My father used to use your sprockets all the time," she said. "He was a craftbuilder. Swore by them."
"Oh, how delightful!" Barger exclaimed. "Come, come, I must introduce you to the others. Can't have you standing around like wet fish."
Crake let this puzzling metaphor pass, and soon they were absorbed into a crowd of a dozen young men and women, all excitedly discussing the prospect of making ever more money in the future.
"It's only a matter of time before the Coalition lifts the embargo on aerium exports to Samarla, and then the money will come rolling in. It's all about who's ready to take advantage."
"Do you think so? I think we'll find that the Sammies don't even need it anymore. Why do you think the last war ended so suddenly?"
"Nobody knows why they called the truce. The Allsoul alone knows what goes on inside that country of theirs."
"Pffft! It was aerium, pure and simple. They fought two wars because they didn't have any in their own country and they couldn't stand buying it from us. Now they've found some. Bet you anything."
"We shouldn't even be trading with those savages. We should have gone in there and flattened them when we had the chance. Mark my words, this is only a lull. They're building a fleet big enough to squash us like insects. There'll be a Third Aerium War, and we won't win this one. New Vardia, that's where I'm going. New Vardia and Jagos."
"The frontier. That's where the money is, alright. Get right in on the ground floor. But I think I'd miss the society. I'd just shrivel out there."
"Oh, you've no sense of adventure!"
After a while, Crake and Jez excused themselves and made their way into an enormous drawing room. Here was the source of the music they'd been hearing ever since they arrived. A quintet of Thacian women played delicate folk songs from their homeland. They were slender, olive-skinned, black-haired, and even the least attractive among them could still be called pretty. They wore colored silks and held exotic, exquisitely made instruments of wood and brass.
"Listen," said Crake, laying a hand on Jez's shoulder.
"Listen to what?"
"Just listen," he said, and closed his eyes.
In the field of the arts—as in science, philosophy, culture, and just about everything else—Thacians were the leaders in the known world. Vardic aristocracy aspired to the heights of Thacian achievement, but usually all they could manage were clumsy imitations. To hear real Thacian players was a treat, which came at a hefty price—but then, Gallian Thade wasn't a man known to be short of money. Crake allowed himself to be swept away in the tinkling arpeggios, the haunting moan of the pipes, the counterpoint rhythms.
This was what he missed. The casual elegance of music and literature. To be surrounded by wonderful paintings and sculpture, perfect gardens and complicated wines. The upper classes insulated themselves against the world outside, padding themselves with beauty. Without that protection, things became ugly and raw.
He wished, more than anything, that he could go back. Back to how it was before everything went bad. Before...
"Excuse me."
He opened his eyes, irritated at the interruption. The man standing before him was taller than he was, broad-shouldered and bull-necked. He was fat but not flabby, bald-headed, and sported a long, thin mustache and expensively cut clothes.
"Sorry to spoil your enjoyment of the music, sir," he said. "I just had to introduce myself. Fredger Cordwain is my name."
"Damen Morcutt. And this is Miss Bethinda Flay." Jez curtsied on cue, and Cordwain kissed her hand.
"Charmed. I must ask you, sir, have we met? Your face seems very familiar to me, very familiar indeed, but I can't place it."
Crake felt a small chill. _Did_ he know this person? He'd been quite confident that nobody who knew his face would be here tonight. His crime had been kept out of the press—nobody wanted a scandal—and the Winter Ball was simply too exclusive for the circles Crake had moved in. Invitations were almost impossible to secure.
"I'm terribly sorry," he said. "I can't quite recall."
"Perhaps we met on business? At a party? May I ask what it is you do?"
"You may well ask, but I'm not sure I could answer!" Crake brayed, falling into his role. "I'm sort of in between occupations at the moment. Father wants me to go into law, but my mother is obsessed with the idea that I should be a politician. Neither of them appeals much to me. I just want to be with my sweetheart." He smiled at Jez, who smiled back dreamily, bedazzled by her rich boyfriend. "May I ask what it is that _you_ do?"
"I work for the Shacklemore Agency."
It took all of Crake's control to keep his expression steady. The news was like a punch in the gut. Suddenly he was certain that Cordwain was watching for a reaction from him, and he was determined to give none.
"And what does the Shacklemore Agency do?" asked Jez innocently, though she must have already known. Crake silently thanked her for the distraction.
Cordwain favored her with a patronizing smile. "Well, Miss, we look after the interests of our clients. We work for some very important people. My job is to deal with those people, keep things running smooth."
"Hired guns and bounty hunters, that's what they are," sniffed Crake. He was quick on his feet in social situations, and he'd already decided on the best tactic for getting away as fast as possible. "I must say, I find it very distasteful."
"Damen! Don't be rude!" Jez said, appalled.
"It's alright, Miss," said Cordwain, with an unmistakable hostility in his gaze. "There's some who don't understand the value of the work we do. The law-abiding man has nothing to fear from us."
"I say, sir, do you dare to imply something?" Crake bristled, raising his voice. People nearby turned and looked. Cordwain noticed the attention their conversation had drawn.
"Not a thing, sir," he said coldly. "I apologize for disturbing you." He bowed quickly to Jez and walked away. The people around them resumed their conversations, glancing over occasionally in the hope of further drama.
Crake felt panicked. Had there been a warning in the man's tone? Had he been recognized? But then, what was the point in confronting him? Was it just monstrous bad luck that he'd run into a Shacklemore here?
The warm sensation of being surrounded by familiar things had faded now. He felt paranoid and uneasy. He wanted to get out of here as soon as possible.
Jez was studying him closely. She was an observant sort, and he had no doubt that she knew something was up. But she kept her questions to herself.
"Let's go find Gallian Thade, hmm?"
Crake found him shortly afterward, on the other side of the room. He was a tall, severe man with a hawk nose and a deeply lined, narrow face. For all his years, his pointed beard and black hair had not a trace of gray. His eyes were sharp and moved rapidly about as he spoke, like an animal restlessly scanning for danger.
"That's him," said Crake, admiring their host's stiff brocaded jacket.
Thade was in conversation with several men, all of them stern and serious-looking. Some of them were smoking cigars and drinking brandy.
"Who's that with him?" murmured Jez, looking at the man next to Thade.
Crake studied Thade's companion with interest. "That's Duke Grephen of Lapin."
Crake knew him from the broadsheets. As ruler of one of the Nine Duchies that formed Vardia, he was one of the most influential people in the land. Only the Archduke held more political power than the dukes.
Grephen was a dour-looking man with a squarish build and a sallow face. His eyes were deeply sunken and ringed with dark circles, making him look faintly ill. His short blond hair was limp and damp with sweat. Though he was thirty-five, and he wore a fine uniform with the Lapin coat of arms on its breast, he looked like a pudgy boy playing at being a soldier.
Despite his less-than-formidable appearance, the others treated Grephen with the greatest respect. He didn't speak often and never smiled, but when he had something to offer, his companions listened intently.
"Bet you never thought you'd see _him_ when you came here tonight," said a voice to their right. They looked over to see a gaunt man with white hair and bushy eyebrows, flushed from alcohol and the heat. He was wearing a Navy uniform, his buttons and boots polished to a high shine.
"Why, no, I hadn't imagined I would," said Jez.
"Air Marshal Barnery Vexford," he said, taking her hand to kiss it.
"Bethinda Flay. And this is my sweetheart, Damen Morcutt."
"Of the Marduk Morcutts," Crake added cheerily, as he shook Vexford's hand. Vexford wasn't quick enough to keep the fleeting, predatory glitter from his eyes. Crake had already surmised what was on his mind. He was after Jez, and that made Crake his competition.
"You know, ferrotypes don't do him justice," Jez twittered. "He's so very grand in real life."
"Oh, he is," agreed Vexford. "A very serious man, very thoughtful. And so devout. A credit to his family."
"Do you know the Duke very well?" Jez asked.
Vexford glowed. "I have had the privilege of meeting the Duke on many occasions. The Archduke is also a personal friend of mine."
"Perhaps you could introduce us to Duke Grephen?" Crake suggested, pouncing. Vexford hesitated. "We'd be honored to meet him and offer our thanks to the host. I know Bethinda would be very grateful."
"Oh! It would be a dream come true!" she gushed. She was getting to be quite the little actress.
Vexford's reservations were obvious. You didn't introduce just _anyone_ to the Duke. But he'd talked himself into a corner, and he'd seem foolish if he backed out now. "How can I refuse such a beautiful lady?" he said, with a hateful smile at Crake. Then he laid his hand on Jez's back, claiming her as his prize, and led her over toward the Duke's group without another look at her "sweetheart." Crake was left to follow, rather amused by the Air Marshal's attempt to snub him.
Vexford's timing was perfect. The conversation had lulled, and his arrival in the group caused everyone to take notice of the newcomers.
"Your Grace," he said, "may I introduce Miss Bethinda Flay." After a pause long enough to be insulting, he added, "And also Damen Morcutt, of the Marduk Morcutts," as if he'd just remembered Crake was there.
On seeing the blank looks of his companions, someone in the group exclaimed knowingly, "The Marduk Morcutts, ah, yes!" The others murmured in agreement, enough to imply that the Marduk Morcutts were indeed a fine family, even if none of them knew who the Marduk Morcutts actually were.
Jez curtsied; Crake bowed. "It's a great honor, Your Grace," he said. "For both of us."
The Duke said nothing. He merely acknowledged them silently with nods, then gave Vexford a look as if to say: _why have you brought these two here?_ The conversation had fallen silent around them. Vexford shifted uncomfortably and sipped his sherry.
"And you must be Gallian Thade!" Crake suddenly exclaimed. He took up Thade's hand and pressed it warmly between his palms, then gave the older man a companionable pat on the hip. "Wonderful party, sir, just wonderful."
Vexford almost choked on his drink. The others looked shocked. Such familiarity with a man who was clearly Crake's social superior was unpardonable. The worst kind of behavior. Nobody expected such oafishness in a place like this.
Thade kept his composure admirably. "I'm so glad you're enjoying it," he said frostily. "You should try the canapés. I'm sure you would find them delicious."
"I will!" said Crake enthusiastically. "I'll do it right now. Come on, Bethinda, let's leave these gentlemen to their business."
He took her by the arm and marched her away toward the canapés, leaving Vexford to face the silent scorn of his peers.
"What was that about?" asked Jez. "I thought you wanted to find out what Thade was up to."
"You remember this?" he said, taking a tiny silver earcuff out of his pocket.
"Of course I do. You showed the Cap'n how they worked. He didn't stop talking about them for two days. I think you impressed him." She watched him affix it to his ear. "Looks a bit tacky for this kind of party," she offered.
"Can't be helped."
"Where's the other one?"
Crake flashed her a gold-toothed grin. "In Thade's pocket. Where I put it when I patted him on the hip."
Jez was agape. "And you can hear him now?"
"Loud and clear," he said. "Now let's get some canapés, settle down, and see what our host has to say."
# _Chapter Nineteen_
CRAKE'S STEREOTYPES—JEZ IS BETRAYED—A DARING SHOW OF CHEEK—DREADFUL INFORMATION
n hour later, and Crake had begun to remember why he'd been so bored with the aristocracy. He seemed to be encountering the same people over and over again. The faces were different, but the bland niceties and insipid observations remained the same. He was yet to meet anyone more interesting than the clothes they wore.
The guests fell neatly into the pigeonholes he made for them. There was the Pampered Adventurer, who wanted to use Father's money to explore distant lands and eventually set up a business in New Vardia. They had no real concept of hardship. Then there was the Future Bankrupt, who talked enthusiastically of investing in dangerous projects and bizarre science, dreaming of vast profits that would never materialize. They were often attached to the Vapid Beauty, whose shattering dullness was tolerable only because they were so pleasant to look at. Occasionally he spotted a Fledgling Harpy, spoiled daughter of a rich family. Unattractive yet intelligent enough to realize that their fiancé was with them only for their money. In revenge for thwarting their fantasies of romance, they intended to make the remainder of his life a misery.
These, and others, he recognized from long experience. A procession of stereotypes and clichés, he thought scornfully. All desperately believing themselves to be unique. They parroted their stupid opinions, plucked straight from the broadsheets, and hoped that nobody disagreed. How had he ever communicated with these people? How could he ever go back among them, knowing what he knew?
They'd moved into the magnificent ballroom, with its swirled marble pillars and copper chandeliers. The floor was busy with couples, some of them lovers but most not. They exchanged partners as they moved, men and women passed around in a political interplay, gossiping and spying on one another. Crake stood to one side with Jez, talking with a pair of brothers who had recently bought an aerium mine and clearly had no idea how to exploit it.
Gallian Thade and Duke Grephen stood on the other side of the room. Crake listened. It was hard to concentrate on two conversations at once, but luckily he needed less than half his attention to keep up with either. Jez was fielding the Aerium Brothers, and Thade and his companions were saying nothing of any interest. Their talk consisted of possible business ventures, witticisms, and pleasantries. He was beginning to wonder if Frey had been wise to believe Thade might give something away.
"We should go elsewhere," he heard Thade murmur, through the silver earcuff. "There are things we must discuss."
Crake's eyes flickered to the host, who was talking to the Duke. Grephen nodded, and they excused themselves and began to move away across the ballroom. This was promising.
"Miss Flay!"
It was Vexford, the rangy old soak who had taken a fancy to Jez. He gave Crake a poisonous glare as they made their greetings. He'd not forgotten his recent embarrassment at Crake's hands. It hadn't embarrassed him enough to keep him from trying to steal his adversary's sweetheart, apparently.
"Air Marshal Vexford!" Jez declared, with false and excessive enthusiasm. "How good to see you again!"
Vexford puffed up with pleasure. "I was wondering if I might have the honor of this dance?"
Jez glanced uncertainly at Crake, but Crake wasn't listening. He was concentrating on the sounds in his ear. Grephen and Thade were exchanging greetings with people as they passed through the ballroom toward a doorway at one end. The greetings were getting fainter and fainter as they moved out of range.
"Damen?" Jez inquired. He noticed her again. "Air Marshal Vexford wishes to dance with me." Her eyes were urgent: _Save me!_
Crake smiled broadly at the Air Marshal. "That would be fine, sir. Just fine," he said. "Excuse me, I must attend to something." He slipped away with rude haste, to spare himself Jez's gaze of horrified betrayal.
He made his way toward the doorway Grephen and Thade were heading for, glancing around nervously as he went. He was searching for a sign of Fredger Cordwain, the man who worked for the Shacklemores. Crake hadn't spotted him since their conversation earlier, and it worried him deeply.
When he was a child, he'd been afraid of spiders. They seemed to like his bedroom, and no matter how the maids chased them out, they always came back. But frightened as he was, he found their presence easier to bear if he could see them hiding in a corner or motionless on the ceiling. It was when he looked away, when the spider disappeared, that the fear came. A spider safely on the far side of the room was one thing; a spider that might already be crawling over the pillow toward his face was quite another. Crake wanted Cordwain where he could see him.
The sound of Thade's voice strengthened in his ear as he drew closer to them. They passed through the grand doorway at the end of the ballroom and away. Crake followed at a distance.
Beyond was a corridor, leading through the manor to other areas: smoking rooms, galleries, halls. Guests were scattered about in groups, admiring sculptures or laughing among themselves. Crake was sweating, and not only because of the heat. He felt like a criminal. The casual glances of the doormen and servants seemed suddenly suspicious and knowing. He sipped his wine and tried to look purposeful.
"Where are we going?" Grephen said quietly to Thade, looking around. "Somewhere more private than this, I hope."
"My study is off limits to guests," Thade replied. He halted at a heavy wooden door with vines carved into its surface and unlocked it with a key. Crake stopped a little way up the corridor, pretending to admire a painting of some grotesque aunt of the Thade dynasty. Thade and Grephen stepped inside and closed the door behind them.
He waited for them to speak again. They didn't. Wait: was that a murmur in his ear? Perhaps, but it was too faint to make out. The study evidently went back some distance into the manor, and they were right at the limit of his range.
_Spit and blood! I knew I should have made these things more powerful_ , he thought, fingering his earcuff in agitation.
He looked both ways up the corridor, but nobody was paying attention to him. He walked across to the door that led to the study. If anyone asked, he could just say he got lost.
He tried the door. It didn't open. He tried again, more forcefully. Locked.
"I don't think you can go in there," said a portly middle-aged man who had spotted his plight.
"Oh," said Crake. "I must be mistaken." He lowered his voice and moved close to murmur, "I thought this was the lavatory. It's quite desperate, you see."
"Other end of the corridor," said the man, giving him a pat on the shoulder.
"Much obliged," he said, and hurried away.
His mind was racing. If Thade had anything worth hearing, he was saying it right now, and Crake was too far away to listen. This whole excursion would be wasted if he couldn't get back in range, and quickly.
Just then he passed the foot of a staircase. It was relatively narrow and simple, with white stone steps and elegant, polished banisters. A manservant stood on the first step, barring entry to guests.
And suddenly Crake had an idea.
"Excuse me," he said. "Would you mind terribly if I had a nose around up there?"
"Guests are not allowed, sir," said the manservant.
Crake grinned hugely. His best grin, his picture grin. His gold tooth glinted in the light of the electric bulb. The manservant's eyes glittered like a magpie's.
"I'd be most grateful if you could make an exception," he said.
THE CORRIDORS UPSTAIRS WERE cool and hushed and empty. The gabble of conversation and music from the ballroom was muted by the thick floors. Crake could hear a pair of maids somewhere nearby, talking in low voices, giggling as they prepared the bedrooms.
He chose a direction that he judged would take him toward Thade's study chambers. Despite the awful thrill of trespass, his limbs were beginning to feel heavy. Using the earcuff, and now his tooth, had sapped his energy. Years of practice had trained him to endure the debilitating effect of employing daemons, but the sustained low-level usage had worn him down.
A man's voice joined the women's. A butler. Chiding. _Get on with your work_. The three of them were up ahead, just around a bend in the corridor. They might step into view at any moment, and Crake would be seen. He could feel his pulse throbbing against his collar. His palms were clammy and wet with the terror of being caught doing something wrong. He marveled at how people like Frey could flout authority with such ease.
Then, a murmur. The faintest of sounds. The daemon thralled to his earcuff was humming in resonance with its twin. He was picking up the conversation again.
Stealthily, holding his breath, he moved down the corridor. The butler was issuing instructions as to how the master wanted his guests' rooms arranged. His voice grew in volume. Frustratingly, Thade's didn't. Crake was skirting around the limit of his earcuff's range. Somewhere on the floor below him, Thade and Grephen were discussing the secret matters he'd come here to learn about. He had to get closer.
Crake crept up to the corner, pressing himself against it. He peered around. The butler was in the doorway of a nearby bedroom, a little way inside. His back was to the corridor, and he was talking to the maids within.
Crake took a shallow breath and held it. He had to do this now, before his nerve failed him. Soft-footed, he padded past the doorway. No voice was raised to halt him. The butler kept talking. Unable to believe his luck, Crake kept going, and the conversation in his ear grew audible.
"... concern that... still haven't caught..."
He opened a plain-looking door and ducked inside, eager to get out of the corridor. Within was a small, green-tiled room, with a shuttered window, a scalloped white sink, and a flush toilet at the end.
_Well_ , he thought. _I found the lavatory after all_.
"It's imperative that Dracken finds him before the Archduke's Knights do," said Grephen in his ear. "It should have been done properly the first time."
Crake felt a guilty shiver, the chill of an eavesdropper who hears something scandalous. They were talking about Frey.
"Nobody expected him to get away," said Thade. "I had four good pilots flying escort."
"So why didn't they do their jobs?"
The lavatory had a lock on the inside, with a large iron key. Crake eased the door closed and quietly turned it, then sat down on the toilet lid. Grephen and Thade were almost directly below him now. He could hear them perfectly.
"The survivor said they launched a surprise attack."
"Well, of course they did! We told them the route the _Ace of Skulls_ would be flying! So why weren't our pilots warned?"
"The pilots were independents, hired through middlemen, that couldn't be connected to you. We needed them to be reliable, untainted witnesses. We could hardly warn them an attack was coming without giving away the fact that we set up the ambush."
_Amalicia Thade was right_ , thought Crake. _Her father wasn't in this alone. This goes all the way up to the Duke_.
"The _Ketty Jay_ had two outfliers—fighter craft," Thade went on patiently. "We didn't even know Frey traveled with outfliers. He's such an insignificant wretch, it's a miracle he keeps his own craft in the sky, let alone three."
"You didn't know?"
"Your Grace, do you have any idea how hard it is to keep track of one maggot amid the swarming cesspool of the underworld? A man like that puts down no roots and leaves little trace when he's gone. The sheer size of our great country makes it—"
"You underestimated him, then."
Crake heard a resentful pause. "I miscalculated," Thade said at last.
"The problem was that you didn't _calculate_ anything," Grephen said. "You allowed your personal hatred of this man to blind you. You saw a chance for revenge because he disgraced your daughter. I should never have listened to you."
"The Allsoul itself thought that Darian Frey was an excellent choice for our scheme."
"The auguries were unclear," said Grephen coldly. "Even the Grand Oracle said so. Do not presume to know the mind of the Allsoul."
"I am saying that I trust in the Allsoul's wisdom," Thade replied. "This is merely a hiccup. We will still emerge triumphant."
Crake couldn't help a sneer and a tut. _Superstition and idiocy_ , he thought. _Strange how your Allsoul can't stop me from using my daemons to listen to every word you say_.
"The survivor told us that the _Ketty Jay_ 's outfliers were fast craft with excellent pilots," Thade explained. "The surprise attack threw them into chaos and took out half of our men. We were lucky that one witness escaped to report to the Archduke."
Nobody spoke for a time. Crake imagined a sullen silence on Grephen's part.
"This is not a disaster," said Thade soothingly. "Hengar is out of the way, and our hands remain clean. Don't you see how things have fallen in our favor? That fool's dalliance with the Samarlan ambassador's daughter gave us the perfect opportunity to remove him and make it look like a pirate attack. If he'd not been traveling in secret, if your spies hadn't discovered his affair, our job would have been that much more difficult."
Grephen grunted in reluctant agreement, allowing himself to be mollified.
"Not only that," Thade went on, "but leaking information about the affair to the public has turned them against Hengar and the archduchy in general. Hengar was the one they loved, remember? He stood aside when his parents began their ridiculous campaign to deprive the people of the message of the Allsoul. His death could have strengthened the family, made them sympathetic in the eyes of the common man, but instead they have never been so unpopular."
"That's true, that's true."
Thade was warming to his own positivity now. "Don't you see how kindly the Allsoul looks on our enterprise? We have cleared the line of succession: the Archduke has no other children to inherit his title. The people will welcome you when you seize control of the Coalition. You will be _Arch_ duke Grephen, and a new dynasty will begin!"
Crake's mind reeled. _This_ was what it was all about? Spit and blood, they were planning a coup! They were planning to overthrow the Archduke!
It was all but inconceivable. Nobody alive remembered what it was like to live without a member of the Arken dynasty ruling the land. The rulers of the duchy of Thesk had been the leaders of the Coalition for almost a century and a half. They'd been the ones who forcefully brought the squabbling Coalition to heel after they deposed the King and threw down the monarchy. The first Archduke of Vardia had been of the family of Arken, as had every one since. The Arkens had been the ultimate power in the land for generations, overseeing the Third Age of Aviation and the Aerium Wars, the discovery of New Vardia and Jagos on the far side of the world, the formation of the Century Knights. They'd abolished serfdom and brought economic prosperity and industry to a land strangled by the stagnant traditions of millennia of royal rule.
Crake felt history teetering. Riveted, he listened on.
"It _... concerns_ me that Darian Frey is still on the run," said the Duke. "He has already been to the whispermonger you employed."
"Don't worry about Quail. Dracken has made sure he won't speak to anyone ever again."
"But Frey is already on the trail. He was spotted near your daughter's hermitage."
"Amalicia has been questioned by the Mistresses, at my request. She swears that he never visited her. Dracken probably caught up to him before he had a chance to—"
"What if she's lying?"
"You know I can't go in there or bring her out. She must stay in isolation. We have to trust her and the Mistresses."
"My point is, he must know about you. That means he may learn about _me._ "
"Peace, Your Grace. Who'll believe him? With Quail dead, there's nothing to link us but the word of a mass murderer."
"It's not a chance I want to take. If he digs deep enough, he might find something. I don't want the Century Knights getting hold of him and giving him the chance to spout his theories to the Archduke."
Crake was sitting atop the toilet, elbows on his knees, one hand on his forehead with his fingers clenched anxiously through his hair. Finally he understood the true seriousness of their situation. Unwittingly, they'd become entangled in a power play for the greatest prize in the land. The only problem was they'd been inconvenient enough not to die when they were supposed to. Now they were hunted, both by those who thought they were responsible and those who wanted them silenced. Small fry dodging the mouths of the biggest fish in the sea.
Thade's voice was soothing again. "Dracken will have him soon. She guessed that he'd go to Quail, and she surmised he'd go after your daughter rather than coming for you. I am learning to respect her intuition where Frey is concerned." He paused. "She also believes he might try something tonight."
"Tonight?"
"It's his best chance of getting close to me, amid all the chaos. But do not fear. She has men undercover all over my manor and in the port on the mainland. The _Delirium Trigger_ itself is hiding up in the night sky, waiting for a signal if the _Ketty Jay_ should arrive."
Crake felt his stomach sink. First the Shacklemores, and now Trinica Dracken was here? One step ahead of them already? This was getting altogether too dangerous. It was only through Amalicia's invitations—and because nobody knew Crake and Jez were part of the crew of the _Ketty Jay_ —that they'd remained undetected thus far. Crake was beginning to wish he'd never gotten involved in the first place.
The lavatory door rattled, making him jump. He looked up. There was a pause, then the door rattled again. A moment later there was a sharp knock on the door.
"Is someone in there?"
It was the butler. Crake was frozen to the spot. He said nothing, in the futile hope that the man outside would go away.
"Hello? Is someone in there?" He sounded angry. There was a knocking again, firmer this time.
The door was locked from the inside. Crake decided that he'd do better to own up, before the butler got really furious.
"I'm in here," he said. "Be out in a minute."
"You'll be out right now, sir!" said the butler. "I don't know how you got up here, but these are the private rooms of Master Thade."
"Do you trust her?" said Grephen, from downstairs. His voice was suddenly faint. They'd moved away, walking into another room. Crake strained to hear over the voice of the butler.
"Dracken? As much as I trust any pirate," Thade replied. "Besides, we need her. She's our only link to—"
"Sir! I must insist you come out here right now!" the butler cried, knocking hard on the door.
"Give a man a moment to finish his business!" Crake protested, delaying his exit as long as he could. He had the sense that something important was being discussed here, but the words were becoming harder and harder to hear as the speakers moved away.
"... we... no one else?" Grephen asked. "I... uneasy about..."
"... Dracken knows the... has charts and... device of some kind. Only way... can find that place. She... our... has to be escorted in... out... secret hideout..."
"Sir!" bellowed the butler.
"I'm coming!" cried Crake. He flushed the toilet and was dismayed when the sound drowned out the last of the conversation from below him. Unable to hold out any longer, he unlocked the door and was immediately seized by the arm. The butler was a short, balding, red-faced fellow, and he was in no mood for Crake's weak excuses. The daemonist was escorted roughly along the corridor and down the stairs, past the startled manservant who was supposed to be guarding them.
"Sir will please stay downstairs from now on, or he shall be thrown from the premises!" the butler snapped, loud enough to draw titters from the guests nearby. Crake blushed despite himself. He hurried back toward the ballroom as the butler began to vent his anger on the hapless manservant who had let Crake pass minutes before.
Once in the ballroom, he looked for Jez and found her with Vexford. The older man was towering over her, drunk on sherry and success, bawling about his outrageous exploits during the Second Aerium War. Crake strode up to them and took Jez by the arm.
"Cra—" Jez began, then corrected herself. "Sweetheart!"
"We're going," he said, pulling her away.
"Here, now, you boor!" protested Vexford, who was still in mid-story; but Crake ignored him, and Jez was propelled away. Vexford grabbed her wrist to stop her.
"Sir!" she exclaimed breathlessly.
Vexford leaned closer and murmured huskily in her ear. "I have a large estate, just outside Banbarr. Anyone in the city will know where it is. If you ever tire of this ruffian, you will be most welcome." Then she was pulled away again by her impatient companion.
"It's been a great pleasure, sir!" Jez called over her shoulder. "I hope to meet again!" Then the crowd closed around them, and she turned to Crake with a narrow glare. "You left me alone with him," she accused. "He smells of sour milk and carrots."
"We'll talk about it later, dear," said Crake.
"I don't think I want to marry you anymore," she sulked.
# _Chapter Twenty_
A GUEST ON THE PATH—THE LETTER KNIFE—A BAD END TO THE EVENING
he crowd on the lawns had thinned out considerably—most of them were in the ballroom now—and the chorus of night insects was in full voice. Crake pulled off his earcuff and threw it into a flower bed as they passed. It was useless without its partner, and he wasn't about to retrieve it from Thade's pocket. He'd make more, and better.
"So I take it you found out what you wanted?"
"I found out more than I wanted," he muttered. "But right now I'd like to get off this island as quickly as possible."
Crake looked up into the moonless sky as they walked, fancying he might see a patch of deeper black in the blackness: the _Delirium Trigger_ , lurking in wait. Jez, having picked up on his obvious agitation, stayed silent.
They crossed the lawns and came to the old path that led to the manor's landing pad. Here, passenger craft ran a shuttle service to the port of Black Seal Bluff on the mainland. The _Ketty Jay_ was hidden in a glade a few kloms out from the port. Shaken by his near-miss with the _Delirium Trigger_ , Frey hadn't dared set down in Black Seal Bluff itself. A sensible precaution, as it turned out. Dracken's undercover spies would have spotted the craft immediately.
They'd been fortunate so far. They'd received more than their share of luck. But the circle was drawing tighter now, and the closer they got to the truth behind the destruction of the _Ace of Skulls_ , the more it constricted.
The path down to the landing pad was wide and deserted, with a knee-high drystone wall on either side. It wound down the hill, occasionally bulging out into small rest areas with carved wooden benches. Weeping bottlebrush and jacarandas overhung the wall, obscuring sections of the path. Electric lamps, set in recesses, lit their faces from below. Bats feasted on insects in the blood-warm darkness overhead.
Crake was so intent on getting down to the pad and away that he was surprised when Jez suddenly tugged him to a halt.
"Someone's there," she said. She was staring intently into the foliage, a distant look in her eyes, as if she was seeing right through the leaves and bark to whoever hid beyond.
"What? Where?" He tried to follow her gaze, but he could see no sign of anyone.
"He's right there," she murmured, still staring. "On the bench. Waiting for us."
They stood there a moment, not knowing what to do. Crake couldn't fathom how she could sense this mysterious man or how she knew his intention. But he didn't doubt the conviction in her voice. They couldn't go forward without passing him, and they couldn't go back. Crake wished they'd tried to smuggle in weapons, but it was forbidden for guests to carry arms.
Yet he couldn't just stand here, trapped, a child afraid to move in case he disturbed the spider. That wasn't the way a man ought to act. So he steeled himself and walked on, Jez following behind.
A dozen paces later, the path twisted and widened into a circular rest area, hidden by the trees. There was an ornamental stone pool, with a weak jet of water bubbling from a spike in its center. Sitting on a bench, contemplating the pool, was Fredger Cordwain. He looked up as Crake and Jez arrived.
"Good night," said Crake, without breaking stride.
"Good night, Grayther Crake," Cordwain replied.
Crake froze at the sound of his name. He tensed to run, but Cordwain surged up from the bench, a revolver appearing in his meaty hand. He must have assumed the rule against carrying arms didn't apply to him.
"Let's not make this difficult," Cordwain said. "You're worth just the same to me dead or alive."
"Who's this?" Jez asked Crake. It took a moment before he realized she was still playing in character. "Sweetheart, what's this about?"
Cordwain walked toward them, his weapon trained on Crake. "Miss Bethinda Flay," he said. "If that is your real name. The Shacklemore Agency has been after your 'sweetheart' for several months now. I'm ashamed to say it took me a little time to recognize him from his ferrotype. It's the beard, I think. I don't have a good memory for faces."
"But he hasn't done anything!" Jez protested. "What did he do?"
Cordwain stared at her levelly. "Don't you know? He murdered his niece. An eight-year-old girl."
Jez looked at Crake, stunned. Crake was slump-shouldered, gazing at the floor.
Cordwain moved around behind Crake, took his wrists, and pulled his arms behind his back. Then he shoved the revolver into his belt and drew out a pair of handcuffs.
"Stabbed her seventeen times with a letter knife," he said conversationally. "Left her to bleed out on the floor of his own daemonic sanctum. That's what kind of monster he is."
Crake didn't struggle. He'd gone pale and cold, and he wanted to be sick.
"His own brother hired us to find him," said Cordwain. "Isn't that sad? It's terrible when families get to fighting among themselves. You should always be able to trust your family."
Tears gathered in Crake's eyes as the handcuffs snapped closed. He raised his head and met Jez's gaze. She stared at him hard, shock on her face. Wanting to be reassured. Wanting to know that he hadn't done this thing.
He had nothing to tell her. She could never condemn him more than he already condemned himself.
"If you don't mind, Miss, I'll have to ask you to come along with me too," said Cordwain as he adjusted the handcuffs. "I'm sure you understand. Just until we establish that you've no connection with this—"
Jez lunged for the pistol sticking out of his belt, but Cordwain was ready for her. He grabbed her by the arm and yanked her off balance, shoving Crake down with his other hand. With his hands cuffed behind his back, Crake was unable to cushion his fall, and he landed painfully on his shoulder on the stony ground.
Jez slapped and punched at Cordwain, but he was a big man, much stronger and heavier than she was.
"As I thought," he said, fending her off. "In on it too, aren't you?"
Jez landed a fist on his jaw, surprising him. But the surprise lasted only a moment. He backhanded her hard across the face: once, twice, three times in succession. Then he flung her away from him. She tripped headlong, flailing as she went, and cracked her forehead against the low stone wall of the pool.
The terrible sound of the impact took all the heat out of the moment. Cordwain and Crake both stared at the small woman in the pretty black dress who now lay motionless on the ground.
She didn't get up.
"What did you do?" Crake cried from where he lay. He struggled to his knees.
Cordwain drew his pistol and pointed it at him. "You calm down."
"Help her!"
"I said cool your heels!" he snapped. He moved over toward Jez, crouched down next to her, and picked up a limp hand, pressing two fingers to her wrist. After a moment, he let it drop, pulled her head aside, and checked for a pulse at her throat.
Crake knew the result by his expression. He felt a surge of unbelievable, irrational hate. "You son of a bitch!" he snarled, getting to his feet. Cordwain immediately thrust his weapon toward him.
"You saw what happened!" Cordwain said. "I didn't mean for that!"
"You killed her! She wasn't anything to do with us!"
Cordwain advanced on him. "You shut your damn mouth! I told you I could take you in dead or alive and I meant it!"
"Well, you'd better take me dead, you bastard! Because even a Shacklemore doesn't get to kill innocent women! And I'm going to make absolutely sure that everyone knows what you've done."
"You need to stop your talking, sir, or I will shoot you like a dog!"
But Crake was out of control. The sight of Jez lying there had freed something inside him. It unleashed all the rage, the guilt, the horror that he kept penned uneasily within. He saw his niece, still and lifeless, her white nightdress soaked in red, her small body violated by vicious wounds. He saw the bloodied letter knife in his hand.
That was the day he began to run, and he hadn't stopped since.
"Why don't you shoot?" he shouted. "Why don't you? Save me the show trial! Pull the trigger!"
Cordwain backed off, his gun raised. He was unsure how to deal with the red-faced, spittle-flecked maniac who was stumbling toward him, his hands cuffed behind his back.
"You stay back, sir!"
"End it, you murderer!" he screamed. "End it! I've had enough!"
And then something moved, quick in the night, and there was a terrible, dull crunch. Cordwain's eyes rolled up into his head and he crumpled, folding onto himself and falling to the ground.
Standing behind him, a rock from the drystone wall in her hand, was Jez.
Crake just stared.
Jez tossed the rock aside and took the keys from the Shacklemore man. She walked over to Crake, turned him around, and undid his handcuffs. By the time they'd fallen free, he'd found words again.
"I thought you were dead."
"So did he," she replied.
"But he... but you _were_ dead."
"Apparently not. Give me a hand."
She began to tug Cordwain toward the trees. After a moment, Crake joined her. As they manhandled him over the drystone wall, his head lolled back, and Crake caught a glimpse of his eyes. They were open, and the whites were dark with blood.
Crake turned away and vomited. Jez waited for him to finish, then said, "Take his legs."
He wasn't used to this merciless tone from her. He did as he was told, and together they carried him out of sight of the path and left him there.
They returned to the clearing, where Jez replaced the rock in the wall and threw Cordwain's gun into the undergrowth. She dusted her dress off as best she could.
"Jez, I—" he began.
"I didn't do it for you, I did it for me," she interrupted. "I'm not being taken in by any damn Shacklemore. Not when half the world still wants us dead." There was a weary disgust in her voice. "Besides, you still haven't told me what you learned in there. The Cap'n will want to hear that, no doubt."
She wasn't the same Jez who had accompanied him to this party. The change was sudden and wrenching. Everything that had happened before, every shared joke and kind word, meant nothing in the face of the crime he'd committed. Crake wished there was something to say, some way he could explain, but he knew that she wouldn't listen. Not now.
"It's better that we don't speak about what happened here tonight," she said, still brushing herself down. She stopped and gave him a pointed look. "Ever."
Crake nodded.
"Right, then," she said, having arranged herself as best she could. "Let's get out of here."
She walked down the path toward the landing pad. Crake cast one last glance into the trees, where Cordwain's body lay, and then followed her.
# _Chapter Twenty-one_
FREY CALLS A MEETING—HOPE—A CAPTAIN'S MEMORIES OF SAMARLA—THE BAYONET
ou want to take on the _Delirium Trigger_?" shrieked Harkins.
Pinn choked on his food, spraying stew across the table and all over Crake's face. Malvery gleefully pounded Pinn on the back, much harder than was necessary, until his coughing fit subsided.
"Thanks," he snarled at the grinning doctor.
"Another day, another life saved," Malvery replied, returning to his position by the stove, where he was working on an artery-clogging dessert made mostly of sugar. Crake dabbed at his beard with a pocket handkerchief.
"So?" prompted Jez. "How do you plan to do it?" Frey surveyed his crew, gathered around the table in the _Ketty Jay_ 's mess hall, and wondered again if he was doing the right thing. His plan had seemed inspired when he came up with it a few hours ago, but now that he was faced with the reality of his situation, he was much less certain. It was fine to imagine a crack squad of experts carrying out their assigned missions with clinical precision, but it was hardly a well-oiled machine he was dealing with here.
There was Harkins, reduced to a gibbering wreck by the mere mention of the _Delirium Trigger_. Malvery, lacing the dessert with rum and taking a couple of swigs for himself as he did so. Pinn, too stupid to even swallow his food properly.
Jez and Crake were trustworthy, as far as he could tell, but they'd barely been able to meet each other's eyes throughout the meal. Something had happened between them at the Winter Ball—perhaps Crake had made an unwelcome move?—and now Jez's loathing for him was obvious, as was his shame.
That left Silo, silently spooning stew into his mouth, unknowable as always. Silo, who had been Frey's constant companion for seven years, about whom he knew nothing. Frey had never asked about his past, because he didn't care. Silo never asked about anything. He was just there. Did he _have_ thoughts like normal men did?
Frey tried to summon up some warm feelings of camaraderie and couldn't.
_Oh, well, damn it all, let's go for it anyway_.
"We all know we can't take on the _Delirium Trigger_ in the air," he said, to an audible sigh of relief from Harkins. "So what we do is we get her on the _ground_. We lure Dracken into port, and when she's down..." He slapped the table. "That's when we do it."
Pinn raised a hand. When Pinn raised a hand, it was only for effect. If he had something to say, he usually just blurted it out.
"Question," he said. "Why?"
"Because she won't be expecting it."
Pinn lowered his hand halfway, then raised it again as if struck by a new idea. "Yes?" Frey said wearily.
"Why don't we do something _else_ she isn't expecting?"
"I liked the running-away plan," said Harkins. "I mean, we've been doing pretty good so far with the running away. Maybe we should, you know, keep on doing it. Just an idea, though, I mean, you're the Cap'n. Only seems to me that, well, if it ain't broke, it doesn't need fixing. Just my opinion. You're the Cap'n. Sir."
The crew fell silent. The only sounds were Malvery quietly stirring the pot and a wet chewing noise coming from the corner of the mess, where Slag was tucking in to a fresh rat. He'd dragged it all the way up from the cargo hold in order to join the crew's dinner.
Frey looked at the faces turned toward him and felt something unfamiliar, a strange weight to the moment. He realized with a shock that they were waiting for him to persuade them. They _wanted_ to be persuaded. In their eyes, he saw the faintest hint of something he'd never thought to see from them. Something he was accustomed to seeing only in the expressions of beautiful girls just before he left them.
Hope.
_Rot and damnation, they're_ hoping! _They're hoping I can save them. They're hoping I know what I'm doing_.
And Frey was surprised to realize he felt a little bit good about that.
"Listen," he said. "Dracken's been catching up with us ever since she set out to get us. She was behind us when she got to Quail, she almost had us at the hermitage, and she was even _ahead_ of us at the Winter Ball. She knows that we know about Gallian Thade, and she'll assume we'll keep on after him. But what she _doesn't_ know is that we know about their secret hideout."
"We don't even know what this secret hideout _is,_ " Jez pointed out.
Pinn looked bewildered. He wasn't sure who knew what anymore.
"Crake heard Thade and the Duke talking about Trinica and some secret hideout," Frey said. "They mentioned charts and a device of some kind. Seems to me that if we get those charts and that device, then we can find our way there too."
Pinn raised his hand. "Question."
"Yes?"
"Why?"
"Because we need proof. We know Duke Grephen arranged Hengar's murder. We know he's planning a coup. But we don't have any way to prove it. If we can prove it, we can shop those bastards to the Archduke."
"What good will that do?" Jez asked. "We still blew up the _Ace of Skulls._ "
"You think they're going to care about the triggerman if they've got the mastermind?" Frey asked. "Look, I'm not saying they'll necessarily forgive, but they might forget. If the Archduke gets his hands on _them_ , he won't worry about _us_. We're small time. And without Duke Grephen putting up that huge reward, Dracken's not going to waste her time chasing us either."
"You think we can actually get ourselves out of this?" Malvery rumbled. He was standing behind Frey, at the stove. He'd stopped stirring and was staring at the pot of dessert.
"Yes!" Frey said firmly. "We play this right, we can do it."
The crew was exchanging glances, as if looking for support from one another. Did their companions feel the same? Were they being foolish to believe that they could win out against all the odds?
"Whatever's going on at this hideout is something to do with all of this," Frey said. "The answers are there, I'm sure of it. There's a way out. But we need to hang on, we need to go a little deeper first. We need to take the risk. Because I'm not spending the rest of my life on the run, and neither are any of you."
"You said we'll lure Dracken into a port," said Jez. "How are we gonna do that?"
"Parley," he said. "I'll invite her to talk on neutral turf. Face-to-face. I'll pretend I want to cut a deal."
"And you think she'll agree?"
"She'll agree." Frey was horribly certain of that.
Nobody spoke for a few moments. Slag looked up, puzzled by the pregnant pause in the conversation, then went back to snacking on his rat.
Frey felt the weight of Malvery's hand on his shoulder. "Tell us the rest of the plan, Cap'n."
FREY STEPPED OFF THE iron ladder that ran from the mess up to the main passageway of the _Ketty Jay_. He stopped there for a moment and took a breath. Explaining his plan had been unusually nerve-racking. For the first time he could remember, he'd actually worried about what his crew thought. There were a few good suggestions, mostly from Jez. Outright shock as he revealed the final part. But they'd liked it. He saw it on their faces.
Well, it was done. Until now he hadn't really been sure they'd go with it. It was frightening to have it all seem so suddenly real.
Because he really, really didn't want a meeting with Trinica Dracken.
Slag scampered up the ladder behind him and thumped down into the passageway, obviously in the mood for some company. He followed Frey into the captain's quarters and waited while Frey shut the door and dug out the small bottle of Shine from the locked drawer in the cabinet. He sat patiently while Frey administered a drop to each eye and lay back on the bed. Then, once he'd determined that Frey was liable to be motionless for a while, he hopped onto the captain's chest, curled up, and fell asleep.
Frey drifted on the edge of consciousness, dimly aware of the warm, crushing weight of the cat on his ribs. He was scared of what was to come. He hated being forced into this position. He hated having to be brave. But in the soothing narcotic haze he felt nothing but peace, and gradually he fell asleep.
In seeking to block out one thing he'd rather forget, he ended up dreaming of another.
THE NORTHWESTERN COAST OF Samarla was a beautiful place. The plunging valleys and majestic mountains were kept lush and green by frequent rains off Silver Bay, and the sun shone all year round this close to the equator. It was a land of sweeping vistas, mighty rivers, and uncountable trees, all green and gold and red.
It was also swarming with Sammies. Or, to be more accurate, it was swarming with their Dakkadian and Murthian troops. Sammies didn't dirty themselves with hand-to-hand combat. They had two whole races of slaves to do that kind of thing.
Frey looked down from the cockpit of the _Ketty Jay_ at the verdant swells beneath him. His navigator, Rabby, was squeezed up close, peering about for landmarks by which to calculate their position. He was a scrawny sort with a chicken neck and a ponytail. Frey didn't much like him, but he didn't have much choice in the matter. The Coalition Navy had commandeered his craft and his services, and since the rest of his crew had deserted rather than fight the Sammies, the Navy had assigned him a new one.
"They're sitting pretty, ain't they? Bloody Sammies," Rabby muttered. "Wish _we_ had two sets of bitches to do our fighting for us."
Frey ignored him. Rabby was always fishing for someone to agree with, constantly probing to find the crew's likes and dislikes so he could marvel at how similar their opinions were.
"I mean, you've got your Murthians, right, to do all your hard labor and stuff. Big strong lot for hauling all those bricks around and working in the factories and what. Good cannon fodder too, if you don't mind the surly buggers trying to mutiny all the time."
Frey reached into the footwell of the cockpit and pulled out a near-empty bottle of rum. He took a long swig. Rabby eyed the booze thirstily. Frey pretended not to notice and put it back.
"And then you've got your Dakkadians," Rabby babbled on, "who are even worse, 'cause they bloody _like_ being slaves! They've, what do you say, _assistimated._ "
"Assimilated," said Frey, before he could stop himself.
"Assimilated," Rabby agreed. "You always know the right word, Cap'n. I bet you read a lot. Do you read a lot? I like to read too."
Frey kept his eyes fixed on the landscape. Rabby coughed and went on.
"So these Dakkadians, they're all dealing with the day-today stuff, administration or what, and flying the planes and commanding all the dumb grunt Murthians. Then what do the actual Sammies do, eh?" He waited for a response, which wasn't going to come. "Sit around eating grapes and fanning their arses, that's what! Calling the bloody shots and not doing a lick of work. They've got it sweet, they have. Really sweet."
"Can you just tell me where I'm setting down, and we can get this over with?"
"Right you are, right you are," Rabby said hastily, scanning the ground. Suddenly he pointed. "Drop point is a few kloms south of there."
Frey looked in the direction that he was pointing and saw a ruined temple complex in the distance. The central ziggurat of red stone had caved in on one side, and the surrounding dwellings, once grand, had been flattened into rubble by bombs.
"How _many_ kloms?"
"We'll see it," Rabby assured him.
Frey took another hit from the rum.
"Can I have some of that?" Rabby asked.
"No."
They came in over the landing zone not long afterward. The hilltop was bald, and where there used to be fields there were now earthworks, with narrow trenches running behind them. Battered stone buildings clustered at the crest of the hill. It was a tiny village, with simple houses built in the low, flat-topped style common in these parts. The trees and grass glistened and steamed as the morning rain evaporated under the fierce sun.
Nothing moved on the hilltop.
Frey slowed the _Ketty Jay_ to a hover. He was surly drunk, and his first reaction was disgust. Couldn't the Coalition even organize someone to meet their own supply craft? Did they _want_ to run out of ammo? Did they think he _enjoyed_ hauling himself all over enemy territory, risking enemy patrols, just so they could eat?
Martley, the engineer, came bounding up the passageway from the engine room and into the cockpit. "Are we there?" he asked eagerly. He was a wiry young carrottop, his cheeks and dungarees permanently smeared in grease as if it was combat camouflage. He had too much energy, that was his problem. He wore Frey out.
Rabby examined the earthworks uncertainly. "Looks deserted, Cap'n."
"These _are_ the right coordinates?"
"Hey!" Rabby sounded offended. "Have I ever failed to get us to our target?"
"I suppose we usually get there in the end," Frey conceded.
"Did the Navy tell us anything about this place?" Martley chirped. "Like maybe why it's so deserted?"
"It's only a drop point," Frey said impatiently. "Like all the others."
Frey hadn't asked. He never asked. Over the past few months, Frey had simply taken whichever jobs paid the most. When the Navy began conscripting cargo haulers into minimum-wage service, the Merchant Guild responded by demanding danger bonuses. Those employed by the big cargo companies were happy to sit out the war ferrying supplies within the borders of Vardia. Freelancers like Frey saw an opportunity.
By taking the most dangerous missions, Frey had all but paid off the loan on the _Ketty Jay_. They'd had some close scrapes, and the crew complained like buggery and kept applying for transfers, but Frey couldn't have cared less. After seven years, she was almost his. That was all that counted. Once he had her, he'd be free. He could ride out the rest of the war doing shuttle runs between Thesk and Marduk, and he'd never again have to worry about the loan companies freezing his accounts and hunting him down. He'd be out on his own, a master of the skies.
"Let's just load out the cargo and get paid," he said. "If there's no one here to collect, that's not our problem."
"You certain?" said Martley uncertainly.
"If there's been a screwup here, it's someone else's fault," said Frey. He took another swig of rum. "We're paid to deliver to the coordinates they give us. We're not paid to think. They've told us _that_ enough times."
"Bloody Navy," Rabby muttered.
Frey lowered the _Ketty Jay_ down onto a relatively unscarred patch of land next to the village. Impatient and drunk, he dumped the aerium from the tanks too fast and slammed them down hard enough to jar his coccyx and knock Martley to his knees. Martley and Rabby exchanged a worried glance they thought he didn't see.
"Come on," he said, suppressing a wince as he got out of his seat. "Quicker we get unloaded, quicker we can go home."
Kenham and Jodd were down in the cargo hold when they arrived, disentangling the crates from their webbing. They were a pair of ugly bruisers, ex-dockworkers drafted in for labor by the Navy. The only people on the crew they respected were each other; everyone else was slightly scared of them.
Jodd was smoking a roll-up. Frey couldn't remember ever seeing him without a cigarette smoldering in his mouth, even when handling crates of live ammunition, as he was now. As captain, Frey made an executive decision to say nothing. Jodd had never blown them all to pieces before. With a track record like that, it seemed sensible to let it ride.
Frey lowered the cargo ramp and they began hauling the crates out. The sun hammered them as they emerged from the cool shadow of the _Ketty Jay_. The air was moist and smelled of wet clay, and there was a lingering scent of gunpowder.
"Where do you want 'em?" Kenham called to Frey. Frey vaguely waved at a clear spot some way downhill, close to the trenches. He didn't want those boxes of ammo too near the _Ketty Jay_ when he took off. Kenham rolled his eyes— _all the way over there?_ —but he didn't protest.
Frey leaned against the _Ketty Jay_ 's landing strut with the bottle of rum in his hand and watched the rest of his crew do the work. Since it took two men to a box, a fifth worker would only get in the way, he reasoned. Besides, it was captain's privilege to be lazy. He swigged from the bottle and surveyed the empty site. For the first time he noted that there were some signs of conflict: burn marks on the walls of the red stone houses; sections where the earthworks had been blasted and soil scattered.
Old wounds? This place had probably seen a lot of action. But then, there was that smell of gunpowder. Weapons had been fired, and recently.
He cast a bleary eye over his crew, to be sure they were getting on with their job, and then pushed off from the landing strut and wandered away from the _Ketty Jay_. He headed toward the village.
The houses were poor Samarlan peasant dwellings, bare and abandoned. Wooden chicken runs and pigpens had fallen into ruin. The windows were just square holes in the walls, some of them with their shutters hanging unevenly, drifting back and forth in the faint breeze. As Frey got closer, he could see more obvious signs of recent attacks. Some walls were riddled with bullet holes.
His skin began to prickle with sweat. He drained the last of the rum and tossed the bottle aside.
The dwellings were built around a central clearing that once had been grassy but was now churned into rapidly drying mud. Frey peered around the corner of the nearest house. Despite the racket from the forest birds, it was unnervingly quiet.
He looked through the window, into the house. The furniture had long gone, leaving a mean, bare shell, dense with hot shadow. The sun outside was so bright that it was hard to see. It took him a few seconds to spot the man in the corner.
He was slumped, motionless, beneath a window on the other side of the house. Frey could hear flies and smell blood.
By now his eyes had adjusted to the gloom. Enough to see that the man was dead, shot through the cheek, his jaw hanging askew and pasted onto his face with dried gore. Enough to see that he was wearing a Vardic uniform. Enough to see that he was one of theirs.
He heard a sound: sharp and hard, like someone stepping on a branch. The voices of his crew, suddenly raised in a clamor.
With a cold flood of nausea, he realized what was happening. Panic plunged in on him, and he bolted, running for the only safety he knew. Running for the _Ketty Jay_.
As he rounded the corner of the house, he saw Kenham lying facedown next to a sundered crate. Jodd was backing away from the trenches, firing his revolver at the men who were clambering out of them. Rifle-wielding Dakkadians: two dozen or more. Small, blond-haired, faces broad and eyes narrow. They'd hidden when they heard the _Ketty Jay_ approaching. Perhaps they'd even had time to throw the bodies of the dead Vards into the trenches. Now they were springing their ambush.
Rabby and Martley were fleeing headlong toward the _Ketty Jay_ , as Frey was. There was fear on their faces.
One of the Dakkadians fell back into the trench with a howl as Jodd scored a hit, but their numbers were overwhelming. Three others sighted and shot him dead.
Frey barely registered Jodd's fate. The world was a bouncing, jolting agony of moment after moment, each one bringing him a fraction closer to the gaping mouth of the _Ketty Jay_ 's cargo ramp. His only chance was to get inside. His only chance to live.
Dakkadian rifles cracked and snapped. Their targets were Rabby and Martley. Several of the soldiers had broken into a sprint, chasing after them. A shout went up in their native tongue as someone spotted Frey racing toward the _Ketty Jay_ from the far side. Frey didn't listen. He'd blocked out the rest of the world, tightened himself to a single purpose. Nothing else mattered but getting to that ramp.
Bullets chipped the turf around them. Martley stumbled and rolled hard, clutching his upper leg, screaming. Rabby hesitated, broke stride for the briefest moment, then ran on. The Dakkadians pulled Martley down as he tried to get up, then began stabbing him with the double-bladed bayonets on the end of their rifles. Martley's shrieks turned to gurgles.
The cargo ramp drew closer. Frey felt the sinister brush of air as a bullet barely missed his throat. Rabby was running up the hill, yelling as he came. Two Dakkadians were close behind him.
Frey's foot hit the ramp. He fled up to the top and pulled the lever to raise it. The hydraulic struts hummed into life.
Outside, he heard Rabby's voice. "Lower the ramp! Cap'n! Lower the bloody ramp!"
But Frey wasn't going to lower the ramp. Rabby was too far away. Rabby wasn't going to make it in time. Rabby wasn't getting anywhere near this aircraft with those soldiers hot on his heels.
"Cap'n!" he screamed. "Don't you leave me here!"
Frey tapped in the code that would lock the ramp, preventing it from being opened from the keypad on the outside. That done, he drew his revolver and aimed it at the steadily closing gap at the end of the ramp. He backed up until he bumped against one of the supply crates that hadn't yet been unloaded. The rectangle of burning sunlight shining through the gap thinned to a line.
_"Cap'n!"_
The line disappeared as the cargo ramp thumped closed, and Frey was alone in the quiet darkness of the cargo hold, safe in the cold metal womb of the _Ketty Jay_.
The Dakkadians had overrun this position. Navy intelligence had screwed up, and now his crew was dead. Those bastards! Those rotting bastards!
He turned to run, to race up the access stairs, through the passageway, into the cockpit. He was getting out of here.
He ran right into the bayonet of the Dakkadian who'd been creeping up behind him.
Pain exploded in his guts, shocking him, driving the breath from his lungs. He gaped at the soldier before him. A boy, no more than sixteen. Blond hair spilling out from beneath his cap. Blue eyes wide. He was trembling, almost as stunned as Frey.
Frey looked down at the twin blades of the Dakkadian bayonet, side by side, sticking out of his abdomen. Blood, black in the darkness, slid thinly along the blades and dripped to the floor.
The boy was scared. Hadn't meant to stab him. When he snuck aboard the _Ketty Jay_ , he probably thought only to capture a crewman for his fellows. He hadn't killed anyone before. He had that look.
As if in a trance, Frey raised his revolver and aimed it point-blank at the boy's chest. As if in a trance, the boy let him.
Frey squeezed the trigger. The bayonet was wrenched from his body as the boy fell backward. The pain sent him to the edge of unconsciousness, but no further.
He staggered through the cargo hold. Up the metal stairs, through the passageway, into the cockpit, leaving smears and dribbles of himself as he went. He slumped into the pilot's seat, barely aware of the sound of gunfire against the hull, and punched in the ignition code—the code that only he knew, that he'd never told anyone and never would. The aerium engines throbbed as the electromagnets pulverized refined aerium into gas, filling the ballast tanks. The soldiers and their guns fell away as the _Ketty Jay_ lifted into the sky.
Frey would never make it back to Vardia. He was going to die. He knew that and accepted it with a strange and awful calm.
But he wasn't dead yet.
He hit the thrusters, and the _Ketty Jay_ flew. North, toward the coast, toward the sea.
# _Chapter Twenty-two_
SHARKA'S DEN—TWO CAPTAINS—A STRANGE DELIVERY—RECRIMINATIONS
he slums of Rabban were not somewhere a casual traveler would stray. Bomb-lashed and tumbledown, they were a mass of junk pits and rubble fields, where naked girders slit the low sunset and the coastal wind smoothed a ceiling of iron-gray cloud over all. In the distance were new spires and domes, some of them still partially scaffolded: evidence of the reconstruction of the city. But here on the edges, there was no such reconstruction, and the population lived like rats on the debris of war.
Sharka's Den had survived two wars and would likely survive two more. Hidden in an underground bunker, accessible only by tortuous, crumbling alleys and an equally tortuous process of recommendation, it was the best place in the city to find a game of Rake. Sharka paid no commission to any guild nor any tax to the Coalition. He offered a guarantee of safety and anonymity to his patrons and promised fairness at his tables. Nobody knew exactly what else Sharka was into, to make the bigwigs so afraid of him, but they knew that if you wanted a straight game for the best stakes, you came to Sharka's Den.
Frey knew this place well. He'd once picked up a Caybery Firecrow in a game here, on the tail end of a ridiculous winning streak that had nothing to do with skill and everything to do with luck. He'd also wiped himself out several times. As he stepped into the den, memories of triumph and despair sidled up to greet him.
Little had changed. There was the expansive floor with its many tables and barely lit bar. There were the seductive serving girls, chosen for their looks but well schooled in their art. Gas lanterns hung from the ceiling, run off a private supply (Sharka refused to go electric; his patrons wouldn't stand for it). The myopic haze of cigarettes and cigars infused the air with a dozen kinds of burning leaf.
Frey felt a twinge of nostalgia. If he didn't count the _Ketty Jay_ , Sharka's Den was the closest thing to a home he had.
Sharka came over to greet him as he descended the iron steps to the gaming floor. Whip-lean, his face deeply lined, he was dressed in an eccentric motley of colors, and his eyes were bright and slightly manic. There was never a time when Sharka wasn't on some kind of drug, usually to counteract the one before. He was overly animated, his face stretching and contorting into grins, smiles, exaggerated poses, as if he were mouthing words to somebody deaf.
"Got you a private room in the back," he said. "She's in there now."
"Thanks."
"You think she was followed?"
"No. I was hiding out there awhile. I watched her go in, checked all the alleys nearby. She came alone."
Sharka grunted and then beamed. "Hope you know what you're doing."
"I always know what I'm doing," Frey lied, slapping Sharka on the shoulder.
Sharka was as much a survivor as his den was. Since the age of fifteen he'd pounded his body with every kind of narcotic Frey had ever heard of, yet somehow he'd made it to fifty-six, and there was no reason to suppose he didn't have thirty more years left. The man's blood must have been toxic by now, but he was tough as a scorpion. You just couldn't kill him.
"Well, I'll leave you to it. You can find your way, eh? Come see me after; I'll make sure you get an escort to wherever you need. Can't have Dracken's men jumping you on the way out."
Perhaps the stress of what was to come had made him over-emotional, but Frey was deeply touched by that. Sharka was a dangerous man, but he had a heart of gold, and Frey felt suddenly unworthy of his kindness. Even if he didn't exactly trust him, it was nice to know that _someone_ didn't want him dead.
"I'm grateful for what you've done, Sharka," he said. "I owe you big."
"Ah, you don't owe me anything," Sharka said. "I like you, Frey. You lose more than you win and you tip big when you score. You don't piss anyone off and you don't re-raise when you're holding dirt and then catch a run on your last card. This place is full of cocky kids with money and old hacks playing percentages. Could do with more players like you at my joint."
Frey smiled at that. He nodded his thanks again and then headed through the tables toward the back rooms. Sharka was a good sort, he told himself. Sharka wouldn't sell him out for the reward on his head. Everyone knew that Sharka's was neutral ground. He'd lose more in custom than he'd gain by the reward if there was the slightest suspicion that he'd turned in a wanted man. Half the people here were wanted by someone.
A serving girl in an appealingly low-cut dress met him at the back rooms and directed him to one of the private gaming areas. Sharka's was all bare brick and brass—not pretty, but Rake players distrusted glitz.
He stepped into a small, dim room. A lantern hung from the ceiling, throwing light onto the black baize of the Rake table. A pack of cards was spread out in suits across it. A well-stocked drinks cabinet rested against one wall. There were four chairs around the table.
Sitting in one of the chairs, facing the door, was Trinica Dracken.
The sight of her was a jolt. She was lounging in the chair, small and slim, dressed head to toe in black: black boots, black coat, black gloves, black waistcoat. But from the buttoned collar of her black shirt upward, everything changed. Her skin was powdered ghost-white. Her hair—so blond it was almost albino—was cut short, sticking up in uneven tufts as if it had been butchered with a knife. Her lips were a red deep enough to be vulgar.
But it was her eyes that shocked him most. Her lashes were almost invisible, but her irises were completely black, dilated to the size of coins. It took him a moment to realize they were contact lenses and not the product of some daemonic possession. Worn for effect, no doubt, but certainly effective.
"Hello, Frey," she said. Her voice was lower than he remembered. "Long time."
"You look terrible," he said as he sat.
"So do you," she replied. "Life on the run must not agree with you."
"Actually, I'm getting to enjoy it. Catching my second wind, so to speak."
She looked around the room. "A Rake den? You haven't changed."
"You have."
"I had to."
He gestured at the cards on the table between them. "Want to play?"
"I'm here to parley, Frey, not play your little game."
Frey sat back in his chair and regarded her. "Alright," he said, "Business it is. You know, there was a time when you liked to sit and talk for hours."
"That was then," she said. "This is now. I'm not the person you remember."
That was an understatement. The woman before him was one of the most notorious freebooters in Vardia. She'd engineered a mutiny to become captain of the _Delirium Trigger_ , and her reputation for utter ruthlessness had earned her the respect of the underworld. Rumor held her responsible for acts of bloody piracy and murder as well as daring treasure snatches and near-impossible feats of navigation. She was feared by some and envied by others, a dread queen of the skies.
Hard to believe he'd almost married her.
RABBAN WAS ONE OF the nine primary cities of Vardia, and, like the others, it bore the same name as the duchy it dominated. Though it had suffered terribly in the Aerium Wars, it was still large enough to need more than a dozen docks for aircraft. These docks were the first things to be repaired after the bombing stopped six years ago. Some were little more than islands in a sea of shattered stone, but even these were busy with passenger shuttles, cargo haulers, and supply vessels. Transport by air had been Vardia's only viable option for more than a century and, even in the aftermath of a disaster, there was no way to do without it.
Only a few of the docks, however, were equipped to deal with a craft the size of the _Delirium Trigger_.
She rested inside a vast iron hangar, alongside frigates and freighters: the heavyweights of the skies. A web of platforms, gantries, and walkways surrounded it at deck height, busy with an ant swarm of engineers, dockworkers, and swabbers. Everything was being checked, everything cleaned, and a complex exchange of services and trade goods was negotiated. A craft like the _Delirium Trigger_ , with a crew of fifty, needed a lot of maintenance.
The _Delirium Trigger_ 's purser was a Free Dakkadian named Ominda Rilk. He had the fair skin and hair typical of his race, the small frame and narrow shoulders, and the squinting eyes that still elicited much mockery in the Vardic press. Dakkadians were famed and ridiculed for their administrative abilities. Education and numeracy were much prized among their kind: it made them useful to their Samarlan masters. But Dakkadians, unlike Murthians, could own possessions, and they could earn their freedom.
It was unusual to find a Dakkadian in Vardia, where there was still much bad feeling toward them after the Aerium Wars. They were seen as persnickety coin-counters and misers by the more-generous souls; the rest thought they were cunning, underhanded, murdering bastards. But still, here was Ominda Rilk. He stood among the crates and pallets waiting to be loaded onto the _Delirium Trigger_ , examining everything and making small notes in his logbook now and again. And his squinty eyes were keen enough to spot two men transporting a very heavy-looking crate in a manner that was frankly quite surreptitious.
"Ho, there!" he cried. The men stopped, and he walked briskly over to them. They were dockworkers, dressed in battered gray overalls. One was large and big-bellied, with a whiskery white mustache; the other was short, stumpy, and ugly, with oversize cheeks and a small thatch of black hair perched atop a small head. They were both flushed and sweating.
"What's this?" he asked, motioning at the crate. It was nine feet tall and six wide, and they'd been rolling it along on a wheeled pallet toward the loading area, where a crane picked up supplies for transport to the deck of the _Delirium Trigger_.
"Don't know," said Malvery, with a shrug. "We just deliver, don't we?"
"Well, who's it from?" snapped Rilk. "Where are the papers? Come on!"
Malvery drew out a battered, folded-up set of papers. Rilk shook them open and checked the delivery invoice. His eyebrows raised a fraction when he read the name of the sender. _Gallian Thade_.
"We weren't expecting this," he said, handing back the papers with a scowl.
Malvery gave him a blank look. "We just deliver," he said again. "This box goes on the _Delirium Trigger._ "
Rilk glared at him and then at Pinn. There was something not right about these two, but he couldn't put his finger on it. Pinn looked back at him mutely.
"Does he speak?" Rilk demanded, thumbing at Pinn.
"Not much," Malvery replied. At least, he'd been told to keep his trap shut, for fear he'd say something stupid and ruin their disguise. Malvery hoped he'd implied enough threat to keep the young pilot in line. "You want us to load this thing on or what?"
Rilk studied the crate for a moment. Then he snapped his fingers. "Open it up."
Malvery groaned. "Aw, come on, don't be a—"
"Open it _up_!" Rilk said, snapping his fingers again in a rather annoying fashion that made Malvery want to break them and then stuff his mangled hand down his throat.
The doctor shrugged and looked at Pinn. "Open it up," he said.
Pinn produced a crowbar. The crate had been nailed shut, but they forced open a gap in the front side with relative ease, then pulled it the rest of the way with brute strength. It fell forward and clattered to the ground.
Rilk stared at the hulking, armored shape inside the box. A monstrosity of metal and leather and chain mail, with a humped back and a circular grille set low between the shoulders. It was cold and silent.
"What _is_ it?" he asked.
Malvery pondered for a moment, studying Bess. "I reckon it's one of those pressure-environment-suit thingies."
Rilk looked it up and down, a puzzled frown on his face. "What does it do?"
"Well, you wear it when you want to work on the deck, see. Like, in arctic environments, or when your craft is really, really high in the sky."
"It's cold as a zombie's tit up there, and the air's too thin to breathe," Pinn added, unable to resist joining in. Malvery silenced him with a glare.
"I see," said Rilk, examining Pinn. "And how is it a dock-worker knows a thing like that?"
Pinn looked lost. "I just do."
"Lot of pilots come to the dockside bars," Malvery said with forced offhandedness. "People talk."
"Yes, they do," said Rilk. He walked up to Bess, put his face to her face grille, and peered inside. "Hello?" he called. The word echoed in the hollow interior.
"He thinks there's somebody in there." Malvery grinned at Pinn, giving him a nudge. Pinn chuckled on cue. Rilk withdrew, his pale face reddening.
"Box it up and load it on!" he snapped, then made a quick note in his logbook and stalked away.
"WHY DID YOU BRING me here, Darian?" asked Trinica Dracken.
"Why did you come?" he countered.
She smiled coldly in the light of the lantern overhead. "Blowing you out of the sky after all this time seemed a little... impersonal," she replied. "I wanted to see you. I wanted to look you in the eye."
"I wanted to see you too," said Frey. He'd scooped up the cards that were laid out on the table.
"You're a liar. I'm the last person you ever wanted to see again."
Frey looked down at the cards and began to shuffle them restlessly.
"I had people watching you," said Trinica. "Did you know that? After you left me."
He was faintly chilled. "I didn't know that."
"The day after our wedding day, I had the Shacklemores looking for you."
"It wasn't our wedding day," said Frey, "because there wasn't a wedding."
"A thousand people turned up thinking otherwise," said Trinica. "Not to mention the bride. In fact, everyone seemed to think they were there for a magnificent wedding, right up until the moment the judge called for the groom." Her expression became comically sorrowful, a sad clown face. "And there was the poor bride, waiting in front of all those people." She blew a puff of air into her hand, opening it out as she did so. "But the groom had gone."
Frey was rather unnerved by her delivery. He'd expected shrill remonstrations, but she was utterly empty of emotion. She was talking as if it had happened to someone else. And those black, black eyes made her seem strangely fey and alien. A little frightening, even.
"What do you want, Trinica?" The words came out angrier than he intended. "An apology? It's a little late for that."
"Oh, that's most certainly true," she replied.
Frey settled back in his seat. The sight of her stirred up all the old feelings. Bad feelings. He'd loved this woman once, back when she was sweet and pretty and perfect. Loved her in a way he'd never loved anyone since. But then he'd broken her heart. In return, she'd ripped his to pieces. He could never forget what she'd done to him. He could never forgive her.
But an argument would do him no good now. He couldn't take the risk that Trinica would storm out. The object of this meeting was to keep her here as long as possible, to let his men do their job on the _Delirium Trigger_.
He cleared his throat and strove to control the bitterness in his voice. "So," he said. "You set the Shacklemores on me." He began cutting the cards and reshuffling them absently.
"You were a hard man to find," she said. "It took them six months. By then... well, you know what had happened by then."
Frey's throat tightened. Rage or grief, he wasn't sure.
"They came back and said they'd found you. You were doing freelance work somewhere on the other side of Vardia at the time. Using what you'd learned from working as a hauler for my father's company, I suppose. Making your own deals."
"It was a living," said Frey neutrally.
She gave him a faint, distracted smile. "They asked me if I wanted them to bring you back. I didn't. Not then. I asked them instead to let you know—discreetly—how I was doing. I was sure you hadn't troubled to inquire."
Frey remembered that meeting well. A stranger in a bar, a shared drink. Casually mentioning that he worked for Dracken Industries. Terrible what had happened to the daughter. Just terrible.
But Trinica was wrong. He _had_ inquired. By then he'd already known what she'd done.
Memories overwhelmed him. Searing love and bilious hate. The stranger before him was a mockery of the young woman he'd almost married. He'd kissed those lips, those whore-red lips that now smiled at him cruelly. He'd heard the softest words pass from them to him.
Ten years. He'd thought that everything would be long ago buried by now. He'd been badly mistaken.
"It didn't seem fair, really," Trinica said, tilting her head like a bird. There was a childish look on her face that said: _Poor Frey. Poor, poor Frey_. "It didn't seem fair that you should be able to turn your back and walk away like that. That you could leave your bride on her wedding day and never have to think about what you'd done, never take any responsibility."
"I _wasn't_ responsible!"
She leaned forward on the card table, deadly serious, those awful black eyes staring out of her white face. "Yes," she said, "you were."
Frey dashed the cards across the table, but his fury died as soon as it had come. He sat back in his chair, his arms folded. He wanted to argue, but he needed to keep things calm. Keep things together.
_Don't let this bitch get to you. Play for time_.
"You had the Shacklemores keep track of me after that?" he asked. Trinica nodded. "Why the interest?"
"I just forgot to call them off."
"Oh, come on."
"It's true. At first, I'll admit, I wanted to see what effect my news would have on you. I wanted to see if you suffered. But then... well, I left home, and other things got in the way. It was only years later that I realized they'd been keeping the file open on you all that time, drawing a fee every month. My father was paying for them, you see. When you've that much money, it's easy to forget about something like that."
"You know I joined the Navy, then?" he said.
"I know they _conscripted_ you when the Second Aerium War began," she said. "And I know you were drinking too much, and you started taking all the most dangerous jobs. I know nobody wanted to fly with you because it was only a matter of time before you self-destructed."
"You must have enjoyed hearing all about that."
"I did, yes," she replied brightly. "But I didn't find out until after you had disappeared."
Frey didn't say anything.
"They tell me the position was overrun by Samarlan troops. My guess is, you landed there and they ambushed you. What happened to the rest of the crew?"
"Dead."
"Naturally."
"Navy intelligence," Frey sneered. "Bunch of incompetent bastards. They sent us out there and the Sammies were waiting."
Trinica laughed: the sound was sharp and brittle. "Same old Darian. Picked on by the world. Nothing's ever your fault, is it?"
"How was it _my_ fault?" he cried. "I landed in a war zone because of information _they_ gave me."
Trinica sighed patiently. "It was a war, Darian. Mistakes happen all the time. You landed in a war zone because you had been flying the most dangerous frontline missions for months. You never used to ask questions; you just took the missions and flew. It was a miracle it didn't happen sooner."
"It was the best chance I had to pay off the loan on the _Ketty Jay,_ " he protested, but it sounded weak even to him. He couldn't forget the desperate tone in Rabby's voice as he closed the cargo ramp. _Don't you leave me here!_
"If you wanted to die, why didn't you kill yourself?" Trinica asked. "Why try to take everyone else with you?"
"I never wanted to die!"
Trinica just looked at him. After a moment she shrugged. "Well, evidently you didn't want it enough, since here you are. Everyone thought you were gone. The Shacklemores closed the file. The loan company wrote off the rest of your repayments on the _Ketty Jay_. And off you went, a corpse to all intents and purposes.
Until one day... one day I hear your name again, Darian. Seems you're alive, and everyone's looking for you. And I just had to throw my hat in the ring."
"You just had to, huh?" Frey said scathingly.
Trinica's demeanor went from casual to freezing in an instant. "That day you disappeared, you cheated me. I thought I'd never get to make you pay. But you're alive, and that's good. That's a wonderful thing." She smiled, the chill smile of a predator, her black eyes glittering like a snake watching a mouse. "Because now I'm going to catch you, my wayward love, and I'm going to watch you hang."
# _Chapter Twenty-three_
BARRICADES—BESS AWAKES—A LESSON IN CARDPLAY—THE MONSTER BELOWDECKS—THIEVES
he _Ketty Jay_ was berthed at a small dock in the outskirts of Rabban, far from the _Delirium Trigger_. The dock was little more than a barely used landing pad set above a maze of shattered and leaning alleyways. Only a few other craft of similar size shared the space. They sat dark and silent, their crews nowhere to be seen. A few dock personnel wandered around, looking for something to do, their presence revealed by a cough or a slow movement in the shadows. All was quiet.
Silo and Jez worked in the white glare of the _Ketty Jay_ 's belly lights, rolling barrels from the cargo hold and manhandling them into rows of five. There were several such rows positioned around the _Ketty Jay_. A haphazard kind of arrangement, an observer might think, unless they guessed what the barrels were really for.
They were building barricades.
Harkins was skirting the edge of the landing pad, scampering along in a crouch, a spyglass in his hand. He stayed out of the light of the electric lampposts that marked out the landing pad for flying traffic. Every so often he'd stop and scan the surrounding alleyways, then run off in a nervous fashion to another location and do it again. The dock personnel paid him no mind. As long as his captain paid the berthing fee, they were happy to tolerate eccentrics.
The night was still new when Harkins straightened, his whole body frozen in alarm. He adjusted his spyglass, shifted it this way and that, counting frantically under his breath. Then he fled back toward the _Ketty Jay_ as if his heels were on fire.
"Here we go," said Jez, as she saw him coming. Silo grunted and levered another barrel of sand into place.
"There's _twenty_ of 'em!" Harkins reported in a quiet shriek. "I mean, give or take a couple, but twenty's near enough! What are we supposed to do against twenty? Or even _nearly_ twenty. _Ten_ would be too many! What's he expect us to do? I don't like this. Not one measly rotting bit!"
Jez studied him, worried. He was even more strung out than usual. The Firecrow and Skylance were not even in the city. They'd been stashed at a rendezvous point far away. Without his craft, he was a snail out of its shell.
"We do what the Cap'n told us to do," she said calmly.
"But we didn't know there'd be _twenty_! That's almost half the crew!"
"I suppose Dracken doesn't want to leave anything to chance," said Jez. She exchanged a glance with Silo, who headed up the cargo ramp and into the _Ketty Jay_.
Harkins watched him go, then turned to Jez with a slightly manic sheen in his eyes. "Here, _that's_ an idea! Why don't we go inside, close up the cargo hold, and lock it? They'd never get in then."
"You don't think they've thought of that? They'll have explosives. Either that or someone who knows how to crack open and rewire a keypad." She motioned toward the small rectangle of buttons nested in the nearby landing strut, used to close and open the cargo ramp from the outside.
The belly lights of the _Ketty Jay_ went out, plunging them into twilight. The barely adequate glow of the lampposts gave a soft, eerie cast to the near-empty dock. Silo emerged carrying an armful of guns and ammo.
Jez gave Harkins a reassuring pat on the arm. He looked ready to bolt. "Twenty men here means twenty less for the others to deal with," she said. "The Cap'n said Dracken would be coming for us. We're ready for it. We just have to hold out, that's all."
"Oh, just that!" Harkins moaned with hysterical sarcasm. But then Silo grabbed his hand and slapped a pistol into his palm, and the glare the Murthian gave him was enough to shut him up.
MALVERY AND PINN REJOINED Crake, who was waiting at a safe remove from the _Delirium Trigger_ with a worried frown on his brow. Together, they watched Bess being loaded on. The arm of the crane was chained to the four corners of one great pallet, on which were secured dozens of crates. It lifted the pallet onto the deck of the _Delirium Trigger_. From there, Dracken's crew carried the crates to a winch, which lowered them through an opening into the cargo hold. Dockers were not allowed aboard. Dracken was wise to the dangers of infiltration that way.
"I don't like this," Crake said to himself for the tenth time.
"She'll be fine," said Malvery, looking at his pocket watch.
"And if she's not," said Pinn, "you can always build a _new_ girlfriend."
Malvery clipped him around the head. Pinn swore loudly.
"She'll be fine," Malvery said again.
Pinn fidgeted and adjusted his genitals inside his trousers. He was dressed in dockworker's overalls, as were his companions, with his regular clothes beneath them. It would be necessary to change in a hurry later. Until then, exertion and multiple layers had left him sweltering. "When can we get on with it? My pods are dripping."
The others ignored him. He smoked a roll-up resentfully as they observed the activity aboard. The pallet, once empty, was lifted off the _Delirium Trigger_ by the crane and returned to the elevated hangar deck, where more crates were loaded on.
"Right-o," said Malvery. "Let's head down there. Crake, keep your mouth shut. Nobody's gonna believe you're a docker with that accent. Pinn... just keep your mouth shut."
Pinn made a face and spat on the ground.
"Now, the Cap'n wants this to go like clockwork," Malvery said. "We all know there's bugger all chance of that, so let's just try not to get ourselves killed, and we'll all be having a drink and a laugh about this by dawn."
They made their way back across the busy dock, weaving between piles of chests and netting and screeching machinery. Huge cogs turned; cage lifts rattled up and down from the lower hangar decks. Cranes swung overhead, and shouts echoed around the iron girders of the roof, where squadrons of pigeons roosted and shat. A massive freighter was easing in on the far side of the hangar, its aerium tanks keeping it weightless, nudging it into place with its gas jets.
Posing as dockworkers, the three impostors were invisible in the chaos. They picked some cargo from a stack of netted crates and barrels that were being loaded onto the _Delirium Trigger_ and made their way toward the huge pallet that was chained to the crane arm. The cargo had been piled high on the pallet by now. They carried their loads on and went around to the far side of the pallet, where they couldn't be seen by the workers on the dock. There, they began unlashing a group of crates, rearranging them to make a space.
Another docker rounded the corner, carrying a heavy-looking chest. Malvery, Pinn, and Crake did their best to look focused and industrious. The docker—a grizzled, burly man with salt-and-pepper hair—watched them in puzzlement for a moment, then decided that whatever they were doing wasn't interesting enough to comment on. He put down the chest, secured it with some netting, and left.
Once they'd dug out a space, they checked that the coast was clear and crammed in. Then they stacked their own crates in front of it, sealing themselves inside.
Their timing was perfect. No sooner had they hushed one another to silence than a steam whistle blew. They heard the footsteps of dockworkers beyond their hiding place evacuating the pallet, and then, with a lurch, it began to lift.
Malvery had to steady the unsecured crates in front of them for fear of being buried, but the crane moved slowly and the pallet was heavy enough to be stable. Though the crates made slight and distressing shifts, nothing moved far enough to fall. Tucked in their little corner, they felt themselves transported across the gap between the hangar deck and the deck of the _Delirium Trigger_.
Crake found himself thinking that this must be how a mouse felt. Hiding in the dark, at the mercy of the world, frightened by every unknown sound. Spit and blood, he hated this. He didn't have it in him to be a stowaway. He was too afraid of getting caught.
But Bess was aboard. He was committed now. He'd committed her.
_Why did you do it? Why did you agree to this?_
He agreed to it because he was ashamed. Because since their encounter with the man from the Shacklemore Agency, he couldn't look Jez in the eye. Absurdly, he felt he _owed_ her something. He felt he owed the crew. He needed to atone, to make amends for being such a despicable, vile monster. To apologize for his presence among them. To make himself worthy.
Anyway, it was too late to turn back now.
"We're nearly there," Malvery said. "Do it."
Crake drew out his small brass whistle. He put it to his lips and blew. It made no sound at all.
"That's it?" asked Pinn, bemused.
"That's it," said Crake.
"So now what happens?"
"Bess has just woken up to find that she's in a box," Crake replied. "I wouldn't want to be in the _Delirium Trigger_ 's cargo hold right now."
By the time the pallet bumped down onto the deck, the howling and smashing had begun.
"I SUPPOSE YOU KNOW I'm innocent, don't you?" Frey asked.
Trinica was pouring two glasses of whiskey from the drinks cabinet. She looked back at him: a moon-white face partially eclipsed by the black slope of her shoulder.
"You're not _innocent_ , Frey. You killed those people. It doesn't matter if you were set up or not."
"The _Ace of Skulls_ was rigged to blow. Those people were going to die anyway, with or without me."
" _Everyone_ is going to die, with or without you. It doesn't mean you're allowed to murder them."
She was needling him and he knew it. It enraged him. She always had a way of pricking at his conscience, puncturing his excuses. She never let him get away with anything.
"You were in on it, then?" he asked. "The plot?"
She handed him his whiskey and sat down again. The card table lay between them, the cards facedown where they'd been thrown by Frey. Skulls, Wings, Dukes, and Aces, all hidden in a jumble.
"No. I didn't set you up. I didn't know you were alive until I heard you were wanted."
"But you know now. You know that Duke Grephen is the man behind it all and that Gallian Thade is in on it too. You know they made me the scapegoat?"
She raised an eyebrow, blond against white. "My. You evidently think you've learned a lot. Was that your sucker punch? Should I be awed at how clever you've been?"
"A little awe would be nice, yes."
She sipped her whiskey. "I assume you're appealing to my better nature? Wondering how I could be part of such a terrible miscarriage of justice? How I could willingly let you take the blame for the death of Hengar when I know it was Grephen's idea?"
"That's about the size of it."
"Because Grephen is paying me a lot of money. And because, frankly, I'd do it for free. You deserve it."
"It doesn't concern you to be an accomplice to the murder of the Archduke's son? Don't you think there might be bigger implications involved?"
"Possibly there are," said Trinica. "But that's none of your concern, since it'll all be over for you very soon."
"Come on, Trinica. Hengar's death is only the start. You must know if Duke Grephen is planning something."
Trinica smiled. "Must I?"
Frey cursed her silently. She wasn't giving anything away. He wanted to push her for more information, but she wouldn't play the game. Telling her that he knew about Grephen was intended to lead her up the wrong path, but he couldn't reveal that he knew about the coup or her mysterious hideout. That would tip his hand.
"One question," he said. "The ferrotype. The one on the _Wanted_ posters. How did they get that, if you didn't give it to them?"
"Yes, I was surprised too," she said. "We had it taken when we were up in the mountains. Do you remember?"
Frey remembered. He remembered a time of romantic adventure, a couple newly in love. He was a lowly cargo pilot and she was the daughter of his boss, one of the heirs to Dracken Industries. He was poor and she was rich, and she loved him anyway. It was breathless, dangerous, and they were both swept giddily along, careless of consequences, armored by their own happiness.
"It was my father who gave it to them, I'd imagine," she said. "I suppose the Navy had no pictures of you, and they knew you had worked for Dracken Industries before that. They were probably hoping for a staff photograph."
"He kept _that_ one?"
"He kept it because I was in it. I imagine that's how he'd like to remember me."
The _Wanted_ posters had shown only Frey's face, but in the full picture, Trinica was clinging to his arm, laughing. Laughing at nothing, really. Laughing just to laugh. He remembered the ferrotype perfectly. Her hair blowing, mouth open and teeth white. A rare, perfect capture; a frozen instant of natural, unforced joy. No one would connect that young girl with the woman sitting in front of him.
In that moment, Frey felt the tragedy of that loss. How cruel it was that things had turned out the way they did.
But Trinica saw the expression on his face and correctly guessed its cause. She always knew his thoughts, better than anyone.
"Look at yourself, Darian. Cursing the fate that brought you here. One day you're going to realize that everything that's happened to you has been your own fault."
"Dogshit," he spat, sadness turning to venom in an instant. "I've tried my damnedest. I tried to better myself."
"And yet here you are, ten years later, barely scraping a living. And I am the captain of a crew of fifty, infamous and rich."
"I'm not like you, Trinica. I wasn't born with a silver spoon shoved up my arse. I didn't have a good education. Some of us don't get the luck."
She looked at him for a long moment. Then her black eyes dropped to the facedown cards scattered on the table.
"I remember when you used to talk about Rake," she said, idly picking up a card and flipping it over. It was the Lady of Crosses. "You used to say everyone thought luck was a huge factor. They said it was all about the cards you were dealt. Mostly luck and a bit of skill." She flipped over another: Ten of Fangs. "You thought they were idiots. You knew it was mostly skill and a bit of luck."
The Ace of Skulls came next. Frey hated that card.
"A good player might occasionally lose to a mediocre one, but in the long run, the good players made money while the bad ones went broke," Trinica continued.
The next card came up: the Duke of Skulls. Any Priest would give her a five-card run to the Ace of Skulls, an unbeatable combination.
She turned the final card: the Seven of Wings. The hand was busted. Her gaze flicked up from the table and met his.
"Over time, luck is hardly a factor at all," she said.
BELOWDECKS, THE DELIRIUM TRIGGER _was_ in chaos. A slow, steady pounding reverberated through the dim passageways. Metal screeched. Men shouted and ran, some toward the sound and some away from it.
"It's in the cargo hold!"
" _What's_ in the cargo hold?"
But nobody could answer that. Those inside the hold had fled in terror when the iron-and-leather monstrosity burst out of its crate and began rampaging through the shadowy aisles. Barrels were flung this way and that. Guns fired, but to no avail. The air had filled with splinters as the intruder smashed through crates of provisions and trade goods. It was dark down there, and the looming thing terrified the crewmen.
Those on the deck above, operating the winch, had peered fearfully through the hatch into the cargo hold at the first signs of a disturbance. The light from the hangar barely penetrated to the floor of the hold. They scrambled back as they caught a glimpse of something huge lunging across their narrow field of view. It was only then that one of them thought to raise the winch.
In the confusion that ensued, nobody noticed three strangers, now dressed in the dirty motley of crew members, making their way belowdecks.
Those who had managed to escape from the cargo hold had slammed the bulkhead door behind them and locked it shut, trapping the monster inside. But the monster didn't like being trapped. It was pounding on the inside of the door, hard enough to buckle eight inches of metal. Enraged bellows came from behind.
"Get your fat stinking carcasses over here!" the burly, dirt-streaked bosun yelled. The men he was yelling at had come to investigate the sound and were now backing away as they saw what was happening. They reluctantly returned at his command. "Weapons ready, all of you! You _will_ defend your craft!"
A rotary cannon on a tripod was being hastily erected in the passageway in front of the door. The bosun knelt down next to the crewman who was assembling the cannon. "When that thing comes through the door, give it everything you've got!"
MALVERY, CRAKE, AND PINN skirted the chaos as best they could, and for a time they were unmolested. The _Delirium Trigger_ was only half crewed, and almost all of them were occupied with the diversion Bess was creating. They did their best to avoid meeting anyone, and when they were seen it was usually at a distance or by somebody who was already hurrying elsewhere. They managed to penetrate some way into the aircraft before they came up against a crew member who got a good look at them and recognized them as impostors.
"Hey!" he said, before Malvery grabbed his head and smashed his skull against the wall of the passageway. He slumped to the floor, unconscious.
"Not big on talking your way out of things, are you?" Crake observed, as they dragged the unfortunate crewman into a side room.
"My way's quicker," he said, adjusting his round green glasses. "No danger of misunderstanding."
The side room was a galley, empty now, its stoves cold. Crake shut the door while Malvery ran some water into a tin cup. The crewman—a young, slack-jawed deckhand—began to groan and stir. Malvery threw the water in his face. His eyes opened and slowly focused on Pinn, who was standing over him, pointing a pistol at his nose.
Malvery squatted down next to the prisoner and tapped him on the head with the base of the tin cup, making him wince. "Captain's cabin," he said. "Where?"
THEY LEFT THE DECKHAND bound and gagged in a cupboard of the galley. Pinn was for shooting him, but Crake wouldn't allow it. Pinn's argument that he was "just a deckhand, no one would miss him" carried little weight.
The captain's cabin was locked, of course, but Crake had come prepared. Given the time and the materials, it was a simple trick for him to produce a daemonic skeleton key. He slipped it into the lock and concentrated, forming a mental chord in the silence of his mind, awakening the daemon thralled to the key. His fingers became numb as it sucked the strength from him. Though small, it was hungry and beyond the power of any but a trained daemonist to handle.
The daemon extended invisible tendrils of influence, feeling out the lock, caressing the levers and tumblers. Then the key turned sharply, and the door was open.
Malvery patted him on the shoulder and grinned. "Good job, mate." Crake felt oddly warmed by that. Then he heard the distant pounding echoing through the _Delirium Trigger_ , and he remembered Bess.
"Let's get this done," he said, and they went inside.
Dracken's cabin was spotlessly clean, but the combination of brass, iron, and dark wood gave it a heavy and oppressive feel. A bookshelf took up one wall, a mix of literature, biography, and navigational manuals interspersed with shiny copper ornaments. Some of the titles were in Samarlan script, Crake noticed. He spotted _The Singer and the Songbird_ and _On the Domination of Our Sphere_ , two great works by the Samarlan masters. He found himself taken by an unexpected admiration for a pirate who would—or even could—read that kind of material.
Pinn and Malvery had gone straight to the desk on the far side of the cabin, which sat next to a sloping window of reinforced windglass. The light from the hangar spilled onto neatly arranged charts and a valuable turtle-shell writing set. Crake had a sudden picture of Dracken looking thoughtfully out that window at a sea of clouds as her craft flew high in the sky.
Pinn pawed through the charts, scattering them about and ruining Crake's moment of reverie. "Nothing," he said.
Malvery's eye had fallen on a long, thin chest on a shelf near the desk. It was padlocked. "Crake!" he said, and the daemonist came over with his skeleton key. The lock was trickier than the one that secured the cabin door, but in the end, it couldn't stand up to the key.
It was full of rolled-up charts. Atop them was what seemed to be a large compass. Malvery passed the compass to Crake, then began scanning through the charts with Pinn. Crake listened to the booming coming from the depths of the _Delirium Trigger_ as he studied Malvery's discovery.
_Keep pounding, Bess_ , he thought. _As long as I hear you, I know you're alright_.
The compass was so big that Crake could barely hold it in one hand. It was also, on closer examination, not a compass at all. It had no north–south–west–east markings, and it had four needles instead of one, all of equal length and numbered. Additionally, there were eight tiny sets of digits, set in pairs, with each digit on a rotating cylinder to allow it to count from zero to nine. These set pairs were also numbered one to four, presumably to correspond with the needles. The needles were all pointing in the same direction, no matter which way he turned it, and the numbers were all at zero.
"I think we found 'em!" Malvery said. He scooped up all the charts from the chest and shoved them inside his threadbare jersey, then looked at Crake. "Is that the device you were after?"
"I believe it is."
Crake had little doubt that what he held was the mysterious device Thade had mentioned. The strangeness of the compass, and the fact that it had been placed in the same chest as the charts, was enough for him.
"We should—" he began, but then he saw a movement in the doorway, and there was the loud report of a gun.
Malvery had seen it too: one of the crew, a black-haired, scruffy man, drawn by the sound of voices and the sight of the captain's door left open. On seeing the intruders, the crewman hastily pulled his gun and fired. The doctor ducked aside, fast enough so that the bullet only grazed his shoulder.
Another gun fired, an instant after the first. Pinn's. The crewman gaped, and a bright swell of blood soaked out from his chest into his shirt. He staggered back and slid down the wall of the passageway outside, disbelief in his eyes.
"We got what we came for," said Malvery, his voice flat. "Time to go."
The crewman lay in the passageway, gasping for air. Pinn and Malvery passed him without looking at him, pausing only to steal his pistol. Crake edged by as if he was contagious, horrified and fascinated. The crewman's eyes followed his, rolling in their sockets with an awful, empty interest.
Crake found himself pinned by that gaze. It was the look of a man unprepared, shocked to find himself at the gates of death so swiftly and unexpectedly. There was bewilderment in that look. The dying man was crushed by the knowledge that, unlike every other desperate moment in his life, there was no second chance, no way that wit or strength could pull him clear. It filled Crake with a terror such as he'd never known before.
Now Crake knew why Malvery and Pinn hadn't looked.
He was trembling as he followed his companions up the corridor. After a moment, he remembered Bess. He put the whistle to his lips, the whistle tuned to a frequency that only she could hear, and he blew. It was a note different from the one he used to wake her up and put her to sleep. This one was a signal.
_Time to come back, Bess_.
"ANY MOMENT NOW, BOYS!" the bosun yelled, as the bulkhead door screeched and lurched forward on its hinges. It was possible to see glimpses of movement through the gap at the top of the door, where the eight-inch steel had bent forward under the assault of the creature in the cargo hold. Enough to see that there was something massive behind, something as fearsome as its roaring suggested.
The crew braced themselves, aiming their revolvers and lever-action shotguns. The man operating the tripod-mounted rotary cannon flexed his trigger finger, wiped sweat from his brow, and sighted. The door had given up the struggle now. Each blow could be the one that brought them face-to-face with the thing in the hold.
Doubt was on their faces. All their guns seemed suddenly pitiful. Only discipline kept them in place, crowded in the dim passageway.
The door buckled inward, its upper hinge coming away completely. One more blow. One more.
But the final blow didn't come. And still it didn't come. And, after a time, it seemed it wasn't going to.
The men let out their pent-up breath, unsure what this new turn of events might mean. Each had been resigned to their fate. Had they been reprieved? They didn't dare to hope.
Some of them began to whisper. What had happened? Why had it stopped? Where had the thing in the hold gone?
From beyond the ruined door, there was only silence.
# _Chapter Twenty-four_
DYNAMITE—JEZ HEARS A CALL—A SWIFT RETREAT—THE CARDS ARE ON THE TABLE
o your left! Harkins, to your left!"
Harkins waved his pistol in the vague direction of the enemy and fired three wild shots before cringing back into the cover of the barrels. The shadowy figure he was aiming for ran behind a parked fighter craft and disappeared from sight.
"Nice shooting," Jez murmured sarcastically, then resumed scanning the dock for signs of movement. She flinched as three bullets pocked the barrels in front of her, searching her out. But the barrels were full of sand, and they were as good as a stone wall for stopping gunfire.
They'd put the _Ketty Jay_ down close to a corner of the elevated landing pad, so as to give themselves only two sides to defend when Dracken's men came for them. The barricades gave them good cover, and the largely empty dock meant that Dracken's men had a lot of open space to deal with. But they had twenty men out there, and on Jez's side there were only three. Two, if you didn't count Harkins, and he wasn't really worth counting. She checked her pocket watch and cursed.
They couldn't hold out. Not against these odds.
Silo was crouched behind a barricade to her right, sighting along a rifle. He fired twice at something Jez couldn't see. An answering salvo chipped the wood inches from his face.
There was one unforeseen disadvantage to their choice of position. Being close to the edge of the landing pad meant that they were near the lampposts that delineated it for the benefit of aerial traffic. Their attackers, on the other hand, had crossed the pad and were shooting from its center, where it was darkest. The landing-pad staff—who would use spotlights to pick out places for craft to land—had fled when the battle began, presumably to rouse the militia.
Jez wasn't hopeful. She doubted help would come through these broken alleys quickly enough. Besides, being arrested by the militia was as sure a death sentence as Dracken's men were. They'd be recognized as fugitives and hung.
Privately, Jez wondered if she'd survive that.
_Don't worry about that now. Deal with the things you can deal with_.
"Silo!" she hissed. "The lights!" She thumbed at the lampposts.
Silo got the message. He sat with his back to the barrels and shot out the nearest lamppost. Jez took out another. In short order, they'd destroyed all the lampposts nearby, and the _Ketty Jay_ sat in a darkness equal to that of their attackers.
But the distraction had let Dracken's men sneak closer. Even in a quiet dock like this, there were hiding places. The need to fuel and restock aircraft meant there was always some kind of clutter, whether it be an idle tractor for pulling cargo, small corrugated sheds for storage, or a trailer full of empty prothane barrels waiting to be taken away.
There was movement everywhere. A shot could come from any angle. Sooner or later, something was going to get through.
Harkins was whimpering nearby. Silo told him to shut up. Jez looked at her pocket watch again. Rot and damnation, this was bad. They hadn't expected twenty. Ten they could have held off. Maybe.
Something skittered across the landing pad, a bright fizz in the gloom. It took Jez only a moment to realize what it was. Dynamite.
"Down!" she cried, and then the stick exploded with a concussion hard enough to clap the air against her ears. The barrels murmured and rattled under the assault, but the throw had fallen short. Dracken's men weren't close enough to get it over the barricades. But it wouldn't be long before they were.
She looked back at the _Ketty Jay_ , rising above them like a mountain. The cargo ramp was open, beckoning them in. She thought about what Harkins had suggested when he first saw Dracken's men coming. How long could they hold out inside? How much damage would a stick of dynamite do to the _Ketty Jay_?
Of course, Dracken's men might have more dynamite. And a lot of sticks of dynamite could do a lot of damage.
She raised her head and looked out over the barrels but was driven down again by a salvo of bullets, coming from all sides. Panic fluttered in her belly. They'd keep her pinned, creeping nearer and nearer until they could fling dynamite over the barricade. There were too many to hold back.
And then, almost unnoticed, she felt the change. It was becoming more natural now, a slight push through an invisible membrane: the tiniest resistance, then a parting. Sliding into elsewhere, easy as thought.
The world altered. The dark was still dark, but it didn't obscure her vision anymore. She sensed them now: eighteen men, two women. Their thoughts were a hiss, like the rushing of the waves along the coast.
Panic swelled and consumed her. She was out of control. Her senses had sharpened to an impossible degree. She _smelled_ them out there. She heard their footsteps. And in the distance, far beyond the range of physical hearing, she heard something else. A cacophony of cries. The engines of a dreadful craft. And its crew, calling her. Calling in one wordless, discordant chorus.
_Come with us. Come to the Wrack_.
She recoiled from them, trying to focus her thoughts on anything other than the beckoning of that nightmarish crew. But instead of snapping out of that strange state, her mind veered away and fixed on something else. She felt herself sucked in, as she had been in Yortland watching predators stalking snow hogs. But this time it was no animal she joined with: it was a man.
She felt his tension, the sweat of him, the thrill of the moment. Comfort and satisfaction at being on the winning side. He knew they had the advantage. Don't slip up, though, you old dog. Plenty of graves full of the overconfident ( _pleased with that line, use it on the boys_ ). Seems like they're keeping their heads down now. That dynamite scared 'em good.
Need to get closer. Get a good shot on 'em then. Cap'n ( _respect awe protectiveness admiration_ ) would love it if you bagged one for her. Come on. Just over there.
_Run for it!_
Suddenly Jez was moving, rising, sighting down her rifle. She was in him and she was herself, two places at once. She knew where he was; she saw through his eyes; she felt his legs pumping as they carried him.
Her finger squeezed the trigger, and she shot him through the head at forty meters in the dark.
His thoughts stopped. All sense of him was gone. He was blanked, leaving only a hole. And Jez was thrust back into herself, her senses all her own again, curled in a fetal ball behind her barricade as she tried to understand what had just happened to her.
_What am I? What am I becoming?_
But she knew what she was becoming. She was becoming one of _them_. One of the nightmare crew. One of the creatures that lived in the wastes behind the impenetrable cloud wall of the Wrack.
_I have to run_ , she told herself, as a fresh volley of gunfire was unleashed. Bullets ricocheted off the side of the _Ketty Jay_. Another stick of dynamite fell close enough to knock over some of the barrels at the end of a barricade.
"We can't hold out no more!" screeched Harkins.
_No_ , she thought grimly. _We can't_.
THE DECK OF THE _Delirium Trigger_ was all but deserted. Most of the skeleton crew were in the guts of the aircraft, anxiously listening to the silence coming from the cargo hold. Others had gone to summon the militia. In the face of such alarm, nobody was loading cargo or swabbing the decks. When Malvery, Pinn, and Crake emerged from the captain's cabin with their plunder, there were no crew to stop them.
They raced across to the winch, now unmanned. A loaded pallet was dangling over the cavernous hatch that led to the cargo hold. Pinn flustered around the controls for a few moments before finding something that he assumed would lower the winch. As it turned out, he was right. There was a loud screech and the pallet began to rattle downward.
Crake scanned the craft nervously. A crowd of dockworkers had gathered around the _Delirium Trigger_ on the hangar deck, but nobody dared cross the gangplank. They'd heard men talking about a monster aboard. Now they followed the activity of the newcomers with keen interest, assuming them to be crew.
Crake didn't even see who shot at them. Pinn threw himself back, spitting a foul oath, as the bullet hit the winch next to his head. They scrambled out of the way, searching for their assailant, but there was no sign of one. Crake tripped and sprawled as another rifle shot sounded. Fear flooded him. He couldn't take shelter if he didn't know which direction the attack was coming from.
That didn't bother Malvery overmuch. "Get to cover!" he yelled, rushing toward an artillery battery, a cluster of massive cannons.
Crake scrambled after him. Another bullet hit. Out of the corner of his eye, Crake saw the dockworkers shouting in consternation. They were unsure who the villain was here. Some were following Crake's plight, but others were looking at a spot above and behind him.
He looked over his shoulder. There, where the deck of the _Delirium Trigger_ rose up toward an electroheliograph mast, he saw movement. A man, crouching, aiming.
Then Crake was behind the cannons, hunkering down next to Pinn and Malvery. "He's up there!" he panted. "By the mast!"
Malvery swore under his breath. "We need to get off this bloody aircraft, sharpish. Before them down below work out what's going on."
There was a sudden whine of strained metal from the winch. The chain swung sharply one way, then another, pulled from below.
Malvery edged along the barrel of the cannon and peered out for an instant, then drew back. "I see the bastard." He drew a pistol from his belt. It looked tiny in his huge hand. His usual shotgun had been too large to smuggle beneath their clothes.
"Wait," said Crake. "Not yet."
The chain pulled restlessly back and forth. The mechanism shrieked in protest at the weight it was carrying. The weight of the golem, clambering up the length of the chain and out of the cargo hold.
An enormous hand grabbed on to the lip of the hatch. Bess pulled herself up with a low bass groan, hauling her enormous bulk onto the deck.
"Now!" said Crake. Malvery swung out of hiding, aimed his pistol, and fired at the crewman hiding near the mast. The crewman, amazed by the sight of Bess, was taken by surprise. The shot missed by inches, but it startled him enough to send him scrambling out of sight.
The dockworkers on the hangar deck were panicking now, beginning to flee as Bess drew herself up to her full height. They'd never seen anything like her, this humpbacked, faceless armored giant. Those who were nearest fought to get out of the way, pushing aside the men at the back, who were crowding closer to see what the fuss was about.
"Bess!" Crake called as they broke from hiding. The golem swung toward him with a welcoming gurgle. He hurried up to her and quickly patted her on the shoulder. "We're getting out of here."
The dockworkers' fear of Bess grew to encompass Crake and the others now: they were friends with the beast!
Malvery sent another blast toward the electroheliograph tower as they ran for the gangplank. There were shouts of alarm from behind them as crewmen were roused by the gunfire. Bullets nipped at their heels. Pinn sent a few back, shooting wild.
Bess thundered down the gangplank and onto the hangar deck, the others close behind. The dockworkers melted away from the _Delirium Trigger_ like ice before a blowtorch, spreading chaos through the hangar as they fled. All activity came to a halt as crewmen on nearby freighters sensed the disturbance.
Malvery took the lead, heading toward the stairs that would take them to ground level, where they could exit the hangar. But he'd barely started in that direction when whistles sounded from below: the Ducal Militia of Rabban. Beige uniforms began to flood up the stairs that Malvery had been running for.
Too many men. Too many guns. Bess could make it through, but her more fragile, fleshy companions wouldn't.
Malvery came to a halt, pulled out his pocket watch, and consulted it. He looked back at the _Delirium Trigger_ , where the angry crew was already marshaling for pursuit. The militia had blocked their escape route. There was no way out.
"Alright," he said. "Now we've got problems."
TRINICA DRACKEN LOOKED AT her pocket watch, snapped it shut, and slipped it back inside the folds of her black coat.
"You need to be somewhere, Trinica?" Frey inquired.
She looked up at him across the card table. She seemed to be weighing a question.
"I think we've beaten around the bush for long enough, Darian. You wanted to parley. Speak your piece."
Her tone was newly impatient. Frey put two and two together.
"Why the hurry, Trinica? You were happy to make small talk until now. You wouldn't have been trying to buy time, would you? Delaying me here for some reason?"
He caught the flicker of anger in her eyes and felt a small satisfaction. She'd had the best of this meeting so far: it was good to score a point on her.
"Make your offer," she said. "Or this meeting is over."
_Might as well try_ , thought Frey. "I want you to give up the chase. Turn your back and leave us alone."
"What good will that do? You'll still be wanted by the Century Knights."
"The Century Knights I can handle. They don't know the underworld. I can scatter my crew, duck my head 'til the worst of it blows over. Maybe I'll get out of Vardia. Sell the _Ketty Jay_ , get a real job. But not with you on my heels. Most of them don't even know my face except from some old ferrotype, but you do. I think you'd find me in the end. So I'm asking you to give it up."
Trinica was waiting for the punch line. "Grephen is paying me a lot of money to track you down. Certainly more than you've ever seen in your life. What can you possibly offer me that would tempt me to give that up?"
"I'll keep your name out of it if I get caught."
"You'll _what_?" She was midway between amusement and astonishment.
"You're a traitor. You're a knowing accomplice in the murder of the Archduke's only son. The Coalition Navy never managed to pin anything on you—maybe because the witnesses have an odd habit of dying—but they know what you are and they'll jump at the chance to see you swing from the gallows. You know Grephen is afraid of the Knights getting me before you do. He's afraid I'll make accusations against him."
"That's the best you've got?" Trinica laughed. "The accusations of a condemned man, without any proof to back them up?"
"Have you thought what's going to happen if whatever Grephen's planning _doesn't_ work?" Frey asked. "My accusations might not save me, but if Grephen makes a move on the Archduke, he'll prove what I said about him is true. And that will mean everything I said about _you_ is true. Now, maybe Grephen will win and everything will be alright for you, but if he loses, you'll have the Navy all over you for the rest of your days. You certainly won't be docking in a place like Rabban anytime soon."
"Why would you believe he's making a move on the Archduke?"
Frey gave her a look. "I'm not stupid, Trinica."
She studied him. Considering. He'd seen that expression a hundred times before at a Rake table, as players stared at their opponents and asked themselves: _do they really have the cards to beat me?_
Then she snorted, disgusted at herself for allowing him to threaten her.
"This is ridiculous, and I don't have time for it anymore. It's all over now. Besides, I've got you." She drained her whiskey and got to her feet. "You're done."
"This is a parley, Trinica. Neutral ground. Sharka guarantees our safety." He grinned at her. "Can't get me here," he added, rather childishly.
"Of course not," she said. "But I can get your craft."
"You don't even know where she is."
"Certainly I do," she replied. "You're berthed in the Southwest Laborers' Quarters. Of course you registered under a false name, but I had every dockmaster in the city keeping an eye out for a Wickfield Ironclad-class cargo–combat hybrid. There aren't many around with the _Ketty Jay_ 's specifications, and I do know that craft quite well. I listened to you talk about her enough."
Frey was unperturbed. Trinica noted his lack of reaction.
"Obviously you guessed I'd do something like this," she said. "It doesn't matter. How many men do you have, Frey? Five? Six? Can you afford to keep that many?" She looked around the room; he bored her now. "I sent twenty."
_Twenty_ , thought Frey, keeping his face carefully neutral, the way he'd learned to at the card table. _Oh, shit_.
"What if I did the same?" he said. "What if my men are on _your_ craft right now?"
Trinica rolled her eyes. "Please, Darian. You never could bluff well. You're too much the coward. You always give in first."
She sighed and looked down at him, as if pitying a dumb animal. "I know you," she said. "You're predictable. That's why I almost caught you at the hermitage. Once Thade told me about you and his daughter, I realized that was the first place you'd go. You always did think with the wrong organ."
Frey didn't reply. She had him there.
"You want to know why I'm a good captain and you're not? Because you don't trust your people. I've earned my men's respect and they've earned mine. But you? You can't keep a crew, Darian. You go through navigators like whores."
Frey kept his mouth shut. He couldn't argue. There was nothing to say.
"And because I know you, I know you'd never trust anyone with your aircraft," she continued, walking past him toward the door. "The _Ketty Jay_ is your life. You'd rather die than give the ignition code to someone who might fly off with her. That means your crew are outnumbered, outgunned, and trapped, defending an aircraft that's nothing more than an armored tomb." She cocked her head. "Perhaps you were thinking of some clever flanking maneuver. Perhaps you're going to bring in reinforcements behind my men. Whatever you try, it makes no difference. You just don't have the numbers."
Frey's shoulders slumped. Twenty men. How long could Jez, Silo, and Harkins hold out against twenty men? Everything had relied on timing, but it was only now he truly realized how desperate the situation was. The plan had sounded so fine coming out of his mouth. But he was the only one _not_ risking his life here.
Trinica saw how it hit him like a hammer. She touched his shoulder in false sympathy and leaned down to whisper in his ear, her lips brushing his lobe. "By now they'll be dead, and my men will have filled the _Ketty Jay_ with so much dynamite, the explosion will be heard in Yortland."
She opened the door and looked back at him. "This will be the second time your crew died because of your hang-ups, Darian. Let's see how far and fast you run without your aircraft."
Then she was gone, leaving the door open behind her. Frey sat at the table, looking down at the mess of cards before him, feeling pummeled and raw and slashed to ribbons. She'd taken him apart with nothing more than words.
That woman. That bloody woman.
# _Chapter Twenty-five_
FLIGHT—"PICK YOUR TARGETS"—NO WAY OUT
rake ran hard. His lungs were burning in his chest and his head felt light, but his legs were tireless, filled with strength lent by adrenaline. Bess lumbered ahead, Malvery and Pinn hot on her heels. Bullets scored the air around them.
But they were only delaying the inevitable. There was nowhere left to go.
The hangar deck was crowded with cranes, portable fuel tanks, and piled cargo. Massive cogs rose out of the floor, part of a mechanism that clamped aircraft in their berths and prevented heavy freighters from drifting. In the distance, elevated platforms for spotlights and a narrow controller's tower rose almost to the roof of the hangar.
They used these obstacles as cover, darting past and around them, blocking the aim of the _Delirium Trigger_ 's crew. Nobody attempted to stop them with Bess leading the way. Dockworkers fled for cover, frightened by the wild gunplay of their pursuers.
The mouth of the hangar opened out to the night and the electric lights of the city. But the hangar deck was forty feet up, and there was no way down. The militia had spread out to block all the stairways. They were trapped, but still they ran, eking every last moment out of their liberty and their lives. There was nothing else left to do.
Bess slowed as they passed another pile of cargo waiting to be loaded onto a frigate. She picked up a crate and lobbed it effortlessly toward their pursuers. They scattered and scrambled away as it smashed apart in their midst. Crake and the others raced past her, and she took up position at the rear. A rifle shot bounced off her armored back, spinning away with a high whine, as she turned to follow them.
_Why did I come here?_ Crake thought. It was the same question he'd been asking himself all night. _Why did I agree to do this? Stupid, stupid, stupid_.
He flayed himself with his own terror as he ran, cursing his idiocy. He could have just refused. He could have stayed out of this and left at any time. But he'd allowed himself to be roped in to Frey's plan, driven by self-loathing and his captain's insidious charm. Back in Yortland, he'd been ready to throw it all in and leave Frey to his fate. Yet somehow he found himself agreeing to join the _Ketty Jay_ 's crew.
He'd made an error. He'd momentarily forgotten that time in the dingy back room of a bar, when Lawsen Macarde held a pistol to his head and told Frey to give up the ignition code to the _Ketty Jay_. He'd forgotten the look on Frey's face, those cold, uncaring eyes, like doll's eyes. He'd allowed himself to believe— _again_ —that Frey was his friend.
And, because of that, he was going to die here.
They dodged around machinery and vaulted over fuel pipes, rushing through the oily metal world of the hangar. Dark iron surrounded them; dim lights glowed; everything was covered with a thin patina of grime. They could expect no quarter here. This wasn't a place for sympathy but for the unforgiving industry of the new world. Crake had grown up on country estates, surrounded by trees, and had rarely ever seen the factories that had made his family rich. Now a grim fatalism swept over him. It seemed a terrible place to live a life and a worse one to end it in.
The deck narrowed as they reached the mouth of the hangar, splitting into long walkways that led to spotlight stations and observation platforms. To their left and right, half submerged below the elevated deck, were freighters and passenger liners, colossal in their shabby majesty. There were people lining the rail, watching their plight with interest, safely remote.
"Up here!" cried Malvery, and they were funneled onto a gantry that projected out to the mouth of the hangar. It was wide enough for three abreast, but at the end there was nothing but a small observation platform. After that, there was only the fatal plunge to the ground.
It didn't matter. They ran until the gantry ran out, and there they stopped.
The crew of the _Delirium Trigger_ slowed, seeing their quarry was trapped. They gathered at the end of the gantry, where there was cover. Between them and the men of the _Ketty Jay_ was a long, open stretch. They'd be easy targets there, and they still feared the golem enough to respect its power.
"Now what?" Pinn asked.
"Now we surrender," said Malvery.
"We _what_?" cried Pinn.
The doctor's grin spread beneath his thick white mustache. Pinn grinned back as he caught on. Crake was appalled to find that he was the only one who seemed nervous at the prospect of imminent death.
"I don't think they're in the mood to take us alive, anyway," said Malvery. "Everyone, get behind Bess. She's our cover."
"Hey, wait a—" Crake began, but they'd already crowded behind the golem, using her bulk as a shield. Bess hunkered down and spread herself out as much as possible. Malvery and Pinn crouched, peering out from either side, their guns ready. Crake, still carrying Dracken's strange compass in his hands, slid in next to them. He listened to the quiet ticks and coos coming from Bess's chest.
"How much ammo do we have?" Malvery asked.
"I got... um... twelve, thirteen bullets?" Pinn replied.
"I'm on about the same. Crake?"
Crake gave Pinn his revolver and a handful of bullets. "You take them. I wouldn't hit anything anyway."
"Right-o," said the doctor, aiming his gun. "Pick your targets."
The men of the _Delirium Trigger_ had swelled in number now. Some held back, studying the situation, while others angrily demanded action. One or two even tried to run up the gantry but were held back by their companions. A chancy long-range shot spanged off Bess's shoulder.
"Look at 'em," Pinn crowed. "Bunch of pussies."
Directed by the bosun, the crew commandeered crowbars from dockworkers and started jimmying nearby bits of machinery. The militia had caught up now—beige uniforms milled in the crowd—but having assessed the situation they seemed happy enough to let the men of the _Delirium Trigger_ handle it. Presumably they'd claim the credit afterward. It was easier than risking any of their own.
"What are they doing out there?" Malvery murmured to himself.
Crake peered out, took one look, and went back into hiding. "They're making a shield."
He was right. Moments later, ten men started to advance up the gantry, holding before them a large sheet of iron pulled from the side of a crane. They crept forward nervously but with purpose, their guns bristling out around the side of the shield.
"Hmm," said Malvery.
"What?" said Pinn. "Soon as they get close enough, we send Crake's girl out to get 'em. She'll squash 'em into paste."
"Ain't quite that easy," said the doctor, nodding toward the hangar deck. "Look."
Pinn looked. Five men had taken position at the edge of the deck and were lying on their bellies, aiming long-barreled rifles at them.
"Sharpshooters," said Malvery. "If Bess moves, we lose our cover, and they kill us." As if to punctuate his statement, a bullet ricocheted off Bess, inches from his face. He drew back a little way.
"Bugger," said Pinn. "Why do _we_ never come up with plans like that?"
"We did," said Malvery. "That's how we ended up here."
The men of the _Delirium Trigger_ crept steadily closer. The narrow angle along the gantry made it impossible to get a good shot at any of them. Malvery tried an experimental salvo with his pistol, but it only rattled their shield. They stopped for a moment, then continued.
Crake was sweating and muttering, calling himself all the names he could think of. He should never have gotten into this situation. He wanted to be sick, but there was nothing in his stomach: he'd been too nervous to eat before they set out on this mission.
The shield, having crossed much of the gantry, stopped. The men hunkered down behind it, becoming invisible. There was an agonizing sense of calm before the inevitable storm.
"Well," said Malvery to Pinn. "I'd say it was nice knowing you, but..." He shrugged. "You know."
"Likewise, you whiskery old fart." Pinn smiled, mistaking genuine distaste for comradely affection. Then the men of the _Delirium Trigger_ popped up out of hiding with their guns blazing, and all thought was lost in the chaos.
The assault was terrifying. They fired until their guns were empty, then ducked down to reload while the men behind them continued the barrage. Bess groaned and howled as she was peppered with bullets. They smacked into her at close range, blasting holes in the chain mail and leather at her joints, chipping her metal faceplate. She swatted at the air as if plagued by bees, cries of distress coming from deep inside her.
Crake had his hands over his ears, yelling over the tumult, a blunt shout of fear and rage and sorrow. The sound of leaden death was bad enough. The sound of Bess's pain was worse.
Malvery managed to point his pistol around the side of Bess's flank and fire off a shot or two, but it did no good. They crammed in behind the golem as best they could, but bullets were flying everywhere and they dared not break cover. Bess was being driven back by the cumulative impacts of the bullets, which punched at her armor, cutting into the softer parts of her. She stumbled backward, roaring now. The others stumbled back with her. Crake saw a spray of blood torn from Pinn's leg. He went down, his pistols falling from his hands, clutching at his thigh.
And suddenly he knew what was behind a dying man's eyes.
He knew what the crewman on the _Delirium Trigger_ had known, the one that Pinn had shot. He knew what it felt like to run out of time, leaving a life incomplete, and so much still to do.
There was blinding light and the bellow of engines. And machine guns, earsplitting machine guns, smashing through the cool night air of the hangar. The men on the gantry were cut to bloodied shreds, jerking as they were pierced, thrown limply over the railings, plunging to the floor of the hangar.
Crake blinked and stared, stunned by his reprieve. But there was no mistake. Hanging in the air, scuffed and scratched and beautiful, was the _Ketty Jay_. And sitting at the controls was Jez.
Malvery guffawed with laughter, waving one arm above his head. Jez waved back through the cockpit window. Pinn, rolling on the ground and shrieking, was largely forgotten.
Harkins sat in the autocannon cupola, and he opened up on the hangar deck as Jez rotated the _Ketty Jay_ into position. The shots were pitched to scare rather than hit anyone, but they caused sufficient panic to keep the sharpshooters busy. The cargo ramp at the rear of the craft was gaping open, and Silo was standing at the top of it, holding on to a rung, beckoning them.
Jez's control of the craft was clumsy: she backed up too hard and swung the lip of the cargo ramp into the gantry rail with a crunch. Metal twisted and screeched, but she managed to stabilize the _Ketty Jay_ again, and now there was an escape route, a ramp leading into the maw of the cargo hold.
Crake was standing as if in a dream, bewildered by all the noise and motion. Bess scooped him up in both arms as if he was a child, holding him close. Then she thumped forward, leaped onto the ramp, and carried him into the cargo hold.
Behind him there was scrambling, voices, men shouting things he didn't understand. The muffled sound of autocannon fire from above; the whine of prothane thrusters on standby; the blessed safety of walls all around him.
Then the hydraulics kicked in, and the cargo ramp began to close. Malvery was shouting, _"Jez! Get us out of here!"_ Pinn was wailing. The whole world swung as the craft moved. There was a wrench of metal from outside as the _Ketty Jay_ tore off part of the gantry rail.
Acceleration.
IT TOOK SOME TIME before the fog of panic cleared and Crake's senses returned. He realized that Bess had put him down on the floor and was squatting next to him. He could see the glimmers of light inside her faceplate, like distant stars. Malvery was telling Pinn to shut up.
"I'm bleeding out, Doc! I'm going cold!"
"It's just a flesh wound, you damn pansy. Stop whining."
"If I don't make it through... you have to tell Lisinda..."
"Oh, her. Sure. I'll tell your sweetheart you died a hero. Come on, hobble your arse to my surgery; I'll give you a couple of stitches. We'll have you fixed by the time we pick up the Cap'n."
There was movement, and the umber-skinned narrow face of the Murthian loomed into Crake's view.
"You alright?" he asked.
Crake swallowed and nodded.
Silo looked up at Bess. "She a fine thing," he said. Then he picked up the compass that was lying next to Crake. The compass he'd taken from Dracken's cabin. Silo weighed it in his hand thoughtfully, then gave Crake a look of approval, stood up, and walked away.
Bess was making echoing coos in her chest. Crake sat up and ran his hand along the metal plating of her arm. It was scored with burn marks and dents.
"I'm sorry, Bess," he murmured. "I'm so sorry."
Bess cooed again and nuzzled him, bumping the cold iron of her faceplate against his cheek.
# _Chapter Twenty-six_
A WELL-EARNED BREAK—SILO LENDS A HAND—THE CAPTAIN IS WOKEN—FROM BAD TO WORSE
rey celebrated his victory in the traditional manner and was roaring drunk by dawn.
He met up with his crew at a rendezvous point outside Rabban, where they'd hidden the Firecrow and the Skylance. They took off and flew for three hours, changing course several times until they were thoroughly sure that any attempts at pursuit would be hopeless. After that they began to search for a place to put down. Frey found a hillside clearing amid the vast moon-silvered landscape of the Vardenwood. There they sallied out, built a campfire, and Frey proceeded to get hammered on cheap grog.
It had been a long, long time since he felt this good.
He looked around at the laughing faces of the men who drank with him: Malvery, Pinn, even Harkins, who had loosened up and joined them after a little bullying. Jez was in her quarters, keeping to herself as usual, deciphering the charts they'd stolen from Dracken's cabin. Crake and Silo were nearby, tending to the damage that Bess had suffered. Nobody wanted to sleep. They were all either too fired up or, in Crake's case, too anxious. He was fretting about his precious golem.
But Frey couldn't worry about Crake for the moment. Right now he was basking in the satisfaction of a job well done. _His_ plan had worked. _His_ crew had triumphed against all the odds. Despite that cold bitch's condescending words, her cruel pity, he'd screwed her over like a master. He imagined her face when she got back to find her crew in disarray and her precious charts missing. He imagined how she'd smolder when she heard of the heroic last-minute rescue in the _Ketty Jay_. He imagined her rage when she realized how badly she'd misjudged him.
_You thought you knew me_ , he gloated. _You said I was predictable. Bet you didn't predict_ that.
And the best thing was that none of his people had gotten hurt. Well, except for Crake's little pet and the scratch on Pinn's leg, but that didn't really count. All in all, it was a brilliant operation.
If this was what success tasted like, he wanted more of it.
The bottle of grog came around to him and he swigged from it deeply. Malvery was telling some ribald story about a high-class whore he used to treat back when he was a big-city doctor. Pinn was already in stitches, long before the punch line. Harkins spluttered and grinned, showing his browned teeth. Their faces glowed warmly, flushed in the firelight and the colors of the breaking dawn. Frey felt a surge of alcohol-fueled affection for them all. He was proud of them. He was proud of himself.
It hadn't been an easy thing, to entrust Jez with the ignition code to the _Ketty Jay_. The code was set during the manufacture of the aircraft, and because it relied on various complex mechanisms it couldn't ever be changed without lengthy and expensive engineering procedures. Jez would forever have the power to activate and fly the _Ketty Jay_. Even now, Frey had to fight the suspicion that Jez might be creeping toward the cockpit, intending to punch in the numbers and run off with his aircraft before anyone could stop her.
_It's done now_ , he thought. _Live with it_.
It had been absolutely necessary for the completion of his plan that someone else fly the _Ketty Jay_. Jez had assured him she could, having grown up flying many types of aircraft. But he'd still found himself unable to give away the code at first. Like marriage, it felt like sacrificing too much of himself to a stranger.
In the end, he'd convinced himself by making an analogy to Rake. He found that most things in life could be related to cards, if only you thought hard enough.
In Rake, it was possible to play _too_ carefully. If you waited and waited for the perfect hand, then the obligatory minimum bets each round would gradually whittle you down. You'd run out of time and money waiting for an opportunity that never came. Sooner or later, you had to take a risk.
So he'd bet on Jez, and thankfully he'd won big. She was an odd fish, but he liked her, and he knew she was competent. He even had to admit to a slight sense of relief at the sharing of the secret code, although he wasn't exactly sure why. It felt as if he'd let out the pressure a little.
Malvery reached the punch line of his story, and they howled with laughter. Frey hadn't been paying attention, but he laughed anyway, caught up in the swell. He passed on the bottle, and Malvery gulped from it. Later, Frey would think of other things: the task they still had ahead of them, the bitter sting that came from seeing Trinica's face again. But for now, drinking with his men, he was happy, and that was enough.
CRAKE WAS ANYTHING BUT happy. Their narrow escape hadn't invigorated him with a sense of triumph but depressed him instead. He was acutely aware that they'd made it out only because Jez had arrived early. She'd been forced to take off sooner than planned, driven back to the _Ketty Jay_ by far superior numbers, and had then headed directly to their pickup point at the hangar. Once there, she'd spotted the disturbance inside and realized there was trouble. Their estimation of the length of the operation had been off: they'd allowed themselves far too much time.
In the end, they got lucky.
Rather to his surprise, Silo had emerged from the engine room to help him patch up Bess. The Murthian was a silent, solid presence around the _Ketty Jay_ , but because he rarely offered an opinion and never socialized, Crake had unconsciously begun to ignore him, as if he was one of the servants back home. He suspected that Silo was simply curious and saw an opportunity to get a closer look at the golem, to work out what made her tick. Whatever his motives, Crake was glad of the help and the quiet company. Between them, they pulled out bullets, stitched up leather, and soldered her wounds.
Though the damage was all superficial, Crake was racked with guilt. He'd allowed Bess to be used as an object. What if they had dynamite? What if they had a really big cannon? Could she have stood up to that? For that matter, what would actually happen to her if she was destroyed?
Bess was a shell, inhabited by a presence. That was as much as Crake knew. A vacant suit of armor, a skin surrounding nothing. Where did the presence truly exist? What exactly was in there? Did it occupy the skin of the suit, or was it somewhere inside? Those glittering eyes in the emptiness—did they mean something?
He didn't know. He didn't even truly know how he'd made her. Bess was an accident and a mystery.
"Does it hurt her?" Silo asked suddenly, rubbing his finger across a bullet hole in her knee. His deep, molten voice was heavily inflected. _Doors eet hoort hair?_
"I don't know," said Crake. "I think so. In a way."
The Murthian stared at him, waiting for more.
"She was... upset," he said awkwardly. "When they were shooting her. So I think she feels it."
Silo nodded to himself and returned his attention to his work. Bess was sitting quietly, not moving. She was asleep, he guessed. Or at least he called it sleep. In these periods of catatonia, she was simply _absent_. There were no glittering lights inside. She was an empty suit. Where the presence had gone, or if it had really gone anywhere at all, he couldn't have said.
The silence between them returned, but Crake felt a pressure to say something now that Silo had. It seemed momentous that the Murthian should be out here alongside him, asking him an unprompted question. He began to feel more and more uncomfortable. The rising chorus of birds from the trees all around seemed unnaturally loud.
"The captain seems in good cheer," he said at length.
Silo only grunted.
"How do you and he know each other?"
Silo stopped and looked up at him. For a few seconds, Silo regarded him in the pale dawn light, his eyes unreadable. Then he went back to his task.
Crake gave up. Perhaps he'd been wrong. Perhaps Silo really didn't want to talk.
"I escaped from a factory," Silo told him suddenly. _Arr scorrpt fram a fack-truh_. He kept working as he talked. "Seven year back. Built aircraft there for the Samarlans. My people are slaves down there. Bet you knew that, yuh?"
"Yes," said Crake. He was shocked to hear such a torrential monologue from Silo.
"The Dakkadians gave up. Stopped fighting long ago, joined their masters. But those of us from Murthia, we never give up. Five hundred year and we never give up." There was a fierce pride in his voice. "So when the time comes, some of us, we kill our overseer and we run. They come after us, yuh? So we scatter. Into the hills and the forest. And pretty soon, there's just me. Starved and lost, but I ain't dead and I ain't no slave.
"Then I see a craft coming down. Ain't damaged, but flies like it is. Pilot look like he don't know a thing. Makes a rough landing, and off I go. That's my way out. And when I get there, I find the Cap'n inside. Stabbed in the guts. In a bad way."
It took Crake a moment to catch on. "Wait, you mean _our_ captain? Frey?"
"Frey and the _Ketty Jay,_ " said Silo.
"How did it happen?"
"Didn't ask, and he didn't say," Silo replied. "Now, there's plenty food and supplies there on that craft, but I can't fly. I know craft on the inside, but I never flew one. So I take care of the Cap'n. I get him his drugs and bandages and I get him well. And in the meantime, I eat, get strong." He shrugged. "When he got better, he said he wasn't never goin' back to the people who sent him there. Said he was goin' to live the life of a freebooter. That was fine by me. He flew us both out, and I been on the _Ketty Jay_ ever since."
"So you saved his life?"
"S'pose. S'pose he saved mine too. Either way, here I am, yuh? We ain't never spoken of it since. I fix his craft, he keeps me in shelter. That's the way it is, and I'm grateful every day I have on board the _Ketty Jay_. Every day, that's one more day I ain't a slave. Lone Murthian wouldn't last long out here in Vardia. Your people ain't exactly fond of us since the Aerium Wars."
Crake looked over at the fire, where Malvery was holding Frey down and pouring grog into his mouth while the other two cheered. Every time he thought he had Frey figured out, he was bewildered anew.
"You never said."
"You never asked," said Silo. "It's a fool that speaks when there ain't no cause to. Too many loudmouths already on this craft."
"On that we agree," said Crake.
Silo got to his feet and stretched. "Well, I done what I can with your lady Bess," he said. "Gonna catch some sleep."
"Thank you for your help," said Crake. Silo grunted and began to walk off.
"Hey," called Crake suddenly, as a new question occurred to him. "Why do they call you Silo?"
"The name Mama gave me is Silopethkai Auramaktama Faillinana," came the reply. For the first time that Crake could remember, he saw the Murthian smile. "Think you can remember it?"
"CAP'N."
Frey was faintly conscious of someone shaking him. He wished with all his heart that they'd go away.
"Cap'n!"
There it was again, dragging him upward from the treacly, grog-soaked depths of sleep. _Leave me alone!_
"Cap'n!"
Frey groaned as it became clear they weren't going to give up. He was aware of a cool breeze and warm sun on his skin, the smell of grass, and the forbidding portents of a dreadful hangover. He opened his eyes and flinched as the eager sun speared shafts of light directly into his brain. He blocked the light with his hand and turned his head to look at Jez, who was kneeling next to him.
"What?" he said slowly, making it a threat.
"I've figured out the charts," she said.
He levered himself upright and groaned again, mashing his face with his palm. His mouth tasted like something had shat in it and subsequently died there. The embers of the fire were still alive, but the sun was high in a blue sky on an unseasonably warm winter's day. Malvery snored like a tractor nearby. Pinn was sucking his thumb, his other hand twitching toward his crotch, around which all his dreams revolved.
"Don't you sleep?" he said.
"Not much," she admitted. "Sorry if it's a bad time. You said you wanted to know straightaway. You said time is—"
"—of the essence, yes, I remember." He deeply regretted those words now. "So you know where Trinica's hideout is?"
"I believe so, Cap'n. The charts weren't easy to work out. It's not just an X-marks-the-spot kind of thing."
"Uh? A chart's a chart, isn't it?"
"Not really. These are very close-detail, marking a route through the mountains. Either we're missing a chart or Trinica already knows the general area where the hideout is. If you don't know where to start, you're just looking at a bunch of mountains." She gave a quirky smile. "Lot of mountains in Vardia."
"But you figured it out?"
"Matched the position of the bigger mountains with my other charts."
"Good work, Jez."
"Thank you, Cap'n."
"Now tell me where we're going."
"You're not gonna like it."
"I rarely do."
"I assume you've heard of Rook's Boneyard?"
"Oh, for shit's sake," he sighed, and then slumped down onto his back again, his eyes closed. He'd expected bad news, just not quite _that_ bad.
Jez patted him on the shoulder. "I'll be in my quarters when you're ready," she said. Then he heard her get up and walk back to the _Ketty Jay_.
Everybody who flew over the south end of the Hookhollows knew Rook's Boneyard. They all knew to avoid it if they possibly could. Aircraft that went into that small, restlessly volcanic area were rarely seen again. Those that ventured into the mists spoke of seeing their companions mysteriously explode. Pilots went mad and flew into mountainsides. Survivors talked of ghosts, terrible spirits that clawed at their craft. It was a cursed place, named after the first man to brave it and survive.
_Why don't I just lie down and die here?_ thought Frey. _It'll save time_.
Time. Time was something they didn't have. There was no telling how long it would take Trinica to replenish her crew and familiarize the newcomers with the complexities of the _Delirium Trigger_. A day? A week? Frey had no idea. It really depended on whether there was anyone vital among the men Jez had machine-gunned on the gantry.
But he knew one thing. As soon as she was up and running, Trinica would be after them with redoubled fury. Without her strange compass and her charts, she wouldn't be able to get to the hideout, but she knew that Frey would be heading that way. She might be able to get word to her allies somehow. He wanted to be in and out before she had a chance to act.
He got to his feet and swayed as his head went light. It took a few moments for everything to stabilize again. He wasn't, he reflected, in good shape for facing certain death anytime soon.
"Alright," he told himself unconvincingly. "Let's do this." And he stumbled off to rouse the crew.
# _Chapter Twenty-seven_
A PERILOUS DESCENT—THE PUZZLE OF THE COMPASS—FREY SEES GHOSTS
he _Ketty Jay_ hung in the white wastes of the Hookhollows, a speck against the colossal stone slopes. There were no other craft to be seen or heard. Below them, there was only the bleak emptiness of the mist. It cloaked the lower reaches, shrouding canyons and defiles, hiding the feet of the mountains. Down there, in Rook's Boneyard, the mist never cleared.
High above them were jagged, ice-tipped peaks. Higher still was a forbidding ceiling of drifting ash clouds, passing to the east, shedding a thin curtain of flakes as they went. A poisonous miasma, seeping from volcanic cracks and vents along the southern reaches of the mountain range. It was carried on the prevailing winds to settle onto the Blackendraft, the great ash flats, where it choked all life beneath it.
Frey sat in the pilot's seat, staring down. Wondering whether it was worth it. Wondering whether they should just turn tail and run. Could he really get them out of this mess? This ragged collection of vagrants, pitted against some of the most powerful people in the land? In the end, did they even have a chance? What lay in that secret hideout that was so important it was worth all this?
Their victory against Trinica had buoyed him briefly, but the prospect of flying blind into Rook's Boneyard had reawakened all the old doubts. Crake's words rolled around in his head.
_As a group, we're rather easy to identify. Apart, they'll probably never catch us. They'll get only Frey_.
Was it fair to risk them all, just to clear his own name? What if he sent them their separate ways, re-crewed, and headed for New Vardia? He might make it there, across the seas, through the storms to the other side of the planet. Even in winter. It was possible.
Anything to avoid going down there, into the Boneyard.
Crake and Jez were with him in the cockpit. He needed Jez to navigate, and he wanted Crake to help figure out the strange compasslike device, which nobody had been able to make heads or tails of yet. He'd banished the others to the mess to keep them from pestering him. Harkins and Pinn had been forced to leave their craft behind again, since it was too dangerous to travel in convoy, and they were insufferable backseat pilots.
"It'll be dead reckoning once we're down in the mist, Cap'n," said Jez. "So keep your course and speed steady and tell me if you change them."
"Right," he said, swallowing against a dry throat. He pulled his coat tighter around himself. He wasn't sure if it was the hangover or the fear, but he couldn't seem to get warm. He twisted around to glance at Crake, who was standing at his shoulder, holding the brass compass in both hands. "Is it doing anything yet?"
"Doesn't seem to be," said Crake.
"Did you turn it on?"
Crake gave him a look. "If you think you know a way to 'turn it on' that all of us have missed, do let me know."
"We don't need your bloody sarcasm right now, Crake," Jez snapped, with a sharp and unfamiliar tone to her voice. Crake, rather than offering a rejoinder, subsided into bitter silence.
Frey sighed. The tension between these two wasn't helping his nerves. It had been slowly curdling the atmosphere on the _Ketty Jay_ ever since they returned from the ball at Scorchwood Heights.
"Where's all this damned mist coming from, anyway?" he griped, to change the subject.
"Hot air from vents to the west blowing over cold meltwater rivers running off the Eastern Plateau," Jez replied absently.
"Oh."
The conversation lapsed for a time.
"Cap'n?" Jez queried, when things had become sufficiently uncomfortable. "Are we going?"
Frey thought about sharing his idea with them. He could offer to cut them loose and go his own way. Wouldn't that be the decent thing? Then nobody had to go down into the Boneyard. Least of all him.
But it all seemed a bit much to try to explain it now. Things had gone too far. He was resigned to it. Easier to go forward than back.
_Besides_ , he thought, in a rare moment of careless bravado, _nothing clears up a hangover like dying_.
He arranged himself in his seat and released aerium gas from the ballast tanks, adding a little weight to the craft. The _Ketty Jay_ began to sink into the mist.
The altimeter on the dashboard ticked steadily as they descended. The world dimmed and whitened beyond the windglass of the cockpit. The low hum of the electromagnets in the aerium engines was the only sound in the stillness.
"Come to one thousand and hold steady," Jez instructed, hunched over her charts at her cramped desk. Her voice sounded hollow in the tomblike atmosphere.
"Crake?"
"Still nothing."
They'd puzzled over the compass for most of the day, but nobody had been able to decipher its purpose. The lack of markings to indicate north, south, east, or west suggested that it wasn't meant for navigation. The four needles, which seemed capable of swinging independently of one another, made things more confusing. And then there were the numbers. Nobody knew what _they_ meant.
They'd established that each pair of number sets corresponded to a different arrow. The pair of number sets marked 1 matched the arrow marked 1. Each number was set on a rotating cylinder, like the readout of the altimeter, and presumably displayed the numbers 0 to 9. The upper set of each pair had two digits, allowing a count from 00 to 99. The lower set had the same but was preceded by a blank digit. All the numbers except this blank were set at 0.
Frey had the sense that this compass was vital to their survival in Rook's Boneyard. They were in danger until they could work out what it did. But right now it didn't seem to be doing _anything_.
Frey brought the _Ketty Jay_ to a hover when his altimeter showed they were a klom above sea level, down among the feet of the mountains. The mist had thickened into a dense fog, and the cockpit had darkened to a chilly twilight. Frey knew better than to use headlamps, which would only dazzle them, but he turned on the _Ketty Jay_ 's belly lights, hoping they'd provide some relief against the gloom. They did, but only a little.
"Alright, Cap'n," said Jez. "Ahead slow, keep a heading of two twenty, stay at this altitude."
"We'll start at ten knots," he replied.
"Right." Jez looked at her pocket watch. "Go."
Frey eased the _Ketty Jay_ forward, angling to the new heading. The sensation of flying blind, even at crawling speed, was terrifying.
He suddenly found a new respect for Harkins, who had chased a Swordwing at full throttle through the mist after the destruction of the _Ace of Skulls_. That nervy, hangdog old beanpole was braver than he seemed.
For long minutes, they moved forward. Nobody said anything. Frey could feel a bead of sweat making its way from his hairline, across his temple. Jez called out a change of heading and altitude. Mechanically, he obeyed.
The pace was excruciating. The waiting was killing him. Something was bound to happen. He just wanted it over with.
"I have something!" Crake announced. Frey jumped in his seat at the sudden noise.
"What is it?"
Crake was moving the compass around experimentally. "One of the needles is moving."
Frey brought the _Ketty Jay_ to a stop and took the compass from Crake. Jez glanced at her pocket watch again, mentally recording how far they had traveled on this new heading.
Crake was right. Though the other needles, numbered 2 to 4, were still dormant, the first needle was pointing in the direction that the _Ketty Jay_ was heading. As Frey twisted it, the needle kept pointing in the same direction, no matter which way the compass was turned.
The number sets corresponding to the first needle had changed too. Whereas all the others were still at 0, these had sprung to life. The topmost set read 91. The bottom set, the one preceded by a blank digit, read 30. They were not moving.
"The top one started counting down from ninety-nine," said Crake. "The bottom one just clicked to thirty and stayed there."
"So what does it mean?" Frey asked.
"He doesn't know what it means," Jez said.
"Do you?" Crake snapped.
Jez turned around in her chair, removed her hairband, and smoothed her hair back into her customary ponytail again. "I've some idea. The topmost digits were counting down when we were moving, and now they're not. I'd guess that they show the distance we are from whatever the arrow is pointing at."
"So what _is_ the arrow pointing at?" Crake asked, rather angry that he hadn't worked it out first.
"Something ninety-one meters ahead of us," Frey replied helpfully. "So now what? Can we go around it?"
"I'd rather not deviate from the charts if we possibly can," said Jez. "They're very precise."
"Alright," Frey replied. "Then we go very, very slowly, and let's see what's up ahead. Crake, read out the numbers."
He settled back into his seat and pushed the _Ketty Jay_ forward at minimum speed. Crake stood behind him, eyes flicking between the compass and the windglass of the cockpit, where there was still nothing but fog to be seen.
"Needle's holding steady. The other set of numbers is still at thirty. The top one is counting down... eighty... seventy... sixty... no change anywhere else... fifty... forty..."
Frey's mind was crowded with possibilities, tumbling over one another in a panic. What was it that waited there for them? The entrance to the hideout? Or something altogether deadlier?
"Thirty... twenty..."
He was so taut that his muscles ached, poised to throw the _Ketty Jay_ into full reverse the instant that anything emerged from the murk.
"Ten... five... zero."
"Zero?" Frey asked.
"Five... ten... The needle has changed direction. Now it's pointing behind us. Twenty... twenty-five."
"Let me have a look," Frey said, and snatched the compass from Crake. The needle was pointing directly behind them, and the numbers were counting up toward 99 again.
"Umm," he said. Then he handed the compass back to the daemonist. "Well. That's a puzzle."
"Perhaps those numbers didn't mean distance after all," Crake suggested churlishly, for Jez's benefit. Jez didn't reply. He went back to reading them off. "Ninety... ninety-five... Now the numbers have reset to zero, and the first needle has joined the other three."
"I suppose that means we've gone out of range," Frey suggested.
"But there wasn't anything _there_!"
"That's fine with me."
Jez called out a new heading, and Frey took it.
"You might see a—" she began, when Frey yelled in alarm as the flank of a mountain emerged from the fog. He banked away from it and it slipped by to their starboard side.
"—mountain," Jez continued, "but there'll be a defile running out of it."
"I didn't see any defile!" Frey complained, annoyed because he'd suffered a scare.
"Cap'n, I'm navigating blind here. Accuracy is gonna be less than perfect. Pull back closer to the mountain flank."
Frey reluctantly did so. The mountain loomed into view again. Jez left her station to look through the windglass.
"There it is," she said.
Frey saw it too: a knife slash in the mountain, forty meters wide, with uneven walls.
"I don't much like the look of that," he said.
"Drop to nine hundred, take us in," Jez told him mercilessly.
Frey eased the _Ketty Jay_ around and into the defile. The mountains pressed in hard, narrowing the world on either side. Shadowy walls lay close enough to be seen, even in the mist. Frey unconsciously hunched down in his seat. He concentrated on keeping a steady line.
"More contacts," said Crake. "Two of them."
"Two needles moving?"
"Yes. Both of them pointing directly ahead."
"Give me the numbers."
Crake licked dry lips and read them off. "First needle: distance ninety and descending. The other number reads fifty-seven and holding steady. Second needle: distance... ninety also now. That's descending too. The other number reads minus forty-three. Holding steady."
" _Minus_ forty-three?" Jez asked.
"A little minus sign just appeared where that blank digit was."
Jez thought for a moment. "They're giving us relative altitude," she said. "The first set of numbers shows the distance we are from the object. The second shows how far it is above or below us."
Frey caught on. "So then, the ones ahead of us... One is fifty-seven meters above us and the other is forty-three meters below?"
"That's why we didn't see anything the last time," Jez said. "We passed by it. It was thirty meters above us."
Frey felt a mixture of trepidation and relief at that. It was reassuring to believe that they'd figured out the compass and could avoid these unseen things, at least. But somehow, knowing where they were made them seem all the more threatening. It meant they were really _there_. Whatever _they_ were.
"Crake, keep reading out the distances," he said.
Crake obliged. "Twenty... ten... zero... needle's swung the other way... ten... twenty..."
Frey had him continue counting until they were out of range and the compass reset again.
"Okay, Cap'n," said Jez. "The bottom's going to drop out of this defile any minute. We come down to seven hundred and take a heading of two eighty."
Frey grunted in acknowledgment. There was enough space between the mountain walls for a much bigger craft to pass through, but the constant need to prevent the _Ketty Jay_ from drifting was grinding away at his nerves and giving him a headache. He dearly wished he hadn't indulged quite so heavily the night before.
As Jez had predicted, the defile ended suddenly. It fed into a much larger chasm, far too vast to see the other end. The fog was thinner here, stained with a sinister red light from below. Red shadows spread into the cockpit.
"Is that lava down there?" Frey asked.
Jez craned over from the navigator's station and looked down. "That's lava. Drop to seven hundred."
"Bringing us closer to the lava."
"I'm just following the charts, Cap'n. You want to find your own way in this mist, be my guest."
Frey was stung by that, but he kept his mouth shut and began to descend. The fog thinned and the red glow grew in strength until they were bathed in it. The temperature rose in the cockpit, drawing sweat from their brows. They could feel the radiant heat of the lava river flowing beneath them. Pinn came up from the mess to complain that it was getting stuffy down there, but Frey barked at him to get out. For once Pinn did as he was told.
Frey added aerium at seven hundred meters to halt their descent and pushed onward along the length of the chasm. Visibility was better now. The mist offered hints of their surroundings. It was possible to see the gloomy immensity of the mountains around them, if only as smudged impressions. To descend a few dozen meters more would bring the lava river into detail: the rolling, sludgy torrent of black and red and yellow. The heat down there would be unimaginable.
"Contacts," said Crake again. "Ahead and to the left a little. We—oh, wait. There's another. Two of them. Three. Three of them."
"There's three?"
"Four," Crake corrected. He showed Frey the compass. The needles were in a fan, all pointing roughly ahead. Frey frowned as he looked at it, and for a moment his vision wavered out of focus. He blinked, and the feeling passed. He swore to himself that he'd never again drink excessively the night before doing anything life-threatening.
"Any of them directly in front of us?"
"One's pretty close. Twenty meters below. Oh!"
"Don't just say 'oh!' " Frey snapped. "Oh _what_?"
"One of the needles moved... now it's changed back... now it's gone back again."
"What you mean, it changed?" Frey demanded. He wiped sweat from his brow. All this tension was making him feel sick.
"It moved! What do you think I mean?" Crake replied in exasperation. "Can you stop a moment?"
"Well, why's it changing? Is there something there or not?" Frey was getting flustered now. He felt a fluttering sensation of panic come over him.
"There's more than four of those things out there," said Jez, who had gotten up from her station and was looking at the compass. "I'd guess it keeps changing the needles to show us the nearest four."
"There's one thirty meters ahead!" Crake cried.
"But is it above us or below us?" Frey said.
"Forty meters above."
"Then _why tell me_?" he shouted.
_"Because you told me to!"_ Crake shouted back. "Will you stop this damn craft?"
But Frey didn't want to. He wanted to get this over with. He wanted to be past these invisible enemies and away from this place. There was a terrible feeling of wrongness stealing over him, a numbness prickling up from his toes. He felt flustered and harassed.
"What the bloody shit is going on, Crake?" he snarled, leaning forward to try to see what, if anything, was above them. "Someone talk to me! Where are they?"
"There's one, there's three in front of us, one behind us now... umm... two above, thirty and twenty meters, there's..." Crake swore. "The numbers keep changing because you're moving! How am I supposed to read them out fast enough?"
"Just tell me if we're going to hit anything, Crake! It's pretty damn simple!"
Jez was staring in bewilderment. "Will you two calm down? You're acting like a pair of—"
But then Frey recoiled from the window with a yell. "There's something out there!"
"What was it?" Jez asked.
"We've got one twenty... ten meters ahead... it's below us though..." Crake was saying.
"It looked like... I don't know, it looked like it had a _face._ " Frey was babbling. His stomach griped and roiled. He could smell his own sweat, and he felt filthy. He wiped at the back of his hands to try to clean them a little, but all it did was smear more dirt into his skin. "The ghosts!" he said suddenly. "It's the ghosts of Rook's Boneyard!"
"There aren't any ghosts, Cap'n," Jez said, but her face was red in the lava light, and her voice sounded strange and echoey. Her plain features seemed sly. Did she know something he didn't? A blast of maniacal laughter came from the mess—Pinn laughing hysterically at something. It sounded like the cackle of a conspirator.
"Of course there are ghosts!" Frey turned his attention back to the windglass, trying to will the mist aside. "Everyone says."
"Two of them are behind us now," Crake droned in the background. "One ahead, one passing to the side."
" _Which_ side?"
"Does it matter?"
Something swept past the windglass, a stir in the mist. Frey saw the stretched shape of a human form and distorted, ghastly features. He shied back from the windglass with a gasp.
"What is it?"
"Didn't you see it?"
"I didn't see anything!"
Frey's vision was slipping in and out of focus and refused to stay steady. He burped in his throat and tasted acid and rotten eggs.
"Cap'n..." said Crake.
"I think something's wrong," Frey murmured.
"Cap'n... the second set of numbers..."
"What second set of—"
"The numbers! They're counting up from minus twenty toward zero! It's coming at us from below!"
"Cap'n! You're drifting off altitude! You're diving!" Jez cried.
Frey saw the altimeter sliding down and grabbed the controls, pulling the _Ketty Jay_ level.
"It's still coming!" Crake shrieked.
"Move!" Jez cried, and Frey boosted the engines. The _Ketty Jay_ surged forward, and a split second later there was a deafening explosion outside, slamming against the hull and throwing Crake and Jez across the cabin. The craft heeled hard, swinging to starboard, and Frey fought with the controls as they were propelled blindly into the red murk. The _Ketty Jay_ felt sluggish and wounded. Frey caught a glimpse of the compass on the floor, its needles spinning and switching crazily.
_They're all around us!_
Crake started shrieking. "Daemons! There are daemons at the windows!" Frey's vision blurred and stayed blurred. There seemed to be no strength in his limbs.
"Cap'n! Above and to starboard!" Jez shouted.
Frey looked and saw a round shadow in the mist. Growing, darkening as it approached. A ghost. A great black ghost.
No. A sphere. A metal sphere studded with spikes.
A floating mine.
Jez grabbed the flight stick and wrenched the _Ketty Jay_ to port. Frey fell bonelessly out of his seat. Crake screamed.
There was another explosion. Then blackness, and silence.
# _Chapter Twenty-eight_
JEZ SAVES THE DAY—LEGENDS COME TO LIFE—THE DOCKMASTER—SOME TACTICAL THINKING—NEWS FROM THE MARKET
rey came to a kind of bleary awareness some time later, to find himself crumpled on the floor of the _Ketty Jay_ 's cockpit. His cheek was pressed to the metal, wet with drool. His head pounded as if his brain were trying to kick its way out of his skull.
He groaned and stirred. Jez was sitting in the pilot's seat. She looked down at him.
"You're back," she said. "How do you feel?"
He swore a few times to give her an idea. Crake was collapsed in the opposite corner, contorted uncomfortably beneath the navigator's desk.
Frey tried to remember how he'd gotten in this state. He was tempted to blame it on alcohol, but he was certain that he hadn't been drinking since last night. The last thing he remembered was flying through the fog and fretting about the numbers on the compass.
"What just happened?" he asked, pulling himself into a sitting position.
Jez had the compass and the charts spread out untidily on the dash. She consulted both before replying. "You all went crazy. Fumes from the lava river, I suppose. It would explain all the ghosts and hallucinations and paranoia." She tapped the compass with a fingernail. "Turns out this thing is to warn us where the magnetic floating mines are. Someone's gone to a great deal of trouble to make sure this secret hideout stays secret."
Frey fought down a swell of nausea. He felt like he'd been poisoned.
"Apologies for taking the helm without permission, Cap'n," said Jez, sounding not very apologetic at all. "Had to avoid that mine, and you were out of action. Close thing. The _Ketty Jay_ took a battering. Anyway, we're nearly there now."
"We are?"
"It's actually pretty easy once you work it out," she said, although he wasn't sure if she meant following the route to the hideout or flying the _Ketty Jay_.
He got unsteadily to his feet, feeling vaguely usurped. The sight of Jez in the pilot's seat disturbed him. It was an unpleasant vision of the future he feared, in which Jez—now possessing the ignition code—stole away with his beloved craft when his back was turned. She looked so damned _comfortable_ there.
Outside, everything was calm and the air had cleared to a faint haze. Though there was still a heavy fog overhead, blocking out the sky, it was possible to see to the rocky floor of the canyon beneath them. A thin river ran along the bottom, hurrying ahead of them, and a light breeze blew against the hull.
Frey rubbed his head. "So how come it didn't affect you?"
She shrugged. "Once I saw what was happening, I held my breath. I took only a few lungfuls before we flew out of it."
Frey narrowed his eyes. The explanation had an overcasual, rehearsed quality to it. As an experienced liar, he knew the signs. So why was his navigator lying to him?
There was a clatter from the passageway behind the cockpit, and Malvery swung around the door. "Allsoul's balls, what were we drinking?" he complained. "They're all comatose down there. Even the bloody cat's conked out."
"You weren't giving the cat rum again, were you?" Frey asked.
"He looked thirsty," Malvery said, with a sheepish smile.
"Eyes front, everyone," said Jez. "I think we're here."
They crowded around her and stared through the windglass as the _Ketty Jay_ droned out of the canyon. And there, down among the fog and the mountains of the Hookhollows, hidden in the dreadful depths of Rook's Boneyard, they found at last what they'd been searching for.
The canyon emptied out into a colossal gloomy sinkhole, a dozen kloms wide, where the ground dropped seventy meters to a waterlogged marsh. Streams from all over the mountains, unable to find another way out, ended up here, tipping over the edge in thin waterfalls. Mineral slurry and volcanic sludge, washed down from distant vents, stained the surface of the marsh with metallic slicks of orange, green, or blue. Ill-looking plants choked the water. The air smelled acidic and faintly eggy.
And yet here, in this festering place, was a town.
It was built from wood and rusting metal, a ramshackle sprawl that had evolved without thought to plan or purpose. Most of it was set on platforms that rose out of the water, supported by a scaffolding of girders. The rest was built on what little land the marsh had to offer: soggy banks and hummocks. Each part was linked by bridges to its neighbors and lit by strings of electric lamps that hung haphazardly across the thoroughfares.
The buildings varied wildly in quality. Some wouldn't have looked out of place on a country estate in the tropical south. Others had been thrown together with whatever could be found or brought from the outside. They were made of wood and stone, with slate or corrugated iron roofs. Parts of the settlement were a cluster of shantytown huts, barely fit for habitation, whereas others were more organized and showed an architect's touch.
Then there were the aircraft. There had to be two hundred or more, crowding around the town. Frigates floated at anchor, secured by strong chains to stop them from drifting. Smaller craft ferried their crews to and from the ground. There was one enormous landing pad, occupying the biggest land mass in the marsh, but even that was nowhere near adequate to cope with the number of craft berthed here. Several large landing pads lay on the surface of the marsh. They were temporary-looking things, buoyed up by flaking aerium tanks filled from portable engines to prevent the pads from sinking.
Frey stared at the multitude. He saw freighters, barques, fighters of all description, double-hulled caravels, monitors, and corvettes. The air above the town was busy with craft taking off and setting down, a restless to-and-fro. A Rainbird-class hunter-killer, sleek and vicious, slipped past them to their starboard and headed into the canyon they'd just exited.
"That's a bit more than just a hideout," Malvery murmured, amazed. "There's a whole bloody port down here."
And suddenly Frey knew where he was. Nothing else matched the picture. He'd always believed this place was a myth, a wistful dream for freebooters all over Vardia. But now it was laid out before his eyes—decaying, shabby, but undoubtedly real. The legendary pirate town, hidden from the Coalition Navy and ruled by the famed pirate Orkmund.
Retribution Falls.
FREY COULD SEE NO indication of where he was supposed to land, no spotlights to guide him in, so he squeezed into a vacant spot on the main pad. When he and his crew opened the cargo ramp to disembark, they found someone waiting for them. He was tall and doughy around the belly and face, with one lazy eye and a gormless smile.
"You signed in yet?" he asked Frey.
Frey was momentarily lost for an answer. The man had just watched them set down. He considered asking how he might possibly have gotten to the dockmaster's office and back while still in midair, but eventually he settled on an easier response.
"No."
"You should sign in. Orkmund's orders."
Frey felt a thrill of excitement at the name. That settled it.
This was Retribution Falls, alright.
"Where's the dockmaster?"
"You the captain?"
"Yeah."
"Follow me, I'll take you."
Frey told the others to wait by the _Ketty Jay_ and then trailed after the man toward the dockmaster's office. It was a grim, low-ceilinged affair, more like a large shed than an administrative building. Dirty windows were divided into small rectangular panes. The door stuck and had to be wrenched open, the frame having warped in the dank air.
Inside, the gloom was barely leavened by a single oil lantern. The dockmaster—a thin old man with a pinched face—was hunched over a desk, writing with a pen. On the other side of the room was a lectern, where a huge book lay open. It was full of names and dates.
Frey waited to be noticed. The man with the lazy eye waited with him. The smell from the swamp lingered in the nostrils, faintly disgusting. Frey suspected that the locals didn't notice it anymore.
After a short time, the dockmaster looked up. "Well, sign in, then!" he snapped, indicating the book on the lectern. "Olric, honestly! Why don't you just tell him to sign in?"
Olric looked shamefaced. Frey went over to the book and picked up the pen that lay next to it. He scanned over the entries. Each line bore the name of a captain, the name of an aircraft, and the date and time of arrival and, in some cases, departure. At the bottom of each double page, the dockmaster had signed his name and title in crabbed script.
He flicked back a few pages, idly searching for someone he knew. Maybe Trinica would be in here.
"Busy recently, aren't you?" he commented. "You usually get this much traffic?"
"Just sign," the dockmaster said impatiently, not looking up from his records.
FREY'S DECISION TO CONFINE most of the crew to the craft wasn't popular with one man in particular.
"You stinking bastard, Frey!" Pinn cried. "You didn't even believe Retribution Falls existed until now! I _told_ you we should come here when we were back in Yortland, but oh, no! You thought: _let's all laugh at Pinn!_ Well, I called it right, and I _deserve_ to come."
"Shut your fat meat-hole, Pinn," Malvery said. "Cap'n's given you an order."
"Oh, really? Well, he can stuff it up his arse with all the other orders he's given me!"
Frey looked at Silo. "If he tries to leave, shoot him," he said, only half joking.
"Cap'n," Silo replied, priming his shotgun with a crunch.
Pinn looked around at the rest of the crew, finding no support, and then stamped back into the depths of the craft, muttering mutinously.
"Jez, Malvery, come on," he said. "We keep a low profile, have a look around, keep our ears open. And don't anybody call me anything but Cap'n, okay? I don't want to hear my name spoken outside the _Ketty Jay._ "
"Right-o."
"Everyone got revolvers? Good. You never know."
They headed across the landing pad toward the bridge to the town. Frey was rather pleased with himself for standing firm against Pinn's outburst. Pinn was envisioning a night out in this pirate haven, but Frey needed to be able to effect a quick escape if necessary, without the need to go searching under bar tables for his drunken crew. Taking the whole group out would be like trying to herd cats.
He reviewed the tactics behind his choice of landing party. Separating Malvery and Pinn was the key. Pinn wouldn't cause any trouble without the doctor's backup, and since Malvery was coming along, he didn't care what happened to Pinn. Malvery was useful muscle and had a bluff charm that would play well, but the two of them together in a place like Retribution Falls would result in alcoholic carnage, sure as bird shit on statues.
Jez would also be useful. She was smart, observant, and she had eyes like a hawk. Plus she was the only sensible one among them. He didn't count Crake. Crake dealt with daemons: nobody could say _that_ was sensible.
But he had an ulterior motive in bringing Jez. He wanted to keep an eye on her. As grateful as he was that she'd saved their lives, he was suspicious. It puzzled him that the fumes hadn't seemed to affect her, and her explanation was weak. He didn't want to leave her alone on his aircraft. Not now that she knew the ignition code. He wasn't so sure he trusted her.
The others wouldn't mind staying on the _Ketty Jay_. Crake, as he was never a freebooter, didn't understand the legend and allure of Retribution Falls. He had no desire to see the place. Harkins didn't like crowds or strangers. He'd rather be secure in his quarters, living in terror of the cat, who would wait for him to fall asleep before trying to suffocate him. And it would be too dangerous to take Silo. A Murthian would attract unwanted and hostile attention in a town like this. Besides, Silo had work to do. He needed to check over the _Ketty Jay_ and repair any damage from the mines.
All in all, he had the whole thing figured out.
_Not bad, Frey_ , he thought. _That's the sort of thinking a real captain does. That's how to handle a crew_.
He was in the mood for self-congratulation, despite his near-catastrophic failure to lead them through Rook's Boneyard. The triumph of finding Retribution Falls outweighed all that. This must have been how Cruwen and Skale felt when they discovered New Vardia. He was an explorer now. Whatever happened after this, he had to admit, he felt more... well, more like a _man_ than he ever had before.
In that moment when he pressed down on his guns and blew the _Ace of Skulls_ into a flaming ruin, his life as he knew it had ended. Every day since then had been one clawed back. He'd been forced to fight every step. It was exhausting, and terrifying, and most of the time he hated it. But just sometimes, when he could snatch a rare instant of peace amid the chaos, he felt different. He felt good about himself. And it had been a long, long time since he'd felt like that.
They took the bridge from the landing pad to the nearest platform and discovered that Retribution Falls was even more unpleasant up close, and a far cry from the legends.
The narrow streets were weathered and worn beyond their years. The marsh air ate through metal, twisted wood, and brought mold to stone. Everything flaked and peeled. Generators buzzed and reeked, providing the power for the lights that hung on wires overhead to stave off the gloom. It was cold, yet their clothes became damp and stuck to them. The smell of the marsh mingled with that of a thousand unwashed bodies.
Retribution Falls was stuffed with every kind of pirate, smuggler, fraudster, and criminal that Frey could imagine. Every pub and inn was crammed to capacity. The streets were choked, the whores hollow-eyed and exhausted. Inside, the humidity and the heat of dozens of bodies made things uncomfortable. Drunken men with short tempers fought hard. Guns were drawn, and bodies fell.
There was a wildness here that he found frightening. It was a jostling, stinking pandemonium of rotted teeth and leering faces. Danger surrounded them. He found he actually missed the specter of the militia. He liked his illegal doings to be conducted within the safety of an orderly civilization. Total lawlessness meant survival based on strength or cunning, and Frey didn't have too much of either.
They passed raucous bars and stepped over men lying in the thoroughfares—rum-soaked, unconscious, and recently robbed. Malvery eyed the bars as they passed, but without Pinn as his accomplice, he behaved himself and stuck close to his captain. Occasionally he'd shove someone out of their path. His size and fierce glare discouraged arguments.
"Not quite the utopia I'd envisioned, Cap'n," Jez murmured.
Frey didn't quite understand what she meant by "utopia"—it sounded like one of Crake's words—but he got the idea.
"All those craft, all these people," he said. "Doesn't it seem like there's far more pirates here than this place was built to hold?"
"Certainly does," she said.
"And what does that say to you?"
"Says they're being gathered here for something."
"That's what I thought," he replied.
THE MARKET WAS A little less crowded than the streets and bars, but not by much. It sat on a platform all its own, linked by bridges to several of its neighbors. Oil lamps hung from the awnings of rickety stalls, adding a smoky tang to the already fouled air. Their flickering light mixed uneasily with the electric bulbs hanging overhead, casting a strange glow on the heaving sea of faces that surged beneath.
Malvery pushed his way through the crowd, with Frey and Jez following in his slipstream. The stalls they passed were guarded by shotgun-wielding heavies. There was all manner of wares for sale: trinkets and knickknacks, hardware, boots and coats, navigational charts. Dubious fried meats were offered to hungry shoppers, and someone was roasting chestnuts nearby. The noise of yelled conversation was deafening.
"You get the impression that this has all gotten a little out of control?" Jez screamed in Frey's ear.
Frey didn't hear what she said, so he nodded as if he agreed, and then replied, "I think whoever's running this show, they've let things get a bit out of control!"
Jez, who also hadn't heard him, said, "Definitely!"
Frey spotted a stall on the edge of the market platform where the traffic wasn't quite so oppressive and it was possible to see the darkening marsh in the background. One of several signs that hung from its pole-and-canvas frontage declared:
_Breathe the Free Air! Filters 8 Shillies!_
He tapped Malvery on the shoulder and steered him over. The storekeeper saw them coming and perked up. He was a thin, ginger-haired man with an enormous puckered patch of scar tissue that ran across one side of his face. It looked as if he'd been mauled by a bear.
"How did you get that?" Frey asked conversationally, indicating the scar.
"How did I get what?" the storekeeper asked, genuinely puzzled.
Frey thought a moment and then let it drop. "These filters you're selling. They'd protect us against the bad air in the canyons?"
The storekeeper grinned. "Guaranteed. Did your old ones let you down?"
"Something like that."
"That's rough, friend. Well, you can rely on these." He pulled one out of a crate behind him and put it on. It was a black metal oval with several breathing slits that fit over the mouth and nose, secured over the head by a strip of leather. "Wo wetter n orb wetwibooshun bawls."
"What?"
The shopkeeper took off the mask. "I said, no better in all Retribution Falls."
"Okay. I need seven."
"Eight," Jez corrected. When Frey and Malvery both looked at her, she said, "The cat."
"Right," said Frey. "Eight. Give me a discount."
"Six bits."
"Three."
"Five."
"Four."
"Four and eight shillies."
"Done."
"You won't regret it," the storekeeper promised, as he began counting out filters from the crate.
"First time in Retribution Falls?"
"How'd you guess?"
"Lot of newcomers recently. You just got the look."
"Why so many?"
The storekeeper dumped an armful of filters on the cheap wooden table that passed as a counter. "Same reason as you, I expect."
"We're only here for the beer and scenery," Malvery grinned. The storekeeper laughed at that, revealing a set of teeth better kept hidden.
"You heard about what's going on tomorrow?" the storekeeper asked, as Frey laid down his coins on the counter.
"Like you noticed, we just got here," Frey replied.
"You know where Orkmund's place is?" He indicated a distant platform. It was too dark to make out anything but a sprinkle of lights. "Ask anyone, you'll find it. Be there tomorrow at midday."
"What's happening?"
"Orkmund's got something to say. Reckon it might be time."
Malvery did a passable job of pretending he knew what the man was talking about. "You think so?"
"Well, look around," said the storekeeper. "Some of these boys are going stir crazy. Can't keep a bunch of pirates cooped up like this. They came to fight, and if they can't fight someone else, they'll fight one another. I reckon he's gonna give the word to start the attack."
"Let me at 'em," said Frey. "Can't wait to show that lot."
"You know who we're fighting?" the storekeeper gasped, which wrong-footed Frey totally.
"Er... what?"
"You know where Orkmund's sending us?"
"Don't _you_?"
"Nobody knows. That's what we're all waiting to find out."
Frey backpedaled. "No, I meant, you know... the _general_ them. Let me at _them_. Whoever _they_ are...." He trailed off lamely.
The storekeeper gave him an odd look, then snatched the coins off the counter and called out to a passerby, trying to lure him over. Dismissed, Frey and the others moved away, distributing the filters among them.
"Orkmund's got himself a pirate fleet," Jez said. "That's how Grephen's going to do it. That's how he'll seize power. He's made a deal with the king of the pirates."
"But there's one last thing I don't understand," Frey replied. "How'd Duke Grephen get Orkmund on his side?"
"Paid him, probably," Malvery opined.
"With what? Grephen doesn't have the money to support an army. Or at least Crake doesn't think so, and he should know."
"Crake could be wrong," Jez said. "Just because he has the accent doesn't mean he has some great insight into the aristocracy. There's a lot you don't know about him."
Frey frowned. He was getting heartily sick of this tension between Jez and Crake. They'd been barely able to work together when he needed them to navigate through the canyons of Rook's Boneyard. Something needed to be done.
"Back to the _Ketty Jay,_ " he said. "We've learned enough for now. Let's see what Orkmund says tomorrow."
"We're not going to have a drink?" Malvery asked, horrified. "I mean, in the interest of gathering information?"
"Not this time. Early start in the morning. I'm not having any trouble tonight."
He started off back toward the landing pad. Malvery trudged behind. "I miss the old Cap'n," he grumbled.
Frey had almost all the information he needed. He was missing only one piece. Someone was backing Duke Grephen, providing the money to build an army of mercenaries big enough to fight the Coalition Navy and take the capital of Vardia. He needed to know who. When that last piece fell into place, he'd understand the conspiracy he was tangled up in. Then he could do something about it.
A serene and peaceful feeling settled on him as they made their way back toward the _Ketty Jay_. Tomorrow would bring an answer. He didn't know how he knew, but he was certain of it.
_Tomorrow. That's when we start turning this around_.
# _Chapter Twenty-nine_
INTERVENTION—THE CONFESSIONS OF GRAYTHER CRAKE—AN EXPERIMENT, AND THE TRAGEDY THAT FOLLOWS
rake was shaken out of sleep by Frey's hand on his shoulder.
"Get up," Frey said.
"What is it?" he murmured.
"Come on," insisted the captain. "I need you in the mess." Crake swung his legs off the bunk. He was still fully clothed, having gone to sleep as soon as Frey left the _Ketty Jay_. He'd hoped to shake off the headache he'd picked up from breathing the lava fumes. It hadn't worked.
"What's so urgent, Frey? Stove making spooky noises? Daemonic activity in the stew?"
"There's just something we need to sort out, that's all." Something in his tone told Crake that Frey wasn't going to let this go, so he got to his feet with a sigh and shambled after his captain, out into the passageway. But instead of going down the ladder to the mess, Frey walked past it and knocked on the door of the navigator's quarters. Jez opened up. She glanced from Frey to Crake and was immediately suspicious.
"Can you come to the mess?" Frey asked, though it sounded less a request than an order.
Jez stepped out of her quarters and shut the door behind her.
They climbed down into the mess. Silo was in there, smoking a roll-up and drinking coffee. He was petting Slag, who was lying flat on the table. At the sight of Jez, the cat jumped to his feet and hissed. As soon as the way was clear, he bolted up the ladder and was gone.
Silo looked up with an expression of mild disinterest.
"How's the _Ketty Jay_?" Frey asked.
"She battered, but she tough. Need a workshop to make her pretty again, but nothing hurt too bad inside. I fixed her best I can."
"She'll fly?"
"She'll fly fine."
Frey nodded. "Can you give us the room?"
Silo spat in his palm and stubbed the roll-up into it. Then he got up and left. Since speaking with Silo, Crake couldn't help seeing the Murthian's relationship with his captain in a new light. They'd been companions so long that they barely noticed each other anymore. They wore each other like old clothes.
"Sit down," Frey said, motioning to the table in the center of the mess. Jez and Crake sat opposite each other. The captain produced a bottle of rum from inside his coat and put it on the table between them.
"She doesn't drink," Crake said. He was beginning to get a dreadful idea what this was about.
"Then you drink it," Frey replied. He straightened, standing over them. "Something's going on between you two. Has been since you went to Scorchwood Heights. I don't know what it is, and I don't want to know, 'cause it's no business of mine. But I need my crew to act like a crew, and I can't have this damned bickering all the time. The only way we're gonna survive is if we work together. If you can't, next port we reach, one of you is getting off."
To his surprise, Crake realized that Frey meant it. The captain looked from one of them to the other to ensure the message had sunk in.
"Don't come out of this room 'til you've settled it," he said, and then he climbed through the hatch and was gone.
There was a long and grudging silence. Crake's cheeks burned with anger. He felt awkward and foolish, a child who had been told off by his tutor. Jez looked at him coldly.
_Damn her. I don't owe her an explanation. She'd never understand_.
He hated Frey for meddling in something that didn't concern him. The captain had no idea what he was stirring up. Couldn't they just let it lie? Let her believe what she wanted. Better than having to think about it again. Better than having to face the memories of that night.
"It's true, isn't it?" Jez said.
He met her gaze resentfully.
"What the Shacklemore said," she prompted. "You stabbed your niece. Seventeen times with a letter knife."
He swallowed against a lump in his throat. "It's true," he said.
_"Why?"_ she whispered. There was something desperate in the way she said it. Some wide-eyed need to understand how he could do something so utterly loathsome.
Crake stared hard at the table, fighting down the shameful heat of gathering tears.
Jez sat back in her chair. "I can take the half-wits and the incompetents, the alcoholics and the cowards," she said. "I can take that we shot down a freighter and killed dozens of people on board. But I can't be on this craft with a man who knifed his eight-year-old niece to death, Crake. I just can't." She folded her arms and looked away, fighting back tears herself. "How can you be how you are and be a child murderer underneath? How can I trust anyone now?"
"I'm not a murderer," Crake said.
"You killed that girl!"
He couldn't bear the accusations anymore. Damn her, damn her, he'd tell her the whole awful tale and let her judge him as she would. It had been pent up inside him for seven months, and he'd never spoken of it in all that time. It was the injustice, the righteous indignation of the falsely accused, that finally opened the gates.
He took a shaky breath and spoke very calmly. "I stabbed her," he said. "Seventeen times with a letter knife. But I didn't murder her." He felt the muscles of his face pulling toward a sob, and it took him a moment to control himself.
"I didn't murder her because she's still alive."
THE ECHO CHAMBER SAT in the center of Crake's sanctum, silent and threatening. It was built like a bathysphere, fashioned from riveted metal and studded with portholes. A small round door was set into one side. Heavy cables ran from it, snaking across the floor to electrical output points and other destinations. It was half a foot thick and surrounded by a secondary network of defensive measures.
Crake still didn't feel even close to being safe.
He paced beneath the stone arches of the old wine cellar. It was cold with the slow chill of the small hours, and his boot heels clicked as he walked. Electric lamps had been placed around the echo chamber—the only source of light. The pillars threw long, tapering shadows, splaying outward in all directions.
_I have it. I have it at last. And yet I daren't turn it on_.
It had taken him months to obtain the echo chamber. Months of wheedling and begging and scraping to the hoary old bastard in the big house. Months of pointless tasks and boring assignments. And hadn't that rot-hearted weasel enjoyed every moment of it! Didn't he relish seeing his shiftless second son forced to run around at his beck and call! He'd strung it out and strung it out, savoring the power it gave him. Rogibald Crake, industrial tycoon, was a man who liked to be obeyed.
"You wouldn't have to do any of this if you had a decent job," he'd say. "You wouldn't need my money then."
But he _did_ need his father's money. And this was Rogibald's way of punishing him for choosing not to pursue the career picked out for him. Crake had come out of university having been schooled in the arts of politics and promptly announced that he didn't want to be a politician. Rogibald had never forgiven him for that. He couldn't understand why his son would take an uninspiring position in a law firm or why it took more than three years for him to "work out what he wanted to do with his life."
But what Rogibald didn't know, what _nobody_ knew, was that Crake had it worked out long ago. Ever since university. Ever since he discovered daemonism. After that, everything else became petty and insignificant. What did he care about the stuffy and corrupt world of politics, when he could make deals with beings that were not even of this world? _That_ was power.
But daemonism was an expensive and time-consuming occupation. Materials were hard to come by. Books were rare and valuable. Everything had to be done in secret. It required hours of study and experimentation every night, and a sanctum took up a great deal of space. He simply couldn't manage the demands of a serious career while pursuing his study of daemonism, and yet he couldn't get the things he needed on the salary of a lawyer's clerk.
So he was forced to rely on his father for patronage. He feigned a passion for invention and declared that he was studying the sciences and needed equipment to do it. Rogibald thought he was being ridiculous, but he was rather amused by the whole affair. It pleased him to let his son have enough rope to hang himself. No doubt he was waiting for Crake to realize that he was playing a fool's game and to come crawling back. To have Crake admit that he was a failure, that Rogibald was right all along—that would be the sweetest prize. So he indulged his son's "hobby" and watched eagerly for his downfall.
Since Crake was unable to afford accommodation grand enough to suit his needs, his father allowed him to live in a house on the family estate, which he shared with his elder brother, Condred, and Condred's wife and daughter. It was a move calculated to humiliate him. The brothers' disdain for each other was scorching.
Condred was the favored son, who had followed his father into the family business. He was a straitlaced, strict young man who always acceded to Father's wishes and always took his side. He had nothing but contempt for his younger brother, whom he regarded as a layabout.
"I'll take him under my roof if you ask me to, Father," he said, in front of Crake. "If only to show him how a respectable family lives. Perhaps I can teach him some responsibility."
Condred's sanctimonious charity had galled him then, but Crake took some comfort in knowing that Condred regretted the offer now. Condred had envisioned a short stay. Perhaps he thought that Crake would be quickly shamed into moving out and getting a good job. But he'd reckoned without his younger brother's determination to pursue his quest for knowledge. Once Crake saw the empty wine cellar, he wouldn't be moved. He could endure anything, if he could have that. It was the perfect sanctum.
More than three years had passed. Three years in which Crake spent all his free time behind the locked door of the wine cellar, underground. Every night he'd come back from work, share an awkward dinner with his disapproving brother and his snooty, dried-up bitch of a wife, then disappear downstairs. Crake would have happily avoided the dinner, but Condred insisted that he was a guest and should eat with the family. It was the proper thing to do, even if all concerned hated it.
How typical of Condred. Cutting off his nose to spite his face, all in the name of etiquette. Moron.
The only thing that made life in the house bearable, apart from Crake's sanctum, was his niece. She was a delightful thing: bright, intelligent, friendly, and somehow unstained by the sour attitude of her parents. She was fascinated by her uncle Grayther's secret experiments and pestered him daily to show her what new creation he was working on. She was convinced that his sanctum was a wonderland of toys and fascinating machines.
Crake found it a charming idea. He began to buy toys from a local toymaker to give to her, passing them off as his own. Her parents knew what he was doing and sneered in private, but they didn't say a word about it to their daughter. She idolized their layabout guest, and Crake loved her in return.
Those three years of studying and experimenting had brought him to this point. He'd learned the basics and applied them. He'd summoned daemons and bid them to do his will. He'd thralled objects, made simple communications, even healed wounds and sickness through the Art. He corresponded often with more-experienced daemonists and was well thought of by them.
All daemonism was dangerous, and Crake had been very cautious all this time. He'd gone step by tiny step, growing in confidence, never overreaching himself. He knew well the kinds of things that happened to daemonists who attempted procedures beyond their experience. But it was possible to be _too_ cautious. At some point, it was necessary to take the plunge.
The echo chamber was the next step. Echo theory was cutting-edge daemonic science, requiring complex calculations and nerves of steel. With it, a daemonist could reach into realms never before accessed, to pluck strange new daemons from the aether. The old guard—the ancient, fuddy-duddy daemonists—wouldn't touch it, but Crake couldn't resist. The old ways had been mapped and explored, but this was new ground, and Crake wanted to be one of the first to the frontier.
Tonight, he was attempting a procedure he'd never tried before. He was going to bring life to the lifeless.
Tonight, he was going to create a golem.
He stopped his pacing and returned to the echo chamber, checking the connections for the twentieth time. The echo chamber was linked by soundproofed tubes to a bizarre armored suit that he'd found in a curio shop. The shopkeeper had no idea what it was. He theorized that it might have been made for working in extreme environments, but Crake privately disagreed. It was crafted to fit a hunchbacked giant, and it wasn't airtight. He guessed it was probably ornamental or a sculptural showpiece made by some deranged metalworker. At any rate, Crake had to have it. It was so fascinatingly grotesque, and perfect for his purposes.
Now it stood in his sanctum, ready to accept the daemon he intended to draw into it. An empty vessel, waiting to be filled. He studied the armored suit for a long time, until it began to unnerve him. He couldn't shake the feeling that it was about to move.
Surrounding the echo chamber and the suit was a circle of resonator masts. These electrically powered tuning forks vibrated at different wavelengths, designed to form a cage of frequencies through which a daemon couldn't pass. Crake checked the cables, following them across the floor of the sanctum to the electrical output he'd had wired into the wall. Once satisfied, he turned them on one by one, adjusting the dials set into their bases. The hairs on his nape began to prickle as the air thickened with frequencies beyond his range of hearing.
"Well," he said aloud. "I suppose I'm ready."
Standing on the opposite side of the echo chamber from the armored suit was a control console. It was a panel of brass dials, waist high, set into a frame that allowed it to be moved around on rollers. Next to the controls was a desk, scattered with open books and notepads displaying procedures and mathematical formulae. Crake knew them by heart, but he scanned them again anyway. Putting off the moment when he'd have to begin.
He hadn't been so terrified since the first time he summoned a daemon. His pulse pounded in his throat. The cellar felt freezing cold. He'd prepared, and prepared, and prepared, but no preparation would ever be enough. The cost of getting this wrong could be terrible. Death would be a mercy if an angry daemon got its hands on him.
But he couldn't be cautious forever. To be a rank-and-file practitioner of daemonism wasn't enough. He wanted the power and renown of the masters.
He went to the console and activated the echo chamber. A bass hum came from the sphere. He left it for a few minutes to warm up, concentrating on his breathing. He had a feeling he might suddenly faint if he didn't keep taking deep breaths.
_It's still not too late to back out, Grayther_.
But that was just fear talking. He'd made this decision long ago. He steeled his nerve and went back to the console. Steadily, he began to turn the dials.
There was an art to catching a daemon. The trick was to match the vibrations of the equipment to the vibrations of the daemon, bringing the entity into phase with what the uneducated called the "real" world. With minor daemons—little motes of power and awareness, possessing no more intelligence than a beetle—the procedure was simple enough. It was rather like fishing: you placed a sonic lure and drew them in.
But the greater daemons were another matter entirely. They had to be caught and forced into phase. A greater daemon might have six or seven primary resonances that all needed to be matched before it could be dragged unwillingly before the daemonist. And, once there, the daemon needed to be contained. It was a foolish man who tried to deal with an entity like that without taking measures to protect himself.
Crake wasn't stupid enough to think he could handle a greater daemon yet. He was aiming lower. Something with a doglike level of intelligence would suit him very nicely. If he could thrall an entity like that into his armored suit, he'd have a golem dull enough to be biddable. And if it proved troublesome, he had procedures in place to drive it out and back into the aether.
But summoning daemons was dangerous in many ways. A man didn't always know exactly what he was getting. He might fish for a minnow and find a shark on the line.
Crake had made calculations, based on the findings of other echo theorists and his own ideas. He'd identified a range of frequencies where he'd be likely to find what he wanted. Then he commenced the hunt proper.
The echo chamber began to vibrate and whine as he searched along the bandwidth. Daemonism was as much about feel and instinct as science. Crake closed his eyes and concentrated, turning the dials slowly.
There it was. That creeping sensation of being watched. He'd found something. Now he had to catch it before it slipped away.
He set up new resonances, starting high and low and then moving them closer together, feeling out the shape of the entity. He stopped when he felt the resistance of it.
The reaction was more pronounced now: a cold shiver, a slight feeling of vertigo and disorientation. He had to keep his eyes open. When he closed them, he started tipping forward.
He looked at the dials. The thing was huge, spread right across the subsonics.
_Let it go_ , he told himself. _Let it go. It's too big_.
He _had_ it now, though. There was no way he could hold on to something like that with his standard equipment. It would simply phase into a different frequency and escape. But with the echo chamber, he could keep it pinned, pounding it with confusing signals that all interfered with one another.
He could _get_ this one. Forget the golem, forget everything else. He just wanted to see it. Then he'd send it back. But just to _see_ it!
Excited, riding on a fear-driven high, he worked the dials feverishly. He set up more vibrations, seeking the daemon's primary frequencies, narrowing and narrowing the bandwidth until he matched them. The daemon was shifting wavelengths, trying to escape the cage, but he shifted with it, never letting it get away from him. The closer he came, the less space the daemon had to wriggle.
The air was throbbing. The echo chamber pulsed with invisible energies.
_Spit and blood, this is working! This is actually working!_
Once he had it fixed as best he could, he stepped away from the console and went to peer inside the echo chamber. Through the porthole in the door, he could see that the sphere was empty. But he wasn't disheartened. Inside, perspectives bent out of shape, and the air warped in eye-watering contortions. Something was coming. He could hardly breathe for terror and fascination. Leaning close to the thick glass, he tried to see farther inside.
A colossal, mad eye stared back at him.
He yelled, falling away from the porthole, his heart thumping hard enough to hurt. That vast eye had surged out of nowhere, surfacing into his reality, burning itself onto his consciousness. He saw it now, impossibly huge, belonging to something far bigger than the echo chamber could contain.
There was a heavy impact, and the echo chamber rocked to one side. Crake sat where he'd fallen, transfixed. Again, the sound of a giant's fist pounding. The echo chamber dented outward.
_Oh, no. No, no_.
He scrambled to his feet and ran for the console. _Get rid of it, get rid of it, any way you can_.
Another impact, sending a shudder through the whole sanctum. The electric lamps flickered. One tipped over, crashing to the ground. Crake lost his footing, stumbled onward.
And then he heard her scream.
The sound froze him to the bone. It was more dreadful than anything he could imagine, more dreadful than the thing in the echo chamber. His world tipped into the primal, inescapable horror of a nightmare as he looked over at his niece, standing there in her white nightdress. She was just outside the circle of resonator poles, transfixed by the scene before her.
He would never know how she'd gotten the key to the wine cellar. Perhaps she'd found an old copy in some dusty, hidden place. Had she been planning this moment ever since? Had she been unable to sleep, so keen was she to see the secret wonderland of toys where her uncle Grayther worked? Had she set her clock to wake her, hoping to sneak down in the dead of night when she thought he wouldn't be there?
He would never know how or why, but it didn't matter in the end. What mattered was that she was here, and the daemon was uncontainable. The door of the echo chamber flew open, and the last thing he knew before his life changed forever was a hurricane wind that smelled of sulfur, and a deafening, unearthly howl.
WHEN HIS SENSES RETURNED to him, the sanctum was dark and silent. A single electric lamp remained unsmashed. It lay on its side near the echo chamber, underlighting the looming shape of the armored suit, which was still connected by cables to the dented metal sphere.
Crake was disoriented. It took him several seconds to understand where he was. His mind felt scratched and sore, as if rodents had been scrabbling at it from the inside, wounding his senses with small, dirty claws. The daemon had been in his head, in his thoughts. But what had it done there?
He realized he was standing. He looked down and saw in his hand a letter knife with the insignia of his university on the hilt. The knife and the hand that held it were slick and dark with blood.
There was a clicking noise from the shadows. Red smears on the stones. He followed them with his eyes, and there he found her.
Her white nightdress was soaked in red. There were slits in her arms and throat, where the knife had plunged. They welled with rich, thick blood, spilling out in pulses. She was gaping like a fish, making clicking noises in her throat. Each breath was a shallow gasp, and her lips and chin were red. Her brown hair was matted into sodden wads.
Her eyes. Pleading. Not understanding. Dazed with incomprehensible agony. She didn't know about death. She'd never thought it could happen. She'd trusted him, with a blind, unthinking love, and he'd turned on her with a blade.
It was the daemon's revenge for daring to summon it from the aether. It had been cruel enough to leave him his life and wits intact.
Crake hadn't known that pain and despair and horror could reach the heights that they now did. The sheer intensity of it was such that he felt he should die from it. If only the darkness would come back, if only his heart would stop! But there was no mercy for him. Realization smashed down upon him like a tidal wave, and he staggered and gagged, the knife falling from numb fingers.
She was still alive. Alive, begging him to make the pain stop, like some half-broken animal ruined under the wheels of a motorized carriage. Begging him to make it better somehow.
_"She's a child!"_ he screamed at the darkness, as if the daemon was still here to be accused. _"She's just a damned child!"_
But when the echoes had died, there was only the wet clicking from his niece as she tried to draw breath.
What overtook him then was a grief so overwhelming that it drowned his senses. He was seized by an idea, mad and desperate, and he acted on it without thought for consequence. Nothing else was important. Nothing except undoing what had been done, in the only way he could think of.
He scooped her up in his arms. She was so light, so thin and pale, white skin streaked with trails of gore. He carried her to the echo chamber and gently placed her inside. He pushed the door shut. Despite the abuse it had suffered, the lock engaged and it sealed itself. Then a weakness took him, and he fell to his knees, his forehead pressed against the porthole in the door, sobs racking his body.
She was lying on her back, her head tilted, looking at him through the glass. Blood bubbled from her lips. Her gaze met his, and it was too terrible to stand. He flung himself away and went to the control console.
There, he did what had to be done.
JEZ HAD SEEN MEN cry before, but never like this. This was heartbreaking. Crake's sobs were deep, wild, dredged up from a depth of pain that Jez couldn't have imagined he held inside him. His story had become almost impossible to understand as he neared the end. He couldn't even form a sentence through the hacking sobs that shook his whole body.
"I didn't know!" he cried, his face blotched and his beard wet with tears. His nose was running, but he didn't trouble to wipe it. He was ugly and shattered before her. It hurt to see him so. "I didn't know what I was doing! Only it... it didn't work like I thought. The tra-... the tra-... transfer wasn't perfect. She's _different_ now, she's not... like she was..." He gasped in a breath. "I just wanted to _save_ her."
But Jez couldn't give him pity or sympathy. She'd hardened herself too much. She saw the tragedy of him now, but if she let herself forgive him, if she gave in even a little, there would be no going back. He could perhaps be excused the crime of stabbing her, if he wasn't in his right mind. But what he'd done next was nothing short of diabolical.
"One thing," she said. Her voice was so tight that it hardly sounded like her. "Her name."
"What?"
"All this time, you never told me your niece's name. You've avoided it."
Crake stared at her with red eyes. "You know her name."
"Say it!" she demanded. Because she needed this final closure before she could walk away.
He swallowed and choked down a sob.
"Bessandra," he said. "Bessandra was her name. But we all just called her Bess."
# _Chapter Thirty_
ORKMUND'S ADDRESS—A FAMILIAR OBJECT—FREY PUTS IT ALL TOGETHER—"GOTCHA!"
y midday, a crowd had gathered outside Orkmund's stronghold.
In a rare moment of architectural forethought, the stronghold had been built in front of a large square, which was employed for the purpose of meetings, markets, and occasional executions or bouts of trial-by-combat. A wooden stage, now groaning under the weight of spectators, stood in the center for just this purpose. Another, more temporary one had been erected just outside the stronghold and was guarded by burly men with cutlasses. This would be Orkmund's podium.
Frey pushed through the press of bodies, with Malvery clearing the way ahead. Pinn and Jez came behind. Pinn had been subdued by his confinement in the _Ketty Jay_ the night before, and Frey had extracted promises of good behavior today. He charged Malvery with enforcing them, knowing how the doctor liked to bully Pinn.
It was fun to torment the young pilot now and then, but Frey knew how much it meant to him to see Retribution Falls before they left. Just so he could say he'd been. Just so he could tell Lisinda of his adventures, on that day when he returned in triumph to sweep her into his arms. Having asserted his authority, Frey was happy to give Pinn a little leash.
The stronghold was constructed in a squared-off horseshoe shape, with two wings projecting forward around a small interior courtyard. It was grim and forbidding, with square windows and iron-banded doors. Its walls were dark stone, streaked with mold. A place built for someone who had no interest in flair or aesthetics. A fortress.
Surrounding the stronghold was a ramshackle barricade of metal spikes and crossed girders, eight feet high and surmounted by wooden watchtowers. The watchtowers were manned by rifle-wielding pirates, who scanned the crowd below them, no doubt deciding who they'd shoot first if they had the chance. In the middle of the barricade was a crude gate, a thick slab of metal on rollers that could be slid back and forth to grant access to the courtyard.
Frey and the others fought their way to a vantage point as the gate began to open and the crowd erupted in ear-pummeling cheers. The floor shook with the stamping of feet. It occurred to Frey that they were standing on a huge platform that was held up by a scaffolding of girders, and that it might not be built to take this kind of weight. It would be an ignominious end to his adventure, to sink to the bottom of a fetid marsh beneath a hundred tons of unwashed pirate flesh.
It wasn't until Orkmund climbed the steps to his stage that Frey caught sight of him. The pirate captain Orkmund, scourge of the Coalition in the years before the Aerium Wars, who disappeared fifteen years ago and was thought by most to be dead. But he wasn't dead: he was building Retribution Falls. A home for pirates, safe from the Navy. A place where they could conduct their business in peace—with a hefty cut for Orkmund, of course.
Though he must have been in his mid-fifties, Orkmund still cut an impressive figure. He was well over two meters high, bald-headed and thickset, with squashed features that gave him a thuggish look. Tattoos crawled over his throat, scalp, and arms. He wore a simple black shirt, tight and unlaced at the throat, to emphasize an upper body and arms that were heavy with muscle. He walked up to the stage with a predator's confidence, surveyed the cheering crowd, and raised his arms for silence. It took some time.
"Some of you know me by sight," he shouted. His voice, though loud, was still faint and thin by the time it reached Frey's ears, and he had to concentrate to hear. "Some don't. For them new to Retribution Falls: welcome. I'm Neilin Orkmund."
The cheer that erupted at that drowned out anything else for a while. When the crowd was relatively quiet again, Orkmund continued.
"I'm proud to see so many men and women here today. Some of the finest pirates in the land. Some of you've known of this place for years. For others, it'd only been legend until recently. But you've come at my call, and I thank you for that. Together, we'll be an unstoppable force. Together, we'll make an army like Vardia's never seen!"
More cheers. Pinn and Malvery cheered along with them, caught up in the moment.
"Now, I know some of you are frustrated. Champing at the bit. You wanna get into action, don't you? You wanna break some bones and smash some skulls!"
Another deafening cheer, accompanied by clapping and jostling that threatened to turn into a riot.
Orkmund held up his hands. "You've enjoyed my hospitality. You've dipped your beaks in the delights of Retribution Falls. And in return, I ask you only one thing: be patient."
The pirates near to Frey groaned and muttered. Suddenly the fervor had gone out of the crowd.
"I know you're disappointed. No one wants to get out there more than me," Orkmund hollered. "But this ain't no small task we're taking on! We ain't here to rob a freighter or steal a few trinkets from some remote outpost. We ain't just a crew of fifty men, or a hundred. We're a crew of thousands! And a crew of thousands takes time to gather and coordinate."
There were reluctant mumbles of concession at this.
"The time's coming very soon. A matter of days," said Orkmund. "But I've brought you here today because I've something to show you all."
As he spoke, a troop of armed pirates sallied out of the stronghold, guarding two dozen men who were carrying a dozen large chests between them. They carried the chests up onto the stage as Orkmund continued.
"I know that there are doubters out there. What are we doing here? Why are we waiting? Who are we attacking, and why's it still a secret?" Orkmund said, prowling back and forth on the stage. "Well, first ask yourself: why'd you come to Retribution Falls? Why'd you answer my call, when you didn't even know who you was fighting? For some, it was loyalty to me. For some, it was the call to adventure. But for most of you... it were this!"
He threw open one of the chests, and a gasp went up from the crowd.
"Loot! Ducats! Money!" Orkmund cried, and the crowd cheered anew, their spirits roused. He went to the next one and threw that open, revealing that it, too, was full of coins. "All this, for you! Booty! A share for every man who survives, and a right generous share it is too!" He threw open another one. "Now, ain't this worth fighting for? Ain't this worth waiting a few more days for?"
The pirates howled with glee, shaking their fists in the air, driven rabid by the sight of so much money. If not for the respect they had for Orkmund and the multiple guns trained on them, they might have tried to storm the stage right then.
But while Pinn and Malvery were yelling themselves hoarse, Frey had spotted something. He turned to Jez. "Can you see the stage?"
She craned to look over the shoulder of the pirate in front. "Not really."
"Come here," he said, and crouched down to offer her a piggyback.
"No, Cap'n, it's really alright."
"I need your eyes, Jez. Help me out."
Since she couldn't think of a good reason to protest, she climbed awkwardly onto his back and he lifted her up.
"You know, my eyesight's not all that great. I mean, it's—"
"The last chest on the right," said Frey. "Describe it to me."
Jez looked. "It's red."
"Describe it _more,_ " he said irritably.
She thought for a moment. "It's very fine," she said. "Dark-red lacquer. Kind of a branch-and-leaf design on the lid. Silver clasp in the shape of a wolf's head. Oh, wait, he's opening it."
Orkmund was throwing open each chest, whipping the pirates into a frenzy with the wealth paraded before them. Frey didn't need Jez to tell him that the red-lacquered chest was full to bursting with ducats.
And that was it. The final piece fell into place.
"Everyone!" he said. "We're leaving."
Pinn whined in complaint. Malvery raised a threatening hand to cuff him. "Fine," Pinn sulked. "Let's go."
Frey let Jez down to the floor. "Seen enough, Cap'n?" she asked.
"Oh, yeah," he said. "I've seen enough."
THE STREETS WERE RELATIVELY quiet on their way back. Retribution Falls seemed cold and bleak without the din of drunken revelry. Frey stepped through the sludge of debris and bodily fluids from the night before, setting a quick pace. He was eager to get to the _Ketty Jay_. There was a purpose in his walk.
"What's the story, Cap'n?" Jez asked. "Are we getting out of here?"
"That's right," he said. "There's no reason to stay anymore."
"I can think of lots," said Pinn. "Most of them come in pints or bottles; the rest have big wobbling tits. Come on, how about a little shore leave?"
"I'm trying to save us all from the noose, Pinn," Frey replied. "Stay chaste for a day. Think of your sweetheart."
"Thinking of her just makes me want to bang a whore even worse." Pinn grinned, then held his hands up in submission. "Okay, okay. Yes, Cap'n. Back to the _Ketty Jay_ like a good little pilot. But I still don't get what's going on."
"Alright, I'll tell you," said Frey. "We knew that Duke Grephen was planning a coup against the Archduke. What he didn't have was an army big enough to take on the Navy, or the money to pay for it. Orkmund's providing the army, and now we know who's providing the money."
"Do we?" Jez asked. "Who?"
"The Awakeners."
"What makes you think that?"
"That chest on the podium. I saw them bringing it out of the hermitage where Amalicia was being kept. I didn't know what was in it then, but now we do. Money. And look where it ended up: here in Retribution Falls."
"The Awakeners are financing the pirates?" Pinn asked.
"Why?"
"Because they want the Archduke out. Him and his wife."
"What's his wife got to do with it?"
"The Archduchess is the one who's got him talking about all these new laws to limit the power of the Awakeners," Frey said. He was aware that he was losing Pinn already. "Look, the Awakeners run themselves like a business. And there's no question they make bucketloads of money from the superstitious. Now, if someone as powerful as the Archduke starts saying that the whole idea of the Allsoul is rubbish, people are going to start listening to him. And that means the Awakeners start going the way of all the other religions they crushed a century ago."
"You're remarkably well informed these days, Cap'n," Jez commented.
"Been talking to Crake," he said.
"You know he's not exactly impartial, don't you?" she said. When she spoke of the daemonist, he noted that her tone wasn't as obviously scornful as it had been yesterday.
"So why are the Awakeners funding Duke Grephen?" Pinn piped up.
Frey sighed. This would require careful explanation for Pinn to understand. "Because Grephen's an Awakener. Just like Gallian Thade. If he becomes the Archduke, then the Awakeners gain power instead of losing it. In fact, they'd become pretty much unstoppable."
Pinn frowned, pondering that for a moment as they hurried through the narrow, filthy lanes, past peeling walls and rusted steps. "And the Awakeners hired Dracken to catch us?"
"No!" Frey and Malvery cried in unison. It was Frey who continued: " _Grephen_ hired her to catch us. Because he didn't want us talking to anyone and blowing his plan before he could put it into action."
Pinn thought some more. Frey had a feeling of dread in his stomach, anticipating the inevitable follow-up question.
"So who hired the Century Knights?"
Malvery covered his face with a hand in despair.
"What?" Pinn protested. "It's complicated!"
"I swear, mate, you have the brains of half a rock."
" _Nobody_ hired the Century Knights," Jez said. "They're loyal to the Archduke. Nothing to do with Grephen. They're after us because they think _we're_ the villains here."
"We did kill the Archduke's son," Malvery pointed out.
"Accidentally!" Frey said. "And, besides, we were set up. That means it doesn't count."
Malvery raised an eyebrow. "I'd like to see you try that line of argument with the Archduke," he said.
"What Grephen wants," Frey told Pinn, before he could ask another question, "is that we get killed, nice and quiet, and he gets to show the bodies to everyone. Hengar's murderers are caught, case closed. That was the idea from the start. We were supposed to die during the ambush."
"What he _doesn't_ want is the Century Knights catching us and giving us a chance to tell our side of the story," Jez continued. "He's afraid that we know enough to make them suspicious, and that will blow his big surprise attack."
"Which is happening in a few days, if you believe that Orkmund feller," added Malvery.
Pinn gave up trying to figure out who was after whom and asked, "So what do we do?"
"What we do is cut a deal," said Frey. "Talk to some people. Set up a safe rendezvous. We'll give them the charts and the compass, let them come see Retribution Falls for themselves. Once they find the army Orkmund's put together, they'll believe us. We'll offer them the big fish and, in return, we demand a pardon."
Pinn stopped dead. The others walked on a few steps before they noticed.
"You're selling this place out?" he said, appalled.
Frey was confused. "What do you mean?"
"I mean, you're going to tell the Coalition Navy where Retribution Falls is?"
"You think you could shout it a bit louder, Pinn?" Malvery cried. "I don't think they heard you in Yortland."
Pinn looked around furtively, suddenly remembering where he was. Thankfully, the alley they were standing in was deserted, and nobody seemed to have heard. He scuttled up closer to Frey and jabbed him in the chest with a finger.
"This place is a legend! This place was built with the sweat and tears of a generation of pirates. It's been the hope of every freebooter since the Aerium Wars that they could one day find Retribution Falls and live out the rest of their days in pirate wonderland. It's a yoo—a yoo—"
"Utopia," Jez said. "Pinn, it's a dump."
Pinn was aghast. "It's Retribution Falls!"
Jez studied her surroundings critically. The sagging roofs, the cracked walls and mildewed corners, the broken bottles and bloodstains. She sniffed, taking in the rank stench of the marsh.
"You know what pirates are really good at, Pinn?" she said. "Being pirates. And that's all. In fact, if you asked me what would happen if you took a thousand pirates and asked them to build a town, I'd say it would look pretty much like this. This place was better as a legend. The real thing doesn't work."
"Let me put it this way, Pinn," said Frey. "Do you want to get hanged, or don't you?"
Pinn examined the question for a trick. "No?" he ventured.
"It's either you or this place. Orkmund's working for Duke Grephen, remember? And Grephen wants all of us dead. You too, Pinn."
Pinn opened his mouth, shut it, opened it again, and then gave up trying to argue. "Lisinda would never get over it if anything happened to me," he said.
Malvery beamed. "Think how proud she'll be when she learns you single-handedly triumphed over an army of pirates."
"I suppose I could dress it up a little," Pinn mused. "Alright, spit on this place. Let's get out of here and stab some backs!"
"That's the spirit!" Frey said cheerily.
BACK AT THE KETTY JAY, Frey issued instructions for takeoff and made sure Slag was trapped in the mess so some unlucky volunteer—Pinn—could force a mouth filter on him during the journey back. Silo was showing Frey some superficial damage to the underwings when Olric, the dockmaster's assistant, wandered up to them.
"Leaving, are you?"
"Just got an errand to run," said Frey. "Orkmund says it'll be a few days yet, so..." He shrugged.
"You gotta sign out."
"I was about to. Be over there in a minute."
Olric ambled away again. Frey asked Silo to fetch Crake from inside, and the daemonist came down the cargo ramp shortly after.
"You needed me?"
"You and Jez sort things out last night?" he asked.
Crake didn't meet his eye. "As best we could."
Frey wasn't encouraged. "Can you come with me to the dockmaster's office? I need to sign out before we fly."
Crake gave him a puzzled look. "Two-man job, is it?"
"Actually, yes. I need you to distract the dockmaster. I mean _really_ distract him. You think you can do the thing with the tooth?"
"I can try," said Crake. "Did he strike you as particularly smart or quick-witted?"
"Not really."
"Good. The less intelligent they are, the better the tooth works. It's the smart ones who cause all the problems."
"Don't they always?" Frey commiserated, as he led the daemonist across the landing pad.
"What are you up to, anyway?" Crake asked.
"Taking out a little insurance," replied Frey, with a wicked smile.
THE JOURNEY OUT WAS less traumatic than the journey there. Now that they had filters to protect against the strange fumes from the lava river, and they knew the trick of the compass and the mines, things were not so daunting. The only drama came from Pinn, who had a miserable time trying to subdue the cat, until Malvery hit on the idea of getting him drunk first. A quarter bottle of rum later, Slag was placid enough to take the mouth filter, after which they headed to Malvery's surgery to apply antiseptic to Pinn's scratched-up arms and hands.
There had been talk of ignoring the charts and flying straight up and out of there, instead of the laborious backtracking through the canyons, but they soon discovered that there was a reason why nobody did that. The area above Retribution Falls was heavily mined, and Jez theorized that these could be more magnetic than the ones they'd encountered, meaning that they'd home in on the _Ketty Jay_ from a greater distance. Frey decided not to push their luck. They'd follow the charts.
Frey had Jez and Crake up in the cockpit again, one to navigate and one to read from the compass while he flew. The atmosphere between them had changed. Instead of sniping, Jez did not talk to Crake at all, beyond what was necessary to coordinate their efforts. Crake also seemed very quiet. Something was different between them, for sure, but Frey had the sense that it wasn't entirely resolved yet.
Well, at least there had been progress. They weren't fighting anymore. It was a start.
Frey was lighthearted as he piloted them through the fog. He was beginning to feel that things were really pulling together for them now. The changes had been slow and subtle, but ever since they'd left Yortland he'd felt more and more like the captain of a crew, rather than a man lumbered with a chaotic rabble. Instead of letting them do whatever they felt like, he'd begun to give them orders, and he'd been surprised how well they responded once he showed a bit of authority. They might gripe and complain, but they got on with it.
The raid on Quail's place had been a complete success. Jez and Crake's infiltration of the Winter Ball had yielded important information. And the theft of the compass and charts from the _Delirium Trigger_ was their crowning glory so far. A month ago, he couldn't have imagined pulling off anything so audacious. In fact, a month ago he couldn't have imagined himself giving anybody orders. He'd have said: _What right do I have to tell someone else what to do?_ He didn't think enough of himself to take command of his _own_ life, let alone someone else's.
But it wasn't about rights, it was about responsibilities.
Whether as passengers or crew, the people on board the _Ketty Jay_ endured the same dangers as he did. If he couldn't make them work together, they all suffered. His craft was the most important thing in the world to him, yet he'd never given a damn about her contents until now. It had always been just him and the _Ketty Jay_ , the iron mistress to whom he was forever faithful. She gave him his freedom, and he loved her for it.
But a craft was nothing without a crew to operate her and pilots to defend her. A craft was made up of people. The _Ketty Jay_ was staffed with drunkards and drifters, all of them running from something—whether it be memories or enemies or the drudgery of a land-bound life—but since Yortland, they'd been running in the same direction. United by that common purpose, they'd begun to turn into something resembling a crew. And Frey had begun to turn into someone resembling a captain.
Damn it, he was getting to _like_ these people. And the thought of that frightened him a little. Because if his crew got hanged, it would be on his account. His fault. He'd gotten them all into this, by taking Quail's too-good-to-be-true offer of fifty thousand ducats. He'd made that desperate gamble, closed his eyes, and hoped for a winning card, but he'd drawn the Ace of Skulls instead.
Jez, Crake, Malvery, Silo... even Harkins and Pinn. They weren't just badly paid employees anymore. Their lives had come to rest on his decisions. He didn't know if he could bear the weight of that. But he did know that he had no choice about it.
"No mines nearby," Crake reported.
"I think we're through, Cap'n," Jez said, slumping back in her seat. "You can start your ascent anytime now."
"Well," Frey said. "That was Rook's Boneyard. I hope you all enjoyed your tour."
They managed weak smiles at that. He cut the thrusters and fed aerium gas into the ballast tanks, allowing the _Ketty Jay_ to rise steadily. The fog thinned, and the mountainsides faded into view.
"Never thought I'd miss daylight quite so badly," Frey said. "It better be sunny up there."
There was no danger of sun, this deep in the Hookhollows, with the clouds and drifting ash high in the sky overhead. But the mist oppressed him. He wanted to be able to see again.
The _Ketty Jay_ rose out of the white haze, and the sky exploded all around them. The concussion threw the _Ketty Jay_ sideways and sent the crew flying from their seats onto the floor. Frey scrambled back into his seat, half blinded by the flash of light, thinking only of escape. _Get out of here, get out of here, get_ —
But the blast had spun the _Ketty Jay_ around, and now he could see their assailant through the windglass of the cockpit. Her black prow loomed before them, a massive battery of guns trained on his small craft.
The _Delirium Trigger_.
Frey slumped forward onto the dashboard. The first shot had been a warning. Her outfliers had surrounded them, waiting for the slightest hint that they were going to run. But Frey wasn't going to run. It was hopeless. They'd be blown to pieces before he had time to fire up the thrusters.
_Not like this. I was so damn close_.
The _Delirium Trigger_ 's electroheliograph mast was blinking.
Jez, who had staggered to her feet and was standing behind the pilot's chair, narrowed her eyes as she watched it.
"What's it say?" Frey asked.
_"Gotcha!"_ Jez replied.
Frey groaned. "Bollocks."
# _Chapter Thirty-one_
ONE IS MISSING—FREY IS PUT TO THE QUESTION—GOOD NIGHT, BESS
_knew I should have gotten out when I had the chance_ , Crake thought, as the men of the _Delirium Trigger_ flooded up the _Ketty Jay_ 's cargo ramp. Six of them covered the prisoners while the others dispersed through the hold, checking corners, moving with military precision. Wary eyes flickered over Bess, who was standing quietly to one side.
"You tell that thing, if it moves, you all get shot," snarled one of the gunmen.
"She won't," said Crake, the words coming out small. "I put her to sleep."
He'd been forced to. He couldn't trust that Bess would behave when their lives were under threat.
The gunman jabbed Crake with the muzzle of his revolver. Bess didn't react. "She'd better not. Or you're the first to go."
The crew of the _Ketty Jay_ stood at the top of the ramp, offering no resistance. All except Jez, anyway. Where Jez was, only the captain knew. Crake had seen her speaking urgently with Frey as they were being escorted out of the mountains. Later, after they were instructed to land in the vast wastes of the Blackendraft, she was gone. When Malvery inquired as to her whereabouts, Frey said, "She's got a plan."
"Oh," said Malvery. "What kind of plan?"
"The kind that won't work."
Malvery harrumphed. "No harm in trying, I suppose."
"That's what I thought."
They were patted down. None of them was carrying weapons, but Crake's heart sank a little further when a crewman pulled his skeleton key from the inside pocket of his greatcoat and held it up in front of his face.
"What's this for?" the crewman demanded.
"My house," Crake lied. The crewman snorted and tossed it away. It skidded across the floor of the cargo hold and into a dark corner. With it went any hope that Crake had of escaping from the _Delirium Trigger_ 's brig and saving their hides.
Once the invaders were satisfied they'd been stripped of anything dangerous, Frey and his crew were herded down the ramp at gunpoint. Crake was sweating and his stomach roiled. The future was closing in on him rapidly, arrowing him toward the gallows. He couldn't see a way out of this one. They were surrounded by overwhelming firepower and completely at Dracken's mercy. There would be no miraculous rescue this time.
Pinn whistled as he walked down the ramp, totally oblivious to the seriousness of their situation. Even now he believed in his own heroic myth enough to trust that a hair-raising escape was just around the corner. Crake hated him for that happy ignorance.
Outside, the world was as bleak as their prospects. The ash flats to the east of the Hookhollows were desolate and grim, featureless in every direction. Even the nearby mountains were invisible beneath the rim of the great plateau. From horizon to horizon was a dreary gray expanse, a dead land choked beneath the blanket of dust and flakes that drifted from the west. A chill wind stirred powdery rills from the ground and harried them into the distance. The sky overhead was the color of slate. The disc of the sun was faint enough to stare at without discomfort.
Looming in the sky to their left was the _Delirium Trigger_ , its massive keel imposingly close, as if it might plunge down and crush them at any moment. Closer by was the small passenger shuttle used to ferry crew from the craft to the ground and back again. The _Delirium Trigger_ was too huge to land anywhere except in specially designed docks.
Their captors halted them at the bottom of the ramp. Standing before them, a short distance away, was a slight figure, dressed head to toe in black. Crake recognized her from Frey's description: the shockingly white skin, the short albino-blond hair torn into clumps, that black, fearsome gaze. She regarded them icily as one of her men walked over to her and whispered something in her ear, then she gave him a short command and he hurried back into the bowels of the _Ketty Jay_. After that, she walked up to Frey. Mutual loathing simmered in their eyes.
"The ignition code, please," she said.
"You know that's not gonna happen," he said. "You've got us. What do you want my aircraft for?"
"Sentimental value. The code?"
"She's not worth anything compared to the reward you'll get for bringing us in. Leave her here."
"She's worth everything to you. Besides, the press will want some ferrotypes of the craft that shot down the _Ace of Skulls_. Perhaps I'll present it to the Archduke as a gift. It may encourage him to overlook certain rumors about my activities elsewhere in the future."
"This is pointless. You won't—"
Dracken pulled a revolver in one quick move and pressed the muzzle against his chest, silencing him. "It wasn't a request. Give me the code."
Frey was shaken; Crake could see it. But he bared his teeth into something approximating a grin and said, "Shoot me if you like. You'll just save the hangman a job."
Dracken and Frey stared at each other: a test of wills. Dracken's finger twitched on the trigger. She was sorely tempted. Then she took the gun away and stepped back.
"No," she said. "You get to live. Duke Grephen will want a signed confession out of you. Besides, there's someone else who may be more willing to talk. I understand there was a woman flying the _Ketty Jay_ that night when you stole my charts. I don't see her here. Where is she, Frey? Won't she know the code?"
Frey didn't reply. Dracken spotted one of her men coming out of the _Ketty Jay_ and heading over to her. "Let's find out," she said. She addressed the crewman, a whiskery, heavyset fellow with a steel ear to replace one that had been cut off. "Anyone else inside?"
"One," he said. "In the infirmary. She's dead, though I ain't sure what of."
Trinica looked at Frey for an instant. "You're sure she's dead?"
"Yes, Cap'n. She don't have a pulse, and she ain't breathing. I listened at her chest, and her heart ain't beating. I seen a lot of dead men and women, and she's dead."
"She hit her head," said Frey. "When you shelled us." He indicated Malvery. "The doc tried to help her, but he couldn't do much. All the damage was inside."
Malvery caught on and nodded gravely. "Terrible thing. Fine young woman," he murmured.
Crake felt a chill go through him. He was remembering that night on the Feldspar Islands when they'd gone to Gallian Thade's ball at Scorchwood Heights. The night when Jez had _really_ fallen and hit her head. Fredger Cordwain, the man from the Shacklemore Agency, had taken her pulse then too. He'd also been convinced she was dead. At the time, Crake had assumed he was mistaken in the heat of the moment, but now he wondered.
How had she managed to fool them both?
"You want us to get rid of her?" the crewman asked Dracken.
"No," she said. "Leave her where she is. We'll need the body to show the Duke. How are they getting on with the golem?"
"Coming out now, Cap'n," he replied, gesturing at the half-dozen men who were manhandling the inert form of Bess down the ramp.
"What are you doing with her?" Crake blurted in distress, before good sense could intervene.
Dracken's black eyes fixed onto him. Crake had a sudden and dreadful feeling that he'd done something very foolish in drawing her attention. "That thing is yours, is it?" she asked. "You're the daemonist?"
Crake swallowed and tasted ash in the back of his throat. Dracken sauntered over toward him, raking her gaze along the line of prisoners as she went.
"Very clever, what you did in Rabban," she murmured. "And surprising too. I'd have expected a daemonist to abandon their golem and make a new one, but you actually _rescued_ it from my cargo hold." She studied him with an intensity that made him squirm. "That's very interesting."
Crake kept his mouth shut. He had the impression that anything he said would only damn him further.
"Still, interesting as it is, I'm not stupid enough to fall for the same trick twice," she said. "And I'm not having that thing wake up on the journey back. So your golem is staying here."
Crake felt weakness flood through him. The horror of it almost made him stagger. He looked around wildly, taking in the endless, trackless expanse of gray that surrounded them. There were no signs of life anywhere. No civilization. Nothing but the tiny smudges of aircraft heading for the coast, hopelessly distant.
To abandon her here would be to lose her forever.
"I've an idea," said Dracken, addressing Frey. "It seems the only other person who knows the ignition code is dead, and I'd rather not kill you until after you've given us a confession. But a daemonist... well, he could be problematic. They have all kinds of... arts. Probably easier to get rid of him now."
Crake saw what was coming. She lifted her gun and pointed it at his forehead in what was becoming a depressingly familiar state of affairs.
"Unless you've something to tell me, Frey?" she prompted.
Frey's face had gone stony. Crake had seen that impassive expression before, when Lawsen Macarde put him in a similar situation. Except, this time, there was little doubt that Trinica's gun was fully loaded.
A strange calm came over him. _Let it end, then_.
"You have until three," said Trinica. "One."
He was tired. Tired of struggling against the grief and shame. Tired of living under the weight of one arrogant mistake, to think that he might summon one of the monsters of the aether and come away unscathed. Tired of trying to understand that awful twist of fortune that had led his niece to his sanctum on that particular night, instead of any other.
Leave her here, amid the ash and dust. If he didn't wake her up, no one ever would. Let her sleep, and perhaps she'd dream of better things.
"Two."
He closed his eyes and, to his faint surprise, dislodged a tear. He felt it trickle down the side of his face, over the hump of his cheekbone, to be lost in his beard.
He'd worked so hard to be great. It had ended in ignominy, disgrace, and failure. What was a world worth that treated its inhabitants so?
"Thr—" Trinica began.
"Stop!" Frey snapped.
Crake's eyes stayed closed. Hovering on the razor-blade edge between existence and oblivion, he dared not tip the balance with the slightest movement.
"Seven sixty-seven, double one, double eight," he heard his captain say.
There was a long pause. His body shook with each thump of his heart. He didn't even hope. He didn't even know if he wanted to be left in the world of the living.
But the choice wasn't his to make. He felt the chill metal of the revolver muzzle leave his forehead. His eyes fluttered open. Dracken had stepped back and was regarding him like a child who has just spared an insect. Then she turned to Frey and raised an eyebrow. Frey looked away angrily.
Crake felt detached from himself, clothed in a dreamlike numbness. He watched as Dracken's crew carried Bess away from the _Ketty Jay_. Then, with obvious glee, they stood her on her feet. A hunched metal statue, a monument to their victory. He heard Dracken order the man with the steel ear to assign two men to fly the _Ketty Jay_ behind them. Frey wouldn't meet anyone's eye: he'd been broken by Dracken and was burning with a hate and fury such as Crake had never seen him show.
But it all seemed far away and inconsequential. He was still alive, somehow, although he wasn't sure he'd fully returned from the brink yet.
Someone patted his shoulder. Malvery. They were being urged toward the nearby passenger shuttle. From there they'd be taken to the _Delirium Trigger_ 's brig. Crake sent a mental message to his feet to get them moving. Dazed, he stumbled along with the group, his boots scuffing up little gray clouds. They were herded up some steps and into the belly of the shuttle, where they sat, surrounded by armed guards.
Crake looked out through the shuttle door at the lonely figure of Bess. The crewmen had deserted her now and were attending to other tasks. The shuttle was powering up its engines, sending veils of dust to coat her.
_Let her sleep_ , he thought. _Good night, Bess_.
Then the door slammed closed, and she was lost from his sight.
# _Chapter Thirty-two_
AN AUDIENCE WITH DRACKEN—BRINGING UP THE PAST—THE UGLY TRUTH OF IT ALL
ut, you."
Frey looked up and saw a thickset, bald man with a bushy black beard on the other side of the bars. "You mean me?"
"You're the cap'n, ain't ya?"
He glanced around at his crew, trying to decide whether there was any advantage in protesting. All six of them had been put in the same cell of the _Delirium Trigger_ 's brig. There were five cells in all, each capable of holding ten men. The walls were metal, and the lights were weak. The smell of oil was in the air, and the sound of clanking machinery and distant engines echoed in the hollow spaces.
Silo met his eyes with a customarily inscrutable gaze. Malvery just shrugged.
"I'm the captain," Frey said at length.
"Cap'n Dracken wants to see you," the bald man informed him.
The gaoler unlocked the door and pushed it open, waving a shotgun to deter any attempts at a breakout. Frey walked through, and the door clanged shut behind him.
"Hey," said Malvery. "While you've got her ear, ask if we can get some rum down here, eh?"
Pinn laughed explosively. Crake didn't stir from where he sat in a corner, drowned in his own misery. Harkins had fallen asleep, tired out by being afraid of everything. Silo was silent.
And Jez? What was Jez doing right now? Frey had turned it over and over in his mind, but he still couldn't work out how she had faked her own death convincingly enough to fool Trinica's man. She'd refused to reveal how she was going to do it when she first told him of her plan. She just said, "Trust me."
Still, he was beginning to wonder if she actually _had_ died.
The bald man took him by the arm and pressed a pistol to his side, then walked him out of the brig and through the passageways of the _Delirium Trigger_. They passed other crew members on the way. Some sneered triumphantly at Frey; others gave him looks of abject hatred. Their humiliation at Rabban—not to mention the deaths of a dozen or so crewmen—hadn't been forgotten.
When they reached the door to the captain's cabin, the bald man brought him to a halt. Frey expected him to knock, but he didn't. He appeared to be deliberating some question with himself.
"Are we going in?" Frey prompted.
"Listen," replied the crewman, turning on Frey with a threatening look in his eyes. "You be careful what you say in there. The Cap'n—she's in one of her moods."
Frey arched an eyebrow. "Thanks for the concern," he said sarcastically. "What's she going to do, kill me?"
"It ain't you I'm concerned about," came the reply, and then he knocked on the door and Trinica called for them to enter.
Trinica's cabin was well ordered and clean, but the dark wood of the bookcases and the brass fittings of the dim electric lamps gave it a close, gloomy feel. Trinica was sitting behind her desk at the far end of the room, on which a large logbook lay open next to a carefully arranged writing set and the brass compasslike device they'd used to navigate the minefields of Retribution Falls. She was looking out of the sloping window. Beyond, night had fallen.
She didn't acknowledge Frey as he was brought in. The bald man stood him in the center of the room. After a moment, without turning from the window, she said:
"Thank you, Harmund. You can go."
"Cap'n," said the big man, and left.
Frey stood uncertainly in the center of the room, but still she didn't speak to him. He decided he'd be damned if he'd feel awkward in front of her. He walked over to a reading chair by one of the bookcases and sat down in it. He could wait as long as she could.
His eyes fell to the compass on the desk. The sight of it inspired a momentary surge of bitterness. That would have been his proof. That device and the charts that came with it would have won him his freedom. He'd been so close.
He fought down the feeling. No doubt Trinica had put it there to inspire just such a reaction. Railing against the injustice of his circumstances would do him no good now. Besides, for the first time he could remember, it felt a little childish.
"You're going to hang, you know," she said at last. She was still staring out of the window.
"I'm aware of that, Trinica," Frey replied scornfully.
She glanced at him then. There was reproach in her eyes. Hurt, even. He found himself regretting his tone.
"I thought we should talk," she said. "Before it's over."
Frey was puzzled by her manner. This wasn't the acerbic, commanding woman he'd met back in Sharka's Den, nor did he recognize her behavior from the years he'd loved her. Her voice was soft, the words sighing out without force. She seemed deeply tired, steeped in melancholy.
Still suspicious of a trick, he resolved not to play into her hands. He'd give her no sympathy. He'd be hard and bitter.
"Talk, then," he said.
There was a pause. She seemed to be seeking a way to begin.
"It's been ten years," she said. "A lot's happened in that time. But a lot of things stayed... unresolved."
"What does it matter?" Frey replied. "The past is the past. It's gone."
"It's _not_ gone," she said. "It never goes." She turned away from the window and faced him across her desk. "I wish I had your talent, Darian. I wish I could walk away from something or someone and it would be as if they never existed. To lock a piece of my life away in a trunk, never to be opened."
"It's a gift," he replied. He wasn't about to explain himself to her.
"Why did you leave me?" she asked.
The question took him by surprise. There was a pleading edge to it. He hadn't expected anything like this when he was led into the room. She was vulnerable, strengthless, unable to defend herself. He found himself becoming disgusted with her. Where was the woman he'd loved, or even the woman he'd hated? This desperation was pitiable.
Why _had_ he left her? The memories seemed distant now. It was hard to summon up the feelings he'd felt then. They'd been tinted by ten years of scorn. Yet he did remember some things. Thoughts rather than emotions. The internal dialogues he had with himself during the long hours alone, flying haulage for her father's company.
In the early months, he'd believed they'd be together forever. He told himself he'd found a woman for the rest of his life. He couldn't conceive of meeting someone more wonderful than she was, and he wasn't tempted to try.
But it was one thing to daydream such notions and quite another to be faced with putting them into practice. When she began to talk of engagement, with a straightforwardness that he'd previously found charming, he began to idolize her a little less. His patience became short. No longer could he endlessly indulge her flights of fancy. His smile became fixed as she played her girlish games with him. Her jokes all seemed to go on too long. He found himself wishing she'd just be sensible.
At nineteen, he was still young. He didn't make the connection between his sudden moodiness and irritability and the impending threat of marriage. He told himself he _wanted_ to marry her. It would be stupid not to, after all. Hadn't he decided she was the one for him?
But the more he snapped at her, the more demanding she became. Tired of waiting—or perhaps afraid to wait too long—she asked him to marry her. He agreed and secretly resented her for a long time afterward. How could she put him in that position? To choose between marrying her, which he didn't want, or destroying her, which he wanted even less? He had no option but to agree at the time and hope to find a way out of it later.
And yet Trinica seemed blissfully unaware of any of this. Though his bad moods were ever more frequent, they didn't seem to trouble her anymore. She was assured that he was hers, and he seethed that she would celebrate her victory so prematurely.
By the time the date of the wedding was announced, Frey's thoughts were mainly of escape. He slept little and badly. Her father's obvious disapproval encouraged him to think that the wedding was a bad idea. A barely educated boy of low means, raised in an orphanage, Frey wasn't a good match for the highly intelligent and beautiful daughter of an eminent aristocrat. Those social barriers, which had seemed laughable in the first flush of love, suddenly rose high in Frey's mind.
He wanted to be a pilot for the Coalition Navy, steering vast frigates to the north to do battle with the Manes or south to crush the Sammies. He wanted to be among the first to land on New Vardia or Jagos after the Great Storm Belt calmed. He wanted to fly free across the boundless skies.
When he looked at Trinica, and she smiled her perfect smile, he saw the death of his dreams.
That was when she became pregnant. The wedding was hastily brought forward, and her father's opposition to it transformed into wholehearted support of their enterprise, backed up by veiled threats if Frey should waver. Frey began to suffer panic attacks in the night.
He remembered the sensation of a vise around his ribs, squeezing a little harder with every day that brought him closer to the wedding. He never seemed to have quite enough breath in his body. The laughter of his friends as they congratulated him became a distressing cacophony, like an enraged brace of ducks. He felt harried and harassed wherever he went. The smallest request was enough to send him into a fluster.
He remembered wondering what it would be like to feel like that forever.
By this point he was absolutely certain he didn't want to marry her. But it didn't mean he didn't want to be with her. Even with all the irritation and buried anger, he still adored this woman. She was his first love, the one who had teased him from his rather cold, uninspiring childhood into a wild world where emotions could be overpowering and deeply irrational. He just wanted things to go back to the way they were before she began to talk about marriage.
But he was terrified of making the wrong choice. What if she _was_ the one for him? Would he be condemning himself to a life of misery? Would he ever meet anyone like her again?
He was paralyzed, trapped, dragged reluctantly into the future like a ship's anchor scoring its way along the seabed. In the end, there was only one way out he could face, and that was to not face it at all. He couldn't make even that decision until the very last minute. He was hoping desperately for some vaguely defined intervention that would spare him from hurting her.
None came, so he ran. He took the _Ketty Jay_ , in which was everything he had in the world, and he left her. He left her carrying his child, standing in front of a thousand witnesses, waiting for a groom who would never come.
After that, it only got worse.
"Darian?" Trinica prompted. Frey realized he'd slipped into reverie and fallen silent. "I asked you a question."
Frey was taken by a sudden surge of anger. What right did she have to make him explain himself? After what she'd done? His love for her had been the most precious thing in his life, and she'd ruined it with her insecurities, her need to tie him down. She'd made him cowardly. In his heart he knew that, but he could never say it. Instead, he attacked her, sensing her weakness.
"You really think I'm interested in a little catch-up to make you feel better?" he sneered. "You think I care if you understand what happened or not? Here's a deal: you let me go and I'll have a nice long chat with you about all the terrible things I did and what an awful person I am. But in case it escaped your notice, I'm going to be hanged, and it's you who's taking me to the gallows. So piss on your questions, Trinica. You can go on wondering what went wrong until you rot."
Trinica's expression was surprised and wounded. She'd not expected such cruelty. Frey found himself thinking that the white-skinned bitch who had taken the place of his beloved might actually cry. He'd expected anger, but instead she looked like a child who had been unjustly smacked for something she didn't do. A profound sadness had settled on her.
"How can you hate me like this?" she asked. Her voice was husky and low. "How can you take the moral high ground, after what you did to me?"
"Broken hearts mend, Trinica," Frey spat. "You murdered our child."
Her eyes narrowed at the blow, but any promise of tears had passed. She turned her face away from him and looked out of the window again. "You abandoned us," she replied, grave-cold. "It's easy to be aggrieved now. But you abandoned us. If it had lived, you'd never have known it existed."
"That's a lie. I came back for you, Trinica. For both of you."
He saw her stiffen and cursed himself. He shouldn't have admitted that, shouldn't have let the words free from his mouth. It weakened him. He'd waited years to throw his hatred in her face, to confront her with what she'd done, but it had always gone so much better during the rehearsals in his head. He wanted her to wreck herself on his glacial indifference to her suffering. He wanted to exact revenge. But his own rage was foiling him.
She was waiting for him to go on. He had no choice now. The gate had been opened.
"I went from place to place for a month. Thinking things through. A bit of time away from you with all your bloody demands and your damn father." He cut himself off. Already he sounded surly and immature. He took a breath and continued, trying not to let his anger overwhelm him. "And I decided I'd made a mistake." He thought about trying to explain further, but he couldn't. "So I came back. I went to see a friend in town, to get some advice, I suppose. That was when I heard. How you'd taken all those pills, how you'd tried to kill yourself. And how the baby... the baby hadn't..."
He put his fist to his mouth, ashamed of the way his throat closed up and his words jammed painfully in the bottleneck. When the moment had passed, he relaxed and sat back in his seat. He'd said enough. There was no satisfaction in this. He couldn't even hurt her without hurting himself.
"I was a stupid girl," said Trinica quietly. "Stupid enough to believe the world began and ended with you. I thought I could never be happy again."
Frey had sat forward in his chair, his elbows on his knees and his fingers tangled in his fringe. His voice was brittle. "I ran out on you, Trinica. But I never gave up on myself. And I never tried to take our child with me."
"Oh, you gave up on yourself, Darian," she replied. "You were just a little more indirect. You spent three years drinking yourself to death and putting yourself in harm's way. In the end, you took your whole _crew_ with you."
Frey couldn't muster the energy to argue. The weary, conversational tone in which she delivered her accusation robbed him of the will to defend himself. Besides, she was right. Of course she was right.
"We're both cowards," he murmured. "We deserved each other."
"Maybe," said Trinica. "Maybe neither of us deserved what we got."
All the fire had gone out of Frey. A black, sucking tar pit of misery threatened to engulf him. He'd imagined this confrontation a thousand ways, but they all ended with him demolishing Trinica, forcing her to face the horror of what she'd done to him. Now he realized there was nothing he could say to her that she hadn't already thought of, nothing he could punish her with that she hadn't already used to punish herself more effectively than he ever could.
The truth was, his position was so fragile that it fell apart when exposed to the reality of an opposing view. While he nurtured his grievances privately, he could be appalled at how she'd mistreated him. But it didn't hold up to argument. He couldn't pretend to be the only one wronged. They'd ruined each other.
Damn it, he hadn't wanted to talk. And now here they were, talking. She always had a way of doing that to him.
"How'd you get this way, Trinica?" he said. He raised his head and gestured at her across the gloomy study. "The hair, the skin..." He hesitated. "You used to be beautiful."
"I'm done with beautiful," she replied. There was a long pause, during which neither of them spoke. Then Trinica stirred in her seat and faced him.
"You weren't the only one who turned away from me after I tried to kill myself," she said. "My parents were disgraced. Bad enough they had a daughter who was going to give birth outside of marriage; now she'd killed their grandchild. They could barely look at me. My father wanted to send me to a sanatorium.
"In the end, I stole some money and took an aircraft. I didn't know where I was going, but I had to get away. I suppose I thought I could be a pilot.
"I was caught by a pirate two weeks later. They must have seen me in port and followed my craft out. They forced me down and boarded me, then took my craft to add to their little fleet. I thought they'd kill me, but they didn't. They just _kept_ me."
Frey couldn't help a twinge of pain. That dainty, elegant young woman he'd left behind hadn't been equipped to survive in the brutal, ugly world of smugglers and freebooters. She'd been sheltered all her life. He knew what happened to people like that.
"I wasn't much more than an animal to them," she went on. Her tone was dead, without inflection. "A pet to use as they pleased. That's what _beautiful_ does for you.
"It took me almost two years to work up the courage to put a dagger in the captain's neck. After that, I stopped being a victim. I signed on as a pilot for another crew, learned navigation on the side. I wanted to make myself indispensable. I didn't want to be dependent on anybody again."
She turned her attention to the window, evading him.
"I'll not bore you with the details, Darian. Let's just say I learned what it takes for a woman to survive among cutthroats."
The omissions spoke more than any description ever could. Frey didn't need to be told about the rapes and the beatings. Physically weak, she'd have needed to use her sexuality to play men off against one another, to ensnare a strong companion for protection. A rich girl who'd never known hardship, she'd been forced into whoredom to survive.
But all that time, she'd been strengthening herself, becoming the woman he saw before him. She could have gone home at any point, back to the safety of her family. They'd have taken her back, of that he was sure. But she never did. She cut out every soft part of herself, so she could live among the scum.
He didn't pity her. He couldn't. He only mourned the loss of the young woman he'd known ten years ago. This mockery of his lover was his own doing. He had fashioned her, and she damned him by her existence.
"By the time I got to the _Delirium Trigger_ , I'd made my way in the underworld. I had a reputation, and they respected me. I knew the crew was troubled and I knew the captain was a syphilitic drunk. It took me a year, building trust, winning them around. I knew he was planning an assault on an outpost near Anduss, I knew it would be a disaster, and I waited. Afterward, I led the survivors against him. We threw him overboard from two kloms up."
She gazed across at him. Her black eyes seemed darker in the faint light of the electric lamps.
"And then you turned yourself into a ghoul," he finished.
"You know how men are," she said. "They don't like to mix desire and respect. They see a beautiful woman in command and they belittle her. It makes them feel better about themselves." She looked away, her face falling into shadow. "Besides, being pretty never brought me anything but pain."
"It kept you alive," he pointed out.
"That wasn't living," she returned.
He had no answer to that.
"So that's the story," she said. "That's what it takes to be a captain. Patience. Ruthlessness. Sacrifice. You're too selfish to make that crew respect you, Darian. You surprised me once, but it won't happen again."
There was a knock at the door. A spasm of irritation crossed her face. "I gave orders that I wasn't to be disturbed!" she snapped.
"It's urgent, Cap'n!" came a voice from the other side. "The _Ketty Jay_ has gone!"
_"What?"_ she cried, surging to her feet. She tore open the door to the cabin. A crewman was outside, obscured from Frey's view by the door.
"She were following us with her lights on," came the breathless report. "All of a sudden the lights go out. By the time we got a spotlight over there, she were nowhere to be seen. She could've gone anywhere in the dark. She's disappeared, Cap'n. Nobody knows where."
Trinica's head swiveled, and she fixed Frey with a glare of utter malice.
Frey grinned. "Surprise!"
# _Chapter Thirty-three_
DELIBERATIONS—BACK IN THE BLIZZARD—THE MANES—A FEAT OF NAVIGATION
ez, in the pilot's seat of the _Ketty Jay_ , flew on into the night.
The craft was dark, inside and out. The light of the moon edged her face in brittle silver. It fell also onto the two bodies on the cockpit floor and glittered in their blood. Dracken's men. The iron pipe that had staved in their heads lay between them.
Jez's jaw was set hard. Navigation charts were spread on the dashboard next to her. She stared through the windglass at the world below, eyes fixed. The _Ketty Jay_ slid through the darkness, high above the clouded mountains, a speck in the vast sky.
She could see the lights of other craft, visible at great distance. A flotilla of fighters surrounded a long, rectangular freighter. A prickle of shining dots signified a Navy corvette, cruising the horizon. And, in between, there were the invisible vessels, like the _Ketty Jay_ , that had reason to stay hidden and wanted to move unobserved. Stealthy shadows in the moonlight. A pilot wouldn't see them unless they were very close, but Jez saw them all.
Even hours later, she was still trembling with the aftershocks of murder. Had there been a gun to hand, she might have used it to threaten the men, then tied them up and kept them prisoner. But they had the guns, and she had only a length of pipe. She crept into the cockpit and brained the navigator before he even knew she was there. The pilot turned in his seat in time to receive the second blow across his forehead.
She'd told herself that she was only going to knock them out, but, as with Fredger Cordwain, the Shacklemore man, it took only one blow to kill them. She was far stronger than her small frame suggested. Just another aspect of the change, along with her penetrating vision, her ability to heal bullet wounds in hours, and the frightening hallucinations.
And the voices. The dissonant voices, the crew of that terrible craft, which loomed out of the endless fog of the Wrack. She could hear them now, their faint cries blowing on the wind that rushed past the hull of the _Ketty Jay_. Calling her. Calling her home.
_Why not? Why not just go to them? Turn this crate to the north and get it all over with_.
She was tired of this life. The last three years had been spent discarding one crew and joining up with another, never putting down roots. She kept her distance from the men and women she worked with because she knew, sooner or later, they'd find her out. It had been the same with the crew of the _Ketty Jay_. Eventually she always had to run. Now that moment had come again.
_Why stay in a world where you're not wanted?_
Every day it got a little harder to resist the call of the Wrack. Every day eroded her willpower a little more. Was it only stubbornness that made her stay among people who would kill her if they knew what she was? Was it simply fear that prevented her from going to the north, where they lamented her absence, where she'd _belong_? Like the distant howl of a wolf pack, their cries stirred her, and she ached to go to them.
_What's stopping you, Jez? What's stopping you?_
What, indeed? Where else could she go from here? Did she imagine she could effect some kind of daring rescue in the _Ketty Jay_? No, that would be suicide. She wasn't even very good at flying her. It would take a long time to get used to the many quirks of a craft as patched together as this. And even if she did somehow save Frey and the others, what then? How would she explain how she'd convinced Dracken's crewman that she was dead?
It was just like all the times before, with all the other crews. The small things were adding up: her fantastically sharp eyesight; the way she never seemed to need sleep or food; how animals reacted around her; the uncanny healing after she got shot in Scarwater; the way she'd been unaffected by the fumes in Rook's Boneyard.
And now there was this new ability to convincingly imitate a corpse. The first time, only Crake had seen it, and he hadn't said a word. It could have been passed off as the Shacklemore man's mistake. But twice?
Now the suspicious glances would begin. She'd start to hear that wary, mistrustful tone in their voices. Even on the _Ketty Jay_ , where you didn't ask about a person's past, questions would be raised. They could accept a daemonist, but could they accept her? How long before Malvery insisted on giving her a checkup to solve the mystery? How long before they found her out?
The reason Fredger Cordwain thought she had no pulse was because she _had_ no pulse.
The reason Dracken's man thought she was dead was because she _was_.
IT HAD HAPPENED THREE years ago.
The first Jez knew of the attack was when she heard the explosion. It was a dull, muffled roar that shook the ground and spilled the soup she was eating, scalding her fingers. A second explosion sent her scurrying to grab her thick fur-and-hide coat. She pulled up the hood, affixed the mask and goggles, and headed out of the warmth of the inn, up the stairs and into the blizzard.
She emerged onto the main thoroughfare of the tiny, remote town in Yortland that had been her home for a month. The dwellings to either side were low domes, built mostly underground, barely visible. The light from the small windows and the smoke from their chimneys pushed through the whirling snow.
There were others already outside: some were Yort locals, others were the Vard scientists who used this town as a base while they worked on the excavation nearby. All eyes were on the bright bloom of fire rising from the far side of the town. From the landing pad.
Her immediate thought was that a terrible accident had occurred, some tragic rupture in the fuel lines. Even before she wondered how many might have died, her stomach sank at the thought of being stranded in this place. The aircraft were their only link to the rest of the world. Here, on the northern tip of Yortland, civilization was scattered and hard to find. There was no other settlement for a hundred kloms in any direction.
She felt a gloved hand on her upper arm and turned. She knew it was Riss, the expedition's pilot, even though his face was hidden behind a fur-lined hood, mask, and goggles. Nobody else touched her arm like that.
"Are you alright?" he shouted over the whistling wind. His voice was muffled.
"Of course I'm alright. The explosion was over there."
But then someone pointed to a dark shape approaching through the gray chaos in the sky, and the cries of alarm began. Jez felt the strength drain out of her as it took on form, huge and ragged and black. The drone of its engines was drowned out by the piercing, unearthly howling coming from its decks. It was a mass of dirty iron, oil, and smoke, all spikes and rivets and shredded black pennants. A dreadnought, come from the Wrack, across the Poleward Sea to the shores of Yortland.
The destruction of the aircraft on the landing pad had been no accident. The attackers wanted to be sure nobody got away.
The Manes were here, searching for fresh victims.
Ropes snaked down as the dreadnought loomed closer, its massive hull swelling as it descended, until its keel was only a few meters above the rooftops. By the time the Manes came slipping and sliding to the ground, people were already scattering in terror. They'd all heard the stories. The appearance of the dreadnought, the sheer _force_ of its presence, panicked them like goats.
Jez panicked with them, fleeing up the thoroughfare, thinking only of escape. It was Riss who grabbed her arm, more forcefully this time, and tugged her into a doorway. He hurried her down some steps and into a circular underground room full of crates of scientific equipment and boxes of food and clothing. It was cold down here, but not as bad as outside. The sound of their boots echoed from the gray stone walls.
As soon as she was released, she bolted into a corner and huddled there, hugging herself and whimpering. She'd always prided herself on being a levelheaded sort, but the sight of the dreadnought was too much for her. The craft exuded terror, an animal sense of wrongness that appealed to the most basic instincts. Whatever the Manes were, her intuition shrieked at their mere presence.
Riss was faring better. He was obviously scared out of his wits, but he was moving with a purpose. He'd grabbed two packs and was shoving dried food and blankets into them.
"We can't stay here," he said, in response to her unspoken question. "They'll go through the whole town. It's what they do."
"We... I'm not... I'm not going out there!" Jez said through juddering lips. She could hear screams and sporadic gunfire from outside.
He pulled the packs tight, hurried over, and shoved one toward her. She could see his eyes through the glass of the goggles. He was staring at her, hard.
"Listen," he said. "When the Manes hit a town, they don't leave people to tell the tale. The ones who aren't taken are killed. You understand? We can't avoid them by hiding down here."
"Where can we go?"
"The excavation. The ice caves. We can survive there for a night. If we get out of town, we can wait 'til they're gone."
Jez calmed a little as his words sank in. Professor Malstrom, their employer, was obsessed with the search for a lost race he'd dubbed the Azryx, who he believed had once possessed great and mysterious technology. Based on slender evidence and some cryptic writings, he'd divined that they died out suddenly, many thousands of years ago, and their civilization had been swallowed by the ice. He'd persuaded the university to fund him on various digs over the past year, hoping to uncover relics of that ancient people. So far, he'd not found a thing. But the excavation would provide them with the shelter they needed, and the Manes might not look there.
"Yes!" she said. "Yes, we can hide out in the caves!"
She clutched at the sanity he offered, soothed by the strength and certainty in his voice. Riss had held a candle for her ever since they'd started working together, as pilot and navigator for Professor Malstrom's expeditionary team. She liked him as a friend but had never been able to summon up any feelings deeper than that.
He'd always been protective of her. It was a trait she found annoying: she interpreted it as possessiveness. But now she was ashamed to realize she _wanted_ a protector. She'd crumbled in the face of the horror bearing down on them, and he hadn't. She clung to him gratefully as he lifted her up and helped her put on her pack.
The thoroughfare was eerily deserted when they emerged. The dreadnought had gone, and the blizzard was closing in, cutting visibility down to fifteen meters. The chill began to seep into them immediately, even through their protective clothing. From somewhere in the skirling mêlée of snowflakes came distant yells and the report of shotguns. Piercing, inhuman howls floated after them.
They stayed close to the buildings. Jez hung on to Riss as he led her toward the edge of town, where a crude trail led up the mountain to the glacier. The excavation site was up there.
They'd not gone far when there was the roar of an engine and a blaze of light up ahead. Gunfire erupted, startlingly close. Riss pulled Jez into the gap between two domed Yort dwellings, and they hid behind a grit bin as a snow tractor came racing up the thoroughfare. The boxy metal vehicles were usually employed to haul supplies and personnel back and forth from the glacier, but someone was trying to escape on one. The Manes had other ideas: there were four of them swarming all over it, trying to drag the doors open or punch their way in through the glass. Jez glimpsed them in the backwash of the headlamps as they passed—ghoulish, feral approximations of men and women—and then the speeding snow tractor fishtailed on the icy ground. It slewed sideways for an instant before its tracks bit and flung it into the wall of a building.
The Manes abandoned the snow tractor as several Yorts, wielding shotguns, came backing up the thoroughfare, firing into the blizzard, where more shadowy figures were darting on the edges of visibility. Manes prowled on all fours along rooftops or slunk close to the ground. They flitted and flickered, moving in fast jerks. They jumped from one spot to another without seeming to pass through the distance between.
Jez cringed as she saw the Manes spread out to encircle their victims. She wanted to run, to break from hiding and flee, but Riss held her tight.
The Yorts wore furs and masks. The Manes wore ragged clothes more suited to a mild spring day in Vardia. The cold, which would kill an unprotected human in minutes, meant nothing to them.
She turned away and burrowed into Riss's arms as the Manes sprang inward as one. She'd closed her eyes to the sight, but she couldn't shut out the screams of men and the exultant howls of the Manes. Mercifully, it was over in seconds.
Once done, there was silence. It was a short while before Riss stirred and looked out. The sounds of conflict still drifted out of the blizzard, but the Manes had moved elsewhere.
"Stay here," he said. "I'll be back in a moment."
Jez obeyed, reluctant to leave the relative safety of the grit bin. His footsteps crunched across the thoroughfare, fading away. For a time, all she heard were faint gunshots and barked commands, carried on the breeze. Then his footsteps came crunching back. She looked out and saw him carrying a cutlass in one hand. There were several dead men scattered across the thoroughfare, their blood stark against the snow. At least three were missing. Not dead but taken. Stolen by the Manes to crew their terrible craft.
Riss hunkered down in front of her. "The man in the snow tractor is dead," he said. He held up the cutlass. "I got this."
"What about a gun? Don't we need a gun?"
He wiggled his fingers inside his thick glove. Unlike the Yort suits, the scientists' gear was built without much consideration for mobility; warmth was their primary concern. The gloves were too clumsy to fit the forefinger inside a trigger guard, but without them his skin would freeze to the metal.
They headed away from the thoroughfare, through the gaps between the close-set dwellings. The snow had collected in drifts here, and they forged on with some difficulty, but at least the buildings hid them from view. Jez followed in Riss's wake, allowing him to carve a path for her. Her breath was loud in her ears, trapped inside her mask. Her fur-lined hood obscured her peripheral vision, forcing her to turn to look behind her every few steps. She was afraid something was sneaking up on them, following their trail through the snow.
Something _was_ sneaking up on them; but the attack, when it came, was from above.
Jez barely saw it. It was a blur of movement in the confusing whirl of the blizzard. Riss reacted with a cry before he was flung aside to crash into the side of a building. Standing in his place, right in front of her, was a Mane. It was the first and last time she ever got a good look at one, and it rooted her to the spot with fear.
The stories said they'd once been human, and they were recognizably so in form and face. But they'd been changed into something else, something that wore human shape uncomfortably, as a skin to contain whatever hid beneath.
The creature before her was scrawny, wearing a tattered shirt and trousers and no shoes at all. Limp black hair was smeared across a pale, wrinkled brow. Its features were twisted out of true. Lips curled to reveal sharp, crooked teeth. It glared at her with eyes that were the yellow and red of bloody pus. Its fingernails were long, dirty, and cracked, and it stood low to the ground in a predator's crouch.
It wasn't what she saw but what she _sensed_ that paralyzed her: the intuitive knowledge that she was in the presence of something not of this world, something that broke all laws and ruined all the certainties of a thousand generations of knowledge. Her body felt that, and rebelled.
Then it pounced and bore her into a snowdrift.
She remembered little of what followed. It didn't seem to make sense when she recalled it later. The Mane had her pinned by the shoulders and stared into her eyes. Her gaze was locked, as if she were a mouse hypnotized by a snake. She could smell the stench of it, a dead scent like damp leaf mold. Her breathing dropped to a shallow pant.
She felt crushed by the weight of the creature's will, oppressed by the force in its gaze. By the time she realized something was being done to her, it was too late to resist it. She struggled to oppose the invader with her thoughts, but she couldn't concentrate. She was losing herself.
She became aware of a change all around her. The blizzard faded, turning ghostly and powerless. The world was darker and sharper all at once. She could see details where there hadn't been details before: the fine jigsaw of creases in the skin of the Mane's face; the shocking complexity of its feathery irises.
There was a whispering in the air, a constant hiss of half-spoken words. Movement all around her. She recognized the movement of the Manes, prowling around the town. She could _feel_ them. She shared their motion. And as she sank deeper and deeper into the trance, she felt the warmth of that connection. A sense of belonging, like nothing she'd experienced before, enfolded her. It was beautiful and toxic and sugary and appalling all at once.
She'd almost surrendered herself to it when she was ripped back into reality.
It took a moment for her senses to cope with the change. She was being pulled to her feet by a faceless man in a hooded fur-and-hide coat. Her initial reaction was to pull away, but he held her firmly and said something to her. When she didn't respond, he said it again, and this time the words got through.
"—re you alright? Jez? _Jez?_ "
She nodded quickly, because she wanted him to shut up. He was frightening her with his urgent inquiries. The Mane was thrashing and squealing on the ground. A cutlass was buried in the base of its neck up to the collarbone, half severing its head. There was little blood, just a clean wound, exposing bone.
But it still wasn't finished. Moving with jerky, spastic movements, it got its feet under it and tried to stand. Riss swore and kicked it in the face, knocking it flat. He wrenched the cutlass free and beheaded it with a second stroke.
Riss turned away from the corpse of the Mane and looked up at her. He held out his hand: _come with me_.
Something snapped inside her. The accumulated horror and shock of the attack broke through. She lost her mind and fled.
She ran, through the passageways between the houses, out into the blizzard. The winds pushed and battered her. Snow stuck to her goggles. She could hear Riss calling her name, but she ignored him. At some point she realized that she could no longer see any houses, just endless, unmarked snow. She kept running, driven by the terror of what lay behind.
Only when exhaustion drove her to her knees did she stop. She was thoroughly lost, and all traces of her passing were being erased by the fury of the snow. She dared not go back, and she couldn't go forward. The cold, which she'd barely noticed during her flight, had set in deep. She began to shiver violently. A tiredness overtook her, every bit as insidious and unstoppable as the power of the Manes.
She curled up into a fetal position, and there, buried in the snow, she died.
EVERY DAY SINCE, JEZ had wondered what might have happened if things had gone another way. If Riss hadn't saved her. If she'd succumbed to the Mane.
Would it have been so bad, in the end? In that brief moment, when she touched upon the world of the Manes, she'd felt something wonderful. An integration, a togetherness above and beyond anything her human life had given her.
She'd never borne children, never been in love. She'd always dreamed of having friends she could call soul mates, but somehow it never happened. She just didn't care about them enough, and they didn't care about her in return. She'd always considered herself rather detached, all in all.
So when she felt the call of the Manes, the primal invitation of the wolf pack lamenting the absence of their kin, she found it harder and harder to think of reasons to resist.
Yes, they killed, but so had she now. Yes, they were fearsome, but a fearsome exterior was no indication as to what was beneath. You only had to know the secret of Bess to understand that.
Would the process have been half so frightening if she'd been invited instead of press-ganged? Might she have gone willingly, if only to know what lay beyond that impenetrable wall of fog to the north? Were there incredible lands hidden behind the Wrack, glittering ice palaces at the poles, as the more lurid pulp novels suggested? Was it a wild place, like Kurg, with its population of subhuman monsters? Or was there a strange and advanced civilization there, like Peleshar, the distant and hostile land far to the southwest?
Whatever had been done to her by the Mane that day was incomplete, interrupted by a cutlass to the neck. She was neither fully human nor fully Mane but somewhere in between. And yet the Manes welcomed her still, beckoned her endlessly, while the humans would destroy her if they knew that she walked their lands without a beating heart.
She never found out what happened to Riss. The morning after she died, she woke up and dug her way out of the snow that had entombed her in the night. The sun shone high in a crystal-blue sky, glittering on distant mounds of white: the roofs of the town. She'd run quite a way in her panic, but it had been in entirely the wrong direction if she'd hoped to reach the safety of the ice caves up on the glacier.
The corpses lay beneath the snow now. Whether Riss was among them or he'd been taken, the result was the same. He was gone.
Numb, she searched for survivors and found none. She stood in front of the snow-covered wreck of the aircraft she'd navigated for a year and felt nothing. Then she found a snow tractor and began to dig it out.
It took her several days to find another settlement, following charts she'd salvaged. Since she felt perfectly healthy, she didn't question how she'd survived at first. She assumed her snowy tomb had kept her warm. It was only when she was far out in the wilderness that she noticed her heart had stopped. That was when she began to be afraid.
By the time she reached the settlement, she had a story, and a plan.
_Keep moving. Keep your secret. Survive, as much as you can be said to live at all_.
But it had been a long and lonely three years since that day.
SHE PASSED OVER THE southern part of the Hookhollows, their glowing magma vents making bright scribbles in the dark. The Eastern Plateau rose up before her, and she took the _Ketty Jay_ down through the black, filthy clouds. Her engines were robust enough to take a little ash. Once she'd broken through, she brought the _Ketty Jay_ to a few dozen meters above ground level and skimmed over the Blackendraft flats. She glanced at the navigational charts she was following—charts that had been meticulously kept by Dracken's navigator since they'd commandeered the _Ketty Jay_.
_Trust me_ , she'd said to Frey, when he demanded to know how she was going to fool Dracken's men into thinking she was dead. The kind of trust he'd shown when he gave her the ignition code to his precious aircraft, the one thing he could be said to love. Even though he was afraid she might steal it and fly off forever, he'd trusted her.
And he trusted her to come back and save him. She wouldn't let him down.
She was under no illusion that she was risking her own life, and she knew that even if she succeeded, she'd probably be despised. They couldn't be her friends. She'd never belong to that crew. If they learned how she was slowly, steadily becoming a Mane, they'd be forced to destroy her. She couldn't blame them for that.
Yet she'd try anyway. Perhaps afterward she'd go to the north, to the Manes; but first, she'd try.
It made no sense. But, sometimes, humans did things that made no sense.
There was one last thing to do before she set off. Though she'd been lying in the infirmary with all the appearance of a corpse, she'd been wide awake. And she'd heard Dracken's men talk. Not _all_ the crew of the _Ketty Jay_ had been taken on board the _Delirium Trigger_.
She slowed the _Ketty Jay_ to a hover and consulted the charts again. She wanted to get this right the first time. It was a small challenge to herself. She adjusted the craft's heading, pushed her on half a klom, then stopped again. When she was satisfied, she engaged the belly lights. The ashen, dusty waste below her was flooded in dazzling light. She smiled.
_Damn it, Jez. You're good_.
There, right where they'd left her, was Bess.
# _Chapter Thirty-four_
MALVERY'S STORY—SOMETHING WORSE THAN CRAMP—FREY GOES TO THE GALLOWS
ortengrace, ancestral home of Duke Grephen of Lapin, stood out white among the trees like an unearthed bone. It was set amid the folds and pleats of heavily forested coastal hills in the western arm of the Vardenwood, overlooking the sparkling blue waters of the Ordic Abyssal to the south. High walls surrounded it, enclosing a landing pad for aircraft, expansive gardens, and the grand manse where the Duke and his family resided. Among the half-dozen outbuildings were an engineer's workshop, a barracks for the resident militia, and a gaol. The latter was rarely used in these more peaceable times, but it had found employment over the last two days, since Trinica Dracken had delivered six of the most wanted men in Vardia.
Crake sat in his cell, with Malvery and Silo, and he waited. It was all that was left to do now. He waited for the noose.
The cell was small and clean, with stone walls plastered off-white. There were hard benches to sleep on and a barred window, high up, that let in the salty tang of the sea. The temperature was mild on the south coast of Lapin, even in midwinter. A heavy wooden door, banded with iron, prevented their escape. There was a flap at the bottom, through which plates of food were occasionally pushed, and a slot their gaoler used to look in on them.
He was a chatty sort, keen to keep them updated on the details of their imminent demise. Through him, they'd learned that Duke Grephen was at an important conference and was on his way back as soon as he could get away and find a judge. "To execute the sentence nice and legal," the gaoler grinned, drawing out the word _execute_ , just in case they missed how clever he was being by using it. "But don't you worry. There ain't no hurry, 'cause not a soul knows you're here. Nobody's coming to your rescue."
There were two guards, in addition to the gaoler, though the prisoners rarely heard them speak. They were there to keep an eye on things. "Just in case you try any foolery," the gaoler said, with a pointed look at Crake. They'd evidently been warned that there was a daemonist among the prisoners. Crake's golden tooth would be useless: he couldn't deal with three men. His skeleton key was lying somewhere in the _Ketty Jay_ 's cargo hold, equally useless.
No way out.
He'd been swallowed by an immense sense of emptiness. It had come upon him in the moment they'd lifted off from the Blackendraft to be taken on board the _Delirium Trigger_. The news that the _Ketty Jay_ had disappeared did little to alleviate it. Bess was gone.
His thoughts went to the small whistle hidden in his quarters aboard the _Ketty Jay_. Only that whistle, blown by the daemonist who had thralled it, had the power to wake her from oblivion. He'd never get to blow that whistle now. Perhaps that was best.
He should never have tried to save her. In attempting to atone for one crime, he'd committed one far greater. And now she'd be left, neither dead nor alive, for an eternity.
Did she sleep? Was she aware? Was she trapped in a metal shell in the endless waste of the ash flats, unable to move or scream? How much was left of the beautiful child he'd ruined? It was so hard to tell. She was more like a faithful dog than a little girl now, muddled and jumbled by his clumsy transfer, prone to fits of rage, insecurity, and animal violence.
He should have let her die, but he couldn't live with the guilt of it. So he'd made her a monster. And, in doing so, made himself one.
A distant howl made Crake, Silo, and Malvery look up as one. The voice was Frey's, coming from the torture room, just beyond the cell he shared with Pinn and Harkins.
"They've started up again," said Malvery. "Poor bastard."
Crake stirred himself. "Why's he bothering to hold out? What does it matter if he signs a confession or not? We're all going to be just as dead with or without it."
Malvery grinned beneath his white walruslike mustache. "Maybe he just likes being an awkward bugger."
Silo actually smiled at that. Crake didn't take up the humor. He felt Malvery put a huge arm around his shoulder.
"Cheer up, eh? You've had a face like a soggy arse since Dracken caught us."
Crake gave him an amazed look. "You know, all my life I've been under the illusion that the fear of death was a common, almost _universal_ part of being human. But recently I've come to think I'm the only one on this crew who is actually worried about it in the slightest."
"Oh, I don't know. I bet the other cell is half full of Harkins's shit by now, he's so scared," Malvery said with a wink. "Then again, he's afraid of just about everything. The only reason he's still a pilot is because he's more afraid of _not_ being a pilot than he is of getting shot down."
"But... I mean, don't you have _regrets_? Thwarted hopes? Anything like that?" Crake was exasperated. He'd never been able to understand how the vagabonds of the _Ketty Jay_ lived such day-to-day lives, never seeming to care about the future or the past.
"Regrets? Sure. I've got regrets like you wouldn't believe, mate," said Malvery. "Told you I was a doc back in Thesk, didn't I? Well, I was good at it, and I got rich. Got a little flush with success; got a little fond of the bottle too.
"One day a messenger from the surgery turned up at my house. There was a friend of mine, been brought in gravely ill. His appendix was what it was. It was early in the morning, and I hadn't gone to bed from the night before. Been drinking the whole time."
Crake noted that the lighthearted tone was draining out of Malvery's voice. He realized suddenly that he was in the midst of something serious. But Malvery kept going, forcing himself to sound casual.
"Well, I knew I was drunk, but I also knew it was my friend and I believed I was the best damn surgeon for the job, drunk or sober. I'd gotten so used to being good that I thought I couldn't do no wrong. Wouldn't trust it to anyone else. Some junior doc tried to stop me, but I just shrugged him off. Wish he'd tried harder now."
Malvery stopped abruptly. He heaved a great sigh, as if expelling something from deep in his lungs. When he spoke again, it was with a deep resignation in his tone. What had been done had been done and could never be undone.
"It should have been easy, but I got careless. Slipped with the scalpel, went right through an artery. He bled out right in front of me, on the table, while I was trying to fix him up."
Even obsessed with his own misery, Crake felt some sympathy for the big man. He knew exactly how he felt. Perhaps that was why they'd instinctively liked each other when they first met. Each sensed in the other a tragic victim of their own arrogance.
Malvery cleared his throat. "I lost it all after that," he said. "Lost my license. Lost my wife. Spent my money. Didn't care. And I drank. I drank and drank and drank, and the money got less and less, and one day I didn't have nothing left. I think that was about when the Cap'n found me."
"Frey?"
Malvery pushed back the round green-lensed spectacles on his broad nose. "Right. We met in some port—I forget which. He bought me some drinks. Said he could use a doctor. I said I wasn't much of a doctor, and he said that was okay, 'cause he wasn't gonna pay me much anyway." He guffawed. "Ain't that just like him?"
Crake cracked a smile. "Yes. I suppose it is."
"I ain't never picked up a scalpel since that day when I killed my friend. I don't think I could. I keep those instruments polished in the infirmary, but I'll never use 'em. I'm good for patching you up and a bit of stitching, but I'd never trust myself to open you up. Not anymore. You wanna know the truth, I'm half a doctor. But that's okay. 'Cause I found a home on the _Ketty Jay_ , and I've got the Cap'n to thank for it." He paused as Frey screamed from down the corridor. A spasm of anger crossed his face but was gone again in an instant. "He's a good man, whatever faults he's got. Been a good friend to me."
Crake remembered how Trinica had put a gun to his head and how Frey had given up the code to his beloved aircraft rather than see the daemonist shot.
"Yes," he said. "To me too."
Crake knotted his fingers behind his head and leaned back against the wall of the cell. Silo, Harkins, and now Malvery: Frey certainly had a thing for picking up refugees. Granted, they were all useful to him in some way, but all owed a debt of gratitude and loyalty to their captain that Crake hadn't detected until recently. Perhaps Frey's intentions had been entirely mercenary—it could be that he just liked cheap crew—but at least half his men viewed him as a savior of sorts. Maybe Frey didn't need them, but they certainly needed him. Without their captain, Silo would end up lynched or sent back to slavery in Samarla, Harkins would be forced to face a life without wings, and Malvery would be a destitute alcoholic once again.
And what of the rest of them? He himself had found a place to hide while he stayed ahead of the Shacklemores. Pinn had found a place that would tolerate him, where he could forever avoid the reality of his sweetheart in his doomed search for riches and fame. And Jez? Well, maybe Jez just liked to be in a place where nobody asked any questions.
Like it or not, Frey gave them all something they needed. He gave them the _Ketty Jay_.
"Ain't one of us who's not running from something," Crake said wryly. Malvery's words, spoken weeks ago, before they'd shot down the _Ace of Skulls_ and all this had begun. Malvery bellowed with laughter, recognizing the quote.
Crake looked up at the ceiling of the cell. "I deserve to be here," he said.
Malvery shrugged. "Then so do I."
"Ain't no deserving or otherwise," Silo said, his bass voice rolling out from deep in his chest. "There's what is and what ain't, and there's what you do about it. Regret's just a way to make you feel okay that you're not makin' amends. A man can waste a life with regrets."
"Wise words," said Malvery, tipping the Murthian a salute. "Wise words."
In the distance, Frey screamed again.
FREY HAD BEEN SHOT twice in his life, beaten up multiple times by members of both sexes, bitten by dogs, and impaled through the gut by a Dakkadian bayonet, but until today he'd always been of the opinion that the worst pain in the world was cramp.
There was nothing quite so dreadful to Frey as waking up in the middle of the night with that telltale sense of tightness running like a blade down the length of his calf. It usually happened after a night on the rum or when he'd taken too many drops of Shine, but on the confined bunk in his quarters, he often lay awkwardly and cut off the circulation to one leg or the other, even when dead sober.
The worst moments were those few seconds before the agony hit. There was always time enough to try to twist out of it in such a way that the pain wouldn't come. It never worked. The inevitable seizure that followed would leave him whooping breathlessly, writhing around in his bunk and clutching his leg. It invariably ended with him knocking multiple items of luggage from the hammock overhead, which crashed down onto him in a tumble of cases and dirty clothes.
Finally, after the chaos of bewildering, undeserved pain, would come a relief so sweet that it was almost worth going through the preceding trauma to get there. He'd lie half buried in the luggage, gasping and thanking whoever was listening that he was still alive.
Frey had learned long ago that the violent clenching of the muscles in his lower leg could send him wild with agony. Today, his torturer had introduced him to the joys of electrocution. Instead of just his leg seizing up, now it happened to his entire body at once.
If he survived this, Frey decided, he'd have to rethink his definition of pain.
_Blinding, shocking torment; his back arching involuntarily; muscles tensed so hard they could break bone; teeth gritted and jaw pulled back in a grimace_.
And then the pain was gone. The joy was enough to make him want to break down and weep. He slumped forward in the chair as much as his restraints would allow, sweat dripping off his brow, chest heaving.
"Do you _want_ to be hurt? Is that it?" the torturer asked.
Frey raised his head with some difficulty. The torturer was looking at him earnestly, wide gray eyes sympathetic and understanding. He was a handsome fellow, square-jawed and neat, wearing a carefully pressed light blue uniform in the ducal colors of Lapin.
"You should have a go at this," Frey said, forcing out a fierce grin. "Gives you quite a kick."
The guard standing by the door—a burly man in an identical uniform to that of the torturer—smiled at that for a moment, before realizing he wasn't supposed to. The torturer tutted and shook his head. He moved over to the machine that stood next to Frey's chair. It was a forbidding metal contraption, the size of a cabinet, with a face of dials and semicircular gauges.
"Obviously it's not kicking hard enough," the torturer said, turning one of the dials a few notches.
Frey braced himself. It did no good.
The pain seemed as if it would never end, until it did. The room swam back into focus. He'd always pictured torture chambers as dank and dungeonlike, but this place was clean and clinical. More like a doctor's surgery than a cell. The electric lights were bright and stark. There were all kinds of instruments in trays and cabinets, next to racks of bottles and drugs. Only the metal door, with a viewing slot set into it, gave away the true nature of this place.
The confession sat on a small table in front of him. A pen waited next to it. The torturer had obligingly read it out to him yesterday, before they began. It was pretty much as he'd expected: _I, Frey, admit every damn thing. I conspired with my crew to kill the Archduke's son because we're greedy and bad, and then we all laughed about it afterward. It was all my idea and certainly nobody_ _else's, especially not Duke Grephen's or Gallian Thade's, who are both spotless and loyal subjects of our revered leader and whose very feces smell of roses and almond, et cetera, et cetera_.
The torturer picked up the pen and held it out to him. "End it, Darian. Why struggle? You know there's no way out of here. Why must you make the last few hours of your life so miserable?"
Frey blinked sweat out of his eyes and stared dully at the pen. Why _didn't_ he sign it? It was only a formality. As soon as Grephen arrived with a judge, they'd be tried and hung anyway, though not necessarily in that order.
But he wouldn't. He wouldn't sign that paper because he didn't want to make it easy on them. Because he'd fight for every moment he had left, eke out every inch of existence there was to be had.
Confessing was giving up. He wasn't resisting in the hope of achieving anything; he was resisting just to resist. It didn't matter how futile it was. He was bitter that he'd gotten so close, that he'd _almost_ managed to get his crew out of the mess he'd gotten them into. It enraged him.
So he relished the small victories that were left. However she did it, Jez had gotten away and taken the _Ketty Jay_ with her. The fact that Grephen wasn't hurrying back immediately to dispose of his prisoners suggested that Trinica Dracken had neglected to mention that she'd lost the _Ketty Jay_ en route. Unwittingly, Dracken had bought them some time.
He'd embarrassed her twice. He took solace in that. He hadn't failed to notice that Trinica kept her compass and charts close to her at all times now. She'd been carrying them as they were shuttled from the deck of the _Delirium Trigger_ to the landing pad at Mortengrace. She was nervous that they might be stolen again and didn't want to leave them in her cabin.
Small victories. But victories nevertheless.
He didn't hold out hope of Jez coming back. Not only would it be stupid, she had no real reason to. They were just a crew, like many she'd taken up with before. Though efficient at her job, she'd always seemed standoffish, keeping to her cabin most of the time. He didn't imagine she held any particular affection for them, and he had no reason to expect loyalty. After all, she'd barely joined before he turned her into an outlaw.
But the _Ketty Jay_ survived, and with a new captain at the helm. That was alright with Frey. If he couldn't have her, he was glad that someone could, and he'd always liked his diminutive navigator. He'd always wonder how Jez did it, though he took consolation in the fact that he wouldn't have to wonder long.
_I suppose Slag made it too_ , he thought. _I wonder how he's going to get on with his new captain_.
"Sign!" the torturer urged, pressing the pen into his hand.
Frey took it. "Give me the paper," he said.
The torturer's eyes lit up eagerly. He moved the table closer, so Frey could write on it. The leather cuffs he wore were attached to straps that gave him a few inches of slack. The torturer presumably thought a man couldn't spasm efficiently without a little room to writhe.
"Bit closer. I can't reach," said Frey. The torturer did as he was asked. "Can you hold the paper steady? This isn't easy with one hand."
The torturer smiled encouragingly as he steadied the paper for Frey to sign. He stopped smiling when Frey stabbed the pen into the soft meaty part between his thumb and forefinger.
A third man in uniform burst through the door and stood bewildered at the sight that faced him. The torturer was wheeling around the room, shrieking, holding his impaled hand, which still had a pen sticking out of it. The guard by the door was in paroxysms of laughter. Frey had crumpled the confession into a ball and was trying to get it into his mouth to eat it, but couldn't quite reach. He paused guiltily as the newcomer stared at him, then let it drop from his hand.
_"What do you want?"_ screamed the torturer, when he got his breath back.
"You can stop now," said the newcomer.
"But he's not confessed!"
"We'll draft a new one and sign it for him. The Duke is back with a judge. He wants this done."
"Can't you give me an hour?" the torturer whined, seeing his chance at revenge slipping away.
"I'm to take charge of him immediately," the newcomer insisted. "Get him out of that chair. He's coming with me."
THE SKY WAS BLUE. Clear, cloudless, and perfect. Frey squinted up at the sun and felt it warm his face. Amazing, he thought, how the north coast of the continent was gripped in ice and yet it was still pleasant here in the south. Vardia was so vast, its northern edge breached the Arctic Circle while its southern side came close to the equator. He'd always thought of winter as the grimmest season, but, like anything, he supposed, it depended on where you were standing.
The spot chosen for his execution was a walled courtyard behind the barracks, where the militia conducted their drills. There was a small raised platform with a wrought-iron lamppost in its center, flying the Duke's flag. Ornamental arms projected out from the lamppost. They were intended for hanging pennants, but the pennants had been removed and a noose thrown over one of the arms to form a crude gallows. The end of the noose lay loosely around Frey's neck. An executioner—a massive, sweaty ogre with a thin shirt stretched over an enormous gut—waited to pull it taut.
A small crowd was assembled before him. There were two dozen militia, a judge, the Duke, and two witnesses: Gallian Thade and Trinica Dracken. Off to one side was a wagon with bars on its sides. Inside this wheeled cage was the remainder of his crew. They were unusually subdued. The seriousness of their situation had sunk in at last. Even Pinn was getting it now. They were going to watch their captain die. Nobody felt like joking.
He'd always wondered how he'd face death. Not the quick, hectic rush of a gun battle but the slow, considered, drawn-out finale of an execution. He'd never imagined that he'd feel quite so serene. The wind stirred a lock of hair against his forehead; the sun shone on his cheeks. He felt like smiling.
The Darian Frey they were about to kill wasn't the same Darian Frey they'd set out to frame for their crime. That man had been a failure, a man who lurched from crisis to disaster at the whim of fate. A man who had prided himself on being better than the bottom-feeding scum of the smuggling world and hadn't desired any more than that.
But he'd surprised them. He'd turned and fought when he should have run. He'd evaded and outwitted them time and again. He'd turned a bunch of dysfunctional layabouts into something approximating a crew. Stories would be told of how they tweaked the nose of the infamous Trinica Dracken in a hangar bay in Rabban. Word would spread. Freebooters all over Vardia would hear of Darian Frey and his craft, the _Ketty Jay_. He'd come close to unearthing a daring conspiracy against the ruling family of the land, involving a duke of Vardia, the legendary pirate captain Orkmund, and the mighty Awakener cult.
Only a final twist of ill fortune had stopped him. Trinica had made copies of the charts he stole. Without the compass, she couldn't make it through the magnetic mines that guarded Retribution Falls, but she could wait at the point where she knew he'd emerge.
One little slipup. But he'd led them a merry chase all the same. They might have caught him, but he still felt as if he'd won.
He looked at the faces behind the bars: Malvery, Crake, Silo, Harkins... even Pinn. He was surprised to find he was sad to be leaving them. He didn't want it all to end now. He'd just begun to enjoy himself.
Frey had stopped listening to the list of crimes and accusations that the judge was reading out. The preliminaries were unimportant. He was thinking only of what was to come. Death was inevitable. He accepted that and was calm. His hands were tied securely before him, and there were two dozen guards with rifles waiting to fill him with bullets if he should try to escape.
But he still had one trick left to play. The world would remember him, alright. Maybe they'd never know the truth, but they'd know his name.
The judge—an ancient, shortsighted relic who was more than half dust—finished his rambling and looked up, adjusting his spectacles.
"Sentence of death has been passed," he droned. "Tradition grants the prisoner the opportunity to make a last request. Does the prisoner have such a request?"
"I do," said Frey. "To be honest, I consider it a bit of an insult that the Duke couldn't even provide a decent gallows to hang me by. I request an alternative method of execution."
Duke Grephen's sallow face colored angrily. Trinica watched the prisoner curiously with her black eyes.
"I'd like to be beheaded with my own cutlass," Frey said.
The judge looked at the Duke. Grephen swiped a strand of lank blond hair from his forehead and huffed.
"I can see no objection," creaked the judge warily, in case the Duke had any objection.
"Fetch his cutlass!" Grephen cried. One of the guards hastened away to obey.
Frey stared at Grephen coolly. Even in his uniform, the Duke looked like a spoiled little boy. His deeply set eyes glittered with childish spite. He was a cold and humorless man; Frey surmised that much. He'd murdered dozens on board the _Ace of Skulls_ just to kill the Archduke's son in such a way that it could be pinned on someone else. Frey didn't believe it bothered him one bit. If there was any warmth in him, it was reserved for the Allsoul.
Next to him stood Gallian Thade. Sharp-faced, beak-nosed, with a pointed black beard. He was all angles and edges, where the Duke was soft and pudgy. Thade watched him with an air of smugness. He'd waited a long time to see the man who had deflowered his daughter receive his punishment.
And then there was Trinica. He couldn't tell what she was thinking. Her ghost-white face revealed nothing. Would she be pleased to see him die? Would she finally be able to close the chapter of her life that had begun with him? Or was she even now remembering fonder moments from their past, wondering if she'd done the right thing in bringing him here?
Grephen had destroyed the _Ace of Skulls_ ; Thade had picked Frey to frame for it; Trinica had caught him.
He had reason to kill them all. But he'd only have time to do one of them. And he'd already chosen his target.
The guard returned from the barracks with his cutlass. Grephen took it and inspected it before passing it to the executioner. The executioner ran his thumb admiringly down the blade, then hissed through his teeth as he slashed the tip open.
"Could you get this thing off me?" Frey asked, jiggling his shoulders to indicate the noose. The executioner thrust the cutlass into his belt and removed the noose with one hand, sucking his bleeding thumb with the other.
"Kneel down, mate," he said. Frey went to his knees on the wooden platform at the foot of the lamppost. He shifted his wrists inside their knots of rope and rolled his neck.
He looked over at the cage, where his crew was imprisoned. Once he was dead, they'd follow him. Pinn seemed bewildered. Crake's gaze was heavy with tragedy. Silo was inscrutable. Harkins was cringing in a corner and looking away. Malvery gave him a rueful smile and a thumbs-up. Frey nodded in silent thanks for his support.
"Sentence of execution by beheading," said the judge, "to be carried out in the sight of these eminent witnesses."
The executioner drew the cutlass and took aim, touching the blade to the back of Frey's neck. "Don't worry, eh?" he said. "One swipe and it'll be done."
Frey took a breath. One swipe. He saw the blade descending in his mind's eye. He saw himself dropping one shoulder, rolling, holding up his hands as the daemon-thralled sword slashed neatly through his bonds. He saw the blade jump from the hands of the executioner and into Frey's grasp. He saw the surprise on Grephen's face as Frey flung it from the podium. He saw it slide point-first into the Duke's fat heart.
The sword always knew his will. He might go down in a hail of bullets, but the author of his misery would go down with him. And all of Vardia would know how Duke Grephen died at the hands of an insignificant little freebooter, who had outwitted him at the last.
"Kill him," said Grephen to the executioner.
The executioner raised the cutlass. Frey closed his eyes.
_Ready..._
The blade quivered, and he fancied he heard the harmonic singing of the daemon within.
_Ready..._
And then a loud voice cried: "STOP!"
# _Chapter Thirty-five_
THE SUSPICIONS OF KEDMUND DRAVE—FREY SAYS HIS PIECE—THE STICKY MATTER OF PROOF—DEATH IN THE COURTYARD
he voice that had halted the execution belonged to Kedmund Drave, the most feared of the Century Knights, whom Frey had last seen lying on a landing pad in Tarlock Cove after he emptied a shotgun into Drave's chest. His molded crimson armor showed no signs of the encounter as he swept across the courtyard toward Duke Grephen, his thick black cape swaying around him.
To either side were Samandra Bree and Colden Grudge. Frey recognized them from their ferrotypes. Samandra was as beautiful as her picture, wearing the outfit she was famous for: battered coat and boots, loose hide trousers, a tricorn hat perched on her head. Grudge, in contrast, looked like something half ape. Shaggy-haired and bristle-faced, he was a hulking mass of dirty armor barely contained inside the folds of a hooded cloak. His autocannon clanked against his back. It was a gun bigger than most men could even carry, let alone fire.
"What exactly is going on here?" Drave demanded, striding up to the Duke. They could scarcely have been more different: the soft, spoiled aristocrat in his neatly pressed uniform and the iron-hard figure of the Knight, his silver-gray hair shorn close to his scalp and his cheek and neck horribly scarred.
Grephen collected himself, overcame the physical intimidation, and attempted to assert his ducal authority. "These men are pirates," he said. "They have been condemned to death. I wasn't aware there was any law forbidding a duke to deal with pirates inside his own duchy. As you can see, I have a judge here to ensure everything is legal."
Drave stared at the old judge, who began to look nervous.
"I see," Drave said slowly. "I imagine the trial has been thorough and fair."
Grephen bristled. "Remember who you're talking to, sir. You may have the Archduke's authority, but even the Archduke knows to respect his dukes."
"I'm not in the business of respect," Drave snarled. He turned to the judge. "There has been a trial, I assume?"
The judge looked shiftily at Grephen and swallowed. "I was brought here to oversee the executions. The Duke assured me that their guilt was not in question."
"You've obtained confessions, then?" Drave asked Grephen.
Frey grinned. There wouldn't have been time to make up and sign another confession after he'd ruined the last one.
"They were caught red-handed in an act of piracy," Grephen declared, flushing angrily. "There was no need for a confession or a trial. I exercised my ducal authority, as is my right. Besides, they admitted it."
"Bollocks we did!" Malvery yelled from the cage. "He's lying!"
"You shut up!" growled Colden Grudge, pointing a meaty finger at the doctor.
"We're innocent!" Pinn cried, joining in happily. For a while his faith in a last-second intervention had wavered, but now here it was, and all was right with the world again.
Drave turned his gaze to Trinica. "Trinica Dracken. You caught these men?"
"Yes."
"You know what crimes they are wanted for?"
"I do."
"And you were hired to catch them by the Duke?"
"I was."
"Then _he_ must know what crimes they are wanted for."
Trinica looked at Grephen, her black eyes emotionless.
"I'd assume so," she said.
Drave turned on Grephen. "Given that, Duke Grephen, why did you see fit to execute these prisoners yourself instead of delivering them to the Archduke for public trial? After all, it wasn't _your_ son they killed."
Grephen had begun to sweat, his limp hair becoming lank. He looked to Gallian Thade, but Thade couldn't help.
"I can answer that," called Frey. He was still kneeling on the platform, with the executioner standing next to him, Frey's cutlass held loosely in his hand.
"You be quiet, criminal!" Grephen snapped.
Drave's eyes narrowed as he looked for the first time at the man who had almost killed him a few weeks earlier. Frey wondered if the malice in that glare would be the death of him or if Drave would give him the chance he needed. For a long instant, Drave said nothing, then he held up a hand.
"Let him speak. I'd like to hear what he has to say."
Grephen looked nauseous with fear. The guards in their light blue uniforms glanced at one another nervously. They'd thought this would be a simple execution; they realized now that there was much more to it.
"Can I get to my feet?" Frey asked. "My knees are getting kind of sore like this."
Drave motioned for him to get up. The executioner backed away a step. "Make it quick," he said. "And make it good. I will get to the bottom of this, but I'll not lie to you, Darian Frey: I'd like to see you dead as much as anyone."
Frey got up. All eyes were on him. He was still possessed of that strange sense of calm that had settled on him with the surety of death. It was as if his body couldn't quite believe there might be a reprieve.
"I'll keep it simple, then," he said. "Duke Grephen plans to overthrow the Archduke. He's being bankrolled by the Awakeners; they want to see the Archduke deposed because of the political measures he and his wife are introducing to limit their power. They know Grephen is devout and that he'll act favorably toward them once he seizes power."
"These are _lies_!" Grephen shouted, but Frey went on anyway.
"The Awakeners don't have an army, and Grephen doesn't command enough troops to challenge the Coalition Navy, so between them they've raised a force of pirates and freebooters, paid for with Awakener gold. This army is at the hidden port of Retribution Falls, waiting for the signal to move on Thesk and unseat the Archduke. As far as I know, that signal is coming any day now."
"And what does any of this have to do with the destruction of the _Ace of Skulls_ and the death of Hengar?" Drave asked.
"Hengar's death was a preliminary. They wanted to be sure there was nobody left for dissenters to rally around. He was the only surviving member of the Arken family who could inherit the title after the Archduke is gone. His secret affair with a Samarlan gave them an opportunity to get him out of the way and make it look like an accident. And Hengar was the popular one; by killing him and then leaking the information about the affair, they made the Archduke's family look dishonest and immoral. All the better for after the coup, when they could claim it was a revolution to depose a corrupt regime, just like the dukes when they overthrew the monarchy."
"This is pure fantasy!" Grephen shrieked. "I will not stand here and listen to this slander from a pirate and murderer."
"I can prove it," said Frey. "I've been to Retribution Falls, and I've seen the army that's waiting there. I know how to find it." He stared hard into the eyes of Kedmund Drave. "I can take you there."
Drave stared back at him. "In exchange for a pardon, no doubt."
"A _pardon_?" cried the Duke, but was ignored.
"For me and my crew," Frey said. "The _Ace of Skulls_ was rigged with explosives. Any engineer would tell you it's nigh on impossible to blow up a craft that size with the guns I have on my craft. We were set up to take the fall for it, so nobody would suspect that it was part of a bigger plot. They hoped we'd be killed before we ever worked out what was going on, so we wouldn't be able to tell anyone." He raised his bound hands and pointed across the courtyard. "The setup was Gallian Thade's doing. He's in on it too."
Thade said nothing, but his gaze was murderous.
"You're going to take his word for what kind of guns he has on his craft?" Grephen spluttered.
"I know what kind of guns he has on his craft," Drave said. "We have it in our possession."
Frey's heart leaped. That could mean only one thing: Jez. Somehow she'd found the Century Knights and told them what was going on. A flicker of real hope ignited in him.
"He's playing for time!" Grephen accused. "He's leading you on a wild-goose chase. You're not really thinking of letting him lead you all over Vardia in search of some mythical pirate port?"
Drave looked at Frey. "Is that what you're doing? Playing for time?"
"If you'll permit me..." said Frey. He reached down into his trousers and began groping around at his crotch. Several guards covered him with guns. Samandra Bree raised an eyebrow.
After a moment, he pulled out a tightly folded piece of paper and proffered it across the podium. Drave looked at it, then nodded at Samandra.
"Me?" she cried in protest. She rolled her eyes. "Fine!" she groaned.
She took the paper delicately from Frey's hand, touching it as little as possible. "That's been down there for days, right?"
"Ever since Dracken captured us," Frey said, with a wink. "Lucky they didn't search us too closely."
Samandra wrinkled her pretty nose. "Ugh."
She handed the paper to Drave, who unfolded it, apparently unconcerned by the moistness and the smell.
"It's a page from the dockmaster's book at Retribution Falls. You can see his name and title signed down there in the bottom corner," Frey told him.
"I see it," said Drave. He turned the paper over. "I don't see the _Ketty Jay_ on here, though."
"We weren't calling ourselves the _Ketty Jay_ at the time. It would have been a bit stupid with half of Vardia trying to catch or kill us."
"How convenient!" Grephen crowed.
"I'm not showing it to you to prove _I_ was there. The fact that you hold it in your hand is proof enough that I was there," Frey replied. "The name you should be looking at is the _Moment of Silence_. If you look up her records, you'll find she's a craft registered to the Awakeners. The signature will also match the captain's. She was the craft shuttling Awakener gold to Retribution Falls to finance the army."
Grephen was becoming short of breath. "That... that piece of paper doesn't prove anything! A forged piece of rubbish!"
There were many tales told about Kedmund Drave. Like all the Century Knights, he had his own kind of legend. One of the less unpleasant stories claimed that he could tell if a man was lying just by looking into his eyes. He looked now: a penetrating gaze, boring into the Duke.
Grephen backed off a step. "You're going to take the word of a convict over that of a Duke?"
"A duke who still hasn't told me why he's attempting to execute these prisoners when he _knows_ they should have gone to the Archduke for trial."
"This is ridiculous!" Grephen cried, flailing. "I'm not answering to you! I don't have to answer to anyone but the Archduke in my own duchy."
"We act for the Archduke," said Drave. "So you answer to me!"
"Come on, Grephen!" Frey jeered. "Tell him why you want me dead! Tell him about Orkmund and all your pirate friends!"
"And you!" Grephen cried, thrusting a shaky finger at him. "I've had quite enough out of you." He looked at the executioner, who was still standing on the podium, holding Frey's cutlass. "Kill him!" Grephen ordered.
Two lever-action shotguns spun out from beneath Samandra Bree's long coat and fixed on the executioner. "Raise that sword and you're the first to die," she said.
The executioner stayed where he was, his gaze flicking between the Duke and the twin barrels aimed at his face. Frey was in no doubt which would prove most persuasive.
The Duke's guards were stirring uneasily now. Their loyalty was to their duke, and they didn't like to see him bullied. Colden Grudge, sensing the tension, flung back his cloak to allow himself easy access to the double-bladed hand axes hanging at his belt.
"Your Grace, I think you had better come with me," said Drave, "until we can verify your innocence."
"You're _arresting_ me?" Grephen gasped. He looked left and right, eyes bulging, a cornered animal searching for a way out. The elderly judge had already retreated, distancing himself from the Duke.
"Your Grace!" Thade snapped, seeing the panic on his companion's face. "Calm yourself!"
"I'm requesting the pleasure of your company on the Archduke's behalf," Drave insisted steadily. "You won't be locked up. We just need to be sure you aren't going anywhere. If these allegations are groundless, you've nothing to fear."
"Nothing to fear?" he screeched. "I'm a duke! Spit and blood, I'm a duke of Vardia! You can't treat me like this in my own house!" He hesitated, gaping, as if shocked by the enormity of what he was about to do. Then he turned to his captain of the guard and shouted, "Seize them! Arrest those Knights!"
Chaos erupted in the courtyard. The militia surged in on the Knights. Samandra Bree's shotguns bellowed, and two men flew backward in a cloud of blood. Colden Grudge swung his axes, severing limbs and fingers. Kedmund Drave moved faster than his bulk and armor suggested he could, slipping out of the grasp of two soldiers, coming up with pistols blazing. In seconds, the space in front of Frey's makeshift gallows became a battlefield, as the militia tried to overwhelm the Century Knights and the Knights retaliated with lethal force.
The executioner was standing agape. Frey turned to him, holding out his hands.
"Cut the ropes!" he said. It was addressed to the cutlass rather than to the man holding it.
The blade moved of its own accord, slashing through the air and dividing the rope between Frey's wrists. As soon as Frey's hands were free, the cutlass somersaulted from the executioner's hands and into his. An instant later, Frey had the tip at the confused executioner's throat. The man's eyes bulged in incomprehension. Frey delivered a good, solid kick square between the legs. The executioner's eyes bulged even further as he sank gently to the ground.
Pinn was cheering from inside the cage. Crake shouted at Frey and pointed. "Dracken's running!" he cried.
Frey looked. The mêlée in the courtyard had become fiercer. The Knights were many times outnumbered, but they still wouldn't go down. Several bloodied bodies lay on the ground. The militia had given up trying to seize anyone and was just trying to kill them now, but their rifles were unwieldy in such close quarters. Some had reverted to pistols and knives. The Knights slipped between the bullets and blades with practiced savagery, and their opponents couldn't lay a hand on them.
Beyond it all, Frey could see Trinica Dracken. She was fleeing toward the door that led into the barracks building, away from the courtyard. Duke Grephen was backing away from the knot of men struggling with the Knights. He looked dazed, startled by the carnage he'd unleashed. Inadvertently, he strayed too close to the cage where the _Ketty Jay_ 's crew was imprisoned, and Malvery reached out and grabbed him with his thick arms, hugging him close to the bars.
"I've got this one, Cap'n!" Malvery yelled, as Frey launched off the platform in pursuit of Trinica. He sprinted across the courtyard as she disappeared through the door. From the corner of his eye, he saw Gallian Thade running for the same door. The aristocrat had obviously decided that Trinica had the right idea and had abandoned his duke in favor of a quick escape.
The two of them raced across the courtyard, and for a moment it looked as if they'd reach their destination at the same time. But then Frey saw Kedmund Drave raise his pistol and fire through the press of bodies that surrounded him. Thade's sprint became a stumble, tripping forward under his own momentum. His face went slack, and he crashed to the ground in a heap of dust, his fine jacket holed and stained with blood.
Frey ran on, fearing a bullet in his own back at any moment, but Drave was too busy saving himself to spare more than a split second to deal with anyone else. Pinn and Malvery cheered him on as he flew through the open doorway, out of the courtyard, and into the cool stone corridors of the barracks.
Trinica was just disappearing around a corner, and he gave chase. Her compass and charts were the only bargaining chips he had; if she got away with them, he and his crew would still hang for their part in the crime. As he rounded the corner, he glimpsed her again—her black-clad figure, her roughly cropped white hair. Hearing his footsteps, she looked back at him. Her eyes showed him nothing, not even surprise. She dodged around another corner and was lost from sight.
Frey sprinted, arms pumping, his cutlass cutting the air. The barracks were deserted, and the walls rang with the hollow echoes of his bootsteps. He swung around the corner after Trinica.
She was standing there, a few meters away, her pistol aimed at his chest. Frey felt a moment of dreadful surprise, and then she shot him.
The gunshots were deafening. He didn't even have time to skid to a halt before she pulled the trigger twice in succession, shooting at virtually point-blank range. Frey's momentum was violently checked. He tottered on his heels and fell onto his back.
Trinica had dismissed him before he'd even hit the floor. She holstered her pistol and ran on, not interested in wasting a moment of her escape on sentiment.
Frey heard her footsteps disappear up the corridor. His chest heaved. His brain and body gradually slipped out of a state of shock.
He got up on his elbows. He felt around his chest in disbelief.
There were no holes in his shirt. He was unharmed. He got to his feet, looking around himself as if there might be an answer lying there.
_I'm not dead_ , he thought dumbly. _Why aren't I dead?_
There was only one thing he could think of. He looked down at his hand, which was still holding the cutlass.
The daemon-thralled blade had deflected the bullets.
"I didn't know it could do that," he murmured, staring at it in wonder. It wasn't even marked. "Crake, you're a bloody genius."
But there was no time for amazement. Matters were too urgent to wallow in good fortune.
The corridor ended in a T-junction, which brought him to a halt. He looked both ways. A door was ajar some way down the left corridor. He crept toward it. As he neared, he heard the sounds of muted rummaging inside and the click of case locks. Suddenly the door flew open and Trinica burst out. His arm snapped up, the edge of the cutlass resting against her throat, and she froze. In one hand was her pistol; in the other was the case he'd seen her carrying when they were shuttled down from the _Delirium Trigger_. The case holding the charts and the compass that would lead him back to Retribution Falls.
"Ah-ah, Trinica," he said chidingly. "You're not going anywhere. Drop the gun."
She stared at him, her eyes black, and said nothing.
"Don't think I'll do it?" he asked. "Try me. After what you just pulled, I'd be glad to be rid of you."
Trinica dropped her gun. Frey kicked it away from her. "Give me the case," he told her. She did so. She didn't seem surprised that he was still alive, and she didn't ask how.
"They'll kill me, Darian," she said. "When Grephen's plan comes to light, they'll hang me as a conspirator."
"Probably," said Frey. He was still angry enough not to care. The fact that she'd pulled the trigger on him had wounded him deeply. Somehow he'd always thought she wouldn't be able to do it. Watching him die was one thing, but this had a whole new level of cold-bloodedness to it. He felt unreasonably betrayed. Their past should have counted for something. You shouldn't be capable of killing someone you once loved.
Trinica stared at him for a long moment. "What now? Are you going to take me back to them?"
Frey didn't answer that. He hadn't thought beyond reclaiming the charts. He hadn't considered what he might do with Trinica.
"You know there's no guarantee they'll pardon you, don't you?" she said. "You know they could force you to cooperate. They might go back on their word after you've done what you said you would. Because, whatever way you cut it, you fired on the _Ace of Skulls_. You were attacking it when it exploded. You think the Archduke is going to want to pardon the man who killed his only son?" The corner of her mouth quirked into a smile. "You're a traitor and a pirate, just like I am."
Frey wanted to deny that intimacy. He wanted to tell her that they were not the same. But he knew she was right. She spoke to all his deepest fears. His whole plan relied on making a deal with the authorities, and he knew how authorities could be. There was no fairness or justice in them. They had the power to go back on any deal they made, if it suited them.
"Come with me, Darian," Trinica said. That shocked him.
"With you?" he sneered automatically.
"I'll drop you at a safe port. You can make your way from there. We'll be under terms of truce, as one captain to another; I'll see you're not harmed."
Frey hesitated, the sneer dropping from his face. He believed her. There was honor among pirates of a kind there never had been among the aristocracy. And yet it enraged him how even this slender invitation made his heart jump. Though he'd loathed her all these years, his body seemed never to forget the love they'd once shared. The merest hint of reconciliation, of alliance, ignited a yearning in his guts that disgusted him. He reacted by hardening his resolve.
_Damn her. Damn her and her terms of truce_.
She was no longer the woman he'd loved. The woman he loved no longer existed. Instead, he was haunted by her ghost.
"Why take the risk, Darian?" she said. "If you go back there, they'll hang you."
"If I don't go back, they'll hang my crew for sure."
"Since when did that matter to you?"
He didn't know the answer to that. It wasn't really important. It had been an accumulation of moments: a clutter of drunken laughter, of triumphant grins, of gunfights and arguments and sarcastic little quips. The feeling had crept up on him stealthily, and by the time he was aware of it, he'd been overtaken.
Maybe he'd decided it when he chose to trust Jez with his ignition code? Or when he'd given it away to Trinica in order to save Crake's life? Maybe it was that he felt the need to repay Jez's loyalty: she'd come back, and he admired her for that.
He didn't know when it had started to matter. He only knew that it did. He wouldn't abandon his crew, no matter what the risks were now.
Trinica saw the decision in his eyes. A faint respect crept into her tone. "Well," she said. "Look at you now."
But Frey was in no mood to be congratulated. He pressed the tip of his cutlass harder under her chin, tipping her head back. A spot of bright red blood bloomed against her white skin. "Give me one reason why I shouldn't kill you."
"There isn't one," said Trinica. "This is your chance, Darian. You take me back, I die anyway. So I promise you, I won't go quietly. You'd better kill me now. I'd rather you did it than them."
Her voice was utterly without fear. It was Frey who was afraid. He had no doubt that she meant what she said. She'd throw herself onto his sword rather than allow herself to be taken prisoner. She didn't just expect death, she welcomed it. At that moment he understood how she'd become one of the most dreaded pirate captains in Vardia. Everything inside her had died with their baby. How could you kill the walking dead?
He looked upon the woman he'd once loved, her chin raised, gazing coolly at him. He knew he'd never be able to do it. Because he owed her. He'd turned her into this creature when he left her so cruelly. Maybe he wasn't entirely responsible for the death of his child, but he bore some of the blame. He'd inspired her to do it. And, bitter as it was, he couldn't lie to himself anymore.
Trinica had suffered enough. It was written all over her.
He lowered the cutlass.
"You'll be hunted now," he said. "Not a freebooter anymore. A straight-out pirate. The Navy will never leave you alone."
Trinica stepped back, one slender hand going to her throat, covering the cut there. She stared at him with a strange, wounded tenderness.
He couldn't bear it. "Get out of here," he told her.
"You're not what I thought you were, Darian," she said, and there was something soft in her voice, something that reminded him of a voice from long ago that had once melted his heart. He dared not let it do so again.
"Goodbye, Trinica," he said. And then she turned and ran down the corridor, and he watched her go until she was lost from sight.
BY THE TIME FREY returned to the courtyard, the battle had ended. Six of the militia had surrendered. The rest lay in various states of death and dismemberment on the floor, their blood turning the dust into red mulch. Of the Century Knights, Colden Grudge had suffered a superficial wound on his brow. He was covering the Duke and the surviving militia with his autocannon. There had been no opportunity to use it earlier, due to the close-quarters fighting, but he looked eager enough to be given the excuse now.
Kedmund Drave looked up as Frey appeared, alerted by the rousing cheer from the caged wagon where his crew was imprisoned. Frey had stashed the charts and compass he'd taken from Trinica, and his cutlass was jammed through his belt. He walked with a tired step.
"Didn't expect to see you back," Drave commented.
"Just eager to help out the Coalition," Frey replied. "Call me a patriot."
"Dracken?"
"She got away."
"You think she might warn the others? Orkmund and his men?"
"I've made sure she can't get to them. But we should move quickly. They won't attack while there's no one to give them a signal, but they'll get wind of what's happened here sooner or later."
"Tell us where they are. We'll deal with them."
Frey laughed sardonically. "No. I'll tell you what'll happen. You assemble a strike force of Navy aircraft. I'll lead them into Retribution Falls. Without me, you won't know where you're going."
Drave stared at him, searching for signs of deceit. Frey wasn't intimidated. Numbed by his recent torture and the shock of facing his own extinction, he'd become impenetrably calm again.
"I'll need my craft and my crew," said Frey. "And I'll need my navigator back too. How did she find you, by the way?"
Samandra Bree had wandered over by this point. She tilted back her tricorn and smiled disarmingly. "Miss Kyte told us she'd made the acquaintance of a very important fellow called Air Marshal Barnery Vexford at a party at Scorchwood Heights. Best not to think about what she had to do to secure an audience with the Archduke's representatives at such short notice. He's quite a filthy old man." She patted him on the shoulder. "You do have an admirably loyal crew, Captain."
Frey could only imagine how loyal Jez had needed to be.
"Once they heard where you were, they sent us," said Drave. He looked around himself at the dead lying on the ground. "By the Duke's reaction, I'd say her story and yours have some truth in it."
"I want pardons," said Frey. "In writing."
"You'll get them," said Drave. "When you've led us to Retribution Falls. Not before." Frey opened his mouth to protest, but Drave held up one metal-gloved hand. "Pardons can be revoked. Makes no difference if you have a piece of paper or not. If you're telling the truth, and you do what you say, then you'll get what you want. But you double-cross me, and there'll be no place in the world that's safe for you."
Frey met his gaze steadily. Threats couldn't faze him now. "Then I suppose we'll just have to trust each other, won't we? Now get my men out of that cage."
# _Chapter Thirty-six_
THE RETURN TO ROOK'S BONEYARD—JEZ IS BROUGHT TO THE FOLD—THE DAEMONS BETWEEN HARKINS AND PINN—FREY TAKES A RISK
urning up ahead, Cap'n. Hold steady 'til you see it."
Frey made a murmur of acknowledgment and Jez went back to her charts. The _Ketty Jay_ slid on through the mists of Rook's Boneyard.
Behind Frey, Crake consulted Dracken's compass and warned them where the deadly floating mines were hiding in the murk. His voice was muffled by the mask he wore. Frey wore one too.
Jez didn't. She'd given up pretending she needed to.
The cockpit was dim and stuffy, and sounds gave back strange echoes. Dew ran down the windglass, and the soft growl of the _Ketty Jay_ 's thrusters filled up the silence. Jez sat in her chair at the navigator's station, plotting their course as efficiently as ever. She absently tapped out a sequence on the electroheliograph with her left hand, warning those who followed of the location of the mines, half her mind still on the calculations.
Frey took off his mask for a moment and yelled, "How we doing back there, Doc?"
"They're still on our tail!" Malvery bellowed back from the cupola, where he had a view of what was going on behind the _Ketty Jay_. Only he could see the huge shapes in the darkness that drifted after them like malevolent phantoms.
"Bet you never thought you'd see the day when you'd be leading a flotilla of Navy craft." Jez grinned, looking over at the captain.
"I never did," he agreed with a wry twitch of the lips, then put his gas filter back on.
There was a dull explosion as a mine was detonated by one of the Navy minesweepers, clearing a path for the fleet behind them. It had been slow progress over many hours, gradually creeping closer and closer to Retribution Falls, removing all threats along the way. Since the other craft didn't have compasses of their own, it was just too risky to try to bring the whole strike force through the mines in single file.
Jez wondered how far the sound carried through the choking mist and deep, sharp canyons. She wondered if they might find the denizens of Retribution Falls waiting for them when they arrived. But despite the danger all around them and the certain knowledge of the conflict to come, she felt content.
The sounds of the _Ketty Jay_ soothed her. She'd come to know its tics and groans, and they were reassuring. The navigator's seat had found her shape, as if it had somehow molded itself to her buttocks and back, and its form seemed natural now. The muggy heat of the cockpit had become cozy, a warm sanctuary from the hostile world that waited outside.
It was a strange experience. So much time had passed since the Manes had attacked that small village in Yortland that she'd forgotten what contentment felt like. Three years she'd been wandering, hiding, always afraid of discovery. She'd never put down roots or allowed herself to care for those around her.
But here, at last, she felt like she was home. She'd found her place. She was here to stay.
Her reunion with the crew had been unexpectedly touching. Malvery had almost crushed her ribs with a hug, before planting a big, whiskery kiss on her cheek. Frey was similarly effusive. Pinn slapped her on the arm; Harkins babbled, jubilant. Silo nodded respectfully, which was as close as he ever came to a joyous outburst. Even Crake seemed happy to see her, though there was a wariness in his eyes, as if he expected her to reject his handshake.
"Thank you," he said simply.
"I brought Bess," she said, thumbing behind her at the open maw of the cargo hold. "She's in there."
Crake's eyes filled with tears, and his face split into an uncontrollable grin that was half a sob; then he hugged her, clutching her tightly to him. She was surprised enough to hug him back. Of all people, Crake had been the one who should have been most enthusiastic in his loathing. He was smart and knowledgeable in the hidden ways. He'd have guessed her nature by now.
And yet he embraced her, as the others did.
She'd hoped that at best they'd let her go on her way. She'd hoped that they'd be grateful enough for their rescue that they'd keep her secret from the Century Knights, no matter how dangerous they knew her to be. The idea of taking her back was ridiculous. They might tolerate an openly practicing daemonist on board, but how could you get on with a woman whose heart didn't beat, who didn't need to breathe or sleep or eat? How could you ever trust someone like that? Robbed of the common vulnerabilities of humanity, how could you ever know what they might do next?
She'd accepted that they might turn her in. Gratitude didn't apply to monsters. They might try to destroy her. She'd been ready for that. It was an acceptable risk.
But they greeted her like an old friend.
She hardly dared believe what was happening. Surely they were just relieved at escaping execution and hadn't had time to think it through? If that was the case, then their suspicions would grow as soon as their happiness faded. She couldn't bear that. She had to know if they accepted her as she was or if they simply hadn't taken in the truth yet.
"I suppose..." she said, once Crake had released her. "I suppose I owe you an explanation."
"No," said Pinn, beaming.
Jez frowned at his abruptness and the twinkle of amusement in his piggy eyes. "No, I mean, you must be wondering how I did it."
Silo shrugged.
"Not really," said Frey.
"Nope," said Harkins.
"Couldn't give a dog's arse, frankly," Malvery added.
She looked at the faces of the crew, and she began to understand. Perhaps they knew exactly what she was, perhaps not. But it didn't matter because they didn't care. She was one of them.
"You?" she asked Crake.
"I already know how you did it," he said. "No need to tell me." His smile was warm. Bringing Bess back had indebted him to her forever. Bringing the _Ketty Jay_ back had won the hearts of the rest of them.
Seeing their grinning faces joined together in a conspiracy of support, she at last let herself believe. The grin spread to her face too.
"Well, then," she said. "That's that."
HARKINS FLEXED HIS FINGERS on his flight stick and tried not to throw up in his own lap. His stomach had knotted into a ball, and his breath came in shallow pants that offered little relief from the crushing anxiety that pressed in on him. He hunkered down in the cockpit of the Firecrow, eyes darting nervously here and there. He wished the mist would clear. He was also afraid of what he'd see when it did.
Only the metal cocoon of the Firecrow kept him together. The sense of safety it afforded stopped him from panicking completely.
It seemed so long ago that they'd left the Firecrow hidden in a remote cave next to Pinn's Skylance. The Cap'n had deemed it too dangerous to travel into Rook's Boneyard in convoy. He'd been right: without masks, the deadly fumes from the lava river would have caused both Harkins and Pinn to crash.
Their fortunes hadn't gone too well since then, though. The Firecrow was Harkins's only security, and without it he was lost. He'd spent most of the subsequent days in blubbering fear: first hiding in the _Ketty Jay_ so as not to venture into Retribution Falls, then trembling in Dracken's brig on the _Delirium Trigger_ , and later waiting to die in his cell at Mortengrace. Superstitiously he blamed his bad luck on his separation from the Firecrow. He should never have deserted her. He wouldn't do so again if he could help it.
Vast, angular shapes glided past to port and starboard like undersea leviathans. Smaller fighters hove between them, their lights bright bruises against the serene fog. Harkins made minute course corrections and fretted about a frigate clipping his wing and sending him spiraling to a fiery death.
The mines petered out after the lava river. Presumably the pirates reasoned that anyone without a compass to detect them would be dead by that point. He'd hoped that leaving the mines behind would ease the tension a little, but he found that it increased it instead. They were on the final leg of the journey. Soon they'd reach the enormous marshy sinkhole where Retribution Falls lay. Soon the fight would begin.
_Survive_ , said Frey. _That's all you have to do. Don't take any risks. Look out for each other_.
The Cap'n had persuaded Kedmund Drave to let them bring the _Ketty Jay_ 's outfliers. They were invaluable pilots, he'd said, and they'd need every craft in the fight. Harkins and Pinn were useless sitting on board the _Ketty Jay_. Since their fighters didn't have Navy markings, they could sow havoc among the pirates, who would be unable to tell them apart from their allies.
Harkins had pointed out that this worked both ways, but Frey had assured him the Navy would know who they were and what they looked like. Harkins wasn't quite so certain. He could just see a Navy frigate firing a shell up his exhaust in the heat of the moment.
The flotilla was packed in tight, a tentative train behind the _Ketty Jay_. Harkins was tucked inside it, with Pinn somewhere nearby. The mist was beginning to thin out noticeably. He could make out the detail on the nearest frigates, their gun turrets and armored keels.
He fingered his silver earcuff. Having a daemon clipped to his ear only added to his unease, but Crake had offered them and Frey had insisted.
"Anybody out there?" he said. "This is... um... this is Harkins. Just wondering if anybody's out there. Say something if you are."
"Clam it, Harkins," said Pinn's voice in his ear, making him jump. "Crake said to use these things only when we had to. They'll drain you if you start gibbering."
"Oh. I was testing it, that's all. You think the Cap'n can hear?"
"He's too far ahead. They've got a short range. Now shut up."
Harkins snapped his mouth closed. His ear was tingling where the cuff touched his skin. He didn't really understand all this daemonism business, but it made him feel a little better to hear a familiar voice.
Ahead, the fleet was beginning to break up and spread out as visibility improved and they dipped below the mist into clear air. Harkins's heart thumped against his thin ribs as craft started to accelerate around him. Beneath them was a river, running along the canyon floor. The last stage of the journey. The moment was imminent. He wanted to curl up and hide.
Then at last the canyon gave out and the river plunged away down the sheer wall of the sinkhole. They'd arrived at Retribution Falls.
It lay as the _Ketty Jay_ had left it, a shabby assemblage of scaffolded platforms and ramshackle buildings, steeped in the rancid marsh air. The great sinkhole, many kloms across, was ribboned in slicks of metallic ooze. Where the earth broke through the water, rotting dwellings grew like scabs.
But Harkins wasn't looking at the town. He was looking at the aircraft. Hundreds of aircraft.
The fleet had grown in their absence. The landing pads were choked with fighters and heavy attack craft. Battered frigates floated at anchor; clusters of caravels and corvettes hung pensively over the town; shuttles and small personal craft hummed through the air.
There must have been three hundred, at least. Harkins felt his stomach clench and his gorge rise. He was suddenly glad he hadn't eaten anything that morning.
A swarm of fighters was already scrambling to meet them as Harkins came out of the canyon. They'd been alerted by the sight of the first Navy craft at the head of the convoy. Retribution Falls kept a standing defense force, it seemed, ready to go at a moment's notice. But those few craft aside, the pirate army had been caught completely by surprise.
The guns of the Navy frigates bellowed in a deafening cascade, making Harkins shriek inside his cockpit. Their opening salvo ripped a flaming scar across the sprawling town.
The primary target was the main landing pad, where the greatest number of smaller craft were clustered. It was obliterated in a cataclysm of fire. The other, more temporary landing pads that floated on the marsh were also struck. Those that weren't destroyed outright began to list as their aerium tanks were holed, sending dozens of craft sliding into the sucking bog beneath.
Two of the nearest pirate frigates, anchored close to each other, were smashed with explosive shells. One of them split along its keel in a smoky red bloom and sank to the ground in two halves. There were enough unpunctured aerium tanks to make the descent slow and terrible, like a ship being pulled to the bottom of the sea.
After the initial assault, there was a pause to reload, and the Navy fighters came racing out of the cover of the fleet. Harkins saw the sleek Windblades shoot past him like darts, heading to meet the fighters rising from Retribution Falls. He gritted his teeth. He wanted, more than anything, to stay concealed behind the flanks of the enormous frigates. This wasn't his fight, after all: the pirates weren't his enemies.
But the heavy guns of the pirate craft would start up soon, blasting at the fleet, and a tiny craft like his would be dashed to pieces in the shell fire.
The safest thing to do was attack.
He heard Pinn whoop in his ear and cursed him for his absurd courage. He could already picture that moron racing ahead of the pack, desperate for the first kill, heedless of the danger. He was the kind who would evade death forever, simply because he didn't realize it was there. The fearless always survived. It was one of the great unfairnesses of life, in Harkins's opinion.
Well, he was damned if he'd let Pinn mock him for being the last one into the battle. The thought of that chubby-cheeked face screwed up in laughter made his blood boil. He hit the throttle and plunged out of the flotilla, pursuing the Navy Windblades into the fray.
The pirate fighters were a motley of different models from different workshops, representing the last thirty years of aviation technology. They came on like a cloud of flies, without discipline or any hint of a formation. The Navy fighters were tighter, punching toward them like an arrow. Harkins slipped in near the back.
The Firecrow's engines roared, encompassing him in sound. The craft shook and trembled. Through the windglass bubble on its nose, Harkins could see the vile colors of the marsh blurring beneath him. Two Windblades hung on his wings, their pilots wearing identical Navy-gray helmets, their attention focused on the attack. Harkins swallowed and hunched forward, his finger hovering over the trigger.
The two sides met as the Navy frigates released another salvo, pounding the town of Retribution Falls, pulverizing those pirate craft that were too slow to react to the surprise attack. Suddenly the world was full of explosions and machine guns, and Harkins yelled in fear as he opened up on the enemy.
The Windblades spread out, spiraling and rolling as they approached. Harkins jinked left to avoid a lashing of tracer fire, picked his target, and sent a long burst back toward them. He aimed where he thought the craft was going rather than where it was, and his guess was accurate. The pilot flew right through the deadly hail of gunfire. The windglass of the cockpit shattered, and the pilot jerked as he was shot through with bullets. The craft tipped into a long, lazy dive toward destruction.
The pirates and Navy fighters broke upon one another like waves onto rocks, spuming in all directions as they scattered. The battle became a mass of individual dogfights.
Harkins threw the Firecrow into a steep climb, raking his guns across the underside of an old Westingley Scout. It corkscrewed out of control and slammed into the tail of another pirate craft as he soared upward. Something thundered past his wing, missing a collision by less than a meter. Dizzy with fear-driven adrenaline, he paid it no mind. He leveled out, letting the G-force off a bit before coming around and onto the tail of a rickety Cloudskimmer.
Pinn screamed with joy in his ear. Harkins gave a scream of a different kind and pressed down on his guns.
"TIME TO GO," SAID FREY, as the first scattered volleys of return fire from the pirate frigates came smashing into the Navy fleet. He vented aerium and dropped the _Ketty Jay_ down beneath the keels of the larger aircraft, then hit the thrusters and sped toward the town.
The pirate frigates had begun to wake up now, shedding their anchor chains and gliding into action, their gun crews finally in position. Frey had hung back to hide as best he could among the heavy craft, but, like Harkins, he knew it would be suicide to stay once the big guns got going. Besides, he'd done his job. He'd led them here. That was enough to earn his pardon, assuming they intended to give it to him.
Now he had a purpose of his own, and it didn't involve getting tangled up in a squabble between the Navy and Orkmund's pirate gang.
Retribution Falls was a mess. Whole areas were flattened as the dwellings, never built for strength, fell apart from the concussion of a single shell. As he watched, one of the platforms at the far end of the town tipped and fell, its gridwork of scaffolding blasted away on one side. Buildings crumbled into landslides of brick, sweeping people with them as they went. Bodies were mangled and ground to bits as an entire district collapsed into the marsh.
Frey heard Malvery start up on the autocannon, blasting away at a pirate fighter as it screamed overhead. He ignored it, steered away from the main conflict, and angled the _Ketty Jay_ toward the platform he wanted. The quality of architecture there was the highest in the town, and Frey was pleased to see it had suffered only superficial damage.
That was good, since he planned to land there.
"You sure you want to do this, Cap'n?" Jez asked doubtfully, peering through the windglass. Large sections of Retribution Falls had been wrecked. Plumes of smoke billowed from their ruins. "There's no telling how long it'll be before someone shells the shit out of that platform too."
Frey was anything but sure. "They're concentrating fire on the pirate frigates now," he said, mostly to convince himself. "The town itself isn't a threat." Malvery cheered in triumph from the cupola. Frey assumed he'd made a hit.
"Your call, Cap'n," she said. "But we can get out of this now if we want to."
"I hear you, Jez," he said. But he was committed in his heart now. He couldn't turn back.
At least this time he'd consulted his crew. He'd outlined his plan and asked them if they wanted to be part of it. Nobody was being forced; nobody was being duped. He wasn't going to order anyone into this.
Some were reluctant. Some thought it would be better to cut their losses. They weren't keen on the risk. But in the end, all of them agreed. Because they trusted him. Because he was their captain.
Frey took the _Ketty Jay_ closer to the platform. Jez leaned over his shoulder and pointed. "There's the square."
"Malvery!" he yelled. "Get out of the cupola and get ready!"
Jez picked up her rifle from beneath the navigator's station as Frey brought the _Ketty Jay_ down in the square. Those few people who were nearby went running as she came in to land, hard and heavy because Frey was too nervous to be careful. She bumped down with a jolt that made Jez stagger.
Frey sat there for a moment. Overhead, shells exploded and pirate fighters weaved through the sky. He should just take off again. He didn't have to do this. Maybe this was just history repeating, another all-or-nothing hand of Rake that might win him everything or lose it all, when he should have just laid down his cards and walked away with what he had.
_You've got a craft, a crew, and the whole world to explore. Nobody's your master. Now, that's not so bad, is it? If you're lucky, the Coalition will pardon you when all this is done. Drave may be a mean bastard, but he doesn't seem like a liar. You'll be free_.
Whether Drave would honor his word or not was a moot point. He wasn't sticking around to see. As soon as he'd done what he came here to do, he planned to run. The Navy would be tied up here for a while. Let them pardon him in his absence.
But first there was the small matter of fifty thousand ducats. Fifty thousand ducats that had been promised him by the brass-eyed whispermonger Quail. Fifty thousand ducats that he felt he'd damn well earned by now.
This was their chance to be rich. To leave the rogue's life behind and allow themselves a bit of comfort. Equal shares for them all, because everyone had done their part.
He looked out of the cockpit at the barricade surrounding Orkmund's stronghold. The square they'd landed in lay right in front of it. A few days ago, they'd stood here to hear the great pirate speak. Somewhere inside that building was a red chest with a silver wolf clasp that he'd first seen being loaded onto the _Moment of Silence_ when he visited Amalicia Thade at the Awakener hermitage.
The thought of Amalicia surprised him. From the moment he left the hermitage, he'd completely forgotten about her. To suddenly encounter her in his memory was a jolt, like rediscovering a discarded trinket that he thought was lost forever.
"Are we going?" Jez asked.
"We're going," said Frey. He got out of his chair and ran down the corridor to the steps that led to the cargo hold, where the rest of the crew was assembled, armed to the teeth.
In the few moments before the cargo ramp opened, he belatedly remembered that Gallian Thade had been killed at Mortengrace. That meant Amalicia was free from the hermitage where she'd been imprisoned. Free, and unbearably rich.
_Damn it, I should have just married her when I had the chance_ , he thought.
Then he remembered that Trinica Dracken had also been the daughter of an enormously rich businessman, and he'd been only moments from making himself a part of that inheritance. He swore under his breath.
_Damn it, I should have married_ her _too!_
By the time they went rushing down the cargo ramp and out into Retribution Falls, Frey was quite eager to shoot someone.
# _Chapter Thirty-seven_
TREASURE HUNT—HARKINS GETS INTO TROUBLE—ORKMUND AGAIN
irates and whores ran in panic across the square, heads covered against the thundering concussions and the threat of falling rubble. With most aircraft already in the air or destroyed on the landing pads, they were helpless witnesses as the Navy pummeled the pirate frigates overhead and fighters wheeled and spat bullets. They fled for what shelter they could and hoped that fate would be merciful.
Frey led his crew down the cargo ramp, cutlass swinging against his leg, pistols raised. The stink of the marsh hit them as they came out into the open air and took up positions around the _Ketty Jay_. He'd been expecting some resistance from the locals, but he found himself pleasantly disappointed. The freebooters who were passing through the square couldn't have cared less why they were landing their craft here. Since they weren't wearing Navy colors, they could do what they liked. The sight of Bess coming down the ramp deterred any thought of further inquiry.
Frey glanced at the Navy fleet, visible in the distance a few kloms away. They were spreading out defensively as the pirate craft increased their assault. Half the pirate army's larger craft were destroyed, but the others were giving as good as they got. Frey saw a Navy frigate slip into a groaning descent, its flanks aflame.
As far as he was concerned, both sides could blow themselves to pieces. He had little love for either. As long as some Navy craft survived to tell the tale and exonerate him, that was fine.
"Alright, let's go!" he cried. Silo closed up the cargo ramp and they hurried toward their target, with the Murthian covering their backs.
There was a barricade surrounding Orkmund's squat gray stronghold. The watchtowers surmounting the mass of crossed girders and spikes were empty, but the gate was still closed. It was an enormous slab of metal on rollers, heavy enough to need three men to move it and presumably secured on the other side.
"Bess! Open that gate!" Frey called.
The golem stamped past him. She dug her massive fingers into the metal and wrenched. The gate shrieked in protest as a bolt on the inside resisted, but Bess's strength was inexorable, and the gate slowly gave way.
Frey could see one or two men who had stopped at the edge of the square and were staring. Clearly, they were puzzled to see several men who looked like pirates breaking into the pirate captain's stronghold. Malvery raised his shotgun and sighted at one of them; Silo took aim at the other.
"Keep moving, lads," said the doctor. "This doesn't concern you."
They decided that it didn't concern them after all. There was a loud snap of metal and the gate rolled out of the way with a screech.
"Nice work, Bess," said Frey. Crake patted her on the arm as they sallied inside.
Orkmund's stronghold wasn't large—certainly not the size of somewhere like Mortengrace—but it was secure. The gray, mold-streaked walls were thick, and the windows were small and deeply set. Too small to climb through.
Once inside the barricade, they were faced with a three-story building with two projecting wings on either side, making a three-sided square. The entrance was set between the wings, at the far end of the square.
Frey led them to the nearest wall, at the tip of one of the wings. He pressed himself against it and looked around the corner. He was sweating with the tension. He expected to be shot at by an unseen foe or obliterated by a shell from above at any moment. But the stronghold was quiet, and the sounds of destruction had retreated temporarily into the distance.
"I don't see anyone, Cap'n," Malvery said at his shoulder.
Frey didn't like the idea of rushing up to the entrance. There were too many windows facing inward on either side. Anyone up there with a gun could pick off attackers with ease.
"We'll make our own way in," he said. He turned to Bess. Her eyes glimmered behind her face grille. "Can you get through this wall?"
Bess could.
There was a pirate standing in the doorway of the room on the other side. The sight of the hulking figure crashing through the wall in a cloud of dust and rubble scared him witless. As with anything that scared him, his first reaction was to shoot it. Bess pushed her way through the debris as bullets ricocheted from her armored torso. She tore a chunk of stone from the wall and flung it at her attacker. It hit him in the forehead hard enough to take his head off. The remains of the pirate staggered a few steps before tipping over.
"Damn good shot!" Malvery exclaimed, climbing through the hole in the wall.
Frey climbed in after him. "Our treasure's in this building somewhere," he said. "Let's get looking."
PINN YOWLED AND WHOOPED like an overexcited monkey as he dodged between swaying trails of tracer fire. The Skylance screamed happily along with him, obedient to his every command, banking and rolling through the chaos of a packed sky. Pinn was flying the fight of his life, and the Skylance was in the best shape she'd ever been, with her tanks full of the finest prothane and aerium, courtesy of the Coalition Navy. Together, nothing could touch them.
It was almost too easy. The pirates never saw him coming. Their eyes were all on the Navy Windblades; they considered Pinn's Skylance as one of their own. The last thing they expected was a fellow pirate to turn against them. Those few seconds of confusion were usually all it took.
He glanced down at the ferrotype of Lisinda that swung on a chain from his dash and grinned fiercely. "You should see me now!" he cried. "Sweetness, you should see me now!"
This was what his world might have been like, day after day, if only those Sammies hadn't pulled out of the Second Aerium War just as he was about to join in. This was living.
His machine guns rattled as he drew a line of puncture marks across the flank of a pirate corvette. He darted away before anyone could see who had done it. The corvette—a medium-size attack craft with two sets of wings and a fearsome battery of guns—was too busy dealing with Windblades to pay him any mind.
_"Pinn!"_ Harkins squealed in his ear, at a pitch high enough to make him wince. "Pinn, where are you? There's three of them on me!"
Pinn scanned the mêlée frantically, but he could find no sign of Harkins amid the swooping tangle of fighters that surrounded him. Belatedly, he remembered that Frey had instructed them to look out for each other. The object was survival, not kill count.
"I can't see you!"
"Pinn! Bloody help me!" Harkins yelled.
"I can't help you if I can't see you!" Pinn yelled back. Then, in a rare and remarkable moment, he had an idea. "Climb up! Climb out of the pack! I'll find you up there!"
He pulled the Skylance into a steep climb, making his way free of the main mass of combat. The higher they got, the fewer aircraft would be in their way to complicate things.
"Pinn! They're right on my tail!"
"I'm coming, you noisy chickenshit! Hold on!"
He spotted the approach of tracers from his port side and rolled the Skylance a moment before a blast of machine-gun fire ripped past the belly of the craft. A quick glance told him that it had come from a Windblade.
"I'm on your side!" he yelled. He could feel a strange tiredness settling into his bones and remembered the daemonic earpiece. Every whoop and comment he made was sucking a little more energy out of him, and now he'd begun to notice it. He nearly cursed, but at the last minute remembered to keep his mouth shut.
The Windblade had realized its mistake and was peeling away to search for fresh targets. Pinn craned around in his seat to look for Harkins and spotted him a klom away, shooting skyward at an angle close to vertical. Three aircraft chased him, sending weaving lines of tracer fire ahead of them.
Pinn hit the throttle and the Skylance responded. He streaked across the dull sky, the battle beneath him and the mists above, his eyes fixed on the steadily ascending quartet of aircraft. Harkins was jinking and twisting as best he could, but the sheer volume of gunfire made it unlikely he could evade them long enough to make it to cover.
Pinn found himself in the grip of an unfamiliar sensation. He was worried. As much as he scorned Harkins, he didn't want to be without him. Harkins was just about the only person on the crew he could push around.
_You better not get shot down, you stuttering old lunatic_.
Smoke began to pour from the Firecrow's wing.
"I'm hit! I'm hit!" Harkins screeched.
Pinn thumbed his trigger and his machine guns clattered, tearing through the foremost of Harkins's pursuers. The aircraft exploded in midair, sending chunks of itself flying away. The others were too close to avoid the debris: a slab of wing, spinning end over end, whipped through the air and into the cockpit of another pirate, smashing him out of the sky. The third aircraft went into evasive maneuvers immediately, searching for the author of the surprise attack, and then decided that the chase wasn't worth it and plunged back down toward the main mass of the fighting.
Pinn whooped and slapped the side of the cockpit, then scooped up the ferrotype of Lisinda and gave her a kiss. "Harkins!" he called. "How bad is it?"
Harkins leveled out and then banked experimentally. He looked wobbly, but the smoke had stopped.
"I... er... I lost one of my thrusters... had to shut it down. Not good, really, then."
Pinn looked regretfully at the combat going on below them. "We're done here. You're not gonna last another skirmish. Let's go help out the Cap'n." He matched Harkins's turn and fell into position behind him.
"Hey, Pinn? Hey?"
"What?"
There was a pause. "Umm... thanks."
Pinn smiled to himself. "Didn't I tell you to clam it?" he said.
"WHERE'S THE TREASURE KEPT?" Malvery demanded. The pirate's reply was incoherent, mouthed as it was around the barrel of a shotgun.
"Take the gun out?" Crake suggested.
Malvery withdrew the shotgun a little way. The pirate—still shocked at being collared by the bulky doctor—bent over and gagged. By the time he'd recovered, there was sullen defiance in his glare.
"The treasure. Where?" Malvery demanded again.
The pirate suggested some anatomically improbable places where Malvery could shove his mother. Malvery broke his nose with the butt of the shotgun, then looked around at his companions and shrugged. "That's me out of ideas," he said.
Silo and Jez were covering either end of the corridor. The stronghold was mostly deserted—the pirates had evidently fled—but Frey was taking no chances. The pounding of the guns outside seemed worryingly close now, echoing through the empty spaces, bouncing off the unadorned walls. Dust shook from the ceiling, bringing new cracks.
"We haven't got time for this," he muttered. He seized the pirate, who was holding his bloodied nose, and pointed at Crake.
"This is my friend Grayther Crake. He's got quite a remarkable smile. Why don't you show him, Crake?"
Crake grinned. The pirate stared at him for a moment. His gore-streaked hands came away from his face, the pain of his nose forgotten, and he craned forward in admiration.
"Here," he said. "That's a nice tooth."
Half a minute later they were on their way, newly furnished with directions. Malvery had insisted on clubbing the pirate once more for that crack about his mother, but afterward they let him go, minus his pistols and several molars.
They hurried through the corridors, keyed up to face resistance, but they found few people to stand against them. One man ignored them completely, presumably running for the exit. Another took a potshot at Bess and was gunned down for his trouble.
A particularly heavy concussion shook the building and sent plate-size flakes of plaster raining from the ceiling. Frey stumbled to his knees, and Silo caught his arm as he fell. As he was helped to his feet, he met the Murthian's eyes. Both of them were thinking the same thing. They should get out of here now, while they still had the _Ketty Jay_ and their lives.
_Just this last thing_ , Frey told himself shakily. _Our luck'll hold_.
Silo saw the resolve in Frey's gaze and gave him the tiniest of nods, then reached out one long-fingered hand and squeezed his shoulder in reassurance.
Frey found himself suddenly grateful for the constant presence of the engineer in his life. Though Frey rarely even noticed him, Silo was always there, a silent strength, working invisibly behind the scenes to keep the _Ketty Jay_ running. Frey realized how important Silo had been to him all these years, a friend who asked for nothing but who would offer unquestioning support whenever it was needed. Silo had saved his life after the ambush in Sammie territory and been with him through all the bitterness that followed. Frey had never wanted a confidant; he wanted someone who he felt would never betray him, no matter what. That was Silo.
Driven by an absurd and overwhelming urge, he hugged his engineer. Silo stiffened in surprise.
"Rot and damnation, Cap'n, this isn't exactly the time!" Malvery cried.
Frey withdrew, his face coloring. "Right," he said. "You're right."
A few more turns brought them to the vault. It was exactly where the pirate had told them it would be. Unfortunately, it was where most of his friends were too.
The vault door was standing open as they arrived, and a dozen pirates were busy carrying out chests full of treasure. Orkmund himself was there, too, directing his men. He was more physically imposing in person than he'd been from a distance: muscular and tattooed, with a bald head and a boxer's face.
Frey had wanted to get the drop on them, but with Bess in tow it was impossible. By the time they'd rounded the corner, the men were alerted. Only the puzzling nature of the metallic clanks and leathery creaks had stopped them from pulling out their guns. But now the golem stepped into sight, with Frey and his crew behind her. Some of the men went white and backed away, dropping their end of the chests. Others let their burden fall and drew guns. But Frey's crew had their guns out already, and at the first sign of violence they started shooting.
The first volley cut down half of Orkmund's men, most of them with their revolvers still half out of their holsters. The crew of the _Ketty Jay_ ducked around the corner as the answering fire came, but it was mostly directed at Bess, who went stamping up the corridor, roaring as she did so. Those who hadn't been killed in the initial volley stumbled backward in the face of the metal giant, tripping over the chests, and scrambled to their feet to flee. Frey could hear Orkmund shouting something incoherent at them, urging them to stand and fight, but then there was a terrific explosion from above and the calamitous sound of falling stone.
Dust billowed out of the corridor and engulfed his crew where they hid. Frey coughed into his fist and looked around the corner. It took some seconds for the dust to clear, but when it did he saw Bess standing there, dirty but unharmed. A section of the ceiling had caved in, burying all but one of the chests. Of Orkmund and his men, there was nothing to be seen. They'd either fled or been buried. Frey didn't care which.
What he did care about was the red-lacquered chest that lay near Bess's feet. A chest with a beautiful branch-and-leaf intaglio on the lid and a silver clasp in the shape of a wolf's head. He ran to it and tugged at the lid. Locked. Stepping back, he blasted the clasp away with his revolver.
There would be no mistakes. He had to be sure.
The others had gathered around him as he knelt down and threw open the chest. Inside was a golden mass of ducats. Thousands upon thousands of coins. Even in the dust-hazed air, it seemed to him that they glimmered.
Bess leaned in over his shoulder to look. She cooed as she saw the wealth within.
Frey could hardly breathe. He had it at last. _They_ had it at last. After all the years of scrabbling in the dirt, they were rich.
He stepped back and looked at the joyous faces of his crew, transfixed by the sight of more money than they'd ever dreamed of.
"Bess, pick that up," he said. "We're getting out of here."
# _Chapter Thirty-eight_
SHELLS—THE DUEL—MALVERY'S HOUR—OUT OF THE MIST
rey didn't hear the explosion.
It took some seconds for his stunned senses to recover, but even then all he could remember was the sensation of being squashed from above by an enormous force, like an insect trodden on by an invisible boot. After that, there was the taste of grit in his mouth, the stinging in his eyes, and the high-pitched whine in his ears, like the squeal of a turbine.
He looked around. Everything was muffled and clouded. The air was gray with pulverized stone. He was on his hands and knees. Ahead of him, what had once been a corridor was now a wall of broken stone.
A shell, he thought numbly. Orkmund's stronghold must have taken a direct hit.
Suddenly he was being pulled to his feet. He looked up dazedly to see Silo holding his arm. The Murthian was saying something, but he couldn't hear. Silo stood him up and spoke with exaggerated volume and clarity, but to Frey it still sounded as if it came from a great distance through the cottony pressure in his ears.
"Cap'n? You hear me?"
"A little bit," he replied. His voice sounded strange in his own head.
"You hurt?"
Frey checked that he had all his arms and legs. "Don't think so."
There was a faint yell. Silo looked toward the rubble that had filled the corridor. Frey followed his gaze.
"Hey!" It was Malvery. Had it not been, Frey probably wouldn't have heard him, but the doctor's bellow could wake the dead.
"Doc!" Frey cried. "You okay?"
"Cap'n! We're fine over here. Cuts and bruises. Silo with you?"
"He's okay."
"Okay!"
The conversation faltered. The dust was settling, and now Frey could see the section of ceiling and wall that had collapsed into the corridor. Frey and Silo had been lagging behind, guarding the rear of the retreating group. Frey stared at the tons of rubble in front of him and thought how lucky they were that nobody had been beneath it.
"Wait there!" cried Malvery. Frey glimpsed him momentarily through a gap in the rubble. "We're going to get Bess to dig through to you!"
Silo grabbed Frey's shoulder and shook his head. He pointed up at the ceiling. "Ain't a good plan, Cap'n."
Frey caught on. "Silo says no!" he cried. "The roof could come down on you."
Malvery considered that for a moment. "I expect that'd hurt quite a bit," he said.
"Go on to the _Ketty Jay_. We'll find another way round."
"You sure?"
"You've got the treasure with you?"
"Safe and sound."
"Get it on board. We'll get there fast as we can."
"Right-o."
"And, Malvery? If they start shelling us again, you tell Jez to get her airborne and get you out of there."
"Without you, Cap'n?"
"Yeah."
"I'd rather choke on my own shit," Malvery replied cheerily. "See you on board."
Frey shook his head to clear it of the ringing. It was about as effective as he'd expected. At least his hearing was getting less muffled with time.
He picked up his revolver from the ground where it had fallen and thumbed in the direction they'd come. "That way, I suppose."
They hurried back down the corridor and through a doorway, into a crude kitchen. They could see an exterior window, but, even though it had been smashed by the explosion, it was too small to get through. Frey led the way into a simple eating hall with benches and a fireplace. He stayed close to the exterior wall, hoping for a door, but room after room confounded him. Eventually they came out into another corridor like the one they had left.
"Damn it, how hard can it be to get out of a building?" he complained, and that was when they ran into Orkmund.
He must have heard them an instant before they came around the corner, and that small warning meant he was faster than they were. He was emerging from a doorway as they came into sight, carrying a small jewelry box in his arms. Frey and Silo skidded to a halt as Orkmund dropped the box and pulled a revolver. By the time their own guns were halfway raised, Orkmund already had his leveled.
"Drop 'em!" he cried, and they froze.
Frey thought desperately, but he couldn't force an idea through the fog in his head. This wasn't a war: there was no question of taking prisoners. If they dropped their guns, he'd shoot them. If they drew, he'd shoot them.
"Drop 'em!" Orkmund shouted again, allowing no time for deliberation.
Frey looked at Silo. Silo looked back at him. And in that moment, Frey realized what the Murthian was thinking.
He could shoot only _one_ of them. And Silo had decided it was going to be him.
"Don't—" Frey began, but it was too late. Silo moved, raising his revolver to fire. Orkmund reacted, shifting his aim to Silo. Frey followed Silo's lead, an instant behind him, but Orkmund had already committed to his target.
Three shots fired, almost simultaneously. Orkmund fired first, and his bullet took Silo in the chest. Silo's own shot went wild. Frey's, hastily aimed, clipped the side of Orkmund's revolver and sent it spinning away with a spark and a metallic whine.
Silo fell to the ground. Orkmund hesitated, surprised to find that his gun was no longer in his hand. Frey aimed square at his head and pulled the trigger.
The hammer fell on an empty chamber. He was out of bullets.
Orkmund lunged at him, drawing a cutlass from his belt. Frey threw his revolver down as his own cutlass leaped from its scabbard, flying into his hand, the blade moving of its own accord. The two cutlasses met hard with a ringing chime. Orkmund swung again, pressing the attack, slicing at Frey's ribs and then his thigh. The daemon-thralled blade parried both, blurringly fast, moving with a speed far beyond anything Frey would have been capable of alone. Orkmund was an expert swordsman; Frey had an expert sword.
There was no time to think of Silo. The necessity of survival wouldn't permit it. All he saw was Orkmund's blunt face twisted in fury, the blades darting between them. He backed away under a flurry of blows, knocking away the pirate's strikes. The cutlass in his hand was doing its work with little help from him, but it could barely manage to keep up with Orkmund's attacks. There was a sharp bite of pain in his shoulder as Orkmund nicked him; a moment later, it was followed by one on his forearm.
The pain set loose the rage. From the corner of his eye, he could see Silo lying motionless on the ground. Possessed by a sudden recklessness, he pushed forward, switching from defense to attack. His cutlass sensed the change, moving with renewed vigor. It felt eager in his grip. Adding his strength to the blade's forced Orkmund to retreat. Suddenly Frey was the one hacking and thrusting while his opponent blocked and parried.
Then an all-consuming roar, bone-shakingly low. The corridor shook as a tremor ran through it from a nearby shell. Orkmund stumbled back, Frey overreached and lost his balance; but it was Frey who went tumbling to the floor and Orkmund who kept his feet. Frey rolled onto his back, parried aside a downward thrust aimed at his heart, and kicked the pirate's legs away. Orkmund went down, and suddenly they were on equal terms again. They rolled apart and sprang to their feet, panting, facing each other.
There was surprise and a little amazement in Orkmund's eyes. "You can _fight_!" he exclaimed.
"Yeah," said Frey hatefully. "I can fight."
He lunged forward again. He was taken by a loathing for this man, a need to eradicate him from existence. The very sight of him was unbearable: the broken planes of his nose, the pattern of tattoos over his neck, skull, and arms. This man had pulled the trigger that sent the bullet into Silo's chest. Maybe the quiet Murthian was already dead. Maybe he was even now gasping his last. The one thing worse than the fact that Orkmund had shot him was the fact that he was now preventing Frey from doing anything about it. Every minute was a minute his friend could be bleeding out. Every minute could be the one that ended him.
Silo had taken a bullet for him. He was damned if he'd have that man's death on his conscience for the rest of his life.
Had it not been for the cutlass Crake had given him, the wildness of his attack would have seen him dead at the hands of a swordsman like Orkmund. But with the blade guiding itself, and his murderous strength behind it, he became formidable. Orkmund parried and blocked, but Frey's blows were so vicious that he could barely hold on to his weapon. Steel rang again and again, punctuating the distant explosions.
Then Frey's hands were wrenched back, and his cutlass withdrew of its own volition, in preparation for a mighty strike. Frey panicked, struggling against the wishes of his own blade: he'd been left wide open. Orkmund, seeing the advantage, thrust inside Frey's guard to skewer him. But then Frey's cutlass twisted impossibly, almost breaking his wrist as it did so, and Frey felt the blade cut through meat and bone.
Orkmund's cutlass clattered noisily to the floor. The pirate captain staggered back a step, dazed, gazing at the severed stump of his forearm. Blood fountained with the pulse of his heart. White-faced, he stared at Frey in disbelief.
Frey gritted his teeth and ran him through.
THE SQUARE IN FRONT of Orkmund's stronghold had become a battleground. Shells pounded the gray sky, and gunfire cracked and snapped all around. A lumbering pirate frigate was cruising slowly overhead, terrifyingly low and close, its cannons bellowing as it fired at distant Navy frigates. The return barrage exploded deafeningly above the square. Stray artillery plowed into the town itself, demolishing whatever it hit. A row of buildings along one side of the square had dissolved into rubble and slumped inward after one such misplaced shell.
In the center of the square sat the _Ketty Jay_. Its cargo ramp was open and guarded by Bess. Its crew had taken cover inside the mouth of the hold or behind the hydraulic struts, firing on anyone who came close. Pinn and Harkins hovered in support, laying down machine-gun fire from their fighter craft, staying just high enough to avoid potshots from below.
An increasingly desperate group of pirates was shooting from the cover of the rubble. The square was scattered with the bodies of those who had already tried to rush the craft, seeing the _Ketty Jay_ as their only hope of escaping the cataclysm around them.
Frey hurried through the gates of Orkmund's stronghold with Silo on his back, heedless of the gunfire crisscrossing the air. The ground shook beneath his feet; there was a vast groan of metal from deep below. It felt as if the platform they were on might collapse at any moment. He ran low and hunched over in an attempt to keep the Murthian from sliding off. Jez called out at the sight of him, and the crew redoubled their fire, keeping the pirates' heads down as their captain came closer.
Frey was exhausted, running on adrenaline alone. The constant noise of the explosions, the effort of carrying almost ninety kilos of dead weight on his back, and the emotional shock of the past few minutes had put him into a shallow trance. He hardly noticed when a shell obliterated one of the buildings nearby, spraying him with stone chips and pushing him sideways with the force. He staggered, corrected himself, and ran doggedly onward.
Bullets whined past. He didn't know if they were meant for him. All he wanted was to get to the _Ketty Jay_.
He stumbled onto the ramp and was met by helping hands from Malvery and Jez, propelling him up into the dim safety of the hold. Bess stepped back onto the ramp, surveying the square threateningly, and Crake pulled the lever to raise it. The pirates screamed with frustration as they saw their chance of escape narrowing, but none of them dared to take on the golem.
All Frey wanted to do was lie down and sleep, but he didn't have that luxury. He hardened his resolve. This wasn't over yet.
He let Malvery take Silo from his back. Warm blood had soaked into his coat where the Murthian lay against him. They laid him on the floor of the hold while the doctor looked at his wound. Malvery's face was pale and fearful.
"Fix him," Frey told Malvery.
"He's losing too much blood," Malvery said.
"Fix him, damn you!" Frey snarled. Then he headed for the stairs that led out of the hold. He went up to the passageway that ran along the spine of the _Ketty Jay_ , then through the doorway into the cockpit, where he flung himself into the pilot's seat and punched in the ignition code. Jez was moments behind him, dropping into her spot at the navigator's station as Frey flooded the aerium tanks to maximum.
Another explosion rocked the _Ketty Jay_ as she began to lift her weight off her landing struts. Frey flinched and ducked as a bullet hit the windglass panel in front of his face, leaving a small circular shatter mark. The pirate frigate loomed above them and to starboard, shells bursting in the air all around it in a pummeling cascade of light and sound. Its keel suddenly came open, unbuttoned in a sequence of detonations that raced along its flank from stern to bow. Frey willed his craft to lift as the frigate tipped sideways toward them with a moan like the death cry of some enormous metal beast.
"Come on, come on!" he murmured under his breath, as the _Ketty Jay_ hauled her bulk into the air and began to ascend. Jez was staring in horror at the black, flaming mass of the frigate as it grew in their vision, threatening to crush them on its way down. The Skylance and Firecrow shot past on either side and streaked away; they couldn't help Frey now, and they had their own safety to consider. Screams from the square could be faintly heard over the concussion of artillery and the sound of the _Ketty Jay_ 's engines. The men and women of Retribution Falls had seen their fate descending on them.
The _Ketty Jay_ had barely cleared the tops of the nearest buildings before Frey kicked the throttle to maximum and the prothane thrusters opened up with a roar. He was thrown back in his seat as they accelerated, their landing struts scraping the roof of an inn, tearing off slates as they powered out of the shadow of the frigate. Frey gritted his teeth as the colossal craft bore down on them from above.
The deck of the frigate plummeted past their stern with a bellow of displaced air, raging with smoke and flame, and collapsed into the square with the force of a landslide. The _Ketty Jay_ carried Frey and his crew away as Orkmund's stronghold was obliterated and the great scaffolded platform cracked in half.
For once, Frey was glad that he couldn't see behind his craft. The appalling destruction in their wake was left to his imagination. A stab of grief surprised him—not for the dead but for Silo, whom he'd dumped into Malvery's care as if he were luggage. He forced himself to be cold. He had a responsibility to the others. Time to wallow in remorse after he'd gotten them safe.
He vented aerium to curb the _Ketty Jay_ 's excessive lift and took her around the rim of the sinkhole, skirting the Navy fleet and avoiding the worst of the fighting. Pinn and Harkins fell into position behind him. To starboard, he had a view of the whole battle. Retribution Falls was a ruin, a half-submerged junkyard. The stern ends of broken pirate craft jutted out of the brackish, rancid water, leaking flaming slicks of fuel. Smoke choked the scene. From within came rapid flashes of guns and the sounds of explosions.
The Navy had blocked off one route into Retribution Falls, but there were evidently other ways out, and the pirates took them. The defense of the town had been abandoned and the pirates were retreating, melting into the mist overhead, vanishing into gullies and canyons. The Navy had taken losses, but the surprise attack had kept them light.
Frey flew the _Ketty Jay_ behind the Navy fleet, who were still looking in toward the town and not out toward the rim. If anyone noticed the three insignificant aircraft sneaking past, then perhaps they recognized them for who they were and held their fire. In any case, the _Ketty Jay_ passed unmolested into the canyon that led out of the sinkhole and away from the battle. The rocky slopes of the Hookhollows closed around them, blocking out the sight of Retribution Falls. Soon they'd left the pirate town behind, and all was quiet again.
MALVERY AND CRAKE CARRIED Silo into the tiny infirmary and laid him on the surgical table. The Murthian was unconscious, his breathing shallow and rapid. His eyes moved restlessly behind their lids. The air smelled of oil and blood, and the floor moved with the tilting of the _Ketty Jay_ as she flew.
Crake's hands were covered in gore. He felt somehow that he should have been sickened by that, but he was too intent on the moment to allow himself weakness. He remembered the Silo who had helped him patch up Bess after the gunfight at Rabban, the one who had talked and joked with him on that grassy hillside. They were no longer strangers to each other. Crake would do whatever he had to.
Malvery ripped open Silo's shirt, exposing the wound. A ragged hole had been punched through one slablike pectoral. Rich blood welled out of it in awful quantity. He swore under his breath.
"He's bleeding inside," said Malvery. "I can't do anything."
"You've got to!" Crake protested. "Open him up. Stop the bleeding!"
"I can't," Malvery said. He adjusted his round green glasses and tugged anxiously at the end of his white mustache. "I just can't."
He opened a drawer and pulled out a bottle of medicinal alcohol. He'd unstoppered it and brought it to his lips before Crake snatched it from him and slammed it angrily down on the operating table.
"You're the only one who can do this, Malvery!" he snapped. "Forget what happened to your friend. You're a surgeon! Do your damn job!"
"I'm not a surgeon anymore," Malvery replied, staring at the man on the operating table in front of him. Blood pumped up from the bullet wound and spilled down Silo's chest in grotesque washes of red. Crake clapped his hands ineffectually over the wound, then began looking around for something better to stanch the flow.
He understood Malvery's pain, but he'd no time for sympathy while Silo lay dying. If only Crake had been a better daemonist, he might have used the Art to heal the Murthian. But he didn't have the equipment, so he couldn't do anything. Silo's only chance was Malvery, and the doctor was paralyzed.
"Spit and blood, you're just going to stand by and watch?" Crake cried.
"What do you want from me?" Malvery bellowed. "A miracle? He's dying! I can't stop that!"
"You can try!" Crake shouted back with equal ferocity. Malvery was shocked at the force in Crake's usually quiet tone. "This isn't like the last time. He's going to die anyway. Nobody will blame you if you fail, I'll make sure of that. But I wouldn't want to be in your shoes when the Cap'n finds out you didn't even _try._ "
Just then, the _Ketty Jay_ yawed to port, making him stumble; he had to put out a hand on the operating table to steady himself. The bottle of alcohol tipped from the table, but Crake caught it before it fell. Malvery's eyes went to it.
"Give me the bottle," he said.
Crake glared at him.
"I'll need a swig to steady my hand!" Malvery insisted.
"Your hand's plenty steady, Doc. Do it. Earn your place."
_"Me?"_ Malvery roared. "You've been barely four months aboard, you arrogant shit!"
"Yes. And I saved you all at Tarlock Cove. Bess saved you from Kedmund Drave. I uncovered Grephen's plans at Scorchwood Heights, and we'd never have taken the charts from Dracken without Bess. We've done our part. Pinn and Harkins fly, Jez navigates, Silo keeps the craft running. What do _you_ do that one of us couldn't? Fire a shotgun? Work the autocannon occasionally? You're a surgeon who doesn't operate, Malvery! You're dead weight!"
Malvery's face twisted in anger. He lunged across the table to grab him, but Crake was too fast and pulled himself out of the way.
"Then prove me wrong!" he cried. "Cut him open! Stop the bleeding and save his life!"
Malvery's huge fists bunched and unbunched. His face was red with rage. For a moment, Crake thought he really would attack, but then he turned away and stamped over to the wooden cabinet that was fixed to the wall. He pulled it open and drew out a scalpel. The surgical instruments were the only clean things in the grimy room. Malvery came back to the table and stared at Crake.
"I'll cut him, damn it," he snarled. "And you're staying to help."
Crake rolled up his sleeves. "Tell me what to do."
FREY SAT IN THE pilot's seat, staring out at the fog. His air filter still hung around his neck, though more than an hour had passed since they crossed the lava river and its noxious fumes. Jez read out directions and coordinates from the navigator's station behind him, and he followed them automatically. Once in a while she consulted the compass and warned of some distant mine that the Navy minesweepers had failed to catch, but it was always too far away to be a threat.
Frey barely paid attention to the job at hand. This was the fourth time he'd flown along this route, and it had lost its fear for him now. He trusted Jez. Harkins and Pinn were following his lights through the murk. As for the Navy, they could find their own way out.
He was thinking of Silo. The black specter of loss hovered over him. It wasn't only the thought that Silo himself might die—Frey had already been surprised by the depth of feeling he had for the taciturn foreigner—but the idea that one of his crew would be lost. With each new peril they survived, he'd thought of them more and more as one indivisible whole. Whereas he'd often dreamed in the past of ditching his crew and flying off alone, now he couldn't stand the thought of it. They'd become a miniature society, the denizens of the _Ketty Jay_ , and they needed one another to survive. Somehow they'd achieved a balance that satisfied them all, and when they were in balance they'd been able to achieve extraordinary things. Frey feared losing any of them now, in case that balance would be upset. He feared a return to the way things were.
All this time he'd armored himself against loss by refusing to care for anyone. Now, somehow, he'd been blindsided. He tried to be angry for allowing himself to become so vulnerable, but he couldn't muster the feeling. It had all been part of a greater change, one that had seen him gain a level of self-respect he'd never had before the destruction of the _Ace of Skulls_. He wouldn't trade the days between that moment and this. Not for anything.
But now Silo lay on the edge of death, left in the hands of an alcoholic doctor and an untrained assistant. Silo, who had taken a bullet so Frey wouldn't have to. Frey didn't want to live with that responsibility.
_What's taking them so long? Aren't they done yet?_
As if in response, he heard the infirmary door slide open. He hunched his shoulders as if expecting a blow. Footsteps came up the passageway, boots rapping on metal. They were too light to be Malvery's, so they had to be Crake's. Frey turned in his seat to be sure and saw the daemonist arrive in the doorway. His hands were gloved with blood, and he was still wearing his filter mask. He pointed to it and said something quizzical that nobody understood.
"You can take it off now," Jez said, guessing his meaning.
Crake pulled off the mask and took a few deep breaths. "That's better," he said. "Those things are so stuffy."
"Mmm," said Jez in mild agreement.
"Everything alright?" Crake inquired.
"Will you bloody well tell me what's going on back there?!" Frey exploded, unable to bear the tension anymore.
"Oh, yes," said Crake. He grinned. "The doc stopped the bleeding and got the bullet out. He says the patient is going to be alright."
Jez gasped and gave a little clap, a surprisingly girlish reaction from someone Frey had come to think of as rather unfeminine. Frey slumped back into his seat with a sigh, and a huge sense of relaxation spread through his body. Exhaustion and relief piled in together. At last, it was over. A broad smile spread across his face. Crake laughed and slapped him on the shoulder, leaving a grotesque handprint there.
"Good work, boys," Frey said. "Bloody good work."
"Well, I've got to go and help Malvery finish up," said Crake. "Just thought I'd let you know." He disappeared down the passageway again and into the infirmary.
"We're here, Cap'n," said Jez. "All stop. You can start ascending now."
Frey brought the _Ketty Jay_ to a halt and set her rising through the fog. The haze gradually thinned and the darkness brightened by degrees. The flanks of the mountains became discernible again as forbidding slabs of shadow.
Frey looked up. A smile was still on his lips. Up there was light and freedom. Up there was the prospect of a new life, a luxurious life, one financed with the chest full of Awakener gold they'd stolen from Orkmund. Up there was a second chance for all of them.
"Never seen you smiling like that, Cap'n," Jez said. "It's just, for once, I really feel that everything's going to be okay."
Then they broke free of the mist, and a shattering explosion hammered the _Ketty Jay_ , filling the cockpit with dazzling light, shaking them about like rag dolls.
When no further explosion came, Frey blinked away the shock and pulled himself back into his seat.
A swarm of Norbury Equalizers surrounded them. Looming ahead of them, with all of its considerable arsenal trained on the _Ketty Jay_ , was the _Delirium Trigger_.
Frey blew out his cheeks and huffed a sigh of resignation. "Bollocks!"
# _Chapter Thirty-nine_
"THIS IS WHERE MERCY GETS YOU"—DRACKEN'S CHOICE—CONCLUSIONS
cold wind chased puffs of gray ash across the Blackendraft flats. Frey's coat flapped restlessly. Bleak horizons encircled them. Overhead, the sky was the color of an anvil. The _Delirium Trigger_ hung at anchor a short distance away, its hard, cruel lines stark against the emptiness.
The crew of the _Ketty Jay_ stood in a row at the bottom of the cargo ramp. Pinn and Harkins had grounded their craft and been rounded up. Silo, Bess, and Slag were missing. Silo was still in the infirmary. Crake had put Bess to sleep to prevent her going berserk and getting them all killed. Slag had vanished into the vents and airways, on some mysterious errand of his own. Nothing would ever separate him from his aircraft.
Facing them was Trinica Dracken and a dozen men from the _Delirium Trigger_. The men covered Frey and his crew with their pistols while Trinica looked down into the red-lacquered chest that sat at her feet. She stared at the wealth within for a long time, but her ghost-white face and unnatural black eyes revealed nothing of what she was thinking. Finally, she looked up.
"You did well, Darian," she said. "Kind of you to carry this all the way from Orkmund's stronghold, just for me."
Pinn muttered something unsavory under his breath. Malvery clipped him around the ear.
"I should have killed you when I had the chance," said Frey. There was no rancor in it; it was simply an observation. "I suppose this is where mercy gets you."
Trinica gave him a dry smile. "Consider this the price of a lesson well learned."
Frey and Trinica gazed at each other across the dusty gap that separated them. The huge silence of the Blackendraft filled the moment.
He couldn't feel hate for her. He couldn't manage to feel much more than a distant disappointment. This felt right, somehow. It had been greed that made him jump at Quail's too-good-to-be-true offer. And while he didn't blame himself for the many deaths aboard the _Ace of Skulls_ —they were doomed with or without him—he'd played a part in it. He might have saved the Archduke and done a great service to his country, but he did it by initiating a massacre at Retribution Falls. It didn't seem fair that he should profit from his own stupidity at the expense of all those lives.
Maybe he owed the world something. For the crew he'd taken into Samarla and left to die. For every Trinica Dracken and Amalicia Thade he'd discarded and forgotten as soon as they showed signs of wanting more than he was prepared to give.
For his baby, that died for its parents' cowardice.
He'd condemned them all when he agreed to take on the _Ace of Skulls_. But since then he'd clawed back all he'd lost, and more besides. He'd forged a crew, and he'd reclaimed himself. Perhaps that was all that was needed, in the end.
"What happens now, Trinica?" he asked her.
"I expect Grephen will hang," she said. "The Awakeners... well, they're too powerful to be brought down, even by this. But I think the Archduke will redouble his efforts to cripple them from now on."
"I mean, what happens to us?"
Trinica gave him a bewildered look. "How would I know? I expect you'll get your pardons, even if you're not there to collect them."
"You're letting us go?"
"Of course I am," she said. "Everyone who put a bounty on your head has either withdrawn it or is in no position to pay anymore. Why would I want you?"
His crew visibly relaxed. Frey brushed away a lock of hair that was blowing across his forehead.
"And you?" he asked.
"I'll be heading off somewhere," she replied, nonchalant. "I suppose I'll have to keep out of the Navy's way from now on, but I'll survive."
She motioned to her bosun, who filled a leather bag with coins from the chest. He tied it with a thin piece of rope and gave it to her. It was almost too big to hold in one hand. She weighed it thoughtfully, then hefted it toward Frey, who barely caught it.
"Finder's fee," she said. "That, and you can keep your craft."
"That's uncommonly merciful of you, Trinica."
She smiled, and this time it wasn't the chilly, guarded smile he'd come to know. It was the smile of the old Trinica, from a time before her world had become full of horrors, and it flooded him with a bittersweet warmth.
"I'm feeling sentimental," she said. "Goodbye, Captain."
She turned her back on them then and walked toward the shuttle that sat a short way distant. Her men closed the chest and gathered it up. Frey and his crew watched as they disappeared inside the craft and it lifted off from the ground, taking them back to the _Delirium Trigger_.
"Well," said Malvery, squinting against the dust as he watched them dwindle into the distance. "That's just about our luck."
"Cheer up!" said Frey. "We've got three aircraft, enough ducats to keep us in the sky for a year, and the world at our feet. I'd say we're the luckiest crew in Vardia right now."
"I'd feel a damn sight luckier if that witch hadn't buggered off with our loot," Malvery griped.
Frey slapped him on the shoulder. "Look on the bright side. She might have killed us."
"There is that," Malvery conceded.
"Is... umm... I was wondering, is anyone else hungry?" Harkins inquired.
"We should probably take Silo to a hospital," Jez suggested. "Get him a good bed and some nurses."
Frey looked at Malvery. "How long before Silo's capable of getting back to work? In your expert opinion?"
"Three weeks, I'd say," Malvery said. "Maybe four."
Frey scratched the back of his neck. "Well, what with all we've been through, we probably deserve a little port time."
Pinn's eyes lit up at the prospect of booze and whores. Frey held up the bag of coins. "Drinks are on me tonight!"
There was a cheer from the men.
"Jez!" he barked.
"Cap'n!"
"Find us a nice out-of-the-way port with a good hospital, lively nightlife, and a place where a man can find a game of Rake."
"Skinner's Gorge?"
"Skinner's Gorge sounds good to me."
There was another cheer, and they slapped one another's backs and shook hands in wild and vague congratulations. The chest full of ducats was forgotten already. They had all they needed. They were glad just to be alive.
Frey couldn't keep down a grin. As he looked at the laughing faces of his crew, he was consumed by a surge of affection for these people, these men and this woman who shared his aircraft and his life. They were happy, and free, and the endless sky awaited them.
It was enough.
# _How to Play Rake_
**H ISTORY**
Rake is a variant of poker, in which the player must make the best five-card hand to win. The game has existed for centuries, since before the fall of the monarchy: the first recorded mention was as far back as 87/29 (UY3069). For most of that time it was confined to the peasantry and was viewed as a rather vulgar pastime by the rich. It was popularized in Vardia during the First Aerium War, when the mingling of conscripted troops allowed the game to spread. Soon, even the aristocrats who commanded them had caught on, and Rake passed from the taverns and dens into the drawing rooms of the wealthy. Since then it has become the most popular card game in Vardia.
**T HE DECK**
Rake is played with a standard fifty-two-card Vardic deck, comprising thirteen cards of each of the four suits: Skulls, Wings, Fangs, and Crosses. Each suit comprises (in order of value) the numbers two through ten, followed by the face cards: Priest, Lady, Duke, Ace. The Ace, the highest card, also functions as the number one for the purpose of runs.
**S CORING**
The object of a hand of Rake is to achieve the best possible combination of five cards. These combinations are scored below, in order of rank. Two pair beats three of a kind, Suits Full beats a run, etc.
**H IGH CARD**
In this case, with no combination possible, only the highest card in the hand is counted. Therefore, a player whose highest card is an Ace beats one whose highest card is only a Priest.
**P AIR**
Two cards of the same value. In the above example, the player has two tens. A higher pair, such as Ladies, will beat him. The remaining cards are disregarded, unless two players have identical pairs, in which case the one with the highest value remaining card wins.
**T WO PAIR**
Two pairs of the same value. If two players should have two pairs, then the player with the highest pair wins. If both players have the same high pair, the player with the next highest pair wins. In the unlikely event that both have the same two pairs, the remaining card comes into play: again, the player with the highest card wins.
**T HREE OF A KIND**
Three cards of the same value. If two or more players have three of a kind, the highest triplet wins. The above hand would be referred to as Three Dukes.
**R UN**
Five cards of sequential value but different suits. An Ace in this case can count as either a one or an Ace. If two or more players have a Run, the one incorporating the highest card wins.
**S UITS FULL**
Five cards of the same suit. They are referred to as Wings Full, Crosses Full, and so on. As in a Run, in the case of two players holding Suits Full, the one whose hand incorporates the highest card wins. If it is the same, they go to the next highest, etc., until the hand is resolved.
**F ULL PACK**
A Full Pack is three of a kind of one value and a pair of another. In the case of two players holding it, the higher value three of a kind wins.
**Q UADS**
Four cards of the same value. The last card is disregarded. The above hand would be referred to as Four Ladies or Quad Ladies.
**R AKE**
Five cards of the same suit in sequential order. This is the highest-scoring hand in Rake. If two people should have it, the one incorporating the highest card wins.
**T HE ACE OF SKULLS**
The Ace of Skulls is the most dangerous card in the game of Rake, both for the player who holds it and their opponents. A player who holds the Ace of Skulls must incorporate that card into a scoring hand of three of a kind or higher, or they will automatically lose the hand. The Ace of Skulls must be part of the scoring combination (e.g., one of the three cards that forms the three of a kind)—it cannot be a redundant card in the hand, or the player loses. Similarly, if the player cannot manage better than high card, pair, or two pair—whether the Ace of Skulls is involved or not—the player loses. However, should the player manage to incorporate the Ace of Skulls into a scoring hand of three of a kind or higher, they automatically win the hand, regardless of what their opponents are holding.
**P ROGRESSION OF PLAY**
Rake is played with two to eight players, though six at a table is considered optimum.
**T HE ANTE**
Players make a minimum bet before the hand commences, the value of which is determined and agreed upon beforehand. This goes into the pot—the money available to be won—and will be taken by the victor of the hand.
**T HE DEAL**
Each player is dealt three cards, facedown. The player does not show these cards to anyone but may look at them.
**F IRST ROUND OF BETTING**
The players bet based on the cards in their hand. A player's bet must be matched or raised by the players to their left, or they may fold, forfeit their ante, and take no further part in the hand. Betting continues until all bets are even, at which point the money goes into the pot. There is no limit or restrictions to the betting in Rake. Players can choose to bet nothing, as long as nobody around the table raises them.
**T HE MIDDLE CARDS**
Now the middle cards are dealt. For each player at the table, one card is dealt faceup and one facedown. So if there are six players, six cards are dealt faceup and six facedown along the middle of the table. These are cards available to be picked up by all the players.
**F IRST PICKUP**
Each player now takes one of the middle cards, hoping to improve their hand. The player to the left of the dealer picks up first, followed by the player to their left, until all players have picked one up. Players may take a faceup card or a facedown card. Faceup cards have the obvious advantage that the player knows their value, but they also give away information about the player's hand to their opponents. They may also be useless to some players, who would rather take an unknown card in the hope of picking up one that helps their hand. Experienced Rake players may use their choice of cards in the pickup to bluff and deceive their opponents by misrepresenting the cards they hold.
**S ECOND ROUND OF BETTING**
Another round of betting, identical to the first.
**S ECOND PICKUP**
Now the remaining cards are picked up, in the same order as before: left of the dealer and clockwise after that. In Rake, the dealer is at a considerable disadvantage, being the last person to pick up a card. However, they are also the last one to bet and have the advantage of studying their opponents' bets before deciding their own.
**T HIRD ROUND OF BETTING**
A final round of betting.
**T HE REVEAL**
If there are still two or more players contesting the hand—those who have not dropped out during one of the rounds of betting—they now reveal their cards. The winner takes all the money in the pot. The position of dealer moves one place to the left, and the sequence of play begins again.
# **ABOUT THE AUTHOR**
CHRIS WOODING is thirty-four years old and is the author of eighteen books. His novels have been published around the world and translated into twenty languages. He also writes for film and television. He lives in London, UK.
www.chriswooding.com
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| {
"redpajama_set_name": "RedPajamaBook"
} | 8,562 |
The NIE is Wrong about Iran, Again
Iran in tatters?
The Wall Street Journal yesterday reported that a new U.S. national intelligence estimate (NIE) says that the bite of international sanctions may be sowing discord among Iranian leaders. The NIE's findings suggest that some Iranian leaders are worried that economic turmoil fueled in part by international sanctions could spur opposition to the regime; and that, as a result, Iran's leaders are locked in an increasingly heated debate over whether to move further toward developing nuclear weapons.
But a range of Israeli intelligence officials and American analysts say that this analysis is simplistic. It's true that economic sanctions are placing the Iranian regime in a difficult place, said a senior-most former Israeli intelligence commander in Jerusalem this week. But this has not diminished their motivation to proceed with the nuclear program. Supreme Leader Ayatollah Ali Khamenei believes that nuclear weapons are his regime's insurance policy. Thus, even though sanctions are adding to ethnic tensions and forcing budgetary cuts, he told the Conference of Presidents of Major American Jewish Organizations, it will take a lot more political and military pressure to stop the Iranians in their tracks – if at all.
In one respect, this top Israeli agreed with the new NIE: There is no smoking gun evidence that the Iranians have yet decided to proceed in assembling a nuclear bomb. "They are enriching uranium to weapons grade levels – yes. They are continuing their weapons design work – yes. They are developing their long-range delivery capabilities – yes. But we know of no order to actually make a bomb. Apparently, the Iranian strategy is to build capacity across a wide range of fields, and stockpile all necessary components in many locations. When they are ready to break out, it will be so fast we'll barely have time to notice."
Mark Dubowitz of the Foundation for the Defense of Democracies and its Iran Energy Project agrees with the Israeli assessment of sanctions. "Sanctions are causing significant economic dislocation to Iran," he told the 2011 Herzliya Conference, "but they can be compared to silver shrapnel. Sanctions are not a silver bullet that will stop the nuclear program." Dubowitz noted that no company has been punished by the US government for violating sanctions, and that crude oil exports from Iran have not been halted. Banks, he said, still need to be discouraged from facilitating oil transfers from Iran and from financing long-term oil contracts for Iran.
Prof. Efraim Inbar, director of the Begin-Sadat Center for Strategic Studies, is even blunter. "There is no historical precedent for the success of sanctions, anywhere, in any conflict in the world, without the credible threat of the use of force," he told the Conference of Presidents on Wednesday. "Unfortunately, the Obama administration says that 'all options are on the table' but it has not convinced Tehran that it is seriously willing to use force if necessary. Economic pain alone is of little consequence to the Iranian leadership elite. It has a high threshold for pain, and in any case – in dictatorships, elites always eat well!"
Meir Javedanfar, an Iranian-born analyst based in Tel Aviv, is one of the few analysts who believes that sanctions may be bringing Tehran close to the brink of collapse. "$60 billion are missing from the Iranian budget," he told the Herzliya Conference, "and $1 billion is missing from Central Bank accounts. This was probably drained away by the nuclear program. Food subsidies have been cut, probably to finance the nuclear program. The press is forbidden from writing about the impact of sanctions, and the government refuses to report to parliament about Iranian oil revenues. This does not bode well for Tehran."
Menashe Amir, who has been broadcasting for Israel Radio to his native Iran for more than half a century, also believes that the ayatollahs' regime in nearing an end — because it is 'artificial'. But in a lengthy interview in today's Jerusalem Post he warns that the regime won't go quietly. "Iran isn't Egypt. The Revolutionary Guard isn't the Egyptian Army that refused to fire on the protestors." Similarly, journalist Yossi Klein Halevy has warned that we could end up with a Middle East where only the 'hard' dictators survive (like Ahmadinejad), while the 'soft' dictators (like Mubarak) who don't have the guts or the ability to violently put down unrest – disappear…
Categories: Israel, Strategic Affairs | Tags: bomb, intelligence, Iran, NIE, nuclear, sanctions
government Palestinian Obama Arafat Arab-Israeli conflict Netanyahu Israeli-Palestinian Israel Orthodox Egypt peace Sderot Jewish Iran Middle East Jerusalem Peace Process democracy terrorism settlements | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,536 |
\section{Synchronization of partial automata}
\emph{Partial finite automaton} (PFA) is an ordered tuple $\mathcal{A} = (\Sigma, Q, \delta)$ where $\Sigma$ is a set of letters, $Q$ is a set of states and $\delta:{Q \times \Sigma}\rightarrow{Q}$ is a transition function, not everywhere defined. For $\emph{w} \in \Sigma^\ast$ and $\emph{q} \in Q$ we define $\delta(\emph{q},\emph{w})$ inductively as $\delta(\emph{q},\epsilon) = q$ and $\delta(\emph{q},\emph{aw}) = \delta(\delta(\emph{q},\emph{a}), \emph{w})$ for $a \in \Sigma$ where $\epsilon$ is the empty word and $\delta(\emph{q}, \emph{a})$ is defined. A word $\emph{w} \in \Sigma^\ast$ is called \emph{carefully synchronizing} if there exists $\overline{q} \in Q$ such that for every $\emph{q} \in Q$, $\delta(\emph{q}, \emph{w}) = \overline{q}$ and all transitions are defined. A PFA is called \emph{carefully synchronizing} if it admits any carefully synchronizing word. Carefully synchronizing automaton $\mathcal{A}_{car}$ is depicted on the Fig. 1 and its shortest carefully synchronizing word $w_{car}$ is $abca^3b^2ca$ what can be easily checked via power automaton construction analogous to the deterministic automata construction. The only difference is that we define only transitions we can define.
\begin{figure}[H]
\centering
\begin{tikzpicture}[shorten >=1pt,node distance=3cm,on grid,auto]
\node[state] (0) {$0$};
\node[state] (1) [right=of 0] {$1$};
\node[state] (2) [below=of 1] {$2$};
\node[state] (3) [left=of 2] {$3$};
\path[->]
(0) edge node {a,c} (1)
(1) edge [loop above] node {a} ()
edge node {b} (2)
(2) edge [loop below] node {a} ()
edge node {b,c} (3)
(3) edge [loop below] node {a} ()
edge node {b,c} (0)
;
\end{tikzpicture}
\caption{A carefully synchronizing $\mathcal{A}_{car}$}
\end{figure}
The concept of careful synchronization of PFA is a generalization of idea of synchronization for deterministic finite automata (DFA) with transition functions defined everywhere. The problem of estimating the value of $d(n)$ was considered first by Ito and Shikishima-Tsuji in [6-7] and later by Martyugin [8]. Ito and Shikishima-Tsuji proved that $2^{\frac{n}{2}} + 1 \leq d(n) \leq 2^n - 2^{n-2} - 1$ and Martyugin improved the lower bound with the construction of automata of length $O(3^{\frac{n}{3}})$. The best known upper bound for $d(n)$ is $O(n^2 \cdot 4^{\frac{n}{3}})$ due to [9].\newline Let $\mathcal{L}_n = \{\mathcal{A} = (\Sigma, Q, \delta): \mathcal{A}\;is\;carefully\;synchronizing\;and\;|Q| = n\}$. We define $d(\mathcal{A}) = min\{|w|:w\; is\; carefully\; synchronizing\:word\; for\; \mathcal{A}\}$ and $d(n) = max\{d(\mathcal{A}) : \mathcal{A} \in \mathcal{L}_n\}$. It can be easily verified from Fig 1. that the \v{C}ern\'y Conjecture is not true for PFAs, since $|w_{car}| = 10 > (4-1)^2 = 9$. We also recall following important facts.
\begin{fact}
Let $\mathcal{A}$ be a PFA and $\mathcal{P(A)}$ be its power automaton. Then $\mathcal{A}$ is synchronizing if and only if for some state $q \in Q$ there exists a labelled path in $\mathcal{P(A)}$ from $Q$ to $\{q\}$. The shortest synchronizing word for $\mathcal{A}$ corresponds to the shortest labelled path in $\mathcal{P(A)}$ as above.
\end{fact}
\begin{fact}
If automaton $\mathcal{A}$ is carefully synchronizing then there exists $a' \in \Sigma$ such that transition under $a'$ is defined for all states and $q_1', q_2' \in Q$ such, that $\delta(q_1',a') = \delta(q_2',a')$.
\end{fact}
Now we are ready to give an example of a PFA with shortest carefully synchronizing word of length $O(d^{n/d})$ for all $d \in \mathbb{N}$.
\section{Automata with long shortest carefully synchronizing words}
This section includes construction of an automaton for which the shortest carefully synchronizing word is of exponential length. \newline
Let $d \in \mathbb{N}$, $d > 1$, $a_1, ..., a_k \in \{0, ..., d-1\}$ and $r = \sum\limits_{i=1}^k a_i \cdot d^{i-1}$. We understand $(a_k, ..., a_1)_d$ as base $d$ representation of $r$.
Let $n = d \cdot k$, $k \in \mathbb{N}$. We define automaton $\mathcal{A}_d(n) = (\Sigma, Q, \delta)$ as follows:
\begin{itemize}
\item $\Sigma = \{a, b_1, b_2, ... b_k, c_k, c_{k-1}, ... , c_2\}$
\item $Q_i = \{q_0^i, q_1^i, ... , q_{d-1}^i\} $
\item $Q = \bigcup\limits_{i = 1}^k Q_i$
\end{itemize}
Let $i \in \{1,..., k\}$, $l \in \mathbb{N}$. we define partial transition function $\delta:{Q \times \Sigma}\rightarrow{Q}$ for $\mathcal{A}_d(n)$ as:
\begin{itemize}
\item $\delta(q_j^i, a) = q_0^i$, $j \in \{0,1, ... , d-1\}$
\item $\delta(q_{j-1}^i, b_i) = q_{j}^i$, $j \in \{1,2, ... , d-1\}$
\item $\delta(q_j^i, b_l) = q_j^i$, $i > l$
\item $\delta(q_{d-1}^i, b_l) = q_0^i$, $i < l$
\item $\delta(q_{d-1}^i, c_i) = q_0^{i-1}$
\item $\delta(q_{d-1}^i, c_l) = q_0^i$, $i < l$
\end{itemize}
Let us remark some facts about the construction useful for further proofs.
\begin{fact}
$\delta(q_j^i, b_l)$ is not defined when $i < l$ and $j \in \{0, ..., d-2\}$.
\end{fact}
\begin{fact}
$\delta(q_{d-1}^i, b_i)$ is not defined.
\end{fact}
\begin{fact}
$\delta(q_j^i, c_i)$ is not defined when $j \in \{0, ..., d-2\}$.
\end{fact}
It is worth noticing that only transitions on letters $c_i$ and $a$, $i \in \{1, ..., k\}$ join two states together and only letter $a$ is defined for all states.\newline
Let $m \in \mathbb{N}$ and $r = (j_m, ..., j_1)_d$. We also define $Q_r^m \subset Q$ such, that:
\begin{itemize}
\item $j_i$ for $i \in \{1,...,m\}$ corresponds to lower index of $q_j^i \in Q_r^m$
\item $|Q_r^m| = m$
\item $|\{q_0^i,q_1^i, ... ,q_{d-1}^i\} \cap Q_r^m| = 1$, $i \in \{1, ... , m\}$
\end{itemize}
In other words each $Q_r^m$ corresponds to m-digit base $d$ representation of $r$. For example if $d=3$, then $Q_{10}^4 = \{q_0^4, q_1^3,q_0^2,q_1^1\}$. Finally we define inductively word $w_i \in \Sigma^i$ as:\newline\newline
$w_i =
\left\{\begin{array}{ll}
\epsilon & \mbox{if } i = 0 \\
(w_{i-1}b_i)^{d-1}w_{i-1} & \mbox{if } i > 0
\end{array}
\right.
$\newline\newline
Now we are ready to formulate first lemma of this section.
\begin{lemma}
Let $n = d \cdot k$ and $\mathcal{A}_d(n)$ be defined as above. For every $i \in \{1, ..., k\}$ there exists a path $(Q_0^i, Q_1^i,Q_2^i, ... ,Q_{d^i - 1}^i )$ in $\mathcal{P}(\mathcal{A}(n))$ and its transitions are labelled with consecutive letters of word $w_i$.
\end{lemma}
\begin{proof}
The result follows by induction on i.\newline
Case $i = 1$ is evident from the definition of $\delta$ since $w_1=b_1^{d-1}$ and $\delta(q_{j-1}^1, b_1)=q_j^1$ for $j \in \{1,...d-1\}$.\newline Now let us assume that the result holds for $i - 1 < k$. From the induction hypothesis we know, that there exists a path
$(Q_0^{i-1}, Q_1^{i-1},Q_2^{i-1}, ... ,Q_{d^{i-1} - 1}^{i-1} )$ whose transitions are labelled with consecutive letters of word $w_{i-1}$. Let $l \in \{0,... , d-1\}$ and notice that for every $r\in \{0, ..., d^{i-1} - 1\}$ we have $Q_{l\cdot d^{i-1} + r}^i = \{q_l^i\} \cup Q_r^{i-1}$ and $\delta$ on any of letters $b_1, ... , b_{i-1}$ maps $q_l^i$ to itself, so for each $l$ there exist a path $(Q_{l\cdot d^{i-1}}^i, Q_{l\cdot d^{i-1} + 1}^i,Q_{l\cdot d^{i-1} + 2}^i, ... , Q_{(l+1)\cdot d^{i - 1} - 1}^i )$ in $\mathcal{P}(\mathcal{A}(n))$ (also labelled with letters of $w_{i-1}$). From the definition of $\delta$ we can see that for each $l \in \{1, ... , d-1\}$ it holds that $\tau(Q_{l \cdot d^{i - 1} - 1}^i, b_i) = Q_{l \cdot d^{i-1}}^i$. Using these two observations and the definition of $w_i$ we conclude that the lemma holds.
\end{proof}
\begin{lemma}
Automaton $\mathcal{A}_d(n)$ is carefully synchronizing and its carefully synchronizing word is $v = aw_kc_kw_{k-1}c_{k-1}...w_2c_2$.
\end{lemma}
\begin{proof}
We must show that $|\tau(Q, v)| = 1$. From the definition of $\delta$ we see that $\tau(Q,a) = Q_0^k$. From Lemma 1. we know that for every $l \in \{2, ..., k\}: \tau(Q_0^l,w_l) = Q_{d^l - 1}^l$. Also from the definition of $\delta$ we see that for every $l \in \{2, ..., k\}: \tau(Q_{d^l - 1}^l,c_l) = Q_0^{l-1}$. Joining those facts together we deduce that $\tau(Q, v) = \{q_0^1\}$.
\end{proof}
\begin{lemma}
Let $v$ be as in Lemma 2. Then $|v| = \frac{1}{d-1}(d^{k+1} + (d-1)k - d^2)$ and $v$ is the shortest carefully synchronizing word for automaton $\mathcal{A}_d(n)$.
\end{lemma}
\begin{proof}
First we will show that $|v| = \frac{1}{d-1}(d^{k+1} + (d-1)k - d^2)$. It's obvious that $|w_ic_i| = d^i$, so $|w_kc_kw_{k-1}c_{k-1}...w_2c_2| = \sum\limits_{i=2}^k (d^i + 1) = \frac{1}{d-1}(d^{k+1} + (d-1)k - d^2 - d + 1)$. We leave that identity as a simple exercise for a reader.\newline\newline
It can be easily verified that $\tau(Q,a) = Q_0^k$ and $a$ is the only letter defined for all states. In order to prove minimality of $v$ it suffices to show that for each state $Q_s^k \subset Q$ in $\mathcal{P}(\mathcal{A}_d(n))$ there is only one transition that leads to a state $Q_{s'}^{k'} \subset Q$ that has not been visited yet. All other transitions are either not defined or lead to states visited earlier. We must investigate two cases:
\newline\newline
Case $Q' = Q_r^m$ for some $r,m \in \mathbb{N}$ and $r \neq d^m - 1$ \newline
From the definition of $\delta$, $\tau(Q', a) = Q_0^m$ (which was visited) and from Fact 6. $\tau(Q',c_j)$ is not defined for any $c_j$. From Lemma 2. it can be seen that for each $Q_r^m$ there exists a letter $b_k$ which leads to an unvisited state $Q_{r+1}^{m}$. In order to show that there exists only one such letter let us assume, that $r = (a_1, a_2, ..., a_{m - k + 1}, d-1, d-1, ... , d-1)_d$, $k \geq 1$ and $a_{m - k + 1} \neq d-1$. It is obvious from Fact 4. that for each $b_l$ such that $l > k$ the transition $\tau(Q', b_l)$ is not defined. If $l < k$ then it follows from Fact 5. that $\tau(Q', b_l)$ is not defined and the statement is true for that case. \newline\newline
Case $Q' = Q_r^m$ for some $r,m \in \mathbb{N}$ and $r = d^m - 1$\newline
From the definition of $\delta$ we see that $\tau(Q',a) = Q_0^m$. Furthermore $\tau(Q',b_j) = Q_0^m$ when $j > m$ otherwise when $j \leq m$, due to Fact 5, transitions are not defined. Notice that $\tau(Q',c_j) = Q_0^m$ for $j > m$. If $j < m$, then $\tau(Q',c_j)$ is not defined. Moreover $\tau(Q',c_m) = Q_0^{m-1}$. Since there is only one letter leading to an unvisited state and transitions under other letters are either undefined or their result is already visited state $Q_0^m$ of $\mathcal{P}(\mathcal{A}_d(n))$ statement holds for that case.\newline\newline
Having that we know by induction that $w$ is minimal and that ends the proof.
\end{proof}
Following theorem is immediate from Lemma 4.
\begin{theorem}
Let $n = d\cdot k$, $k \in \mathbb{N}$. The shortest carefully synchronizing word for $\mathcal{A}_d(n)$ has length $O(d^\frac{n}{d})$.
\end{theorem}
Using that theorem we can simply reproduce result obtained by Martyugin [8] as follows:
\begin{corollary}
If $n > 3$ then there exist a PFA with $n$ states and minimal carefully synchronizing word of length $O(3^\frac{n}{3})$.
\end{corollary}
\begin{proof}
We construct automaton $\mathcal{A}_3(m)$ with $m = n - (n\mod 3)$ and denote rest of states as $Q'$. Now we can add a letter to the automaton, say $d$, and add a transition over that letter to $\tau$, resulting with $\tau'$, such that it acts like identity on $Q \setminus Q'$ and $\tau'(Q', d) \in Q$.
\end{proof}
\section{Further improvements}
Define $\sigma_a$ a relation on the set of states $Q$ for an automaton $\mathcal{A}$ and a given letter $a \in \Sigma$ such that $q_1 \sigma_a q_2$ if, and only if $\delta(q_1,a) = \delta(q_2,a)$. It is obvious that, for any $a \in \Sigma$, $\sigma_a$ is an equivalence relation on the set of states. We also define $\sigma_a$-\textit{transversal} as $Q' \in Q$ such that each equivalence class has at most one representative in $Q'$. Let $Q_1, ..., Q_l$ be equivalence classes of $\sigma_a$ on $Q$. Finally we say that letter $b \in \Sigma$ is $\sigma_a$-\textit{preserving} with respect to $\tau: 2^Q \times \Sigma \rightarrow 2^Q$ if for any $\sigma_a$-transversal $Q' = \{q_{i_1}, ... ,q_{i_k} \}$, such that lower index of $q_i$ corresponds to $i$-th equivalence class, $\tau(Q',b) = \{q'_{i_1}, ... ,q'_{i_k}\}$ such that lower index of $q'_i$ corresponds to $i$-th equivalence class. It can be easily seen that letter $a$ in automaton $\mathcal{A}_d(n)$ defines $\sigma_a$ on the $Q$, resulting with partition of it on $k$ pairwise disjunctive sets, and letters $b_i$ for $i = 1, ..., k$ are $\sigma_a$-preserving. In Section 2. we defined the automaton that first creates $\sigma_a$ on $Q$ and then traverses some of transversals of that relation. Specifically, after applying letter $c_i$ on $A_d$ we reducing number of equivalence classes possible to traverse by one. It is natural question to ask if we can traverse more transversals than we have shown in Section 2. We give universal construction that can be used to improve the lower bound obtained by Martyugin.\newline
Main idea is to immediately reduce $Q$ to $k$ equivalence classes and then treat those classes as states of some synchronizable DFA or carefully synchronizable PFA. First we define construction sufficient to construct carefully synchronizing automaton with long shortest carefully synchronizing word for any given synchronizable DFA or carefully synchronizable PFA and next we apply that construction to \v{C}ern\'{y} automata $\mathcal{C}_n$ in order to give an upper bound for shortest carefully synchronizing word for such created automaton.\newline Let $\mathcal{B} = (S, \Delta, \gamma)$ be a finite automaton. Let $S = \{q_1, ..., q_k\}$ and $\Delta = \{c_1, ..., c_s\}$. We define PFA $\mathcal{A}_d(\mathcal{B}) = (Q, \Sigma, \delta)$ as follows:
\begin{itemize}
\item $\Sigma = \{a, b_1, b_2, ... b_k, c_1, c_2, ... , c_l\}$
\item $Q_i = \{q_0^i, q_1^i, ... , q_{d-1}^i\} $
\item $Q = \bigcup\limits_{i = 1}^k Q_i$
\end{itemize}
Let $i \in \{1,..., k\}$, $l \in \mathbb{N}$. we define partial transition function $\delta:{Q \times \Sigma}\rightarrow{Q}$ for $\mathcal{A}_d(\mathcal{B})$ as:
\begin{itemize}
\item $\delta(q_j^i, a) = q_0^i$, $j \in \{0,1, ... , d-1\}$
\item $\delta(q_{j-1}^i, b_i) = q_{j}^i$, $j \in \{1,2, ... , d-1\}$
\item $\delta(q_j^i, b_l) = q_j^i$, $i > l$
\item $\delta(q_{d-1}^i, b_l) = q_0^i$, $i < l$
\item $\delta(q_{d-1}^i, c_l) = q_0^j$, for all $i,l$ such that $\gamma(q_i, c_l) = q_j$
\end{itemize}
We start with following simple observations.
\begin{fact}
Let $\sigma_a$ be defined as above on $Q$. Letters $a, b_1, ..., b_k \in \Sigma$ are $\sigma_a$-preserving.
\end{fact}
Before moving further we prove following lemma.
\begin{lemma}
Let $\mathcal{P}(\mathcal{A}_d(\mathcal{B})) = (2^{Q}, \Sigma, \tau)$ be a power automaton for automaton $\mathcal{A}_d(\mathcal{B})$. Let $\{i_1, i_2, ... , i_s\} \subset \{1, ..., k\}$. Let also $Q_0 = \{q_0^{i_1},q_0^{i_2},... ,q_0^{i_s}\}$ and $Q_{3^s - 1} = \{q_{d-1}^{i_1},q_{d-1}^{i_2},... ,q_{d-1}^{i_s}\}$ . Then the shortest path $p$ from $Q_0$ to $Q_{d^s - 1}$ in $\mathcal{P}(\mathcal{A}_d(\mathcal{B}))$ is of length $d^s - 1$.
\end{lemma}
\begin{proof}
Since showing that path $p$ of desired length exists is similar to the proof of Lemma 1. and the word which traverses all sets on $p$ is analogous to the word in the proof of Lemma 1. we omit that part of proof and focus on proving that at any point on $p$ there exist only one transition to state not visited before in order to show minimality of $p$.\newline
Notice that we can treat any $Q_r$ on $p$ as $d$-ary representation of $r$ just like in Lemma 3, the only difference is that we omit "empty spots" in $Q_r$. Another similarity is that after $Q_{r+1}$ is directly after $Q_r$ on $p$. Assume that $r = (a_1, a_2, ..., a_{s - k + 1}, d-1, d-1, ... , d-1)_d$, $k \in \{1, ..., s\}$ and $a_{s - k + 1} \neq d-1$. It is obvious from Fact 4. that for each $b_l$ such that $l > k$ the transition $\tau(Q_r, b_l)$ is not defined. If $l < k$ we must investigate two cases. If $q_{d-1}^l \in Q_r$ then from Fact 5. $\tau(Q_r, b_l)$ is not defined. Else notice from definition of $\delta'$ that transition $\tau(Q_r, b_l)$ zeros some of positions younger than $s - k + 1$ and maps older positions and position $s - k + 1$ to itself, so the result of that transition is such $Q_{r'}$ that $r > r'$ which was visited earlier since all numbers between $0$ and $r$ must be on $p$. None of letters $c_i$ is defined. That concludes the proof.
\end{proof}
Having that construction we may prove main theorem of that section.
\begin{theorem}
Let $d > 1$. $\mathcal{B}$ is a synchronizing DFA(carefully synchronizing PFA) if, and only if $\mathcal{A}_d(\mathcal{B})$ is carefully synchronizing.
\end{theorem}
\begin{proof}
Let $\mathcal{P(B)} = (2^S, \Delta, \rho)$ be a power automaton for automaton $\mathcal{B}$ and $\mathcal{P}(\mathcal{A}_d(\mathcal{B})) = (2^{Q}, \Sigma, \tau)$ be a power automaton for automaton $\mathcal{A}_d(\mathcal{B})$. Fix $d > 1$. First we prove right implication. Since $\mathcal{B}$ is (carefully) synchronizing there exist a (carefully) synchronizing word $w = c_{k_1}...c_{k_s} \in \Sigma^\ast$. We now construct carefully synchronizing word $w'$ for $\mathcal{A}_d(\mathcal{B})$. Let $w' = \epsilon$. Since only letter $a$ is defined for all states we append letter $a$ to $w'$. Notice that $\tau(Q, a) = Q_0^k$. From Lemma 1. We know that there exist $u_0 \in \Sigma^\ast$ of length $d^k-1$ such that $\tau(Q_0^k) = Q_{d^k-1}^k$. Notice that for each $i \in \{1, ..., k\}$ $\tau(Q_{d^k-1}^k, b_i) = Q_0^k$ or is undefined. Let $P = \{q_{i_1}, ... ,q_{i_n} \}$. It is easy to observe that if $\rho(P, c_{k_i}) = P'$ such that $P' = \{q_{j_1}, ... ,q_{j_m}\}$, then $\tau(\{q_{d-1}^{i_1}, ... , q_{d-1}^{i_n},\},c_{k_i}) = \{q_0^{j_1}, ... , q_0^{j_m}\} = Q_{c_k}$. Any such $Q_{c_k}$ is subset of $Q_0^k$, so from Lemma 4. there exist word $w_k$ (of length $3^{|Q_{c_k}|} - 1$), such that $\tau(Q_{c_k}, w_k) = \{q_{d-1}^{j_1}, ... , q_{d-1}^{j_m}\} $. From that it is easy to notice that word $w' = aw_kc_{k_1}w_{k_1}...c_{k_{s-1}}w_{k_{s-1}}c_{k_s}$ carefully synchronizes $\mathcal{A}_d(\mathcal{B})$.\newline In order to prove left implication suppose, for the sake of contradiction, that there exist non-synchronizable DFA (non-carefully synchronizable PFA) $\mathcal{B}$ such that its $\mathcal{A}_d(\mathcal{B})$ is carefully synchronizable. Any (carefully) synchronizing word $w$ for $\mathcal{A}_d(\mathcal{B})$ must start with letter $a \in \Sigma$, which defines $\sigma_a$ on $Q$. Notice that for any $\sigma_a$-transversal $Q' \neq \{q_{d-1}^{i_1}, ... , q_{d-1}^{i_l}\}$, $\tau(Q', c_i)$ is not defined for $i = 1, ..., s$ and, due to Fact 7, there is no letter that can change traversed equivalence classes. That leads to contradiction, since $\mathcal{B}$ must be (carefully) synchronizing so to $\mathcal{A}_d(\mathcal{B})$ be carefully synchronizing.
\end{proof}
\begin{corollary}
If $\mathcal{B}$ is a synchronizing DFA(carefully synchronizing PFA) then the shortest carefully synchronizing word for $\mathcal{A}_d(\mathcal{B})$ is $\Omega(d^{n/d})$
\end{corollary}
\begin{proof}
Since $|w_k| = d^k-1$ and it labels the shortest path from $Q_0^k$ to $Q_{d^k-1}^k$ we conclude corollary holds.
\end{proof}
Now we are ready to bound shortest carefully synchronizing word for $\mathcal{A}_d(\mathcal{C}_n)$. Let $\mathcal{C}_n = (S, \Delta, \gamma)$ be a DFA such that $S = \{q_0, ..., q_{n-1}\}$, $\Delta = \{c_1, c_2\}$ and $\gamma$ is defined as follows:
\begin{itemize}
\item $\gamma(q_0,c_1) = q_1$
\item $\gamma(q_m,c_1) = q_m$ for $m \in \{1, ..., n-1\}$
\item $\gamma(q_m,c_2) = q_{m+1 (mod\;n)}$
\end{itemize}
Despite the shortest synchronizing word for $\mathcal{C}_n$ is $(c_1c_2^{n-1})^{n-2}c_1$ of length $(n-1)^2$ [1], we find another synchronizing word, more appropriate word to bound $d(\mathcal{A}_d(\mathcal{C}_n))$.
\begin{lemma}
If $n > 2$ is even, then word $w_1 = (c_1c_2^2)^\frac{n}{2}(c_1c_2^{n-1})^{n-3}c_1$ synchronizes $\mathcal{C}_n$, otherwise $w_2 = (c_1c_2^2)^\frac{n+1}{2}(c_1c_2^{n-1})^{n-4}c_1$ synchronizes $\mathcal{C}_n$.
\end{lemma}
\begin{proof}
We prove for even $n$ since proof for odd $n$ is similar. Let $\mathcal{P}(\mathcal{C}_n) = (2^S, \Delta, \rho)$ be a power automaton for automaton $\mathcal{C}_n$. Denote $c_1c_2^2 = u$ and $c_1c_2^{n-1} = v$. It is easy to check by induction on $k$ that if $k \leq n/2$, then $|\rho(S,u^k)| = \{q_0, q_2, ...,q_{2k-2}, q_{2k}, q_{2k+1}, ..., q_{n-2} \}$. So $\rho(S, u^\frac{n}{2}) = \{q_0, q_2, ...,q_{n-2}\} = S_{\frac{n}{2}}$. Now consider action of $v^2$ on set $S_{\frac{n}{2}}$. We will proof by induction on $k$ that if $k < \frac{n}{2} - 1$, then $\rho(S_{\frac{n}{2}}, v^{2k}) = \{q_0, q_2, ..., q_{n -2k -2}\}$. If $k = 1$ then, from definition of $\gamma$, $\rho(S_{\frac{n}{2}}, c_1) = \{q_1, q_2, ..., q_{n-2}\}$. So it is easily seen that $\rho(S_{\frac{n}{2}}, c_1c_2^{n-1}c_1) = \{q_1, q_3, ..., q_{n-3}\}$, and $\rho(S_{\frac{n}{2}}, c_1c_2^{n-1}c_1c_2^{n-1}) = \{q_0, q_2, ..., q_{n-4}\}$. Assume that lemma holds for every $i < k$, then $\rho(S_{\frac{n}{2}}, v^{2k}) = \{q_0, q_2, ..., q_{n -2k - 2}\}$. Similarly as in $k=1$ case, applying word $v^2$ results with $\{q_0, q_2, ..., q_{n -2k - 4}\}$ so the statement holds. That implies $\rho(S_{\frac{n}{2}}, (c_1c_2^{n-1})^{n-4}) = \{q_0, q_2\}$. Notice that $\rho(\{q_0, q_2\}, c_1c_2^{n-1}c_1) = \{q_1\}$ and that ends proof.
\end{proof}
Having that we prove following theorem.
\begin{theorem}
If $n$ is even, then $d(\mathcal{A}_d(\mathcal{C}_n)) \leq \frac{1}{d-1}[d^{n+1} + 2 \cdot d^n + (n-4)\cdot d^{\frac{n}{2} + 1} + (n-1)(d^{\frac{n}{2}} - d^2 -d^3) - 1]$. Otherwise $d(\mathcal{A}_d(\mathcal{C}_n)) \leq \frac{1}{d-1}[d^{n+1} + 2\cdot d^n + (2n-5)\cdot d^\frac{n}{2} - (n-1)(d+1)d^2] - (n-1)\cdot d^{\frac{n} + 1} + 2$.
\end{theorem}
\begin{proof}
Let $n$ be even. We construct for a given automaton a reset word of desired length. From Lemma 5 we know that $(c_1c_2^2)^{n/2}(c_1c_2^{n-1})^{n-3}c_1$, so we know that there exist carefully synchronizing word $w$ for $\mathcal{A}_d(\mathcal{C}_n)$ of form $aw_n \cdot \prod\limits_{i=1}^{\frac{n}{2}} c_1u_1^ic_2u_2^ic_2u_3^i \cdot (\prod\limits_{i=1}^{n-3} c_1w_1^ic_2w_2^i ... c_2w_{n-1}^i )\cdot c_1$. Denote $\prod\limits_{i=1}^{\frac{n}{2}} c_1u_1^ic_2u_2^ic_2u_3^i = v_1$ and $\prod\limits_{i=1}^{n-3} c_1w_1^ic_2w_2^i ... c_2w_{n-1}^i = v_2$. It is obvious that $|aw_n| = 3^n$. Now we calculate $|v_1|$. It is easy to notice that after applying letter $c_1$ number of equivalence classes traversed by $u_i^j$ reduces by one, so from Lemma 4. $|v_1| = \sum\limits_{i=1}^{\frac{n}{2} - 1} 3\cdot d^{n-i}$. Consider $|v_2|$. From the proof of Lemma 5 we can deduce that we reduce number of equivalence classes by one after two letters $c_1$ in $w_2$. Thus, from Lemma 4 we obtain $|v_2| = \sum\limits_{i=\frac{n}{2}}^{n - 2} ((2n-2) \cdot d^{n - i}) - (n-1)\cdot d^{\frac{n}{2}} - (n-1) \cdot d^2 + 1$. Because we do not traverse all $(2n-2)$ equivalence classes when $i = n/2$ and $i = n-2$ (see proof of Lemma 5), we substract $(n-1)\cdot d^{\frac{n}{2}} + (n-1) \cdot d^2$. After simple calculations we obtain $|w| = \frac{1}{d-1}[d^{n+1} + 2 \cdot d^n + (n-4)\cdot d^{\frac{n}{2} + 1} + (n-1)(d^{\frac{n}{2}} - d^2 -d^3) - 1]$. Similar analysis of word $w$ when $n$ is odd results with $|w| = \frac{1}{d-1}[d^{n+1} + 2\cdot d^n + (2n-5)\cdot d^\frac{n}{2} - (n-1)(d+1)d^2] - (n-1)\cdot d^{\frac{n} + 1} + 2$.
\end{proof}
We are now able to formulate following corollary.
\begin{corollary}
Let $d > 1$, $n > 2$. Then $d(\mathcal{A}_d(\mathcal{C}_n)) \in O(d^n + n\cdot d^\frac{n}{2})$ and $d(\mathcal{A}_d(\mathcal{C}_n)) \in \Omega(d^n )$.
\end{corollary}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 286 |
Last weekend was my first public event where I exhibited my sculptures. It was a three day exhibition/fair called Parallax Art Fair .
Despite unusually hot weather (London is not famous for its hot summers) there was a fair amount of people attending the event, - not all chose to do BBQ despite my expectations; although I remember the venue to be more crowded when I visited it in February.
One of the most interesting observations I made is that my works triggered deeper interest in the visitors, who either artists themselves or work in art sphere (e.g. critics, curators, gallery owners). Typical feedback I heard from them that my work is rather mature and deep - indeed something that I would like to hear, as it does summarise my general moto in art.
Lessons learned. Next time I should test my exposition equipment more and in advance; one of the approached I planned to use to lift the table to be 110 cm high (from standard 70 cm) did not work, and in order to keep my stand stable I had to revert to default 70 cm to stabilise it - that was rather unfortunate.
One of the most interesting observations I made is that my works triggered deeper interest in the visitors, who either artists themselves or work in art sphere (e.g. critics, curators, gallery owners). | {
"redpajama_set_name": "RedPajamaC4"
} | 6,584 |
class MiqReport::Formats
private_class_method def self.format_hash
@format_hash ||= YAML.load_file(ApplicationRecord::FIXTURE_DIR.join('miq_report_formats.yml')).freeze
end
private_class_method def self.formats
format_hash[:formats]
end
private_class_method def self.defaults_and_overrides
format_hash[:defaults_and_overrides]
end
def self.available_formats_for(column, suffix, datatype)
is_break_sfx = (suffix && MiqReport.is_break_suffix?(suffix))
formats.each_with_object({}) do |(format_name, properties), result|
# Ignore formats that don't include suffix if the column name has a break suffix
next if is_break_sfx && (properties[:suffixes].nil? || !properties[:suffixes].include?(suffix.to_sym))
next unless (properties[:columns] && properties[:columns].include?(column)) ||
(properties[:sub_types] && properties[:sub_types].include?(sub_type(column))) ||
(properties[:data_types] && properties[:data_types].include?(datatype)) ||
(properties[:suffixes] && properties[:suffixes].include?(suffix.to_sym)) ||
format_name == sub_type(column) ||
format_name == defaults_and_overrides[:formats_by_sub_type][sub_type(column)]
result[format_name] = properties[:description]
end
end
def self.default_format_for_path(path, datatype)
column = path.split('-').last.to_sym
suffix = column.to_s.split('__').last.try(:to_sym)
# HACK: formats for columns are unqualified, so we need a
# temporary way to avoid collisions
defaults_and_overrides[:formats_by_path].fetch(path.to_sym) do
defaults_and_overrides[:formats_by_suffix][suffix] ||
defaults_and_overrides[:formats_by_column][column] ||
defaults_and_overrides[:formats_by_sub_type][sub_type(column)] ||
defaults_and_overrides[:formats_by_data_type][datatype]
end
end
def self.default_format_details_for(path, column, datatype)
format = formats[default_format_for_path(path, datatype)]
if format
format = format.deep_clone # Make sure we don't taint the original
if defaults_and_overrides[:precision_by_column].key?(column.to_sym)
format[:precision] = defaults_and_overrides[:precision_by_column][column.to_sym]
end
end
format
end
def self.details(format_name)
formats[format_name.try(:to_sym)]
end
def self.sub_type(column)
defaults_and_overrides[:sub_types_by_column][column]
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,860 |
Dynamix Physical Therapy is West Tennessee's leader in elite physical therapy and sports medicine, and is the recipient of the 2022 Ascend National Practice of the Year Award. With the mission of being the bright spot in customers' day while pursuing excellence in healthcare and service, Dynamix offers multiple services, including manual therapy, a hands-on based treatment unique to this area, sports medicine, fitness, free injury assessments and work conditioning. Follow Dynamix on Facebook and Instagram for the latest news.
Dynamix Physical Therapy Named Inc. 5000 Company for Third Consecutive Year
MILAN, TENN. - Dynamix Physical Therapy has been named an Inc. 5000 company for the third year in a row.
The company made the Inc. 5000 list in 2019, 2020 and now 2021. This means that Dynamix is in the top 5,000 fastest-growing companies in the nation.
"We were just named the 2022 'Ascend National Practice of the Year' back in May, and now to be named an Inc. 5000 company for a third year is just the icing on top of the cake," said Russ Huffstetler, co-founder of Dynamix. "Our team has put in a lot of work over the last 13 years, and it is exciting that they are seeing all of their hard work be recognized nationally."
Dynamix continues to grow its team of professionals throughout the West Tennessee region, including the southwest areas. In the Millington and Covington communities, Dynamix's southwest regional director, Cory Wilcox, has just hired six new physical therapists, two physical therapy assistants, and three certified athletic trainers for the schools in that region.
"We have the momentum and potential to grow in the southwest region, including the greater Memphis area, and having Cory as our regional director gives us the confidence that Dynamix will have a presence in those communities in the coming years," said Heath Ladd, co-founder of Dynamix. "We want to continue to be on the Inc. 5000 list year after year, and to do so, we must keep pushing and finding those talented team members who are passionate about their profession and who want to serve their communities with the best physical therapy and sports medicine services."
Dynamix has 12 locations throughout West Tennessee, offering outpatient orthopedic services, advanced dry needling, workers' compensation and sports medicine. Dynamix clinicians specialize in manual therapy, a hands-on based treatment that is unique to this region.
For more information on the services offered by Dynamix or to schedule your free injury assessment, visit dynamix.life.
« Dynamix Physical Therapy of Jackson Celebrates One Year of Serving the Community
Dynamix Physical Therapy Launches New Website » | {
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HUDSON, New Hampshire — The U.S. Navy has placed a $225 million order for additional precision guidance kits from BAE Systems, the company announced in a March 19 release.
The combat-proven Advanced Precision Kill Weapon System (APKWS) kits transform standard 2.75-inch (70mm) Hydra rockets into guided munitions that provide warfighters with a precision-strike capability with limited collateral damage.
The APKWS guidance kits are the U.S. government's only program of record for 2.75-inch laser-guided rockets and are available to all four military branches and to allied nations via Foreign Military Sales.
The warhead of the Hydra rocket combined with precision guidance lets warfighters strike light targets while minimizing the risk of harm to friendly forces or civilian buildings nearby — making the rockets ideal for dense urban combat.
BAE Systems continues to accelerate production of APKWS guidance kits at its production facilities in Hudson, New Hampshire, and Austin, Texas, as it builds toward an annual production level of more than 20,000 units.
The company's supply chain, advanced manufacturing capabilities and infrastructure investments exceeding $100 million have enabled it to meet increasing production rates for its top precision munition and electronic warfare programs. | {
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} | 3,172 |
2014 Historcal Fiction Releases (Part III)
I've got a bevy of beautiful historical fiction titles to share with you! Break out the wishlists, folks!
The Fortune Hunter
by Daisy Goodwin
US Publication Date: July 29, 2014
UK Publication Date: April 24, 2014
In 1875, Sisi, the Empress of Austria is the woman that every man desires and every woman envies.
Beautiful, athletic and intelligent, Sisi has everything - except happiness. Bored with the stultifying etiquette of the Hapsburg Court and her dutiful but unexciting husband, Franz Joseph, Sisi comes to England to hunt. She comes looking for excitement and she finds it in the dashing form of Captain Bay Middleton, the only man in Europe who can outride her. Ten years younger than her and engaged to the rich and devoted Charlotte, Bay has everything to lose by falling for a woman who can never be his. But Bay and the Empress are as reckless as each other, and their mutual attraction is a force that cannot be denied.
Full of passion and drama, THE FORTUNE HUNTER tells the true story of a nineteenth century Queen of Hearts and a cavalry captain, and the struggle between love and duty.
by Anne Clinard Barnhill
From Anne Barnhill, the author of At the Mercy of the Queen, comes the gripping tale of Mary Shelton, Elizabeth I's young cousin and ward, set against the glittering backdrop of the Elizabethan court.
Mistress Mary Shelton is Queen Elizabeth's favorite ward, enjoying every privilege the position affords. The queen loves Mary like a daughter, and, like any good mother, she wants her to make a powerful match. The most likely prospect: Edward de Vere, Earl of Oxford. But while Oxford seems to be everything the queen admires: clever, polished and wealthy, Mary knows him to be lecherous, cruel, and full of treachery. No matter how hard the queen tries to push her into his arms, Mary refuses.
Instead, Mary falls in love with a man who is completely unsuitable. Sir John Skydemore is a minor knight with little money, a widower with five children. Worst of all, he's a Catholic at a time when Catholic plots against Elizabeth are rampant. The queen forbids Mary to wed the man she loves. When the young woman, who is the queen's own flesh and blood, defies her, the couple finds their very lives in danger as Elizabeth's wrath knows no bounds.
by Sally Beauman
Publication Date: July 1, 2014
Based on a true story of discovery, The Visitors is New York Times bestselling author Sally Beauman's brilliant recreation of the hunt for Tutankhamun's tomb in Egypt's Valley of the Kings-a dazzling blend of fact and fiction that brings to life a lost world of exploration, adventure, and danger, and the audacious men willing to sacrifice everything to find a lost treasure.
In 1922, when eleven year-old Lucy is sent to Egypt to recuperate from typhoid, she meets Frances, the daughter of an American archaeologist. The friendship draws the impressionable young girl into the thrilling world of Lord Carnarvon and Howard Carter, who are searching for the tomb of boy pharaoh Tutankhamun in the Valley of the Kings.
A haunting tale of love and loss, The Visitors retells the legendary story of Carter and Carnarvon's hunt and their historical discovery, witnessed through the eyes of a vulnerable child whose fate becomes entangled in their dramatic quest. As events unfold, Lucy will discover the lengths some people will go to fulfill their deepest desires-and the lies that become the foundation of their lives.
Intensely atmospheric, The Visitors recalls the decadence of Egypt's aristocratic colonial society, and illuminates the obsessive, daring men willing to risk everything-even their sanity-to claim a piece of the ancient past. As fascinating today as it was nearly a century ago, the search for King Tut's tomb is made vivid and immediate in Sally Beauman's skilled hands. A dazzling feat of imagination, The Visitors is a majestic work of historical fiction.
Cavendon Hall
by Barbara Taylor Bradford
Cavendon Hall is home to two families, the aristocratic Inghams and the Swanns who serve them, just as their ancestors did over the centuries. Charles Ingham, the sixth Earl of Mowbray, lives there with his wife Felicity and their six children: Guy, the heir, who is studying at Cambridge; their younger son Miles, attending Eton; and their four daughters Diedre, Daphne, DeLacy and Dulcie, affectionately called the Four Dees by the staff. Walter Swann, the premier male of the Swann family, is valet to the earl. His wife Alice, a clever seamstress, who is in charge of the countess's wardrobe, also makes clothes for the four daughters. For centuries, these two families have lived side-by-side, beneath the backdrop of the imposing Yorkshire manor. But now, with World War I looming, these two families will find themselves tested in ways they never thought possible. Loyalties are tested and betrayals are set into motion. In this time of uncertainty, one thing is sure: these two families will never be the same again. Set over a period of sixteen years (from 1913 to 1929), Cavendon Hall is Barbara Taylor Bradford at her very best.
The Beautiful American
by Jeanne Mackin
From Paris in the 1920s to London after the Blitz, two women find that a secret from their past reverberates through years of joy and sorrow....
As recovery from World War II begins, expat American Nora Tours travels from her home in southern France to London in search of her missing sixteen-year-old daughter. There, she unexpectedly meets up with an old acquaintance, famous model-turned-photographer Lee Miller. Neither has emerged from the war unscathed. Nora is racked with the fear that her efforts to survive under the Vichy regime may have cost her daughter's life. Lee suffers from what she witnessed as a war correspondent photographing the liberation of the Nazi concentration camps.
Nora and Lee knew each other in the heady days of late 1920s Paris, when Nora was giddy with love for her childhood sweetheart, Lee became the celebrated mistress of the artist Man Ray, and Lee's magnetic beauty drew them all into the glamorous lives of famous artists and their wealthy patrons. But Lee fails to realize that her friendship with Nora is even older, that it goes back to their days as children in Poughkeepsie, New York, when a devastating trauma marked Lee forever. Will Nora's reunion with Lee give them a chance to forgive past betrayals…and break years of silence to forge a meaningful connection as women who have shared the best and the worst that life can offer?
A novel of freedom and frailty, desire and daring, The Beautiful American portrays the extraordinary relationship between two passionate, unconventional women.
Labels: 2014 Release, Anne Clinard Barnhill, Barbara Taylor Bradford, Daisy Goodwin, Jeanne Mackin, Sally Beauman
Oh my goodness! I want to read all of them !! You give us the best books to review!!
What a wonderful selection of books. They all sound interesting. THE VISITORS, QUEEN ELIZABETH'S DAUGHTER, and CAVENDON HALL would be at the top of my list.
Connie January 19, 2014 at 9:46 AM
I just cannot wait to read these! They are at the top of my Wish List.
Sharon January 19, 2014 at 6:32 PM
They all sound great AND have beautiful covers! Thanks!
Kailana January 20, 2014 at 1:46 PM
I hope I can get back in the swing of things for historical fiction this year. I missed out on so much last year!
On My Wishlist: Somerset by Leila Meacham
Facebook Party for History Lovers: Stephanie Dray'...
Jeannie Ruesch's Cloaked in Danger Launch Party, J...
Guest Post by Ella March Chase & Giveaway of The Q...
Win a copy of Isabella: Braveheart of France by Co...
Win a copy of The Winter Siege by D.W. Bradbridge ...
On My Wishlist: Bittersweet by Collen McCullough
Post by Sophie Schiller + Giveaway of Spy Island
Guest Post by Patricia Bracewell + Giveaway of Sha...
Susanna Kearsley & Friends Tour + Giveaway of The ...
HFVBT Blog Tour: Win a copy of The Harlot's Tale b...
The Secret Daughter of the Tsar by Jennifer Laam: ...
Guest post by Jennifer Robson + Giveaway of Somewh...
Win a copy of The Gods of Heavenly Punishment by J...
Win a copy of Becoming Josephine by Heather Webb | {
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} | 4,048 |
{-# LANGUAGE TemplateHaskell #-}
{-# LANGUAGE QuasiQuotes #-}
--
-- Ivory pre/post conditions quasiquoter.
--
-- Copyright (C) 2014, Galois, Inc.
-- All rights reserved.
--
module Ivory.Language.Syntax.Concrete.QQ.CondQQ where
import Prelude hiding (exp)
import Language.Haskell.TH hiding (Stmt, Exp, Type)
import qualified Language.Haskell.TH as T
import Language.Haskell.TH.Quote()
import qualified Ivory.Language.Cond as I
import Ivory.Language.Syntax.Concrete.ParseAST
import Ivory.Language.Syntax.Concrete.QQ.BindExp
import Ivory.Language.Syntax.Concrete.QQ.Common
--------------------------------------------------------------------------------
-- Assertions, e.g.,
--
-- requires (checkStored (pid ~> pid_err) (\err -> err <? 1)) ensures (\res ->
-- checkStored (pid ~> pid_err) (\err -> err <? res))
--------------------------------------------------------------------------------
mkPrePostConds :: [PrePost] -> T.Exp -> Q T.Exp
mkPrePostConds conds procBody = do
condFns <- mapM mkCond conds
-- Apply conditions to the proc body in the Q monad.
return (foldr AppE procBody condFns)
where
mkCond :: PrePost -> Q T.Exp
mkCond cond = case cond of
PreCond exp -> appE (varE 'I.requires) (runExp exp)
PostCond exp -> appE (varE 'I.ensures) (lamE [varP $ mkName "return"]
(runExp exp))
runExp :: Exp -> Q T.Exp
runExp exp = do
(e, derefs) <- runToQ (fromExpCond exp)
return (foldr go e derefs)
where
go :: (T.Exp, T.Name) -> T.Exp -> T.Exp
go (deref, nm) acc =
AppE (AppE (VarE 'I.checkStored) deref) (LamE [VarP nm] acc)
--------------------------------------------------------------------------------
fromExpCond :: Exp -> QStM (T.Exp, Name) T.Exp
fromExpCond = fromExp insertDerefCond
insertDerefCond :: Insert (T.Exp, Name)
insertDerefCond key nm exp
| isArea key
= insert (exp, nm)
| otherwise
= error $ "Cannot insert " ++ show exp ++ " in pre/post conditions!"
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,140 |
Category Archives: FPHPS News
Posted in Cookbook Committee • FPHPS News
Holiday Shopping – Federal Point History Center Gift Shop
Does everyone in your extended family have one of our Local Flavor Cookbooks? How about friends and neighbors? At $25.00 it's the perfect homegrown gift for every cook you know.
It is full of "cookable" recipes mostly built from ingredients you already have in your pantry or can pick up at any local grocery store. And, it has a section with historic highlights of well known restaurants of Federal Point.
Don't forget our t-shirts are a real bargain at $12.00 each. We've got plenty of the Society shirts in every size and color. We're also well stocked with the Ocean Plaza BIRTHPLACE of the SHAG shirts. Anyone with a history of the Boardwalk would love this reflection of our history.
Books, Books, Books! We have lots of books that relate to the history and culture of our area. The two most important are
Elaine Henson's Carolina Beach in Postcards and Brenda Coffey's Images of America: Kure Beach. Both are well researched and would be a great present to anyone who's interested in the history of our local area.
Posted in FPHPS News
March Meeting: Celebration of our 25th Anniversary
Monday, February 18, 2019 – 7:30 PM
The Federal Point Historic Preservation Society will hold its monthly meeting on Monday, March 18, 2019, at 7:30 pm at the Federal Point History Center, 1121-A North Lake Park Blvd., adjacent to Carolina Beach Town Hall.
Founded March 28, 1994
Join us for a trip down memory lane as our founding members talk about the early years of the Society.
There will be special refreshments and time to talk about all the projects the Society has been involved with over the years.
Newton Homesite and Cemetery
Beauregard Shipwreck Overlook
Sedgeley Abbey
Posted in FPHPS News • From the President
President's Letter — March, 2019
By Elaine Henson
FPHPS 25th Anniversary, Gazebo/Picnic Shelter
This month we celebrate the 25th Anniversary of the Federal Point Historic Preservation Society. The organization was incorporated on March 28, 1994. In those early days the Society met at various places including Fort Fisher State Historic Site, but after a few years they were eyeing the gazebo/picnic shelter next to the Town Hall complex in the 1100 block of North Lake Park Boulevard.
The Town of Carolina Beach had purchased the former Blockade Runner Museum in 1989 to remodel and expand into the present day town complex. The property included a replica of a 19th Century open air public market which was used as a picnic shelter for school groups and visitors to the museum.
In the late 1990s, FPHPS approached the town about converting the picnic shelter into a meeting space. After a couple of years, the town gave the go ahead and the fund raising and gathering of materials began.
There were generous donations from many individuals from the Federal Point area, Wilmington and New Hanover County. Many donated money, materials, services, talents and man hours. Just to name a few, the HVAC was donated by Taylor Heating and Air; M & M Plumbing donated their labor and got a vendor to donate fixtures; EWE Electrical donated their labor; Hanover Iron Works donated the shingles and Lowes gave a discount on all the building materials and other purchases.
Many organizations donated their time such as the Junior Sorosis who donated and installed the ceiling tiles and the North Carolina Aquarium employees who helped with the display cases. FPHPS members, their families and other volunteers worked tirelessly to complete enclosing the picnic shelter and adding a 16-foot addition to the back to make the almost 1600 square foot History Center.
Upon completion they held a grand opening celebration on March 30, 2001. The guest speaker was Lisbeth Evans, Secretary of North Carolina Department of Cultural Resources.
The Early Years of Federal Point History Center
♦ June 22, 1994: First Speaker, Catherine Bishir of the North Carolina Department of Cultural Resources, Historic Preservation Section
♦ October 21, 22, 23, 1994: First fundraiser "Ocean Plaza Ballroom Blast" Featuring Chicken Hicks
♦ Fall 1994: First Newsletter, editor Sandy Jackson
Ballroom Blast, 1994
♦ December 1994 – October 1995: First Preservation Campaign – Protection and preservation of the historic plantation ruins of Sedgeley Abbey
♦ March 1995: Lighthouse logo, created by Martin Peebles, adopted
♦ Spring 1995: Agreement with Town of Carolina Beach for the construction of the Beauregard Shipwreck Overlook
♦ April 1995: Bingo fundraiser
♦ Spring 1995: Ocean Plaza and Joy Lee Apartments nominated to the National Register of Historic Places
♦ July 1995: Fort Fisher Revetment Project, advocacy, support, and ground breaking
♦ October 20-22, 1995: Second Annual Ocean Plaza Reunion
♦ Received $10,000 grant from North Carolina Department of Cultural Resources for compiling an inventory of known historic sites and cartographic inventory of Federal Point, directed by Sandy Jackson
♦ May 26, 1996: Hosted a celebration marking the 50th Anniversary of the Ocean Plaza Building. Wilmington Concert Band performed, followed by a fashion show in keeping with the original opening in 1946
♦ August 18, 1996: Participated in Belk "Preservation Celebration" fundraiser
♦ June 22, 1997: Oakdale Cemetery guided tour by E. F. "Gene" Risley Jr.
♦ Saturday October 18, 1997: Barbeque fundraiser
♦ November 15, 1997: Traditional Holiday Decorating Workshop, hosted by Fort Fisher State Historic Site, with demonstrations by staff members of Tryon Palace
♦ February, 1998: First Cookbook
♦ February, 1998: House Plaque Committee was formed and drafted guidelines for plaquing historic buildings
♦ March 1998: Published Monuments & Markers of Federal Point, North Carolina compiled by Sandy Jackson
♦ May, 1998: Fundraiser: Raffle of framed art print of the Federal Point Lighthouse by Kay Robbins
♦ Summer, 1998: Entered into an agreement with MOTSU to maintain, prepare signage and protect the Newton Homesite and Graveyard. Work began with construction of a wooden fence
♦ September, 1998: The first historic plaques were awarded to the Loughlin Cottage, Burnett Beach Cottage, and Ocean Plaza Ballroom, all over 50 years old and of significance to the community
♦ December 5, 1998: "Down East" Barbecue fundraiser
♦ February, 1999 – Entered into a lease with the Town of Carolina Beach for the Gazebo structure to be converted into the Federal Point History Center
♦ April 1999: Sugar Loaf Battle marker moved from Dow Rd. to Federal Point History Center
♦ May, 1999: First Student Essay Contest open to fifth grade classes at Carolina Beach Elementary School was won by Waverly Jones
♦ May 23, 1999: First fundraising Cruise – Aboard Pirate IV
♦ June 27, 1999: Commemorative Ceremony held celebrating the listing of Newton Homesite and Graveyard in the National Register of Historic Places
♦ October 22, 1999: Ground breaking for renovation of the Gazebo structure to become the Federal Point History Center
Posted in Fort Fisher • FPHPS News • Monthly Meeting Reports
The Story of Blackbeard's Shipwreck: Queen Anne's Revenge
by Nancy Gadzuk
Mark Wilde-Ramsing, former Director of the Underwater Archaeology Unit at Fort Fisher, spoke at the January 21, 2019 meeting of the Federal Point Historic Preservation Society.
Mark and Leslie Bright, Director of the History Center, worked together as a team for many years at Fort Fisher, and the Underwater Archaeology Unit there is the oldest in the country. Mark spoke on The Story of Blackbeard's Shipwreck: Queen Anne's Revenge. He was also promoting his new book, Blackbeard's Sunken Prize: The 300-Year Voyage of Queen Anne's Revenge.
Edward Teach, better known as the pirate Blackbeard, was notorious in the early 1700's, a prime time for privateers and pirates.
In 1717 he commandeered the French frigate the Concorde and renamed it Queen Anne's Revenge. Fast and well-armed, it became Blackbeard's flagship, and he and his crew stole as much bounty as they could from other less notorious privateers and pirates.
But not for too long, as Blackbeard ran the ship aground in 1718 outside of Beaufort, North Carolina, possibly to evade capture by the British. There the ship sat underwater until the wreckage was discovered in 1997.
It took almost ten years of environmental review and geological research to determine if bringing up these relics from the past was important enough to warrant disrupting the ocean floor. Apparently it was.
Full recovery took from 2006 to 2015, as salt and water made recovering artifacts difficult. Each item had to be kept wet until it could be cleaned, documented, and preserved in a laboratory. More than 400,000 artifacts were recovered, including pieces of fine glassware, jewelry, intricate weapons, pewter plates, medical tools, and more.
These artifacts came from all around the world: England, France, Germany, Sweden, Spain, Italy, the Netherlands, China, and Africa. Thirty cannons were also recovered, which explains how Queen Anne's Revenge was able to amass such a trove of riches in only six months.
Leg shackles were also recovered, suggesting that Blackbeard and his crew may have been slave traders as well as upscale, high-end thieves.
Mark shared pictures of some of the artifacts from the recovery and entertained a short question and answer before signing copies of the book he'd brought and made available for sale.
Upcoming Programs: Winter & Spring 2019
Winter & Spring 2019
image Posted in FPHPS News
December Meeting – Christmas Potluck
6:30 pm – One Hour Early!
The Federal Point Historic Preservation Society will hold its monthly meeting on Monday, December 17, 2018 at 6:30 pm at the Federal Point History Center, 1121-A North Lake Park Blvd., adjacent to Carolina Beach Town Hall.
The Federal Point Historic Preservation Society will hold its annual holiday potluck on Monday, December 17 at 6:30 pm. This year we will be back at the History Center as it's a lot easier for the hospitality committee. Please join us for food, fun and festivities.
Again this year Judge Jay Hockenbury and his wife Deborah, will have another trivia contest for us and a timely Christmas story.
John Golden will round out the evening with his wonderful Christmas sing-along.
Posted in Carolina Beach • FPHPS News • Kure Beach
Holiday Shopping – FPHPS Gift Shop
Does everyone in your extended family have one of our Local Flavor Cookbooks? How about friend and neighbors!
At $25.00 it's the perfect homegrown gift for every cook you know. It is full of "cookable" recipes mostly built from ingredients you already have in your pantry or can pick up at any local grocery store. And, it has a section with historic highlights of well known restaurants of Federal Point.
Don't forget our t-shirts are a real bargain at $12.00 each.. We've got plenty of the Society shirts in every size and color. We're also well stocked with the Ocean Plaza BIRTHPLACE of the SHAG shirts. Anyone with a history of the Boardwalk would love this reflection of our history.
Books, Books, Books! We have lots of books that relate to the history and culture of our area. The two most important are Elaine Henson's Carolina Beach in Postcards and Brenda Coffey's new Images of America: Kure Beach. Both are well researched and would be a great present to anyone who's interested in the history of our local area.
Carolina Beach, North Carolina, has been a destination for beachgoers, boaters, and fishermen since the 1880s. Visitors came first by the combination of river steamers and a train and later by automobiles to seek respite from the summer's heat and the daily grind. This book shares the history of this seaside community through the postcards its visitors sent home. From the early hand colored cards printed in Germany to the modern chrome cards of today, we see the people and places of Carolina Beach.
Kure Beach derived its name from a Danish immigrant named Hans Anderson Kure, Sr. He began acquiring land in the area in 1891, and by 1900, he had purchased 900 acres just south of Carolina Beach to Fort Fisher.
He established the Kure Land and Development Company and in 1913 produced a map of Fort Fisher Sea Beach, which would later become Kure's Beach and eventually Kure Beach. In 1923, the first wooden fishing pier on the Atlantic coast was constructed by Lawrence Kure.
DAN PRI, one of the first surfboard companies on the East Coast, was also established at Kure Beach.
The area is rich in historical significance from Verrazzano's discovery to Cape Fear Indians, pirates, lighthouses, the "Rocks," the Ethel Dow Chemical Plant and the community's role in both the Civil War and World War II.
Posted in Carolina Beach • FPHPS News
Tooting Our Own Horn?
Several months ago, we were asked to present a program on our Historical Society by the Kiwanis Club in Wilmington for their monthly meeting. Don't know why we had not thought of doing that before, but we got a program together and presented it at their August meeting. A few days after that we were invited to come to the Wilmington Civitan Club meeting in September and then by the Men's Breakfast Group at Carolina Beach Presbyterian in October.
On January 8, 2019, we will be presenting a shorter version at the Carolina Beach Town Council Meeting having been invited by Mayor Joe Benson. So, it seems that we have taken our show on the road.
The presentation begins with an overview of who we are, where we are, what we do and what we collect. It talks about our monthly meetings, our exhibits, our newsletter, our special programs like the Historic Boardwalk Tour, and our awesome website that has so much of our archives online.
Then, there are two short history lessons. One is about the beginning of Carolina Beach as a resort in the early1880s, the Winners, Capt. John Harper, the Steamers and Shoo Fly train, first Pavilion, etc. The other is how our Federal Point Peninsula became an island with the coming of the Intracoastal Waterway 1926-1932.
If you know of an organization that has monthly meetings or any group that looks for programs, please tell them about us. We would love to visit them!
Posted in FPHPS News • Society Notes
Society Notes – December, 2018
By Darlene Bright, History Center Director
Thanks to Steve Arthur for doing our Boardwalk tour for a group of local Island Women!
The History Center recorded 80 visitors in November. There were 36 people in attendance at the November meeting. The History Center was used by the Got-Em-On Live Bait Fishing Club, Step Up for Soldiers, and the United Daughters of the Confederacy (UDC).
And don't forget!! If you take a trip with Wilmington Water Tours, please tell them you are a member of the FPHPS! If you do, we get a portion of your ticket price. Call us at 910-458-0502 or them at 910-338-3134. wilmingtonwatertours.net
Welcome to new member, Roger Brix from Wilmington.
Thanks to John Gregory for taking great pictures of the walking tour for the Island Women.
Thanks to our volunteers in November; James Kohler, Andre Blouin, Steve Arthur, Darlene Bright and Leslie Bright. Refreshments were provided by Steve Arthur and Cheri McNeill. Nancy Gadzuk has graciously agreed to take minutes at the meetings until we find a Secretary.
Rebecca, Cheri and Darlene greatly appreciate Andre Blouin's help getting the November newsletter laid up and posted on our web site!!!
Thanks so much to Juanita Winner's grandson, Skylar Slaughter, who volunteered for us this month. He helped decorate the History Center for Christmas. Also, he washed the windows and help neaten everything up.
Congratulations to Judge Jay Hockenbury on his retirement from the bench after almost 24 years of distinguished service. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,898 |
Thapa and Lingden in the fray for RPP chairmanship
Published On: December 3, 2021 04:00 PM NPT By: Republica
KATHMANDU, Dec 3: The current Chairman Kamal Thapa and leader Rajendra Lingden have registered their candidacies for the chairmanship of Rastriya Prajatantra Party (RPP) in its Unity General Convention underway in the capital.
Chairman Thapa and leader Lingden reached the national assembly on Friday with their panels to register their candidacies. Speaking with media personnel after registering his candidature, Thapa claimed that he will emerge victorious with a majority of votes. He said that he will stand together with candidate Lingden among other leaders to move the party forward after winning the election. Stating that the rise of young leaders is a good sign for the party, he said that he had announced his candidacy for a second time to maintain the party's unity, strengthen the party based on the principles of nationalism, democracy, and liberalism.
Meanwhile, candidate Lingden claimed that he did not announce his nomination for any personal ambition or aspiration for the post. He said that steps would be taken under his leadership to raise the level of Rastriya Prajatantra Party and establish it as the leading party of the country.
Lingden said that the current change of leadership would be a starting point for the party cadres to carry out the task of uplifting the party.
Thapa and Lingden are running for the post of the party's chairman. Likewise, Lila Devi Shrestha and Rosan Karki have registered their names from the women quota for the post of the party's vice-chairperson while Jagat Prasad Gauchan, Ramchandra Raya Yadav, Bikram Pandey, Dhruv Bahadur Prashan, Budhiman Tamang, and Durga Devi Shrestha have also registered their candidacies for the post.
Similarly, while Rekha Thapa, Kunti Kumari Shahi have registered their candidacies for the post of General Secretary through the women's quota, Dhawal Sumsher Rana, Shyan Prasad Timalsina, Bhuwan Kumar Pathak, and Bhaskar Bhadra have registered their names from the men's quota.
Rastriya_Prajatantra_Party
General_Convention
Kamal_Thapa
Rajendra_Lingden
Voting begins to elect new working committee
KATHMANDU, Dec 4: The voting for the election of new 199-member working committee of Rastriya Prajatantra Party (RPP) has started...
LIVE: RPP Unity General Convention begins
KATHMANDU, Dec 1: The General Convention of the Rastriya Prajatantra Party (RPP) is being held in Kathmandu today. ...
Inaugural ceremony of RPP Unity General Convention today, Thapa and Lingden in the fray for chairmanship
KATHMANDU, Dec 1: The Unity General Convention of the Rastriya Prajatantra Party (RPP) is being held in Kathmandu on Wednesday.... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,355 |
\section{Introduction}
\subsection{} For finite groups there are well known local-global principles which enable
the study of their representation theory and cohomology via that of proper subgroups. For example if $G$ is a
finite group, $k$ is field of characteristic $p>0$, and $P$ is a $p$-Sylow subgroup of $G$, then
the restriction map induces an embedding $\text{res}:\text{Ext}^{\bullet}_{kG}(M,N)\hookrightarrow
\text{Ext}^{\bullet}_{kP}(M,N)$ where $kG$ (resp. $kP$) is the group algebra of $G$ (resp. $P$).
We therefore say that the cohomology is ``detected by'' the $p$-Sylow subgroups.
Another collection of subgroups which detect the cohomology is the set ${\mathcal E}$
of elementary abelian $p$-subgroups.
Here the restriction map
induces an inseparable isogeny ($F$-isomorphism):
$$\text{H}^{\bullet}(G,k)\rightarrow \lim_{E\in {\mathcal E}}\text{H}^{\bullet}(E,k).$$
Moreover, a cohomology class $\zeta\in \text{Ext}^{\bullet}_{kG}(M,M)$ is nilpotent if and only if
$\text{res}(\zeta)$ is nilpotent in $\text{Ext}^{\bullet}_{kE}(M,M)$ for every $E\in {\mathcal E}$.
Such cohomological ``detection theorems'' may be used to deduce properties
of support varieties of finite groups. Let $M$ be a finite-dimensional module for $kG$ and
write ${\mathcal V}_{kG}(M)$ (resp. ${\mathcal V}_{kP}(M)$) for its support variety over $kG$ (resp.
$kP$). The restriction map induces a morphism of algebraic varieties
$\text{res}^{*}:{\mathcal V}_{kP}(M)\rightarrow {\mathcal V}_{kG}(M)$
which is finite to one. Moreover,
$${\mathcal V}_{kG}(M)=\bigcup_{E\in {\mathcal E}}\text{res}^{*}({\mathcal V}_{kE}(M)).$$
Detectability on small subgroups is a rather special feature of modular group algebras. For
other finite-dimensional cocommutative Hopf algebras, like the restricted enveloping algebra of
a restricted Lie algebra, one can define cohomology and support varieties, but cohomology is rarely
detected on a {\em finite} set of smaller (proper) subalgebras.
\subsection{} We shall be concerned in this work with the
representation theory of finite-dimensional classical
Lie superalgebras ${\mathfrak g}={\mathfrak g}_{\bar 0}
\oplus {\mathfrak g}_{\bar 1}$ over the complex numbers, which has strong
analogies with the finite group case. Boe, Kujawa, and Nakano \cite{BKN1}
recently initiated the study of local-global principles in the setting of Lie superalgebras.
Using natural properties of the action of the reductive
group $G_{\bar 0}$ (where $\text{Lie }G_{\bar 0}=\ensuremath{\mathfrak{g}}_{\bar 0})$ on
${\mathfrak g}_{\bar 1}$, they proved the existence of two types of detecting subalgebras,
${\mathfrak f}={\mathfrak f}_{\bar 0} \oplus {\mathfrak f}_{\bar 1}$ and ${\mathfrak e}=
{\mathfrak e}_{\bar 0} \oplus {\mathfrak e}_{\bar 1}$. These subalgebras were used to study
representations in the category ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$ of
finite-dimensional $\ensuremath{\mathfrak{g}}$-modules which are semisimple over $\ensuremath{\mathfrak{g}}_{\bar 0}$.
The present work is a continuation of that program.
In this situation, the
restriction maps induce isomorphisms:
$$\text{H}^{\bullet}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},{\mathbb C})\cong
\text{H}^{\bullet}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0},{\mathbb C})^{N/N_{0}}\cong
\text{H}^{\bullet}(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0},{\mathbb C})^W$$
where $N/N_{0}$ is a reductive group and $W=W(\ensuremath{\mathfrak{e}})$ is a finite pseudoreflection group. These
relative cohomology rings may be identified with the invariant ring
$S^{\bullet}({\mathfrak g}_{\bar 1}^{*})^{G_{\bar 0}}$, where $S^{\bullet}$ denotes the symmetric
algebra, and so are
finitely generated. This property was used to construct support varieties for
$M$ in ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$. The restriction maps in cohomology induce embeddings
of support varieties:
\begin{equation}\label{supportemb}
{\mathcal V}_{(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0})}(M)/W\hookrightarrow
{\mathcal V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M)/(N/N_{0})\hookrightarrow
{\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M).
\end{equation}
It was suspected that these embeddings are in fact isomorphisms, but when these varieties were introduced
there was no reasonable theory of cohomological detection for arbitrary modules in
${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$.
An analogous theory has been developed for the Lie superalgebras $W(n)$ and $S(n)$ of Cartan type
in \cite{Ba, BaKN}. These Lie superalgebras are ${\mathbb Z}$-graded and detecting subalgebras
were constructed using the reductive group corresponding to the zero component of the Lie superalgebra.
\subsection{} The main goal of this paper is to develop the remarkable
theory of cohomological detection for arbitrary Type I classical Lie superalgebras and the
Lie superalgebras of Cartan type, $W(n)$ and $S(n)$. The class of classical Lie superalgebras under
consideration
includes the general linear Lie superalgebra ${\mathfrak g}=\mathfrak{gl}(m|n)$.
In Section 2, we review the fundamental definitions of classical Lie superalgebras and the
constructions of the detecting subalgebras $\ensuremath{\mathfrak{f}}$ and $\ensuremath{\mathfrak{e}}$. We also indicate how detecting subalgebras are
constructed for $W(n)$ and $S(n)$. In the following section, we
prove that for Type I classical Lie superalgebras, the restriction map
\begin{equation}\label{E:injectf}
\text{res}:\text{H}^{n}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},M)\hookrightarrow \text{H}^{n}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0},M)
\end{equation}
is injective for all $n\geq 0$ and $M$ in ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$.
The same arguments may be used to verify cohomological embedding results for
$W(n)$ and $S(n)$.
In Section 4, we show by example that (\ref{E:injectf}) does not hold when $\ensuremath{\mathfrak{f}}$ is
replaced by $\ensuremath{\mathfrak{e}}$. However, one can describe a relationship between support varieties
of $\ensuremath{\mathfrak{f}}$ and $\ensuremath{\mathfrak{e}}$ by using an auxiliary sub Lie superalgebra $\overline{\ensuremath{\mathfrak{f}}}$.
These cohomological embedding results are then applied to the theory of
support varieties, and used to prove that the embeddings given in (\ref{supportemb})
are indeed isomorphisms of varieties. One consequence of this result is the concrete
realization of the support variety ${\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M)$ as a quotient of
a rank variety over $\ensuremath{\mathfrak{e}}_{\bar 1}$ by the finite (pseudo) reflection group $W=W(\ensuremath{\mathfrak{e}})$
(cf. Theorem~\ref{T:supportprop}(a)).
Our results indicate the importance of the subalgebras $\ensuremath{\mathfrak{f}}$ and $\ensuremath{\mathfrak{e}}$ for the
theory of classical Lie superalgebras.
Finally, in Section 5, we apply our results to show that these support varieties can be viewed as support
data as defined by Balmer \cite{Bal}. We also indicate how the support theory fits
into the classical combinatorial
notion of atypicality as defined by Kac, Wakimoto and Serganova.
This paper was completed while the second author was visiting the University of Sydney during
Spring 2010. The second author would like to express his gratitude for the hospitality and
support of the University of Sydney and the University of New South Wales during
his residence in Australia.
\section{Detecting subalgebras}
\subsection{Notation: } We will use and summarize the conventions developed in
\cite{BKN1, BKN2, BKN3}. For more details we refer the reader to \cite[Section 2]{BKN1}.
Throughout this paper, let ${\mathfrak g}$ be a Lie superalgebra over the complex numbers ${\mathbb C}$.
In particular, ${\mathfrak g}={\mathfrak g}_{\bar 0}\oplus {\mathfrak g}_{\bar 1}$ is a ${\mathbb Z}_{2}$-graded vector space
with a supercommutator $[\;,\;]:\ensuremath{\mathfrak{g}} \otimes \ensuremath{\mathfrak{g}} \to \ensuremath{\mathfrak{g}}$. A finite dimensional Lie superalgebra
${\mathfrak g}$ is called \emph{classical} if there is a connected reductive algebraic group $G_{\bar 0}$ such
that $\operatorname{Lie}(G_{\bar 0})=\ensuremath{\mathfrak{g}}_{\bar 0},$ and the action of $G_{\bar 0}$ on $\ensuremath{\mathfrak{g}}_{\bar 1}$ differentiates to
the adjoint action of $\ensuremath{\mathfrak{g}}_{\bar 0}$ on $\ensuremath{\mathfrak{g}}_{\bar 1}.$ We say that ${\mathfrak g}$ is a \emph{basic classical}
Lie superalgebra if it is a classical Lie superalgebra with a nondegenerate invariant supersymmetric
even bilinear form.
Let $U(\ensuremath{\mathfrak{g}})$ be the universal enveloping superalgebra of ${\mathfrak g}$. We will be interested in
supermodules which are ${\mathbb Z}_{2}$-graded left $U(\ensuremath{\mathfrak{g}})$-modules. If $M$ and $N$ are ${\mathfrak g}$-supermodules
one can use the antipode and coproduct of $U({\mathfrak g})$ to define a ${\mathfrak g}$-supermodule
structure on the dual $M^{*}$ and the tensor product $M\otimes N$. For the remainder of the
paper the term ${\mathfrak g}$-module will mean a ${\mathfrak g}$-supermodule. In order to
apply homological algebra techniques, we will restrict ourselves to the
\emph{underlying even category}, consisting of ${\mathfrak g}$-modules with the degree preserving morphisms.
In this paper we will study homological properties of the category
${\mathcal F}_{({\mathfrak g},{\mathfrak g}_{\bar 0})}$ which is the full subcategory of
finite dimensional ${\mathfrak g}$-modules which are finitely semisimple over ${\mathfrak g}_{\bar 0}$ (a
${\mathfrak g}_{\bar 0}$-module is \emph{finitely semisimple} if it decomposes into a direct sum of
finite dimensional simple ${\mathfrak g}_{\bar 0}$-modules).
The category ${\mathcal F}:={\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$ has enough injective
(and projective) modules. In fact, ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$
is a Frobenius category (i.e., where injectivity is equivalent to projectivity) \cite{BKN3}.
Given $M, N$ in ${\mathcal F}$, let $\operatorname{Ext}_{\mathcal{F}}^{d}(M,N)$ be the degree $d$
extensions between $N$ and $M$. In practice, there is a concrete realization for these
extension groups via the relative Lie superalgebra cohomology for the pair
$({\mathfrak g},{\mathfrak g}_{\bar 0})$:
$$\text{Ext}_{\mathcal{F}}^{d}(M,N)\cong \text{H}^{d}({\mathfrak g},{\mathfrak g}_{\bar 0}; M^{*}\otimes N).$$
The relative cohomology can be computed using an explicit complex (cf. \cite[Section 2.3]{BKN1}).
Moreover, the cohomology ring
$$
R:=\text{H}^{\bullet}({\mathfrak g},{\mathfrak g}_{\bar 0}; {\mathbb C})=
S^{\bullet}({\mathfrak g}_{\bar 1}^*)^{G_{\bar 0}}.
$$
Since $G_{\bar 0}$ is reductive it follows that $R$ is finitely generated.
\subsection{Classical Lie superalgebras: } In this section we review the construction of the two
families of (cohomological) detecting subalgebras for classical Lie superalgebras
defined in \cite[Section 3,4]{BKN1}
using the invariant theory of $G_{\bar 0}$ on ${\mathfrak g}_{\bar 1}$.
First we consider the case when $G_{\bar 0}$ has a {\em stable} action on ${\mathfrak g}_{\bar 1}$
(cf. \cite[Section 3.2]{BKN1}). That is, there is an open dense subset of $\ensuremath{\mathfrak{g}}_{\bar 1}$ consisting of
semisimple points. Recall that a point $x\in {\mathfrak g}_{\bar 1}$ is called
{\em semisimple} if the orbit $G_{\bar 0}\cdot x$ is closed in ${\mathfrak g}_{\bar 1}$.
Let $x_{0}$ be a generic point in ${\mathfrak g}_{\bar 1}$; that is, $x_0$ is semisimple and
regular, in the sense that its stablizer has minimal dimension.
Let $H=\text{Stab}_{G_{\bar 0}}x_0$ and $N:=N_{G_{\bar 0}}(H)$.
In order to construct a detecting subalgebra, we let
${\mathfrak f}_{\bar 1}={\mathfrak g}_{\bar 1}^{H}$, ${\mathfrak f}_{\bar 0}=\text{Lie }N$, and set
$${\mathfrak f}={\mathfrak f}_{\bar 0}\oplus {\mathfrak f}_{\bar 1}.$$
Then ${\mathfrak f}$ is a classical Lie superalgebra and a sub Lie superalgebra
of ${\mathfrak g}$. The stability of the action of $G_{\bar 0}$ on ${\mathfrak g}_{\bar 1}$
implies the following properties.
\begin{itemize}
\item[(2.2.1)] The restriction homomorphism $S(\ensuremath{\mathfrak{g}}_{\bar 1}^{*}) \to S(\ensuremath{\mathfrak{f}}_{\bar 1}^{*})$ induces an isomorphism
\[
\operatorname{res}: \text{H}^{\bullet}({\mathfrak g},{\mathfrak g}_{\bar 0},{\mathbb C}) \rightarrow
\text{H}^{\bullet}({\mathfrak f},{\mathfrak f}_{\bar 0},{\mathbb C})^{N/N_{0}}.
\]
Here $N_{0}$ is the connected component of the identity in $N$.
\item[(2.2.2)] The set $G_{\bar 0}\cdot {\mathfrak f}_{\bar 1}$ is dense in $\ensuremath{\mathfrak{g}}_{\bar 1}.$
\end{itemize}
Next we recall the notion of {\em polar action} introduced by Dadok and Kac \cite{dadokkac}.
Let $v \in {\mathfrak g}_{\bar 1}$ be a semisimple element, and set
\begin{equation*}
{\mathfrak e}_{v}=\left\{x \in \ensuremath{\mathfrak{g}}_{\bar 1} \:\vert\: \ensuremath{\mathfrak{g}}_{\bar 0}.x \subseteq \ensuremath{\mathfrak{g}}_{\bar 0}.v \right\},
\end{equation*}
where $\ensuremath{\mathfrak{g}}_{\bar 0}$ is the Lie algebra of $G_{\bar 0}$. The action of $G_{\bar 0}$ on ${\mathfrak g}_{\bar 1}$
is called {\em polar} if for some semisimple element $v \in {\mathfrak g}_{\bar 1}$ we have
$\operatorname{dim} {\mathfrak e}_{v} = \operatorname{dim} S({\mathfrak g}_{\bar 1}^{*})^{G_{\bar 0}},$
where $\operatorname{dim} S({\mathfrak g}_{\bar 1}^{*})^{G_{\bar 0}}$ is the Krull dimension of this ring.
The vector space ${\mathfrak e}_{v}$ is called a \emph{Cartan subspace}; let
${\mathfrak e}_{\bar 1}$ denote a fixed choice of a Cartan subspace.
If the action of $G_{\bar 0}$ on $\ensuremath{\mathfrak{g}}_{\bar 1}$ is both stable and polar, we
can further assume
\begin{equation*}\label{E:containments}
x_{0} \in \ensuremath{\mathfrak{e}}_{x_{0}}=\ensuremath{\mathfrak{e}}_{\bar 1} \subseteq {\mathfrak f}_{\bar 1},
\end{equation*}
where $x_{0}$ and ${\mathfrak f}_{\bar 1}$ are as above. Furthermore,
the Cartan subspace is unique up to conjugation by $G_{\bar 0}$ (cf. \cite[Theorem 2.3]{dadokkac}).
Set $\ensuremath{\mathfrak{e}}_{\bar 0}=\text{Lie}(H)$. Then the detecting subalgebra ${\mathfrak e}$ is the classical
Lie sub-superalgebra of ${\mathfrak g}$ defined by:
$${\mathfrak e}={\mathfrak e}_{\bar 0}\oplus {\mathfrak e}_{\bar 1}.$$
Assume $\ensuremath{\mathfrak{g}}$ is a classical Lie superalgebra where the action of $G_{\bar 0}$ is both
stable and polar on ${\mathfrak g}_{\bar 1}$. Then by \cite[Theorem 3.3.1]{BKN1} we have
the following two facts.
\begin{itemize}
\item[(2.2.3)] The restriction homomorphism $S(\ensuremath{\mathfrak{g}}_{\bar 1}^{*}) \to S(\ensuremath{\mathfrak{e}}_{\bar 1}^{*})$ induces an isomorphism
$$
\operatorname{res}: \text{H}^{\bullet}({\mathfrak g},{\mathfrak g}_{\bar 0},{\mathbb C}) \rightarrow
\text{H}^{\bullet}({\mathfrak e},{\mathfrak e}_{\bar 0},{\mathbb C})^{W},
$$
where $W=W({\mathfrak e})$ is a finite pseudoreflection group. In particular, $R$ is a polynomial algebra.
\vskip .15cm
\item[(2.2.4)] The set $G_{\bar 0}.\ensuremath{\mathfrak{e}}_{\bar 1}$ is dense in $\ensuremath{\mathfrak{g}}_{\bar 1}$.
\end{itemize}
For the Lie superalgebras ${\mathfrak g}=W(n)$ and $S(n)$ detecting families (analogous
to the $\ensuremath{\mathfrak{f}}$'s) were also
constructed using stable actions. We will describe a basis for these subalgebras below.
\subsection{ Type I Lie superalgebras:} A Lie superalgebra is said to be of {\em Type I} if
it admits a ${\mathbb Z}$-grading ${\mathfrak g}=\ensuremath{\mathfrak{g}}_{-1}\oplus {\ensuremath{\mathfrak{g}}}_{0}\oplus {\ensuremath{\mathfrak{g}}}_{1}$ concentrated
in degrees $-1,$ $0,$ and $1$ with ${\mathfrak g}_{\bar 0}={\mathfrak g}_{0}$ and
${\mathfrak g}_{\bar 1}=\ensuremath{\mathfrak{g}}_{-1}\oplus {\ensuremath{\mathfrak{g}}}_{1}$. Otherwise, ${\mathfrak g}$ is of Type II.
Examples of Type I Lie superalgebras include: $\mathfrak{gl}(m|n)$ and simple Lie superalgebras of
types $A(m,n)$, $C(n)$ and $P(n)$.
The simple modules for ${\mathfrak g}$, a Type I classical Lie superalgebra, can be constructed in the
following way. Let ${\mathfrak t}$ be a Cartan subalgebra of ${\mathfrak g}_{\bar 0}$ and
$X^{+}_{0} \subseteq {\mathfrak t}^{*}$ be the set of dominant integral weights (with
respect to a fixed Borel subalgebra). For $\lambda \in X^{+}_{0},$ let $L_{0}(\lambda)$
be the simple finite dimensional ${\mathfrak g}_{\bar 0}$-module of highest weight $\lambda$. Set
$$
{\mathfrak g}^{+}= {\mathfrak g}_{0} \oplus {\mathfrak g}_{1}
\qquad \text{and} \qquad {\mathfrak g}^{-}= {\mathfrak g}_{0} \oplus {\mathfrak g}_{-1} .
$$
Since ${\mathfrak g}$ is a Type I Lie superalgebra ${\mathfrak g}_{\pm 1}$ is an abelian
ideal of ${\mathfrak g}^{\pm}$. We can therefore
view $L_{0}(\lambda)$ as a simple ${\mathfrak g}^{\pm}$-module via inflation.
For each $\lambda\in X^{+}_{0}$, we construct the \emph{Kac module} and the \emph{dual Kac module}
by using the tensor product and the Hom-space in the following way:
$$
K(\lambda)=U({\mathfrak g})\otimes_{U({\mathfrak g}^{+})} L_{0}(\lambda)
\qquad \text{and} \qquad K^{-}(\lambda) =
\operatorname{Hom}_{U({\mathfrak g}^{-})}\left(U({\mathfrak g}), L_{0}(\lambda) \right).
$$
The module $K(\lambda)$ has a unique maximal submodule. The head of $K(\lambda)$ is the simple
finite dimensional ${\mathfrak g}$-module $L(\lambda)$. Then $\{L(\lambda):\ \lambda \in X_{0}^{+}\}$
is a complete set of non-isomorphic simple modules in $\mathcal{F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$. Let $P(\lambda)$
(resp.\ $I(\lambda)$) denote the projective cover
(resp.\ injective hull) in $\mathcal{F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$ for the
simple ${\mathfrak g}$-module $L(\lambda)$. These are all finite-dimensional. Moreover, the projective
covers admit filtrations with sections being Kac modules and the injective hulls
have filtrations whose sections are dual Kac modules. These filtrations also respect the
ordering on weights and thus ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$ is a highest
weight category (cf. \cite[Section 3]{BKN3}) as defined in
\cite{CPS}.
\subsection{General Linear Superalgebra:} The prototypical example of a Type I classical Lie superalgebra
admitting both a stable and polar action of $G_{\bar 0}$ on $\ensuremath{\mathfrak{g}}_{\bar 1}$ is $\ensuremath{\mathfrak{g}}=\mathfrak{gl}(m|n)$,
which as a vector space is the set of $m+n$ by $m+n$ matrices. As
basis one may take the matrix units $E_{i,j}$ where $1\leq i,j \leq m+n$. The degree zero component
${\mathfrak g}_{\bar 0}$ is the span of $E_{i,j}$ where $1\leq i,j \leq m$ or $m+1 \leq i,j \leq m+n$.
As a Lie algebra ${\mathfrak g}_{\bar 0}\cong \mathfrak{gl}(m)\times \mathfrak{gl}(n)$, and
the corresponding reductive group is $G_{\bar 0}=GL(m)\times GL(n)$. Note that $G_{\bar 0}$ acts on
$\mathfrak g_{\bar 1}$ via the adjoint representation. A basis for ${\mathfrak g}_{\bar 1}$ is
given by the $E_{i,j}$ such that $m+1\leq i \leq m+n$ and $1\leq j \leq n$ or
$1\leq i \leq m$ and $m+1 \leq j \leq m+n$.
Observe that ${\mathfrak g}_{\bar 1}={\mathfrak g}_{-1}\oplus {\mathfrak g}_{1}$ where ${\mathfrak g}_{-1}$
(resp. ${\mathfrak g}_{1}$) consists of the lower triangular matrices (resp. upper triangular matrices)
in ${\mathfrak g}_{\bar 1}$. The action of $G_{\bar 0}$ on $\ensuremath{\mathfrak{g}}_{-1}$ is given by $(A,B).X=BXA^{-1}$ so
the orbits are the matrices of a given rank in $\ensuremath{\mathfrak{g}}_{-1}$. By results from
elementary linear algebra, $G_{\bar 0}\cdot \ensuremath{\mathfrak{f}}_{\bar 1}$ and $G_{\bar 0}\cdot \ensuremath{\mathfrak{e}}_{\bar 0}$ are dense in $\ensuremath{\mathfrak{g}}_{\bar 1}$.
For simplicity of exposition, let us assume that
$m=n=r$. With an appropriate choice of $x_{0}$ the detecting
subalgebras have the following descriptions. The detecting subalgebra $\ensuremath{\mathfrak{f}}=\ensuremath{\mathfrak{f}}_{\bar 0}\oplus \ensuremath{\mathfrak{f}}_{\bar 1}$ where
$\ensuremath{\mathfrak{f}}_{\bar 1}$ is the span of $\{E_{i,i+r}:\ i=1,2,\dots,r\}\cup \{E_{i+r,i}:\ i=1,2,\dots,r\}$
and $\ensuremath{\mathfrak{f}}_{\bar 0}$ is the span of $[\ensuremath{\mathfrak{f}}_{1},\ensuremath{\mathfrak{f}}_{\bar 1}]$. Here $H\cong T^{r}$ where $T$ is a one-dimensional
torus, and $N\cong W\ltimes T^{r}$ where $W=\Sigma_{r}\ltimes ({\mathbb Z}_{2})^{r}$ (hyperoctahedral
group). The detecting subalgebra $\ensuremath{\mathfrak{e}}=\ensuremath{\mathfrak{e}}_{\bar 0}\oplus e_{\bar 1}$ where $\ensuremath{\mathfrak{e}}_{\bar 1}$ is the span of
$\{E_{i,i+r}+E_{i+r,i}:\ i=1,2,\dots,r\}$ and $\ensuremath{\mathfrak{e}}_{\bar 0}$ is the span of $[\ensuremath{\mathfrak{e}}_{1},\ensuremath{\mathfrak{e}}_{\bar 1}]$.
Constructions of detecting subalgebras for other classical Lie superalgebras are explicitly described in
\cite[Section 8]{BKN1}.
\subsection{The Witt algebra $W(n)$ and $S(n)$:} Let $n\geq 2$, and
$\Lambda^{\bullet}(V)$ be the exterior algebra of the vector space $V={\mathbb C}^n$. The Lie superalgebra
$W(n)$ is the set of all superderivations of $\Lambda^{\bullet}(V)$,
and the superalgebra structure is provided via the supercommutator bracket.
The Lie superalgebra $W(n)$ inherits a ${\mathbb Z}$-grading,
$${\mathfrak g}:=W(n)=W(n)_{-1}\oplus W(n)_0\oplus \dotsb \oplus W(n)_{n-1},$$
from $\Lambda^{\bullet}(V)$ by letting ${\mathfrak g}_{k}:=W(n)_k$ be the
superderivations which increase the degree of
a homogeneous element by $k$. Furthermore, the ${\mathbb Z}_2$-grading on
$W(n)$ is obtained from the ${\mathbb Z}$-grading by taking
$W(n)_{\bar 0}= \oplus_{k} W(n)_{2k}$ and $W(n)_{\bar 1}= \oplus_{k} W(n)_{2k+1}$.
One can give an explicit basis for $W(n)$ in the following way. We have
$$W(n)\cong \Lambda(n)\otimes V^\ast.$$
We can fix an ordered basis $\{\xi _1, \dotsc , \xi _n\}$ for $V$. For each ordered subset $I=\{i_1, \dotsc , i_s\}$ of
$N=\{1, \dotsc , n\}$ with $i_1<i_2< \dotsb <i_s$, let $\xi _I=\xi _{i_1}\xi _{i_2}\dotsb \xi _{i_s}$.
The set of all such $\xi_I$ forms a basis for $\Lambda(n)$. If $1\leq i\leq n$ let $\partial _i$ be the
element of $W(n)$ given by partial differentiation: i.e., $\partial _i(\xi_j)=\delta _{ij}$. Write
$\xi_I\partial _i$ for $\xi_I\otimes \partial _i$. Then an
explicit basis for $W(n)$ is given by $\xi_{I}\partial_{i}$ where $I$
runs over all ordered subsets of $N$ and
$i=1,2,\dots,n$. Observe that $W(n)_{k}$ is spanned by the basis
elements $\xi_I\partial _i$ with $|I| =k+1$.
In particular, $\ensuremath{\mathfrak{g}}_{0}$ is spanned by the elements $\xi_i\partial_j$, from which one
easily sees that $\ensuremath{\mathfrak{g}}_0=\mathfrak{gl}(n)$.
Let ${\mathcal C}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}$ be the category of $\ensuremath{\mathfrak{g}}$-supermodules
which are completely reducible over $\ensuremath{\mathfrak{g}}_{0}$.
In \cite[Section 5.5]{BaKN}, a detecting subalgebra was constructed as follows. There
exists a generic point $x_{0}$ in ${\mathfrak g}_{-1}\oplus {\mathfrak g}_{1}$
for the action of $G_{0}=GL(n)$.
Set $H=\text{Stab}_{G_{0}}\ x_{0}=T_{n-1}$, an $(n-1)$-dimensional torus.
The normalizer $N=N_{G_{0}}(H)$ is
$\Sigma_{n-1}\ltimes T$ where $\Sigma_{n-1}$ is the symmetric group of degree
$n-1$ and $T$ is the torus of $G_{0}$ consisting
of diagonal matrices. Set $\ensuremath{\mathfrak{f}}_{\bar 1}=(\ensuremath{\mathfrak{g}}_{-1}\oplus \ensuremath{\mathfrak{g}}_{1})^{H}$. Then $\ensuremath{\mathfrak{f}}_{1}$ is the span of the vectors
$\{\partial_{1},\xi_{1}\xi_{i}\partial_{i}:\ i=2,3,\dots,n\}$.
Set $\ensuremath{\mathfrak{f}}_{\bar 0}=\text{Lie }(N)$. Then $\ensuremath{\mathfrak{f}}=\ensuremath{\mathfrak{f}}_{\bar 0}\oplus \ensuremath{\mathfrak{f}}_{\bar 1}$
detects cohomology (as in the classical stable case), that is the restriction map:
$$
\text{res}:\text{H}^{\bullet}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0},{\mathbb C})\rightarrow
\text{H}^{\bullet}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0},{\mathbb C})^{N}
$$
is an isomorphism.
In \cite{BaKN} a support variety theory was developed for finite dimensional
modules in ${\mathcal C}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}$ (and
${\mathcal C}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}$), it was shown that for such a module $M$ there exists an injection of
algebraic varieties:
\begin{equation} \label{Cartaninjection}
{\mathcal V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M)/\Sigma_{n-1}\hookrightarrow {\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}(M).
\end{equation}
In Theorem~\ref{T:isosupports2} we will prove that this is indeed an isomorphism of varieties.
A similar development of cohomology and support varieties has been investigated by
Bagci for the Lie superalgebra
of Cartan type $S(n)$. Detecting subalgebras analogous to $\ensuremath{\mathfrak{f}}$ have been constructed.
Details are omitted here and
left to the interested reader (see \cite{Ba}).
\section{Cohomological Embedding}
\subsection{The Stable Case: } The goal in this section is to prove that the
relative $({\mathfrak g},{\mathfrak g}_{\bar 0})$
cohomology for $M\in {\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$ embeds in the relative cohomology for
$({\mathfrak f},{\mathfrak f}_{\bar 0})$. Our first result uses a dimension shifting argument to reduce to
looking at cohomology in degree one.
\begin{prop}\label{P:reducecoho1} Let ${\mathfrak g}$ be a classical Lie
superalgebra which is stable. Suppose that
$\operatorname{res}:\operatorname{H}^{1}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},M)\rightarrow \operatorname{H}^{1}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0},M)$
is an injective map for every $M\in {\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$. Then
$\operatorname{res}:\operatorname{H}^{n}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},M)\rightarrow \operatorname{H}^{n}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0},M)$
is an injective map for all $n\geq 0$ and $M\in {\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$.
\end{prop}
\begin{proof} For $n=0$ the statement of the theorem
is clear because $\operatorname{H}^{0}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},M)=M^{\mathfrak g}$ (fixed
points under ${\mathfrak g}$) and $\operatorname{H}^{0}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0},M)=M^{\mathfrak f}$. By assumption the
result holds for $n=1$. Now assume by induction the result holds for $n<t$, and consider the
short exact sequence
$$0\rightarrow M \rightarrow I \rightarrow \Omega^{-1}(M)\rightarrow 0$$
where $I$ is the injective hull of $M$ in ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$. Note that
$I$ is also injective in ${\mathcal F}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}$.
Now applying the long exact sequence to the short exact sequence above and using these facts, we
have the following commutative diagram:
\begin{equation}
\CD
\text{H}^{t-1}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},\Omega^{-1}(M)) @>\text{res}>> \text{H}^{t-1}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0},\Omega^{-1}(M))\\
@VVV @VVV\\
\text{H}^{t}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},M) @>>\text{res}> \text{H}^{t}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0},M)
\endCD
\end{equation}
where the vertical maps are isomorphisms. The top horizontal map is injective by induction.
Therefore, the bottom $\text{res}$ map is also injective.
\end{proof}
\subsection{} Let ${\mathfrak g}={\mathfrak g}_{-1}\oplus {\mathfrak g}_{0}\oplus {\mathfrak g}_{1}$
be a Type I Lie superalgebra. The detecting subalgebra ${\mathfrak f}$ has a triangular decomposition
which is compatible with the ${\mathbb Z}$-grading: ${\mathfrak f}={\mathfrak f}_{-1}\oplus
{\mathfrak f}_{0}\oplus {\mathfrak f}_{1}$ with $\ensuremath{\mathfrak{f}}_{\pm 1}=\ensuremath{\mathfrak{g}}_{\pm 1}^{H}$. Note that
${\mathfrak f}^{\pm}={\mathfrak f}_{0}\oplus {\mathfrak f}_{\pm 1}$ and
${\mathfrak f}^{\pm}={\mathfrak f}\cap {\mathfrak g}^{\pm}$.
In order to analyze the question of embedding of cohomology of Type I Lie superalgebras we first
investigate the case $({\mathfrak g}^{\pm},{\mathfrak g}_{\bar 0})$ and
$({\mathfrak f}^{\pm},{\mathfrak f}_{\bar 0})$.
\begin{thm}\label{T:embedplusminus} Let ${\mathfrak g}$ be a classical Type I Lie superalgebra which is
stable. Then for all $M$ in ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$ and
$n\neq 0$, the restriction map
$$\operatorname{H}^{n}({\mathfrak g}^{\pm},{\mathfrak g}_{\bar{0}},M)\rightarrow
\operatorname{H}^{n}({\mathfrak f}^{\pm},{\mathfrak f}_{\bar{0}},M)$$
is injective.
\end{thm}
\begin{proof} Without loss of generality we can consider the case
${\mathfrak g}^{+}={\mathfrak g}_{0}\oplus
{\mathfrak g}_{1}$. Since $\ensuremath{\mathfrak{g}}_{1}$ is an ideal of $\ensuremath{\mathfrak{g}}^{+},$ one has the
Lyndon-Hochschild-Serre (LHS) spectral sequence
for the pair $(\ensuremath{\mathfrak{g}}_{1}, \{0 \})$ in $(\ensuremath{\mathfrak{g}}^{+}, \ensuremath{\mathfrak{g}}_{0})$ (cf.\ proof of \cite[Theorem 3.3.1]{BKN3}):
$$
\widetilde{E}_{2}^{i,j}=\text{H}^{i}(\ensuremath{\mathfrak{g}}_{0},\ensuremath{\mathfrak{g}}_{0},\text{H}^{j}(\ensuremath{\mathfrak{g}}_{1},M))\Rightarrow
\text{H}^{i+j}(\ensuremath{\mathfrak{g}}^{+},\ensuremath{\mathfrak{g}}_{0},M).
$$
This spectral sequence collapses because modules are completely reducible over ${\mathfrak g}_{0}$ and
yields:
\begin{equation}\label{E:giso}
\text{H}^{n}(\ensuremath{\mathfrak{g}}^{+},\ensuremath{\mathfrak{g}}_{0},M)\cong \text{H}^{n}(\ensuremath{\mathfrak{g}}_{1},M)^{\ensuremath{\mathfrak{g}}_{0}}
\end{equation}
for all $n\geq 0$. Similarly, for $n\geq 0$
\begin{equation}\label{E:fiso}
\text{H}^{n}(\ensuremath{\mathfrak{f}}^{+},\ensuremath{\mathfrak{f}}_{0},M)\cong \text{H}^{n}(\ensuremath{\mathfrak{f}}_{1},M)^{\ensuremath{\mathfrak{f}}_{0}}.
\end{equation}
Next observe that $\ensuremath{\mathfrak{f}}_{1}$ is an abelian Lie superideal in $\ensuremath{\mathfrak{g}}_{1}$. Conseqently, we have
another LHS spectral sequence:
$$
E_{2}^{i,j}=\text{H}^{i}(\ensuremath{\mathfrak{g}}_{1}/\ensuremath{\mathfrak{f}}_{1},\text{H}^{j}(\ensuremath{\mathfrak{f}}_{1},M))\Rightarrow
\text{H}^{i+j}(\ensuremath{\mathfrak{g}}_{1},M).
$$
This gives rise to an exact sequence (i.e., the first three terms of the standard
five term exact sequence):
\begin{equation}\label{E:5exactseq}
0\rightarrow \text{H}^{1}(\ensuremath{\mathfrak{g}}_{1}/{\ensuremath{\mathfrak{f}}_{1}},M^{\ensuremath{\mathfrak{f}}_{1}}) \rightarrow \text{H}^{1}(\ensuremath{\mathfrak{g}}_{1},M)
\rightarrow \text{H}^{1}(\ensuremath{\mathfrak{f}}_{1},M)^{\ensuremath{\mathfrak{g}}_{1}/\ensuremath{\mathfrak{f}}_{1}}\rightarrow \dots
\end{equation}
Under the restriction map we have
$$\text{res}:\text{H}^{1}(\ensuremath{\mathfrak{g}}_{1},M)^{\ensuremath{\mathfrak{g}}_{0}}\rightarrow \text{H}^{1}(\ensuremath{\mathfrak{f}}_{1},M)^{\ensuremath{\mathfrak{f}}_{0}}.$$
In order to prove the theorem it suffices by Proposition~\ref{P:reducecoho1}, (~\ref{E:giso}), and
(~\ref{E:fiso}) to demonstrate that the restriction map above is injective.
The sequence (~\ref{E:5exactseq}) arises by looking at the following exact sequence
at the cochain level:
$$0\rightarrow (\ensuremath{\mathfrak{g}}_{1}/\ensuremath{\mathfrak{f}}_{1})^{*}\otimes M \xrightarrow{\alpha}
(\ensuremath{\mathfrak{g}}_{1})^{*}\otimes M \xrightarrow{\beta} (\ensuremath{\mathfrak{f}}_{1})^{*}\otimes M \rightarrow 0.$$
Consider $\text{Hom}_{G_{0}}(\ensuremath{\mathfrak{g}}_{1},M)\cong [(\ensuremath{\mathfrak{g}}_{1})^{*}\otimes M]^{G_{0}}$ as a subspace of
$(\ensuremath{\mathfrak{g}}_{1})^{*}\otimes M$. The restriction of the original map
$\beta$ to this subspace $\beta:\text{Hom}_{G_{0}}(\ensuremath{\mathfrak{g}}_{1},M)\rightarrow
[(\ensuremath{\mathfrak{f}}_{1})^{*}\otimes M]^{\ensuremath{\mathfrak{f}}_0}$ is given by $\beta(\psi)=\psi\mid_{\ensuremath{\mathfrak{f}}_{1}}$. Now $\beta$ is
an injective map. This can be seen as follows. If $\psi\mid_{\ensuremath{\mathfrak{f}}_{1}}=0$ then $\psi(\ensuremath{\mathfrak{f}}_{1})=0$.
The fact that
$\psi$ is $G_{0}$-invariant shows that $\psi(G_{0}\cdot\ensuremath{\mathfrak{f}}_{1})$. Finally using the
density of $G_{0}\cdot \ensuremath{\mathfrak{f}}_{1}$ in $\ensuremath{\mathfrak{g}}_{1}$ implies that $\psi=0$.
This means that $\text{Im }\alpha\cap [(\ensuremath{\mathfrak{g}}_{1})^{*}\otimes M]^{G_{0}}=0$. The first
map in (~\ref{E:5exactseq}) is induced by $\alpha$, thus restricting
the second map to $\text{H}^{1}(\ensuremath{\mathfrak{g}}_{1},M)^{G_{0}}$ yields an embedding
$\text{H}^{1}(\ensuremath{\mathfrak{g}}_{1},M)^{G_{0}}\hookrightarrow \text{H}^{1}(\ensuremath{\mathfrak{f}}_{1},M)^{\ensuremath{\mathfrak{g}}_{1}/\ensuremath{\mathfrak{f}}_{1}}$.
Note that the representative cocycles in $\text{H}^{1}(\ensuremath{\mathfrak{g}}_{1},M)^{G_{0}}$ can be chosen to
be in $[(\ensuremath{\mathfrak{g}}_{1})^{*}\otimes M]^{G_{0}}$ because the fixed point functor $(-)^{G_{0}}$ is exact.
\end{proof}
\subsection{} We now combine information from both sides of the triangular decomposition of
${\mathfrak g}$ and ${\mathfrak f}$ to prove that the relative cohomology for $(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})$
embeds in the relative cohomology for $(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})$.
\begin{thm}\label{T:stablecase} Let ${\mathfrak g}$ be a classical Type I Lie superalgebra which
is stable. Then for all $M\in {\mathcal F}_{({\mathfrak g},{\mathfrak g}_{\bar{0}})}$ and
$n\neq 0$ the restriction map
$$\operatorname{H}^{n}({\mathfrak g},{\mathfrak g}_{\bar{0}},M)\rightarrow
\operatorname{H}^{n}({\mathfrak f},{\mathfrak f}_{\bar{0}},M)$$
is injective.
\end{thm}
\begin{proof} For an explicit definition of relative Lie superalgebra cohomology via a
cochain complex we refer the reader to
\cite[Section 2.3]{BKN1}. By Proposition~\ref{P:reducecoho1} it suffices to verify the case $n=1$.
Consider cochain differentials whose images are respectively used to define
$\text{H}^{1}(\ensuremath{\mathfrak{f}}^{\pm},\ensuremath{\mathfrak{f}}_{\bar 0},M)$,
$\text{H}^{1}(\ensuremath{\mathfrak{g}}^{\pm},\ensuremath{\mathfrak{g}}_{\bar 0},M)$, $\text{H}^{1}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0},M)$, and $\text{H}^{1}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},M)$:
\begin{equation*}
d_{\ensuremath{\mathfrak{f}}^{\pm}}:M^{\ensuremath{\mathfrak{f}}_{\bar 0}}\rightarrow [(\ensuremath{\mathfrak{f}}_{\pm 1})^{*}\otimes M]^{\ensuremath{\mathfrak{f}}_{\bar 0}},
\end{equation*}
\begin{equation*}
d_{\ensuremath{\mathfrak{g}}^{\pm}}:M^{\ensuremath{\mathfrak{g}}_{\bar 0}}\rightarrow [(\ensuremath{\mathfrak{g}}_{\pm 1})^{*}\otimes M]^{\ensuremath{\mathfrak{g}}_{\bar 0}},
\end{equation*}
\begin{equation*}
d_{\ensuremath{\mathfrak{f}}}:M^{\ensuremath{\mathfrak{f}}_{\bar 0}}\rightarrow [(\ensuremath{\mathfrak{f}}_{\bar 1})^{*}\otimes M]^{\ensuremath{\mathfrak{f}}_{\bar 0}},
\end{equation*}
\begin{equation*}
d_{\ensuremath{\mathfrak{g}}}:M^{\ensuremath{\mathfrak{g}}_{\bar 0}}\rightarrow [(\ensuremath{\mathfrak{g}}_{\bar 1})^{*}\otimes M]^{\ensuremath{\mathfrak{g}}_{\bar 0}}.
\end{equation*}
In Theorem~\ref{T:embedplusminus}, we proved that the restriction map
embeds $\text{H}^{1}(\ensuremath{\mathfrak{g}}^{\pm},\ensuremath{\mathfrak{g}}_{\bar 0},M)$ into $\text{H}^{1}(\ensuremath{\mathfrak{f}}^{\pm},\ensuremath{\mathfrak{f}}_{\bar 0},M)$.
At the cochain level, we have the following commutative diagram:
\begin{equation}
\CD
M^{\ensuremath{\mathfrak{g}}_{\bar 0}} @>>> M^{\ensuremath{\mathfrak{f}}_{\bar 0}}\\
@Vd_{\ensuremath{\mathfrak{g}}^{\pm}}VV @Vd_{\ensuremath{\mathfrak{f}}^{\pm}}VV\\
[(\ensuremath{\mathfrak{g}}_{\pm 1})^{*}\otimes M]^{\ensuremath{\mathfrak{g}}_{\bar 0}} @>>\sigma_{\pm}> [(\ensuremath{\mathfrak{f}}_{\pm})^{*}\otimes M]^{\ensuremath{\mathfrak{f}}_{\bar 0}}
\endCD
\end{equation}
where $\sigma_{\pm}$ is the map obtained by restriction of functions.
The embedding of $\text{H}^{1}(\ensuremath{\mathfrak{g}}^{\pm},\ensuremath{\mathfrak{g}}_{\bar 0},M)$ into $\text{H}^{1}(\ensuremath{\mathfrak{f}}^{\pm},\ensuremath{\mathfrak{f}}_{\bar 0},M)$
implies that
\begin{equation}
\sigma_{\pm}^{-1}(\text{Im }d_{\ensuremath{\mathfrak{f}}^{\pm}})=\text{Im }d_{\ensuremath{\mathfrak{g}}^{\pm}}
\end{equation}
or
\begin{equation}
\text{Im }d_{\ensuremath{\mathfrak{f}}^{\pm}}=\sigma_{\pm}(\text{Im }d_{\ensuremath{\mathfrak{g}}^{\pm}}).
\end{equation}
Next note that $\text{Im }d_{\ensuremath{\mathfrak{f}}^{\pm}}\cong M^{\ensuremath{\mathfrak{f}}_{\bar 0}}/\text{Ker }d_{\ensuremath{\mathfrak{f}}^{\pm}}$ and
$\text{Im }d_{\ensuremath{\mathfrak{g}}^{\pm}}\cong M^{\ensuremath{\mathfrak{g}}_{\bar 0}}/\text{Ker }d_{\ensuremath{\mathfrak{g}}^{\pm}}$. Since $G_{\bar 0}\cdot {\mathfrak f}_{\bar 1}$ is
dense in ${\mathfrak g}_{\bar 1}$ the map obtained by restriction of functions
$$\sigma:[(\ensuremath{\mathfrak{g}}_{\bar 1})^{*}\otimes M]^{\ensuremath{\mathfrak{g}}_{\bar 0}} \hookrightarrow [(\ensuremath{\mathfrak{f}}_{\bar 1})^{*}\otimes M]^{\ensuremath{\mathfrak{f}}_{\bar 0}}$$
is injective. In order to prove that the induced map in cohomology
from $\text{H}^{1}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},M)\rightarrow
\text{H}^{1}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0},M)$ is injective, it suffices (using reasoning similar to that above) to
prove that $\sigma(\text{Im }d_{\ensuremath{\mathfrak{g}}})=\text{Im }d_{\ensuremath{\mathfrak{f}}}$.
From the definition of the differential note that
\begin{equation}
K_{\ensuremath{\mathfrak{f}}}:=\text{Ker }d_{\ensuremath{\mathfrak{f}}}=\text{Ker }d_{\ensuremath{\mathfrak{f}}^{-}}\cap \text{Ker }d_{\ensuremath{\mathfrak{f}}^{+}},
\end{equation}
\begin{equation}
K_{\ensuremath{\mathfrak{g}}}:=\text{Ker }d_{\ensuremath{\mathfrak{g}}}=\text{Ker }d_{\ensuremath{\mathfrak{g}}^{-}}\cap \text{Ker }d_{\ensuremath{\mathfrak{g}}^{+}}
\end{equation}
because $\text{Ker }d_{ \ensuremath{\mathfrak{f}}^\pm}=M^{f^\pm}$ and $\text{Ker }d_{\ensuremath{\mathfrak{g}}^\pm}=M^{ g^\pm}$.
We have the following commutative diagram
\begin{equation}
\CD
0 @>>> \text{Ker }d_{\ensuremath{\mathfrak{g}}^{+}}/K_{\ensuremath{\mathfrak{g}}} @>>> M^{\ensuremath{\mathfrak{g}}_{\bar 0}}/K_{\ensuremath{\mathfrak{g}}} @>>> M^{\ensuremath{\mathfrak{g}}_{\bar 0}}/\text{Ker }d_{\ensuremath{\mathfrak{g}}^{+}} @>>> 0 \\
@. @V\text{$\sigma$}VV @V\text{$\sigma$}VV @V\text{${\sigma}_{+}$}VV @. \\
0 @>>> \text{Ker }d_{\ensuremath{\mathfrak{f}}^{+}}/K_{\ensuremath{\mathfrak{f}}} @>>> M^{\ensuremath{\mathfrak{f}}_{\bar 0}}/K_{\ensuremath{\mathfrak{f}}} @>>> M^{\ensuremath{\mathfrak{f}}_{\bar 0}}/\text{Ker }d_{\ensuremath{\mathfrak{f}}^{+}} @>>> 0
\endCD
\end{equation}
From our analysis above the map $\sigma_{+}$ is an isomorphism. Therefore, in order to prove the
theorem it suffices to show that
$$\sigma: \text{Ker }d_{\ensuremath{\mathfrak{g}}^{+}}/K_{\ensuremath{\mathfrak{g}}}\rightarrow \text{Ker }d_{\ensuremath{\mathfrak{f}}^{+}}/K_{\ensuremath{\mathfrak{f}}}$$
is an isomorphism because the five lemma would imply that the map in the middle is an isomorphism.
This would show that $\sigma(\text{Im }d_{\ensuremath{\mathfrak{g}}})=\text{Im }d_{\ensuremath{\mathfrak{f}}}$.
By the second isomorphism theorem we have the following isomorphisms:
\begin{equation}
\text{Ker }d_{\ensuremath{\mathfrak{f}}^{+}}/K_{\ensuremath{\mathfrak{f}}}\cong (\text{Ker }d_{\ensuremath{\mathfrak{f}}^{+}}+\text{Ker }d_{\ensuremath{\mathfrak{f}}^{-}})/\text{Ker d}_{\ensuremath{\mathfrak{f}}^{-}},
\end{equation}
\begin{equation}
\text{Ker }d_{\ensuremath{\mathfrak{g}}^{+}}/K_{\ensuremath{\mathfrak{g}}}\cong (\text{Ker }d_{\ensuremath{\mathfrak{g}}^{+}}+\text{Ker }d_{\ensuremath{\mathfrak{g}}^{-}})/\text{Ker d}_{\ensuremath{\mathfrak{g}}^{-}}.
\end{equation}
Now observe that from the relationship between
$(\ensuremath{\mathfrak{g}}^{-},\ensuremath{\mathfrak{g}}_{\bar 0})$ and $(\ensuremath{\mathfrak{f}}^{-},\ensuremath{\mathfrak{f}}_{\bar 0})$, we have a commutative diagram
\begin{equation}
\CD
(\text{Ker }d_{\ensuremath{\mathfrak{g}}^{+}}+\text{Ker }d_{\ensuremath{\mathfrak{g}}^{-}})/\text{Ker }d_{\ensuremath{\mathfrak{g}}^{-}} @>>> M^{\ensuremath{\mathfrak{g}}_{\bar 0}}/\text{Ker }d_{\ensuremath{\mathfrak{g}}^{-}}\\
@V\text{$\sigma_{-}$}VV @V\text{$\sigma_{-}$}VV \\
(\text{Ker }d_{\ensuremath{\mathfrak{f}}^{+}}+\text{Ker }d_{\ensuremath{\mathfrak{f}}^{-}})/\text{Ker }d_{\ensuremath{\mathfrak{f}}^{-}} @>>> M^{\ensuremath{\mathfrak{f}}_{\bar 0}}/\text{Ker }d_{\ensuremath{\mathfrak{f}}^{-}}
\endCD
\end{equation}
The horizontal maps are embeddings and the rightmost vertical map is an isomorphism
(using the fact that $\text{Im }d_{\ensuremath{\mathfrak{f}}^{-}}=\sigma_{-}(\text{Im }d_{\ensuremath{\mathfrak{g}}^{-}})$). Therefore, the map
$$\sigma_{-}:(\text{Ker }d_{\ensuremath{\mathfrak{g}}^{+}}+\text{Ker }d_{\ensuremath{\mathfrak{g}}^{-}})/\text{Ker }d_{\ensuremath{\mathfrak{g}}^{-}}\hookrightarrow
(\text{Ker }d_{\ensuremath{\mathfrak{f}}^{+}}+\text{Ker }d_{\ensuremath{\mathfrak{f}}^{-}})/\text{Ker }d_{\ensuremath{\mathfrak{f}}^{-}}$$
is injective. In order to finish the proof we need to show that $\sigma_{-}$ is surjective.
Suppose that
$y+\text{Ker }d_{\ensuremath{\mathfrak{f}}^{-}}\in (\text{Ker }d_{\ensuremath{\mathfrak{f}}^{+}}+\text{Ker }d_{\ensuremath{\mathfrak{f}}^{-}})/\text{Ker d}_{\ensuremath{\mathfrak{f}}^{-}}$ with
$y=y_{-}+y_{+}$ where $y_{\pm}\in \text{Ker }d_{\ensuremath{\mathfrak{f}}^{\pm}}$. From the isomorphism given by $\sigma_{-}$ above
we have $g(y_{\pm}+\text{Ker }d_{\ensuremath{\mathfrak{f}}_{-}})=y_{\pm}+\text{Ker }d_{\ensuremath{\mathfrak{f}}^{-}}$ for all $g\in G_{\bar 0}$.
Moreover, $d_{\ensuremath{\mathfrak{f}}^{\pm}}(y_{\pm})=0$ so in particular $\ensuremath{\mathfrak{f}}_{1}.y_{+}=0$ and $\ensuremath{\mathfrak{f}}_{-1}.y_{-}=0$. Now
$0=g.(f.y_{\pm})=(g.f).(g^{-1}.y_{\pm})=(g.f).y_{\pm}$ for all $g\in G_{\bar 0}$ and $f\in \ensuremath{\mathfrak{f}}_{\pm 1}$.
Since $G_{\bar 0}\cdot \ensuremath{\mathfrak{f}}_{\pm 1}$ is dense in $\ensuremath{\mathfrak{g}}_{\pm 1}$ it follows that $x.y_{\pm}=0$ for all
$x\in \ensuremath{\mathfrak{g}}_{\pm 1}$, thus $y_{\pm}\in \text{Ker }d_{\ensuremath{\mathfrak{g}}^{\pm}}$.
\end{proof}
\subsection{} Let ${\mathfrak h}$ be a classical Lie subsuperalgebra of ${\mathfrak g}$ with the
property that
\begin{equation}\label{E:propertyinj}
\text{res}:\text{H}^{n}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},M)\hookrightarrow \text{H}^{n}(\ensuremath{\mathfrak{h}},\ensuremath{\mathfrak{h}}_{\bar 0},M)
\end{equation}
is an injective map for all $n\geq 0$ and $M$ in ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$.
Let $M$ be a module in ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$ which is projective when
considered as a module in ${\mathcal F}_{({\mathfrak h},{\mathfrak h}_{\bar 0})}$. By
(~\ref{E:propertyinj}), we have
\begin{equation*}
\text{res}:\text{H}^{1}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},M\otimes S^*)\hookrightarrow \text{H}^{1}(\ensuremath{\mathfrak{h}},\ensuremath{\mathfrak{h}}_{\bar 0},M\otimes S^*)
\end{equation*}
for all simple modules $S$ in ${\mathcal F}_{({\mathfrak g},{\mathfrak g}_{\bar 0})}$. By the
projectivity of $M$ in ${\mathcal F}_{({\mathfrak h},{\mathfrak h}_{\bar 0})}$, we further have
$\text{H}^{1}(\ensuremath{\mathfrak{h}},\ensuremath{\mathfrak{h}}_{\bar 0},M\otimes S^*)=0$. Consequently,
$$0=\text{H}^{1}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},M\otimes S^*)\cong \text{Ext}^{1}_{{\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}}(S,M)$$
for all simple modules $S$ in ${\mathcal F}_{({\mathfrak g},{\mathfrak g}_{\bar 0})}$, whence
$M$ is projective in ${\mathcal F}_{({\mathfrak g},{\mathfrak g}_{\bar 0})}$. For Type I classical
Lie superalgebras when ${\mathfrak h}={\mathfrak f}$ this fact about projectivity
was earlier deduced using geometric methods involving support varieties
(cf. \cite[Theorem 3.5.1, Theorem 3.7.1]{BKN3}). Theorem~\ref{T:stablecase} can be viewed as
a strong generalization of this projectivity result between modules in ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$ and
${\mathcal F}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}$
\subsection{The Polar Case: } In this section we will consider the example
when ${\mathfrak g}=\mathfrak{gl}(1|1)$ and
demonstrate that the analogue Theorem~\ref{T:stablecase}
does not hold when the detecting subalgebra $\ensuremath{\mathfrak{f}}$ is
replaced by $\ensuremath{\mathfrak{e}}$. The simple modules in the principal block
of ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$ are
labelled by $L(\lambda)$ where $\lambda$ denotes the highest weight
$(\lambda|-\lambda)$. Consider the two-dimensional
dual Kac module $K^{-}(\lambda)$. The module $K^{-}(\lambda)$ remains indecomposable when restricted to
$\ensuremath{\mathfrak{e}}$, and the only 2-dimensional indecomposable $\ensuremath{\mathfrak{e}}$-modules are the projective indecomposable
modules in the category ${\mathcal F}_{(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0})}$. Consequently, $K^{-}(\lambda)$ is a projective
module in ${\mathcal F}_{(\ensuremath{\mathfrak{e}},e_{\bar 0})}$, thus $\text{H}^{n}(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0},K^{-}(\lambda))=0$ for $n>0$.
It is well known that the Kac modules in the principal block are not
projective in ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$,
thus by the argument in Section 3.4, the property (~\ref{E:propertyinj})
cannot hold for ${\mathfrak h}=
{\mathfrak e}$.
We also see that the restriction map is not injective by direct computation. By Frobenius reciprocity
\begin{equation}
\text{H}^{n}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},K^{-}(\lambda))\cong \text{H}^{n}(\ensuremath{\mathfrak{g}}^{-},\ensuremath{\mathfrak{g}}_{\bar 0},\lambda)
\cong [\text{H}^{n}(\ensuremath{\mathfrak{g}}_{-1},{\mathbb C})\otimes \lambda]^{G_{\bar 0}}\cong
[S^{n}((\ensuremath{\mathfrak{g}}_{-1})^*)\otimes \lambda]^{G_{\bar 0}}.
\end{equation}
In this instance $G_{\bar 0}$ a torus (specifically, the set of invertible diagonal matrices).
The subspace $\ensuremath{\mathfrak{g}}_{-1}$ is one-dimensional
and is spanned by a weight vector having weight $(-1|1)$. Consequently,
\begin{equation}
\text{H}^{n}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},K^{-}(\lambda))\cong \begin{cases} {\mathbb C} & \text{$\lambda=-n$}\\
0 & \text{otherwise}.
\end{cases}
\end{equation}
In particular, we have $\text{H}^{1}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},K^{-}(-1|1))\cong {\mathbb C}$, and
$\text{H}^{1}(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0},K^{-}(-1,1))=0$. Therefore,
$$\text{res}:\text{H}^{1}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},K^{-}(-1|1))\rightarrow \text{H}^{1}(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0},K^{-}(-1|1))$$
is not an injective map. However,
$$\text{res}:\text{H}^{n}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},K^{-}(-1|1))\rightarrow \text{H}^{n}(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0},K^{-}(-1|1))$$
is an injective map for $n\geq 2$.
It was shown in \cite{BKN1} that when $\dim \ensuremath{\mathfrak{e}}_{\bar 1}=1$ the
restriction map is injective for $n$ sufficiently large. An interesting problem would be
to determine whether this occurs for arbitrary $\ensuremath{\mathfrak{e}}$.
\subsection{Types $W(n)$ and $S(n)$: } The techniques used to prove Theorem~\ref{T:embedplusminus}
and ~\ref{T:stablecase} can
be used with the triangular decomposition given by the ${\mathbb Z}$-grading for $W(n)$ and $S(n)$ to
prove the following detection theorem.
Let $\ensuremath{\mathfrak{f}}$ be as in Section 2.5.
\begin{thm}\label{T:WSembedplusminus} Let ${\mathfrak g}=W(n)$ or $S(n)$.
Then for all $M$ in ${\mathcal C}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}$ and
$n\neq 0$. The restriction map
$$\operatorname{H}^{n}({\mathfrak g}^{\pm},{\mathfrak g}_{\bar{0}},M)\rightarrow
\operatorname{H}^{n}({\mathfrak f}^{\pm},{\mathfrak f}_{\bar{0}},M).$$
is injective.
\end{thm}
\section{Support Varieties}
\subsection{} We first recall the definition of the support variety
of a finite dimensional $\ensuremath{\mathfrak{g}}$-supermodule $M$ (cf. \cite[Section 6.1]{BKN1}).
Let $\mathfrak{g}$ be a classical Lie superalgebra,
$R:=\operatorname{H}^{\bullet}(\mathfrak{g}, \mathfrak{g}_{\bar 0};{\mathbb C})$, and $M_{1}$, $M_{2}$
be in ${\mathcal F}:={\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$. According to \cite[Theorem 2.5.3]{BKN1},
$\operatorname{Ext}_{\mathcal{F}}^{\bullet}(M_{1},M_{2})$ is a finitely generated $R$-module.
Set $J_{({\mathfrak g},{\mathfrak g}_{\bar 0})}(M_{1},M_{2})=
\operatorname{Ann}_{R}(\operatorname{Ext}_{\mathcal{F}}^{\bullet}(M_{1},M_{2}))$
(i.e., the annihilator ideal of this module). The \emph{relative support variety of the pair $(M,N)$} is
\begin{equation}
\mathcal{V}_{(\mathfrak{g},\mathfrak{g}_{\bar 0})}(M,N)=
\operatorname{MaxSpec}(R/J_{({\mathfrak g},{\mathfrak g}_{\bar 0})}(M,N))
\end{equation}
In the case when $M=M_{1}=M_{2}$, set $J_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M)=J_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M,M)$, and
$$\mathcal{V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M):=\mathcal{V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M,M).$$
The variety $\mathcal{V}_{(\mathfrak{g},\mathfrak{g}_{\bar 0})}(M)$ is called the \emph{support variety} of $M$.
In this situation, $J_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M)=\text{Ann}_{R} \ \text{Id}$ where $\text{Id}$ is the identity
morphism in $\text{Ext}^{0}_{\mathcal F}(M,M)$.
\subsection{} We will now compare support varieties for the classical Lie superalgebras ${\mathfrak g}$,
${\mathfrak f}$, and ${\mathfrak e}$. Assume that $\ensuremath{\mathfrak{g}}$ is both stable and polar. Without
the assumption that $\ensuremath{\mathfrak{g}}$ is polar, the statements concerning cohomology and
support varieties for ${\mathfrak g}$ and ${\mathfrak f}$ remain true.
First there are natural maps of rings given by restriction:
$$
\text{res}:\operatorname{H}^{\bullet}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0}; {\mathbb C}) \rightarrow \operatorname{H}^{\bullet}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0}; {\mathbb C})
\rightarrow \operatorname{H}^{\bullet}(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0},{\mathbb C}).
$$
which induce isomorphisms
$$
\text{res}:\operatorname{H}^{\bullet}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0}; {\mathbb C}) \rightarrow
\operatorname{H}^{\bullet}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0}; {\mathbb C})^{N/N_{0}}
\rightarrow \operatorname{H}^{\bullet}(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0},{\mathbb C})^{W}.
$$
The map on cohomology above induces morphisms of varieties:
$$
\text{res}^{*}:\mathcal{V}_{(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0})}({\mathbb C})\rightarrow
\mathcal{V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}({\mathbb C})\rightarrow \mathcal{V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}({\mathbb C})
$$
and isomorphisms (by passing to quotient spaces)
$$
\text{res}^{*}:\mathcal{V}_{(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0})}({\mathbb C})/W\rightarrow
\mathcal{V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}({\mathbb C})/(N/N_{0})\rightarrow \mathcal{V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}({\mathbb C}).
$$
Let $M$ be a finite dimensional $\ensuremath{\mathfrak{g}}$-module. Then $\ensuremath{\operatorname{res}}^{*}$ induces maps
between support varieties:
$$\mathcal{V}_{(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0})}(M) \rightarrow \mathcal{V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M) \rightarrow
\mathcal{V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M).
$$
Since $M$ is ${\mathfrak g}_{\bar 0}$-module, the first two varieties are stable under the action of
$W$ and $N/N^{0},$ respectively. Consequently, we obtain the following induced maps of
varieties:
$$\mathcal{V}_{(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0})}(M)/W \hookrightarrow \mathcal{V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M)/(N/N_{0})
\hookrightarrow \mathcal{V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M).
$$
These maps are embeddings because if $x\in R$ annihilates the identity in
$\text{H}^{0}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},M^{*}\otimes M)$ then
it must annihilate the identity elements in $\text{H}^{0}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0},M^{*}\otimes M)$ and
$\text{H}^{0}(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0},M^{*}\otimes M)$.
\subsection{The Intermediate Subalgebra $\overline{\ensuremath{\mathfrak{f}}}$: } We next define an
intermediate subalgebra between $\ensuremath{\mathfrak{e}}$ and $\ensuremath{\mathfrak{f}}$ which will be useful for our purposes. Let
$\overline{\ensuremath{\mathfrak{f}}}$ be defined as follows. Given $\ensuremath{\mathfrak{f}}$, let $\overline{\ensuremath{\mathfrak{f}}}_{\bar 1}:=\ensuremath{\mathfrak{f}}_{\bar 1}$ and
$\overline{\ensuremath{\mathfrak{f}}}_{\bar 0}=\text{Lie}(H)$. Set $\overline{\ensuremath{\mathfrak{f}}}=\overline{\ensuremath{\mathfrak{f}}}_{\bar 0}\oplus {\ensuremath{\mathfrak{f}}}_{\bar 1}$.
From the proof of \cite[Theorem 4.1]{BKN1}, we know that $\overline{\ensuremath{\mathfrak{f}}}$ is a Lie subsuperalgebra of
$\ensuremath{\mathfrak{f}}$ and contains $\ensuremath{\mathfrak{e}}$ (in the case that we have a polar action). Moreover,
$[\overline{\ensuremath{\mathfrak{f}}}_{\bar 0},\overline{\ensuremath{\mathfrak{f}}}_{\bar 1}]=0$. This implies that we have a rank variety description for
${\mathcal V}_{(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}(M)$ when
$M\in {\mathcal F}_{(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}$.
The \emph{rank variety} of $M$ is
$$
{\mathcal V}_{\overline{\ensuremath{\mathfrak{f}}}}^{\text{rank}}(M)=\left\{ x \in\overline{\ensuremath{\mathfrak{f}}}_{\bar 1}\:\vert\: M
\text{ is not projective as $U(\langle x \rangle)$-module} \right\} \cup \{0 \}.
$$
It was shown in \cite[Theorem 6.3.2]{BKN1} that there is an isomorphism
$${\mathcal V}_{(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}(M)\cong {\mathcal V}_{\overline{\ensuremath{\mathfrak{f}}}}^{\text{rank}}(M).$$
\subsection{Comparing support varieties over $\ensuremath{\mathfrak{f}}$ and $\overline{\ensuremath{\mathfrak{f}}}$: } In this section we
compare support varieties for modules over $\overline{\ensuremath{\mathfrak{f}}}$ and $\ensuremath{\mathfrak{f}}$. Let
$M \in \mathcal{F}_{({\ensuremath{\mathfrak{f}}},{\ensuremath{\mathfrak{f}}}_{\bar 0})}$. There is a map of varieties
induced by the restriction map in cohomology:
\begin{equation}
\text{res}^{*}:{\mathcal V} _{(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}(M) \rightarrow
{\mathcal V} _{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M).
\end{equation}
Using the fact that
$$\text{H}^{\bullet}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0},{\mathbb C})\cong \text{H}^{\bullet}(\overline{\ensuremath{\mathfrak{f}}},
\overline{\ensuremath{\mathfrak{f}}}_{\bar 0},{\mathbb C})^{N_{0}/H},
$$
it is clear that the above map is the restriction of the orbit map:
\begin{equation}\label{E:trivialresequality}
{\mathcal V} _{(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}({\mathbb C})
\longrightarrow{\mathcal V} _{(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}({\mathbb C})/(N_{0}/H)
\overset{\sim}{\longrightarrow}
{\mathcal V}_{({\ensuremath{\mathfrak{f}}},{\ensuremath{\mathfrak{f}}}_{\bar 0})}({\mathbb C}).
\end{equation}
The following theorem demonstrates that support varieties in this context
are natural with respect to taking quotients.
\begin{thm}\label{T:supportquotient}
Let $M$ be a finite dimensional object in $\mathcal{F}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}$. Then
\begin{equation}\label{E:fres}
{\mathcal V}_{(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}(M)/(N_{0}/H)\cong \operatorname{res}^{*}
({\mathcal V}_{(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}(M))={\mathcal V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M).
\end{equation}
\end{thm}
\begin{proof} The proof, which we include
for the convenience of the reader, will follow the same lines as \cite[Theorem 6.4.1]{BaKN}.
Observe that the first isomorphism holds by \eqref{E:trivialresequality}. To show that the
second isomorphism holds we need to prove that $\text{res}^{*}$ is a surjective map.
The group $N_{0}/H$ is reductive, and therefore has completely reducible module category.
Let $X_+$ be a parametrizing set for the finite-dimensional
simple $N_{0}/H$-modules, and for $\lambda\in X_+$, let $S_{\lambda}$ be the corresponding simple module.
Let $Q$ be a module in $\mathcal{F}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}$. Then $N_{0}/H$ acts on the cohomology
$\text{H}^{\bullet}(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0},Q)$, and by complete reducibility,
\begin{eqnarray*}
\text{H}^{\bullet}(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0},Q) &\cong &
\text{H}^{\bullet}(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0},Q)^{N_{0}/H} \oplus
\bigoplus_{\lambda \in X_{+}:\ \lambda \neq 0} \text{Hom}_{N_{0}/H}(S_{\lambda},
\text{H}^{\bullet}(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0},Q))\otimes S_{\lambda} \\
&\cong & \text{H}^{\bullet}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0}; Q) \oplus
\bigoplus_{\lambda \in X_{+}:\ \lambda \neq 0} \text{Hom}_{N_{0}/H}(S_{\lambda},
\text{H}^{\bullet}(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0},Q))\otimes S_{\lambda}.
\end{eqnarray*}
From this isomorphism, one sees that for all $Q$ in $\mathcal{F}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}$
\begin{equation}\label{E:embedgeneralcohom}
\operatorname{res} : \text{H}^{\bullet}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0}; Q) \xrightarrow{\cong}
\text{H}^{\bullet}(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0}; Q)^{N_{0}/H} \subseteq
\text{H}^{\bullet}(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0}; Q).
\end{equation}
Let $\text{id}_{\ensuremath{\mathfrak{f}},M}$ (resp. $\text{id}_{\overline{\ensuremath{\mathfrak{f}}},M}$) denote the identity element in
$\text{H}^{\bullet}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0}, M^*\otimes M)$ (resp. $\text{H}^{\bullet}(\overline{\ensuremath{\mathfrak{f}}},
\overline{\ensuremath{\mathfrak{f}}}_{\bar 0}, M^*\otimes M)$). The ideal
$\operatorname{res}^{-1}( J_{(\overline{\ensuremath{\mathfrak{f}}}, \overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}(M))$ defines the
variety $\text{res}^{*}({\mathcal V}_{(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}(M))$.
We need to prove that
\begin{equation}
\operatorname{res}^{-1}(J_{(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}(M))=J_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M).
\end{equation}
If $x \in J_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M)$, then
\begin{equation*}
0=\operatorname{res}(x.\text{id}_{\ensuremath{\mathfrak{f}}, M})=
\operatorname{res}(x).\operatorname{res}(\text{id}_{\ensuremath{\mathfrak{f}} ,M})=
\operatorname{res}(x).\text{id}_{\overline{\ensuremath{\mathfrak{f}}},M}.
\end{equation*}
Therefore, $\operatorname{res}(x) \in J_{(\overline{\ensuremath{\mathfrak{f}} },\overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}(M)$, and so
$x \in \operatorname{res}^{-1}(J_{(\overline{\ensuremath{\mathfrak{f}} },\overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}(M)).$
Conversely, if $x \in \operatorname{res}^{-1}(J_{(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}(M))$, then
\begin{equation*}
0=\operatorname{res}(x).\text{id}_{\overline{\ensuremath{\mathfrak{f}}},M}=\operatorname{res}(x).
\operatorname{res}(\text{id}_{\ensuremath{\mathfrak{f}},M})
=\operatorname{res}(x.\text{id}_{\ensuremath{\mathfrak{f}},M}).
\end{equation*}
Since the restriction map is injective, $0=x.\text{id}_{\ensuremath{\mathfrak{f}} ,M}$ and so $x \in J_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M)$.
\end{proof}
Using the identification of ${\mathcal V}_{(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}(M)$ as a rank variety,
Theorem~\ref{T:supportquotient} provides a concrete realization of
${\mathcal V}_{({\ensuremath{\mathfrak{f}}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M)$. Since the tensor product theorem holds for rank varieties
it follows that it also holds for support varieties in this context. The proof
of \cite[Theorem 6.5.1]{BaKN} adapted to our setting yields the following result.
\begin{cor}\label{T:fvarietyprops}Let $M$, $N$ be modules in
${\mathcal F}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}$. Then
\begin{itemize}
\item [(a)] ${\mathcal V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M)\cong
{\mathcal V}^{\operatorname{rank}}_{\overline{\ensuremath{\mathfrak{f}}}}(M)/(N_{0}/H)$,
\item[(b)] ${\mathcal V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M\otimes N)={\mathcal V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0} )}(M)\cap
{\mathcal V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0} )}(N)$.
\end{itemize}
\end{cor}
\subsection{Comparing support varieties over $\ensuremath{\mathfrak{f}}$ and $\ensuremath{\mathfrak{e}}$: } In this section we compare the
varieties for the detecting subalgebras $\ensuremath{\mathfrak{f}}$ and $\ensuremath{\mathfrak{e}}$. In general $\ensuremath{\mathfrak{f}}$ does not have a simple
rank variety description. This necessitates the use of the auxiliary algebra $\overline{\ensuremath{\mathfrak{f}}}$
to make the transition between $\ensuremath{\mathfrak{f}}$ and $\ensuremath{\mathfrak{e}}$.
\begin{thm}\label{T:isosupportsfe} Let $\ensuremath{\mathfrak{g}}$ be a classical Lie superalgebra which is
stable and polar. If $M\in {\mathcal F}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}$ then
we have the following isomorphism of varieties:
$$\operatorname{res}^{*}:{\mathcal V}_{(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0})}(M)/W\rightarrow
{\mathcal V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M)/(N/N_{0}).$$
\end{thm}
\begin{proof} Let $M\in {\mathcal F}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}$. We have the following commutative diagram of varieties:
\begin{equation*}
\CD
{\mathcal V}_{(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0})}(M) @>\text{res}^{*}>> {\mathcal V}_{(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{\bar 0})}(M)
@>\text{res}^{*}>> {\mathcal V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M) \\
@VVV @VVV @VVV \\
{\mathcal V}^{\text{rank}}_{\ensuremath{\mathfrak{e}}}(M) @>>> {\mathcal V}^{\text{rank}}_{\overline{\ensuremath{\mathfrak{f}}}}(M)
@>{\beta} >> {\mathcal V}^{\text{rank}}_{\overline{\ensuremath{\mathfrak{f}}}}(M)/(N_{0}/H)\\
@VVV @VVV @VVV \\
{\mathcal V}^{\text{rank}}_{\ensuremath{\mathfrak{e}}}({\mathbb C}) @>\alpha>> {\mathcal V}^{\text{rank}}_{\overline{\ensuremath{\mathfrak{f}}}}({\mathbb C})
@>{\beta} >> {\mathcal V}^{\text{rank}}_{\overline{\ensuremath{\mathfrak{f}}}}({\mathbb C})/(N_{0}/H)
\endCD
\end{equation*}
It suffices to show that the composition of the top (horizontal) maps
$$\operatorname{res}^{*}:{\mathcal V}_{(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0})}(M)\rightarrow {\mathcal V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M)$$
is surjective.
Let $\sigma$ denote the middle (horizontal) composition of maps from
${\mathcal V}^{\text{rank}}_{\ensuremath{\mathfrak{e}}}(M)$ to ${\mathcal V}^{\text{rank}}_{\overline{\ensuremath{\mathfrak{f}}}}(M)/(N_{0}/H)$.
Since the first row of vertical maps are all isomorphisms,
it suffices to prove that $\sigma$ is surjective.
Take $y\in {\mathcal V}^{\text{rank}}_{\overline{\ensuremath{\mathfrak{f}}}}(M)/(N_{0}/H)$.
There exists $x\in {\mathcal V}^{\text{rank}}_{\overline{\ensuremath{\mathfrak{f}}}}(M) $ with $\beta(x)=y$,
and by naturality of rank varieties, we may consider
$x$ to be an element of ${\mathcal V}^{\text{rank}}_{\overline\ensuremath{\mathfrak{f}}}({\mathbb C})$.
Since the composition $\beta\circ\alpha$ of the two lowest horizontal arrows is an isomorphism, there is
an element $z\in {\mathcal V}^{\text{rank}}_{\ensuremath{\mathfrak{e}}}({\mathbb C})$ such that $\beta(x)=\beta(z)$,
where, since $\alpha$ is an inclusion, we identify $z$ as an element of
${\mathcal V}^{\text{rank}}_{\overline\ensuremath{\mathfrak{f}}}({\mathbb C})$. But then $z=gx$ for some $g\in N_0/H$,
and it follows that
$z\in {\mathcal V}^{\text{rank}}_{\ensuremath{\mathfrak{e}}}(M)$, and clearly $\sigma(z)=y$.
\end{proof}
\subsection{} In the case of ${\mathfrak g}=W(n)$ or $S(n)$ there is an auxilliary subalgebra
$\widetilde{\ensuremath{\mathfrak{f}}}$ which is analogous to $\overline{\ensuremath{\mathfrak{f}}}$. In this setting we have the following result
(cf. \cite[Theorem 6.4.1, (6.3.4)]{BaKN}).
\begin{thm}\label{T:supportquotient2} Let ${\mathfrak g}=W(n)$ or $S(n)$ and
let $M$ be a finite dimensional object in $\mathcal{C}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{0})}$. Then there exists a
torus $T$ such that
\begin{itemize}
\item[(a)] ${\mathcal V}_{(\widetilde{\ensuremath{\mathfrak{f}}},\widetilde{\ensuremath{\mathfrak{f}}}_{0})}(M)/T\cong \operatorname{res}^{*}
({\mathcal V}_{(\overline{\ensuremath{\mathfrak{f}}},\overline{\ensuremath{\mathfrak{f}}}_{0})}(M))={\mathcal V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{0})}(M)$.
\item[(b)] ${\mathcal V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{0})}(M)/T\cong {\mathcal V}^{\operatorname{rank}}_{\widetilde{\ensuremath{\mathfrak{f}}}}(M)$.
\end{itemize}
\end{thm}
\section{Applications}
\subsection{Isomorphism Theorems} In \cite{BKN1,BKN2} there was convincing theoretical and
computational evidence of direct relationships between
the support varieties for a classical Lie superalgebra ${\mathfrak g}$
and those of its detecting subalgebras. The cohomological embedding
theorem provided in Section 3 enables us to provide such a relation, which we now proceed
to do.
\begin{thm}\label{T:isosupports} Let $\ensuremath{\mathfrak{g}}$ be a classical Type I Lie superalgebra
with ${\mathfrak f}$ etc. as above, and let $M$ be in the module category ${\mathcal F_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}}$.
\begin{itemize}
\item[(a)] If $\ensuremath{\mathfrak{g}}$ is stable then the map on support varieties
$$\operatorname{res}^{*}:{\mathcal V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M)/(N/N_{0})
\rightarrow {\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M)$$ is an isomorphism.
\item[(b)] If $\ensuremath{\mathfrak{g}}$ is stable and polar then the maps on support varieties
$$\operatorname{res}^{*}:
{\mathcal V}_{(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0})}(M)/W({\ensuremath{\mathfrak{e}}})\rightarrow {\mathcal V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M)/(N/N_{0})
\rightarrow {\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M)$$
are isomorphisms, where $W=W(\ensuremath{\mathfrak{e}})$ is the pseudoreflection group of (2.2.3).
\end{itemize}
\end{thm}
\begin{proof} (a) We have seen that $\text{res}^{*}$ is an embedding. Therefore,
it suffices to show that this map is surjective. Observe that by Theorem~\ref{T:stablecase},
the restriction map
$$\text{H}^{\bullet}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},M^{*}\otimes M)\hookrightarrow
\text{H}^{\bullet}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0},M^{*}\otimes M)$$
is an injection. Therefore, if $x\in R$ annihilates $\text{H}^{\bullet}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0},M^{*}\otimes M)$
then it annihilates $\text{H}^{\bullet}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},M^{*}\otimes M)$, and it follows that
$$\text{Ann}_{R}\ \text{H}^{\bullet}(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0},M^{*}\otimes M)=\text{Ann}_{R}\
\text{H}^{\bullet}(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0},M^{*}\otimes M).$$
The statement (b) follows using part (a) and Theorem~\ref{T:isosupportsfe}.
\end{proof}
For ${\mathfrak g}=W(n)$, $S(n)$ we have the following result, which verifies the conjecture
made at the end of \cite[Section 6.2]{BaKN}.
\begin{thm}\label{T:isosupports2} Let $\ensuremath{\mathfrak{g}}$ be $W(n)$ or $S(n)$ and let $M$ be
a finite dimensional module in
${\mathcal C}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}$. The induced map of support varieties
$$\operatorname{res}^{*}:{\mathcal V}_{(\ensuremath{\mathfrak{f}},\ensuremath{\mathfrak{f}}_{\bar 0})}(M)/W
\rightarrow {\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M)$$
is an isomorphism where $W=\Sigma_{n-1}$ (resp. $\Sigma_{n-2}$) for $W(n)$ (resp. $S(n)$).
\end{thm}
\begin{proof} This follows by using the analogue of Theorem~\ref{T:stablecase} for $\ensuremath{\mathfrak{g}}=W(n)$ or
$S(n)$ and applying the argument given above in the proof of Theorem~\ref{T:isosupports}(a).
\end{proof}
\subsection{Realizability and the Tensor Product Theorem: } Theorem~\ref{T:isosupports}
allows us to provide a concrete realization for the variety ${\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M)$
when $M\in {\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$. The
next theorem provides a rank variety description of ${\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M)$ when
$M$ is in ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$.
\begin{thm} \label{T:supportprop} Let ${\mathfrak g}$ be a Type I classical Lie superalgebra
which is both stable and
polar, and let $M_{1}$, $M_{2}$ and $M$ be in ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$. Then, writing
$W(\ensuremath{\mathfrak{e}})$ for the pseudoreflection group associated with $\ensuremath{\mathfrak{e}}$,
\begin{itemize}
\item[(a)] ${\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M)\cong {\mathcal V}^{\operatorname{rank}}_{(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0})}(M)/W(\ensuremath{\mathfrak{e}})$;
\item[(b)] ${\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M_{1}\otimes M_{2})=
{\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M_{1})\cap {\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M_{2})$.
\item[(c)] Let $X$ be a conical subvariety of ${\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}({\mathbb C})$. Then
there exists $L$ in ${\mathcal F}$ with $X={\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(L)$.
\item[(d)] If $M$ is indecomposable then $\operatorname{Proj}({\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M))$
is connected.
\end{itemize}
\end{thm}
\begin{proof} (a) This statement follows from the realization of ${\mathcal V}_{(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0})}(M)$
as a rank variety and Theorem~\ref{T:isosupports}(b). (b) This follows by using part (a) and
the fact that the tensor product property for support varieties holds for
${\mathcal V}^{\operatorname{rank}}_{(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0})}(-)$. Parts (c) and (d) are proved using the
same arguments as those for support varieties of finite groups (cf. \cite{Car:83,Car:84}).
\end{proof}
We remark that since the stable module category of ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$ is a symmetric monoidal
tensor category, one can consider the spectrum ``$\text{Spc}(\text{Stab }{\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})})$'' as in
\cite{Bal}. Our results show that the
assignment $(-)\rightarrow {\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(-)$ satisfies the properties
stated in Balmer's paper for support datum.
Using Theorem~\ref{T:supportquotient2} and the same arguments as in Theorem~\ref{T:supportprop}, we can
similarly realize the
support varieties for the Cartan type Lie superalgebras $W(n)$ and $S(n)$, and prove that they satisfy the
tensor product property.
\begin{thm} \label{T:supportprop2} Let ${\mathfrak g}=W(n)$ or $S(n)$ and let $M_{1}$, $M_{2}$, $M$
be finite dimensional modules in ${\mathcal C}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}$.
\begin{itemize}
\item[(a)] ${\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}(M)\cong {\mathcal V}^{\operatorname{rank}}_{(\widetilde{\ensuremath{\mathfrak{f}}},\widetilde{\ensuremath{\mathfrak{f}}}_{0})}(M)/[W\ltimes T]$;
\item[(b)] ${\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}(M_{1}\otimes M_{2})=
{\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}(M_{1})\cap {\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}(M_{2})$.
\item[(c)] Let $X$ be a conical subvariety of ${\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}({\mathbb C})$. Then
there exists $L$ in ${\mathcal F}$ with $X={\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}(L)$.
\item[(d)] If $M$ is indecomposable then $\operatorname{Proj}({\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}(M))$
is connected.
\end{itemize}
\end{thm}
\subsection{Connections with Atypicality} Let $\ensuremath{\mathfrak{g}}$ be a basic classical Lie superalgebra
with a non-degenerate invariant supersymmetric even bilinear form $(-,-)$.
For a weight $\lambda\in {\mathfrak t}^{*}$, the
\emph{atypicality} of $\lambda$ is the maximal number of linearily independent, mutually orthogonal,
positive isotropic roots $\alpha \in \Phi^{+}$ such that $ (\lambda+\rho,\alpha)=0$ where
where $\rho=\frac{1}{2}(\sum_{\alpha\in \Phi_{\bar 0}^{+}} \alpha-\sum_{\alpha\in \Phi_{\bar 1}^{+}} \alpha)$.
For a simple finite dimensional ${\mathfrak g}$-supermodule $L(\lambda)$ with
highest weight $\lambda$, the atypicality of $L(\lambda)$ is defined to be $\text{atyp}(L(\lambda)):=
\text{atyp}(\lambda)$. In \cite[Conjecture 7.2.1]{BKN1}, a conjecture was stated relating the
atypicality of $L(\lambda)$ to the dimension of ${\mathcal V}_{(\ensuremath{\mathfrak{e}},\ensuremath{\mathfrak{e}}_{\bar 0})}(L(\lambda))$. This
conjecture was verified for ${\mathfrak g}=\mathfrak{gl}(m|n)$ in \cite{BKN2}. In light of
the results of the previous section, it seems reasonable to modify the ``Aypicality Conjecture''
to a statement which does not involve detecting subalgebras.
\begin{conj}\label{C:atypconjecture} Let $\ensuremath{\mathfrak{g}}$ be a simple basic classical Lie superalgebra
and let $L(\lambda)$ be a finite dimensional simple $\ensuremath{\mathfrak{g}}$-supermodule.
Then $$\operatorname{atyp}(L(\lambda))=\dim {\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(L(\lambda)).$$
\end{conj}
In \cite{BKN2} this modified version of the conjecture has been also verified, and the support
varieties for the simple modules have been completely described. This leads one to believe that the
atypicality for any classical Lie superalgebra ${\mathfrak g}$ should be defined for all modules $M$
in ${\mathcal F}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}$ (in a functorial way) as the dimension of ${\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{\bar 0})}(M)$.
For the Lie superalgebras of Cartan type, Serganova \cite{Se} defined the notion of typical and atypical
weights. In \cite[Theorem 7.3.1]{BaKN}, it was shown that for typical weights the support varieties for
simple $W(n)$-modules is $\{0\}$, and for atypical weights the support varieties are equal to
${\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}({\mathbb C})$. In this case it also makes sense to define the atypicality
of a finite dimensional module in ${\mathcal C}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}$ as $\dim{\mathcal V}_{(\ensuremath{\mathfrak{g}},\ensuremath{\mathfrak{g}}_{0})}(M)$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,006 |
Q: Laravel 5 Auth is non object in Controller construct In app\Http\Controller there is a master controller class named Controller that all other controllers inherits. In any other controller the following code works fine:
class OtherController extends Controller{
public function index(){
dd(\Auth::user()->id);
}
}
However, in the Controller.php i.e the parent class, when I try to do something like that in the construct method I have got Trying to get property of non-object error:
class Controller extends BaseController {
use AuthorizesRequests,
DispatchesJobs,
ValidatesRequests;
public $foo = 'null';
public function __construct() {
$this->middleware('auth');
dd(\Auth::user()->id);//Error is here.
dd(\Route::getCurrentRoute()->action['controller']);
}
}
How could I solve this issue?
A: Edit to add an actual answer
Try declaring this method and putting your code in it, before the parent::call_action() line:
public function callAction($method, $parameters)
{
// Put your code right here
// then do this last
return parent::callAction($method, $parameters);
}
Taken from the bottom of this page. The short of it is that callAction is always called right before the destination method is called, but it is also called after the middleware runs, so you will be able to properly access your auth facade. In all likelihood whatever you were trying to do in the constructor will work as you wanted it to in this method.
Original Answer
It's been a bit since I've poked around in laravel 5.4, but if I recall correctly the middleware doesn't fire until after the controller constructor, which means that you specifically can't do what you are trying to do. The auth middleware doesn't populate the Auth facade with the logged in user until after the controller is constructed, but before the controller method is called (which is why your first example works).
I believe that this is the intended behavior, so it is unlikely to change.
The short of it is (unless someone more knowledgeable corrects me) that you will have to find another way to do whatever it is you are trying to do.
Edit to add
Indeed, this question makes it clear that my suspicions were correct. Middleware fires after the controller constructor, which means that you cannot access the logged in user inside the controller constructor. In retrospect this is obviously the case because one thing you can do from within the controller constructor is specify which middlewares the system should run. If the middleware had already run, then you wouldn't be able to do that. So the short of it is that you can't (practically) access the Auth facade from within the controller constructor. This is intentional on the part of the laravel team, so it won't change anytime soon. You will have to find another way to do it.
A: I have update my answer on the basis of your error... make sure on the top of file add use Illuminate\Support\Facades\Auth;
Now with laravel 4.2+ it is easy to get user's id:
$userId = Auth::id();
that is all.
But to retrieve user's data other than id, you use:
$email = Auth::user()->email;
For more details, check security part of the documentation
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,482 |
Téra is a department of the Tillabéri Region in Niger. Its capital lies at the city of Téra. As of 2011, the department had a total population of 579,658 people.
History
Téra Department covers most of the historic territory of Liptako. Today primarily a home to speakers of Songhay Cine, a Southern Songhai language. Prior to the Songhai Empire, the area was populated by Gourmantche and Mossi peoples. By the 16th century Tera was one of the many small states which survived the destruction of the Songhai Empire, with many Songhai communities resettling into what is today Niger from the north. The Dendi Songhai state of the 17th century ruled the area, before dividing into several small states, each ruled by an Askia of Songhai noble lineage. In the late 19th century, the Songhai city state around Tera was in almost continual conflict with Tuareg groups in the north and east, with the city of Tera sacked and destroyed as late as 1885. Forces from Usman dan Fodio's Sokoto Caliphate took the area several times in the early 19th century as well, but were pushed back by Djerma forces in the regions to the southeast, though semi-nomadic Fula communities were present in the area from at least the 18th century.
Population
With a majority of Songhay - Djerma peoples, large Fula, Gourmantche and Buzu populations live in the area as well. Tera is one of the few places where the Songhoyboro Ciine dialect is still spoken, although the people are more likely to call themselves (and their language) "Songhay" or "Zarma".
Geography
The Tera department slopes down to the east, bordered by the Niger River. To the west, the area laterite plateaus broken by occasional mesas. Tera Department lies entirely within the Sahel region, and receives between 400mm-500mm of rain a year. The vegetation is sparse, with dry scrub forests and grasses, except in the land along the Niger River, which is lush with wild vegetation as well as vegetable and fruit gardens. In the northern section of Tera Department the Niger River forms a broad channel, where Hippopotamus and other wildlife are still common.
The Kokoro and Namga Wetlands in Tera Department was designated a Ramsar site in 2001. Covering 668km2 the wetland hosts migratory birdlife and is important to the local ecology.
The Departmental seat, a town of almost 20,000, lies on a tributary of the Niger, dammed to provide a small lake. Its main tarred road—the RN4—passes south from Tera to Diagourou (a Fula town 15 km away), Dargol and Gotheye, crossing the Niger at Bac Farie by ferry. The RN1 road continues to Niamey, 180 km from Tera. To the north of the department lie the small towns of Yatakala, Labezanga (where the Niger crosses to Mali), Bankilare (a mainly Tuareg town) and Kokoro in the center of the department. The cities of Ayorou and Tillaberi lie just across the Niger in Tillaberi Department to the east.
The department is bordered to the south by Say Department and Kollo Department, along the course of the Sirba River, a seasonal tributary of the Niger.
Economy
Most of the population is engaged in agriculture, with Millet being the primary crop.
The area contains a major road connecting Niger with Dori Burkina Faso, making Tera a transport and trade hub. Tera also has an airport, a 3900 ft/1189m strip designated DRRE.
The Samira Hill Gold Mine (Namaga concession) began producing gold in 2004 from the Tera greenstone belt which surrounds Koma Bangou, the country's largest artisanal mining site. The mine is operated by SML (Societe des Mines du Liptako), a joint venture between Moroccan Societe SEMAFO Inc and Canadian ETRUSCAN, along with a Nigerien 20% stake.
The traditional and distinctive Tera-Tera fabric is produced by Djerma artisans in Tera. Strips of striped, dyed handloomed fabric are sewn into larger blankets and are used in traditional marriage ceremonies and exported abroad.
Contested international border
Prior to 1910, portions of what are now Ansongo Mali were administered from Tera, while prior to 1922, Dori in modern Burkina Faso was administered from Tera, as part of the Niger colonial military territory of French West Africa. When the French divided French Upper Volta colony between its neighbors in 1932, the districts of Fada and most of Dori
(excluding the canton of Aribinda) were added to Niger, most in the (then) Cercle of Tera. Upper Volta was reconstituted in 1946, but the Tera Cercle's boundary remained a source of dispute until the Niger-Upper Volta protocol of agreement on their common boundary was signed at Niamey on June 23, 1964, fixing the present western border of the Tera administrative entity.
References
James Decalo. Historical Dictionary of Niger. Scarecrow Press/ Metuchen. NJ - London (1979)
Finn Fuglestad. A History of Niger: 1850-1960. Cambridge University Press (1983)
Jolijn Geels. Niger. Bradt UK/ Globe Pequot Press USA (2006)
Portions of this article were translated from the French language Wikipedia articles :fr:Tillabéri (région) and :fr:Téra (département). 2008-06-20
Further reading
Hammadou Soumalia. Traditions des Songhay de Tera, Niger (Hommes et sociétés) Editions Karthala, Niger (1998)
External links
Photographs and family stories of students from Tera at Africa Speaks: Life Stories (1998)
Worldvision and Islamic Relief projects in Tera during the 2004 famine.
Catholic Relief Services Tera project, working with sesame farming.
brighton2capetown:Niger. Story and photos of a trip through Tera on motorcycle.
Departments of Niger
Tillabéri Region | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 304 |
Q: Deserializing a null value in an object C# WPF I am serializing an object of employee class. Some of the properties in the class might be null. I need to deserialize the object with the null values, so that the null properties remail null. I am getting TargetInvocationException when I try to deserialize null values. Please Help me out
public class Employee
{
public string Name {get; set;}
public string Id{get;set;}
}
public mainclass
{
public void MainMethod()
{
Employee emp = new Employee();
emp.ID = 1;
//Name field is intentionally uninitialized
Stream stream = File.Open("Sample.erl", FileMode.Create);
BinaryFormatter bformatter = new BinaryFormatter();
bformatter.Serialize(stream, sample);
stream.Close()
stream = File.Open("Sample.erl", FileMode.Open);
bformatter = new BinaryFormatter();
sample = (Employee)bformatter.Deserialize(stream); //TargetInvocationException here
//It works fine if Name field is initialized
stream.Close();
Textbox1.text = sample.Name;
Textbox2.text = sample.ID;
}
}
A: I tried your code in LinqPad (with a few mods, the code above would never compile or work). This works fine:
void Main()
{
Employee emp = new Employee();
emp.Id = "1";
const string path = @"C:\Documents and Settings\LeeP\Sample.erl";
emp.Dump();
using(var stream = File.Open(path, FileMode.Create))
{
var formatter = new BinaryFormatter();
formatter.Serialize(stream, emp);
}
using(var stream = File.Open(path, FileMode.Open))
{
var formatter = new BinaryFormatter();
var sample = (Employee)formatter.Deserialize(stream);
sample.Dump();
}
}
// You need to mark this class as [Serializable]
[Serializable]
public class Employee
{
public string Name {get; set;}
public string Id{get;set;}
}
A: You only need to apply the required attribute as mentioned above.
namespace SO
{
using System;
using System.IO;
using System.Runtime.Serialization.Formatters.Binary;
[Serializable]
public class Employee
{
public string Name { get; set; }
public string Id { get; set; }
}
public class Test
{
public static void Main()
{
var emp = new Employee { Id = "1" };
//Name field is intentionally uninitialized
using (var stream = File.Open("Sample.erl", FileMode.Create))
{
var bformatter = new BinaryFormatter();
bformatter.Serialize(stream, emp);
}
using (var stream = File.Open("Sample.erl", FileMode.Open))
{
var bformatter = new BinaryFormatter();
var empFromFile = (Employee)bformatter.Deserialize(stream);
Console.WriteLine(empFromFile.Id);
Console.ReadLine();
}
}
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,983 |
\section{Introduction}
Geometrical interpretations of quantum mechanics have attracted much attention in recent years~\cite{Shojai_Article,QMGeometry,GabayJoseph1,GabayJoseph2}.
Dualities between quantum field theory and general relativity must results from the more fundamental fact that quantum mechanics can be formulated purely in-terms of geometry. Extending Maldacena's ER=EPR towards Susskind's more general GR=QM proposal is strongly suggesting that quantum theory must be entirely based on geometry. One can also put-forward a curious question about the nature of the geometry of quantum theory. Nottale had pointed out that the quantum theory naturally arises from the the most general geometry of nature, the fractal geometry \cite{NottaleBook1,NottaleBook2}. Due to the non-differentiable nature of
fractals, it is hard to find an extended version of differential geometry incorporating fractal manifolds.
One can use an infinite sum of smooth manifolds to approximate a fractal manifold.
It is already known that quantum-potential is associated to the fractality of the space-time~\cite{NottaleBook1,NottaleBook2}. In a more simplified scenario, one can take into account the scale symmetry of fractals on a differential manifold to incorporate the quantum mechanical nature of matter field and it will appear as an effective theory. The fractal contributions can appear as a small correction to the smooth Riemannian manifold, hence Einstein's Gravitational field contribution must contain small quantum mechanical corrections due to the micro-curvature or fractality of space-time; or in other words the conformal fluctuation of the metric.
In order to incorporate these micro-curvatures, one needs to look into extended versions of Einstein's theory~\cite{BeyondEinsteinGravity,WeylReview2017,Bergmann1968,fR_HamFormlation,fofT1,*fofT2,Bekenstein2011_TeVeS,
*BekensteinPRD_TeVeS,Moffat_STVG,*MOG_Moffat,Shojai_Article}.
Using the deBroglie-Bohm version of Quantum theory~\cite{Bohm1975,BohmI,BohmII,Bohm1975,PHollandBook,SheldonReview}, one can combine Quantum Theory with General Theory of Relativity and reaches to a special class of Scalar-Tensor Theory which is explored in ~\cite{Shojai_Article,ShojaiCnstrAlgebra,ShojaiBohmianQM,Shojai_ScalarTensor,GabayJoseph1,GabayJoseph2} and this special class of Scalar-Tensor Theory is termed as Bohmian-Quantum Gravity. It is to be noted that, aforementioned theory is just a classical field theory in its geometric form; space-time is not quantized like in Canonical Quantum Gravity
~\cite{Thiemann_2007} or like in Loop Quantum Gravity~\cite{rovelli_vidotto_2014,Gambini2011,AshtekarBook2017},
gravity is still treated as classical; space-time is continuous but the quantum nature of matter field on a smooth manifold is the main aspect of the theory. In the continuum limit of a sucessfull Quantum-Gravity theory, one should reproduce Bohmian Quantum Gravity kind of theory. Exploring such an interesting theory reveals an unexplored fluctuating backround field $\lambda$ which appears as the Lagrage multiplier in the theory. It is already shown that quantum mechanical uncertainty relation can be deduced from this fluctuating field $\lambda$ ~\cite{GabayJoseph1}. It was Nelson who proposed a stochastic mechanics approach to quantum mechanics \cite{NelsonBook}. In his approach, quantum particles are driven by a kind of Browninan motion resulting from quantum fluctuation. Physical orgin of such a fluctuation is still mysterious. It was S. Roy who suggested that the random zero point field induces the probabilistic aspect in the geometry of background space-time~\cite{Sisir_Roy1}. G. S. Asanov, S. P. Ponomarenko and S. Roy derived four dimensional quantum mechical conformal factor using five-dimensional probabilistic Finsler metric~\cite{Sisir_Roy2}. According to them, any microscopic system will have stochastic behavior due to the presence of fluctuating vacuum characterized by the probabilistic Finsler metric. According to them,
underlying Finsler manifold plays a great role to understand the quantum behavior in a consistent and complete way.
The nature of quantum fluctuations and its possible relationship with the gravitational field is also explored by L. Smolin~\cite{Smolin_QFluct1}. Bohmian-Quantum gravity also hints towards the idea that quantum particle might be immersed in some kind of fluctuating vacuum field.
In Bohmian-Quantum Gravity, such a fluctuating vacuum field appears from the unification of gravitation and quantum mechanics using deBroglie-Bohm version of Quantum theory~\cite{Bohm1975,BohmI,BohmII,PHollandBook,SheldonReview}. For such a unification, one needs to take three crucial steps, (a) Extend Einstein's Tensor Theory to Scalar-Tensor Theory (b) Accept the generalization of deBroglie-Bohm picture of Quantum theory (Note that deBroglie-Bohm theory is only linear order in quantum potential $Q$) (c) Accept the presence of a fluctuating vacuum field which appears as the Lagrange multiplier in the theory.
Taking all these factors into account the following action is proposed ~\cite{GabayJoseph1,GabayJoseph2},
\begin{eqnarray}
& & A[g_{\mu\nu},{\Omega}, S, \rho, \lambda]=\nonumber\\
& & \frac{1}{2\kappa}\int{d^4x\sqrt{-g}\left(R\Omega^2-6\nabla_{\mu}\Omega\nabla^{\mu}\Omega\right)} \nonumber \\
& & +\int{d^4x\sqrt{-g} \left(\frac{\rho}{m}\Omega^2 \nabla_{\mu}S \nabla^{\mu}S-m\rho\Omega^4\right)} \nonumber \\
& & +\int{d^4x\sqrt{-g}\lambda\left[\ln{\Omega^2}-\left(\frac{\hbar^2}{m^2}\frac{\nabla_{\mu}\nabla^{\mu}\sqrt{\rho}}{\sqrt{\rho}}
\right)\right]} \label{Actioneq}.
\end{eqnarray}
and Lagrange multiplier $\lambda$ is identified as a scalar vacuum field.
It is already shown that, minimizing the action with respect to $\rho$ and $S$
leads to real and imaginary parts of the Generalized Klein-Gordon Equation.
It can be easily seen that the real part of the Klein-Gordon Equation obeys the following equation
\begin{equation}
\begin{split}
\nabla_{\mu}S \nabla^{\mu}S-m^2\Omega^2+\frac{\hbar^2}{2m
\Omega^2\sqrt{\rho}}\Bigl[\Box{\Bigl(\frac{\lambda}{\sqrt{\rho}}\Bigr)}-\lambda\frac{\Box\sqrt{\rho}}{\rho}\Bigr]=0. \label{EqMotion}
\end{split}
\end{equation}
while the imaginary part gives the generalized continuity equation
\begin{eqnarray}
\nabla_{\mu}(\rho\Omega^2\nabla^{\mu}S)=0 \label{ContiEqn}.
\end{eqnarray}
Combining these two equations (Eq.~\ref{EqMotion} and Eq.~\ref{ContiEqn}) into a single complex wave equation, the generalized Klein-Gordon Equation can be obtained.
\section{Modified Klein-Gordon Equation to Modified Dirac Equation}
Bohmian equation of motion and continuity equation (Eq.~\ref{EqMotion} and Eq.~\ref{ContiEqn}), without vacuum field contributions (taking $\lambda=0$), can be expressed in terms of the density ${\rho}$, classical action $S$, and conformal factor $\Omega^2$ as already explored in the previous works~\cite{Shojai_Article,QMGeometry,GabayJoseph1,GabayJoseph2},
\begin{eqnarray}
\nabla_{\mu}S\nabla^{\mu}S - m^2\Omega^2=0 \label{EoMnoVac}\\
\nabla_{\mu}(\rho\Omega^2\nabla^{\mu}S)=0 \label{EoMcnteqn}.
\end{eqnarray}
Here, $\Omega^2=\exp(\frac{\hbar^2}{m^2} \frac{\nabla_{\mu} \nabla^{\mu}\sqrt{\rho}}{\sqrt{\rho}})=\exp(Q)$, where $Q=\frac{\hbar^2}{m^2} \frac{\nabla_{\mu} \nabla^{\mu}\sqrt{\rho}}{\sqrt{\rho}}$, is the Klein-Gordon quantum potential.
One can combine Eq.~\ref{EoMnoVac} and Eq.~\ref{EoMcnteqn} in order to obtain the modified Klein-Gordon Equation. Hence the generalized Klein-Gordon equation is given by,
\begin{eqnarray}
\Box\phi+\frac{1}{2}\frac{\nabla_{\mu}\Omega^2}{\Omega^2}\Bigl(\frac{\nabla^{\mu}\phi}{\phi}-\frac{\nabla^{\mu}\phi^{*}}{\phi^{*}}\Bigr)\phi+ \frac{m^2}{\hbar^2}\phi=0 \label{WFeqnp1}
\end{eqnarray}
Using the fact that,
\begin{eqnarray}
\frac{1}{2}\Bigl(\frac{\nabla^{\mu}\phi}{\phi}-\frac{\nabla^{\mu}\phi^{*}}{\phi^{*}}\Bigr)=\frac{i}{\hbar}\nabla^{\mu}S,\label{SDer}
\end{eqnarray}
Eq.~\ref{WFeqnp1} can be written as,
\begin{eqnarray}
\Box\phi+\frac{i}{\hbar}\Bigl(\frac{\nabla_{\mu}\Omega^2}{\Omega^2}{\nabla^{\mu}S}\Bigr)\phi+ \frac{m^2}{\hbar^2}\phi=0.
\label{WFeqn1}
\end{eqnarray}
The expression $\frac{\nabla_{\mu}\Omega^2}{\Omega^2}$ can be perceived as the quantum force for an exponential constraint
($\Omega^2=e^{Q}$) of the conformal factor $\frac{\nabla_{\mu}\Omega^2}{\Omega^2}= \nabla_{\mu}\ln{\Omega^2}=\nabla_{\mu}Q$. Hence,
\begin{eqnarray}
\Box\phi+\frac{i}{\hbar}\Bigl({\nabla_{\mu}Q\,\nabla^{\mu}S}\Bigr)\phi+ \frac{m^2}{\hbar^2}\phi=0.
\label{WFeqn1}
\end{eqnarray}
It is pretty straight forward to see that Eq.~\ref{EoMnoVac} and Eq.~\ref{EoMcnteqn} can give Eq.~\ref{WFeqn1}. In the usual Klein-Gordon equation, the continuity equation is given by $\nabla_{\mu}(\rho\nabla^{\mu}S)=0$ while the conformal gravity action (see Eq.~\ref{Actioneq}) generalizes the usual Klein-Gordon continuity equation $\nabla_{\mu}(\rho\nabla^{\mu}S)=0$ to $\nabla_{\mu}(\rho\Omega^2\nabla^{\mu}S)=0$ with an extra factor $\Omega^2$.
Continuity equation is usually derived from the imaginary part of the Klein-Gordon equation, hence this extra contribution should appear either as the part of $\nabla_{\mu}\phi$ or with a complex number times $\phi$. Since the equation of motion (Eq.~\ref{EoMnoVac}) didn't get corrected by the conformal gravity, $\Box\psi+\frac{m^2}{\hbar^2}\psi$ remains the same.
Due to the extra conformal factor $\Omega^2$ in the continuity equation, an additional imaginary term appears as the conformal gravity correction to the Klein-Gordon equation. One can see that the middle term $+\frac{i}{\hbar}\Bigl({\nabla_{\mu}Q\,\nabla^{\mu}S}\Bigr)\phi$ arises as a gravitational correction to the Klein-Gordon equation and it is interesting to note that such a correction appears as a dissipative contribution to the wave-equation.
Here the aim is to further simplify Eq.~\ref{WFeqn1}, for that purpose, one need to have a good understanding about the operator nature of the quantum potential $Q$. It is a matter of switching between quantum density formalism (Bohmian formalism ) and wave-function formalism. After some simple algebra involving $\phi$ and $\sqrt{\rho}$, one finds an interesting relationship between quantum potential $Q$ and scalar field $\phi$. Starting from the very basic relation, we get,
\begin{eqnarray}
\phi&=&\sqrt{\rho}\,\exp{\Bigr(\frac{i}{\hbar}S\Bigl)}\\
\nabla_{\mu}\phi&=&\nabla_{\mu}\sqrt{\rho}\,\exp{\Bigr(\frac{i}{\hbar}S\Bigl)}+\frac{i}{\hbar}\nabla_{\mu}S\,\phi\\
\Bigr(\nabla_{\mu}-\frac{i}{\hbar}\nabla_{\mu}S\Bigl) \phi&=&\nabla_{\mu}\sqrt{\rho}\,\exp{\Bigr(\frac{i}{\hbar}S\Bigl)}.
\end{eqnarray}
As a more general statement, it can be proven that
$f(\nabla_{\mu})\phi=f\Bigr(\overset{+}{\mathcal{D}}_{\mu}\sqrt{\rho}\Bigl)\exp{\Bigr(\frac{i}{\hbar}S\Bigl)}$ where $\overset{+}{\mathcal{D}}_{\mu}=(\nabla_{\mu}+\frac{i}{\hbar}\nabla_{\mu}S)$. One can also define ${\mathcal{D}}_{\mu}=(\nabla_{\mu}-\frac{i}{\hbar}\nabla_{\mu}S)$ and the following operator relationships are obtained,
\begin{eqnarray}
\frac{f(\overset{+}{\mathcal{D}}_{\mu})\sqrt{\rho}}{\sqrt{\rho}}=\frac{f(\nabla_{\mu})\phi}{\phi}\label{DplusEq}
\end{eqnarray}
\begin{eqnarray}
\frac{f(\nabla_{\mu})\sqrt{\rho}}{\sqrt{\rho}}=\frac{f({\mathcal{D}}_{\mu})\phi}{\phi}\label{DminusEq}.
\end{eqnarray}
These operator relationships are really useful to switch between the quantum mechanical density $\rho$ and the wave-function $\phi$ formalism. Ignoring $\frac{\hbar^2}{m^2}$ factor from the quantum potential, define a new quantity $\widetilde{Q}=\frac{\Box{\sqrt{\rho}}}{\sqrt{\rho}}$, one can
explore the operator correspondence.
Rearranging Eq.~\ref{DminusEq} and considering operator $\nabla_{\mu}\nabla^{\mu}$, Eq.~\ref{DminusEq3} can be deduced,
\begin{eqnarray}
\Bigl(\frac{f(\nabla_{\mu}\nabla^{\mu})\sqrt{\rho}}{\sqrt{\rho}}\Bigl)\phi=f({\mathcal{D}}_{\mu}{\mathcal{D}}^{\mu})\phi\label{DminusEq3}.
\end{eqnarray}
Equation~\ref{DminusEq3} is just another statement of the exponential shift theorem for the linear differential operators $f(D)(e^{ax}\sqrt{\rho})\equiv e^{ax}f(D+a)\sqrt{\rho}$.
Applying Eq.~\ref{DminusEq3}, the quantity $\widetilde{Q}\phi$ can be seen as an operator acting on $\phi$. Note that, in the Bohmian perspective quantum potential $Q$ is just a scalar function of $\sqrt{\rho}$.
\begin{eqnarray}
\widetilde{Q}\phi=\Bigr(\frac{\Box{\sqrt{\rho}}}{\sqrt{\rho}}\Bigl)\phi={\mathcal{D}}_{\mu}{\mathcal{D}}^{\mu}\phi\label{Qpsi}
\end{eqnarray}
After a straight forward expansion of the expression $\widetilde{Q}\phi=0$, it is found that the expression is equivalent to Eq.~\ref{WFeqn1}, once the classical energy momentum relation
$\nabla_{\mu}S\nabla^{\mu}S=m^2$ is assumed. Hence the equation can be written in a more compact form,
\begin{eqnarray}
\Box\phi+\frac{i}{\hbar}\Bigr(\nabla_{\mu}Q\nabla^{\mu}S\Bigl)\phi+ \frac{m^2}{\hbar^2}\phi=0 \implies {\mathcal{D}}_{\mu}{\mathcal{D}}^{\mu}\phi=0 \nonumber\\ \label{DisipKGEq}
\end{eqnarray}
This quantum gravity modified generalized Klein-Gordon equation, looks like a massless equation in terms of the gauge covariant derivative ${\mathcal{D}}_{\mu}$.
\begin{eqnarray}
{\mathcal{D}}_{\mu}{\mathcal{D}}^{\mu}\phi=0, \label{KleinNice}
\end{eqnarray}
where ${\mathcal{D}}_{\mu}=(\nabla_{\mu}-\frac{i}{\hbar}\nabla_{\mu}S)$.
Finding the Dirac equation from Eq.~\ref{KleinNice} is straight forward and it is given by,
\begin{eqnarray}
i\gamma^{\mu}{\mathcal{D}}_{\mu}\Psi=0, \label{DiracNice}
\end{eqnarray}
Expanding ${\mathcal{D}}_{\mu}$ operator, the following equation is obtained,
\begin{eqnarray}
(i\gamma^{\mu}{\nabla}_{\mu}+\frac{1}{\hbar}\gamma^{\mu}\nabla_{\mu}S)\Psi=0, \label{DiracNice}
\end{eqnarray}
where $\gamma^{\mu}$ describe Dirac matrices and $\Psi$ is a spinor field.
One can assign $\gamma^{\mu}\nabla_{\mu}S=-m I_{4}$ to match it with the Dirac equation
\begin{eqnarray}
(i\hbar\gamma^{\mu}{\nabla}_{\mu}-m)\Psi=0. \label{DiracNice2}
\end{eqnarray}
One can make a curious observation that $\nabla_{\mu}S$ looks like electro-magnetic gauge connection $A_{\mu}$, if $\nabla_{\mu}S\propto e A_{\mu}$, even the rest mass $m$ can be seen as an electro-magnetic quantity $m \propto e\gamma^{\mu}A_{\mu}$.
Note that, in the Dirac equation given in Eq.\ref{DiracNice2}, rest mass $m$ is a function of space-time variables, hence $\nabla_{\mu}m$ can give an imaginary contribution while squaring the Dirac equation and it should match with the imaginary part appearing the generalized Klein-Gordon equation given in Eq.~\ref{DisipKGEq}. One need to be aware that, squaring Eq.\ref{DiracNice2} should be done without using the conjugate of the complex differential ${\mathcal{D}}_{\mu}$ to re-obtain the Klein-Gordon Equation given in Eq.\ref{KleinNice}.
In order to get the complete form of Eq.\ref{KleinNice}, incorporating an extra mass term, one needs to take into account the vacuum field $\lambda$ contribution. So far we have dealt with the action $A$, while ignoring the $\lambda$
contribution (Eq.~\ref{Actioneq} with $\lambda=0$). Considering only the conformal gravity correction a dissipative term appears in Eq.~\ref{DisipKGEq}, in order to balance the dissipation, one must incorporate a forcing contribution in the equation. This contribution naturally appears when we take into account the quantum vacuum contribution arising from $\lambda$ field.
\section{Generalized Klein-Gordon with extra mass via Lagrange multiplier}
Taking into account the $\lambda$ contribution, as in the previous case, two real field equation can be obtained
\begin{eqnarray}
& & \nabla_{\mu}S \nabla^{\mu}S-m^2\Omega^2+\frac{\hbar^2}{2m
\Omega^2\sqrt{\rho}}\Bigl[\Box{\Bigl(\frac{\lambda}{\sqrt{\rho}}\Bigr)}-\lambda\frac{\Box\sqrt{\rho}}{\rho}\Bigr]=0. \nonumber \\ \label{EqMotionGen}
\\& & \nabla_{\mu}(\rho\Omega^2\nabla^{\mu}S)=0 .\label{ContiEqGen}
\end{eqnarray}
Wave-function equation can be defined in the same manner as before. After substituting Eq.~\ref{ContiEqGen} into Eq.~\ref{EqMotionGen}, more general wave-function equation coupled to the quantum vacuum can be obtained,
\begin{eqnarray}
\mathcal{D}_{\mu}\mathcal{D}^{\mu}\phi
- \Bigl[\frac{1}{2m\Omega^2\rho}\Bigl(\Box
- \frac{2m^2(1-Q)}{\hbar^2}\Bigr)\lambda\Bigr]\phi =0 \label{WFeqnVacuumF1}.
\end{eqnarray}
The vacuum density $\lambda$ obeys a first order differential equation coupled to the density via $\sqrt{\rho}$
\begin{eqnarray}
\lambda=\frac{\hbar^2}{m^2(1-Q)}\nabla_{\mu}\Bigl(\lambda\frac{\nabla^{\mu}\sqrt{\rho}}{\sqrt{\rho}} \Bigr) \label{LambdaNiceEq1}
\end{eqnarray}
Eq.~\ref{WFeqnVacuumF1} and Eq.~\ref{LambdaNiceEq1} together yields, quantum gravity corrected Klein-Gordon equation with vacuum contribution.
From Eq.~\ref{WFeqnVacuumF1} and Eq.~\ref{LambdaNiceEq1}, it can be seen that $\lambda$ gives a mass like contribution to the equation.
Defining a vacuum mass contribution $M(\lambda,\rho)$ in the following way,
\begin{eqnarray}
\Bigl[\frac{1}{2m\Omega^2\rho}\Bigl(\Box
- \frac{2m^2(1-Q)}{\hbar^2}\Bigr)\lambda\Bigr]\phi= -\frac{M(\lambda,\rho)^2}{\hbar^2}\phi \label{WFeqnVacuumF} ;
\end{eqnarray}
Modified Klein-Gordon equation can be written as,
\begin{eqnarray}
\Bigr(\mathcal{D}_{\mu}\mathcal{D}^{\mu}+\frac{M(\lambda,\rho)^2}{\hbar^2}\Bigl)\Phi=0 \label{WFeqnVacuumF}.
\end{eqnarray}
Here the gravitationally corrected Klein-Gordon equation written in terms of ${\mathcal{D}}_{\mu}=(\nabla_{\mu}-\frac{i}{\hbar}\nabla_{\mu}S)$ looks like the usual Klein-Gordon equation $(\Box+m^2/\hbar^2)\Phi=0$.
\section{Vacuum corrections to Dirac equation}
Its is already shown that the Quantum-Gravity Modified Klein-Gordon equation can be written as,
\begin{eqnarray}
\Bigr(\mathcal{D}_{\mu}\mathcal{D}^{\mu}+\frac{M(\lambda,\rho)^2}{\hbar^2}\Bigl)\Phi=0 \label{GenKGM}.
\end{eqnarray}
Now applying the Dirac's technique of taking the square root of the operator $\Bigr(\mathcal{D}_{\mu}\mathcal{D}^{\mu}+M(\lambda,\rho)^2\Bigl)$, the corresponding gravity corrected fermionic equation can easily be found. Corresponding modified
Dirac equation is given by,
\begin{eqnarray}
\Bigl(i\hbar\gamma^{\mu}\mathcal{D}_{\mu}-M(\lambda,\rho)\Bigr)\Psi=0 \label{WDiracGen}.
\end{eqnarray}
Expressing above equation in terms of the usual space-time curvilinear ordinates $\nabla_{\mu}$,
\begin{eqnarray}
\Bigl(i\hbar\gamma^{\mu}\nabla_{\mu}+\gamma^{\mu}\nabla_{\mu}S- M(\lambda,\rho)\Bigr)\Psi=0 \label{WFeqnVacuumF}.
\end{eqnarray}
Taking $\gamma^{\mu}\nabla_{\mu}S=-m I_{4}$, the usual Dirac equation is obtained with an extra vacuum mass contribution $M(\lambda,\rho)$,
\begin{eqnarray}
\Bigl(i\hbar\gamma^{\mu}\nabla_{\mu}-m-M(\lambda,\rho)\Bigr)\Psi=0 \label{WFeqnVacuumF}.
\end{eqnarray}
\begin{eqnarray}
\lambda=\frac{\hbar^2}{m^2(1-Q)}\nabla_{\mu}\Bigl(\lambda\frac{\nabla^{\mu}\sqrt{\rho}}{\sqrt{\rho}} \Bigr) \label{LambdaNicePsibarEq}
\end{eqnarray}
Here $\Psi$ is the usual Dirac Spinor, $M(\lambda,\rho)= \pm\Bigl[\Bigl(\frac{m(1-Q)\lambda}{\rho}-\frac{\hbar^2}{2m\rho}\Box\lambda\Bigr)\Bigr]^{1/2}$ and $\rho=\bar{\Psi}\Psi$. Here vacuum contribution should obey an extra condition $\nabla_{\mu}M(\lambda,\rho)=0$, to match it with the generalized Klein-Gordon equation (Eq.~\ref{GenKGM}).
It is important to notice that Eq.~\ref{LambdaNicePsibarEq} is obtained from the Scalar-Tensor Einstein equation
and the scalar curvature equation, for a detailed derivation see Ref.~\cite{GabayJoseph1,GabayJoseph2}.
In the case of the mass of the particle or antiparticle, $M(\lambda,\rho)$ can take positive or negative values (it can also be complex). Hence there are four ways to combine the rest mass of the particle with vacuum mass $M(\lambda,\rho)$ contribution. It is to be noted that the vacuum field may affect the particle and anti-particle differently. In order to get the Klein-Gordon equation one need to assign equal but opposite vacuum mass contributions $M(\lambda,\rho)$ to particle and antiparticle respectively.
\begin{eqnarray}
\bar{\Psi}\Bigl(i\hbar\gamma^{\mu}\nabla_{\mu}+m+M(\lambda,\rho)\Bigr)=0 \label{WFeqnVacEplus}
\end{eqnarray}
There exist another set of Dirac-equation for particle and anti-particle by flipping the sign of $M(\lambda,\rho)$, and they are given by,
\begin{eqnarray}
\Bigl(i\hbar\gamma^{\mu}\nabla_{\mu}-m+M(\lambda,\rho)\Bigr){\Psi}=0 \\
\bar{\Psi}\Bigl(i\hbar\gamma^{\mu}\nabla_{\mu}+m-M(\lambda,\rho)\Bigr)=0
\end{eqnarray}
\section{Proposal for a neutrino equation}
Taking $m=0$, the neutrino equation can be proposed as,
\begin{eqnarray}
\Bigl(i\hbar\gamma^{\mu}\nabla_{\mu}-M(\lambda,\rho)\Bigr)\Psi=0 \label{WFeqnVacEminus}
\end{eqnarray}
\begin{eqnarray}
\bar{\Psi}\Bigl(i\hbar\gamma^{\mu}\nabla_{\mu}+M(\lambda,\rho)\Bigr)=0 \label{WFeqnVacEplus}
\end{eqnarray}
where $M(\lambda,\rho)= \pm \lim_{m\to 0}\Bigl(\frac{m(1-Q)\lambda}{\rho}-\frac{\hbar^2}{2m\rho}\Box\lambda\Bigr)^{1/2}$.
\begin{eqnarray}
\lambda=\lim_{m\to 0} \frac{\hbar^2}{m^2(1-Q)}\nabla_{\mu}\Bigl(\lambda\frac{\nabla^{\mu}\sqrt{\rho}}{\sqrt{\rho}} \Bigr) \label{LambdaNiceEq2}
\end{eqnarray}
Its is to be noted that, even in the massless fermion case ($m=0$) the equations give a mass like contribution from the vacuum interaction term $M(\lambda,\rho)$ if the limiting quantity is non-zero.
This interesting mass contribution is not a constant, but a function of vacuum field $\lambda$ and quantum mechanical density $\rho$. Hence it can be real or complex depending upon the dynamical behavior of $\lambda$ and $\rho$.
Taking into account quantum gravity correction, the extra term appears $M(\lambda,\rho)$ is no-longer a constant,
in some specific scenarios it may behave like a constant. One can think about vacuum-mass as a quantity depends on
space-time coordinate and even the rest-mass is a function of space-time coordinate.
This implies an interesting possibility that, we might be
able to modify the mass of an object by properly adjusting the vacuum interaction.
A.Sakharov had suggested that the gravitation might be electromagnetic phenomenon induced by the presence of matter in the quantum vacuum~\cite{Sakharov1968}. In that perspective, gravity cannot be considered as a fundamental interaction, it must arise from the interaction of the matter field with the quantum vacuum in the accelerating frames.
Bohmian quantum gravity reveals the nature of inertia in a more interesting manner along the similar lines of thought
as introduced in the quantum vacuum inertia hypothesis ~\cite{GrvaccInertia}. It was H. E. Puthoff
who suggested that the quantum vacuum might be responsible for inertia ~\cite{PuthoffZPEFluct1989}.
Later A. Rueda, B. Haisch and H. E. Puthoff indicated that the origin of inertial reaction forces can be explained as a zero-point field Lorentz force~\cite{InertiaZPFLorentz}.
It is proposed that the interaction of electrically charged elementary particles with the vacuum electromagnetic zero-point
field results into inertial forces. These ideas needs to be understood in the usual quantum field theory context. Results presented here will shed light on these kinds of theoretical extensions of quantum field theory. In the usual quantum field theory mass is just considered as a constant while taking into account the quantum-gravity idea, we need to extend the concept of mass from a constant to a quantity which depends on the vacuum field variable $\lambda$ and quantum mechanical probability density $\rho$. The vacuum interaction is not restricted to charged particles only, it can appear for any matter and the
interaction happens through the probability density $\rho$ of the particle with the vacuum field $\lambda$. One should distinguish the mass fluctuation due to the conformal factor $\Omega^2$ and the vacuum mass contributions appears in the theory through $\lambda$.
\section{Conclusion}
In this manuscript, a more general Dirac equation is derived using Bohmian Quantum Gravity consideration.
It is found that the Dirac equation contains an effective mass contribution arising from the vacuum.
This vacuum mass term persist even for massless fermions, hence the equation might be useful to explain some of the mysterious properties of neutrinos. Dirac equation is coupled with a scalar vacuum field equation $\lambda$ via the particle density $\bar{\Psi}\Psi$. The appearance of vacuum-mass term to be taken as an important feature arising from the coupling of scalar vacuum field $\lambda$ with fermionic matter field.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,388 |
Гнилиця — колишнє селище в Україні, підпорядковувалося Великобурлуцькій селищній раді Великобурлуцького району Харківської області.
1987 року в селі проживало 20 людей. 1997 року приєднане до села Буряківка.
Селище знаходилося на правому березі річки Великий Бурлук, поміж селами Буряківка та Михайлівка, неподалік знаходиться залізнична станція Гнилиця.
Принагідно
Сайт ВРУ
Колишні населені пункти Харківської області
Великобурлуцький район
1997 у Харківській області
Населені пункти Куп'янського району | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,374 |
<?php
/**
* The template used for displaying page content in page.php
*
* @package asu-wordpress-web-standards-theme
*/
?>
<article id="post-<?php the_ID(); ?>" <?php post_class(); ?>>
<header class="entry-header">
<?php
//the_title( '<h1 class="entry-title">', '</h1>' );
?>
</header><!-- .entry-header -->
<div class="entry-content">
<?php the_content(); ?>
<?php
wp_link_pages(
array(
'before' => '<div class="page-links">' . __( 'Pages:', 'asu-wordpress-web-standards-theme' ),
'after' => '</div>',
)
);
?>
</div><!-- .entry-content -->
<footer class="entry-footer">
<?php edit_post_link( __( 'Edit', 'asu-wordpress-web-standards-theme' ), '<span class="edit-link">', '</span>' ); ?>
</footer><!-- .entry-footer -->
</article><!-- #post-## -->
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,144 |
{"url":"https:\/\/randomwalk.eu\/contact\/","text":"# On the subtle art of getting in touch\n\nEmail. The best way to contact me is to drop an email to: oscar@randomwalk.eu (click to copy to clipboard).\n\nWhile I like receiving thoughtful and interesting emails, I am only (one) human. Hence response time, however annoyingly, \u00abtends to be an exponential function of message length\u00bb.1 Writing properly helps make the curve less steep.\n\nBefore emailing me, please read the rest of this page. Especially if I don\u2019t know you! I may silently ignore your message if you don\u2019t\u2014and I will be able to tell from your message if you didn\u2019t!\n\nRule number 1. Here is the first rule of the now defunct Email Charter,2 the very first principle:\n\nRespect Recipients\u2019 Time. This is the fundamental rule. As the message sender, the onus is on YOU to minimize the time your email will take to process. Even if it means taking more time at your end before sending.\n\nI expect all my correspondents to honour it.\n\nTwo other things to keep in mind:\n\n\u2022 I strongly dislike HTML email and attachments in proprietary formats (especially Microsoft\u2019s Word and PowerPoint!!).3 Sending me such things is guaranteed to make me less willing to write a reply!\n\n\u2022 Email encryption is one royal pain in the arse. There is no doubt about it. But in this 2.0 world of ours, where everyone, including myself, uses and abuses of \u201cthe cloud\u201d, encryption is more necessary than ever. If for no other reason, then at least because in the end, \u00abthere is no cloud, only somebody else\u2019s computer\u00bb.4\nBut PGP being a nightmare to use, I\u2019m afraid I no longer use it regularly. Maybe this is an interesting project in the loom\u2026\n\ntwitter. You can also ping me on twitter, although the preferred method of getting in touch is still email.\n\n1. In the words quoted by Frank Stajano.\n\n2. It used to be found at http:\/\/www.emailcharter.org\/.","date":"2023-02-07 06:15:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.21209776401519775, \"perplexity\": 1842.9149467062396}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500384.17\/warc\/CC-MAIN-20230207035749-20230207065749-00412.warc.gz\"}"} | null | null |
Q: How make an array only selected input text values and send array via ajax? i want to add values in array and want to send on click
i want to store value in array only those user input text i have click not all values store in array.
when i store all value in array its tacking time to update
so i want to update only those value on which user have click or change in usert input text
*
*This is my html code. When I click in one input text then it shows only that value and then I click on another input text then it store first input text value and second input text value.
*Meaning, I want an array which can store input text value in an array() whenever I try to store value in array it shows undefined of array
*Now I want this array to send via ajax. How can I do this so that all array data goes via ajax
*I know array is a collection of similar datatype. So I want to store id in which all ids, reps_value of array store all reps_value and reps_percent of array store all reps_percent value
*And I want to submit my form using submit button on click
this is not working on click using seralizeArray() its giving me all input values when i click only two user input text
$(document).on('Click', '.submit_prescription,input', function (e) {
var x = $(this).parent().find("input").serializeArray();
var val[] = $(this).val();
var g[] = $(this).attr("name");
var id[] = $(this).attr("id");
//i am trying to store data in an array i using each loop but its not working properly how to set this loop to sore in array
/* alert(g);
alert(val);
alert(id);
*/
// $.each(x, function(key, values) {
// console.log(key, values.value);
// alert(key, values.value);
// });
// }
// var val = $("input:id").val();
// var val = $("input").val();
var g = $(this).attr("name");
//alert(g);
if (g == 'reps_value') {
// alert('reps_valwwue');
var f = $(this).attr("id");
var d = $(this).val();
var g = $(this).attr("name");
var data2 = "id=" + f + "&reps_value=" + d + "&type=" + g;
$.ajax({
type: "POST",
url: "prescriptionUpdate1",
data: data2,
cache: false,
success: function (data) {
$("#message").html(data);
$("p").addClass("alert alert-success");
},
error: function (err) {
// //alert(err);
}
});
} else {
// alert('reps_percentage');
var f = $(this).attr("id");
var d = $(this).val();
//var data : { reps_percent : d, id : f },
var data2 = "id=" + f + "&reps_percent=" + d + "&type=" + g;
$.ajax({
type: "POST",
url: "prescriptionUpdate1",
data: data2,
cache: false,
success: function (data) {
$("#message").html(data);
$("p").addClass("alert alert-success");
},
error: function (err) {
//alert(err);
}
});
}
});
============================================================
public function prescriptionUpdateData1($data) {
@$id = $data['id'];
@$reps_percent = $data['reps_percent'];
@$reps_value = $data['reps_value'];
$type = $data['type'];
if ($data['type'] == 'reps_value') {
$result = $this - > db - > query("UPDATE dev_prescription SET `reps_value`= '".$reps_value.
"'
WHERE id = '".$id."'
");
}
else {
$result = $this - > db - > query("UPDATE dev_prescription SET `reps_percent`= '".$reps_percent.
"'
WHERE id = '".$id."'
");
}
if ($result) {
return TRUE;
} else {
return $this - > db - > _error_message();
}
}
<td align="center" valign="middle" bgcolor="#F2F2F2" sdval="20" sdnum="1033;"> <font face="Agency-Roman" color="#000000">
<input style="width:90%" type="text" name="reps_value" id="<?php echo $location[2] ?>" value="<?php echo $location[0] ?>" class="" /> </font></td>
<td align="center" valign="middle" bgcolor="#F2F2F2" sdnum="1033;0;0.0%">
<font face="Agency-Roman" size="1" color="#FF0000"> <input style="width:38px" type="text" name="reps_percent" id="<?php echo $location[2] ?>" value="<?php echo $location[1] ?> "class="" /><input style="width:38px" type="hidden" name="id" id="id" value="<?php echo $location[2] ?>" class="" /> </font>
</td> <td align="left" valign="bottom" bgcolor="#FFFFFF"><font color="#000000"><br></font></td>
<td align="center" valign="middle" bgcolor="#F2F2F2" sdval="20" sdnum="1033;">
<font face="Agency-Roman" color="#000000">
<input style="width:90%" type="text" name="reps_value" id="<?php echo $location[5] ?>" value="<?php echo $location[3] ?>"
class="" />
</font>
</td>
<td align="center" valign="middle" bgcolor="#F2F2F2" sdnum="1033;0;0.0%">
<font face="Agency-Roman" size="1" color="#FF0000">
<input style="width:38px" type="text" name="reps_percent" id="<?php echo $location[5] ?>" value="<?php echo $location[4] ?>"
class="" />
<input style="width:38px" type="hidden" name="id" id="id" value="<?php echo $location[5] ?>" class="" />
</font>
</td>
<td align="left" valign="bottom" bgcolor="#FFFFFF">
<font color="#000000"><br></font>
</td>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 330 |
Swami Rakshananda (Gary Wager) is a disciple of and an ordained Swami in Goswami Kriyananda's Yoga lineage for nearly 2 decades as well as a graduate of both the Meditation and Hatha teacher programs. He currently teaches Meditation at the Temple and has been teaching hatha yoga, meditation and astrology privately and in various yoga centers in the Chicagoland area as well as other mystical traditions. He enjoys the challenge of integrating the teachings into all areas of life. He currently works as a detective with the Chicago police department and has a commercial real estate appraisal business. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,529 |
Q: Using values from an ArrayList in a where clause without loads of queries? I'm trying to accomplish basically this:
Select * FROM x WHERE id != for(int i : arraylist)
So basically I have an arraylist with say 10 ids, I want the sql not to return any of the rows that have an ID the same as in my arraylist, and I'd rather not do it with like 10 queries. Is there any way to do this?
A: You're looking for the SQL IN keyword:
SELECT * FROM x WHERE id NOT IN ( 1, 2, 3, etc. )
A: in this way:
SELECT * FROM x WHERE id not in ( list of value ) // i.e. 'A', 'B', ...
Pay attention:
You can execute this statement only if yuo array has been filled.
If you have an array with no items NOT IN can cause problems.
So, you must check your condition before put NOT IN clause.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,492 |
package com.lightcrafts.app.menu;
import com.lightcrafts.app.ComboFrame;
import com.lightcrafts.ui.editor.Document;
import javax.swing.*;
class RedoMenuItem extends ActionMenuItem {
RedoMenuItem(ComboFrame frame) {
super(frame, "Redo");
}
Action getDocumentAction() {
Document doc = getDocument();
if (doc != null) {
return doc.getRedoAction();
}
else {
return null;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,877 |
Table of Contents
Title Page
Dedication
Dreadlock Man,
I Almost Forgot
There's This Whole
Third Street Promenade
Nobody Knew Anything
Sticky Swipes Gear
Current Foster Lady,
It's Tied Sevens
I Could Tell
Baby Dressed Up
Sometimes I Think
Francine Was All
Jimmy Comes Running
On the Way
Cheerleaders Screamed Out
The Fellas All
Carmen Rolled Up
Things Are Heating
Fat Chuck Is
Mrs. Smith Brought
Dave Was Snapping
Check This Move,
Sticky and Anh-thu
Sticky Tapped Every
It's All Set,
Pop Songs Echo
Two Old Cops
My Name's Sticky,
It's Just Hoops
Lincoln Rec Shuts
Wong and Rolando,
In the Heart
After the Good
Dreadlock Man
Acknowledgments
Copyright Page
_For Al and Roni de la Peña_
Dreadlock Man,
with his fierce fists and suspect jump shot, sets his stuff ($1.45 sandals, key to bike lock, extra T-shirt) on the bleachers and holds his hands out for the ball. It's ten in the morning and Lincoln Rec has just opened. Sticky's at the free-throw line working out his routine, while all the regulars come swaggering in. _Come on, little man,_ Dreadlock Man says. _Give up the rock_.
Sticky throws an around-the-back, no-look dime. Watches Dreadlock Man rise into the air with his awful form—calves tightening, dreads scattering, eyes poised on the goal—and let go of a sorry-looking line drive. Before he comes back down to the dusty old hardwood, he yells out: _Peanut Butter!_ Says it every time he takes a jumper. _Peanut_ _Butter!_ That's what he wants everyone to call him, but nobody does.
When the ball ricochets off the side of the backboard, entirely missing the rim, he says what any man would say: _Hey, yo, Stick, let me get one more_.
Hawk passes through the door, from sunny day into old dark gym. A big black man. Wears bright wraparound shades and baggy shorts, the new Jordans on his size-sixteen feet. Hawk has a little money to his name. He's one of the few Lincoln Rec ballers who does. Some of the regulars say he made a few movies a couple years back. A stunt double maybe or security on the set. If you look quickly, get a fast profile shot, you might think he looks like someone.
_Hey, yo, Dreadlock Man_ , he says, megaphoning a hand around his mouth. _I got five says you brick that shot_. The whole side of his shaved head flexes as he chews hard at his gum.
Dreadlock Man takes a couple awkward dribbles and rises again. _Peanut Butter!_ This time his ball arcs through the air without backspin. A Phil Niekro knuckleball that thuds off the back of the rim and drops into Sticky's waiting hands.
_Damn, Dreadlock Man, your shot's straight broke_. Hawk falls into the bleachers laughing, goes to lace up his new sneaks.
Other dudes come strutting into the gym. Slapping hands. Slinging their bags onto the bleachers and talking trash.
Sticky high-dribbles to the other end of the court, spins in an acrobatic reverse. He points up at an invisible crowd.
Dreadlock Man watches, hands on hips. Yells out: _Come_ _on, Stick, we tryin to shoot down here_.
A couple other balls get tossed into the rotation. Everybody shooting short set-shots to get warm, stretching out stiff shoulders and legs. Most of these cats are just out of bed. A couple have pulled themselves off a piece of cardboard on court two, having spent the night where all the homeless stay.
Lincoln Rec functions both as a great place to hoop and a small-time homeless shelter.
Sometimes things overlap.
Sticky comes dribbling down from the other side of the court with his left hand. He goes right up to Dante, who's just walked in carrying a duffel bag, the best player in the gym, and shoots a soft twenty-footer over his outstretched hand. Dante and Sticky watch the ball smack both sides of the rim and bounce off toward the east sideline.
_Go get that brick, Stick,_ Dante says. _Bring it back my way_ _so you could watch a real shooter._
Dante played ball overseas for six or seven seasons; he's slick with both the rock and his mouth. Some cats say, _Watch_ _it, man,_ to newcomers, _dude will beat you two times_. Then they sit back and clown those who brush off their warning:
_Told ya, dawg. Didn't I tell him, Big J, when he walked his_ _sorry ass in here?_
_Yeah, I heard it, OP. I was sitting right there when you_ _said it._
Dante's skin shines black as night, and his hair is scare-crow wild. The devil's growth fingers out from his chin.
Sticky skips a bounce pass to Dante, who pats it around his back a little, through his legs some, close to the ground with his tips like a magician, and then fires up a twenty-five-footer that nestles in the gut of the net. _You see how I play the_ _strings, young Stick?_ He laughs a little and nods his head: _Just_ _like that, baby boy. That's string music._
Dante struts off the court with hip-hop rhythm, brushes past a businessman (who's stopped in to watch these black guys play: arms folded, subtle smile) and lies down near the bleachers to stretch his thirty-seven-year-old back.
This is Lincoln Rec on a Thursday, midsummer.
It's the best place in L.A. to ball. Some sports mag even did a cover story about it a few years back. Gym manager Jimmy's gold-tooth smile spread right across pages seventy-two and seventy-three. The article talked about how one court houses the homeless and the other accommodates the fearless. How Michael Cage sometimes shows up. Cliff Levingston. Eddie Johnson. Bill Walton was quoted saying: "It's the sweetest run in all of Southern California." The gym is in the middle of a pretty good-sized park, adjacent to some museums and business offices. The place gets so dark that when you've been in there a while and you go to peek your head outside to check your car, your eyes freeze up and hide like you've just stared in the sun.
Games go to eleven straight up. No win by two here. Fouls are called by the offense. The ball they use is dead weight. The leather has soaked up so much sweat from so many different dudes over the years, it takes a lot of legs just to get it up to the rim.
Other than that, there's a constant sour smell in the air, a NO DUNKING sign that _nobody_ pays attention to, and an unwritten rule that all who step foot through the gym doors with the intention of getting on the court better come with their A-game. "If you're gonna run with the big dogs," the article reads, "you can't pee like a puppy."
Sticky does what he does every day. He stands on the free-throw line with his ball. Simple as that. It doesn't matter who says what to him, if a ball caroms out his way, or nothing: He's not moving. He puts his rock between his knees and goes to tuck his shirt in. Pulls his shirt back out and retucks. Pulls it out and retucks. Ball between his knees, watching everybody shoot warm-up jumpers. Pulls out and retucks. Pulls out and retucks. There are eighteen, nineteen guys by this time—shooting around, running a quick game of twenty-one to get loose—this is the only way Sticky can make sure he finds himself balling in the first game. Pulls his shirt out and retucks. Pulls out and retucks. He's seventeen and white; these guys are men. Even though his game has improved from here to the 405, and most regulars swear they'd make room on their squad, there's still that thought in the back of his head that he might not get picked up.
He can't kick the aftertaste of that first month he started showing up, way back before he was a sophomore. It's only been a year and some change, but any baller would swear it's been longer. He'd cruise into the gym wearing all his state-issued gear, a bottle of tap water and bag of granola in his backpack, and the kid wouldn't get in one game all day. He'd just sit up in the bleachers like the thirteenth man, feeling like a scrub, headphones on his ears and basketball in his hands, figuring out on the sly who he could take. Absorbing the rhythms of squeaking sneaks and slapping hands, mouths left running all day and the rap of body against body in the paint.
_Shoot em up!_ Dante yells from the side, touching his toes with both hands.
And here's Sticky, already on the free-throw line with his ball. Simple fifteen-footer. Shot ninety-two percent this past season on JV. Ninety-four percent in league. He bounces twice with the left, wipes right-hand sweat off on his sock and cradles the ball. Same deal, different day. Pulls in a deep breath, runs an index finger across his bead necklace three times (just like that, peeking up at the rim: one, two, three). Middle finger in the groove of his rock, thumb between the felt-penned 7 and F. Shot's up and it rips through the bottom of the net.
First two who knock down free throws are captains. Sticky's got first pick.
OP follows and misses way short.
Hawk has one go in and out.
Dallas shoots an air ball. _I just got here, man. My first shot_.
Finally Trey gets a generous bounce and the ball rolls in.
_I got Dante,_ Sticky says.
_All right then,_ Trey says, _gimme Slim_. . . .
I Almost Forgot
to tell you about Sticky. . . .
How he keeps his raisin-brown hair cropped close. Faded up on the sides with some fancy-ass clippers he snatched from Macy's. How he's long and thin like somebody's stick-figure sketch, scissored off lined paper and Scotch-taped to a basketball court. How he goes everywhere with his duct-taped Walkman cranked up—loud rips of Jay-Z and Tupac before hoops, old-school Alexander O'Neal when it's time to chill. Sometimes the right-side phone goes out for a sec, but the kid knows just where to slap to get it going again. Only time he pulls the headphones off is when he's about to hoop or get with some little honey.
Sticky places the deck and phones in his bag, along with his flea-bitten ball, and zips up. Then he slides his stuff under the first bleacher. Slides it back out and back in. Back out and back in. Back out and back in. When something in the process fits just right he wraps the straps around the side support three times and ties up.
There's twelve bucks in that bag, and Sticky knows to watch it all day. Some pretty shady dudes roll through Lincoln Rec. Guys that'll thumb through stray bags when everybody's head is in the game, looking like rats in a trash bag for something quick to jack. A watch or earring. A fat gold chain. Somebody's hard-earned twelve bucks.
Some guys would steal from their best friend's bag if it came right down to it.
Sticky scrounged up all that cash this past week at the Third Street Promenade, after extended sessions of hard-core hoop. He panhandled with a bowl like back when he was a kid. Same spot and everything.
Hard-earned currency.
Twelve bucks to buy the stuffed bear he spotted in some old lady's card shop about a week ago. His girl Anh-thu's sixteenth birthday is today and he's set on hooking it up right. Figures he'll put the gold bracelet he's planning on swiping from a department store on one of the little brown bear's arms. Give it to her first thing tonight when they meet.
But check it out, Sticky would never steal from an old lady's card shop, he's got morals about things like that. Some gold from a department store, though? That's ripe for the taking.
He has the exact gold bracelet all picked out and everything. Saleslady called it a snake because of the way it wraps around itself. So slick and shiny and it never kinks. _You could_ _crumple it all up in your hand,_ the lady said, spinning and pulling at it in her palm. _And it still won't kink. See?_
Sticky spied it a few days back after his boy Sin sat him down and told him how it goes. _Dude, you gotta score your_ _girl a gift,_ he said as they rolled through Foot Locker, lifting new hoop shoes off the rack, checking them from every angle and then sticking them back up. _Not somethin all dumb,_ _either. It's like this: if you tryin to get with some little lady, but_ _you ain't sure, flowers are cool. Candy. But if you hangin with_ _her consistently, you gotta step up big_.
He looked Sticky right in the eyes when he said it, a pair of the new Iversons hanging in his right hand. _There's no gettin around it, man. It's mandatory_.
A couple minutes later Sticky spotted the snake in Macy's and stopped Sin cold. _This is it, dude,_ he said, running his fingers along the glass set up to keep grubby hands away. _This is the one I gotta get her_.
Sin stepped up next to him, squinted his eyes to get a better look. _All right,_ he said, nodding. _OK_.
_Yeah, she's gonna dig this,_ Sticky said. _I just got this feelin_ _about it._
Sin touched the glass, and Sticky called the lady over.
Lift the bracelet, pay for the bear: He's had it all worked out like this for almost a full week.
It's been a crazy hot summer in Los Angeles. Hot and dry. Like a tray of crackers just pulled from the oven. Like the whole city was pulled up by the roots and set down in the Mojave. Overheated cars line the freeway shoulder.
Abandoned. The parking lot in front of the gym is an afternoon mirage. The sun pushing its hot rays so deep into the newly laid pavement, it feels like you're walking on grainy-black bread dough. Packs of businessmen slip two fingers between their throat and collar on the lunchtime march back from their cars, waiting for a cool breeze that never shows up. After midday games, all the guys jockey for the water fountain like a group of camels looking for a two-week fix.
Some of the old-timers go on and on about this global warming thing people are talking about, how all the concrete and packed freeways are going to be the end of this crazy city. _Feel like I'm in hell today,_ Old-man Perkins says to no one in particular. _Man, I done felt like this almost every_ _day this summer_.
_You said it, OP,_ Slim says as a ball falls into his hands. He banks one in off the glass and holds his hands out for the change. _Already told you I'm out. I'm gonna go back where I_ _got most my peoples, up in Carolina. Soon as I get my last_ _check, poof, I'm yesterday's news_.
Others sit in front of the huge gym fan and pass around cold beer in a water bottle. Jimmy posted a sign that says NO ALcOHOL right above the entrance, did it just this summer, but a little beer in a water bottle doesn't bother him much. Not when the gym's as stuffy as it is today. Not when a guy can break a sweat just talking. Everybody's mouths cracked and full of cotton.
The blazing sunlight sneaks in through the slightly propped door. About seven or eight feet of warped rectangle glare that the team shooting at the south bucket has to deal with all game. But there's no getting around it. Gym policy. Jimmy tells everyone it's something handed down to him, something his boss made completely clear, door has to stay open all day. Guys sitting out who know the deal stand in the door out of courtesy. And it helps. Sometimes Jimmy himself stands there, arms folded, watching the games. Sticky tells everybody he can figure out what time it is by looking at the way the rectangle is folding up on itself.
_Check ball,_ Old-man Perkins yells.
_Yeah, y'all, it's already two games down, man_. Johnson counts all the guys, one at a time, with a slow index finger. _Three games down, matter a fact_.
Sticky ties the drawstrings on his shorts, waiting for the first ten to walk out onto the court so they can get things started. He unties his drawstring and ties again. Unties and ties. He's a thin 6 **'** 3'' at the top of the key, with skinny legs poking out of old hoop shorts. A fat homemade tattoo on his left shoulder, BABY, and deep brown eyes that pierce through anyone who catches his stare.
He unties and ties again.
Sticky's face is chiseled and tan. He doesn't talk much, but when he smiles all the girls seem to like what he's saying.
Unties and ties.
A couple uneven scars map his face. One that zigs and zags above his right eye from a knife (got cornered in an alley back in Long Beach and wouldn't give up his three bucks. Slash). Another circles up behind his left ear. The skin charred and purple about the size of a pencil eraser. He says he has no idea how he got that one.
Finally Dante slips off his platinum watch and tosses it into his bag. Shady characters know Dante's platinum is off-limits. His fists come down hard, anybody will tell you. There are silent understandings in even the most messed-up settings. A delicate balance. He stands up and struts out onto the long dusty court. _Let the runs begin,_ he says, and gives daps to Dallas. He straightens out his tucked-in shirt and starts in on Trey. _You didn't ask for all this, did you, Treydaddy? To have to come out here and try to check a big baller_ like myself?
Trey pushes out an uncomfortable laugh, wipes hands on his shorts. He looks down at his shoes and kicks his heel into the floor a couple times. _We'll see what up, D,_ he says, and pushes out another laugh.
Dante makes ten and everybody matches up on defense. Rob snatches the ball out of New York's hands and bounces it hard off the ground with both hands. Power dribble. _I_ _got white boy,_ he says, and shoves the ball into Sticky's chest, glaring.
Rob is: faded Malcolm X T-shirt covering faded yellow skin. Thick gold rope and dookie yellow cornrows. _I'm_ _gonna slap the handcuffs on you, white boy_.
_Slim, you match up with Dallas,_ Trey says, pointing to Dallas on the wing. _Big Mac, you d-up New York_. He rolls up and slaps Big Mac on his big butt. _Keep this reboundin fool off_ _the boards_.
_I got this light-in-the-ass cat,_ Big Mac says, and throws a couple playful jabs into New York's ribs.
I got Dante, Trey says. Y'all help out in the middle when he _drives._
Rob spits on the ground, runs his sneaks through for grip. He turns to Sticky and holds his hands out for the check.
Sticky tosses a bounce pass into his waiting hands. Rob spins around with the ball on his hip. Y'all good? he says to his squad. But just as he is about to check the ball in play, Slim throws his hands in the air.
_Hold up, hold up, Rob. I got a rock in my shoe._ Slim jogs over to the bleachers and starts unlacing his shoe.
Before anybody can start in on Slim, Crazy Ray staggers up to the sideline from the homeless court. You lazy sons a _bitches,_ he says. _Look at ya. Don't know nuthin about no goddamn game of basketball. A bunch a gang-bangin thugs is all_ _you is_. Ray's words spew out forty proof, and he points an old bony finger at everybody on the court. Like he does every day. A dark green plastic bag, outdoor size, wrapped around his thin upper body. Hanging down like a dress. He takes a few more unbalanced steps until he's standing at half-court. Armholes cut out on both sides. Pulled over a dull orange T-shirt.
Halloween in late July.
His black face mashes up and he yells: _Ain't like it was_ _back in my day. We had respect for the game_.
Everybody starts in on Ray to get off the court.
_Go on, old man, get back over where you belong._
_Ain't nobody tryin to hear about no 1940._
_Somebody get that decrepit, cryin-ass bum off the court._
_I got em, I got em,_ Dallas says, hustling over to Ray. _Let's_ _go, old man, come on_. He takes hold of both arms and backs him up slow.
Ray shakes a fist over Dallas's shoulder as he retreats. Y'all got no heart, he says in a weaker voice. No heart. And then the tears start rolling down his cheeks. Big heavy tears that launch off his quivering chin. The second he wipes all the wet away with a big right hand, six or seven more make the dive.
Same script, different day.
Dallas settles the Crazy Ray situation whenever it pops up. Most of the time it's just once a day. He'll walk out onto the court each morning with the same rap, get pulled off by Dallas and break down as he shouts a last line or two. But some days the old cat has a little more energy and comes staggering out again before closing. When most guys have taken off and all that's left are a couple ragged games of three-on-three.
Sticky spits on his right hand, watching. Lifts his right foot up and wipes the dust off his sole. Spits and wipes. Spits and wipes. He's watching Dallas handle Crazy Ray, but he's thinking about that smooth-looking gold bracelet. Figuring out the different ways he can go about snatching it. Trying to picture Anh-thu's face when he drops it on her tonight. Never thought he'd actually be excited to get a girl a gift. But Anh-thu's different. Anh-thu's his lady.
He spits again and wipes his right sole. Spits and wipes. Does the same thing again and again and then starts in on the left. Spits and wipes.
Spits and wipes.
Spits and wipes.
Dante watches the whole deal from a few feet away, says to him: _Hey, yo, Stick,_ as Slim jogs back out onto the court claiming he's cool now, he's ready. _Yo, Stick, for real, what the_ _hell you doin over there?_
There's This Whole
thing where Sticky sets a cup down fifteen, twenty times when nobody's looking. Thirty times. Until the heavy base meets the garage-sale coffee table with just the right feel. One of the dudes who lives with him in his current foster home, the only other white kid, calls him "Sticky Two Times."
A couple years back, a friend's mom went to pick her boy and Sticky up from a park when thick gray clouds opened up and drenched everything in sight. Heavy drops slapped against the blacktop, streaked down the metal backboard, gathered at the ends of the ragged chain net. The mom sat with her wagon running as her boy jumped into the front seat all wet. But Sticky couldn't leave until he hit the perfect shot. No rim. He kept shooting it over and over.
_Come on, Stick!_ his buddy yelled out the window.
Middle finger in the groove, thumb between the 7 and F. Another shot went through but grazed iron.
Middle finger in the groove, thumb between the 7 and F. Shot bounced off the back rim and caromed out toward the sideline.
_Please, son,_ the woman pleaded, _you've got to get out of_ _this rain!_ She sat on her horn.
Sticky kept chasing down his ball, though, carrying it back out to the arc to try again. Couldn't stop. He swallowed down hard at the lump in his throat. His chest burned. The salty rain dripped off his hanging bottom lip, ran down his neck and into his soaked tank top. He squinted his eyes and spit.
There was the crashing sound of thunder.
_Let's just go, Mom,_ the boy said. _Sticky's foster place is only_ _a couple blocks down_. Another shot floated through the air again and rimmed out. _I told you how he is. We'll be here_ _all night_.
Sticky ran to get another rebound, clawing at the skin on his forearm, elbowing himself in the stomach.
_We can't just leave him here,_ the mother said. _Goddamn it!_
Back even further, when Sticky was six and still lived with his mom in Long Beach, he already had this weird thing with change.
Baby would get them both on a bus going out to Santa Monica on certain weekends. When the government check she picked up every two weeks didn't stretch far enough. She'd set him up in the middle of the Third Street Promenade with a big white bowl. She'd dress him in dirty rags and rub dirt into his paper-thin cheeks, put a construction-paper sign around his neck that said: MY BOY NEEDS MONEY TO EAT.
This is back before he could ride a two-wheeler.
Back when Baby was still calling him Travis.
An older lady might tilt her head when she saw him there, sitting Indian style next to the freshly trimmed acacia, make that cooing sound older women make. She might go straight to her change purse and fish out a few quarters. Drop them in the bowl one at a time.
A guy in a too-cool suit might pimp by and look back over his shoulder. Reach into deep pockets for a dime or nickel and lob it the couple feet underhand. He'd slap hands with a fellow suit-buddy if the coin went straight in.
And every time Sticky got a new coin he would pull it out and toss it back in. Pull out and toss back in. He'd keep doing it again and again until something in the sound felt right.
Pull out and toss back in.
Pull out and toss back in.
It annoyed Baby on certain days. When there was nothing in her blood and all she could do was scowl. She'd slap at his hand and snarl. _Jesus Christ! Leave the money_ _alone!_
And he'd stop. Like a good boy. But the urge to reach back in the bowl was torture. A thousand drips on the forehead, tied up. He'd stare at it. Heart racing. Everything around him shutting down. It was just his body and the bowl. And his trembling hands.
Another woman would walk by and pat Sticky on the head. _Oh, here, honey_. Toss a quarter in.
Now he was behind. Two quarters in wrong. Or one quarter and one nickel. Two dimes. He'd look up at Baby standing above him.
She'd glare down. _Please, for the sake of my sanity, leave_ _the goddamn change alone_.
Back to the bowl. Everything crashing in. His limbs warming. Eyes burning.
Everything else in the world turning off.
Third Street Promenade
is blowing up. A hundred shopping bags swinging in rhythm. Marching, marching, marching. Plastic people with plastic hair, pushing and shoving. Plastic world. Stop in the middle of traffic on a cell phone. _Dana, I'm like_ _already here. I'm just outside Crate & Barrel_. A near pileup just outside Gap Kids.
There are stairs and elevators in Santa Monica Place. An escalator shooting up to the third floor with blinking lights. There are open doors and red Sale signs. This-is-who-we-are music oozing out of every mall shop.
_Can I help you?_
_Is there something in particular you're looking for?_
_Hi, my name is David._
_My name is Veronica._
_I'm Stuart, if you have any questions don't hesitate to ask._
Anh-thu and Laura are sitting in the middle of the food court, finishing up teriyaki bowls. Their first break in a marathon day. Everything in Millers Outpost is at least twenty percent off. The super-sale table is knocked down fifty percent. Since the doors first swung open at eight-thirty, Millers has been madness. Hundreds of hands pulling out shirts, unsnapping snaps, unzipping zippers, reaching into pockets, unfolding and throwing back, holding up against a body in the mirror and spinning around. Anh-thu's on until nine this evening. Even if it is her birthday. Laura's scheduled to leave at nine, too. Both have been dreading this sale for weeks.
The food court is slammed throughout the day during the summer. European tourists with shiny cameras and black socks, folding greasy slices of pizza. Japanese businessmen looking every blonde up and down, staring them around a corner or up an escalator. Quarter Pounders with a knife and fork. A mother and daughter shoveling Chinese into open mouths. Leaning over plastic plates with mouthfuls of chow mein. Cutting through excess noodle with bright white teeth and lifting smiling faces to one another. Fancy bags full at their feet. Groups of bored kids run in packs between the crowded tables. From the valley. From the South Bay. From East L.A. and West L.A. and Culver City.
The lines at each food stop extend out into the walkway like crooked trails of ants.
Anh-thu pinches a piece of broccoli with her chopsticks, slips it into her mouth. She notices a group of Mexican guys staring as she chews. One of the guys flips out a long tongue and wiggles it around. His friends laugh and throw playful punches at each other.
_Hey, Laura,_ Anh-thu says, _your friends are trying to get_ _your attention_.
Laura swings her head around. Gets the same tongue wiggle from the same guy. _Hey, why don't you go hang out at_ _the junior high!_ she yells, and turns back around.
The guys all laugh at their boy. _Ahhh, she punked you,_ _man_.
_She's clowning._
Laura pulls a piece of fat from her mouth with two fingers, wraps it in a white napkin. _Why guys gotta do that shit,_ _Annie? What they think, we gonna go right over there and give_ _em some?_
Anh-thu forces a smile and picks up her drink. She focuses on a piece of rice that has tumbled onto the table between the Styrofoam bowl and her cotton tank top. She rubs a hand across her cramping stomach and sucks her Coke through a straw.
_What's the matter, girl?_ Laura says.
_Nothing._
_Nah, I could tell when somethin be wrong with you, Annie._ _Your lips go all tight in a circle_. She points to Anh-thu's lips, tries to make her own into the same shape as an example.
Anh-thu frowns. _I don't know,_ she says. _It's nothing_.
_What, you can't tell your girl nothin no more?_
_It's stupid, Laura. . . . I mean, I'm stupid._
_Hey, heina!_ the Mexican guy yells. _Yo!_
Laura doesn't bother turning around this time, gives an over-the-shoulder flip-off. _Pendejos. Anyways, go on with_ _what you was saying, girl_.
_Forget it. It's nothing_. Anh-thu moves the food around in her bowl, grabs hold of a piece of chicken, lets it go.
Anh-thu's half Vietnamese. She's tall and thin with straight black hair falling down a pretty brown back. Green eyes like two tiny mirrors when the sun's in a certain spot in the sky. Sticky tells her she's his jujube when they meet at night in the park. Jujube like the candy she's always chewing on between classes at school. And when somebody asks how to say her name right, he tells them: _You know, Anh-thu. Like,_ _not on one, but on two. Anh-thu_.
They've been hanging together almost a year now and everybody says they match good together.
The crew pull out chairs and stand up cool. They leave dirty trays on dirty tables and walk smooth slow toward the girls. White wife-beaters and baggy khakis hanging off skinny bodies. Fifteen-year-old gangsta style. Hair slicked. Earrings and tats. Gold chains.
Anh-thu sees them coming but pretends she doesn't.
Laura pulls out a mini mirror from her bag, checks her made-up face from different angles. Long lashes and penciled-in eyebrows. Thick cherry lips. Fake mole. She lightly pats each cheek with powder.
The guy with the tongue leads his boys to their table. _Hey, homegirl,_ he says, _ain't no need for no attitude_.
Laura brings a frown up to him slow. Puts her stuff away.
_For real,_ he says. _We gotta just start over and shit_. He smooths out four or five peach-fuzz mustache hairs and looks at Anh-thu. He leans his hands on the table and turns back to Laura. _We about to go smoke some a this chronic_ _right now_. He peeks a baggy partially out of his pocket, looks around the mall for somebody spying his petty crime. _You down?_
The crew hang back a little, hands deep in pockets. One kicks a cigarette butt across the black and white tile. It comes to a spinning stop under a table of businesswomen.
Laura turns to Anh-thu with wide eyes, then back to the guy. _First of all, dude, that tongue thing you did was rude. Like_ _the biggest turnoff. Secondly, you and your friends look like_ _you're in junior high and we both in college_. She motions between herself and Anh-thu. _And lastly, we got serious_ _boyfriends who go to UCLA_. Having said all that, Laura tilts her head a little to the side and smiles. _So, uh, probably not_.
The guy's sneaky little smile runs away. He pulls his hands off the table and smooths his stash again. Looks at his boys and pulls up his sagging pants. _You got an attitude, you_ _know that? Better watch it before someone puts their foot in_ _your mouth._
One of the other guys pulls a small knife from his pocket and flips open the blade.
Anh-thu moves her eyes off the guy and stares down at the table.
Laura slides her bag over to Anh-thu, pulls her chair out and stands up. She gets right up in the tongue guy's grill with her finger. _Yo, why you coming over to me and my girl's_ _table anyway, man? Somebody invite you?_
_It's OK,_ Anh-thu says. _Laura, we'll just move somewhere else_.
_Screw that, Annie_. Laura kicks her chair over on its side. Everybody around them turns to look. Veins push out on her forehead. Teeth lock. When the guy takes a baby step back, she takes one forward. _Nobody said for this punk to_ _come over here_.
She turns to Anh-thu. _We ask them over here?_
Anh-thu shakes her head quick and folds her arms. Laura puts her palms in the air and tells the guy: _Nobody_ _wants you over here, man_.
Crazy broad! the guy says. He turns to his boys and backs up. Look at this crazy psycho broad, yo!
Laura puts her palm in his face and shakes her head. _Weak little cholo_.
The guy with the knife moves forward, toward Laura. His boy spots the blade and stops him short. _Be cool, yo,_ he says. _Not here_.
Anh-thu watches the guy fold the knife back up and drop it in his pocket. Watches him put his hands in his pockets too, then take them back out.
_Go play your little games with kids your own age,_ Laura says, setting her chair back upright. She sits down slowly, keeps an eye on them the whole time.
The tongue guy tries to get all hard again. Mad-dogs a couple seconds. He runs a hand through his greasy hair and motions to his boys. _Come on, man, forget about this psycho-broad, let's roll_. And they start off.
He looks back halfway through the food court and yells so everybody can hear: Yo, that's a crazy bitch over there! He points at the girls' table.
Everybody drops their small talk again and turns to look. All the bangers glare back at different times. They strut through the food court, chains swinging beside thighs, into the center of the mall and out of sight.
_Fakers!_ Laura yells out. Everybody turns toward her again, and she breaks into laughter.
Anh-thu forces another smile and smooths her hair behind each ear. _I didn't know what they were gonna do,_ she says.
_Don't worry, Annie. They wasn't nothin but posers. I could_ _tell you that right now_. Laura touches Anh-thu's arm. _You_ _see his face, though, girl? You see how serious he got when I_ _told em what's up?_ She imitates his look and then bends over laughing.
Anh-thu puts her chopsticks back in her bowl, sifts around for a piece of broccoli.
A few Japanese men at the table in front of them spin around straight-faced. Their thick ties stuffed inside stiff white dress shirts while they eat. Laura looks up at them and puts on her serious face, tells them: _Sorry._
When they turn back she starts laughing again.
She slows down and picks up her fork. Moves a strand of curly brown hair out of her face with the back of her hand. _Like we was gonna go smoke out with them fools_.
Anh-thu pinches a piece of broccoli and slips it into her mouth. She pulls at her bra through her tank top.
Laura shovels a forkful of rice into her mouth and puts her fork down. As she chews, she reaches into her bag and pulls out a small wrapped gift. Sets it up on the table. Anh-thu places her chopsticks in her bowl and wipes her mouth. She rolls her eyes. _Laura, you didn't have to get me something_.
_It's your sweet sixteen, of course I'm gonna get you_ _somethin_. Laura pushes the gift across the shiny plastic table. _You my girl, ain't you?_
Anh-thu picks up the gift and slips her hand under one of the folds. She pulls it apart by the tape without ripping the red and blue paper. Laura watches, smiling and sitting on her hands. Anh-thu folds the paper back up nice and sets it to the side. Pulls the box open and carefully peels the tissue paper. _Oh, my God, Laura, this frame's totally beautiful_. She reaches over the table and hugs Laura. _I love it_.
_Happy birthday, Annie._
They let go and Anh-thu holds the frame up: sparkling black plastic with white roses in the upper right corner. Laura takes it from Anh-thu and turns it around. _It's one of_ _them ones you can put like twenty pictures in. See?_ She pulls out the bottom lever and the sample picture is replaced by another.
Anh-thu takes it back and pulls the bottom out. Pictures switch places again. _That's so cool,_ she says. _I've never seen_ _one of these_. Pulls the bottom again. Pictures switch.
Laura picks up her fork, moves the rice and vegetables around until she uncovers a piece of chicken. She stabs and slides it into her mouth. _I was thinking you could have a_ _bunch of pictures of you and Sticky in there,_ she says with her mouth full. _You guys are so cute together_.
Anh-thu holds the frame to the side before putting it back in the box. She puts the folded paper on top and then closes the box back up. _If we had any._ She unzips her backpack, puts the box inside and zips it closed. _You know, I don't_ _even have one picture of us together? We went in that booth_ _outside Sam Goody like a month ago, but Sticky says he lost_ _them all._ She looks inside her bowl and pushes it to the side.
Laura smiles at a group of white guys walking by. One of them looks back over his shoulder. _Yum, he looks good,_ she says. _He's a total UCLA guy, you can tell by the way he dresses_. Laura always talks about landing a guy from UCLA. On weekends she and her two cousins use fake IDs to get into Westwood bars and flirt with the best-looking Bruins.
She stares in their direction until they round a corner, out of sight. _What was you sayin again, Annie? Oh, yeah, you_ _guys don't got no pictures. Take a camera tonight. Just get one_ _of them disposable ones_.
Anh-thu folds her fingers and lifts her head to think. _I_ _could do that._
_What are you guys doing, anyway?_ Laura looks up at Anh-thu and frowns. _Sticky better take you somewhere_.
_He won't tell me. Just said to meet him at the Colorado entrance at nine. He says he has everything figured out._
Anh-thu stares blankly at the table for a few seconds and then rubs her eyes with the back of her hands. All this talk about Sticky and now her eyes are glassy. She crosses her legs and folds up her napkin.
_For real, what's the matter?_ Laura says. She puts her fork down and dabs at her lips.
_I might have messed up._
_How?_
_I might have messed up bad, Laura,_ Anh-thu says, and holds a folded napkin against her closed eyes.
_What are you talkin about, girl?_
Anh-thu lowers her head. A heavy tear starts down her right cheek and she wipes it with the napkin.
_Oh, my God, Annie_. Laura shoves her bowl from in front of her. Walks around the table and sits next to Anh-thu.
_It's just somethin I gotta deal with._
_You don't wanna talk?_
Anh-thu shakes her head.
Laura reaches into her purse and pulls out a pack of cigarettes. Slips one out, puts it between her lips and flicks on a lighter. She pulls in the smoke. Exhales. _You don't wanna_ _talk, all right,_ she says. _But maybe I could help_.
They both sit quiet. Anh-thu balls her napkin up and tosses it in her bowl. Laura stares at the table and smokes.
Anh-thu looks down at her watch. _Oh, my God,_ she says, _it's already past eleven-thirty_.
Laura stabs her cigarette into her bowl and grabs her bag strap. _Sergio's gonna kill us,_ she says.
Anh-thu fumbles with her backpack zipper, opens up and pulls out her Millers name tag. Pins it on her shirt. She dabs at her eyes one last time with another napkin as they push away from their chairs and throw their bowls in the trash.
Laura stops Anh-thu before they leave, holds her by the shoulders. _I'm serious, Annie, you wanna talk, I'm here_.
_I know,_ Anh-thu says. _I appreciate it._
_Ain't nobody supposed to be sad on they birthday,_ Laura says. _For real_.
They both take off running through the food court, in the same direction the Mexican guys had gone, dodging packs of people on their way back to Millers.
Nobody Knew Anything
about Sticky turning seventeen. Except his foster lady, Georgia, who put three candles in a Hostess Cup Cake that morning before going to work.
This was back on the day of the big play-off game, four months ago. Anh-thu didn't know. Sticky's foster siblings didn't know. Coach Reynolds, who finally subbed him into the game with only seven minutes remaining in the third quarter, he didn't have a clue.
Georgia set the Cup Cake on the edge of the coffee table next to where Sticky sleeps, out on the green and brown couch in the living room. Pieces of dull orange foam bulging through thin rips in the worn fabric, swelling up between his arms and legs.
And Sticky didn't exactly offer up any info himself. Almost made it through the entire day without saying a word. The only time it was even mentioned was when Sticky, Dave, and Sin were spread-eagled against a cop car, getting searched for weapons. Hands on the roof, feet wide, as a pissed-off, out-of-breath cop patted them down one at a time.
Three sets of eyes on the jagged pavement.
The cop with the radio came walking around the front of the car with a grin on his face, shaking his head. _Hey, listen to_ _this one,_ he said, tapping his partner on the shoulder with the radio antenna.
Dave and Sin looked at Sticky all confused after the cop said what he said.
_It's your birthday today?_ Dave said.
Sticky shrugged.
_For real?_ Sin said.
But that's as far as it went. Nobody felt like singing happy birthday to you, cuffed in the back of some cop car on the way to the station.
Sticky Swipes Gear
like he shoots hoops. Shuts off his mind and rolls instinct. Every move in a department store is a SportsCenter highlight:
You go in with an empty Double Gulp cup and walk out drinking your favorite new shirt through a straw.
You try on those new Nikes with a hat pulled low, walk around Fragrances three, four times to check the feel, and cruise right out of the store. Get your sprint on if the high-pitched alarm starts screaming.
You walk past the register and out the front doors with three new shirts draped over the shoulder. In front of everyone. Bar codes dangling like fish in a cotton waterfall. If the blue-hair behind the counter says, _Excuse me, sir, have you_ _paid for that?_ (which they never do, because they don't really give a damn either) you say, _No, I'm OK, ma'am. I'm OK, sir._ Maybe add a quick _It's all taken care of,_ like you know exactly what you're doing. And you do. Then you cruise out calm like some suit guy with a heavy wallet.
It's the same high that's floating around a hoop court. Lift a product and don't give up a dime.
Makes you feel alive.
Take, for example, the day Sticky met Anh-thu in Millers Outpost last summer. He dropped in thinking layer scam (go in wearing your baggiest jeans, pull six or seven pants off the rack and hop into an empty dressing room. Leave the two best pair on under jeans, dump the rest in the saleschick's arms and roll out cool). But when Sticky was busy sifting through the overstocked rack, beats jumping through Walkman headphones, pulling a pair of khakis off the rack and putting them back, pulling off and putting back, pulling and putting, Anh-thu came up on his blind side.
_Can I help you find something?_ she said, tapping him on the shoulder.
Sticky jumped two feet.
Even when you shoplift with a slow heart, somebody's voice behind you can sound like a pair of heavy handcuffs rattling. Sticky turned around thinking undercover security, but what he found was Anh-thu.
Something clicked.
He knew the face, dark-type skin in the center of long black hair. Straight. Yeah, he'd seen her before, coming in or out of the caf, standing with her girls outside econ class, but never up close like this. Where he could see the green in her eyes. To be straight up, his stomach dropped. All he could do was stare. It was like the time his cart strained up to the highest peak on Space Mountain during some big foster outing to Disneyland. He could feel it in his stomach: Something crazy was about to go down.
_Um, hel-lo,_ Anh-thu said, waving a hand in front of his daydream face. _Can I help you?_
Sticky hadn't been with too many ladies up to that point. You have to understand that. Some of the sluttier chicks at school, like Angelica and Chloe. A rich white girl everybody called Grand Slam (cause one, her dad owned a local Denny's, and two, if you paid her a little attention, three, four days in a row, you were bound to take a trip around the bases) who used to buy Sticky lunch at school every Friday when the caf came through with pizza. There were a couple episodes with girls he lived with in previous foster homes. Sneak in each other's room after lights-out. But he never felt anything special with any of them. It was just something you were supposed to do, like cheating on tests and drinking forties with the boys.
_Yeah,_ Sticky said, shaking out of it. _I need some new pants._
You have to realize, Sticky was a baller first and foremost. There wasn't any time for chicks. He was too busy working on the Iverson crossover, or trying to sneak the rock over the rim with two hands-behind-the-back and call it a reverse.
Fire one off before a big game and legs go to rubber by halftime.
_Come on,_ Anh-thu said. _We just got this cool new line of_ _Bugle Boys. I'll show you._
Anh-thu, on the other hand, she's always had a pack of dudes on her heels. Blacks, whites, Mexicans, Asians—even grown men who come in shopping for their kids scribble down digits on a business card, try and slip it to her while she rings them up at the register. It's been this way since she moved to L.A. from Modesto. Her pops, who refuses to speak English in their house, has tried everything: ten o'clock curfews, big brother chaperoning her Saturday nights, checking skirt length with a ruler before letting her leave the house for school. But it's all hopeless. The more he tries, the more they blow up her cell phone.
_American boys,_ he always mumbles in disgust.
_Here, how do you like these?_ Anh-thu said, holding up a pair of khakis with white stitching. What size are we looking _for? Yeah, these will look way cool on you_.
In the dressing room Sticky experienced his first episode of shoplifting jitters. He stared at the pants like they were a pile of undercover cop bait.
He took off his baggy jeans and slipped the first Bugle Boy khakis over his boxers. Snapped, unsnapped, and snapped again. Perfect fit in the mirror. Kinda smooth, too. Unsnapped and snapped. Unsnapped and snapped. Unsnapped and snapped. He spun around to check the sag. Belt loops hung just below the boxer label, like they're supposed to. Unsnapped and snapped. Unsnapped and snapped. Slipped on the second pair, a little lighter shade with deeper pockets. But they were cool too.
_How we doing in there?_ Anh-thu said through the door. You need a different size or anything?
_Nah,_ Sticky said.
He sat down on the bench and spent a few seconds looking at his face in the mirror. The scars. The dark brown eyes and long eyelashes. The closely cut brown hair. He had a pretty face, according to his old foster friend, Maria. _You're a_ _boy with a pretty face,_ she'd always tease him. He looked away from the mirror.
He could still see Anh-thu in his head, and he didn't know what to do about it. He could still smell her smell through the door. Something wasn't right in his stomach.
The kid was shook.
He went to slip his jeans over the khakis but stopped himself cold. Tried again but couldn't come through. It wasn't fear so much, that maybe this girl was secretly keeping a head count of all his items. Nothing like that. And he dug the pants all right. Needed some khakis for school and everything. But he couldn't finish the play. There'd be no stealing this time. Do the right thing out of respect to this super-pretty Vietnamese girl. The one that had his mind doing some crazy jig.
He stripped and zipped up his jeans alone. He even folded the Bugle Boys up super nice to give back.
When he broke out of the dressing room, holding a pile of pants in his arms, he found Anh-thu helping some other buster. A black kid from the varsity football squad, one of the starting defensive backs. He watched the guy's mouth move as he walked up on them. Anh-thu bent over to let out a laugh.
_I smacked the crap outta my puppy,_ the guy said, working the scene. _He's gotta learn about that. I shoved his nose right_ _in my ripped-up shirt and told him, No, little Pepper! You can't_ _be messin up my gear like that_.
Anh-thu seemed all jazzed by what the guy was saying so Sticky hung back. On the sly he spied the defensive back lightly touching Anh-thu on the shoulder, Anh-thu looking up into the guy's face with her full attention.
Sticky decided to say screw it and turned to take off.
But just as he was about to set the pile down and flee the scene, Anh-thu dropped the hammer on the defensive back. _You know,_ she said, _it sounds fun and everything, but I have_ _a boyfriend_.
The guy smiled big, pushed out a laugh. _For real?_ he said. He put the flashy shirt back up on the rack.
_But thanks for asking._
_Who's the lucky Jack?_
Anh-thu turned around and pointed at Sticky. _Him!_
Sticky did a double take on Anh-thu's finger pointing at him. He looked over his shoulder. Nobody.
_Well, man,_ the guy said, putting his hand up on a shelf of folded white T-shirts, then taking it off and slipping it into his pocket. _My moms probably has the pot roast all ready. I_ _should probably, you know . . . I'll most likely just come in_ _sometime next week to pick up that shirt_.
_Sounds good,_ Anh-thu said. _Oh, and try to keep your_ _puppy away from your clothes_.
The guy worked up another laugh. _Yeah, we'll see,_ he said, and made his move for the exit.
Anh-thu walked over to Sticky with a big embarrassed smile on her face. _I'm sorry I did that,_ she said, taking the pile of pants out of Sticky's hands. _I didn't know what else to say._ _It just totally popped in my head_.
Sticky stuck hands in his pockets, real cool-like, and looked to the floor. He brought his head up to check those green eyes again but quickly cut away. He leaned against a shelf of white cotton V-necks and said: Them pants wasn't _really my thing_.
Anh-thu laughed a little and fingered one of the price tags. _It's OK,_ she said. _They make us try to sell these. Maybe_ _some other pants, though? You might like our Anchor Blue_ _stuff, they're super-baggy and comfortable_.
Sticky pulled his hands from his pockets. He locked them up behind his back for a sec, linked fingers, then stuck both hands back in his pockets. _Nah,_ he said.
_Yeah, forget it._ Anh-thu put the pants on a pile of T-shirts and pulled her long black hair behind her ears. _You know, I_ _go on break in like ten minutes, you wanna get some hot chocolate or something? Down in the food court?_ She smoothed loose hair behind both her ears.
Sticky tapped his right foot against the bottom of the shelf a couple times and looked up at her. _Nah, I can't,_ he said.
Then he walked out of the store.
Current Foster Lady,
Georgia, pulled up to Sticky's foster care pad in almost the same meat-and-potato minivan as his previous lady, Mrs. Smith. Same dull white paint job and sloping hood. Same snail-like movement across the crumbling road.
Could have been a déjà vu situation if it wasn't for Georgia's bumper-to-bumper sports stripe. Red.
Sticky was sixteen, and he promised the old Mexican director he'd try harder this time.
He spied her through the game room window: creeping along the curb out front, double-checking a scribbled-down address. She was in the middle of the road and two or three cars had to swing into the other lane to make the pass. He spied her screwing up a simple parallel park, shutting off that familiar minivan-engine hum, stepping out (fat arms and fat legs poking out of a two-man-tent-like summer dress), and slamming the door shut behind her.
She stood there duck-footed and stared up at his run-down foster care pad, the place where she'd agreed to take on another kid only two weeks previous.
Add a fifth to her pack of strays. Three hundred and sixty dollars a month per stray from the state.
Do the math.
Sticky spied all this through the game room window while at the same time kicking Counselor Julius's ass in foosball again.
Earlier that morning, Julius had laid down the challenge. _Let's run a quick game, Stick,_ he said, his dark blue Duke cap pulled cool-like crooked over his smooth black face. All the residents shoveled spoonfuls of cereal into their mouths. Warehouse-size sacks lying between them, generic flakes spilling onto the table. _For old times' sake_.
Sticky took one last bite before he set the spoon in his bowl. He pulled the spoon out and set it back in. Pulled out and set back in. Pulled out and set back in. Julius knew the routine and rolled his eyes.
Pulled out and set back in.
Pulled out and set back in.
When the dull tap of metal on plastic sounded like Sticky thought it should, he hustled over to the foosball table and grabbed two of the worn-out handles.
Back in those days, Sticky spent all his free time playing foosball. Couldn't get enough. The other residents said he took it way too far. Always spinning the handles around. Always pulling and pushing the rusting arms, stopping the ball on a dime and firing it into the box. Aerial shots, crank shots, simple bank shots off either wall, snake shots where a heavy spin shoots the colorful ball out like a rainbow across the blue face of the game. All this even if there was nobody on the other side to play to eleven.
_You're obsessed with a stupid game,_ Maria always said, on nights she'd sneak into his room after lights-out. She'd look him right in the eyes after they messed around in his bed for a while, tell him: _I don't get it, why play a game when you can_ _talk to a real person?_
Sticky would pick at the calluses on the insides of his fingers. Shrug her off. Sometimes he'd peel a callus off completely and toss it in her lap.
But if anybody needed to find Sticky in his foster care pad, all they had to do was follow these simple directions: move straight through the living room, past the always shouting TV, hang a sharp left before the long dark hall, swing on through the kitchen, past the locked-up counselors' office and into the tight-quarter game room. That was where he'd be. After chores. After lunch. Before lights-out. Sometimes even in the middle of the night, when he'd sneak out of his room to duct-tape a defenseman he dreamed was starting to crack.
And maybe Maria had it right, maybe he was obsessed with playing a dumb game, but when he beat somebody good, got them all flustered and defeated, the blood ran through his veins all smooth like melted butter. Warm and fast. The rush of heat moving along his skin giving him a reason to spring out of bed in the morning.
It wasn't so much the foosball he was addicted to anyway, it was the beating people.
He'd beat the old Mexican director, who rolled in with boxes full of frozen foods twice a week. The director, who'd already sent him off with a packed bag three separate times, only to eventually have him returned like a shirt that didn't fit. A pair of pants. He'd beat all the residents at least twice a day, including Maria, who he'd play with one hand behind his back. He'd beat any new buster that got dropped off before they had time to unpack their bag. He'd beat the night watch. He'd beat the college girls who showed up to give presentations about sexually transmitted diseases. But the sweetest wins of all came against Counselor Julius. With all his hooting and hollering whenever he scored a goal. His intense facial expressions, all eyebrows and teeth. Sticky knew once he got Julius to the table, he'd have somebody to beat for hours.
Julius would get so pissed at all the losing, he'd end up dragging Sticky down to a local park with the scuffed-up house basketball. This was back before Sticky could even dribble with his left hand. Julius strutting out onto the blacktop in his Scottie Pippen jersey and new Air Jordans. His fake diamond earrings. Sticky with his standard white T-shirt issued by the state. Standard old blue jeans, ripped in the knees by the resident who'd handed them down. Sorry-looking white Wilson low-tops.
Julius would kick Sticky's butt in one-on-one, game after game. Back him up in the post. Block his shots. Swipe the ball when he dribbled. The whole time talking trash. The whole time giving Sticky this new game to fixate on. To obsess about. To do over and over and over until he got good. A reason to dribble back down on solo missions. Every day.
Return to the scene of the crime, where Julius had kept in on him until foosball frustration faded from his mind.
With all those jumpers Sticky shot by himself, sometimes long after the moon replaced the sun above the backboard, it wasn't long before he was getting Julius in hoops, too.
Then Counselor Julius had nowhere to turn.
_All right, Stick_ , Julius said, cupping the white ball in his hands and blowing on it. _One more foosball game before this_ _lady shows up, and I swear to God I'm gonna get you this time_. _We'll see._
But you know the routine, how one game turns into ten and before you know it, Sticky's two points away from taking thirteen straight.
He ran off the first seven or eight in a flash, mixed in a couple shutouts. And then a frustrated Counselor Julius came up with the idea of switching sides. Sticky went from whipping Julius with his back to the window, to whipping him facing it. Like that. And to top it all off, Sticky didn't even give his full attention to that thirteenth game. He had one eye on the action and the other spying this new foster lady making her way up the driveway.
Georgia finally got to the door and knocked three times hard. Julius, playing the part of responsible counselor, abandoned foosball. _That's her, Stick,_ he said, pulling his hands from the handles and straightening his cap. He reached back and took one last whack, but Sticky blocked the shot with a quick shift of his goalie.
_You lucky, too,_ Julius said, _I was about to catch my rhythm_. He hustled to the front door.
Julius pulled the heavy door open and painted on his best smile. _Hello, ma'am,_ he said, shaking her hand. _I'm_ _Julius. If you'll just follow me into the office, I'll have you sign a_ _few things. It won't take long_.
_That's him, right?_ the woman said, pointing through the game room door at Sticky.
_Yes, ma'am, Travis Reichard. That's his given name. But he_ _goes by Sticky around here._
_Oh, that name,_ the lady said in a voice so Sticky couldn't hear. She moved under the game room door frame. _We'll_ _have to do something about that awful name._
_Like I said, ma'am, real name is Travis. I'm not even sure_ _how he got the name Sticky._
She watched Sticky slap the ball around a couple more times and then piped up: _Hello, Travis_. She spoke in a long and drawn-out voice, as if Sticky was retarded. How are you today?
There were a few awkward seconds of dead air.
_Tell the lady hi,_ Julius said. _Where's your manners, big guy?_
Sticky put his head up, then his hand up. _But my name's_ _Sticky,_ he said, and went back to hitting the ball around.
Counselor Julius smiled.
Georgia smiled.
_We're gonna give you a fine place to live,_ she said. _Me and_ _my husband. I think you'll be very happy in our home_.
Minivans and this same opening line, _Our happy_ _home_. . . There must be a book these ladies check out from the county library.
She stood watching Sticky a couple more seconds, then turned and followed Julius into the office, to get the details worked out.
It's Tied Sevens
and Sticky's handling the rock up top. Back and forth with the left hand. In front of his glazed body. Rhythm pats. Type of dribbles that get you in the groove to cut and slash, body loose and quick to make somebody look like a fool.
Rob's weight is on the back of his heels on defense. Waiting.
The face rattles off truth in situations like this. Fear flickering in Rob's wide eyes: Get too close and Sticky zips by for a layin, give too much room and Sticky sticks a jumper in his eye. Too many possibilities when the man with the ball gets to say which way and when, how fast and for how long. And you can multiply all that by ten if the guy can play. Get busted on in front of everybody. Get dragged all game by the skinny white kid everybody talks about.
All the loudmouths on the sideline are at full attention.
Sticky jab-steps right and pulls back, keeps his dribble.
Rob retreats.
Sticky is: through the legs, around the back, playing hoops with a yo-yo. Walk the dog when everybody calls for a trick. Hold the ball too long.
He is: stolen Nike shoes, stolen mesh shorts, ankle socks. Back and forth handling the ball, knees bent, his eyes in Rob's eyes. Piss off the old purists who cry for a return to fundamentals. The ones who've lost so much vision they're blind to the dance of it all. The spin move like a skirt lifting pirouette on callused toes. The dip. Jump shot splashing through the net like a perfect dismount.
There's spirituality here. On this court. With these guys. Holding this ball.
Dallas clears out of the lane. _Go on, Stick,_ he says, pointing to the open lane, backpedaling. _Take him, boy_.
Sticky is: in neutral down a mountain hill without sound. There's no little voice saying where to go or how. Everything is in slow motion:
_You could go to that one nasty spin when the defender's vulnerable._
_Could cross somebody over and pull up for the short_ _midrange jumper._
_Could skip an around-the-back pass to a cutter for the_ _cram._
_Could spin the rock around the right side of this clown and_ _cut around the left, meet up on the other side for a slick-looking_ _finger roll._
_You could shoot from twenty-four feet out. From twenty-five feet out._
_You could knock it in off the glass._
_Could bust it straight through._
_Or you could just hold the rock at your side for a quick sec,_ _watch everybody watching you. . . ._
Every pair of eyes is watching as Sticky makes his move on Rob—goes through his legs to the right, hesitates, the ball spinning like a top in the palm of his hand, weightless, stutter-steps one way (Rob's feet go right, body left, have a nice trip), crosses him back the other way and blasts to the basket, dropping a sweet little no-look dime to Dante when the big man comes over to help.
Dante rises up uncontested and flushes in a two-hand jam.
_That's on you,_ New York says in Big Mac's face.
_Eight-seven!_ Dallas yells out on the way back downcourt. _Good guys!_
Everybody on the side goes crazy, falling all over each other. Laughing and frowning at the same time. _Goddamn,_ _Rob_ , Old-man Perkins says, his fist raised to his lips. _That_ _white boy just took you. I ain't gonna lie_.
_He clowned you!_ Johnson hollers, slapping five with Old-man Perkins and then taking a couple steps onto the court.
Dante points at Sticky on the way back down the court. _Good look, boy_.
_That kid's got a bag full of highlights, don't he, J?_ Old-man Perkins says, falling back onto the dull bleachers. Between somebody's carved-in initials. _Goddamn, he got a bag_ _a highlights_.
_And Dante just flushes it through, OP,_ Johnson says back. _He don't barely even touch the rim when he dunks. Straight_ _through like Dominique use to do for the Hawks_.
And that's the thing about this game. Go back on your heels and somebody's gonna spin you around. Break you down. Shake left and have you falling all over yourself. Walk down the lane for two or drop a highlight on you. It's all about the oohs and aahs from the sidelines. Turn heads. Do something that makes them stand up and slap fives, something that has them still talking on the way out, when Jimmy shuts and locks the doors to go home for the night.
Everybody from Dreadlock Man to all the three-piece suits who roll in during their lunch hour, even Rob (though he would never give up the information), they all watch a little harder when Sticky gets the ball on the wing. There's something natural about the kid. Something authentic. Simple pass to the post comes in a wraparound no-look. Sticky either comes out with some spectacular mouth-dropping demonstration or he messes up. Nothing in between. But when he hooks up something slick, some Pistol Pete hide-the-ball trick shot during a game, and this happens more often than not, everybody on the sideline falls all over each other slapping bleachers and providing commentary:
_That white boy can ball, huh, Heavy?_
_I said the whole time he could play, KP._
_I mean, he can_ ball, _though._
_He don't play like no regular white boy, that's why._
Trey pops out on the wing and Slim hits him with a chest pass. He sizes up Dante, jab-steps at him. Big Mac comes out of the post and sets a 280-pound screen. Trey slides right and Dante struggles to get through. Enough time to fire up a long jump shot. But the ball hits the front of the rim and pops straight up.
Dallas springs up to get it, high-dribbles back the other way. He gets trapped in the corner and throws out a desperation hook pass that Dante somehow tracks down. Everybody's out of position when Dante starts left, switches right and buries a fifteen-foot jumper over Trey's outstretched hand.
_Nine-seven,_ Dallas yells out.
Sticky picks up Rob as he dribbles down the court. Payback is in Rob's eyes. Sideline comments can get under your skin. Especially when the white boy you hate is doing all the damage. He isolates Sticky on the left wing, dribbling with his back to the basket. _Clear out, Slim,_ he yells. _I got a_ _mouse in the house_.
Sticky gets low and pushes with his legs. He's down forty pounds and a stack of strength. And Rob always takes him into the post. Throws his big butt into Sticky's middle and takes up the slack. Slaps hands away when they come onto the small of his back.
Rob starts baseline, brings up his head, stutters his steps and pushes the ball in front of his body toward the middle of the court. Sticky retreats on defense and keeps his position. Rob spins quickly back toward the baseline but loses the ball. It leaks out toward the sideline, where Sticky is quick to scoop it up and head the other way.
He races downcourt toward the bucket for an uncontested flush. But just as he's about to take his two and a half steps—guys on the sidelines laughing at Rob lying on the ground, Dante and Dallas jogging behind the play—Rob goes to his knees and yells out, _Foul!_ The sound of his deep voice echoing throughout the gym.
_Foul!_
Everybody on the sidelines goes crazy:
_Nah, Rob, white boy ripped you clean!_
_That's an embarrassment call!_
_You weak, Rob!_
Dallas runs clear around the court with his hands on his head. _Oh, hell no!_ he keeps yelling. _Hell no!_
Rob gets to a squatting position and looks down the court. _That's my ball! My ball!_
Sticky sits on the ball under the far basket while everybody yells at everybody else. Dante holds his hands out for the rock and tells him: _It wasn't no foul, boy_.
_I didn't touch em,_ Sticky says.
_Come on,_ Dante says, and helps Sticky up. Takes the ball and tucks it under his arm.
Sticky shakes his head and stands alongside Dante. _I_ _won't even get three games cause of this dude_.
_What you gotta do, boy?_ Dante shoots a look down the other end of the court where the argument is building.
_It's my girl's birthday._
_What I tell you about messin with them tricks,_ Dante says, and starts walking toward the commotion.
_It'll mess with my game,_ Sticky says, following Dante.
_That's right._
_I just gotta handle something, though,_ Sticky says.
Dante laughs and shakes his head. He steps into the middle of the argument and yells over everybody: _That's a bitch-ass call, Rob!_
_What, I can't get a call?_ Rob says.
_Shoot for ball, man,_ Johnson says from the sideline. _Gotta_ _shoot for that one_.
_We ain't shootin for nuthin!_ Trey yells. _My man made_ _a call._
_Respect the man's call!_ someone else yells from the side.
Y'all know that wasn't no foul, Dallas says.
And nobody backs down in situations like these. There's too much at stake. Fifteen guys swelling up the sidelines means the team that loses will be waiting three games minimum. Street ball debates are part of the game; sometimes it's the team with the biggest mouths that holds court all day.
Carlos, a five-foot toothless Mexican, rolls off his bag on the homeless court and walks up to the pack. _There is no foul_ _here,_ he says with a heavy accent, pointing at Rob. _I watch_ _and this is very bad call_.
_Get off my court,_ Rob says, puffing up. _Before somebody_ _knocks your little midget-ass out_.
_This is bad call_. Carlos walks to the side a bit, doesn't look Rob in the eyes. _No way, your ball. This is very much_ _bad call_.
_I ain't playin,_ Rob says. He clenches his fists, takes a few steps forward.
The few businessmen eating lunch near the door drop their forks. These guys show up to see one of two things: a nasty dunk or a big-time altercation. And with Rob in Carlos's face and Slim pulling Trey away from Dallas, with everybody yelling stuff out at the same time, they have their altercation.
It's twelve-thirty on a Thursday. Pale businessmen watching black bodies posture and toss threats. The guys on the sidelines are black. The motionless bodies scattered across the homeless court are black. The two boys reaching skinny arms up the mouth of the soda machine are black. The Mexicans are black. Even Sticky, with his flashy passes and through-the-legs-around-the-back strut, is black.
Pale businessmen will take this story and hold everyone's attention back in the office. They'll all congregate around somebody's desk. The water cooler. In the men's room. _It's so_ _wild, man. You have to go check it out sometime_.
Fat Chuck comes down out of the bleachers with his sagging gray sweats. An overweight, always-smelling-like-tequila mulatto who shows up almost every day to watch but never plays. He goes right up to Rob and tries to talk reason. _Come on now, guys_. He places his hand on Rob's left bicep. _Now you know Jimmy gonna come runnin out his office if you_ _all keep it up_.
Rob pulls his arm out of Chuck's grasp. Glares. He turns his attention to the pack that's moving to the other side of the court. _You ain't gettin that,_ he says. _It's my rock. I ain't_ _movin one step from here_.
Fat Chuck backs up and watches him.
I Could Tell
you a lot about this game. . . .
How a dark gym like Lincoln Rec is a different world. Full of theft and dunk, smooth jumpers and fragile egos. Full of its own funky politics and stratification. Music bleeding out of old rattling speakers from open to close. Old rhythm and blues. Stevie Wonder. Aretha Franklin. Funk. Motown. Marvin Gaye. Sometimes Jimmy gets talked into hard-core rap on weekends. Or Trey sneaks in his three-year-old demo tape.
Always music.
There are fat rats that scurry through the lane on game point. Beady eyes on the man with the ball. There are roaches congregating under the bleachers.
There is so much dust on the slick floor that sometimes a guy will go to stop and slide right out of the gym. Every time there's a break in the action, ten guys put palm to sole for grip.
There are a hundred different ways of talking and a thousand uses of the word _motherfucker._
There are no women.
In the winter there are so many homeless bodies spread out across court two you can hardly see the floor. There are leaks when it rains. Rusted pots are set out to collect heavy drops. Sometimes a guy will track in mud and delay the games. Jimmy sets out a twenty-five-dollar heater and everybody puts their hands up to it before they play.
In the summer you can hear the foundation cracking. The walls, the ceiling. Like the old gym is stretching out its stiff arms and legs.
There are faded bloodstains and tooth marks in the wood. Arguments that end with a gun being pulled. Like a year ago when Old-man Perkins couldn't get his call one crowded Saturday. Guy laughed right in his face. Perkins calmly walked over to the sideline and pulled a forty-five out of a gym bag. _Now, whose ball is it?_ he said, holding the gun limp at his side. Drips of sweat running down his wrinkled forehead.
_Your ball, old man,_ the guy said, backing up with his hands in the air.
And everybody shows up for a different reason. A potpourri of ballers:
Some guys come because they're regulars. Used to seeing all the fellas on a daily basis.
Some show for the first time on a tip from a friend. Try their skills in the best pickup around to see if they can hang.
A couple NBA cats roll through when it's their off-season.
Some jokers walk through the doors looking for nothing more than a sweat. They come in wearing wet suit–looking wraps around bulging stomachs. Keep love handles away without hopping on a treadmill. They get run out of the gym after one game.
Some guys come to drop rainbow jumpers from deep.
Some come to throw their bodies around down low. To bang with the big boys.
Some guys pull in every day because they love talking trash. Barbershop talk in high-tops. They always have something to say when they score. They have something to say when anybody scores.
Some guys show up because they have nothing better to do.
Some guys come because they didn't play much in college. Get the sour taste out of their mouth by busting somebody up.
Some cause they didn't play much in high school.
Some guys show up drunk. High. Tweaking.
Some of the best ballers roll in wearing a work shirt and jeans. Some of the worst have top-of-the-line sneakers, top-of-the-line gym shorts, the most effective and smooth-looking knee braces. Basketball runway show.
Some guys come to dunk on somebody. They come to hype up all the loudmouths on the sideline with a rim-rocking two-hand bash.
Some don't mind being one of the loudmouths that gets hyped when the guy who comes to dunk on somebody, dunks on somebody.
Sticky shows up cause the game's his life and the guys are like family.
Some guys stay behind when the gym closes, curl up on their spot on court two with the rest of the homeless.
Some come to score enough junk to soothe junky bones. Chronic. Ups. Downs. Meth. Crack. X. Or to score shiny watches. Gold bracelets. Platinum hoop earrings. Heavy ropes.
Some come to sell.
Some feel like they're part of something. Like a book club or church.
Some show up because they just got off work. Doing all-night security or hustling on the streets.
Sometimes a cop is guarding a robber. Everybody has a joke when that happens.
Some guys roll in because they're addicted to competition. Gotta beat somebody in something to get happy.
Some cause it's the only place in the world they get respect. The only place they have any real control.
But no matter who they are, or why they come, every one of them squints their eyes when they step foot out of the dark gym and back into the bright world that waits outside.
Baby Dressed Up
sexy on Friday and Saturday nights after dinner. All through the time Mico was hanging around the apartment.
She'd high-heel through the bedroom door frame like a movie star, creeping slow-motion like a cat sneaking up on a mouse. Black skirt riding high, slits on both sides. Or the sequined pink deal with the silver zipper down the front. Legs shooting out six laps around the track when she plopped down on the couch with makeup and a mirror.
Sticky couldn't take six-year-old eyes off his mom when she dressed like that.
On the way out the door one Saturday night, the day Mico moved all his stuff in, Baby butterfly-kissed Sticky on the cheek. Like she always did. Fluttering eyelashes that tickled his skin. _Mommy's gotta go to work now,_ she said. _My little Sticky Boy_. She squeezed his shoulders and smiled so big her eyes almost shut. And no matter how uneasy Sticky was about staying with some random guy again, he couldn't help himself: when she smiled, he smiled. Even if they were just playing it as a game, see who could keep a straight face longest, he was always the first to fold.
_Bye, Baby,_ Sticky whispered when she hugged him.
_Bye-bye, little boy_. She touched a finger to his nose. _My_ _little dirt-faced Sticky Boy._
She went over to Mico and kissed him all over his face. _I'm so glad you're here, Mico. Little Sticky Boy is too_. She stood back and put her hands on her hips. _This is just going to work_ _out so great. You'll see._
_Just go make us some money,_ Mico said. _Then it'll be all_ _good_. He glanced over at the fridge and pointed. _You gonna_ _bring over that six-pack before you leave?_
Baby hopped over to the fridge like a bunny, giggling. She opened the door and reached her hands in for the beer.
Mico and Baby were a six-month team. They counted twenties every Sunday morning while Sticky watched cartoons. Mico had connections on the street, made sure Baby was safe and took his cut. Baby liked eating good meals and getting her hair done at a salon. She also liked having a man in the house. A big strong man with a confident walk. If it had been up to her, Mico would have stayed a lot longer than the six months. Even if he did punch her in the mouth once when she'd talked back. Even if when he got super-high he sometimes thought it was funny to toss empty beer cans at her sleeping boy.
But Mico ran off with a girl even younger and prettier than Baby. Took off in the middle of the night and left half his stuff.
Back before Mico showed up it was just Baby and Sticky living together. Eating tuna out of a can or noodles with butter. Cold hot dogs. Both walking around the dim, run-down apartment with bare feet. Plates of ceiling paint would flake off at night, float to the ground like little flying saucers. There were trails of ants. Roaches. Daddy longlegs sleeping in every corner. Dust balls spinning across the dull kitchen tile when a gust of wind came through the rusty screen.
Baby didn't work, and Sticky didn't go to school.
They slept together on a broken futon bed in the middle of the room. Their apartment in a shady part of Long Beach. Rumbling trains would wake them throughout the night. A loud earthquake of power rattling their thin windows.
During the day, Baby was either dancing around the place on her toes or sobbing under the covers. There was nothing in between. She either rolled up a magazine like a mike and sang with her favorite radio songs, or she sat in the open window with tears streaming down her face, saying: _I_ _swear um gonna do it this time. I swear to God um gonna_ _jump_. She'd look down at the sidewalk with both hands white-knuckled against the window frame. _You'll be better off_ without your crazy mom.
Before falling asleep each night, Baby would tell Sticky stories about his dad. And every night he was a different person. An actor. A construction worker. The head of some prestigious company overseas. Sticky would snuggle in close to Baby, shut his eyes and try to picture it all in his head.
Some nights his dad loved sports. Lettered in everything back in high school. During big games on TV, he would sit Baby down and explain all the rules. Other nights he hated the violence of football, preferred sinking into a comfortable chair with a thick Russian novel. Sipping gin and puffing a cigar.
Sometimes he'd sailed across the Pacific in record time. Battled high winds and monster whitecaps. No lifejacket. Then she'd turn it all around a week later, say the man's only real fear in the world was the Lord's dark oceans.
A lot of times she told Sticky his dad was dead. Shot down in a foreign country. The medals still in a box back in Virginia. Or he was taken out during a big-time drug bust while working for the FBI. One time he'd died when his car spun out of control and launched off the Golden Gate Bridge. Sticky would picture the car flipping over, again and again, then the giant splash.
But other nights Baby would claim his dad had placed a call that very day. That he was thinking about swinging by for a quick visit.
The story Sticky really believed, though, was the one Baby told most often. His dad was a country-western singer she'd met only once. _He had the voice of an angel,_ she'd say all dramatic, staring into the flickering light hanging from the ceiling. _And those boots, little boy. When you get old enough_ _you have to buy yourself a pair of boots_.
Sticky had this story memorized. How when his dad came out to L.A. from Virginia, to try to make the jump into movies, Baby packed Sticky up and followed her singer out by bus. How she and Sticky stayed six months at a YMCA in East L.A., shared a bathroom with the entire third floor. How at night she would put him in an old TV box so he wouldn't crawl away. Cover him with two or three pillowcases from the Salvation Army. All that effort and her singer didn't return even one of her phone calls. Didn't answer even one of her letters.
Sticky believed this story because Baby would lower her eyes when she told it. She'd get all quiet and stare at the floor.
Baby stood at the door waving before she left. She did a twirl and waved some more. Mico shook his head and turned to the TV. He pulled a can from the plastic rings. Sticky watched Baby and laughed.
She laughed with Sticky as she waved. Got on one knee and began blowing dramatic kisses.
_Don't I look pretty, Mico?_ she said, standing up, resting a hand on her twenty-four-year-old hip and spinning around like a fashion model. Subtly covering her birthmark cheek with her free hand. _Don't you think I'll be the prettiest one on_ _the block?_
_You probably blowin a big score with all this messin_ _around,_ Mico said.
Baby made a face, and Sticky laughed.
That afternoon Sticky had watched Mico show up with a pickup truck full of his stuff. Watched him take load after load into the apartment and dump it. As Mico brought stuff in, Baby took Sticky's stuff out of their one bedroom and piled it up next to the TV.
_This will be so great for you,_ she said as Sticky watched her. _Every boy needs a daddy. Hey, maybe he'll wanna throw_ _the football around if you ask him_. She skipped back into the room singing with a B-52 song playing on the clock radio.
In a few minutes she skipped back out with Sticky's pillow, his two blankets folded. _We're gonna be a real family_ _now,_ she told him.
Soon as Baby was out the door, Sticky ducked into a corner of the living room and sat with his back against the wall.
He stood up and sat back down again.
Back up and back down.
Back up and back down.
Back up and back down.
Mico looked over and Sticky stopped. They sized each other up for a sec, then Mico tipped his beer and went back to the TV. Sticky went up and down a half dozen more times until it felt right. When he sat for good he wrapped both arms around his knees and spied the room: Mico's jacket hanging off the kitchen table like a leather waterfall. The bedroom door, closed for the first time ever. Keep Out. Mico kicking his feet up on the table and picking something from his teeth.
Sticky shifted around a bit. He pulled his legs in tighter and rested his chin on his knees.
_Why you so quiet, kid?_ Mico said. He took a healthy pinch of something out of his black smoke box and started rolling it up like a cigarette. _Come sit up here with me and I'll show_ _you how you roll em_. He licked the Zig-Zag and pulled a lighter out of his pocket. Aimed and sucked in. He held the smoke in and talked at the same time. _All the little kids in_ _your school would be impressed_. He looked at Sticky and let the thick gray smoke snake through his lips.
Sticky looked down and picked at the rug between his feet.
_What, you ain't wanna be the big man on campus?_ He pulled in another long drag and held it. _Couple years,_ _man . . . You gonna start chasin the ladies_.
Mico cracked open a can of beer and scooped up the remote. He blew the smoke through his wide nostrils and took a long swig. Flipped through channels.
Sticky kept his mouth closed. He stared at Mico's sharp brown face. His nappy black hair down to the shoulders. The green words and pictures scribbled up and down both muscled arms. The way he sank into the couch and rested his beer can between his legs.
_Your time's gonna come,_ Mico said, flipping through channels. _Grab one a them little cheerleaders and pull her behind the lockers_. He pulled in a drag and blew it out. _Them's_
_the best days of your life, kid. Chasin after the fine girls. Even_ _the not-so-fine ones_.
Mico laughed and turned his head to look at Sticky. _I was_ _a equal-opportunity type of dude_.
Sticky watched Mico tilt the can against his mouth again. Watched his lips work the can like a baby with a bottle. The front of his neck driving up and down as he swallowed.
That night, Mico stayed up late watching late-night comics and _I Love Lucy_ reruns. He talked on the phone with a deep voice and kicked his dirty boots up on the end of the couch.
Sticky tried to stay awake too. Gave everything to keep heavy lids from sliding down tired eyes. He went back and forth between the TV and Mico. Watched guests wave to a cheering crowd before taking a seat next to Leno, and Mico crumple up empty beer can after empty beer can. Watched singer Bette Midler extend her free hand out whenever she went after a high note, watched Mico flick cigarette ashes onto one of Baby's beauty magazines. Sticky listened to Lucy's jokes and Mico's laugh. Ricky's heavy accent and the long, deep belches Mico blew up at the ceiling.
The longer Sticky watched Mico, the more he warmed up to him. He liked how Mico laughed at everything. A deep manly laugh. And he was so big and strong. Like he could beat most people in a fight. Sticky pictured Mico walking with him down the street to the market. He pictured everybody making room as they walked past, not wanting any trouble.
At one point Sticky even thought: _Maybe this guy could_ _be my dad_.
The later it got, though, the more Sticky lost the battle with his tired eyes. And soon he drifted into sleep.
Sticky woke up an hour later with Mico tapping him on the forehead.
_What?_ he yelled, shooting to his feet. _What?_ His blurry eyes darting around the room.
Mico pulled a cigarette from his mouth, let it hang between two fingers. _You should go to sleep, kid_. He unfolded one of Sticky's blankets and spread it out on the rug. _Sleep on_ _this,_ he said, pointing to the blanket.
Sticky walked over to the blanket and went to his knees. Mico tossed another blanket on his lap and walked to the fridge. He reached in and pulled out another beer, cracked it open.
Sticky smoothed the blanket over his legs, went to lay his head down but realized he didn't have his pillow.
_Pillow's over there,_ Mico said, pointing toward the table. He tilted the can back and sucked down a few swigs. He leaned his elbows on the blotchy counter and looked all around the tiny apartment. His eyes drooped, head swayed. He laughed and shook his head, then stumbled back to his spot on the couch.
Sticky was up fast. He grabbed the pillow with one hand and pulled it behind him. But as he passed the table, his pillow knocked Mico's black smoke box to the ground. The box tumbled and landed upside down. Clumps of pot scattering everywhere.
_What the hell you doin?_ Mico yelled, quickly reaching for the box, turning it right side up.
_Sorry,_ Sticky said, nervously trying to pick the crumpled green out of the thick, dark rug.
Mico crushed his cigarette into Baby's magazine and got down on his knees. He pushed Sticky's little hands away and sifted through the rug himself, trying to rescue some of the bigger clumps. _Man, I just bought this shit_.
_I'm sorry,_ Sticky said.
Mico quickly realized it was useless, that most of his stuff had been swallowed by long tentacles of rug, and then he flipped.
He jumped up and gripped Sticky by the ear. Shoved his nose into the rug and told him: _Sorry don't do nothin for me_ _now, do it?_ He pushed Sticky's face into the rug so hard that his cheeks and lips smashed. _See that? Huh? I just bought all_ _that weed yesterday_.
_I'm sorry,_ Sticky said. _I'm sorry._
Mico jerked Sticky's head again and let go.
Sticky sat up quick like he'd just been held under water. Almost drowned. He sucked air through his nose. Fought the lump in his throat, swallowed at it a couple times and made a frown out of his eyebrows.
Mico stuck a finger in Sticky's face, told him: _I'm gonna_ _tell you how this is gonna go, kid_. He took a few deep breaths, trying to calm down. _Now that I'm stayin here, we go by my_ _rules. I don't care how it went with your moms. All that's over._ _From now on we go by my rules_.
He pulled a new cigarette out of his pack and lit it. Sucked in hard and threw the pack back onto the table. _Now, I ain't_ _never had to mess with no kid before, but I'm gonna do like my_ _pops did with me. When I messed up, my pops whipped me good_.
Mico pulled in another drag and blew it to the side of Sticky's red face. _That's how my pops made me a man. Now, if_ _I'm gonna stay here in this crappy apartment_. . . He circled his finger around the room. _If I gotta live like this, man, you_ _damn for sure we doin things my way_.
He held the cigarette in his left hand, reached around with his right and scratched his shoulder blade. Transferred the cigarette back into his right hand. _I'm gonna make you_ _grow up to be a man_.
Mico reached out quick and grabbed the back of Sticky's head, pushed his face into the couch. He held Sticky's head still with a strong left hand and stuck the burning cigarette against the thin skin behind his left ear.
The sizzle on skin made Sticky swallow everything. Gasping and sucking. Choking. Pushing with his hands to get away. He was sucking in all the pain and dry-heaving through his mouth and nose.
Mico held his head tight, screwed the embers in.
The skin melting and dripping. The smell like burning rags. Everything snapping and cracking and breaking in his ear.
He held the cigarette there until Sticky's charred skin pulled every last bit of the red out.
Warm piss ran down Sticky's jeans. Darkened the faded blue denim. Pooled on the rug in front of his bare curled toes.
_Sometimes I Think_
_if I don't make it to the NBA I'll kill myself. I know_ _it don't sound so good when I say it, Annie, but that's how I_ _feel. There ain't nothin else I wanna do. Just play ball. I mean, I_ _hear them people talkin bout how hard it is to make it and all_ _that, but I know I could do it. Dallas says if I keep workin on_ _my game I got a good chance. Slim, too. They said I got the intangibles. Old-man Perkins says if he was startin up a squad_ _from scratch he'd be lookin for a point guard just like me._ _Someone who could score points_ and _get assists_.
_It's like this, Annie, God puts us here for a reason. We all_ _born with somethin we could do good, but it's up to us to make_ _sure we use it. That's why I play ball so much. I ain't gonna lie,_ _I think God put me here to play ball. And when I go to pray at_ _night, I pray so I could get better and better. That's why I grew_ _so much last year. That's why I could shoot so good. It ain't just_ _me doin it._
_This lady I used to stay with told me all about what could_ _happen when you pray. And she was true. I know she was now._
_Last week I was walkin back from Lincoln Rec, you know,_ _and I just started thinkin about all this. It was after I had one a_ _my best days ever. I couldn't miss a jumper, my dribbles were_ _super tight, I was swipin the ball left and right from everybody._ _I remember right where I was when I started thinkin about it:_ _corner of Washington and Grand View, right outside the_ Foster's Freeze where this crazy white dude was strummin his _guitar. I remember it was gettin all dark and there wasn't too_ _many cars out. I sat down at the bus stop there and thought_ _about it. I couldn't believe what's happenin to me. How good I_ _could ball now. How I can take almost any guy I play against_ _now. And I know it ain't just me, Annie, but God, too. I know I_ _couldn't ball like that just by myself._
_Do you understand what a sweet life they got in the NBA?_ _They got fat bank accounts and big-ass houses. They got three,_ four cars each. BMWs and Expeditions. Range Rovers. And all _they gotta do is just play ball all day._
They get paid to play ball, Annie. That's crazy.
_Sometimes when I'm sitting in class I picture what it would_ _be like if I got there. The announcer sayin my name over the_ P.A., the crowd holding up signs, me chillin out back at the _hotel after a big game, watchin some highlight I did on_ SportsCenter. When I'm walkin through the airport people _pointin at me and sayin, "Is that Sticky Reichard? Nah, for_ _real, is that Sticky Reichard?"_
_When I think about that too much, my stomach starts gettin all messed up._
_I know it don't sound good, Annie, but I think if I couldn't_ _make it I wouldn't wanna be around no more. Cause it's all I_ _got in my life, you know? Playin ball. It's all I got in the whole_ _world. And if I couldn't make it, I woulda been wrong all this_ _time about God's plan._
_But you ain't gotta worry about all that, girl. Cause I swear_ _to you, man, one day I'm gonna make it to the NBA. . . ._
Francine Was All
smiles when she drove up to Sticky's foster care pad in her old-school Volkswagen van. Bumper stickers about Greenpeace and the Dodgers. Christian fish. Shiny black cross hanging like a pendulum from the rearview mirror.
Francine was the first of the foster ladies.
She had long red-gray hair and freckles. A silver cross dangling from her fragile neck. She showed up for Sticky three days after he turned nine.
The night before the pickup, all the counselors horseshoed around Sticky in the TV room. Told him how lucky he was.
_This is a perfect match,_ Counselor Jenny said, and everybody agreed.
_She picked you out of everyone_ , Counselor Amy said.
Sticky yanked his socks up and scrunched them back down. Yanked up and scrunched down.
_Yeah, how often does somebody looking to adopt pick a_ _nine-year-old?_ Jenny said, looking to the old Mexican director for the facts. _Most are looking for babies, right?_
_It's rare,_ the director said.
_She must think Sticky's pretty special,_ Jenny said.
Amy stroked Sticky's hair and smiled at him. _Plus you're_ _such a tough little guy,_ she said. She looked to Jenny, told her: _He didn't even cry when he first came here. Most kids do, you_ _know_. She made a playful face to Sticky. _Do you even have_ _tear ducts in those eyes, mister tough guy?_
_But it's OK to cry, Sticky,_ Jenny said. _In fact, it's healthy to_ _cry. It can make you feel better._
Turned out Francine's husband had passed away, leaving her alone in their big house in Pasadena. All three of her own kids had grown up and graduated college. Moved away. She told the adoption agency that such a big lonely house should be shared with a child. _What better way for an old_ _lady like me to give back?_ she said, after pulling out Sticky's picture from a stack of thirty. _What could be better than giving a child like him an opportunity?_
And Francine wasn't just blowing smoke, she gave the situation everything she had. Hooked up three meals a day in the kitchen, told Sunday school stories by Sticky's bed until he fell asleep at night. She took him to movies and museums and amusement parks. Held up multiplication flash cards when she found out he bombed a math test. Every afternoon she'd be there to pick Sticky up from school, her van pulled along the curb just like any other kid's mom.
One Friday after school, Sticky pulled open the van door and spotted a wrapped package sitting on the passenger seat. _What's this?_ he said.
_It's for you,_ Francine said.
Sticky stood there a sec, ran through possible holidays in his head. He picked the box up and set it back down. _But it_ _ain't my birthday or nothin_.
_I know that,_ Francine said, and she laughed. _It's just because I like you. Now, go on and open it_.
Sticky climbed into the seat and ripped through the baseball wrapping paper. Tossed it to his feet. He opened the box and pulled out a brand-new black suede jacket, held it out in front of his excited face.
Francine took her hands off the wheel, folded them in her lap. Her face was frozen in a smile.
Sticky reached back in the box, pulled out a white collared shirt and a pair of black pants.
_You have to have nice clothes where we're going tonight,_ Francine said.
They drove straight to Santa Monica from the school. Sat in heavy traffic on the 110 with everybody else. Traffic on the 10 West. They listened to talk radio and the sound of cars gassing and breaking. The smell of exhaust floated in through their open windows.
When Francine finally pulled off the 10 at Lincoln, she headed west on Broadway. They inched through Third Street Promenade foot traffic and cars waiting to pull into parking garages. Out the window Sticky spied the exact spot he used to beg for change with Baby. Pictured himself holding out the white bowl and making the sad face Baby taught him. The felt-penned sign around his neck blowing into his face when the wind picked up. Pictured Baby right behind him, sitting Indian style and humming to herself.
_Here we are,_ Francine said as she pulled up to the Loews Hotel lobby, shut off the engine and handed the keys to the valet guy. _This is the place_.
Up in the fancy room, Francine came out of the bathroom wearing a long black dress and lipstick. High heels. Long silver earrings that dangled over her bare freckled shoulders. She helped Sticky tuck his new shirt into his new pants. Held the jacket out so he could put one arm in and then the other.
When Sticky was all set she took out a blow-dryer and ran a brush through her wet hair. _We're going to eat at a place_ _called Ivy at the Shore,_ she shouted over the hot air. _It's a really nice place. My husband took me there for every one of our_ _anniversaries_. She flipped off the blow-dryer and set it down. She spun around in the mirror and then turned her attention to Sticky. _Now I'm taking you._
At dinner Francine taught Sticky about table manners: where to place the napkin in his lap, where to keep hands and elbows, how to hold the menu, which fork to use and at what time. Sticky sat stiff and listened to everything she said.
In the dim light, and with his new gear, he wondered if he looked like he belonged. Or could people tell it was his first time inside a restaurant. Ever. That it felt like a foreign country to him.
He watched a boy sitting three tables down wearing a tie. Watched the way the boy talked to adults and ordered for himself, the way he sipped soup from a spoon and dabbed at his mouth with a napkin. Every move seemed so natural. Sticky swore to himself right then and there that when he got older, had money of his own, he'd be eating at places like this every single night.
Before the food came out Francine reached over and took Sticky's hands. She closed her eyes and began a prayer: _Thank you, Lord, for this wonderful night, thank you for_ _bringing Sticky and me into one another's lives. Lord, one day_ _Sticky, too, will come to you. . . ._
As Francine went on, Sticky kept his eyes open. He watched the wrinkles in her chin stretch and fall as she spoke, her eyelids twitch. She always talked to him about God. Read Bible passages each morning while he wolfed down eggs and toast. She told him about Jesus and heaven, how to lead life like a true Christian. He could never figure out what to make of all that talk, but he liked that her words were aimed at him and nobody else.
Just as Francine released Sticky's hands and opened her eyes, the waitress set down their plates.
_Let's eat,_ Francine said.
But a year into things, Francine was diagnosed with cancer. Told she had to undergo immediate and intensive treatment just to have a chance at pulling through.
Her daughter flew in from New York two days after they found out, drove the van when they took Sticky back to his foster care pad.
They dropped him off late at night.
_This is only temporary,_ Francine said outside the van, tears running down her face. Her daughter stayed inside the van, left the motor running. _I promise,_ Francine said. _The_ _Lord will make sure of it_. Her face was outlined by a glowing sliver of moon and Sticky felt bad for her. _When I get better_ _I'm going to rush back here and take you home._
And as she stared at him, Sticky thought it was true what she was saying. This lady. She would come back for him.
They looked at each other for a while, neither of them moving or saying a word. Then Francine smiled through her tears and took both of his hands in hers. She kneeled so they were eye-level and told him: _I love you, Sticky._
She hugged him tight.
Sticky didn't cry when her old Volkswagen van pulled out of the driveway and into the street. The old Mexican director's hand on his shoulder. The cold wind on the back of his neck. _Is this when you're supposed to cry?_ he wondered as the van moved slowly down the long, busy road and mixed with other taillights. _Is this when you're supposed to feel sad_ _and cry?_ Because his eyes were as dry as a Santa Ana.
Francine died three months later in a hospital just outside Manhattan. Sticky found out when he overheard some counselors whispering in the office.
When he heard it for a second time later that week, a big sit-down kinda conversation with the old Mexican director, he acted like he didn't know.
Jimmy Comes Running
out the office when he hears all the racket. _Everyb-b-b-b-body g-g-g-get out!_ he says, and points at the door.
Nobody notices.
He moves up to the core of the pack, invisible. Takes quick strained breaths.
Jimmy is: eyes the size of golf balls in thick Coke-bottle glasses, overgrown crop that starts a thumb's width from his bushy eyebrows, old beat-up flea-bitten sweatshirt zipped up to the throat: ARMY FOOTBALL. He yanks the rock from Trey's grasp and stomps his foot on the ground, yells: _I s-s-s-s-said, everyb-b-b-b-body out!_
Ballers stop dead and turn to check him this time.
Jimmy's already shut Lincoln Rec down twice this summer. Stood by the soda machine with his arms crossed while everybody grabbed their stuff and filed out slow. First time after Big Mac blasted some first-timer in the mouth and wouldn't stop kicking after he hit the ground. Guy's teeth went through his bottom lip. Blood all over the low post area. The second time when Old-man Perkins pulled a gun and dudes hit the ground, ducked behind bleachers. But Jimmy's bluffed on a handful of other occasions. When arguments build up like volcanoes and everybody blows at once. A chorus of over-the-top cursing and street ball threatening.
_We ain't done nuthin,_ New York yells.
_Nah,_ Rob says. _I ain't movin one step_.
_G-g-g-g-get out!_ Jimmy says again, swinging an arm through the air and almost knocking off his glasses. He straightens himself out, adds: _N-N-N-N-NOW!_
_Aah, come on, Jimmy,_ Dallas says.
_It don't gotta be like all that,_ Johnson says.
Old-man Perkins jumps off the bleachers and throws somebody's towel onto the court. _I ain't even played one_ _game yet_.
Dante walks up cool as a cat and puts a hand on Jimmy's shoulder. _Just a simple misunderstanding, Jimmy,_ he says. _We_ _about to shoot for it right now, as a matter of fact_. He looks in Jimmy's magnified eyes and smiles white teeth.
Jimmy takes a deep breath to slow himself down. He looks back at Dante and shakes his head. _You kn-kn-kn-kn-know th-th-this ain't r-right, D-D-D_.
_I know it, Jimmy. I know it. The question is: What are we_ _gonna do about it now?_
_All the c-c-c-c-cursin and h-h-h-h-h-h-hollerin_. Jimmy squints his eyes and scrunches up his face to get all the words out.
_You know how brothers be actin sometimes,_ Dante tells him.
Jimmy puts a hand on a hip and looks around at all the guys' faces. Shakes his head, disgusted. _B-b-b-but th-th-they's_ _imp-p-p-portant off-ff-ff-ff-ff-ffices next door,_ he says, and points at the east wall.
_I know it, Jimmy. I know about all that_. Dante reaches in slowly, takes the ball from Jimmy's hands and bounces it a couple times off the hardwood. He palms the ball with his left hand and fingers his beard with the right. _But we about_ to settle all this jazz right now.
Dante spins around and yells out: _Hey, yo, Rob!_
_What?_ Rob yells back, sitting at half-court with his legs sticking straight out, weight on the palms of his hands behind him.
_You made a call, shoot for it_. Dante rolls the ball to Rob. Rob gets up slow, dribbles a few times and struts to the top of the key. _Messed up I gotta shoot,_ he mumbles under his breath. _White boy tries to tackle me and now I gotta shoot_ _for it_.
Rob's the light-skinned black dude who preaches non-stop, up in the bleachers after games, about Marcus Garvey and Malcolm X. About going back to Africa and taking back from the "white devil." All that passionate preaching and the very next day he'll bring in a white chick to watch him play.
He steps up to the top of the key and takes a couple more bounces. Wipes hands on shorts and lofts up a high-arcing knuckleball that gets a good bounce on the rim and rolls in. _Water!_ he says, and holds his right-hand follow-through in the air so everyone can check it. _My rock!_
_You right, baby,_ Dante says. _Your rock_.
_Ball don't lie,_ Trey says. He picks the ball up and sticks it in New York's face. New York slaps it out of his hands.
_What's the count?_ Slim says.
_Anybody got the score?_ Dallas says.
_My jumper's like water,_ Rob says.
_This is very bad call,_ Carlos says, and he gradually retreats back to his spot on the homeless court.
New York stares at Rob and laughs. _Worst call I ever seen,_ he says. He shakes his head and walks toward the baseline.
_Shut up and check ball,_ Rob says.
Dante puts his hand on Jimmy's shoulder again and spins him back toward the office. _See, it's all good, Jimmy. My_ _main man in charge. Boss man. Check it, we about to play_ _straight up now. Some good old-fashion ball and no more_ _quarrelin_. He walks Jimmy as far as the bleachers, pats him on his back and lets him go.
Jimmy takes a few steps and turns around. _N-n-no more_ _qu-qu-qu-qu-qu-qu-quarrelin, though, D-D-D-D,_ he says, shaking his finger.
_We done with all that, Jimmy,_ Dante says, with his hands in the air. _Good old-fashion ball from now on. The way Dr._ _James Naismith intended it to be played all them years ago_ _when he invented the game_.
Jimmy stands nodding his head at Dante for a few seconds. Then he turns around and walks into his office, reaches a hand back and pulls the door shut behind him.
On the Way
back to Millers, just a week after he met Anh-thu there last summer, Sticky made himself a promise: A dude like him wasn't leaving Millers empty-handed a second time.
He worked it all out in the back of a dark Number 3 bus. Next to some old lady who smelled like a wet sandbox. He came up with what route to take and how to keep his head in the game. Told himself over and over in his head: _Just stay_ _cool, man, stay cool_.
But it wasn't even a question, this trip he was jacking some khakis.
He busted in the open doors all business, stupid pop songs turned way too loud. The smell of cheap sample colognes. He steered straight up to the Anchor Blues and started searching for three or four pairs he could pull into a dressing room.
Walkman turned low enough this time to hear footsteps behind his back.
_Hey,_ Anh-thu said, coming at him from the side. _You're_ _back._ She was excited to see him.
Sticky gave her a nod.
She put her elbows on the metal rack and watched Sticky's mad search. _You need some help?_ she said.
Sticky shook her off. Pulled out one pair, checked tags twice (price and size) and then stuck them back on the rack. When the sticking-back sound didn't sound right, he pulled them off and stuck them back again. Pulled them off and stuck them back. He started to panic inside. Started sweating. Last thing he wanted to do in front of this pretty girl was act all retarded. But he couldn't stop himself. He pulled them off and stuck them back again.
Pulled them off and stuck them back.
Pulled them off and stuck them back.
Pulled them off and stuck them back.
Anh-thu didn't understand what was happening, but she had to do something. She ignored Sticky's refusal for help and stuck her hands in the mix. Right next to his. Acted like she didn't even notice Sticky's repetition. She pulled out a pair herself, looked at them, then stuck them back on the rack.
Sticky eventually got his cool back and they worked together, side by side.
_Listen,_ Anh-thu said as they continued going through the rack. I have to apologize about last week. About saying you _were my boyfriend. I just didn't know what I could say at_ _the time._
_I ain't worried about it,_ Sticky said.
_I just didn't want you to think I was weird_. A pair of pants slipped off the hanger and dropped to the floor. Anh-thu picked them up and smoothed out the creases. _I've seen you_ _around school and stuff, though. You're in Mrs. Edelson's econ_ _for third period, right?_
Sticky nodded, held a pair of khakis up to his jeans.
_I have her right after you. I've seen you walking out when_ _I'm walking in._
Sticky kept his eyes on the cotton, told her: _She boring._
_Tell me about it. And she's super scatterbrained. People say_ _she puts vodka in her orange juice every morning_. Anh-thu moved in closer to Sticky. She sifted through the pants shoulder to shoulder with him. _Thirty-four in the light ones, right?_
Sticky nodded.
_Plus I've seen you play basketball, too._
And that was what did it, man. Sticky dropped his hands from the search and looked right at her. This green-eyed girl with long black hair. This perfect face floating through his head all week between runs at Lincoln Rec. Just like that: Anh-thu mentions one thing about hoops and Sticky's at full attention.
_Yeah, you're number seven on JV,_ she said. She smiled when she saw she had Sticky's attention. _I totally go to all the_ _games. Unless I have to work or something. I love basketball_.
_You check out our games?_ Sticky said, dropping his cool for a couple beats. Trying to picture her up in the stands and him shooting free throws with the game on the line.
_Yeah, and you're the best one on the team. You make all the_ _points_. She made a face and put her hands on her hips. _Hey,_ _just cause I'm a girl doesn't mean I don't know what's up_.
_Nah, I didn't say all that._
Anh-thu pulled a thirty-four off the rack and held it out. _Is this kinda what you're looking for?_
Sticky took the pants. _Yeah, these are smooth_.
She pulled keys from her pocket and motioned for Sticky to follow.
They weaved in and out of intense late-night shoppers, through racks of shirts, hats, socks, freshly dressed mannequins, and slipped into the dressing room area, where Anh-thu unlocked one of the doors and let Sticky in. _Let me_ know if you need another size or anything, she said.
She smiled big and closed the door behind her.
Sticky planted himself on the bench and stared at the pair of khakis. Not even a dime in his pockets. This girl had been to games, man. She knew his jersey number. And she was fine. Smelled good too. Sticky sat there awhile, in the dressing room, thinking about Anh-thu and the pair of pants in his lap.
Finally, he walked out of the dressing room without trying them on. He handed the khakis off to Anh-thu and told her: _These didn't really fit too good_.
Really? Anh-thu said. You wanna try a different size?
_Nah,_ Sticky said. He scratched his head, leaned against a shelf of shirts, and when it wobbled, stood up straight again. _I don't think I want em no more_.
They stood next to each other in silence for a few seconds. Anh-thu folded the pants up perfect, let them fall loose and started folding all over. She watched Sticky out of the corner of her eye.
Sticky pulled a T-shirt off the rack, stood staring at the design for a little bit and then pulled out a different one.
Anh-thu put the pants on a hanger and hung them on the wrong rack. She turned and fished out Sticky's eyes, cleared her throat. _I don't know what you have to do or whatever, but I was gonna walk through the promenade after work._ _Before I catch the bus home_. She reached down and adjusted the strap on one of her flip-flops. _I don't know, maybe you'd_ _wanna go with me. I mean, if you're going that way or_ _something?_
Sticky shrugged his shoulders and buried his hands in his pockets. _That's cool,_ he said, keeping away from her eyes.
_Great_ , Anh-thu said. _Let me check out_.
They moved through the packed mall without saying a word. Through the food court and into the well-lit promenade. Waited for the green Walk sign with everybody else, and then walked across Broadway.
They strolled past Hear Music, Borders, the new Rip Curl store, the movie theater with its two-story list of films and times. They walked through a crowd that had gathered around a guy finger-picking his guitar and singing a James Taylor cover.
The night air was cool. The moon glowed through a thin patch of clouds. Sometimes Sticky would think up a question to ask, about classes or kids they both knew from school, but they all seemed dumb so he kept them to himself.
They passed Urban Outfitters and Mario's Pizza, the glass walls of World Gym and the long curving line coming out of Starbucks. A young black kid dressed in an all-glitter suit busted fancy dance steps to Michael Jackson's greatest hits. He had only one glove on and everything. People cheered. Anh-thu kept pulling her hair behind her ears, out of her face, only to have it slip forward again.
Sticky's hand accidentally brushed against Anh-thu's a couple times, so he stuck it in his pocket. There was a subtle squeak coming from one of his Nikes, so he tried to step soft with that foot to make it go away.
It was across a crowded Santa Monica Boulevard and then west on Arizona. Sticky led the way and Anh-thu followed. They crossed Ocean Street to the beach side and had to high-step through a pack of Venice Beach overflows pounding bongos. They stopped at the bridge that goes over the PCH to the sand. Leaned elbows against the wood railing and stared out at the ocean.
_I have to admit one thing to you,_ Anh-thu said, breaking a long silence.
_OK_ , Sticky said.
_Just so you know, my girlfriends made something up when_ _we watched your games_. The wind was strong and Anh-thu had to keep pushing her hair out of her face. _They kinda pretended like you and me were together. Like boyfriend-girlfriend. I'm sure it's cause I always talk about what a great_ _player you are. And cause I told em I thought you were cute_. She pulled a rubber band from her pocket and put it in her mouth. Gathered her hair for a ponytail and double-wrapped. _I guess that's why I came up to you in the store_ _like that._
_I never seen you at no games._
_You're probably just concentrating. Like you're supposed to._
Sticky tossed a piece of ice plant over the cliff. _We're first_ _place in league._
_I know._
_And I'm getting called up to varsity for play-offs. Coach_ _said he'd get me some time, too. I know it's ways off, but I_ _can't wait._
_I'll totally be going to the play-offs,_ Anh-thu said, and pushed Sticky, all jazzed. _My girlfriend Laura and me already_ _talked about it. But I didn't know you were gonna be playing_ _too. That's so cool_.
Her face went straight and she said: _Hey, why didn't you_ _go with me to get hot chocolate last time?_
Sticky put his hands in his pockets and shrugged his shoulders. He didn't feel like getting into the whole thing about him never having money. He barely knew this girl.
Cars whizzed by on the PCH below. A trail of red lights going north, white coming south. All the different motors blending like the hum Baby used to make pushing around her broken-down vacuum. Sometimes a group of people would walk by on their way to the beach. Swinging bags full of blankets and wine. They'd disappear around the bend for a few minutes and then come out smaller on the other side of the bridge. In the distance the Santa Monica Ferris wheel was still spinning tourists around and around. Little arms and legs poking out of old-style seats. There were the faint smells of popcorn and dying seaweed in the air. The muted sound of waves rolling in across the sand.
When Sticky didn't say anything for a while, Anh-thu wondered if he was getting bored. _Maybe I should let you go,_ she said, straightening up. _I have to catch the bus anyway_.
They looked at each other for an awkward second. Sticky opened his mouth to say something but decided to keep it put away.
_My dad gets worried when I take the bus too late._
Sticky made the move when Anh-thu looked to the ground. Stuck his face in hers. Touched his lips on her lips and wrapped hands around her back.
Anh-thu pressed against the railing and placed her soft hands on his face.
They looked at each other. Anh-thu giggled a little. She reached for his hand.
Sticky led her down to where the bridge starts and helped her climb over the railing. _I can't believe we're doing_ _this,_ she said as they crept along a narrow stretch of cliff and ducked underneath the bridge, out of sight.
There were abandoned fast-food bags at their feet. Styrofoam cups. A soiled blanket. Pieces of cardboard. Beer bottles that had settled in a ditch by one of the thick concrete pillars. Sticky kissed Anh-thu again. They tugged at each other's clothes.
_What are we doing?_ Anh-thu said whenever they separated to deal with a stubborn button or snap.
They sat on the dirt, half dressed.
Sticky reached a hand up her blouse. Anh-thu fumbled with Sticky's zipper. No layer scam meant no khakis underneath. No stop in the action because of a crime.
There were voices of people walking over their heads. Spanish. English. French and Japanese. Someone dropped a glass and the shattering sound echoed under the bridge. When one of their feet slipped a little, slid across the loose dirt, a small cloud of dust would rise up into the bottom of the bridge and separate.
Then it was over. Sticky stood up quick and pulled his jeans over himself. Zipped up. Anh-thu straightened her skirt and stood up too. They both put themselves back together in silence.
_OK, I think I have to go now,_ Anh-thu said, giggling. _I_ _have to catch the bus_.
Sticky stepped over two faded Pepsi cans and an abandoned flannel. He got in close to Anh-thu, looked right in her eyes and pulled the loose rubber band from her hair. Anh-thu's black hair spilled down her shirt, covered her name tag. When she leaned her head back and shook her hair out, Sticky got a weird feeling in his stomach. Like everything was the way it was supposed to be: the cool breeze, the sound of the highway and the beach, the bridge and cliff covering them like they were in their own little world. He'd never had this feeling before. _You wanna be my_ _girl?_ he said, slipping his hands into his pockets.
Anh-thu looked right back into Sticky's eyes, caught her lips breaking into a smile and made her face go straight. _Yes,_ she said. _I totally do_. She reached up and put her hands on the back of Sticky's head. Went up on her toes and kissed his cheek. She looked up into his eyes and let herself smile this time.
_I should really go,_ she said.
_I'll walk you to the bus stop,_ Sticky said.
When they climbed back onto the bridge Sticky put his headphones on without sound. Walked slow through the homeless bodies curled up on the grass. Anh-thu picked a little yellow flower from a bush and put it behind her ear. She walked a little ahead of Sticky up to the crosswalk that would take them back into the promenade, pushed the button.
_Damn,_ Sticky said to himself, _I guess I didn't get me_ _no pants._
He stared at the flower stem sticking out the back of Anh-thu's long black hair and felt happy.
Cheerleaders Screamed Out
chants and posed with blue and white pom-poms, whipped thin arms and legs around like little windup toys. The football squad leaned in close to the action on the court, sprayed venom and pointed bench-press fingers in the other team's faces. The ten-piece band broke into hype-up-the-crowd tunes the second a ref's whistle stopped play.
Sticky's seventeenth birthday may have ended in a crowded holding cell, but it kicked off in a sold-out forty-year-old high school gym.
Dominguez Hills rolled into town with twenty-two wins and a bus full of hype. Players filling out purple jerseys like men. "Too much experience," all the papers said. "Too many athletes. Two of their starters already committed to big-time colleges." All this and it was Venice High's first play-off game in five years.
Sticky was the hotshot sophomore who gets called up from JV for play-offs. The outsider at the end of the bench with his warm-ups still buttoned all the way.
The kid pinched out of the huddle during time-outs.
Players called up don't see much run in the play-offs. It's a pat on the back just being on the bench. Sticker on the helmet. But all that went out the window in the middle of the third quarter, when Dominguez Hills went up by twelve. Coach Reynolds shook his head at every face he scanned on the bench. When he got to Sticky he pointed a shaky finger, told him: _Get the hell in there, kid_. Grabbed Sticky under the arm and damn-near threw his ass toward the scorer's table. _Run the point_.
Sticky jogged to the table and pulled off his warm-up jacket, tossed it behind the bench. He reached over and picked it back up, threw it down. Picked it up and threw it down.
Picked it up and threw it down.
Picked it up and threw it down.
It was a crazy time to have an episode, with all the varsity guys on the bench, watching, but he knew all he had to do was get on the court. That was when everything would disappear.
It took three or four more tries before he got the perfect toss. Then he slid a hand across both soles for grip.
The buzzer sounded and the ref waved Sticky into the game.
Reynolds put a hand on his shoulder and yelled something, but Sticky didn't hear a word. He didn't hear anything, in fact. Not his coach. Not the crowd. Not the announcer calling his name out over the loudspeaker or his teammates telling him who to take on defense. He strutted out onto the stage with nothing but a blank mind.
When the ref whistled the ball back in play, Sticky made like it was just another street ball game down at Lincoln Rec.
See, I have this theory about hoops. About what makes one dude smooth under pressure and another fold.
Sticky picked off a cross-court pass right off the bat, high-dribbled down the sideline like Deion Sanders and stuck a deep three-pointer.
The crowd rustled.
The more a player thinks about the game—what setting they're in, who they're running against, what folks will say depending on whether or not they hook up a decent showing—the more messed up that player is gonna play. It's unnatural.
Sticky ripped the other squad's point guard clean, like he was wrapped and on a shelf at half-court, took three quick dribbles and dropped in a sweet one-handed finger roll over the rim. His face broke a smile on the way back downcourt. He pointed to the crowd and pumped his fists.
He was like a showman at the circus.
The guy in the cage with the whip.
Go ahead and pick out the smartest dude in the house, and I'll promise you he's the most weak-minded baller. All that analyzing. Examining. Calculating. Man, you gotta stay clear out there. There's no time for reflection when you need reaction to a situation.
The crowd started catching on to this new guy up from JV. Running the squad. Flashy passes and slick attacks on the bucket. Slashing and bombing away. Hoops on autopilot.
Sure, the game with refs is supposed to be different from the game on the street. More under control. Less razzle-dazzle. Fundamentals like they teach in clinics all across the country. How to play hoops for $425 per week. But Sticky plays with the same flavor no matter what the setting.
Every time you turned around in the second half, the announcer was calling Sticky's name over the loudspeaker: _STICKYYYYYYY REICHARD FOR ANOTHER TWO._ _COUNT IT_.
White space.
Then the whine of the school band's trumpet, a couple thuds from the bass drum. The crowd stepping up its volume another notch. The weight of all the energy testing the old gym's tired bleachers.
And some movie writer couldn't have made it up any better. The way it all came down in the end. With eleven seconds left, Dominguez Hills' star guard was at the free-throw line shooting two. Score tied 85–85. Crowd booming. Band banging through sets during a time-out Reynolds called to ice the shooter.
_No matter what,_ Reynolds yelled over screeching horns. The whole squad was huddled around him with blank faces, ready to gobble up whatever he fed them. _No matter what, if_ _he makes them or misses them, we call time out_.
When the kid stepped up to the line, the crowd was so out of control, stomping their feet and screaming, the rim actually started vibrating. The bottom of the net started flipping back and forth. The ref handed the kid the rock and he went into his routine: three dribbles, tuck the ball under the chin, deep breath. He lofted the first one up soft and it fell through.
The crowd died.
Just like that. Ball hits nylon, no more noise. Like someone in the control room flipped a switch. Dominguez Hills 86, Venice 85.
Purple jerseys went up and slapped their guy on the back, told him: _One more, baby. One more_.
The crowd topped out for the second free throw. Feet pounded bleachers like a tank rolling through. Both teams snuck over-the-shoulder glances at the wave of screaming fans. Felt deep vibrations swim through the floor and into their shoes, scale up weary legs and unfold in the pit of their stomachs.
Three dribbles, tuck the ball under the chin, deep breath. Kid lofted up another soft one, but this time it rattled around the rim and fell out. Venice's starting center, Sinclair, ripped down the board with two hands and made a quick T around the ball. _Time out, ref! Time out!_
Down one, nine seconds to play.
No time to analyze.
Venice huddled around Coach Reynolds again. A pocket of concentration. All the guys gave everything to ignore the cries of the crowd, the thumping of the band.
Reynolds reached for his stick of chalk and stared at the ground. All eyes were glued to a blank chalkboard. _All right,_ _here's what we do!_ he yelled, but then he fell silent again.
The crowd locked into a rhythm of sound. Two stomps and a clap. All at once. Boom boom clap. Boom boom clap. Sticky stood pinched out of the huddle with a water bottle, squirting an arc of tap into his mouth and trying to listen.
Reynolds reached through the huddle for Sticky's arm. Pulled him into the middle of everything.
_Sinclair,_ Reynolds said, just as the buzzer sounded. End of time-out. End of brainstorm. _You inbound to Sticky._ _Sticky, you penetrate and look for the open man. Nothing's_ _there, pull up for the shot._
Everybody looked at Sticky all crazy as they broke the huddle and stepped back onto the floor. Coach put the rock in the hands of a JV kid. A sophomore. Skinny white boy who didn't even have a name on the back of his jersey.
Sinclair put a big mitt on Sticky's head. _Come on, youngster. Make something happen out there_.
_Spread the court,_ Reynolds yelled, following his team halfway out onto the hardwood. He reached out for Sticky's shoulder but missed. _Don't do nothing stupid, kid!_
And a sold-out gym fell silent for Sticky.
The ref blew his whistle and handed the ball to Sinclair. The movements of the crowd without sound. Every kid on the court in super-slo-mo. Ticks of the clock farther and farther away. He jab-stepped at his defender and broke for the ball. Hands out. Sinclair whipped a pass in to him and the seconds started rolling:
_Nine seconds on the clock . . . eight seconds. And, see, this is_ _what you do . . ._
You size up the purple jersey in your face, man. Some num _ber 23-be-like-Mike black face with straight teeth. Braces._ _Beads of sweat dribbling down his forehead. Baby Afro. The_ _triangle of small moles on his right cheek, calling out. Down in_ _defensive stance like basketball camp demonstration says._
_Scared eyes._
_Seven seconds . . . six seconds._
_All your guys clear out. Give space so you can break it_ down. Do your thing. Lay out crazy beats cause you're the man _on the mix. Official game ball leather is soft in your hands,_ _man, like Anh-thu's smooth face up in the crowd, watching._ _Cup it between your fingers and forearm and feel alive._
_Aware._
_Necessary._
_Cause, man, this is your jam they're waiting for. And this is_ _your world they're waiting in._
_See Sinclair trailing the play, his big high-top sneakers like_ _fists against a soundproof wall._
_See your path to the promised land. Without looking. Left_ side of the lane, where a dozen possibilities flash through your _head. See your red carpet. See your yellow brick road. Hesitate._ _Get a split-second survey._
_Feel the electricity, man, of two thousand faces burning_ _on YOU. Four thousand eyes in the back pocket of YOUR_ _hoop shorts._
_Revel in it._
_Five seconds . . . four seconds. Take off with your head_ _down. . . ._
_Know the statues around you. Guys' empty faces._
_Know the power in your legs and feet. The spring in_ _your step._
_Know the ticking of the clock._
_Know what purple jerseys will do before they do it. It's in_ the way they lean.
_Know the six inches of open lane that will be cut off by_ _which guy and at what point._
_Know your coach's crinkled leather face on the sideline. The_ _ref with the whistle in his mouth, backpedaling._
_Know your defender's wide eyes as a pathway to his mind._
_Know your body inside and out. That it will do exactly as_ _it's told._
_Know the ball in your hands as you put it on the floor._
_Know your third move before you make your first._
_Know quickness._
_Know stopping on a dime._
_Know nothing._
_Three seconds. When you blast past the slo-mo purple jersey with straight teeth, pouring out of a jar thick like syrup, the_ _biggest purple jersey leaves his guy to cut you off. Like you knew_ _he would. Like a stray dog after a fake toss._
_You stutter-step around his tree-trunk legs and cross over._
_You feel the brush of another purple jersey, like a rush of_ _wind across your left side. But you dance by that, too. When_ _your Nikes get in the paint, you lift into the air with the ball_ _cupped like a football. Like a running back going over the top_ _at the goal line. Dirty work before an end-Zone shuffle._
_You feel the weight of everybody in the gym holding their_ _breath. Out of their seats and balanced on flexed toes. Bodies_ frozen and useless.
_Purple jersey arms swing like they might block your shot,_ _but here's the thing: It ain't nothing but a street ball game to_ _you. Down at Lincoln Rec. Old-man Perkins in the bleachers._ _Fat Chuck. Dante and his rainbow jumpers. Everybody talking_ _trash and cheating on the score. It ain't nothing but a game to_ _eleven with two full squads on the sideline, waiting._
_Defender arms swing, but they don't get nothing._
_Hands full of empty air._
_And when you got them all committed like that, exposed_ _and in the air, that's when you pull it out of your pocket. That's_ _when you break out the around-the-back flip, no-look style, to_ _a wide-open Reggie. Purple arms get sucked back down by_ _gravity, and your guy Reggie is laying it up off the glass for the_ _game winner._
_And it's nothing but that white-space thing again._
Sticky watched his coach leap up and down like a clown. Watched him hold back the assistant coaches with an arm bar. He watched the guys on his bench grab each other around the waist and point into the crowd. Slap fives and pump fists. He watched the biggest Dominguez Hills player in-bound the ball to their star guard. Watched him loft up a weak three-quarter-court prayer. Watched the way their bench crumbled when the ball fell twenty feet short as time expired.
_All this. It happens for you in silence._
The final buzzer went off and the home crowd erupted. Everybody stomping their feet and yelling for the other team to get the hell out of the gym. Slapping hands with whoever stood on the right or left. As the band sounded off, all the guys on the bench sprang into the air, charged the court and dog-piled on Reggie. A pile of Venice hoops at midcourt. Sticky stood next to them, breathing fast, putting his hand on one of their shoulders, then taking it off.
A group of Venice football players charged the court with lettermen's jackets on. They ripped big banners off the walls and paraded around the court, holding them high above their heads. Everybody on the Dominguez Hills squad sat still on their bench, watched Venice celebrate. Some had white towels spilling off lowered heads.
Coach Reynolds pulled Sticky aside in the middle of all the mayhem. _Shoulda had you up here all year, son,_ he said, trying to catch his breath. Twenty-three years on a sideline in his leather-black face. He palmed the back of Sticky's sweaty head and shook it around. _Goddamn, boy! That was_ _one hell of a pass you just made!_
Fat Jay, the squad's big backup center, and Sinclair picked Reggie up and carried him into the stands. _Reggie's_ _good for the game winner!_ Fat Jay kept chanting. _Good for the_ _game winner! Sent em on their way!_
Dave and Sin, Sticky's boys from the JV team, ran up into the stands after Sticky and jumped all over him. _MVP!_ they kept yelling. _MVP!_
The school's longtime gym custodian, Manuel, came up and hugged Sticky. A small old Mexican man with a chewed-up beard. _One of the best wins I've ever seen,_ he said, letting go of Sticky and wiping his face with a towel. Sticky used to walk with Manuel while he mopped the floor before JV practices and games. Listened to him talk about old-time hoops: Jerry West and Pistol Pete. Dr. J., Magic and Bird. Sticky reached out for Manuel's hand in the middle of the celebration, shook it firm. Manuel would let Sticky hang in the gym solo on weeknights. And Sticky appreciated it. He'd spend hours working on his shooting and ball handling. Manuel would play dumb, pretend he didn't hear a basketball tapping the hardwood or rattling the rim when he closed it down for the night. And they'd never spoken a word of it, even to each other. But this handshake, Sticky thought, this was saying it all.
_Yo, Stick,_ Dave said. _Let's roll. We gotta celebrate._
_Come on,_ Sin said.
_Cool,_ Sticky said. He let go of Manuel's hand and looked him in the eyes. He nodded.
_One of the best wins ever,_ Manuel said again, then he made his way back down the bleachers, weaving through the thick packs of hyped-up students, and stood by his cart.
Anh-thu and her friend Laura were standing at the end of the bleachers. Sticky signaled for Dave and Sin to hold on and he hustled over to her.
Oh, my God! Laura said. You were amazing!
_Thanks,_ Sticky said.
Anh-thu stood staring at Sticky, a huge smile painted on her face.
Sticky kissed her cheek, told her: _It was a good game,_ _huh?_
_Oh, my God, it was the best game ever!_ Anh-thu said. She covered her face with her free hand, looked at the ground. _Oh, my God, I can't stop smiling_.
_Come on, Stick!_ Sin yelled out from near the gym exit.
_MVP!_ Dave yelled, cupping his hand around his mouth. _MVP!_
_Go on,_ Anh-thu said. _Celebrate with your boys. But call_ _me tonight_. And when Sticky nodded, she snuck him a quick little hug.
The fellas all hopped in Sin's '67 Impala after Sticky grabbed his bag from the locker room. Dave was about 6' 6'' so he rode shotgun. Sticky slid in back and pulled his jeans on over his hoop shorts. Pulled a clean sweatshirt on right over his sweaty jersey. He leaned forward and they all talked over each other about the comeback.
Sin turned down his reggae beats. _You see their coach's_ _face, man? When we locked em up at eighty? Serious, you see_ _that dude's look?_
Dave and Sticky laughed and pounded the roof.
Dave slapped a hand onto Sin's shoulder, said: _Nah, man,_ _you see him throw his hands in the air when Stick took dude_ _three straight times? He didn't know who the hell Stick was_.
Sin said: _Wasn't no scouting report on no number seven_.
Dave spun around to Sticky. _They didn't even know who_ _you was, Stick_.
Sin pounded his steering wheel and yelled a grown man's yell: _Hell yeah!_ Veins rising in his neck. He cranked up his beats again, so loud the whole car vibrated. Then he peeled out of the school parking lot, laying on the horn a few times with the heel of his hand, and headed north toward Santa Monica.
The Fellas All
started for the JV squad (Sticky, Dave and Sin) before Sticky got called up to varsity for the play-offs.
Sticky ran the point, led the team in both scoring and assists (25.9 ppg, 5.6 apg). Sin and Dave operated down low, controlled things in the paint.
Sin's a muscular first-generation Puerto Rican American who was also the star running back on the JV football squad. He's dark skinned with blue eyes. Not an ounce of body fat. The ladies tend to go wherever he's going.
Dave's a tall, skinny white kid from deep Venice. A section of the neighborhood people used to call Ghost Town due to the number of unsolved shootings. Everybody thinks he's a shade crazy because he's always mumbling to himself. He lives in a one-bedroom apartment on Fifth with his mom and three sisters. Man of the house.
Things got off to a rough start when Sticky first came in at the beginning of the school year. Sin and Dave tried to be cool after open gyms, talked to him about the team and the coach and the best-looking cheerleaders, but Sticky wouldn't look anybody in the eye. It was his third school in two years, and he wasn't sure these punks by the beach were worth his time.
Sure, he showed up every time the coaches opened the gym. He listened when they went over pick-and-rolls, various zone defenses, and the half-court trap. He ran the sprints hard during conditioning. Never once complained. He even showed up for the big fund-raiser on Main Street, washed and dried cars all day like everybody else. But he never said a word to any of his new teammates. No jokes. No boasting. No talk of the past. He simply kept his mouth shut, his head down.
Sin finally grabbed Sticky by the neck before the first official practice of the season. They were in the locker room and Sticky'd gone up without saying a word and shut off Sin's reggae. He flipped it to his favorite hip-hop station and went back to lacing up his kicks. Sin froze, beanie in hand. He looked at a couple guys on the team, confused. They shrugged.
_Yo, go put my tape back on,_ Sin said.
Sticky didn't look up.
_Yo!_ Sin yelled this time. _Go put my damn tape back on!_
When Sticky didn't answer him that time, Sin stood up and walked toward him. See, there's one thing most people don't know about Puerto Ricans. You don't mess with their music. When Sin was halfway to Sticky he tossed his beanie to the side and charged.
Sticky sensed it was coming and fired one of his Nikes. Sin ducked and grabbed for Sticky's neck.
Sticky reached back and threw a series of wild punches, none of which landed, and then threw an elbow that caught Sin in the side of his shaved head. Sin grabbed Sticky's arms and put him in a tight headlock. They wrestled around on the floor, clawing at each other's faces, until the coaches came running in and pulled Sin off. Separated the two teammates.
_What the hell's going on?_ Coach Reynolds demanded.
_What are you doing?_ Coach Wilkins said.
Sticky and Sin didn't answer, they just stared at each other with fire in their eyes. The right side of Sticky's face was all scratched up and red. Both of their chests were moving in and out quick. Their fists were still clenched.
That was when the coaches pulled Sin into their office and explained about Sticky. How he had just moved into some house off Rose with five other strays. How it was the fourth foster home he'd lived in since the age of seven. Coach Wilkins, the JV coach, leaned forward and rested his elbows on his knees. _So cut the kid some slack, big guy,_ he said. _You can do that, right?_
Sin shifted around in his chair, touched his fingers to a red spot on his cheek, checked them for blood. Nothing.
Coach Reynolds opened up a file he pulled from his desk drawer and cleared his throat. _Listen, I know there are some_ _major discipline issues we're facing with this kid_. He thumbed through some of the paperwork. Scanned one of the pages with his finger. _But he gives y'all a legitimate point guard_.
_He's gonna make your life so much easier, Sin,_ Coach Wilkins said. _His penetration will lead to easy buckets for_ _you. Plus the kid can shoot the lights out. He's gonna stretch_ _defenses out and you and Dave will have a goddamn field_ _day inside_.
_Listen, son,_ Coach Reynolds said. _I want you to cut this_ _stuff out right now, OK? Just squash it_. He leaned back in his chair, worked a toothpick in between his teeth. _In fact, I don't_ _wanna hear nothing else about you two ever again_. He looked over at Coach Wilkins. _Right, Coach?_
_Right,_ Coach Wilkins said.
Sin shifted around in his chair, touched his fingers to the red spot on his cheek again. Nothing.
Coach Reynolds folded up the file and put it back in his desk drawer. Coach Wilkins lifted a whistle from the floor and put it around his neck.
Sin looked to Coach Wilkins, said: _All I'm saying, Coach,_ _is that he ain't got no respect, and if he keeps on—_
_What I'm saying, Sin,_ Coach Reynolds interrupted, pulling the toothpick from his mouth. _Is we don't need you_ _adding fuel to the fire. Got it?_
_Got it,_ Sin said.
_Good,_ Coach Reynolds said.
_Now go stretch out,_ Coach Wilkins said. _We'll start practice in five minutes_.
After that first practice Sin waited for Sticky in the parking lot. He didn't say one word to him until after he knocked him to the pavement with an overhand right to the ear.
_That's right, boy!_ he said, pouncing on Sticky.
In the scramble, Sticky kept yelling out: _I'll kill you, man!_ _I'll kill you!_ He tried as hard as he could to roll over and get up, but Sin was too strong.
Sin put a knee in Sticky's chest and stared down at him with this wild look in his eyes. Told him: _I don't give a shit_ _how many foster homes you been in. And you can believe that_.
After a few more minutes, Sticky stopped struggling and let his eyes come up to Sin's. In them he saw two tiny reflections of himself. Then he turned his head and let all his muscles relax.
It was over.
Carmen Rolled Up
to Sticky's foster care pad in a beat-up Chevy Nova with the backseat ripped out. The passenger seat was piled high with roses, tulips and daises. Sunflowers. Sticky was twelve.
Carmen was Sticky's second foster lady, and she was much younger than Francine. Prettier, too. She stepped out of the car wearing cutoff jean shorts and a tight black half-shirt. Her wavy brown hair was pulled back in a ponytail except for the wispy bangs that framed her dark brown eyes. As Carmen walked up the driveway, Counselor Amy said she looked like a movie star. Counselor Jenny argued she looked more like a runway model. But throughout the whole pickup process, Carmen never once cracked a smile. She kept a serious face through all the paperwork. Through the introductions. Through the awkward session in the driveway when everybody hugged and waved goodbye.
She didn't have a whole lot to say, either. On the long drive through traffic to Costa Mesa she kept her mouth shut and lips sealed. Hands gripped the steering wheel at ten and two. She turned on the radio, tuned in a Spanish station and tuned this new foster kid out.
Sticky sat Indian style in the empty back of the car, leaned against his bag. Around corners he'd put a hand out to keep from falling over. To pass the time he stared out the back window and kept a running head count of all the cars they passed.
When Carmen pulled up to a run-down apartment complex under the freeway overpass, security bars on every window, a good-looking light-skinned Hispanic dude came rushing out of the corner apartment with an electric guitar around his neck. He swung open the back door and helped Sticky get out. _Hey, bro, I'm Ruben,_ he said. _I'm gonna be your_ _new dad_. He smiled so big you could see all his teeth.
Ruben picked up Sticky's bag and carried it around to his wife. He kissed her on the cheek and said: _Thanks_ _for going for me, baby. We couldn't quit until we worked out_ _that chorus._
Ruben carried Sticky's bag inside and slid it next to a buzzing amp, which he flipped off. He stuck his guitar on an empty guitar stand and turned his attention to Sticky. _Bro,_ _you don't even know. I'm totally stoked to have a kid_. He pulled a couple picks out of his pocket and set them on the amp. _We found out six months ago that my old lady can't have_ _no babies. She can get pregnant and all that, but somethin always happens before it gets born_.
Carmen overheard Ruben and stormed out of the room. She slammed their bedroom door.
Ruben looked at Sticky, said: _That's why you here, bro_.
Before Sticky could even hit the bathroom, Ruben lobbed him a baseball glove and pulled him into the street to toss a ball around. He squatted like a catcher and told Sticky to fire it in there. _I'll let you know if it's a ball or a strike,_ he said. _I played ball in high school_.
The shadow from a tall tree hung over Ruben like a giant tarp. Sticky stood in the sun. There was a line between them on the pavement where shade was creeping in on sunlight. Sticky stared at that line, tossing the baseball into the heel of his glove, trying to make the perfect popping sound. He tossed the ball into his glove, again and again, and thought about how much different it was with Francine on the first day. Tossed the ball into his glove. Pulled it out and tossed the ball into his glove. How Francine smiled so much on their drive to her house, stopped off at a mall and let him pick stuff out for his bedroom.
Tossed the ball into his glove. Pulled it out and tossed the ball into his glove.
Finally Ruben called out: _Come on, bro! Just peep the target and let it go!_
The street smelled like _carne asada_ grilling on a barbecue. Like refried beans. Like fresh-rolled tortillas bubbling up with steam on some old Mexican lady's griddle.
Ruben smacked his glove, waiting.
When the popping sound sounded right Sticky wound up and threw a wild ball that Ruben had to pick up on the short hop.
Sticky threw pitch after pitch that first afternoon. He threw a few strikes, but they were mostly balls. A couple bounced off the pavement and dug into Ruben's shins. One sailed wide right and smacked somebody's truck. Ruben grabbed the spinning baseball after that wild pitch and scoped the neighborhood for witnesses. He fingered the fresh dent on the side of the pickup and said, _It ain't nothing,_ _bro. Just a little scratch. No worries_. He walked back toward the makeshift home plate and tossed the ball to Sticky.
At one point, when Sticky struck out an invisible batter on three straight pitches down the middle, Ruben went crazy and made crowd sounds with his mouth.
_You lucky, bro,_ he told Sticky after he walked out to the mound and wrapped an arm around him. The tree shadow was now only inches from Sticky's feet. _My pops didn't never_ _wanna play catch with me when I was a kid_.
After catch it was a couple horror flicks and microwave popcorn. Carmen went to snuggle in close to Ruben, but he thought Sticky should sit in the middle. Said it would make him feel more like a part of the family. Ruben turned off all the lights and turned up the volume.
After every scary scene Ruben would turn to Sticky and whisper: _That's some scary stuff, right, bro?_
When the credits rolled on the second flick, Ruben ran through a few songs on his acoustic guitar and sang. He sat on a stool in front of the TV, strummed and picked soft at the shiny bronze strings. He closed his eyes and tilted his head back when he was really into it. Sticky and Carmen sat on opposite ends of the couch, watching.
When Ruben's voice got tired he put away his guitar and set up a bed for Sticky on the couch. _Good to have you, bro,_ he said, and gave Sticky an awkward little hug. Then he pulled his wife into the bedroom, and she shut and locked the door behind her.
Through the door Sticky heard Carmen yelling about how having some random kid in the house wasn't the same as having a kid of their own. How he was a complete stranger and it creeped her out. He heard Ruben telling her over and over to calm down and give the situation a chance.
Sticky fell asleep that first night to a chorus of sirens outside the front room window and the Hispanic couple arguing in the bedroom.
The next day they worked off the same script: burritos, catch, a couple scary flicks and then some singing and fighting. The next week was like that too, in fact, except Carmen was gone during the day, Monday through Friday. It turned out Ruben played bass in a couple salsa bands but was taking time off to write new songs. Carmen worked part-time at a flower shop. Made lunchtime deliveries to all the businesses around South Coast Plaza. When she went to work, Ruben stayed home with Sticky.
On the one-week anniversary of Sticky's stay, Carmen came out for breakfast with a big smile. It was the first time Sticky had ever seen her smile, and he decided she looked prettier that way. Happy. Ruben followed her out and went right up to the couch where Sticky was lying under a blanket watching cartoons. He stuck his hands in his pockets. _We're_ _goin on a trip today, bro_. He paused a second and then added: _To celebrate_.
Carmen laughed when he said that.
They had Sticky pack his bag back up and set it out by the car.
_Where we goin?_ Sticky said between bites of cereal.
_It's a secret,_ Carmen said, and she gave him a wink.
_Just hold on, bro_ , Ruben said, and he shook his head at Carmen. _And remember, bro, it wasn't me who thought this_ _whole thing up_.
On the road Ruben talked about his own dad. How he worked all day in a factory and then went straight to the bar. _We barely ever saw my old man,_ he said. _When we did he was_ _always wasted and yelling for us to do something_.
Sticky sat in the back again, leaning against his bag. He stared out the window and wondered where these people were taking him.
Ruben moved in and out across the four freeway lanes as he talked. Carmen sat in the passenger seat with her arms folded. She didn't say a word.
_So, one day my old man comes in and moves all his things_ _out,_ Ruben went on. _Just like that, bro. Right there in front of_ _me and my two brothers. My mom_. He turned almost all the way around so he could see Sticky's face. When he swerved a little, started riding braille, he spun his face back to the road and straightened up. _But you know what?_ he said, looking at Sticky in the rearview this time. _Everything was better when_ he was gone. It took me a while to realize it, but I didn't need _him_.
Ruben directed his attention to the road again. He didn't say anything for a few minutes. Carmen was now filing her nails, sitting with one foot up on the dash.
_I still don't need him,_ Ruben said. He raced by a station wagon and looked in at the woman driver. He looked at Carmen to see if he'd been caught, but she was busy pulling nail polish out of her purse. _I'm just sayin, bro, people can_ _turn out cool, you know? Without no dad_. He coasted down a familiar off ramp and stopped at a familiar red light. He turned around and looked Sticky in the eyes, told him: _I_ _mean, just look at me_.
The long drive led them right back to the foster care pad, where the old Mexican director was waiting outside with his arms folded.
Ruben pulled the parking brake and hopped out of the car. Carmen stayed inside.
Ruben unloaded Sticky's bag and ran him through the whole hand-shaking thing. _Sorry about this, bro,_ he said. _It_ _just didn't work out_.
Before Sticky had time to say anything back, the old Mexican director was pulling him away.
Things Are Heating
up at Lincoln Rec. Sticky cuts through the lane and Rob busts an elbow in his ribs. Knocks him to the ground and scowls.
Sticky springs back up and continues through the lane.
Dallas swings the rock over to New York on the right wing. New York holds it against his hip, surveys the situation, then chucks it over to Sticky at the head of the key.
Sticky sizes Rob up with a couple jab steps and spies the lane. He makes a quick hesitation and slashes past. Rob reaches out to hold on, but Sticky's too slippery. As he scoots into the lane, Trey steps over to cut Sticky off, but this leaves Dante all alone up top. Wrong answer. Sticky instinctively whips the ball over his head and hits Dante in perfect rhythm for the jumper.
Ball rips through the net.
Dante points at Sticky as he backpedals down the court.
Sticky points back.
_Tied up,_ Dallas says.
_Next bucket wins,_ Trey says.
_Now y'all playin some ball,_ Old-man Perkins yells from the side. He reaches down and starts lacing up his old-school Nike Airs.
_It don't matter which one of y'all wins, though,_ Johnson says. _Cause we on next, and we ain't losin_. He nods his head at Perkins. Reaches out a fist for some daps.
It's one-thirty in the afternoon and Lincoln Rec is bursting at the seams with guys waiting to play. Everybody dribbling around on the sidelines to get warmed up. Throwing bounce passes to each other and sneaking up jumpers when the action's at the other end of the court.
Next team that scores sits the other five down. Sends them to the back of the line. A five- or six-game wait means a good two hours sitting up in the bleachers, watching.
Watching instead of playing.
Only guys with heart look for the rock when it's game point. When everything's on the line. Guys confident enough to put a team on their shoulders and good enough to bring it home. This is where you find out who came to win and who's happy just playing. Who's willing to rip somebody's head off when the pressure's on, and who's likely to cower in the corner like a puppy.
_Bring it down to me,_ Rob says in the post, digging an elbow into Sticky's middle. Holding and grabbing. Pushing. _Come on, Slim, bring it down_.
Slim dumps it in, clears out to the top of the key.
Rob backs Sticky in, frees up enough space to work his unpolished post moves.
Trey comes over to set a screen but Rob waves him off.
He spins into the lane and makes his move to the cup, muscles toward the basket like a linebacker. Lowers a shoulder and blasts Sticky in the face as he powers the ball up to the rim. The skin under Sticky's right eye splits from the blow and blood starts zigging and zagging down his face. He goes down on one knee and watches Rob's shot toilet-bowl around the rim and fall through.
_Game over!_ Trey yells, and fires wild fists through the air.
_Get off my court!_ Big Mac says.
Rob stands over Sticky and flexes his biceps. _Too strong,_ _white boy!_
Sticky runs his fingers across his cut and stares at the blood.
The next five are already making their way out onto the court, stretching arms and legs, jogging in place. Shooting warm-up jump shots and talking matchups.
Dante scoops the ball up and fires it against the backboard. When it ricochets back at him he punches it with a closed fist. The ball caroms past Dallas, who's squatting next to the door, covering his face with his hands. _I can't believe_ _we lost to them fools,_ he says to no one in particular.
Trey and Slim give each other daps and head for the drinking fountain.
Sticky presses his shirt against his cut and studies the red blotch of blood. He gets up and walks toward the wall behind the basket, leans against it. Dabs his shirt against his cut again and studies the red blotch of blood.
Fat Chuck takes earthquake steps up to Sticky and touches a fat hand to the back of his head. He pulls Sticky's shirt away from the cut and investigates. _Damn, Stick. Looks_ _kinda deep_. He turns Sticky's face into the light, investigates some more.
Sticky shakes out of Chuck's grasp and presses his shirt against his cut, studies the red blotch of blood. _It ain't that_ _bad,_ he says, talking more to himself than Chuck. And what flashes through his mind at that point is Anh-thu. He was planning on hooking up her birthday perfect. Walking up to her at nine tonight with a big shiner wasn't exactly what he had in mind.
_Check it out,_ Fat Chuck says. _I got some Neosporin in the_ _car._ He holds his hands up in the air. _At least you gotta let me_ _help you clean up in the bathroom, Stick. It ain't like you_ _gonna be playin anytime soon anyway_. He points to all the guys on the sideline, waiting to play.
Sticky looks at all the guys, does some quick math in his head and figures he won't be playing again for another hour and a half, minimum. For a second he considers taking off. Handling the stuff he's gotta handle for Anh-thu's birthday early and worrying about ball tomorrow.
Rob struts by with a devil-like smile on his face. He ducks into the drinking fountain for a few seconds and comes up gargling. He spits.
On the strut back he leans in near Sticky's ear, tells him in a quiet voice: _You know you can't handle all this muscle in_ _the post, white boy_. He flexes his guns and laughs when Sticky shoves him out of his face. Then he pimps back out on the court for the next game.
Chuck wraps a meat hook around Sticky's neck. _Don't_ _worry about him, Stick,_ he says. _Rob ain't nothing but a fool._ _You can trust me on that one_.
Sticky presses his shirt against his cut, studies the red blotch of blood. He looks up just as Old-man Perkins checks the ball into play and another game starts rolling.
It digs in Sticky's stomach when he thinks about Rob hitting the game winner on him. How he let Dante down. Dallas. How because he didn't play good enough defense they're all sitting on the sideline now. Waiting. Watching. Not a worse feeling in the world, he thinks. And then Anhthu's face flashes through his head again and it makes him feel better. Maybe his day of hoops is off to a bad start and he has a cut on his face, but at least he gets to see his girl later on. At least he gets to chill with her. And the fact that these thoughts make him feel better surprises him. He wonders if it's a good thing or a bad thing in terms of his dedication to hoops. In terms of the fact that hoops has to be number one in his life. Always. The truth is, maybe he shouldn't be so excited. Maybe this feeling is wrong. Especially considering he just lost the game for his team.
_Stick?_ Fat Chuck says. _You comin or what?_
Sticky looks up at Chuck. He presses his shirt against his cut again, studies the red blotch of blood. Then he pushes off the wall and follows Chuck out the gym door and across the parking lot, toward the run-down public restrooms.
Fat Chuck Is
dabbing at Sticky's face with a wadded-up rag. One he pulled out of the bag he wears around his waist. (Big red blotches framed by white cloth.) _Hold still, boy,_ he says, and moves Sticky's head to the side.
Wet concrete is cracked under their feet. Jagged fault lines running into cement walls full of graffiti: thick black ink, spray-painted vato letters, blood. The kind of thing you might find in any kept-up-by-the-city restroom. Saturated toilet paper clogs both floor drains, forming ankle-high puddles you have to step over to get where you're going. And there's no getting around the sour smell.
Chuck's breath is tequila. Fingers salt. He puts pressure on the cut and Sticky flinches. _Yeah,_ he says. _Just like I suspected. Looks like you gonna need a couple stitches, Stick_.
Sticky pulls away to check himself in the eroding mirror. Moves his face right up to the glass for a closer look: A thin uneven cut jetting across the skin under his right eye. He takes the rag from Chuck and wipes away the blood. Watches it quickly pool back up.
Chuck takes Sticky's head, tilts it to the side again and squints: _Maybe a butterfly, though_. He moves him into better yellow light. _But I'm thinking . . . Yeah, I'd say most likely_ _some stitches_.
Sticky leans against the sink and flips through daydream channels: clipboard forms, explanations, rides to and from, insurance numbers, fancy doctors, rubber-gloved fingers, long needles in his cheek, a sun-bright dentist light. He puts his fingers to the cut and cringes when the salty sweat stings. _Nah, man,_ he says. _I ain't got no time for no stitches_.
Chuck stares at Sticky's reflection and shakes his head, puts a fist to his mouth to catch a cough.
Chuck is: fat boy licking double-scoop all grown up. Gray sweatpants, gray sweatshirt, grass-stained high-top Converse. Celtic colors. Chuck is: Lincoln Rec's team mom or resident die-hard fan. Never even tossed up one jumper in all the months he's been showing up. Old-man Perkins warns everybody: _Don't let the man plop down in front of you up in them_ _bleachers. That cat's so fat he'll cause a total eclipse of the court_.
Chuck catches another cough and spits in the sink. _All_ _right, Stick,_ he says, wiping his mouth. _It's your call. I'm just_ _sayin, you could use about three or four_. He folds up his fat arms and leans against the wall.
_I was takin Rob, too,_ Sticky says.
_I saw the game,_ Chuck says.
Sticky pushes away from the mirror and goes for one of the stalls. The first three are occupied by homeless. Green trash bags next to callused streetfeet. None of the stalls have doors. Homeless dudes pick heads up slow as Sticky passes, show empty eyes. The fourth is wide open and Sticky quickly slips down his hoop shorts and reads the walls.
Chuck steps up to the mirror, plays with the ends of his chewed-up mustache. Twirls uneven hairs together and then smooths them out. He tries on six or seven different facial expressions and then laughs at himself. _Where you from anyway, Stick? I mean, where was you born?_
Sticky is: hands on knees, back straight. Defensive stance so that none of him is touching the metal bowl. He says through the wall: _I was born in Virginia, I think. That's what_ _my papers say. But I don't remember it none_.
Chuck runs nubby fingers across the gray stubble of his cheeks and neck. _Well, you gotten pretty good at ball,_ he says. _I_ _been watchin, and you gotten pretty damn good_.
Sticky pulls up his hoop shorts and flushes.
A guy with flies comes staggering up to Sticky's stall and knocks on the wall twice. Politely. Sticky whips around wide-eyed.
This cat's a rotting burrito. Greasy gray hair and beard sticking out of a tightly wrapped Mexican blanket. Half-dead eyes. Callused clay feet under nappy frill. _Excuse me,_ _sir,_ he says, drawing out each word. _You sharing this room_ _with anyone?_
Sticky smooths out his shirt and shorts on the scoot-by, tells him: _Go ahead, man, take it. I'm done_. He walks the six or seven steps back to the sink and thuds the faucet with the heel of his hand. Water shoots out strong and he washes up. He turns and watches the blanket dude slowly stagger into the stall until it's just his clay feet under the stall wall.
The water shuts off and Sticky dries wet hands on hoop shorts.
_Gotta watch them vagrants,_ Chuck says. _They'll creep up_ _on you sometimes_. He moves closer and puts a hand on the back of Sticky's head. Positions salt fingers next to the cut and pulls it open a little.
_Like I said,_ Chuck says. _It's up to you. It's my cut, I have_ _em stitch me up so it don't leave no scar. Don't mess me up_ _down the road with the ladies_. He elbows Sticky in the ribs and laughs.
_I already got a girl,_ Sticky says, checking himself in the mirror again. The blood is coming back molasses-slow now. It's caking up everywhere except in one little spot. _Matter of_ _fact, I gotta go get some stuff for her birthday tonight. That's_ _why I gotta hurry_.
Chuck shows the whites of his eyes. He finds Sticky's eyes in the mirror and says: _So you already gots you a little_ _honey then?_
Sticky nods.
_And you tryin to get her a little somethin?_ Chuck reaches both hands behind the back of his head and links his fingers. He looks up at the ceiling. _I see,_ he says.
Sticky's mind is hoop channels again: three-game wait, max. Dante and Dallas probably got picked up already, but he could hook up his own squad. Get one more shot at Rob. Redeem himself. Anh-thu's birthday stuff can wait. He's still got business to attend to. His heart picks up its pace and he holds the rag out to Chuck (white framed by red at this point). _What should I do with this?_
Chuck looks at the rag. _Don't give it to me, Stick. Go on_ _and throw it away_.
Sticky dabs one more dab. Nothing. He throws the rag on top of the overflowing trash. As he turns to leave the bathroom, Chuck wraps a meat hook around his thin elbow. _Hold up, Stick,_ he says. _I know you all anxious to get back and_ _ball, but let's first figure out this birthday thing_.
Chuck releases Sticky's elbow and folds up his arms. He glances at the door, looks over at the stalls. _How much money_ _you got?_
_Twelve bucks._
_Twelve bucks?_
Sticky nods his head.
_Shit, Stick,_ Chuck says. _You can't do nothing with no_ _twelve bucks_. Chucks looks at the floor, takes his right mitt and adjusts himself a little. _Well, hell, you ever snatch some_ _lady's purse?_
Sticky leans back against the sink and starts messing with the empty soap dispenser. _I ain't gonna take no lady's_ _purse,_ he says. _I'll swipe somethin from a store, you know, but I_ _can't be rippin off no lady's purse_.
_Oh, I see, you some sorta moral thief, right?_ Chuck throws his hands in the air. He reaches up to scratch the top of his head and looks Sticky right in the eyes. _Stealin is stealin,_ _Stick. Don't matter if it's from a store or some little old lady, it's_ _the exact same state of condition_.
Sticky hops up on the sink and stares at the floor. He gets his legs swinging like a little kid might.
Chuck walks over in front of the door, puts his hands on the overhang and looks out. Fat-man sweat stains under both arms. Shirt raised where you can see his stretch-mark stomach climbing up over his sweatpants drawstring. _You_ _need money, Stick. I'm gonna tell you that right now_.
Sticky gently touches his cut with his fingers.
Chuck lets his eyes wander outside again, looks both ways. Brings a hand down to adjust his sweatpants a little. He turns and lumbers back into the bathroom. _That's your_ _only option, the way I see it_. He puts a round hand on Sticky's shoulder.
Sticky slides off the sink. Feels warm Fat Chuck energy pass through his shirt and skin.
First-stall resident pipes up. At first he's whining and coughing. Both Chuck and Sticky turn to the sound. Chuck drops his hand. Then the homeless guy starts slurring out some crazy political statement. _Down with the white man,_ he says. _It's the white devil that done it to us,_ he says.
_Shut up, old man!_ Chuck yells.
_It's the white devil,_ the guy says again.
_Shut up!_
Sticky gets the water shooting out again, splashes it on his face. This type of talk never gets to Sticky simply because he's never seen himself as white. He hears it all the time. The antiwhite stuff. It's up in the bleachers. It's out by the hot dog stand. It's in Jimmy's office. Guys always talk a little lower when they spot him coming. Or they say things like _We don't mean you, Stick_. Or _You're different, Stick_. But the truth is, it never would have crossed his mind. That they might group him with the whites. It's something that has never even occurred to him.
Sticky cranks out a few paper towels and rubs his face dry.
_Anyways,_ Chuck says. _What you gotta do is find some old_ _rich-looking broad walking an empty street. Come up from behind her and snatch her purse. Simple as that. If she tries to_ _scream, smack her over the head_.
_I hear what you sayin,_ Sticky says, hoping the lecture is over. He nods a couple times and tosses the damp paper towel on top of the rag. But as he starts toward the door, Fat Chuck picks him off again.
_Only one other idea I could come up with,_ Chuck says, running a fat left hand up Sticky's inner thigh.
Sticky fights to get away, but Chuck has too much bulk. Too much power. Like being posted up by Rob on game point. _Do me this one favor,_ Chucks says, struggling to keep Sticky still. _Do this one thing for me and I'll personally drive_ _you there. Buy whatever she wants_.
Sticky jerks his arm back, yells: _Come on, man! Lemme_ _go!_ But before he can gain any leverage, Chuck shoves Sticky's hand into his lap.
Sticky fights even harder. He pushes and pulls, kicks, scratches, bites. But Chuck won't let go.
_That's it, white boy,_ he says. _I'll buy your little girlfriend_ _whatever she wants_. He grabs Sticky's head with one hand, pulls down the front of his sweatpants with the other.
The homeless guy starts whining again. Sticky's Nikes squeak against the concrete. Something pops when Chuck leans all his weight against the sink.
Sticky finally spins out and pulls away. He boots Chuck in his sloppy stomach two times quick and darts into the parking lot.
Chuck doubles forward and holds himself. He goes down on one knee and then hurriedly grabs the sink and pulls himself back up. Before both feet are even on the ground he is sprinting out into the parking lot toward his car. Duck-footed. Stomach bouncing.
Sticky rushes into the gym and goes straight up to Dallas, who's sitting in the bleachers. _That faggot Chuck,_ he says, sucking in breaths. _He tried to . . ._
Dallas straightens up, says: _What, boy?_
_Chuck tried to make me . . ._ Sticky shoots a look out the gym door and tries to wipe Fat Chuck off his hand.
New York and Dreadlock Man stop shooting. Dollar Bill looks up from tightening his laces.
Dallas stands up.
Sticky tries to catch his breath. Tries to wipe Fat Chuck off his hand. Onto his shorts. He points out the door, toward the parking lot. _Chuck tried to make me suck him off_.
Dallas looks down the barrel of Sticky's finger. Outside the gym. He spots Chuck lumbering through the parking lot and takes off sprinting. New York takes off too. Dollar Bill. The game stops and Trey and Slim ask what's going on.
A group of Lincoln Rec regulars drop everything and take off after Dallas and Dollar Bill.
Jimmy hears the rumbling outside his office window and rushes out. _Wh-wh-wh-what is it?_ He brings up the rear of the pack, yelling the whole time for someone to tell him what's going on.
Sticky races after the pack.
When they reach the parking lot, New York spots Chuck squeezing into his old, paint-chipped Buick. Everybody charges after him.
The suits on the sidewalk stop walking to watch this pack sprinting through the lot, cutting and leaping over cars.
Chuck slams his heavy door shut and fumbles through his bag for his keys. When he finally gets ahold of them he frantically shoves the car key into the ignition and cranks it. As the big boat coughs and turns over a few times, he looks over his shoulder. It finally starts and he pulls it into reverse, tires squealing as he hastily backs out of the parking spot.
New York is the first to arrive just as Chuck is slamming it into drive. He pounds on Chuck's hood and kicks the door in. He reaches for the handle but misses, yells: _Faggot!_
Chuck floors it. Tires spinning to grip pavement. Smoke lifting into the air.
Dollar Bill and Dallas catch up to New York. Dreadlock Man picks up an empty forty bottle and heaves it at Chuck's car. It shatters against the back door. Old-man Perkins and Johnson catch up. They both pick up rocks and fire them at the Buick as it speeds through the lot. One of the rocks crashes through the driver's side window. Chuck ducks, puts a meaty hand up to save his head.
They all stop running when Chuck rounds the last island in the parking lot and peels onto the street.
Sticky watches as Chuck's Buick blazes down the road and out of sight.
Heads shake. Language flies. Old-man Perkins says he knew there was something not right about that cat. They all underline fierce words with forceful hand motions. Declare what they would have done if they were just a couple seconds quicker.
Sticky kicks the tire of somebody's Pathfinder. He lowers his head and wishes he was invisible. The Fat Chuck thing was bad, yeah, hell yeah, but this is even worse. All the guys asking him to explain, asking him to review the situation. In detail. The sun pounding off the pavement. His knees all weak like a sissy. This is the worst part of it. He messed up the games by running in there like a little bitch and now everybody's looking at him, waiting for him to speak, telling him it's OK. But it's not OK. It's all messed up.
Everything's messed up.
The suits that were walking the sidewalk are now in one big group, speculating. There are curious faces pressed against most of the business windows. Sticky reaches up to his cut and discovers that it's bleeding again. It's dripping blood onto his shirt, his shorts, the pavement. He starts back toward the gym, tries to wipe Fat Chuck off his hand.
Everybody asks him question after question, about what happened, where it happened, why it happened, and Sticky keeps his cool. He answers clinically. Detached. But the minute Dante comes up, something changes. Sticky's face looks like a little kid's and he stares at the ground. He doesn't want Dante to see him like this. Like a victim. Like he can't take care of himself.
_What happened?_ Dante says.
_It was Chuck,_ Sticky says. And then he kicks another SUV tire as hard as he can. He clenches his fists and slams the hood of an SUV. His face turns into a vicious frown and he quickens the pace of his walking.
New York and Dallas come jogging up. New York tells Dante: _Chuck tried to fuck with Sticky_.
Dallas says: _We gonna find that fat-ass, though. Trust me_.
New York says: _Might not be today, tomorrow, but we'll get_ _him_.
Dante looks at the blood streaming down Sticky's face. Nods his head.
Dollar Bill adds: _We'll get em soon enough. Don't you_ _worry_.
Big Mac comes jogging out of the gym all sweaty and asks what happened. Carlos comes out. Even Rob peeks his head through the doors.
Jimmy marches past the others and goes right up to Sticky. He takes him by the arm and marches with him toward the gym. L-l-l-l-let's g-g-g-g-go, Sticky! We g-g-g-gotta _c-c-c-call the p-p-p-p-p-po-po-po-po—armed forces!_
Mrs. Smith Brought
her whole family to Sticky's foster care pad for the pickup: Mr. Smith with his Coke-bottle glasses; Tammie and Jamie, their two well-developed and wide-eyed daughters (seventeen and fifteen); and Johnny, their seven-year-old son with the two missing front teeth. Sticky was fourteen when they pulled along the curb in a sparkling white minivan. When they filed out wearing big smiles and hustled up the driveway together.
This was Sticky's third try at finding a family that fit. Mr. and Mrs. Smith shook hands with all the counselors. Mrs. Smith placed her hand on the old Mexican director's shoulder like they were old friends and told him: _We both have degrees in social work_. She turned and smiled at her husband. _So don't worry, we know what we're_ _getting into_.
All the counselors nodded their heads. They smiled, too.
When Sticky came walking into the office with his bag, Mrs. Smith gave him a long tight hug. She pressed his cheek against hers and told him: _Welcome to our family._ Then she tousled his hair with her hand.
Sticky bristled under all that touch.
Mr. Smith walked up when it was his turn, wrapped hairy arms around Sticky's stiff frame and squeezed. _We_ _know about the troubles you've had finding the right home,_ he said, pulling his face away and fishing for Sticky's eye. _Well,_ _that search is over now_. He placed a soft hand on the back of Sticky's head and took a deep breath. _Young man,_ he said, _I'm going to let you know this right up front, the only way_ _you're going to see this place again is if you want to visit a_ _friend. Either that or I drop dead of a heart attack_.
_Honey!_ Mrs. Smith said, poking her husband in the arm. _That's a terrible thing to say_.
_Well, that's how strongly I feel about this, sweetheart._
Mrs. Smith blushed and looked at the old Mexican director. _Hopefully it's just to see a friend, right?_ she said. _I_ _mean, my God_.
The director nodded.
_Let's hope,_ Mr. Smith said, and he laughed.
Tammie walked up and gave Sticky a hug, told him: _We're really happy to have you_.
Jamie giggled a little and hugged him next. _Nice to meet_ _you,_ she said.
Mrs. Smith shot a look at little Johnny, and he timidly stepped up for the hug too. _Do you like the Dodgers?_ he asked as they separated, and everybody laughed. Sticky nodded.
Then the Smiths loaded all their own kids, plus Sticky, into the van and drove off toward their home in Oxnard.
The only problem was, by the time the Smiths got their hands on Sticky he'd already started to figure out who he was supposed to be. How he was supposed to fit in. And for the first time in his life he was determined to play the role.
The Smiths tried their hardest to treat Sticky like one of their own. Bought him new shirts, new jeans, shoes and socks. Mrs. Smith cooked dinner every night. But Sticky was trying even harder to be a thug. On the way to those dinners he would cruise past a crowded park and ride home on somebody's unlocked ten-speed.
The Smiths set up "family hour." Every night they'd all sit around the living room with the TV off and talk about their days. Something good that happened and something not so good. But the first week Sticky missed a meeting. He was hanging out in a baseball dugout on the other side of town with some kids from school. A pack of black dudes who stayed in a local group home. Two crazy Mexican cats who somehow managed to smuggle beers from their dads' stashes.
The second week Sticky missed two meetings. One of the black dudes showed up with a stack of old _Playboy_ s. One of the Mexican dudes brought smokes. They started setting up meetings of their own after school. Told people they were forming a new gang. Every night, as the sun set over the right-field fence, they'd file into the dugout with new recruits and try to one-up each other. The shadier the show-and-tell, the bigger the impression.
After a month Sticky stopped showing up to "family hour" completely.
He started skipping certain classes to smoke out behind the track-and-field shed. A couple hits off a buddy's blunt and he'd be flying above the goalposts. Or he'd bail out on school altogether and cruise down to the arcade with his boys. Street Fighter and Mortal Kombat, old-school Mr. Do! and Police Trainer.
Foosball.
One night when Sticky walked into the house, Mr. and Mrs. Smith were sitting at the kitchen table waiting for him. _Sticky,_ Mrs. Smith said, _could we talk to you for a minute?_
_Grab a chair, son,_ Mr. Smith said.
Sticky took a seat and Mrs. Smith held up his report card: five Fs and a C in PE.
_Now, this is just a progress report, son,_ Mr. Smith said. _You_ _have a chance to turn this around if you put forth the effort_.
_But it's unacceptable,_ Mrs. Smith said. _We called the_ _school and your attendance is horrific. Why aren't you going_ _to class?_
Mr. Smith took his wife's hand and told her: _All right_ _now, sweetheart, this is about turning things around from here._ _We have to remember to look forward_.
_I know it, honey. OK._
_I think the boy needs to know that we believe in him. That_ _we love him._
_And that we're here no matter what._
_Exactly._
They both turned to Sticky.
_You do know that, don't you, son?_ Mr. Smith said. _That we_ _love and support you one hundred percent?_
The following Saturday Sticky was brought home by the cops. He got caught swiping a pair of binoculars from Kmart by undercover security. (This was back when he was still refining his skills.)
Mrs. Smith was stunned when she opened the front door and found her foster boy in handcuffs. _Oh, my God,_ she said, and yelled for her own kids to hurry upstairs.
Mr. Smith tried tough love after that situation. Restriction. Extra chores. An early curfew on Friday and Saturday nights.
_Son,_ Mr. Smith said a couple nights later, as he moved Sticky's bed, piece by piece, out of little Johnny's room and into the sewing room. _I want you to know that I understand_ _what you're feeling inside_. He lifted one side of the single mattress and Sticky picked up the other. They moved awkwardly through the door frame and into the hall. _I've seen it_ all before, you have to realize. They leaned the mattress against the sewing room wall and walked back into Johnny's room, picked up the headboard. _I know how awful you must_ _have felt after what happened with your mother_.
Sticky dropped his side of the headboard after that comment. Just dropped it on the rug and walked out of little Johnny's room without a word. One mention of Baby and he was gone. He cruised straight down the stairs and out the front door.
At around four in the morning, Sticky staggered back into the house through the side door. He was so high he passed out in the middle of the kitchen floor with an open half-gallon of orange juice in his hand.
Jamie found him like that when she walked into the kitchen for a drink.
_Sticky?_ she said. _Oh, my God. Sticky!_
Sticky lifted his head to look at her. There was blood all over his face and hands, and he was smiling.
Jamie dragged him into the upstairs bathroom and gently cleaned his face with a warm washcloth. _What happened_ _to you?_ she said.
Sticky shrugged his shoulders.
_You're, like, on drugs, aren't you?_
He reached up and touched her face. _You look good,_ he said, and ran a finger through her long blond hair.
_Well, you better go back to your room and act like you're_ _asleep,_ she said. _I'll tell them you're home. They've been, like,_ _freaking out all night_. She wrapped Sticky's arm around her neck and helped him walk to his room. She put him on his bed, pulled off his shoes and socks and covered him with a blanket.
Sticky was asleep before Jamie even left the room.
This prompted a big family meeting the following night before dinner. There were discussions about hard-core counseling and antidepressants. Mentor programs and outpatient drug rehab centers.
But Sticky talked his way out of everything when Mr. and Mrs. Smith brought it up again over dessert. _I'll change,_ he said. _I swear I'll change_. He put down his fork and gave them both his most honest expression. _I wanna be better_ _now_.
Mr. Smith turned to Mrs. Smith, told her: _You know,_ _honey, I think he means it_.
_I do too,_ Mrs. Smith said.
_I think it's about trust right now._
It's a trust issue.
_I wanna be better,_ Sticky said.
Mrs. Smith turned her attention back to Sticky and reached over the table for his hand. _All right, son, we're going_ _to trust you on this one_.
Mr. Smith nodded his head and smiled at them both.
But two weeks later, Sticky took things to another level.
It was a Sunday afternoon and Mr. Smith was putting in extra hours at work. Mrs. Smith was out running errands. Sticky flipped off the TV and headed upstairs. Knocked on Jamie's door. _Hey, Jamie,_ he said, and knocked again. _You_ _in there?_
Jamie pulled the door open and waved him in. _Of course_ _I am,_ she said. She was listening to Incubus and filing her nails. There were posters all over her walls: Bob Marley, Blink-182, Pearl Jam. Sticky was standing at the door, looking around the room, when Jamie told him: _Um, you can,_ _like, come in, you know. I'm only doing my nails_. She stood up and pulled him by his arm, closed the door behind them. She sat Indian style at the edge of her bed, and Sticky lounged into the big purple beanbag on the floor.
Jamie handed Sticky a photo album of all her friends, told him he could look through it if he felt like it. While Sticky turned the pages, she excitedly played parts of all her favorite songs. _I love Rage,_ she said, pointing to a series of posters on her wall. _But I love Radiohead and Coldplay, too. I_ _think there are, like, times you wanna hear slow songs and_ _times you wanna hear fast songs. I like any type of music that_ _inspires me_.
Sticky laughed at her energy.
They were both quiet for a while, listening to the first few tracks off the Toadies album. Jamie pulled a glittery pink nail polish from inside a drawer and started applying careful strokes. When Sticky was done looking at all the pages of pictures in her photo album, he started at the front again.
_So, you got a boyfriend?_ Sticky said, studying a picture of Jamie posed with some guy at a dance.
_Not really,_ she said. _That guy, like, thinks we're together, I_ _guess. His name's Ricky. But I'm over it_. She blew on her wet nails and said: _What about you? You have a girlfriend?_
_Nah,_ Sticky said.
Jamie screwed the nail polish cap back on and tossed the bottle behind her on the bed. She stretched out on her stomach and faced Sticky. _Have you ever, like, been with somebody, though?_
_Huh?_ Sticky said. He set the photo album on the floor and sat up in the beanbag.
_You know, have you ever . . ._
_Yeah, I been with somebody._
_Who?_
_This girl Maria._
_Was it before or after you started living at our house?_ Jamie leaned her chin in the palms of her hands and stared at Sticky, fascinated.
_She used to stay at the same place I stayed,_ Sticky said.
_Did you like her?_
_I mean, we was friends and all that._
Jamie covered her face with her hands and giggled. She looked at Sticky and told him: _God, that's so weird if you_ _think about it_.
Sticky stood up from the beanbag and sat next to Jamie on the bed. _What about you?_ he said.
_No. Never._
_You'd probably like it._
_I've never even been to second base_. Jamie rolled her eyes and laughed at herself.
Sticky reached over to the stereo. He turned off her Nirvana and tuned in his favorite hip-hop station.
Jay-Z filled the room with rhyme.
Mrs. Smith arrived home in good spirits.
She walked into the house leafing through a stack of junk mail and humming under her breath. She tossed the mail on the end table by the couch and headed upstairs. When she opened the bedroom door, singing out Jamie's name, she found Sticky and Jamie half naked on the bed. Sticky's hands all over her daughter. Shorts and skirt thrown recklessly on the floor.
Mrs. Smith freaked out.
She slapped her daughter across the face and called her a whore. She reached over the bed and punched Sticky in the back, in the neck, on the shoulder. When he ducked out of the way she got him in the leg.
She swung a few more wild fists at Sticky and then ran out of the room holding her hands over her face. She rumbled down the stairs, picked up the phone and called Mr. Smith at work. Through hysterical tears, she told him to come home right away. Then she hung up the phone and rumbled back up the stairs to slap her daughter again and try and hit Sticky some more.
Jamie was crying too. She screamed at the top of her lungs when she caught another one of her mom's hands across the face.
Sticky managed to dodge most of Mrs. Smith's flailing while at the same time pulling on his shirt and shorts.
An hour later, Mr. Smith came rushing through the front door and immediately locked Sticky out in the garage with the dog. Sticky spent that night sleeping on a cold cot next to a rusty tool bench.
The next morning Mr. Smith packed up Sticky's bag and stuck it in the back of the van. He opened the passenger door and let Sticky in. Then he took his seat at the wheel. He stared out the window for a few minutes before starting the motor. For this leg of the trip it was just him and Sticky. There would be no more big family affairs. No more upbeat conversations. No more hugging and talk of trust.
Sticky sank into the passenger seat and hung his head. He had succeeded in playing a role.
Mrs. Smith stayed inside with her kids. She watched her husband through the living room window as he turned the key, flipped the car into reverse and rolled down the driveway.
Tammie and Johnny sat at the kitchen table eating oatmeal with their heads down. Neither said a word.
Jamie watched from her bedroom window with glassy eyes as the white van made its way down their street, stopped at the stop sign and then turned out of sight.
When Mr. Smith pulled up outside the foster care pad again, not even a year after he'd come to pick Sticky up, the old Mexican director was once again standing out on the curb, waiting.
_We've got young children,_ Mr. Smith said after he shut off the engine and hopped out of the van.
_I understand,_ the old Mexican director said, and he crossed his arms.
_You know how it is,_ Mr. Smith said, and he stood there a second, slipped his hands in his pockets. _They're impressionable_. He watched Sticky walk around the van, then pulled open the sliding side door and reached in for Sticky's bag. Set it down on the sidewalk and shook his head.
The director picked up the bag.
_I have to think about my own kids right now,_ Mr. Smith said, and then he climbed back into the van and drove away.
Dave Was Snapping
his fingers to the rhythm of the reggae when Sin pulled his Impala along the curb and shut off the engine. They were about a block from Milo's Liquor in Venice, the place they always ended up on the nights they hung together. Where the forties were cheap and nobody asked for ID.
The buzz of the big play-off win was still spinning in their heads, and the fellas were set on prolonging their night.
Sin reached underneath his seat for his beanie, pulled it on over his shaved head and checked himself in the rearview. Sticky slipped out of the backseat. He leaned back in and slipped out again.
Leaned back in and slipped out.
Leaned back in and slipped out.
Leaned back in and slipped out.
When something in the process was precise, and just before Dave and Sin turned to look, he slammed the door shut cool and joined his boys.
In all the commotion, an old white man lifted trash-can eyes from his sleeping bag. He was lying between rusted fence posts near an old abandoned shed. All the windows were busted out and the rotting plywood served as a practice canvas for up-and-coming taggers.
It was a cool spring night. The streetlamps looked like little yellow suns hanging from concrete posts.
Dave pushed Sticky in front of the homeless dude. _Hey, old-timer,_ he said. _See this kid right here? Yo, he just_ single-handedly beat Dominguez Hills, man. Goddamn single- _handedly_.
_And dude's girl was right there to see it all,_ Sin said.
_And Annie's fine, too,_ Dave said. _Especially for an_ _Oriental_.
Sin and Sticky laughed when Dave started popping without music. The homeless dude shifted around his sweatshirt-pillow, dropped his head and closed his eyes.
They walked along the vacant sidewalk three deep, pulled open Milo's security-barred door and headed straight for the beer section. Sticky swung open the spidered glass door and reached in for three cold forties.
_I got the beer,_ Sin said, pulling out his Velcro wallet on the way up to the counter.
When Milo handed Sin his change, Dave hopped up on the counter and pumped his fist into the security camera. _Venice High, yo! Venice fucking High!_ He twisted the cap with his shirt, put the bottle up to the camera and let the cold beer run down his throat.
Sin and Sticky watched Dave dance to the strange music coming out of Milo's old transistor radio. All 6 **'** 6'' of him grooving on the counter, ducking his head slightly to avoid the ceiling.
_Yo, this is a jam,_ Dave said, and he turned to Sticky and Sin. _You all don't know about this_. He got his arms into it, snapped in rhythm with his empty left hand. _I could make_ _love to this beat_.
Milo tugged at the bottom of Dave's pants. _Must come_ _down,_ he said in his jacked-up English. He grinned at Sticky and Sin, pulled on Dave's pants again and repeated himself: _Must come down_.
Dave put his forty up to the camera again and then went to get down. My bad, Milo. We just beat Dominguez Hills _tonight and I'm super up_. He hopped down to the floor with the grace of a power forward and wiped off the counter with a bare hand.
Milo smiled.
The fellas loved Milo. Not just because he sold to them, but because he was always smiling. The liquor store was all he had. No wife. No kids. No car. Whenever Sticky used the bathroom in back, he'd spy the unmade futon-bed where Milo slept. But still, he always had a smile. Even for three punks that came into his little store looking like trouble.
Dave grabbed a roll of paper towels from behind the register. He wadded up three and wiped away the giant footprints his Nikes had left.
_Let's roll,_ Sin said, and he took a long swig.
Milo pulled out an open box of Tootsie Pops and pointed for the guys to take some.
Dave shook his head.
Sin politely told him: _No thanks_.
But Sticky's eyes lit up. _Man, I dig these,_ he said, unwrapping one and popping it in his mouth. _We used to get em as a_ _reward back at the place I used to stay_.
Sticky took a couple little hits off his lollipop and then a big hit off his forty. The candy made him remember that he was a year older today. Seventeen instead of sixteen. Almost a man. He wondered if maybe he was too old to eat candy now. Maybe it wasn't cool anymore. He considered this for a quick sec and then reached his hand in the box for a couple backups. A blue one and an orange one. Stuck them both in his pocket, waved to Milo and followed his boys out of the store.
The fellas rolled down Broadway, buzzing, recapping the big game. Dave said his favorite play was when Sticky took two guys baseline and spun in a reverse. He made Sticky and Sin stop so he could show how the move looked from his seat.
_Say this beam was the hoop,_ he said, pointing to the makeshift sidewalk that had been put in front of a construction site. _Sticky came in from the side like this_. Dave ducked his shoulder and pretended to dribble low to the ground. _He_ _barely got by em and I was like, oh, man, big guy might swat_ _this. But thing is, Sticky used the rim to protect the ball_. Dave took the ball up on one side of the beam, ducked and came up on the other side. _That was a smooth-ass layin right there,_ he said. _Most guys don't know how to do it, protect the ball_ with the rim.
A block later, Sin came up with his. _I know what the play_ _of the day was, though. Besides the game winner_. Sin set his forty down so he could involve his hands in his story. _You_ _know that little guard they got? Number twenty-three or whatever? The play of the day was when he came flyin through the_ _lane and Fat Jay straight put em to the ground_.
_Oh, damn, that was a nasty foul,_ Dave said, holding a fist to his mouth.
_Jay had to slap em down, though,_ Sticky said. _He had to let_ _em know whose lane it was_.
_Fat Jay did just like this,_ Sin said, and he put his arm out clothesline style and swung it through the air. _That's the_ _kinda foul that could set the tone. Like Sticky said. After Fat Jay_ _did that, wasn't no little number twenty-three flyin through_ _the lane no more, right?_
Sin yelled out at a car speeding by: _Ain't no free shit in_ _our house, boy!_
They all laughed and kept walking.
The wind increased slightly, picked up loose grocery bags, scraps of paper, then set them back down. Background sound. Broadway is a sad gray face after the sun goes down, with Santa Monica Boulevard and Wilshire shining bright only a couple blocks north. Dull light from fading streetlamps, peeling billboards, tired and stained sidewalks that crumble in places under the weight of heavy feet. Lifeless homeless bodies curled up sleeping under old-school CLOSED signs. As they walked, a range of different cars sprayed music out from open windows as they zipped past. Sticky booted a rock ahead of them. It rebounded off a cement wall and Sin kicked it.
Dave stopped in front of Corona Imports and pointed inside the window. _Yo, I wish I could just roll a Ferrari around_ _one time,_ he said. He put his hands and face up to the glass. _Look at that car, man_.
Sin stepped up to the glass too. _For real, that's a smooth_ _ride right there,_ he said. _I'd just wanna sit in it one time_. A black convertible Ferrari sat inside the glass sparkling like jewelry under a security light.
_Yo, just one time,_ Dave said.
Sticky listened to Dave and Sin go on and on about the car, how fast it goes from zero to sixty, how much rich cats are willing to pay to drive right off the showroom floor. He looked at the way his boys stood there, hands against the glass, on the outside looking in. Drooling over something completely out of their reach. He set down his forty, picked up a chunk of concrete from the broken sidewalk, shooed Sin and Dave out of the way: _Look out, look out_.
_What?_ Dave said.
_What the hell you doin?_ Sin said.
Sticky took a couple running steps and heaved the concrete through the glass. Just like that. Without thinking. Chucked it right through the window so shards of thick glass spilled onto the showroom floor on volume ten.
_Go ahead!_ Sticky yelled over the screaming alarm. _Go on_ _and sit in it!_
Sin's and Dave's eyes bugged out of their heads. Their mouths hung open like old socks.
People in passing cars slowed and stuck heads out of rolled-down windows.
_Holy shit!_ Dave said.
_Damn!_ Sin said. _Let's get the hell outta here!_ And they took off running through an alley behind Lincoln.
It wasn't four blocks before the first cop car spotted them running.
Dave and Sin followed Sticky as he made a sharp right behind Denny's and Ralphs. They shoved their way through a group of hipsters smoking outside and ducked behind a boarded-up flower stand. The cop screeched around the corner and turned his lights and siren on.
The fellas raced back up to Lincoln, past the art supply shop, the Italian deli, the long string of antique shops. There was nothing but the sound of heavy air whipping past their ears, the loud siren gaining on them by the second.
Sticky led them west down Santa Monica Boulevard and then north along Seventh. He sprinted with his hands fixed sharp like Carl Lewis. Whizzed past Al's Guitar Shop, Primavera Pizza, the building where a Nike Outlet once stood. At the 7-Eleven on the corner of Wilshire and Seventh, he shot west past Wahoos and Chevron and Houstons. The avant-garde bagel shop. Then to throw the cops completely off their trail he led them through the crowded promenade. Weaved in and out of thick packs of people.
Dave and Sin followed their point guard wherever he went. Sin directly behind. Dave a couple steps off the pace.
Sticky swung into an alley between Third and Fourth and ducked behind some trash bins. The cop car screamed by. They all huddled there for a while, laughing nervously and sucking in quick, choppy breaths.
_What the hell?_ Sin said, grabbing for air.
_Jesus Christ,_ Dave said. _You outta your mind?_
Sticky shrugged, hands on knees. He brought a devil grin up to both of them.
They started sneaking back through the Promenade in silence, but just as they rounded the corner onto Third, another black-and-white spotted them.
As they took off running again, the car screeched to a stop and two cops hopped out to pursue on foot. The cops ran back the other way, down the same streets, past the same businesses. Keys rattling, clubs swinging, leather boots clopping against the pavement.
Sticky, Dave and Sin tried jumping the fence to the Boys and Girls Club back up on Lincoln, to hide behind the skate-board ramps, but one of the officers grabbed Dave by an ankle and pulled him back down.
Sticky and Sin made their way back over too, and all three put their hands up on the fence like they were told.
Both cops were sucking hard for breath as they patted the fellas down: slapped hard on the chest, hard on the hips, hard on the back of the legs, grabbed hard around the ankle. They got names and ages and one of the cops talked into his radio.
_Dumb decision you gentlemen just made,_ the other cop said.
_Which one of you threw the rock?_ the radio cop said.
Nobody moved.
_Simple question, guys. Which one of you threw the rock?_
Sticky put his eyes on the pavement. Drips of sweat ran down the back of his neck. Sin and Dave picked up their heads and snuck glances at him.
Another car rolled onto the scene, lights flashing without sound. It slowed to a stop and a woman cop stepped out. She stayed back and talked into her radio.
_So, what's it gonna be?_ one of the cops said. _We gonna_ _have to do this the hard way?_
Just as the cop finished his sentence, Sticky raised his hand off the fence. He turned around and looked at the cop.
_That's more like it. A little cooperation would be a good_ _thing right about now_. The cop moved over to Sticky and pulled his arms down one at a time, slapped on the cuffs. _Some pretty dumb shit you just pulled, kid_.
The cop with the radio came walking over with a grin on his face. _You're not gonna believe this,_ he said, tapping his partner on the shoulder with the radio antenna. _The kid who_ _threw it, it's his goddamn birthday_.
_Say what?_
_This kid here_. He pointed at Sticky with his radio. _Travis_ _Reichard. It's his birthday today_.
_Well, what a way to bring in a new year._
They both laughed a little and then one of them asked: _So, why'd you do it, Travis?_
Sticky cringed at the sound of his real name. He didn't look up.
The other cop put his right thumb through his belt loop and stared down at Sticky. _I don't get it, ace. Why not just get_ _a piñata like normal kids_.
Both cops thought that was pretty funny, too.
The woman cop walked over and they gave her the scoop.
_It's your birthday?_ Dave said while the cops were huddled together, talking.
The lights spun around the cop car without sound. A flash of bright blue light spinning around methodically. Cars slowed as they passed, stared out at the fellas as they were led from the fence toward one of the cop cars, guided into the backseat. A motorcycle cop rolled onto the scene. He shut off his engine and flipped out his kickstand. Took off his helmet and started walking toward the other cops.
An old homeless black woman rolled her cart by without looking. She had a black bandana covering her hair, a soiled USC sweatshirt and boots without laces. She kept her head down and hummed to herself as she passed.
There were muffled voices coming over the cop's radio. The tinny echo of the rattling cart. The soothing sound of the woman humming as she slowed by. Never once did she look up to see what it was all about.
_Check This Move,_
Sticky said, and Anh-thu turned around to watch. _This is for when some dude gets all up on me_. Sticky jab-stepped with his right foot, hesitated, and took off baseline. Cupped the ball between his right hand and forearm and spun it in cool off the glass. _Just like that, Annie. These fools_ _can't hold me_.
_Too bad nobody's really guarding you,_ Anh-thu said, and she went back to reading her book.
It was Saturday night back in mid-April, and the park Sticky'd pulled Anh-thu into was dark. She had to put her book right up to a yellow floodlight to make out the words.
_But if there was,_ Sticky said. _That's how I'd do em_.
On their way to some high school party on Second Street, at some guy named Cyrus's house, Sticky spotted the baskets and veered Anh-thu onto the empty blacktop with pleading eyes. He just couldn't resist it. Needed to fix. It was his last Saturday of community service, the court-ordered 100 hours he'd been given for the window-breaking incident, and he was hyped. No more ten-hour weekend days cleaning up graffiti. No more orange vest.
He pulled his beat-up ball out of his backpack and ran his fingers along the grooves. _Just a couple quick jumpers,_ he'd said. But Anh-thu knew better and rolled her eyes. Sticky ripped off his sweatshirt and dribbled over to the nearest hoop. Anh-thu sat with their stuff and pulled out the novel she was reading for English class.
Over an hour spiraled by and here they were, still holding down the same spots.
_I'll post this dude up,_ Sticky said, crab-dribbling with his back to the basket, digging a couple elbows into an imaginary defender's chest. He spun quick and rose up off two feet. Dunked the ball with two hands and hung on the rim. _Too strong,_ he said, and let himself drop. _These fools gotta get_ _in the weight room_.
He picked up his ball and dribbled out to the wing. _This_ _ain't no fair, Annie. These dudes ain't got no defense. I need_ _someone out here who could at least put on a little defense_.
Anh-thu, without looking up from her book, told him: _What happened to you taking me to that big party?_
_I am,_ Sticky said. _I just gotta finish off these fools_.
The park had four hoops, all without nets. One of the rims was bent so far it pointed right back down at the blacktop. The old-school wooden backboards were all tagged up with different-style gang markings. Sticky knew how to take care of stuff like that, though. He'd even kept some of the community service solvent thinking he might one day clean up the outside of Lincoln Rec. Figured there shouldn't be any taggings on a dude's home away from home.
Anh-thu looked up at Sticky for a second. Watched him spin in a reverse layup and point at an invisible crowd. She shook her head and reached into her bag for a pack of Starburst. She unwrapped a red one and popped it in her mouth.
_You better get up on me,_ Sticky said to a new defender. _I'll_ _knock em down all day_. He was dribbling around at the top of the key, through his legs and close to the ground. Sizing up. Then he blasted to the basket and dunked the ball with two hands. _Too slow, man_.
Sticky kept on like that, talking trash and clowning, no matter who was guarding him. Most of the time he took the ball to the bucket, but sometimes he'd pull up for a deep jumper.
After a while, Anh-thu got tired of reading in such bad light. She closed her book and slipped it back into her bag. She watched Sticky for a bit and then lay down with her back against the blacktop, balling up Sticky's sweatshirt and using it as a pillow. There was a slight break in the night clouds and she could see the faint light of a couple stars peeking through.
It wasn't long before she closed her eyes and drifted into sleep.
_Jumpers is all about rhythm,_ Sticky started chanting. _Point your feet at the bucket. Elbow at the bucket. Middle_ _finger in the groove, thumb between the 7 and the F, bend_ _knees, and put it up_. He spied the writing on his ball to check his fingering and lofted up a shot from the free-throw line.
Ball went straight through.
He retrieved his rock and marched back out to the free-throw line. _Middle finger in the groove, thumb between the 7_ _and the F, bend knees, and put it up_.
Ball went straight through.
Retrieved his rock and marched back out to the free-throw line. _Middle finger in the groove, thumb between the 7_ _and the F, bend knees, and put it up_.
Ball went straight through.
Once he started getting in a rhythm like this, he could go on for hours without missing. Like a robot.
Cabs started to fill up along Wilshire. They crept along the curb outside Renee's waiting to gobble up the first group of people who waved or gave a loud whistle.
_Middle finger in the groove, thumb between the 7 and the_ _F, bend knees, and put it up._
Ball went straight through.
Anh-thu looked like a little girl lying on the concrete. Her fingers linked on her stomach, feet crossed. Breaths long and deep. The dull yellow glow highlighting her pretty face.
_Middle finger in the groove, thumb between the 7 and the_ _F, bend knees, and put it up._
Ball went straight through.
Deep thumping from a nearby club filled the black night with bass. A black-and-white flipped on its siren, screamed past Fifth, Sixth and Seventh, and headed south down Lincoln. Another black-and-white flew after without a siren, for backup.
Eventually Sticky stopped chanting and shot with his mouth shut. The sprinklers came on in the grass field beside the courts, making the cool air smell like rain. In the distance he could hear two women yelling at each other outside a bar.
Sticky lofted another shot up and the ball went straight through.
There were the faint sounds of the faraway sirens, a few honking cabs and the hollow tap of rubber on blacktop as Sticky dribbled back out to the free-throw line for another shot.
Sticky and Anh-thu
made six months just before school let out for summer. Sticky was already seventeen, but Anh-thu was still two months away from her sixteenth birthday.
Sticky rolled into Midnight Liquor on Rose with his backpack strapped tight. Slipped two bottles of black-label champagne in beside his books and took a pack of Hostess Cup Cakes to the counter.
_This it?_ the old man said, looking Sticky up and down.
_This is it,_ Sticky said, looking right back at him.
The old man bagged the Cup Cakes and dumped the change in Sticky's palm.
Sticky took the back way to Paradise Park, the place he and Anh-thu had agreed to meet. He was a shade early, but he didn't mind. It was a Friday night and the school year was coming to a close. The air was warm. The sky was just starting to change colors. June bugs were starting to show up, flying around all loud and clinging to everybody's screen door.
Earlier in the day, Sticky had pulled a letter out of the mail that informed him he'd been invited to some prestigious summer basketball camp. "Only the top players in the country will attend," the letter read. Every college coach would be up in the stands, scribbling things down in notebooks. Kentucky. UCLA. Duke and Indiana. Sticky's heart picked up its pace every time he thought about it. This was his chance. This was his dream. Two more weeks yawning in the back of a classroom, scanning sports page box scores, and then he could spend all-day-every-day at Lincoln Rec working on his game.
Sticky unwrapped his cupcakes on the way, took giant bites. Left uneven tooth marks in chocolate icing. He kept a running head count of all the BMWs that cruised by. Tried to picture what model he'd hook up if he was getting paid. What color. How fast he'd drive late at night when the roads were empty. As he walked, he made sure to step over every line. Sometimes he had to take a triple jumper's stride, sometimes he had to shorten up. But never on lines, and the whole time in his head keeping a BMW count.
The closer Sticky got to the park, the bigger the houses got. Dark green lawns, baskets set up over garages, wind chimes. Shiny cars pulled into sloping driveways. There were kids out front chasing each other around with spurting hoses. Laughing. A dad holding his baby on his shoulders. Somebody's mom snipping red and white roses out of a perfectly manicured flower bed, laying them side by side on a spread-out towel. Little kids racing around on tricycles and older kids playing Wiffle ball in the street. All of them watched Sticky out of the corner of their eyes as he took another bite of cupcake and stutter-stepped to avoid a line.
In the park, he watched couples stroll by holding hands as he tossed chocolate crumbs to the pigeons waddling around at his feet. He watched the birds dart in for the grab and then back off quick. Watched them set their little dinners on the cement path and then pick pick pick.
He had his best gear on, the blue collar shirt he swiped from the Foot Locker Outlet in Venice, a pair of ironed jeans. His newest hoop shoes. It was a weird deal to think about, having an anniversary with a girl. Six whole months. But Anh-thu made him feel happy.
When he made it to their spot he found Anh-thu already hanging out on a bench. Her long black hair falling down her back. White tank top. Short khaki skirt and flip-flops. Always flip-flops. She was reading a book.
He snuck up from behind and touched his hands lightly on her shoulders. Kissed her hair.
Anh-thu turned around, smiling. She closed her book and set it down. Grabbed Sticky's face and kissed him on the lips. _I can't believe you're on time,_ she said.
The sun was drifting below the rooftops. Dark colors trapped in clouds. A slight breeze played tree leaves quiet. Anh-thu reached down and picked up her book. She stuck it in her bag and pulled out some nicely folded blue tissue paper, handed it over. _Happy anniversary, Sticks_.
Sticky ripped the paper apart and let it fall to the grass, held out a brown bead necklace. _I made it myself last week,_ Anh-thu said.
Sticky held the necklace up to his eyes to check it good. He wrapped it around his neck and hooked the clasp. _This_ _thing's smooth,_ he said.
_Six months,_ she said. _That's a big deal_.
Sticky pushed his backpack over to Anh-thu, told her: _Go ahead. See what's in there_.
She unzipped the pack and reached in her hand. She grinned as she pulled out one of the bottles. _Wow, Sticks,_ _champagne. Two bottles_.
Sticky took the first bottle and tried to figure out how to open it. _A half year we been together,_ he said. _We gotta celebrate_. He pushed hard on the cork and it shot off. A few of the suds ran down the neck of the bottle and onto his hand. _There's some glasses in there too_. He wiped his hand on his pants.
Anh-thu reached in and pulled out two cream-colored coffee mugs. _You just thought of everything, didn't you?_ she said. _I bet you think you're being pretty romantic, don't you?_
_I'm like that one cat. The one you're always talkin about_ _you read in English class._
_Romeo?_
_Yeah, I'm like him_. Sticky poured the champagne slow, first in Anh-thu's mug and then in his own.
_I guess you're some kind of modern-day version,_ Anh-thu said. _How'd you get this champagne anyway?_
Sticky smelled inside his mug. He pulled back and smelled again. Pulled back and smelled again. He shrugged his shoulders. Then he smelled the mug again.
_You stole it, didn't you? Why do I even have to ask?_ Anh-thu took a sip, said: _Watch, someday you're gonna get caught_ _and then what?_
_Ain't nobody can catch me, girl._
_Oh, yeah?_
_Trust me, Annie,_ Sticky said. _I know what I'm doing._
_I just don't want anything to happen to you, Sticky. That's_ _all_. They looked at each other and Sticky laughed. They both put their mugs to their lips.
As the sun slowly dipped below the neighborhood, the automatic park lights clicked on and started buzzing. They took a few minutes to come to life. At this exact moment, a strange thing happened—most of the nice middle-class couples instinctively headed back to their safe houses and cars, while packs of street zombies seemingly came out of the bushes to take their place. But that's the way it is in a place like Paradise Park. Sunlight fades and the whole face of the park changes. One well-off couple gets up to go, two shopping cart women come out from hiding and stretch out their stiff arms and legs.
It's like two completely different worlds exist in L.A.: one that operates under the sun and another that slinks out under the shadows of the moon.
Sticky tipped back and emptied his first mugful. When Anh-thu followed his lead, he picked up the bottle and poured them both full again. He shook out the last couple drops over her mug.
_Let's play a game,_ Anh-thu said. She held her mug up to her face and looked closely at the chipped rim. Took a sip. _Let's play five questions. We each get to ask five questions_ _about anything, and the other person has to tell the truth no_ _matter what_.
Sticky shrugged his shoulders.
_You go first,_ Anh-thu said. _No, wait, I'll go first_. She took a sip and stared at the ground. _I know. What in the whole world_ _makes you the most happy?_
_What do you mean?_ Sticky said. _Like anything?_
_Yeah, what makes you feel the most happy inside?_
_You do._
_You can't say me, silly_. Anh-thu reached out and pushed Sticky's shoulder. _Besides, that's something we already know_.
Sticky ran a finger along a skinny crack on the outside of his mug. He took a sip. _I know what it is,_ he said.
_What?_
_Sometimes I get on this streak playin ball. I make like five,_ _six straight buckets and I get this feelin inside. I'm, like, locked_ _in or something. I feel like I could make any shot I try. Falling_ _away from twenty-three feet, driving in on a big man, off the_ glass, left-handed, whatever. It's crazy. It's like I'm doin magic _or somethin. And everybody on the sideline gets all louder_ _every time I hit another one. And I get this feeling all through_ _my body. My skin gets hot. Yeah, it's like magic. Like I could_ _just cast a spell and make anything happen. And I get all this_ _energy in my chest when it's like that. Cause I feel so alive._
Sticky stopped talking and tossed a stick against a tree. He took a quick sip of his champagne. _I don't know._ _Sometimes I wish I could just keep on playin when I got it like_ _that. I wish I didn't never have to stop._
Anh-thu stared at Sticky but didn't say anything.
Sticky looked at her, then quickly put his eyes on the grass. _Anyways,_ he said, and he reached up to scratch his head.
She took his face and made him look her in the eyes, said: _I swear to God, Sticks, I know you're gonna make it to college playing ball. You can see it in your eyes when you talk_.
_We gotta break out bottle number two,_ Sticky said, reaching into his pack. He pulled the bottle out and shot off the cork. He peeked into Anh-thu's mug and it was still half full. _You gotta finish this off, Annie. You drinkin like you ain't never_ _drank before_.
She took the mug and gunned the rest in one motion, handed it back to Sticky with a sour face. _Better?_
Sticky poured both mugs full again.
_OK, now you ask me one,_ Anh-thu said.
Sticky watched a homeless guy digging through the trash for cans. _What makes you the most sad?_ he said, turning back to Anh-thu. _You know, when you think about it_.
Anh-thu took a long sip with both hands on her mug. _That's easy,_ she said. She took one of her hands off the mug and started playing with the tips of her hair. _It's my mom. I_ _think every time I think about her I get sort of sad. I mean, especially for my dad. He was always so nice to her and then she_ _just leaves_. She pulled her hair behind her ear and stared into her drink. She took a sip.
Sticky picked up another stick and tossed it against a tree. He took a sip.
_But it's weird,_ Anh-thu continued. _Why should I even_ _worry about something that's completely out of my control? I_ _mean, I was only six when she left. How could I have really_ _even known her, right? But still, I have this picture of her in my_ _drawer. Under my pajamas. I mean, I could just throw the_ _thing away and try harder to forget about her. But, you know,_ _for whatever reason I know I won't do that._
_That ain't right,_ Sticky said, touching his knee against hers.
Anh-thu smiled at Sticky. _I know one thing, though,_ she said. _I'm gonna be such a better mom when I have kids_. She brought the mug to her lips, let the cool champagne run onto her tongue and then down her throat. _What about your_ _real mom, Sticks? You still haven't told me about her_.
Sticky shifted his weight on the bench. He reached down and picked up another stick, twirled it around in his fingers.
_That's my second question. And remember the rules, you_ _have to answer._
Sticky twirled the stick faster and then switched hands. _I_ _ain't playin this game no more,_ he said.
_Come on, Sticky. We've been together six months now, why_ _won't you ever talk about your mom? I mean, I'm your girlfriend. I want to know everything—_
_I said I'm done playin!_ Sticky shot back. _All right? I don't_ _have nothin to say about it!_ Sticky fired his stick at the tree and scooped up another one.
Anh-thu smoothed her hair behind her ears and then looked up at him. _Don't yell at me, Sticky_.
Sticky dropped the new stick and picked up his champagne, took a sip. He didn't want to talk anymore. He didn't understand why everybody always wanted to talk. Why? What good did it do? He wished he could just live his life. Do his thing. And never have to talk. But he also felt bad for snapping at Anh-thu.
_I didn't mean to get all loud,_ he said.
Anh-thu took a drink, said: _You have so much stuff bottled up inside, and, I don't know, sometimes I worry_.
Sticky looked at his girl. He felt the buzz of the champagne spreading into the tips of his fingers, all through his chest and back. She actually cared about him. How strange, he thought. To be around somebody who cares. Who worries. He wanted to tell her how much he liked her. Right then and there. But when he tried to think something up in his head, it didn't sound so good. Nothing cool was coming. Instead he set his mug on the grass and moved in on her, knocking over the empty bottle. He kissed her and said in her ear: _I hope we don't never get messed up_.
Anh-thu kissed him back. It was the first time she'd ever heard Sticky say something like that and it made her skin tingle. _I love you, Sticky,_ she whispered in his ear. _I mean it_.
They both pulled away and looked at each other. Anh-thu giggled and a smile grew on her face. She touched her mouth and told Sticky: _I think my lips are numb_.
Sticky picked up his backpack and Anh-thu's bag and pulled her by the hand. He spied the public restrooms up ahead and hurried through the grass. Anh-thu giggled again as she allowed herself to be pulled. Everything blurred around her except the back of Sticky's favorite shirt.
Sticky set both packs down in the women's restroom and checked the stalls for homeless. Nobody. He moved back in on Anh-thu, put his hands on her face, ran his fingers through her hair. He kissed her and could smell the scent of fruit on her skin. She kissed him back. It was so dark in the bathroom it didn't matter if their eyes were open or closed. But they'd been together six months now. They didn't need to see anymore. And as the rest of the park fell asleep outside the restroom, Sticky and Anh-thu were alive inside. Together.
Outside, they fell against the tree beside their bench and tried to find faces in the night clouds. Sticky leaned his head back against the tree. Anh-thu rested her head on Sticky's shoulder. They watched a woman walk by rolling a duct-taped suitcase and talking to herself in Spanish. Watched a cat slinking underneath a bench. Neither one of them said much, their heads still buzzing from the champagne and all, but they held hands.
Sticky let his eyes close up shop. He thought about how he was happy where he was. With Annie. He squeezed her hand.
Anh-thu kissed his cheek, then let her eyes slide shut too.
A few homeless bodies were curled up around them. Sleeping.
There was the sound of a long train grinding by slow in the distance. A Santa Fe train maybe. Its muted whistle blowing subtly under the purr of a thousand crickets.
When Anh-thu woke up it was already past midnight. She looked at her watch and shook Sticky's arm.
Sticky jumped out of sleep with his fists raised. _What are_ _you doing?_ he yelled. He sucked in quick breaths as his eyes darted around the park.
_It's OK, Sticky,_ Anh-thu said, backing her face away. _It's_ _only me_.
He dropped his fists and looked at her through blurry eyes.
_I didn't mean to scare you_. Anh-thu stood up next to him and put her hands on his cheeks.
_You didn't scare me,_ Sticky said.
_I just have to go,_ Anh-thu said.
They both reached down to pick up their bags. _You_ _gonna get in trouble?_ Sticky said, putting his backpack straps on his shoulders.
Anh-thu smoothed out the creases in her shirt. _No. I told_ _my dad I was staying at Laura's house. And she said I could go_ _over there at whatever time_.
Sticky walked with Anh-thu the twelve blocks to Laura's place. Down the middle of the quiet side streets lined by pale streetlamps. They didn't hold hands, but they kept their bodies close. Arms sometimes brushing. It was late and all the kids Sticky had watched on the way were tucked safely in their houses now. Manicured lawns were abandoned. Doors were shut and locked. Alarms were set.
_That's the kinda house I'd want,_ Anh-thu said, pointing to a big cabin-style pad with a steep roof and huge windows. _I'd put in secret rooms and stuff. Secret passages_.
Sticky kicked a rock. Walked up to it and kicked it again. _I got some letter today,_ he told her.
_About what?_
_It said for me to come to some basketball camp this summer. It's for the best players in the country. You have to be invited_. Sticky slipped his hands in his pockets. _It said there's_ _gonna be all these college coaches watching us_.
_Wow, sounds like a big deal,_ Anh-thu said.
_You know, them colleges cut you checks every month to live_ _on. And you fly big-ass jets to away games_. Sticky took his hands out of his pockets, moved them all around with his story. _And I heard the pilot tells everybody that a college team_ _is on the plane. They say it over the loudspeaker. And then all_ _the passengers cheer and turn around to look at you_.
_The letter said all that?_
_Nah, I heard it, though. From the guys at Lincoln Rec. And_ _if you play good at a school like UCLA or Kentucky, you're almost guaranteed a shot at the NBA. That's what Dante says_.
Sticky reached down and picked up a rock, threw it up into a tree. It sliced through the dense leaves, cracked against branches and fell onto somebody's perfect lawn.
_I wonder if we could go to the same college,_ Anh-thu said, slipping her arm through Sticky's. _I took my transcript to the_ _counselor and he said I'll have an excellent chance to get in_ most schools. Even the UCs.
_That'd be cool,_ Sticky said. _Going to the same school. I'd_ _ball and you could keep up with your studies. We could get an_ _apartment with the money they give me_.
_See, Sticky, I told you there's a reason why I study so much_. Anh-thu gave a little I-told-you-so face. _I'm not just being_ _some nerd like you always say_.
_You still kinda nerdy,_ he said.
They crossed to the other side of Rose and left the nice neighborhood behind. Walked right into the shady side of town. It happens that fast in Venice: One street you're surrounded by nice big houses with driveways holding fancy cars, and a block away all the streetlights have been shot out. Houses shrink up and are bunched closer together. Lawns shorten and turn brown. Cars lose their luster. Old Chevy Novas and pickup trucks. Impalas with crushed front fenders. Newspapers cover cracked windows. Mailboxes are tagged by gang signs and all the sidewalks are crumbling.
_There's her apartment,_ Anh-thu said, pointing to a run-down complex with sagging wood steps.
Anh-thu and Sticky sat down on the curb in front of Laura's place. Sticky shot rocks across the street like marbles. Anh-thu crossed her legs and played with her sandal. _Hey,_ _Sticks,_ she said.
_Yeah?_
_You're not gonna be at that basketball thing all summer,_ _are you?_
_Nah,_ he said. _It only goes for like a week_.
_Cause I was just thinking maybe we could hang out like_ _this more when the summer starts._
Sticky put a hand on her leg, told her: _That's cool._
They stayed sitting on the curb awhile longer in silence. Anh-thu leaned her head against Sticky's shoulder. Sticky carved his name into the concrete with a sharp rock. The street was quiet aside from the two crackheads standing on the corner talking.
Then Anh-thu stood up. She brushed off the back of her skirt and told Sticky: _I should probably go. I don't even know_ _if she's still awake or what_.
_I'll stay here till you get in,_ he said.
_Bye, Sticky,_ Anh-thu said. She wrapped her arms around his neck and squeezed. She kissed him. _We had the best anniversary._
Anh-thu walked up the wood steps and knocked on door #17. She stood there waiting and looked back.
You could hear locks being fumbled with, and then the heavy door creaked open. Laura popped her head out and hugged Anh-thu. She waved to Sticky.
Anh-thu blew Sticky a kiss, and then she and Laura both stepped into the apartment, pulling the door shut behind them.
Sticky Tapped Every
mailbox he passed twice on his long walk home from dropping off Anh-thu. One tap on the firm corner for a stale sound, the other in the hollow middle. Bass. When the sounds didn't sound right he'd slap twice more. Then he'd back up and do it all over. And he still stepped over lines.
As he walked he was surrounded by the nighttime rhythms of his world: old dirty women wrapped in plastic bags, pushing rattling shopping carts full of their lives down a side street; thick-bearded men curled up on bus stop benches, sleeping, faces like old leather shoes; a crackhead stumbling out of an abandoned building having just scored a fix, talking crazy: _A fine night, huh, boy? Marvelous night,_ _boy. You wanna have this dance, boy?_ Sticky shook his head and kept on walking.
An occasional car full of dudes coming back from a bar would slow up to flip Sticky off. Music thumping out of open windows. Drooping faces. They'd yell out: _Hey!_ And then when Sticky turned around to look: _Fuck you!_ Their laughter trailing off as they screeched away. Sticky always handled this by flipping them right back. There was the beat of an after-hours club, tucked nameless between an empty warehouse and a condemned apartment building. The sturdy black bouncer out front gave Sticky a what's up with his head and watched him walk by.
A slightly overweight white girl, tight spandex shorts cut so high you could see all of her legs, whistled for Sticky from across the street. _Hey, guy, you need a date tonight?_ She swung her purse and smacked on gum.
Sticky started across the street toward her.
_You need some company tonight?_ she said.
Sticky walked up to the girl and slipped his hands in his pockets. _You ain't cold?_ he said.
The girl's face was a third-grade finger-paint project gone bad. Bright red lipstick, pink blush. Eyelashes clumped together with black gunk. Blue eye shadow and hair a bad-perm blond. When she looked at Sticky, something in her eyes made him think of Baby. _I'm a little cold,_ she said. _Not_ _that much_.
An old Honda Civic slowed. The driver leaned himself across the passenger's seat to get a good look. She motioned with a finger for the guy to come get her. He stopped, rolled his window down and stuck his head out. He looked her up and down and shook his head. Then he sped off down the road.
The girl made a face and turned back to Sticky. _You seem_ _real young. How old are you?_
Sticky looked straight in her eyes. _I'm about to be_ _twenty-one_.
_You ain't look that old to me._
_Well, I am._
The girl pulled out a cigarette and lighter. Cupped her hand over the cig and flicked the lighter on. She sucked in a long drag and said: _I'm twenty-two. By the way, you got_ _any money?_
Sticky shook his head. _Nah_.
The girl stepped out of her heels, reached down and rubbed one foot, then slipped the shoe back on, rubbed the other and then slipped that shoe back on. _These shoes are_ _killing me._
Another car rolled by. There was a long silence and Sticky sat down on the curb. The night clouds had cleared and all the stars were naked. A McDonald's cup started rolling on its own down the road with a gust of wind. An old club flyer chased after.
_One thing I learned out here,_ the girl said, breaking the silence. _Men don't know how to treat no woman_. She frowned as she spoke. Took a drag and let the smoke leak out her nostrils. _I'm talkin about the men I dated, too. Not only tricks. My_ _baby's daddy didn't know the first thing about how to kiss no_ _woman's hand_.
Sticky looked up at her.
She pulled him up by his arm, said: _I'll bet you don't_ _know the right way to kiss a woman's hand neither_.
Sticky shrugged his shoulders.
_Figures,_ she said. _Most men don't_. The girl took Sticky's right hand in hers. _By the way, you got a girlfriend?_
_Nah,_ Sticky said.
_Well, I'm gonna show you something for when you get one._
She dropped her cig and ground it out with the toe of her shoe. Told Sticky: _First, here goes the wrong way_. She leaned her head down and kissed soft, her blond hair spilling across Sticky's wrist. _That's the wrong way. Ya got it?_ She paused for a second so Sticky would have time to let it sink in. _OK, now here goes the right way_. Same thing except she kept her eyes on Sticky's eyes through the whole deal. _See the_ _difference?_ she said, and she held Sticky's hand and waited for his answer.
_I don't know, maybe you was lookin at me._
_That's exactly right_. She let go of Sticky's hand and reached in her purse for another cig. She lit up and held it in her fingers. _So, now you know. Girls will be impressed if you_ _know how to kiss they hands right_. She put the cigarette to her lips and sucked in.
Another car rolled up slow to check the girl. The window came down and an old gray-haired guy stuck his head out. The girl twirled around. She flicked her ashes all sexy.
The guy pulled it to the curb.
Sticky leaned back against a pole, put his free hand in his pocket and watched the deal go down. When the girl hopped in the guy's car Sticky continued down the road.
Three blocks later Sticky was at Lincoln Rec.
He cut through the grass and cruised into the dark parking lot. Thought he'd check it out before cruising back home. He went right up to the gym doors and shook the handles. Locked. He peeked through the crack between the doors and was able to make out all the lifeless bodies lying on the floor. He thought about how soon he'd be back to ball. Only five hours. He imagined sneaking in the gym somehow and curling up next to all of them, along the three-point line.
Sticky startled when he heard a rustling in the bushes. _Hey, yo, Stick,_ somebody said. _Yo, Stick, that you, boy?_
Sticky turned around and found Dallas and Dreadlock Man sitting against the dark gym wall behind some bushes, holding paper-bagged bottles.
Dallas stumbled getting to his feet but caught himself on the wall. _Hey, yo, Stick, my brother_. He held out his fist and Sticky gave him daps. _What you doin out here, boy?_
Sticky glanced at Dreadlock Man. His eyes were half shut. _I was just walking around,_ he said. _What are you guys doin?_
Dallas fell back a little but caught his balance by grabbing a handful of bush. He peeped down at Dreadlock Man and then looked at Sticky. _Yo, you ballin in the mornin?_ He put a fist to his mouth and laughed. _What I'm talkin about, I_ _know you ballin. Tomorrow's Saturday. You always ball on_ _Saturday. Boy, you might as well stay here with us, man. We_ _gots blankets and stuff up in the van_. He teetered when he put hands on his hips. _It's a nice night out_.
_Nah, I gotta get back to the house,_ Sticky said.
_Come on, Stick,_ Dallas said. _If you look at it a certain way,_ _this sorta_ is _the house_. He cleared his throat and spit into the bushes. He held out his bottle, said: _Plus you gotta try this_ _juice we got for you. I know you be drinkin juice sometimes_.
Sticky stared at the bottle.
Dreadlock Man laughed through his teeth.
A siren wailed in the distance.
Sticky took the bottle and sniffed the mouth. Then he poured a healthy swig down his throat. He coughed and spit some of it out. _Damn!_ He wiped his face on the bottom of his shirt. _That's strong as hell!_
Dallas and Dreadlock Man started howling with laughter and slapping fives. _Shit, boy_ , Dallas said. _This juice right_ _here could make you a man. Put some hair on your chest_.
Sticky cringed as he swallowed a smaller sip.
Dallas took the bottle and tilted back. He took a super long swig and shook his head: _Goddamn it, that's what I'm_ _talkin about!_ He passed the bottle to Dreadlock Man, who did the same.
_I better take off,_ Sticky said.
_Nah,_ Dallas said. _Hang with your peeps_.
_Chill with your boys tonight,_ Dreadlock Man said.
_We's hidin out from my old lady,_ Dallas said. _She after me_ _again. Been up and down this parkin lot three, four times already lookin for me_. Dreadlock Man grinned and shut his eyes completely. He leaned his head against the wall. His green ten-speed was leaning against the wall behind him. He took the bottle from Dallas and tipped back, held it out for Sticky.
Dallas watched to see if Sticky would take it.
Sticky turned around and looked at the empty parking lot, the quiet road that would take him back to his couch in Georgia's living room. He looked at the bottle.
Nobody moved for those few seconds, when Sticky was thinking. Dreadlock Man held the bottle out. Sticky stared at it, looked back at the road again. Dallas watched Sticky.
Finally Sticky reached for the bottle.
Dallas started talking about his girl again. Dreadlock Man flipped three or four dreads out of his face and yawned. Sticky slid down the gym wall and put the bottle to his mouth.
_It's All Set,_
Counselor Julius said as he and current foster lady, Georgia, walked out from the office together.
Sticky took his hands off the foosball handles and reached down for his bag. By the fourth episode he had the whole checkout process down: prepacked, papers signed, ready to roll.
_Looks like you're gonna be living out in Venice Beach, big_ _guy,_ Julius said, and he shot a smile at Georgia. _You'll probably turn into a surfer or something_.
Georgia laughed and told Sticky: _Be sure to tell your_ _friends goodbye. And take your time. I'll be waiting outside in_ _the car_. She shook hands with Julius again and walked duck-footed out the front door.
Sticky strolled over to the TV room, stuck his head in.
The other residents were glued to the couch watching a video on MTV. After Sticky's checkout, Julius had to load everybody up in the foster pad van and make the long drive out to the Getty, their scheduled outing for the weekend.
_I'm out,_ Sticky said, and gave the group a head nod.
All the residents got up and circled around him, took turns saying their goodbyes. Long-haired Tommy shook Sticky's hand. Angie and Lisa, fourteen-year-old twins from the valley, stuck out their bottom lips all sad-like and gave tight hugs. _We'll miss you, Sticky,_ they both said at the exact same time.
_Be cool,_ Jerome said in a Barry White voice. He slapped Sticky five and pulled him in for a quick dude-to-dude-style embrace.
Six-year-old Pedro, who was new to the house and spoke only Spanish, didn't say anything. He wrapped his skinny arms around Sticky's legs and clung tight. Wouldn't let go. Two tears fell down his brown cheeks as Sticky patted him on the top of the head. After a couple minutes, Counselor Julius had to peel Pedro away.
Everybody loved Sticky because he was OG. The seasoned veteran of the house. The resident who knew the system inside and out and could go into detail to some new kid with questions.
What was strange about the whole goodbye scene, though, was that it didn't include Maria. She was Sticky's closest companion. His first lady of the foster world. But when everybody else got up to see him off, she stayed with the video. And this was something Sticky didn't understand. He leaned his head forward, tried to make eye contact and told her: _Hey, Maria, I'm takin off_.
_Oh, bye,_ Maria said, pretending to be distracted by the video. She put her hand up no-look style and then let it fall back into her lap.
Sticky shrugged and turned to take off, but Maria spoke up: _They're just gonna bring you back here in a couple months,_ _Sticky. You know that by now. They always bring us back_.
Sticky shook his head and walked away.
Julius was waiting in the hall with his arms folded. He pulled Sticky into the counselor's office. _I wanna give you_ _something,_ he said, and he walked around the desk and reached into the closet. Pulled out an Old Navy bag and set it up on the desk. _It's not that big of a deal, but I thought you_ _might want it_.
Sticky pulled open the bag and found the old beat-up house basketball. The one he'd learned to play the game with.
Julius pulled the ball out and looked it over. _This thing's_ _seen better days, man_. He spun the ball on his finger and looked at Sticky. _Listen, the supervisors are always trippin_ _about house equipment, so don't go tellin everybody_.
_You could really just give it to me?_ Sticky said.
_I can't,_ Julius said. _But I just did_. He faked a pass and Sticky flinched.
_That's real cool, Julius. Thanks._
Julius reached out and shook Sticky's hand. _I'm not_ _gonna lie, dude, you turned out to be a pretty good baller. Now_ _hurry up and get out of here. That fat broad is sittin out there_ _waitin for you_.
Sticky cruised out of the house with the Old Navy bag over his shoulder and a small bag of clothes in his left hand. As he went to pull open the passenger door of Georgia's minivan, the old Mexican director came speeding around the corner in his little Honda Civic. He pulled up along the curb and cut off the engine. Jumped out. _I barely caught you,_ he said, shutting his door. He walked over to Sticky's side of the van and waved through the window to Georgia. She waved back.
_Julius did all the paperwork,_ Sticky said.
_I'm not worried about the paperwork,_ the director said. He looked Sticky in the eye and gave him a firm handshake. _You and I have known each other a long time_.
_Yeah,_ Sticky said.
_I'd like to think we've become friends._
_Yeah,_ Sticky said.
_Well, I wanted to tell you this, Sticky, as a friend: Over the_ _past eight years I've seen a few families pick you up only to_ _turn around and bring you right back. And that's confusing._ It's real confusing. But you need to realize that it's not your _fault. That it has nothing to do with you. In fact I consider myself the lucky one. I've had the pleasure of watching Sticky-the-boy become Sticky-the-young-man. And what's special about_ _you, son, is not the way you play foosball or basketball or any_ _other game—it's who you are_. He pointed to Sticky's chest. _You're a good person, Sticky. A good human being._
Sticky didn't know what he was supposed to say to that so he didn't say anything. Instead he nodded his head and stared at the pavement.
The director smiled and drummed his fingers on the roof of the van, said: _I hope you never lose sight of that. You_ _mean a lot to me_. As Sticky climbed into the passenger's seat, the director exchanged waves with Georgia again. Then he shut the door and motioned her back onto the road.
Georgia merged onto the 10 heading west. She pulled open a big bag of chips, set it in her lap and reached inside. _You're my fifth foster kid,_ she said, shoving a couple chips into her mouth. _This makes two whites, a Oriental, a black and a_ _little Mexican girl_. She reached back into the bag. _I tell_ _friends: It's like the flippin United Nations at my house_.
Sticky pulled the house basketball out of the Old Navy bag and set it in his lap. He examined the tiny rips in the synthetic leather.
_Now, I run a pretty laid-back house,_ Georgia went on. _You_ _kids do your chores and don't give me any headaches, everything's fine. It really boils down to this: You make life easy for_ _me, I make life easy for you. I like to think of it as a kind of_ _business arrangement._
Sticky ran his fingers along the grooves of the ball. He spun it around and stared at the thick black initials of his foster care pad: 7 FLOW. Stuck his fingers in the groove, thumb between the 7 and F, and imagined lofting up a soft twenty-footer over the outstretched hands of some over-matched defender.
_My husband's gone all day. He works like sixty-hour weeks_ _so you'll hardly see him. Of course, I work just as hard as he_ _does. We got in a big fight about that just last night. He thinks_ _all I do is sit around the house watching TV. I told him: Uh, no,_ _honey, I don't think so. I told him: I got a full-time job just the_ _same as you do—I take care of other people's kids._
Georgia kept on talking, but Sticky wasn't listening. He had his daydream channel set on more important things. Like, where was he gonna play ball in Venice? He'd seen some famous court by the beach in a movie. And Julius told him about some gym called Lincoln Rec. He started thinking about other things too. Like, what was the old Mexican director trying to tell him when he was leaving? That he was a good person? And how strange was that? He'd never had anybody talk about him like that. It didn't make any sense. But maybe that was just part of his job. Something he was supposed to say.
Georgia's voice turned into background sound. Like the wind rushing in through the rolled-down windows. Like the sound of the traffic report coming in over the AM radio. Sticky traced the letters on the ball and did some other kinds of thinking too. He thought about what Maria said. How he'd probably get dropped back off in a couple months. How if that came true he just wished the director hadn't said what he said. It seemed like the kind of thing someone says to someone when they know they'll never see that person again. And he figured if it was true, that this chip-eating lady would one day bring him back too, like the rest of them, then he should at least know in advance. That way he could think up something good to say to the director. Something that might make him feel right about what he said. About Sticky being a good person. Something that might make him feel like it wasn't a mistake.
Sticky stared out the window and tried to remember all the billboards they passed: Chevron Gas, Gateway Computers, In-N-Out Burger, Staples, The Sports Club/LA. That way he could know what's up if they ever passed them again, going back the other way.
Pop Songs Echo
through the tiny staff bathroom in Millers. Britney Spears drops bubblegum beats that bounce off the stall walls and into Anh-thu's throbbing head. When her tune fades, Justin Timberlake takes over. OutKast. Matchbox Twenty. Jennifer Lopez. Their melodies filling the blue-tiled box of a bathroom with cotton candy.
Anh-thu leans on her elbows over the toilet bowl. One hand gripping white porcelain, the other holding back her long black hair. She spits and stares into the water: a rippling reflection of her puffy brown face. She heaves again and coughs. Flushes. Everything is pushing at the back of her watery eyes.
She spits again and stares.
The summer music mix bumps into an old-school Rob Base jam: "It Takes Two." It's the third time Anh-thu's heard this song today and her ears anticipate every shift in melody. She pictures the way customers always react, busting a couple quick dance steps near a mirror or keeping time with a subtle head bop.
She wipes away forehead sweat with the back of her hand.
Shift leader Dori creeps up to the locked bathroom door and leans in with an ear, taps her knuckles. _Everything OK,_ _Annie?_ She fingers the end of her long blond ponytail.
Anh-thu spins around, says through the door in her best smiling voice: _Everything's fine_.
_All right,_ Dori says. _Just checking_. She listens at the door a few seconds longer and then heads back out onto the floor.
Anh-thu turns back to the bowl. She digs her fingers into her stomach again and starts to cry. She's picturing Sticky's face if she really is pregnant. She's so nervous her stomach feels nauseous again. She heaves and coughs. She spits. Flushes.
Ten minutes ago Anh-thu was folding clothes with the rest of the girls. Folding and talking about some guy that gave Laura his cell number. They were gathered around the fifty-percent-off table, listening to Laura and cleaning up the two-story mess left by thoughtless customers—people who pull every item off a sale stack, unfold and throw back. Laura was dropping serious insight about UCLA dudes, what a girl has to do to catch their eye. She was doing heavy analysis, but Anh-thu had stopped listening.
Anh-thu was thinking about Sticky again. How her situation might mess everything up. It was her birthday, she was sixteen today, and Sticky would show up with a gift. He'd want to touch her and kiss her. But what if everything was different now?
She tossed an unfolded shirt on a stack of sweaters and hustled for the staff bathroom holding her stomach.
_Somethin up with Annie,_ Laura said, watching Anh-thu hurry off.
_She's not being normal,_ a girl named Julie said.
Shift leader Dori finished folding a sweater and watched Anh-thu turn the corner into the break room. She figured she'd give her a couple minutes before she went over to investigate.
Anh-thu picks herself up from the toilet and moves to the sink. She turns the water on full blast, cups her hands and splashes her face. She rinses out her mouth. She shuts the water off and pulls down a clean towel from the cupboard. As she dries her face, she stares at herself in the mirror. Her hair matted to her forehead. Her swollen eyes and puffy cheeks. This is the way her face looks after a long night of crying.
But when she locks in on her own eyes for too long, starts thinking about her situation, her and Sticky's situation, that nervous sick feeling comes spinning back into her stomach. She plants a hand against the sink and looks away.
Eminem starts flowing through the speakers: "Lose Yourself." The song Sticky made Anh-thu listen to over and over a few days back, on the tape deck of a borrowed car.
He drove them up to a small empty lot between two giant houses with tall fences. Somewhere in the Pacific Palisades. There were dense trees and bushes so nobody could look in. Signs that warned in big black writing: KEEP OUT. There were construction postings and idle tractors, a streetlight dug out of the ground and lying on its side. Sticky maneuvered the car past all that stuff and up to the edge of the cliff, where he cranked the parking brake. He and Anh-thu looked out at the stars hovering above the big black ocean. There wasn't a single cloud in the sky. Sticky bopped his head with the beat and pulled Anh-thu in close. They held hands and kissed.
At one point, he motioned to the tape deck and told her: _This dude got skills, Annie_.
Anh-thu agreed.
He said: _You know what? I wanna be the Eminem of_ _hoops_.
Anh-thu laughed.
When Eminem spilled his last line of lyrics and the beat trailed off, Sticky hit rewind and started the song all over again.
Anh-thu feels like crying again but instead she stomps her foot to stop herself. _Quit acting like a little girl,_ she says to her image in the mirror, and she grits her teeth. _Just stop it_ _already_.
She takes a few deep breaths and tries to pull herself together. She tosses the towel in a bin, runs a finger under each eye and straightens her clothes. She takes another deep breath and unwraps a stick of gum. As she pops the gum in her mouth she devises a plan. What's done is done, she thinks. All she can do is deal. She'll tell Sticky the situation tonight, and then go from there based on how he reacts. It might not go perfect at first, but they'll figure out what to do.
A Jewel song comes on, one she doesn't really dig, but Anh-thu feels okay about her plan. She takes another deep breath and unlocks the door. Then she heads back out onto the floor.
Two Old Cops
climb into the bleachers next to Sticky and pull out notepads.
The white cop has a salt-and-pepper crop of hair starting halfway up his head. A beer gut that lips over the buckle of his belt. He plops down on one side of Sticky and strains for breath.
The black cop is a tall lanky dude, 6' 5'' at least. He has juicy Jheri Curls spiraling down the back of his neck. He sits on the other side and pulls a pen from behind his ear, says: _Now the suspect, you say he's a light-skinned brother?_
Sticky nods, eyes glued to the court. Sticky's been a prisoner since the Fat Chuck thing happened a couple hours earlier. Sure, the games are rolling again, Dallas and Dante are back on against Rob's squad, but right now Sticky is Stuck in the bleachers.
Out on the court, Old-man Perkins sets a hard screen for Johnson, who dribbles toward the wing and dishes to Dreadlock Man in the corner. Dreadlock Man lofts up a prayer that somehow hits nothing but net. He yells out _Peanut_ _Butter!_ three times as he runs back on defense. And then, in case someone still isn't sure who scored, he yells it out again: _Peanut Butter!_ His raspy voice cutting through an otherwise silent game. The whole gym quiet and cuss-free due to the two busters with badges sitting up in the bleachers with Sticky.
Off the court, every couple minutes a different guy will cruise by the bleachers and pat Sticky on the back. Tell him in his ear: _We gonna find him, Stick. Don't even trip about_ _that_. Then walk away.
_And he's been in here before?_ the white cop says.
Sticky nods.
_What about his clothing?_ the black cop says. _What's this_ _brother wearin?_
Sticky shrugs his shoulders. He kicks at the bleacher in front of him and leans back. Kicks again, only softer this time, and puts on a mad-dog glare. He can't believe this has happened. Can't believe he's stuck up here going over the play-by-play again with a couple of cops.
_Now don't get all sassy, kid,_ the white cop says. _That certainly won't get us anywhere_.
_Just play it cool and answer the questions,_ the black cop says. _We'll take care of the rest_.
Sticky leans forward with his elbows on his knees, rests his head in his hands.
_I can't say I understand the attitude,_ the white cop says. He fingers the sweat off his forehead and wipes it on his cop pants. _We're here to help you. We're on your side_.
_Know what, Tom_? the black cop says to his partner. _This_ _kid looks kinda familiar_. He turns to Sticky, says: _Why do I get_ _the feelin I've slapped some cuffs on you before?_
Sticky stays with the game, but his mind floats back in time. He pictures leaning against the chain-link fence next to the Boys and Girls Club in Santa Monica, the black cop patting him down and calling him an asshole. He remembers the sound the metal cuffs made behind his back when they clinked together. The feel of the cold metal digging into his wrists as the cop pulled him to the car, shoved him in the backseat and slammed the door.
_I'll be damned if he don't look familiar,_ the black cop says.
_He who is the victim one minute, is the perpetrator the_ _next,_ the white cops says to his partner. He laughs a little, says: _You know how this city works, Sam_.
_That's poetic, man,_ the black cop says.
The white cop turns back to Sticky, says: _Now let's try this_ _again, kid. What type of clothing was the suspect wearing?_
_Sweatpants,_ Sticky says, still staring at the game. _Gray,_ _I think_.
After Chuck sped out of the parking lot, Sticky had begged Jimmy to keep the cops out of it. He walked through the office door and explained that it wasn't that big of a deal. Bringing cops in would be an overreaction. He claimed he and the guys in the gym could handle business on their own. It's called street justice, he said. But Jimmy told Sticky he had to make the call, that it was part of his job. Something goes down on his watch and he makes the call. That's the way he was trained to do it. Somebody finds out he didn't follow protocol and then it would be _his_ job on the line. After he said that, he nudged Sticky out of the way and pulled the office door closed. Picked up the phone.
Through the window, Sticky watched Jimmy put the receiver to his ear and punch in the numbers. Watched his mouth lock and twist and twitch as he tried to form the words that would explain what Fat Chuck had tried to do to him in the public restroom.
The two cops fire question after question at Sticky, they want to know everything: what did the suspect say? when did he try to touch? was there actual sexual contact? was there a scuffle? They take turns, one after the other like they're singing a duet. And they barely allow time for an answer. When Sticky speaks up they scribble his words down on notepads. Sticky thinks back to his court appearance for throwing the concrete through the glass at that car dealership. He had to answer the same questions posed the same way. He wonders how this, being the innocent person, is any different from being guilty: where do you live? how long have you lived there? who needs to be contacted? how will you get home?
At one point, the white cop leans in close and tells Sticky: _You really think this is the type of place you should be_ _hanging around, kid?_
Sticky turns and looks him in the face for the first time, says: _I always play ball in here_.
_Oh yeah?_ the cop says.
_Yeah,_ Sticky says. _And I always will_.
The cop scans the gym, laughs a little under his breath and says: _All right, kid. All right. But look around a little bit._ _Open your eyes. I mean, I'm just saying_.
Jimmy walks up and hands the two cops ice-cold Cokes, says: _Such a h-h-h-hot d-d-day_. Both cops crack open their cans.
The black cop puts the can up to his forehead, says: _Ah,_ _man. It's a hot one, all right_.
Sticky watches their necks vibrate as they suck down soda. Checks out the dark blue of their uniforms. The club and gun on either hip. The radio. The shiny leather shoes and cuffed pants. And right then he realizes something: cops are just normal people dressed up. They get hot and drink soda and clear their throats just like everybody else. They used to be seventeen, same as he is. And this is a strange idea for Sticky. Strange because cops have always just been blue robots to him. Nothing more, nothing less. Blue uniforms he had to avoid on the streets. Like a video game.
Jimmy folds his arms and leans against the side of the bleachers. Checks out the action on the court. _N-n-not m-m-much t-t-t-t-trouble in here m-most d-d-d-days,_ he says.
_No, you do a hell of a job, Jimmy,_ the white cop says, and he sets his empty can on the bleacher in front of him. _Hardly_ _ever get calls about this place_.
_You run a tight ship with these characters,_ the black cop says. _Don't you, Jimmy?_
Jimmy cracks a baby smile and buries his hands in his pockets. He stands there awhile longer, watches Dante pick off a pass at half-court and streak down the right sideline all by himself. Watches Dante take off just outside the key, lean through the air and flush in a one-hand jam.
The sideline erupts.
_Oh, man!_ the white cop says.
_D-d-d-d-damn, D-D-Dante,_ Jimmy says.
The black cop sets his Coke can down and stands up, yells through megaphoned hands: _Somebody tell that brother_ _it ain't right to get up that high! It just ain't right!_
Sticky feels a surge of energy rush through his skinny frame. All that action on the floor and he's stuck in the bleachers, yapping. All the highlights are going down without him. Everybody's running up and down the court making music with their squeaking sneaks, and he's no more than a spectator. No more a part of the show than the business guys watching in ties. He squeezes his fists together and sits up straight. Watches Rob inbound to Trey and everybody rumble back down the other way. He punches at his own legs a couple times. Feels the burn in his muscles and then sits on his hands.
Jimmy picks up the empty cans and adjusts his glasses. _Well,_ he says, _I g-g-g-g-got s-s-some p-p-p-paperw-w-work in_ _the off-ff-ff-ff-ffice_. He reaches out a hand and both cops shake it.
_All right then, Jimmy,_ the black cop says.
_Be good,_ the white cop says.
Jimmy nods his head, his magnified eyes gleaming with pride, and walks back toward his office.
The cops put away their notepads and pens, their questions. They chill in the bleachers like everybody else for a while, watching the game.
Sticky's eyes follow the ball. Trey's handling it at the top of the key. He swings it over to Slim and cuts through the lane. Slim holds the ball above his head while he surveys the scene. Sticky's eyes are on the action, but his mind is somewhere else. It's almost three in the afternoon according to the clock above the door. Not much time before he's gotta break for the house, drop his bag off and grab a shower. Then he's gotta pick out some smooth gear to wear and review the different ways he might swipe the bracelet.
He spies the cops out of the corner of his eye.
And then there's the stuffed bear he's gotta buy. Eleven bucks and some change at the old lady's card shop. He peeks through the bleachers for his bag. Still there. That's a lot of stuff to do before nine tonight. He touches the back of his fingers against his cut and checks them for blood. Nothing.
The white cop nudges the black cop with an elbow, tells him: _Used to be pretty good back in my day, Sam. Second leading scorer on the team in high school_.
The black cop shoots out a deep belly laugh and slaps a hand on his partner's shoulder. _About thirty years and fifty_ _pounds ago, right?_
Both cops laugh.
Sticky spies the clock again and does some minor math in his head. He should probably leave the gym by five-thirty. That would give him enough time to walk home, shower and figure out some gear. Then he'd bus it out to the promenade and size up the bracelet. According to his calculations that leaves him barely enough time for two more games. And he's gotta get on it right now if he wants to make even that happen.
_We done?_ he asks the cops politely, standing up.
_I think we got the information we need,_ the black cop says.
_Don't worry, kid,_ the white cop says. _We'll get right on_ _this one_.
_We'll do everything we can,_ the black cop says.
_In fact,_ the white cop says, nudging his partner, _it's about_ _time we start heading out._
_Let's do it,_ the black cops says.
Sticky nods and hops down from the bleachers. He walks over to a pack of guys waiting against the wall, asks them who has next. Does the guy have five? Who's got the game after that? Does he need one? Yes? Cool.
When his slot is secure he glances back up at the bleachers and discovers that neither cop has moved.
_My Name's Sticky,_
Sticky said, first time a group of Lincoln Rec regulars got him in a corner and came with their questions.
_Nah, man, what's your real name?_ Johnson said.
_Sticky_ is _my real name._
_What kinda name is Sticky?_ Dallas said. He turned and laughed with his boys, told them: Y'all ever heard some crazy _stuff like that?_
_Sticky?_ Old-man Perkins said, trying hard to put his finger on it. _Like, I spilled beer on my dashboard and now the_ _shit's all sticky, Sticky?_ Dallas nodded his head at the question. Johnson stopped laughing and waited for an answer, his mouth hanging open in a little _o_.
But Sticky had nothing more to say. He was new at the gym and didn't know these guys. He leaned his weight against the wall by the door. Pushed off with his fingertips and let his weight fall back to the wall again. Pushed off and fell back.
Pushed off and fell back.
_I ain't know no dude named Sticky before,_ Dreadlock Man mumbled. He was sitting on the seat of his beat-up green ten-speed, rolling a little forward and then stopping himself with the hand brake. Rolling back and then the hand brake.
It was the sixth straight day Sticky had shown up at Lincoln Rec, and these guys had a right to their interview. They stood around him in a horseshoe. Old-man Perkins with his arms crossed; Dallas with a bag slung over his right shoulder and a little white stuff in the corners of his mouth; Johnson's feet shoulder width apart, Raiders cap backwards; Dreadlock Man rolling back and forth on his bike. Slim hung behind a few feet. He sat with his back turned, stretching out his long legs—whenever somebody said something clever, you could hear his deep Carolina laugh. They were all digging for some background info on this new white kid coming in every day. Where'd he come from? What's he thinking, trying to ball with a bunch of black folks? They asked all kinds of different questions that day, one after another, when the games had died down and Jimmy started going up and down the floor with a wet mop, but they couldn't seem to get past the whole name thing.
_That the name that's on your papers?_ Johnson asked.
Sticky nodded. He stared down at their high-tops, said: _It's been the same thing since I was born_.
_Know what, kid?_ Old-man Perkins took a couple steps forward, got right in Sticky's face this time with his dead-serious look. _Know what I got to say about that?_
There were a few seconds of heavy silence in the gym. Sticky snuck a quick look up at Perkins's scowl and quickly went back to his shoes, Dreadlock Man stopped his bike cold, Slim turned his body all the way around.
Dallas peeked around the gym to see who was watching.
Perkins pulled a knife from his pocket and flipped the blade out, puffed up over Sticky and swung a violent knife hand through the air. The blade wooshed by Sticky's chin and Perkins yelled: _That's a stupid-ass name!_ leaning in extra hard on the word _stupid._
Everybody laughed.
Dallas put a fist to his mouth, stomped the floor with his heel. Dreadlock Man made a whistling sound through his gold teeth. Perkins folded the knife up and stuck it back in his pocket. He pointed a finger at the side of Sticky's head, told him: _Your momma must of been all up in the pipe when_ _you was born_.
Johnson pumped his fist at that line. His laugh was high and loud.
Old-man Perkins reached out a fist and gave Dallas daps.
But Sticky was telling lies.
He started out Travis Reichard back in Virginia. Travis after Randy Travis, Baby's favorite singer. But when they moved all the way out to Los Angeles, the name Travis quickly disappeared. Their apartment was above a Hostess wholesale shop. In the beginning, Baby would go downstairs and buy Cup Cakes every Sunday evening when they were half price. It was a once-a-week treat, and Sticky looked forward to their dessert. He had a system for eating the Cup Cakes: cakes first, the creamy filling, and then the sweet frosting. Baby would look at her boy after he'd polished off a pack, the chocolate all over his face, and she'd shake her head. _You're my little sticky boy,_ she'd say.
But eventually Baby started making more frequent runs downstairs. Whenever she wasn't in the mood to cook she'd go get a bunch of cupcakes and serve them on paper plates. And before long, she was always calling him Sticky Boy.
_For real, though, kid,_ Old-man Perkins said. _Anybody_ _can't come in here tellin people what to call em. Look at this_ _Dreadlock Man fool, he been tryin to get people to call em_ _Peanut Butter since he first showed up two years ago. Peanut_ _Butter, like the goddamn sandwich. You heard em yellin that_ _out every time he shoot that broke-ass jumper he got_.
_Come on, man,_ Dreadlock Man said. He laughed a little and shook his head. _It ain't like all that. I could shoot sweet_ _when I gets my rhythm. For real. It's just y'all ain't never gonna_ _pass me the ball enough_. He looked at Sticky and pulled a thick dread out of his face. _I ain't never said to call me it. They_ _just ain't figure out when I gets my rhythm, that ball come_ _bustin through that net like—_
_D-man,_ Dallas interrupted. _You sittin here lyin to that_ _white boy. You know you came up to me and Dante way back_ _and said you go by Peanut Butter. I remember like it was yesterday, right over there_ (he pointed and everybody looked) _by_ _the soda machine. You straight-up lyin to the kid_.
_Nah, Dallas, nah. I ain't never said to call me it, nah, it's_ _just when I gets my rhythm, that ball look like peanut butter_ _goin through the net. The creamy kind with no nuts_. Dreadlock Man held his hands up in the air. It all seemed pretty obvious to him. _That's what I said back when you_ _talkin about, Dallas. I swear on my moms on that one_.
Dallas raised his eyebrows at him and shook his head. _All_ _right, if that's how you wanna say it then_.
_What I'm sayin,_ Old-man Perkins said, looking at Sticky, _is you can't be comin in here tellin people what to call you by._ _They may call you Sticky where you from, but that don't apply_ _here. This is a whole nother world_.
Sticky shrugged his shoulders and avoided Perkins's eyes.
_Two, three weeks down the road and we'll have somethin_ _for you,_ Dallas said. _Maybe sooner_.
_I don't care what you all call me,_ Sticky said. _Cause I know_ _my name's still Sticky_. He broke through their horseshoe and made a beeline for the bleachers where he had his bag tied up.
The guys all looked at each other as he walked away. Dallas and Dreadlock Man laughed.
_You ain't gotta go runnin off when grown people tryin to_ _talk,_ Johnson said.
Old-man Perkins fingered his stubble. He looked perplexed. _I don't think this dude understand who he talkin to_.
_He think this place like Disneyland or some shit,_ Johnson said all excited, proud of his analogy. _He think anybody can_ _just come in here whenever they want and ride the rides and_ _watch the shows and then just go home_.
All the guys nodded in agreement.
_And it sure ain't no Disneyland,_ Dallas said.
_No Mickey Mouse,_ Dreadlock Man mumbled.
_Ain't no line for no Space Mountain,_ Johnson added.
Sticky reached under the bleachers for his bag. He unzipped the zipper and pulled out his headphones, carefully slipped them over his ears, making sure the duct tape was still holding strong. He pulled them off and put them back on.
Pulled off and put back on.
Pulled off and put back on.
These dudes were coming at him and he wasn't quite sure how to handle it. He didn't wanna go home, but he wasn't gonna let anybody call him something other than Sticky.
Pulled off and put back on.
Pulled off and put back on.
Pulled off and put back on.
When the worn-out pads on both phones were up against his ears just right, he pressed a thumb against the Play button and Tupac came to life.
Slim stood up and slung his bag over his shoulder. _Why_ _don't you all just leave the kid alone. I mean damn, he's just_ _here cause he wants to play ball. He didn't show up lookin for_ _no lecture about nicknames_.
_There's more to playin ball than just playin ball,_ Dallas said. _You know that just as much as I do, Slim_.
Old-man Perkins walked over to the bleachers and swiped the headphones off Sticky's ears. _Hey, white boy,_ he said. _We about to get this straightened out right now_.
Sticky stood up and yanked his phones out of Perkins's hands, shot his eyes toward the open door. His escape. He glanced back at Perkins, who now held his hands up in the air, like he was trying to show some invisible referee it wasn't him that fouled anybody. _Easy, kid,_ he said.
_Don't get stupid,_ Johnson said.
Sticky eyed the door again, then Perkins.
_Easy._
Slim walked over with his bag and climbed into the bleachers, sat a couple feet to Sticky's left. He folded his arms and stared at Perkins.
Sticky, trusting Slim for some reason, slowly sat back down and left the phones off his ears.
_I ain't tryin to mess with you, kid,_ Perkins said. _I'm just_ _lettin you know_.
_We tryin to teach you somethin,_ Johnson said.
_This the way things been,_ Old-man Perkins said, _since_ _long before you stumbled in here from wherever you stumbled_ _in here from_.
_Know why we call Slim, Slim?_ Dallas said, walking closer to Sticky. He looked at Slim and started laughing. _We call this_ _cat Slim cause even if he stood right between you and the clock,_ _you could still tell what time it was_.
Everybody laughed. Slim laughed.
Johnson sat up in the bleachers, too. Dreadlock Man coasted over on his ten-speed, one foot on a pedal, the other pushing off the wood floor. Jimmy leaned the mop against the wall near the homeless court and started going around the gym tearing off old outdated announcements.
Slim stood up in the bleachers and pointed at Dallas. _See_ _this dude right here?_ he said, his eyes stuck on Sticky. _You_ _probably thinkin he must be from Dallas or somethin, with a_ _name like Dallas. But he ain't. New York was in here earlier,_ _now we call him New York cause he legit. Born and raised in_ _Brooklyn. My man Dallas, though, he's from right down the_ _road. Right over there on Washington next to the ninety-nine-_ cent store. Pure L.A., this dude. We call him Dallas cause he _used to come in here talkin about that stupid-ass TV show_ _called_ Dallas. Slim winked at Sticky, and that wink eased Sticky's mind.
_Ain't that right, big D?_ Old-man Perkins said, laughing. _You black as night and watchin some show with a bunch of_ _white folks sippin on champagne_.
_Don't make no kind of sense,_ Johnson said.
_One time I mention that show in here_. Dallas crinkled his face and rolled his head around. _One time and all these dudes_ _jump on it_.
Slim pounded the bleachers, laughing, and sat back down. Dallas waved him off.
_Same thing happened to me, though_ , Johnson said. _My_ _real name ain't no Johnson. I came walkin in here three years_ _ago as Dimarcus Jackson, same name as my granddaddy. A_ _distinguished name. Jackson, not Johnson. And then cause I occasionally wore this one specific T-shirt I got for Christmas—_
_Lemme stop you right there,_ Old-man Perkins interrupted, putting his hand on Johnson's shoulder. _Yeah, you_ _came in Dimarias Jackson or some shit like that—_
_Dimarcus!_
_But for the first three months I seen you, no lie, you wore_ _the same Big Johnson T-shirt every damn day. Never washed_ _it, neither. That's how we started callin em Big Johnson. Like_ _the T-shirt_. Old-man Perkins wrapped Johnson up in a bear hug and they stumbled out of the bleachers. _Tell em bout_ _that Big Johnson shirt. All them rips under the arms. Like you_ _was some kinda homeless cat._ They both laughed and wrestled around a little. _And you supposed to be some big man_ _workin for the city_.
_Always talkin about his damn benefits, too,_ Dallas said. _We_ _was like, yo, forget Blue Shield, what they need to do is buy_ _your ass a new shirt_.
Slim stepped off the bleachers and acted like he was gonna drop a WWF elbow on Johnson. Old-man Perkins let go when Johnson's Raiders cap fell off his head. Johnson reached down and picked it up, pulled it on his head and adjusted the bill.
_And what about Jimmy over there,_ Dallas said. He pointed toward the homeless court where Jimmy was kneeling down next to an old Indian-looking woman. _His real_ _name wasn't even Jimmy, right?_ He looked over at Old-man Perkins for him to pick the story up from there.
_That's right,_ Perkins said, and he paused a sec to catch his breath.
Sticky unzipped his bag and threw his Walkman inside, zipped it back up. He set the bag on the bleacher next to his feet.
Dreadlock Man started rolling back and forth on his ten-speed again.
Johnson took off his cap and checked out the bill. He brushed off some dust from the floor and pulled it back tight on his head.
_His real name is Sam,_ Old-man Perkins said. _Me, Johnson_ _and this guy Shotgun was all sittin right here in these bleachers_ _talkin about that movie_ Hoosiers. _Jimmy was walkin by_ _and heard what we was talkin about. He says to us, Oh, I l-l-l-l-love that m-m-movie, man_. Everybody laughed at Old-man Perkins's version of Jimmy stuttering. _Then Shotgun_ _says, You like that movie, Sam? What you like about that_ _movie, Sam?_
_I remember that Shotgun cat,_ Dallas said. _He was always_ _startin up fights on the court_.
_Yeah, Shotgun was a real mean cat,_ Old-man Perkins said. _Cops came in one day and slapped the cuffs on em, too._ _Grabbed him right in the middle of a fast break and pulled him_ _out by his elbows. Ain't nobody seen em since_.
_Damn,_ Slim said.
_Guess you could figure out why they call him Shotgun,_ Johnson said.
_So anyways,_ Perkins said. _He starts tellin Shotgun what he_ _liked about the movie. He says, Ah man, I th-th-th-think that_ _Jimmy ch-character c-c-could sure sh-sh-sh-sh-shoot the ball._ _Then he sits up in the bleachers with us cause he all happy we_ _ain't talkin about nothin illegal. You remember all this,_ _Johnson?_
Johnson nodded his head and laughed.
Perkins continued: _Sam says, M-makes em all d-day on_ _that d-dirt c-court, that Jimmy. So Shotgun says, You kinda_ _shoot like Jimmy yourself, Sam. I seen you out there knockin_ _down jumpers between games when nobody be payin attention. And Shotgun's one hell of a lie about all that, cause you all_ _know that dude Jimmy couldn't throw a rock in the ocean if he_ _was standin on a pier_. He pointed over at Jimmy, who was kneeling next to some homeless man now. _But it pumps him_ _all up inside, see. He says back, F-f-for r-real, y-you s-s-s-s-s-s-seen me? Shotgun nods his head up and down like that's the_ _straight truth what he said_.
Old-man Perkins stared at the floor for a few seconds and shook his head. _From that point on everybody called Sam_ _Jimmy. He liked it so much he crossed out his own name_ Sam _on his office door with a black felt pen and wrote in_ Jimmy _right above it_.
_He was so hyped about his new name,_ Johnson said. He pulled his cap off again, scratched the top of his head, put the cap back on. _Next few days he was comin up to everybody,_ _telling em to call him Jimmy_.
_Still does,_ Slim said.
_Ain't no Sam no more,_ Dreadlock Man said.
_Plus he don't stutter none when he says Jimmy,_ Dallas said, his eyes all big like he'd stumbled onto something important. _I ain't never even heard him slip up on it once_.
_That's gotta tell you somethin,_ Johnson said.
_Hell yeah it does!_ Old-man Perkins said, looking right in Sticky's eyes. _Tells me something, all right. Tells me we gave_ _that cat a whole new life when we changed his name_.
When Baby yelled out from the tub—her voice piercing like boiling water from a teapot, the late-night sound of two cats screeching outside the bedroom window—Sticky was standing next to the one window of their sixth-story apartment, holding a chunk of government cheese and trying to spit into the bed of a dirty red pickup parked below.
STICKKKYYYYYY! she screamed, over and over. STICKKKYYYYY!
It was the first time she'd said his name like that, just Sticky, without the _Boy_ coming after, but it wasn't the first time she'd screamed out for him. Baby was always getting loud about some crazy thing: She screamed if there was a trail of roaches running around the fridge, or when the fat rat that lived in their bedroom peeked its ugly head out; she screamed at the TV if she was unhappy with something going down in her soap opera, or if a bill came that she couldn't possibly pay; and she always screamed when she needed a fix. _Oh, my_ _God! Baby needs her medicine! Baby needs her medicine!_
The screaming was nothing new, and Sticky spit a few more globs of orange saliva down at the red pickup before he even thought about moving.
Even back then, at age seven, Sticky knew how he was about things. How if he spit once he might be stuck there all day. He'd definitely been down that road before, spitting at some target outside his window. An empty Burger King bag or some stray cat. There were days he'd be in the window spitting for hours, until there was no more spit in his mouth and his throat went sore. He'd slap at himself to stop, pull at his own hair, but nothing could slow his momentum. He had to keep on going until something clicked. Baby would eventually give up and close the curtain around him, saying: _I don't understand. What's wrong with you? Acting all weird_ _all the time. Normal boys don't do things like this_.
And this time was no different. Sticky spit again and missed wide right.
_STICKKKYYYY!_ Baby yelled.
He spit again and grazed the bumper. He knew he should at least check on Baby, but the pull to get one perfectly in the truck's bed was too strong.
He knew the next one would hit perfect, and when that missed, the next one.
_STICKKKYYYY!_
He knew one of the next five at least, cause he had a knack for stuff like that. He gathered saliva up and spit again: windshield. He glanced back at the half-closed bathroom door. Listened for his mom. Baby. She was screaming. He was trapped again, and he had no idea how to break free. He felt like crying, but he wouldn't let himself do that, either.
_STICKKKYYYY!_
Just _Sticky_ she's yelling out. He leaned out the window this time for a better angle, felt the cool breeze against his face, held on to the wall for support and spit again: wide left. Just _Sticky._ Why just _Sticky_?
Spit again: left door handle. His mom. Baby.
Spit again: missed everything.
Just _Sticky._
Spit again: missed.
Spit again: missed.
Spit again: missed.
When the screaming stopped, the apartment went silent. His mind went silent. His life went silent. There was only the sound of what was outside: the wind in his ears, a distant siren, the honk of an impatient car, the guy walking by holding his daughter's hand and whistling. His life went silent. Everything that was inside him went silent.
There was nothing left inside.
And outside, it was just the sound he made as he spit again.
_We gotta think up somethin else for this kid to go by,_ Old-man Perkins said. He stared at Sticky and thought hard.
_What about Little Bird?_ Dallas said. _He got that nice little_ _jumper from the outside_.
_Pistol Pete,_ Slim said, jumping off the bleachers, pointing at Sticky's socks. _Look at them limp-ass socks. Same as Pistol_ _Pete use to wear_.
_How about Eminem?_ Old-man Perkins said, and all the guys laughed.
_Yeah,_ Johnson said. _Or maybe Little Norm. For Norm_ _Nixon_. He laughed hard at his idea. Old-man Perkins, Dreadlock Man, Slim and Dallas looked at Johnson with straight faces, trying to make out the connection.
_What's that mean?_ Slim finally said.
_You know: Norm Nixon._
_We know who goddamn Norm Nixon is, J,_ Old-man Perkins said. _But why you wanna call a white boy Little Norm?_
_It just sound right, man. Plus Nixon was a light-skinned_ _brother._
_That's just plain stupid, J,_ Perkins said. _It don't make no_ _kinda sense_.
Sticky grabbed his bag in his hand again. The frustration bubbling in his stomach was becoming too much. He clenched the straps as hard as he could and shouted: _I_ _don't need some stupid name! I already got a name! My_ _name's Sticky!_
Dallas shook his head. _Nah, kid, that ain't no kinda name_.
_That just ain't natural,_ Perkins said.
_I can't be callin nobody Sticky,_ Johnson said. _I could tell_ _you that right now_.
_Sticky sound like Dicky, make em sound like some little_ _homo from West Hollywood,_ Dreadlock Man mumbled. He lifted his front tire and smiled gold teeth. Sometimes the guys couldn't quite make out what Dreadlock Man was trying to say, but half the time they laughed anyway.
Sticky stood up to leave. He started stomping down the bleachers, but Old-man Perkins reached out and snatched his forearm. _Where you goin, kid? I ain't done talkin to you_.
Sticky dropped his bag.
_He been disrespectin all day,_ Johnson said.
Old-man Perkins glared right in Sticky's face. Dreadlock Man got off his bike, set it on the floor. The back wheel was still spinning.
_Let em go,_ Slim said.
Nobody moved.
_Let the kid go!_
Jimmy's head whipped around from the homeless court. _H-h-hey!_ he yelled out. He dropped the blanket he was holding and stood up.
Sticky ripped his arm from Old-man Perkins's grasp and jumped down from the bleachers. When he reached back to grab his bag, Johnson moved behind him so he couldn't go anywhere.
_You gonna keep playin ball here,_ Old-man Perkins said, _then you gotta start showin some respect_.
Jimmy started walking over. _Wh-wh-what's the p-p-p-problem?_
Sticky slipped by Johnson and made his move for the door.
Johnson and Old-man Perkins threw their hands up at the same time and said, _Ain't no problem, Jimmy. There ain't_ _no problem here_.
_It's all good,_ Dallas said.
Sticky turned around and looked all the guys in their eyes, each one of them, first time he'd done that, and shouted: _I already got a name! My name's Sticky!_
_Come on, kid,_ Dallas said. _It ain't gotta be like all that_. He walked toward Sticky, said: _Lemme buy you a hot dog or_ _somethin. From the food cart_.
_Let us buy you lunch,_ Perkins said, eyeing Jimmy.
_There's a food cart right out front,_ Johnson said.
But Sticky shook them all off. His name was Sticky and nobody was going to call him anything different. Nobody. He turned away from them and walked out the gym door.
Whenever Sticky pictures himself walking into the bathroom that night, to check on Baby, his memory locks up. It stops him cold and spins him back around the other way. He remembers her yelling, _Sticky!_ He remembers spitting over and over out the window at a truck. He remembers feeling something strange in his chest when her voice dropped off. The sudden quiet of the apartment making him nervous. But when he finally pulls himself away from the window and walks through the bathroom door, that's the part he can't get to. Everything crumbles in his head. It slips away. The images disappear. But he remembers her calling him Sticky. That's the last thing she said. _Sticky._ So that's who he is now, Sticky. That's what his mom called him. And he's never gonna answer to anything else.
It's Just Hoops
now. Ten guys, two buckets and a basketball. Everything else shut off like the beats on Sticky's duct-taped Walkman. No more Fat Chuck. No more two old cops in the bleachers posing questions. No more sunlight-glare sneaking through the open doors. No more waiting on a sorry sideline, watching. No more cleaning out a cut or coming up with a birthday game plan.
No more thinking.
It's just Sticky and a game now. His game. Hoops. Ball. Pickup. Fives. It's Sticky in his secret world. His haven. And he's making plays that look like magic tricks. He's clowning whoever's guarding him so bad the guys on the sideline are on their knees, laughing. Whistling. Hooting and hollering.
What does it matter how the rest of the day goes? Getting home on time, swiping a bracelet, buying a bear? Sure, there's a time and a place for all that. Of course there is. Real life always comes whipping back around at you like a boomerang. But right now there's one last game to play. And Sticky's right here. In the zone. Flowing. Every shot ripping through a nylon net and playing the same song. And it's almost mean the way he does it, making people look so bad. So sad. So human. But this game is Sticky's drug. It's his stage. This court is Sticky's home. It's his hiding place. It's his church. And he's the one who gets to talk to God.
It took two games and forty-five minutes for Sticky to get back on the court, and even then he had to get a lucky break. TJ and Daway from the winner's squad had to hustle off to their jobs at Chevron, leaving two open slots. Dreadlock Man picked up Dante first and then yelled out: _Hey, yo, Stick! You wanna run with us? We need a point guard!_
Sticky hopped down from the bleachers, pulled off his Walkman and stuck it deep inside his bag. Then he ran out onto the court and the game began.
At first Boo was checking him, a long-limbed, light-skinned brother who plays for Santa Monica College. But that was no match. Boo face-guarded Sticky all around the court, nose to nose. He was oblivious to everything else. His breath like the bottom of a soup can. Eyes bugged like he just got tapped on the shoulder by the devil. Boo played defense so tight that if Sticky had run out of the gym midgame, to score a quick dog and Coke at the snack cart, Boo would have run right out of the gym with him.
Sticky handled Boo like a puppet. The game wasn't ten minutes old and he had all four of his squad's points. That's when Rob called for the switch. _I got white boy,_ he yelled out, and he shoved Boo out of the way.
Boo backed off and sought out Rob's man with his head down and his tail between his legs.
Rob checked ball and stuck a forearm in Sticky's chest, told him: _Forget all that, white boy_. Said: _You ain't scorin not_ _one more bucket_.
But Sticky played deaf to all that. He dropped in two quick jumpers and asked Rob how bad it hurt to have a skinny white boy school him like that. Asked could he at least get a hand up in his face. Call for a double team. Anything so it wasn't so damn easy.
Rob kept his mouth shut and ran back the other way. But a man's shifty eyes are like a window to his busted-up ego.
The only thing that stopped Sticky's barrage on Rob was when Crazy Ray came stumbling out onto the court for a second time. He lifted himself off his piece of cardboard on court two, started launching into his typical tirade, pointing his finger at the guys in the game and walked right into a fast break. Three players trampled over him on their way to the basket.
New York, tripped up on the play, let the ball sail out of bounds and got up pissed. _Now that's some dangerous shit_ _right there,_ he yelled. He stood over Ray. _I ain't tryin to get_ _hurt out here, now. I got my kids to feed_.
Ray lay flat on his back, holding his head.
Dallas had to come to his rescue again. He helped Ray up, wrapped an arm around his shoulder and pretty much dragged his ass back to the homeless court.
That's when Sticky first looked up and saw it on Rob's face. The confusion. The helplessness. The look of a trapped animal. And he knew it might turn bad.
Sticky gets the pass out on the wing and sizes up. He swoops by on the drive, but Rob reaches out two tree-trunk arms and wraps up. Holds on tight so Sticky can't go anywhere.
_Check ball,_ Sticky says.
Dante checks the rock up top. He flips it back in to Sticky and tells him: _Go to work, boy_.
Sticky stutter-steps and spins into the lane, but just as he's about to let it go, Rob swings his arm out and cracks Sticky in the face.
Sticky springs off the ground and throws an unpolished right hook that thuds against the side of Rob's neck.
Rob stumbles back, puts a hand to his neck. He looks back at Sticky, stunned.
The gym goes silent.
_I'm sick a them fouls!_ Sticky yells, and he steps up and wings another wild right. Rob ducks it and wraps Sticky in a headlock. Slams him to the ground and pounces. He throws muted blows at the top of Sticky's head. He knees him in the ribs. Sticky reaches back and claws for Rob's face.
Dante and Trey move quick to pull Rob off.
_We don't need no fightin in here,_ Old-man Perkins says as he steps in on the action too.
Dante gets Rob's arms locked up behind him and pulls him backward. He doesn't say a word.
Rob tries to yank free but can't. _I'm gonna kill you, white_ _boy!_ he yells.
Dallas pulls Sticky away and holds him by his elbows, tells him: _Be cool, kid. Be cool_.
Sticky rubs the back of his head. Touches the cut from earlier and checks his fingers. Blood.
Rob breaks away from Dante and charges. Pushes Dallas and Perkins out of the way and cracks Sticky in the ear with a solid right. He wrestles him to the ground again, holds Sticky's head still and fires quick jabs to the mouth and chin.
Sticky puts his hands up to try and muffle some of the blows.
Dante strides up from behind and pulls Rob off by his face. When Rob stands up, Dante busts him in the mouth twice with a quick left-right combination. He doesn't say a word. Rob's knees buckle and he spills back to the hardwood. Blood trickles out of a tear in his bottom lip.
Jimmy comes hustling out of the office. W-wh-wh-what _the h-h-hell's g-g-g-going on?_ he yells. A couple guys turn and watch him marching toward the scuffle.
Rob touches at his lip gently and eyeballs the red on his fingertips. _This wasn't none a your business, D,_ he says, and looks up at him. _Wasn't nothin to do with you_. He clenches his fists and goes to get up again, but Dante blasts him twice more, above the right eye and on the chin.
Rob crumbles to the hardwood again. Blood is oozing from above his eye now and branching down his face.
_Just stay down,_ Trey says.
Rob turns and looks at Trey. He frowns and goes to get up again, but when he's on his feet Dante cracks him in the ribs with an uppercut and smacks him on the chin again. Rob spills back to the ground and with a dazed expression on his face looks up at Dante.
_I said to stay down!_ Trey yells.
_Don't get up no more,_ New York says.
Dante stands over Rob and doesn't say a word.
Dreadlock Man steps up and puts a hand on each of Rob's shoulders. _You done fightin, dawg,_ he says. _You done_.
Jimmy walks right between Rob and Dante and yells at everybody: _G-g-g-get outta here!_ He's so mad, spit flies from his mouth.
Nobody moves at first. They all stare as Jimmy looks down at a trail of blood splotches that dot-to-dot through the free-throw lane. He yells again: _G-g-get outta here, all_ _y-y-y-y-y'all! G-g-g-g-get out!_
Everybody scatters.
Sticky goes over to the bleachers holding the bottom of his shirt against his cut. He unties his bag and checks inside for his twelve bucks: still there. Then Dallas and Dante lead him out of the gym. New York and Slim head over to the bleachers and grab their bags. Old-man Perkins tosses the ball toward the homeless court and heads for the door. Big Mac rumbles out shaking his head. Three newcomers who were waiting for next cruise out of the gym together.
Trey and Dreadlock Man help Rob to his feet. Johnson tosses him an old gray towel. Rob presses the towel against his lip, his eye. When he sees the towel is full of blood he punches the wall.
Trey leads Rob through the doors, out of the gym.
Outside, Trey digs the keys out of Rob's bag and unlocks the passenger side door. He pulls the door open and helps Rob in. As Trey walks around to the driver's side, Rob pulls down the sun visor and checks the damage in the mirror. There is blood all over his face and running down the front of his sweaty shirt. He slams the heel of his hand against the dash.
Trey gets in and pushes the key in the ignition, starts the car up. He looks over his shoulder and backs up, flips it into drive and moves slowly through the parking lot.
The rest of the guys aren't so quick to leave. They huddle around Big Mac's Caddy under the blazing sun and replay the highlights of the fight.
_You see the quick combo D laid on em?_ New York says. _He_ _said bam bam and put Rob's ass right back on the ground_. New York whips his fists through the air to show what it looked like from his angle.
_Used to box Golden Gloves,_ Old-man Perkins says. _I been_ _telling people all along, that's the wrong brother to mess with_.
_I heard that,_ Johnson says, and wipes the sweat off his forehead. _But, damn, you see how quick he threw_ _them punches?_
_Rob went down hard, too,_ New York says. _He said just like_ _this_. . . . New York falls to the hot pavement cross-eyed, arms and legs spread, tongue hanging out of his mouth.
The guys all laugh.
Boo laughs hard and hits his palm against Big Mac's hood.
Big Mac's face goes serious and he stares Boo down. _I_ _don't know about all that, dawg,_ he says.
_What?_ Boo says.
_That's my car you just slapped, boy._
_Oh, man,_ Boo says. _My bad_. And he takes a couple steps back from the Caddy.
_What about Sticky, though?_ Old-man Perkins says. _White_ _boy gots some heart, don't he?_
_I knew it was comin,_ Johnson says. _You could see it brewin_ _all day_.
_Socked em right in the neck, too,_ Dreadlock Man says.
New York nods his head, says: _Socked em right in his neck_.
_Bound to happen,_ Big Mac says, and pulls a sixty-four-ounce Gatorade from a cooler full of ice. _Rob always be tryin_ _to mess with that white boy_.
_He makes Rob look silly,_ New York says.
_Ain't Stick's fault he can play,_ Perkins says. _Hell, I can't_ _guard em either, but it don't mean I'm gonna foul em every_ _time he touches the ball_.
Everybody nods their head in agreement.
Big Mac unscrews the cap to the Gatorade and tilts back for a long swig. When he's done he passes the bottle to Johnson and wipes his mouth on his shoulder. Johnson holds the bottle inches above his mouth and lets the cool green liquid spill down his dry throat. They all lean against different parked cars and talk some more about the fight. The air is warm and thick, humid. The afternoon sun is like fire against their shiny black skin.
Lincoln Rec Shuts
down at eight on most nights. That's what the sign on the door says: HOURS OF OPERATION—10 AM TO 8 PM. But when closing time rolls around and the games are still solid, Jimmy's pretty flexible. He'll handle other business first: tally up the books or post new gym announcements. Help pass out bused-in meals to the homeless. Some nights he doesn't start kicking guys out until well after ten.
But today's a different situation.
It's a quarter past five and Jimmy's taping a cardboard sign on the door that says in thick black marker: GYM CLOSED UNTIL FURTHER NOTICE. When he gets the sign up sturdy, he turns around and scans the parking lot. He spies all the guys jiving around Big Mac's Caddy and shakes his head, says under his breath: _B-b-bunch a kn-kn-kn-knuckleheads_.
Then he slips back into the gym, shuts and locks the door behind him.
Dallas steps up to the snack cart and tells the Mexican vendor he wants three waters. When the vendor says how much, he pulls a soggy five out of his sock and hands it over. He takes the waters and his change and tells Sticky and Dante he has some chairs stashed behind the gym. Then he hands them both a bottle and leads them around the corner.
There are four plastic chairs stacked between some overgrown bushes and the back wall of the gym. Dallas pulls out three and brushes off the leaves and spiderwebs. Sets them up as far away from the big trash receptacles as he can get.
Sticky sits down and tosses his bag on the ice plant, listens to the buzzing Rob's fist has left in his ear. Runs a couple fingers over the three or four lumps on the back of his head.
Dante swallows some of his water and sets the bottle down next to his chair. He checks his right hand, shakes it out and checks again: there are a couple small nicks on his knuckles from Rob's teeth.
Sticky sets his water down and looks at Dante's hand too. _Your hand messed up?_ he says.
_Nah,_ Dante says, and he stretches out his fingers. _This is_ _what you gonna deal with when you dot a man in his mouth_. He picks up his bag and pulls open the zipper. _I just hope_ _that dude don't got AIDS_.
The tall gym has the sun blocked out, but the air is still warm and thick. It brings out the sour smell of the overflowing trash bins, where buzzing flies dip in and out in clusters. Dallas wipes beaded sweat off his forehead with the back of his hand. Dante pulls a towel from his bag and wipes down his face and arms. Through a seam in the buildings, Sticky watches all the businesspeople filing out of their big glass building. Loosening their ties and marching toward their shiny cars. Women fan themselves with magazines and everybody chirps their alarm before reaching for a door handle.
_You did good today, young Stick,_ Dante says. _You stood_ _up for yourself, and I'm proud of you_. He throws a playful little jab that glances off Sticky's chin. Sticky smiles and looks at the ground. He takes a sip of water and screws the cap back on.
_Hit em right in his neck,_ Dallas says. _Last place somebody_ _wants to get hit_.
_That's right,_ Dante says.
An old homeless white woman staggers past pushing a cart. Her eyes half-closed and empty. Mouth moving without sound. She holds a hand out as she passes. When nobody makes the move to give up any change, she continues by, parks her cart near the trash receptacles and reaches two hands into the mess. As she digs around, her soiled jeans slip down her backside.
_Ah, man,_ Sticky says, pointing at the spectacle.
_Come on, lady,_ Dante says, and he puts a hand up to shield his eyes. _Get em up!_
The woman reaches back with one hand and pulls her jeans up. But when she goes back to the trash, they immediately start the slow slide back down.
_Go on, old lady,_ Dante says, laughing. _Ain't nobody tryin_ _to look at that old flea-bitten ass_.
She pulls two cans from the trash and sets them in her cart. Continues forward. Holds her jeans up with one hand and steers with the other. The sound of her rattling cart becomes more and more faint after she rounds the corner of the gym, out of sight.
_She don't know no better,_ Dallas says. _That's all_. He reaches into his bag and pulls out some foil and a pack of Zigs. Opens the foil and drops a clump of brown weed into his palm. He rolls it, lines it and licks it. _She just tryin to get_ _herself some cans,_ he says.
Dante throws his towel back in his bag. He takes out a baseball cap and pulls it low over his forehead. _You can get a_ _couple cans without showin everybody the goods, right?_
_It ain't easy livin on the street,_ Dallas says.
_No, I ain't arguin that,_ Dante says.
People get all weird, Sticky says. Look at Crazy Ray. He takes in another sip of water and swooshes it around in his mouth, spits it out. _Why you always helpin that old dude_ _out, Dallas?_
Dallas shrugs his shoulders and pulls out a yellow lighter. He flicks the fire on, cups his free hand to block the wind and lights the joint. He sucks in deep and passes to Dante.
Dante pulls in a drag, holds the smoke in long and then lets it slip out over his lips.
Sticky stares at all the years of graffiti spray-painted up and down the back wall of the gym. Ten years of gang names, he thinks. Fifteen. All the different colors on top of each other turning the white wall brown. Anything over six months old has been crossed out or covered up by something new. And then Rob's face flashes though his mind. The glimpse he got of Rob's expression through the flurry of punches coming at the back of his head, his face. And to get Rob out of his mind he thinks back on all those hours he spent painting over graffiti walls just like this one when he did his community service. The burn in his shoulders, the ache in his feet. It didn't sound so bad when the judge gave his ruling: 100 hours of community service instead of jail time. But those ten-hour Saturdays were no joke. Painting up and down, side to side, standing on ladders all day, the brush turning so heavy in his hand he could hardly keep it above his head.
He reaches down to untie his laces. When loose laces don't feel quite right he ties up and unties again. Turns a little to the side so Dante and Dallas won't recognize his process. Ties up and unties.
Ties up and unties.
Ties up and unties.
Ties up and unties.
Dante passes back to Dallas and pulls a clean shirt out of his bag. He peels off his sweaty shirt and says: _It's survival of_ _the fittest out here, man. Too many people_. He slips the clean shirt over his muscled-up black shoulders. Takes his shoes and socks off and slides his feet into Nike sandals.
_You say you wanna make it playin ball, right?_ Dante says. _And you good, Stick. I ain't gonna take nothin away, your_ _game is real tight. But it's more than that. Every one a these_ _guys wanted to make it playin ball. What makes you any different? What separates you?_
Sticky nods his head.
_Look at Dallas here,_ Dante says. _How much you wanna bet_ _he thought he was gonna make it too. And you seen his_ _broke game_.
_Now wait a minute, D,_ Dallas says, waving his hand in the air and laughing. _I didn't play no overseas like you, but I_ _gets mine out there. You know that_.
_I'm playin, money. You all right_. Dante takes the joint, lights and sucks in. He holds the hit in his lungs and passes to Dallas. Blows out. _I'm just lettin you know, Stick, you can't_ _back down from nobody on the court. I used to fight three or_ _four times a week when I was comin up. And I was skinny, too._ _Like you. Sometimes cats would beat my ass, man. I'm not_ _gonna lie. But them same cats found out quick, if anybody ever_ _came at me, there was definitely gonna be some fightin_.
Sticky nods his head.
_That fight with Rob,_ Dante says. _I'm gonna tell you right_ _now, there's gonna be more where that came from_.
_Especially cause you white,_ Dallas says. _Brothers don't like_ _no white boy makin em look bad playin ball_.
_That's right,_ Dante says.
_Damn,_ Dallas says, fumbling what's left of his joint. _That's the only stuff I got_. He gets on his hands and knees and digs through the ice plant looking for it.
_Look at you, man,_ Dante says. _Like a damn crackhead_. He and Sticky both laugh.
Dallas finds the joint and sits back in his chair. He flicks off a little mud hanging from the tip and pulls a roach clip from his bag. _What?_ he says, looking up at Sticky and Dante. _I don't even care_. He lights up and pulls in as much smoke as he can get.
Dante reaches into his bag and pulls out his watch. When Sticky sees the watch a wave of panic rushes over him and he straightens up quick in his chair. _D, what time_ _you got?_
_Almost six,_ Dante says. _And, that's right, Stick, you still_ _ain't done nothin about no birthday gift_.
_Whose birthday?_ Dallas says.
_His old lady's,_ Dante says.
_Stick, you messin up, man_. Dallas laughs and tosses the lighter back in his bag.
Sticky stands up, pulls his Walkman out of his bag and puts the phones around his neck. _I gotta go,_ he says. _I gotta_ _jog home right now_.
_Don't sweat it,_ Dante says. _I'll give you a ride_. He leans back in his chair. _But chill a minute. Sit down. Let's discuss_ _what options you got_.
_How much time's left?_ Dallas says.
_Until she gets off work at nine,_ Sticky says.
_How long you been together?_ Dante says.
_Over six months. I got it handled, though._
_What you got handled, boy?_ Dante says.
_I'm swipin this bracelet from Macy's._
_Macy's?_ Dallas says. _You know department stores got all_ _kinda security, right?_
Sticky nods.
_I'm just sayin,_ Dallas says, _my dawg just got busted at a department store. He was tryin to make off with a toaster oven_ _and security tackled him just as he was steppin into the parking lot_.
_I hear you,_ Sticky says. _But I got a plan_.
Dallas rolls his eyes and laughs. _Oh, I see, Stick, you got_ _a plan_.
While Sticky and Dallas go back and forth a little, Dante leans forward in his chair straight-faced and kicks Sticky in his leg, tells him: _Yo, it's me, I might mess around and rob_ _somebody_.
_What?_ Sticky says.
_Back in my day I'd have probably hunted down some_ _rich cat, stuck a knife to his throat and told him to give me_ _his wallet._
You crazy, Dallas says. Sticky don't know nothin about _muggin nobody. He ain't got no experience_.
_I ain't sayin for him to do it,_ Dante says. _I'm just sayin_ _what I'd do. If I had some cash I'd figure I could buy the_ _bracelet and take my girl out to a nice little dinner somewhere._ _Italian. And some cats, man, they got enough extra cash that_ _they could fund a little somethin like that_.
_Nah,_ Sticky says. _I wouldn't wanna rob somebody. That_ _ain't right. Stores, man, they don't even know the difference,_ _but robbin a person ain't right_.
_What's the difference between stealin from a store and_ _stealin from some rich cat?_ Dante says. _Huh, Stick? Explain_ _your logic behind that last statement_.
Sticky looks up at Dante and thinks hard about it for a minute. He says: _Cause stealin from a store isn't as bad_.
_But why?_ Dante says. _I want to hear your reasoning_.
_I don't know,_ Sticky says. _It just isn't_.
_Damn, boy,_ Dante says, shoving his shoes and socks in his bag. _You ain't listened to a word I said since I met you_.
_Yeah, I have,_ Sticky says.
_You heard me talkin, but was you listenin to my words?_
Sticky looks at Dante, but he keeps his mouth closed this time. He can tell Dante's getting frustrated, and that's the last thing he wants. He thinks about the question again: What's the difference between stealing from a store and stealing from some rich guy? Dante must think it's the same thing, but why? It doesn't make any sense.
Dallas sips his water quietly, glances back and forth between Sticky and Dante.
Dante picks up a stick and lobs it against the back of the gym, says: _I'm not sayin for you to do the shit. In fact, I'll tell_ _you this right now, Stick: Don't do it. For real. It ain't in your_ _nature. But just hear me out for a minute. No matter how you_ _look at it, this ain't no righteous world. It just ain't. I mean,_ _there's no debatin about that. The laws we operate under are_ _set up by those who have everything, in order to protect themselves from those who have nothing. That makes sense, right?_ _Now, let's take me for example. When I was comin up on these_ _same Westside streets, I was one of the ones who had nothing._ _Just the same as you. So it was up to me to find ways to acquire_ _the basic things that other people already had. That was my reality, and I understood the situation. Now, when you don't got_ _enough to live an adequate life you can do one of two things:_ _either you can sit there and accept your fate, or you can do_ _somethin about it_. He shakes his head and leans back in his chair. _I chose to do somethin about it_.
_I know what you sayin, D,_ Sticky says.
_No, Stick, you don't know nothin about what I'm sayin._ _That's the problem. You ignorant to your own circumstances._
Dante reaches down and grabs a couple stones off the ground. _See that wall in front a you?_ he says. _In America, life's_ _like a race to that wall. That's the way I see it_. He sets the first stone less than a foot from the wall, points and says: _If you_ _born white and got money then you start the race way up here._ _Ahead of everybody. These cats got nice clothes and eat at nice_ _restaurants. Their parents send em to private high schools and_ _expensive colleges so they can one day be in a position to get the_ _best jobs. And when they make it they'll do for their kids just_ _like their parents did for them. It's a cycle_.
Dante stares at Sticky. He waits for it to sink in for a bit and then sets the second stone a couple feet behind the first. _But say you ain't white and you ain't rich. Say you poor and_ _black. Or you Mexican. Puerto Rican. Well, guess what? You_ _don't get to go to that nice private high school, that expensive_ _college. In fact, you may not even have enough food to eat a_ _balanced meal every night. You suffer from a lack of nutrition_ _and that ain't no good for a young mind. In this case you_ _startin the race of life way back here_. He points to the second stone. _Only a fool would think someone who starts here has the_ _same opportunities as cats startin at the first stone_.
Sticky feels Dante's eyes burning through the side of his face, but he doesn't look up. He just stays staring at the two stones and their different distances from the wall.
_Now I didn't make all this stuff up,_ Dante says. _This life-being-a-race thing. America did. But I sure as hell gotta deal_ _with it, don't I?_
_God knows it,_ Dallas says, nodding his head. _We all gotta_ _deal with it_.
_And let me tell you something. If you some scrubby white_ _boy who's been moved in and out of different foster homes since_ _you was little, then you off the charts, boy_. Dante physically lifts Sticky's face up to his so he can look in his eyes. _How_ _many foster homes you been in?_
Sticky looks Dante in the eyes but doesn't say anything.
_Answer me, boy. How many houses?_
_Four._
_That means three of em, plus your real momma, didn't_ _want your ass no more. They straight up gave you away like_ _you wasn't nothin. I gotta be real with you, brother. I gotta tell_ _it like it is cause that's_ my _nature. All these people, Stick, they_ _decided you wasn't worth a damn thing. They decided you was_ _a nothing. A Zero. Add to the fact you got that mental thing,_ _where you gotta do stupid stuff over and over and over._. . . (Dante snatches up another stone and puts it even further back. Points at it. Moves Sticky's face so he has to look at it) . . . _and fuck it, boy, you startin out way back here. You_ _three stones back_.
Sticky stares down at that third stone. He refuses to look up. He's hearing what Dante's saying, about people giving him back, about the stuff he does over and over, but he doesn't want to think about it. That's the last thing he wants to think about. The only thing he wants to think about right now is hurrying home and getting ready. Picking out some gear to wear and catching the bus. Getting the bracelet and the bear and meeting up with Anh-thu. He only wants to think about the next thing he has to do. The next hour. The next day maybe. But all this other stuff, what Dante is talking about, this is exactly the kind of thing he's tried to put out of his mind all his life.
Dallas nods his head, staring at the three stones. He looks up at Dante, glances at Sticky and then looks down at the stones again.
Dante turns Sticky's face back to his, digs into his chest with crazy eyes and tells him: _You the nigger, too, boy_.
Sticky jerks his chin out of Dante's grasp and goes back to the third stone. He concentrates on the way it looks. How it's small and chipped on the side facing him. How it's caked with dirt and a couple blades of dead grass. He realizes that his phones are unplugged from his tape deck and he reaches down to connect them back up. When they don't snap back in with the perfect pop, he goes to do it over again but stops himself short. He goes back to that third stone, stares at it. The chip on the side. The brown blades of grass. He battles the urge to pull the plug back out and snap it in right. He fights with everything he has to leave it alone. To leave it the way it is.
_And we supposed to worry about rules?_ Dante says. _What_ _rules? The ones set up to keep us way back here?_
Sticky shifts in his chair and looks up at Dante, he opens his mouth to say something but decides to keep it put away. He shifts in his chair again and then finally breaks down: he unplugs the phones and plugs them back in. He unplugs and plugs back in.
Unplugs and plugs back in.
Unplugs and plugs back in.
Unplugs and plugs back in.
Dallas sits back in his chair, watching Sticky. He folds his arms up and shakes his head.
Sticky unplugs and plugs back in.
Unplugs and plugs back in.
Unplugs and plugs back in.
Dante picks up all three stones and tosses them against the gym wall. He picks up his bag, zips it closed and fingers the edge of the zipper. _I ain't gotta do that stuff now,_ he says. _I_ _played ball overseas and made some money. I invested. I'm_ _successful now. But when I was comin up, man, I'm gonna tell_ _you right now: I did what I had to do_.
When the popping sound finally sounds perfect, Sticky stops unplugging and hangs his head.
All three of them remain quiet for the next few minutes. There's only the hum of the hundreds of cars starting and stopping on the nearby highway. Dante stares at Sticky, pulls in a few deep breaths to slow himself down. He shakes his head at all this stuff he's just said. Maybe he shouldn't have done it. Maybe he shouldn't have launched into all that. It wasn't the time or place. And he doesn't want Sticky to rob anybody. Definitely doesn't want that. He just wants him to see the world for what it is. For how it works. Because even though Sticky's white and he's black, there are obvious similarities: the passion for playing ball, the grace with the rock, the way every move on the court comes from some inherent instinct. He looks at Sticky and he knows basketball is all he has. A game. A sport. He knows there's nobody looking out for him. Nobody talking to him about life or waiting for him to come home at night. Sticky's completely alone. Just like he was when he was a kid. Sometimes just looking at Sticky brings back painful things about his own past. Things he thought he'd long since put away.
The stones are gone, but Sticky's still staring at the ground where they were. Where Dante had put them. Everything he has just heard, all the words and phrases out of Dante's mouth, all the important things he was meant to learn, rattle around in his head. But all of it messes up when he tries to come up with meaning. He knows he can't rob anybody. He knows that's something he wouldn't be able to do. But he wishes he could. He wishes he could bum rush some business dude and bring the wallet back to Dante for proof. Because even though what Dante has said is not the kind of stuff he wants to think about, he knows Dante said it because he cares. And the fact that Dante cares about him is incredibly important. There may not be anything more important. There's a low whistle in Sticky's ear now. And it's sore. Really sore. He can't stop touching his fingers against the part that hurts worst. The cartilage just above the earlobe. He flips daydream channels to the way it would go down if he could do it: putting a knife to some rich dude's neck, snatching his wallet, shoving him down to the pavement and taking off running.
_It's the truth,_ Dallas says, staring at the ground. He nods his head and wipes a hand down his face, _It's the truth what_ _he's sayin_.
Dante and Sticky look at Dallas.
Dallas brings his eyes up and stares at Sticky. He scratches his head and chuckles a little to himself, says: _That's my pops, man_.
_Who?_ Sticky says.
Crazy Ray, Dallas says. From the gym. That's my daddy.
Sticky returns Dallas's stare but doesn't say anything more.
Dante nods his head and looks at the side of Dallas's face. _I know it is,_ he says.
Wong and Rolando,
two of Sticky's foster bros, are at the TV playing Madden. Wong is fifteen and Korean; he walks around the neighborhood all day in thick army fatigues, clutching a water gun and pretending it's the real thing. Rolando is fourteen and black, but he looks like he's twenty and Samoan. One of his eyes is set a little lower than the other, and there's a two-nickel gap between his front two teeth, but nobody ever mentions these things. Both stop arguing when Sticky walks through the door.
_Hey, Stick,_ Wong says.
Sticky gives him a what's up with his head.
_Yo, you been ballin?_ Rolando says.
_Yup._
Wong pauses the game and walks over to the kitchen counter, grabs a letter and hands it over. _This came for you_ _today,_ he says. _I think it's for college_.
Sticky peeps the return address: UNIVERSITY OF SOUTHERN CALIFORNIA TROJANS. He rips it open and reads the words: it says one of their scouts saw him play against Dominguez Hills. Says they liked his skills. Says they wanted to contact him early to let him know they're extremely interested in recruiting him. Says if he fills out the enclosed questionnaire and sends it back, that they'll be in touch real soon about having him as a guest at one of their home games next season. It's signed by the head coach.
Sticky folds the letter up and shoves it back in the envelope. Coach Reynolds told him the letters might start rolling in soon. Do work in the play-offs and college coaches will track you down. He said by the time Sticky's a senior it will be like a circus. Calls at all hours of the night. Coaches knocking on his door. Letters from prestigious alumni and offers to host him on recruiting trips. But all that was just talk. It was nothing like this. Nothing like having that first letter from a college right there in your hands.
_What they said?_ Rolando says. _They gonna give you a_ _scholarship or what?_
_That's what they want,_ Sticky says, and he tosses the letter in his bag like it's no big deal. _But I'm waitin to see what_ _UCLA says. And all them Big Ten schools. The ACC_.
_I'd go to Duke,_ Rolando says. _All the best basketball players_ _go to Duke_.
_Yeah, Duke's all right,_ Sticky says.
_Hey, what happened to your face?_ Wong says.
_Nothin,_ Sticky says. _Just playin ball_.
Rolando starts the game back up while Wong is still standing by Sticky. He maneuvers his linebacker through Wong's offensive line and crushes his quarterback. _That's_ _right, boy!_ he yells. _Fifteen-yard loss_.
Wong sprints back around the table and grabs his joystick, finds his quarterback lying motionless on the turf. _What the hell you doin?_ he yells. _I wasn't ready_.
_You the one ain't been playin by the rules,_ Rolando says, laughing.
_I am playing by the rules,_ Wong says. _You just mad cause_ _I'm beating your ass three games in a row_.
Rolando throws his joystick to the side and tackles Wong. They wrestle around on the dirty rug, laughing and talking trash. Knocking things over with erratic arms and legs.
Sticky cruises into the bathroom and shuts the door. When he flips the light on, roaches scurry back into corners of the room. Underneath the cracked linoleum or behind somebody's tossed-to-the-side towel. He turns the shower on and rips off his gear. Sets everything on top of the cracked toilet seat. He steps in with a foot and when it doesn't feel right, steps back out. He thinks about what Dante just said. Reaches in to touch the cool water with his hand and wonders if everybody knows. If everybody sees it. And if so, do they look at him like he's some kind of freak?
_I ain't doin that no more,_ he says under his breath, and he steps back in the shower. And even though it feels all messed up the way he went in, he stays put. Even though just standing there feels completely wrong, totally unbalanced, he won't let himself move. He clenches up his fists and fights the urge to do it over. Grabs for the soap and forces his head under the water. But it's only a couple seconds before he gives in and backs everything up. He puts the soap down and steps a drenched foot back out of the shower. He drops his head as he steps in and out again, tells himself he's gonna fix this about himself. Starting tomorrow he's gonna practice at doing things regular. Doing things like a normal person. But for now he lets himself do what he's gotta do. He steps a foot in the shower and then steps back out. Tomorrow he's gonna start fresh. Change his ways. Be more normal.
He steps a foot in the shower and steps back out.
Steps in, steps out.
Steps in, steps out.
Steps in, steps out.
It's a shade past seven when Sticky finally finishes his shower. He hops out and stands in front of the mirror toweling off. Moves his face in close to get a good look at his cut. It's still open a little, but it's not that bad. Definitely doesn't need stitches like Fat Chuck said. He fingers the lumps on the back of his head. Listens to the hollow fuzz filling his sore and bright red ear.
While brushing his teeth, Sticky stares at his face as a whole. His eyes, ears, lips, cheeks, chin. His color. He looks at the way everything comes together. Anh-thu says he has a beautiful face. She says a lot of the girls think that about him. But why? he wonders. He imagines Dante looking at this face when he was talking about the stones. Telling him how nobody wants him. This face. Telling him how everybody keeps giving him back. Dropping him back off cause he's nothing. This face. These dark eyes. These cheekbones. These lips. At some point in their life, he thinks, maybe everybody looks at their face like this. Wishes they could change one or two things. But has anybody ever experienced this situation? Feeling that none of it makes sense? Cause he's looking closely at his face, closer than he ever has before, and he doesn't recognize himself. He doesn't see himself in himself. This isn't the Sticky he's always imagined in his head. It's a picture of somebody else. A mask. Something off a TV show. He doesn't know this face. It's a complete stranger. And the whole thing freaks him out to the point that he has to look away.
He finishes brushing his teeth by staring at a big hole in the wall. Thinking of nothing. He stares at the hole he always stares at instead of looking in the mirror. An ex–foster brother made it a while back. He locked himself in the bathroom one night for almost two hours and clawed at his own skin. Ripped down the moldy shower curtain. Put a fist through the wall. The cops had to break the door down to finally get him out. It was Sticky's first week in the house, and he didn't ask any questions. Nobody offered up any info on their own, either. Not even a few days later, after the kid was shipped back to wherever he came from and a new kid was brought in to take his place.
Sticky walks into the hall and pulls out his bag full of gear from the closet. He always keeps his clothes stuffed in a bag now. Georgia offered him a couple shelves in one of the bedrooms, but Sticky told her he wanted to stick with the bag. Call it superstition or reverse psychology or whatever you want, but he's always ready for the next time somebody tells him it's time to move again.
He pulls out some boxers and socks, the fresh retro Nike Airs he swiped last week from a shoe shop in Culver City. He pulls out some baggy jeans and a wife-beater and then puts everything on in a particular order: first the drawers and the jeans, second the wife-beater, third the socks. He goes to one knee and pulls on the left Nike first, makes loose laces look perfect, and then does the same deal with the right. He grabs his bag and rolls into his brothers' bedroom, pulls open a drawer and snatches Rolando's favorite shirt. Light blue button-up with a collar. He sticks the shirt in his bag on top of his ball and his letter and then zips up.
Out in the living room, Wong and Rolando are still battling on the video-game football field. And little Julia has come back from class. She's sitting on the far end of the couch reading the funny pages. Julia started out as a temporary in the house. Way back before Sticky arrived. A temporary is a kid who's scheduled to stay in the system only a couple months or so on paper, until a mom or dad can get a handle on things financially or a group of counselors working the case give the go-ahead. But like so many other temporaries, Julia's two months has turned into two years and now nobody mentions her real parents anymore. Including Julia. But Sticky has spotted her a couple times, writing long letters in the middle of the night on the back of old home-work assignments.
Julia spends her summer days in a science class for eighth graders, even though she's only eleven.
_What up, Jules?_ Sticky says, and pats the top of her shiny black hair.
When she turns around her face lights up. _Sticky,_ she says, all long and drawn out, and slugs him in the arm.
_You have fun in class today?_ Sticky says.
_Yeah,_ she says. _I learned how astronauts use this stuff_ _called polymers in space that they grow food in,_ Julia says. _It_ _absorbs four hundred and eight times more water than dirt_.
_Say what?_ Sticky says as he cruises into the kitchen. _They_ _can grow stuff up in space?_
Wong stands up and leans into his long pass downfield. Rolando speed-taps his Run button to catch up. When the receiver drops the pass, they both yell out at the same time: _Ahhhhh!_
Rolando shoves Wong. _You can't mess with my DB's, man_.
_You just lucky,_ Wong says.
Julia follows Sticky into the kitchen and leans against the overflowing trash. _They can grow all kinds of food up there,_ she says. _They gotta eat somethin, you know_.
Sticky opens the cupboard and reaches in for the big bag of granola. He digs his hand in and grabs a fistful. _You just_ _like Annie,_ he says, chomping through a mouthful and swallowing, reaching into the bag for more. _She's always learnin_ _about stuff like that_. He opens the drawer next to him and sifts through the silverware. Grabs hold of an old steak knife and flips it around in his fingers. He gives a quick peek over his shoulder and when he sees Julia's not paying attention, he slips it into his pocket on the sly.
Julia reaches her hand in the granola and takes out a handful of her own. She crunches through a mouthful and swallows, looks up at Sticky and smiles little kid teeth.
_I don't know why,_ Sticky says, _but I just can't listen that_ _good when I'm sittin in a classroom_. He palms his hand on Julia's head and shakes it around. _But you and Annie got it totally different. You all actually like learnin that stuff_.
_When we gonna go shoot baskets?_ Julia says as she digs her hand in for more granola.
_I can't tonight, Jules,_ Sticky says. _It's Annie's birthday_.
_It's Annie's birthday?_
_Yeah. I'm about to go meet her right now_. Sticky puts the granola away and scoops up his bag.
_I wish I could tell her happy birthday,_ Julia says.
_Maybe she'll swing by this weekend,_ Sticky says. _And then_ _you could tell her_.
_Yeah, yeah. Tell her, please? I'll get her a present and everything. Please? Please?_
_OK, I'll tell her. I promise_. Sticky picks up his bag. _I gotta_ _break, Jules. We'll shoot some hoops tomorrow, all right?_
Julia leans forward and hugs Sticky. She wraps her skinny arms around him and squeezes tight. _Bye, Sticky,_ she says.
Sticky goes stiff in her arms. He's never been good at hugs. Even with Anh-thu. The feel of somebody's body next to his is always awkward. When she lets go he pats her on the head a couple times.
Outside, the sun is finally losing its grip on the day. It sits low in the sky like a giant orange ball, resting just above some stores on a patch of clouds. As Sticky walks down his street he stares at it, amazed at how big it seems. It's as if you could just reach out and snatch it in your hand, start dribbling it around the block or spin it on your finger. The air's cooled down a bit, too. The pleasant breeze smells like salt and seaweed and the exhaust of cars all mixed up.
Sticky cruises a couple blocks and sits down on his bus-stop bench next to a black lady dressed in a Ralphs uniform. He sets his bag down. She looks over at him and shakes her head, glances down at her watch and tells him: _Number three_ _bus was supposed to be here ten minutes ago_.
_Yeah?_ Sticky says.
_Yeah,_ the lady says. _This damn bus driver is ten minutes_ _late every single day. Like clockwork. But don't go tryin to work_ _your schedule around him bein ten minutes late, now. Nah,_ _you try to coordinate your schedule and that's the one day he_ _comes on time. Trust me_.
Sticky laughs with her a little and fumbles with the zipper on his bag. He reaches past Rolando's shirt and his ball and pulls out the letter from USC. The lady says something else, about how hot it was today or how her air-conditioning unit doesn't pump out any cool air anymore, and Sticky nods his head at her. Two cars almost smack into each other in the nearby intersection during a yellow light. They both slam their horns and flip each other off. They drive around each other, cursing out of rolled-down windows, and then continue on their way. A group of Mexicans walks by with shovels on their shoulders, shirts soaked with sweat. Dirt-covered work boots. They laugh and say things back and forth in Spanish.
Sticky holds the envelope in his hands and stares at the return address: UNIVERSITY OF SOUTHERN CALIFORNIA TROJANS. It's so official with the school name on the envelope like that. It makes it seem totally legit. Important.
He smiles at the lady, who is pointing at her watch again and shaking her head. Then he pulls the letter out of the envelope and unfolds it in his lap. He reads it again.
In the Heart
of Santa Monica, the beautiful people come out at night. They step out of fancy rides in hip gear, stroll into trendy clubs with no name above the door. You see dudes in black leather jackets, black leather pants, black leather boots. Surfers in sandals wearing a curvy blond girlfriend around their waist. Spiky-haired hipsters wearing cool throwback shirts plucked off a retro rack in some pseudo-thrift shop on Melrose. You see ladies in short skirts and backless tops. Tight jeans. Pointy high heels. Cowboy hats. You see some forty-year-old dude escorting a twenty-year-old model-type into some bar off Second Street. See a little white honey holding a big black cat's hand, checking out of one club and rolling into another two doors down. See two six-foot French-looking ladies stepping out of a stretch limo and walking hand in hand into some crowded café. Smoking long fancy cigarettes and giggling at a crowd of gawking guys.
It's a little before eight and a bright moon has nudged the sun completely offstage. The air is warm and full of salt. Sticky watches all the different people from a patch of grass outside Sanwa Bank on Colorado. He takes big bites of the Hostess Cup Cake he swiped when he got off the bus and chews with his mouth slightly open.
Some suit guy walks past him to the ATM, slips his card in the mouth of the machine. Sticky wads up his wrapper and holds it in his hand, watches the guy from behind: punching in numbers and waiting, pulling out seven or eight twenties and shoving them deep in his wallet. The guy snatches his receipt and heads off toward a nearby bar.
Sticky reaches in his bag and pulls out his own stash of cash. Counts out the bills in his hand over and over. Slowly. Straightens out each buck and sets it on the grass in front of him. Licks a finger and peels off the next bill.
He does this repeatedly, but each time the total remains the same: twelve bucks.
Anh-thu stands next to Sergio as he totals up the sales from her drawer. He does the numbers, but she's staring across the store at a family. A young black man holding a baby on his shoulders, his wife's arm wrapped snugly around his back. They stop at the T-shirt rack and while the dude sifts through the extra-large section, his wife makes faces at their baby and pulls at its toes. The baby laughs. Anh-thu feels her stomach tighten up as she stares. She feels the anxiety of her and Sticky's meeting climbing into her lungs, making it tough to get a deep breath. She turns her eyes away from the family and goes back to Sergio's counting.
When everything comes up even, Sergio points over to the broom. _Just the dressing rooms and the back, Annie,_ he says. _Don't worry about the store floor, I wanna get you out of_ _here a little early for your birthday_.
Anh-thu forces a smile and grabs the broom. As she walks over to the dressing rooms, she scans the store. Her tired legs carry her across the floor while her green eyes sweep over every detail in the store. The place is almost empty now—aside from the black family, a couple kids from the valley rifling through what's left of the half-off table, a guy with a bad flattop holding up a pair of baggy jeans and sizing himself up in the mirror—but the store looks like a tornado has just rumbled through. Shirts are scattered everywhere, pants are hanging from the jacket rack, sweat-shirts are turned inside out and lying on top of the hat table. For the girls on the floor the real work starts not when Sergio flips the OPEN sign on in the morning, but when he cuts it off at the end of the day.
Julie is in the middle of the store sorting out a pile of swim trunks that somehow ended up on the floor by the jackets. She's folding and stacking according to size. Laura is at the far register ringing up an elderly guy. She smiles at the old-timer as she takes his credit card. She swipes it through and they both stand there waiting.
Anh-thu pushes open the door to the first dressing room and thinks about Sticky again. How he'll act if she tells him how late she is. She runs through the different scenarios as she sweeps pins, tiny threads, ripped-off tags and dust into a little pile. As she reaches down and pushes the pile over the plastic lip of the dustpan.
Sticky watches a couple ladies stick their cards into the ATM one at a time. The first has long pasty legs shooting out of short khaki shorts. The second is shorter and heavier and is wearing a tacky pink skirt. They both have fake blond hair with awful dark roots. They stand next to each other and talk in voices so loud that everyone on the block knows their business.
Sticky's watching these ladies, but he's brainstorming about hoops. He figures if Lincoln Rec is shut down tomorrow, because of the fight, he might have to suck it up and ball with those scrubs on Twentieth and Pico. On that beat-up outdoor court with chain nets. Or maybe he could track down his school custodian, Manuel. Ask him to leave the back door of the gym open a crack. If he had the gym to himself all night he'd flip on the court spotlights and make like the dark bleachers were filled with screaming fans. Or maybe he'd smuggle in the new OutKast CD and slip it into the gym's beat-up old boom box. He'd shoot five hundred jumpers while jamming to his beats. A thousand jumpers. And after that he'd get in a little ball handling work. Do some passing drills off the wall. It's time to buckle down, he thinks. Time to get his game right, with that camp only a couple weeks away. And the mere idea of the camp hypes him up even more. Matter of fact, he'll shoot _more_ than a thousand jumpers. He'll keep on shooting and shooting until his arms feel like they're gonna fall off. Dribble so many dribbles the ball will turn into an extension of his hand. Run home so hard from the gym that the muscles in his legs will feel like they're catching fire. He's gotta go into that camp with every part of his game clicking. Everything perfect. Dante says the most important hours of hoops you'll ever spend are the hours you spend alone.
Sticky decides only a few things really matter to him right now: his rhythm on the court, his performance at the camp, the college coaches who will be watching, the letter from USC in his bag, and Annie's birthday. Everything else is secondary, he thinks. Everything else is a million miles away.
As Anh-thu is diligently sweeping through the back of the store, Laura comes walking up to her with a brown paper bag. She hands it over and winks.
_What's this?_ Anh-thu says, leaning the broom against the wall.
_Just somethin I thought you might need,_ Laura says. _Don't_ _open it here, though. Serious. Wait until you get off_.
_OK,_ Anh-thu says, confused.
_Trust me, girl. I been there._
_OK._
_And no matter what,_ Laura says, _know that I'm totally_ _here for you_. Anh-thu nods her head. Laura gives her a long hug and tells her in her ear: _Later on tonight, it doesn't matter_ _what time, call me. OK?_
_I will,_ Anh-thu says back.
After Laura takes off, Anh-thu stares at the brown paper bag in her hands, folded over at the top. She presses through the outside and determines it's a box of some kind. But a box of what? And is it for her birthday? Because Laura's already given her the picture frame. _Strange,_ she thinks, setting the bag on top of a shirt bin and grabbing for the broom.
She begins sweeping again, though she can't help continually glancing over at the bag. What could be inside? What did Laura mean when she said she's been there? Been where? She rolls a rack of returned jeans to the side so she can sweep underneath, and a tiny spider scurries across the floor. Anh-thu lunges back and smacks at it with the broom. She picks it up with a paper towel, throws it in the trash and continues with her sweeping.
A pack of tourists walks up to the ATM, talking all loud in a foreign language. Sticky can't figure out what language it is, but he knows it's not English. One guy goes to the machine at a time while the group hangs back. Each guy pulls out a wad of twenties from the slot and shoves it deep in his pocket. When all of them have had a turn at the ATM, and they're armed with enough money for their big night out in Santa Monica, they move quickly down the street as a group, like a pack of excited dogs. They hook around the big parking structure and head toward the Third Street Promenade. Even when they're completely out of his sight, little pieces of their loud language still funnel back to Sticky's ears.
A cab rolls to a stop in front of Sticky and a light-skinned black dude hops out of the back. He hands over some money to the driver and slams the door shut. The cab rolls off, and the guy heads for the ATM.
Sticky watches this guy and for some reason he thinks back on the incident with Fat Chuck. The way Chuck seemed cool at first, like he was trying to help, and then all of a sudden tried some crazy shit. Sticky feels himself getting pissed all over again. He should have kicked Chuck in his teeth, man. Blasted him with a heel to the back of the head. Booted him in the ribs over and over until he couldn't breathe. And Sticky wonders why he ran off to tell all the guys. Why didn't he just handle business on his own? Like Dante told him to. Like he did with Rob. And Dante's right about what he said. About him taking care of himself. Dante's right about a lot of things.
He leans his head back on his bag and stares up into the sky. He stares at the bright moon and watches his fight with Rob play out in his head. The way he leaped up off the hardwood and clocked Rob in the neck. The way Rob tackled him to the ground. Everything happening so fast, like when you're on some pitch-black roller coaster at Disneyland. Space Mountain. Everything flashing by before you even know what's happening. All you have time to feel is the exhilaration, the twists and turns, your heart climbing into your throat. And then suddenly it's over.
Sticky's eyes slide shut.
An old woman screaming at no one pulls Sticky out of his dreams. He wakes up as Dante's laying out that last stone. The third one. Look how far away he is from the wall. Look. He's the bottom of the barrel, man. The last rung on the ladder. He wakes up and acknowledges what Dante is explaining to him. That people keep giving him away. That they keep sending him back. But then Sticky sees himself playing ball. Dominating guys twice his age, twice his size. He sees his grace on the court. His beauty. His secret refuge. Those foster parents never saw him play ball, he thinks. They never saw him on the wing with the rock in his hands. Something incredible happens out there, man. Something he can't even explain. If only they'd come to watch. Like Anh-thu did. Then they'd see it for themselves. That he's not retarded. That he's actually really good at something. Great, even. That he's been blessed.
Maybe then they would have wanted to keep him.
When Anh-thu is finished sweeping she puts the broom and dustpan back into the storage closet, picks up the brown paper bag and carries it with her into the employee bathroom. She takes off her work clothes, a brown Millers T-shirt and jean skirt, and pulls her summer dress out of her bag. She lifts it above her head and slips it on. She changes her shoes, too, all the while staring at the brown paper bag in front of her.
Anh-thu finally takes the bag in her hands, unfolds the top and slides the box into her palm. Her stomach drops. It's a home pregnancy test. How could Laura have known? Does everybody know? Does she even want to know what it will tell her? She's too young. Sticky's too young. He has his basketball, and there's college. Her dad and brothers would kill her. But then to have the chance to do right what her own mom did so wrong. Even if it started today, on her sixteenth birthday.
Anh-thu tries to push all of these thoughts out of her head as she follows the directions. Then she waits.
A chorus of honking pulls Sticky out of his head.
He looks over at the road and spies three cars stopped at a green light while an old man is trying to pull off a tight U-turn. The people in the waiting cars have no patience; they throw their hands in the air all dramatic and shout out their windows.
When Sticky snaps his attention back to the ATM, he finds a skinny-looking white dude in a suit standing alone. A briefcase at his side. He watches the guy pull out a chunk of money and take his card from the machine, then slip another card out of his wallet. He shoves the new card in, punches in a series of numbers and pulls out another stack of cash. He peeks quickly over both shoulders, as if aware that someone's watching him, and then reaches for the second receipt. He stands there organizing himself, slipping a thick stack of twenties into a gold money clip, then slipping the money clip into his back pocket.
Sticky sits up, leans his weight on his hands behind him and checks the big clock above the parking structure: 8:15. He's gotta go get that bracelet now. Before the store closes. The bear. He's only got forty-five minutes before he's supposed to meet Anh-thu, and a feeling of failure spreads through his stomach.
The guy walks away from the machine and cuts through the alley between Third and Fourth. Sticky gets up and grabs his bag. He follows the guy.
The alley behind Third Street is dark and grimy. The asphalt slick from all the years of trash bags being left outside the back doors of fast-food joints to rot. There are small cars parked against the wall, wet mops leaning against giant trash receptacles and mysterious warm smells drifting out of exotic restaurants. Sticky follows behind the guy like a private detective. He keeps him in his vision but stays far enough back that nobody would ever know. He wants to see where the guy's going, and then he'll go get the bracelet and the bear and meet Anh-thu.
Sticky slides his right hand into his pocket, fingers the twelve bucks, fingers the steak knife. He looks up as the guy walks out across Broadway and into the alley on the other side. _The thing about Annie,_ Sticky thinks as he crosses the street, _is she wouldn't care if all she got for her birthday was a_ _stuffed bear. She'd love it. That's just the way Annie is._ And maybe he doesn't need to be out stealing stuff so close to his basketball camp. What if he got nabbed? The judge warned him last time to straighten up. He brought up juvenile hall and military-style work camps. And how could he show those coaches what's up if he was standing behind bars? Annie just wants them to spend time together. That's what she's always telling him. She's doesn't trip on all that other stuff. Material things. That's not the kind of stuff she thinks is important.
And that settles it, Sticky thinks. He's not stealing anything today. He's gonna go get the bear and maybe take Anh-thu down to the Santa Monica Pier. They can watch all the different people circling around on the Ferris wheel and talk. The most important thing is to be with her on her birthday.
And without even thinking about it, Sticky starts kind of jogging through the alley. He's excited now because he's settled on a plan. The weight of decision has lifted from his shoulders. He's kind of jogging behind the suit guy, slowly cutting into his lead, while at the same time thinking about Anh-thu's face when she sees the bear. When he sings her "Happy Birthday" in her ear and kisses her sixteen kisses on the lips. And without even thinking about it, Sticky reaches into his pocket and pulls out the steak knife. He pulls the knife out and jogs through the dark alley with it clutched at his side.
The suit guy whips his head around when he hears Sticky's footsteps, but it's too late. Sticky slams into him like a free safety. He lowers his shoulder, lunges at the guy and sends him flying into some plastic trash cans. He thrusts the jagged blade against the guy's neck and grabs a fistful of his hair.
The guy's eyes are wide. His teeth are long and yellow, lips thin and white. His jacket is ripped at both elbows and tiny drops of dark red blood are starting to soak through. The tip of a dark green tattoo juts out above his collar.
Sticky spies the briefcase, which has sprung open in the fall. It's full of little white baggies of powder. Drugs. The guy's a drug dealer. He's tackled a drug dealer. Sticky goes back to his man, opens his mouth to talk but nothing comes out. Instead of talking he yanks at the guy's hair and watches his face cringe.
_What the hell you want?_ the guy says, his voice altered by the pressure of the knife against his throat.
_Gimme the money!_ Sticky says, pressing the knife harder against the guy's throat. _The money in your back pocket!_
_OK, OK, OK,_ the guy says, and he holds his hands out to his sides. _All right. Just hold on_. He reaches behind his back slowly, the whole time staring into Sticky's darting eyes, and into his back pocket. He pulls out the wad of cash and sets it gently on the pavement. Then he holds his hands out to his sides again. _All right, buddy,_ he says, almost in a calm voice now, _there it is. There's the money. But you don't know who I_ _am, buddy. You have the money, but I'm just telling you, you_ _don't know me_.
Sticky releases the guy's hair long enough to pick up the money clip and push it into his own back pocket. He has no idea what to do next and this makes him panic. He cracks the guy in the back of the head with the butt of the steak knife and takes off running. He races down the alley as fast as he can. Fists pumping, mouth sucking in air. Chest pounding, burning. He sprints away from what he's just done as fast as he can, still clutching the steak knife in his hand.
Sticky flies out into Santa Monica Boulevard. He dodges a couple slow-moving cars and ducks into the alley on the other side. He whips his arms at his side, barreling through the length of the alley, and then pops out onto Arizona. A woman in a minivan has to slam on her brakes to avoid hitting him. Her eyes grow huge and she covers her chest with her hand. Sticky slips into the alley on the other side and hurdles a homeless man, an empty crate. On the run, he anxiously looks back over his shoulder but nobody's there. He pops out onto Wilshire and barely slips past a bus speeding west toward the PCH onramp. The driver sounds his deep horn, swerves slightly, and all the people out walking turn to look. But Sticky's already halfway down the alley behind the big Catholic church. When another anxious glance behind him reveals nobody he slows down, ducks behind a big trash receptacle and bends over, hands on knees. He begs for breath. Salty sweat rushes down his face and neck. It runs into his eyes and ears and mouth and he can't get his wind. He leans his head against the church wall. His hands and knees are shaking. When he thinks about what he's done a wave of panic rushes over him. And then guilt. And then shame. And then incredible excitement. He looks up at the stained-glass windows of the church and prays for a deep breath. Just one good deep breath and he'll figure out what's happened.
Sticky peers over his shoulder again, down the dark alley: nobody. He closes his eyes for a second and tries to swallow. He wipes his nose on his shirt and realizes his entire body is shaking uncontrollably. His teeth are chattering. He can't control any part of himself. Then he notices the steak knife in his hand. He chucks it down the alley and feels a wave of nausea wash over him.
He pulls the wad of money out of his pocket with trembling fingers and slips it out of the money clip. He peeks over his shoulder: nobody. He goes to one knee and quickly counts the twenties in his hands:
Twenty-forty-sixty-eighty-a hundred . . .
twenty-forty-sixty-eighty-two hundred . . .
twenty-forty-sixty-eighty-three hundred . . .
twenty-forty-sixty-eighty-four hundred.
Four hundred bucks, man. Four hundred. It's the most money he's ever seen at once. And it's in his hands. _His_ hands. He peeks over his shoulder: nobody. He picks up the four stacks of twenties and counts the money out again:
Twenty-forty-sixty-eighty-a hundred . . .
Twenty-forty-sixty-eighty-two hundred . . .
Twenty-forty-sixty-eighty-three hundred . . .
Twenty-forty-sixty-eighty-four hundred.
He wipes the sweat out of his eyes. The sweat off of his forehead. His shirt is completely soaked through, and his heart is still racing. He peers down the dark alley both ways: nobody. He closes his eyes to try and calm down and pulls in his first deep breath. He tries to think about what he's just done. What it means. Whether or not he's crossed some invisible line he told himself he'd never cross. He doesn't know what to think so he stops thinking and pulls in another deep breath. But his body is still trembling. His heart is still racing. He picks up the four stacks of twenties and counts the money out again:
Twenty-forty-sixty-eighty-a hundred . . .
Twenty-forty-sixty-eighty-two hundred . . .
Twenty-forty-sixty-eighty-three hundred . . .
Twenty-forty-sixty-eighty-four hundred.
All this money. In _his_ hands. Four hundred. He could take Anh-thu anywhere she wants to go. Let her order anything she wants to order. He could walk her into Macy's and buy her whatever bracelet she wants. He thinks of Dante. Wonders what he'll say when he hears about this. _You gotta do what you gotta do,_ is what he'll say. But then when he runs through it again in his head, the tackle and the knife to the throat and the blow to the back of the head, the panic comes back. The nausea. The uncontrollable feeling of falling. He swallows hard and looks down the dark alley both ways: nobody. _Get out of here!_ he tells himself. _Come on! Go!_ He looks down the alley again: nobody. Picks up the four stacks of twenties and counts the money out again:
Twenty-forty-sixty-eighty-a hundred . . .
twenty-forty-sixty-eighty-two hundred . . .
twenty-forty-sixty-eighty-three hundred . . .
twenty-forty-sixty-eighty-four hundred.
Four hundred bucks. In his hands. _Get out of here, man!_ _Go find Annie!_ He picks up the four stacks of twenties and counts the money out again:
Twenty-forty-sixty-eighty-a hundred . . .
twenty-forty-sixty-eighty-two hundred . . .
twenty-forty-sixty-eighty-three hundred . . .
twenty-forty-sixty-eighty-four hundred.
_That's it!_ he tells himself. But as he stands up to leave, he freezes. He can't move. He hasn't counted right. He hasn't stacked the bills right. He hasn't done anything the way it needs to be done, and his body won't let him move on to the next step. The next stage. And as he stands there cursing himself, fighting with his body, his mind, all these images come crashing down on him at once: stepping in and out of the shower, tucking and retucking his shirt, tying and re-tying the laces of his shoes, brushing and rebrushing his teeth, washing and rewashing his hands, snapping and resnapping his warm-ups, zipping and rezipping his bag, tossing and retossing change into the bowl with Baby hovering over him, spitting and respitting into the bed of the truck with Baby yelling for him in the background. The cash doesn't feel right in his hands. It's off. He's off. He's leaning to one side, like after you spin around and around on a merry-go-round and then get off and try to walk. Like that.
He's counted wrong and now he's off. And he can't move. Can't do anything. He peeks over his shoulder: nobody. He gives in to his body and goes to one knee, starts counting the twenties in his hands again:
Twenty-forty-sixty-eighty-a hundred . . .
twenty-forty-sixty-eighty-two hundred . . .
twenty-forty-sixty-eighty-three hundred . . .
twenty-forty-sixty-eighty-four hundred.
And again:
Twenty-forty-sixty-eighty-a hundred . . .
Twenty-forty-sixty-eighty-two hundred . . .
Twenty-forty-sixty-eighty-three hundred . . .
Twenty-forty-sixty-eighty-four hundred.
And again:
Twenty-forty-sixty-eighty-a hundred . . .
Twenty-forty-sixty-eighty-two hundred . . .
Twenty-forty-sixty-eighty-three hundred . . .
Twenty-forty-sixty-eighty-four hundred.
And before Sticky can pick up the four stacks of twenties and count them out yet again, a dark shadow slinks in from the side and sticks something to the back of his head.
_You messed up, buddy,_ a voice says. Sticky goes to turn around but two taps to the back of the head make him stop short. The voice says: _Just keep on lookin at that wall, buddy._ _This will be over soon_.
The end of the alley is only twenty yards away and Sticky can hear the sound of a car rushing past on California. He can smell the ocean in the air. But all he can feel is the breath of this man on the back of his neck.
_Pick up the money, buddy, and hand it up to me slow. Nice_ _and easy_. Sticky reaches for the cash and slowly brings it up over his head.
_That's it._
When the guy takes the handoff, Sticky spins around and knocks the object out of the guy's hand. The gun. It tumbles to the ground. The guy staggers back and loses hold of his money. He catches his balance against the trash receptacle and he and Sticky both stare at the gun lying on the ground between them. Sticky jumps at the guy, tries to smack him with a closed fist, but the guy slips it. He shoves Sticky against the wall and reaches down for his gun, cracks Sticky in the mouth. Sticky puts a hand to his bleeding lip. When he looks up the gun is pointed right at his face. He instinctively lunges to the side and sticks his right hand in front of the barrel.
The gun goes off.
The bullet explodes into Sticky's right hand.
The bullet goes straight through the skin between the thumb and forefinger of his shooting hand, ricochets off the church wall and disappears down the alley.
The guy looks both ways, shoves the gun back in his pants. He reaches down to collect the money and grabs the handle of his briefcase. Then he quickly steps over Sticky and takes off running the other way.
Sticky lays his face down flat against the filthy asphalt. Sweat is streaming down his neck. He rolls over clutching his hand. Rolls back the other way. He opens his mouth wide enough to yell but there's no sound. He opens his eyes, cheek mashed against asphalt, and from this strange angle watches the guy running away. Watches the boots of this man lifting and falling in silence. He rolls over and looks the other way, sees two older dudes looking at him from the edge of the alley. One of them is pointing. Sticky closes his eyes and opens them. He closes and opens them again and settles his stare on one of the filthy trash receptacle wheels. He stares at the wheel and keeps his face completely straight and then he passes out from the pain.
Before Anh-thu leaves Millers, Sergio checks her bag. Like he always does. _OK, birthday girl,_ he says, zipping it open, looking in for less than a second and then zipping it back up. Do your best to forget about this crazy place and go _have some fun_.
_Bye, Annie,_ Laura and Dori say in a girl-like harmony. Laura winks. Anh-thu smiles, waves to everybody and then walks out into the quiet mall, alone.
All around her, store doors are being shut and locked for the night. Neon signs are being flipped off. Trash bags are being taken out and tossed. Vendors are breaking down their stands and wheeling them away. Security guards, manning the mall exits, fumble with their keys and nod to all the familiar faces of mall employees who head for the parking garage and the freedom of their cars. Anh-thu smiles at one particular guard, Manny, the old Mexican man she always passes on her way out the Colorado exit.
Outside she looks around for Sticky, but there's no sign of him. She leans against the wall and checks her watch: 9:10. He's late, she thinks. But he's always late. It's possible he's never once been on time in his life. And besides, she thinks, how appropriate that he be late today, after she has just discovered that late is all that she is. Ten days late. Nothing more. There will be no big talk tonight. No discussion about the future. No weighty decision to make. Everything is still the same, and she's relieved. She and Sticky are just two high school kids going together.
All of the nervousness Anh-thu has felt for the past couple days has left her exhausted. She hopes Sticky doesn't want to do anything major. Something that might require her having more energy than she has. Mellow sounds better right about now. Some fish tacos at the park or a hot chocolate on Abbot Kinney Boulevard. Something like that, she thinks. Maybe a slice of pizza on the Santa Monica Pier, where they can sit and watch the tourists spinning around on the Ferris wheel. She'd sit and hear about Sticky's day. About the crazy guys at the gym. Dallas, Dreadlock Man, Old-man Perkins, Dante, New York, Crazy Ray. Sticky always comes back to her with some sort of story involving one of those guys, and she likes listening.
A rattling sound coming from behind Anh-thu makes her turn around quick, but it's only Manny shutting and locking the glass doors to the mall. He waves, and she waves back.
A group of high school guys in a red Mustang stare at Anh-thu while they wait for the light to turn green. One of them points and the rest of them laugh. It's late and Anhthu's starting to feel a little anxious. It's not the guys, though. Guys like that are everywhere. It's more the dress. For the most part, Anh-thu's a jeans-and-sweatshirt kind of girl. But she decided to wear a dress tonight. For Sticky. She peeks down at her watch again: 9:25. Still no sign of him.
There's a cool breeze blowing in from the ocean, and the seaweed smell makes Anh-thu feel calm. She's always loved the smell of the ocean. The breeze kicks up a little and blows her hair into her face. She grabs a rubber band from inside her bag.
Anh-thu looks down at her watch: 9:30. Still no sign of Sticky.
After the Good
Samaritans leave, the two who found Sticky and fired off the 9-1-1 call on a cell phone, followed the ambulance to Emergency in a dinged-up Chevy Cavalier; after the cop leaves, taking his twenty-two unanswered questions with him, his breath like the bottom of a coffee mug; after the tall Indian doctor is out the door, the man who came in holding an X-ray and offering heavily accented words of encouragement, who proceeded to stick needles and tweezer out metal shards and tug and blot and stitch, who disclosed in the breathy voice of a woman that the situation would remarkably be devoid of any long-term complications because of where the bullet entered the hand (this diagnosis meaning absolutely nothing to Sticky); after Georgia hands off the necessary paperwork and runs out the door, having spent her entire fifteen-minute visit listing all the reasons she couldn't stay, never once looking down at her foster boy laid up in a hospital bed; after three different nurses, two ladies and a dude, walk out the door, promising to check back within the hour; after everybody has fled the scene, gone on to other parts of their lives having fulfilled their role in the room, it's just Sticky and Anh-thu left, the two of them sitting under a suffocating silence that has spread through the room like a gas.
Anh-thu sits on the edge of a chair next to Sticky's bed. She has tears in her eyes. Puffy cheeks. A cottony mouth. Every question she could think of to ask she has asked. But Sticky's hand is still a mystery. He's been shot. She knows that. But why? And how? And when? The problem is, Sticky isn't talking. He hasn't said a word since she's arrived. Won't even look anybody in the eye. His face is a blank, like the simple oval outline of a face in some kid's coloring book, precrayons.
Anh-thu is running her fingers through Sticky's hair, but he isn't there. He's absent. He's missing. He's an empty vessel. This is his way of dealing with the hurt, she thinks. It's not personal. This is a defense mechanism. This is shock. This is post-traumatic stress disorder. There's a lump in her throat as she runs through terms learned in psych class, trying to make sense of it all. Today is her sixteenth birthday. It's supposed to be a good day. A rite of passage. How did it end like this? She looks at Sticky again—sitting propped up in his hospital bed, hoop shoes still stuck to his feet, white wife-beater still wet with sweat, right hand wrapped in gauze and set in a sling above his chest—and it seems impossible to her how much she's hurting right along with him.
It's two in the morning. A sterile black and white clock counts the seconds. A small fan spins a subtle breeze through the room from left to right and back again. There are a couple of laminated posters above the door that warn whoever's paying attention about the harmful effects of secondhand smoke.
Anh-thu takes Sticky's good hand, the left one, and lays her head on his forearm. The image of Sticky being held up at gunpoint flashes through her head again, but she manages to push it away this time. No use speculating. He'll explain it all soon enough. She wipes her eyes on Sticky's forearm, picks her head back up and looks in his face, says: _I'll take_ _care of you, Sticky. You know I will_. The words coming out thin and hollow.
When Sticky never showed up at the place they'd planned to meet, Anh-thu panicked and called everybody she could think of to call. She called Sticky's house, her dad's work, her brothers, the high school gym, Lincoln Rec, the police, and finally all the local hospitals. When the new UCLA hospital in Santa Monica confirmed that Sticky had indeed been brought into Emergency, she flagged a cab and told the driver to get her there as fast as he could. When they pulled up she paid the guy, rushed to the front desk, asked for Sticky's room number, sprinted through the halls and up the stairs and around the first corner and pushed through the door, where she found Sticky lying in a hospital bed staring at the ceiling. She wrapped her arms around him and started crying and asked question after question and begged for somebody to tell her what had happened. And even after learning that he would be OK, that he was lucky, that he would make a full recovery, she still felt an overwhelming pain in her chest. She'd never seen Sticky that way. Hurt and helpless. Vulnerable. With a complete emptiness in his eyes. And she lay there on him for quite some time, squeezing his shoulders, trying to ignore the fact that he wasn't talking.
Anh-thu stares at Sticky and recalls the earlier exchange between nurses about the basketball they'd pulled out of his bag. The guy nurse asked what _7 FLOW_ stood for. And when Sticky didn't answer, one of the lady nurses who had his file open cited the name of Sticky's old foster care pad: Foster Living of the West. House number seven. She said it must be short for that, and the guy nurse nodded his head in agreement.
Anh-thu's ears perked up when she heard that information. Sticky had told her it was a gift from his mom. Something he found under the Christmas tree way back when he was just a kid. And sitting here now, she wonders how well she even knows Sticky.
It's three in the morning. The TV in the upper corner of the room is on without sound. Sticky's right hand is a constant throbbing pain, one that crawls up his arm and into his shoulder, settles in the base of his neck. On the other side of the curtain, an old man's snoring gets louder and louder until he almost chokes on his own breath and wakes up. The springs in his bed crunch and moan as he rolls over and starts the process again.
A nurse walks into the room and glances at Anh-thu sleeping with her head on Sticky's bed, her hand on his thigh. She smiles at Sticky, tiptoes past his bed and around the other side of the curtain. She pulls the old man out of his snore by telling him something in a soft voice. He answers in a slur. In a few seconds she comes back around the curtain, smiles again and leaves the room.
Sticky listens to Anh-thu's breaths get slower and deeper. Feels her heavy hand slip off his leg.
It's four in the morning and Sticky is completely alone. The entire hospital is asleep. Anh-thu's asleep. The old man on the other side of the curtain is asleep. The TV, having turned into bars of color, is asleep. Sticky finally looks down at his right hand. At this point he has to. Everybody else is out of the picture, and now he can try to figure things out.
He reaches up with his left hand and pokes at the gauze. He traces the outline of his right hand and presses harder. A few sparks of sharp pain shoot up through his arm. It hurts. And he can't even move a finger when he goes to clench a fist. There's nothing.
He looks back at the wall, his tired heart sagging in his chest, and lets his left hand drop back into his lap.
And for the first time, Sticky thinks maybe all that magic in his right-hand fingertips might be gone. Stolen away when he put his shooting hand up to the gun. And if that's the case, maybe his whole life is gone too. Who the hell is he without basketball? He's nobody. Without basketball maybe his life is completely meaningless.
Sticky's head is dancing from the morphine the nurses have running into his veins. It's tough to focus. The room is fuzzy and dark, aside from the dull shine of the overhead lightbulb. And there's a relentless warm hum inside his head.
For a second he forgets where he is. He's lying on a patch of grass outside Sanwa Bank, and the light above him is the moon. He's dreaming about the guys in the gym and the letter in his bag. He's lying on his back in the park under the sun. He can smell the fruit shampoo in Annie's hair as they drift in and out of sleep. He's walking home late at night and a woman in heels is asking him if he knows how to kiss a woman's hand. He's walking home from the gym after playing ball and it's raining. But it feels nice, like a hundred fingertips touching soft as lips. And he's happy because he played so well. Old-man Perkins told him it was like he was operating at a different speed than everybody else. But he said he was graceful, too, like a dancer. He's parked in Dante's car outside Georgia's house saying his goodbyes and just as he's about to get out Dante reaches for his wrist and tells him: _The only reason I come down on you so hard, Stick,_ _is because I care. I care about you like I do my own sons. My_ _own flesh and blood_. And Sticky's nodding and slapping Dante's hand and walking away, but all the while something's growing inside his chest. Something meaningful, important. This strange sense of belonging that he's spent his entire life without. He's walking toward the house thinking about what that means: Dante caring about him. He doesn't have to. And this makes him feel bigger. Much bigger. He's holding himself completely upright and he feels as big as the biggest big man to ever post somebody up on a Lincoln Rec low block. But then he's curled up along the three-point line on a piece of cardboard, starving. And there's a throbbing pain in his hand. And the humming in his head is the sound of all the other homeless waking up beside him. They're all mumbling street mumbles and he realizes he's mumbling too. He's one of them. And when he looks down at his shooting hand all of his fingers are missing. In fact, his entire hand's been amputated. . . .
He wakes up suddenly and finds his hand in the sling. He remembers where he is. Who he is. He's in a hospital bed cause he messed up. And he's been hurt. And when he remembers everything that has happened his stomach drops and he has to swallow down hard on the lump growing in his throat. He has to squeeze his eyes tight to keep everything inside him locked behind closed doors.
It's five in the morning and Sticky looks down at Anh-thu. Her eyes closed, lips barely apart. Breaths long and drawn out. Heavy. A few strands of her long black hair are in her face. He reaches out with his left hand and moves the hair away. She looks so pretty when she's sleeping, he thinks. He studies her face and notices the contrast—her dark skin against his milky white skin.
When he feels a sharp spasm of pain rip through his right hand, he looks at the gauze and wishes he could take it all back. What he's done. He's made a mistake. He wishes he could go back and erase it. Do it over. He would leave the steak knife in the drawer. He would leave everybody alone. Buy Annie the bear and take her to the pier. That's all she wanted. But when he feels the tears coming he does his best to stop thinking altogether. He swallows down hard on his hurt. Because he can't go back. He swallows it like poison, like he always does, and he stares at the bare wall in front of him.
He has to get away from it.
But this is when all the fragile walls finally come crumbling down around Sticky. He's lying in a hospital bed, his throbbing hand in a sling, and everything splits open. Cracks in two. Tears apart. He can no longer pretend he's someone else. He has to give that up, shed his cool. He lets ten years' worth of pretending he doesn't exist come pouring out of his eyes. Streams of heavy tears rush down his face and he refuses to wipe them away. He's been shot in the hand and he's scared he'll never play ball again.
And right then Annie raises her sleepy head. The second she opens her eyes, though, Sticky closes his. He pretends to be asleep. And she kisses his cheek and shifts around in her chair. When she lays her head back on the bed and falls asleep again, Sticky cries even harder. Everything he's had stored up in his chest comes rushing out through his swollen eyes. Annie is still right here with him. He hasn't said a word to her all night, he won't even look at her, but she refuses to give up on him.
And then Sticky goes back to the moment his whole life changed. When he was in the window spitting into the bed of the truck and his mom was in the bathroom screaming his name. _Sticky!_ Something in the way he's crying so hard triggers the images to come flooding back.
He finally pulls himself away from the window. He walks toward the bathroom. He steps through the door and there's his mom. She's slumped over in the tub. Cloudy red water spilling onto the floor and still running. He cups his hands over his ears and stares. His face scrunches up and then goes normal again. It feels like he's choking. He's so small. He's just a boy. He picks Baby's head up, tries to balance it straight on her neck, only to have it slump back forward or to the side. He looks at her wrists. He turns the water off, pulls the plug and watches the red start slowly sucking down the drain. He gets up. Zips around the bathroom: to the sink, the toilet, opens the medicine cabinet and brushes all the prescription bottles off the shelves. Bangs his head against the wall. Races back to the tub and puts a hand on Baby's shoulder. Shakes her. Nothing. Shakes her. Nothing. Shakes her. Nothing. Then he falls to a sitting position in the middle of the broken-up tile next to the tub and rocks himself. Back and forth and back and forth and back and forth and back and forth. He covers his ears with his hands and rocks himself. Back and forth and back and forth.
Two cops bust through the front door, yelling.
Sticky continues rocking himself.
The cops tear around the house, yelling: _Anybody here?_ _Anybody here? A neighbor reported screaming!_
They find their way into the bathroom and swing open the door. _Oh, no,_ one cop says under his breath. The other steps over Sticky and reaches for Baby's arm. The water fully drained now. The bottom of the tub pink. They check her for a pulse. Check her neck. They take out special tools and check again.
Other cops show up. One scribbles things down on a pad of paper. _What's your name, son?_ he wants to know. _What's_ _your name?_
They pull Baby's naked body out of the tub and lay her on a stretcher.
_What's your name, son?_
There are five, ten, fifteen blue suits with badges staring down at Sticky. At Baby. They lay a blanket on top of her. They pull the blanket up over her face. They wheel her out of the bathroom. They wheel her away from him.
Sticky holds his ears tight and rocks himself. Back and forth and back and forth and back and forth and back and forth.
A cop lifts a steak knife out of the tub with two fingers and places it in a plastic bag.
_What's your name, son?_
They take fingerprints from the sink, from the tub, from the broken cabinet. They say things to each other in low voices and then one of them makes a call on his radio.
The cop with the pad reaches down and puts his hand on Sticky's shoulder. _What's your name, son?_
Sticky continues rocking, but he brings up his scrunched-up face. He's crying. He looks up at this cop, this blue suit, this face now burned into his mind and tells him through his tears:
_My name's Sticky._
The hospital room is dark. The old guy is snoring on his side of the curtain. The fan is humming. And Sticky's crying. He's crying for the first time since that night, and his entire body is shaking. Baby's gone. But he's starting to feel better too. He's starting to feel like a real person. It actually feels good to cry. It's like he can actually feel himself settling into his own body. He feels the beating of his own heart. He no longer wants to hide, pretend he doesn't exist. He is here. In this hospital bed. Breathing. This is who he is. Sticky Reichard.
Travis Reichard.
He stares through his tears at Anh-thu's blurry fingers and he remembers something the old Mexican director said. That maybe it wasn't his fault that people didn't want to keep him. And he thinks about Baby. His mom. Maybe _that_ wasn't his fault either. Maybe she had to go to heaven to get better. And the director said what was special about him wasn't the way he played basketball but that he was a good person. And he thinks that if a girl like Annie cares about him, if she's willing to sit next to him all night even after he's messed up so bad, then maybe the director was right. Maybe he really _is_ a good person.
Sticky reaches out and puts his hand on top of Annie's. The tears are still coming and his entire body is trembling, but this feels good to Sticky. Reaching out. Letting go. This feels more than good, it feels like life.
Dreadlock Man
rolls right into Lincoln Rec on his ten-speed, dreads pulled back and wrapped in rope, and holds a hand out for the pass. He coasts onto the court, where Slim and Heavy are shooting free throws, says: _Yo, man, lemme get_ _a shot_.
Slim reluctantly delivers a bounce pass, which Dreadlock Man scoops up on the flyby. He circles around at half-court, heads back toward the bucket and tosses up an awkward one-handed fifteen-footer on the move.
_Peanut Butter!_
The ball bangs violently off the side of the rim, caroms off toward the homeless court and smacks Crazy Ray in the back of the head. A stunned Ray keeps perfectly still for a few seconds, refusing to turn around, and then cautiously lays his head back down on his piece of cardboard and pulls a blanket over his face.
Dallas, stretching out near the far sideline, springs up off the hardwood and retrieves the loose ball. He tucks it under his arm, taps his dad on the shoulder to see if he's okay. Failing to get any kind of response, he shrugs his shoulders. He races toward the basket on the dribble, slows slightly to time his jump, rises up off one foot and rattles home a questionable dunk. _Still got it,_ he says to Slim and Heavy on his way back down to the ground.
_Come on, Dallas,_ Slim says. _That wasn't no dunk. That_ _was a power layup_.
Dreadlock Man bikes back on the court holding his hand out again. _Yo, Dallas,_ he says, _let your boy get one_ _more shot_.
Old-man Perkins, walking into the gym with Johnson, yells out, _Yo, man, don't give that cat nothin_.
He says, _Dreadlock, you couldn't score if you was the only_ _man in the gym. What you think you gonna do on a bicycle_.
Johnson laughs, says, _And Dallas, man, you couldn't dunk_ _if Jimmy hooked you up with a pogo stick_.
Trey rolls into the gym with New York. They both sling their bags into the bleachers, pull their hoop shoes out and start lacing up. Big Mac rumbles in sipping a Red Bull. Boo cruises in. Hawk. Rob. It's another Saturday morning and Lincoln Rec is slowly filling up. Everybody's stretching out along the sideline and shooting their mouths back and forth.
Jimmy steps out of his office, folds his arms and shakes his head, watching Dreadlock Man, who's now riding laps around the perimeter of the gym. Jimmy puts two fingers in his mouth and lets go of a piercing whistle. When Dreadlock Man looks up, spots Jimmy, he quickly hops off his bike and walks it over to the bleachers, where he locks up.
Old-man Perkins yells out: _Let's shoot em up! Get this_ _thing goin!_
_Yeah,_ Johnson says. _Can't be here all day. I got city business to attend to_.
But before anybody can make a move for the free-throw line, Sticky walks through the Lincoln Rec doors with Sin in tow. He has his duct-taped headphones around his neck and a UCLA cap pulled slightly crooked.
It's Sticky's first day back at Lincoln Rec after a mini hiatus. He's fresh off the all-star summer camp circuit—for the last three weeks he's been up and down the West Coast with every other high school hotshot, playing in front of coaches and scouts from every big-time college in the country.
Sticky gives daps to some of the guys, pushes his bag under the bleachers.
Sin quietly follows Sticky's lead.
Dante comes strutting into the gym with his bag on his shoulder and goes right up to Sticky. He reaches for Sticky's right hand and holds it up to get a good look. Sticky's scar looks like a purple spider now, three months after the incident—there's a thick blotch of shiny new tissue growing over his wound with several leg-looking scars underneath. Sticky watches Dante's face as he studies his hand.
_Lookin a lot better, boy._
Sticky nods.
_You let everybody know you the man at them camps?_
_I held it down,_ Sticky says.
_That's what I like to hear_. Dante turns to the rest of the guys and says, _Let's go, y'all. Let the runs begin_.
A few of the guys head for the free-throw line and shoot for captains. Trey has one go in and out. Slim knocks his shot down. Dreadlock Man barely grazes iron, but Big Mac rattles home a line drive and he and Trey immediately start picking their squads. As usual, Dante is the first to go, but today Sticky is selected second. Secretly listening for his draft position, Sticky smiles inside. He's moving on up the food chain.
While the rest of the teams are picked, Dallas says, _I see_ _you brought your boy with you today, Stick_.
Sticky pulls his cap and headphones off, reaches under the bleachers for his bag. _Yeah, this is my buddy Sin_.
Sin nods his head at the regulars.
_Sin?_ Johnson says.
_Yeah,_ Sin says. _My pops nicknamed me that when—_
S-I-N? Dallas interrupts. He turns to Old-man Perkins, says, _Yo, that shit's kind of blasphemous, ain't it?_
_How a youngster expect to be right with the good Lord,_ Perkins says, _when he walkin around callin himself Sin?_
New York, Hawk and Big Mac join the group now circling around a noticeably nervous Sin.
_Nah, first-timer,_ Old-man Perkins adds, _that shit ain't_ _gonna fly around here_.
_It'd be a sin on our part,_ Johnson points out, _to call the_ _boy Sin._ Everybody nods their head with Johnson's logic.
Sin looks over at Sticky as the guys continue to light into him about his name. Sticky laughs under his breath. He thinks about how far he's come since _his_ first day. How much he's learned. And the guys see him as one of their own now. Dallas said it best just before the end of school. They were all sitting up in the bleachers after a full day's run and out of nowhere he turned to the rest of the guys and said: _Yo, I don't_ _know about y'all, but when I look at Stick now, I don't even see_ _white. I see family_. All the guys nodded their heads when Dallas said that.
Sticky zips open his bag and stashes his headphones and cap inside. He spots the letter from UCLA he received in the mail yesterday and pulls it out. He has the urge to run it over to Dante, tell him: See, D? This is how much I was lettin them know at them camps. But he'll mention it later. When they get a minute alone. Right now all he wants to do is get back on the court. His home court. He wants to run up and down and hear the familiar voices of the guys. Doesn't matter how important all those camps were for his future, there's still nothing that beats a Saturday at Lincoln Rec.
Sticky slips the letter back into his bag and pushes his bag under the bleachers. He pulls the bag out and pushes it under. Does it three more times and then heads out onto the court, where everybody is matching up. Rob motions Dante over to Sticky, tells him, _Yo, you got white boy, D. I ain't feel_ _like chasin his ass around all day_.
Dallas smacks the ball and says over everybody, _Ball's in!_ He checks the ball into New York. New York passes to Boo, who immediately swings it over to Sticky on the wing. Sticky holds the ball for a second and stares into Dante's eyes. A hundred possibilities flash through his head. He sees everybody on the court and knows exactly how they'll react to whatever he does. He knows they're watching. Waiting. He jab-steps a couple times. Smiles as Dante goes back on his heels.
Then he makes his move.
Acknowledgments
I'd like to thank the following people for helping make this novel possible: my patient professors, Harold Jaffe and Lydia Yuknavitch; my creative confidants, Spencer Figueroa, Rob Jones, and Brin Hill; my support system, Sean Kim, Melissa Marconi, Tracee Lee, Matt Van Buren, and Dan Hooker; my incredibly talented editor, Krista Marino; my heart, all the de la Peñas; and my best friend in the world, Kristin Foote.
Published by Delacorte Press
an imprint of Random House Children's Books
a division of Random House, Inc.
New York
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.
All rights reserved.
Delacorte Press and colophon are registered trademarks of Random House, Inc.
www.randomhouse.com/teens
Educators and librarians, for a variety of teaching tools, visit us at
www.randomhouse.com/teachers
First Trade Paperback Edition
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eISBN: 978-0-307-43316-9
v3.0
| {
"redpajama_set_name": "RedPajamaBook"
} | 9,902 |
import datetime
YEARS = [(x, x) for x in reversed(xrange(1926,datetime.date.today().year +1))]
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,147 |
\section{Introduction}
Chirality is a key signature of life since many biological molecules owe their specific function to their molecular shape and symmetry. Examples range from molecules possessing at least one carbon atom with four different substituents to more complicated helical molecular assemblies such as the DNA double helix.
Apart from the microscopic domain, chirality is expressed abundantly in the macroscopic world, for example in the left- and right-handed helical conformation of snail shells, the spiral patterns of leaves on a plant stem (phyllotaxis), and other biological and inert structures.
The structure of condensed phases are naturally affected by the chirality of the building blocks. For example, liquid crystals composed of chiral constituents may form extraordinary structures, ranging from cholesterics, chiral smectics and blue phases \cite{gennes-prost,hiltrop}. The particular meso-structure imparted by chiral correlations on the micro-scale, such as a helical director field in cholesterics or a cubic arrangement of defect lines in blue phases, endow these materials with extraordinary optical and mechanical properties. Important applications of chiral mesostructures reside in domains such as non-linear optics \cite{rodrigues2014}, photonics \cite{shibaev_kopp2003} and stereoselective catalysis \cite{palmer2008}.
Helical structures may also arise in systems of achiral particles. Illustrious examples are spontaneous chiral symmetric breaking of homopolymers composed of attractive non-chiral monomers \cite{magee2006}, the formation of Bernal spiral clusters in patchy colloidal particles \cite{campbell2005, morgan2013} and the self-assembly of achiral magnetic nanoclusters into helical superstructures \cite{singh2014}. Helical meso-structures also emerge from packing colloidal particles into cylindrical tubes at high pressure \cite{snir_kamien, oguz_messina} where transitions between different cluster morphologies may occur upon varying the packing load \cite{lohr_yodh}. In fact, coiled, toroidal and helical architectures seem to be a recurring feature in systems of (block co-)polymers \cite{sevink_jcp2001,yu_confinedpolymers2008} and other soft matter compounds confined in nanopores \cite{wu_naturematerials2004,alcoutlabi2005}.
Although the formation of the helical meso-structure occurs by means of spontaneous self-assembly, there should be no preference for a particular helix sense (left- or right-handed) if the constituent molecules are achiral. Spontaneous breaking of the chiral symmetry usually happens at the initial stage of the self-assembly process by some combination of fluctuations (of e.g. thermal nature) favoring one chiral symmetry over the other. The local bias toward a particular chiral symmetry then proliferates to mesoscopic length scales by means of some self-sorting and amplification mechanism which secures the uniform mesoscale symmetry of the assembly \cite{dressel2014,garcia_jacs2012}.
In this work we investigate the possibility of helical meso-structures in columnar assemblies of discotic particles with non-chiral pair interactions. Columnar meso-phases are characterized by a two-dimensional crystalline arrangement of columns, each consisting of a quasi one-dimensional stack of particles with a disordered internal structure. In discotic liquid crystals supramolecular chirality manifests itself along the columnar direction and can be imparted by chiral intramolecular interactions \cite{hirschberg2000}, or by non-chiral ones via some type of directionality in the face-to-face interactions (so-called $\pi$-stacking), or a helical organization of the orientational or translation degrees of freedom of the discotic particles along the backbone of the stack \cite{vera_columnar}.
The focus of this paper is to explore the latter type of chirality, namely the one expressed by an effective helical shape of the columnar stack. To this end we devise a microscopic model and use simple mean-field theory to identify generic expressions for the free energy associated with weak conformational changes from a linear stack to a helically buckled assembly of stacked particles subject to a confining external field.
We subsequently apply the theory to columnar meso-phases of charged discotic colloidal particles, a common building block for many natural \cite{kaolin} and synthetic clays \cite{Brown98,vanderkooijcolumnar, davidson-overview}. The dominant particle interactions stem from electrostatic repulsions and the stabilization of columnar and other liquid crystal structures can be rationalized on entropic grounds alone \cite{onsager1949, frenkel_nature88}. Particle density, rather than temperature therefore constitutes the chief thermodynamic parameter in these lyotropic systems.
In order to establish a link between the one-dimensional fluid and the three-dimensional columnar meso-phase, we construct a self-consistent mean-field theory by invoking a simple Lennard-Jones cell-theory for columnar liquid crystals \cite{lennardjones1937,kramerherzfeld-pre2000,graf1999} and identifying the externally imposed confining potential acting on the stack with an effective internal cell potential imparted by adjacent stacks in the columnar lattice. Expressions for the typical amplitude and pitch of the helical undulations as well as the corresponding free energy differences between the various buckling modes can then be readily established in relation to the particles density and electro-chemical properties of the individual disks via the inter-disk potential. We believe that the results of this study provide an essential first step in assessing collective conformational fluctuations and the possibility of spontaneous chiral symmetry breaking in lyotropic columnar assemblies.
\section{Model}
Let us consider a one-dimensional (1D) fluid of $N$ discotic particles at constant temperature $T$ confined to a cylindrical potential trap $V_{c}(r)$. We focus on dense stacks where the average interparticle distance is very small so that the disks are assumed to be perfectly aligned along the $ {\bf \hat{z}} $ direction of the laboratory frame (see \fig{fig1}). Their centres-of-mass are constrained to lie on a one-dimensional contour parameterized by $ {\bf s} (l) = \{ s_{x}(l), s_{y} (l), s_{z}(l) \} $ with $0 < l < \ell$. The contour length is given by
$ \ell_{c} = \int_{0}^{\ell} dl \| \partial_{l} {\bf s} \| $
and $\ell_{c} > \ell$ for any contour except for a trivial linear one $ {\bf s} (l) = \{0,0,l \}$ ($ \ell_{c} = \ell$) which corresponds to an `unbuckled' system. Under certain conditions the linearly stacked column may buckle into e.g. a helical shape.
\begin{figure}
\includegraphics[width=0.8 \columnwidth]{fig1}
\caption{A columnar fluid of stacked parallel soft disks (in red) confined in a cylindrical parabolic trap with potential $V_{c}(r)$. The backbone of the stacks may adopt a linear (left) or buckled (right) configuration. The typical interaction range between adjacent disks is quantified by a (Debye) screening length $\kappa^{-1}$.}
\label{fig1}
\end{figure}
The central idea of this study is to investigate the optimal buckling shape of the stacked discotic fluid in relation to various control parameters such as particle density, the choice of the pair interaction, and the external potential. The equilibrium shape of the buckled state is not known {\em a priori} but can be established self-consistently using a simple variational approach based on classical density functional theory (DFT). To this end we define the Helmholtz free energy $F$ of a 3D fluid with one-body density $\rho( {\bf r} )$ defined as the ensemble average $\rho( {\bf r} ) = \langle \hat{\rho}( {\bf r} ) \rangle$ of the microscopic density
$\hat{\rho}( {\bf r} ) = \sum_{i=1}^{N} \delta ( {\bf r} - {\bf r} _{i}) $ with normalization $N$. The Helmholtz free energy consists of an ideal, external field and excess contribution, and is formally given by:
\begin{align}
F[\rho] &= k_{B}T \int d {\bf r} \rho ( {\bf r} ) \left ( \ln \Lambda^{3} \rho( {\bf r} ) -1 \right ) \nonumber \\
&+ \int d {\bf r} \rho ( {\bf r} ) V_{c} ( {\bf r} ) + \frac{1}{2}\int d {\bf r} \rho( {\bf r} ) \int d {\bf r} ^{\prime} \rho( {\bf r} ^{\prime}) \Phi( {\bf r} - {\bf r} ^{\prime} ),
\label{hform}
\end{align}
with $k_{B}T$ the thermal energy ($k_{B}$ denotes Boltzmann's constant) and $\Lambda$ the de Broglie length of each particle. While the first two contributions are exact, the excess term is not. The disk-disk correlations are approximated by a simple {\em mean-field} term featuring the pair potential $\Phi( {\bf r} )$ between parallel disks. This potential has an intricate directional dependence through the centre-of-mass distance vector $ {\bf r} $ which we will specify later on. Let us assume the disk centre-of-masses to be distributed uniformly along the strictly one-dimensional contour $ {\bf s} (l)$ so that we may parameterize $\rho( {\bf r} ) = \int_{0}^{\ell} d l \| \partial_{l} {\bf s} \| \delta ( {\bf r} - {\bf s} (l))$ in terms of a contour-dependent line density $\bar{\rho} = N/\ell_{c}$. Substituting the parameterization into the free energy yields:
\begin{align}
F &= -TS_{\text{id}} + \bar{\rho} \int_{0}^{\ell} d l \| \partial_{l} {\bf s} \| V_{c} ( {\bf s} (l)) \nonumber \\
& + \frac{\bar{\rho}^{2}}{2}\int_{0}^{\ell} d l \| \partial_{l} {\bf s} \| \int_{0}^{\ell} d l^{\prime} \| \partial_{l^{\prime}} {\bf s} \| \Phi( {\bf s} (l) - {\bf s} (l^{\prime}) ).
\end{align}
Let us consider a helical parameterization $ {\bf s} (l) = \{ R \cos ql, R \sin ql, l \} $ in terms of radius $R$ and pitch $q$. The contour length of a helix is $ \ell_{c} = \ell \sqrt{1+ (qR)^{2}}$. Since a helix has constant curvature, $ \| \partial_{l} {\bf s} \|$ is independent of $l$ and the deformation free energy per particle simplifies to:
\begin{align}
\frac{F}{N} &= \frac{-TS_{\text{id}}}{N} + V_{c} (R) + \frac{\rho}{2}\int_{0}^{\ell} d (l - l^{\prime}) \Phi( {\bf s} (l) - {\bf s} (l^{\prime}) ),
\end{align}
now featuring a linear density $\rho = N/\ell$ independent of the underlying contour shape. We emphasize that the confining external potential is symmetric in the cylindrical reference frame of the helix and only depends on the lateral extent of the helix via the radius $R$.
Due to the one-dimensional constraint imposed on the particle contour, the ideal entropy $S_{\text{id}}$ cannot immediately be reproduced from its 3D phase-space definition given by the first contribution in \eq{hform}. In fact, substitution of the delta-distribution for the one-body density into \eq{hform} would lead to a divergence of the free energy. Nevertheless, a plausible expression for the ideal entropy compatible with 3D translational motion in real systems can be recovered by factorizing the translational degrees of freedom of each disk into a one-dimensional contribution representing particle motion along the helical contour and a bi-dimensional one originating from infinitesimally small excursions the disk centres-of-mass are allowed to make across the plane normal to the contour. This will become apparent later on in this text. We will proceed by specifying a suitable form for the inter-disk potential $\Phi$.
\section{Generalized Yukawa potential for charged disks}
A natural question that arises is what type of pair potential could give rise to stacked fluids with a stable helical meso-structure? A promising candidate is the effective electrostatic interaction between charged disks of diameter $D$ which in the far-field limit takes the form of an effective Yukawa form \cite{trizac-weis}:
\begin{equation}
\Phi ( {\bf r} _{12}, {\bf \hat{u}}_{1} , {\bf \hat{u}}_{2} ) = u_{0} \xi( \theta_{1} ; \kappa D) \xi(\theta_{2} ; \kappa D) \frac{\exp ( - \kappa r_{12})}{r_{12}},
\label{yukpotential}
\end{equation}
with amplitude $u_{0} = Z^{2} \lambda_{B}$ in terms of the total number of elementary charges $Z$ per disk, the Debye screening constant $\kappa$ and $\lambda_{B}$ the Bjerrum length. The angles $\theta$ featuring in the prefactor $\xi$ denote the disk normal orientation relative to the centre-of-mass distance vector $ {\bf r} _{12}$ via $ \cos \theta = {\bf \hat{z}} \cdot \hat{ {\bf r} }_{12} $. The angular prefactor reads in explicit form:
\begin{equation}
\xi(\theta; \kappa D) = 2 \frac{ I_{1} \left (\frac{1}{2} \kappa D \sin \theta \right ) } { \frac{1}{2} \kappa D \sin \theta },
\label{yuk}
\end{equation}
with $I_{1}(x)$ a modified Bessel function. The Yukawa potential for discotic colloids favors stacked pair configurations over co-planar ones \cite{trizac-weis}. As a consequence, the inherently electrostatic anisometry drives self-assembly into stacked meso-phases, including columnar structures \cite{jabbari2013}. Conformational changes from a linear to a buckled state can be anticipated from a trade-off between the external and excess free energy. While the external free energy associated with a confining trap potential increases upon buckling, the cost is offset by a simultaneous reduction of the excess free energy due to the fact that the centres-of-mass between adjacent disks will, on average, be further apart in an undulated stack. This we will explore in detail in the next Section.
\section{Free energy of a helically deformed columnar fluid}
Let us now attempt to express the directional Yukawa potential $\Phi$ (\eq{yukpotential}) in terms of the helical vector $ {\bf s }(l)$. Since the disks normals are all pointing along the $z$-axis, the only relevant coordinate is the centre-of-mass distance between a pair of particles located at position $l$ and $l^{\prime}$ along the helical contour, i.e., $\Phi(\Delta r) =\Phi( {\bf s }(l) - {\bf s }(l^{\prime}))$. Without loss of generality, the centre-of-mass distance ${\bf \Delta r} = {\bf r}^{\prime} - {\bf r} $ between two particles located on a helical contour can be expressed in the cylindrical coordinate frame of one of the particles. Straightforward algebra then leads to:
\begin{equation}
{\bf \Delta r} = R (\cos (q \Delta l) -1) {\bf \hat{r}} + R \sin (q \Delta l ) \hat{\bm \varphi} + \Delta l {\bf \hat{z}} ,
\label{delr}
\end{equation}
with $\Delta l = l^{\prime} - l$. In order to simplify the orientation-dependent prefactor of the Yukawa potential we Taylor expand the Bessel function \eq{yuk} for small $\theta$. Some manipulation then leads to:
\begin{equation}
\Phi (\Delta {\bf s} ) = \tilde{u}_{0}(\cos \theta) \frac{\exp ( - \kappa | \Delta {\bf r} | )}{| \Delta {\bf r} |},
\label{phipar}
\end{equation}
where the amplitude depends on the angle $\theta$ between the centre-of-mass distance and the disk orientation unit vectors aligned along helix axis:
\begin{equation}
\tilde{u}_{0}(\cos \theta) = u_{0} \left( 1+ \frac{(\kappa D)^{2}}{16} (1 - \cos \theta) \right )^{2},
\end{equation}
Up to leading order in $q$ the dot product $\cos \theta$ between the centre-of-mass distance and disk orientations can be written as:
\begin{align}
\cos \theta & = \frac{| s_{z}(l) - s_{z}(l^{\prime}) | }{\Delta r} \approx \frac{1}{\varepsilon},
\label{cost}
\end{align}
in terms of the contour length $\varepsilon = \sqrt{1 + (qR)^{2}}$. The amplitude then becomes:
\begin{align}
\tilde{u}_{0} (\varepsilon) = u_{0} \left( 1+ \frac{(\kappa D)^{2}}{16} (1 - \varepsilon^{-1}) \right )^{2}.
\label{u0}
\end{align}
The effective Yukawa amplitude increases with $q$ and $R$ and thus clearly disfavors buckling. With this the deformation free energy takes the following form:
\begin{align}
\frac{F}{N} &= \frac{-TS_{\text{id}}}{N} + V_{c}(R) + \frac{\rho }{2} \tilde{u}_{0} (\varepsilon ) \int_{0}^{\ell} d \Delta l \frac{\exp [ - \kappa \Delta r ] }{ \Delta r },
\end{align}
with the inter-disk distance $\Delta r$ depending implicitly on the contour parameters $\Delta l$ via \eq{delr}. The last term confronts us with a double contour integral that does not converge for unbounded potentials since the disk centre-of-masses are forced to reside on a one-dimensional contour. The divergence can be lifted by allowing the disks to perform small spatial fluctuations $\delta {\bf r} _{\perp}$ in the $\left \{ {\bf \hat{r}}, {\hat{\bm \varphi}} \right \} $-plane perpendicular to the main helical direction $ {\bf \hat{z}} $. The centre-of-mass distance then takes the form:
\begin{equation}
{\bf \Delta r} = \delta {\bf r}_{\perp} + R (\cos (q \Delta l ) -1) {\bf \hat{r}} + R \sin ( q \Delta l ) \hat{\bm \varphi} + \Delta l {\bf \hat{z}} ,
\label{delr_fluc}
\end{equation}
and the contour integral can then be converted to a three-dimensional via a straightforward weighted average:
\begin{equation}
\frac{\rho}{2}\tilde{u}_{0} (\varepsilon) \int d \delta {\bf r} _{\perp} f_{G} (\delta {\bf r} _{\perp}) \int_{0}^{\ell} d \Delta l \frac{\exp [ - \kappa \Delta r ] }{\Delta r}.
\end{equation}
The transverse fluctuations obey a 2D-Gaussian distribution with variational parameter $\alpha$:
\begin{equation}
f_{G} (\delta r_{\perp}) = \left ( \frac{\alpha}{\pi} \right ) e^{-\alpha \delta r_{\perp}^{2}}.
\end{equation}
In principle, these fluctuations will also have an effect on the angular contribution $\cos \theta$. However, their impact is expected to be less critical than on the centrally symmetric part and we will retain the form \eq{u0} for simplicity. It is expedient to recast the expression in Fourier space in terms of the 2D wavevector $ {\bf k} $. This allows us to express the integral in factorized form:
\begin{align}
& \frac{\rho}{2} \tilde{u}_{0}(\varepsilon) \int d \delta {\bf r} _{\perp} \int \frac{d {\bf k} }{(2 \pi)^{3}} f_{G} (\delta {\bf r} _{\perp}) e^{-i {\bf k} \cdot \delta {\bf r} _{\perp}} \hat{U}(k) P( {\bf k} ),
\end{align}
where $ \hat{U}( {\bf k} ) = 4 \pi / ( k^{2} + \kappa^{2}) $ is the FT of the spherically symmetric Yukawa potential and $P( {\bf k} )$ a shape factor accounting for the helical shape of the buckled fluid. The Fourier transform only acts on the Gaussian displacement distribution so that:
\begin{align}
\frac{\rho}{2} \tilde{u}_{0}(\varepsilon) \int \frac{d {\bf k} }{(2 \pi)^{3}} \hat{f}_{G}(k ) \hat{U}(k) P( {\bf k} ),
\label{ftex}
\end{align}
where the FT of a Gaussian is itself a Gaussian, namely $\hat{f}_{G}(k_{\perp}) = e^{-k_{\perp}^{2}/4\alpha}$.
The shape factor is formally given by a FT of the helical backbone parameterized in \eq{delr_fluc}:
\begin{equation}
P( {\bf k} ) = \int_{-\ell/2}^{\ell/2} d \Delta l \exp [i {\bf k} \cdot ({\bf \Delta r} - {\bf \delta r}_{\perp})],
\label{form}
\end{equation}
and depends implicitly on the helix shape via $\Delta r$.
Reverting to the ideal entropy in \eq{hform}, as of yet unspecified, we may now associate the transversal Gaussian fluctuations with an entropic term of the following form:
\begin{align}
S^{(G)}_{\text{id}}(\alpha) &\sim -Nk_{B} \int d\delta {\bf r} _{\perp} f_{G}(\delta r_{\perp}) [ \ln \Lambda^{2} f_{G} (\delta r_{\perp}) -1 ] \nonumber \\
&\sim -Nk_{B} (\ln(\alpha \Lambda^{2} /\pi) - 2),
\end{align}
indicating a reduction of ideal entropy upon increased localization along the contour. As anticipated, $S_{\text{id}}$ diverges if the particle centres-of-mass follow a strictly 1D contour in the limit $\alpha \rightarrow \infty$.
The two-dimensional contribution needs to be supplemented with the entropy arising from the one-dimensional degrees of translational freedom along the contour. This leads to a contribution $-Nk_{B}( \ln \Lambda \rho - 1)$ so that the total entropic contribution to the free energy takes the following approximate form:
\begin{equation}
S_{\text{id}} \sim -Nk_{B} (\ln(\bar{\rho} \alpha \Lambda^{3} /\pi) - 3).
\end{equation}
The relevant density variable $\bar{\rho} = \rho /\sqrt{1+ (qR)^{2}}$ reflects the notion that the available one-dimensional volume should increase upon buckling. In the ideal gas limit ($\Phi \downarrow 0$, $V_{c} \downarrow 0$) a linearly stacked system will trivially maximize its configurational entropy by reducing its transverse localization ($\alpha$ decreases) and maximizing its fictitious contour length (although the shape need no longer be a simple helix).
\eq{form} cannot be worked out in analytical form for arbitrary $R$. Instead we will resort to a series expansion up to quadratic order in $R$ which should be of sufficient accuracy for weakly buckled fluids. The details of this calculation are shown in the Appendix. The generic expression for the deformation free energy per particle reads:
\begin{align}
\frac{F}{N} &\sim \ln (\rho \Lambda^{3} \alpha / \pi) - 3 - \frac{1}{2}(qR)^{2} + V_{c}(R) \nonumber \\
& + \frac{\rho }{2} \tilde{u}_{0} (\varepsilon ) {\mathcal U}(R,q),
\label{wtot}
\end{align}
For notational brevity we shall implicitly use the thermal energy $k_{B}T$ as energy unit and $\kappa^{-1}$ as length unit. An approximate expression for the interaction free energy ${\mathcal U}$ of a weakly buckled helical fluid in the strong localization limit ($\alpha \gg 1$) is derived in the Appendix and reads:
\begin{align}
{\mathcal U}(R,q) & = \int \frac{d {\bf k} }{(2 \pi)^{3}} \hat{f}_{G}(k ) \hat{U}(k) P( {\bf k} ) \nonumber \\
& \sim (\pi \alpha)^{1/2} \left ( 1 + R^{2} \alpha p_{2} (q \ell) + {\mathcal O} ( (\kappa R)^{4}) \right ),
\label{uquad}
\end{align}
up to quadratic order in $R$. At this point it is instructive to broaden our scope a little by making a distinction between lyotropic systems where Yukawa interactions are repulsive, and thermotropic ones where an {\em attractive} Yukawa potential could be used as a rough approximation for the ($\pi$-)stacking or hydrogen-bonding forces between discotic molecules \cite{brandt_michels1995,delrio2005}. In the lyotopic case $\rho u_{0} > 0$ the density is the main thermodynamic variable, whereas
for thermotropics ($\rho u_{0} <0 $) the potential amplitude plays the role of an effective temperature $T_{\rm eff} = | \rho u_{0} |^{-1} $. From \eq{wtot} and \eq{uquad} we infer that an unbuckled fluid of repulsive disks can reduce its excess free energy by decreasing $\alpha$ and delocalizing from the contour, while attractive disks tends to favor increased localization along the contour with $F$ growing more negative with $\alpha$. As a consequence, the buckling scenario for thermotropics will be quite different from that of lyotropics as we will see shortly.
Contrary to the ideal contribution, which increases with $\alpha$ due to a loss of configurational entropy, the confinement free energy must be inversely proportional with particle localization. The free energy penalty associated with disks moving away from the helical backbone can be made explicit by considering a parabolic form $V_{c}(r) = v_{0}r^{2}$ with $v_{0} > 0$. The transverse delocalization enters via a simple Gaussian average:
\begin{align}
V_{c}(R, \alpha ) &= v_{0} \int_{0}^{\infty} \frac{ d \delta r_{\perp}^{2}}{2} \int_{0}^{2 \pi} d \varphi f_{G}( \delta r_{\perp} ) (R + \delta r_{\perp} \cos \varphi )^{2} \nonumber \\
& = v_{0} \left ( R^{2} + \frac{1}{2 \alpha } \right ).
\label{vext}
\end{align}
and is inversely proportional to $\alpha$.
In the weak buckling limit ($R \ll 1$) we assume the equilibrium value of $\alpha$ to be unaffected by the undulation so that $\left . \partial F / \partial \alpha \right |_{R=0} =0 $. For the lyotropic case $\rho u_{0} >0 $ an approximate analytical solution of the minimization condition can be found in the limit of strong confinement $v_{0} \gg 1$ in which case the equilibrium localization strength is governed by a trade-off between the external and excess free energy. The extremum value reads:
\begin{align}
\alpha \sim \left ( \frac{2v_{0}}{ \pi^{1/2} \rho u_{0}} \right )^{2/3} \hspace{0.2cm} \rho u_{0} > 0,
\label{extrema_lyo}
\end{align}
showing that the disks become progressively delocalized with increasing particle concentration. For the thermotropic case the linear fluid is naturally stabilized by the strongly attractive face-to-face interactions and the confining potential will only have a marginal effect on the localization strength. Irrespective of $v_{0} >0 $, the minimal free energy is obtained in the limit $ \alpha \rightarrow \infty $. In order to systematically assess the buckling free energy in powers of $R$ we expand the curvature-dependent Yukawa amplitude \eq{u0} up to quartic order in $qR \ll 1$:
\begin{equation}
\frac{\tilde{u}_{0}( \varepsilon )}{u_{0}} \approx 1 + c_{2}^{(L)}(qR)^{2} + c_{4}^{(L)}(qR)^{4} + {\mathcal O}(R^{6}),
\end{equation}
with $c_{2}^{(L)} = D^{2}/16$ and $c_{4}^{(L)} = (D^{4} - 48 D^{2})/1024$.
It is obvious that the orientation-dependent Yukawa amplitude in \eq{yuk} only makes sense in the context of charge-stabilized colloidal interactions where the Debye screening length is clearly defined. For the thermotropic case the prefactors $c_{n}$ must have a different origin as their amplitude and sign will depend on whether face-to-face configurations are energetically favorable compared to edge-to-edge ones or vice versa.
Going back to the lyotropic case we may now formulate the free energy difference between the buckled and the unbuckled columnar fluid of up to quartic order in the buckling amplitude:
\begin{align}
\frac{\Delta F(q)}{NR^{2}} & \sim - \frac{1}{2}q ^{2} + \frac{1}{4}R^{2}q^{4} + v_{0} + \frac{\rho u_{0}}{2} (\pi \alpha)^{1/2} \nonumber \\
& \times \left (c_{2}^{(L)} q^{2} + \alpha p_{2} (q \ell) + c_{4}^{(L)} R^{2} q^{4} + \alpha^{2} R^{2} p_{4}(q \ell) \right ).
\label{flyo}
\end{align}
In the limit of strong confinement $\alpha \gg 1$ the free energy is fully dominated by the external and excess free energy.
Minimization with respect to $R$ yields the equilibrium value for the buckling radius $R^{\ast}$:
\begin{equation}
R^{\ast} \propto f(q \ell) \left ( \frac{\rho u_{0}}{v_{0}} \right )^{1/3} \ll 1,
\label{rmin}
\end{equation}
in terms of the pitch-dependent function $ f(q \ell) = \sqrt{-2 (1 + p_{2}(q \ell) )/ p_{4}(q \ell)}$ (see \eq{p2n} in the Appendix). The $q$ values that minimize the free energy turn out to be independent of the parameter combination $\rho u_{0}/v_{0}$ and follow from $f^{\prime}(q^{\ast} \ell) =0$. The corresponding buckling free energy is uniformly negative:
\begin{equation}
\frac{\Delta F(q^{\ast})}{N} \propto -\frac{v_{0}^{5/3}}{(\rho u_{0})^{2/3}} \frac{(1+ p_{2}(q^{\ast} \ell)) )^{2}}{p_{4}(q^{\ast} \ell) } < 0,
\end{equation}
indicating that helical buckling always leads to a reduction of the free energy. On the contrary, for thermotropic systems buckling is generally disfavored. Up to lowest order in radius
it follows that $ \Delta F(q^{\ast})/NR^{2} \propto \alpha^{3/2} |p_{2} (q^{\ast} \ell)|/ 2 T_{{\rm eff}} \gg 0 $. This outcome is, of course, not surprising given that the attractive stacking interactions prevent any deformation of the linear stack at low temperatures.
\begin{figure}
\includegraphics[width=0.9 \columnwidth]{fig2}
\caption{Renormalized buckling radius and associated deformation free energy (on a log scale) for a linear stack of soft discotic particles as a function of the optimal pitch values $q^{\ast} \ell$ . The energetically most favourable mode is $q^{\ast}\ell \approx 9.132$, with corresponding radius $R^{\ast} \approx 0.333 ( \rho u_{0}/v_{0} )^{1/3} $. Higher order pitch `harmonics' correspond to increasingly weaker buckling radii $R^{\ast}$ and free energy reductions. }
\label{fig3}
\end{figure}
\section{Connection to lyotropic columnar assemblies}
As an alternative to the parabolic trap we consider a confining potential of the Yukawa form:
\begin{equation}
V_{Y}(x) = u_{0} \left ( \frac{1}{2} \frac{ e^{ -\kappa \left ( \sigma- x \right ) }}{\sigma -x } + \frac{1}{2} \frac{ e^{-\kappa \left ( \sigma + x \right ) } }{\sigma + x} - \frac{e^{-\kappa \sigma}}{\sigma} \right ), \hspace{0.2cm} |x| < \sigma,
\end{equation}
which can be thought of as a simple effective potential acting on each particle along the backbone imparted by its neighbouring columns at a typical distance $ \sigma $ from the target particle. The last term provides the correct offset so that $V_{Y}(0) = 0$. Contrary to the parabolic form, the Yukawa potential thus features a typical interaction range $\sigma$ whose value can either be varied as an independent parameter (cf. the spatial extent of an optical trap), or it can be coupled self-consistently to the thermodynamic state of the columnar assembly. Analogous to \eq{vext} the potential is modified by the transverse displacements perpendicular to the column direction.
For small radii an expansion up to quadratic order is appropriate so that:
\begin{equation}
V_{Y}(R, \alpha) = u_{Y} \left ( R^{2} + \frac{1}{2 \alpha} \right ) + {\mathcal O} \left ( \frac{R^{4}}{\alpha} \right ).
\label{vcell}
\end{equation}
We may now simply repeat the steps outlined in the previous section. The localization parameter $\alpha$ still obeys \eq{extrema_lyo} but with $v_{0}$ replaced by an effective value associated with the cylindrical Yukawa potential (denoted ``Y''), namely
$u_{Y} = u_{0} \sigma^{-3} e^{-\sigma }(1 + \sigma + \sigma^{2}/2)$. In line with the previous section we use $\kappa^{-1}$ as an implicit length unit.
A connection with a three-dimensional columnar phase can be made by coupling the one-dimensional free energy discussed thus far to a simple cell theoretical expression for a (hexagonal) packing of columns. Lennard-Jones cell theory \cite{lennardjones1937} tells us that, approximately:
\begin{equation}
Q_{\rm cell}(N) \approx \left ( \int \frac{d^{2} {\bf r} }{\Lambda^{2}} \exp \left [- \frac{ V_{Y}( {\bf r} )}{ 2 k_{B}T } \right ] \right )^{N}.
\end{equation}
Within this framework the line fluids are localized in $N$ cylindrical cells centered on the sites of a fully occupied lattice of some prescribed symmetry. Each particle experiences a potential energy $V_{Y}(R)$ generated by its nearest neighbors. The term between brackets is therefore equivalent to the lateral `free volume' each particle explores within its cell.
In its simplest version, the theory presupposes each cell to contain only one stacked fluid behaving independently from its neighbors \cite{lennardjones1937,kramerherzfeld-pre2000,graf1999}. Moreover, the free volume term does not reflect the precise nature of the 2D Bravais lattice of the columnar phase which we assume to be a simple hexagonal arrangement. Using \eq{vcell} and some algebra then yields for the partition sum:
\begin{equation}
Q_{\rm cell}(N) \approx \left ( \frac{\pi e^{-\frac{u_{Y}}{2 \alpha}} \left ( 1 - e^{-\sigma^{2} u_{Y}} \right ) }{u_{Y} \Lambda^{2}} \right )^{N},
\label{qcell}
\end{equation}
Single occupancy of each cell implies that the three-dimensional density $\varrho = N/V$ is linked to the linear density via $ \varrho = \rho /\sigma^{2} $.
Recalling that $F = -k_{B}T \ln Q$ we obtain the free energy per particle of the columnar phase by superimposing the contributions from the stacked fluid and the cellular confinement (ignoring irrelevant constants) \cite{kramerherzfeld-pre2000}\footnote{The external field term featuring in the fluid free energy must be omitted given that the effect of confinement is accounted for by the cell free energy \eq{qcell}. }:
\begin{align}
\frac{F(\varrho)}{N } &\sim \ln (\varrho \sigma^{2} \alpha ) + \frac{u_{Y}}{2 \alpha} + \frac{\varrho \sigma^{2}}{2} u_{0} (\pi \alpha)^{1/2} + \ln u_{Y} \nonumber \\
& - \ln \left (1 - e^{-\sigma^{2} u_{Y}} \right ).
\label{wcol}
\end{align}
In the limit of very strong confinement we expect $\sigma \ll 1$ so that $u_{Y} \sim u_{0} /\sigma^{3} \gg 1$. Minimization of the columnar free energy with respect to $\alpha$ yields, as expected, an inverse proportionality of the localization strength with density and amplitude, i.e. $\alpha \sim (4/\pi)^{1/3} \varrho^{-2/3} \sigma^{-10/3} $. Subsequent minimization of the free energy with respect to $\sigma$ gives a similar proportionality but with different exponents, namely $ \sigma \sim \sigma_{0} / u_{0}^{3} \varrho^{2} $
with constant $\sigma_{0} = 70304/27\pi$. With this we can establish an explicit relation between the density $\varrho$ of the columnar phase and the localization strength, namely:
\begin{equation}
\alpha \sim \alpha_{0} u_{0}^{10} \varrho^{6},
\end{equation}
with prefactor $\alpha_{0} = (4/\pi)^{1/3} \sigma_{0}^{-10/3}$. Similarly, the effective amplitude of the self-consistent cell potential reads:
\begin{equation}
u_{Y} \sim \frac{u_{0}^{10} \varrho^{6}}{\sigma_{0}^{3}}.
\end{equation}
In order to get a feeling of the magnitude of the variables above we may identify $u_{0} \approx Z^{2} \kappa \lambda_{B} $ with $Z \gg 1$ the number of bare or effective elementary charges per disk \cite{rowan_mp2000,trizac-weis} and $\lambda_{B}$ the Bjerrum length (which is typically about 1 nm in aqueous conditions at room temperature). The effective columnar packing fraction can be estimated from $\phi \sim N D^{2} \kappa^{-1} /V $ so that $\varrho \sim \phi (\kappa^{-1}/D)^{2}<1$.
The typical buckling radius then scales as:
\begin{equation}
R^{\ast} \propto \frac{ f(q^{\ast} \ell)}{\varrho^{3} u_{0}^{5}} \ll 1,
\label{rmincol}
\end{equation}
and the deformation free energy associated with the pitch optima (see \fig{fig3}) scales as follows:
\begin{equation}
\frac{\Delta F(q^{\ast})}{N} \propto - u_{0}^{20} \varrho^{12} \frac{(1+ p_{2}(q^{\ast} \ell)) )^{2}}{p_{4}(q^{\ast} \ell) } \ll 0.
\label{minfree}
\end{equation}
For infinitesimally weak buckling deformations, a Taylor expansion of \eq{flyo} up to the lowest order contribution in the helix parameters gives an expression which could be interpreted as the typical bend or twist elastic modulus of a single column:
\begin{equation}
\frac{\Delta F}{N(qR)^{2}} \propto - \frac{u_{0}^{10} \varrho^{6} \ell^{2} }{24 \sigma_{0}^{3}},
\end{equation}
The modulus is negative because internal stresses along the backbone lead to buckling instabilities and `floppy' modes whose amplitude increases quadratically with length $\ell$.
We finally wish to point out that the buckling radius and free energy depend implicitly on the Debye screening length. Rearranging the scaling expressions \eqs{rmincol} and (\ref{minfree}) in terms of the (fixed) disk diameter $D$ reveals that $R^{\ast}/D \propto (D/\kappa^{-1})^{3} $ and $\Delta F/ N \propto (\kappa^{-1}/D)^{16}$ and shows that the buckling characteristics depend sensitively on the range of the stacking potential.
\section{Concluding remarks}
We have proposed a microscopic theory to probe helical buckling deformations of a columnar fluid of discotic particles. Contrary to the more commonly studied case of buckling under an external load, the stresses that stabilize helical confirmations of the columnar backbone are purely internal and are transmitted through a strongly repulsive stacking potential between adjacent disks located along the backbone. The approach is particularly suited to investigate conformational fluctuations in lyotropic columnar assemblies of e.g. clay platelets at low salt concentrations where the effective charge-mediated particle interactions are strongly repulsive. Using a simple mean-field theory based on microscopic principles we derive scaling relations for the optimal pitch, which sets the typical buckling length scale, and amplitude in relation to the density of the columnar mesophase and the strength of the electrostatic interactions. Our findings indicate that helical deformations of stacked fluids can be stabilized by long-ranged soft interactions.
Our analysis is based on the assumption that the columnar deformations are independent from each other. However, we consider a detailed knowledge of the single-column buckling behaviour as an essential precursor to tackling the more complicated problem of collective conformational fluctuations. In this context, we plan to extend the present model such as to include chiral intercolumnar couplings, e.g., by introducing a weakly chiral self-consistent cell potential, in an effort to understand if and how helical conformational fluctuations of an individual column proliferate to larger length scales. This could open a route to studying spontaneous chiral symmetry breaking and chiral amplification in columnar assemblies of achiral discotic particles under the influence of internal stresses generated by repulsive long-ranged stacking potentials. The results of the current model are expected to facilitate the construction of coarse-grained models for studying the chiral intra-columnar meso-structure of columnar liquid crystals. These models could provide a useful intermediate route between fully particle-resolved simulations \cite{duncanPRE2009,martinezhayaPRE2010} and phenomenological continuum theories that
have been fruitfully employed to study columnar bundles of helical fibres \cite{grason_bruinsma2007} and defects in chiral columnar phases \cite{kamien_nelsonPRE1996}. Research efforts along these lines are currently pursued.
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One thing we are keen on here at Cooking for the Sensitive Gut is the idea of 'managing' the foods in the diet that may trigger symptoms. One group of foods which many patients overly restrict are those containing FODMAPS (Fermentable oligosacchararides, disaccharides, monosacahrides and polyols). Many foods that contain FODMAPs such as fruit vegetables and whole grain cereals are foods which are otherwise very nutritious and good for the gut. They provide the substrate for gut bacteria to produce short chain fatty acids that enhance the healthy composition of gut microbiota. They are also a good source of fibre that keeps the content of the gut moving along. So it is important to manage, rather than overly restrict, FODMAP containing food in the long term.
It is always worth remembering that other factors can trigger symptoms such as tiredness, stress and generally feeling under the weather. So if you are fine eating a particular food one day and the next you are not – something else may have triggered the symptoms.
The idea of managing these foods in your diet means being knowledgeable about which FODMAPs are contained in different foods and understanding, the portion size you can eat without triggering symptoms.
Just because a food or ingredient contains one or more of a FODMAP does not mean you should not eat it but that you should be cautious. Eat a little of it – see how you get on and then if it does not trouble you make a note of it and eat it in a similar portion on another time. This strategy works very well for most people.
One good way of learning which foods contain high levels of these fermentable carbohydrates is to check the database of a reliable smartphone app. We use two apps to do this – The MONASH University Low FODMAP diet app and Food Maestro. Both are available form App stores.
When we devise and test a recipe we always consult the food database in these apps and then highlight any foods which contain high levels of FODMAPs. We also flag up the portion size of the food which is low in FODMAPs and then leave it up to you to adjust this according to what you can manage. People vary a lot in which foods they can and cannot tolerate.
These week's cooking has been inspired by the garden which is full of tender, courgettes with their delicate bright yellow flowers which remind me of a blackbird's yellow gape. They are great steamed but they are also lovely stuffed with minced beef, herbs and cheese which can easily be adapted for vegetarians. It is one of our 'forgiving recipes' because it can be left gently cooking in the oven without spoiling and leftovers can be eaten the next day. Courgettes are low in FODMAPS.
Preheat the oven to 200C. Cut the courgettes in half lengthways. Cut a V shape along the length of each courgettes half and gouge out the seeds with the point of a teaspoon. This forms an indent for the filling. Place the courgettes on a baking tray and set to one side. Discard the courgette seeds.
Place the remaining ingredients in a bowl except the egg. Mix the ingredients together well and gradually add enough beaten egg to form a stiff mixture. If you are making a vegetarian version of this stuffing – leave out the meat and add another tablespoon of grated cheese.
Fill the courgettes with the stuffing and don't worry if any spills over onto the baking tray. Cover the baking tray and courgettes with foil and place in the oven for 30 minutes. Insert a pointed knife into the courgettes to check they are cooked. If not return the tray to the oven for a few more minutes.
Remove the foil and cook for a further 5 – 10 minutes or until the filling begins to brown.
While the courgettes are baking make this simple tomato sauce. Fry the sliced garlic in a little oil until golden brown. This flavours the oil with garlic. Remove the garlic flesh from the oil and discard. Remember the flesh of garlic can cause symptoms in some people.
Add the passata, bayleaf and dried oregano and seasoning and cook gently for about 10 minutes until a thick sauce has formed.
Serve the stuffed courgettes with the tomato sauce and diced potato baked in the oven at the same time as the courgettes. The courgettes are also lovely with rice – either brown or white.
14 Aug This entry was written by Joan Ransley and published on August 14, 2016 at 4:46 pm. It's filed under Basic methods, Dinner, Side dish, Useful information and tagged cooking, Diet tips, healthy eating, IBS, IBS Network, Irritable bowel syndrome, low FODMAP, low FODMAP diet, vegetables, vegetarian. Bookmark the permalink. Follow any comments here with the RSS feed for this post.
Hi, new visitor to your site. I'm a bit surprised to see garlic in this recipe – isn't it a high FODMAP food?
The FODMAPs (fructo-oligosaccharides) in garlic are soluble in water, whereas the allicin and other volatile compounds that give garlic its distinctive aroma are soluble in oil. Thus the gorgeous, aromatic flavour of garlic can be extracted by steeping sliced garlic in oil and then straining the oil before using it in cooking.
You can just make garlic oil as and when you need it. Either fry a couple of cloves of sliced garlic in enough olive oil for a recipe, discard the garlic and use the garlic flavoured oil within a day. Or just slice two cloves of fresh garlic and leave it covered with olive oil for an hour. Strain the garlic from the oil and use it within a day. It does not keep for long.
Both methods capture the wonderful flavour of garlic but each oil tastes slightly different. The raw garlic infused oil might be nice to use in salad dressings whilst the cooked version would be better suited to add to tomato sauce in pasta dishes. But you could experiment to see which you prefer. Either way the flavour in the home made oil does not last very long so it is best to make it up in small batches and use the same day or the day after.
Thanks for the quick and informative reply. I'm planning the start of my FODMAP elimination phase and was saddened to hear that garlic and onions (2 of my favourite foods) were off the list. Knowing I can still capture the taste of garlic in this way makes me very, very happy! 😀 Thanks so much! | {
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\section{Introduction}
Layered compounds of transition-metal elements frequently show interesting and novel electrical and magnetic properties and have been studied extensively. Among them, La$_2$CuO$_4$
\cite{muller} and LaFeAsO \cite{kamihara} as the representatives are widely known because of
their superconductivity with high critical temperatures after doping or under high pressures,
in which the CuO$_2$ square planar layers and the FeAs tetrahedral layers play a substantial
role respectively. Now another class of layered compounds built from alternatively stacking
special octahedral layers M$_2$Q$_2$O (M=Fe or Ti; Q=As, Sb, S, or Se) with otherwise layers are attracting much attention.\cite{adam,mayer} Here an M$_2$Q$_2$O layer includes an
anti-CuO$_2$-type M$_2$O square planar layer with two Q atoms respectively located above and
below the center of each M$_2$O square unit. As shown in Fig. \ref{figa}, the M$_2$Q$_2$O layer is in an octahedral layer structure, bridging or a combination of the CuO$_2$ square planar
layer and the FeAs tetrahedral layer structures, which is thus expected to show distinct
electronic and magnetic properties and possibly superconductivity by doping or under pressure.
Fe$_2$Q$_2$O layer-based (Q=S or Se) oxychalcogenides were first synthesized in 1992 \cite{mayer} and recently characterized to be antiferromagnetic insulators with a magnetic moment of $4\mu_B$ on each Fe ion experimentally\cite{kabbour} and with strong correlation effects suggested theoretically.\cite{zhujianxin} To our knowledge, there has been no any structural distortion found experimentally in these compounds.
Ti$_2$Q$_2$O layer-based (Q=As or Sb) oxypnictides have been also studied for a quite long time. Compounds Na$_2$Ti$_2$Q$_2$O were first synthesized in 1990,\cite{adam} which subsequently have been intensively investigated both experimentally and theoretically.\cite{ozawa1,ozawa2,ozawa3,ozawa4} The temperature-dependent powder
neutron diffraction measurement showed that there is a structural distortion in the Ti$_2$Sb$_2$O layer at a temperature of about 120K,\cite{ozawa2,ozawa3} which corresponds well to the observed anomalies in the temperature-dependent magnetic susceptibility and
resistivity measurements.\cite{adam,ozawa1} However, no neutron scattering due to magnetic spin ordering was detected. Meanwhile, the measurement further found that the bond distance ratio of O-Ti-O/Sb-Ti-Sb increases below the transition temperature, but the crystal space group symmetry remains unchanged after the structural distortion.\cite{ozawa2,ozawa3} This is different from the tetragonal-orthorhombic structural distortions observed in the iron pnictides.\cite{cruz}
Recently another new Ti$_2$As$_2$O layer-based oxypnictide BaTi$_2$As$_2$O was successfully synthesized,\cite{chenxianhui1,chenxianhui2} which shows the similar characteristics as Na$_2$Ti$_2$Q$_2$O, namely there are anomalies observed as well in the temperature-dependent magnetic susceptibility, resistivity, and heat capacity measurements at a temperature of about 200K.\cite{chenxianhui1,chenxianhui2} Considering that the similar anomalies are observed in the iron pnictides, in which the superconductivity takes place when the anomaly is suppressed, one thus speculates that the Ti$_2$Q$_2$O layer-based oxypnictides may also be another parent compounds for the high T$_c$ superconductivity. In the iron pnictides, those anomalies are ascribed to the tetragonal-orthorhombic structural transitions accompanied by the collinear antiferromagnetic ordering.\cite{cruz} The underlying mechanism is still in debate, either the As-bridged antiferromagnetic superexchange interactions between fluctuating Fe local moments \cite{ma1,si,yildirim} or spin density wave instability induced by the Fermi surface nesting.\cite{mazin} There are now more and more evidences in favor of the fluctuating Fe local moment picture. Especially, the neutron inelastic scattering experiments have shown that the low-energy magnetic excitations can be well described by the spin waves based on the quantum Heisenberg model.\cite{dai-0,dai-1}
Likewise, the scenario in Ti$_2$Q$_2$O layer-based oxypnictides is very confusing. The experimentally observed anomalies suggest that there is an antiferromagnetic transition, but no magnetic ordering has been detected. Theoretically, no ordered magnetic state has been either found in the first-principles electronic structure calculations,\cite{pickett} which however show that there is a Fermi surface nesting. It was thus suggested that the observed anomalies would be due to charge density wave or spin density wave instability induced by such a Fermi surface nesting.\cite{pickett,fabrizia}
In order to clarify the issue, we performed the thorough first-principles electronic structure calculations upon compounds Na$_2$Ti$_2$Q$_2$O (Q=As, Sb). We find that the ground state of Na$_2$Ti$_2$As$_2$O is a blocked checkerboard antiferromagnetic semiconductor, in contrast, the ground state of Na$_2$Ti$_2$Sb$_2$O is a bi-collinear antiferromagnetic semimetal. And we verify that these two antiferromagnetic states both are indeed induced by the Fermi surface nesting rather than based on local magnetic moments.
\section{Computational details}
In our calculations the plane wave basis method was used.\cite{baroni} We used the PW91-type\cite{perdew} generalized gradient approximation for the exchange-correlation functionals. The ultrasoft pseudopotentials \cite{vanderbilt} were used to model the electron-ion interactions. After the full convergence test, the kinetic energy cut-off and the charge density cut-off of the plane wave basis were chosen to be 600eV and 4800eV, respectively. The Gaussian broadening technique was used and a mesh of $20\times 20\times 10$ k-points were sampled for the Brillouin-zone integration in nonmagnetic state. The lattice parameters with the atomic positions were fully optimized by the energy minimization.
It was found that the calculated lattice parameters have an excellent agreement with the experimental ones (less than $ 1.0 \%$).
\section{Results and analysis}
\begin{figure}
\includegraphics[width=6cm]{fig1.eps}
\caption{(Color online) Crystal structure of Na$_2$Ti$_2$As$_2$O: (a) A conventional unit cell composed of two formula cells; (b) A primitive unit cell composed of one formula cell.} \label{figa}
\end{figure}
Compound Na$_2$Ti$_2$Pn$_2$O (Pn=As or Sb) crystallizes in tetragonal (I4/mmm) symmetry and has a layered structure, as shown in Fig. \ref{figa}(a). Its conventional tetragonal unit cell
consists of two formula unit cells. However, its primitive unit cell is constructed by
considering Na$_2$Ti$_2$Pn$_2$O as a triclinic crystal, in which only one formula unit cell is
included as shown in Fig. \ref{figa}(b). On the other hand, Na$_2$Ti$_2$Pn$_2$O can be
considered geometrically with an anti-K$_2$NiF$_4$ type structure, where a Ti atom is located
between two oxygen atoms in a plane to form a square planar layer of Ti$_2$O, which is an
anti-configuration to the CuO$_2$ layer in the high $T_c$ cuperates. Meanwhile a Ti atom is
also four-coordinated by As atoms, forming a Ti$_2$As$_2$O layer where two As atoms are
respectively located above and below the center of a Ti$_2$O square unit. The crystal is formed by stacking such Ti$_2$Pn$_2$O layers in a body-centered manner along the $c$ axis with the
separation by a double-layer of Na atoms (Fig. \ref{figa}(a)).
\subsection{Nonmagnetic States}
We first studied the nonmagnetic state of compound Na$_2$Ti$_2$Pn$_2$O (Pn=As or Sb), which would describe the high temperature phase of these materials. The electronic band structure of
this nonmagnetic state also provides a reference to further studying of the possible low
temperature magnetic phases, by analyzing which we can better understand the underlying
mechanism or interactions that drive possible magnetic phase transitions and the related
structural transitions. In the calculations, we adopted a primitive unit cell including only
one formula cell (Fig. \ref{figa}(b)).
\subsubsection{Na$_2$Ti$_2$As$_2$O}
\begin{figure}
\includegraphics[width=7cm]{fig2.eps}
\caption{(Color online) Electronic structure of Na$_2$Ti$_2$As$_2$O in the nonmagnetic state: (a) Electronic Band structure; (b) Hole-type Fermi surface sheets; (c) Electron-type Fermi surface sheets. The Fermi energy is set to zero.}\label{figb}
\end{figure}
After the full structural optimization, we find the tetragonal crystal lattice parameters of $a=b=4.060$~\AA~ and $c=15.3892$~\AA~ for Na$_2$Ti$_2$As$_2$O, in good agreement with the experimental values of $a=b=4.0810$~\AA~ and $c=15.311$~\AA~ measured at a temperature of 310K. \cite{ozawa2} The calculated electronic band structure and Fermi surface in the primitive unit cell are presented in Fig. \ref{figb}. As we see, there are three Fermi surface sheets, contributed from the two bands crossing the Fermi energy level (Fig. \ref{figb}(a)). Among them, the one sheet is of the hole-type, forming an approximate square-shaped pipe centered at each $N$ point in the Brillouin zone (Fig. \ref{figb}(b)); the other two are of the electron-type, one of them is a $\Gamma$-centered box with the windows at its top and bottom, the other is a $X$-centered perfectly fluted pipe around the corner of the Brillouin zone (Fig. \ref{figb}(c)). The volumes enclosed by these Fermi surface sheets are 0.186 electrons/cell and 0.186 holes/cell respectively. The electron carrier concentration is the same as the hole carrier concentration. Both are equal to $2.94 \times 10^{21}/{cm}^3 $. Compound Na$_2$Ti$_2$As$_2$O is thus a semimetal with a low carrier concentration between the ones in normal metals and semiconductors, similar to the recent iron pnictides, for example, LaFeAsO and BaFe$_2$As$_2$.\cite{ma,ma2} The density of states at the Fermi energy is 3.84 states per eV per formula unit. The corresponding electronic specific heat coefficient is $\gamma$ = $9.05mJ/(K^2\ast mol)$ and the Pauli paramagnetic susceptibility is $\chi_p$=$1.56 \times 10^{-9} m^3/mol$ (1 mole formula unit), in reasonable agreement with the measured value of $\chi = 3.01 \times 10^{-9} m^3/mol$.\cite{ozawa3}
\begin{figure}
\includegraphics[width=7cm]{fig3.eps}
\caption{(Color online) Total and atomic orbital-resolved partial density of states of
Na$_2$Ti$_2$As$_2$O per formula cell in the nonmagnetic state. The Fermi energy is set to
zero.}\label{figc}
\end{figure}
Inspection of the distribution of the density of states will help to clarify the atomic bonding character. Figure \ref{figc} plots the calculated total and atomic orbital-resolved partial
density of states of Na$_2$Ti$_2$As$_2$O in the nonmagnetic state respectively. As expected, the calculations show that most electrons of the Na $3s$-orbitals transfer into the O $2p$-orbitals so that the occupied O $2p$-orbitals are far below the Fermi energy level while the empty Na $3s$-orbitals are much above the Fermi energy level (centered at +5.0 eV). Although the density of states at the Fermi level is dominated by the Ti $3d$-orbitals, the density of states around the Fermi energy (from -3.0 to 2.0eV) consist mainly of both the Ti $3d$-orbitals and As $4p$-orbitals. Especially, the two peaks in the distribution of the As $4p$-orbitals coincide with those of the Ti $3d$-orbitals, as specifically shown in Fig. \ref{figd}, which indicates that there is the hybridization between $4p_x$ ($p_y$) orbitals and $d_{xz}$ ($d_{xy}$) orbitals, namely there is covalent bond formed between them. Figure \ref{figd} also specifies that it is the remaining three $3d$-orbitals ($d_{xy}$, $d_{x2-y2}$, and $d_{z^2}$) that dominate in the density of states at the Fermi energy.
\begin{figure}
\includegraphics[width=7cm]{fig4.eps}
\caption{(Color online) Density of states of Na$_2$Ti$_2$As$_2$O in the nonmagnetic state projected onto Ti-$3d$ atomic orbitals (upper panel), As-$4p$ atomic orbitals and O-$2p$ atomic orbitals (lower panel), respectively.}\label{figd}
\end{figure}
\subsubsection{Na$_2$Ti$_2$Sb$_2$O}
\begin{figure}
\includegraphics[width=7cm]{fig6.eps}
\caption{(Color online) Electronic structure of Na$_2$Ti$_2$Sb$_2$O in the nonmagnetic state: (a) Electronic band structure; (b) Hole-type Fermi surface sheets; (c) Electron-type Fermi surface sheets. The Fermi energy is set to zero.}\label{figf}
\end{figure}
Likewise, we performed the calculations on Na$_2$Ti$_2$Sb$_2$O. The optimized tetragonal
crystal lattice parameters are found to be of $a=b=4.1369$~\AA~ and $c=16.6428$~\AA~, also in good agreement with the experimental values of $a=b=4.160$~\AA~ and $c=16.558$~\AA~ measured at a temperature of 150K.\cite{ozawa2} Figure \ref{figf} shows the electronic band structure and Fermi surface calculated in a primitive unit cell, similar to those of Na$_2$Ti$_2$As$_2$O. From the volumes enclosed by these Fermi sheets, we determine that the electron and hole concentrations are 0.140 electrons/cell and 0.140 holes/cell respectively. The corresponding electron (or hole) carrier density is about $1.97\times 10^{21}/cm^3$. The density of states at the Fermi energy is 4.40 states per eV per formula unit. The corresponding electronic specific heat coefficient and Pauli paramagnetic susceptibility are $\gamma$ = $10.37mJ/(K^2\ast mol)$ and $\chi_p$=$1.79\times 10^{-9} m^3/mol$ (1 mole formula unit), respectively. The latter is in excellent agreement with the experimental measurement on a high-quality single crystal ($\chi = 1.51\times 10^{-9} m^3 /mol$).\cite{ozawa3}
\begin{figure}
\includegraphics[width=7cm]{fig7.eps}
\caption{(Color online) Total and atomic orbital-resolved partial density of states of Na$_2$Ti$_2$Sb$_2$O per formula cell in the nonmagnetic state. The Fermi energy is set to zero.}\label{figg}
\end{figure}
In Fig. \ref{figg}, we plot the calculated total and atomic orbital-resolved partial density of states for Na$_2$Ti$_2$Sb$_2$O in the nonmagnetic state, which are overall similar to the ones
of Na$_2$Ti$_2$As$_2$O (Fig. \ref{figc}) except that the Sb $5p$-orbitals have more
contribution to the density of states at the Fermi energy than the As $4p$-orbitals do. This is easily understandable because the Sb $5p$-orbitals are more diffusive and extended than the As
$4p$-orbitals. Nevertheless, the atomic bonding character of Na$_2$Ti$_2$Sb$_2$O is basically
the same as the one of Na$_2$Ti$_2$As$_2$O, i.e. O and Na atoms are in ionic state while there
are covalent bonds formed between Ti and Sb atoms.
\subsubsection{Fermi Surface Nesting}
In order to well analyze the Fermi surface symmetry of compounds Na$_2$Ti$_2$Pn$_2$O (Pn=As, Sb) in connection with the crystal symmetry, we replot the calculated Fermi surfaces in the tetragonal conventional Brillouin zone as shown in Fig. \ref{fige}, which clearly reflects the two dimensional character of the electronic structures by the nearly $k_z$-dispersionless Fermi surface sheets, being consistent with the layered crystal structure. As we see, the Fermi surfaces in these two compounds are very similar in shape to each other. There are now six Fermi surface sheets for each Fermi surface, among which there are the two nearly degenerate square-box-like Fermi-surface sheets centered around X contributed from the hole-type bands in conjugation with another two similar Fermi surface sheets centered around M but from the electron-type bands. Obviously these two sets of Fermi surface sheets between X and M are strongly nested with each other. The corresponding nesting vector is $(\frac{\pi}{a},0,0)$ or $(0,\frac{\pi}{a},0)$. The early studies reported in Refs. \onlinecite{pickett} and \onlinecite{fabrizia} first suggested that such a Fermi surface nesting in Na$_2$Ti$_2$Sb$_2$O would likely lead to a charge- or spin-density wave instability in association with the observed anomalies in the temperature-dependent susceptibility and resistivity.\cite{pickett,fabrizia,ozawa2}
\begin{figure}
\includegraphics[width=7.5cm]{fig5.eps}
\caption{(Color online) Na$_2$Ti$_2$As$_2$O in the nonmagnetic state: (a) and (b) are respectively the hole-type and electron-type Fermi surface sheets, and (c) is the top view of the whole Fermi surface in the Brillouin Zone denoted by the dotted square; Na$_2$Ti$_2$Sb$_2$O in the nonmagnetic state: (d) and (e) are respectively the hole-type and electron-type Fermi surface sheets, and (f) is the top view of the whole Fermi surface in the Brillouin Zone denoted by the dotted square.}\label{fige}
\end{figure}
\subsection{Magnetic States}
The powder neutron diffractions have not yet detected any magnetic spin ordering in compounds Na$_2$Ti$_2$Pn$_2$O (Pn=As, Sb) even though the observed anomalous transitions in the temperature-dependent magnetic susceptibility and electric resistivity strongly suggested an antiferromagnetic transition in the compounds. Nevertheless the neutron diffractions indeed found a commensurate structural distortion but without the space group symmetry breakdown.
Right now we still cannot exclude any weak magnetic ordering just based on the powder neutron
diffraction because of its low resolution. The issue that we meet here is still whether or not there exists a magnetic long-range order related to the structural distortion, based on either local moments formed around Ti atoms or spin-density wave induced by the Fermi surface nesting; Otherwise, whether or not there exists a spin-unpolarized charge-density wave related to the structural distortion, induced still by the Fermi surface nesting. In order to clarify this issue, we have carried out the systematical and extensive calculations on compounds Na$_2$Ti$_2$Pn$_2$O by constructing a wide variety of unit cells and magnetic orders.
We first performed the spin-unpolarized calculations with different unit cells and possible distortions, especially by elaborately constructing new unit cells in consideration of the nesting vector between X and M points in the Brillouin zone. It turns out that all the distortions and structures different from the original one are unstable. We thus exclude spin-unpolarized charge-density waves in compounds Na$_2$Ti$_2$Pn$_2$O.
We then investigated whether or not any magnetic order exists, based on local moments around Ti atoms. If there are local moments formed around Ti atoms, the exchange interactions
between these local moments would be either Pn-bridged or O-bridged superexchange interaction
with the direct exchange interaction between the nearest neighbor Ti atoms. There would be a
variety of stable or metastable magnetic orders resulting from the competition among these
exchange interactions, such as the ferromagnetic, checkerboard antiferromagnetic, and collinear antiferromagnetic orders, all of which were found to be stable or metatable in Fe-based superconducting parent compounds.\cite{ma,ma1,ma2,ma3} Our spin-polarized calculations show that none of these magnetic orders is stable or metastable in compounds Na$_2$Ti$_2$Pn$_2$O, similar to the previous calculations that show no magnetic ordering found in Na$_2$Ti$_2$Sb$_2$O.\cite{pickett} We thus exclude magnetic orders based on local moments.
To the end, we came to study possible magnetic orders based on spin-density waves induced by the Fermi surface nesting.
\subsubsection{Blocked checkerboard antiferromagnetic order and bi-collinear antiferromagnetic order}
As presented above, there are two independent and orthogonal but equivalent nesting vectors $\vec{Q_1}=(\frac{\pi}{a},0,0)$ and $\vec{Q_2}=(0,\frac{\pi}{a},0)$, respectively. A possible spin density wave will oscillate as $\vec{M}\cos(\vec{Q}\cdot\vec{R})$ on the Ti-Ti square lattice ($\vec{R}$ being a lattice site vector and $\vec{M}$ magnetization vector), with a wave vector $\vec{Q}$ equal to a commensurate linear combination of $\vec{Q_1}$ and $\vec{Q_2}$, namely $\vec{Q}=n\vec{Q_1}+m\vec{Q_2}$ ($n$ and $m$ being integers). We elaborately constructed several kinds of magnetic orders with magnetic unit cells based on such wave vectors $\vec{Q}$. Through the calculations, we eventually find two stable peculiar magnetic orders with wave vectors $\vec{Q_1}+\vec{Q_2}$ and $\vec{Q_1}$ (or $\vec{Q_2}$) respectively, called as blocked checkerboard antiferromagnetic order and bi-collinear antiferromagnetic order respectively.
\begin{figure}
\includegraphics[width=7.5cm]{fig8-two-order.eps}
\caption{(Color online) Schematic top view of the Ti$_2$Pn$_2$O (Pn=As, Sb) layer. The large blue circles represent the projected sites of Pn atom into the Ti$_2$O plane, the dark olive circles represent the O atoms, the small black circles with arrows represent Ti atoms and the green and red arrows indicate the two opposite directions of Ti moments. The blocked checkerboard antiferromagnetic order (a) and the bi-collinear antiferromagnetic order (b) are shown. Here a small blue dashed square represents an $a \times a$ unit in (a) and (b). The large blue dashed square $2a \times 2a$ unit in (a), and the blue dashed rectangle $2a \times a$ unit in (b) correspond to a magnetic unit cell in the blocked checkerboard antiferromagnetic order and bi-collinear antiferromagnetic order respectively. For the blocked checkerboard antiferromagnetic order, the Ti$_2$Pn$_2$O layer is divided into square blocks denoted by gray dash-dotted lines. The four Ti moments in a block are in the same direction and those between two nearest neighbor blocks have opposite directions.}
\label{fig8-two-order}
\end{figure}
Figure \ref{fig8-two-order}(a) schematically shows the blocked checkerboard antiferromagnetic order, in which the Ti$_2$Pn$_2$O (Pn=As, Sb) layer is divided into $\sqrt{2}\times\sqrt{2}$-squares represented by the dash-dotted thin lines. Each square is a magnetic block with the moments of four Ti atoms being parallel. Between the nearest neighbor blocks, the block moments are in anti-parallel. But the magnetic unit cell is a $2a\times 2a$ square. It should be emphasized that in each magnetic block the four Ti moments are centered around a Pn atom rather than O atom. For the latter case, the calculations show that it is unstable or energetically unfavorable. Thus in the blocked checkerboard anitferromagnetic order any pair of the next nearest neighbor Ti moments bridged by an O atom is always in anti-parallel. This is also consistent with the O-bridged superexchange effect being more stronger than the Pn-bridged superexchange effect, even though this magnetic order is induced by the Fermi surface nesting.
In Fig. \ref{fig8-two-order}(b), we schematically show the bi-collinear antiferromagnetic order, in which the Ti moments align in parallel along a diagonal direction and in anti-parallel along the other diagonal direction on the Ti-Ti square lattice. In other words, if the Ti-Ti square lattice is divided into two square sublattices $A$ and $B$, the Ti moments on each sublattice take their own collinear antiferromagnetic order. Here the magnetic unit cell is a $2a\times a$ rectangle. Unlike in the blocked checkerboard antiferromagnetic order, here half of pairs of the next nearest neighbor Ti moments bridged by O atoms are in parallel while the other half are in anti-parallel, likewise with Pn atoms.
For compounds Na$_2$Ti$_2$Pn$_2$O the moment around each Ti atom is found to be about $0.5\mu_B$ in both blocked checkerboard and bi-collinear antiferromagnetic orders. This is consistent with the spin density wave mechanism, in which the moment should be small since its formation is only attributed to those states nearby the Fermi energy. Considering there is a large amount of antiferromagnetic quantum fluctuation in low dimensional systems, the effective ordering moment around each Ti atom will be further much reduced. This is likely the reason why no magnetic ordering in compounds Na$_2$Ti$_2$Pn$_2$O has so far been detected by the powder neutron diffraction.\cite{bao}
A bi-collinear antiferromagnetic order was first introduced in the study of Fe-based superconductors to describe the ground state of $\beta$-FeTe,\cite{ma3} very interestingly as a strong evidence to against the view of spin density waves in the Fe-based superconducting parent compounds, since there is no any Fermi surface nesting vector to correspond to such a magnetic order, moreover there is a large magnetic moment of about 2.5$\mu_B$ formed on each Fe atom.
The substantial difference between the two cases lies in the fact that there is only one electron in 3$d$-orbital for a Ti$^{3+}$ ion, in contrast there are five electrons in 3$d$-orbital for an Fe$^{2+}$ ion. Accordingly it is not expected that there is strong electronic correlation effect in Na$_2$Ti$_2$Pn$_2$O. We have performed a GGA+$U$ calculation. We find that when $U$ is larger than 2~eV, the ferromagnetic state becomes the lowest energy state for both compounds. This contradicts the experimental observation that there is an antiferromagnetic-like transition around 300 K for Na$_2$Ti$_2$As$_2$O and around 120 K for Na$_2$Ti$_2$Sb$_2$O.\cite{ozawa2,ozawa3,ozawa4}
\subsubsection{Na$_2$Ti$_2$As$_2$O}
As aforementioned, the calculations show that any magnetic state, except the blocked checkerboard and bi-collinear antiferromagnetic states, is unstable and will relax into a nonmagnetic state for compound Na$_2$Ti$_2$As$_2$O.
\begin{figure}
\centering
\includegraphics[width=7.5cm]{fig9-As-blockneel-bandfs.eps}
\caption{(Color online) Na$_2$Ti$_2$As$_2$O in the blocked checkerboard antiferromagnetic state: (a) Band structure; (b) Magnetic unit cell; (c) Brillouin Zone with selected high symmetric points. Here the top of the valence band sets to zero.}\label{fig9}
\end{figure}
To calculate the blocked checkerboard antiferromagnetic (AFM) order in Na$_2$Ti$_2$As$_2$O, we construct such a crystal unit cell consisting of four formula cells, which is triclinic with three base vectors of (2$a$, 0, 0), (0, 2$a$, 0), and (0.5$a$, 0.5$a$, 0.5$c$) respectively, as shown in Fig. \ref{fig9}(b). For the bi-collinear antiferromagnetic order, the crystal unit cell is also constructed as a triclinic cell but with three base vectors of (2$a$, 0, 0), (0, $a$, 0), and (0.5$a$, 0.5$a$, 0.5$c$) respectively (see Fig. \ref{fig11}(b)).
\begin{figure}
\includegraphics[width=7.5cm]{fig11-As-bicoll-bandfs.eps}
\caption{(Color online) Na$_2$Ti$_2$As$_2$O in the bi-collinear antiferromagnetic state: (a) Band structure; (b) Magnetic unit cell; (c) Brillouin Zone with selected high symmetric points. Here the top of the valence band sets to zero.}\label{fig11}
\end{figure}
After the full structural optimization, we find that compound Na$_2$Ti$_2$As$_2$O in the blocked checkerboard AFM order is energetically lower by about 4.5 meV/Ti than in the bi-collinear AFM order. Figures \ref{fig9} and \ref{fig11} show the electronic band structures of compound Na$_2$Ti$_2$As$_2$O in the blocked checkerboard AFM and bi-collinear AFM orders respectively. In both AFM orders, Na$_2$Ti$_2$As$_2$O is transformed into a semiconductor from a semimetal, with a small band gap of about 0.15 and 0.05 eV respectively; moreover the moment around each Ti atom is found to be 0.56 and 0.53 $\mu_B$ respectively. These calculated results are summarized in Table \ref{table1}.
\begin{table} \renewcommand{\arraystretch}{1.3}
\addtolength{\tabcolsep}{+1pt}
\caption{Calculated energies and magnetic moments of the nonmagnetic, ferromagnetic, checkerboard, collinear, bi-collinear, and blocked checkerboard antiferromagnetic states respectively, for Na$_2$Ti$_2$As$_2$O and Na$_2$Ti$_2$Sb$_2$O. The energy in the nonmagnetic state is set to zero. The bar ``--" in table means that the corresponding magnetic state is not stable and will relax into a nonmagnetic state.}
\begin{tabular}{l c c c c}
\hline
\hline
Magnetic order & \multicolumn{2}{c}{Na$_2$Ti$_2$As$_2$O}
& \multicolumn{2}{c}{Na$_2$Ti$_2$Sb$_2$O} \\
\hline
&energy & moment & energy&moment \\
& (meV/Ti) & ($\mu_B$) & (meV/Ti) & ($\mu_B$) \\
\hline
nonmagnetic & 0 & 0 & 0 & 0 \\
ferromagnetic & -- & -- & -- & -- \\
checkerboard & -- & -- & -- & -- \\
collinear & -- & -- & -- & -- \\
bi-collinear & -20.8& 0.53 & -19.5 & 0.53 \\
blocked checkerboard & -25.3& 0.56 & -11.7 & 0.50 \\
\hline
\hline
\end{tabular}
\label{table1}
\end{table}
Table \ref{table2} lists the calculated lattice parameters with the selected bond lengths. There is a structural distortion found for compound Na$_2$Ti$_2$As$_2$O in the blocked checkerboard AFM state, in which the four Ti atoms in a magnetic block slightly gather towards their center, an As atom. This gathering leads to the Ti-O bond slightly elongating. Correspondingly, the bond distances among the four Ti atoms in a magnetic block are reduced. These four Ti atoms will form a compact square. As a result, the Ti-Ti and Ti-As chemical bonds within each magnetic block become stronger. Such a tetramer lattice distortion is certainly due to the spin-lattice coupling since it cannot happen in any nonmagnetic state. And it can be quantitatively described by the bond distance ratio of O-Ti-O/As-Ti-As, as reported in Table \ref{table2}. Such a distortion is in good agreement with the powder neutron diffraction observation that there is a commensurate structural distortion but the space group symmetry is retained after the anomaly transition.\cite{ozawa3}
\begin{table}
\caption{\label{table2} Calculated lattice parameters with some selected bond lengths in the blocked checkerboard antiferromagnetic state versus nonmagnetic state for Na$_2$Ti$_2$As$_2$O. }
\begin{tabular}{l c c}
\hline
\hline
& nonmagnetic & blocked checkerboard \\
\hline
a & 4.0597 & 4.0701 \\
c & 15.389 & 15.364 \\
Ti-O & 2.0299 & 2.0347/2.0352 \\
Ti-As & 2.7214 & 2.7175/2.7281 \\
O-Ti-O/(As-Ti-As) & 0.7459 & 0.7474 \\
\hline
\hline
\end{tabular}
\end{table}
There is also a structural distortion found for compound Na$_2$Ti$_2$As$_2$O in the metastable bi-collinear AFM state, which will be discussed in detail below in association with compound Na$_2$Ti$_2$Sb$_2$O.
\subsubsection{Na$_2$Ti$_2$Sb$_2$O}
We performed the similar calculations on compound Na$_2$Ti$_2$Sb$_2$O. However, we find that compound Na$_2$Ti$_2$Sb$_2$O in the bi-collinear AFM order is the lowest in energy, about 8 meV/Ti lower than in the blocked checkerboard AFM order. In Figs. \ref{fig12} and \ref{fig10}, we respectively show the calculated electronic band structures of Na$_2$Ti$_2$Sb$_2$O in the bi-collinear and blocked checkerboard AFM orders. In Fig. \ref{fig12} (b), the high symmetry directions $N'-Z$ and $M'-Z$ correspond to the moment parallel and anti-parallel alignments respectively. The shapes of the Fermi surface along these two directions are distinct. The volume enclosed by the electron-type Fermi surface sheets is larger than that enclosed by the hole-type Fermi surface sheets, which means that the electron carrier density is larger than the hole carrier density. It is noticed that the shapes of Fermi surface in both AFM orders are quite similar.
Unlike Na$_2$Ti$_2$As$_2$O, Na$_2$Ti$_2$Sb$_2$O remains semi-metallic in both AFM orders. This is consistent with the experimental observation that the electric resistivity of Na$_2$Ti$_2$As$_2$O is higher than that of Na$_2$Ti$_2$Sb$_2$O by approximately a factor of 10 after the anomaly happening.\cite{ozawa3} Examining the Fermi surfaces of Na$_2$Ti$_2$Pn$_2$O in the nonmagnetic state (Fig. \ref{fige}), we see that Na$_2$Ti$_2$As$_2$O has nearly perfect two-dimensional isotropic nesting between the Fermi surface sheets centered around X and M, which is expected to induce a full energy gap opened at the Fermi energy after the spin density wave transition. In contrast, Na$_2$Ti$_2$Sb$_2$O shows obvious anisotropic nesting between the Fermi surface sheets centered around X and M. Such an anisotropic nesting can only induce a partial energy gap opened at the Fermi energy. This is the underlying physics that Na$_2$Ti$_2$As$_2$O becomes semiconducting while Na$_2$Ti$_2$Sb$_2$O remains semi-metallic after the spin density wave transition, and may explain why the former in the isotropic blocked checkerboard AFM order is more favorable energetically while the latter in the anisotropic bi-collinear AFM order.
\begin{figure}
\includegraphics[width=7.5cm]{fig12-Sb-bicoll-bandfs.eps}
\caption{(Color online) Na$_2$Ti$_2$Sb$_2$O in the bi-collinear antiferromagnetic state: (a) Band structure; (b) Hole-type Fermi surface in the Brillouin Zone; (c) Electron-type Fermi surface. The Fermi energy is set to zero.}\label{fig12}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=7.5cm]{fig10-Sb-blockneel-bandfs.eps}
\caption{(Color online) Na$_2$Ti$_2$Sb$_2$O in the blocked checkerboard antiferromagnetic state: (a) Band structure; (b) Hole-type Fermi surface in the Brillouin Zone; (c) Electron-type Fermi surface in the Brillouin Zone. The Fermi energy is set to zero.}\label{fig10}
\end{figure}
Table \ref{table3} summarizes the calculated lattice parameters with selected bond lengths. The calculations show that there is also a small structural distortion for compound Na$_2$Ti$_2$Sb$_2$O in the bi-collinear AFM state, similar to the case of $\beta$-FeTe in the bi-collinear AFM state. The lattice constant becomes slightly longer along the spin anti-parallel alignment to lower AFM energy and shorter along the spin-parallel alignment to lower further ferromagnetic energy. As a result, the crystal primitive unit cell on Ti$_2$Sb$_2$O layer deforms from a square to a rectangle.
Compound Na$_2$Ti$_2$Sb$_2$O in the metastable blocked checkerboard AFM state shows a similar tetramer structural distortion as found in Na$_2$Ti$_2$As$_2$O above.
\begin{table}
\caption{\label{table3} Calculated lattice parameters with some selected bond lengths in the bi-collinear antiferromagnetic state versus nonmagnetic state for Na$_2$Ti$_2$Sb$_2$O. }
\begin{tabular}{l c c}
\hline
\hline
& nonmagnetic & bi-collinear \\
\hline
a (b) & 4.1369 & 4.1059 (4.1851) \\
c & 16.6428 & 16.6166 \\
Ti-O & 2.0684 & 2.0486/2.0924 \\
Ti-Sb & 2.8928 & 2.8605/2.9059 \\
O-Ti-O/(Sb-Ti-Sb) & 0.7150 & 0.7162/0.7201 \\
\hline
\hline
\end{tabular}
\end{table}
\section{Summary}
We have presented the first-principles calculations of the electronic structures of layered pnictide-oxide Na$_2$Ti$_2$Pn$_2$O (Pn=As, Sb). We find that the ground state of Na$_2$Ti$_2$As$_2$O is a blocked checkerboard antiferromagnetic semiconductor with a small gap of about 0.15 eV, while the ground state of Na$_2$Ti$_2$Sb$_2$O is a bi-collinear antiferromagnetic semimetal; and both have a small moment of about 0.5$\mu_B$ around each Ti atom. Moreover we find that there is a tetramer structural distortion in the blocked checkerboard antiferromagnetic state, in which the four Ti atoms at each magnetic block gather towards their center of an As atom, in good agreement with the experimental observation of a commensurate structural distortion but the space group symmetry retained after the anomaly happening. We confirm that there is a strong Fermi surface nesting in Na$_2$Ti$_2$Pn$_2$O. And we verify that the two antiferromagnetic states both are induced by the Fermi surface nesting rather than based on local magnetic moments. Actually these two antiferromagnetic states are close in energy for each of Na$_2$Ti$_2$Pn$_2$O, in which one is the ground state, then the other is the metastable state. We may consider layered pnictide-oxides Na$_2$Ti$_2$Pn$_2$O as a paradigm for spin density waves.
\begin{acknowledgements}
We wish to thank Professor Xian-Hui Chen for bringing the layered pnictide-oxides to our notice. This work is supported by National Program for Basic Research of MOST of China (Grant No. 2011CBA00112) and National Natural Science Foundation of China (Grant Nos. 11190024 and 91121008).
\end{acknowledgements}
{\it Note added.} While in the preparation of this manuscript, we learnt of a paper by David Singh (arxiv1209.4668), which reports that the instability at $X$ point associated with the Fermi surface nesting leads to a double striped antiferromagnetic (namely bi-collinear AFM) state in compound BaTi$_2$Sb$_2$O. We have subsequently examined a variety of magnetic states for compounds BaTi$_2$Pn$_2$O (Pn=As, Sb) and find that none of them is stable except the blocked checkerboard and bi-collinear antiferromagnetic states, which are however nearly degenerated with the nonmagnetic state with a lower energy of no more than 0.5 meV/Ti and the moment around each Ti atom is just about 0.2~$\mu_B$. The Fermi surfaces that we obtained for BaTi$_2$Pn$_2$O are similar to those shown in Figs. 6 and 7 reported in the arxiv1209.4668 paper, in which the Fermi surface nesting clearly is dramatically weaker than those in compound Na$_2$Ti$_2$Pn$_2$O reported in the present paper (see Fig. \ref{fige}).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,331 |
dbtoyaml - Database to YAML
===========================
Name
----
dbtoyaml -- extract the schema of a Postgres database in YAML format
Synopsis
--------
::
dbtoyaml [option...] dbname
Description
-----------
:program:`dbtoyaml` is a utility for extracting the schema of a
Postgres database to a `YAML <http://yaml.org>`_ formatted
specification. By default, the specification is output as a single
output stream, which can be redirected or explicitly sent to a file.
As an alternative, the ``--multiple-files`` option allows you to break
down the specification into multiple files, in general, one for each
object (see `Multiple File Output`_).
Note that `JSON <http://json.org/>`_ is an official
subset of YAML version 1.2, so the :program:`dbtoyaml` output should
also be compatible with JSON tools.
A sample of the output format is as follows::
schema public:
owner: postgres
privileges:
- postgres:
- all
- PUBLIC:
- all
table t1:
check_constraints:
t1_c2_check:
columns:
- c2
expression: (c2 > 123)
columns:
- c1:
not_null: true
type: integer
- c2:
type: smallint
- c3:
default: 'false'
type: boolean
- c4:
type: text
foreign_keys:
t1_c2_fkey:
columns:
- c2
references:
columns:
- c21
schema: s1
table: t2
owner: alice
primary_key:
t1_pkey:
columns:
- c1
schema s1:
owner: bob
privileges:
- bob:
- all
- alice:
- all
table t2:
columns:
- c21:
not_null: true
type: integer
- c22:
type: character varying(16)
owner: bob
primary_key:
t2_pkey:
columns:
- c21
privileges:
- bob:
- all
- PUBLIC:
- select
- alice:
- insert:
grantable: true
- delete:
grantable: true
- update:
grantable: true
- carol:
grantor: alice
privs:
- insert
The above should be mostly self-explanatory. The example database has
two tables, named ``t1`` and ``t2``, the first --owned by user
'alice'-- in the ``public`` schema and the second --owned by user
'bob'-- in a schema named ``s1`` (also owned by 'bob').
The ``columns:`` specifications directly under each table list each
column in that table, in the same order as shown by Postgres. The
specifications ``primary_key:``, ``foreign_keys:`` and
``check_constraints:`` define PRIMARY KEY, FOREIGN KEY and CHECK
constraints for a given table. Additional specifications (not shown)
define unique constraints and indexes.
User 'bob' has granted all privileges to 'alice' on the ``s1`` schema.
On table ``t2``, he also granted SELECT to PUBLIC; INSERT, UPDATE and
DELETE to 'alice' with GRANT OPTION; and she has in turn granted
INSERT to user 'carol'.
:program:`dbtoyaml` currently supports extracting information about
nearly all types of Postgres database objects. See :ref:`api-ref`
for a list of supported objects.
The behavior and options of ``dbtoyaml`` are patterned after the
`pg_dump utility
<https://www.postgresql.org/docs/current/static/app-pgdump.html>`_
since it is most analogous to using ``pg_dump --schema-only``.
Multiple File Output
--------------------
.. program:: dbtoyaml
The :option:`--multiple-files` option breaks down the output into
multiple files under a given root directory. The root is created if
it does not exist. The root directory name defaults to ``metadata``
in the system configuration file. The location of the root directory
defaults to the configuration item ``repository.path`` or can be
specified using the `--repository` option (see :doc:`config`
and :doc:`cmdargs` for further details).
The first level contains ``schema.<name>`` subdirectories,
``schema.<name>.yaml`` files and ``<objtype>.<name>.yaml`` files,
where ``<name>`` is the name of the corresponding objects and
``<objtype>`` is the type of top-level (non-schema) object. Note that
non-schema refers to Postgres extensions, casts, languages or
foreign data wrappers.
The second level, i.e., the ``schema.<name>`` subdirectories contain
``<objtype>.<name>.yaml`` files for each object in the particular
schema (but see below for caveats).
Object Name Conflicts
~~~~~~~~~~~~~~~~~~~~~
The names of Postgres objects can include characters that are not
allowed in filesystem object names. The most common example is the
division operator ('/'), but even table names can include
non-alphanumeric characters, if the identifiers are quoted.
In addition, one can define two or more objects with the same base
name, e.g., function ``foo(integer)`` and function ``foo(text)``, or a
table named ``"My Table"`` and another named ``"my table"`` or
``"MY TABLE"``. On certain operating systems, i.e., Windows, it is not
possible to create two files in the same directory that differ only in
the case of their characters.
In order to deal with the aforementioned issues, ``dbtoyaml`` places
certain objects in common files and transforms object identifiers so
that they are suitable for use in files and directories. For example,
the information for all user-defined casts are written to the file
``cast.yaml`` in the root directory. Functions with the same name but
different arguments are written to a single file, e.g.,
``function.foo.yaml`` in the first example above. Identifiers are
also converted to all lowercase, non-alphanumeric characters
(excluding underscore) are converted to underscores and, by default,
schema object names are truncated to 32 characters.
If two object names, thus transformed, map to the same string, then
the objects' information is written to the same file, e.g.,
``table.my_table.yaml`` in the second example above. If you prefer to
change the default truncation length, please define the environment
variable ``PYRSEAS_MAX_IDENT_LEN`` to some integer value (up to 63).
Version Control and Dropped Objects
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
It is expected that the output of ``dbtoyaml --multiple-files`` will
be placed under version control. Further invocations should then
update the files in the same directory tree. However, if an object is
dropped from the database ``dbtoyaml`` would normally only output
files for new or changed objects--and thus keep the dropped object
file under version control. To deal with dropped objects, ``dbtoyaml
-m`` outputs a special YAML "index" file, named
``database.<dbname>.yaml`` in the root directory. When ``dbtoyaml
-m`` is run a second time, it looks for this "index" file and if
found, proceeds to delete the previous run's ``.yaml`` files before
outputting new ones.
Options
-------
:program:`dbtoyaml` accepts the following command-line arguments (in
addition to the :doc:`cmdargs`):
dbname
Specifies the name of the database whose schema is to be extracted.
.. cmdoption:: -m, --multiple-files
Extracts the schema to a two-level directory tree. See `Multiple
File Output`_ above.
.. cmdoption:: -n <schema>
--schema <schema>
Extracts only a schema matching `schema`. By default, all schemas
are extracted. Multiple schemas can be extracted by using multiple
``-n`` switches. Note that normally all objects that belong to the
schema are extracted as well, unless excluded otherwise.
.. cmdoption:: -N <schema>
--exclude-schema <schema>
Does not extract schema matching `schema`. This can be given more
than once to exclude several schemas.
.. cmdoption:: -O, --no-owner
Do not output object ownership information. By default, as seen
in the sample output above, database objects (schemas, tables,
etc.) that can be owned by some user, are shown with an "owner:
*username*" element. The :option:`-O` switch suppresses all those
lines.
NOTE: If you specify `--no-owner`, you will most likely also want
to specify :option:`--no-privileges`. If the former is used
without the latter the resulting YAML output will have privilege
information without user data, which will cause errors if the YAML
is then fed to :doc:`yamltodb`.
.. cmdoption:: -t <table>
--table <table>
Extract only tables matching `table`. Multiple tables can be
extracted by using multiple :option:`-t` switches. Note that
selecting a table may cause other objects, such as an owned
sequence, to be extracted as well
.. cmdoption:: -T <table>
--exclude-table <table>
Do not extract tables matching `table`. Multiple tables can be
excluded by using multiple :option:`-T` switches.
.. cmdoption:: -x, --no-privileges
Do not output access privilege information. By default, as seen
in the sample output above, if specific GRANTs have been issued on
various objects (schemas, tables, etc.), the privileges are shown
under each object. The :option:`-x` switch suppresses all those
lines.
See also the NOTE under :option:`--no-owner`.
Examples
--------
To extract a database called ``moviesdb`` into a file::
dbtoyaml moviesdb > moviesdb.yaml
To extract only the schema named ``store``::
dbtoyaml --schema=store moviesdb > moviesdb.yaml
To extract the tables named ``film`` and ``genre``::
dbtoyaml -t film -t genre moviesdb -o moviesdb.yaml
To extract objects, to standard output, except those in schemas
``product`` and ``store``::
dbtoyaml -N product -N store moviesdb
To extract objects to a directory under version control::
dbtoyaml moviesdb -m movies/dbspec
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,638 |
package org.jetbrains.plugins.scala
package lang
package psi
package impl
package toplevel
import api.base.ScModifierList
import com.intellij.lang.ASTNode
import com.intellij.psi.{PsiElement, PsiClass}
import com.intellij.psi.tree._
import com.intellij.navigation.NavigationItem
import com.intellij.openapi.editor.colors._
import org.jetbrains.plugins.scala.lang.parser._
import org.jetbrains.plugins.scala.lang.psi.ScalaPsiElement
import org.jetbrains.plugins.scala.lang.lexer._
import psi.api.toplevel.packaging._
import psi.api.toplevel.templates._
import com.intellij.extapi.psi.ASTWrapperPsiElement
import com.intellij.psi.PsiReferenceList.Role
;
import com.intellij.lang.ASTNode;
import com.intellij.openapi.util.Pair;
import com.intellij.psi._;
import com.intellij.navigation._;
import com.intellij.psi.javadoc.PsiDocComment;
import com.intellij.psi.meta.PsiMetaData;
import com.intellij.util.IncorrectOperationException;
import _root_.java.util.Collection;
import _root_.java.util.Collections;
import _root_.java.util.List;
/**
* @author ilyas
*/
trait PsiClassFake extends PsiClass with PsiReferenceList {
//todo: this methods from PsiReferenceList to avoid NPE. It's possible for asking different roles, so we can
//todo: have problems for simple implementation of them
def getRole: Role = Role.EXTENDS_LIST
def getReferencedTypes: Array[PsiClassType] = PsiClassType.EMPTY_ARRAY
def getReferenceElements: Array[PsiJavaCodeReferenceElement] = PsiJavaCodeReferenceElement.EMPTY_ARRAY
def isInterface: Boolean = false
def isAnnotationType: Boolean = false
def isEnum: Boolean = false
def getExtendsList: PsiReferenceList = this //todo: to avoid NPE from Java
def getImplementsList: PsiReferenceList = this //todo: to avoid NPE from Java
def getExtendsListTypes: Array[PsiClassType] = PsiClassType.EMPTY_ARRAY
def getImplementsListTypes: Array[PsiClassType] = PsiClassType.EMPTY_ARRAY
def getSuperClass: PsiClass = null
def getInterfaces: Array[PsiClass] = PsiClass.EMPTY_ARRAY
def getSupers: Array[PsiClass] = PsiClass.EMPTY_ARRAY
def getSuperTypes: Array[PsiClassType] = PsiClassType.EMPTY_ARRAY
def getFields: Array[PsiField] = PsiField.EMPTY_ARRAY // todo
def getConstructors: Array[PsiMethod] = PsiMethod.EMPTY_ARRAY // todo
def getInnerClasses: Array[PsiClass] = PsiClass.EMPTY_ARRAY // todo
def getInitializers: Array[PsiClassInitializer] = PsiClassInitializer.EMPTY_ARRAY
def getAllFields: Array[PsiField] = getFields
def getAllMethods: Array[PsiMethod] = getMethods
def getAllInnerClasses: Array[PsiClass] = getInnerClasses
def findFieldByName(name: String, checkBases: Boolean): PsiField = null
def findMethodBySignature(patternMethod: PsiMethod, checkBases: Boolean): PsiMethod = null
def findMethodsBySignature(patternMethod: PsiMethod, checkBases: Boolean): Array[PsiMethod] = PsiMethod.EMPTY_ARRAY
def findMethodsAndTheirSubstitutorsByName(name: String, checkBases: Boolean): List[Pair[PsiMethod, PsiSubstitutor]] = Collections.emptyList[Pair[PsiMethod, PsiSubstitutor]]
def findMethodsAndTheirSubstitutors: List[Pair[PsiMethod, PsiSubstitutor]] = Collections.emptyList[Pair[PsiMethod, PsiSubstitutor]]
def getAllMethodsAndTheirSubstitutors: List[Pair[PsiMethod, PsiSubstitutor]] = Collections.emptyList[Pair[PsiMethod, PsiSubstitutor]]
def findInnerClassByName(name: String, checkBases: Boolean): PsiClass = null
def getLBrace: PsiJavaToken = null
def getRBrace: PsiJavaToken = null
def getScope: PsiElement = null
def isInheritor(baseClass: PsiClass, checkDeep: Boolean): Boolean = false
def isInheritorDeep(baseClass: PsiClass, classToPass: PsiClass): Boolean = false
def getVisibleSignatures: Collection[HierarchicalMethodSignature] = Collections.emptyList[HierarchicalMethodSignature]
def getModifierList: PsiModifierList = ScalaPsiUtil.getEmptyModifierList(getManager)
def hasModifierProperty(name: String): Boolean = name.equals(PsiModifier.PUBLIC)
def getDocComment: PsiDocComment = null
def isDeprecated: Boolean = false
def getMetaData: PsiMetaData = null
def isMetaEnough: Boolean = false
def hasTypeParameters: Boolean = false
def getTypeParameterList: PsiTypeParameterList = null
def getTypeParameters: Array[PsiTypeParameter] = PsiTypeParameter.EMPTY_ARRAY
def findMethodsByName(name: String, checkBases: Boolean): Array[PsiMethod] = Array[PsiMethod]()
def getMethods = Array[PsiMethod]()
def getQualifiedName: String = null
def getContainingClass: PsiClass = null
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 6,599 |
Apresentando o espaço aéreo de rota livre nos céus do Reino Unido
Last week, on 2nd December 2021, NATS implemented the biggest airspace change ever undertaken in the UK and introduced Free Route Airspace for the first time into UK skies. It's an exciting milestone in a project which has been underway for over five years and will enable huge fuel, flight time and CO2 savings. But what is Free Route Airspace (FRA) and what does it mean for airlines?
What is Free Route Airspace?
Free Route Airspace (FRA) means that airspace structures are removed above 25,000 ft, allowing aircraft to fly the route they want to between a defined entry and exit point. It allows airlines the freedom to plan and fly their optimal route considering things such as weather and wind speed. This level of flexibility hasn't previously been possible, with our airspace structure requiring aircraft to follow predefined routes.
This traditional way of flying was determined several decades ago when technology was significantly less advanced than it is today. At the time, it made sense to think of routes like fixed motorways in the sky where to get from A to B you would travel along one corridor until you got to a junction that let you change direction towards your destination. It meant that aircraft often fly further than necessary at these higher altitudes. Now, with new technology and navigational techniques, we can make this high level flying much more efficient.
What were the changes?
The changes implemented last week removed the long-established routes over Scottish airspace (the airspace over the North Sea, Scotland, North Atlantic, Northern Ireland and a small portion of northern England). This area of airspace accounts for roughly 1/3 of UK airspace, handling up to 2,000 flights a day and around 80% of all transatlantic flights, so the changes will significantly help to reduce cost, fuel burn, flight time and CO2 emissions.
This is one of many changes that NATS will undertake over the next few years to help the aviation industry achieve its target of reducing its CO2 emissions by at least 15% by 2030 (15% of 2019 levels) and reaching net-zero by 2050. With the introduction of sustainable fuels and new engine and aircraft technology still some years away, managing our airspace in a way which allows more efficient flying is already playing a major role in helping the aviation industry reduce its environmental impact. The introduction of FRA in Scottish airspace alone will save CO2 equivalent to the power used by 3,500 family homes every year.
The introduction of FRA in Scottish airspace marks the first of four planned deployments to introduce FRA across the UK over the next five years and it should be fully implemented across Europe by the end of 2027.
Uma abordagem inteligente para proporcionar crescimento futuro do aeroporto em tempos de incerteza
Planos de fundo #avgeek festivos para sua próxima videochamada
5 dúvidas sobre 5G no Brasil
Astrônomos registram explosão de estrela em 3D, assista
Estante robô, Sony no metaverso; veja as novidades do 1º dia da CES 2022 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,640 |
wget https://github.com/mozilla/geckodriver/releases/download/v0.16.1/geckodriver-v0.16.1-linux64.tar.gz
sudo sh -c 'tar -x geckodriver -zf geckodriver-v0.16.1-linux64.tar.gz -O > /usr/bin/geckodriver'
sudo chmod +x /usr/bin/geckodriver
rm geckodriver-v0.16.1-linux64.tar.gz
pip install -r requirements.txt
apt-get install xvfb -y
apt-get install tor -y
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,963 |
{"url":"https:\/\/projecteuclid.org\/euclid.agt\/1513796707","text":"Algebraic & Geometric Topology\n\nConnective $\\mathrm{Im}(J)$\u2013theory for cyclic groups\n\nKarlheinz Knapp\n\nAbstract\n\nWe study connective $Im(J)$\u2013theory for the classifying space $B\u2124\u2215pa$ of a finite cyclic $p$\u2013group and compute the $Im(J)$\u2013cohomology groups completely. We also compute the $Im(J)$\u2013homology groups, with the exception of a finite range of dimensions.\n\nArticle information\n\nSource\nAlgebr. Geom. Topol., Volume 7, Number 2 (2007), 797-828.\n\nDates\nAccepted: 20 March 2007\nFirst available in Project Euclid: 20 December 2017\n\nhttps:\/\/projecteuclid.org\/euclid.agt\/1513796707\n\nDigital Object Identifier\ndoi:10.2140\/agt.2007.7.797\n\nMathematical Reviews number (MathSciNet)\nMR2336242\n\nZentralblatt MATH identifier\n1175.19005\n\nCitation\n\nKnapp, Karlheinz. Connective $\\mathrm{Im}(J)$\u2013theory for cyclic groups. Algebr. Geom. Topol. 7 (2007), no. 2, 797--828. doi:10.2140\/agt.2007.7.797. https:\/\/projecteuclid.org\/euclid.agt\/1513796707\n\nReferences\n\n\u2022 J F Adams, Lectures on generalised cohomology, from: \u201cCategory Theory, Homology Theory and their Applications, III (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Three)\u201d, Springer, Berlin (1969) 1\u2013138\n\u2022 J F Adams, Stable homotopy and generalised homology, University of Chicago Press, Chicago, Ill. (1974) Chicago Lectures in Mathematics\n\u2022 A K Bousfield, The localization of spectra with respect to homology, Topology 18 (1979) 257\u2013281\n\u2022 A K Bousfield, On the homotopy theory of $K$-local spectra at an odd prime, Amer. J. Math. 107 (1985) 895\u2013932\n\u2022 M C Crabb, K Knapp, Applications of nonconnective ${\\rm Im}(J)$-theory, from: \u201cHandbook of algebraic topology\u201d, North-Holland, Amsterdam (1995) 463\u2013503\n\u2022 D Eisenbud, Commutative algebra, Graduate Texts in Mathematics 150, Springer, New York (1995) With a view toward algebraic geometry\n\u2022 S Hashimoto, A note on some periodicity of ${\\rm Ad}$-cohomology groups of lens spaces, Osaka J. Math. 23 (1986) 307\u2013312\n\u2022 A Jankowski, Splitting of $K$-theory and $g\\sb\\ast$ characteristic numbers, from: \u201cStudies in algebraic topology\u201d, Adv. in Math. Suppl. Stud. 5, Academic Press, New York (1979) 189\u2013212\n\u2022 S J\u00e4schke, Stabil sph\u00e4rische Elemente in der Bild-$J$-Homologie des klassifizierenden Raumes der zyklischen Gruppe mit $p$ Elementen, PhD thesis, Wuppertal (1992)\n\u2022 K Knapp, Connective $Im(J)$-theory for torsion-free spaces, the complex projective space as an example, preprint\n\u2022 K Knapp, On the $K$-homology of classifying spaces, Math. Ann. 233 (1978) 103\u2013124\n\u2022 K Knapp, Stably spherical classes in the $K$-homology of a finite group, Comment. Math. Helv. 63 (1988) 414\u2013449\n\u2022 K Knapp, Introduction to nonconnective ${\\rm Im}(J)$-theory, from: \u201cHandbook of algebraic topology\u201d, North-Holland, Amsterdam (1995) 425\u2013461\n\u2022 K Knapp, Anderson duality in $K$-theory and ${\\rm Im}(J)$-theory, $K$-Theory 18 (1999) 137\u2013159\n\u2022 T Kobayashi, S Murakami, M Sugawara, Note on $J$-groups of lens spaces, Hiroshima Math. J. 7 (1977) 387\u2013409\n\u2022 P Leiverkus, Diplomarbeit, Wuppertal\n\u2022 R Weth, Zur $A$-Theorie von $B\\mathbb{Z}\/p^{2}$, Diplomarbeit, Wuppertal (1992)","date":"2019-08-24 21:55:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 5, \"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\": 0.5405857563018799, \"perplexity\": 4892.345253591153}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027321786.95\/warc\/CC-MAIN-20190824214845-20190825000845-00351.warc.gz\"}"} | null | null |
Q: Use medium weight Fira Font I would like to use the medium weight Fira font in Latex but textmd doesn't seem to do anything. What am I doing wrong?
\usepackage[defaultsans]{FiraSans}
Hello \textmd{World}.
Hello \textbf{World}.
Here is what it looks like. As you can see, the first "World" is not any bolder than the regular text.
A: If you can switch to either XeLaTeX or LuaLaTeX to compile your document, you could download the OpenType version of the FiraSans font family, which features 16 [!!] separate font weights. In comparison, the FiraSans package (which works with pdfLaTeX) offers only 9 font weights.
As may be seen from the following screenshot, the weight difference between "Medium" and "SemiBold" is rather subtle. To avoid any uncertainty among your readers as to what constitutes "bold", it's a good idea to emply a weight that's 2 or even 3 steps heavier than the chosen default font weight. E.g., if "Fira Sans Book" were chosen as the base font weight, then either "Fira Sans Medium" or "Fira Sans SemiBold" would be suitable candidates for the 'bold' font variant. (What's selected as the "regular" font weight for a document does not have to be the same as the weight that's labelled "Regular" by the font designer.)
Thus, one might write
\setsansfont{Fira Sans Book}[%
ItalicFont = {Fira Sans Book Italic},
BoldFont = {Fira Sans SemiBold},
BoldItalicFont = {Fira Sans SemiBold Italic}]
to choose font weights "9" and "12" from the following list as the 'regular' and 'bold' font weights of one's document.
\documentclass{article}
\usepackage{fontspec}
\defaultfontfeatures{Ligatures={TeX,Common,Rare}}
\setmainfont{Fira Sans Regular} % set as default
\newfontfamily{\FSTwo}{Fira Sans Two} % 1
\newfontfamily{\FSFour}{Fira Sans Four} % 2
\newfontfamily{\FSEight}{Fira Sans Eight} % 3
\newfontfamily{\FSHair}{Fira Sans Hair} % 4
\newfontfamily{\FSThin}{Fira Sans Thin} % 5
\newfontfamily{\FSUltraLight}{Fira Sans UltraLight} % 6
\newfontfamily{\FSExtraLight}{Fira Sans ExtraLight} % 7
\newfontfamily{\FSLight}{Fira Sans Light} % 8
\newfontfamily{\FSBook}{Fira Sans Book} % 9
\newfontfamily{\FSRegular}{Fira Sans Regular} % 10
\newfontfamily{\FSMedium}{Fira Sans Medium} % 11
\newfontfamily{\FSSemiBold}{Fira Sans SemiBold} % 12
\newfontfamily{\FSBold}{Fira Sans Bold} % 13
\newfontfamily{\FSExtraBold}{Fira Sans ExtraBold} % 14
\newfontfamily{\FSHeavy}{Fira Sans Heavy} % 15
\newfontfamily{\FSUltra}{Fira Sans Ultra} % 16
\begin{document}
\obeylines % just for this example
{\FSTwo Hello World --- Two 1}
{\FSFour Hello World --- Four 2}
{\FSEight Hello World --- Eight 3}
{\FSHair Hello World --- Hair 4}
{\FSThin Hello World --- Thin 5}
{\FSUltraLight Hello World --- UltraLight 6}
{\FSExtraLight Hello World --- ExtraLight 7}
{\FSLight Hello World --- Light 8}
{\FSBook Hello World --- Book 9}
{\FSRegular Hello World --- Regular 10}
{\FSMedium Hello World --- Medium 11}
{\FSSemiBold Hello World --- SemiBold 12}
{\FSBold Hello World --- Bold 13}
{\FSExtraBold Hello World --- ExtraBold 14}
{\FSHeavy Hello World --- Heavy 15}
{\FSUltra Hello World --- Ultra 16}
\end{document}
A: The FiraSans package uses mdweight to set regular text, so calling \textmd explicitly has no effect whatsoever. If you want to use a specific weight explicitly, you can use one of the commands defined in FiraSans.sty:
*
*\firathin
*\firaultralight
*\firaextralight
*\firalight
*\firabook
*\firamedium
*\firasemibold
*\firaextrabold
*\firaheavy
Example:
\documentclass{article}
\usepackage[sfdefault]{FiraSans}
\begin{document}
Hello {\firathin World}.
Hello {\firaultralight World}.
Hello {\firaextralight World}.
Hello {\firalight World}.
Hello {\firabook World}.
Hello {\firamedium World}.
Hello {\firasemibold World}.
Hello {\firaextrabold World}.
Hello {\firaheavy World}.
\end{document}
A: Here is another alternative that might be useful if you either want to support a font family with no LaTeX package, or use the same extended font weight across different font families.
\documentclass[varwidth]{standalone}
\usepackage{fontspec}
\setmainfont{FiraSans}[
Scale = 1.0 ,
Ligatures = {Common, Discretionary, TeX} ,
Numbers = OldStyle ,
UprightFont = *-Regular ,
BoldFont = *-Bold ,
ItalicFont = *-RegularItalic ,
BoldItalicFont = *-BoldItalic ,
FontFace = {mb}{n}{*-Medium} ,
FontFace = {mb}{it}{*-MediumItalic} ,
FontFace = {sb}{n}{*-SemiBold} ,
FontFace = {sb}{it}{*-SemiBoldItalic} ,
Extension = .otf
]
\DeclareRobustCommand\mbseries{\fontseries{mb}\selectfont}
\DeclareTextFontCommand\textmb{\mbseries}
\DeclareRobustCommand\sbseries{\fontseries{sb}\selectfont}
\DeclareTextFontCommand\textsb{\sbseries}
\begin{document}
\textmd{Fira Sans} \textmb{Medium} \textsb{Semibold} \textbf{Bold} \\
{\itshape\mdseries Fira Sans Italic \mbseries Medium \sbseries Semibold \bfseries Bold}
\end{document}
This defines two new font weights, mb and sb, with commands analogous to textbf and \bfseries. Although the fontspec package lets you give a new series or shape an arbitrary name, those are the conventional series names from the LaTeX documentation, and a number of packages define commands such as \textsb.
Here, I went with \mdseries, but a number of existing packages use \mdweight instead. In theory, a series should set both the font weight and its width (such as Condensed or Expanded). So, if you implement a command to set the font weight, and also support more than one width, authors will expect to be able to change the width independently.
| {
"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"http:\/\/mathhelpforum.com\/calculus\/40726-differential-equations.html","text":"1. ## differential equations\n\nSolve the following differential equations.\n\ny' = ((y^2)+xy)\/(x^2)\n\nWhats the method of solving this? What theorem am i using?\n\n2. First divide by $y^2$ both sides : $\n\\tfrac{{y'}}\n{{y^2 }} = \\tfrac{1}\n{{x^2 }} + \\tfrac{1}\n{x} \\cdot \\tfrac{1}\n{y}\n$\n\nBut note that: $\n\\tfrac{{y'}}\n{{y^2 }} = - \\left( {\\tfrac{1}\n{y}} \\right)^\\prime\n$\n\nLet: $\n\\tfrac{1}\n{y} = z\n$\nwe have: $\n- z' = \\tfrac{1}\n{{x^2 }} + \\tfrac{1}\n{x} \\cdot z\n$\n\nYou can solve this last one by the integrating factor method\n\n3. Why do you divide by Y^2?\n\n4. $\\frac{y^{2}+xy}{x^{2}}=\\bigg( \\frac{y}{x} \\bigg)^{2}+\\frac{y}{x}.$\n\nThen let $y=xz.$","date":"2017-05-30 09:44:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 7, \"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\": 0.9585970640182495, \"perplexity\": 1459.7014222553826}, \"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-2017-22\/segments\/1495463614620.98\/warc\/CC-MAIN-20170530085905-20170530105905-00134.warc.gz\"}"} | null | null |
{"url":"http:\/\/www.r-bloggers.com\/page\/477\/","text":"## Cherry Picking to Generalize ~ NASA Global Temperature Trends ~ enhanced w\/\u00a0ggplot2\n\nApril 12, 2010\nBy\n\nIn a prior article, I tried to visualize the linear global temperatures trends for a grid of start and end years. The visual I created was confusing in that the specification of color scale was interdependent with the data values. I wanted a blue -> white -> red scale of the temperatures indicating cool ->\n\n## Using MKL-Linked R in Eclipse\n\nApril 12, 2010\nBy\n\nSetting up Eclipse to use MKL-Linked RIn my previous post, I showed how to compile R 2.10.1 using Intel's Math Kernel Library for the BLAS\/LAPACK interface. Even though it takes a bit of time to setup, I think the noticeably improved calculation speed justifies the effort. Although I'm happy to use R from the command line for basic stuff,...\n\n## Using MKL-Linked R in Eclipse\n\nApril 12, 2010\nBy\n\nSetting up Eclipse to use MKL-Linked RIn my previous post, I showed how to compile R 2.10.1 using Intel's Math Kernel Library for the BLAS\/LAPACK interface. Even though it takes a bit of time to setup, I think the noticeably improved calculation speed justifies the effort. Although I'm happy to use R from the command line for basic stuff,...\n\n## Jeroen Ooms\u2019s ggplot2 web interface \u2013 a new version released (V0.2)\n\nApril 12, 2010\nBy\n\nGood news. Jeroen Ooms released a new version of his (amazing) online ggplot2 web interface: yeroon.net\/ggplot2 is a web interface for Hadley Wickham\u2019s R package ggplot2. It is used as a tool for rapid prototyping, exploratory graphical analysis and education of statistics and R. The interface is written completely in javascript, therefore there is no need to install anything on the...\n\n## pgfSweave version 1.0.5 released\n\nApril 12, 2010\nBy\n\nVersion 1.0.5 is now on CRAN. This version brings some bug fixes as well as two new features: Unlabeled code chunks are now allowed. The correct version of PGF is now checked for on startup. If the version is < 2.00, the package will fail to load....\n\n## Arizona court rules statistical sampling is legal\n\nApril 12, 2010\nBy\n\nA court in Arizona has ruled that statistical sampling is legal for determining damages awarded to individual claimants when there are thousands of similar cases to be assessed simultaneously. In a case where 30,000 claims were filed Maricopa County, AZ by hospitals for improper reimbursement, the trial judge appointed a former judge as a special master in the case...\n\n## Working with themes in Lattice Graphics\n\nApril 12, 2010\nBy\n\nThe Trellis graphics approach provides facilities for creating effective graphs with a consistent look and feel and one of the good things about the system is the use of themes to define the colour, size and other features of the components that make up a graph. The lattice package in R is an implementation of\n\n## Example 7.32: Add reference lines to a plot; fine control of tick marks\n\nApril 12, 2010\nBy\n\nSometimes it's useful to plot regular reference lines along with the data. For a time-series plot, this can show when critical values are reached in a clearer way than simple tick marks.As an example, we revisit the empirical CDF plot shown in Example...\n\n## Anecdotal Evidence that Facebook Stores all Clicks?\n\nApril 11, 2010\nBy\n\nThis is not really news. A few months ago, news broke that Facebook recorded each user\u2019s clicks and profile views in a database. Of course, I am not at all surprised. I would be more surprised if they didn\u2019t store every single click. By now, most people have some sense as to how Facebook\u2019s recommendation system works. It typically performs...\n\n## Significant Figures in R and Info Zeros\n\nApril 11, 2010\nBy\n\nThe other day, I stumbled upon the signif function in R, so I thought I'd take a look at what it does and compare it with some results discussed in Chap. 3 \"Damaging Digits in Capacity Calculations\" of my GCaP book, viz., Example 3.5 on page 31. The m...\n\n## R frustration of the day\n\nApril 11, 2010\nBy\n\nWhenever you take a 1 column slice of a matrix, that gets automatically converted into a vector. But if you take a slice of several columns, it remains a matrix. The problem is you don\u2019t always know in advance how big the slice will be, so if you do this: newMatrix\n\n## Historical \/ Future Volatility Correlation Stability\n\nApril 11, 2010\nBy\n\nMichael Stokes, author of the MarketSci blog recently published a thought-provoking post about the correlation between historical and future volatility (measured as the standard deviation of daily close price percentage changes). This post is intended...\n\nApril 11, 2010\nBy\n\nThere is a central notion in Time Series Econometrics, cointegration. Loosely it refers to finding the long run equilibrium of two non-stationary series. As the most know non-stationary series examples comes from finance, cointegration is nowadays a tool for traders (not a common one though!). They use it as the theory behind pairs trading (aka\n\n## Summarising data using histograms\n\nApril 11, 2010\nBy\n\nThe histogram is a standard type of graphic used to summarise univariate data where the range of values in the data set is divided into regions and a bar (usually vertical) is plotted in each of these regions with height proportional to the frequency of observations in that region. In some cases the proportion of\n\n## Compiling 64-bit R 2.10.1 with MKL in Linux\n\nApril 10, 2010\nBy\n\nThe rationale for compiling R using the Intel Math Kernel LibraryRecently, there has been a surge in the use of Intel's Math Kernel Library (MKL; http:\/\/software.intel.com\/en-us\/intel-mkl\/) among data analysis packages. MKL is a highly optimized set of...\n\n## Compiling 64-bit R 2.10.1 with MKL in Linux\n\nApril 10, 2010\nBy\n\nThe rationale for compiling R using the Intel Math Kernel LibraryRecently, there has been a surge in the use of Intel's Math Kernel Library (MKL; http:\/\/software.intel.com\/en-us\/intel-mkl\/) among data analysis packages. MKL is a highly optimized set of...\n\n## Where do you sit? Author position and the h-index\n\nApril 10, 2010\nBy\n\nI was recently introduced to the concept of the h-index and was compelled to find out my own h-index via Scopus.\u00a0 Numbers don't matter, but discussion with my colleagues turned to the issue of author position.\u00a0 We quickly decided that there are three important \"positions\" in the list of authors for a publication: first, last and everywhere else...\n\n## Where do you sit? Author position and the h-index\n\nApril 10, 2010\nBy\n\nI was recently introduced to the concept of the h-index and was compelled to find out my own h-index via Scopus.\u00a0 Numbers don't matter, but discussion with my colleagues turned to the issue of author position.\u00a0 We quickly decided that there are three important \"positions\" in the list of authors for a publication: first, last and everywhere else...\n\n## Because it\u2019s Friday: Pixels invade New York\n\nApril 9, 2010\nBy\n\nPosted for no other reason than it warms my gamer-geek heart to see NYC taken over by 8-bit video game characters. The Tetris sequence is particularly cool. Update: The original video was deleted from YouTube, I'm guessing because of copyright issues with the music. This version has no music. (Thanks to reader MB in the comments for the heads-up.)\n\n## REvolution R Community 3.2 now available\n\nApril 9, 2010\nBy\n\nREvolution R Community, REvolution's free distribution based on R from the R Project, has been updated to version 3.2 and is now available for download for Windows and MacOS. Some features of this release include: Upgraded R engine. This release is based on R 2.10.1, the latest release (as of this writing). This brings many new features to the...\n\n## Chicago R User Group\u2026 It\u2019s for the sexy people!\n\nApril 9, 2010\nBy\n\nI think we all know that Morris Day was talking about when he wrote the lyrics to \u201cThe Bird\u201d: Yes! Hold on now, this dance ain\u2019t for everybody. Just the sexy people. White folks, you\u2019re much too tight. You gotta shake your head like the black folks. You might get some tonight. Look out! That\u2019s right, he was talking about the new\n\n## The Future of Math is Statistics\n\nApril 9, 2010\nBy\n\nThe future of math is statistics\u2026 and the language of that future is R: I\u2019ve often thought there was way too little \u201cstatistical intuition\u201d in the workplace. I think Author Benjamin would agree.\n\n## Maximum Probability of Profit\n\nApril 9, 2010\nBy\n\nTo continue with the LSPM examples, this post shows how to optimize a Leverage Space Portfolio for the maximum probability of profit. The data and example are again taken from The Leverage Space Trading Model by Ralph Vince. These optimizaitons take ...\n\n## GLMM using DPpackage\n\nApril 9, 2010\nBy\n\nI was able to fit a\u00a0semi-parametric Bayesian\u00a0GLMM model using DPpackage. It took me many hours to sample from the posterior distribution (DPM prior):MCMC scan 1000 of 5000 (CPU time: 18950.080 s)MCMC scan 2000 of 5000 (CPU time: 22510.100 s)M...\n\n## Gravity Game in R\n\nApril 8, 2010\nBy\n\nSo why should R only be used for \u2019serious\u2019 stuff? No longer! I\u2019ve written the following code in R which executes a little gravitational physics game. The goal of the game is simple. You supply a velocity and direction to a spaceship with the goal of getting the ship to the\n\n## R: heatmaps with gplots\n\nApril 8, 2010\nBy\n\nI use heatmaps quite a lot for visualizing data, microarrays of course but also DNA motif enrichment, base composition and other things. I particular like the heatmap.2 function of the gplots package. It has a couple of defaults that are a little ugly ...\n\n## R: heatmaps with gplots\n\nApril 8, 2010\nBy\n\nI use heatmaps quite a lot for visualizing data, microarrays of course but also DNA motif enrichment, base composition and other things. I particular like the heatmap.2 function of the gplots package. It has a couple of defaults that are a little ugly ...\n\n## New R User Group in Dallas\n\nApril 8, 2010\nBy\n\nWow, it's a big week for new R User Groups. Larry D'Agostino has started up a new R user group based in Dallas, Texas (USA). It's just getting started, and Larry posted the following request on the r-help mailing list: I would like to know if there is anyone like me interested in an R User Group in Dallas,...\n\nMake sure to click on the image to see the large version. Code for this graph: moxbuller = function(n) { u = runif(n) v = runif(n) x = cos(2*pi*u)*sqrt(-2*log(v)) y = sin(2*pi*v)*sqrt(-2*log(u)) r = list(x=x, y=y) return(r) } r = moxbuller(50000) par(bg=\"black\") par(mar=c(0,0,0,0)) plot(r$x,r$y, pch=\".\", col=\"blue\", cex=1.2)","date":"2014-09-03 07:01:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.29578399658203125, \"perplexity\": 3050.215926634865}, \"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-2014-35\/segments\/1409535925433.20\/warc\/CC-MAIN-20140901014525-00345-ip-10-180-136-8.ec2.internal.warc.gz\"}"} | null | null |
200 Southeast Oak Street E.
It's all about people, product and technology. We've all heard it many times before… But let's be honest - as we build products and create innovation, we (the business, stakeholders and partners) tend to direct our focus heavily (some would even say manically) on how we will define and measure success.
For this year's ProdConf, we'll examine some of the best ways to successfully put the "people" piece back into the product development equation and leverage the enormous power it can yield for team/organizational health and success. Using the "people" lens, we'll delve into topics in management, product thinking, leadership and DevOps to gain a holistic view of how and why it's time to put people first.
There will be plenty of time for people to engage the speakers directly as well as interact with other participants.If you are interested in speaking, please email us at hello@devjam.com with your topic, talk title, and a brief description. | {
"redpajama_set_name": "RedPajamaC4"
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Préveranges ist eine französische Gemeinde mit Einwohnern (Stand ) im Département Cher in der Region Centre-Val de Loire. Préveranges ist die südlichste Gemeinde des Départements Cher. Sie liegt etwa 50 Kilometer südwestlich von Bourges. Die Gemeinde gehört zum Kanton Châteaumeillant im Arrondissement Saint-Amand-Montrond.
Bevölkerungsentwicklung
Siehe auch
Liste der Monuments historiques in Préveranges
Literatur
Le Patrimoine des Communes du Cher. Band 1. Flohic Editions, Paris 2001, ISBN 2-84234-088-4, S. 323–324.
Weblinks
Ort in Centre-Val de Loire | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,202 |
\section{Background}
\jbbo{\section{Introduction}}
\label{sec:problem}
Linear equality constrained minimization problems are formulated as
\begin{equation}
\label{eq:main}
\underset{\b{x} \in \mathbb{R}^n}{ \text{ minimize } } f(\b{x})
\quad \text{subject to} \quad \b{A} \b{x} = \b{b},
\end{equation}
where $ f:\mathbb{R}^n \to \mathbb{R} $ and $ \b{A} \in \mathbb{R}^{m \times n} $.
We assume that the number of variables $n$ is large,
$g(x) = \nabla f(x)$ is available,
$ A $ is sparse,
and that the initial guess $ x_0 $ is feasible: $Ax_0=b$.
\jbr{If $ \b{A} $ has low rank, one can obtain a full-rank matrix
by deleting rows in $ \b{A} $ that correspond to small
diagonals of the triangular matrix in a sparse QR factorization of $ \b{A}\tp $.
Our methods here
use the rank information contained in sparse QR factors, and
thus we assume
that $A$ has full rank until
implementation details are described in
section~\ref{sec:appB}.}
\jbbo{For large problems,} computing the Hessian $ \nabla^2 f(\b{x}) \in \mathbb{R}^{n \times n} $ is \jbbo{often} not practical, \jbbo{and} we approximate this matrix using
a limited-memory BFGS (Broyden-Fletcher-Goldfarb-Shanno, \cite{Broyden70,Fletcher70,Goldfarb70,Shanno70}) quasi-Newton matrix
$ \bk{B} \approx \nabla^2 f(\bk{x}) $. Starting from $x_0$,
we update iterates according to $ \bko{x} = \bk{x} + \bk{s} $. The
step $ \bk{s} $ is computed as the solution of a quadratic trust-region subproblem, in which
the quadratic objective is defined as
$ q(s) \equiv \b{s}\tp \bk{g} + \frac{1}{2} \b{s} \tp \bk{B} \b{s} $ with $\bk{g} \equiv g(\bk{x})$. For a given trust-region radius $\Delta > 0$ and norm $\norm{\cdot}$, the
trust-region subproblem is
\begin{equation}
\label{eq:trsub}
\underset{ \| \b{s} \| \le \Delta }{ \text{ minimize } } q(\b{s})
\quad \text{subject to} \quad \b{A} \b{s} = \b{0},
\end{equation}
which ensures that each search direction is in the nullspace of \jbrt{$A$}, and thus each iterate $x_k$ \jbrt{is} feasible.
\subsection{Background}
\label{sec:bkg}
\jbbo{Large problems of the form \eqref{eq:main} are the focus of recent research
because large statistical- and machine-learning problems can be cast in this way. As such,
\eqref{eq:main} constitutes the backbone of the Alternating Direction Method of Multipliers (ADMM)
\cite{BoydEtAl11}, with applications to optimal exchange problems, consensus and sharing problems, support-vector machines, and more. Recent work \cite{FuZhangBoyd20} emphasizes methods that use
gradients
of $\jbrt{f}$
and suggest accelerations via quasi-Newton approximations.}
Quasi-Newton methods
estimate Hessian matrices using low-rank updates at each iteration (typically \hbox{rank-1} or \hbox{rank-2}).
Starting from an initial matrix, the so-called \emph{compact representation} of quasi-Newton matrices \cite{ByrNS94} is a matrix
representation of the recursive low-rank updates. Because the compact representation
enables effective limited memory implementations, which update a small number of previously stored
vectors, these methods are well suited to large problems.
\jbr{Trust-region and line-search
methods are standard
for determining search directions for smooth problems,
and each approach has its own merits.
Combinations of trust-region methods and quasi-Newton compact representations have
been developed in \cite{BruBEM19,B18,BruEM15,BurdakovLMTR16}. Widely
used quasi-Newton line-search methods are \cite{Byrd2006,LiuNocedal89,ZhangHager04,ZhuByrdLuNocedal97}. The main ideas
in this article are applicable to both trust-region and line-search methods.}
\subsection{Compact representation}
\label{sec:compact}
A storage-efficient approach to quasi-Newton matrices is
the compact representation of Byrd et al.\ \cite{ByrNS94},
which represents the BFGS matrices in the form
\begin{equation}
\label{eq:Bk}
\bk{B} = \gamma_k \b{I} + \bk{J} \bk{M} \bk{J} \tp,
\end{equation}
with scalar $\gamma_k > 0$.
The history of vectors $\jbrt{\{}\bk{s}\jbrt{\}} = \jbrt{\{}\bko{x} - \bk{x}\jbrt{\}}$ and $\jbrt{\{}\bk{y}\jbrt{\}} = \jbrt{\{}\bko{g} - \bk{g}\jbrt{\}}$
is stored in rectangular
$\bk{S} \equiv \bmat{\b{s}_0, \dots, \b{s}_{k-1}} \in \mathbb{R}^{n \times k}$
and $\bk{Y} \equiv \bmat{\b{y}_0, \dots, \b{y}_{k-1}} \in \mathbb{R}^{n \times k}$.
\jbrt{The} matrices
\begin{align}
\bk{J} &\equiv \bmat{\bk{S} & \bk{Y} },
\\ \bk{S}\tp \bk{Y} &\equiv \bk{L} + \bk{D} + \bar{\b{T}}_k,
\\ \bk{M} &\equiv
-
\bmat{
\delta_k \bk{S}\tp \bk{S} & \,\delta_k \bk{L}
\\ \delta_k \bk{L}\tp & \,- \bk{D}}^{-1}
\end{align}
are defined with $\delta_k = 1/\gamma_k$,
where $\bk{L}$ \jbr{and} $\bar{\b{T}}_k$ are the strictly lower and upper triangular parts
of $\bk{S}\tp \bk{Y}$ and $\bk{D}$ is the diagonal. For large problems, limited-memory versions
store only a small subset of recent pairs $\{ \b{s}_i, \b{y}_i \}_{i=k-l}^{k-1} $,
resulting in storage-efficient matrices $ \bk{J} \in \mathbb{R}^{n \times 2l} $
and $ \bk{M} \in \mathbb{R}^{2l \times 2l} $ \jbr{where $ l \ll n $}.
\jbk
Following Byrd et al.\
\cite[Theorem 2.2]{ByrNS94},
the inverse BFGS matrix has the form
\begin{equation}
\label{eq:iBk}
\bik{B} = \delta_k I + \bk{J} \bk{W} \bk{J} \tp,
\end{equation}
where
$ \bk{W} \in \mathbb{R}^{2l \times 2l} $ is given by
\begin{equation}
\label{eq:Wk}
\bk{W} =
\bmat{
\bk{T}^{-\top} (\bk{D} + \delta_k \bk{Y} \tp \bk{Y}) \bk{T}^{-1} & -\delta_k \bk{T}^{-\top} \\
-\delta_k\bik{T} & 0_{l \times l}}.
\end{equation}
The diagonal matrix $\bk{D}$ (and hence the upper triangular matrix $\bk{T} \equiv \bk{D} + \bar{\b{T}}_k$) are nonsingular as long as $\bk{B}$ is also.}
\subsection{Outline}
\label{sec:outline}
\jbbo{Section~2 describes our contributions in the context of large problems, while section~3 motivates our proposed representations.
Section~4 develops the reduced compact representation and updating techniques that enable efficient implementations.
Section~5 describes computations of orthogonal
projections, and the trust-region strategy for optimization.
Section~6 gives an efficient method
when an $\ell_2$-norm trust-region subproblem is used.
Sections~7 and~8 develop an effective factorization, and a method that uses a shape-changing norm in the trust-region subproblem.
Numerical experiments are reported in section~9,
and conclusions are drawn in section~10.}
\section{Contributions}
\label{sec:contrib}
\jbr{The first-order necessary conditions for the solution of problem \eqref{eq:trsub}
without the norm constraint
are characterized by the linear system}
\begin{equation}\label{eq:KKTsystem}
\bmat{\b{B}_k & \b{A}\tp
\\ \b{A} & \b{0}_{m \times m}}
\bmat{\jbr{\b{s}_E} \\ \jbr{\bs{\lambda}_E}}
=
\bmat{-g_k \\ \b{0}_m},
\end{equation}
where $ \jbr{\bs{\lambda}_E} \in \mathbb{R}^m $ is a vector of Lagrange multipliers
\jbr{and $ \b{s}_E $ denotes the ``equality'' constrained minimizer of \eqref{eq:trsub}.
Adopting the naming convention of \cite[Sec.~16.1, p.~451]{NocW06}, we refer to
\eqref{eq:KKTsystem} as the KKT system
(a slight misnomer, as use of the system for the equality constrained setting
predates the work of Karush, Kuhn, and Tucker).}
For large $n$, compact representations
of the (1,1) block in the inverse KKT matrix were recently proposed by Brust et al.\ \cite{BMP19}.
Two limited-memory trust-region algorithms, \rfm{{\small LTRL2-LEC} and {\small LTRSC-LEC} (which we refer to as \TR1 and \TR2 in the numerical experiments in Sec.~\ref{sec:numex})},
use these representations to compute search directions efficiently when $\b{A}$ has relatively few rows.
This article develops efficient algorithms when the number of equality constraints is large and the constraint matrix is sparse. In particular,
by exploiting the \jbr{property} that part of the solution to the KKT system
\jbbo{is unaltered} when it is projected onto the nullspace of \jbrt{$A$},
we develop \emph{reduced compact representations (RCR)}, \jbbo{which need
a small amount of memory and} lead to efficient
methods for \jbrt{solving} problems with many constraints
\Red{(large $m$ and $n$) and possibly many degrees of freedom (large $n-m$)}.
In numerical experiments \jbrt{when solving} large problems, the proposed methods
are often significantly more efficient than both our previous implementations and IPOPT \cite{WaechterBiegler06}.
\section{Motivation}
\label{sec:motiv}
The solution $\jbr{\b{s}_E}$ in \eqref{eq:KKTsystem} can be computed from only the (1,1)
block of the inverse KKT matrix, as opposed to both the (1,1) and (1,2) blocks,
because of the zeros in the right-hand side. Let $ \bk{V}$ be the (1,1) block
of the inverse KKT matrix (obtained for example
from a block LDU factorization). It is given by
\begin{equation}
\label{eq:invKKT11}
\bk{V} \equiv (\bik{B} - \bik{B}\b{A}\tp(\b{A} \bik{B} \b{A}\tp)^{-1}\b{A}\bik{B}),
\end{equation}
and then $ \jbr{\b{s}_E} = -\bk{V}\bk{g} $.
At first sight the expression in \eqref{eq:invKKT11}
\jbr{appears to be expensive to compute because of the multiple inverse operations
and matrix-vector products.}
However, as
$
\bik{B} = \delta_k I + \bk{J} \bk{W} \bk{J} \tp
$,
we can exploit computationally useful structures. Specifically, with
$
\bk{G} \equiv (\b{A} \bik{B} \b{A} \tp)^{-1}
$
and
$
\bk{C} \equiv \b{A} \bk{J} \bk{W},
$
\cite[Lemma 1]{BMP19} describes the expression
\begin{equation}
\label{eq:cmpVk_L1}
\bk{V} = \delta_k I +
\bmat{\b{A}\tp & \bk{J} }
\bmat{ -\delta_k^2 \bk{G} & - \delta_k \bk{G} \bk{C} \\
-\delta_k \bk{C} \tp \bk{G} & \bk{W} - \bk{C} \tp \bk{G} \bk{C}}
\bmat{ \b{A} \\
\bk{J}\tp}.
\end{equation}
For large $n$, once the components of the middle matrix in \eqref{eq:cmpVk_L1} are available, this compact
representation of $ \bk{V} $ enables efficient computation
of a matrix-vector product $ \bk{V} \bk{g} $, \jbbo{hence the solution of \eqref{eq:KKTsystem}}, and an economical eigendecomposition $ \bk{V} = \b{U} \b{\Lambda} \b{U}\tp $.
However, unless $m$ is small (there are few rows in $ \b{A} $), \jbrt{multiplying}
with the $ (m+2l) \times (m+2l) $ middle matrix is not practical.
With large $n$ and $m$ in mind, we
\jbr{note} that the solution $ s_E $ is unchanged if instead of
$\bk{g}$ a projection of this vector onto the nullspace of $\b{A}$ is
used, or if $\jbr{\b{s}_E}$ is projected onto the nullspace of $ \b{A} $.
This is a consequence of the properties of $ \bk{V} $.
To
formalize these statements, let the orthogonal projection matrix onto $ \text{null}(\b{A}) $ be
$
\b{P} = \b{I}_n - \b{A}\tp (\b{A} \b{A}\tp)^{-1}\b{A}.
$
\jbr{Since the} \jbrt{columns of the} (1,1) \jbr{block of the inverse from} \eqref{eq:KKTsystem}
(namely \jbrt{columns of} $V_k$) \jbr{are in the nullspace of $A$, the orthogonal projection onto $\text{null}(A) $ acts as an identity operator on the vector space spanned by $V_k$:}
\begin{equation}
\label{eq:projKKT1}
\bk{V} = \bk{V}\b{P}
= \b{P}\tp\bk{V}
= \b{P}\tp \bk{V} \b{P}.
\end{equation}
\jbr{Relation \eqref{eq:projKKT1} can equivalently be derived from
\eqref{eq:invKKT11}, the expression for $P$, and the equality $ V_k A \tp = 0 $.}
The methods in this article are based on representations of projected matrices $ \b{P}\tp \bk{V} \b{P}$ $ \in \mathbb{R}^{n \times n}$,
whose properties enable desirable numerical advantages for large $ n $ and $m$.
Instead of \jbrt{multiplying}
with the possibly \jbrt{large} $ \bk{G} \in \mathbb{R}^{m \times m} $ and
$ \bk{C} \in \mathbb{R}^{m \times 2l} $ in \eqref{eq:cmpVk_L1},
we store the matrices $ \bk{S} \in \mathbb{R}^{n\times l}$ and
$\bk{Z} \equiv \b{P}\bk{Y} \in \mathbb{R}^{n\times l} $ and
\jbr{small square matrices that depend on the memory parameter $l$ but
not on $m$. The columns of \bk{Z} are defined as
$
\bk{z} = \b{P}\bk{y} = \b{P}(\bko{g}-\bk{g}),
$ and they are contained in the nullspace of $A$.}
\jbr{With \eqref{eq:invKKT11} and \eqref{eq:cmpVk_L1}} we motivated the solution of \eqref{eq:trsub} without the norm
constraint (giving the equality-constrained step $s_E$). Computing $s_E$ is important for the implementation of practical algorithms, but it is even more important to solve \eqref{eq:trsub} efficiently with the norm constraint. In Sec.~\ref{sec:l2}, using the $ \ell_2 $ norm, we develop a modified
version of $ \bk{V} $ as a function of a scalar parameter $\sigma>0$, i.e., $ \bks{V} $. In Secs.~\ref{sec:eigen} and~\ref{sec:SCNorm},
we describe how the structure of
$ \bk{V} $ can be exploited to compute an inexpensive eigendecomposition
that, when combined with a judiciously chosen norm \jbr{(the shape-changing infinity norm
from \cite[Sec. 4.2.1]{BurdakovLMTR16})}, provides a search direction by
an analytic formula.
\jbk{Note that the representation of $ \bk{V} $ is not specific to the L-BFGS matrix,
and other compact quasi-Newton matrices could be used
(Byrd et al.\ \cite{ByrNS94}, DeGuchy et al.\ \cite{DeGuchyEM16}).
}
\section{Reduced compact representation (RCR)}
\jbr{This section describes a computationally effective representation of \eqref{eq:projKKT1},
which we call the \emph{reduced compact representation} (RCR). In section~\ref{sec:redHessian},
the RCR is placed into historical context with reduced Hessian methods. Subsequently, sections \ref{sec:redComp}--\ref{sec:compComplx} develop the specific formulas that enable effective computations.}
\subsection{Reduced Hessian}
\label{sec:redHessian}
\jbk{The name \emph{reduced compact representation} is related to the term
\emph{reduced Hessian} \cite{GillMurray74},
where $ \jbr{\bh{Z}} \in \mathbb{R}^{n \times (n-m)} $ denotes a basis for the
nullspace of $\b{A}$ (satisfying $A\jbr{\bh{Z}}=0$). In turn, $\jbr{\bh{Z}}$ defines the so-called reduced Hessian matrix
as $ \jbr{\bh{Z}}\tp \nabla^2 f_k \jbr{\bh{Z}} $ or $ \jbr{\bh{Z}}\tp \bk{B} \jbr{\bh{Z}} $.
In order to compute an equality-constrained step
$ \jbr{\b{s}_E} $, a reduced Hessian method solves $ (\jbr{\bh{Z}}\tp \bk{B} \jbr{\bh{Z}})\jbr{\bh{s}_E} = -\jbr{\bh{Z}}\tp \bk{g} $ and computes $ \jbr{\b{s}_E} = \jbr{\bh{Z}}\jbr{\bh{s}_E} $. Known computational challenges
with reduced Hessian methods are that a desirable basis $ \jbr{\bh{Z}} $ may be expensive to
compute, the condition number of the reduced linear system may be larger than the original one, and
the product $ \jbr{\bh{Z}}\tp \bk{B} \jbr{\bh{Z}} $ is not necessarily sparse even if the
matrices themselves are. For large-scale problems, these challenges can result in
significant computational bottlenecks. In the sequel we refer to $ \b{P}\tp \bk{V} \b{P} $
as a \emph{reduced compact representation} because it has a reduced memory footprint compared to
$ \bk{V} $ in \eqref{eq:cmpVk_L1}
(although the matrices have the same dimensions). We also note that $ \bk{V} $
and $ \b{P}\tp \bk{V} \b{P} $ have the same condition}, \jbbo{and $ \b{P}\tp \bk{V} \b{P} $
has structure that enables efficient implementations.}
\subsection{Reduced compact representation}
\label{sec:redComp}
To simplify \eqref{eq:cmpVk_L1}, we note
that $ \bk{V} = \b{P}\tp \bk{V} \b{P} $, \jbr{that} $ \b{P}\tp\! \b{A} \tp = 0 $,
and $ \b{P} \tp\! \bk{J} = \bmat{\bk{S} & \bk{Z}} $ \jbr{(where $\b{P}\tp \bk{Y} \equiv \bk{Z}$ by definition)}, so that
\begin{equation*}
\b{P}\tp \bk{V} \b{P}
= \delta_k \b{P} +
\bmat{\bk{S} & \bk{Z} }
( \bk{W} - \bk{C} \tp \bk{G} \bk{C} ) \bmat{\bk{S} & \bk{Z} }\tp\!.
\end{equation*}
In Appendix A we show that $ \bk{C} \tp \bk{G} \bk{C} $
simplifies to
$
\bk{C} \tp \bk{G} \bk{C} =
\big[
\begin{smallmatrix}
(\bk{C} \tp \bk{G} \bk{C})_{11} & 0 \\
0 & 0
\end{smallmatrix}
\big]
$ with $(\bk{C} \tp \bk{G} \bk{C})_{11} = \delta_k \bk{T}^{-\top} \bk{Y} \tp \b{A} \tp (\b{A} \b{A} \tp)^{-1} \b{A} \bk{Y} \bik{T} $. Based on this, we derive a \emph{reduced compact representation} of $ \bk{V} $.
\smallskip
{Lemma 1:} The \emph{RCR of $\bk{V}$
in \eqref{eq:cmpVk_L1} for the L-BFGS matrix is given by}
\begin{equation}
\label{eq:redComp}
\bk{V} = \delta_k I +
\bmat{\b{A}\tp & \bk{S} & \bk{Z} }
\bmat{ - \delta_k (\b{A} \b{A} \tp)^{-1} & \\
& \bk{N}
}
\bmat{\b{A} \\ \bk{S} \tp \\ \bk{\jbr{Z}} \tp },
\end{equation}
\emph{where}
\begin{equation*}
\bk{N} = \bmat{ \bk{T}^{-\top} (\bk{D} + \delta_k \bk{Z} \tp \bk{Z}) \bk{T}^{-1} & -\delta_k \bk{T}^{-\top} \\
-\delta_k\bik{T} & 0_{k \times k} }.
\end{equation*}
\smallskip
\begin{proof} Multiplying $ \bk{V} $ in \eqref{eq:cmpVk_L1} from the left and right
by $ \b{P} \tp $ and $ \b{P} $ yields
$ \bk{V} = \delta_k \b{P} + \bmat{\bk{S} & \bk{Z} } ( \bk{W} - \bk{C} \tp \bk{G} \bk{C} ) \bmat{\bk{S} & \bk{Z} }\tp $. Since only the (1,1) block in $ \bk{C} \tp \bk{G} \bk{C} $
is nonzero, we consider only the (1,1) blocks, namely
\begin{equation*}
(\bk{W})_{11} - (\bk{C}\tp \bk{G} \bk{C})_{11} =
\bk{T}^{-\top} (\bk{D} + \delta_k( \bk{Y} \tp \bk{Y} -
\bk{Y} \tp \b{A} \tp (\b{A} \b{A} \tp)^{-1} \b{A} \bk{Y} )) \bk{T}^{-1}.
\end{equation*}
\jbr{Since}
$ \bk{Y} \tp \b{P}\tp \bk{Y} = \bk{Y} \tp \b{P}\tp \b{P} \bk{Y} = \bk{Z} \tp \bk{Z} $, we
obtain the (1,1) block in $ \bk{N} $. Subsequently, by factoring
\jbr{$P$ as
\begin{equation*}
P = \b{I} - \bmat{\b{A}\tp & \bk{S} & \bk{Z} }
\bmat{ - \delta_k (\b{A} \b{A} \tp)^{-1} & \\
& \b{0}_{2k \times 2k}
}
\bmat{\b{A} \\ \bk{S} \tp \\ \bk{\jbr{Z}} \tp },
\end{equation*}
we see that
\begin{equation*}
\b{P}\tp \bk{V} \b{P}
= \delta_k \b{I} + \bmat{\b{A}\tp & \bk{S} & \bk{Z} }
\bmat{ - \delta_k (\b{A} \b{A} \tp)^{-1} & \\
& \bk{W} - \bk{C}\tp \bk{G} \bk{C}
}
\bmat{\b{A} \\ \bk{S} \tp \\ \bk{\jbr{Z}} \tp }.
\end{equation*}
Because all blocks of $ \bk{W} - \bk{C}\tp \bk{G} \bk{C} $
except for the (1,1) block are equal to those in $\bk{W}$,
all blocks in $ \bk{N} $ are fully specified
and representation \eqref{eq:redComp}
is complete.}
\end{proof}
\smallskip
Note that
$ \bk{S} \tp \bk{Y} = \bk{D} + \bk{L} + \bar{\b{T}}_k = \bk{S} \tp \bk{Z} $,
which means \jbr{that} $ \bk{D} $ and $ \bk{T} = \bk{D} + \bar{\b{T}}_k $ can be
computed from $ \bk{S} $ and $ \bk{Z} $ alone, and \jbr{that} \bk{G} \jbr{and} \bk{C} \jbrt{need not be explicitly computed}. Therefore, for the RCR,
only $ \bk{S} $, $ \bk{Z} $, $ \bk{T} $ \jbr{and} $ \bk{D} $ are stored.
An \jbbo{addition} is the scalar $ \delta_k $,
which is typically set to be
$ \delta_k = \bk{s} \tp \bk{y} \big / \bk{y}\tp \bk{y} = \bk{s} \tp \bk{z} \big / \bk{y}\tp \bk{y} $ and may depend on the most recent $ \bk{y} $.
\jbbo{As $ \b{P} \bk{J} = \bmat{ \bk{S} & \bk{Z} } $,
we \jbr{note} a key advantage of the RCR: that
\eqref{eq:redComp} can be written as
\begin{equation}
\label{eq:redCompConcise}
\bk{V} = \delta_k \b{P} + \b{P} \bk{J} \bk{N} \bk{J}\tp \b{P} \tp =
\delta_k \b{P} + \bmat{ \bk{S} & \bk{Z} } \bk{N}
\bmat{\bk{S} \tp \\[4pt] \bk{Z}\tp}.
\end{equation}
By storing a few columns of $ \bmat{ \bk{S} & \bk{Z} } \in \mathbb{R}^{n \times 2l} $
(as described in section \ref{sec:compComplx}), which in turn define
a small matrix $ \bk{N} \in \mathbb{R}^{2l \times 2l} $ (cf.\ Lemma 1), we can separate the solves with $ \b{A} \b{A}\tp $ from other
calculations. Concretely, \jbr{note} that solves with $ \b{A} \b{A} \tp $
only occur \jbrt{as part of} the orthogonal projection $ \b{P} $, which can be represented
as a linear operator and does not need to be explicitly formed. Also \jbr{note}
that \eqref{eq:iBk} and \eqref{eq:redCompConcise} are related, with the
difference being that $ \bk{Y} $ and $ \delta_k \b{I} $ in \eqref{eq:iBk}
are replaced by $ \bk{Z} $ and $ \delta_k \b{P} $ in \eqref{eq:redCompConcise}.
Hence for large $ n $ and $ m $, computation with
\eqref{eq:redCompConcise} is efficient and requires little memory,
provided orthogonal projections with $ \b{P} $ are handled effectively (as described in section \ref{sec:Proj}).
On the other hand, the compact representation in \eqref{eq:cmpVk_L1}
does not neatly decouple solves with $ \b{A} \b{A} \tp $, and results
in perhaps prohibitively expensive computations for large $m$. In particular,
$ \bk{G} $ in the middle matrix of \eqref{eq:cmpVk_L1} is defined
by $ \bk{G} \equiv ( \b{A} \bik{B} \b{A} \tp )^{-1} \in \mathbb{R}^{m \times m} $, which interleaves
solves with $ \b{A} \b{A} \tp $ and other terms.
Therefore, the RCR in \eqref{eq:redComp}--\eqref{eq:redCompConcise} is recognizably more practical for large $ n $
and $m$ than \eqref{eq:cmpVk_L1}.}
We apply $ \bk{V} $ \jbbo{from \eqref{eq:redCompConcise}} to a vector $ \b{g} $ as
\begin{equation}
\label{eq:Vg}
h = \bmat{ \bk{S} \tp \\[4pt] \bk{Z} \tp } \b{g}, \qquad
\bk{V} \b{g} =
\bmat{ \bk{S} & \bk{Z} } \bk{N}h + \delta_k \b{P} \b{g}.
\end{equation}
\subsection{Computational complexity}
\label{sec:compComplxNew}
\jbbo{With adequate \jbrt{precomputation and storage}, the cost of the matrix-vector product \eqref{eq:Vg} is often inexpensive. If the columns of $ \bk{Z} $ are stored, updating the small $ 2l \times 2l $ matrix $ \bk{N} $ does not depend on
solves with $ \b{A} \b{A} \tp $. Moreover, factors of $ \b{P} $ can be precomputed
once at $k=0$ and reused. In particular, suppose that a (sparse) QR factorization
$
\b{A}\tp =
\big[
\begin{smallmatrix}
\biidx{1}{Q} & \biidx{2}{Q}
\end{smallmatrix}
\big]
\big[
\begin{smallmatrix}
\b{R} \\
\b{0}
\end{smallmatrix}
\big]
$
is obtained once, with $\b{Q} = \big[
\begin{smallmatrix}
\biidx{1}{Q} & \biidx{2}{Q}
\end{smallmatrix}
\big] $ \jbrt{being sparse}, such that the product $ \b{Q}\tp \!\b{g} $ takes
$ \mathcal{O}(rn) $ multiplications, where $r$ is constant. Subsequently, the projection
$ \b{P}\b{g} = \b{g} - \biidx{1}{Q}\biidx{1}{Q}\tp \b{g} $ can be computed in
$ \mathcal{O}(n + 2rn) $ multiplications
(or $ \b{P}\b{g} = \biidx{2}{Q} \biidx{2}{Q}\tp \b{g} $ in
$ \mathcal{O}(2rn) $ multiplications). Thus, we summarize
the multiplications in \eqref{eq:Vg} as: $ \b{h} $ with
$ 2nl $, $ \bk{N}\b{h} $ with negligible $(2l)^2$,
$ \bmat{\bk{S} & \bk{Z}} \bk{N}\b{h} $ with $2nl$, and
$ \b{P} \b{g} $ with, say, $ 2nr $. The total, without negligible terms,
is $ \mathcal{O}( 2n(2l + r)) $. The multiplications scale linearly with $n$,
are related to the sparsity in $ \b{A} $, and are thus suited for large problems.}
\subsection{Updating}
\label{sec:compComplx}
We store and update the columns of $ \bk{Z} = \bmat{ \biidx{k-l}{z} & \cdots & \biidx{k-1}{z} } $
one at a time and \jbr{recall} that $ \biidx{k}{z} = \b{P} \biidx{k+1}{g} - \b{P} \bk{g} $.
Based on this, no additional solves with $ \b{A} \b{A} \tp $ are required to \jbrt{represent}
the matrix $ \biidx{k+1}{V} $. Specifically, suppose that we computed and stored
$ \b{P} \bk{g} $ at the end of the previous iteration, and that we compute
$ \b{P} \biidx{k+1}{g} $ at the end of the current iteration. We can use this vector
in two places: first to \jbrt{represent} $ \biidx{k+1}{Z} $ with $ \bk{z} = \b{P} \biidx{k+1}{g} - \b{P} \bk{g}$ and hence $ \bko{V} $, and secondly in the
computation of $ \bko{V} \bko{g} $. Thus only one solve with $ \b{A} \b{A} \tp $
per iteration is necessary to update $ \bko{V} $ and to compute a step of the
form $ \b{s} = - \bko{V} \bko{g} $.
For large problems, the limited-memory representation in \eqref{eq:redComp} is obtained
by storing only the last $l$ columns of $ \bk{S} $ and $\bk{Z} $. With $1 \le l \ll n$, limited-memory strategies enable computational efficiencies and lower storage requirements \cite{Noc80}. Updating $ \bk{S}$ and $ \bk{Z} $ requires replacing or inserting one column at each iteration. Let an underline below a matrix represent the
matrix with its first column removed. That is, $ \underline{\b{Z}}_k $ represents $ \bk{Z} $ without its first column.
With this notation, a column update of a matrix $ \bk{Z} $ by a vector $ \bk{z} $ is defined as
\begin{equation*}
\text{colUpdate}\left(\bk{Z},\bk{z} \right) \equiv
\begin{cases}
[\: \bk{Z} \: \bk{z}\: ] & \text{ if } k < l, \\
[\: \underline{\b{Z}}_k \: \bk{z}\: ] & \text{ if } k \ge l. \\
\end{cases}
\end{equation*}
Such a column update either directly appends a column to a matrix or first removes a column and then appends one.
This column update will be used, for instance, to obtain $ \bko{Z} $ from $ \bk{Z} $ and $ \bk{z} $, i.e., $\bko{Z}= \text{colUpdate}( \bk{Z}, \bk{z} )$. Next, let an overline above a matrix represent
the matrix with its first row removed. That is, $ \overline{\b{S}\tp_k \b{Z}}_k $ represents $ \b{S}\tp_k \bk{Z} $ without its first row.
With this notation, a product update of $ \bk{S}\tp\bk{Z} $ by matrices $ \bk{S} $ \jbrt{and} \bk{Z}
and vectors $ \bk{s} $ \jbrt{and} $\bk{z}$ is defined as
\begin{equation*}
\text{prodUpdate} \left( \bk{S}\tp\bk{Z}, \bk{S}, \bk{Z}, \bk{s}, \bk{z} \right) \equiv
\begin{cases}
\left[
\begin{array}{ c c }
\bk{S}\tp\bk{Z} & \bk{S}\tp\bk{z} \\
\bk{s}\tp\bk{Z} & \bk{s}\tp \bk{z}
\end{array}
\right] & \text{ if } k < l, \vspace{0.1cm} \\
\left[
\begin{array}{ c c }
\left(\underline{\overline{\b{S}\tp_k \b{Z}_k}}\right) & \underline{\b{S}}_k\tp\bk{z} \\
\bk{s}\tp \underline{\b{Z}}_k & \bk{s}\tp \bk{z}
\end{array}
\right] & \text{ if } k \ge l.\\
\end{cases}
\end{equation*}
This product update is used to compute matrix products such as $ \bko{S}\tp \bko{Z} $ with
$ \mathcal{O}(2ln) $ multiplications, instead of $ \mathcal{O}(l^2n) $ when the product
$ \bk{S}\tp \bk{Z} $ is stored and the vectors $ \bk{s} $ \jbrt{and} $ \bk{z} $ have been computed. Note that a diagonal matrix can be updated in this way by setting the rectangular matrices $\bk{S} $ \jbrt{and} $ \bk{Z}$ to zero
and $ \bko{D} = \text{prodUpdate}(\bk{D},\b{0},\b{0},\bk{s},\bk{z})$. An upper triangular matrix
can be updated in a similar way, e.g., $ \bko{T} = \text{prodUpdate}(\bk{T},\bk{S},\b{0},\bk{s},\bk{z}) $.
To save computation, products with zero matrices are never formed explicitly.
\noindent
\section{Computing projections}
\label{sec:Proj}
With $P = I_{\jbrt{n}} - A\tp (AA\tp)^{-1} A$, projections $z=Py$ can be computed by direct or
iterative methods. Their efficiency depends on the sparsity of $A$.
\subsection{QR factorization}
\label{sec:QR}
When $A$ has full row-rank and the QR factorization
\begin{equation}
\label{eq:QR}
A\tp = Q\bmat{R \\ 0} = \bmat{Q_1 & Q_2} \bmat{R \\ 0} = Q_1 R
\end{equation}
is available, the projection operator becomes
$P = I - Q_1 Q_1\tp = Q_2 Q_2\tp$.
Thus, $z = Py$ can be computed stably as $z = Q_2(Q_2\tp y)$.
With $m<n$, the QR factors are best obtained using a product of Householder
transformations \cite{GV4}:
\begin{equation}
\label{eq:HQR}
Q\tp A\tp = H_m \dots H_3 H_2 H_1 A\tp = \bmat{R \\ 0} = \bmat{Q_1\tp \\ Q_2\tp} A\tp.
\end{equation}
Thus $Q = H_1 H_2 H_3 \dots H_m$ and the operators $Q_1$ and $Q_2$ are available from
\begin{align}
Q_1 = Q \bmat{I \\ 0} \qquad \jbrt{\textnormal{and}} &\qquad Q_2 = Q \bmat{0 \\ I}.
\end{align}
When $A$ is sparse, the SuiteSparseQR software \cite{SuiteSparseQR}
permutes the columns of $A\tp$ in \eqref{eq:HQR} to retain sparsity in $H_k$ and $R$.
The projection $z = Py = Q_2(Q_2\tp y)$ can then be computed efficiently.
\jbrt{One can avoid storage of $Q_1$ by noting} that $Q_1 = A\tp R^{-1}$.
The projection can be computed as $z = (I - Q_1Q_1\tp)y = y - A\tp R^{-1} R\mtp Ay$, though with lower precision than $z = Q_2(Q_2\tp y)$.
\subsection{Iterative computation of \texorpdfstring{$z$}{z}}
\label{sec:LSQR}
\jbr
Computing QR factors is sometimes not practical
because $A$ contains one or more relatively dense columns.
(In the numerical experiments of section \ref{sec:numex}, this occurred with only 2 out of 142 sparse constrained problems.)
The multifrontal QR solver SuiteSparseQR \cite{SuiteSparseQR} then has to handle dense factors,
slowing computing times. For problems with thousands of constraints
we regard column $j$ as relatively dense if
$ \text{nnz}(A_{:j}) / m > 0.1 $. When one expects \jbrt{the} QR \jbrt{factorization to be slow} because of dense columns,
an alternative is to solve the least-squares problem}
\begin{equation}
\label{eq:LS}
\min_w \norm{A\tp w - y}
\end{equation}
and compute the residual $z = Py = y - A\tp w$.
Suitable iterative solvers for \eqref{eq:LS} are CGLS \cite{HS1952}, LSQR \cite{PS82a}, and LSMR \cite{FonS2011}.
If $\tilde A$ is the same as $A$ with any relatively dense columns deleted,
the factor $\tilde R$ from a sparse QR factorization of ${\tilde A}\tp$ (again with suitable
column permutation) should be a good right-preconditioner to accelerate the
iterative solvers. If $\tilde A$ does not have full row-rank, the zero or small
diagonals of $\tilde R$ can be changed to 1 before $\tilde R$ is used as a preconditioner.
\subsection{Implementation}
Appendix B describes the implementation of the
two preceding projections. We refer to these operations through the definition
\begin{equation*}
\b{z} \equiv
\text{compProj}(\b{A},\b{y},\texttt{P}) \equiv
\begin{cases}
\text{Householder QR} & \text{if } \texttt{P} = 1, \\
\text{Preconditioned LSQR} & \text{if } \texttt{P} = 2.
\end{cases}
\end{equation*}
Note that the implementations do not require $ \b{A} $ to have full \jbrt{row} rank.
\subsection{\texorpdfstring{\jbbo{Trust-region algorithm}}{TR algorithm}}
\label{sec:TR}
\jbbo{To solve \eqref{eq:main} we use the trust-region strategy},
which is regarded as a robust minimization method \cite{ConGT00a}.
At each iteration, the method measures progress using the ratio of
actual over predicted reductions:
\begin{equation*}
\rho_{\jbrt{k}} = \frac{f(\bk{x}) - f(\bk{x} + \jbrt{\bk{s}})}{q(\b{0})-q(\jbrt{\bk{s}})},
\end{equation*}
where \jbrt{$ \bk{s} $} is an intermediate search direction\jbr{,
in the sense that $ \bk{s} $ will ultimately be used as an update only
if \jbrt{$\rho_k$ is greater than a threshold.}}
By accepting steps that fulfill the so-called sufficient decrease condition
$ \rho > c_1 $ (\jbrt{suppressing the subscript $k$ on $ \rho_k $}) for a constant $ c_1 > 0 $, the method successively
moves towards \jbrt{a local minimizer (though there is no guarantee that a minimizer
will be reached)}. The trust-region radius $ \Delta > 0 $
controls the norm of the search direction by means
of the constraint $ \| \b{s} \|_2 \le \Delta $. There are two possible
cases for the solution of the TR subproblem: either the search direction
is in the interior of the constraint ($ \| \b{s} \| < \Delta $)
or it is on the boundary ($ \| \b{s} \| = \Delta $). Since the L-BFGS matrix $ \bk{B} $ is positive definite, the solution of
\eqref{eq:trsub} is given by the unconstrained minimizer $ \b{s} = \jbr{\b{s}_E} $
from \eqref{eq:KKTsystem} if $ \| \jbr{\b{s}_E} \| \le \Delta $.
Otherwise, if $ \| \jbr{\b{s}_E} \| > 0 $, then \eqref{eq:trsub}
is solved with the active norm constraint $ \| \b{s} \| = \Delta $.
Note that even if $ \| \jbr{\b{s}_E} \| \le \Delta $, the condition $ \rho > c_1 $
\jbr{might} not hold. In this situation, or in any case when
$ \rho \le c_1 $, the radius $\Delta$ is reduced and a new problem \eqref{eq:trsub}
(with smaller $ \Delta $) and constraint $ \| \b{s} \| = \Delta $ is solved.
The overall trust-region strategy for one iteration is given next, with radius
$ \Delta > 0 $ and $ c_1 > 0 $ and iteration counter suppressed.
\begin{center}
\begin{tabular}{ll}
\multicolumn{2}{l}{Trust-Region Strategy:}
\\ 1. & Compute the unconstrained step $ \b{s} \leftarrow \jbr{\b{s}_E} $ from \eqref{eq:KKTsystem} (using \eqref{eq:Vg})
\color{white}{XXXX}
\\ 2. & \jbbo{While ($\norm{s}_2 > \Delta$ or $ \rho \le c_1 $)}
\\ & \quad 2.1. Solve \eqref{eq:trsub} with $ \norm{\b{s}} = \Delta $
\\ & \quad 2.2. Reduce $\Delta$
\\ & end
\\ 3. & \jbr{Increase (or at least do not decrease) $\Delta$}
\\ 4. & Update iterate $ \b{x} \leftarrow \b{x} + \b{s} $
\end{tabular}
\end{center}
Practical aspects of an implementation include the setting of constants
and starting the method. Detailed procedures are described in
sections \ref{sec:l2}, \ref{sec:eigen}, \ref{sec:SCNorm} \jbrt{and \ref{sec:numex}}.
\section{\texorpdfstring{\jbb{$\ell_2$-norm trust-region constraint}}{L2-norm}}
\label{sec:l2}
With an $\ell_2$-norm trust-region constraint in \eqref{eq:trsub}, the search direction
is given by
\begin{equation*}
\biidx{L2}{s} =
\underset{ \| \b{s} \|_{2} \le \Delta_k }{\text{ arg min }}
q(\b{s})
\quad \text{subject to} \quad \b{A} \b{s} = \b{0}.
\end{equation*}
\jbrt{With} $\sigma \ge 0$ denoting a scalar Lagrange multiplier, the search
direction is a feasible solution to a shifted KKT system \jbrt{including the norm constraint}:
\begin{equation}
\label{eq:KKTshifted}
\bmat{\bk{B} + \sigma \b{I} & \b{A} \tp
\\ \b{A} & \b{0}}
\bmat{\biidx{L2}{s} \\ \biidx{L2}{\lambda}}
=
\bmat{ -\bk{g} \\ \b{0}},
\qquad
\| \biidx{L2}{s} \|_2 \le \Delta_k.
\end{equation}
By computing the (1,1) block of the shifted inverse KKT matrix,
we \jbr{note} that a necessary condition for the solution is $ \biidx{L2}{s}(\sigma) = - \bks{V} \bk{g} $, where
\[
\bks{V} = (\bk{B} + \sigma \b{I})^{-1} -
(\bk{B} + \sigma \b{I})^{-1} \b{A} \tp
(\b{A} (\bk{B} + \sigma \b{I})^{-1} \b{A} \tp )^{-1}
\b{A} (\bk{B} + \sigma \b{I})^{-1}.
\]
For the L-BFGS matrix, with $ \tau_k = \tau_k(\sigma) = (1/ \delta_k + \sigma) $
we have
$
(\bk{B} + \sigma \b{I})^{-1} =
\tau_k^{-1} \b{I} + \bk{J} \bks{W} \bk{J} \tp,
$
where the small $ 2l \times 2l $ matrix is
\[
\bks{W} =
- \bmat{
\theta_k \bk{S} \tp \bk{S}
& \theta_k \bk{L} + \tau_k \bk{T} \\
\theta_k \bk{L} \tp + \tau_k \bk{T} \tp
& \tau_k (\tau_k \bk{D} + \bk{Y} \tp \bk{Y})
}^{-1}
\]
with $ \theta_k = \tau_k(1- \delta_k \tau_k) $.
In terms of $ \bks{C} \equiv \b{A} \bk{J} \bks{W} $ and
$ \bks{G} \equiv (\b{A} (\bk{B}+\sigma \b{I})^{-1} \b{A} \tp)^{-1} $,
the compact representation of $ \bks{V} $ \cite[Corollary 1]{BMP19} is
\begin{align}
\label{eq:cmpVks_L1}
&\bks{V} =
\\ &\frac{1}{\tau_k} I +
\bmat{\b{A}\tp & \bk{J}}
\bmat{-\frac{1}{\tau_k^2}\bks{G} & - \frac{1}{\tau_k}\bks{G} \bks{C}
\\ -\frac{1}{\tau_k}\bks{C} \tp \bks{G} & \bks{W} - \bks{C} \tp \bks{G} \bks{C}}
\bmat{ \b{A} \\[4pt] \bk{J}\tp}. \nonumber
\end{align}
Once the middle matrix in \eqref{eq:cmpVks_L1} is formed, the compact representation
can be used to compute matrix-vector products efficiently. However, when
$ m $ is large (many equality constraints), computing terms such
as $ \bks{G} $ become expensive. Therefore, we describe a reduced representation
similar to \eqref{eq:redComp}, based on \jbr{the property} that
$ \b{P} \tp \bks{V} \b{P} = \bks{V} $ and by storing $ \bk{S} $ and $ \bk{Z} $.
Lemma 2 summarizes the outcome.
\smallskip
{Lemma 2:} \emph{The RCR of $ \bks{V} $
in \eqref{eq:cmpVks_L1} for the L-BFGS matrix is given by}
\begin{equation}
\label{eq:redComps}
\bks{V} = \frac{1}{\tau_k} I +
\bmat{\b{A}\tp & \bk{S} & \bk{Z} }
\bmat{ - \frac{1}{\tau_k} (\b{A} \b{A} \tp)^{-1} & \\
& \bks{N}
}
\bmat{\b{A} \\[2pt] \bk{S} \tp \\[2pt] \bk{\jbr{Z}} \tp },
\end{equation}
\emph{where} $ \jbbo{\tau_k = \tau_k(\sigma) = (1/\delta_k + \sigma)} $, $ \jbbo{\theta_k = \theta_k(\sigma) = \tau_k(\sigma)(1-\delta_k \tau_k(\sigma))} $,
\emph{and}
\begin{equation*}
\bks{N} = -\bmat{ \jbbo{\theta_k(\sigma)} \bk{S} \tp \bk{S}
& \jbbo{\theta_k(\sigma)} \bk{L} + \jbbo{\tau_k(\sigma)} \bk{T} \\
\jbbo{\theta_k(\sigma)} \bk{L} \tp + \jbbo{\tau_k(\sigma)} \bk{T} \tp
& \jbbo{\tau_k(\sigma)} (\jbbo{\tau_k(\sigma)} \bk{D} + \bk{Z} \tp \bk{Z})}^{-1}.
\end{equation*}
\smallskip
\begin{proof} To simplify notation, we suppress the explicit dependence on $ \sigma $
in this proof, so that
$ \bk{V} \equiv \bks{V}$, $\bk{C} \equiv \bks{C}$, and $\bk{W} \equiv \bks{W}$. Multiplying $ \bk{V} $ in \eqref{eq:cmpVks_L1} from the left and right
by $ \b{P} \tp $ and $ \b{P} $ yields
$$
\bk{V} = \frac{1}{\tau_k} \b{P} + \bmat{\bk{S} & \bk{Z} } ( \bk{W} - \bk{C} \tp \bk{G} \bk{C} ) \bmat{\bk{S} & \bk{Z} }\tp.
$$
Observe that
$ \bk{C} = \b{A} \bk{J} \bk{W} = \bmat{ 0 & \b{A} \bk{Y}} \bk{W} $ is block-rectangular and that
$
\bk{G} = (\b{A} (\frac{1}{\tau_k} \b{I} + \bk{J} \bk{W} \bk{J} \tp) \b{A} \tp)^{-1}
$
depends on $ \bk{W} $.
Defining $ \bk{F} \equiv \tau_k (\b{A} \b{A} \tp)^{-1} $, we \jbr{show that}
the Sherman-Morrison-Woodbury (SMW) inverse gives the simplification
\begin{align*}
&\bk{W} - \bk{C} \tp \bk{G} \bk{C} \\&=
\bk{W} - \bk{W} \bmat{ 0 \\ \bk{Y}\tp \b{A}\tp} \bk{G} \bmat{ 0 & \b{A} \bk{Y}} \bk{W} \\
&= \bk{W} - \bk{W} \bmat{ 0 \\ \bk{Y}\tp \b{A}\tp}
\big(\b{I} + \bmat{ 0 & \bk{F} \b{A} \bk{Y}} \bk{W} \bmat{ 0 \\ \bk{Y}\tp \b{A}\tp} \big)^{-1}
\bmat{ 0 & \bk{F}\b{A} \bk{Y}} \bk{W} \\
&= \left( \bk{W}^{-1} + \bmat{ 0 \\ \bk{Y}\tp \b{A}\tp} \bmat{ 0 & \bk{F} \b{A} \bk{Y}} \right)^{-1},
\end{align*}
where the third equality is obtained by applying the SMW formula in
reverse. Since only the (2,2) block in the low-rank matrix
of the third equality is nonzero, and since
$ \bk{F} = \tau_k (\b{A} \b{A} \tp)^{-1} $, \jbr{note} that
$$
(\bk{W}^{-1})_{22} + \bk{Y} \tp \b{A} \tp \bk{F} \b{A} \bk{Y}
=-(\tau_k(\tau_k \bk{D} + \bk{Y}\tp \bk{Y} -
\bk{Y}\tp \b{A} \tp (\b{A} \b{A} \tp)^{-1} \b{A} \bk{Y}) ),
$$
which corresponds to the $(2,2)$ block $ \bks{N} $ in \eqref{eq:redComps}. Because
all other blocks are unaffected, it holds that $ \bk{W} - \bk{C} \tp \bk{G} \bk{C} = \bks{N} $.
Subsequently, by factoring $ \b{P} = I - \b{A} \tp (\b{A} \b{A} \tp )^{-1} \b{A} $ we deduce the compact representation \eqref{eq:redComps}.
\end{proof}
\smallskip
Note that
$ \bk{S}\tp \bk{Z} = \bk{S}\tp \bk{Y} = \bk{L} + \bk{D} + \bar{\b{T}}_k $,
with $ \bk{T} = \bk{D} + \bar{\b{T}}_k $, means that the RCR
for $ \bks{V} $ is fully specified by storing $ \bk{S} $ and $ \bk{Z} $. An exception
is the scalar $ \delta_k $, which may depend on the most recent $ \bk{y} $. Also
when $ \sigma = 0 $, the representations
\eqref{eq:redComp} and \eqref{eq:redComps} coincide. We apply $ \bks{V} $ to a vector $ \b{g} $ as
\begin{equation*}
h = \bmat{ \bk{S} \tp \\ \bk{Z} \tp } \b{g}, \qquad
\bks{V} \b{g} =
\bmat{ \bk{S} & \bk{Z} } \bks{N}h + \frac{1}{\tau_k} \b{P} \b{g}.
\end{equation*}
\subsection{\texorpdfstring{$\ell_2$}{L2}-norm search direction}
\label{sec:l2search}
To compute the $\ell_2$ TR minimizer we first set $ \sigma = 0 $
and $ \biidx{L2}{s}(0) = - \bk{V}(0) \bk{g} $. If $ \| \biidx{L2}{s}(0) \|_2 \le \Delta_k $,
the \jbbo{minimizer with the \jbrt{$\ell_2$-norm}} is given by $ \biidx{L2}{s}(0) $. Otherwise
($ \| \biidx{L2}{s}(0) \|_2 > \Delta_k $) we define the so-called secular equation
\cite{ConGT00a} as
$$ \phi(\sigma) \equiv \frac{1}{\| \biidx{L2}{s}(\sigma) \|_2} - \frac{1}{\Delta_k}. $$
To solve the secular equation we apply the 1D Newton iteration
\begin{equation*}
\sigma_{j+1} = \sigma_j - \frac{\phi(\sigma_j)}{ \phi'(\sigma_j) },
\end{equation*}
where $ \phi'(\sigma_j) =
- (\biidx{L2}{s}(\sigma_j)\tp \biidx{L2}{s}(\sigma_j)') / \| \biidx{L2}{s}(\sigma_j) \|^3_2 $ and
$ \biidx{L2}{s}(\sigma_j)' = -\bk{V}(\sigma_j) \biidx{L2}{s}(\sigma_j) $
(with prime `` $'$ " denoting the derivative).
\jbr{Note that $ \biidx{L2}{s}(\sigma_j)' $ can be derived from the shifted system
\eqref{eq:KKTshifted} by differentiation with respect to $\sigma$. Applying the
product rule in \eqref{eq:KKTshifted} and regarding the solutions as functions of
$\sigma$, i.e., $ \biidx{L2}{s}' \equiv \biidx{L2}{s}(\sigma)' $ and
$ \biidx{L2}{\lambda}' \equiv \biidx{L2}{\lambda}(\sigma)' $, one obtains the
differentiated system
\begin{equation*}
\bmat{\bk{B} + \sigma \b{I} & \b{A} \tp
\\ \b{A} & \b{0}}
\bmat{\biidx{L2}{s}' \\ \biidx{L2}{\lambda}'}
=
\bmat{ -\biidx{L2}{s} \\ \b{0}}.
\end{equation*}
Since the system matrix is the same as in \eqref{eq:KKTshifted} (only the
right-hand side differs), $\biidx{L2}{s}(\sigma_j)'$ is fully determined by
$\bk{V}(\sigma_j)$ and $ \biidx{L2}{s}(\sigma_j)$.}
Starting from $ \sigma_0 = 0 $, we terminate the Newton iteration if
$ | \phi(\sigma_{j+1}) | \le \varepsilon $ or an iteration limit is reached.
The search direction is then computed as
$ \biidx{L2}{s}(\sigma_{j+1}) = - \bk{V}(\sigma_{j+1}) \bk{g} $.
Our approach with the $ \ell_2 $ norm is summarized in Algorithm \ref{alg:LTRL2-LEC}.
\jbbo{This algorithm is based on storing and updating $ \bk{S}, \bk{Z} $,
and the small blocks of $ \bks{N} $ in \eqref{eq:redComps}. Suppose that
$ \biidx{0}{s} $ and $ \biidx{0}{z} $ are obtained by \jbrt{an}
initialization procedure (\jbrt{for instance, Init.~1 from section \ref{sec:numex}}). With $k=0$, the
initial matrices that define $ \bks{V} $ are given as
\begin{align}
\label{eq:initSZ}
\bk{S} = \bmat{ \bk{s} }, &\quad \bk{Z} = \bmat{ \bk{z} },
\\
\label{eq:initVks}
\bk{D} = \bmat{ \bk{s} \tp \bk{z} }, \quad
\bk{T} = \bmat{ \bk{s} \tp \bk{z} }, &\quad
\bk{Z}\tp \bk{Z} = \bmat{ \bk{z} \tp \bk{z} }, \quad
\bk{L} = \bmat{ \b{0} }.
\end{align}
Once the iteration starts, we update
\begin{equation}
\label{eq:upSZ}
\bko{S} = \text{colUpdate}(\bk{S},\bk{s}), \quad
\bko{Z} = \text{colUpdate}(\bk{Z},\bk{z}),
\end{equation}
\begin{align}
\label{eq:upVks}
\bko{D} &= \text{prodUpdate}(\bk{D},\b{0},\b{0}, \bk{s}, \bk{z})\jbr{,}
\\ \bko{T} &= \text{prodUpdate}(\bk{T},\bk{S},\b{0}, \bk{s}, \bk{z})\jbr{,} \nonumber
\\ \bko{Z}\tp \bko{Z} &= \text{prodUpdate}(\bk{Z}\tp \bk{Z},\bk{Z},\bk{Z}, \bk{z}, \bk{z}), \text{ \jbr{and}} \nonumber
\\ \bko{L} &= \text{prodUpdate}(\bk{L},\b{0},\bk{Z}, \bk{s}, \b{0}) \nonumber.
\end{align}
Note that we store and update matrices like $ \bk{Z}\tp \bk{Z} \in \mathbb{R}^{l \times l} $ instead of recomputing them. Because of the limited memory
technique (typically $ 3 \le l \le 7 $ \cite{ByrNS94}), such matrices
are very small relative to large $n$.
Subsequently, $ \bks{N} \in \mathbb{R}^{2l \times 2l} $, defined by the
blocks in \eqref{eq:upVks}, remains very small compared to $n$.
}
\begin{algorithm}
\caption{{\small LTRL2-SLEC} (Limited-Memory Trust-Region 2-norm for Sparse Linear Equality Constraints)}
\label{alg:LTRL2-LEC}
\begin{algorithmic}[1]
\ENSURE
$ 0 < c_1$, $0 < c_2, c_3,c_4,c_5,c_6 < 1 < c_7 $, $0 < \varepsilon_1, \varepsilon_2 $,
$ 0 < i_{\text{max}} $, $ k=0 $, $ \jbrt{0< l} $, $ \Delta_k = \| \bk{x} \|_2$,
$\bk{g} = \nabla f(\bk{x})$, $\texttt{P} \in [0,1]$,
$ \bk{g}^P = \text{compProj}(\b{A},\bk{g},\texttt{P}) $,
$ \bko{g}^P, \bk{s}, \bk{z}, \bk{y} $ (from initialization),
$ \bk{S}, \bk{Z}, \bk{D}, \bk{T}, \bk{L}, \bk{Z}\tp \bk{Z} $ from
\eqref{eq:initSZ} and \eqref{eq:initVks},
$\delta_k = \bk{s}\tp \bk{z} / \bk{y} \tp \bk{y} $,
$ \sigma = 0 $, $\tau_k = (1/\delta_k + \sigma)$, $ \theta_k = \tau_k(1-\delta_k \tau_k) $, $ \bk{N}(\sigma) $ from \eqref{eq:redComps}, $k = k + 1$
\WHILE{$ (\varepsilon_1 \le \| \b{g}^P_k \|_{\infty} ) $}
\STATE{$ h = -\bmat{ \bk{S} & \bk{Z} } \tp \bk{g} $}
\STATE{$
\bk{s} = \bmat{ \bk{S} & \bk{Z} } \bk{N}(0)h - \delta_k \b{g}^P_k
$; $ \rho_k = 0 $\;} \COMMENT{Equality constrained step}
\IF{$ \| {s}_k \|_2 \le \Delta_k $}
\STATE{$\rho_k = (f(\bk{x}) - f(\bk{x}+\bk{s}))/ (q(\b{0}) - q(\bk{s})) $}
\ENDIF
\WHILE{ $ \rho_k \le c_1 $ }
\STATE{$ \sigma = 0 $, $ i = 0 $; $\tau_k = (1/\delta_k + \sigma)$, $ \theta_k = \tau_k(1-\delta_k \tau_k) $}
\STATE{$ h' = -\bmat{ \bk{S} & \bk{Z} } \tp \bk{s} $}
\STATE{$\bk{s}' = \bmat{ \bk{S} & \bk{Z} } \bks{N}h' - \delta_k \bk{s}$;
}
\WHILE{ $ \varepsilon_2 < |\phi(\sigma)| $ \AND $ i < i_{\max} $ }
\STATE{$ \sigma = \sigma - \phi(\sigma) / \phi'(\sigma) $}
\STATE{$\tau_k = (1/\delta_k + \sigma)$, $ \theta_k = \tau_k(1-\delta_k \tau_k) $}
\STATE{$ h = -\bmat{ \bk{S} & \bk{Z} } \tp \bk{g} $;
$
\bk{s} = \bmat{ \bk{S} & \bk{Z} } \bks{N}h - \frac{1}{\tau_k} \b{g}^P_k
$}
\STATE{$ h' = -\bmat{ \bk{S} & \bk{Z} } \tp \bk{s} $;
$
\bk{s}' = \bmat{ \bk{S} & \bk{Z} } \bks{N}h' - \frac{1}{\tau_k} \bk{s}
$; }
\STATE{$i = i+1$}
\ENDWHILE
\COMMENT{Newton's method}
\STATE{$\rho_k = 0$}
\IF{$ 0 < (f(\bk{x}) - f(\bk{x}+\bk{s})) $}
\STATE{$\rho_k = (f(\bk{x}) - f(\bk{x}+\bk{s}))/ (q(\b{0}) - q(\bk{s})) $}
\ENDIF
\IF{ $\rho_k \le c_2 $}
\STATE{$ \Delta_k = \min(c_3 \| \bk{s} \|_2, c_4 \Delta_k)$}
\ENDIF
\ENDWHILE
\STATE{$\b{x}_{k+1} = \bk{x} + \bk{s}$}
\COMMENT{Accept step}
\IF{$c_5\Delta_k \le \| \bk{s} \|_2$ \AND $ c_6 \le \rho_k $}
\STATE{$\Delta_k = c_7 \Delta_k$}
\ENDIF
\STATE{$\bko{g} = \nabla f(\bko{x}) $, $ \bko{g}^P = \text{compProj}(\b{A},\bko{g},\texttt{P}) $,
$ \bk{z} = \bko{g}^P - \bk{g}^P $,
$ \bk{y} = \bko{g} - \bk{g} $,
$ \bko{S}, \bko{Z}, \bko{D}, \bko{T}, \bko{L}, \bko{Z}\tp \bko{Z} $ from
\eqref{eq:upSZ} and \eqref{eq:upVks}
$ \delta_{k+1} = \bk{z}\tp \bk{s} / \bk{y}\tp \bk{y} $,
$ \sigma = 0 $, $\tau_k = (1/\delta_k + \sigma)$, $ \theta_k = \tau_k(1-\delta_k \tau_k) $}
\STATE{Update $ \bks{N}$ from \eqref{eq:redComps}, $k = k+1$}
\ENDWHILE
\end{algorithmic}
\end{algorithm}
\section{Eigendecomposition of \texorpdfstring{$ \bk{V} $}{Vk}}
\label{sec:eigen}
We describe how to exploit the structure of the RCR \eqref{eq:redComp}
to compute an implicit eigendecomposition of $ \bk{V} $, and how to
combine this with a shape-changing norm. The effect is that the trust-region subproblem solution
is given by an analytic formula.
Since the RCR is equivalent to representation
\eqref{eq:cmpVk_L1}, we can apply previous results. However,
using representation \eqref{eq:redComp} is computationally more efficient.
First, note that $ \bk{N} \in \mathbb{R}^{2l \times 2l} $ is a small symmetric
square matrix.
Therefore, computing the
nonzero eigenvalues and corresponding eigenvectors of the matrix
$
\bmat{ \bk{S} & \bk{Z} } \bk{N} \bmat{ \bk{S} & \bk{Z} }\tp
= \biidx{2}{U} \biidx{2}{\Lambda} \biidx{2}{U} \tp
$
is inexpensive. In particular, we compute the thin QR factorization
$ \bmat{\bk{S} & \bk{Z}} = \biidxh{2}{Q}\biidxh{2}{R} $ and the small eigendecomposition
$
\biidxh{2}{R} \bk{N} \biidxh{2}{R} \tp = \biidxh{2}{P} \biidx{2}{\Lambda} \biidxh{2}{P} \tp.
$
The small factorization is then
$$
\bmat{ \bk{S} & \bk{Z} } \bk{N} \bmat{ \bk{S} & \bk{Z} }\tp =
\biidxh{2}{Q} (\biidxh{2}{R} \bk{N} \biidxh{2}{R} \tp) \biidxh{2}{Q} \tp
= \biidxh{2}{Q} (\biidxh{2}{P} \biidx{2}{\Lambda} \biidxh{2}{P} \tp) \biidxh{2}{Q} \tp
\equiv \biidx{2}{U} \biidx{2}{\Lambda} \biidx{2}{U} \tp,
$$
where the orthonormal matrix on the right-hand side is defined as
$ \biidx{2}{U} \equiv \biidxh{2}{Q} \biidxh{2}{P} $.
Since $ \b{A} \tp (\b{A} \b{A} \tp)^{-1} \b{A} = \biidx{1}{Q} \biidx{1}{Q} \tp $
from \eqref{eq:QR}, we express $ \bk{V} $ as
\begin{equation*}
\bk{V} = \delta_k \b{I} +
\bmat{ \biidx{1}{Q} & \biidx{2}{U} }
\bmat{ - \delta_k \b{I}_m & \\
& \biidx{2}{\Lambda}}
\bmat{ \biidx{1}{Q} \tp \\ \biidx{2}{U} \tp },
\end{equation*}
where $ \biidx{1}{Q} \in \mathbb{R}^{n \times m} $ and
$ \biidx{2}{U} \in \mathbb{R}^{n \times 2l} $ are orthonormal,
while $ \biidx{2}{\Lambda} \in \mathbb{R}^{2l \times 2l} $ is diagonal.
Defining the orthogonal matrix
$
\b{U} \equiv
\bmat{\biidx{1}{Q} & \biidx{2}{U} & \biidx{3}{U}},
$
where $\biidx{3}{U} \in \mathbb{R}^{n \times n-(m+2l)}$ represents
the orthogonal complement of $ \bmat{\biidx{1}{Q} & \biidx{2}{U}} $,
we obtain the implicit eigen\-decomposition of $ \bk{V} $ as
\begin{equation}
\label{eq:eigVk}
\bk{V} =
\bmat{ \biidx{1}{Q} & \biidx{2}{U} & \biidx{3}{U} }
\bmat{ \b{0}_{m} & & \\
& \delta_k \b{I}_{2l} + \biidx{2}{\Lambda} & \\
& & \delta_k \b{I}_{n-(m+2l)}}
\bmat{ \biidx{1}{Q} \tp \\ \biidx{2}{U}\tp \\ \biidx{3}{U} \tp}
\equiv \b{U} \b{\Lambda} \b{U} \tp.
\end{equation}
Note that
we do not
explicitly form the potentially expensive \jbrt{to compute} orthonormal matrix $ \biidx{3}{U} $,
as only scaled projections $ \delta_k \biidx{3}{U} \biidx{3}{U} \tp $ are needed.
We therefore refer to factorization \eqref{eq:eigVk} as being implicit. In particular,
from the identity $ \b{U} \b{U} \tp = \b{I} $, we obtain that
$
\biidx{3}{U} \biidx{3}{U} \tp = \b{I} - \biidx{1}{Q} \biidx{1}{Q} \tp - \biidx{2}{U} \biidx{2}{U} \tp = \b{P} - \biidx{2}{U} \biidx{2}{U} \tp.
$
\jbbo{Note here and above that $ \biidx{2}{U} $ is a thin rectangular matrix with only $2l$ columns.}
\section{\texorpdfstring{\jbb{Shape-changing-norm trust-region constraint}}{Shape-changing-norm}}
\label{sec:SCNorm}
To make use of the implicit eigensystem \eqref{eq:eigVk},
we apply the so-called shape-changing infinity norm introduced
in \cite{BurdakovLMTR16}:
\begin{equation*}
\| \b{s} \|_{\b{U}} \equiv
\text{max}\left\{ \Norm{\bmat{ \biidx{1}{Q} & \biidx{2}{U} }\tp \! \b{s}}_{\infty}, \Norm{\biidx{3}{U} \tp \b{s}}_2 \right\}.
\end{equation*}
With this norm, the trust-region subproblem has \jbrt{a computationally efficient} solution that
can be \jbrt{obtained from}
\begin{equation*}
\biidx{SC}{s} =
\underset{ \| \b{s} \|_{\b{U}} \le \Delta_k }{\text{ arg min }}
q(\b{s})
\quad \text{subject to} \quad \b{A} \b{s} = \b{0}.
\end{equation*}
Since the RCR is equivalent to
\eqref{eq:cmpVk_L1}, we invoke \cite[section 5.5]{BMP19} to obtain
an \jbrt{direct} formula for the search direction:
\begin{equation*}
\biidx{SC}{s} = \biidx{2}{U}(\biidx{2}{v} - \beta \biidx{2}{U}\tp \bk{g} )
+ \beta \b{P} \bk{g},
\end{equation*}
where with
$ \biidx{2}{U} \tp \bk{g}
= \biidxh{2}{P}\tp \biidxh{2}{R}^{-\top} \bmat{ \bk{S} & \bk{Z} }\tp \bk{g} \equiv \bk{u} $, and
$ \mu_i = (\delta_k + (\biidx{2}{\Lambda})_{ii})^{-1} $,
\begin{alignat}{2}
\label{eq:SCanalytic1}
&&(\biidx{2}{v})_i =&
\begin{cases}
\frac{-(\bk{u})_i}{\mu_i}
& \text{ if } \left| \frac{(\bk{u})_i}{\mu_i} \right| \le \Delta_k, \\
\frac{-\Delta_k (\bk{u})_i}{|(\bk{u})_i|}
& \text{ otherwise},
\end{cases} \\
\label{eq:SCanalytic2}
&&\beta =&
\begin{cases}
-\delta_k
& \text{ if } \norm{\delta_k \biidx{3}{U} \tp \bk{g}}_2 \le \Delta_k,
\\ \frac{-\Delta_k}{ \|\biidx{3}{U}\tp \bk{g} \|_2}
& \text{ otherwise},
\end{cases}
\end{alignat}
for $ 1 \le i \le 2l $. More details for the computation of $ \b{s}_{SC} $
are in Appendix C.
Note that the norm $ \| \biidx{3}{U}\tp \bk{g} \|_2 $ can be computed without explicitly
forming $ \biidx{3}{U} $, since
$
\| \biidx{3}{U}\tp \bk{g} \|^2_2 = \bk{g} \tp (\b{P} - \biidx{2}{U}\biidx{2}{U}\tp) \bk{g}
= \| \b{P} \bk{g} \|_2^2 - \| \biidx{2}{U}\tp \bk{g} \|_2^2.
$
The trust-region algorithm using the RCR and the shape-changing norm
is summarized in Algorithm \ref{alg:LTRSC-LEC} below.
\jbbo{Like Algorithm \ref{alg:LTRL2-LEC}, this algorithm is based on
storing and updating $ \bk{S}, \bk{Z} $ and the small blocks of
$ \bk{N} $ in \eqref{eq:redComp}. Therefore, the initializations
\eqref{eq:initSZ}--\eqref{eq:initVks} and updates
\eqref{eq:upSZ}--\eqref{eq:upVks} can be used. In addition,
since in the thin QR factorization
$ \bmat{ \bk{S} & \bk{Z} } = \bh{Q}_2 \bh{R}_2 $ the triangular
$ \bh{R}_2 $ is computed from a Cholesky factorization of
$ \bmat{ \bk{S} & \bk{Z} }\tp \bmat{ \bk{S} & \bk{Z} } $, we
initialize the matrices
\begin{equation}
\label{eq:initVk}
\bk{S}\tp \bk{S} = \bmat{ \bk{s}\tp \bk{s} }, \quad
\bk{S}\tp \bk{Z} = \bmat{ \bk{s}\tp \bk{z} },
\end{equation}
with corresponding updates
\begin{align}
\label{eq:upVk}
\bko{S}\tp \bko{S} &= \text{prodUpdate}(\bk{S}\tp \bk{S},\bk{S},\bk{S}, \bk{s}, \bk{s}), \textnormal{ \jbr{and} } \\
\bko{S}\tp \bko{Z} &= \text{prodUpdate}(\bk{S}\tp \bk{Z},\bk{S},\bk{Z}, \bk{s}, \bk{z}) \nonumber.
\end{align}
As before, with a small memory parameter $ l $, these matrices are very
small compared to large $n$, and computations with them are inexpensive.
}
\begin{algorithm
\caption
{\small LTRSC-SLEC} (Limited-Memory Trust-Region Shape-Changing Norm for Sparse Linear Equality Constraints)}
\label{alg:LTRSC-LEC}
\begin{algorithmic}[1]
\ENSURE
{
$ 0 < c_1$, $0 < c_2, c_3,c_4,c_5,c_6 < 1 < c_7 $, $0 < \varepsilon_1 $, $ \jbrt{0 < l} $, $ k=0 $, $ \Delta_k = \| \bk{x} \|_2$, $\bk{g} = \nabla f(\bk{x})$, $\texttt{P} \in [0,1]$,
$ \bk{g}^P = \text{compProj}(\b{A},\bk{g},\texttt{P}) $,
$ \bko{g}^P, \bk{s}, \bk{z}, \bk{y} $ (from initialization),
$ \bk{S}, \bk{Z}, \bk{D}, \bk{T}, \bk{Z}\tp \bk{Z},
\bk{S}\tp \bk{S}, \bk{S}\tp \bk{Z}$ from
\eqref{eq:initSZ}, \eqref{eq:initVks} and \eqref{eq:initVk},
$\delta_k = \bk{s}\tp \bk{z} / \bk{y} \tp \bk{y} $,
$ \bk{N} $ from \eqref{eq:redComp}, $k = k + 1$
}
\WHILE{$ (\varepsilon_1 \le \| \b{g}^P_k \|_{\infty} ) $}
\STATE{$ h = -\bmat{ \bk{S} & \bk{Z} } \tp \bk{g} $}
\STATE{$
\bk{s} = \bmat{ \bk{S} & \bk{Z} } \bk{N}h - \delta_k \b{g}^P_k; $ \bk{\rho} = 0 $;
$}
\COMMENT{Equality constrained step}
\IF{$ \| {s}_k \|_2 \le \Delta_k $}
\STATE{$\rho_k = (f(\bk{x}) - f(\bk{x}+\bk{s}))/ (q(\b{0}) - q(\bk{s})) $;
$ \| {s}_k \| = \| {s}_k \|_2 $}
\ENDIF
\IF{ $ \rho_k \le c_1 $ }
\STATE{$
\bh{R}_2\tp\bh{R}_2 =
\bigg[
\begin{smallmatrix}
\bk{S}\tp \bk{S} & \bk{S}\tp \bk{Z} \\
\bk{Z}\tp \bk{S} & \bk{Z}\tp \bk{Z}
\end{smallmatrix}
\bigg]
$}
\COMMENT{Cholesky factorization}
\STATE{$ \bh{P}_2 \b{\Lambda}_2 \bh{P}_2\tp = \bh{R}_2 \bk{N} \bh{R}_2\tp $}
\COMMENT{Eigendecomposition}
\STATE{$\bk{u}= \bh{P}_2\tp \bh{R}_2^{-\top} \bmat{ \bk{S} & \bk{Z} } \tp \bk{g} $}
\STATE{$ \xi_k =
( \| \bk{g}^P \|_2^2 - \| \bk{u} \|_2^2 )^{\frac{1}{2}}$}
\WHILE{ $ \rho_k \le c_1 $ }
\STATE{Set $ \b{v}_2 $ from \eqref{eq:SCanalytic1} using $ \bk{u} $, $ \b{\Lambda}_2 $}
\STATE{Set $ \beta $ from \eqref{eq:SCanalytic2} using $ \xi_k = \| \biidx{3}{U}\tp \bk{g} \|_2 $}
\STATE{$ \bk{s} = \bmat{ \bk{S} & \bk{Z} }
\bh{R}_2^{-1} \bh{P}_2 (\b{v}_2 - \beta \bk{u}) + \beta\bk{g}^P$;
\quad $\rho_k = 0$}
\IF{$ 0 < (f(\bk{x}) - f(\bk{x}+\bk{s})) $}
\STATE{$\rho_k = (f(\bk{x}) - f(\bk{x}+\bk{s}))/ (q(\b{0}) - q(\bk{s})) $}
\ENDIF
\IF{ $\rho_k \le c_2 $}
\STATE{$ \Delta_k = \min(c_3 \| \bk{s} \|_{\b{U}}, c_4 \Delta_k)$}
\ENDIF
\ENDWHILE
\STATE{$ \| \bk{s} \| = \| \bk{s} \|_U $}
\ENDIF
\STATE{$\b{x}_{k+1} = \bk{x} + \bk{s}$
\COMMENT{Accept step}
\IF{$c_5\Delta_k \le \| \bk{s} \|$ {\bf and} $ c_6 \le \rho_k $}
\STATE{$\Delta_k = c_7 \Delta_k$}
\ENDIF
\STATE{$\bko{g} = \nabla f(\bko{x}) $, $ \bko{g}^P = \text{compProj}(\b{A},\bko{g},\texttt{P}) $,
$ \bk{z} = \bko{g}^P - \bk{g}^P $,
$ \bk{y} = \bko{g} - \bk{g} $,
$ \bko{S}, \bko{Z}, \bko{D}, \bko{T}, \bko{Z}\tp \bko{Z},
\bko{S}\tp \bko{S}, \bko{S}\tp \bko{Z}$ from
\eqref{eq:upSZ}, \eqref{eq:upVks} and \eqref{eq:upVk};
$ \delta_{k+1} = \bk{z}\tp \bk{s} / \bk{y}\tp \bk{y} $}
\STATE{Update $ \bk{N}$ from \eqref{eq:redComp}; $k = k+1$}
\ENDWHILE
\end{algorithmic}
\end{algorithm}
\section{Numerical experiments}
\label{sec:numex}
The numerical experiments are carried out in MATLAB 2016a on a
MacBook Pro @2.6 GHz Intel Core i7 with 32 GB of memory. For comparisons, we use the implementations of
\rfm{Algorithms 1 and 2 from \cite{BMP19}, which we label \TR1 and \TR2.}
All codes are available in the public domain:
\begin{center}
\url{https://github.com/johannesbrust/LTR_LECx}
\end{center}
For \TR1, \TR2 we use the modified stopping criterion
$ \| \b{P} \bk{g} \|_{\infty} \le \epsilon $ in place of $\| \b{P} \bk{g} \|_2 / \text{max}(1, \b{x}_k ) \le \epsilon $ in order to compare consistently across solvers. Unless otherwise specified,
the default parameters of these two algorithms are used.
\rfm{We use the following names for our proposed algorithms:}
\begin{center}
\begin{tabular}{ll}
\jbbo{\TR1H}: &Alg.~\ref{alg:LTRL2-LEC} with representation \eqref{eq:redComps} and Householder QR
\\ \jbbo{\TR1L}: &Alg.~\ref{alg:LTRL2-LEC} with representation \eqref{eq:redComps} and preconditioned LSQR
\\ \jbbo{\TR2H}: & Alg.~\ref{alg:LTRSC-LEC} with representation \eqref{eq:redComp} and Householder QR
\\ \jbbo{\TR2L}: &Alg.~\ref{alg:LTRSC-LEC} with representation \eqref{eq:redComp} and preconditioned LSQR
\end{tabular}
\end{center}
Note that \TR1 and \TR2 were developed for low-dimensional linear equality constraints. In addition, we include {\small IPOPT } \cite{WaechterBiegler06} with an L-BFGS quasi-Newton matrix (we use a precompiled Mex file with {\small IPOPT } 3.12.12 that includes MUMPS and MA57 libraries). \jbr{We note that a commercial state-of-the-art quasi-Newton trust-region solver that uses a projected conjugate gradient solver is implemented in the {\small K{\footnotesize NITRO}-I{\footnotesize NTERIOR}/CG} \cite[Algorithm 3.2]{Byrd2006}.}
For \jbr{the freely available} {\small IPOPT } we specify the limited-memory BFGS option using the option
\texttt{hessian\_approximation=`limited memory'} with
\texttt{tol=1e-5}.
\jbr{(The parameter \texttt{tol} is used by {\small IPOPT } to ensure that the (scaled) projected gradient in the infinity norm and the constraint violation are below the specified threshold.
The default value is \texttt{tol=1e-8}.)}
All other parameters in {\small IPOPT } are at their default values unless otherwise
specified. The parameters in \TR{1\{H,L\}} and \TR{2\{H,L\}} are set to $ c_1 $ (as machine epsilon), $ c_2 = 0.75 $, $ c_3 = 0.5 $, $ c_4 = 0.25 $, $ c_5 = 0.8 $, $ c_6 = 0.25 $, $ c_7 = 2 $, and $i_{\text{max}}=10$.
The limited-memory parameter of all compared TR solvers is set to $l=5$ ({\small IPOPT }'s default is $l=6$). Because the proposed
methods are applicable to problems with a large number of constraints, problems with large dimensions such as $m \ge 10^4$, $n \ge 10^5$ are included. Throughout the experiments, $\b{A} \in \mathbb{R}^{m \times n}$ with $m < n$.
\jbrt{To initialize the algorithm, we distinguish two main cases.} If $ \biidx{0}{x} $ is not available, it
is computed as the
\jbbo{minimum-norm solution
$
\biidx{0}{x} = \text{argmin}_{\b{x}} \norm{\b{x}}_2 \text{~s.t.~} \b{A}\b{x} = \b{b}.
$
(e.g., $\biidx{0}{x}=\b{A}\tp(\b{A}\b{A}\tp)^{-1}\b{b} $ when
$\b{A}$ is full rank.)}
If $ \bh{x}_0 $ is provided but is infeasible, the initial
vector can be computed from \jbbo{
$
\biidx{0}{p} = \text{argmin}_{\b{p}} \norm{\b{p}}_2 \text{~s.t.~}
\b{A}\b{p} = \b{b} -\b{A} \bh{x}_0
$}
and $ \biidx{0}{x} = \bh{x}_0 + \biidx{0}{p}$.
To compute the initial vectors
$ \biidx{0}{s} = \biidx{1}{x} - \biidx{0}{x} $,
$ \biidx{0}{z} = \b{P}\biidx{1}{g} - \b{P}\biidx{0}{g} $,
and
$ \biidx{0}{y} = \biidx{1}{g} - \biidx{0}{g} $
we determine an initial $ \biidx{1}{x} $ value also.
Suppose that at $ k = 0 $, all of
$ \bk{x} $, $ \bk{g} = \nabla f(\bk{x}) $ and
$ \bk{g}^{P} = \b{P} \bk{g} $ are known. An initialization for
$ \bk{s}$, $\bk{z}$ and $ \bk{y} $ at $ k =0 $ is the following:
\begin{center}
\begin{tabular}{ll}
\multicolumn{2}{l}{Init.~1:}
\\ 1. & Backtracking line-search: $ \bko{x} = \bk{x} - \alpha \bk{g}^P / \|\bk{g}^P \|_2 $ (cf.\ \cite[Alg.\ 3.1]{NocW06})
\\ 2. & $ \bko{g} = \nabla f(\bko{x}) $, $ \bko{g}^P = \text{compProj}(\b{A},\bko{g},\texttt{P}) $
\\ 3. & $ \bk{s} = \bko{x} - \bk{x} $
\\ \textcolor{white}{3.} & $ \bk{z} = \bko{g}^P - \bk{g}^P $
\\ \textcolor{white}{3.} & $ \bk{y} = \bko{g} - \bk{g} $
\end{tabular}
\end{center}
Once $\biidx{0}{s}$, $\biidx{0}{z}$ and $\biidx{0}{y}$ have been initialized (with initial radius $ \Delta_0 = \| \biidx{0}{s} \|_2 $), all other updates are done automatically within the trust-region
strategy
The \jbr{outcomes from the subsequent Experiments I--III} are summarized in Figures \ref{fig:EX_I}--\ref{fig:EX_III} as performance profiles (Dolan and Mor\'{e} \cite{DolanMore02}, extended in \cite{MahajanLeyfferKirches11} and often used
to compare the effectiveness of various solvers). \jbr{Detailed information for each
problem instance is in Tables \ref{tab:exp1}--\ref{tab:exp3}.}
Relative performances are displayed in terms of iterations and computation times.
The performance metric $\rho_s(\tau)$ on $n_p$ test problems is given by
\begin{equation*}
\rho_s(\tau) = \frac{\text{card}\left\{ p : \pi_{p,s} \le \tau \right\}}{n_p} \quad \text{and} \quad \pi_{p,s} = \frac{t_{p,s}}{ \underset{1\le i \le S,\ i \ne s}{\text{ min } t_{p,i}} },
\end{equation*}
where $ t_{p,s}$ is the ``output'' (i.e., iterations or time) of
``solver'' $s$ on problem $p$, and $S$ denotes the total number of solvers for a given comparison. This metric measures
the proportion of how close a given solver is to the best result. Extended performance profiles are the same as the classical ones but include the part of
the domain where $\tau \le 1$.
In the profiles we include a dashed vertical grey line to indicate $ \tau = 1 $.
\jbbo{We note that although the iteration numbers are recorded
differently for each solver, they correspond approximately to the number of KKT systems solved.}
\jbbo{Overall, we observe that the number of iterations used by the respective
solvers is relatively similar across different problems. However, the differences
in computation times are large. In particular, the RCR implementations use the least time in almost
all problem instances. This is possible because RCR
enables an efficient decoupling of computations with the constraint matrix
$ \b{A} $ and remaining small terms.}
\subsection{Experiment I}
\label{sec:numex1}
This experiment uses problems with sparse and possibly low-rank $ \b{A} \in \mathbb{R}^{m \times n} $. The objective is the Rosenbrock function
\begin{equation*}
f(\b{x}) = \sum_{i=1}^{n/2} (\b{x}_{2i}-\b{x}_{2i-1})^2 + (1-\b{x}_{2i-1})^2,
\end{equation*}
where $ n $ is an even integer. The matrices $ \b{A} \in \mathbb{R}^{m \times n} $ are obtained from the Suite\-Sparse Matrix Collection \cite{SuiteSparseMatrix}. Because \TR1 and \TR2
were not developed for problems with a large number of constraints, these solvers are only applied to problems for which $ m \le 2500 $. All other solvers were run on all test problems. Convergence of an algorithm is determined when two conditions are satisfied:
\begin{equation} \label{eq:converge}
\norm{\b{P} \bk{g}}_{\infty} < 10^{-5} \quad \text{and} \quad
\norm{\b{A} \bk{x} - \b{b}}_2 < 10^{-7}.
\end{equation}
We summarize the outcomes in Figure\jbr{~\ref{fig:EX_I}} and Table \ref{tab:exp1}.
\begin{figure*}[t!]
\begin{minipage}{0.48\textwidth}
\includegraphics[trim=0 0 20 0,clip,width=\textwidth]{EXPERIMENT_III_ITER.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.48\textwidth}
\includegraphics[trim=0 0 20 0,clip,width=\textwidth]{EXPERIMENT_III_TIME}
\end{minipage}
\caption{\jbbo{Comparison of the 7 solvers from Experiment I using performance profiles \cite{DolanMore02} on 50
test problems from \cite{SuiteSparseMAT}.}
\jbbo{\TR{2H} and \TR{1H} converge on all problem instances (100\%).
\TR{2L}, \TR{1L} and {\small IPOPT} converge on 47 problems (94\%).
\TR{2} and \TR{1} are not applied to 9 large problems.
In the right plot, \TR{2L} and \TR{1L} are the fastest
\jbr{(as seen from their curves being above others)}, while
\TR{2H} and \TR{1H} are the most robust \jbr{(as seen from their curves ultimately
reaching the top of the plot)}.
Overall, \TR{2\{H,L\}} and \TR{1\{H,L\}} are faster than the other solvers.}}
\label{fig:EX_I}
\end{figure*}
In this experiment we observe that our proposed algorithms (any of \TR{1\{H,L\}}, \TR{2\{H,L\}})
perform well in terms of computing time. Both ``H" versions of the proposed algorithms converged to the prescribed tolerances on all problems.
On the other hand, the ``L" versions are often the overall fastest, yet they did
not converge on 3 problem instances (\texttt{beacxc}, \texttt{lp\_cre\_d}, \texttt{fit2d}).
\jbk{After rerunning the 3 problems for which {\small IPOPT } did not converge, we note that {\small IPOPT } did converge to its own (scaled) tolerances on one of these problems (\texttt{beacxc}), yet the computed solution did not satisfy
\eqref{eq:converge}.
On the other two problems (\texttt{lp\_cre\_d}, \texttt{fit2d}), {\small IPOPT } returned a message such as info.status=$-2$, which is caused by an abort when the ``restoration phase" is called at an almost feasible point.}
\subsection{Experiment II}
\label{sec:EX2}
In a second experiment, we compare the 7 solvers on large problems from the CUTEst collection
\cite{GouOT03}. The dimension $n$ is determined by the size of the corresponding CUTEst problem,
while we set $m$ to be about $25\%$ of $n$, i.e., $ \texttt{m=ceil(0.25n)} $. The
matrices $\b{A}$ are formed as $ \texttt{A=sprand(m,n,0.1)} $, with $ \texttt{rng(090317)} $. Convergence is determined
by each algorithm internally. For \TR1, \TR{1H}, \TR{1L}, \TR2, \TR{2H}, \TR{2L} the conditions
$ \| \b{P} \bk{g} \|_{\infty} < 1 \times 10^{-5} $ and $ \| \b{A} \bk{x} - \b{b} \|_2 < 5 \times 10^{-8} $ are explicitly enforced, while for {\small IPOPT } we set $ \texttt{options\_ipopt.ipopt.tol=1e-5} $. We use the iteration limit of $100,000$ for all solvers. The limited-memory parameter is $l=5$
for all TR solvers and $l=6$ (default) for {\small IPOPT }. We summarize the outcomes in
\jbr{Figure~\ref{fig:EX_II}} and Table \ref{tab:exp2}.
\begin{figure*}[t!]
\begin{minipage}{0.48\textwidth}
\includegraphics[trim=0 0 20 0,clip,width=\textwidth]{EXPERIMENT_IV_ITER.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.48\textwidth}
\includegraphics[trim=0 0 20 0,clip,width=\textwidth]{EXPERIMENT_IV_TIME.pdf}
\end{minipage}
\caption{\jbbo{Comparison of the 7 solvers from Experiment II using performance profiles
on 62 test problems from \cite{GouOT03}.}
\jbbo{\TR{1L} converged on 58 problems.
All other solvers except {\small IPOPT} converged on 57 problems.
In the left plot, the iteration numbers for \TR1, \TR{1\{H,L\}},
\TR2 and \TR{2\{H,L\}} are similar, as seen by the tight clustering
of the lines. However, the computational times of \TR1 and \TR2 are
markedly higher than those of \TR{1\{H,L\}} and \TR{2\{H,L\}},
as seen from the widening gap in the right plot.
} }
\label{fig:EX_II}
\end{figure*}
\subsection{Experiment III}
\label{sec:EX3}
\jbr{In a third experiment we compare the 7 solvers on 31 linear equality constrained problems
from CUTEst. Four of these problems (\texttt{AUG2D}, \texttt{AUG2DC}, \texttt{AUG3D}, \texttt{AUG3DC}) directly correspond
to the problem formulation \eqref{eq:main}. The remaining problems have additional
bound constraints, which are relaxed in this experiment.
Problems 1--19 in Table \ref{tab:exp3} are convex and can immediately be attempted by
the solvers (with bounds released). Problems 20--31 are not convex when the bounds
are relaxed,
but adding the term $ \frac{\delta}{2} \| \b{x} \|_2^2 $ with $ \delta = 10 $
to the objective functions
produced finite solutions for these problems.
As in the previous experiment, convergence is determined
by each algorithm internally. For \TR1, \TR{1H}, \TR{1L}, \TR2, \TR{2H}, \TR{2L} the conditions
$ \| \b{P} \bk{g} \|_{\infty} < 1 \times 10^{-5} $ and $ \| \b{A} \bk{x} - \b{b} \|_2 < 5 \times 10^{-8} $ are explicitly enforced, while for {\small IPOPT} we set $ \texttt{options\_ipopt.ipopt.tol=1e-5} $. We use the iteration limit of $100,000$ for all solvers. The limited-memory parameter is $l=5$
for all TR solvers and $l=6$ (default) for {\small IPOPT}. Since \TR1 and \TR2 are not
designed for large $m$, they are applied to problems with $ m < 2500 $,
with the exception of 3 problems (\texttt{BLOWEYA}, \texttt{BLOWEYB}, \texttt{BLOWEYC})
that did not terminate within hours using \TR1 and \TR2. All other solvers are applied
to all problems. The results
are in Figure \ref{fig:EX_III} and
Table \ref{tab:exp3}}.
\begin{figure*}[t!]
\begin{minipage}{0.48\textwidth}
\includegraphics[trim=0 0 20 0,clip,width=\textwidth]{EXPERIMENT_Va_ITER}
\end{minipage}
\hfill
\begin{minipage}{0.48\textwidth}
\includegraphics[trim=0 0 20 0,clip,width=\textwidth]{EXPERIMENT_Va_TIME}
\end{minipage}
\caption{\jbr{Comparison of the 7 solvers from Experiment III using performance profiles
on 31 large linear equality constrained test problems from \cite{GouOT03}.
\TR1 and \TR2 are applied to 6 problems
(they are not practical on the remaining problems because of their size). \TR{2H}
(also \TR{1H}) converged on all 31 instances. \TR{1L}, \TR{2L}, and {\small IPOPT } converged
on 30 problems. In the {\small ITER} plot the number of iterations is relatively
similar across the solvers that converged. In the {\small TIME } plot
there is a gap between \TR1\{H,L\},\TR2\{H,L\} and {\small IPOPT}. \TR{2L} can
have computational advantages, but appears slightly less robust than
\TR{2H}, as seen from the final staircase in the {\small TIME } plot.}
}
\label{fig:EX_III}
\end{figure*}
\section{Conclusion}
\label{sec:conclusion}
For subproblem \eqref{eq:trsub},
this article develops the reduced compact representation~(RCR) of the (1,1)
block in the inverse KKT matrix, when the objective Hessian
is approximated by a compact quasi-Newton matrix.
The representation is based on the \jbr{fact} that part of the solution
to the KKT system is unaffected when it is projected onto the nullspace
of the constraints. An advantage of the RCR is that it enables
\jbbo{a decoupling of solves with the constraint matrix and remaining small terms.
Moreover,}
a projected gradient can be used in two places:
once as part of the matrix update, and second as part of the new step.
By effectively handling
orthogonal projections,
in combination with limited memory techniques, \jbbo{we can compute search directions efficiently}.
\jbk{We apply the orthogonal projections with a sparse
QR factorization or a preconditioned LSQR iteration}, \jbbo{including
large and potentially rank-deficient constraints}.
The RCRs are implemented in two trust-region algorithms, one of which
exploits the underlying matrix structures in order to compute the search
direction by an analytic formula.
\jbk{The other is based on an $\ell_2$ norm and uses the RCR
within a 1D Newton iteration to determine
the optimal scalar shift.}
In numerical experiments on large problems,
our
implementations of the RCR yield
often significant improvements in the computation time,
\jbk{as a result of the advantageous structure of the proposed matrices}.
\Red{Applications of problem \eqref{eq:main} often include bounds $\ell \le x \le u$. When second derivatives of the objective function are available, the problem is best handled by an interior method. Otherwise, a barrier function could be added to the objective, and the methods here may sometimes be effective on a sequence of large equality-constrained subproblems.}
\section*{Appendix A}
\label{sec:appA}
Here we describe a simplified expression for the matrix $ \bk{C} \tp \bk{G} \bk{C} $
from section~\ref{sec:redComp}. Recall that the L-BFGS inverse
$
\bik{B} = \delta_k I + \bk{J} \bk{W} \bk{J} \tp
$
is defined by
\begin{equation*}
\bk{J} = \bmat{ \bk{S} & \bk{Y} }, \quad
\bk{W} =
\bmat{ \bk{T}^{-\top}(\bk{D} + \delta_k \bk{Y} \tp \bk{Y})\bik{T} & -\delta_k \bk{T}^{-\top} \\
-\delta_k\bik{T} & 0_{l \times l}}.
\end{equation*}
First, \jbr{note} that
\begin{equation*}
\bk{C} \equiv \b{A} \bk{J} \bk{W}
= \bmat{ 0 & \b{A} \bk{Y} } \bk{W}
= \bmat{ -\delta_k \b{A} \bk{Y} \bik{T} & 0 }.
\end{equation*}
Second, \jbr{it holds} that
\begin{equation*}
\bik{G} \equiv \b{A} \bik{B} \b{A} \tp
= \delta_k \b{A} \b{A}\tp + \b{A} \bk{J} \bk{W} \bk{J} \tp \b{A} \tp
= \delta_k \b{A} \b{A}\tp + \bk{C}
\bmat{ 0 \\ (\b{A}\bk{Y}) \tp },
\end{equation*}
so that $ \bik{G} = \delta_k \b{A} \b{A} \tp $, because the last term in
the above expression for $ \bik{G} $ vanishes. Multiplying $ \bk{C} \tp $, $ \bk{G} $ and
$ \bk{C} $ we see that
\begin{equation*}
\bk{C} \tp \bk{G} \bk{C} =
\bmat{\delta_k
\bk{T}^{-\top} \bk{Y} \tp \b{A} \tp (\b{A} \b{A} \tp )^{-1} \b{A} \bk{Y} \bik{T} & 0_{l \times l}\\
0_{l \times l} & 0_{l \times l} }.
\end{equation*}
\section*{Appendix B}
\label{sec:appB}
This appendix describes how we apply the functions from the SuiteSparse
library \cite{SuiteSparseMAT} in our implementations. We use
SuiteSparse version 5.8.1 from
\url{https://github.com/DrTimothyAldenDavis/SuiteSparse/releases}.
\subsection*{B.1: Householder QR projection}
\label{sec:appB1}
The Matlab commands to compute the projection $ P \bk{g} $ using a
Householder QR factorization are listed in Table \ref{tab:HHQR}.
\begin{table}[ht]
\caption{Matlab commands to use SparseSuite functions for computing projections $z=Py$
using a Householder QR factorization.}
\label{tab:HHQR}
\qquad
\begin{tabular}{l}
\texttt{\% Options}
\\ \texttt{opts.Q = `Householder';}
\\ \texttt{opts.permutation = `vector';}
\\
\\ \texttt{\% QR factorization using SPQR}
\\ \texttt{[Q,\~{},\~{},info] = spqr(A',opts);}
\\ \texttt{rankA = info.rank\_A\_estimate;}
\\
\\ \texttt{\% Projection}
\\ \texttt{ztmp = spqr\_qmult(Q,y,0);}
\\ \texttt{zrkA = zeros(rankA,1);}
\\ \texttt{z = [zrkA;ztmp(rankA+1:end)];}
\\ \texttt{z = spqr\_qmult(Q,z,1);}
\end{tabular}
\end{table}
\subsection*{B.2: Preconditioned LSQR projection}
\label{sec:appB2}
The Matlab commands to compute the projection $ P \bk{g} $ using
preconditioned LSQR \cite{PS82a} are listed in Table \ref{tab:LSQR}.
\begin{table}[ht]
\caption{Matlab commands for computing projections
$z=Py$ \hbox{using} preconditioned LSQR
(where $P = I - A\tp (AA\tp)^{-1} A$).
\Red{If $A$ has full row rank ($\texttt{rankA}=m$),
LSQR should need only 1 iteration.}
Notes: SPQR uses all of $\b{A}\tp$ in the QR factorization
$\b{A}\tp \b{P}_{\text{msk}} = \b{Q}\b{R}$,
where $\b{P}_{\text{msk}}$ is a column permutation of
$\b{A}\tp$ and $\b{R}$ is upper trapezoidal. We store the
permutation in the vector $\texttt{maskA}$. If $\b{A}\tp$ does not have full row rank,
we use the first $\texttt{rankA}$ columns
of $\b{A}\tp\b{P}_{\text{msk}}$
(the command $\texttt{A(maskA(1:rankA),:)'}$).
If $A$ contains some relatively dense columns, we should partition
$\b{A}\b{P}_{\text{prt}} = [\: \b{A}_{S} \: \b{A}_{D} \:]$ into sparse and dense columns,
then use $\b{A}_{S}$ in place of $\b{A}$
in the call to $\texttt{spqr}$.
}
\label{tab:LSQR}
\qquad
\begin{tabular}{l}
\texttt{\% Options}
\\ \texttt{opts.econ = 0;}
\\ \texttt{opts.Q = `Householder';}
\\ \texttt{opts.permutation = `vector';}
\\ \texttt{tol = 1e-15;}
\\ \texttt{maxit = m;}
\\
\\ \texttt{\% Preconditioner using a triangular}
\\ \texttt{\% factor from SPQR}
\\ \texttt{[\~{},R,maskA,info] = spqr(A',opts);}
\\ \texttt{rankA = info.rank\_A\_estimate;}
\\
\\ \texttt{\% Projection}
\\ \texttt{x = lsqr(A(maskA(1:rankA),:)',y,...}
\\ \quad \quad \quad \quad \ \ \texttt{tol,maxit,R(1:rankA,1:rankA));}
\\ \texttt{z = y - A(maskA(\jbbo{1:rankA}),:)'*x(1:rankA,1);}
\end{tabular}
\end{table}
\section*{Appendix C}
\label{sec:appC}
This appendix overviews the subproblem solution with the shape-changing norm.
Note that $ \b{U} = \bmat{ \b{Q}_1 & \b{U}_2 & \b{U}_3 } \in \mathbb{R}^{n \times n} $
(from section \ref{sec:eigen}) represents an orthogonal matrix, and that the quadratic function is
\begin{equation*}
q(\b{s}) = \b{s} \tp \bk{g} + \frac{1}{2} \b{s} \tp \bk{B} \b{s}
= \b{s} \tp \b{U} \b{U} \tp \bk{g} + \frac{1}{2} \b{s} \tp \b{U} \b{U} \tp \bk{B} \b{U} \b{U} \tp \b{s}.
\end{equation*}
We introduce the change of variables
$ \b{v} \tp = \bmat{ \b{v}_1\tp & \b{v}_2\tp & \b{v}_3\tp } \equiv \b{s}\tp \b{U} $. Moreover, it
holds that
$$
\b{U} \tp \bk{B} \b{U} =
\bmat{
\b{Q}_1 \tp \bk{B} \b{Q}_1 & \b{Q}_1 \tp \bk{B} \b{U}_2 & \b{Q}_1 \tp \bk{B} \b{U}_3 \\
\b{U}_2\tp \bk{B} \b{Q}_1 & (\delta_k \b{I} + \b{\Lambda}_2)^{-1} & \\
\b{U}_3\tp \bk{B} \b{Q}_1 & & \delta_k^{-1} \b{I}}
$$
(cf.\ \cite[Lemma 2]{BMP19}), and that
$$
\b{A} \b{U} \b{U} \tp \b{s} = \b{A} \b{U} \b{v}
= \bmat{\b{R} & \b{0} & \b{0}} \bmat{\b{v}_1 \\ \b{v}_2 \\ \b{v}_3}
= Rv_1.
$$
With the constraint $ \b{A} \b{s} = \b{0} = \b{A} \b{U} \b{v} $, this implies $ \b{v}_1 = \b{0} $
(for $\b{R}$ nonsingular).
Therefore, the trust-region subproblem defined by the
shape-changing norm decouples into a problem with $ \b{v}_2 $ and $ \b{v}_3 $
only (once $ \b{v}_1 = \b{0} $ is fixed):
\begin{align*}
\underset{
\tiny
\begin{array}{c}
\| \b{s} \|_U \le \Delta_k \\
\b{A} \b{s} = \b{0}
\end{array} }{ \text{ minimize } } q(\b{s})
=
\bigg\{&
\underset{
\| \b{v}_2 \|_{\infty} \le \Delta_k
}{ \text{ minimize } } \biidx{2}{v} \tp \biidx{2}{U} \tp \bk{g} +
\frac{1}{2} \biidx{2}{v} \tp (\delta_k \b{I} + \Lambda_2)^{-1} \biidx{2}{v} \\
& +
\underset{
\| \b{v}_3 \|_{2} \le \Delta_k
}{ \text{ minimize } } \biidx{3}{v} \tp \biidx{3}{U} \tp \bk{g} +
\frac{\| \biidx{3}{v} \|^2_2}{2\delta_k}
\bigg\}.
\end{align*}
This reformulated subproblem can be solved analytically and the component-wise
solution of $ \biidx{2}{v} $ is in \eqref{eq:SCanalytic1}. The analytic solution
of $ \biidx{3}{v} $ is $ \biidx{3}{v} = \beta \biidx{3}{U} \tp \bk{g} $
with $ \beta $ from \eqref{eq:SCanalytic2}. Subsequently, $\b{s}$ is
obtained by transforming variables as
$ \b{s} = \b{U} \b{v} = \biidx{2}{U} \biidx{2}{v} + \biidx{3}{U} \biidx{3}{v} $.
The orthonormal matrix $ \biidx{2}{U} $ is computed as
$ \biidx{2}{U} = \bmat{ \bk{S} & \bk{Z} } \bh{R}_2^{-1} \bh{P}_2 $,
and since $ \biidx{3}{U} \biidx{3}{U} \tp = \b{P} - \biidx{2}{U} \biidx{2}{U} \tp $, the optimal step with the shape-changing norm is as in \eqref{sec:SCNorm}:
\begin{equation*}
\b{s}_{SC} = \biidx{2}{U}(\biidx{2}{v} - \beta \biidx{2}{U}\tp \bk{g})
+ \beta \b{P} \bk{g}.
\end{equation*}
With $ \bk{u} \equiv \biidx{2}{U} \tp \bk{g} $, the step is then
computed as in Algorithm \ref{alg:LTRSC-LEC} (line 15):
\begin{equation*}
\b{s}_{SC} = \bmat{ \bk{S} & \bk{Z} } \bh{R}_2^{-1} \bh{P}_2(\biidx{2}{v} - \beta \bk{u})
+ \beta \b{P} \bk{g}.
\end{equation*}
\section*{Appendix D}
\label{sec:appD}
\subsection*{D.1: \jbr{Detailed Table for Experiment I}}
\label{sec:appD1}
\begin{table}[p]
\captionsetup{width=1\linewidth}
\caption{Experiment I compares 7 solvers on problems from the SuiteSparse Matrix Collection \cite{SuiteSparseMatrix}.
Entries with $\texttt{N/A}^{*}$ denote problems to which \TR1 and \TR2 were not applied, because they are too large.
$\texttt{NC}^{\dagger}$ means the solver did not converge to tolerances. \jbbo{\TR{2H} and \TR{1H} converged on all problem
instances. Overall, the computational times of \TR{2\{H,L\}}
and \TR{1\{H,L\}} were lower by a
significant factor compared to the times of \TR1, \TR2, and {\small IPOPT}.
The number of iterations for each solver is similar across all problems.}}
\label{tab:exp1}
\setlength{\tabcolsep}{2pt}
\scriptsize
\hbox to 1.00\textwidth{\hss
\begin{tabular}{|l|c|c| c c | c c | c c | c c | c c | c c | c c |}
\hline
\multirow{2}{*}{Problem} & \multirow{2}{*}{$m$/$n$} & \multirow{2}{*}{$\text{rank}(\b{A})$} &
\multicolumn{2}{c|}{\jbbo{TR2}} &
\multicolumn{2}{c|}{\jbbo{TR2H}} &
\multicolumn{2}{c|}{\jbbo{TR2L}} &
\multicolumn{2}{c|}{\jbbo{TR1}} &
\multicolumn{2}{c|}{\jbbo{TR1H}} &
\multicolumn{2}{c|}{\jbbo{TR1L}} &
\multicolumn{2}{c|}{IPOPT} \\
\cline{4-17}
& &
& It & Sec
& It & Sec
& It & Sec
& It & Sec
& It & Sec
& It & Sec
& It & Sec \\
\hline
$\texttt{beacxc}$ & 497/506 & 449/0.2 &73 &0.52&25 &\emph{0.044}&25 &0.15&419 &3.8&25 &\textbf{0.041}&25 &0.15&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{lp\_25fv47}$ & 821/1876 & 820/0.007 &60 &0.82&60 &0.21&60 &\textbf{0.14}&62 &0.85&62 &0.22&62 &\emph{0.14}&61 &0.73\\
$\texttt{lp\_agg2}$ & 516/758 & 516/0.01 &40 &0.21&40 &\emph{0.054}&40 &\textbf{0.052}&42 &0.21&42 &0.056&42 &0.055&41 &0.22\\
$\texttt{lp\_agg3}$ & 516/758 & 516/0.01 &39 &0.21&39 &\textbf{0.051}&39 &\emph{0.051}&39 &0.2&39 &0.052&39 &0.051&44 &0.24\\
$\texttt{lp\_bnl1}$ & 643/1586 & 642/0.005 &70 &0.57&70 &0.14&70 &\emph{0.079}&67 &0.6&67 &0.14&67 &\textbf{0.078}&62 &0.59\\
$\texttt{lp\_bnl2}$ & 2324/4486 & 2324/0.001 &69 &11&69 &0.62&69 &\emph{0.28}&69 &11&69 &0.52&69 &\textbf{0.27}&67 &2.2\\
$\texttt{lp\_cre\_a}$ & 3516/7248 & 3428/0.0007 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &83 &0.65&83 &\textbf{0.37}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &88 &0.71&88 &\emph{0.38}&87 &3.3\\
$\texttt{lp\_cre\_d}$ & 8926/73948 & 6476/0.0004 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &556 &1.2e+02&510 &\textbf{24}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &503 &1e+02&552 &\emph{25}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{lp\_czprob}$ & 929/3562 & 929/0.003 &17 &0.27&17 &0.059&17 &\emph{0.032}&17 &0.25&17 &0.049&17 &\textbf{0.028}&18 &0.34\\
$\texttt{lp\_d6cube}$ & 415/6184 & 404/0.01 &35 &0.22&35 &0.4&35 &\textbf{0.17}&36 &0.21&36 &0.44&36 &\emph{0.18}&38 &1.1\\
$\texttt{lp\_degen3}$ & 1503/2604 & 1503/0.006 &39 &2.2&39 &0.25&39 &\emph{0.25}&39 &2.3&39 &0.27&39 &\textbf{0.24}&40 &1.5\\
$\texttt{lp\_dfl001}$ & 6071/12230 & 6071/0.0005 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &226 &\textbf{16}&231 &19&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &226 &\emph{16}&238 &20&207 &1.2e+02\\
$\texttt{lp\_etamacro}$ & 400/816 & 400/0.008 &78 &0.27&78 &0.12&78 &\textbf{0.085}&86 &0.29&86 &0.13&86 &\emph{0.09}&68 &0.44\\
$\texttt{lp\_fffff800}$ & 524/1028 & 524/0.01 &$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &61 &\textbf{0.095}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &59 &\emph{0.097}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &57 &0.45\\
$\texttt{lp\_finnis}$ & 497/1064 & 497/0.005 &150 &0.72&151 &0.2&156 &\textbf{0.13}&159 &0.69&155 &0.2&155 &\emph{0.14}&167 &1.2\\
$\texttt{lp\_fit2d}$ & 25/10524 & 25/0.5 &266 &1.3&266 &\emph{0.9}&258 &\textbf{0.88}&247 &1.4&261 &0.91&279 &0.99&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{lp\_ganges}$ & 1309/1706 & 1309/0.003 &41 &1.3&41 &0.11&41 &\textbf{0.067}&41 &1.4&41 &0.12&41 &\emph{0.073}&37 &0.43\\
$\texttt{lp\_gfrd\_pnc}$ & 616/1160 & 616/0.003 &$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &54 &0.054&54 &\emph{0.043}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &54 &0.052&54 &\textbf{0.042}&48 &0.36\\
$\texttt{lp\_greenbea}$ & 2392/5598 & 2389/0.002 &149 &47&149 &1.2&149 &\textbf{0.63}&157 &33&153 &1.3&150 &\emph{0.72}&181 &6.8\\
$\texttt{lp\_greenbeb}$ & 2392/5598 & 2389/0.002 &149 &45&149 &1.2&149 &\emph{0.65}&157 &31&153 &1.3&150 &\textbf{0.65}&181 &6.5\\
$\texttt{lp\_grow22}$ & 440/946 & 440/0.02 &79 &0.24&79 &0.079&79 &\emph{0.071}&79 &0.24&79 &0.08&79 &\textbf{0.069}&65 &0.36\\
$\texttt{lp\_ken\_07}$ & 2426/3602 & 2426/0.001 &34 &12&34 &0.091&34 &\textbf{0.067}&34 &7.4&34 &0.093&34 &\emph{0.07}&31 &0.85\\
$\texttt{lp\_maros}$ & 846/1966 & 846/0.006 &74 &0.87&74 &\emph{0.21}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &74 &0.86&74 &\textbf{0.21}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &71 &0.92\\
$\texttt{lp\_maros\_r7}$ & 3136/9408 & 3136/0.005 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &57 &\emph{2.1}&57 &2.2&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &57 &\textbf{2.1}&57 &2.3&51 &25\\
$\texttt{lp\_modszk1}$ & 687/1620 & 686/0.003 &71 &0.51&71 &0.13&71 &\textbf{0.07}&71 &0.51&71 &0.14&71 &\emph{0.071}&70 &0.53\\
$\texttt{lp\_osa\_30}$ & 4350/104374 & 4350/0.001 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &46 &8.5&46 &\textbf{1.9}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &45 &8.8&45 &\emph{2}&43 &30\\
$\texttt{lp\_osa\_60}$ & 10280/243246 & 10280/0.0006 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &47 &24&47 &\textbf{5.9}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &44 &23&44 &\emph{5.9}&42 &1.1e+02\\
$\texttt{lp\_pds\_02}$ & 2953/7716 & 2953/0.0007 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &25 &0.25&25 &\emph{0.14}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &25 &0.24&25 &\textbf{0.099}&26 &1.3\\
$\texttt{lp\_pds\_10}$ & 16558/49932 & 16558/0.0001 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &61 &13&61 &\emph{8.1}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &60 &13&60 &\textbf{7.9}&59 &62\\
$\texttt{lp\_perold}$ & 625/1506 & 625/0.007 &58 &0.39&58 &0.16&58 &\textbf{0.084}&58 &0.38&58 &0.16&58 &\emph{0.087}&57 &0.59\\
$\texttt{lp\_pilot}$ & 1441/4860 & 1441/0.006 &105 &13&105 &1.3&105 &\textbf{0.83}&109 &6.2&109 &1.3&109 &\emph{0.97}&117 &5.8\\
$\texttt{lp\_pilot87}$ & 2030/6680 & 2030/0.006 &102 &17&102 &2.6&102 &\textbf{1.9}&104 &15&104 &2.7&104 &\emph{1.9}&110 &15\\
$\texttt{lp\_pilot\_we}$ & 722/2928 & 722/0.004 &73 &0.69&73 &0.24&73 &\emph{0.11}&73 &0.65&73 &0.21&73 &\textbf{0.11}&81 &1.4\\
$\texttt{lp\_pilotnov}$ & 975/2446 & 975/0.006 &77 &1.3&77 &\textbf{0.35}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &77 &1.3&77 &\emph{0.35}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &78 &1.3\\
$\texttt{lp\_qap12}$ & 3192/8856 & 3192/0.001 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &27 &\emph{3.3}&27 &3.7&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &26 &\textbf{3.1}&26 &3.6&25 &1.4e+02\\
$\texttt{lp\_qap8}$ & 912/1632 & 912/0.005 &20 &0.42&20 &0.15&20 &\emph{0.11}&22 &0.31&22 &0.15&22 &\textbf{0.1}&21 &1.9\\
$\texttt{lp\_scfxm1}$ & 330/600 & 330/0.01 &45 &0.1&45 &0.042&45 &\emph{0.036}&44 &0.098&44 &0.041&44 &\textbf{0.036}&44 &0.19\\
$\texttt{lp\_scfxm2}$ & 660/1200 & 660/0.007 &52 &0.42&52 &0.079&52 &\textbf{0.056}&57 &0.43&57 &0.094&57 &\emph{0.06}&55 &0.46\\
$\texttt{lp\_scfxm3}$ & 990/1800 & 990/0.005 &45 &0.8&45 &0.096&45 &\textbf{0.061}&45 &0.76&45 &0.097&45 &\emph{0.062}&48 &0.54\\
$\texttt{lp\_scsd1}$ & 77/760 & 77/0.04 &74 &0.035&74 &0.035&74 &0.044&74 &\textbf{0.033}&74 &\emph{0.034}&74 &0.043&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{lp\_scsd6}$ & 147/1350 & 147/0.02 &84 &0.077&84 &\textbf{0.054}&84 &0.06&92 &0.084&92 &\emph{0.06}&92 &0.065&75 &0.34\\
$\texttt{lp\_scsd8}$ & 397/2750 & 397/0.008 &66 &0.21&66 &0.07&66 &\textbf{0.066}&65 &0.21&65 &0.069&65 &\emph{0.066}&66 &0.54\\
$\texttt{lp\_sctap1}$ & 300/660 & 300/0.009 &107 &0.25&107 &0.088&107 &\emph{0.075}&102 &0.24&102 &0.081&102 &\textbf{0.07}&100 &0.45\\
$\texttt{lp\_sctap2}$ & 1090/2500 & 1090/0.003 &145 &5.5&146 &0.43&146 &\textbf{0.18}&145 &3.6&143 &0.46&146 &\emph{0.19}&157 &2.3\\
$\texttt{lp\_sctap3}$ & 1480/3340 & 1480/0.002 &204 &27&205 &0.75&201 &\emph{0.39}&199 &13&197 &0.74&202 &\textbf{0.39}&220 &4.3\\
$\texttt{lp\_ship04l}$ & 402/2166 & 360/0.007 &84 &0.25&84 &0.12&84 &\textbf{0.077}&84 &0.27&84 &0.12&84 &\emph{0.08}&92 &0.84\\
$\texttt{lp\_ship04s}$ & 402/1506 & 360/0.007 &74 &0.17&74 &0.063&74 &\textbf{0.053}&74 &0.16&74 &0.065&74 &\emph{0.055}&71 &0.48\\
$\texttt{lp\_stair}$ & 356/614 & 356/0.02 &47 &0.11&47 &0.047&47 &\emph{0.046}&47 &0.11&47 &0.047&47 &\textbf{0.045}&47 &0.23\\
$\texttt{lp\_standata}$ & 359/1274 & 359/0.007 &78 &0.22&78 &0.072&78 &\emph{0.058}&79 &0.21&79 &0.067&79 &\textbf{0.057}&80 &0.65\\
$\texttt{lp\_standmps}$ & 467/1274 & 467/0.007 &52 &0.21&52 &0.06&52 &\textbf{0.042}&52 &0.21&52 &0.065&52 &\emph{0.043}&58 &0.48\\
\hline
\end{tabular}
\hss}
\end{table}
In this experiment the degree of difficulty in solving a problem depends largely
on handling $ \b{A} $, because the structure of the
objective function is the same for all instances.
We observe that our proposed algorithms (any of \TR{1\{H,L\}}, \TR{2\{H,L\}}) always use less
computation time (often significantly), except for two problem instances.
On problem $ \texttt{lp\_d6cube} $, \TR2 used less time than \TR{2H}, as did \TR1 over
\TR{1H}. However, the ``L" versions were fastest overall on this problem.
On problem $ \texttt{lp\_scsd1} $, \TR1 used the least time. In these
two problems the number of constraints is not large, and one can expect
that \TR1, \TR2 do comparatively well.
However, for all other 48
problems the new methods used the least time. We observe that
both ``H" versions converged to the prescribed tolerances on all problems.
On the other hand, the ``L" versions are often the overall fastest, yet they did
not converge on 3 problem instances (\texttt{beacxc}, \texttt{lp\_cre\_d}, \texttt{fit2d}).
\subsection*{D.2: \jbr{Detailed Table for Experiment II}}
\label{sec:appD2}
\begin{table}[p]
\captionsetup{width=1\linewidth}
\caption{Experiment II compares 7 solvers on 61 large problems from the CUTEst collection \cite{GouOT03}.
$\texttt{NC}^{\dagger}$ means the solver did not converge to tolerances. $\texttt{MX}^{\dagger}$ means the iteration limit was reached.
\jbbo{\TR{1L} converged on 58 problems, the largest number of problems
amongst the solvers. \TR{2H} was faster than \TR2 on 51 problems,
and \TR{2L} was faster than \TR2 on 46 problems (the differences are often significant). \TR{1H} was faster than \TR{1} on 49 problems and
\TR{1L} was faster than \TR{1} on 41 problems (often significantly).
All of \TR{1\{H,L\}} and \TR{2\{H,L\}} were faster than
{\small IPOPT}.}}
\label{tab:exp2}
\setlength{\tabcolsep}{1.5pt}
\scriptsize
\hbox to 1.00\textwidth{\hss
\begin{tabular}{|l | c | c c | c c | c c | c c | c c | c c | c c|}
\hline
\multirow{2}{*}{Problem} & \multirow{2}{*}{$m$/$n$} &
\multicolumn{2}{c|}{\jbbo{TR2}} &
\multicolumn{2}{c|}{\jbbo{TR2H}} &
\multicolumn{2}{c|}{\jbbo{TR2L}} &
\multicolumn{2}{c|}{\jbbo{TR1}} &
\multicolumn{2}{c|}{\jbbo{TR1H}} &
\multicolumn{2}{c|}{\jbbo{TR1L}} &
\multicolumn{2}{c|}{IPOPT} \\
\cline{3-16}
&
& It & Sec
& It & Sec
& It & Sec
& It & Sec
& It & Sec
& It & Sec
& It & Sec \\
\hline
$\texttt{ARWHEAD}$ & 1250/5000 &343 &1.7e+02&349 &19&372 &19&264 &72&304 &\textbf{16}&315 &\emph{16}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{BDQRTIC}$ & 1250/5000 &181 &50&174 &\textbf{8.1}&187 &9.9&174 &31&186 &8.9&160 &\emph{8.4}&78 &1.2e+02\\
$\texttt{BOX}$ & 2500/10000 &240 &1.5e+03&280 &63&281 &79&218 &2.1e+02&258 &\textbf{54}&208 &\emph{58}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{BROYDN7D}$ & 1250/5000 &355 &20&370 &\emph{18}&367 &18&355 &20&370 &\textbf{17}&381 &19&432 &6.5e+02\\
$\texttt{BRYBND}$ & 1250/5000 &897 &1.5e+02&883 &\textbf{45}&1273 &64&1396 &1.2e+02&1177 &\emph{60}&1421 &70&1027 &1.7e+03\\
$\texttt{COSINE}$ & 2500/10000 &$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &5028 &1e+03&4527 &1.2e+03&4755 &2e+03&7318 &1.6e+03&3292 &\emph{910}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{CRAGGLVY}$ & 1250/5000 &373 &63&371 &\textbf{18}&369 &\emph{19}&400 &45&390 &20&397 &21&205 &3.4e+02\\
$\texttt{CURLY10}$ & 2500/10000 &1563 &7.2e+02&2498 &5.3e+02&1496 &\emph{429}&1512 &4.5e+02&1549 &\textbf{347}&1759 &4.9e+02&1775 &3e+04\\
$\texttt{CURLY20}$ & 2500/10000 &1951 &9.5e+02&2015 &\textbf{455}&1993 &\emph{552}&3149 &9.5e+02&4110 &8.7e+02&3836 &1.1e+03&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{CURLY30}$ & 2500/10000 &4457 &2.8e+03&4210 &\emph{952}&3669 &1e+03&2744 &\textbf{783}&6940 &1.6e+03&6145 &1.7e+03&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{DIXMAANA}$ & 750/3000 &10 &0.53&10 &\textbf{0.43}&10 &0.51&10 &0.5&10 &0.47&10 &\emph{0.46}&13 &8.3\\
$\texttt{DIXMAANB}$ & 750/3000 &9 &0.55&9 &0.5&9 &\emph{0.5}&9 &0.59&9 &\textbf{0.5}&9 &0.55&11 &8.1\\
$\texttt{DIXMAANC}$ & 750/3000 &12 &0.73&12 &0.67&12 &0.72&12 &\emph{0.65}&12 &\textbf{0.63}&12 &0.69&14 &10\\
$\texttt{DIXMAAND}$ & 750/3000 &23 &1.7&23 &1.1&23 &1.2&22 &\emph{0.93}&22 &\textbf{0.83}&22 &1&27 &16\\
$\texttt{DIXMAANE}$ & 750/3000 &35 &1.1&35 &1&35 &1.1&35 &\textbf{0.83}&35 &\emph{0.88}&35 &1.1&41 &18\\
$\texttt{DIXMAANF}$ & 750/3000 &183 &5.2&194 &\textbf{3.9}&194 &5.7&194 &6.6&195 &\emph{4.9}&203 &6.7&297 &1.3e+02\\
$\texttt{DIXMAANG}$ & 750/3000 &434 &19&397 &\textbf{8.3}&439 &12&435 &13&408 &\emph{9.8}&404 &11&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{DIXMAANH}$ & 750/3000 &433 &14&470 &11&454 &13&459 &\emph{11}&421 &\textbf{9.3}&443 &12&422 &1.8e+02\\
$\texttt{DIXMAANI}$ & 750/3000 &82 &2&82 &\emph{1.8}&82 &2.4&82 &\textbf{1.6}&82 &1.8&82 &2.5&103 &46\\
$\texttt{DIXMAANJ}$ & 750/3000 &1054 &41&1506 &35&1023 &\emph{27}&1415 &42&1490 &34&944 &\textbf{24}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{DIXMAANK}$ & 750/3000 &2971 &1e+02&3026 &65&3082 &71&2831 &80&2870 &\textbf{61}&2691 &\emph{62}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{DIXMAANL}$ & 750/3000 &1461 &\textbf{38}&3198 &69&2609 &60&2690 &66&2728 &\emph{58}&2597 &59&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{DIXON3DQ}$ & 2500/10000 &51 &17&51 &\emph{12}&51 &17&51 &17&51 &\textbf{12}&51 &16&56 &6.7e+02\\
$\texttt{DQDRTIC}$ & 1250/5000 &13 &1.7&7 &0.85&7 &0.77&13 &1.5&7 &\textbf{0.75}&7 &\emph{0.75}&7 &13\\
$\texttt{DQRTIC}$ & 1250/5000 &63 &\emph{4.6}&107 &6.7&107 &7&63 &\textbf{4.5}&107 &5.7&107 &6.1&93 &1.5e+02\\
$\texttt{EDENSCH}$ & 500/2000 &32 &\emph{0.33}&32 &0.4&32 &0.38&32 &\textbf{0.32}&32 &0.39&32 &0.36&34 &5\\
$\texttt{EG2}$ & 250/1000 &423 &2.2&504 &\emph{1.3}&439 &\textbf{1.3}&514 &5.2&624 &2&502 &1.9&908 &23\\
$\texttt{ENGVAL1}$ & 1250/5000 &31 &2.7&31 &\textbf{1.8}&31 &2&31 &2.6&31 &\emph{1.9}&31 &2&38 &61\\
$\texttt{EXTROSNB}$ & 250/1000 &148 &\emph{0.44}&148 &0.45&148 &0.49&145 &0.53&145 &0.46&145 &\textbf{0.39}&129 &3\\
$\texttt{FLETCHCR}$ & 250/1000 &150 &\emph{0.4}&150 &0.46&150 &0.51&150 &\textbf{0.37}&150 &0.42&150 &0.41&137 &3.1\\
$\texttt{FMINSRF2}$ & 1407/5625 &122 &10&122 &\emph{9.3}&122 &10&122 &10&122 &\textbf{7.8}&122 &9.6&167 &4e+02\\
$\texttt{FREUROTH}$ & 1250/5000 &287 &1e+02&247 &\textbf{12}&235 &\emph{13}&274 &37&255 &13&234 &13&202 &3.2e+02\\
$\texttt{GENHUMPS}$ & 1250/5000 &2215 &1.2e+02&1762 &99&1829 &\textbf{93}&2215 &1.3e+02&1762 &98&1829 &\emph{95}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{LIARWHD}$ & 1250/5000 &3854 &1.6e+03&3998 &4.4e+02&2726 &\emph{196}&2638 &1.2e+03&2408 &2.6e+02&1591 &\textbf{128}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{MOREBV}$ & 1250/5000 &151 &23&151 &22&151 &20&151 &19&151 &\emph{16}&151 &\textbf{16}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{MSQRTALS}$ & 256/1024 &$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &78461 &6.6e+02&99724 &\emph{620}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{MSQRTBLS}$ & 256/1024 &$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{NCB20}$ & 1253/5010 &345 &47&348 &18&349 &18&314 &33&317 &\textbf{16}&307 &\emph{16}&252 &3.9e+02\\
$\texttt{NONCVXU2}$ & 1250/5000 &185 &20&185 &\textbf{9}&185 &9.5&186 &14&187 &\emph{9.2}&186 &9.4&120 &1.9e+02\\
$\texttt{NONCVXUN}$ & 1250/5000 &282 &33&283 &\textbf{14}&282 &\emph{14}&360 &31&354 &17&370 &19&199 &3.1e+02\\
$\texttt{NONDIA}$ & 1250/5000 &1612 &6.9e+02&1600 &88&1734 &\emph{88}&2764 &7.3e+02&1407 &\textbf{78}&1907 &98&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{NONDQUAR}$ & 1250/5000 &897 &4.3e+02&865 &47&811 &\textbf{42}&816 &2.1e+02&876 &47&857 &\emph{44}&332 &8.1e+02\\
$\texttt{PENALTY1}$ & 250/1000 &8 &0.051&2 &0.018&2 &\emph{0.017}&8 &0.056&2 &0.019&2 &\textbf{0.016}&1 &0.043\\
$\texttt{POWELLSG}$ & 1250/5000 &88 &6.1&88 &\textbf{4.4}&88 &4.6&88 &5.6&88 &\emph{4.4}&88 &4.6&99 &1.5e+02\\
$\texttt{POWER}$ & 2500/10000 &51 &\textbf{17}&$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &51 &\emph{17}&$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &62 &6.9e+02\\
$\texttt{QUARTC}$ & 1250/5000 &70 &\emph{4.9}&104 &5.3&104 &5.4&70 &\textbf{4.5}&104 &5.1&104 &5.6&89 &1.4e+02\\
$\texttt{SCHMVETT}$ & 1250/5000 &$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &70882 &3.9e+03&$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &$\texttt{MX}^{\dagger}$ & $\texttt{MX}$ &96572 &5.1e+03&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{SINQUAD}$ & 1250/5000 &236 &56&282 &15&214 &\textbf{11}&247 &32&216 &\emph{12}&277 &14&116 &1.8e+02\\
$\texttt{SPARSQUR}$ & 2500/10000 &35 &13&43 &\emph{10}&43 &14&35 &13&43 &\textbf{9.9}&43 &14&31 &3.5e+02\\
$\texttt{SPMSRTLS}$ & 1250/4999 &2222 &2.7e+02&1791 &\textbf{95}&2377 &1.2e+02&2792 &2e+02&2475 &1.3e+02&1834 &\emph{98}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{SROSENBR}$ & 1250/5000 &5561 &4.1e+02&8211 &4.3e+02&4814 &\textbf{235}&6400 &4.3e+02&6747 &3.6e+02&5280 &\emph{270}&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{TOINTGSS}$ & 1250/5000 &39 &3.1&39 &\textbf{2.2}&39 &2.3&39 &3&39 &2.3&39 &\emph{2.3}&49 &76\\
$\texttt{TQUARTIC}$ & 1250/5000 &2069 &8.7e+02&1155 &\textbf{64}&1508 &\emph{78}&1494 &3.7e+02&1867 &1e+02&1871 &98&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{TRIDIA}$ & 1250/5000 &147 &9&82 &\textbf{4.2}&82 &4.3&147 &9.1&82 &\emph{4.2}&82 &4.4&66 &1e+02\\
$\texttt{WOODS}$ & 1000/4000 &1192 &45&1157 &38&1077 &\textbf{27}&1236 &44&1167 &37&1132 &\emph{29}&971 &1.3e+03\\
$\texttt{SPARSINE}$ & 1250/5000 &1504 &1.7e+02&1476 &79&1464 &\textbf{74}&2188 &1.6e+02&1407 &\emph{74}&3999 &2e+02&2294 &5.6e+03\\
$\texttt{TESTQUAD}$ & 1250/5000 &10988 &\textbf{623}&14186 &7.3e+02&13357 &6.5e+02&10988 &\emph{643}&14186 &7.3e+02&13357 &6.6e+02&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{JIMACK}$ & 888/3549 &$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{NCB20B}$ & 1250/5000 &57 &4.1&56 &3.2&56 &3.2&57 &4.2&56 &\textbf{3.1}&56 &\emph{3.2}&47 &73\\
$\texttt{EIGENALS}$ & 638/2550 &202 &\emph{3.2}&204 &3.7&203 &4.1&202 &\textbf{3}&204 &3.6&203 &4&161 &43\\
$\texttt{EIGENBLS}$ & 638/2550 &28 &0.59&28 &0.65&28 &0.6&28 &\textbf{0.51}&28 &\emph{0.52}&28 &0.62&28 &7.7\\
\hline
\end{tabular}
\hss}
\end{table}
In Experiment II, the objective functions for each problem are defined by
a large CUTEst problem, whereas the corresponding $ \b{A} $ matrices are not meant
to be overly challenging. We observe that the proposed algorithms
(the ones including ``\{H,L\}") improve the computation times on the majority of
problems. For the 10 instances in which \TR2 used less time than \TR2H, the
differences are relatively small. An exception is $ \texttt{DIXMAANL} $, where
the difference amounts to 31s. However, for the other 51 problems, \TR2H resulted
in often significant improvements in computation time. For instance, in
$ \texttt{LIARWHD} $ this difference amounts to 1182s (more than 19 minutes).
These observations carry over when comparing \TR1 with \TR{1H}. The ``L" versions
exhibit similar outcomes as the ``H" ones, with occasional increases in computation
times. Overall, \TR{1L} converged to the specified tolerances on the largest number of problems. \jbk{The problems reported as ``\texttt{NC}" in {\small IPOPT }'s column correspond to status flags other than ``0, 1, 2" $\equiv$ ``solved, solved to acceptable level, infeasible problem detected".}
\subsection*{D.3: \jbr{Detailed Table for Experiment III}}
\label{sec:appD3}
\jbr{In Experiment III, \TR{2H} and \TR{1H} converged on all 31 problems,
while all other solvers (besides \TR1 and TR2) converged on all problems except one:
\texttt{CVXQP2}. \TR{2H} was the fastest on 10 problems (the best outcome among the solvers),
while \TR{1L} was the fastest on 9 problems (the second best outcome). Problems
\texttt{A0ESDNDL} and \texttt{A0ESINDL} appear noteworthy:
they contain dense columns (satisfying the condition $ \textnormal{nnz}(A_{:,j}) \big / m > 0.1 $).
Sparse QR factorization
is expensive because
of fill-in. However, the iterative method LSQR (with the preconditioning technique
from section \ref{sec:LSQR}) can overcome these difficulties.}
\begin{table}[p]
\captionsetup{width=1\linewidth}
\caption{\jbr{Experiment III compares 7 solvers on 31 linear equality constrained problems from the CUTEst collection \cite{GouOT03}.
$\texttt{NC}^{\dagger}$ means the solver did not converge to tolerances.
$\texttt{N/A}$ means that \TR1 and \TR2 were not applied because the problem size rendered them not practical. \TR{2H} and \TR{1H} converged on all 31 problems.
\TR{2L}, \TR{1L}, and {\small IPOPT} converged on 30 problems (the exception is
\texttt{CVXQP2}). The fastest and second fastest solvers for each problem
are highlighted in bold and italic fonts, respectively. Overall,
\TR{2H} was fastest on 12 problems (the best outcome on this experiment), while
\TR{1L} was fastest on 11 problems (the second best outcome). Problems
\texttt{A0ESDNDL} and \texttt{A0ESINDL} contain dense columns in $A$, and the sparse QR
factorization takes additional time as seen from the entries of \TR{2H} and \TR{1H}.
However, preconditioned {\small LSQR} can overcome
this difficulty, as observed in the entries for \TR{2L} and \TR{1L} for these problem
instances.
}}
\label{tab:exp3}
\setlength{\tabcolsep}{2pt}
\scriptsize
\hbox to 1.00\textwidth{\hss
\begin{tabular}{|l | c | c c | c c | c c | c c | c c | c c | c c|}
\hline
\multirow{2}{*}{Problem} & \multirow{2}{*}{$m$/$n$} &
\multicolumn{2}{c|}{\jbbo{TR2}} &
\multicolumn{2}{c|}{\jbbo{TR2H}} &
\multicolumn{2}{c|}{\jbbo{TR2L}} &
\multicolumn{2}{c|}{\jbbo{TR1}} &
\multicolumn{2}{c|}{\jbbo{TR1H}} &
\multicolumn{2}{c|}{\jbbo{TR1L}} &
\multicolumn{2}{c|}{IPOPT} \\
\cline{3-16}
&
& It & Sec
& It & Sec
& It & Sec
& It & Sec
& It & Sec
& It & Sec
& It & Sec \\
\hline
$\texttt{AUG2D}$ & 10000/20200 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &7 &0.26&7 &\emph{0.15}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &7 &0.24&7 &\textbf{0.13}&12 &1.4\\
$\texttt{AUG2DC}$ & 10000/20200 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &0.11&2 &\emph{0.067}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &0.1&2 &\textbf{0.067}&1 &0.15\\
$\texttt{AUG2DCQP}$ & 10000/20200 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &0.11&2 &\emph{0.072}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &0.11&2 &\textbf{0.07}&1 &0.16\\
$\texttt{AUG2DQP}$ & 10000/20200 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &7 &0.23&7 &\textbf{0.13}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &7 &0.24&7 &\emph{0.13}&12 &1.4\\
$\texttt{AUG3D}$ & 8000/27543 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &10 &0.68&10 &\emph{0.52}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &10 &0.6&10 &\textbf{0.51}&11 &2.6\\
$\texttt{AUG3DC}$ & 8000/27543 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &0.3&2 &\emph{0.28}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &0.3&2 &\textbf{0.26}&1 &0.31\\
$\texttt{AUG3DCQP}$ & 8000/27543 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &0.3&2 &\emph{0.27}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &0.33&2 &\textbf{0.26}&1 &0.33\\
$\texttt{AUG3DQP}$ & 8000/27543 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &10 &0.74&10 &\emph{0.55}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &10 &0.64&10 &\textbf{0.5}&11 &2.6\\
$\texttt{CVXQP1}$ & 5000/10000 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &827 &7.8&805 &\textbf{3.8}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &827 &7.3&805 &\emph{3.8}&740 &51\\
$\texttt{CVXQP2}$ & 2500/10000 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &39596 &1.5e+02&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &47572 &1.8e+02&$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ \\
$\texttt{CVXQP3}$ & 7500/10000 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &169 &2.8&169 &\emph{1.4}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &169 &2.4&169 &\textbf{1.4}&118 &8.9\\
$\texttt{STCQP1}$ & 4095/8193 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &88 &\textbf{0.15}&88 &0.42&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &88 &\emph{0.18}&88 &0.36&75 &6.8e+02\\
$\texttt{STCQP2}$ & 4095/8193 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &142 &\textbf{0.25}&142 &0.8&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &144 &\emph{0.28}&144 &0.72&136 &4.8\\
$\texttt{DTOC1L}$ & 3996/5998 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &13 &\textbf{0.073}&13 &0.13&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &13 &\emph{0.075}&13 &0.14&16 &0.41\\
$\texttt{DTOC3}$ & 2998/4499 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &5 &\textbf{0.025}&5 &0.059&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &5 &\emph{0.03}&5 &0.033&4 &0.09\\
$\texttt{PORTSQP}$ & 1/100000 &2 &0.09&2 &0.064&2 &0.067&2 &\emph{0.062}&2 &\textbf{0.059}&2 &0.062&1 &0.42\\
$\texttt{HUES-MOD}$ & 2/5000 &1 &0.0028&1 &\textbf{0.0018}&1 &0.0027&1 &0.0026&1 &\emph{0.0018}&1 &0.0026&1 &0.024\\
$\texttt{HUESTIS}$ & 2/5000 &2 &0.0073&2 &\textbf{0.0042}&2 &0.011&2 &0.0061&2 &\emph{0.0047}&2 &0.0094&2 &0.072\\
$\texttt{A0ESDNDL}$ & 15002/45006 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &5 &69&5 &\emph{0.13}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &5 &71&5 &\textbf{0.12}&6 &1.8\\
$\texttt{A0ESINDL}$ & 15002/45006 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &5 &73&5 &\emph{0.12}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &5 &70&5 &\textbf{0.11}&6 &1.8\\
$\texttt{PORTSNQP}$ & 2/100000 &$\texttt{NC}^{\dagger}$ & $\texttt{NC}$ &2 &\textbf{0.092}&2 &0.11&14 &0.47&2 &\emph{0.095}&2 &0.1&2 &0.88\\
$\texttt{BLOWEYA}$ & 2002/4002 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &\textbf{0.011}&2 &0.031&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &\emph{0.015}&2 &0.021&2 &0.082\\
$\texttt{BLOWEYB}$ & 2002/4002 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &\textbf{0.015}&2 &0.019&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &\emph{0.016}&2 &0.019&2 &0.082\\
$\texttt{BLOWEYC}$ & 2002/4002 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &\textbf{0.015}&2 &0.017&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &\emph{0.015}&2 &0.021&2 &0.15\\
$\texttt{CONT5-QP}$ & 40200/40601 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &\emph{0.51}&2 &0.79&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &\textbf{0.49}&2 &0.8&2 &1.3\\
$\texttt{DTOC1L}$ & 3996/5998 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &5 &\textbf{0.03}&5 &0.09&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &5 &\emph{0.043}&5 &0.045&4 &0.12\\
$\texttt{FERRISDC}$ & 210/2200 &2 &0.084&2 &0.083&2 &0.077&2 &\emph{0.076}&2 &0.078&2 &0.079&0 &\textbf{0.021}\\
$\texttt{GOULDQP2}$ & 9999/19999 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &0.038&2 &\textbf{0.025}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &2 &0.038&2 &\emph{0.026}&2 &0.2\\
$\texttt{GOULDQP3}$ & 9999/19999 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &6 &0.076&6 &\emph{0.054}&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &6 &0.077&6 &\textbf{0.053}&7 &0.69\\
$\texttt{LINCONT}$ & 419/1257 &5 &0.058&5 &\emph{0.02}&5 &0.031&5 &0.05&5 &\textbf{0.019}&5 &0.03&5 &0.055\\
$\texttt{SOSQP2}$ & 2501/5000 &$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &3 &\textbf{0.017}&3 &0.04&$\texttt{N/A}^{*}$ & $\texttt{N/A}$ &3 &0.022&3 &\emph{0.019}&4 &0.11\\
\hline
\end{tabular}
\hss}
\end{table}
\clearpage
\section*{Acknowledgments}
We would like to acknowledge the valuable discussions initiated by Ariadna Cairo Baza and
spurred by the 9th ICIAM conference at the Universidad de Valencia.
\jbb{R. Marcia's research was partially supported by NSF Grant IIS 1741490.}
\jbrt{We thank two referees for
their extremely detailed and helpful comments.}
\bibliographystyle{siamplain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 23 |
relations {
HTMLElement: <: Element:
}
expressions {
succeed function(doc) :: HTMLDocument: -> String {
return doc.title;
};
succeed function() :: (-> Undefined) {
document.title = "this should work";
};
fail function() :: (-> Undefined) {
document.title = 234234;
};
succeed function() :: (-> Undefined) {
document.write("Calling a method succeeds.");
};
fail function() :: (-> Undefined) {
// This code may work in the browser. However, it is always safe to reject
// code.
var m = document.write;
m("Failure because this is wrong.");
};
succeed function(element) :: HTMLElement:
-> { left:: Double, top:: Double } {
var valueT = 0.0, valueL = 0.0;
while (element) {
valueT += element.offsetTop || 0;
valueL += element.offsetLeft || 0;
element = element.offsetParent;
}
return { left: valueL, top: valueT };
};
succeed function(element) :: HTMLElement:
-> { left:: Double, top:: Double } {
var valueT = 0.0, valueL = 0.0;
do {
valueT += element.offsetTop || 0;
valueL += element.offsetLeft || 0;
element = element.offsetParent;
} while (element);
return { left: valueL, top: valueT };
}
}
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Home \ Archives \ Youth gather in the name of sexual reproductive health rights
Youth gather in the name of sexual reproductive health rights
Alvine Kapitako SWAKOPMUND - Young people from the country's 14 regions convened at Swakopmund over the weekend to discuss sexual reproductive health rights. The National Youth Conference was organised by the Ministry of Education, Arts and Culture and the Ministry of Sport, Youth and National Service in collaboration with the United Nations Population Fund (UNFPA). The other stakeholders are the African Youth and Adolescents Network (AfriYAN), Star for Life, Namibia Planned Parenthood Association (NAPPA) and the German Agency for International Cooperation (GIZ). The conference was held under the theme 'Young people putting Sexual Reproductive Health Rights knowledge into practice'. The power of young people to exercise their rights and make correct choices will determine the health and well-being of present and future generations, said the mayor of Swakopmund, Pauline Nashilundo, at the opening of the two-day conference on Friday. Nashilundo said that young people in Namibia below the age of 35 make up 66 percent of the population while those younger than 15 years make up more than 39 percent of the population. The mayor noted that these young people daily face challenges such as unemployment, teenage pregnancy, dropping out of school, inequality and gender-based violence. There is a need to find innovative ways on addressing the challenges, said the mayor. "Adolescents are diverse and programmes addressing adolescents and young people need to take note of the cultural diversity, differences in age and education levels while responding to their healthcare needs." She added that a sustainable future depends on having a resilient population, which cannot be achieved without investments in the youth. "Young people deserve their fair share as a matter of equity but are also in a critical stage of their lifecycle that will determine their future and those of their families, communities and societies at large," said the mayor. The president of AfriYAN Eastern and Southern Africa, Lorence Kabasele, also addressed the young people, encouraging them that they have the potential to position matters affecting young people at the forefront of Africa's development agenda. This, she highlighted, can be done through meaningful advocacy based on strong and coordinated collective youth voices. "AfriYAN's mission is to innovatively package data as a tool to amplify intergenerational dialogue which influence strategic action to deliver upon the best sexual reproductive health rights outcome for young people," said Kabasele. This means that young people need to understand their challenges as far as sexual reproductive health rights is concerned. "Our common dream is to see the Namibian young people enjoying the most basic universal rights and we are convinced that we are called to work hard today to prepare a better generation for Africa tomorrow," remarked Kabasele. Young people who spoke to New Era at the end of the two-day conference said that they acquired knowledge that they will pass on to their peers using various means including social media, the mass media, as well as youth meetings in their respective regions. Jacob Hanghuwo, a 31-year-old, said he learned about his sexual behaviour, his sexual reproductive health rights and how to behave in a relationship, amongst others. "I learned how ignorant one can be regarding their health. From the group discussions regarding males' feelings we heard how men depend on their partners for their HIV results. I was also like that. The only time I used to find out about my HIV results was when I impregnated my partner but someone's result is not your own," said Hanghuwo. "It is useless to gain knowledge and not act upon it," said 15-year-old MaBianca Dax from the Omaheke Region.
Home \ Archives \ Youth gather in the name of sexual reproductive health rights - New Era Live
Is One Namibia, One Nation losing its meaning? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 108 |
Q: Fastest way to 'ignore' a row I am writing a PL/SQL function that processes table rows individually. I pass it a key. What is the fastest way to check whether or not that row has been processed, and if so ignore it? It may sound stupid but please assume that it always tries to process all the rows in the table (mainly because it does other things too).
One solution I had was to create a flag column on that table(fastest I can think of), another was to insert a record into another table and check if the row is not in that table (probably slower).
A: Assuming you need to be using a PL/SQL function, you should only pass into it the rowset that it needs to handle. That means using plain SQL to select the rows from the table you need and pass that to the function. In any case though, you should look very carefully at what you're doing whenever you end up having to use a cursor in a database environment, because that's not really what databases are optimized for.
| {
"redpajama_set_name": "RedPajamaStackExchange"
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import org.specs2.mutable._
import org.specs2.runner._
import org.junit.runner._
import play.api.test._
import play.api.test.Helpers._
@RunWith(classOf[JUnitRunner])
class IntegrationSpec extends Specification {
sequential
"Application" should {
"send 404 on a bad request" in new WithApplication {
route(app, FakeRequest(GET, "/boum")) must beSome.which (status(_) == NOT_FOUND)
}
"render the index page" in new WithApplication {
val home = route(app, FakeRequest(GET, "/")).get
status(home) must equalTo(OK)
contentType(home) must beSome.which(_ == "text/html")
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,935 |
{"url":"https:\/\/wiki.hattrick.org\/index.php?title=Help:Editing&diff=70031","text":"# Difference between revisions of \"Help:Editing\"\n\n## Wiki Editing\n\nWikis are easy to edit, giving the user a wide array of options without having to learn any programming or markup languages.\n\n## Editing Resources\n\n1. Use the Hattrick wiki sandbox to try out your editing skills.\n3. You can see more generic details in Help:Contents.\n4. Detailed help for Editors can be found on the Wikimedia site: Wikimedia for Editors.\n\n## Short examples\n\nThe best example is to click EDIT at the top of any page to see the source of the content to see how things are done.\n\nSome basic examples to get you going (click EDIT to see the source of how this is done):\n\n\u2022 To create a list (like this one) just start every line with an asterisk: *\n\u2022 To create a numbered list, start each line with a hash: #\n\u2022 Italic text is created by enclosing your text with two apostrophes: ''Italic text''.\n\u2022 Bold text is created by enclosing your text with three apostrophes: '''Bold text'''.\n\u2022 To create a link to something inside the Wiki, just enclose the page name in 2 sets of square brackets My Hattrick: [[My Hattrick]]\n\u2022 To create a link to something outside the Wiki, enclose the URL 1 set of square brackets: Hattrick\n\u2022 To write titles, sections... put 2 or more equals signs at the start and the end of the line: ==Short Examples==.\n\n## The wiki markup\n\nIn the left column of the table below, you can see what effects are possible. 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This header has the h4 font, but is NOT in the Table of Contents This effect is obtained by the following line of code. <h4> This header has the h4 font, but is NOT in the Table of Contents <\/h4> Variables (See also m:Help:Variable) Code Effect {{CURRENTMONTH}} 08 {{CURRENTMONTHNAME}} August {{CURRENTMONTHNAMEGEN}} August {{CURRENTDAY}} 25 {{CURRENTDAYNAME}} Sunday {{CURRENTYEAR}} 2019 {{CURRENTTIME}} 06:39 {{NUMBEROFARTICLES}} 19,446 {{PAGENAME}} Editing {{NAMESPACE}} Help {{localurl:pagename}} \/wiki\/Pagename {{localurl:Wikipedia:Sandbox|action=edit}} \/index.php?title=Wikipedia:Sandbox&action=edit {{SERVER}} https:\/\/wiki.hattrick.org {{ns:1}} Talk {{ns:2}} User {{ns:3}} User talk {{ns:4}} Hattrick {{ns:5}} Hattrick talk {{ns:6}} File {{ns:7}} File talk {{ns:8}} MediaWiki {{ns:9}} MediaWiki talk {{ns:10}} Template {{ns:11}} Template talk {{ns:12}} Help {{ns:13}} Help talk {{ns:14}} Category {{ns:15}} Category talk {{SITENAME}} Hattrick NUMBEROFARTICLES is the number of pages in the main namespace which contain a link and are not a redirect, i.e. number of articles, stubs containing a link, and disambiguation pages. 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<?xml version="1.0" encoding="UTF-8"?>
<!-- XBRL 2.1 Tests -->
<!-- Copyright 2003 XBRL International. All Rights Reserved. -->
<?xml-stylesheet type="text/xsl" href="../../testcase.xsl"?>
<testcase xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" name="Preferred Label" description="Section 5.2.4.2.1 The preferredLabel attribute (optional)" owner="mg@us.fujitsu.com" minimal="true" xsi:noNamespaceSchemaLocation="../lib/test.xsd" outpath="out">
<variation id="V-1" name="pref01">
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<result expected="valid"/>
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<linkbase readMeFirst="false">prefLabel-single-language-invalid-presentation.xml</linkbase>
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<result expected="invalid"/>
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<variation id="V-3" name="pref03">
<description>The presentationArc's 'from' concept has label resource with a role value which is equal to the value of the preferredLabel attribute. But the presentationArc's 'to' concept does not have label resource with a role value which is equal to the value of the preferredLabel attribute. </description>
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<result expected="invalid"/>
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<data>
<xsd readMeFirst="true">prefLabel-incomplete-multi-languageset.xsd</xsd>
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<linkbase readMeFirst="false">prefLabel-incomplete-multi-languageset-presentation.xml</linkbase>
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<data>
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| {
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A couple weeks ago I was asked to emcee and speak at an event called Practical Federalism. The event's significance was so important it featured several presidential candidates and garnered national press attention.
In preparing for the day and what I would say, one phrase kept swirling around in my mind: Do your job. In the middle of this vote harvesting season, with talk of dysfunction, partisan bickering and ever increasing dissatisfaction with government at all levels, the words of Patriots coach Bill Belichick provided clarity as to why "We the People" have found ourselves in this state of disappointment. Much like a failing business or dysfunctional family, government stumbles when people are not focused on the true intent of their jobs.
Too many public servants have lost sight of their responsibilities, do not truly understand what those responsibilities are or don't care. It is time for us as a country to recommit ourselves to doing our job and holding people at all levels of public service accountable.
The simplicity of "Do your job" is not lost on most responsible people who possess a healthy dose of commonsense. Many of us have seen success and failure. Those who value life's lessons develop a keen sense of what works and what doesn't. It is not necessary to fully embrace the Pavlovian approach to learned behavior, but simply learning fire burns and "biting off more than you can chew" are good lessons more politicians and public servants should embrace. Drunk with power is a perfect analogy to describe many of our elected servants and their bureaucratic allies. Our professional political class lacks self-control, stumbles about, interjects themselves into situations where they don't belong or don't fully understand, and the more they consume, the fewer brain cells they seem to possess.
Most of us are introduced to the basic concepts of governing by being part of a family. If our family is functional, and generally speaking most are, we learn about economics, security/defense, interpersonal relationships, education and the benefits of a strong social commitment by simply watching our parents and other families. A family is a government unto itself, responsible for the same things, although in a microcosm, that every government should be responsible for.
Our parents understood economics is no more complicated than making sure the priority is making enough money to feed, clothe and provide shelter for the family first. Once you took care of the basic needs, you tried to save for a "rainy day," and if the fruits of your labor permitted, you helped those in need. You avoided spending beyond your needs because you knew that would place your family in jeopardy, and that charity above luxury was your duty. You ensured your home was safe, that doors and windows could be secured, and you had emergency equipment such as smoke detectors and fire extinguishers, and you had an escape plan in the case of an emergency.
You watched for any strange activity on your street, and when your neighbors were away you watched their home. You respected the boundaries of your property and expected the same from your neighbors, but if there was a problem you chatted with them. This was easy as you had a rapport with them, watching their kids as they played in the street or lending them tools or a cup of sugar. You honored your neighbor's privacy, but understood some activities should not be tolerated, and to ignore them would eventually affect the harmony of the neighborhood.
Your family understood education began at home, and so your parents made sure you could read and write, and that you understood the basic concepts of time, how to tie your shoes and be self-reliant. As you went to school, you built upon that foundation, yet your family was still involved in your education helping you with homework, participating in school activities and seeking additional help for matters beyond your family's means or skills. Your parents understood they were not raising kids, but adults, future citizens.
I know some will dismiss my "Leave it to Beaver" analogy, asking what about this or that. Some will miss the comparison between how a family functions and how a government should function. The point goes back to "Do your job" and applies from the federal government down to city and town governments and families. We have a tiered approach to governing for a reason, and the biggest reason for the dysfunction, partisan bickering and ever increasing dissatisfaction with government is because we have surrendered too much power to the federal government. Our government lacks the simplicity of bringing solutions closer to the issues.
This entry was posted in History, National News, News, Uncategorized by jeffchidester. Bookmark the permalink. | {
"redpajama_set_name": "RedPajamaC4"
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Huge Earnings Miss Sends Helix Energy Solutions Group Inc's Stock Down 17%
By Matthew DiLalloFool.com
What: Helix Energy Solutions Group Inc reported much weaker than expected second-quarter results after the market closed yesterday. That poor showing fueled a deep sell-off on Tuesday, with the stock down more than 17% as of this writing. That said, volatility is the norm when Helix reports quarterly results, as the company's stock price made double-digit moves the last two times it reported.
So what: For the quarter, Helix reported revenue of $166 million, which was not only down 45.7% year over year, but missed analysts estimates by $27.7 million. The company didn't only miss big on revenue -- it was way off from analysts' earnings estimates, turning in a loss of $0.03 per share, when analysts were expecting a profit of $0.15 per share. Investors don't like big misses, and are showing their displeasure by selling off the stock.
Helix blamed the weak oil market for its poor showing. The company noted that its well intervention business was negatively affected by a longer-than-planned Q4000 regulatory dry-dock, while it also experienced customer delays on the H534. In addition, the company's sales, general, and administrative expenses spiked to 10% of revenue, which was up from just 6.7% last quarter. This was partially due to an extra $2.5 million in charges associated with what now looks to be an uncollectable receivable. Meanwhile, interest expenses nearly doubled as the company took delivery of the Q5000 using proceeds from a term loan to fund the final payment to the shipyard. Suffice it to say that what could go wrong did go wrong.
Now what: Helix Energy Solutions endured a rough quarter, as two of its key vessels were out of work longer than expected. To make matters worse, the company also had higher expenses -- the downturn in the energy market is impacting its customers, leading to uncertainty regarding collection of a big receivable.
That said, the company is working to reduce its costs and has substantial liquidity, which puts it in a position to weather the downturn better than many of its competitors.
The article Huge Earnings Miss Sends Helix Energy Solutions Group Inc's Stock Down 17% originally appeared on Fool.com.
Matt DiLallo has no position in any stocks mentioned. The Motley Fool has no position in any of the stocks mentioned. Try any of our Foolish newsletter services free for 30 days. We Fools may not all hold the same opinions, but we all believe that considering a diverse range of insights makes us better investors. The Motley Fool has a disclosure policy. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,760 |
Q: How to do this loop in java I have n number of objects which have a property called status.
So I have to check this property on all the objects and come up with a single value for all of the objects.
Let me explain.
The status property can have values like this (Strings) - "1","2","3","4","5","7","completed","not completed",......(any string other than 1..7)
*
*I will go through all the objects and if the status property has the value of 1-7 for any of the objects, the higher one will be selected then that value will be selected.
example
obj1.status="1", obj2.status="6". Value Selected="6"
obj1.status="4", obj2.status="2". Value Selected="4"
obj1.status="7", obj2.status="6". Value Selected="7"
*
*If none of the objects have status property as 1-7 , then value selected will be "XYZ"
example
obj1.status="completed", obj2.status="abc". Value Selected="XYZ"
obj1.status="abc", obj2.status="def". Value Selected="XYZ"
obj1.status="jkl", obj2.status="mno". Value Selected="XYZ"
If atleast one of the objects has status property as 1-7 and all the rest have other strings then number value will be selected.
obj1.status="abc", obj2.status="1". Value Selected="1"
obj1.status="abc", obj2.status="5",obj3.status="4",. Value
Selected="5"
A: It will be something like this:
String regex = "[1-7]";
int max = 0;
int number;
for (Object obj : objectList){
if(obj.status().matches(regex)){
number = Integer.valueOf(obj.status());
if(number > max){
max=number;
}
}
}
if(max==0){
System.out.println("XYZ");}
else{
System.out.println(String.valueOf(max));
}
A: Maybe something like that:
Optional<YourObject> optional = test.stream().
filter(object -> (object.status.matches("[1-7]"))).sorted((o1, o2) -> Integer.valueOf(o2.status).compareTo(Integer.valueOf(o1.status))).findFirst();
String value = optional.isPresent() ? optional.get().status : "XYZ";
A: Try this
String newStatus = list.stream()
.map(Obj::getStatus)
.map(x -> {
try {
return Integer.parseInt(x);
} catch (NumberFormatException ignore) {
return null;
}
})
.filter(Objects::nonNull)
.max(Comparator.naturalOrder())
.map(x -> Integer.toString(x))
.orElse("XYZ");
list.forEach(x -> x.setStatus(newStatus));
Obj is type of your objects.
A: That's a classic reduce algorithm. (No casts and regular expressions nessecary.)
BinaryOperator<String> reducer = (v1, v2) -> {
if (v1.length() == 1 && v1.compareTo("1") >= 0 && v1.compareTo("7") <= 0) {
// v1 is a number - return v2 if higher number, else v1
return v2.length() == 1 && v2.compareTo("1") >= 0 && v2.compareTo("7") <= 0 && v1.compareTo(v2) < 0 ? v2 : v1;
} else {
// v1 is not a number - return v2 if number, else "XYZ"
return v2.length() == 1 && v2.compareTo("1") >= 0 && v2.compareTo("7") <= 0 ? v2 : "XYZ";
}
};
Test:
Stream<String> of1 = Stream.of("3", "1", "2");
Stream<String> of2 = Stream.of("1", "incomplete", "2");
Stream<String> of3 = Stream.of("incomplete", "status");
System.out.println(of1.reduce(reducer).get()); // 3
System.out.println(of2.reduce(reducer).get()); // 2
System.out.println(of3.reduce(reducer).get()); // XYZ
A: import java.util.Arrays;
import java.util.Comparator;
import java.util.List;
import java.util.Optional;
import java.util.regex.Pattern;
import org.junit.Assert;
import org.junit.Test;
public class LoopTest {
private class Item {
private String status = "";
public Item(String status) {
if(status != null)
this.status = status;
}
public String getStatus() {
return status;
}
}
private static final Pattern MY_REXP = Pattern.compile("[1-7]");
private String getValue(List<Item> listValues) {
if(listValues == null)
return "XYZ";
Optional<Item> opItem = listValues.stream().filter(item -> MY_REXP.matcher(item.getStatus()).matches()).max(new Comparator<Item>() {
@Override
public int compare(Item o1, Item o2) {
if(o1 == null || o2 == null)
return -1;
return o1.getStatus().compareTo(o2.getStatus());
}
});
return opItem.isPresent() ? opItem.get().getStatus() : "XYZ";
}
@Test
public void test1() {
try {
List<Item> listValues = Arrays.asList(new Item("1"), new Item("6"));
String result = getValue(listValues);
Assert.assertTrue("6".equals(result));
} catch (Exception ex) {
Assert.fail("Exception " + ex);
}
}
@Test
public void test2() {
try {
List<Item> listValues = Arrays.asList(new Item("completed"), new Item("8"));
String result = getValue(listValues);
Assert.assertTrue("XYZ".equals(result));
} catch (Exception ex) {
Assert.fail("Exception " + ex);
}
}
@Test
public void test3() {
try {
List<Item> listValues = Arrays.asList(new Item("abc"), new Item("5"), new Item("4"));
String result = getValue(listValues);
Assert.assertTrue("5".equals(result));
} catch (Exception ex) {
Assert.fail("Exception " + ex);
}
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 29 |
{"url":"https:\/\/git.rockbox.org\/cgit\/rockbox.git\/tree\/manual\/configure_rockbox\/display_options.tex?id=17e03e75a43ff75b2cc20ba794b0f4d3131f4a36","text":"summaryrefslogtreecommitdiffstats log msg author committer range\n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 % $Id$ % \\section{\\label{ref:Displayoptions}Display} \\begin{description} \\nopt{player}{ \\item[Browse fonts:] Browse the fonts that reside in your \\fname{\/.rockbox\/fonts} directory. Selecting one will activate it. See \\reference{ref:Loadingfonts} for further details about fonts. } % \\nopt{player} \\item[Browse WPS files:] Opens the \\setting{File Browser} in the \\fname{\/.rockbox\/wps} directory and displays all \\fname{.wps} files. Selecting one will activate it, stop will exit back to the menu. For further information about the WPS see \\reference{ref:WPS}. For information about editing a .wps file see \\reference{ref:ConfiguringtheWPS}. \\opt{HAVE_REMOTE_LCD}{ \\item[Browse RWPS files:] Opens the \\setting{File Browser} in the \\fname{\/.rockbox\/wps} directory and displays all \\fname{.rwps} files. Selecting one will activate it, stop will exit back to the menu. \\note{ A \\fname{.rwps} file is a special \\fname{.wps} file for the remote display. } } \\item[LCD Settings:] This sub menu contains settings that relate to the display of the \\dap. \\begin{description} \\nopt{ondiofm,ondiosp}{ \\item[Backlight:] The amount of time the backlight shines after a key press. If set to \\setting{Off}, the backlight will not light when a button is pressed. If set to \\setting{On}, the backlight will never shut off. If set to a time (1 to 90 seconds), the backlight will stay lit for that amount of time after a button press. \\item[Backlight on When Plugged:] This setting is equivalent to the Backlight setting except it applies when the \\dap\\ is plugged into the charger. \\item[Caption Backlight:] This option turns on the backlight a number of seconds before the start of a new track, and keeps it on for the same number of seconds after the beginning so that the display can be read to see song information. The amount of time is determined by the value of the backlight timeout setting, but is no less than 5 seconds. \\opt{h1xx,ipodmini,ipodnano,ipodvideo}{ \\item[Backlight fade in:] The amount of time that the backlight will take to fade from off to on after a button is pressed. If set to \\setting{Off} the backlight will turn on immediately, with no fade in. Can also be set to \\setting{500ms}, \\setting{1s} or \\setting{2s}. \\item[Backlight fade out:] Like Backlight fade in, this controls the amount of time that the backlight will take to fade from on to off after a button is pressed. If set to \\setting{Off} the backlight will turn off immediately, with no fade out. Other valid values: \\setting{500ms}, \\setting{1s}, \\setting{2s}, \\setting{3s}, \\setting{4s}, \\setting{5s} or \\setting{10s}. } \\item[First Keypress Enables Backlight Only:] With this option enabled the first keypress while the backlight is turned off will only turn the backlight on without having any other effect. When disabled the first keypress will \\emph{also} perform its appropriate action. \\opt{h300,x5}{ \\item[Brightness:] Changes the brightness of your LCD display. } } % \\nopt{ondiofm,ondiosp} \\opt{archos,h1xx,ipodmini,ipod3g,ipod4g,x5}{ \\item[Contrast:] Changes the contrast of your LCD display. \\warn{Setting the contrast too dark or too light can make it hard to find this menu option again!} \\nopt{HAVE_LCD_COLOR,player}{ \\item[LCD Mode:] This setting lets you invert the whole screen, so now you get a black background and light text and graphics. } % \\opt{HAVE_LCD_BITMAP} } % \\opt{archos,h1xx,ipodmini,ipod4g,x5} \\opt{HAVE_LCD_BITMAP}{ \\nopt{ipodcolor,ipodnano,ipodvideo}{ \\item[Upside Down:] Displays the screen so that the top of the display is nearest the buttons. This is sometimes useful when carrying the \\dap\\ in a pocket for easy access to the headphone socket. } % \\nopt{ipodcolor,ipodnano.ipodvideo} \\item[Line Selector:] This option allows you to select whether the line selector is a bar of inverted text (\\setting{Bar (inverse)} option) or a small arrow to the left of the menu text (\\setting{Pointer} option). \\nopt{archos}{% \\item[Clear Backdrop:] Rockbox allows you to select bitmap pictures to use as backdrops, see \\reference{ref:LoadingBackdrops} for further information. This option allows you to clear the backdrops that you set. }% \\opt{HAVE_LCD_COLOR}{ \\item[Background Colour:] Sets the background colour for the LCD display. \\item[Foreground Colour:] Sets the colour used for text and icons. \\item[Reset Colours:] Resets the LCD display to Rockbox's default colours. } } % \\opt{HAVE_LCD_BITMAP} \\end{description} % \\opt{HAVE_REMOTE_LCD}{ \\item[Remote-LCD Settings:] This sub menu contains settings that relate to the display of the remote. \\begin{description} \\item[Backlight:] Similar to the main unit backlight this option controls the backlight timeout for the remote control. The remote backlight is independent from the main unit backlight. \\item[Backlight on When Plugged:] This controls the backlight when the \\dap\\ is plugged into the charger. \\item[Caption Backlight:] This option turns on the backlight a number of seconds before the start of a new track, and keeps it on for the same number of seconds after the beginning so that the display can be read to see song information. The amount of time is determined by the value of the backlight timeout setting, but is no less than 5 seconds. \\item[First Keypress Enables Backlight Only:] This controls what happens when you press a button on your remote while the backlight is turned off. Like for the main unit, if this setting is set to \\setting{Yes}, the first keypress will light up the remote backlight, but have no other effect. If set to \\setting{No}, the first keypress will light up the remote backlight \\emph{and} engage the function of the key that is pressed. \\item[Contrast:] Changes the contrast of your remote's LCD display. \\warn{Setting the contrast too dark or too light can make it hard to find this menu option again!} \\item[LCD Mode:] This setting lets you invert the whole screen, so now you get a black background and light text and graphics. \\item[Upside Down:] Displays the screen so that the top of the display is nearest the buttons. This is sometimes useful when carrying the \\dap\\ in a pocket for easy access to the headphone socket. \\opt{h1xx,h300}{ \\item[Reduce Ticking:] Enable this option if you can hear a ticking sound in your headphones when using your remote. } \\end{description} } % \\item[Scrolling] This feature controls how text will scroll in Rockbox. You can configure the following parameters: \\begin{description} \\item[Scroll Speed:] Controls how many times per second the scrolling text moves a step. \\item[Scroll Start Delay:] Controls how many milliseconds Rockbox should wait before a new text begins scrolling. \\opt{HAVE_LCD_BITMAP}{ \\item[Scroll Step Size:] Controls how many pixels the text scroll should move for each step. } \\opt{HAVE_REMOTE_LCD}{ \\item[Remote Scrolling Options:] The options here have the same effect on the remote LCD as the options mentioned above have on the main LCD. } \\item[Bidirectional Scroll Limit:] Rockbox has two different scroll methods: always scrolling the text to the left and when the line has ended beginning again at the start, or moving to the left until you can read the end of the line and scroll right until you see the beginning again. Rockbox chooses which method it should use depending of how much it has to scroll left. This setting lets you tell Rockbox where that limit is, expressed in percentage of line length. \\opt{HAVE_LCD_BITMAP}{ \\item[Screen Scrolls Out of View:] On lists with long entries that don't fit on the screen using \\opt{recorder,recorderv2fm,h1xx,h300}{\\ButtonOn+\\ButtonRight\/ \\ButtonLeft}\\opt{ondio}{\\ButtonMenu+\\ButtonRight\/\\ButtonLeft} the complete content will be scrolled right\/left. With this option set to \\setting{Yes} the lines can scroll out of view. Otherwise the entries will only scroll as far as they align to the margins. \\item[Screen Scroll Step Size:] Determines how many pixels the text should advance in every click when scrolling the screen. } \\opt{player}{ \\item[Jump Scroll:] This setting makes text scroll a page at a time instead of a character at a time. If set to \\setting{One time}, \\setting{2}, \\setting{3} or \\setting{4} it will scroll a line in paged mode that many times and then scroll it a character at a time. If set to \\setting{Always} lines will always scroll in paged mode. \\item[Jump Scroll Delay:] Controls how long the delay is before a page is scrolled. } \\item[Paged Scrolling:] When enabled scrolling will page up\/down instead of changing lines. This can be useful on slow displays. \\end{description} % \\opt{HAVE_LCD_BITMAP}{ \\item[Status\/Scrollbar:] Settings related to on screen status display and the scrollbar. \\begin{description} \\item[Scroll Bar:] Enables or disables the scroll bar at the left. \\item[Status Bar:] Enables or disables the status bar at the upper side. \\opt{RECORDER_PAD}{ \\item[Button Bar:] Enables or disables the button bar prompts for the F''-keys at the bottom of the screen. } \\item[Volume Display:] Controls whether the volume is displayed as a graphic or a numeric value on the Status Bar. If you select a numeric display, volume is displayed in decibels. \\fixme{cross-reference to volume setting.} \\item[Battery Display:] Controls whether the battery charge status is displayed as a graphic or numerical percentage value on the Status Bar. \\end{description} } % \\opt{HAVE_LCD_BITMAP}{ \\item[Peak Meter:] The peak meter can be configured with a number of parameters. \\begin{description} \\item[Peak Release:] This determines how fast the bar shrinks when the music becomes softer. Lower values make the peak meter look smoother. Expressed in scale units per 10ms. \\item[Peak Hold Time:] Specifies the time after which the peak indicator will reset. For example, if you set this value to 5s, the peak indicator displays the loudest volume value that occurred within the last 5 seconds. Larger values are useful if you want to find the peak level of a song, which might be of interest when copying music from the \\dap\\ via the analogue output to some other recording device. \\item[Clip Hold Time:] The number of seconds that the clipping indicator will be visible after clipping is detected. \\item[Scale:] Select whether the peak meter displays linear or logarithmic values. The human ear perceives loudness on a logarithmic scale. If the Scale setting is set to \\setting{Logarithmic} (dB) scale, the volume values are scaled logarithmically. The volume meters of digital audio devices usually are scaled this way. On the other hand, if you are interested in the power level that is applied to your headphones you should choose \\setting{Linear} display. This setting cannot be displayed in units like volts or watts because such units depend on your headphones. \\item[Minimum and maximum range:] These two options define the full value range that the peak meter displays. Recommended values for the \\setting{Logarithmic} (dB) setting are {}-40 dB for minimum and 0 dB for maximum. Recommended values for \\setting{Linear} display are 0 and 100\\%. Note that {}-40 dB is approximately 1\\% in linear value, but if you change the minimum setting in linear mode slightly and then change to the dB scale, there will be a large change. You can use these values for zooming' into the peak meter. \\end{description} } \\item[Default Codepage:] A codepage describes the way extended characters that aren't available within the ASCII character set are encoded. ID3v1 tags don't have a codepage encoding contained so Rockbox needs to know what encoding has been used when generating these tags. This should be ISO-8859-1'' but to support languages outside Western Europe most applications use the setting of your operating system instead. If your operating system uses a different codepage and you're getting garbled extended characters you should adjust this settings. 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\section{Introduction}
Recent developments in trapping techniques for cold atomic systems \cite{blo08}, along with the possibility of visualizing individual atoms using quantum Fermi microscopes \cite{che15,hal15,par15} have lead to a resurgence of theoretical interest in the equilibrium and dynamical properties of confined systems of fermions.
In particular the rather idealised case of spinless noninteracting fermions is experimentally relevant as magnetic traps are based on polarising the spin degrees of freedom of cold atoms and furthermore in the spin polarised state interactions are weak due to the suppression of s-wave scattering. The interactions can further be suppressed experimentally by suitably tuning Feshbach resonances.
This noninteracting case, at zero temperature, still presents interesting theoretical challenges since nontrivial statistical behavior arises because of
{the Pauli} exclusion principle.
The statistics of spinless non interacting fermionic systems can be encoded in a two point kernel from which the local density, as well as all the higher order correlation functions, can be expressed, in essence via Wick's theorem for fermions. For bulk systems, where a large number of fermions are confined by an external potential, the regions where the local fermionic density is large can be studied using the local density approximation or LDA \cite{gio08,cas06} which is based on the approximation that the trapping potential can be treated locally as constant in space. The LDA allows the calculation of the bulk density.
Together with more controlled approaches, the LDA also predicts
that, in the bulk and at scales of the order of the typical inter-particle distance, the kernel takes a universal form, independent of the details of the potential, given by the sine-kernel \cite{gio08,cas06,eis13,dea15,dea15b,dea16}. The LDA can be used to predict its own downfall in regions where the density of fermions becomes small. This occurs, by definition, at the edge of the trapped atomic cloud. Here, the form of the density and kernel is modified and one finds that the {\em edge} physics is described by fermions in a linear potential, as the trapping potential can in general be expanded as a locally linear potential near the edge \cite{koh98,eis13,dea15,dea15b,dea16}. The associated kernel near a locally linear edge is called the Airy kernel and the fermions in this region referred to as the {\em Airy gas} \cite{koh98,eis13,dea15,dea15b,dea16}. It can be shown to be universal for a broad class of smooth potentials. However, other edge regimes and edge universality classes exist, notably when the trap has an infinite hard wall or a continuous but divergent wall potential \cite{cal11,lac17}, but also when the Fermi energy coincides with the maximum of a double well potential \cite{smi20}. Interestingly, while the kernels appearing in the aforementioned problems arise from quantum mechanical problems, most of them also arise in the context of random matrix theory, where they describe the eigenvalue statistics of certain random matrix ensembles \cite{eis13,dea15,dea15b,dea16,dea19}.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.8 \linewidth]{new_schema.pdf}
\caption{Square well potential of height $V_0$ within an overall confining potential $V(x)$. The critical regime where the Fermi energy $\mu$, the height of blue region, coincides with $V_0$ is shown.
The physics near the edge can be described by zooming at the edge, as shown on the right side of the figure.
Two examples of barrier potentials are shown: {\bf a)} The square step barrier studied in Section \ref{sec:step}, {\bf b)} a smooth barrier, varying over a scale $\lambda$, such as studied in Section \ref{smooth}.}\label{schema}
\end{center}
\end{figure}
The LDA is based on the assumption that the trapping potential is locally constant, i.e. that it varies very slowly on the scale of the typical inter-particle distance. If the potential has fast variations on this scale, e.g. if it is discontinuous, the LDA will fail, even if the fermion density is
large. Similarly, we can expect the edge universality classes to be modified. The main goal of this paper,
and of a companion paper \cite{inprep}, is to analyse the statistical properties of non interacting fermions when the trapping potential exhibits a local singularity on top of an overall smooth confining shape. An important question is what replaces the sine-kernel in the bulk near the singularity, and how far the effect of this singularity can be felt. A related question is what effect does it have on the counting statistics, such as the fluctuations of the number of fermions in a given region.
The main goal of this paper is to address these questions. For this, we consider $N$ noninteracting fermions in a globally confining
trapping potential and focus on the large $N$ limit. On top of this, we assume that the potential has a singularity at some point in space (see Fig. \ref{schema}). We zoom in near the singularity and study a class of fast-varying potentials (see Fig. \ref{schema}), which include the square step barrier as well as other continuous barrier potentials, such as the Woods-Saxon potential \cite{woo54}, well known in nuclear physics. Our goal is to describe how the quantum correlations in the ground state are modified in the vicinity of the singularity -- as opposed to the smooth potential case. In order to carry out the analysis of such fast-varying potentials we develop a method based on the single particle Green's function associated to the Schr\"odinger equation in the presence of a general trapping potential. As a preliminary benchmark, we first show how this method can be used to derive well known properties of smooth potentials such as the LDA and the Airy gas physics. This already allows us to discuss the limitation of the LDA which, as we find, fails when the potential varies too fast in space.
We then apply this Green's function method to obtain the exact form of the kernel, and the statistics of fermions, near a square step barrier of height $V_0$, as shown in Fig. \ref{schema}. For Fermi energies $\mu>V_0$ it describes the interpolation between two regions of different densities in a Fermi gas, each described by a differently-scaled sine kernel. We examine in particular the case where the Fermi energy coincides with the top of the step potential, $\mu=V_0$. This mimics a macroscopic system of fermions confined in a finite square well potential within an overall trapping potential - an everyday analogy being that of a swimming pool of fermions which is full to the edge - see Fig. \ref{schema}. Said otherwise, the swimming pool is on the point of overflowing a bit like in an {\em infinity pool}. The statistics of the number of particles in the region outside the square well potential (on the pool edges) have rather interesting properties when $V_0=\mu$. In particular, we analyse the mean, variance and third cumulant of the number of particles $N_{\rm out}$ to the right outside the pool is independent of $\mu$ and can be computed exactly.
In fact, we find that a critical behavior emerges as a function of the dimensionless control parameter $r=V_0/\mu > 0$ which is summarised in Fig. \ref{ph_diag}. We find a sub-critical behavior for $r>1$ where the local density profile decays exponentially $\rho(x)\sim e^{-x/\xi_r}$ for $x \to +\infty$. The decay length diverges as $\xi_r \sim 1/\sqrt{r-1} $ as $r$ approaches the critical point $r=1$ from above. Exactly at the critical point we find an algebraic decay $\rho(x) \sim 1/x^2$ for $x \to +\infty$. On the super-critical side $r < 1$ the density approaches a non-zero constant $\sqrt{2\mu(1-r)}$ for $x \to + \infty$.
\begin{figure}[t]
\includegraphics[width = 0.9\linewidth]{Fig_Ph_Diag.pdf}
\caption{Density profile $\rho(x)$ vs $x$ for different values of the dimensionless parameter $r = \frac{V_0}{\mu}$. For $r>1$, in the sub-critical phase, $\rho(x) \sim e^{-x/\xi_r}$ vanishes exponentially as $x \to + \infty$ on a scale $\xi_r \sim 1/\sqrt{r-1}$. For $r < 1$, the super-critical phase, the density approaches a constant $\rho(x) \sim \sqrt{2\mu(1-r)}$ as $x \to + \infty$. Exactly at the critical point $r=1$, $\rho(x)$ vanishes algebraically as $1/x^2$, as $x \to +\infty$.}\label{ph_diag}
\end{figure}
To explore the universality of this critical behaviour with respect to the shape of the barrier, we explore more general smooth barriers of the type $V(x) = V_0\,v(x/\lambda)$ where $v(z)$ smoothly interpolates between $0$ and $1$ as $z$ increases from $-\infty$ to $+\infty$. Here $\lambda$ is the characteristic length scale describing how fast the barrier varies in space (see Fig. \ref{schema} b)). We find that indeed this critical behavior near $r=1$ is universal and sets in whenever $\xi_r \gg \lambda$. For example, the density profile in the vicinity of the transition is described by a universal scaling function, up to a non-universal amplitude that depends on $\lambda$. This result is obtained by an exact solution in the case of the Woods-Saxon potential as well as through an analysis valid for
more general barrier potentials. This analysis unveils a general relation between the scattering amplitudes of the single particle wave functions and the asymptotic behaviour of the kernel at large distances {from the barrier.}
The rest of the paper is organised as follows. In Section \ref{sec:green} we introduce the Green's function method to compute the kernel for arbitrary potentials. In Section \ref{sec:smooth_pot} we study the case of a smooth trapping potential (without singularity) and show how to recover the standard scaling forms for the kernel both in the bulk (LDA and sine-kernel) and {at} {near} the edge of the Fermi gas (Airy-kernel). In Section \ref{sec:step}, we obtain the exact expression of the kernel for the square step barrier and discuss the critical behaviour. We also obtain the first three cumulants of the total number of particles to the right of the barrier. In Section \ref{smooth} A., we obtain asymptotic formula for the kernel for {a} general barrier potential in terms of the corresponding scattering amplitudes. In Section \ref{smooth} B., we study the Woods-Saxon potential, for which exact formula can be derived, before we conclude in Section \ref{sec:conclusion}. Some technical aspects are presented in Appendices.
\section{Green's function formalism to compute the kernel}
\label{sec:green}
\subsection{The kernel - basic definitions}
We consider $N$ non interacting spinless fermions confined by a potential $V(x)$.
The single particle Hamiltonian is $H = \frac{p^2}{2 m} + V(x)$, where $m$ is the particle mass. We denote by $\psi_k(x)$ the eigenstates of $H$ and $\epsilon_k$ the associated energies.
The zero temperature kernel can be written in terms of the Fermi energy $\mu$ as
\begin{equation}
K_\mu(x,y) = \sum_k\theta(\mu - \epsilon_k) \psi_k^*(x)\psi_k(y).\label{k1}
\end{equation}
Here $\theta(x)$ denotes the Heaviside function, and $\mu$ is considered as a continuous parameter, the total number of fermions being related to $\mu$ as $N=\sum_k \theta(\mu-\epsilon_k)$ \cite{com1}.
By construction, the kernel \eqref{k1} is real and symmetric.
, {\em i.e.} $K_\mu(x,y) = K_\mu(y,x)$ \cite{comreal}.
We also consider below cases of non confining potentials $V(x)$, i.e. fermions on the whole line with a continuous spectrum for $H$, as limiting cases of
\eqref{k1} for large system sizes. In this case $N$ is infinite and $\mu$ is the control parameter.
The kernel encodes all of the statistical properties of an $N$ body system as all $n$-point correlation functions can be constructed from it \cite{dea16,dea19}. These correlations can be computed using Wick's theorem for fermionic fields or equivalently by noting that the particle positions are described by a determinantal point process \cite{bor11}. In particular, for the purposes of the current work, we note that the number density of fermions is given by
\begin{equation}
\rho(x)=K_{\mu}(x,x),
\end{equation}
which means that the number of fermions in a region ${\cal I}$, denoted by $N_{\cal I}$, has the average value
\begin{equation}
\langle N_{\cal I}\rangle =\int_{\cal I} dx\ K_\mu(x,x),\label{ns}
\end{equation}
while the variance of $N_{\cal I}$ is given by \cite{dea16}
\begin{equation}
{\rm{Var}}( N_{\cal I}) = \langle N_{\cal I}\rangle-\int_{\cal I}\int_{\cal I} dxdy\ [K_\mu(x,y)]^2.\label{vs}
\end{equation}
In what follows we explain, formally, how the single particle Green's function can be used
to compute the kernel. Some of these results were previously derived by other methods (such as by direct evaluation of the kernel by computing and summing the wave functions or by using the
Euclidean propagator associated with $H$ \cite{dea16}), but here we use a new Green's function method that turns out to be technically advantageous compared to other methods, in particular in computing the limiting kernels for discontinuous potentials, as demonstrated later in the paper.
\subsection{Kernels via Green's function}
Differentiating Eq. ({\ref{k1}) with respect to $\mu$ gives
\begin{equation}
{\frac{\partial}{\partial \mu}}K_\mu(x,y) = \sum_k\delta(\mu - \epsilon_k) \psi_k^*(x)\psi_k(y),
\end{equation}
whose diagonal part is the local density of states of $H$ at energy $\mu$.
Now we use the well known formula
\begin{equation}
{ \frac{1}{z-i 0^+} := \lim_{\varepsilon\to 0^+} } \frac{1}{z-i\varepsilon} = \pi i \delta(z) + P \frac{1}{z},
\end{equation}
interpreted in terms of distributions, where $P$ indicates that one should use
the Cauchy principle part in any integrals. { We will use everywhere below the notation $z - i 0^+$ to denote
the limit $\varepsilon\to 0^+$ at the end of the calculation.} We thus find
{
\begin{equation}
\delta(\mu-\epsilon) =\frac{1}{\pi} \rm {Im} \frac{1}{\mu-i 0^+ -\epsilon} .
\end{equation}
}
Now, as the terms involving the wave function are real, we can write
\begin{equation}
\frac{\partial}{\partial \mu} K_\mu(x,y) =\frac{1}{\pi}
{\rm Im}
\sum_k \frac{1}{\mu -i 0^+ -\epsilon_k} \psi_k^*(x)\psi_k(y).
\end{equation}
This gives
\begin{equation}
\frac{\partial}{\partial \mu} K_\mu(x,y) = \frac{1}{\pi} { \rm Im} \, G_\mu(x,y)\label{dmu}
\end{equation}
where $G_\mu$ is the resolvent of the operator $H$, evaluated at $\mu - i 0^+$ just below the real axis,
in operator notation
\begin{equation}
G_\mu = (\mu -i 0^+ -H)^{-1} .
\end{equation}
Hence it is in general a complex quantity. The imaginary part
of its diagonal component gives the local density of states of $H$ at energy $\mu$.
Equivalently, $G_\mu(x,y)$ is the solution of
\begin{equation}
(\mu -i 0^+ -H)G_\mu (x,y) = \delta(x-y),
\end{equation}
with proper decay at infinity. Hence $G_\mu$ is the Green's function corresponding to the single particle Schr\"odinger equation of Hamiltonian
\begin{equation}
H= -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} +V(x),
\end{equation}
with $V(x)$ the trapping potential.
In other words, $G_\mu(x,y)$ is the solution of
\begin{equation}
\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}G_\mu(x,y) +(\mu-i0^+ -V(x))G_\mu(x,y)=
\delta(x-y).\label{gfunction}
\end{equation}
It is important to note that when integrating Eq. (\ref{dmu}) to recover the kernel we have the boundary condition, or completeness condition,
\begin{equation}
\lim_{\mu\to\infty} K_\mu(x,y) =\delta(x-y),
\end{equation}
as in this limit the sum in Eq. (\ref{k1}) is over a complete set of states. We also note
the trivial identity
\begin{equation}
K_\mu(x,y) =\delta(x-y) - \sum_k\theta(\epsilon_k-\mu) \psi_k^*(x)\psi_k(y),
\end{equation}
which yields
\begin{equation}
K_\mu(x,y) = \delta(x-y) -\int_{\mu}^\infty \frac{\partial}{\partial \mu'} K_{\mu'}(x,y) d\mu'
= \delta(x-y) -\int_{\mu}^\infty d\mu' \frac{1}{\pi} { \rm Im} \, G_{\mu'}(x,y)
\label{bigmu}
\end{equation}
which will be useful in what follows. An alternative integration formula is
\begin{equation}
K_\mu(x,y)= \int_{-\infty}^\mu \frac{\partial}{\partial \mu'} K_{\mu'}(x,y) d\mu'=\int_{-\infty}^\mu d\mu' \frac{1}{\pi} { \rm Im} \, G_{\mu'}(x,y)
\label{bigmu2}
\end{equation}
which obviously holds as long as the ground state of the system is bounded from below. In what follows we will always denote by $\mu$ the Fermi energy (which we assume is fixed) and denote by $\mu'$ the running Fermi energy used in the integrands of the representations given in Eq. (\ref{bigmu}) and (\ref{bigmu2}). These two representations can also be used to represent the kernel in terms of a kernel corresponding to a locally constant potential (so exact far away from the step) plus a term due to the variation of the potential. The derivation is rather technical and is relegated to appendix \ref{contsec}.
\section{Smooth potentials}\label{sec:smooth_pot}
Before applying {the Green's function} method to obtain new exact solutions (for any $\mu$)
for discontinuous potentials in the next section, we show how the method can be applied to
analyse the well studied case of smooth potentials. In the bulk we recover the prediction
of the LDA (which is exact for potentials constant in space) and identify the validity of the LDA via this method. We then examine, the again well known, edge {\em Airy gas} behavior, based on a local linear approximation to the potential
(the method being again exact for purely linear potentials).
\subsection{The bulk regime and the local density approximation}
\label{sec:LDA}
Here we use the Green's function to derive the LDA or Thomas-Fermi approximation which is the standard theoretical tool used to study the bulk thermodynamics behavior of free fermionic systems. In our derivation we identify the two key regimes where the approximation fails, the first regime is where the conditions for bulk behavior do not apply, notably near the edge of the system in continuous potential where the density becomes small. The second case occurs when the potential is not continuous, or varies too fast in space.
The basic approximation consists of computing the Green's function at two points
$x=x_0+z$ and $y=x_0+z'$. {We focus here on the bulk, hence one has $V(x_0) < \mu$.}
Assuming that $z$ and $z'$ are small, we can make the approximation $V(x) \approx V(x_0)$ in Eq. (\ref{gfunction}) and write
\begin{equation} \label{s1}
\frac{1}{2}\frac{\partial^2}{\partial z^2} G_{\mu'}(x_0+z,x_0+z') +[{ \mu' - i 0^+} - V(x_0)]G_{\mu'}(x_0+z,x_0+z')=\delta(z-z'),
\end{equation}
where we have introduced the running Fermi energy $\mu'$ which will be integrated over.
To simplify notation we have set $\hbar=1$ and $m=1$. The dependence on $\hbar^2/m$ can be reintroduced by making the rescalings $G\to m/\hbar^2 \times G$ and $(\mu-V)\to m/\hbar^2\times(\mu-V)$.
{Let us consider $\mu' > V(x_0)$.} For $z<z'$ we have
\begin{equation}
G_{\mu'}(x_0+z,x_0+z') = A_- \exp\left(i\sqrt{2\mu' - i 0^+ -2V(x_0)}\ z\right) \;.
\end{equation}
We have used that for $a>0$, $\sqrt{a - i 0^+} \equiv \sqrt{a} - i 0^+$, hence
the r.h.s. tends to zero as $z\to -\infty$ as required, due to the small imaginary part regulator $- i 0^+$. Similarly
for $z>z'$
\begin{equation}
G_{\mu'}(x_0+z,x_0+z') = A_+ \exp\left(-i\sqrt{2\mu' - i 0^+ - 2V(x_0)}\ z\right).
\end{equation}
Matching at $z=z'$, {taking into account the delta function on the r.h.s. of \eqref{s1},}
we obtain
\begin{equation}
G_{\mu'}(x_0+z,x_0+z') = \frac{i\exp\left(-i\sqrt{2\mu' - i 0^+ -2V(x_0)}|z-z'|)\right)}{\sqrt{2\mu' -2V(x_0)} } \, ,
\label{gflocal}
\end{equation}
where the small imaginary part {has been} neglected in the denominator.
This result is sufficient
to derive the kernel in the bulk.
For the computation of kernels away from the bulk it is useful to derive an equivalent representation for the Green's function.
For $y>x$, (so $z'>z$) Eq. (\ref{gflocal}) can be rewritten in terms of $x$ and $y$ for $|z-z'|$ small
as
\begin{eqnarray}
G_{\mu'}(x_0+z,x_0+z')
&=&\frac{i\exp\left(-i\sqrt{2\mu' - i 0^+ -2V(x_0)}(z'-z)\right)}{\sqrt{2\mu' -2V(x_0)} }\nonumber \\
&\approx& \frac{i\exp\left(-i\int_{z}^{z'}\sqrt{2\mu' - i 0^+ -2V(x_0+s)}\ ds\right)}{\sqrt{2\mu'-2V(x_0)}} \nonumber\\
&\approx& \frac{i\exp\left(-i\int_{z}^{z'}\sqrt{2\mu' - i 0^+ -2V(x_0)- 2V'(x_0)s}\ ds\right)}{\sqrt{2\mu' -2V(x_0)} }\nonumber \\
&=& \frac{i\exp\left(i\left[\frac{1}{3V'(x_{0})}\left(2\mu'-i0^{+}-2V(x_{0})-2V'(x_{0})s\right)^{3/2}\right]_{z}^{z'}\right)}{\sqrt{2\mu'-2V(x_{0})}}\,,
\end{eqnarray}
and thus to the same order this can be written as
\begin{equation}
G_{\mu'}(x_0+z,x_0+z') = Y_-(z)Y_+(z'),
\end{equation}
where
\begin{equation}
Y_-(z)=\frac{\sqrt{i}\exp\left(-i\frac{1}{3V'(x_{0})}\left(2\mu'-i0^{+}-2V(x_{0})-2V'(x_{0})z\right)^{3/2}\right)}{\left(2\mu'-2V(x_{0})-2V'(x_{0})z\right)^{1/4}} \, ,
\label{bcag}
\end{equation}
and
\begin{equation}
Y_+(z)=\frac{\sqrt{i}\exp\left(i\frac{1}{3V'(x_{0})}\left(2\mu'-i0^{+}-2V(x_{0})-2V'(x_{0})z\right)^{3/2}\right)}{\left(2\mu'-2V(x_{0})-2V'(x_{0})z\right)^{1/4}} \, .
\label{bcad}
\end{equation}
Adding the extra $z$ dependence in the numerator above does not change the basic small $z$ approximation and extends the basic plane wave approximation to the more general WKB form which will be useful when we will discuss the behaviors of kernels near a linear edge and matching plane wave solutions with Airy functions.
Returning to Eq. (\ref{gflocal}) we find that the kernel in the bulk can be written as
\begin{equation}
\frac{\partial}{\partial\mu'}K_{\mu'}\left(x_{0}+z,x_{0}+z'\right)=\frac{1}{\pi}{\rm Im}\frac{i\exp\left(-i\sqrt{2\mu'-i0^{+}-2V(x_{0})}|z-z'|\right)}{\sqrt{2\mu'-2V(x_{0})}} {= \frac{\cos(\sqrt{2\mu'-2V(x_0)}|z-z'|)}{\pi\sqrt{2\mu'- 2V(x_0)} } \, .}
\end{equation}
Integrating over $\mu'$ then gives
\begin{equation}
K_\mu(x_0+z,x_0+z') = \frac{1}{\pi |z-z'|}\sin(\sqrt{2\mu-2V(x_0)}|z-z'|) \, .
\end{equation}
The constant of integration has been determined by using the identity
\begin{equation}
\lim_{k\to\infty}\frac{1}{\pi x}\sin(kx) = \delta(x) \, .
\end{equation}
Note that strictly speaking the analysis is only valid in the case where $\mu'>\mu$ (assuming that already $\mu-V(x_0)$ is large enough to justify the analysis), however one sees that the analysis can be restricted to the case $\mu'>\mu$ by using the representation of $K_\mu$ given in Eq. (\ref{bigmu}) as opposed to Eq. (\ref{bigmu2}).
We have thus recovered the well known sine kernel for the
kernel in the bulk. It is often written as
\begin{equation} \label{sk}
K_\mu(x_0+z,x_0+z') = \frac{1}{\pi |z-z'|}\sin(p_\mu(x_0)|z-z'|),
\end{equation}
where
\begin{equation}
p_\mu(x_0)= \sqrt{2\mu-2V(x_0)} \, ,\label{fmom}
\end{equation}
is the local Fermi momentum at $x_0$. Finally the density of fermions at the point $x_0$ is obtained as
\begin{equation}
\label{eq:densityLDA}
\rho_\mu(x_0) = K_\mu(x_0,x_0)= \frac{p_\mu(x_0)}{\pi} \, .
\end{equation}\\
Let us now discuss the validity of the LDA. Its validity depends on the accuracy of the approximation where the spatial dependence of the potential $V(x)$ is neglected about the point $x_0$. The correction to this potential to give the exact one is $\Delta V(x) = V(x_0 +z)-V(x_0) \approx
zV'(x_0)$ for small $z$. The exact Green's function is given by
\begin{equation}
G_\mu = (\mu - H_{LDA}- \Delta V)^{-1}
\end{equation}
where
$H_{LDA} = -\frac{1}{2}\frac{\partial^2}{\partial x^2} + V(x_0)$ is the approximate form of the Hamiltonian used in the LDA and so $G_\mu^{LDA} = (\mu-H_{LDA})^{-1}$ is the corresponding Green's function. The first order correction to the LDA Green's function is thus (in operator product notation)
\begin{equation}
\Delta G_\mu = G_\mu^{LDA} \Delta V G_\mu^{LDA}.
\end{equation}
Now making the approximation $\Delta V(x) \simeq z V'(x_0)$, so assuming that the derivative of $V(x)$ exists, we find
\begin{equation}
\Delta G_\mu(z,z') = V'(x_0) \int du \, u \ G_\mu^{LDA}(z,u)G_\mu^{LDA}(u,z').
\end{equation}
From this we obtain the estimate $|\Delta G(z,z')| \approx |V'(x_0)|/ p_\mu(x_0)^4$. However we have seen that $G_\mu^{LDA}\approx 1/p_\mu(x_0)$ and so the condition that the relative error in using the LDA is small can be written as \cite{comment}
\begin{equation}
\frac{|\Delta G(z,z')|}{|G_{LDA}(z,z')|} \approx \frac{|V'(x_0)|}{p_\mu(x_0)^3 }\ll1
\label{ldaval}
\end{equation}
Note that this condition agrees with the more heuristic argument that the
potential should vary little on the scale of the inter-particle distance, which is $1/{p_\mu}(x_0)$,
i.e. that $|V'(x_0)/{p_\mu}(x_0)| \ll V(x_0) \sim \mu = \frac{{p_\mu}(x_0)^2}{2}$ (since in the bulk
$V$ is of the order of $\mu$).
We thus see that the LDA is valid in the bulk where the density $\rho(x_0) = p_\mu(x_0)/\pi$ is large. The LDA fails at the {\em edge} where the bulk density vanishes. The behavior of the kernel near this edge has been derived in several works by various methods \cite{dea16,eis13,dea15}.
In the following section we will show how the edge behavior can be derived using the Green's function method. As one might anticipate, we will see that the LDA also fails when the potential is discontinuous.
What happens in this case has been much less studied, and in Section \ref{sec:step} we will provide an analysis of the kernel for the cases of the step like potentials.
This analysis is exact, as we can obtain exact expressions for the Green's function.
\subsection{The edge regime and the Airy kernel}
\label{sec:airy}
Let us now study the Green's function near the edge points $x_e$ which are defined via the vanishing of the LDA prediction of the density, {\em i.e.} as the solutions of the equation
\begin{equation}
\mu - V(x_e) = 0,
\end{equation}
and we note that in general the function $V(x)-V(x_e)$ will vanish linearly near $x_e$. Thus the Green's function will locally obey the equation
\begin{equation}
\frac{1}{2}\frac{\partial^2}{\partial z^2} G_{\mu'}(z,z') +(\mu'-V(x_e)-z V'(x_e) )G_{\mu'}(z,z')=\delta(z-z'),\label{gfe}
\end{equation}
for $z$ and $z'$ in the neighborhood of {$x_0$} {$x_e$}.
Here, for notational simplicity we denoted $\mu' - i 0^+$ by $\mu'$ hence we
temporarily assume that $\mu'$ has a small negative imaginary part.
Without loss of generality we assume that $V'(x_e)>0$ (so we are considering
a right edge). From Eq. (\ref{gfe}) we see that when $\mu'\gg V(x_e)$ we can ignore the term linear in $z$ and repeat the bulk calculation. However when $\mu'\approx \mu$ the conditions necessary for the validity of the bulk calculation no longer hold and so we keep the linear correction which can be larger than or of the same order as the constant term.
We define the variable
\begin{equation} \label{cv}
\mu' - V(x_e) -z V'(x_e) = -\alpha \zeta,
\end{equation}
where $\alpha$ is a constant determined below. The equation \eqref{gfe} becomes
\begin{equation}
\frac{\partial^2}{\partial \zeta^2}G_{\mu'}(\zeta,\zeta) - \frac{2\alpha^3}{V'^2(x_e)}\zeta G_{\mu'}(\zeta,\zeta) =
2\frac{\alpha}{V'(x_e)}\delta(\zeta-\zeta').
\end{equation}
We choose $\frac{2\alpha^3}{V'^2(x_e)} =1$, which fixes $\alpha$ as
\begin{equation} \label{defalpha}
\alpha=\left[V'(x_{e})\right]^{2/3} \! / \, 2^{1/3}\,.
\end{equation}
This yields
\begin{equation}
\frac{\partial^2}{\partial \zeta^2}G_{\mu'}(\zeta,\zeta) - \zeta G_{\mu'}(\zeta,\zeta) =
\frac{2\alpha}{V'(x_e)}\delta(\zeta-\zeta').
\end{equation}
We therefore have
\begin{equation} \label{Gs}
G_{\mu'}(\zeta,\zeta') = \frac{2\alpha}{V'(x_e)} g(\zeta,\zeta')= \frac{V'(x_e)}{\alpha^2} g(\zeta,\zeta'),
\end{equation}
where $g(\zeta,\zeta')$ is the solution of
\begin{equation}
\frac{\partial^2}{\partial \zeta^2}g(\zeta,\zeta') - \zeta g(\zeta,\zeta') =
\delta(\zeta-\zeta'), \label{airyeq}
\end{equation}
i.e. it is the Green's function of the Airy operator. Note that the resolvent of
the Airy operator has a branch cut on the real axis, and here we consider its
value, $g(\zeta,\zeta')$,
for infinitesimal negative imaginary part (corresponding to $\mu' \to \mu' - i 0^+$ above).
The derivation is given in Appendix \ref{app:airygreen}. The final result is
\begin{equation}
\label{resg}
g(\zeta,\zeta')=\begin{cases}
-\pi{\rm Ai}(\zeta)[-i{\rm Ai}(\zeta')+{\rm Bi}(\zeta')] & {\rm for}\ \zeta>\zeta'\\[0.1cm]
-\pi{\rm Ai}(\zeta')[-i{\rm Ai}(\zeta)+{\rm Bi}(\zeta)] & {\rm for}\ \zeta<\zeta'.
\end{cases}
\end{equation}
Hence, for any $\zeta,\zeta'$ we have
\begin{equation}
{\rm Im}\left[g(\zeta,\zeta') \right]=\pi {\rm Ai}(\zeta){\rm Ai}(\zeta').
\end{equation}
Using the general relation \eqref{dmu} between the kernel and the Green's function,
together with \eqref{cv}, \eqref{defalpha}, \eqref{Gs}, and the result \eqref{resg}, we obtain
\begin{equation}
\frac{\partial}{\partial \mu'} K_{\mu'}(x,y) = \frac{V'(x_e)}{\alpha^2}{\rm Ai}(\zeta) {\rm Ai}(\zeta') = \frac{V'(x_e)}{\alpha^2}
{\rm Ai} \left(\frac{(x-x_e) V'(x_e)-\mu' + V(x_e)}{\alpha}\right){\rm Ai} \left(\frac{(y-x_e) V'(x_e)-\mu' + V(x_e)}{\alpha}\right),
\end{equation}
where here $\mu'$ is real.
Using Eq. (\ref{bigmu}) now gives
\begin{equation}
K_\mu(x,y)= \delta(x-y) -
{\frac{V'(x_e)}{\alpha^2} } \int_{\mu}^\infty d\mu' {\rm Ai} \left(\frac{(x-x_e) V'(x_e)-\mu' + V(x_e)}{\alpha}\right){\rm Ai} \left(\frac{(y-x_e) V'(x_e)-\mu' + V(x_e)}{\alpha}\right).\end{equation}
Recalling that $x_e$ is the edge, such that $\mu = V(x_e)$, we can change variables to
$\mu' = \mu + \alpha u$ and obtain
\begin{equation}
K_\mu(x,y) = \delta(x-y)-\frac{1}{w_\mu}\int_{-\infty}^0 du\ {\rm Ai}\left(\frac{x-x_e}{w_\mu}+ u\right){\rm Ai}\left(\frac{y-x_e}{w_\mu}+u\right), \nonumber \\
\end{equation}
where we have introduced the width of the edge regime \cite{dea16}
\begin{equation}
w_\mu = \alpha/V'(x_0) = (2 V'(x_0))^{-1/3} \;.
\end{equation}
Finally, using the completeness identity for Airy functions
\begin{equation}
\int_{-\infty}^\infty du \ {\rm Ai}(u+x){\rm Ai}(u+y)=\delta(x-y),
\end{equation}
we recover that the kernel near the edge takes the following scaling form in terms of the Airy kernel
$K_{\rm Ai}$
\begin{equation} \label{Kedge}
K_{\mu}(x,y)=\frac{1}{w_{\mu}}K_{{\rm Ai}}\left(\frac{x-x_{e}}{w_{\mu}},\frac{y-x_{e}}{w_{\mu}}\right)\quad,\quad K_{{\rm Ai}}(a,b)=\int_{0}^{+\infty}du\ {\rm Ai}(a+u){\rm Ai}(b+u).
\end{equation}
\subsection{General fast varying potential: perturbation theory}
One way of treating rapidly varying potentials which cause a breakdown of the LDA is via perturbation theory, valid when they are rapidly varying but weak. We take $H=H_0 + \delta V(x)$ where $\delta V(x) \ll H_0$ is a rapidly varying potential of a general form. If it varies notably on scales of the inter-particle distance its effect will be in general difficult to calculate. However we can study its effect via perturbation theory for the Green's function assuming that $\delta V(x)$ is sufficiently weak. Here we find to first order
\begin{equation}
G_\mu(x,y) = G_{0\mu}(x,y) + \int dx' G_{0\mu}(x,x') \delta V(x') G_{0\mu}(x',y) + O(\delta V^2) .
\end{equation}
Hence the change in the kernel due to the perturbation is given by
\begin{equation}
\Delta K_\mu(x,y) = - \frac{1}{\pi} \int_\mu^{+\infty} d\mu' \, {\rm Im} \int dx' G_{0\mu'}(x,x') \delta V(x') G_{0\mu'}(x',y) + O(\delta V^2) .
\end{equation}
Let us study it in the bulk, as in Section \ref{sec:LDA}. Using \eqref{gflocal} and $\frac{d\mu}{d p_\mu(x_0)}= p_\mu(x_0)$, we obtain for $x,y$ near {a given} $x_0$, the general formula
\begin{equation}
\Delta K_\mu(x,y) = - \frac{1}{\pi} \int_{p_\mu(x_0)}^{+\infty} \frac{dp}{p}
\int dx' \delta V(x') \exp\left(- i p |x-x'| - i p |y-x'|\right)
+ O(\delta V^2) .
\end{equation}
We thus see that the result is quantitatively significant if $\delta V(x)$ varies on scales of order or shorter than $1/p_\mu(x_0)$.
\section{Square step barrier, exact results}
\label{sec:step}
\subsection{General setting}
Here we apply the Green's function method presented above to analyse fermion statistics
in the presence of a finite square step potential located at $x=0$ which can be written as
\begin{equation} \label{step1}
V(x) = 0\ {\rm for}\ x<0;\ V(x)= V_0 \ {\rm for}\ x>0,
\end{equation}
and is shown in Fig. \ref{square}. For the potential \eqref{step1} the basic formalism of section \ref{sec:green}
can be implemented without any approximation for any $\mu$ without resorting to any approximation, because the Green's function can be computed exactly.
Let us introduce
\begin{equation}
p_F^L = p_F^L(\mu) = \sqrt{2 \mu} \quad , \quad p_F^R = p_F^R(\mu) = \sqrt{(2 \mu- 2 V_0)_+}
\end{equation}
the Fermi momentum in the regions $x<0$ and $x>0$ respectively, where $(x)_+=\max(x,0)$,
and we work here at fixed $\mu$.
There are two main cases:
(i) $\mu > V_0$, which we will refer to as supercritical in what follows. Here
very far from the barrier, at distances much greater that $ 1/p_F^{L,R}$, the
Fermi gas has two different uniform mean densities for $x<0$ (left -L) and $x>0$ (right -R), given by
\begin{equation} \label{densLR}
\rho_L=\frac{p_F^L}{\pi} = \frac{\sqrt{2 \mu}}{\pi} \quad , \quad \rho_R = \frac{p_F^R}{\pi} = \frac{\sqrt{2 (\mu-V_0)}}{\pi} < \rho_L.
\end{equation}
(ii) $0< \mu < V_0$, which we refer to as sub-critical, in which case only the left half space is filled for all $x$,
with, far from the barrier, the uniform mean density $\rho_L$. The mean density
vanishes for $x>0$ at distances $x \gg 1/\sqrt{2 V_0- 2\mu}$.
These are however only the behavior of the system far from the step
and we wish to calculate the behavior at distances $\sim 1/p_F^{L, R}$ from the
step. Hence we are interested in the crossover between the behaviors
of the two bulk Fermi gases.
The cases (i) and (ii) are separated by a transition, for $\mu=V_0$, which we call the critical case, where the Fermi gas on the left is at the brink of overflowing. It has a very interesting behavior, that we analyse below.
The general scaling form that the kernel takes can be obtained by using simple dimensional analysis. The units of the kernel are $\text{length}^{-1}$. The most general way to construct such a quantity in this system is
\begin{equation}
\label{eq:kernel_scaling}
K_{\mu}(x,y)=\frac{1}{\ell}\kappa_{r}\left(\frac{x}{\ell},\frac{y}{\ell}\right)\quad,\quad\frac{1}{\ell}=\rho_{L}=\frac{\sqrt{2\mu}}{\pi} \quad , \quad r=V_0 / \mu
\end{equation}
where
$\kappa_r$ is a dimensionless function. The cases $r<1$, $r=1$ and $r>1$ correspond to the supercritical, critical and subcritical cases respectively. As a result, the density takes the scaling form
\begin{equation}
\label{eq:density_scaling}
\rho(x)=\frac{1}{\ell}\,n_{r}\left(\frac{x}{\ell}\right)\quad,\quad\frac{1}{\ell}=\rho_{L}=\frac{\sqrt{2\mu}}{\pi}\quad,\quad n_{r}(a)=\kappa_{r}(a,a)
\end{equation}
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.30]{squarewell.pdf}
\caption{Locally square well potential with barrier of height $V_0$ situated at $x=0$. The Fermi energy $\mu$ for $\mu=\mu_c=V_0$ as well as the sub-critical case $\mu=\mu_{sub}<V_0$ and the super-critical case $\mu= \mu_{sup}>V_0$.}\label{square}
\end{center}
\end{figure}
\subsection{Green's function}
The computation of the Green's function is long but straightforward.
It is performed in Appendix \ref{app:green} for completeness
and also to detail the needed analytic continuations. One can also
find a formula in the reference book on path integrals \cite{gro98}. Here we give
the results for the imaginary part, which is what we need to compute the kernel.
For $\mu'<0$ one has ${\rm Im}\ G_{\mu'}(x,y) =0$, which is natural
since there are no energy eigenstates for negative energy.
Next there are two cases, either $0< \mu'<V_0$ or $\mu'> V_0$.
We start with the case where $0< \mu'< V_0$. In the region $x,y<0$ we find
\begin{equation}
{\rm Im}\,G_{\mu'}(x,y)=\frac{1}{V_{0}\sqrt{2\mu'}}\!\left[V_{0}\cos\left(\!\sqrt{2\mu'}\ (x-y)\right)+(2\mu'-V_{0})\cos\left(\!\sqrt{2\mu'}\ (x+y)\right)-2\sqrt{\mu'(V_{0}-\mu')}\sin\left(\!\sqrt{2\mu'}\ (x+y)\right)\right], \label{vbigleft}
\end{equation}
which obeys the symmetry $G(x,y)=G(y,x)$ of the Green's function. For $x>0>y$ we find
\begin{equation}
{\rm Im}\ G_{\mu'}(x,y)=\frac{\sqrt{2}\exp\left(-\sqrt{2V_{0}-2\mu'}\ x\right)}{V_{0}}\left[\sqrt{\mu'}\cos\left(\sqrt{2\mu'}\ y\right)-\sqrt{V_{0}-\mu'}\sin\left(\sqrt{2\mu'}\ y\right)\right], \label{vbigright}
\end{equation}
and the behavior for $y>0>x$ is obtained from the symmetry of the Green's function.
In region $x,y>0$ one finds the decaying solution
\begin{equation}
{\rm Im}\ G_{\mu'}(x,y) = \frac{\sqrt{2\mu'}}{V_0}\exp\left(-\sqrt{2V_0-2\mu'}\ (x+y)\right).
\label{g2}
\end{equation}
We now consider the case where $\mu'> V_0$. For $x,y<0$ we find
\begin{equation}
{\rm Im}\ G_{\mu'}(x,y)=\frac{1}{V_{0}\sqrt{2\mu'}}\left[V_{0}\cos\left(\sqrt{2\mu'}\ (x-y)\right)+\left(\sqrt{\mu'}-\sqrt{\mu'-V_{0}}\right)^{2}\cos\left(\sqrt{2\mu'}\ (x+y)\right)\right],
\label{e1}
\end{equation}
while for $y<0<x$ we find
\begin{equation}
{\rm Im}\ G_{\mu'}(x,y) =
\frac{\sqrt{2\mu'}-\sqrt{2\mu'-2 V_0}}{V_0}\cos\left(\sqrt{2\mu'-2V_0}\ x- \sqrt{2\mu'}y\right).
\label{e2}
\end{equation}
Finally, for $x,y>0$ we find
\begin{equation}
{\rm Im}\ G_{\mu'}(x,y)=\frac{1}{V_{0}\sqrt{2\mu'-2V_{0}}}\left[V_{0}\cos\left(\sqrt{2\mu'-2V_{0}}\ (x-y)\right)-\left(\sqrt{\mu'}-\sqrt{\mu'-V_{0}}\right)^{2}\cos\left(\sqrt{2\mu'-2V_{0}}\ (x+y)\right)\right].
\label{e3}
\end{equation}
These forms for $G_{\mu'}$ can now be used to compute the kernel.
\subsection{Infinite barrier and the hard wall kernel}
Let us start with the simplest case of an infinite barrier $V_0\to\infty $. In this limit we see
that when both points are to the left of the wall, $x,y<0$, one obtains from Eq. (\ref{vbigleft})
\begin{equation}
{\rm Im}\ G_{\mu'}(x,y) = \frac{1}{\sqrt{2\mu'}}
\left[\cos(\sqrt{2\mu'}\ (x-y))-\cos(\sqrt{2\mu'}\ (x+y))\right].
\end{equation}
By integrating Eq. (\ref{e3}) over $\mu'$ one finds from Eq. (\ref{dmu})
\begin{equation} \label{reflectedkernel}
K_{\mu}(x,y) = \frac{\sin(\sqrt{2\mu}\ (x-y))}{\pi (x-y)}-\frac{\sin(\sqrt{2\mu}\ (x+y))}{\pi (x+y)},
\end{equation}
which is the reflected sine-kernel that describes the infinite, or hard wall, potential barrier.
This result is well known and has been derived in the literature using different methods \cite{cal11,lac17}.
\subsection{{Super-critical case (or overflow) $\mu>V_0$}}
Let us study the case $\mu>V_0$ where the mean densities far on both sides are
both positive, $\rho_{L,R} >0$. The square barrier thus acts as a perturbation inside
the bulk of the Fermi gas. Using \eqref{e1} and \eqref{e3}, it is easy to see that the
integration of Eq. (\ref{dmu})
leads to
\begin{eqnarray} \label{2eq}
&& K_\mu(x,y) = \frac{\sin(\sqrt{2\mu}|x-y|)}{\pi|x-y|} - \int_\mu^\infty d\mu'
\frac{(\sqrt{\mu'} - \sqrt{\mu'-V_0})^2 \cos(\sqrt{2\mu'}\ (x+y))}{\pi V_0\sqrt{2\mu'}} \quad , \quad x, y < 0 \\
&&
K_\mu(x,y) = \frac{\sin(\sqrt{2\mu-2V_0}\ (x-y))}{\pi(x-y)} + \int_\mu^\infty d\mu'
\frac{(\sqrt{\mu'} - \sqrt{\mu'-V_0})^2 \cos(\sqrt{2\mu'-2V_0}\ (x+y))}{\pi V_0\sqrt{2\mu'-2V_0}} \quad , \quad x, y > 0 \nonumber \;.
\end{eqnarray}
The result in the region $x>0>y$ can be obtained from integration over $\mu'$ of
\eqref{vbigright} from $0$ to $V_0$, and of \eqref{e2} from $V_0$ to $\mu$, but
is not displayed here.
The results in \eqref{2eq} describes the deviations from the sine kernel forms,
which hold deep in the left and right bulks, due to the barrier at $x=0$. These deviations are written as integrals, which are convergent at large $\mu'$ since
$(\sqrt{\mu'} - \sqrt{\mu'-V_0})^2 \simeq \frac{V_0^2}{4 \mu'}$ for $\mu' \gg V_0$,
and which decay far from the barrier.
The Eq.~\eqref{2eq} can be written in the scaling form \eqref{eq:kernel_scaling} where for $r<1$,
\begin{numcases}{\kappa_{r}\left(a,b\right)=}
\frac{\sin\left(\pi\left|a-b\right|\right)}{\pi\left|a-b\right|}-\sqrt{r}\int_{1/r}^{\infty}dw\frac{\left(\sqrt{w}-\sqrt{w-1}\right)^{2}\cos\left(\pi\sqrt{rw}\left(a+b\right)\right)}{2\sqrt{w}}, & $a,b<0$\\% \label{a}\\
\frac{\sin\left(\pi\sqrt{1-r}\left(a-b\right)\right)}{\pi\left(a-b\right)}+\sqrt{r}\int_{1/r}^{\infty}dw\frac{\left(\sqrt{w}-\sqrt{w-1}\right)^{2}\cos\left(\pi\sqrt{r\left(w-1\right)}\ \left(a+b\right)\right)}{2\sqrt{w-1}} & $a,b>0.$
\end{numcases}
We recall that the mean density is given by $\rho(x)=K_\mu(x,x)$. We find that the density at $x=0$
can be expressed in terms of the densities $\rho_L$ and $\rho_R$ far from the barrier
given in \eqref{densLR} as
\begin{eqnarray}
\label{eq:rho_at_zero_supercritical}
&& \rho(0)= \frac{2}{3} \frac{\rho_L^2 + \rho_R^2 + \rho_L \rho_R}{\rho_L + \rho_R}
= \frac{\rho_L+\rho_R}{2} + \frac{1}{12 \rho_L} (\rho_L-\rho_R)^2 + O((\rho_L-\rho_R)^3) .
\end{eqnarray}
Far from the barrier the density decays to its asymptotic values as
\begin{eqnarray} \label{asymptdens}
&& \rho(x) \simeq \rho_L + \frac{(\sqrt{\mu} - \sqrt{\mu-V_0})^2 \sin(\sqrt{2\mu}\ 2 |x|)}{2 \pi V_0 |x|}
= \rho_L + \frac{\rho_L-\rho_R}{\rho_L+\rho_R} \frac{\sin(2 \pi \rho_L |x|)}{2 \pi |x|} \quad , \quad x \to - \infty \\
&& \rho(x) \simeq \rho_R - \frac{(\sqrt{\mu} - \sqrt{\mu-V_0})^2 \sin(\sqrt{2\mu-2 V_0}\ 2 x)}{2 \pi V_0 x}
= \rho_R - \frac{\rho_L-\rho_R}{\rho_L+\rho_R} \frac{\sin(2 \pi \rho_R x)}{2 \pi x} \quad , \quad x \to + \infty \;.
\end{eqnarray}
\subsection{Kernel in the critical -- just overflowing -- case}
For the square barrier one obtains a new form of edge when the Fermi energy intersects the potential, for $\mu=\mu_c =V_0$, see Fig. \ref{square}. The picture is that the bulk of the Fermi gas on inside the potential well for $x<0$ is on the point of overflowing out of the well for $x>0$.
Let us first study the region $x,y>0$. Setting $\mu=V_0$
in the second equation in \eqref{2eq}, we see that the first term vanishes and we find
\begin{eqnarray}
K_{\mu=V_0}(x,y)\equiv K_c(x,y) &=& \int_\mu^\infty d\mu' \frac{(\sqrt{\mu'} - \sqrt{\mu'-\mu})^2 \cos(\sqrt{2\mu'-2\mu}\ (x+y))}{\pi \mu\sqrt{2\mu'-2\mu}}\nonumber \\
&=& \frac{\sqrt{\mu}}{\pi}\int_0^\infty dv \frac{(\sqrt{1+v}-\sqrt{v})^2 \cos(\sqrt{2v}\sqrt{\mu} (x+y))}{\sqrt{2v}}
\quad , \quad x,y >0
\end{eqnarray}
where we have set $\mu'=\mu(1+v)$. This integral can be evaluated and one find that the kernel takes the scaling form \eqref{eq:kernel_scaling}
where the critical reduced kernel $\kappa_{c}(a,b)\equiv\kappa_{r=1}(a,b)$ is given by
\begin{equation}
\kappa_c(a,b) = \frac{2}{3} + \frac{\pmb{L}_2(\pi (a+b))-I_2(\pi(a+b))}{(a+b)}
\label{crit+} \quad , \quad a,b >0
\end{equation}
where $\pmb{L}_\alpha$ denotes the modified Struve function and $I_\alpha$ the modified
Bessel function \cite{abr65}.
It is worth noticing that Eq. (\ref{crit+}) can also be obtained, using \eqref{bigmu2} and \eqref{dmu},
by integrating Eq. (\ref{g2}) between
$\mu'=0$ and $\mu'=\mu$, since there are no states for $\mu'<0$
hence ${\rm Im}\ G_{\mu'<0}(x,y)=0$. Upon inserting $V_0=\mu$ and setting $\mu'=\mu(1-v)$
we obtain again \eqref{eq:kernel_scaling}
together with the alternative, more useful, representation of the kernel
\begin{equation}
\kappa_c(a,b) = \int_0^1 dv\ \sqrt{1-v}\exp(-\sqrt{v}\pi (a+b))\label{rep2} \quad , \quad a,b >0 \;.
\end{equation}
Consider now the region $x, y<0$. Setting $\mu=V_0$
in the first equation in \eqref{2eq}, we find
\begin{equation}
K_c(x,y) = \frac{\sin(\sqrt{2\mu}(x-y))}{\pi(x-y)} -
\int_\mu^\infty d\mu' \frac{
(\sqrt{\mu'} - \sqrt{\mu'-\mu})^2 \cos(\sqrt{2\mu'}\ (x+y))}{\pi \mu\sqrt{2\mu'}}.
\end{equation}
The integral can be evaluated and the result takes again the scaling form \eqref{eq:kernel_scaling}
with now
\begin{eqnarray} \label{kcleft}
\kappa_c(a,b) &=& \frac{\sin(\pi(a-b))}{\pi(a-b)} +\frac{\sin(\pi(a+b))}{\pi(a+b)} \\
&+&
\frac{1}{\pi(a+b)}\left[4\ \frac{ \pi(a+b)\cos(\pi(a+b))-\sin(\pi(a+b)) }{\pi^2(a+b)^2} - \pi J_2(\pi(a+b))\right]
\quad , \quad a,b < 0 \nonumber
\end{eqnarray}
where $J_\alpha$ is the Bessel function of the first kind \cite{abr65}. Finally the kernel for
$x>0>y$, not displayed here, is obtained upon integrating \eqref{vbigright} between
$\mu'=0$ and $\mu'=\mu$.
The mean fermion density is thus given by Eq.~\eqref{eq:density_scaling} with $n_{c}\left(a\right)\equiv n_{r=1}\left(a\right)=\kappa_{c}\left(a,a\right)$.
On the left side $x<0$ it is obtained from \eqref{kcleft} and it reaches its uniform
limit $\rho(x)=\rho_L$ for $x \to - \infty$, i.e. for $a \to - \infty$, by oscillating as
\begin{equation}
n_{c}(a)=1+\frac{\sin(2\pi a)}{2\pi a}+O\left(\frac{1}{|a|^{3/2}}\right)\quad,\quad a\to-\infty\;.
\end{equation}
On the right side $x>0$ the density is obtained from \eqref{rep2}
and we find that it decays to zero as a power law
\begin{equation} \label{alg1}
n_{c}(a)=\frac{1}{2\pi^{2}a^{2}}-\frac{3}{8\pi^{4}a^{4}}+O\left(\frac{1}{a^{6}}\right).
\end{equation}
The behavior of $n_c(a)$ is shown in Fig. (\ref{densc}). At the barrier we find that $n_c(a)$ and its derivative are continuous (as they must be). In addition we have $n_c(0)=2/3$.
It is interesting to study the total number of fermions $N_R$ outside the well, i.e. on the right side $x>0$, together
with its fluctuations. Using Eq. (\ref{ns}) we see that its average is given by
\begin{equation}
\langle N_R \rangle = \int_0^{+\infty} \rho(x) dx = \int_0^\infty n_c(a) da =
\int_0^1 \frac{dv}{2 \pi \sqrt{v}} \sqrt{1-v}
= \frac{1}{4},\label{avenout}
\end{equation}
where we used the representation \eqref{rep2} of the kernel.
It is finite, and independent of $\mu$, which may be surprising a priori. This independence on $\mu$ is an interesting consequence
of the scaling form \eqref{eq:density_scaling}
together with the convergence of the integral of $n_c(a)$ at large $a$. Recall that as soon as $\mu>V_0$, the mean number of fermions on the right,
$\langle N_R \rangle$, becomes infinite (with a density $\rho_R = \sqrt{2\mu-2V_0}/\pi$).
Using Eq. (\ref{vs}), and again the kernel representation Eq. (\ref{rep2}), we find that the variance of $N_R$ is given by
\begin{eqnarray}
\label{varnout}
{\rm Var}(N_{R}) &=& \langle N_{R}\rangle -\int_0^\infty da db \ \kappa_c(a,b)^2
= \frac{1}{4} - \frac{1}{\pi^2} \int_0^1 dv \int_0^1 du \frac{\sqrt{1-v}\sqrt{1-u}}{(\sqrt{u}+ \sqrt{v})^2} = \frac{2}{\pi^2},
\end{eqnarray}
and where computation of the last integral above is explained in Appendix \ref{integral}.
Given the small value of $\langle N_R\rangle$ it would be tempting to assume that the number of fermions outside the well has a Bernoulli distribution, {\em i.e.} $N_{R}=1$ with probability $p$ and $N_{R}=0$ with probability $q=1-p$. This would imply from Eq. (\ref{avenout}) that $p=1/4$ and so ${\rm Var}(N_{R})=pq=3/16=0.1875$. However our exact calculation finds
${\rm Var}(N_{R})=\frac{2}{\pi^2} = 0.202642$. Hence the actual random variable $N_R$ fluctuates more than a Bernoulli random variable, and the above result shows that the probability that $N_R \geq 2$ is strictly nonzero. {A similar calculation for the third cumulant (see Appendix \ref{appendixThirdCumulant} for the details) gives $\left\langle N_{R}^{3}\right\rangle _{c}=0.1281169\dots$. Here too one can see the difference from the Bernoulli distribution, whose third cumulant is $p\left(1-p\right)\left(1-2p\right)$, which for $p=1/4$ would give $3/32=0.09375$.}
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.50]{criticaldensity.pdf}
\caption{The behavior of $n_c(a)$ vs $a$. The density normalised by the bulk density, at the critical point where $\mu=\mu_c=V_0$.}\label{densc}
\end{center}
\end{figure}
\subsection{Kernel in the sub-critical regime}
We now consider the case where $\mu<V_0$. The physics here has similarities with the critical case, with additional off-critical features near the transition, which we now analyse. We will
focus on the region $x,\ y>0$.
We integrate Eq. (\ref{g2}) over $\mu'$ from $\mu'=0$ to $\mu'=\mu$ and obtain
that the kernel takes the scaling form \eqref{eq:kernel_scaling}
in terms of the reduced kernel, for $r = \frac{V_0}{\mu}>1$
\begin{equation} \label{kappar}
\kappa_r(a,b) = \frac{1}{r}\int_0^1 dv\ \sqrt{v}\exp\left(-\pi\sqrt{r-v}(a+b)\right) \quad , \quad a,b >0 .
\end{equation}
The critical kernel $\kappa_c(a,b)$ in \eqref{rep2} is recovered setting $r=1$.
The density takes the scaling form \eqref{eq:density_scaling} where
\begin{equation}
n_r(a)= \kappa_r(a,a) = n_{r}\left(a\right)=\kappa_{r}\left(a,a\right)=\frac{1}{r}\int_{0}^{1}dv\ \sqrt{v}\exp\left(-2\pi a\sqrt{r-v}\right).
\end{equation}
Remarkably, $\kappa_r(a,b)$ is only a function of the sum $a+b$. In particular, it satisfies
$\kappa_r(a,b) = n_r((a+b)/2)$.
The rescaled density at $a=0$ is given by $n_r(0)=\frac{2}{3r}$.
One can write this result together with Eq.~\eqref{eq:rho_at_zero_supercritical}, in the form
\begin{equation}
\label{eq:g_of_r}
\rho\left(0\right)=\frac{2\rho_{L}g\left(r\right)}{3}, \qquad
g\left(r\right)=\begin{cases}
1/r & r>1\\
\frac{2-r+\sqrt{1-r}}{1+\sqrt{1-r}} & 0<r<1
\end{cases}.
\end{equation}
In the vicinity of $r=1$, the function $g(r)$ exhibits the singular behavior
\begin{equation} \label{gofr}
g\left(r\right)=\begin{cases}
1+\left(1-r\right)+\left(1-r\right)^{2}+\dots & r-1\ll1\\[0.1cm]
1+\left(1-r\right)-\left(1-r\right)^{3/2}+\left(1-r\right)^{2}+\dots & 1-r\ll1
\end{cases}.
\end{equation}
The function $g(r)$ is plotted in Fig.~\ref{fig:g_of_r}.
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.50]{g_of_r.pdf}
\caption{The function $g(r)$ that describes the density at $x=0$, see Eq.~\eqref{eq:g_of_r}. The marked dot is the point $g(r=1)=1$ which corresponds to the critical case. {Note that $g(r)$ is non-analytic at this point $r=1$ [see Eq. (\ref{gofr})].}}\label{fig:g_of_r}
\end{center}
\end{figure}
For large $a>0$ and fixed $r>1$ one finds that the density decays exponentially
\begin{equation} \label{decay1}
n_{r}(a)\simeq\left(\frac{\sqrt{r-1}}{\pi ar}+O\left(\frac{1}{a^{2}}\right)\right)\exp\left(-2\pi a\sqrt{r-1}\right)\quad,\quad a\to+\infty.
\end{equation}
Hence for $r=V_0/\mu>1$ the density $\rho(x)$ decays exponentially for $x>0$, with a decay length $\xi_r=\ell/(2 \pi \sqrt{r-1})$ which diverges at the transition
with a square root singularity. Around the transition, in the double limit $r \to 1$ and $a \to +\infty$, with
$a \sqrt{1-r}$ fixed, it is easy to see by performing the change of variable
$v = 1 - (r-1) w$ in the integral \eqref{kappar}, that the reduced density takes the scaling form
\begin{equation} \label{nu}
n_r(a) \simeq (r-1) \, \nu(2 \pi a \sqrt{1-r}) \quad , \quad \nu(\tilde a)= \frac{2}{\tilde a^2} (1+\tilde a) \exp\left(- \tilde a\right)
\end{equation}
which describes the crossover between the exponential decay \eqref{decay1} and the
algebraic decay \eqref{alg1} at criticality.
In this regime $V_0=r \mu$, $r>1$, we find that the average number of particles outside the well is given by
\begin{equation}
\label{eq:NR_subcritical}
\langle N_{R}\rangle=\int_{0}^{\infty}n_{r}(a)da=\frac{1}{2\pi r}\left[r\sin^{-1}\left(\frac{1}{\sqrt{r}}\right)-\sqrt{r-1}\right]=\begin{cases}
\frac{1}{4}-\frac{\sqrt{r-1}}{\pi}+O((r-1)^{3/2})\quad, & r\to1^{+}\\[0.2cm]
\frac{1}{3\pi r^{3/2}}+O(r^{-5/2})\quad, & r\to+\infty
\end{cases}
\end{equation}
which tends to zero as a power law for large $r$ and to $1/4$ as $r\to1$.
Eq.~\eqref{eq:NR_subcritical} is plotted in Fig. \ref{fig:mean_and_var_NR} together with its asymptotics.
The variance is given
by
\begin{equation}
\label{eq:var_NR_subcritical}
{\rm Var}\left(N_{R}\right)=\left\langle N_{R}\right\rangle -\frac{1}{\pi^{2}r^{2}}\int_{0}^{1}dv\int_{0}^{1}du\frac{\sqrt{v}\sqrt{u}}{\left(\sqrt{r-u}+\sqrt{r-v}\right)^{2}} =\left\langle N_{R}\right\rangle -F\left(\frac{r-1}{r}\right)
\end{equation}
where
\begin{equation}
F\left(A\right)=\frac{\cos^{-1}\left(\sqrt{A}\right)\left[2\left(1-2A\right)\sqrt{A\left(1-A\right)}+\cos^{-1}\left(\sqrt{A}\right)\right]-\left(A-1\right)\left(3A+2A\ln A-2\right)}{\pi^{2}}\label{fint},
\end{equation}
and the computation is explained in Appendix \ref{integral}.
As $r \to +\infty$ the second term {in Eq. (\ref{eq:var_NR_subcritical})} behaves as $\simeq -1/(9 \pi^2 r^3)$ and is thus small
compared to the first one. Consequently we see that $ N_R$ becomes a Bernoulli random variable in this limit, because it satisfies ${\rm Var}\left(N_{R}\right)\simeq\left\langle N_{R}\right\rangle -\left\langle N_{R}\right\rangle ^{2}$ {[see the second line of Eq. (\ref{eq:NR_subcritical})]}.
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.48]{meanNR.pdf}
\includegraphics[scale=0.48]{varNR.pdf}
\caption{ (a) The mean number of particles on the right side of the step $N_R$, see Eq.~\eqref{eq:NR_subcritical} (solid line), plotted together with its $r\to1^+$ and $r\gg1$ asymptotic behaviors (dotted and dashed lines respectively).
(b) The variance of $N_R$, see Eq. \eqref{eq:var_NR_subcritical} (solid line), together with its $r\gg1$ approximation $\text{Var}\left(N_{R}\right)\simeq\left\langle N_{R}\right\rangle -\left\langle N_{R}\right\rangle ^{2}$ (dashed line) that corresponds to a Bernoulli random variable. Inset is a zoom in on the regime $r\simeq1$ where the result clearly deviates from the large-$r$ approximation.
At $r=1$ the mean and variance are $1/4$ and $2/\pi^2$ respectively.}
\label{fig:mean_and_var_NR}
\end{center}
\end{figure}
The kernel $K_\mu(x,y)$ in the other regions $x,y<0$ and $x>0>y$, not displayed here, are obtained
by integrating \eqref{vbigleft} and \eqref{vbigright}, respectively, for $\mu'$ from
$0$ to $\mu$, and can be studied similarly.
\section{Smooth barrier, universality and scattering amplitudes}
\label{smooth}
{In this section we consider potentials which become asymptotically constant far from the origin. Without loss of generality we write
\begin{equation}
V(x)=0 \ {\rm as}\ x\to -\infty; \ V(x) = V_0\ {\rm as}\ x\to \infty,
\end{equation}
where the convergence occurs beyond a typical scale which we call the {\em barrier width}. The discontinuous step potential is thus a special case which corresponds to zero barrier width. In what follows we show how the kernel can be obtained from scattering solutions, first via the Green's function method and then by a direct summation of the eigenfunctions.
We obtain formulas for the kernel valid at distances larger than the barrier width,
and for a general barrier potential, in terms of the coefficients of the scattering solutions.
These formula recover the exact result in the case of the discontinuous barrier.
In a second part we give the exact solution for a special {smoothened} step potential, known as the Woods-Saxon potential in the context of nuclear physics, and show the convergence to the aforementioned large distance
formula. In that part we identify which features of the transition at $\mu=V_0$
are universal, i.e. independent of the details of the shape of the barrier.}
\subsection{General representation for a barrier in terms of scattering solutions}\label{gfg}
Here we determine the Green's function for a general barrier, and from it we derive the kernel.
In general in one dimension the Green's function can be written as
\begin{equation}
G_{\mu'}(x,y) = A_{R{\mu'}}(y)\phi_{R{\mu'}}(x) \ {\rm for }\ x>y,\ \ \ \ G_{\mu'}(x,y) = A_{L{\mu'}}(y)\phi_{L{\mu'}}(x) \ {\rm for }\ y>x;
\end{equation}
where $\phi_{R{\mu'}}(x)$ and $\phi_{L{\mu'}}(x)$ and are solutions to the homogeneous equation $({\mu'}-i0^+-H)\phi_{{\mu'}}(x)=0$ respecting the boundary conditions $\phi_{R/L{\mu'}}(x)\to 0$ as $x\to-\infty$ (L) and as $x\to+\infty$ (R).
Now matching the solutions at $x=y$ gives the result
\begin{equation}
G_{\mu'}(x,y) =-\frac{ 2\phi_{R{\mu'}}(x)\phi_{L{\mu'}}(y)}{W} \ {\rm for }\ x>y,\ \ \ \ G_{\mu'}(x,y)=-\frac{ 2\phi_{R{\mu'}}(y)\phi_{L{\mu'}}(x)}{W} \ {\rm for }\ y>x, \label{match}
\end{equation}
where $W \equiv W[\phi_{R{\mu'}},\phi_{L{\mu'}}]= \phi_{R{\mu'}}(x)\phi'_{L{\mu'}}(x)- \phi_{L{\mu'}}(x)\phi'_{R{\mu'}}(x)$ is the Wronskian and is constant for differential equations having no first derivative term as is the case here.
However as the potential becomes constant away from the barrier, we must find the bulk solutions which are given up to a constant prefactor by
\begin{equation}
\phi_{L{\mu'}}(x)\sim\exp\left(i\sqrt{2{\mu'}-i0^{+}}\ x\right),\ {\rm as}\ x\to-\infty,\label{as0}
\end{equation}
and
\begin{equation}
\phi_{R{\mu'}}(x)\sim\exp\left(-i\sqrt{2{\mu'}-i0^{+}-2V_{0}}\ x\right),\ {\rm as}\ {x\to+\infty,} \label{as1}
\end{equation}
in the case ${\mu'}>V_0$ and
\begin{equation}
\phi_{R{\mu'}}(x)\sim\exp\left(-\sqrt{2V_{0}-2{\mu'}+i0^{+}}\ x\right),{\rm as}\ {x\to+\infty,}
\end{equation}
in the case { ${\mu'}<V_0$.} This means that if we write $k=\sqrt{2{\mu'}-i0^+-2V_0}$, the region $2{\mu'}-i0^+-2V_0>0$ gives values of $k$ along the positive real axis whereas when $2{\mu'}-2V_0<0$ we obtain $k=-ik'$ where $k'>0$.\\
{\noindent \bf The Green's function for ${\mu'}>V_0$ and the supercritical case}.
\vspace*{.3cm}
In the case ${\mu'}>V_0$, we can use the solutions $\psi_{k_1}(x)$ of the Schr\"odinger equation $(k_1^2/2 - H)\psi_{k_1} = 0$ corresponding to the
energy $\mu'=k_1^2/2$ which have the asymptotic form of plane waves far away from the barrier. For potential barriers, eigenstates are often expressed in terms of the scattering of a plane wave coming from the left of the barrier. The incoming momentum to the left of the barrier is $k_1=\sqrt{2{\mu'}-i0^+}$ and the outgoing momentum to the right is $k_2=\sqrt{2{\mu'}-i0^+-2V_0}$ (we added the $-i 0^+$
for future convenience when discussing the Green's function). Such a plane wave is partially transmitted and one writes
\begin{eqnarray} \psi_{k_1}(x) =
\begin{cases}
&\exp\left(i k_1 x\right)+ \frac{C_2(k_1,k_2)}{C_1(k_1,k_2)} \exp\left(-ik_1x\right) \;, \; x \to - \infty \\
& \\
& \frac{1}{C_1(k_1,k_2)} \exp\left(i k_2 x\right) \;, \;\hspace*{1.1cm} x \to +\infty \;,
\label{asleft}\end{cases}
\end{eqnarray}
with ${\mu'}=\frac{ k_1^2}{2}= \frac{ k_2^2}{2}+V_0$. Here $\frac{C_2(k_1,k_2)}{C_1(k_1,k_2)}$ is the reflection amplitude and $\frac{1}{C_1(k_1,k_2)}$ the transmission amplitude, both of which depend on the precise form of the barrier. The reflection probability is then given by
\begin{equation}
R(k_1,k_2) =\left|\frac{C_2(k_1,k_2)}{C_1(k_1,k_2)} \right|^2,
\end{equation}
and the transmission probability is $T(k_1,k_2)=1-R(k_1,k_2)$.
The above wave function clearly does not satisfy the correct boundary conditions for either $\phi_{R{\mu'}}$ or $\phi_{L{\mu'}}$ however, comparing with Eq. \eqref{as1}, we see that we can write
\begin{equation}
\phi_{R{\mu'}}(x)=\psi_{k_1}^*(x),
\end{equation}
as, if $\psi_{k_1}(x)$ is an eigenfunction then so is $\psi^*_{k_1}(x)$, one just has to verify that it is not the same eigenfunction. Now if we write $\phi_{L{\mu'}}(x)= \psi_{k_1}(x)
+B \psi^*_{k_1}(x)$, the asymptotic condition given in Eq. \eqref{as0}, i.e. $\phi_{L{\mu'}}(x)\sim \exp\left(i k_1 x\right)$, as $x\to-\infty$ then gives
\begin{equation}
\left[\exp\left(ik_{1}x\right)+\frac{C_{2}(k_{1},k_{2})}{C_{1}(k_{1},k_{2})}\exp\left(-ik_{1}x\right)\right]+B\left[\exp\left(-ik_{1}x\right)+\frac{C_{2}^{*}(k_{1},k_{2})}{C_{1}^{*}(k_{1},k_{2})}\exp\left(ik_{1}x\right)\right]=A\exp\left(ik_{1}x\right),
\end{equation}
where $A$ is a constant. Note that since $k_1$ and $k_2$ are real we denote $C^*_j(k_1,k_2) := ( C_j(k_1,k_2) )^*$ [see the discussion around Eq. (\ref{conjugate})]. This yields $B=-C_2(k_1,k_2)/C_1(k_1,k_2)$ and thus
\begin{equation}
\phi_{L{\mu'}}(x)=\psi_{k_1}(x)-\frac{C_2(k_1,k_2)}{C_1(k_1,k_2)}\psi^*_{k_1}(x).
\end{equation}
From this we find that the Wronskian is given by $W[\phi_{{\mu'} R},\phi_{{\mu'} L}] = W[\psi^*_{k_1},\psi_{k_1}]$. However as this is a constant we can evaluate it in the regions $x\to+\infty$ where its is known, i.e.,
\begin{equation}
W[\psi^*_{k_1},\psi_{k_1}]= \frac{2ik_2}{C_1(k_1,k_2)C_1^*(k_1,k_2)}.
\end{equation}
On the other hand, the evaluation of the Wronskian as $x\to-\infty$ yields
\begin{equation}
W[\psi_{k_{1}}^{*},\psi_{k_{1}}]=2ik_{1}\left[1-\frac{C_{2}(k_{1},k_{2})C_{2}^{*}(k_{1},k_{2})}{C_{1}(k_{1},k_{2})C_{1}^{*}(k_{1},k_{2})}\right] \, .
\end{equation}
Equating the two expressions for the Wronskians yields the Wronskian identity, which is the well known formula corresponding the the conservation of current,
\begin{equation}
\frac{k_{2}}{|C_{1}(k_{1},k_{2})|^{2}}=k_{1}\left[1-\frac{|C_{2}(k_{1},k_{2})|^{2}}{|C_{1}(k_{1},k_{2})|^{2}}\right]\, ,\label{wid1}
\end{equation}
which can be written as
\begin{equation}
T(k_1,k_2)=1-R(k_1,k_2) = \frac{k_2}{k_1|C_1(k_1,k_2)|^2} \, .\label{tr}
\end{equation}
From the above relation we see that
when $k_2$ vanishes $\psi_{k_1}^*$ and $\psi_{k_2}$ correspond to the same wave function as the Wronskian vanishes - physically this is due to total reflection of the incoming wave. The case $k_2^2<0$ must thus be treated separately and will be considered at the end of this section.
Putting all the above results together and using Eq. \eqref{match}, we obtain, for $x>y$,
\begin{equation}
G_{\mu'}(x,y) = \frac{i C_1(k_1,k_2) C^*_1(k_1,k_2)}{k_2}\psi_{k_1}^*(x)\left[\psi_{k_1}(y) - \frac{C_2(k_1,k_2)}{C_1(k_1,k_2)}\psi^*_{k_1}(y)\right],\label{gsoft}
\end{equation}
where we emphasize again that here, for $\mu'>V_0$, $k_1,k_2$ are both real and positive.
This is an explicit formula for the Green's function in terms of the scattering eigenstates.
Although the function {$\psi_{k_1}(x)$} is not necessarily known everywhere, its asymptotics can be read of from
Eq. (\ref{asleft}). As $x,\ y\to +\infty$, with $x>y$, we find
\begin{equation} \label{Gapprox}
\ G_{\mu'}(x,y)\approx \frac{i}{k_2}\exp(-ik_2(x-y))-i\frac{C_2(k_1,k_2)}{k_2C^*_1(k_1,k_2)}\exp(-ik_2(x+y)) \;.
\end{equation}
This gives, using Eq. (\ref{repi})
\begin{equation} \label{Kmuprime}
K_\mu(x,y) \approx \frac{\sin(k_{2F}(x-y))}{\pi (x-y)}+\frac{1}{\pi}{\rm Im}\int_{k_{2F}}^{\infty} idk_2 \ \frac{C_2(\sqrt{k_2^2+2V_0},k_2)}{C^*_1(\sqrt{k_2^2+ 2V_0},k_2)}\exp(-ik_2(x+y))
\end{equation}
where $k_{2F}= \sqrt{2\mu-2V_0}$ and where we have used $d \mu' = k_2 dk_2$.
\eqref{Kmuprime} should be valid for $x,y>0$ much larger than the {\em barrier width} when the asymptotics for above the wave-functions hold. The notion of a {\em barrier width} will be quantified in the next Section (and called $\lambda$) in the concrete example of the Woods-Saxon potential. In addition, one can perform a second asymptotics if furthermore $x+y \gg 1/k_{2F}$
(a scale which can become much larger than the barrier width near criticality, here the integral is dominated by the vicinity of $k_2 \approx k_{2F}$ and we obtain (see Appendix \ref{contsec} for details)
\begin{equation} \label{Kmu_asympt}
K_\mu(x,y) \approx \frac{\sin(k_{2F}(x-y))}{\pi (x-y)}-\frac{|C_2(k_{1F},k_{2F})|}{|C_1(k_{1F},k_{2F})|}\frac{\sin(k_{2F}(x+y)-\phi_{12})}{\pi(x+y)},
\end{equation}
where $\phi_{12}=\arg(C_2(k_{1F},k_{2F})/C^*_1(k_{1F}, k_{2F}))$. We thus see that at large distances from the barrier, the kernel takes the bulk sine-kernel form, plus an oscillatory correction with an amplitude that decays algebraically with distance from the barrier.
{
In the region to the left of the barrier as $x,\ y\to-\infty$ one finds
\begin{eqnarray}
G_{\mu'}(x,y) &=& i\frac{C_1^*(k_1,k_2) C_1(k_1,k_2)}{k_2}\left[1 -\frac{C_2^*(k_1,k_2) C_2(k_1,k_2)}{C_1^*(k_1,k_2) C_1(k_1,k_2)}\right]\left(\exp(-ik_1(x-y)) + \frac{C_2^*(k_1,k_2)}{C_1^*(k_1,k_2)}\exp(ik_1(x+y))\right)\nonumber \\
&=&
\frac{i}{k_1}\left(\exp(-ik_1(x-y)) + \frac{C_2^*(k_1,k_2)}{C_1^*(k_1,k_2)}\exp(ik_1(x+y))\right),\label{negxy}
\end{eqnarray}
where we have used the Wronskian identity Eq. (\ref{wid1}). Using Eq. (\ref{repi}), it leads to
\begin{equation} \label{Kmuprime2}
K_\mu(x,y) \approx \frac{\sin(k_{1F}(x-y))}{\pi (x-y)}- \frac{1}{\pi}{\rm Im}\int_{k_{1F}}^{\infty} idk_1 \ \frac{C^*_2(k_1,\sqrt{k_1^2 + 2 V_0})}{C^*_1(k_1,\sqrt{k_1^2+ 2V_0})}\exp(ik_1(x+y))
\end{equation}
where $k_{1F}=\sqrt{2 \mu}$. A similar calculation as above (see Appendix \ref{contsec}) then yields
\begin{equation}
K_\mu(x,y) \approx \frac{\sin(k_{1F}(x-y))}{\pi (x-y)}- \frac{|C_2(k_{1F},k_{2F})|}{|C_1(k_{1F},k_{2F})|}\frac{\sin(k_{1F}(x+y)-\phi'_{12})}{\pi|x+y|}.\label{kas1}
\end{equation}
where $\phi'_{12}= \arg(C_2(k_{1F},k_{2F})/C_1(k_{1F},k_{2F}))$.
The above formula agrees with Eq.
\eqref{asymptdens} for the density of the square well when one sets $x=y$ and uses the corresponding scattering coefficients given in Eq. (\ref{G4}).}
\medskip
{\noindent \bf The Green's function for ${\mu'}<V_0$ and the subcritical case.}
\vspace*{0.3cm}
We now turn to the case where $k_2^2<0$, here two solutions can be found by analytic continuation of the previous solutions. Recalling that
$k_2=\sqrt{2{\mu'}-i0^+-2V_0}$ where the positive root is taken we see that when $V_0>{\mu'}$ the continuous branch of square root is $k_2= -i\kappa_2=-i\sqrt{ 2V_0 -2{\mu'} +i0^+}$ as this choice has a positive real part and
negative imaginary part in the region where ${\mu'}\approx V_0$. The analytic continuation of the Green's function given in Eq. (\ref{gsoft}) is then, for $x>y$
\begin{equation}
G_{\mu'}(x,y) = \frac{- C^*_1(k_1,-i\kappa_2)C_1(k_1,-i\kappa_2)}{\kappa_2}\psi_{k_1}^{(2)}(x)\left[\psi_{k_1}^{(1)}(y) - \frac{C_2(k_1,-i\kappa_2)}{C_1(k_1,-i\kappa_2)}\psi^{(2)}_{k_1}(y)\right],\label{cgf}
\end{equation}
where here and below we denote the complex conjugate function
\begin{equation} \label{conjugate}
f^{*}(k_{1},-i\kappa_{2}):=\left.\left(\left.f(k_{1},k_{2})\right|_{k_{1},k_{2}\in\mathbb{R}}\right)^{*}\right|_{k_{2}\to-i\kappa_{2}}\;.
\end{equation}
In other words $f^*(k_1,k_2)$ is calculated by taking first the complex conjugate of $f(k_1,k_2)$ with $k_1,k_2$ real, and then performing analytical continuation in $k_2$. Note that this is the
usual definition, which satisfies $( f(z,w) )^* = f^*(z^*,w^*)$, e.g if $f(z,w)=a z + b w$ then
$f^*(z,w)=a^* z + b^* w$.
In Eq. (\ref{cgf}) we have denoted the analytic continuation of $\psi_{k_1}$ by $\psi_{k_1}^{(1)}$ and $\psi_{k_1}^*$ by $\psi_{k_1}^{(2)}$. The analytic continuations have the asymptotic forms
\begin{equation}
\psi_{k_{1}}^{(1)}(x)=\begin{cases}
\exp\left(ik_{1}x\right)+\frac{C_{2}(k_{1},-i\kappa_{2})}{C_{1}(k_{1},-i\kappa_{2})}\exp\left(-ik_{1}x\right)\;, & x\to-\infty\\[0.2cm]
\frac{1}{C_{1}(k_{1},-i\kappa_{2})}\exp\left(\kappa_{2}x\right)\;, & x\to+\infty\;,
\end{cases}
\end{equation}
and
\begin{equation}
\psi_{k_{1}}^{(2)}(x)=\begin{cases}
\exp\left(-ik_{1}x\right)+\frac{C_{2}^{*}(k_{1},-i\kappa_{2})}{C_{1}^{*}(k_{1},-i\kappa_{2})}\exp\left(ik_{1}x\right)\;, & x\to-\infty\\[0.2cm]
\frac{1}{C_{1}^{*}(k_{1},-i\kappa_{2})}\exp\left(-\kappa_{2}x\right)\;, & x\to+\infty\;.
\end{cases}. \label{psi2}
\end{equation}
It is easy to see that these two functions used to construct the Green's function have the Wronskian
identity
\begin{equation}
W[\psi_{k_{1}}^{(2)},\psi_{k_{1}}^{(1)}]=\frac{2\kappa_{2}}{C_{1}(k_{1},-i\kappa_{2})C_{1}^{*}(k_{1},-i\kappa_{2})}=2ik_{1}\left[1-\frac{C_{2}(k_{1},-i\kappa_{2})C_{2}^{*}(k_{1},-i\kappa_{2})}{C_{1}(k_{1},-i\kappa_{2})C_{1}^{*}(k_{1},-i\kappa_{2})}\right].\label{w2}
\end{equation}
Now returning to the Green's function, in region $x\ ,y \to \infty$ we find, for $x>y$,
\begin{equation}
G_{\mu'}(x,y) \approx -\frac{1}{\kappa_2}\exp(-\kappa_2(x-y))+ \frac{1}{\kappa_2}\frac{C_2(k_1,-i\kappa_2)}{C_1^*(\kappa_1, - i\kappa_2)}\exp(-\kappa_2(x+y)),
\end{equation}
where we recall that here $k_1=\sqrt{2{\mu'}- i 0^+}$ and $\kappa_2=\sqrt{ 2V_0 -2{\mu'} +i0^+}$.
From this we find
\begin{equation}
{\rm Im}\ G_{\mu'}(x,y)\approx \frac{\exp(-\kappa_2(x+y))}{\kappa_2}{\rm Im}\frac{C_2(k_1,-i\kappa_2)}{C_1^*(k_1, - i\kappa_2)} \, .
\end{equation}
This then gives the kernel as
\begin{equation}
K_\mu(x,y) \approx \frac{1}{\pi}\int_0^{k_{1F}} k_1 dk_1 \frac{\exp(-\kappa_2(x+y))}{\kappa_2}{\rm Im}\frac{ C_2(k_1,-i\kappa_2)}{C_1^*(k_1,- i\kappa_2)},\label{ksubplus}
\end{equation}
where $k_{1F}=\sqrt{2 \mu}$ and $\kappa_2=\sqrt{2V_0- k_1^2}$ in the integrand above.
An important identity is derived in Appendix \ref{identityap}
\begin{equation}
{\rm Im}\frac{C_2(k_1,-i\kappa_2)}{C^*_1(k_1, - i\kappa_2)}= \frac{\kappa_2}{2k_1}\frac{1}{|C_1(k_1,i\kappa_2)|^2},\label{idformula}
\end{equation}
Hence we see that an alternative formula for the kernel (at distances much larger than the
barrier width) is
\begin{equation}
K_\mu(x,y) \approx \int_0^{k_{1F}} \frac{dk_1}{2 \pi}
\frac{\exp(-\kappa_2(x+y))}{|C_1(k_1,i\kappa_2)|^2} \, .
\label{ksubplus2}
\end{equation}
This form is the one naturally obtained in the alternative method which uses the summation
over the eigenstates, as we will see in the next Section, see formula \eqref{KmuWood}.
Further asymptotics can be performed if $x+y \gg 1/\kappa_{2F}$,
where $\kappa_{2F} = \sqrt{2 V_0- 2 \mu}$. Again this scale can be much
larger than the barrier width if one is near criticality $\mu \approx V_0$. In the
region $x+y \gg 1/\kappa_{2 F}$ the integral in Eq. (\ref{ksubplus}) is dominated by the region near $k_1=k_{1F}$ (equivalently $\kappa_2$ near $\kappa_{2F}$), due to the exponential decay of the integrand. Expanding the integral about $k_1=k_{1F}$ using
$k_1 dk_1 = \kappa_2 d\kappa_2$, yields the asymptotics for the kernel as $x,y \to +\infty$ as
\begin{equation} \label{asp}
K_\mu(x,y) \approx \frac{\exp(-\kappa_{2F}(x+y))}{\pi (x+y)}{\rm Im} \frac{C_2(k_{1F},-i\kappa_{2F})}{C_1^*(k_{1F},- i\kappa_{2F})},
\end{equation}
where we recall that $k_{1F} = \sqrt{2\mu}$ and $\kappa_{2F} = \sqrt{2V_0-k_{1F}^2}= \sqrt{2 V_0- 2 \mu}$. Let us emphasize again that the above calculation requires that $\kappa_{2F}>0$. The critical point where $\kappa_{2F}=0$ will be discussed below.
In the region $x\ ,y \to -\infty$, for $x>y$, we can analytically continue Eq. (\ref{negxy}) to find
\begin{eqnarray}
G_{\mu'}(x,y)
&\approx&
\frac{i}{k_1}\left(\exp(-ik_1(x-y)) + \frac{C^*_2(k_1,- i\kappa_2)}{C^*_1(k_1,- i\kappa_2)}\exp(ik_1(x+y))\right).
\end{eqnarray}
Note that $C^*_j(k_1,- i\kappa_2)=( C_j(k_1, i \kappa_2) )^*$, $j=1,2$. Using Eq. (\ref{repii}), we obtain
\begin{equation} \label{asympt_subcrit}
K_\mu(x,y) \approx \frac{\sin(k_{1F}(x-y))}{\pi (x-y)} + \frac{1}{\pi}\int_0^{k_{1F}} dk_1 {\rm Im} \left(i
\frac{C^*_2(k_1,- i\kappa_2)}{C^*_1(k_1,- i\kappa_2)}\exp(ik_1(x+y))
\right) \;,
\end{equation}
where $k_{1F} = \sqrt{2 \mu}$. This can also be written using Eq. (\ref{repi}) (and analytically continuing he coefficients $C_1$ and $C_2$) as
\begin{equation} \label{asympt_subcrit2}
K_\mu(x,y) \approx \frac{\sin(k_{1F}(x-y))}{\pi (x-y)} - \frac{1}{\pi}\int_{k_{1F}}^\infty dk_1 {\rm Im} \left(i
\frac{C^*_2(k_1,- i\kappa_2)}{C^*_1(k_1,- i\kappa_2)}\exp(ik_1(x+y))
\right) \;.
\end{equation}
Using the same method as in Appendix \ref{contsec}, the asymptotic behavior of $K_{\mu}(x,y)$ in (\ref{asympt_subcrit2}) for $x, y \to - \infty$ is given by
\begin{equation}
K_\mu(x,y) \approx \frac{\sin(k_{1F}(x-y))}{\pi (x-y)}+\frac{|C_2(k_{1F},-i\kappa_{2F})|}{|C_1(k_{1F},-i\kappa_{2F})|}\frac{\sin(k_{1F}(x+y)-\phi'_{12})}{\pi(x+y)}.\label{kas2}
\end{equation}
where $\phi'_{12}= \arg(C_2(k_{1F},i \kappa_{2F})/C_1(k_{1F}, i \kappa_{2F}))$.
\vspace*{0.3cm}
{\bf \noindent The critical case.}
\vspace*{0.3cm}
Here the critical case corresponds to $\kappa_{2F}=0$. If we use the representation in Eq.
\eqref{ksubplus2}
the kernel at the critical point, for $x,\ y>0$ much larger than the barrier width, can be written as
\begin{equation}
K_\mu(x,y) \approx \frac{1}{2\pi}\int_0^{k_{1F}} dk_1 \frac{1}{|C_1(k_1,i\kappa_2)|^2}\exp(-\kappa_2(x+y)),
\end{equation}
with $\kappa_2 = \sqrt{k_{1F}^2 - k_1^2}$ in the integrand. This kernel has the
following asymptotics as $x+y \to +\infty$. The dominant contribution in that limit comes from
$\kappa_2\approx 0$ and so $\kappa_{1F} \approx k_{1F}$ and using this we find
\begin{equation} \label{GFres}
K_\mu(x,y) \approx \frac{1}{2\pi |C_1(k_{1F},0)|^2 k_{1F}(x+y)^2},
\end{equation}
and so we see that the large distance $1/(x+y)^2$ decay of the kernel at the critical point is universal. This will be confirmed below from an exact solution for the Woods-Saxon potential \cite{woo54}.
\subsection{Direct summation of eigenfunctions and the Woods-Saxon potential}
We now analyze a fully solvable model of a smooth barrier, described by the Woods-Saxon potential \cite{woo54}
\begin{equation} \label{smoothstep}
V(x) = \frac{V_0}{1 + \exp(- \frac{x}{\lambda}) }
\end{equation}
such that $V(x) \to 0$ for $x \to - \infty$ and $V(x) \to V_0$ for $x \to + \infty$.
The length scale $\lambda$ thus controls the width of the step, and for $\lambda \to 0$ one
recovers the square step barrier. The method we use here is a direct summation
over the eigenstates. This will allow us to (i) connect with the previous
subsection where we obtained the asymptotics far from the barrier using the Green's function
method (we will see how these asymptotics emerge in this second method) (ii) explore the universality of the transition at $\mu=V_0$ with respect to the shape and width of the barrier. We will investigate here the situation where $\lambda$ and the typical inter-particle distance, $\ell$, are of the same order.
In this subsection we restrict to the case $\mu \leq V_0$, i.e. subcritical and critical.
We will only sketch the main results, the details are given in the Appendix \ref{app:smooth}.
Using the standard solution \cite{Landau} for the eigenstates for energies $\epsilon_k< V_0$, one finds (see Appendix \ref{app:smooth}) the exact expression of the kernel for any $x,y$ and for $\mu \leq V_0$
\begin{equation} \label{kernelsmooth}
K_\mu(x,y) = \int_0^{\sqrt{2 \mu}} \frac{dk_1}{2 \pi}
B(\lambda k_1, \lambda \kappa_2) \phi^*_{\lambda k_1,\lambda \kappa_2}(x) \phi_{\lambda k_1,\lambda \kappa_2}(y) \exp\left(- \kappa_2 (x+y)\right) \quad , \quad \kappa_2= \sqrt{2 V_0 - k_1^2}
\end{equation}
with
\begin{equation}
\phi_{k_{1},\kappa_{2}}(x)={}_{2}F_{1}\left(ik_{1}+\kappa_{2},-ik_{1}+\kappa_{2},1+2\kappa_{2},-\exp\left(-\frac{x}{\lambda}\right)\right)~,~B(k_{1},\kappa_{2})=\left|\frac{\Gamma(-ik_{1}+\kappa_{2})\Gamma(1-ik_{1}+\kappa_{2})}{\Gamma(-2ik_{1})\Gamma(1+2\kappa_{2})}\right|^{2}
\end{equation}
and ${}_2F_1$ is the standard hypergeometric function.
For $\lambda \to 0$ one has
$\phi_{\lambda k_1,\lambda \kappa_2}(x) \to 1$ and $B(\lambda k_1,\lambda \kappa_2)
\simeq 4 k_1^2/(k_1^2 + \kappa_2^2)$ and one recovers the result for the
square step barrier, in the form given in the appendix in \eqref{crit_kernel_app1}.
Let us first study the region $x,y>0$, i.e. the penetration of the fermions in the classically forbidden region.
We note that for $x \to +\infty$ the function
$\phi_{k_1,\kappa_2}(x)$ approaches unity exponentially fast, i.e.
$\phi_{k_1,\kappa_2}(x) = 1 - \frac{2 V_0}{1 + 2 \lambda \kappa_2} \exp(- x/\lambda) + O(\exp({- 2 x/\lambda}))$.
Hence for $x,y \gg \lambda$ the kernel takes the form
\begin{eqnarray} \label{KmuWood}
K_\mu(x,y) \simeq \int_0^{\sqrt{2 \mu}} \frac{dk_1}{2 \pi}
B(\lambda k_1, \lambda \kappa_2) \exp\left(- \kappa_2 (x+y)\right) \, .
\end{eqnarray}
Since $B(\lambda k_1, \lambda \kappa_2) = 1/|C_1(k_1,i \kappa_2)|^2$
in terms of the scattering amplitudes given in \eqref{scattWood},
the formula \eqref{KmuWood} is thus consistent with the result obtained in \eqref{ksubplus}
and \eqref{ksubplus2} by the Green's function method,
for a general barrier in terms of their associated scattering amplitudes.
At criticality $\mu=V_0$, one can shift to $\kappa_2=\sqrt{2 \mu-k_1^2}$ as integration
variable, and one sees that the kernel decays as a power law at large distance, as
\begin{equation}
\label{kernelsmoothTail}
K_{\mu}(x,y)\simeq\int_{0}^{\sqrt{2\mu}}\frac{d\kappa_{2}}{2\pi}\frac{\kappa_{2}}{\sqrt{2\mu-\kappa_{2}^{2}}}B\left(\lambda\sqrt{2\mu-\kappa_{2}^{2}},\lambda\kappa_{2}\right)\exp\left(-\kappa_{2}(x+y)\right)\simeq\frac{2\lambda}{(x+y)^{2}}\coth\left(\pi\lambda\sqrt{2\mu}\right)+O\left(\frac{1}{(x+y)^{3}}\right)
\end{equation}
since for large $x+y$ the integral is dominated by $\kappa_2 \approx 0$ and we used
that $B(\lambda \sqrt{2 \mu},0) = 4 \pi \lambda \sqrt{2 \mu} \coth(\pi \lambda \sqrt{2 \mu})$.
Hence, comparing with \eqref{alg1}, we see that the inverse square power law decay at large distance appears to be universal, but that the overall amplitude depends on the width of the barrier in units of inter-particle distance $\lambda \sqrt{2 \mu} = \pi \lambda/\ell$. Again this is consistent
with the general result obtained by the Green's function method in \eqref{GFres}. The exact formula Eq. \eqref{kernelsmooth} allows one to also obtain all sub-leading corrections.
We now show that, up to this overall amplitude, the scaling function $\nu$ defined in \eqref{nu} is
universal.
It describes the decay of the kernel and of the density at large distance in the critical region,
i.e. $r-1 \ll 1$ and $x,y$ large of the order of the decay length
$\xi_r = \ell/(2 \pi \sqrt{r-1})$ introduced above \eqref{nu}. Note that as $r \to 1$, $\xi$ becomes much larger than the
width of the barrier, hence it is natural to expect universality.
Let us define $k_1=\sqrt{v} \sqrt{2 \mu}$, $\kappa_2=\sqrt{r-v} \sqrt{2 \mu}$ with $r=V_0/\mu \geq 1$ and
we see that the kernel can be put in the scaling form \eqref{eq:kernel_scaling}, with $\ell = \frac{\pi}{\sqrt{2\mu}}$ the typical inter particle distance, and $\tilde \lambda = \lambda \sqrt{2 \mu} = \pi \lambda/\ell$
\begin{equation}
K_{\mu}(x,y) \simeq \frac{1}{\ell}\kappa_{r,\tilde \lambda}\left(\frac{x}{\ell},\frac{y}{\ell}\right)
\quad , \quad
\kappa_{r,\tilde \lambda}(a,b) = \int_0^1 \frac{dv}{4 \sqrt{v}}
B(\tilde \lambda \sqrt{v}, \tilde \lambda \sqrt{r-v}) e^{- \pi \sqrt{r-v} (a+b) }
\quad , \quad a,b \gg \lambda/\ell
\end{equation}
If we take $\tilde \lambda \to 0$ we have $B(\tilde \lambda \sqrt{v}, \tilde \lambda \sqrt{r-v})\simeq \frac{4 v}{r}$ and one recovers \eqref{kappar}. Let us now write $v=1- (r-1) w$ and perform
an expansion in $r-1$. We obtain
\begin{eqnarray}
\kappa_{r,\tilde \lambda}(a,b) & \simeq & (r-1)
\int_0^{1/(r-1)} dw
\left( \pi \lambda \coth(\pi \lambda) + O(\sqrt{r-1}) \sqrt{1+w} \right)
e^{- \pi \sqrt{r- 1} \sqrt{1+w} (a+b)} \\
& \simeq & \pi \lambda \coth(\pi \lambda) (r-1) \,
\nu(\pi (a+b) \sqrt{r-1}) \quad , \quad \nu(\tilde a)= \frac{2}{\tilde a^2} (1+\tilde a) \exp\left(- \tilde a\right)
\label{nu2}
\end{eqnarray}
hence in the critical region $x,y = O(\xi_r)$ the kernel and the density take the same
scaling form as in \eqref{nu}. Note that inside the subcritical phase the decay is exponential with a
rate predicted by \eqref{nu2} for any $r$. However, away from the critical region, that is
for large $a,b>0$ and fixed $r>1$, the amplitude of the
large distance decay is given by
\begin{equation}
\kappa_{r,\tilde \lambda}(a,b) \simeq \frac{\sqrt{r-1}}{2 \pi (a+b)} B(\tilde \lambda, \tilde \lambda \sqrt{r-1})
\exp\left(- \pi (a+b) \sqrt{r-1}\right) .
\end{equation}
The above is the analog of
\eqref{decay1}, but exhibits a non-universal $r$-dependent amplitude.
Let us now discuss the region $x,y<0$ for $\mu \leq V_0$. From \eqref{psi_step22}
the wave functions are oscillating for $x,y \to -\infty$, and for $-x,-y \gg \lambda$,
the kernel takes the form
\begin{equation} \label{left}
K_\mu(x,y) \simeq \int_0^{\sqrt{2 \mu}} \frac{dk_1}{\pi} \left( \cos k_1 (x-y)
+ {\rm Re} \beta_{k_1,\kappa_2}^* e^{i k_1 (x+y)} \right)
= \frac{\sin \sqrt{2 \mu} (x-y)}{\pi (x-y)}
+ \int_0^{\sqrt{2 \mu}} \frac{dk_1}{\pi} {\rm Re} \beta_{k_1,\kappa_2} e^{- i k_1 (x+y)}
\end{equation}
where we recall that $\kappa_2= \sqrt{2 V_0 - k_1^2}$ and
\begin{equation}
\beta_{k_1,\kappa_2} = \frac{C_2(k_1,i\kappa_2)}{C_1(k_1,i\kappa_2)}
\end{equation}
is the reflection amplitude (whose modulus square $|\beta|^2$ is the reflection coefficient of the barrier,
here equal to unity). In \eqref{left} the second term (the reflected kernel) gives the far away
correction to the sine kernel of the bulk due to the presence of the barrier.
Note that the asymptotic form \eqref{left} for $x,y \to -\infty$, and the expression of the reflected part of the kernel, is very general for any barrier. It is in perfect agreement with the formula
\eqref{asympt_subcrit} obtained by the Green's function method.
In the case of the Woods-Saxon potential
\begin{equation}
\beta_{k_1,\kappa_2} = \frac{ \Gamma(2 i \lambda k_1) \Gamma(- i \lambda k_1+ \lambda \kappa_2 ) \Gamma(1- i \lambda k_1+ \lambda \kappa_2) }{\Gamma(-2 i \lambda k_1) \Gamma(i \lambda k_1 + \lambda \kappa_2) \Gamma(1+ i \lambda k_1 + \lambda \kappa_2)}
\end{equation}
The limit of the square barrier is recovered for $\lambda \to 0$
in which case $\beta_{k_1,\kappa_2} = \frac{k_1- i \kappa_2}{k_1 + i \kappa_2}$.
Note that for any $\lambda$ in the limit $V_0 \to +\infty$ one has $\beta_{k_1,\kappa_2} \to -1$
and one recovers the infinite wall reflected kernel \eqref{reflectedkernel}. From \eqref{left}
we can extract the asymptotics of the mean fermion density as $x \to - \infty$. In that limit the integral
is dominated by the vicinity of $k_1=k_F=\sqrt{2 \mu}$ leading to the general formula for $\mu \leq V_0$
\begin{equation}
\rho(x) \simeq \rho_L + \frac{1}{2 \pi |x|} {\rm Im} \left[ \beta_{\sqrt{2 \mu},\sqrt{2 (V_0- \mu)}}\exp(2 i k_F |x|) \right]
\quad , \quad x \to - \infty
\end{equation}
which, for the square barrier gives
\begin{equation}
\rho(x)\simeq\rho_{L}+\frac{1}{2\pi V_{0}|x|}{\rm Im}\left[\left(\sqrt{\mu}-i\sqrt{V_{0}-\mu}\right)^{2}\exp\left(2ik_{F}|x|\right)\right]\quad,\quad x\to-\infty
\end{equation}
i.e. the continuation for $\mu \leq V_0$ of the formula \eqref{asymptdens} (which was
valid for $\mu \geq V_0$). The similar asymptotics were derived for a general barrier in terms of scattering amplitudes for
$\mu > V_0$ using the Green's function method in the previous subsection.
{
The density $\rho(x)=K_\mu\left(x,x\right)$ for the Woods-Saxon potential in the critical case, with $\mu=V_0=1$, is plotted in Fig.~\ref{fig:woods_saxon_density} together with its $x \to \infty$ asymptotic behavior \eqref{kernelsmoothTail}. For a broad barrier (which, for the Woods-Saxon potential corresponds to large $\lambda$), the density is described correctly by the LDA, because the potential varies slowly in space. This is seen in the figure even for the moderately large value $\lambda=2$. Likewise, the density correlations are described by the sine kernel \eqref{sk}. Note that even for large $\lambda$, there is ultimately a power law decay of the density a large distance, as seen in the inset of Fig. \ref{fig:woods_saxon_density}.
}
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.6]{critical_density_smooth_barrier_with_tails.pdf}
\caption{Solid lines: The density $\rho(x)/\rho_{L}=K_\mu\left(x,x\right)/\rho_{L}$ vs. $x$, for the Woods-Saxon potential \eqref{smoothstep} with $\mu=V_0=1$ (so $r=1$, the critical case) and three different values of $\lambda$: 1/2, 1 and 2, obtained via a numerical evaluation of Eq.~\eqref{kernelsmooth}. Dashed line: The prediction of the LDA, Eq.~\eqref{eq:densityLDA}, for $\lambda=2$. Inset: The large-$x$ tails of the density (solid lines) compared to the asymptotic \eqref{kernelsmoothTail} (dashed lines).}
\label{fig:woods_saxon_density}
\end{center}
\end{figure}
\section{Conclusions}\label{sec:conclusion}
In this paper we studied the quantum correlations of spinless non-interacting fermions
in their ground state. We
introduced an alternative Green's function method to compute the kernel
which does not rely on an explicit summation over eigenstates.
We first showed how it allows to recover the known results for smooth potentials.
It allows one to derive, in a compact way, the kernel in the bulk, which is given by the LDA approximation, and to ascertain the validity of this approximation.
We also showed how the same basic method can be adapted
to study the properties of the Airy gas at the edges of the Fermi gas.
The method is particularly useful when one has exact results for the single particle Green's function.
This is the case for a system in the presence of a finite step in the potential for which we have
obtained the kernel and the fermion density analytically. We have analyzed the cases where the Fermi energy is below the height of the step (the sub-critical case) and above the height of the trap (the super-critical case). Of particular interest is the critical case where the step height coincides with the Fermi energy. Here the kernel takes a particularly simple form and one can show that the number of fermions $N_R$ to the right of the edge is of order $1$, even though the system itself is macroscopic. The analysis of the second moment shows that the distribution is not Bernoulli in most cases, showing that more that than one fermion may {\em leak} over the edge. However as the Fermi energy is lowered below the step height, we find that the distribution of the number of fermions becomes Bernoulli, but with a probability of presence $p$ that becomes very small. For completeness, we have shown in the Appendix B how to recover the kernel for the step potential from a direct summation over eigenstates, focusing for illustration on the simplest case $\mu \leq V_0$. This method, which also allowed us to analyze the case of a step of finite width, has its advantages, but it requires a careful treatment of the normalization factors and selection of the proper eigenstates which contribute, technical details which are automatically taken into account in the Green's function method. Furthermore, in a companion paper, this Green's function method will be applied to treat the
case of delta impurities for which again it turns out to be particularly well adapted.
Next we considered the case of a general smoothed potential and showed how the asymptotics of the Green's function (far from the region where the potential varies) can be written in terms of generic scattering coefficients of plane waves arriving from the left. From this we were able to derive asymptotic results for the kernel and density far from the step in the potential, in particular we were able to show that the algebraic decay $1/x^2$ of the density, far to the right of the step, is universal at the critical point $\mu=V_0$. The behavior of the density close to the step does however depend on the shape of the step. For the Woods-Saxon potential we have given an integral expression for the kernel and density in the subcritical and critical regimes and explicitly verified that for this potential, at distances greater than the step width $\lambda$, the general asymptotic results derived here hold.
This study opens up a number of perspectives for further research. In particular achieving close to zero temperatures in experiments is still an on going challenge. The effect of finite temperature can be incorporated
in a straight forward manner at finite temperature and in the grand canonical ensemble. Here the kernel is given by \cite{dea15b,dea16,dea19}
\begin{equation}
K_{\tilde \mu}(x,y)=\sum_k \frac{1}{1 + \exp\left(\beta( \tilde \mu-\epsilon_k)\right)}\psi_k^*(x)\psi_k(y),
\end{equation}
where $\tilde \mu$ is the chemical potential, which becomes the Fermi energy in the zero temperature limit.
Applying the results presented here it is straightforward to see that
\begin{equation}
K_{\tilde \mu}(x,y)=
\frac{1}{\pi} \int d\mu' \frac{1}{1 + \exp\left(\beta (\mu'-\tilde \mu)\right)} { \rm Im} \, G_{\mu'}(x,y),
\end{equation}
From this formula the effect of a finite temperature for a step potential can be analysed, in particular one can ask how the distribution of fermions to the right of the step in the critical and sub-critical regimes
will depend on the temperature.
It would also be interesting to investigate the properties of the Wigner function \cite{wig32,cas08} for stepwise potentials. Existing methods based on the extraction of the Wigner function from the kernel have revealed interesting and universal properties at the edges of trapped systems both for statics \cite{dea18} and dynamics \cite{dea19b} and the methods proposed here might facilitate further studies.
In a similar vein, the determinantal properties of trapped fermionic systems allow one to study extreme value statistics, typically answering questions such as what is the distribution of the farthest fermion from the center of a trap \cite{dea16,dea17,dea19}. For one dimensional smooth traps these statistics are given by the celebrated Tracy-Widom \cite{tra94} distribution \cite{dea16,dea19}, while in higher dimensional systems with eigenstate degeneracy Gumbel type distributions are found \cite{dea17}. In principle extreme value statistics can be determined from knowledge of a Fredholm determinant involving the kernel \cite{bor11,dea16,dea19}, however the computation of the Fredholm determinant presents a daunting mathematical task. It is however possible that the simple form of the critical kernel given by Eq. (\ref{rep2}) may allow further analytical progress.
\vspace*{0.5cm}
{\it Acknowledgments:}
NRS acknowledges support from the Yad Hanadiv fund (Rothschild fellowship). This research was supported by ANR grant ANR-17-CE30-0027-01 RaMaTraF.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 50 |
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MTC New Play Program
West Coast Premiere
By J. T. Rogers
Directed by Jasson Minadakis
"A thrilling, rich drama"
Extended thru October 28
Overview Schedule & Prices Tickets Artists Reviews Learn More
Overview Schedule & Prices Artists Reviews Learn More
This smash-hit political thriller tells the true and widely unknown story of how Norwegian diplomat Mona Juul, and her husband, social scientist Terje Rød-Larsen, planned and orchestrated top-secret meetings between the State of Israel and the Palestine Liberation Organization, culminating in the historic 1993 Oslo Accords. Through back-channel talks, unlikely friendships and quiet heroics, common ground between the Israeli and Palestinian envoys is carefully unearthed. Oslo is a deeply personal story set against a complex historical canvas, a story about the individuals behind world history and their all too human ambitions.
Generous support for Oslo provided by The Shubert Foundation, William & Flora Hewlett Foundation, and The Bernard Osher Foundation.
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27 7:30pm
Ashkon Davaran*
Hassan Asfour
Ashkon Davaran is an actor and songwriter originally from the San Francisco Bay Area. Earlier this year he played various roles in the world premiere of One Thousand Nights and One Day(Off-Broadway). Past credits include the role of Balaga in Natasha, Pierre and The Great Comet of 1812 (Off-Broadway), and the title roles in Bloody Bloody Andrew Jackson (San Francisco Playhouse, SF Premiere) and Beardo, composed by Dave Malloy (Shotgun Players). As a musician, Ashkon has released multiple albums/singles and been widely recognized for his viral music videos, including his 2010 SF Giants inspired remake of Journey's 'Don't Stop Believing'.
Aaron Davidman*
Yossi Beilin
Aaron Davidman is an actor, playwright, director and producer. He is drawn to stories of ethnic history and cultural complexity that challenge our assumptions of the "other". He served as Artistic Director of Traveling Jewish Theatre from 2002-2011. His play WRESTLING JERUSALEM, a solo performance about the Israel/Palestine story, toured North America from 2014 - 2018 and was made into a feature film. His new play, THE SHOOTING GALLERY, A Play About Guns in America will premiere at Mosaic Theater of DC in April, 2019. More at aarondavidman.com.
Joe Estlack*
Thor Bjornevog/American Diplomat
This is Joe Estlack's MTC debut. His last performance was in The Effect at SF Playhouse. He's appeared in Shotgun Players' Blasted, Bonnie & Clydeand Woyzeck. He's also worked at Magic Theater, Aurora, San Jose Stage and received a TBA award for his performance inStupid Fucking Birdat SF Playhouse. He is a founding member of Mugwumpin. www.joseph-estlack.squarspace....
Corey Fischer*
Corey Fischer is an actor, writer and director who has been creating and performing theatre for several decades. In 1978, with Albert Greenberg and Naomi Newman, he co-founded the award-winning Traveling Jewish Theatre and served the company as writer, actor and director for its 34 years of existence. Robert Hurwitt, drama critic emeritus for the Chronicle, once wrote that Fischer is "one of the Bay Area's acting treasures." In 2000, he adapted renowned Israeli author David Grossman's novel "See Under: Love" which won a Kennedy Center New American Plays award. Before the founding of TJT, Fischer worked with several visionary directors and companies including The ProVisional Theater, Robert Altman and Joseph Chaikin. He is also a published writer of fiction and non-fiction, and works as a developmental editor/dramaturg with playwrights. He is married to writer China Galland, has three adult step-children and six grandchildren. www.coreyfischer.com.
Brian Herndon*
Yair Hirschfeld
Brian Herndon happily returns to MTC, having appeared in Shakespeare in Love, Swimmers, Failure: A Love Story, The Good German and As Thousands Cheer. Favorite roles not at MTC include Shylock, Caliban, Dromio of Syracuse, Leo Bloom in The Producers and Edward Gant in Edward Gant's Amazing Feats of Loneliness. Brian has played Mr. Elton in the musical adaptation of Jane Austen's Emma, from its first staged reading and the world premiere at TheatreWorks, to productions all over the country, from San Diego to Cincinnati to New York City. Brian is a proud member of Actors' Equity and the PlayGround Company. Eternal love and gratitude to his wife, Jocelyn, and his daughter, Gwen.
Peter James Meyers*
Joel Singer
PETER JAMES MEYERS most recently performed the role of De Giche in Cyrano at TheatreWorks where he has also performed in Old Moneythe Heidi Chronicles and The Old Boy. Meyers has performed and directed throughout the Bay Area including The Magic Theatre, Berkeley Repertory Theatre and American Conservatory Theater. As artistic director of The Vector Theater Company, Meyers directed the premieres of Seattle Speaks, The Gypsy Marker, Museum and a musical adaption of Henrik Ibsen's Peer Gynt. Peter is the founder of Stand & Deliver - a global consulting group dedicated to inspiring great performance in organizations. He is the author of the book As We Speak - How to Make Your Point and Make it Stickand has led seminars for Fortune 500 Companies in 31 countries bringing the art of performance to leaders throughout the world.
J Paul Nicholas*
Ahmed Qurie
J Paul Nicholas is thrilled to work on JT's amazing play at this amazing theatre. He was most recently in Disgraced at the Chester Theater in MA, and twice in The Invisible Hand -- at Clevelnad Play House and at Theater Exile in Philadelphia. He was also recently on a national tour with the Broadway phenomenon The Curious Incident Of The Dog In The Night-Time. Some of his other recent credits include The Butcher at Gulfshore Playhouse, Ciphers at Inis Nua Theater, and JT's Blood and Gifts at the Lincoln Center. Paul has also performed regionally at The Shakespeare Theater; The Wilma; The Olney; Interact Theater; Alabama Shakespeare Festival; Geva Theatre; Seattle Repertory; Woolly Mammoth; The Alliance; and Everyman Theater. He has also performed Off-Broadway at 59E59. He can be seen on several TV series, and for 8 seasons he was Attorney Linden Delroy on TV's LAW & ORDER: SVU. Paul earned an MFA from the Academy for Classical Acting at GWU. JPaulNicholas.com.
Adam Niemann
Trond Gundersen/German Husband
Adam Niemann is making his debut with Marin Theatre Company. Adam's credits include Berkeley Repertory Theatre (u/s for Angels in America, Macbeth and It Can't Happen Here), Cutting Ball Theater (Timon of Athens), Ray of Light Theatre (Reefer Madness), Custom Made Theatre Co. (Isaac's Eye and Vonnegut's Mother Night), Curtain Theatre (The Comedy of Errors), Douglas Morrisson Theatre (Book of Days), FaultLine Theater (Oreo Carrot Danger), and Berkeley Playhouse (Beauty and the Beast, Peter Pan and Tarzan). Adam is currently writing and developing a two-man musical entitled JFK 'n Me. Adam is a graduate of UC Berkeley and studied at the A.C.T. Summer Training Congress. www.AdamNiemann.com
Paris Hunter Paul*
Uri Savir
Born a Jew, Paris had espoused the merits of a Jewish homeland, where Jews finally had the right to live in peace and sovereignty; that is, until he traveled to that homeland to see first hand Israel's neighbors being denied the right to live in peace and sovereignty. Paris believes it a privilege to now have a nuanced perspective of the conflict facing Israel and Palestine, and a privilege to engage in the dialectics of this conflict by acting this play. Thank you, Jasson, for the opportunity. Theatre credits include: the Guthrie Theater, Geffen Playhouse, Berkeley Repertory Theatre, California Shakespeare Theater, Cutting Ball Theater, The Jungle, Great River Shakespeare Festival, etc. University of Minnesota/Guthrie Theater B.F.A. Actor Training Program.
Mark Anderson Phillips*
Terje Rød-Larsen
Mark Anderson Phillips* (Terje Rød-Larsen) was last seen in MTC's productions of Shakespeare in Love, and Thomas & Sally. Other MTC credits include Waiting for Godot, Tiny Alice, and Good People. His work at other theaters includes roles with ACT, Berkeley Repertory Theatre, ACT Theatre of Seattle, San Jose Repertory Theatre, TheatreWorks, The Magic Theatre, Aurora Theatre, Repertory Theatre of St. Louis, SF Playhouse, California Shakespeare Theatre, San Francisco Shakespeare Festival, Center REP, Arizona Theatre Company, Word for Word, Santa Cruz Shakespeare and California Theatre Center. He recently directed The Siegel, and Ideation at CityLights Theater. Other directing credits include The 39 Steps!, Sleuth, Vanya & Sonia & Masha & Spike at CenterREP, Dr. Jekyll & Mr. Hyde at CityLights, The King & I, Cinderella, Joseph & The Amazing Technicolor Dreamcoat with Children's Musical Theater of San Jose, The Birds, and The Trojan Women at California Theatre Center, and Othello at Calaveras Rep. www.markandersonphillips.com
Marcia Pizzo*
Marianne Heiberg/Toril Grandal/Swedish Hostess/German Wife
Marcia Pizzo is delighted to return to Marin Theatre Company where she appeared in The Women. Other Bay Area credits include Round and Round the Garden, Rock'n'Roll, Opera Comique, You Never Can Tell andA Mother with Olympia Dukakis ( A.C.T),TheCountry House and the West Coast premiere of The Pitman Painters(TheatreWorks), Wilder Timesand Eccentricities of aNightingale (Aurora Theatre Company) Restoration Comedy, Two Gentlemen of Verona, The Merchant of Venice, King Lear, andThe Tempest(CaliforniaShakespeare Theater),An Ideal Husband, Romeo and Juliet,Antony and Cleopatra, Taming of the Shrew, As You Like It, Macbeth, Cyrano de Bergerac, and The Servant of Two Master(Marin Shakespeare Company). She has also performed at the Magic Theatre, Center REP, SF Playhouse, Pacific Repertory Theatre, The Jewell and in many musicals for the Mountain Play including The King And I,Oklahoma!, My Fair Lady, Anything Goes, and Annie.Marcia received her MFA from the American Conservatory Theatre.
Charles Shaw Robinson*
Johan Jorgen Holst/Finn Grandal
"There is no way to peace along the way of safety. For peace must be dared."
"Do I not destroy my enemies when I make them my friends?"
Erica Sullivan*
Mona Juul
Erica Sullivan's most recent theatre credits come from spending five seasons with the Oregon Shakespeare Festival performing in such productions as Fingersmith, As You Like It, All The Way, The Sign In Sidney Brustein's Window, and The Two Gentleman Of Verona. She's also worked at Lincoln Center, Yale Repertory Theatre, The Studio Theatre, Cal Shakes, Williamstown Theater Festival and Long Wharf Theatre. Recent television and film credits include Netlfix's The OA, TNT's The Librarians, NBC's Grimm, Jackie Katzman's The Strangeness You Feel, and Paul Thomas Anderson's Inherent Vice. She holds an M. F. A. from the Yale School of Drama.
Ryan Tasker*
Ron Pundak/Jan Egeland
Ryan Tasker has previously appeared at MTC in Straight White Men (Matt), Anne Boleyn (Simpkin/Barrow) and August: Osage County (Deon Gilbeau). Other recent roles include Cyrano in Cyrano de Bergerac with the Livermore Shakespeare Festival, where he has also appeared in Much Ado About Nothing (Benedick), Romeo and Juliet (Benvolio), The Real Thing (Max), and Sense and Sensibility (Colonel Brandon). He is an Associate Artist with Word for Word Performing Arts Company in San Francisco, with whom he has originated roles in Three on a Party and You Know When the Men Are Gone, as well as acting and directing for their Off the Page staged reading series. Tasker's other credits include work with the San Francisco Shakespeare Festival, American Conservatory Theatre, Aurora Theatre Company, San Jose Repertory, TheatreWorks, Shotgun Players, Pacific Repertory, Theatre Rhinoceros, TheatreFIRST, theatre Q, and Just Theater, among others."
J.T. Rogers
J.T. Roger's plays include Oslo (Lincoln Center Theater, then Broadway; National Theatre, London, then West End); Blood and Gifts (Lincoln Center Theater; National Theatre); The Overwhelming (National Theatre, then UK tour with Out of Joint; Roundabout Theatre); White People (Off Broadway with Starry Night Productions); and Madagascar (Theatre 503, London; Melbourne Theatre Company). For Oslo he won the Tony, New York Critics, Outer Critics, Drama Desk, Drama League, Lortel, and Obie awards. As one of the playwrights for the Tricycle Theatre of London's The Great Game: Afghanistan he was nominated for an Olivier Award. His works have been staged throughout the United States and in Germany, Canada, Australia, and Israel. He is a Guggenheim fellow and has received three NYFA fellowships in playwriting. Rogers is a member of the Dramatists Guild, where he is a founding board member of the Dramatists Legal Defense Fund. He is an alum of New Dramatists and holds an honorary doctorate from his alma mater, the University of North Carolina School of the Arts.
Jasson Minadakis
Jasson Minadakis is in his 13th season as artistic director of Marin Theatre Company, where he has directed Shakespeare In Love, Thomas and Sally, Guards at the Taj, August: Osage County, The Invisible Hand, Anne Boleyn, The Convert, The Whale, Failure: A Love Story, the world premiere of Lasso of Truth, The Whipping Man (San Francisco Bay Area Critics Circle Awards for Best Production and Best Acting Ensemble), Waiting for Godot, Othello: the Moor of Venice, The Glass Menagerie, Edward Albee's Tiny Alice, the world premiere of Libby Appel's adaptation of Chekhov's Seagull, Happy Now?, Equivocation(SFBATCC Award, Best Director), the world premiere of Sunlight, Lydia, The Seafarer, Frankie and Johnny in the Clair de Lune, A Streetcar Named Desire, said Saïd, Love Song, and The Subject Tonight is Love. As artistic director of Actor's Express Theatre Company, he directed The Pillowman; Bug; The Love Song of J. Robert Oppenheimer; Echoes of Another Man;Killer Joe; Burn This; The Goat or, Who is Sylvia?; Blue/Orange; and Bel Canto. As producing artistic director of Cincinnati Shakespeare Festival, he directed Jesus Hopped the 'A' Train, Chagrin Falls (2002 Cincinnati Entertainment Award for Best Production), and numerous others, including 19 productions of Shakespeare. Regional credits include The Whipping Man at Virginia Stage Company, Cat on a Hot Tin Roof and Hamlet at Georgia Shakespeare, Copenhagen at Playhouse on the Square (2003 Ostrander Theatre Award for Best Dramatic Production), and Bedroom Farce at Wayside Theatre.
Betsy Norton
Betsy Norton most recently stage managed Shakespeare In Love after four seasons (and over 15 shows!) as Production Assistant for the company. Other works include: stage managing Mike Birbiglia: The New One (Berkeley Repertory Theatre), PAing Monsoon Wedding (Berkeley Repertory Theatre) and Amélie: A New Musical (Berkeley Repertory Theatre and Center Theatre Group), and stage managing for the MTC summer camps. Betsy is a proud new member of Actors' Equity.
Sean Fanning
Scenic Designer
Sean Fanning is a Southern California-based designer who previously did the scenic design for Thomas & Sally and My Mañana Comes at MTC. San Diego credits include Full Gallop, A Doll's House, Kingdom, Plaid Tidings (The Old Globe), Evita, Manifest Destinitis, The Oldest Boy, Everybody's Talkin', Honky, In the Heights (San Diego Repertory Theatre), On the Twentieth Century, Animal Crackers, Bad Jews, Seven Guitars, King Hedley II, Gypsy, The Whale, Sons of the Prophet (Cygnet Theatre Company), Silent Sky, Equivocation (Lamb's Players); Damn Yankees, West Side Story (San Diego Musical Theatre). He received the first Craig Noel award for Designer of the Year for his body of work in 2016, and is a resident artist of Cygnet Theatre. In 2016, he had the pleasure of designing the Old Globe/ SDPL's exhibit of Shakespeare's First Folio in honor of the 400th anniversary of Shakespeare's death. Sean holds an M.F.A. in Scene Design from San Diego State University. seanfanningdesigns.com
Fumiko Bielefeldt+
Fumiko Bielefeldt has designed costumes for MTC's productions of Straight White Men, Guards at the Taj, The Oldest Boy, The Convert, The Beauty Queen of Leenane, Othello, the Moor of Venice, Edward Albee's Tiny Alice, Equivocation, What the Butler Saw, Lovers & Executioners, Frozen, Displaced, Fugitive Kind, Indiscretions, and Candida. Other credits include Splendour, The Soldier's Tale, Miss Julie, Antigone, and The Persians at Aurora Theatre; Evie's Waltz, Territories, The God of Hell, and Fautus at Magic Theatre; The Bridges of Madison County, Rags, Cyrano, Emma, Fallen Angels, Sweeny Todd, Silent Sky, The Little Women,Being Earnest, The Secret Garden, Light in the Piazza, Caroline, or Change, M. Butterfly, and Pacific Overtures at TheatreWorks; The Understudy at San Jose Rep. Her designs also appeared at American Conservatory Theatre, California Shakespeare Festival, Cincinnati Playhouse, Kansas City Repertory Theatre, the Repertory Theatre of St. Louis among others. In Japan, she has designed for Tokyo Theatre of Children and Tokyo Shitamachi Festival. She has received many awards, including the Barbara Bladen Porter Award, San Francisco Bay Area Theatre Critics Circle, Dean Goodman Choice, Back stage West Garland and Drama-Logue awards.
York Kennedy
York Kennedy is a Lighting Designer for the Performing Arts and Architecture. His designs for MTC include The Invisible Hand, Failure: A Love Story, The Beauty Queen of Leenane, Waiting for Godot, the world premiere of Bellwether, In the Red & Brown Water (scenic and lighting designer), My Name is Asher Lev. His designs have been seen in theatres and attractions across America, Europe and Asia in companies including Arena Stage, Chicago Shakespeare, Hartford Stage, Sibiu International Arts Festival, Seattle Repertory, American Conservatory Theatre, Macedonian Arts Festival, Polish National Opera, Los Angeles Opera, The Alley Theatre, Dallas Theatre Center, Brooklyn Academy of Music and the Denver Center. He has designed over 45 productions for The Old Globe in San Diego, CA and spent six seasons as resident lighting designer for the company's Shakespeare Festival. In the dance world he has designed for Malashock Dance, Brian Webb and Tracey Rhodes. Architectural lighting designs include both national and international themed environments, theme park, residential, retail, restaurant and museum projects. Current projects include Moscow Dream Island (Russia), Motiongate (Dubai), Ocean Flower (China) and Rear Window (Broadway). He is a graduate of the California Institute for the Arts and the Yale School of Drama.
Chris Houston
Chris Houston is a pianist, composer, and sound designer. He has composed music and/or designed sound for over 30 productions at Marin Theatre Company, including The Legend of Georgia McBride, Guards at the Taj, The Invisible Hand; The Oldest Boy; The Convert; The Whale;Failure: A Love Story; August Wilson's Fences; Jacob Marley's A Christmas Carol; The Whipping Man; Waiting for Godot; It's a Wonderful Life: A Live Radio Play; Topdog/Underdog; Othello, the Moor of Venice; The Glass Menagerie; the world premiere of Bellwether; Seven Guitars; and In the Red and Brown Water. Locally, his designs and compositions have been featured at American Conservatory Theater, Aurora Theatre Company, SF Playhouse, Center REP, Magic Theatre, and the San Francisco Shakespeare Festival.
Sara Huddleston
For MTC, Sara has previously designed sound for Straight White Men, Shakespeare in Love, Miss Bennet: Christmas at Pemberley and Gem of the Ocean. Other favorite Bay Area sound design credits include: The Gangster of Love, Reel to Reel, Grandeur, Fool for Love, Dogeaters, And I And Silence, Every Five Minutes, Hir, Terminus, Se Llama Cristina, Any Given Day, Mrs. Whitney and Goldfish (Magic Theatre); Octopus (Magic/Encore Theatre Company); I Call My Brothers, 410 [Gone] and Invasion! (Crowded Fire), Autobiography of a Terrorist (Golden Thread Productions); In On It and T.I.C (Encore Theatre Company); The Shaker Chair (Encore Theatre Company/Shotgun Players); Kiss and Macbeth (Shotgun Players) and Word for Word's 25th Anniversary! Show (Word for Word). Next up: The Resting Place (Magic Theatre) and We Swim, We Talk, We Go to War (Golden Thread Productions). she/her/hers
Danny Osburn
Assistant Lighting Designer
Danny Osburn is an electrician and lighting designer native to the East Bay. Prior to joining MTC, Danny spent seven years in Chicago. There he worked as the Master Electrician of Lifeline Theatre and of Adventure Stage Chicago, and as the Assistant ME for Northwestern University's Cherubs program for four summers. In addition, he has worked as a freelance electrician for companies ranging from Steppenwolf to the Museum of Contemporary Art. Danny cut his teeth as a professional electrician as an intern for MTC in 2010. He earned a BA in Theatre and Philosophy from Northwestern in 2011.
Dori Jacob
Dori Jacob joined Marin Theatre Company as the casting director in May 2015. For the previous four seasons, she served as the director of new play development for Magic Theatre in San Francisco, and dramaturged its world premieres of Octavio Solis' Se Llama Cristina, Linda McLean's Every Five Minutes, Christina Anderson's PEN/MAN/SHIP, and John Kolvenbach's Sister Play. As resident producer for Magic Theatre's developmental programming, Ms. Jacob's credits include 2011-2015 Virgin Play Series, the 2012 Asian Explosion Reading Series, and the 2013 Costume Shop Festival. Further Bay Area dramaturgy/producing/casting credits include: Assassins at Shotgun Players, Marilee Talkington's The Creative Process at SOMArts, Laura Schellhardt's The Comparables, and Elizabeth Hersh's Shelter in Place at Playwrights Foundation. Ms. Jacob previously served on the executive board and literary committee for the National New Play Network, is a current member of Literary Managers and Dramaturgs of the Americas, and is a graduate of U.C. Santa Cruz and N.Y.U.'s Tisch School of the Arts.
Laura A. Brueckner
Production Dramaturg
Laura A. Brueckner has been supporting productions and playwrights with her dramaturgical work for over 20 years, with an emphasis on digital dramaturgy, world premieres, and commissions. During this time, she has been proud to count among her collaborators stellar artists such as MTC Playwright in Residence Lauren Gunderson, Christopher Chen, Frances Ya-Chu Cowhig, Mina Morita, Marissa Wolf, Idris Goodwin, Lachlan Philpott, and Dominique Serrand, as well as groundbreaking companies Berkeley Repertory Theatre, Crowded Fire Theater, The New Harmony Project, Playwrights Foundation, and, now, Marin Theatre Company. As an artist, she is committed to theatre as a path of social action, critical inquiry, discovery, and delight. Her journalistic writing on artistic process and audience engagement has been published by HowlRound and Theatre Bay Area; her dramaturgical writing has been published by Berkeley Rep, California Shakespeare Theater, and Crowded Fire. A current member of Literary Managers and Dramaturgs of the Americas, she holds a B.A. in English dramatic literature (magna cum laude) from U.C. Berkeley and a Ph.D. in dramaturgy from U.C. San Diego.
Trevor Scott Floyd
Artistic Producer
Trevor (he/him/his) started as an Artistic Direction intern with MTC in 2015 and, prior to his current role as Artistic Producer, spent two seasons as Director of Ticketing, Artistic Associate, and Company Manager. As Artistic Producer, Trevor serves as MTC's local Casting Director. Originally from the beaches of South Carolina, he graduated with a dual major in Theatre and Political Science from Clemson University before trading in the Palmetto trees for the Redwoods. In addition to his role at MTC he is a freelance director and writer. You can read his latest work on the New Play Exchange.
Eran Kaplan
Cultural Consultant
Eran Kaplan is the Rhoda and Richard Goldman Chair in Israel Studies at SFSU. He received his B.A. (magna cum laude) from Tel Aviv University and his PhD in Modern Jewish History from Brandeis University. Before coming to San Francisco, he taught at Princeton, Cincinnati and Toronto. He is the author of The Jewish Radical Right: Revisionist Zionism and its Ideological Legacy and of The Origins Of Israel: A Documentary History with Derek Penslar (both published by the University of Wisconsin Press). Most recently, he authored Beyond Post-Zionism, published in 2016 with SUNY Press. In addition to his scholarly publications, he contributed articles to Haaretz and Tikkun. At SFSU, Professor Kaplan teaches courses on Modern Israel, the Arab-Israeli Conflict, Israeli Cinema, Modern Hebrew Culture and on the History of Jerusalem.
Omar Dajani
Professor Omar Dajani, co-director of the law school's Global Center for Business & Development, is recognized as a leading expert on legal aspects of the Palestinian-Israeli conflict. In 1999, Professor Dajani was recruited to serve as a legal adviser to the Palestinian negotiating team in peace talks with Israel, ultimately participating in the summits at Camp David and Taba. He then joined the office of the United Nations Special Coordinator for the Middle East Peace Process (UNSCO), where he worked on peacebuilding initiatives and played a lead role in marshaling and organizing international efforts to support Palestinian legal and political reforms. Professor Dajani has continued since that time to work as a consultant on a variety of legal infrastructure development and conflict resolution projects in the Middle East and elsewhere – for institutions including the U.S. Department of State, the Norwegian Peacebuilding Resource Center (NOREF), and the Center for Humanitarian Dialogue.
Jessica Berman
Dialect Coach
Jessica Berman is a dialect, voice, and text coach. With Marin Theatre Company, she has coached Shakespeare in Love, Thomas and Sally, The Legend of Georgia McBride, Native Son, Miss Bennet: Christmas at Pemberley, and August: Osage County. Other recent dialect and vocal coaching credits include: Angels in America: Parts One and Two, What the Constitution Means to Me, Monsoon Wedding, An Octoroon, Watch on the Rhine, and Hand to God (Berkeley Rep), An American in Paris (North American Tour, 2017), The Eva Trilogy, The Baltimore Waltz, Sojourners, and runboyrun, (Magic Theatre), and The War of the Roses and Fences (Cal Shakes). Jessica currently teaches at U.C. Berkeley, A.C.T. (STC and SF Semester), and Academy of Art University. She holds an M.A. from the Birmingham School of Acting, and an M.F.A. from the Royal Central School of Speech and Drama.
* Denotes member of Actors Equity Association
+ Member, United Scenic Artists
^ Member, Society of Stage Directors and Choreographers
Israel-Palestine conflict goes mano-a-mano in Marin Theatre Company's 'Oslo'
For the three hours of "Oslo," the stakes remain unchanged. Accomplish an impossible task, somehow, magically, or all the secret negotiations leading to the 1993 Oslo Accords will fall apart, sending the Israel-Palestinian peace process back to square one, or making it even worse off than it was before.
At the same time, the stakes of J.T. Rogers' Tony-winning play are also constantly changing, because each impossible task only breeds another one. Find someone who can credibly represent Israel when its laws bar its officials from meeting with the Palestine Liberation Organization. Get two men from opposite sides to walk into the negotiation room when one can't even make an innocuous remark about the weather without dredging up decades-old wrongs. Conceal the whole process from top Norwegian officials who would think it was crazy, and persuade everyone who needs to know about it to risk losing their jobs.
The show's Marin Theatre Company West Coast premiere, which opened Tuesday, Oct. 3, is a master class in the procedural thriller. In "Oslo," the Oslo Accords are never an ungainly, endlessly complicated operation, spanning many months and countless names, as they are in the historical record. Rather, the show sticks ruthlessly to the single, intimate, human interaction that matters in the present moment: de-escalating a blowhard, convincing a skeptic, deflecting an intruder, soothing a wounded ego.
The mano-a-mano also drives the diplomatic philosophy of Terje (Mark Anderson Phillips), the Norwegian behind the accords, even though he's not even a government official. But his wife, Mona (Erica Sullivan), is, and Terje (pronounced Tie-yuh), has little compunction — at times, too little — about combining her savvy and resources with his ambition and moxie as well as the altruism they share. His process "is rooted not in the organizational but in the personal," he says. That means adversaries work behind closed doors without a facilitator; it means they drink together afterward and learn about each other's families.
Rogers defines his array of characters as crisply as his situations, with single lines so finely chiseled they etch out the entire lives behind them. The cast, directed by Jasson Minadakis, bring aching, tremulous humanity to the text. When Israeli Professor Yair Hirschfeld (Brian Herndon) greets PLO Finance Minister Ahmed Qurie (J. Paul Nicholas) — as the first Israeli Ahmed has ever seen in the flesh — Herndon offers each phrase as if it's an infant he's leaving at the mercy of a deity. Nicholas gives Ahmed an exquisite, I-knew-that-already cool, but one whose facade cracks at opportunities for righteous indignation or sentimental antics.
Norwegian mediators Mona Juul (Erica Sullivan, left) and husband Terje Rød-Larsen (Mark Anderson Phillips) speak with Israel and the PLO.
Photo: Kevin Berne, Marin Theatre Company
As Mona, Sullivan has to act the part of a saint for most of the play, but even without the character flaws that make roles so juicy, she finds dynamism as the play's moral center, her eyes fathomless wells of both terror and resignation as she stands in judgment of everyone around her and of herself.
Corey Fischer effortlessly commands as Shimon Peres, taking hold of a room as if it all fit in the palm of his hand. Charles Shaw Robinson makes delectable the narcissism of Johan Jorgen Holst, Norway's foreign minister; when Mona and Terje stroke his ego, he accepts the tribute as if it's the most natural and obvious thing in the world. As Uri Savir, the first Israeli official with whom the Palestinians negotiate, Paris Hunter Paul brings bombast worthy of an action movie. When he shuts the door on the Norwegians, he narrows his eyes as if to say, "No more Mr. Nice Guy."
"Oslo" doesn't exaggerate the positive impact of the Oslo Accords. Rogers is careful to point out the violence that erupted almost immediately after the treaty was signed. But nor does the show wholly despair of peace, even as the promise of the accords remains unfulfilled 25 years later. It is clear and unsentimental, balanced and methodical. Yet it also slyly asserts that our best shot at peace rests on the opposite — on humankind's eternal, heedless optimism and ability to connect with one another, no matter what the cold, hard facts say.
— Lily Janiak, San Francisco Chronicle Read full review
Stressed are the peacemakers in gripping 'Oslo' at MTC
They often say you wouldn't want to see sausage being made, and the same could be said of diplomacy. Peacemaking can be a messy, messy business. It is, however, extremely entertaining to watch in "Oslo," the Tony Award-winning 2016 play by J.T. Rogers now making its West Coast premiere at Marin Theatre Company.
"Oslo" is about the top-secret back-channel negotiations in Norway between Israeli and Palestine Liberation Organization representatives that led to the 1993 Oslo Peace Accords, which didn't resolve the long-standing conflict but was a landmark agreement hammered out between the PLO and the Israeli government.
Even the initiation of the negotiations itself is portrayed in Rogers' fictionalized telling as a rogue operation by one Norwegian married couple, sociologist Terje Rød-Larsen (Mark Anderson Phillips, amusingly brash, egotistical and charismatic) and diplomat Mona Juul (coolly composed Erica Sullivan). On their own initiative they get the separate parties beginning to talk about talking and provide a safe and remote space to simply get people in a room together to try to find common ground.
It's fascinating to watch the Palestinian and Israeli representatives quickly go from nervous semi-hostility to jovial camaraderie as they negotiate and socialize together in MTC artistic director Jasson Minadakis' tense and often very funny production. It helps that we mostly see them outside the meeting room, unwinding together in the spacious sitting room of Sean Fanning's spare and versatile set.
J. Paul Nicholas is charming but prickly as the PLO finance minister, meeting first with Brian Herndon as a nervously friendly professor of economics. They're soon joined by Ashkon Davaran as a scowling Palestinian communist and Ryan Tasker as another, boyishly timid Israeli professor. Pair Hunter Paul is boorish at first as the first Israeli official in the room, and Peter James Meyers is bullheaded and overbearing as Israeli legal adviser Joel Singer.
Charles Shaw Robinson is imperious and volatile as the Norwegian foreign minister, who has to be carefully eased into the loop. Former Traveling Jewish Theatre artistic director Aaron Davidman is cagey and low-key as the Israeli deputy foreign minister who keeps his distance, and TJT cofounder and longtime Marin resident Corey Fischer exudes amiability and tremendous presence as foreign minister Shimon Peres. Joe Estlack and Adam Niemann are amusingly stony-faced as a pair of security guys.
The excellent cast of 14 is made up of a whole lot of men in suits, and Rogers seems to make an effort to give women a presence in the play with limited success. Mill Valley's Marcia Pizzo deftly plays an array of women from a dubious colleague of Terje's to an officious Swedish hostess, but the closest any of them come to playing a real role in the story is the kindly housekeeper whose waffles everybody rhapsodizes about.
More curiously, everybody talks up how essential Mona is to the process, but we're never shown that in the play. It's her "beautiful, powerful Rolodex" that makes setting the meetings up possible, and she occasionally serves as narrator just to explain who some people are in fill-in-the-blanks asides to the audience. In terms of what we actually see in scenes, however, the negotiations seem to be entirely Terje's show.
When Norway's deputy foreign minister (Tasker, agitated and enthusiastic) says he's going to need to drop out of the proceedings, it's confusing because as far as we've seen he hasn't been doing anything besides being one of the very few people who knows this is going on. But that's true of Mona as well. We mainly see her talking her husband down and cautioning him when he's going too far. Although we're often reminded that she's the one with the government career on the line, she's cast here in the oft-seen role of the long-suffering wife who supports her husband's big dreams.
There are occasional moments where the playwright shows his hand a little too conspicuously, especially in the final monologue, but on the whole it's a smartly constructed and thoroughly engaging piece that gives a real sense both of the sweep of history and the small, personal moments that somehow kick it into motion.
— Sam Hurwitt, Marin IJ Read full review
MTC Articles
QR Code Gallery - Full list of accessible articles
WATCH: Playwright J.T Rogers discusses his play, prior to winning the 2017 Tony Award for Best Play! Interview at 692 Broadway in NYC for BUILD Series.
Politico Magazine interview with J.T. Rogers: How a Play About Mideast Peace Became A Trump-Era Sensation
NY Times Review & Critic's Pick: A Byzantine Path to Middle East Peace in 'Oslo' (2016)
WATCH: 'The Oslo Diaries' Sundance Clip: How To Make A Secret Accord
Hours: Until further notice, the physical Box Office at 397 Miller Avenue is operating remotely (no in-person sales) Tues-Fri 12-5pm; and on performance days, 12pm-Intermission (actual hours may vary). On performance days during a performance, the box office will be staffed for in-person sales.
Email: boxoffice@marintheatre.org
Marin Theatre Company
397 Miller Avenue
Mill Valley, CA 94941-2885 | Directions
Email: info@marintheatre.org
Casting inquiries: casting@marintheatre.org
MTC is a 501(c)3 nonprofit organization. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,771 |
package io.protostuff.generator.html.markdown;
import org.assertj.core.api.Assertions;
import org.junit.jupiter.api.BeforeEach;
import org.junit.jupiter.api.DisplayName;
import org.junit.jupiter.api.Test;
import static org.assertj.core.api.Assertions.assertThat;
import static org.junit.jupiter.api.Assertions.*;
class FlexmarkMarkdownProcessorTest {
FlexmarkMarkdownProcessor processor;
@BeforeEach
void setUp() {
processor = new FlexmarkMarkdownProcessor();
}
@Test
@DisplayName("Simple text")
void test() {
assertThat(processor.toHtml("Hello, world"))
.isEqualTo("<p>Hello, world</p>");
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,090 |
/* ###
* IP: GHIDRA
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package ghidra.trace.database.symbol;
import java.io.IOException;
import java.util.*;
import org.apache.commons.lang3.StringUtils;
import org.apache.commons.lang3.tuple.ImmutablePair;
import org.apache.commons.lang3.tuple.Pair;
import com.google.common.collect.Collections2;
import db.DBRecord;
import generic.CatenatedCollection;
import ghidra.program.model.address.*;
import ghidra.program.model.listing.CircularDependencyException;
import ghidra.program.model.listing.Program;
import ghidra.program.model.symbol.*;
import ghidra.program.util.ProgramLocation;
import ghidra.trace.database.DBTrace;
import ghidra.trace.database.address.DBTraceOverlaySpaceAdapter;
import ghidra.trace.database.address.DBTraceOverlaySpaceAdapter.DecodesAddresses;
import ghidra.trace.database.program.DBTraceProgramView;
import ghidra.trace.database.symbol.DBTraceSymbolManager.DBTraceSymbolIDEntry;
import ghidra.trace.database.symbol.DBTraceSymbolManager.MySymbolTypes;
import ghidra.trace.model.Lifespan;
import ghidra.trace.model.Trace.TraceSymbolChangeType;
import ghidra.trace.model.TraceAddressSnapRange;
import ghidra.trace.model.symbol.TraceSymbol;
import ghidra.trace.model.thread.TraceThread;
import ghidra.trace.util.TraceAddressSpace;
import ghidra.trace.util.TraceChangeRecord;
import ghidra.util.LockHold;
import ghidra.util.database.*;
import ghidra.util.database.annot.DBAnnotatedColumn;
import ghidra.util.database.annot.DBAnnotatedField;
import ghidra.util.exception.DuplicateNameException;
import ghidra.util.exception.InvalidInputException;
import ghidra.util.task.TaskMonitor;
public abstract class AbstractDBTraceSymbol extends DBAnnotatedObject
implements TraceSymbol, DecodesAddresses {
private static final byte SOURCE_MASK = 0x0F;
private static final int SOURCE_SHIFT = 0;
private static final byte SOURCE_CLEAR = ~(SOURCE_MASK << SOURCE_SHIFT);
private static final byte PRIMARY_MASK = 0x10;
@SuppressWarnings("unused")
private static final int PRIMARY_CLEAR = ~PRIMARY_MASK;
static final String NAME_COLUMN_NAME = "Name";
static final String PARENT_COLUMN_NAME = "Parent";
static final String FLAGS_COLUMN_NAME = "Flags";
@DBAnnotatedColumn(NAME_COLUMN_NAME)
static DBObjectColumn NAME_COLUMN;
@DBAnnotatedColumn(PARENT_COLUMN_NAME)
static DBObjectColumn PARENT_COLUMN;
@DBAnnotatedColumn(FLAGS_COLUMN_NAME)
static DBObjectColumn FLAGS_COLUMN;
@DBAnnotatedField(column = NAME_COLUMN_NAME, indexed = true)
String name;
@DBAnnotatedField(column = PARENT_COLUMN_NAME, indexed = true)
long parentID;
@DBAnnotatedField(column = FLAGS_COLUMN_NAME)
byte flags;
protected DBTraceNamespaceSymbol parent;
protected final DBTraceSymbolManager manager;
public AbstractDBTraceSymbol(DBTraceSymbolManager manager, DBCachedObjectStore<?> store,
DBRecord record) {
super(store, record);
this.manager = manager;
}
/**
* {@inheritDoc}
*
* NOTE: If the IDs match, then the symbols are considered equal, regardless of their other
* attributes. This mechanic seems required to support the whole "placeholder" idea. See
* {@link SymbolTable#createSymbolPlaceholder(Address, long)}.
*/
@Override
public boolean equals(Object obj) {
if (!(obj instanceof AbstractDBTraceSymbol)) {
return false;
}
if (obj == this) {
return true;
}
AbstractDBTraceSymbol that = (AbstractDBTraceSymbol) obj;
if (this.getID() == that.getID()) {
return true;
}
if (this.getSymbolType() != that.getSymbolType()) {
return false;
}
if (!this.getName().equals(that.getName())) {
return false;
}
if (!this.getAddress().equals(that.getAddress())) {
return false;
}
if (!Objects.equals(this.getParentSymbol(), that.getParentSymbol())) {
return false;
}
return true;
}
@Override
public int hashCode() {
return Long.hashCode(getID());
}
@Override
public String toString() {
return name;
}
protected void assertNotGlobal() {
if (isGlobal()) {
throw new UnsupportedOperationException("Cannot modify the global namespace");
}
}
protected DBTraceNamespaceSymbol assertIsNamespace(AbstractDBTraceSymbol symbol) {
assert symbol != null;
if (!(symbol instanceof DBTraceNamespaceSymbol)) {
throw new AssertionError(
"Trace database corrupted. Symbol has a non-namespace parent.");
}
return (DBTraceNamespaceSymbol) symbol;
}
@Override
protected void fresh(boolean created) throws IOException {
if (created) {
return;
}
parent = parentID == -1 ? null : assertIsNamespace(manager.getSymbolByID(parentID));
}
@Override
public DBTraceOverlaySpaceAdapter getOverlaySpaceAdapter() {
return manager.overlayAdapter;
}
@Override
public DBTrace getTrace() {
return manager.trace;
}
@Override
public TraceThread getThread() {
return null;
}
protected TraceAddressSpace getSpace() {
return null;
}
@Override
public long getID() {
if (isGlobal()) {
return GlobalNamespace.GLOBAL_NAMESPACE_ID;
}
return DBTraceSymbolManager.packID(getSymbolType().getID(), getKey());
}
@Override
public String getName() {
return name;
}
@Override
public Address getAddress() {
return SpecialAddress.NO_ADDRESS;
}
protected Collection<? extends TraceAddressSnapRange> getRanges() {
return new CatenatedCollection<>(Collections2.transform(manager.idMap.getActiveSpaces(),
space -> Collections2.transform(
space.getUserIndex(long.class, DBTraceSymbolIDEntry.ID_COLUMN).get(getID()),
ent -> ent.getShape())));
}
// Internal
public Lifespan getLifespan() {
// TODO: Cache this computation and/or keep it as transient fields?
long min = Long.MAX_VALUE;
long max = Long.MIN_VALUE;
for (TraceAddressSnapRange range : getRanges()) {
min = Math.min(min, range.getLifespan().lmin());
max = Math.min(max, range.getLifespan().lmax());
}
if (min > max) {
return null;
}
return Lifespan.span(min, max);
}
protected void doCollectAddressSet(AddressSet set) {
for (TraceAddressSnapRange range : getRanges()) {
set.add(range.getRange());
}
}
// Internal
public AddressSet getAddressSet() {
AddressSet result = new AddressSet();
doCollectAddressSet(result);
return result;
}
@Override
public String[] getPath() {
try (LockHold hold = LockHold.lock(manager.lock.readLock())) {
checkIsValid();
if (isGlobal()) {
return new String[] { getName() };
}
ArrayList<String> list = new ArrayList<>();
if (parent != manager.globalNamespace) {
parent.doGetPath(list);
}
list.add(getName());
return list.toArray(new String[list.size()]);
}
}
@Override
public String getName(boolean includeNamespace) {
if (!includeNamespace) {
return getName();
}
return StringUtils.join(getPath(), "::");
}
@Override
public DBTraceNamespaceSymbol getParentNamespace() {
return parent;
}
@Override
public DBTraceNamespaceSymbol getParentSymbol() {
return parent;
}
@Override
public boolean isDescendant(Namespace namespace) {
for (AbstractDBTraceSymbol s = this; s != null; s = s.parent) {
if (s == namespace) {
return true;
}
}
return false;
}
@Override
public Collection<? extends DBTraceReference> getReferenceCollection() {
return manager.trace.getReferenceManager().getReferencesBySymbolId(getID());
}
@Override
public int getReferenceCount() {
return getReferenceCollection().size();
}
@Override
public boolean hasMultipleReferences() {
// TODO: Could be slightly more efficient by just iterating twice?
return getReferenceCount() > 1;
}
@Override
public boolean hasReferences() {
return !getReferenceCollection().isEmpty();
}
@Override
public DBTraceReference[] getReferences(TaskMonitor monitor) {
Collection<? extends DBTraceReference> refs = getReferenceCollection();
// NOTE: Size computation is just iteration over address spaces. Should be snappy.
DBTraceReference[] result = new DBTraceReference[refs.size()];
int i = 0;
for (DBTraceReference r : refs) {
result[i++] = r;
if (monitor.isCancelled()) {
break;
}
}
return result;
}
@Override
public DBTraceReference[] getReferences() {
return getReferences(TaskMonitor.DUMMY);
}
@SuppressWarnings("hiding")
void rawSet(String name, long parentID) {
this.name = name;
this.parentID = parentID;
update(NAME_COLUMN, PARENT_COLUMN);
}
protected void set(String name, DBTraceNamespaceSymbol parent, SourceType source) {
this.name = name;
this.parentID = parent.getID();
doSetSource(source);
update(NAME_COLUMN, PARENT_COLUMN, FLAGS_COLUMN);
this.parent = parent;
}
protected TraceChangeRecord<?, ?> doSetNameWithEvent(String newName)
throws InvalidInputException {
String oldName = name;
if (oldName.equals(newName)) {
return null;
}
this.name = newName;
return new TraceChangeRecord<>(TraceSymbolChangeType.RENAMED, getSpace(), this, oldName,
newName);
}
/**
* Checks and sets the parent
*
* The caller must still call {@link #update(DBObjectColumn...)} for {@link #PARENT_COLUMN}.
*
* @param newParent the parent namespace
* @throws CircularDependencyException
*/
protected TraceChangeRecord<?, ?> doSetParent(DBTraceNamespaceSymbol newParent)
throws CircularDependencyException {
DBTraceNamespaceSymbol oldParent = parent;
if (oldParent == newParent) {
return null;
}
if (!isValidParent(newParent)) {
throw new IllegalArgumentException(
"This symbol type cannot be a child of the given namespace type");
}
DBTraceNamespaceSymbol checkedParent = checkCircular(newParent);
this.parent = checkedParent;
this.parentID = parent.getID();
return new TraceChangeRecord<>(TraceSymbolChangeType.PARENT_CHANGED, getSpace(), this,
oldParent, checkedParent);
}
protected void doSetSource(SourceType newSource) {
flags =
(byte) ((flags & SOURCE_CLEAR) | (newSource.ordinal() & SOURCE_MASK) << SOURCE_SHIFT);
}
/**
* Sets the flags for the given source.
*
* The caller must still call {@link #update(DBObjectColumn...)} for {@link #FLAGS_COLUMN}. The
* update should be called before the returned event, if applicable, is fired.
*
* @param newSource the source type
* @return the appropriate change event, if a change was actually made
*/
protected TraceChangeRecord<?, ?> doSetSourceWithEvent(SourceType newSource) {
SourceType oldSource = getSource();
if (oldSource == newSource) {
return null;
}
doSetSource(newSource);
return new TraceChangeRecord<>(TraceSymbolChangeType.SOURCE_CHANGED, getSpace(), this,
oldSource, newSource);
}
@Override
public boolean isValidParent(Namespace ns) {
DBTraceNamespaceSymbol dbns = manager.checkIsMine(ns);
if (dbns == null) {
return false;
}
return MySymbolTypes.values()[this.getSymbolType().getID()].isValidParent(dbns);
}
protected DBTraceNamespaceSymbol checkCircular(DBTraceNamespaceSymbol newParent)
throws CircularDependencyException {
return newParent;
}
protected Pair<String, SourceType> validateNameAndSource(String newName, SourceType newSource)
throws InvalidInputException {
if ((newSource == SourceType.DEFAULT) ^ (getSource() == SourceType.DEFAULT)) {
throw new IllegalArgumentException("Cannot create or remove DEFAULT symbols");
}
DBTraceSymbolManager.assertValidName(newName);
return new ImmutablePair<>(newName, newSource);
}
@Override
public void setName(String newName, SourceType newSource)
throws DuplicateNameException, InvalidInputException {
assertNotGlobal();
Pair<String, SourceType> validated = validateNameAndSource(newName, newSource);
newName = validated.getLeft();
newSource = validated.getRight();
try (LockHold hold = LockHold.lock(manager.lock.writeLock())) {
TraceChangeRecord<?, ?> nameEvent = doSetNameWithEvent(newName);
TraceChangeRecord<?, ?> sourceEvent = doSetSourceWithEvent(newSource);
if (nameEvent != null || sourceEvent != null) {
update(NAME_COLUMN, FLAGS_COLUMN);
}
if (nameEvent != null) {
manager.trace.setChanged(nameEvent);
}
if (sourceEvent != null) {
manager.trace.setChanged(sourceEvent);
}
}
}
protected void validateNameAndParent(String newName, DBTraceNamespaceSymbol newParent)
throws DuplicateNameException {
manager.assertNotDuplicate(this, newName, newParent);
}
@Override
public void setNamespace(Namespace newNamespace)
throws DuplicateNameException, InvalidInputException, CircularDependencyException {
// TODO: Why InvalidInputException?
assertNotGlobal();
try (LockHold hold = LockHold.lock(manager.lock.writeLock())) {
DBTraceNamespaceSymbol dbnsParent = manager.assertIsMine(newNamespace);
validateNameAndParent(getName(), dbnsParent);
TraceChangeRecord<?, ?> parentEvent = doSetParent(dbnsParent);
if (parentEvent != null) {
update(PARENT_COLUMN);
manager.trace.setChanged(parentEvent);
}
}
}
@Override
public void setNameAndNamespace(String newName, Namespace newNamespace, SourceType newSource)
throws DuplicateNameException, InvalidInputException, CircularDependencyException {
assertNotGlobal();
Pair<String, SourceType> validated = validateNameAndSource(newName, newSource);
newName = validated.getLeft();
newSource = validated.getRight();
try (LockHold hold = LockHold.lock(manager.lock.writeLock())) {
TraceChangeRecord<?, ?> parentEvent = doSetParent(manager.assertIsMine(newNamespace));
TraceChangeRecord<?, ?> nameEvent = doSetNameWithEvent(newName);
TraceChangeRecord<?, ?> sourceEvent = doSetSourceWithEvent(newSource);
if (parentEvent != null || nameEvent != null || sourceEvent != null) {
update(NAME_COLUMN, PARENT_COLUMN, FLAGS_COLUMN);
}
if (parentEvent != null) {
manager.trace.setChanged(parentEvent);
}
if (nameEvent != null) {
manager.trace.setChanged(nameEvent);
}
if (sourceEvent != null) {
manager.trace.setChanged(sourceEvent);
}
}
}
@Override
public void setSource(SourceType newSource) {
assertNotGlobal();
try {
Pair<String, SourceType> validated = validateNameAndSource(getName(), newSource);
newSource = validated.getRight();
}
catch (InvalidInputException e) {
throw new AssertionError(e);
}
try (LockHold hold = LockHold.lock(manager.lock.writeLock())) {
TraceChangeRecord<?, ?> sourceEvent = doSetSourceWithEvent(newSource);
if (sourceEvent != null) {
update(FLAGS_COLUMN);
manager.trace.setChanged(sourceEvent);
}
}
}
@Override
public SourceType getSource() {
assertNotGlobal();
return SourceType.values()[(flags >> SOURCE_SHIFT) & SOURCE_MASK];
}
@Override
public boolean delete() {
assertNotGlobal();
try (LockHold hold = LockHold.lock(manager.lock.writeLock())) {
return doDelete();
}
}
protected boolean doDelete() {
return manager.doDeleteSymbol(this);
}
@Override
public boolean isDynamic() {
return false;
}
@Override
public boolean isGlobal() {
return parentID == -1;
}
@Override
public DBTraceProgramView getProgram() {
return manager.trace.getProgramView();
}
@Override
public ProgramLocation getProgramLocation() {
return new ProgramLocation(getProgram(), getAddress());
}
/**
* {@inheritDoc}
*
* Since blocks cannot be relocated as they can in a {@link Program}, it's tempting to say all
* symbols are pinned; however, this presents in the UI and is a bit confusing and/or
* distracting.
*/
@Override
public boolean isPinned() {
return false;
}
@Override
public void setPinned(boolean pinned) {
// Nothing
}
@Override
public boolean isExternal() {
return false;
}
@Override
public boolean isExternalEntryPoint() {
return false;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 965 |
The 203-249-XXXX Phone Number is located in our around Fairfield County, CT and other near by areas.
Sick of having those telphone calls which you won't take because you do not recognize the phone number? This site provides a FREE Phone Search on any number if it is private or not. You now have the tools to locate any number call you receive, obtain data related the owner of that number, and find out if it was a phone call you needed to take, or if it was just another phone spammer trying to waste your time. Simply insert the incoming number into the search bar, and find out exactly who is calling you. Answering your phone has never been so stress free. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,973 |
Q: how to check if office document excel is readonly office.js addin i want to know if there is any API/function which can be used to detect if office document is read-only. I am creating an office add-in using office.js library. the document is shared to other user in readonly mode. if i use following property, it shows readWrite but actually document is in readonly mode. following property is not helping
var docMode = Office.context.document.mode; // did not work, it gives value of manifest file i guess
I am trying following thing, it worked but not sure if that is the right approach. if the file is in one drive it give url as http://d.doc.onedrive..something like this.
if (Office.context.document.url.indexOf("https://excel.officeapps.live.com") > -1) {// its in read-only mode-this is the unique folder name where the file is
$("#container").show();
$("#readOnlyArea").show();
} else {
$("#mainArea").show();}
A: It is a really good question. Since Excel has the very interesting scratch mode, basically even though Excel is in read only mode. Both users and add-ins are still able to write to the workbook. Therefore this API always returns ReadWrite meaning the add-in is able to write. The current behavior is by design. However, these changes can't be saved for read only files. We are aware this is a gap and we have added it to our backlog. Thanks for raising this issue.
Thanks,
Sky
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,381 |
\section{Introduction}
Geometric phase originates due to cyclic time evolution of an
Hamiltonian. This phase can be distinguished from the usual
dynamical phase by its dependence on the quantum level structure of
the system and the form of the time evolution of the involved
Hamiltonian. Although geometric phase depends on the quantum level
structure, it is independent of the energy eigenvalues. Therefore it
is largely conceived as immune to external perturbations and hence a
good quantum computation resource
\cite{Zanardi99},\cite{Ekert00},\cite{Duan01},\cite{Shi03},\cite{Cosmo09}.
Any quantum computation at its basic level requires quantum gates
for both single and two qubits. In order to perform reliably as a
component of a quantum computation system, these quantum gates needs
to have an error generation probability of $10^{-4}$, which is known
as fault tolerant threshold (FTT)\cite{Knill10}. So far the only
single qubit operation that has shown such kind of FTT is based on
an ion trap experiment \cite{Brown11} . Lemmer {\it et. al}
\cite{Lemmer13} proposed a protocol based on spin phonon interaction
which has the potential to beat the FTT. However, their analysis is
specific to the proposed qubit system. Geometric phase, being
inherently robust to external field fluctuations, provide a likely
resource which can be exploited to create quantum gates having such
FTT.\\
In general, the robustness of geometric phase is an inherent
property of the mechanics itself. Therefore understanding the origin
of such robustness in the context of geometric phase in spin system
can allow more controllability on such geometric phase gates. Any
universal quantum computation will need single and two qubit gates
which are both Abelian and non-Abelian in nature\cite{Zanardi99}. In
most cases, studies on geometric phases are restricted to either of
these two regimes
\cite{Jacob07},\cite{Vaya11},\cite{Wang12},\cite{Jonas09}. Here we
put forward a study in which we can go from Abelian to non-Abelian
regime in the same spin system by slowly breaking the symmetry of
the system. Analysis of such a system not only provides insights
into the underlying physical processes governing each of the limits
but also effectively probes the limits of the adiabatic theorem
\cite{Marz04},\cite{Bould08} and the relation of adiabaticity and
non-abelian behaviour. This is relevant as most of the quantum
states are operated in the super-adiabatic regime \cite{Zu14}. Our
analysis also provides room for identifying the relevant parameters
which influence the phase fluctuation in different regimes.
\section{Theory}
In \cite {Berry84}, Berry formulated the form of geometric phase
under adiabatic approximation as
\begin{equation}
\mathcal{G}_{n}=i\oint\bra{\psi_{n}}\nabla_{r}\ket{\psi_{n}}dR
\end{equation}
for the $n$th eigenstate. The quantity
$\gamma_{n}=\bra{\psi_{n}}\nabla_{r}\ket{\psi_{n}}$ is called the
gauge of the evolution because it remains invariant under any
similarity transformation, except those involving the variable of
the evolution themselves. This definition of the 'scalar' gauge
holds only for non-degenerate levels. For degenerate levels, the
definition is generalized to a matrix gauge
$\gamma_{mn}=i\bra{\psi_{m}}\nabla_{r}\ket{\psi_{n}}$, where $m$ and
$n$
belong to the degenerate subspace \cite{Wilc84}.\\
The adiabatic form of geometric phase can also be derived from the
adiabatic theorem. The probability amplitude of an eigenstate
belonging to a time dependent Hamiltonian varies as
\begin{equation}
\dot{C_{m}}=-C_{m}\braket{\psi_{m}}{\dot{\psi_{m}}}-\sum_{n\neq m}
C_{n}\frac{\bra{\psi_{m}}\dot{H}\ket{\psi_{n}}}{E_{n}-E_{m}}~
e^{i(\xi_{n}-\xi_{m})},
\label{basic_eq}
\end{equation}
where the states $m$ and $n$ are non-degenerate and $\xi_{m}$ and
$\xi_{n}$ are the dynamical phases. The above equation governs the
time dependence of the amplitudes of the states, beyond the
dynamical contribution. Under adiabatic approximation,
$\bra{\psi_{m}}\dot{H}\ket{\psi_{n}} \ll (E_{n}-E_{m})$ and hence
the second term in Eq. \ref{basic_eq} can be neglected in comparison
to the first term. Thus for adiabatic evolution, there is no
'mixing' of the different eigenstates. However, as a consequence of
the time dependence of the Hamiltonian, there is an additional
phase, on top of the dynamical phase, governed by
$\braket{\psi}{\dot{\psi}}$. Under the conditions of implicit time
dependence, this term leads to the underlying gauge of geometric
phase as derived by Berry. In the following sections we expand the
ideas of Abelian and non-Abelian evolution and
their corresponding gauges.\\
{\bf Abelian:} Abelian evolution corresponds to evolution without
any population transfer. For a non-degenrate set of levels, under
adiabatic condition, all evolutions are Abelian, since adiabaticity
guarantees lack of population transfer or mixings. Degenerate levels
can also have Abelian evolutions, if the underlying gauge matrix
corresponding to the degenerate subspace is diagonal.\\
{\bf Non-Abelian:} On the contrary, non-Abelian evolutions
inherently introduces mixing of states or population transfer
between the states. For a degenerate subspace, if the off-diagonal
elements of the gauge matrix are nonzero, then the evolution is
considered as non-Abelian \cite{Wilc84}. However, degeneracy itself
doesn't guarantee non-zero off diagonal elements. A subspace of
non-degenerate levels, under certain evolutions can have non-zero
off diagonal elements. However, conditions imposed by adiabaticity
doesn't leave room for population transfer in such cases and the
off-diagonal elements have no physical significance under such
conditions. This can be easily demonstrated from Eq. \ref{basic_eq}
as the coupling term drops off because of finite strength of the
oscillatory function and the relatively smaller coupling strength in
the adiabatic limit.\\
{\bf Non-Abelian to Abelian Transition:} The primary goal of this
work is to see how a system responds if it is taken in a continuous
manner from non-Abelian to Abelian evolution and vice versa. To
achieve this, we need an evolution with degeneracies and focus on
non-Abelian degenerate subspaces. Now, if we can introduce
non-degeneracy into the system, without changing the underlying
geometry and hence the gauge matrix of the evolution, then we can
observe the physical significance of the off-diagonal elements
slowly diminishing and vanishing in the Abelian regime. Thus the
primary goal is to study the dynamics of the off-diagonal elements
with respect to the symmetry breaking field. This also allows us to
probe the Adiabatic theorem the limits of which, under the influence
of a symmetry breaking field has long been
debated\cite{Marz04},\cite{Bould08}\\.
\section{System}
The system we work with is a spin system interacting with
electro-magnetic fields. Even though we are interested in the
$D_{3/2}$ state of Ba$^{+}$ ion interacting with a rotating electric
field gradient \cite{Ddm13}, all the discussion made here holds true
for any spin $3/2$ system. We choose the electric field to be zero
to avoid monopolar interaction. The dipole moment of this state is
zero because of the definite parity of the states and hence we
choose the electric quadrupole moment. Electric quadrupole moment
interact with electric field gradient. To generate geometric phases,
the electric field gradient is taken to be time dependent. The time
dependence is such that the principle axes describe a conical path
about the degeneracy point as shown in Fig. 1. The spin $3/2$
interacting with the electric field gradient maintains a time
reversal symmetry and hence there are Kramer's degeneracies in the
system, i.e., $\ket{\pm1/2}$ and $\ket{\pm3/2}$ states form two
pairs of degenerate subspaces. The Hamiltonian of quadrupole
interaction has $S_{+}^2$ terms and hence couples the $\ket{\pm1/2}$
substates. Thus the $\ket{\pm1/2}$ subspace undergoes non-Abelian
evolution. The Hamiltonian however cannot couple $\ket{\pm3/2}$
states, which have a $\Delta m=3$ and thus cannot be coupled by the
quadratic terms in angular momentum operator. Hence the
$\ket{\pm3/2}$ subspace undergoes Abelian evolution. This however is
true only for the quadrupole approximation which limits the field
expansion to $S^{2}$ terms only.\\
To induce the non-Abelian to Abelian transition, we now apply a time
dependent magnetic field along with the electric field gradient. The
magnetic field lifts the degeneracy of the system and hence drives
the system away from non-Abelian behaviour. However, for a true
transition from non-Abelian to Abelian, the underlying gauge is
required to be the same in the presence of the magnetic field. This
is achieved by making the magnetic field rotate along with the
electric field gradient. This preserves the 'geometry' of the system
even in the presence of the magnetic field as the quantization axis
remains unchanged. 'Geometry' in this context implies the
transformation which connects the diagonal basis of the Hamiltonian
with the stationary basis. Because of this constancy of 'geometry',
the underlying gauge of the evolution remains invariant. In the
following sections a detailed theoretical description of the
system's evolution under the influence of the
time varying field will be derived.\\
\begin{figure}
\includegraphics[scale=.4]{non_ad_ab_trans.eps}\
\label{field_fig} \caption{Configuration of the applied electric
field gradient and magnetic field for inducing non-Abelian to
Abelian transition. In the adiabatic regime, for the $\ket{\pm 1/2}$
substates, absence of magnetic field is non-Abelian while non-zero
magnetic field gives rise to Abelian geometry.}
\end{figure}
{\bf Non-Abelian Regime:} The quadrupole moment of the spin $3/2$
system interacting with an electric field gradient gives rise to the
non-Abelian regime for the $\ket{\pm1/2}$ subspace. The Hamiltonian
for a quadrupole moment- electric field gradient interaction is
given as
\begin{equation}
H_{Q}=\frac{1}{6} Q_{ij}\frac{\partial E_{i}}{\partial x_{j}},
\end{equation}
where $Q_{ij}$ is the $ij$th component of the quadrupole moment and
is defined for spin systems as $Q_{ij}=c (\frac{1}{2}(S_i S_j + S_j
S_i) -\frac{1}{3} \overrightarrow{S}^2\delta_{ij})$. $\frac{\partial
E_{i}}{\partial x_{j}}$ is the $ij$the component of the electric
field gradient tensor.\\
In our case, because of a suitable choice of the principle axes,
only the $\frac{\partial E_{z}}{\partial z}$ component of the
electric field gradient tensor contributes to the Hamiltonian of the
system. Thus, in the non-Abelian scenario, we obtain an effective
Hamiltonian given by,
\begin{equation}
H_{NA}=c(S_{z^{'}}^2-\frac{1}{3}S^{2}),
\end{equation}
where
$S_{z^{'}}=S_{x}\sin{\theta}\cos{\phi}+S_{y}\sin{\theta}\sin{\phi}+S_{z}\cos{\theta}$,
$c$ being the strength of interaction and $\phi=\omega t$, $\omega$
being the rotational frequency of the electric field gradient.\\
The eigenstates of this Hamiltonian are doubly degenerate. The two
doubly degenerate subspaces consists of $\ket{\pm\frac{3}{2}}$
corresponding to eigenvalue $c$ and $\ket{\pm\frac{1}{2}}$
corresponding to eigenvalue $-c$. Now to obtain the gauge matrices
corresponding to these sets of states through the relation
$\gamma_{mn}=i\bra{\psi_{m}}\nabla_{r}\ket{\psi_{n}}$, the
wavefunctions in the stationary frame are required. However, for the
ease of calculation, we use the Wigner D matrices to obtain the
wavefunctions in the stationary basis from the rotating basis, which
is also the diagonal basis for the Hamiltonian. The Wigner D
matrices are the transformations which connect these two bases. The
respective gauge matrices are-
\[
\gamma_{\pm 3/2}=
\begin{pmatrix}
\frac{3}{2} \cos{\theta} & 0\\
0 & -\frac{3}{2} \cos{\theta}
\end{pmatrix}
\]
and
\[
\gamma_{\pm 1/2}=
\begin{pmatrix}
\frac{1}{2} \cos{\theta} & \sin{\theta}\\
\sin{\theta} & -\frac{1}{2} \cos{\theta}\\
\end{pmatrix}
\].
From the matrices, we can see that for the subspace
$\ket{\pm\frac{1}{2}}$, we have non-zero off-diagonal elements.
Hence the degenerate subspace $\ket{\pm\frac{1}{2}}$ follows
non-Abelian evolution. In the non-Abelian regime, the eigengauge is
given by the eigenvalues of the gauge matrix,
$\pm\frac{1}{2}\sqrt{4-3\cos^{2}{\theta}}$.
{\bf Abelian Regime:} To transfer the system from non-Abelian regime
to Abelian regime, we apply a degeneracy lifting magnetic field.
However, to make any comparison between the two situations, we
require the 'geometry' of the system to remain invariant in the
presence of the magnetic field. More precisely, the connection
between the diagonal basis and the stationary frame, which is the
Wigner D matrices, should remain the same in the presence or absence
of magnetic field.
The effective Hamiltonian in the Abelian regime is given by
\begin{equation}
H_{A}=c(S_{z^{'}}^2-\frac{1}{3}S^{2})-b S_{z^{'}}.
\end{equation}
The eigenvalue of the $\ket{\pm\frac{1}{2}}$ subspace now becomes
$-c \mp \frac{1}{2}b$. However, the gauge matrix does not change, as
the geometry is kept invariant.\\
In the Abelian configuration, the adiabatic theorem leads us to
conclude that the off diagonal elements of the gauge matrix
corresponding to these set of states, do not contribute in the
physical manifestation of the phase. In this regime, the underlying
gauge is simply $\pm\frac{1}{2}\cos{\theta}$ corresponding to the
two states.
{\bf Non-Abelian to Abelian Transition:} In the two regimes of
evolution, the states, $\ket{\pm1/2}$, are governed by two different
underlying gauges given by
$\pm\frac{1}{2}\sqrt{4-3\cos{\theta}^{2}}$ for non-Abelian and
$\pm\frac{1}{2}\cos{\theta}$ for Abelian. The physical manifestation
of the gauges is obtained through phase dependent energy shifts
given by $A_{n}\omega$. Thus the variation of the energy level
shifts, on top of the energy eigenvalue, while going from
non-Abelian to Abelian is the primary signature of such a
transition.
In the true adiabatic regime ($\omega \rightarrow 0$), even the
smallest value of $b$ will drive the system from non-Abelian to
Abelian. However, for finite values of $\omega$, the system is
governed by two timescales, one depends on $\omega$, the rotational
frequency of the fields and the other depends on $b$, which
determines the splitting between the $\ket{\pm1/2}$ states. For
$\omega$ finite, Abelian regime can only be achieved for
$b\gg\omega$.
\section{Dressed State Calculations}
Unlike the systems studied so far, the system we constructed above
allows us to move continuously between the non-Abelian and Abelian
regimes. In the previous section we saw that by moving the system
from non-Abelian to Abelian regime, the off-diagonal elements loses
their physical significance. Continuous tunability of our system
allows us to investigate the dynamics of the off diagonal elements
with respect to symmetry breaking field,
in this case the magnetic field.\\
For studying the dynamics of the off diagonal elements, we begin at
the basic equation Eq. (\ref{basic_eq}) governing the evolution of
the two states. Here we work with the $\ket{\pm1/2}$ subspace. We
assume that $\omega$ is small enough compared to $c$ so that we can
neglect the coupling of the $\ket{\pm1/2}$ subspace with
$\ket{\pm3/2}$ subspace. By plugging in the values of the variables,
the governing equation for the $\ket{\pm1/2}$ substates is obtained
as
\begin{equation}
\begin{pmatrix}
\dot{C_{1}}\\
\dot{C_{2}}
\end{pmatrix}
=i
\begin{pmatrix}
\frac{\omega}{2}\cos{\theta} & \omega\sin{\theta}~e^{ibt}\\
\omega\sin{\theta}~e^{-ibt} & -\frac{\omega}{2}\cos{\theta}
\end{pmatrix}
\begin{pmatrix}
C_{1}\\
C_{2}
\end{pmatrix}.
\label{a_eq}
\end{equation}
For $b=0$, this equation governs the behaviour in the non-Abelian
regime.\\
The matrix
\begin{equation}
\mathcal{H}=
\begin{pmatrix}
\frac{\omega}{2}\cos{\theta} & \omega\sin{\theta}~e^{ibt}\\
\omega\sin{\theta}~e^{-ibt} & -\frac{\omega}{2}\cos{\theta}
\end{pmatrix}
\label{eff_ham}
\end{equation}
is like an effective Hamiltonian governing the evolution of the
states $\ket{\pm\frac{1}{2}}$ for a given value of $\omega$ and
$b$.\\
{\bf Application of Dressed State Method:} The dressed state
approach, which is a derivative of the Floquet Theorem of
differential equations with periodic co-efficients, takes into
account the full time dependence and allows us to obtain the true
eigenvalues, considering all the effects of the time dependence.
Even though it was first developed to deal with atom photon
interaction, it can be generalized to any equation with periodic
co-efficients. For a detailed description of the mathematical
algorithm applied here to
obtain the eigenvalues of Eq. (\ref{eff_ham}), please refer to \cite{Meyer09}.\\
The eigenvalue obtained using the dressed state algorithm provides
the complete picture including the effect of phase dependent energy
shifts of the level. It also allows us to obtain the complete
dependence of the geometric phase on $b$ and $\omega$ and thus
letting us probe not only the non-Abelian and Abelian
limit but the entire behaviour of the system.\\
\begin{figure*}[t]
\subfigure[]{\includegraphics[scale=1.7]{3D_t_57.eps}}
\subfigure[]{\includegraphics[scale=.4]{w_1_t_57_3.eps}}
\subfigure[]{\includegraphics[scale=.4]{w_10_t_57_3.eps}}
\subfigure[]{\includegraphics[scale=.4]{w_100_t_57_3.eps}}
\label{delwn}
\caption{Figure demonstrates the non-Abelian to Abelian transition.
Figure (a) depicts the three dimensional dependence of the evolution
gauge on $b$ and $\omega$. Figures (b),(c) and (d) show the
behaviour of the system moving away from the non-Abelian point for
angular frequencies of $1$, $10$ and $100$ Hz. As can be seen, with
increase of rotational frequency, the transition of the system is
much more slow. The non-abelian behaviour is 'retained' for higher
values of magnetic field for a higher angular frequency. The figure
also demonstrates that by selecting appropriate values of $b$ and
$\omega$, we can bring control the behaviour of the system
precisely, with the phase acquired defined only by the pair of
values of $b$ and $\omega$. For these graphs, the value of $\theta$
is kept as $57.3^{o}$. Magnetic field $b$ and angular frequencies
$\omega$ are expressed in $Hz$.}
\end{figure*}
To apply the dressed state method, we assume an ansatz of the form
\begin{equation}
\begin{pmatrix}
\dot{C_{1}}\\
\dot{C_{2}}
\end{pmatrix}
=
\begin{pmatrix}
\alpha_{1}(t) e^{-i\omega_{+}t}\\
\alpha_{2}(t) e^{-i\omega_{-}t}
\end{pmatrix}
.
\label{alp_eq}
\end{equation}
Now by inserting Eq. \ref{alp_eq} into Eq. \ref{a_eq}, we obtain the
following equation for $\alpha_{1}$ and $\alpha_{2}$
\begin{widetext}
\begin{equation}
i
\begin{pmatrix}
\dot{\alpha_{1}}\\
\dot{\alpha_{2}}
\end{pmatrix}
=
\begin{pmatrix}
-\frac{\omega}{2}\cos{\theta}-\omega_{+} & -\omega\sin{\theta}~e^{i(b+\omega_{+}-\omega_{-})t}\\
-\omega\sin{\theta}~e^{i(b+\omega_{+}-\omega_{-})t} &
\frac{\omega}{2}\cos{\theta}-\omega_{-}
\end{pmatrix}
\begin{pmatrix}
\alpha_{1}\\
\alpha_{2}
\end{pmatrix}.
\label{aa_eq}
\end{equation}
\end{widetext}
Now if we choose $\omega_{\pm}=\mp\frac{b}{2}$, then the above
$2\times2$ matrix becomes time independent and all the information
about the time dependence of the system becomes encoded in behaviour
of $\alpha_{1}(t)$ and $\alpha_{2}(t)$. Such a choice of the values
of $\omega_{\pm}$ converts the above equation into a time
independent problem, with an effective Hamiltonian given by
\begin{equation}
\mathcal{H}_{D}=
\begin{pmatrix}
-\frac{\omega}{2}\cos{\theta}+\frac{b}{2} & -\omega\sin{\theta}\\
-\omega\sin{\theta} & \frac{\omega}{2}\cos{\theta}-\frac{b}{2}
\end{pmatrix}.
\label{dress_ham}
\end{equation}
The Hamiltonian in Eq. (\ref{dress_ham}) is the dressed form of the
effective Hamiltonian given by Eq. (\ref{eff_ham}). The advantage is
that we converted the time dependent problem into an effective time
independent problem thus allowing us to capture the complete
behaviour of the system through the
eigenvalues of the dressed Hamiltonian.\\
The eigenvalues of $\mathcal{H}_{D}$ are given by $\pm
\frac{\omega}{2}\sqrt{4
\sin{\theta}^2+\cos{\theta}^{2}+(\frac{b}{\omega})^{2}-2
\cos{\theta}\frac{b}{\omega}}$. The complete solution for $C_{1}$
and $C_{2}$ is given by
\begin{equation}
\begin{pmatrix}
{C_{1}}\\
{C_{2}}
\end{pmatrix}
=
\begin{pmatrix}
C_{1}(0) e^{-i\lambda t}\\
C_{2}(0) e^{+i\lambda t}
\end{pmatrix},
\label{alp_eq_1}
\end{equation}
where,
\begin{equation}
\lambda=\frac{\omega}{2}(\sqrt{4
\sin{\theta}^2+\cos{\theta}^{2}+(\frac{b}{\omega})^{2}-2
\cos{\theta}\frac{b}{\omega}}-\frac{b}{\omega}).
\end{equation}
Here $\lambda$ represents the phase dependent energy shift of the
levels. As can be seen, this shift is of the form
$\gamma_{n}\omega$. In the pure non-Abelian or Abelian regime, the
value of $\gamma_{n}$ is independent of $b$ or $\omega$. However, in
intermediate region, this gauge of the system depends on both the
value of the degeneracy lifting field
as well as the frequency of evolution.\\
To obtain the non-Abelian limit, we put $b=0$ and obtain the
familiar non-Abelian gauge eigenvalues given by
\begin{equation*}
\pm \sqrt{4 - 3 \cos^2{\theta}}.
\end{equation*}
We can reach the Abelian limit by putting $\omega\rightarrow0$ for
any value of $b\neq0$. However, physically, the exact value of
$\omega$ required to reach the Abelian limit, depends on the value
of $b$ and hence a more suitable limit for the Abelian regime is
$\frac{b}{\omega}\gg1$. In this limit, the gauge tends to
\begin{equation}
\pm\frac{1}{2}\cos{\theta}.
\end{equation}
It should be mentioned here that the choice of
$\omega_{\pm}=\pm\frac{b}{2}$ is also a valid choice of the ansatz.
However, this choice represents a the opposite sense of rotation of
the fields. In principle, these two choices physically corresponds
to a difference of $\pi$ of the angle between $\omega$ and $b$.
\section{Perturbative analysis of different contributions}
In the previous section, we have obtained the complete behavior of
the system. However, the dressed state approach did not reveal the
underlying physical phenomenon governing the transition region. The
physical processes controlling the two limits is however known. Now
to obtain the physics of the transition region, we approach
perturbatively from the two extremes and try to figure out the
physical processes driving the system away from the two limits.\\
In this section our goal is to capture the response of the system
due to small changes in system parameter($b$ or $\omega$) from its
two extreme limits. We apply perturbation theory to achieve this.
The key point is to choose unperturbed Hamiltonian in the two
regimes. As the non-Abelian and Abelian regimes are very different
in nature one should not expect to use the same unperturbed
Hamiltonian to describe both. We work with the dressed state
Hamiltonian, where the problem is reduced to a time independent
situation. We use this 'dressed' Hamiltonian, to identify the
unperturbed Hamiltonians governing the behaviour of the system in
the two limits. Other than the unperturbed Hamiltonian, whatever is
left, is treated as the perturbation.\\
{\bf Perturbation in the Abelian Limit:} The Abelian limit
corresponds to the situation where $b\gg\omega$. In this condition,
the phase dependent energy shift is given by
$\frac{\omega}{2}\cos{\theta}$. The effective time independent
Hamiltonian which can describe this system, including the phase
dependent energy shifts, can be written as
\begin{equation}
H_{D}^{A}=
\begin{pmatrix}
\frac{1}{2}\cos{\theta}\omega -\frac{b}{2} & 0\\
0 & -\frac{1}{2}\cos{\theta}\omega +\frac{b}{2}
\end{pmatrix}
\label{h_dress_o}
\end{equation}
Now to study the deviation of the system from the Abelian limit, we
rewrite the total dressed Hamiltonian as
\begin{equation*}
\mathcal{H}_{D}={H}_{D}^{A}+\delta H_{D}^{A},
\end{equation*}
where $\delta H_{D}^{A}=\mathcal{H}_{D}-{H}_{D}^{A}$ is the
perturbing Hamiltonian. The form of $\delta H_{D}$ comes out as
\begin{equation}
\delta{H}_{D}^{A}=
\begin{pmatrix}
0 & \omega \sin{\theta}\\
\omega \sin{\theta} & 0
\end{pmatrix}.
\label{h_dress_p}
\end{equation}
Now we calculate the terms of the perturbation series using this
$\delta H_{D}^{A}$. The first order contribution of this
perturbation being zero, the leading term of the perturbation series
is the second order contribution which is
\begin{equation}
E_{D}^{''}=-\frac{\sin^{2}{\theta}}{\cos{\theta}-\frac{b}{\omega}}.
\end{equation}
We can thus have a handle on the underlying physical processes
governing the level shift of the system from the Abelian behaviour.
As the perturbation series reveals, the 'non-Abelian' perturbation
does not effect the energy levels of the unperturbed states.
However, it causes a population transfer between the eigenstates as
is given by a non-zero second order term. We also notice that
decreasing the value of $\frac{b}{\omega}$, which we know takes the
system away from Abelian behaviour, also increases the coupling
between the two
states.\\
This perturbation however fails when $\frac{b}{\omega}$ approaches
$\cos{\theta}$. This is because for $b=\omega\cos{\theta}$, the
second order contribution has a singularity. This point indicates a
deviation of the guiding physics from the Abelian behaviour. For
$b>\omega|\cos{\theta}|$, the Hamiltonian decomposition used above
holds true.\\
\begin{figure}[t]
\centering
\includegraphics[scale=.8]{pert_w_10_t_57_3.eps}
\label{device} \caption{The figure depicts the results of the
perturbative analysis with respect to the exact behaviour. The
dashed line corresponds to the perturbation in the abelian regime.
As can be seen, the perturbation mimics the exact behaviour till it
reaches close the $b=\omega\cos{\theta}$ limit of the perturbation.
The dotted line denotes the perturbation in the non-Abelian regime.
As can be seen, the second order perturbation in this regime does
not have a strong correspondence with the exact behaviour. This
hints towards significant contribution from higher order
perturbation terms, which is beyond the scope of this work. For this
graph $\theta=57.3^{0}$ and $\omega=10$ Hz.}
\end{figure}
{\bf Perturbation in the Non-Abelian Limit:} The non-Abelian limit
corresponds to $b\ll\omega$. At the non-Abelian point, that is
$b=0$, the phase dependent energy shift is given by
$\frac{\omega}{2}\sqrt{4-3\cos^2{\theta}}$. As in the Abelian limit,
in the non-Abelian limit, the Hamiltonian is given by
\begin{equation}
H_{D}^{NA}=
\begin{pmatrix}
\frac{1}{2}\cos{\theta}\omega & \sin{\theta}\omega\\
\sin{\theta}\omega & -\frac{1}{2}\cos{\theta}\omega
\end{pmatrix}
\label{h_dress_o}
\end{equation}
We again write the total dressed Hamiltonian as
\begin{equation*}
\mathcal{H}_{D}={H}_{D}^{NA}+\delta H_{D}^{NA}
\end{equation*}
where $\delta H_{D}^{NA}=\mathcal{H}_{D}-{H}_{D}^{NA}$ is the
perturbing Hamiltonian in the Non-Abelian limit. $\delta H_{D}^{NA}$
is given as
\begin{equation}
\delta H_{D}^{NA}=
\begin{pmatrix}
-\frac{b}{2} & 0\\
0 & +\frac{b}{2}
\end{pmatrix}
.
\label{h_dress}
\end{equation}
Now we calculate the perturbating terms using this $\delta
H_{D}^{NA}$. We find now that unlike the Abelian limit, in this
case, the first order perturbation is non-zero whereas the second
order contribution is zero. The first order contribution is given as
\begin{equation}
E_{D}^{'}=\frac{b}{\omega}\frac{\cos{\theta}}{2\sqrt{4-3\cos^2{\theta}}}.
\end{equation}
This series illuminate the fact that in the deviation of the system
from non-Abelian behaviour, level shift plays the major role. The
increasing energy gap between the levels however leads to a reduced
rate of population transfer and thus drives the system away from the
non-Abelian behaviour.
\section{Sensitivity of Geometric Phase to Parameter Fluctuations}
Having explored both the regions perturbatively, we can now analyze
the influence of fluctuations of different external parameters on
the geometric phase.\\
To develop a general framework, let us assume that $b^{'}= b+\delta
b$ and $\omega^{'}=\omega+ \delta \omega$, where $\delta b$ and
$\delta \omega$ are the fluctuations of $b$ and $\omega$
respectively. The phase at the two extremes, that is the Abelian and
non Abelian limits are independent of the values of $b$ and
$\omega$. However in the intermediate regions, it is dependent on
these parameters. In general, we can write the phase as follows
\begin{equation}
\gamma(b,\omega)=\gamma_{0}+\gamma^{'}(b,\omega)
\end{equation}
where $\gamma_{0}$ can be either the Abelian or non-Abelian phase
and $\gamma_{'}$ is the perturbative deviation in each limit.
$\gamma_{0}$ is independent of $\omega$ and $b$ in both the
limits.\\
Now to study the effect of fluctuation of each parameter on the
geometric phase, we apply the above mentioned substitution and
obtain
\begin{equation}
\gamma(b^{'},\omega^{'})=\gamma_{0}+\gamma^{'}(b,\omega)+\frac{\partial
\gamma^{'}}{\partial \omega}\delta \omega + \frac{\partial
\gamma^{'}}{\partial b}\delta b
\end{equation}
The effects of the fluctuations of these parameters on the geometric
phase are obtained through $\frac{\partial \gamma^{'}}{\partial
\omega}$ and
$\frac{\partial \gamma^{'}}{\partial b}$.\\
In the Abelian limit, we have
\begin{equation}
\frac{\partial \gamma ^{'}}{\partial
\omega}=\frac{\sin^{2}{\theta}}{b}
\end{equation}
and
\begin{equation}
\frac{\partial \gamma ^{'}}{\partial
b}=\frac{\omega\sin^{2}{\theta}}{b^{2}}.\\
\end{equation}
In the above equations it is assumed $\frac{b}{\omega}\gg 1$ which
holds true in the Abelian regime.\\
For Non-Abelian limit,
\begin{equation}
\frac{\partial \gamma ^{'}}{\partial
\omega}=\frac{b\cos{\theta}}{2\omega^{2}\sqrt{4-3\cos^{2}{\theta}}}\\
\end{equation}
and
\begin{equation}
\frac{\partial \gamma ^{'}}{\partial
b}=\frac{\cos{\theta}}{2\omega\sqrt{4-3\cos^{2}{\theta}}}.
\end{equation}
From the above four equations, depicting the effect of fluctuation
of parameters on the geometric phase, it is evident that the role of
$\omega$ and $b$ are interchanged in the non-Abelian and the Abelian
limits. While the non-Abelian limit is more sensitive to magnetic
field fluctuations, the Abelian limit on the other hand is more
sensitive to fluctuations of the angular frequency.\\
Experimentally usually magnetic field noise is one of the biggest
source of dephasing for quantum systems. From that point of view, we
can say that the phase fluctuations due to magnetic field noise will
be much higher in the non-Abelian regime than compared to the
Abelian regime and thus the Abelian limit is much more robust
against fluctuation of magnetic field. However, if in a certain
situation, robustness against rotational frequency is required, then
non-Abelian limit is a much better choice than
Abelian limit.\\
\section{Discussion and Conclusion}
In this work we have performed an extensive study of a system as it
is continuously moved from Abelian regime to Non-Abelian regime.
Although it was known that Abelian regime is signified by non
transfer of population and non-Abelian system by population
transfer, we have for the first time shown the dynamics of the
system with respect to a symmetry breaking field, driving the system
from one regime to another.\\
The dressed state approach revealed the exact dynamics of the system
and at the same time allowed us to probe the underlying mechanisms a
play using the method of perturbation.\\
The perturbative approach also helped us to gain insight into
robustness of geometric phase to external parameter fluctuations. As
can be seen the non-Abelian limit is more susceptible to magnetic
field fluctuations whereas Abelian limit is prone to fluctuations
arising from angular frequency fluctuations. Thus a detailed
understanding of Abelian and non-Abelian evolutions and the
behaviour off the system in between can in turn lead to better
designing of architecture for implementation of quantum computation
protocols. The two field system provides a greater handle on phase
engineering requirements for quantum technology purposes.\\
\section{Acknowledgements}
The authors would like to thank Mr. Sanjib Ghosh for extensive
discussion both on the scientific as well as presentation aspect of
this paper. We would also like to thank CQT for financial support.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,620 |
La Copa Femenina de la DFB (DFB-Pokal Frauen en alemany) és la principal competició nacional de Copa de futbol femení a Alemanya, per tant la contrapartida femenina de la Copa DFB. Va ser creada el 1980, i des del 1991 també inclou equips de l'Est. Els darrers títols els ha guanyat el VfL Wolfsburg (cinc títols consecutius). El FFC Frankfurt ha guanyat més títols amb nou. La final, a excepció de la final de 1983, es celebrava sempre el dia anterior a la final masculina. Des de 1985 la final se celebrava així a Berlín. Només l'any 1983 es va celebrar a la ciutat de Frankfurt. Aquesta rutina va canviar el 2010 quan el final es va disputar a la ciutat de Colònia. Alguna vegada es va quedar a la ciutat i es va celebrar al RheinEnergieStadion. La final sol tenir lloc un cap de setmana o festiu a principis de maig, independentment de la final masculina, per tal de guanyar més atenció.
Format
Participació
Tots els clubs de la Bundesliga i de la 2a Bundesliga poden competir a la copa, així com els clubs que van aconseguir l'ascens a la 2a Bundesliga. També els guanyadors de les competicions de copa regional competeixen a la copa. Com a excepció a aquestes regles, els segons equips dels clubs no poden participar a la Copa DFB. Quan un segon equip guanya la seva copa regional, l'associació regional d'aquest equip pot enviar un altre equip a la Copa DFB només si el segon equip guanyador de la copa no ha aconseguit l'ascens a la 2a Bundesliga.
Classificació
Dels equips classificats, no tots han de competir a la primera volta. A la segona ronda del torneig han de competir exactament 32 equips, de manera que a la primera volta el nombre de partits està determinat pel nombre d'equips en excés, donant lloc a un partit per a cada equip després del 32è. Els equips que no han de competir a la primera ronda són els millors classificats de la temporada anterior de la Bundesliga, el nombre de nou determinat pel nombre d'inscrits al torneig.
Els aparellaments per a la primera, segona i tercera ronda no són del tot aleatoris ja que hi ha una comissió que assigna els clubs a dos o quatre grups segons els convingui. Aquests grups es corresponen amb la procedència regional dels clubs. En la tercera ronda la comissió pot decidir no assignar els participants a cap grup. Dins d'aquests grups els clubs tornen a estar separats, aquesta vegada en funció de la lliga en què juguen. Per al sorteig, els clubs de la Bundesliga es posen en un pot i la resta en un segon pot. Els clubs que no són de la Bundesliga tenen automàticament avantatge a casa contra els clubs de la Bundesliga.
Regles dels partit
Tots els jocs es desenvolupen en dues parts de 45 minuts amb el guanyador avançant a la següent ronda. En cas d'empat, el partit s'allargarà en dues parts de 15 minuts. Si el marcador segueix igualat després de 120 minuts, el guanyador es decideix en una tanda de penals. A la final no s'afegeix cap pròrroga en cas d'empat al cap de 90 minuts, sinó que segueix immediatament la tanda de penals.
Guanyadors
Abans de la reunificació d'Alemanya, la competició de copa només incloïa equips d'Alemanya Occidental.
Palmarés per equip
(*) Nota: Subcampió del VfL Wolfsburg com a VfR Eintracht Wolfsburg
Referències
Enllaços externs
List of Cup Finals at rsssf.com
Vegeu també
Bundesliga Femenina de Futbol
Copa alemanya de futbol
Competicions futbolístiques a Alemanya
Competicions futbolístiques femenines
Campionats estatals femenins | {
"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"http:\/\/msp.org\/agt\/2007\/7-4\/p05.xhtml","text":"#### Volume 7, issue 4 (2007)\n\n Recent Issues\n The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear Contacts ISSN (electronic): 1472-2739 ISSN (print): 1472-2747\nA parametrized Borsuk\u2013Ulam theorem for a product of spheres with free $\\mathbb{Z}_p$\u2013action and free $S^1$\u2013action\n\n### Denise de Mattos and Edivaldo L dos Santos\n\nAlgebraic & Geometric Topology 7 (2007) 1791\u20131804\n##### Abstract\n\nIn this paper, we prove parametrized Borsuk\u2013Ulam theorems for bundles whose fibre has the same cohomology (mod $p$) as a product of spheres with any free ${\u2124}_{p}$\u2013action and for bundles whose fibre has rational cohomology ring isomorphic to the rational cohomology ring of a product of spheres with any free ${S}^{1}$\u2013action. These theorems extend the result proved by Koikara and Mukerjee in [A Borsuk\u2013Ulam type theorem for a product of spheres, Topology Appl. 63 (1995) 39\u201352]. Further, in the particular case where $G={\u2124}_{p}$, we estimate the \u201csize\u201d of the ${\u2124}_{p}$\u2013coincidence set of a fibre-preserving map.\n\n##### Keywords\nparametrized Borsuk\u2013Ulam theorem, characteristic polynomials, free action, equivariant map, product of spheres\n##### Mathematical Subject Classification 2000\nPrimary: 55M20\nSecondary: 55R91, 55R25","date":"2016-10-21 20:10:32","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 5, \"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\": 0.7136179208755493, \"perplexity\": 2392.2444695016184}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-44\/segments\/1476988718303.21\/warc\/CC-MAIN-20161020183838-00200-ip-10-171-6-4.ec2.internal.warc.gz\"}"} | null | null |
Weicheng District () is a district of Xianyang, Shaanxi, China.
The district is notable for a number of Zhou and Han era tombs.
History
The area belonged to the , also known as the state of Ying (), during the Shang dynasty. The Cheng state, which included northern portions of present-day Weicheng District, was settled by the descendants of sometime between 16th and 14th centuries BCE. Present-day was the site of a fief of the Cheng state.
During the Zhou dynasty, King Wen oversaw the expansion of the dynasty to the west of the , into present-day Xianyang. Him and King Wu were buried in a complex known as the , located in present-day . The complex contains two ancestral halls dedicated to the kings, as well as over 40 steles, which were erected in their honor during the Song dynasty.
Administrative divisions
Weicheng District administers the following 10 subdistricts:
()
()
()
()
()
()
()
()
()
()
References
External links
County-level divisions of Shaanxi
Xianyang | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,990 |
Pseudocolus fusiformis (E. Fisch.) Lloyd – gatunek grzybów z rodziny sromotnikowatych (Phallaceae).
Systematyka i nazewnictwo
Pozycja w klasyfikacji według Index Fungorum: Pseudocolus, Phallaceae, Phallales, Phallomycetidae, Agaricomycetes, Agaricomycotina, Basidiomycota, Fungi.
Po raz pierwszy takson ten zdiagnozował w 1891 r. Eduard Fischer nadając mu nazwę Colus fusiformis. Obecną, uznaną przez Index Fungorum nazwę nadał mu w 1909 r. Curtis Gates Lloyd, przenosząc go do rodzaju Pseudocolus.
Charakterystyka
Jest najszerzej rozpowszechnionym przedstawicielem rodzaju przęślikowiec, z zasięgiem występowania obejmującym Stany Zjednoczone, Australię, Japonię, Jawę i Filipiny.
W Stanach Zjednoczonych określany popularnie jako "stinky squid" z powodu charakterystycznego kształtu i zapachu.
Przypisy
Sromotnikowate | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,044 |
\section{Introduction}
Meson physics is a very broad field, and this workshop covers a very large part of it, with about 25 plenary and 50 parallel talks. I can not possible attempt to summarize all that. So, I will pick and choose from what was presented, a process which is admittedly very subjective, and add some new and interesting results which were not talked about. I will also update some results which have appeared in print since my talk.
Before I go into physics, let me acknowledge the laboratories and experiments which have contributed to this field, and to this workshop, and briefly mention what is, to the best of my knowledge, their current status. In the following, `ended' means that data taking has ended, but physics analysis and publications continue.
\begin{quote}
BEPC (BES II ended, BES III operational in 2007?); BNL (E852 ended, PHOENIX and STAR at RHIC have minor spectroscopy programs); CELSIUS (WASA ended); CERN (Crystal Barrel and OBELIX ended; COMPASS has potential); COSY (ANKE, GEM, TOF operational); DESY (H1, ZEUS, HERMES operational), Fermilab (SELEX and FOCUS ended, Tevatron operational until 2009), Frascati (at DA$\Phi$NE; KLOE and DEAR ended, FINUDA and SIDDHARTA operational); GSI (HADES operational, PANDA in 2014?); JLab (CLAS operational, 12 GeV upgrade?); JPARC (operational in 2008?), KEK (PS and Belle (high potential) operational; Bonn (ELSA operational); Mainz (MAMI II operational); SLAC (BaBar (high potential) operational until 2010).
\end{quote}
The sum total of the above summary is that while of the many very productive older programs, some have ended, many continue, and new programs (BEPC II, GSI (FAIR), and JPARC) are in the offing. We can therefore have high expectations for continued exciting discoveries in meson physics in the future.
I will begin my summary in the order of the mass hierarchy. Light quark ($n (=u,d)~\mathrm{and}~s$) mesons, heavy quark ($c,b$) mesons, and then light/heavy ($c/n,b/n$) mesons. For good measure, I will throw in a gluon or two, and say a few words about glueballs. Finally, I will talk about the recently discovered exotics, about which everybody is excited, but whose nature nobody really knows.
\section{Pions and Kaons}
Pions and kaons are ancient particles, and one would think that there could not possible by anything new to learn about them. Actually, there is! One of the most interesting and elementary questions one can ask about a particle is ``what does it look like?'' To put it in terms of the simplest physical observable, the question becomes, ``what do the particle's electromagnetic form factors look like?'' To relate the meson's form factors to its constituent quarks one needs to invoke perturbative QCD (at least at present), and that means that one needs experimental measurements of form factors at large enough momentum transfers at which pQCD might be expected to be valid. Let us see where we stand both experimentally and theoretically on this issue.
\begin{figure}[!tb]
\begin{center}
\includegraphics[width=2.4in]{thesis-pifinaltl.eps}
\includegraphics[width=2.4in]{thesis-kfinaltl.eps}
\end{center}
\caption{Timelike electromagnetic form factors of pion and kaon. The new CLEO results for $|Q^2|=13.48~\mathrm{GeV}^2$ are shown with square symbols.}
\end{figure}
\subsection{Electromagnetic Form Factors of Pions and Kaons}
Let me recall the great debate about ``when is $Q^2$ large enough to validate pQCD?'' which took place in the mid-1980s between Brodsky and colleagues\cite{lepagebrodsky}, who believed $Q^2\approx10~\mathrm{GeV}^2$ was enough, and Isgur and Llwellyn Smith\cite{isgursmith}, who claimed that $Q^2>100~\mathrm{GeV}^2$ would be required. The experimental data for pion form factors used in the debate consisted of barely three data points for $Q^2>2~\mathrm{GeV}^2$ with $\pm50\%$ errors. We would all agree that perhaps this was ``much ado about nothing''. So, what is new?
What is new is the first precision measurements at large timelike momentum transfer ($Q^2=-s\approx13.48~\mathrm{GeV}^2$) of pion and kaon form factors at CLEO\cite{cleoff} as shown in Fig. 1\footnote{The $Q^2F_\pi$ point marked ``$J/\psi$'' is not a direct measurement, but dervied from the measured branching ratio for $J/\psi\to\pi^+\pi^-$ by Milana et al.\cite{milana}. Using the same arguments, I have derived $Q^2F_K$\cite{seth}. The agreement of these $Q^2F_{\pi,K}$ at $Q^2=9.6~\mathrm{GeV}^2$ with the measured values at $Q^2=13.5~\mathrm{GeV}^2$ lends justification to the procedure.}. The solid curves in the figure show the arbitrarily normalized variation of $\alpha_{\mathrm{Strong}}$ with $Q^2$. Also shown for the pion are pQCD predictions and QCD sum rule predictions, neither of which come even close to the measured values, which says that here is now something real for theorists to chew on.
\section{The S--Wave Mesons}
Most of the activity in this category has been confined to the pseudoscalar $\eta$ and $\eta'$ mesons. The obvious reason is that accelerators in the $1-3$ GeV region can produce them copiously. At this conference, a lot of activity was reported on production of $\eta$ and $\eta'$. Detailed reports were presented on photo-- and electro--production measurements being made at Mainz, JLab, and ELSA. We also heard about pion-induced production at BNL, production in $e^+e^-$ collisions at Frascati, in $pp$ collisions at Julich, and in heavy ion collisions at GSI.
We have an interesting development in the measurements of the mass of $\eta$. A recent COSY measurement\cite{cosyetamass} reported $M(\eta)=547.311\pm0.028(\mathrm{stat})\pm0.032(\mathrm{syst})~\mathrm{MeV}$, which differs by $\sim8\sigma$ from an earlier NA38 measurment\cite{na38etamass}. $M(\eta)=547.843.0.030(\mathrm{stat})\pm0.041(\mathrm{syst})$ MeV. Now we have the report from KLOE\cite{kloeetamass} of $M(\eta)=547.822\pm0.005(\mathrm{stat})\pm0.069(\mathrm{syst})$, which agrees with NA38 within $0.25\sigma$. The resolution of the $\sim500$ keV difference in $M(\eta)$ between COSY and NA38/KLOE would normally be only of academic interest, but in this era of meson--meson molecules, it could be important if somebody comes up with a weakly bound molecule of $(\eta+X)$!
The mass of $\eta'$ is known with a $\pm140$ keV uncertainty\cite{pdg}, but so far we have no reports from COSY or KLOE on improved measurements.
KLOE\cite{kloeetamass} has also reported the long--sought value of the ratio $R\equiv\mathcal{B}(\phi\to\gamma\eta')/\mathcal{B}(\phi\to\gamma\eta)=(4.74\pm0.09\pm0.20)\times10^{-3}$, and used it to address the question of the possible glue content of $\eta'$. They find it to be $1-(0.92\pm0.06)=(8\pm6)\%$. While the numerical value is not significant at this level, it does indicate that the glue content of $\eta'$ is quite small.
The spectroscopy of the low--mass vectors, the $\rho,~\omega,~\phi$ has been dominated by the precision measurements of the Novosibirsk SND and CMD2 groups. Measurements of comparable precision are now being reported by KLOE. Hopefully, long--standing problems, like the $>5$ MeV uncertainty in the $\rho$ mass can be resolved by these new data, although it is my belief that the problem actually lies with the different formalisms for resonance analysis being used.
\section{The P--Wave Mesons}
The P--wave singlets, $h_1$ and $b_1$, have remained inactive for the last fifteen years. The focus of the activity has been on the P--wave triplets, and among them the greatest attention has been drawn by the isosinglet scalars, primarily because $I=0,~J^{PC}=0^{++}$ are also the quantum numbers of the lowest mass glueball. It behooves us, however, to take a look at all three, $^3P_0(0^{++})$, $^3P_1(1^{++})$, $^3P_2(2^{++})$.
The problem common to all the states with masses larger than $\sim1$ GeV is excessive crowding. In the 1 GeV interval of mass, $1.25-2.25$ GeV, according to Godfrey and Isgur\cite{godfreyisgur}, 75 ($u,d,s$) mesons are expected, the average spacing being $\sim15$ MeV. The ones which have been observed have widths ranging from $100-500$ MeV, which means that states overlap, mix, and interfere horribly, and even the most sophisticated partial width analyses of experimental data have a tough time identifying and characterizing (i.e. $J^{PC}$, branching ratios) the states. The problem is best illustrated by the isospin singlet scalars, the $0^{++}~f_0$'s.
\subsection{The Isosinglet $^3P_0(0^{++})$ or $f_0$ States}
With $u,d,s$ quarks, one expects two isoscalars, $f_0$ and $f_0'$, and an isovector $a_0$. The trouble is that there are \textit{five} $f_0$'s known: $f_0(600)$ or $\sigma$, $f_0(980)$, $f_0(1370)$, $f_0(1500)$, and $f_0(1710)$. It has become very important to examine these very carefully because one expects only one more isosinglet $0^{++}$ state beyond $f_0$ and $f_0'$, and that third state is to be identified with the interesting, and ever elusive, glueball. But we have five! So, any new experimental information on these is most welcome.
\subsubsection{The Sigma, or $f_0(600)$: non--$q\bar{q}$ (?)}
The sigma has been there for a long time, albeit in the shadows. Its most recent observation is by BES in $J/\psi\to\omega\sigma,~\sigma\to\pi^+\pi^-$, with the result $M(\sigma)=541\pm35~\mathrm{MeV}$, $\Gamma(\sigma)=502\pm84~\mathrm{MeV}$\cite{bessigma}. Leutwyler and colleagues\cite{leutwyler} interpret the $\sigma$ as a pole in the $\pi\pi$ scattering amplitude and obtain $M(\sigma)=441^{+16}_{-8}~\mathrm{MeV}$, $\Gamma(\sigma)=544^{+18}_{-15}~\mathrm{MeV}$, and consider its physics to be governed by the Goldstone pions, i.e., the sigma is not a simple $q\bar{q}$ meson. This is consistent with the fact that typical potential model calculations (e.g. Godfrey and Isgur\cite{godfreyisgur}) predict the lowest mass $f_0$ to have a mass of $\sim1.09$ GeV. Lattice has so far nothing to say about the $\sigma$; it is still struggling with 250 MeV pions. The fact is that the nature of the $\sigma$ remains as cloudy as ever, althought Pennington\cite{pennington} claims that its two photon width precludes its being a tetraquark ($\bar{q}\bar{q}qq$) or a glueball.
Parenthetically, let me mention that BES\cite{beskappa} has confirmed the observation of sigma's strange partner, the \textbf{kappa}, previously reported by Fermilab E791\cite{e791kappa} in the reaction $D^+\to(K^-\pi^+)\pi^+$ with $M(\kappa)=797\pm19\pm43~\mathrm{MeV}$ and $\Gamma(\kappa)=410\pm43\pm87~\mathrm{MeV}$. In the reaction $J/\psi\to\overline{K^*}(890)+\kappa,~\kappa\to K^+\pi^-$, BES obtains $M(\kappa)=878\pm23^{+64}_{-38}~\mathrm{MeV}$ and $\Gamma(\kappa)=499\pm52^{+55}_{-87}~\mathrm{MeV}$. It will be interesting to see if the $\kappa$ also emerges from the analysis of $K\pi$ scattering length, a la Leutwyler.
\subsubsection{The $f_0(980)$: non--$q\bar{q}$ (?)}
We had five scalars, and we wanted to weed out two in order to have left two $|q\bar{q}>$ and one glueball. Well, we appear to have put $f_0(600)$ or $\sigma$ aside as a pole in the $\pi\pi$ scattering amplitude. We need to remove one more. The axe usually falls on $f_0(980)$. Because $M(K\bar{K})=985$ MeV, and $f_0(980)$ strongly couples to $K\bar{K}$, $f_0(980)$, and its cousin $a_0(980)$, are often dismissed as $K\bar{K}$ molecules. This conjecture, or, for example, the recent conjecture\cite{beveren} that like the $\sigma$, $f_0(980)$ is a dynamically generated resonance, make $f_0(980)$ non--$|q\bar{q}>$. However, there is no hard proof, theoretical or experimental, for this assertion. As a matter of fact, as described later, I believe that the systematics of $^3P_J$ states argues against this prejudice.
Despite the fact that $f_0(980)$ is one of the oldest known (since 1973) resonances, and has been observed in numerous experiments, very little quantitative is known about it. An average of the latest measurements by E791\cite{e791f0}, Belle\cite{bellef0}, BES\cite{besf0}, and KLOE\cite{kloef0} gives $M(f_0(980))=979\pm2$ MeV, and the average of the E791\cite{e791f0} and Belle\cite{bellef0} results gives $\Gamma(f_0(980))=45\pm5~\mathrm{MeV}$. Besides this, we know very little. PDG06 summarizes the decays of $f_0(980)$ to $2\pi$, $2K$, and $2\gamma$ as \textit{seen}, \textit{seen}, \textit{seen}, which can be seen as a very unsatisfactory situation. The only quantitative measure we have is that according to BES\cite{besf0}, the partial width for $K^+K^-$ is about $1/3$ that for $\pi^+\pi^-$, and when the relative phase space differences are taken into account, the coupling to kaons is about 3 to 4 times larger than that to pions\cite{besf0,kloef0}.
\subsubsection{The $f_0(1370)$, $f_0(1500)$, and $f_0(1710)$: Glueballs, anyone?}
With $f_0(600)$ and $f_0(980)$ banished as non--$|q\bar{q}>$, the scalar field is reduced to the required three, two $|q\bar{q}>$ and one $|gg>$ glueball. Most of the glueball games are played between these three. Despite early over--optimistic searches for `pure' glueballs, it is now generally accepted that `pure' glueballs do not exist. Quantum mechanics dictates that the scalar glueball must mix with the scalar mesons. So, the game now is \textit{how much} $|gg>$ exists in each of the three candidates. Is there a winner? Perhaps not by a landslide, but even so! The answer depends on who you ask, and how they order the \textit{unmixed} $|n\bar{n}>$, $|s\bar{s}>$ and $|gg>$ states. Not much new has been forthcoming in this debate which used to be the talk of the town just a few years ago. The $f_0(1710)$ is now firmly established as being $J^{PC}=0^{++}$, and it has been found to be healthily populated, along with $f_0(1500)$ in $\bar{p}p$ annihilation at 5.2 GeV/$c$\cite{uman}.
\begin{figure}[!tb]
\begin{center}
\includegraphics[width=4.in]{f0_levels.eps}
\end{center}
\caption{Systematics of $^3P_J$ levels, (left) $|c\bar{c}>$ charmonium $^3P_J$ states in experiment and theory, (right) low--lying light quark $^3P_J$ states.}
\end{figure}
\subsection{The isoscalar $^3P_J$ and the $\vec{l}\cdot\vec{s}$ problem}
There is a vexing problem with the spin--orbit splitting of the lowest light quark $^3P_J$ states, and I believe that it relates to the non--$|q\bar{q}>$ assignment of $f_0(980)$. Let me elaborate.
The QCD interaction does not depend on quark flavor. The spin--orbit splitting has the same heirarchy for $|n\bar{n}>$, $|c\bar{c}>$ and $|b\bar{b}>$, and it is in the order: $0^{++}$ (lowest), $1^{++}$, $2^{++}$. As shown in Fig.~2, it is the order predicted for, and experimentally observed for $|c\bar{c}>$ charmonium, and $|b\bar{b}>$ bottomonium. It is what is predicted for $|n\bar{n}>$, for example by Godfrey and Isgur\cite{godfreyisgur}. In assigning $f_0(1370)$ as the $0^{+}$ member of the lowest $^3P_J$, the PDG would have us believe in the completely inverted order, $2^{++}$ (lowest), $1^{++}$, $0^{++}$. This, of course, becomes necessary because the $f_0(980)$ has been dismissed as a non--$|q\bar{q}>$. It is far easier for me to believe that the theoretically predicted $0^{++}$ at 1090 MeV has been shifted down to 980 MeV by non--$|q\bar{q}>$ admixtures, than to believe that it moves up by $\sim300$ MeV to 1370 MeV. In other words, I want to keep alive the possibility of $f_0(980)$ having a substantial component of $|q\bar{q}>$. In doing so, I am entirely aware of the width arguments advanced by Godfrey and Isgur against $f_0(980)$ being $|q\bar{q}>$.
Having dwelt so long on the $0^{++}$ light--quark mesons, let me simply mention that the situation about $2^{++}$ tensors is no better. Below 2.4 GeV in mass, six $f_2$ are predicted by Godfrey and Isgur\cite{godfreyisgur}, twelve are claimed by different experiments, and of these only six are considered as well established\cite{pdg}.
\begin{figure}[!tb]
\begin{center}
\includegraphics[width=3.in]{ccbar_spect_all.eps}
\end{center}
\caption{Spectrum of $|c\bar{c}>$ charmonium states. The dashed lines indicate states predicted by Godfrey and Isgur\protect\cite{godfreyisgur}.}
\end{figure}
\section{The Heavy--Quark Mesons}
As stated earlier, the light--quark mesons, have the experimental problem of numerous closely spaced overlapping states with very large widths, and the theoretical problems of being highly relativistic and having too large a value of $\alpha(\mathrm{strong})$ ($\gtrsim0.6$) to allow application of pQCD. In contrast, the heavy quark mesons are far more tractable, both because they have well--resolved, narrow--width bound states (e.g., for $|c\bar{c}>$ typical level spacing and width are 100 MeV and $<10$ MeV, respectively), and tractable relativistic problems ($(v/c)^2\approx0.2~\mathrm{for}~|c\bar{c}>$, $0.1~\mathrm{for}~|b\bar{b}>$), and small enough $\alpha(\mathrm{strong})(\approx0.3~\mathrm{for}~|c\bar{c}>,$ $\approx0.2~\mathrm{for}~|b\bar{b}>$) to permit use of pQCD. It follows that if you want to relate meson spectroscopy to the fundamental theory of QCD, you must turn to heavy--quark mesons, and this series of workshops must evolve more and more in that direction.
Let me begin by drawing your attention to the level diagram for $|c\bar{c}>$ charmonium states (in Fig.~3). There are eight bound states before the system becomes unstable with respect to decay into open charm $D\bar{D}$ mesons ($D\equiv|c\bar{n}>$) at 3.73 GeV. The $|b\bar{b}>$ bottomonium system is similar, but because of the three--times heavier mass of the $b$--quark ($m_b\approx4.5~\mathrm{GeV}$, $m_c\approx1.5~\mathrm{GeV}$), it has fourteen bound states before it becomes unstable with respect to decay into open--beauty $B\bar{B}$ mesons ($B\equiv|b\bar{n}>$) at 10.5 GeV.
Let me talk about the experimental situation. The bad news is that many of the facilities which produced most of the recent high precision hadron spectroscopy data have ended. This includes: BNL (E852), CERN (CBarrel), FERMILAB (E760/E835). The good news is that new and promising facilities are just coming on, or are on the horizon. These include: DAFNE, GSI (FAIR), JPARC, BEPC II, and CEBAF Upgrade. The (unexpected) good news is that the SLAC and KEK $B$ factories, not built for spectroscopy, have accumulated such huge luminosities that they seem to be capable of making excellent contributions to hadron spectroscopy, almost wherever they choose to.
Progress is made as much by improved precision as by discoveries. So, let me give you a few examples of precision measurements in heavy quark meson spectroscopy. At Novosibirsk, the mass of $J/\psi$, $M(J/\psi)=3096.917\pm0.02~\mathrm{MeV}$, has been measured, i.e., with an unprecedented precision of 1 part in 3 million\cite{novosibirsk}. At CLEO, in a single measurement, $\mathcal{B}(J/\psi\to e^+e^-)/\mathcal{B}(J/\psi\to\mu^+\mu^-)=0.997\pm0.013$, lepton universiality is confirmed to $\sim1\%$\cite{cleodilep}. In another measurment, isospin conservation (violation) is confirmed to $\sim2\%$ (0.5\%) by measuring $\psi'$ decay to $\pi^+\pi^-J/\psi$ versus $\pi^0\pi^0J/\psi$ (to $\pi^0J/\psi$ versus $\eta J/\psi$)\cite{cleopsitrans}. In the bottomonium sector new measurements at CLEO\cite{cleodielec} of the leptonic branching ratios of $\Upsilon(1S,2S,3S)$ states with a precision of $1-2\%$ have resulted into up to 55\% changes in long--accepted values. The precision result, $R\equiv \Gamma_{ee}(1S)/\Gamma_{ee}(2S)=2.19\pm0.03$ can be now used to validate the latest result of unquenched lattice calculation, $R=1.91\pm0.05$. It appears that lattice is getting there. Today $|b\bar{b}>$, tomorrow $|c\bar{c}>$?
Let me move on from improved precision to real discoveries. In the charmonium sector, these have consisted of the discovery of the spin--singlet states $\eta_c'(2^1S_0,0^{-+})$ and $h_c(1^1P_1,1^{+-})$. These bound states have long defied attempts to identify them, the main reson being that they are nearly impossible to populate via the vector $\psi'(2^3S_1,1^{--})$ formed in $e^+e^-$ annihilation. Radiative decay to $\eta_c'$ is expected to be a very weak M1 transition, and that to $h_c$ is forbidden by charge conjugation. Yet the importance of these spin--singlet states can not be exaggerated. Together with the known spin--triplet states (or their centroid), these provide measures of the QCD hyperfine splitting, and therefore the $q\bar{q}$ \textbf{spin--spin interaction}. So, our only hope of learning about the spin--spin interaction rests on the identification of these states of charmonium.
\subsection{The Discovery of the $\eta_c'(2^1S_0)$ State of Charmonium}
The only hyperfine splitting known in the heavy quarkonia is the mass difference of $1S$ charmonium states, $J/\psi$ and $\eta_c$:
$$\Delta M_{hf}(1S)=M(J/\psi)-M(\eta_c)=3096.9-(2980.4\pm1.2)=116.5\pm1.2~\mathrm{MeV}$$
A model--independent prediction for the hyperfine splitting of the $2S$ states is that
$$\Delta M_{hf}(2S)=M(\psi')-M(\eta_c')=\Delta M_{hf}(1S)\times\frac{M^2(\psi')}{M^2(J/\psi)}\times\frac{\Gamma(\psi'\to e^+e^-)}{\Gamma(J/\psi\to e^+e^-)}=68\pm2~\mathrm{MeV}$$
In 1982, the Crystal Ball Collaboration at SLAC claimed identification of $\eta_c'$ and reported $M(\eta_c')=3594\pm5~\mathrm{MeV}$, which would correspond to $\Delta M_{hf}(2S)=92\pm5$ MeV, about 35\% \textbf{larger} than expected. Fortunately, or unfortunately, the Crystal Ball observation was never confirmed, and $\eta_c'$ remained unidentified for 30 years, until now.
\begin{figure}[!tb]
\begin{center}
\includegraphics[width=2.5in]{etacp_cleo_sum.eps}
\includegraphics[width=2.4in]{etacp_babar.eps}
\end{center}
\caption{The invariant mass $M(K_SK\pi)$ spectra from two--photon fusion measurements by CLEO (left) and BaBar (right). The $\eta_c(2S)$ peak is prominent in both spectra.}
\end{figure}
In 2002, Belle claimed identification of $\eta_c'$ in the decay of 45 million $B$ mesons, $B\to K(K_SK\pi)$ and reported $M(\eta_c')=3654\pm10$ MeV, which would correspond to $\Delta M_{hf}(2S)=32\pm10$ MeV, a factor \textbf{two smaller} than expected and a factor \textbf{four} smaller than $\Delta M_{hf}(1S)$. Could this be possible? Well, perhaps! There are two important ways $2S$ states differ from $1S$ states. $1S$ states, with $r\approx0.4~\mathrm{f}$, lie in the Coulombic region (prop. to $1/r$) of the $q\bar{q}$ potential, $V=A/r+Br$, whereas the $2S$ states, with $r\approx0.8~\mathrm{f}$, lie in the confinement part (prop. to $r$) of the potential. The spin--spin potential in the two regions could be different. The second difference is that the $2S$ states, particularly $\psi'$, lie close to the $D\bar{D}$ breakup threshold at 3730 MeV, and can be expected to mix with the continuum as well as higher $1^{--}$ states. All in all, it is important to nail down $\eta_c'$ experimentally, and measure its mass accurately.
This was successfully done by CLEO\cite{cleoetacp} and BaBar\cite{babaretacp} in 2004 by observing $\eta_c'$ in two--photon fusion, $\gamma\gamma\to\eta_c'\to K_SK\pi$. The two observations are shown in Fig.~4. The average of all observations is $M(\eta_c')=3638.7\pm2.0$ MeV, which leads to $\Delta M_{hf}(2S)=47.4\pm2.0$ MeV, which is almost a factor \textbf{2.5 smaller} than $\Delta M_{hf}(1S)$. Explaining this large difference is a challenge to the theorists. The challenge for the experimentalists lies in completing the spectroscopy of $\eta_c'$, now that its mass is known.
\subsection{The Discovery of the $h_c(1^1P_1)$ State of Charmonium}
The interest in identifying $h_c$ is also related to the hyperfine or $\vec{s_1}\cdot\vec{s_2}$ interaction, but in a different way. In the conventional $q\bar{q}$ potential, the confinement part is scalar, and therefore there is no long--range spin--spin interaction. The $\vec{s_1}\cdot\vec{s_2}$ interaction is only due to the Coulombic part, and is a contact interaction proportional to the wave function at the origin. In other words, it is finite only for $S$--states. For $P$--wave, or higher $L$ states, the interaction is zero. Thus, for $1P$ states, the $^1P_1$ state should be degenerate with the centroid of the $^3P_J$ states, or
$$\Delta M_{hf}(1P)\equiv \left<M(^3P_J)\right>-M(^1P_1)=0$$
There are many ways this prediction could be wrong, viz the simple way of calculating the centroid as $\left<M(^3P_J)\right>=[5M(^3P_2)+3M(^3P_1)+M(^3P_0)]/9$ could be (and is) wrong, or the $q\bar{q}$ potential could have a long--range $\vec{s_1}\cdot\vec{s_2}$ part in it. It is therefore important to experimentally determine $\Delta M_{hf}(1P)$ by identifying the $h_c(1^1P_1)$ state of charmonium. Numerous attempts to identify $h_c$ made during the last 35 years had failed. Now, two attempts have succeeded. The Fermilab E835 $\bar{p}p$ annihilation experiment\cite{e835hc} has claimed $h_c$ observation at $\sim3\sigma$ level\cite{cleohc} in the reaction $\bar{p}p\to h_c \to \gamma\eta_c$ and reported $\Delta M_{hf}(1P)=-0.4\pm0.2\pm0.2$ MeV. CLEO has reported $h_c$ observation at $>6\sigma$ level in the reaction $\psi(2S)\to\pi^0h_c,~h_c\to\gamma\eta_c$. Both inclusive ($\pi^0$ and $\gamma$ detected) and exclusive analyses were made, with the result $\Delta M_{hf}(1P)=+1.0\pm0.6\pm0.4$ MeV. The exclusive spectrum is shown in Fig.~5. It appears that the $1P$ hyperfine splitting is very small, perhaps even zero.
\begin{figure}[!tb]
\begin{center}
\rotatebox{270}{\includegraphics[width=2.in]{fithcmass.ps}}
\end{center}
\caption{The spectra of $\pi^0$ recoil energies in the exclusive analysis of the reaction $\psi'\to\pi^0,~h_c\to\gamma\eta_c$, as measured by CLEO. The peak corresponding to $M(h_c)$ is prominent.}
\end{figure}
With the discovery of $\eta_c'$ and $h_c$, a milestone has been reached in charmonium spectroscopy. The bound state spectrum is now complete.
\section{New Results in Bottomonium Spectroscopy}
CLEO is now the only producer of $|b\bar{b}>$ bottomonium spectroscopy results. Besides improving the precision of the branching ratio measurements for the radiative\cite{cleoupsrad} and leptonic \cite{cleodielec} decays of the Upsilon states, CLEO has made the first observation of a non--$\pi\pi$ transition between bottomonium states\cite{cleoupsomega}, and the first observation of a $D$--state in bottomonium\cite{cleoupsdstate}. Since no bottomonium physics was presented at the Workshop, I will dwell no further on it, and return to meson spectroscopy below 5 GeV.
\section{The Unanticipated Discoveries}
The renaissance in hadron spectroscopy to which I referred earlier has been triggered by the discovery, or claimed discovery, of a number of unanticipated states. This is very exciting, and also confusing, because the nature of these states is still not understood. In chronological order, the series consists of the pentaquark (Spring8---Jan. 2003), $D_{sJ}$ (BaBar---March 2003), X(3872) (Belle---March 2003), X,Y,Z(3940) (Belle 2005), and Y(4260) (BaBar 2005).
\subsection{The Pentaquark}
There have been many reports of the sightings of the pentaquark $\Theta^+(1540)$ and its heavy quark siblings, and even a greater number of reported failures in finding them. Google tells me that there were 123,000 entries for the pentaquark by March 28, 2006, 135,000 by April 10, and 151,000 by the time of this writing in July 25, 2006. This tells you how exciting and intoxicating the idea is, or rather was.
See, for example, Refs. 34 and 34.
The pentaquark has been on ``life--support'' for too long. We should let it go---may it rest in peace!
\subsection{The Veteran X(3872)}
Discovered by Belle\cite{bellex}, and quickly confirmed by CDF\cite{cdfx}, D\O\cite{d0x}, and BaBar\cite{babarx}, the X(3872) state definitely exists. Its average mass and width are $M(\mathrm{X})=3871.5\pm0.4~\mathrm{MeV}$, and $\Gamma(\mathrm{X})\le2.3~\mathrm{MeV}$, and it decays prominently in $\pi^+\pi^-J/\psi$. What is debatable, however, is ``What is it?'' Numerous speculations have been made. Is it a charmonium ($C=-$, $1^3D_{2,3}$ or $2^1P_1$) or ($C=+$, $1^1D_2$ or $2^3P_1$)\cite{xcharm}? Is it a $C=+$ charmonium hybrid\cite{xhybrid}? Is it a vector glueball mixed with vector charmonium\cite{xglueball}, or is it a $\overline{D^0}D^{0*}$ molecule\cite{xmolecule}? A firm determination of the $J^{PC}$ of X(3872) would go a long way in sifting through the above models. Unfortunately, it is difficult to say that any of the claims made so far pin the $J^{PC}$ down unambiguously. A number of reported decays, e.g., $\mathrm{X}\to\gamma J/\psi$, $\mathrm{X}\to \omega J/\psi$, $\mathrm{X}\to \overline{D^0}D^0\pi^0$, which would pin down some of the quantum numbers, suffer from limited statistics, and angular distribution studies and the dipion mass distribution in $\mathrm{X}\to J/\psi\pi^+\pi^-$ appear to be inconclusive. The present toss--up appears to be between $J^{PC}=1^{++}$ and $2^{-+}$, with $1^{++}$ being a slight favorite. With the goal of pinning down the binding energy of $\overline{D}D^*$ (if it is bound), CLEO has announced\cite{cleod0mass} a preliminary precision determination of the $D^0$ mass, $M(D^0)=1864.847\pm0.150\pm0.200~\mathrm{MeV}$, so that the best estimate of the binding energy of the $\overline{D}D^*$ molecule as $\mathrm{X}$ is now $E_b\equiv M(\overline{D^0}D^{0*})-M(\mathrm{X})=+0.31\pm0.64~\mathrm{MeV}$, which constrains the molecular model considerably more than before. Further constraint would require a more precise measurement of the mass of X.
\begin{table}[!tb]
\tbl{Summary of Experimental data for the X, Y, Z states of Belle.}
{
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline \hline
& M (MeV) & N (evts) & $\Gamma$ (MeV) & Formed in/ & No decays to & Suggested? \\
& & & & Decays to & & \\
\hline
X\cite{xxbelle} & 3936(14) & 266(63) & 39(26) & $e^+e^-\to J/\psi(\mathrm{X})$ & $\mathrm{X}\nrightarrow \overline{D}D$ & $\eta_c''(2^1S_0)$ \\
& 3943(10) & 25(7) & 15(10) & $\mathrm{X}\to\overline{D}D^*$ & $\mathrm{X}\nrightarrow\omega J/\psi$ & \\
Y\cite{yybelle} & 3943(17) & 58(11) & 87(34) & $B\to K\mathrm{Y}$ & & $c\bar{c}$ hybrid \\
& & & & $\mathrm{Y}\to \omega J/\psi$ & $\mathrm{Y}\nrightarrow \overline{D}D^*$ & \\
Z\cite{zzbelle} & 3929(5) & 64(18) & 29(10) & $\gamma\gamma\to\mathrm{Z}$ & & $\chi_{c2}'(2^3P_2)$\\
& & & & $\mathrm{Z}\to D\overline{D}$ & $\mathrm{Z}\nrightarrow \overline{D}D^*$ & \\
\hline \hline
\end{tabular}
}
\end{table}
\subsection{Ringing in Belle: The X, Y, Z States}
With huge luminosities ($>350$ fb$^{-1}$) available to it, Belle has been announcing the discovery of resonance after resonance. By now we have X($3940\pm10$), Y($3943\pm17$), and Z($3929\pm5$)), with all masses being consistent with 3935 MeV. The latest results for these states are summarixed in Table~I. As shown in the table, the errors in the parameters are large enough, and the evidence at least for \textit{non-observations} is tentative enough so that not all three, X, Y, Z, need be distinct and separate states. This is particularly true for X and Z whose widths are similar, and both of them could accomodate different levels of $D\overline{D}$ and $\overline{D}D^*$ decays. It is obvious that lots more experimental data will be needed before the dust settles down. In this respect, it is very disappointing that BaBar, with comparable data sets available to it, has maintained an ominous silence on X, Y, Z. With the excitement these states have generated, one would think that BaBar would jump right in, and prove or disprove the Belle claims!
\begin{figure}[!tb]
\begin{center}
\includegraphics[width=2.3in]{babar_y426.eps}
\includegraphics[width=2.05in]{cleo_y4260_cleoiii_isr.ps}
\end{center}
\caption{The Y(4260) state observed in ISR production, and decay, $\mathrm{Y}\to\pi^+\pi^-J/\psi$; (left) by BaBar, (right) by CLEO.}
\end{figure}
\subsection{BaBar Chimes in: The Y(4260) State}
While BaBar has regretably stayed away from Belle's X, Y, Z states, it has exploited its expertise in ISR (initial state radiation), or RR (radiative return), studies to present another surprise\cite{babary}. It has presented evidence for the observation of a broad resonance formed in $e^+e^-$ annihilation following ISR, and decaying into $\pi^+\pi^-J/\psi$ (see Fig.~6). The parameters of the resonance are $M(\mathrm{Y})=4259\pm8^{+2}_{-6}~\mathrm{MeV}$, and $\Gamma(\mathrm{Y})=88\pm23^{+6}_{-4}~\mathrm{MeV}$. The formation of $Y(4260)$ in ISR and the S--wave dipion mass distribution essentially ensure that $J^{PC}(\mathrm{Y}(4260))=1^{--}$. This comes as a great surprise, because there is a deep minimum in the $R\equiv\sigma(e^+e^-\to\mathrm{hadrons})/\sigma(e^+e^-\to\mathrm{leptons})$ measurement at 4260 MeV, which would militate against the presence of a vector with this mass. It is therefore extremely important to confirm BaBar's observation. This has been done in two new measurements at CLEO.
In the first measurement, CLEO\cite{cleoy4260} has confirmed a strong (factor $\sim7$) enhancement of the $\pi\pi J/\psi$ yield at 4260 MeV compared to the yield at 4160 MeV and 4040 MeV. Lest one argue that enhancement at a single energy point does not confirm a resonance, in a second measurement, CLEO has confirmed the Y(4260) in ISR, exactly the same measurement as BaBar (see Fig.~6). They obtain $M(Y)=4283^{+17}_{-16}\pm4~\mathrm{MeV}$, and $\Gamma(Y)=70^{+40}_{-25}~\mathrm{MeV}$. The measurement has very small background, and a very clean ($\sim5\sigma$) signal. So, the Y(4260) $1^{--}$ state is very real. But, what is it? No charmonium vector is expected with this mass, which deepens the mystery. The usual speculations, such as misplaced charmonium, a $c\bar{c}$ hybrid, a molecule, etc., abound. Clearly, lots more measurements are needed to sift through these speculations.
As a postscript to the discovery of all these new and unanticipated states, and the uncertainty about their nature, let me recall that the discovery of $J/\psi$\cite{jpsiexp} was followed by a spate of theoretical papers about its nature\cite{jpsitheory}, most of which, even by famous Nobelists, turned out to be wrong. That is how physics develops.
Let me now move to the non--quarkonium results presented at the workshop.
\section{Heavy--Light ($Q\bar{q},~\overline{Q}q$)}
Here the heavy quark can be either the charm or bottom quark, and the light quark can be $n(=u,d)$, or $s$. The convention for labeling the quantum numbers of heavy--light mesons is to couple the spin $s_Q$ of the heavy quark with the $j_q=l+s_q$ of the light quark to give the total spin $J=j_q+s_Q$.
\subsection{Open--Charm, or $D(=c\bar{n},~c\bar{s})$ Mesons}
We have had considerable activity in this sector, with BaBar, Belle and CLEO (particularly in its reincarnation as CLEO-c) making important contributions.
The first exciting news was BaBar's discovery\cite{babarnewd} of a new narrow width $D^*_s(c\bar{s})$ meson with $M=2317~\mathrm{MeV}$, $J^P=0^+$, unexpectedly below the $DK$ threshold. This was followed by the discovery of its $J^+=1^+$ partner with $M=2483~\mathrm{MeV}$, below the $D^*K$ threshold, by CLEO\cite{cleonewd}. These mesons were expected to lie above their respective thresholds, and to be therefore quite wide. Theoretical suggestions to explain the observed characteristics of these mesons again range from simple $|c\bar{s}>$ to $DK^*$ molecules, to $|c\bar{s}>$ mixed with $DK$ and $D^*K$ continua, and to tetraquark, with no resolution so far. The corresponding non-strange $|c\bar{d}>$ states would be expected at 2217 MeV ($0^+$) and 2363 MeV ($1^+$). These are expected to be wide, and have not been firmly identified so far.
Admittedly, the main interest in heavy--light mesons rests in the study of their weak interaction properties, and CLEO-c has made this their major goal. Hopefully precision measurements in this sector will go a long way towards validating the lattice QCD calculations. CLEO has reported preliminary form factor results, $f(D^+)=222.6\pm16.7\pm^{+2.8}_{-3.4}~\mathrm{MeV}$\cite{cleodff} and $f(D_s^+)=282\pm16\pm7~\mathrm{MeV}$\cite{cleodsff}. The corresponding predicitions from unquenched lattice calculations are $f(D^+)=201\pm3\pm17~\mathrm{MeV}$ and $f(D^+_s)=249\pm3\pm16~\mathrm{MeV}$\cite{latticedff}. CLEO has also measured a large number of hadronic and weak decays of $D$ and $D_s$ mesons.
\subsection{Open--Beauty, or $B(=b\bar{n},~b\bar{s},~b\bar{c}$ Mesons}
This is the domain of CDF and D\O~contributions. The latest of their contributions relate to the identification and mass measurements of the excited $B$--mesons. In the notation $J(j_q)$ the masses are:\\
\begin{tabular}{lll}
CDF: & $M(B_1^0)=5734\pm3\pm2$ MeV, & $M(B_2^{*0})=5738\pm5\pm1$ MeV, \\
D\O: & $M(B_1^0)=5721\pm3\pm5$ MeV, & $M(B_2^{*0})=5746\pm5\pm5$ MeV, \\
\end{tabular}
\\
and the tour-de-force measurement by CDF is $M(B_c)=6275.2\pm4.3\pm2.3~\mathrm{MeV}$,
which compares with the lattice prediction of $M(B_c)=6304\pm12^{+18}_{-0}~\mathrm{MeV}$.
\subsubsection{The \textbf{Triumph} of the $B$-Factories}
Since we had a presentation of the latest results on $CP$ violation, let me summarize what we know now. The best values of the angles of the unitary triangle are ${\alpha=105^{+15}_{-9}}^\circ$, ${\beta=21.7^{+1.3}_{-1.2}}^\circ$, and $\gamma=65\pm20^\circ$. One set of CKM matrix elements is: $|V_{ud}|=0.97363(26)$, $|V_{us}|=0.2252(18)$, $|V_{ub}|=0.004455(33)$, and $|V_{td}|/|V_{ts}|=0.205(15)$, and, of course, $\sin2\beta=0.685\pm0.032$.
\section{Epilogue}
As I mentioned in the beginning, hadron spectroscopy has had a renaissance during the last five years. We have heard about exciting new discoveries at this Workshop, and the future promises many more to come.
On a personal note, I want to thank the organizers and the hosts of this Workshop for a very successful and productive meeting in this wonderful town.
This work was supported by the U.S. Department of Energy.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,379 |
\section{Conclusion}
We have shown that the Matrix-Exponential Problem is undecidable in
general, but decidable when the matrices $A_{1}, \ldots, A_{k}$ commute.
This is analogous to what was known for the discrete version
of this problem, in which the matrix exponentials $e^{At}$ are
replaced by matrix powers $A^n$.
A natural variant of this problem is the following:
\begin{definition}[Matrix-Exponential Semigroup Problem]
Given square matrices $A_{1}, \ldots, A_{k}$ and $C$, all of the
same dimension and all with real algebraic entries, is $C$ a member
of the matrix semigroup generated by
\begin{align*}
\lbrace \exp(A_{i} t_{i}) : t_{i} \geq 0 , i=1,\ldots,k \rbrace ?
\end{align*}
\end{definition}
When the matrices $A_1,\ldots,A_k$ all commute, the above problem is
equivalent to the Matrix-Exponential Problem, and therefore decidable. In the non-commutative case, the following result holds:
\begin{theorem}
The Matrix-Exponential Semigroup Problem is undecidable.
\end{theorem}
A proof will appear in a future journal version of this paper. This can be done by reduction from the Matrix-Exponential Problem, using a set of gadgets to force a desired order in the multiplication of the matrix exponentials.
It would also be interesting to look at possibly decidable
restrictions of the MEP/MESP, for example the case where $k=2$ with a
non-commuting pair of matrices, which was shown to be decidable for
the discrete analogue of this problem in \cite{MEHTP}. Bounding the dimension of the ambient vector space could also yield decidability, which has been partly accomplished in the discrete case in \cite{CK05}. Finally, upper bounding the complexity of our decision procedure for the commutative case would also be a worthwhile task.
\section{Decidability in the Commutative Case}
We start this section by reducing the Generalised MEP with commuting
matrices to LEP. The intuition behind it is quite simple: perform a
change of basis so that the matrices $A_{1}, \ldots, A_{k}$, as well
as $C$, become block-diagonal matrices, with each block being upper
triangular; we can then separate the problem into several
sub-instances, corresponding to the diagonal blocks, and finally make
use of our uniqueness result concerning strictly upper triangular
logarithms of upper unitriangular matrices.
\begin{theorem}
The Generalised MEP with commuting matrices reduces to LEP.
\end{theorem}
\begin{proof}
Consider an instance of the generalised MEP, as given in Definition~\ref{def:MEP},
with commuting $n\times n$ matrices $A_1,\ldots,A_k$ and target
matrix $C$.
We first show how to define a matrix $P$ such that each matrix
$P^{-1}A_iP$ is block diagonal, $i=1,\ldots,k$, with each block
being moreover upper triangular.
By Theorem~\ref{subspace_decomposition} we can write $\mathbb{C}^n$
as a direct sum of subspaces $\mathbb{C}^n = \oplus_{j=1}^b \mathcal{V}_j$
such that for every subspace $\mathcal{V}_j$ and matrix $A_i$, $\mathcal{V}_j$ is an
invariant subspace of $A_i$ on which $A_i$ has a single eigenvalue
$\lambda_i^{(j)}$.
Define a matrix $Q$ by picking an algebraic basis for each
$\mathcal{V}_j$ and successively taking the vectors of each basis to
be the columns of $Q$. Then, each matrix $Q^{-1} A_{i} Q$ is
block-diagonal, where the $j$-th block is a matrix $B^{(j)}_i$ that
represents $A_{i} \restriction{\mathcal{V}_j}$, $j=1,\ldots,b$.
Fixing $j\in\{1,\ldots,b\}$, note that the
matrices $B_1^{(j)},\ldots,B_k^{(j)}$ all commute. Thus we
may apply Theorem \ref{simultaneous-triangularisation} to obtain an
algebraic matrix $M_j$ such that each matrix $M_j^{-1} B^{(j)}_{i} M_j$
is upper triangular, $i=1,\ldots,k$. Thus we can write
\[ M_j^{-1} B^{(j)}_{i} M_j = \lambda_i^{(j)}I + N_i^{(j)} \]
for some strictly upper triangular matrix $ N_i^{(j)}$.
We define $M$ to be the block-diagonal matrix with blocks $M_1,\ldots,M_b$.
Letting $P=QM$, it is then the case
that $P^{-1} A_{i} P$ is block-diagonal, with the $j$-th block being
$\lambda_i^{(j)}I + N_i^{(j)}$ for $j=1,\ldots,b$. Now
\begin{align}
\prod \limits_{i=1}^{k} \exp(A_{i} t_{i}) = C \Leftrightarrow \prod \limits_{i=1}^{k} \exp(P^{-1}A_{i}P t_{i}) = P^{-1}CP .
\label{eq:block}
\end{align}
If $P^{-1}CP$ is not block-diagonal, with each block being upper
triangular and with the same entries along the diagonal, then Equation
(\ref{eq:block}) has no solution and the problem instance must be
negative. Otherwise, denoting the blocks $P^{-1}CP$ by $D^{(j)}$ for
$j \in \lbrace 1, \ldots, b \rbrace$, our problem amounts to
simultaneously solving the system of matrix equations
\begin{align}
\prod\limits_{i=1}^{k} \exp\big(\big(\lambda_i^{(j)}I + N_i^{(j)}\big)t_{i}\big) = D^{(j)}, \quad j \in \lbrace 1, \ldots, b \rbrace
\label{eq:main1}
\end{align}
with one equation for each block.
For each fixed $j$, the matrices $N_{i}^{(j)}$ inherit commutativity from
the matrices $B^{(j)}_{i}$, so we have
\begin{align*}
\prod\limits_{i=1}^{k} \exp((\lambda_i^{(j)}I + N_i^{(j)})t_{i}) &=
\exp\big(\sum_{i=1}^k (\lambda_i^{(j)}I +
N_i^{(j)}) t_i \big)\\
&= \exp\big(\sum_{i=1}^k \lambda_i^{(j)} t_i\big) \cdot
\exp\big(\sum_{i=1}^k N_i^{(j)} t_i \big) .
\end{align*}
Hence the system (\ref{eq:main1}) is equivalent to
\begin{align}
\exp\big(\sum_{i=1}^k \lambda_i^{(j)} t_i\big) \cdot
\exp\big(\sum_{i=1}^k N_i^{(j)} t_i \big) = D^{(j)}
\label{eq:main2}
\end{align}
for $j=1,\ldots,b$.
By assumption, the diagonal entries of each matrix $D^{(j)}$ are
equal to a unique value, say $c^{(j)}$.
Since the diagonal entries of
$\exp\left(\sum_{i=1}^kN^{(j)}t_i\right)$
are all $1$, the equation system (\ref{eq:main2}) is equivalent to:
\begin{align*}
\exp\big(\sum_{i=1}^k \lambda_i^{(j)} t_i\big)
= c^{(j)} \mbox{ and }\exp\big(\sum_{i=1}^k N_i^{(j)} t_i \big)
=\frac{1}{c^{(j)}} D^{(j)}
\end{align*}
for $j=1,\ldots,b$.
Applying Theorem \ref{logarithm_uniqueness}, the
above system can equivalently be written
\begin{align*}
\exp\big(\sum_{i=1}^k \lambda_i^{(j)} t_i\big)
= c^{(j)} \mbox{ and } \sum_{i=1}^k
N_i^{(j)} t_i =
S^{(j)}
\end{align*}
for some effectively computable matrix
$S^{(j)}$ with algebraic entries, $j=1,\ldots,b$.
Except for the additional linear equations, this has the form of an
instance of LEP.
However we can eliminate the linear equations by
performing a linear change of variables, i.e., by computing the
solution of the system in parametric form. Thus we finally arrive at
an instance of LEP.
\end{proof}
In the following result, we essentially solve the system of equations \ref{single_eqn_form}, reducing it to the simpler problem that really lies at its heart.
\begin{theorem}
\label{reference-for-log}
LEP reduces to ALIP.
\end{theorem}
\begin{proof}
Consider an instance of LEP, comprising a system of equations
\begin{align}
\exp\left(\sum_{\ell=1}^k \lambda_\ell^{(j)} t_\ell \right) = c_j \exp (d_j)
\quad j=1,\ldots,b,
\label{eq:LEPinstance}
\end{align}
and polyhedron $\mathcal{P}\subseteq \mathbb{R}^{2k}$, as described in
Definition~\ref{def:LEP}.
Throughout this proof, let $\log$ denote a fixed logarithm branch that
is defined on all the numbers $c_j, \exp(d_j)$ appearing
above, and for which $\log(-1) = i \pi$. Note that if any $c_j=0$ for
some $j$ then (\ref{eq:LEPinstance}) has no solution. Otherwise, by
applying $\log$ to each equation in (\ref{eq:LEPinstance}),
we get:
\begin{align}
\sum_{\ell=1}^k \lambda_\ell^{(j)} t_\ell = d_j+\log(c_j) + 2i\pi n_j \quad j=1,\ldots,b,
\label{eq:logs}
\end{align}
where $n_j \in \mathbb{Z}$.
The system of equations (\ref{eq:logs}) can be written in matrix form as
\begin{align*}
A \boldsymbol{t} \in \boldsymbol{d}+\log(\boldsymbol{c}) +
2i\pi \mathbb{Z}^b \, ,
\end{align*}
where $A$ is the $b\times k$ matrix with $A_{j,\ell} = \lambda_\ell^{(j)}$ and $\log$
is applied pointwise to vectors.
Now, defining the convex polyhedron $\mathcal{Q}\subseteq \mathbb{R}^{2b}$ by
\begin{align*}
\mathcal{Q} = \lbrace &(\Re(A\boldsymbol{y}), \Im(A\boldsymbol{y})) :
\boldsymbol{y}\in\mathbb{C}^k, (\Re(\boldsymbol{y}), \Im(\boldsymbol{y})) \in \mathcal{P} \rbrace \, ,
\end{align*}
it suffices to decide whether the affine lattice
$\boldsymbol{d} + \log(\boldsymbol{c})+ 2i\pi \mathbb{Z}^b$
intersects
$\lbrace \boldsymbol{x} \in \mathbb{C}^b : (\Re(\boldsymbol{x}),
\Im(\boldsymbol{x})) \in \mathcal{Q} \rbrace$.
Define $f:\mathbb{R}^b \rightarrow \mathbb{C}^b$ by
$f(\boldsymbol{v})= \boldsymbol{d} + \log(\boldsymbol{c}) +
2i\pi \boldsymbol{v}$,
and define a convex polyhedron $\mathcal{T}\subseteq \mathbb{R}^b$ by
\[\mathcal{T}=\{ \boldsymbol{v}\in\mathbb{R}^b : (\Re(f(\boldsymbol{v})),
\Im (f(\boldsymbol{v}))) \in \mathcal{Q} \} \, . \]
The problem then amounts to deciding whether the convex polyhedron
$\mathcal{T}$ intersects contains an integer point. Crucially, the
description of the convex polyhedron $\mathcal{T}$ is of the form
$\pi B\boldsymbol{x} \leq \boldsymbol{b}$, for some matrix $B$ and
vector $\boldsymbol{b}$ such that the entries of $B$ are real
algebraic and the components of $\boldsymbol{b}$ are real linear forms
in logarithms of algebraic numbers. But this is the form of an
instance of ALIP.
\end{proof}
We are left with the task of showing that ALIP is decidable. The argument essentially consists of reducing to a lower-dimensional instance whenever possible, and eventually either using the fact that the polyhedron is bounded to test whether it intersects the integer lattice or using Kronecker's theorem to show that, by a density argument, it must intersect the integer lattice.
\begin{theorem}
ALIP is decidable.
\end{theorem}
\begin{proof}
We are given a convex polyhedron $\mathcal{P}=\lbrace \boldsymbol{x} \in \mathbb{R}^{d} : \pi A \boldsymbol{x} \leq \boldsymbol{b} \rbrace$, where the coordinates $\boldsymbol{b}$ are linear forms in logarithms of algebraic numbers, and need to decide whether this polyhedron intersects $\mathbb{Z}^{d}$. Throughout this proof, $\log$ denotes the logarithm branch picked at the beginning of the proof of Theorem \ref{reference-for-log}. We start by eliminating linear dependencies between the logarithms appearing therein, using Masser's Theorem. For example, suppose that
\begin{align*}
b_{i} = r_{0} + r_{1} \log(s_{1}) + \cdots + r_{k} \log(s_{k}), r_{0}, r_{1}, s_{1}, \ldots, r_{k}, s_{k} \in \overline{\mathbb{Q}}.
\end{align*}
Due to Baker's theorem, there exists a non-trivial linear relation with algebraic coefficients amongs $\log(-1), \log(s_{1}), \ldots, \log(s_{k})$ if and only if there is one with integer coefficients. But such relations can be computed, since
\begin{align*}
&n_{0} \log(-1) + n_{1} \log(s_{1}) + \cdots + n_{k} \log(s_{k}) = 0 \Leftrightarrow \\
&(-1)^{n_{0}} s_{1}^{n_{1}} \cdots s_{k}^{n_{k}} = 1
\end{align*}
and since the group of multiplicative relations $L(-1, s_{1}, \ldots, s_{k})$ can be effectively computed. Whenever it contains a non-zero vector, we use it to eliminate an unnecessary $\log(s_{i})$ term, although never eliminating $\log(-1)$. When this process is over, we can see whether each term $b_{i}/\pi$ is algebraic or transcendental: it is algebraic if $b_{i} = \alpha \log(-1), \alpha \in \overline{\mathbb{Q}}$, and transcendental otherwise.
Now, when $\boldsymbol{x} \in \mathbb{Z}^{d}$, $A \boldsymbol{x}$ is a vector with algebraic coefficients, so whenever $b_{i} / \pi$ is transcendental we may alter $\mathcal{P}$ by replacing $\leq$ by $<$ in the $i$-th inequality, preserving its intersection with $\mathbb{Z}^{d}$. On the other hand, whenever $b_{i} / \pi$ is algebraic, we split our problem into two: in the first one, $\mathcal{P}$ is altered to force equality on the $i$-th constraint (that is, replacing $\leq$ by $=$), and in the second we force strict inequality (that is, replacing $\leq$ by $<$). We do this for all $i$, so that no $\leq$ is left in any problem instance, leaving us with finitely many polyhedra, each defined by equations of the form
\begin{align*}
K \boldsymbol{x} &= \boldsymbol{k} \quad &(\boldsymbol{k} \in \overline{\mathbb{Q}}^{d_{1}}) \\
M \boldsymbol{x} &< \boldsymbol{m} \quad &(\boldsymbol{m} \in \overline{\mathbb{Q}}^{d_{2}}) \\
F \boldsymbol{x} &< \boldsymbol{f} \quad &(\boldsymbol{f} \in \mathbb{R} \setminus \overline{\mathbb{Q}}^{d_{3}})
\end{align*}
where $K,M,F$ are matrices with algebraic entries. Before proceeding, we eliminate all such empty polyhedra; note that emptiness can be decided via Fourier-Motzkin elimination, as shown in Theorem \ref{thm:fme}.
The idea of the next step is to reduce the dimension of all the problem instances at hand until we are left with a number of new instances with full-dimensional open convex polyhedra, of the same form as the original one, apart from the fact that all inequalities in their definitions will be strict. To do that, we use the equations $K \boldsymbol{x} = \boldsymbol{k}$ to eliminate variables: note that, whenever there is an integer solution,
\begin{align*}
K \boldsymbol{x} = \boldsymbol{k}, \boldsymbol{x} \in \mathbb{Z}^{d} \Leftrightarrow \boldsymbol{x} = \boldsymbol{x}_{0} + M \boldsymbol{z},
\end{align*}
where $M$ is a matrix with integer entries, $\boldsymbol{x}_{0}$ is an integer vector and $\boldsymbol{z}$ ranges over integer vectors over a smaller dimension space, wherein we also define the polyhedron
\begin{align*}
\mathcal{Q} = \lbrace \boldsymbol{y} : \boldsymbol{x}_{0} + M \boldsymbol{y} \in \mathcal{P} \rbrace .
\end{align*}
Having now eliminated all equality constraints, we are left with a finite set of polyhedra of the form $\mathcal{P} = \lbrace \boldsymbol{x} \in \mathbb{R}^{d}: \pi A \boldsymbol{x} < \boldsymbol{b} \rbrace$ that are either empty or full-dimensional and open, and wish to decide whether they intersect the integer lattice of the corresponding space (different instances may lie in spaces of different dimensions, of course). Note that, when $\mathcal{P}$ is non-empty, we can use Fourier-Motzkin elimination to find a vector $\boldsymbol{q} \in \mathbb{Q}^{d}$ in its interior, and $\varepsilon > 0$ such that the $l_{1}$ closed ball of radius $\varepsilon$ and centre $\boldsymbol{q}$, which we call $\mathcal{B}$, is contained in $\mathcal{P}$.
The next step is to consider the Minkowski-Weyl decomposition of $\mathcal{P}$, namely $\mathcal{P} = \mathcal{H} + \mathcal{C}$, where $\mathcal{H}$ is the convex hull of finitely many points of $\mathcal{P}$ (which we need not compute) and $\mathcal{C} = \lbrace \boldsymbol{x} \in \mathbb{R}^{d}: A \boldsymbol{x} \leq \boldsymbol{0} \rbrace$ is a cone with an algebraic description. Note that $\mathcal{P}$ is bounded if and only if $\mathcal{C} = \lbrace \boldsymbol{0} \rbrace$, in which case the problem at hand is simple: consider the polyhedron $\mathcal{Q}$ with an algebraic description obtained by rounding up each coordinate of $\boldsymbol{b} / \pi$, which has the same conic part as $\mathcal{P}$ and which contains $\mathcal{P}$, and therefore is bounded; finally, compute a bound on $\mathcal{Q}$ (such a bound can be defined in the first-order theory of reals), which is also a bound on $\mathcal{P}$, and test the integer points within that bound for membership in $\mathcal{P}$. Otherwise,
\begin{align*}
\mathcal{C} = \lbrace \lambda_{1} \boldsymbol{c}_{1} + \cdots + \lambda_{k} \boldsymbol{c}_{k}: \lambda_{1}, \ldots, \lambda_{k} \geq 0 \rbrace,
\end{align*}
where $\boldsymbol{c}_{1}, \ldots, \boldsymbol{c}_{k} \in \overline{\mathbb{Q}}^{d}$ are the extremal rays of $\mathcal{C}$. Note that $\boldsymbol{q} + \mathcal{C} \subseteq \mathcal{P}$ and that $\mathcal{B} + \mathcal{C} \subseteq \mathcal{P}$.
Now we consider a variation of an argument which appears in \cite{KP}. Consider the computable set
\begin{align*}
\mathcal{L} = \mathcal{C}^{\perp} \cap \mathbb{Z}^{d} = \bigcap\limits_{i=1}^{k} A(\boldsymbol{c}_{i}) ,
\end{align*}
where $A(\boldsymbol{v})$ denotes the group of additive relations of $\boldsymbol{v}$.
If $\mathcal{L} = \lbrace \boldsymbol{0} \rbrace$ then due to Kronecker's theorem on simultaneous Diophantine approximation it must be the case that there exists a vector $(n_{1}, \ldots, n_{k}) \in \mathbb{N}^{k}$ such that
\begin{align*}
\operatorname{dist} \left(\boldsymbol{q} + \sum\limits_{i=1}^{k} n_{i} \boldsymbol{c}_{i}, \mathbb{Z}^{d} \right) \leq \varepsilon ,
\end{align*}
and we know that $\mathcal{P} \cap \mathbb{Z}^{d} \neq \emptyset$ from the fact that the $l_{1}$ closed ball $\mathcal{B}$ of radius $\varepsilon$ and centre $\boldsymbol{q}$ is contained in $\mathcal{P}$.
On the other hand, if
$\mathcal{L} \neq \lbrace \boldsymbol{0} \rbrace$, let
$\boldsymbol{z} \in \mathcal{L} \setminus \lbrace \boldsymbol{0}
\rbrace$.
Since $\mathcal{H}$ is a bounded subset of $\mathbb{R}^n$, the set
\[ \{ \boldsymbol{z}^T\boldsymbol{x} : \boldsymbol{x}\in \mathcal{P} \} =
\{ \boldsymbol{z}^T\boldsymbol{x} : \boldsymbol{x}\in \mathcal{H} \} \]
is a bounded subset of $\mathbb{R}$.
Therefore there exist
$a,b \in \mathbb{Z}$ such that
\begin{align*}
\forall \boldsymbol{x} \in \mathcal{P}, a \leq \boldsymbol{z}^{T} \boldsymbol{x} \leq b ,
\end{align*}
so we can reduce our problem to $b-a+1$ smaller-dimensional instances
by finding the integer points of
$\lbrace \boldsymbol{x} \in \mathcal{P} :
\boldsymbol{z}^{T} \boldsymbol{x} = i \rbrace$,
for $i \in \lbrace a, \ldots, b \rbrace$. Note that we have seen
earlier in the proof how to reduce the dimension of the ambient space
when the polyhedron $\mathcal{P}$ is contained in an affine
hyperplane.
\end{proof}
\section{Example}
Let $\lambda_{1}, \lambda_{2} \in \mathbb{R} \cap \bar{\mathbb{Q}}$ such that $\lambda_{1} > \lambda_{2}$ and consider the following commuting matrices $A_{1}, A_{2} \in (\mathbb{R} \cap \bar{\mathbb{Q}})^{2 \times 2}$:
\begin{align*}
A_{i} = \begin{pmatrix} \lambda_{i} && 1 \\ 0 && \lambda_{i} \end{pmatrix}, i \in \lbrace 1, 2 \rbrace .
\end{align*}
One can easily see that
\begin{align*}
\exp(A_{i} t_{i}) &= \exp(\lambda_{i} t_{i} I) \exp(t_{i} (A_{i} - \lambda_{i} I)) \\
&= \exp(\lambda_{i} t_{i})
\exp \begin{pmatrix} 0 && t_{i} \\ 0 && 0 \end{pmatrix} \\
&= \exp (\lambda_{i} t_{i})
\begin{pmatrix} 1 && t_{i} \\ 0 && 1 \end{pmatrix}, i \in \lbrace 1, 2 \rbrace .
\end{align*}
Let $c_{1}, c_{2} \in \mathbb{R} \cap \bar{\mathbb{Q}}$ such that $c_{1}, c_{2} > 0$, and let
\begin{align*}
C = \begin{pmatrix} c_{1} && c_{2} \\ 0 && c_{1} \end{pmatrix} .
\end{align*}
We would like to determine whether there exists a solution $t_{1}, t_{2} \in \mathbb{R}$, $t_{1}, t_{2} \geq 0$ to
\begin{align*}
\exp(A_{1} t_{1}) \exp(A_{2} t_{2}) = C
\end{align*}
This amounts to solving the following system of equations:
\begin{align*}
&\begin{cases}
\exp(\lambda_{1} t_{1} + \lambda_{2} t_{2}) = c_{1} \\
(t_{1} + t_{2}) \exp(\lambda_{1} t_{1} + \lambda_{2} t_{2}) = c_{2}
\end{cases}
&\Leftrightarrow \\
&\begin{cases}
\exp(t_{1} (\lambda_{1} - \lambda_{2}) + \frac{c_{2}}{c_{1}} \lambda_{2}) = c_{1} \\
t_{2} = \frac{c_{2}}{c_{1}} - t_{1}
\end{cases}
&\Leftrightarrow \\
&\begin{cases}
t_{1} = \frac{\log(c_{1}) - \frac{c_{2}}{c_{1}} \lambda_{2}}{\lambda_{1} - \lambda_{2}} \\
t_{2} = \frac{\frac{c_{2}}{c_{1}} \lambda_{1} - \log(c_{1})}{\lambda_{1} - \lambda_{2}}
\end{cases}
\end{align*}
Then $t_{1}, t_{2} \geq 0$ holds if and only if
\begin{align*}
\lambda_{2} \leq \frac{c_{1}}{c_{2}} \log(c_{1}) \leq \lambda_{1} .
\end{align*}
Whether these inequalities hold amounts to comparing linear forms in logarithms of algebraic numbers.
\section{Undecidability of the Non-Commutative Case}
In this section we show that the Matrix-Exponential Problem is
undecidable in the case of non-commuting matrices. We show
undecidability for the most fundamental variant of the problem, as
given in Definition~\ref{def:MEP}, in which the matrices have real
entries and the variables $t_i$ range over the non-negative reals.
Recall that this problem is decidable in the commutative case by the
results of the previous section.
\subsection{Matrix-Exponential Problem with Constraints}
The proof of undecidability in the non-commutative case is by
reduction from Hilbert's Tenth Problem. The reduction proceeds via
several intermediate problems. These problems are obtained by
augmenting MEP with various classes of arithmetic constraints on the
real variables that appear in the statement of the problem.
\begin{definition}
We consider the following three classes of arithmetic constraints
over real variables $t_1,t_2,\ldots$:
\begin{itemize}
\item $\mathcal{E}_{\pi\mathbb{Z}}$ comprises constraints of the form
$t_i\in\alpha+\beta\pi\mathbb{Z}$, where $\alpha$ and $\beta\neq 0$
are real-valued constants such that $\cos(2\alpha\beta^{-1})$,
$\beta$ are both algebraic numbers.
\item $\mathcal{E}_{+}$ comprises linear equations of the form
$\alpha_1 t_1 + \ldots + \alpha_n t_n = \alpha_0 $, for
$\alpha_0,\ldots,\alpha_n$ real algebraic constants.
\item $\mathcal{E}_{\times}$ comprises equations of the form
$t_\ell=t_it_j$.
\end{itemize}
\end{definition}
A class of constraints $\mathcal{E} \subseteq \mathcal{E}_{\pi\mathbb{Z}} \cup
\mathcal{E}_{+}\cup \mathcal{E}_{\times}$
induces a generalisation of the MEP problem as follows:
\begin{definition}[MEP with Constraints]
Given a class of constraints
$\mathcal{E} \subseteq \mathcal{E}_{\pi\mathbb{Z}} \cup
\mathcal{E}_{+}\cup \mathcal{E}_{\times}$,
the problem MEP$(\mathcal{E})$ is as follows. An instance consists
of real algebraic matrices $A_1,\ldots,A_k,C$ and a finite set of
constraints $E\subseteq\mathcal{E}$ on real variables
$t_1,\ldots,t_k$. The question is whether there exist non-negative
real values for $t_1,\ldots,t_k$ such that
$\prod_{i=1}^ke^{A_it_i}=C$ and the constraints $E$ are all
satisfied.
\label{def:contraintMEP}
\end{definition}
Note that in the above definition of MEP$(\mathcal{E})$ the set of
constraints $E$ only mentions real variables $t_1,\ldots,t_k$
appearing in the matrix equation $\prod_{i=1}^ke^{A_it_i}=C$.
However, without loss of generality, we can allow constraints to
mention fresh variables $t_i$, for $i>k$, since we can always define a
corresponding matrix $A_i=0$ for such variables for then
$e^{A_it_i}=I$ has no effect on the matrix product. In other words,
we effectively have constraints in $\mathcal{E}$ with existentially
quantified variables. In particular, we have the following
useful observations:
\begin{itemize}
\item[\textbullet] We can express inequality constraints of the form
$t_i\neq \alpha$ in $\mathcal{E}_{+}\cup \mathcal{E}_{\times}$ by
using fresh variables
$t_j,t_\ell$. Indeed $t_i \neq \alpha$ is satisfied whenever there
exist values of $t_j$ and $t_{\ell}$ such that $t_i=t_j+\alpha$ and
$t_jt_\ell=1$.
\item[\textbullet] By using fresh variables,
$\mathcal{E}_{+}\cup \mathcal{E}_{\times}$ can express polynomial
constraints of the form $P(t_1,\ldots,t_n)=t$ for $P$ a polynomial
with integer coefficients.
\end{itemize}
We illustrate the above two observations in an example.
\begin{example}
Consider the problem, given matrices $A_1,A_2$ and $C$, to decide
whether there exist $t_1,t_2 \geq 0$ such that
\[ e^{A_1t_1}e^{A_2t_2}=C \,\mbox{ and }\, t_1^2-1=t_2, t_2\neq 0 \, .\]
This is equivalent to the following instance of
MEP$(\mathcal{E}_{+}\cup\mathcal{E}_{\times})$: decide whether there exist $t_1,\ldots,t_5\geq 0$ such that
\[ \prod_{i=1}^5 e^{A_it_i}=C \,\mbox{ and }\, t_1t_1=t_3, t_3-1=t_2, t_2t_4=t_5, t_5=1\]
where $A_1,A_2$ and $C$ are as above and $A_3=A_4=A_5=0$.
\end{example}
We will make heavy use of the following proposition to combine several
instances of the constrained MEP into a single instance by combining
matrices block-wise.
\begin{proposition}\label{prop:remark}
Given real algebraic matrices $A_1,\ldots,A_k,C$ and
$A_1',\ldots,A_k',C'$, there exist real algebraic matrices
$A_1'',\ldots,A_k''$, $C''$ such that for all $t_1,\ldots,t_k$:
\[\prod_{i=1}^ke^{A_i''t_i}=C''\qquad\Leftrightarrow\qquad\prod_{i=1}^ke^{A_it_i}=C\wedge\prod_{i=1}^ke^{A_i't_i}=C'.\]
\label{prop:combine}
\end{proposition}
\begin{proof}
Define for any $i\in\{1,\ldots,k\}$:
\[A_i''=\begin{bmatrix}A_i&0\\0&A_i'\end{bmatrix},\qquad
C''=\begin{bmatrix}C&0\\0&C'\end{bmatrix}.\]
The result follows because the matrix exponential can be computed
block-wise (as is clear from its power series definition):
\[\prod_{i=1}^ke^{A_i''t_i}=\prod_{i=1}^k\begin{bmatrix}e^{A_it_i}&0\\0&e^{A_i't_i}\end{bmatrix}=
\begin{bmatrix}\prod_{i=1}^ke^{A_it_i}&0\\0&\prod_{i=1}^ke^{A_i't_i}\end{bmatrix}.\]
\end{proof}
We remark that in the statement of Proposition~\ref{prop:remark} the
two matrix equations that are combined are over the same set of
variables. However, we can clearly combine any two matrix equations
for which the common variables appear in the same order in the
respective products.
The core of the reduction is to show that the constraints in
$\mathcal{E}_{\pi\mathbb{Z}},\mathcal{E}_{+}$ and
$\mathcal{E}_{\times}$ do not make the MEP problem harder: one can
always encode them using the matrix product equation.
\begin{proposition}
MEP$(\mathcal{E}_{\pi\mathbb{Z}} \cup \mathcal{E}_{+} \cup
\mathcal{E}_{\times})$ reduces to MEP$(\mathcal{E}_{+} \cup
\mathcal{E}_{\times})$.
\label{lem:pi}
\end{proposition}
\begin{proof}
Let $A_1,\ldots,A_k,C$ be real algebraic matrices and
$E\subseteq \mathcal{E}_{\pi\mathbb{Z}} \cup \mathcal{E}_{+} \cup
\mathcal{E}_{\times}$
a finite set of constraints on $t_1,\ldots,t_k$. Since $E$ is
finite it suffices to show how to eliminate from $E$ each constraint
in $\mathcal{E}_{\pi \mathbb{Z}}$.
Let $t_j\in\alpha+\beta\pi\mathbb{Z}$ be a constraint in $E$. By definition
of $\mathcal{E}_{\pi\mathbb{Z}}$
we have
that $\cos(2\alpha\beta^{-1}),\sin(2\alpha\beta^{-1})$ and
$\beta\neq0$ are real algebraic. Now define the following extra
matrices:
\[A'_j=\begin{bmatrix}0&2\beta^{-1}\\-2\beta^{-1}&0\end{bmatrix},
C'=\begin{bmatrix}\cos(2\alpha\beta^{-1})&
\sin(2\alpha\beta^{-1})\\-\sin(2\alpha\beta^{-1})&\cos(2\alpha\beta^{-1})\end{bmatrix}.\]
Our assumptions ensure that $A_j'$ and $C'$ are both real algebraic.
We now have the following chain of equivalences:
\begin{align*}
e^{A'_j t_j}=C'
&\Leftrightarrow\begin{bmatrix}\cos(2t_j\beta^{-1})&
\sin(2t_j\beta^{-1})\\-\sin(2t_j\beta^{-1})&\cos(2t_j\beta^{-1})\end{bmatrix}
=C'\\
&\Leftrightarrow\cos(2t_j\beta^{-1})=\cos(2\alpha\beta^{-1})\\
&\qquad\wedge\sin(2t_j\beta^{-1})=\sin(2\alpha\beta^{-1})\\
&\Leftrightarrow2\beta^{-1}t_j=2\alpha\beta^{-1}\mod 2\pi\\
&\Leftrightarrow t_j\in\alpha+\beta\pi\mathbb{Z}.
\end{align*}
Thus the additional matrix equation $e^{A_j't_j}=C'$ is equivalent to
the constraint $t_j\in\alpha+\beta\pi\mathbb{Z}$. Applying
Proposition~\ref{prop:combine} we can thus eliminate this constraint.
\end{proof}
\begin{proposition}
MEP$(\mathcal{E}_{+}\cup\mathcal{E}_{\times})$ reduces to
MEP$(\mathcal{E}_{+})$.
\label{lem:times}
\end{proposition}
\begin{proof}
Let $A_1,\ldots,A_k,C$ be real algebraic matrices and
$E \subseteq\mathcal{E}_{+}\cup\mathcal{E}_{\times}$ a finite set of
constraints on variables $t_1,\ldots,t_k$. We proceed as above,
showing how to remove from $E$ each constraint from
$\mathcal{E}_{\times}$. In so doing we potentially increase the
number of matrices and add new constraints from $\mathcal{E}_{+}$.
Let $t_l=t_i t_j$ be an equation in $E$. To eliminate this equation
the first step is to introduce fresh
variables $x,x',y,y',z$ and add the constraints
\[ t_i=x,\, t_j=y,\, t_\ell = z,\]
which are all in $\mathcal{E}_{+}$. We now add a new matrix equation
over the fresh variables $x,x',y,y',z$ that is equivalent to the
constraint $xy=z$. Since this matrix equation involves a new set of
variables we are free to the set the order of the matrix products,
which is crucial to express the desired constraint.
The key gadget is the following matrix product equation,
which holds for any $x,x',y,y',z\geqslant0$:
\begin{align*}
&\begin{bmatrix}1&0&-z\\0&1&0\\0&0&1\end{bmatrix}\begin{bmatrix}1&0&0\\0&1&-y'\\0&0&1\end{bmatrix}
\begin{bmatrix}1&x&0\\0&1&0\\0&0&1\end{bmatrix}\\
&\qquad\times\begin{bmatrix}1&0&0\\0&1&y\\0&0&1\end{bmatrix}
\begin{bmatrix}1&-x'&0\\0&1&0\\0&0&1\end{bmatrix}=
\begin{bmatrix}1&x-x'&z-xy\\0&1&y-y'\\0&0&1\end{bmatrix}.
\end{align*}
Notice that each of the matrices on the left-hand side of the above
equation has a single non-zero off-diagonal entry. Crucially each
matrix of this form can be expressed as an exponential. Indeed we can
write the above equation as a matrix-exponential product
\[ e^{B_1z} e^{B_2y'} e^{B_3x} e^{B_4y} e^{B_5x'} = \begin{bmatrix}1&x-x'&z-xy\\0&1&y-y'\\0&0&1\end{bmatrix} \]
for matrices
\[\begin{array}{r@{}cp{.1cm}r@{}c}
B_1=&\begin{bmatrix}0&0&-1\\0&0&0\\0&0&0\end{bmatrix} &&
B_2=&\begin{bmatrix}0&0&0\\0&0&-1\\0&0&0\end{bmatrix} \\
B_3=&\begin{bmatrix}0&1&0\\0&0&0\\0&0&0\end{bmatrix} &&
B_4=&\begin{bmatrix}0&0&0\\0&0&1\\0&0&0\end{bmatrix} \\
B_5=&\begin{bmatrix}0&-1&0\\0&0&0\\0&0&0\end{bmatrix} &&
\end{array}\]
Thus the constraint $xy=z$ can be expressed as
\begin{gather}
e^{B_1z} e^{B_2y'} e^{B_3x} e^{B_4y} e^{B_5z'} = I \, .
\label{eq:fresh}
\end{gather}
Again, we can apply Proposition~\ref{prop:combine} to combine the
equation (\ref{eq:fresh}) with the matrix equation from the original
problem instance and thus encode the constraint $x=yz$.
\end{proof}
\begin{proposition}
MEP$(\mathcal{E}_{+})$ reduces to MEP.
\label{lem:plus}
\end{proposition}
\begin{proof}
Let $A_1,\ldots,A_k,C$ be real algebraic matrices and
$E \subseteq\mathcal{E}_{+}$ a set of constraints.
We proceed as above, showing how to eliminate each constraint from
$E$ that lies in $\mathcal{E}_{+}$, while preserving the set of
solutions of the instance.
Let $\beta+\sum_{i=1}^k\alpha_it_i=0$ be an equation in $E$.
Recall that $\beta,\alpha_1,\ldots,\alpha_k$ are real algebraic.
Define the extra matrices $A_1',\ldots,A_k'$ and $C'$ as follows:
\[A_i'=\begin{bmatrix}0&\alpha_i\\0&0\end{bmatrix},
\qquad C'=\begin{bmatrix}1&-\beta\\0&1\end{bmatrix}.\]
Our assumptions ensure that $A_1',\ldots,A_k'$ and $C'$ are all real algebraic.
Furthermore, the following extra product equation becomes:
\begin{align*}
\prod_{i=1}^ke^{A_i't_i}=C
&\Leftrightarrow\prod_{i=1}^k\begin{bmatrix}1&\alpha_it_i\\0&1\end{bmatrix}=\begin{bmatrix}1&-\beta\\0&1\end{bmatrix}\\
&\Leftrightarrow\sum_{i=1}^k\alpha_it_i=-\beta \, .
\end{align*}
\end{proof}
Combining Propositions~\ref{lem:pi}, \ref{lem:times}, and \ref{lem:plus} we
have:
\begin{proposition}
MEP$(\mathcal{E}_{\pi\mathbb{Z}}\cup\mathcal{E}_{+}\cup\mathcal{E}_{\times})$
reduces to MEP.
\end{proposition}
\subsection{Reduction from Hilbert's Tenth Problem}
\begin{theorem}
MEP is undecidable in the non-commutative case.
\end{theorem}
\begin{proof}
We have seen in the previous section that the problem
MEP$(\mathcal{E}_{\pi\mathbb{Z}}\cup\mathcal{E}_{+}\cup\mathcal{E}_{\times})$
reduces to MEP without constraints. Thus it suffices to reduce
Hilbert's Tenth Problem to
MEP$(\mathcal{E}_{\pi\mathbb{Z}}\cup\mathcal{E}_{+}\cup\mathcal{E}_{\times})$.
In fact the matrix equation will not play a role in the
target of this reduction, only the additional constraints.
Let $P$ be a polynomial of total degree $d$ in $k$ variables with
integer coefficients. From $P$ we build a homogeneous polynomial $Q$, by
adding a new variable $\lambda$:
\[Q(\mathbf{x},\lambda)=\lambda^dP\left(\frac{x_1}{\lambda},\ldots,
\frac{x_k}{\lambda}\right).\]
Note that $Q$ still has integer coefficients. Furthermore, we have the relationship
\[Q(\mathbf{x},1)=P(\mathbf{x}).\]
As we have seen previously, it is easy to encode $Q$ with constraints,
in the sense that we can compute a finite set of constraints
$E_Q \subseteq \mathcal{E}_{+}\cup\mathcal{E}_{\times}$ mentioning variables
$t_0,\ldots,t_m,\lambda$
such that $E$ is
satisfied if and only if $t_0=Q(t_1,\ldots,t_k,\lambda)$. Note that
$E_Q$ may need to mention variables other
than $t_1,\ldots,t_k$ to do that. Another
finite set of equations
$E_{\pi\mathbb{Z}}\subseteq\mathcal{E}_{\pi\mathbb{Z}}$ is used to
encode that $t_1,\ldots,t_k,\lambda\in\pi\mathbb{Z}$. Finally,
$E_{=}\subseteq\mathcal{E}_{+}\cup\mathcal{E}_{\times}$ is used to
encode $t_{0}=0$ and $1\leqslant \lambda\leqslant4$. The
latter is done by adding the polynomial equations $\lambda=1+\alpha^2$ and
$\lambda=4-\beta^2$ for some $\alpha$ and $\beta$. Finally we have the
following chain of equivalences:
\begin{align*}
&\exists t_0,\ldots,\lambda\geqslant0\text{ s.t. }E_Q\cup E_{\pi\mathbb{Z}}\cup E_{=}\text{ is satisfied }\\
&\qquad\Leftrightarrow\exists t_1,\ldots,\lambda\geqslant0\text{ s.t. } 0=Q(t_1,\ldots,t_k,\lambda)\\
&\qquad\qquad\wedge t_1,\ldots,t_k,\lambda\in\pi\mathbb{Z}\wedge 1\leqslant\lambda\leqslant4\\
&\qquad\Leftrightarrow\exists n_1,\ldots,n_k\in\mathbb{N}\text{ s.t. } 0=Q(\pi n_1,\ldots,\pi n_k,\pi)\\
&\qquad\Leftrightarrow\exists n_1,\ldots,n_k\in\mathbb{N}\text{ s.t. } 0=\pi^d Q(n_1,\ldots,n_k,1)\\
&\qquad\Leftrightarrow\exists n_1,\ldots,n_k\in\mathbb{N}\text{ s.t. } 0=P(n_1,\ldots,n_k).\\
\end{align*}
\end{proof}
\section{Introduction}
Reachability problems are a fundamental staple of theoretical computer
science and verification, one of the best-known examples being the
Halting Problem for Turing machines. In this paper, our motivation
originates from systems that evolve continuously subject to linear
differential equations; such objects arise in the analysis of a range
of models, including linear hybrid automata, continuous-time Markov
chains, linear dynamical systems and cyber-physical systems as they
are used in the physical sciences and engineering---see,
e.g.,~\cite{Alu15}.
More precisely, consider a system consisting of a finite number of
discrete locations (or control states), having the property that the
continuous variables of interest evolve in each location according to
some linear differential equation of the form $\dot{\boldsymbol{x}} = A
\boldsymbol{x}$; here $\boldsymbol{x}$ is a vector of continuous
variables, and $A$ is a square `rate' matrix of appropriate
dimension. As is well-known, in each location the closed form solution
$\boldsymbol{x}(t)$ to the differential equation admits a
matrix-exponential representation of the form $\boldsymbol{x}(t) =
\exp(At)\boldsymbol{x}(0)$. Thus if a system evolves through a series
of $k$ locations, each with rate matrix $A_i$, and spending time $t_i
\geq 0$ in each location, the overall effect on the initial continuous
configuration is given by the matrix
\begin{align*}
\prod \limits_{i=1}^{k} \exp(A_{i} t_{i}) \, ,
\end{align*}
viewed as a linear transformation on $\boldsymbol{x}(0)$.\footnote{In
this motivating example, we are assuming that there are no discrete
resets of the continuous variables when transitioning between
locations.}
A particularly interesting situation arises when the matrices $A_i$
commute; in such cases, one can show that the order in which the
locations are visited (or indeed whether they are visited only once or
several times) is immaterial, the only relevant data being the total
time spent in each location. Natural questions then arise as to what
kinds of linear transformations can thus be achieved by such systems.
\subsection{Related Work}
Consider the following problems, which can be seen as discrete analogues of the question we deal with in this paper.
\begin{definition}[Matrix Semigroup Membership Problem]
Given $k+1$ square matrices $A_{1}, \ldots, A_{k}, C$, all of the same dimension, whose entries are algebraic, does the matrix $C$ belong to the multiplicative semigroup generated by $A_{1}, \ldots, A_{k}$?
\end{definition}
\begin{definition}[Solvability of Multiplicative Matrix Equations]
Given $k+1$ square matrices $A_{1}, \ldots, A_{k}, C$, all of the same dimension, whose entries are algebraic, does the equation
\begin{align*}
\prod\limits_{i=1}^{k} A_{i}^{n_{i}} = C
\end{align*}
admit any solution $n_{1}, \ldots, n_{k} \in \mathbb{N}$?
\end{definition}
In general, both problems have been shown to be undecidable, in
\cite{Paterson} and \cite{MEHTP}, by reductions from Post's
Correspondence Problem and Hilbert's Tenth Problem, respectively.
When the matrices $A_{1}, \ldots, A_{k}$ commute, these problems are
identical, and known to be decidable, as shown in
\cite{MultiplicativeMatrixEquations}, generalising the solution of the
matrix powering problem, shown to be decidable in \cite{KL}, and the
case with two commuting matrices, shown to be decidable in \cite{ABC}.
See \cite{HalavaSurvey} for a relevant survey, and \cite{CK05} for
some interesting related problems.
The following continuous analogue of \cite{KL}'s Orbit Problem was
shown to be decidable in \cite{Hainry}:
\begin{definition}[Continuous Orbit Problem]
Given an $n \times n$ matrix $A$ with algebraic entries and two
$n$-dimensional vectors $\boldsymbol{x}, \boldsymbol{y}$ with
algebraic coordinates, does there exist a non-negative real $t$ such
that $\exp(At) \boldsymbol{x} = \boldsymbol{y}$?
\end{definition}
The paper \cite{ContinuousOrbitIPL} simplifies the argument of
\cite{Hainry} and shows polynomial-time decidability. Moreover, a
continuous version of the Skolem-Pisot problem was dealt with in
\cite{ContinuousSkolem}, where a decidability result is presented for
some instances of the problem.
As mentioned earlier, an important motivation for our work comes from
the analysis of hybrid automata. In addition to~\cite{Alu15},
excellent background references on the topic are
\cite{HenzingerSTOC,HenzingerLICS}.
\subsection{Decision Problems}
We start by defining three decision problems that will be the main
object of study in this paper: the \emph{Matrix-Exponential Problem},
the \emph{Linear-Exponential Problem}, and the
\emph{Algebraic-Logarithmic Integer Programming} problem.
\begin{definition}
An instance of the Matrix-Exponential Problem (MEP) consists of
square matrices $A_{1}, \ldots, A_{k}$ and $C$, all of the same
dimension, whose entries are real algebraic numbers. The problem
asks to determine whether there exist real numbers
$t_1,\ldots,t_k \geq 0$ such that
\begin{align}
\label{MEP}
\prod \limits_{i=1}^{k} \exp(A_{i} t_{i}) = C \, .
\end{align}
\label{def:MEP}
\end{definition}
We will also consider a generalised version of this problem, called
the \emph{Generalised MEP}, in which the matrices $A_1,\ldots,A_k$ and
$C$ are allowed to have complex algebraic entries and in which the
input to the problem also mentions a polyhedron
$\mathcal{P}\subseteq\mathbb{R}^{2k}$ that is specified by linear
inequalities with real algebraic coefficients. In the generalised problem
we seek $t_1,\ldots,t_k \in \mathbb{C}$ that satisfy (\ref{MEP}) and
such that the vector
$(\Re(t_1),\ldots,\Re(t_k),
\Im(t_1),\ldots,\Im(t_k))$ lies in $\mathcal{P}$.
In the case of commuting matrices, the Generalised Matrix-Exponential
Problem can be analysed block-wise, which leads us to the following
problem:
\begin{definition}
An instance of the Linear-Exponential Problem (LEP) consists of a system
of equations
\begin{align}
\label{single_eqn_form}
\exp\left(\sum_{i \in I} \lambda_i^{(j)} t_i \right) = c_j \exp (d_j)
\quad (j \in J),
\end{align}
where $I$ and $J$ are finite index sets, the $\lambda_i^{(j)}$, $c_j$
and $d_j$ are complex algebraic constants, and the $t_i$ are complex
variables, together with a polyhedron
$\mathcal{P} \subseteq \mathbb{R}^{2k}$ that is specified by a system
of linear inequalities with algebraic coefficients. The problem asks
to determine whether there exist $t_1,\ldots,t_k\in \mathbb{C}$ that
satisfy the system (\ref{single_eqn_form}) and such that
$(\Re(t_1),\ldots,\Re(t_k),\Im(t_1),\ldots,\Im(t_k))$
lies in $\mathcal{P}$.
\label{def:LEP}
\end{definition}
To establish decidability of the Linear-Exponential Problem, we reduce
it to the following
\emph{Algebraic-Logarithmic Integer Programming}
problem. Here a \emph{linear form in logarithms of algebraic numbers}
is a number of the form
$\beta_{0} + \beta_{1} \log(\alpha_{1}) + \cdots + \beta_{m}
\log(\alpha_{m})$,
where
$\beta_{0}, \alpha_{1}, \beta_{1}, \ldots, \alpha_{m}, \beta_{m}$ are
algebraic numbers and $\log$ denotes a fixed branch of the complex
logarithm function.
\begin{definition}
An instance of the Algebraic-Logarithmic Integer Programming Problem (ALIP) consists of a finite system of equations of the form
\begin{align*}
A \boldsymbol{x} \leq \frac{1}{\pi} \boldsymbol{b}
\end{align*}
where $A$ is an $m\times n$ matrix with real algebraic entries and
where the coordinates of $\boldsymbol{b}$ are real linear forms in
logarithms of algebraic numbers. The problem asks to determine whether
such a system admits a solution $\boldsymbol{x} \in \mathbb{Z}^{n}$.
\end{definition}
\subsection{Paper Outline}
After introducing the main mathematical techniques that are used in
the paper, we present a reduction from the Generalised Matrix
Exponential Problem with commuting matrices to the Linear-Exponential
Problem, as well as a reduction from the Linear-Exponential Problem to
the Algebraic-Logarithmic Integer Programming Problem, before finally showing that the Algebraic-Logarithmic Integer Programming Problem is decidable. By way of hardness, we will prove that the Matrix-Exponential Problem is
undecidable (in the non-commutative case), by reduction from Hilbert's
Tenth Problem.
\section{Mathematical Background}
\label{background}
\subsection{Number Theory and Diophantine Approximation}
A number $\alpha \in \mathbb{C}$ is said to be \emph{algebraic} if
there exists a non-zero polynomial $p \in \mathbb{Q}[x]$ for which
$p(\alpha) = 0$. A complex number that is not algebraic is said to be
\emph{transcendental}. The monic polynomial $p \in \mathbb{Q}[x]$ of
smallest degree for which $p(\alpha) = 0$ is said to be the minimal
polynomial of $\alpha$. The set of algebraic numbers, denoted by
$\overline{\mathbb{Q}}$, forms a field. Note that the complex
conjugate of an algebraic number is also algebraic, with the same
minimal polynomial. It is possible to represent and manipulate
algebraic numbers effectively, by storing their minimal polynomial and
a sufficiently precise numerical approximation. An excellent course
(and reference) in computational algebraic number theory can be found
in \cite{Cohen}. Efficient algorithms for approximating algebraic
numbers were presented in \cite{Pan}.
Given a vector $\boldsymbol{\lambda} \in \overline{\mathbb{Q}}^{m}$, its \emph{group of multiplicative relations} is defined as
\begin{align*}
L(\boldsymbol{\lambda}) = \lbrace \boldsymbol{v} \in \mathbb{Z}^{m} : \boldsymbol{\lambda}^{\boldsymbol{v}} = 1 \rbrace .
\end{align*}
Moreover, letting $\log$ represent a fixed branch of the complex logarithm function, note that $\log(\alpha_{1}), \ldots, \log(\alpha_{m})$ are linearly independent over $\mathbb{Q}$ if and only if
\begin{align*}
L(\alpha_{1}, \ldots, \alpha_{m}) = \lbrace \boldsymbol{0} \rbrace .
\end{align*}
Being a subgroup of the free finitely generated abelian group $\mathbb{Z}^{m}$, the group $L(\boldsymbol{\lambda})$ is also free and admits a finite basis.
The following theorem, due to David Masser, allows us to effectively determine $L(\boldsymbol{\lambda})$, and in particular decide whether it is equal to $\lbrace \boldsymbol{0} \rbrace$. This result can be found in \cite{Masser}.
\begin{theorem}[Masser]
The free abelian group $L(\boldsymbol{\lambda})$ has a basis $\boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{l} \in \mathbb{Z}^{m}$ for which
\begin{align*}
\max\limits_{1 \leq i \leq l, 1 \leq j \leq m} \lvert v_{i,j} \rvert \leq (D \log H)^{O(m^{2})}
\end{align*}
where $H$ and $D$ bound respectively the heights and degrees of all the $\lambda_{i}$.
\end{theorem}
Together with the following result, due to Alan Baker, Masser's theorem allows us to eliminate all algebraic relations in the description of linear forms in logarithms of algebraic numbers. In particular, it also yields a method for comparing linear forms in logarithms of algebraic numbers: test whether their difference is zero and, if not, approximate it numerically to sufficient precision, so as to infer its sign. Note that the set of linear forms in logarithms of algebraic numbers is closed under addition and under multiplication by algebraic numbers, as well as under complex conjugation. See \cite{Baker75} and \cite{BakerPaper}.
\begin{theorem}[Baker]
Let $\alpha_{1}, \ldots, \alpha_{m} \in \overline{\mathbb{Q}} \setminus \lbrace 0 \rbrace$. If
\begin{align*}
\log(\alpha_{1}), \ldots, \log(\alpha_{m})
\end{align*}
are linearly independent over $\mathbb{Q}$, then
\begin{align*}
1, \log(\alpha_{1}), \ldots, \log(\alpha_{m})
\end{align*}
are linearly independent over $\overline{\mathbb{Q}}$.
\end{theorem}
The theorem below was proved by Ferdinand von Lindemann in 1882, and later generalised by Karl Weierstrass in what is now known as the Lindemann-Weierstrass theorem. As a historical note, this result was behind the first proof of transcendence of $\pi$, which immediately follows from it.
\begin{theorem}[Lindemann]
If $\alpha \in \overline{\mathbb{Q}} \setminus \lbrace 0 \rbrace$, then $e^{\alpha}$ is transcendental.
\end{theorem}
We will also need the following result, due to Leopold Kronecker, on simultaneous Diophantine approximation, which generalises Dirichlet's Approximation Theorem. We denote the \emph{group of additive relations} of $\boldsymbol{v}$ by
\begin{align*}
A(\boldsymbol{v}) = \lbrace \boldsymbol{z} \in \mathbb{Z}^{d} : \boldsymbol{z} \cdot \boldsymbol{v} \in \mathbb{Z} \rbrace .
\end{align*}
Throughout this paper, $\operatorname{dist}$ refers to the $l_{1}$ distance.
\begin{theorem}[Kronecker]
\label{Kronecker}
Let $\boldsymbol{\alpha}_{1}, \ldots, \boldsymbol{\alpha_{k}} \in \mathbb{R}^{d}$ and $\boldsymbol{\beta} \in \mathbb{R}^{d}$. The following are equivalent:
\begin{enumerate}
\item For any $\varepsilon > 0$, there exists $\boldsymbol{n} \in \mathbb{N}^{k}$ such that
\begin{align*}
\operatorname{dist}(\boldsymbol{\beta} + \sum\limits_{i=1}^{k} n_{i} \boldsymbol{\alpha}_{i}, \mathbb{Z}^{d}) \leq \varepsilon .
\end{align*}
\item It holds that
\begin{align*}
\bigcap\limits_{i=1}^{k} A(\boldsymbol{\alpha}_{i}) \subseteq A(\boldsymbol{\beta}) .
\end{align*}
\end{enumerate}
\end{theorem}
Many of these results, or slight variations thereof, can be found in \cite{HardyAndWright} and \cite{Cassels}.
\subsection{Lattices}
Consider a non-zero matrix $K\in\overline{\mathbb{Q}}^{r\times d}$ and vector
$\boldsymbol{k} \in \overline{\mathbb{Q}}^r$. The following proposition shows
how to compute a representation of the affine lattice
$\{ \boldsymbol{x}\in\mathbb{Z}^d : K\boldsymbol{x} = \boldsymbol{k}
\}$.
Further information about lattices can be found in \cite{LatticeBook}
and \cite{Cohen}.
\begin{proposition}
There exist $\boldsymbol{x}_{0} \in \mathbb{Z}^{d}$ and
$M \in \mathbb{Z}^{d \times s}$, where $s < r$, such that
\begin{align*}
\{ \boldsymbol{x}\in\mathbb{Z}^d : K\boldsymbol{x} =
\boldsymbol{k} \} =
\boldsymbol{x}_{0} + \{ M \boldsymbol{y} : \boldsymbol{y} \in \mathbb{Z}^s \} \, .
\end{align*}
\end{proposition}
\begin{proof}
Let $\theta$ denote a primitive element of the number field
generated by the entries of $K$ and $\boldsymbol{k}$. Let the degree
of this extension, which equals the degree of $\theta$, be
$D$. Then for $\boldsymbol{x} \in \mathbb{Z}^d$ one can write
\begin{align*}
K \boldsymbol{x} = \boldsymbol{k} &\Leftrightarrow \left( \sum \limits_{i=0}^{D-1} N_{i} \theta^{i} \right) \boldsymbol{x} = \sum \limits_{i=0}^{D-1} \boldsymbol{k}_{i} \theta^{i} \\
&\Leftrightarrow N_{i} \boldsymbol{x} = \boldsymbol{k}_{i}, \forall i \in \lbrace 0, \ldots, D-1 \rbrace ,
\end{align*}
for some integer matrices
$N_{0}, \ldots, N_{D-1} \in \mathbb{Z}^{r \times d}$ and integer
vectors
$\boldsymbol{k}_{0}, \ldots, \boldsymbol{k}_{D-1} \in \mathbb{Z}^{r}$.
The solution of each of these equations is clearly an affine lattice, and
therefore so is their intersection.
\end{proof}
\subsection{Matrix exponentials}
Given a matrix $A \in \mathbb{C}^{n \times n}$, its exponential is defined as
\begin{align*}
\exp(A) = \sum \limits_{i=0}^{\infty} \frac{A^{i}}{i!} .
\end{align*}
The series above always converges, and so the exponential of a matrix is always well defined. The standard way of computing $\exp(A)$ is by finding $P \in \mathit{GL}_{n}(\mathbb{C})$ such that $J=P^{-1}AP$ is in Jordan Canonical Form, and by using the fact that $\exp(A) = P \exp(J) P^{-1}$, where $\exp(J)$ is easy to compute. When $A \in \overline{\mathbb{Q}}^{n \times n}$, $P$ can be taken to be in $GL_{n}(\overline{\mathbb{Q}})$; note that
\begin{align*}
\mbox{if } J &= \begin{pmatrix}
\lambda && 1 && 0 && \cdots && 0 \\
0 && \lambda && 1 &&\cdots && 0 \\
\vdots && \vdots && \ddots && \ddots && \vdots \\
0 && 0 && \cdots && \lambda && 1 \\
0 && 0 && \cdots && 0 && \lambda
\end{pmatrix} \mbox{ then } \\
\exp(Jt) &= \exp(\lambda t) \begin{pmatrix}
1 && t && \frac{t^{2}}{2} && \cdots && \frac{t^{k-1}}{(k-1)!} \\
0 && 1 && t && \cdots && \frac{t^{k-2}}{(k-2)!} \\
\vdots && \vdots &&\ddots && \ddots && \vdots \\
0 && 0 && \cdots && 1 && t \\
0 && 0 && \cdots && 0 && 1
\end{pmatrix} .
\end{align*}
Then $\exp(J)$ can be obtained by setting $t=1$, in particular $\exp(J)_{ij} = \frac{\exp(\lambda)}{(j-i)!}$ if $j \geq i$ and $0$ otherwise.
When $A$ and $B$ commute, so must $\exp(A)$ and $\exp(B)$. Moreover, when $A$ and $B$ have algebraic entries, the converse also holds, as shown in \cite{MatrixExps}. Also, when $A$ and $B$ commute, it holds that $\exp(A)\exp(B) = \exp(A+B)$.
\subsection{Matrix logarithms}
The matrix $B$ is said to be a logarithm of the matrix $A$ if $\exp(B) = A$. It is well known that a logarithm of a matrix $A$ exists if and only if $A$ is invertible. However, matrix logarithms need not be unique. In fact, there exist matrices admitting uncountably many logarithms. See, for example, \cite{MatrixLogs1} and \cite{MatrixLogs2}.
A matrix is said to be unitriangular if it is triangular and all its diagonal entries equal $1$. Crucially, the following uniqueness result holds:
\begin{theorem}
\label{logarithm_uniqueness}
Given an upper unitriangular matrix $M \in \mathbb{C}^{n \times n}$, there exists a unique strictly upper triangular matrix $L$ such that $\exp(L)=M$. Moreover, the entries of $L$ lie in the number field $\mathbb{Q}(M_{i,j}: 1 \leq i,j \leq n)$.
\end{theorem}
\begin{proof}
Firstly, we show that, for any strictly upper triangular matrix $T$ and for any $1<m<n$ and $i<j$, the term $(T^{m})_{i,j}$ is polynomial on the elements of the set $\lbrace T_{r,s} : s-r<j-i \rbrace$. This can be seen by induction on $m$, as each $T^{m}$ is strictly upper triangular, and so
\begin{align*}
(T^{m})_{i,j} = \sum\limits_{l=1}^{n} (T^{m-1})_{i,l} T_{l,j} = \sum\limits_{l=i+1}^{j-1} (T^{m-1})_{i,l} T_{l,j} .
\end{align*}
Finally, we show, by induction on $j-i$, that each $L_{i,j}$ is polynomial on the elements of the set
\begin{align*}
\lbrace M_{i,j} \rbrace \cup \lbrace M_{r,s} : s-r < j-i \rbrace .
\end{align*}
If $j-i \leq 0$, then $L_{i,j}=0$, so the claim holds. When $j-i>0$, as $L$ is nilpotent,
\begin{align*}
M_{i,j} &= \exp(L)_{i,j} = L_{i,j} + \sum\limits_{m=2}^{n-1} \frac{1}{m!} (L^{m})_{i,j} \\ \Rightarrow L_{i,j} &= M_{i,j} - \sum\limits_{m=2}^{n-1} \frac{1}{m!} (L^{m})_{i,j} .
\end{align*}
The result now follows from the induction hypothesis and from our previous claim, as this argument can be used to both construct such a matrix $L$ and to prove that it is uniquely determined.
\end{proof}
\subsection{Properties of commuting matrices}
We will now present a useful decomposition of $\mathbb{C}^{n}$ induced by the commuting matrices $A_{1}, \ldots, A_{k} \in \mathbb{C}^{n \times n}$. Let $\sigma(A_{i})$ denote the spectrum of the matrix $A_{i}$. In what follows, let
\begin{align*}
\boldsymbol{\lambda} = (\lambda_{1}, \ldots, \lambda_{k}) \in \sigma(A_{1}) \times \cdots \times \sigma(A_{k}) .
\end{align*}
We remind the reader that $\ker(A_{i} - \lambda_{i})^{n}$ corresponds to the generalised eigenspace of $\lambda_{i}$ of $A_{i}$. Moreover, we define the following subspaces:
\begin{align*}
\mathcal{V}_{\boldsymbol{\lambda}} = \bigcap \limits_{i=1}^{k} \ker(A_{i} - \lambda_{i} I)^{n}.
\end{align*}
Also, let $\Sigma = \lbrace \boldsymbol{\lambda} \in \sigma(A_{1}) \times \cdots \times \sigma(A_{k}) : \mathcal{V}_{\boldsymbol{\lambda}} \neq \lbrace \boldsymbol{0} \rbrace \rbrace$.
\begin{theorem}
\label{subspace_decomposition}
For all $\boldsymbol{\lambda} = (\lambda_{1}, \ldots, \lambda_{k}) \in \Sigma$ and for all $i \in \lbrace 1, \ldots, k \rbrace$, the following properties hold:
\begin{enumerate}
\item $\mathcal{V}_{\boldsymbol{\lambda}}$ is invariant under $A_{i}$.
\item $\sigma(A_{i} \restriction_{\mathcal{V}_{\boldsymbol{\lambda}}}) = \lbrace \lambda_{i} \rbrace$.
\item $\mathbb{C}^{n} = \bigoplus \limits_{\boldsymbol{\lambda} \in \Sigma} \mathcal{V}_{\boldsymbol{\lambda}} .$
\end{enumerate}
\end{theorem}
\begin{proof}
We show, by induction on $k$, that the subspaces $\mathcal{V}_{\boldsymbol{\lambda}}$ satisfy the properties above.
When $k = 1$, the result follows from the existence of Jordan Canonical Forms. When $k > 1$, suppose that $\sigma(A_{k}) = \lbrace \mu_{1}, \ldots, \mu_{m} \rbrace$, and let $\mathcal{U}_{j} = \ker(A_{k} - \mu_{j} I)^{n}$, for $j \in \lbrace 1, \ldots, m \rbrace$. Again, it follows from the existence of Jordan Canonical Forms that
\begin{align*}
\mathbb{C}^{n} = \bigoplus \limits_{j = 1}^{m} \mathcal{U}_{m} .
\end{align*}
In what follows, $i \in \lbrace 1, \ldots, k-1 \rbrace$ and $j \in \lbrace 1, \ldots, m \rbrace$. Now, as $A_{k}$ and $A_{i}$ commute, so do $(A_{k}-\mu_{j} I)$ and $A_{i}$. Therefore, for all $\boldsymbol{v} \in \mathcal{U}_{j}$, $(A_{k} - \mu_{j} I)^{n} A_{i} \boldsymbol{v} = A_{i} (A-\mu_{j} I)^{n} \boldsymbol{v} = \boldsymbol{0}$, so $A_{i} \boldsymbol{v} \in \mathcal{U}_{j}$, that is, $\mathcal{U}_{j}$ is invariant under $A_{i}$. The result follows from applying the induction hypothesis to the commuting operators $A_{i} \restriction_{\mathcal{U}_{j}}$.
\end{proof}
We will also make use of the following well-known result on simultaneous triangularisation of commuting matrices. See, for example, \cite{CommutingMatrices}.
\begin{theorem}
\label{simultaneous-triangularisation}
Given $k$ commuting matrices $A_{1}, \ldots, A_{k} \in \overline{\mathbb{Q}}^{n \times n}$, there exists a matrix $P \in \mathit{GL}_{n}(\overline{\mathbb{Q}})$ such that $P^{-1}A_{i}P$ is upper triangular for all $i \in \lbrace 1, \ldots, k \rbrace$.
\end{theorem}
\subsection{Convex Polyhedra and Semi-Algebraic Sets}
A convex polyhedron is a subset of $\mathbb{R}^{n}$ of the form $\mathcal{P} = \lbrace \boldsymbol{x} \in \mathbb{R}^{n} : A \boldsymbol{x} \leq \boldsymbol{b} \rbrace$, where $A$ is a $d \times n$ matrix and $\boldsymbol{b} \in \mathbb{R}^{d}$. When all the entries of $A$ and coordinates of $\boldsymbol{b}$ are algebraic numbers, the convex polyhedron $\mathcal{P}$ is said to have an algebraic description.
A set $S \subseteq \mathbb{R}^{n}$ is said to be semi-algebraic if it is a Boolean combination of sets of the form $\lbrace \boldsymbol{x} \in \mathbb{R}^{n}: p(\boldsymbol{x}) \geq 0\rbrace$, where $p$ is a polynomial with integer coefficients. Equivalently, the semi-algebraic sets are those definable by the quantifier-free first-order formulas over the structure $(\mathbb{R}, <, +, \cdot, 0, 1)$.
It was shown by Alfred Tarski in \cite{Tarski} that the first-order theory of reals admits quantifier elimination. Therefore, the semi-algebraic sets are precisely the first-order definable sets.
\begin{theorem}[Tarski]
The first-order theory of reals is decidable.
\end{theorem}
See \cite{Renegar} and \cite{BPR06} for more efficient decision procedures for the first-order theory of reals.
\begin{definition}[Hilbert's Tenth Problem]
Given a polynomial $p \in \mathbb{Z}[x_{1}, \ldots, x_{k}]$, decide whether $p(\boldsymbol{x}) = 0$ admits a solution $\boldsymbol{x} \in \mathbb{N}^{k}$. Equivalently, given a semi-algebraic set $S \subseteq \mathbb{R}^{k}$, decide whether it intersects $\mathbb{Z}^{k}$.
\end{definition}
The following celebrated theorem, due to Yuri Matiyasevich, will be used in
our undecidability proof; see \cite{HTP} for a self-contained proof.
\begin{theorem}[Matiyasevich]
Hilbert's Tenth Problem is undecidable.
\end{theorem}
On the other hand, our proof of decidability of ALIP makes use of some techniques present in the proof of the following result, shown in \cite{KP}:
\begin{theorem}[Khachiyan and Porkolab]
It is decidable whether a given \emph{convex} semi-algebraic set $S \subseteq \mathbb{R}^{k}$ intersects $\mathbb{Z}^{k}$.
\end{theorem}
\subsection{Fourier-Motzkin Elimination}
Fourier-Motzkin elimination is a simple method for solving systems of
inequalities. Historically, it was the first algorithm used in solving
linear programming, before more efficient procedures such as the
simplex algorithm were discovered. The procedure consists in isolating one
variable at a time and matching all its lower and upper bounds. Note
that this method preserves the set of solutions on the remaining
variables, so a solution of the reduced system can always be extended
to a solution of the original one.
\begin{theorem}
\label{thm:fme}
By using Fourier-Motzkin elimination, it is decidable whether a
given convex polyhedron
$\mathcal{P} = \lbrace \boldsymbol{x} \in \mathbb{R}^{n} : \pi
A\boldsymbol{x} < \boldsymbol{b} \rbrace$,
where the entries of $A$ are all real algebraic numbers and
those of $\boldsymbol{b}$ are real linear forms in logarithms of
algebraic numbers, is empty. Moreover, if $\mathcal{P}$ is
non-empty one can effectively find a rational vector
$\boldsymbol{q} \in \mathcal{P}$.
\end{theorem}
\begin{proof}
When using Fourier-Motzkin elimination, isolate each term $\pi x_{i}$, instead of just isolating the variable $x_{i}$. Note that the coefficients of the terms $\pi x_{i}$ will always be algebraic, and the loose constants will always be linear forms in logarithms of algebraic numbers, which are closed under multiplication by algebraic numbers, and which can be effectively compared by using Baker's Theorem.
\end{proof} | {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,640 |
They call him 'Mr. Mentor'
Hyunjin Seo on November 26, 2010
"I've mostly got A's and B's but only one D in AP Bio. I struggled with this subject…"
Paola Portillo, a senior and cheerleader at Richmond High School, starts to jabber as soon as she closes the door of Mr. Mentor's office. Another 10th grader, Magnolia Lopez, also occupying his cramped office smelling somewhat of musky old papers and Febreeze, chimes in with chatter about how to start a "toy drive project" aimed at gathering toys to help the Oakland Children's Hospital. Of course this is Mr. Mentor's idea. "He's cool. If you need somebody to talk, he's always there for you."
Amongst Richmond High School students "Mr. Mentor" is more familiar than his real name, Sammie Lee Hill; though perhaps some students aren't aware of his surprisingly diverse background as a police officer, lyricist and TV show extra.
Mr. Mentor believes that education is the key for young generations' success as role models.
Before working at Richmond High School, Hill spent 25 years as a California State prison guard and then as a police officer in the San Pablo Police Department. Throughout his law enforcement career, he particularly enjoyed working with youth between ages 8 to 17 through the "Police Activity League." So after retiring, officer Hill began working as an educator at Richmond High School. "I've seen a lot of troubled kids during my police officer years. But if you give them some love and respect, you'll get the same thing as return. It's very simple."
Living up to his nickname, Hill has been trying to point students in the right direction throughout his 11 years at Richmond High School. If students start to rack up absences, Hill has been known to pay a visit to their homes and "bring them back." The mediation program that Hill sponsors gives students a chance to resolve conflicts, and according to Magnolia "the program really works, it stops making problems get bigger."
As a co-founder of the 21st Century Mentors Foundation (CMF), a program that provides guidance, counseling and scholarships for highly motivated students, Hill seeks to inspire emerging young leaders. Last May, 40 Richmond High students from the program traveled to Universal Studio in Los Angeles, entirely funded by the CMF as a reward for their demonstrated initiative and dedication. For some students the trip offered a chance to broaden perspectives. "Our mediation students need to be rewarded, they never been to Universal Studios, they even never been to Southern California."
Cluttering the walls of his small office are mementos from former students.
With his background as a lyricist—songwriter of "Mentor" and "We can change the world"—he now routinely gives workshops on song writing and publishing for students interested in a music career. "There are a lot of types of mentors out there like truck drivers or a friend in jail," he said, "but these are not good mentors." Hill hopes to show students how to be positive role models.
"I want to see every one of these kids graduate from high school and go to college and become productive role models of our community." when asked his dream, Mr. Mentor pondered for a moment and answered with a smile. He asked to stop the interview and hurried out of his office to get the delivery food he ordered an hour ago for two students who didn't have time to get food because of their busy schedules. This is Mr. Mentor's other job.
Company Town | Education | Front | Richmond Faces
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Poway single mother killed while walking to work
By WebMaster - On Nov 30th 2009 - in Legal Press Release
Legal news for California criminal lawyers. A woman was hit by a car while walking to work on Monday.
California criminal lawyers alert-A single mother is dead after being hit by a car while walking on the sidewalk.
San Diego, CA—A single mother, who was trying to save money by walking to work, was fatally struck by an intoxicated driver on Monday, November 23, 2009. A car jumped onto the sidewalk and struck the woman along Community Road at Aubrey Street, as reported by NBC San Diego.
Kristen Ann Bedard, 36, who is a single mother of two children was fatally struck while walking to work because she couldn't afford the gas. Becky Anderson, 44, was driving the vehicle that jumped the sidewalk and struck Bedard. Anderson was arrested and charged with vehicular manslaughter and driving under the influence. Bedard was pronounced dead at the scene by responding emergency medical services (EMS) teams, when her injuries proved fatal. It is currently unknown what exactly caused Anderson to drive her car up onto the sidewalk. Witnesses reported seeing police officials removing several objects from Anderson's vehicle, which included a pill bottle. Investigators stated Anderson was under the influence of Valium and methadone. Bedard leaves behind two children, 14-year-old Tyler, and 13-year-old Kelsey. Police officials are reportedly conducting a full investigation into the fatal wreck.
A memorial college fund has been set up for Bedard's two children at Point Loma Credit Union under the name of KTK Bedard Memorial Fund.
Legal News Reporter: Nicole Howley-Legal news for California criminal lawyers.
Indiana 63 semi-school bus crash injured 2 people
3 mentally disabled passengers injured in Akron van crash | {
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Public access to charter school spending is "just not going to happen," says House Ed Committee chairman
By Patrick O'Donnell, The Plain Dealer
ohio-statehouse.jpg
Proposals to make charter schools provide more financial information to the public are not getting support in the Ohio House.
(Rich Exner, Northeast Ohio Media Group)
COLUMBUS, Ohio -- The Ohio House won't force companies that make a profit from running charter schools to show the public any more of how they spend tax money than they do now, Education Committee Chairman Bill Hayes says.
House Democrats want the financial books of charter schools to be as open as those of a traditional school district. State Auditor Dave Yost won't go that far, but wants for-profit operators to report a lot more than they do now.
But not enough House Republicans agree, Hayes said.
"That's just not going to happen," Hayes said. "With the folks you have in the majority now. We're not going to make private companies' records public."
It's a controversial issue for charter schools -- public schools that are privately-run, but which receive most of their money from state taxes.
Critics are upset that some charter schools pay more than 90 percent of their money to private companies to run the school. Those companies then only report their spending to the public in very broad categories. On the other side, charter school supporters say, the companies are paid to provide a service and they should be judged on their performance alone.
The dueling sides, and lack of clarity about how much a charter school is a public school and how much it is a private entity, can lead to long legal battles over to whom the buildings, books and computers bought with those tax dollars belong.
The Ohio Supreme Court is now deciding in a case involving charter school operator White Hat Management and a former charter school board over who owns the computers, books and other materials bought with management fees that came from tax dollars.
Hayes said his committee will likely approve House Bill 2, a charter school reform bill, on Wednesday and send it to the full House for a vote. Though the bill makes several changes that even charter school critics are happy with, committee members today rejected -- again -- language that would make charter school finances more open.
After rejecting several amendments last week -- including some that Yost wanted -- members voted this morning to table new bill amendments that would have called for more charter openness.
Click here for an example of how current financial reports look for private charter operators, and what Yost wants them to look like.
Rep. Teresa Fedor, the ranking Democrat on the committee, was upset that amendments proposed by Democrats had no Republican support.
"The basic components of good government were rejected ... all with the fallacy that private companies cannot provide public access," Fedor said.
Hayes said there are many changes in the bill already that charter schools are unhappy with. He doubts enough House members would pass a bill that went further and required more financial disclosure.
"I don't think the bill would pass if that were in the bill," Hayes said.
"(Yost) got another shot at that place we call the Senate," Hayes added. "It may fly there, but I don't think it will in the House."
Rep. Kristina Roegner, a Hudson Republican and co-sponsor of HB 2, disagrees. She twice proposed adding some of Yost's proposals to the bill, but they did not make it into the first round of amendments to the bill last week and were again voted down today.
"That's unfortunate," said Roegner, who could not make today's committee meeting because of a previous conflict. "I thought it was a good idea. I support additional transparency when it comes to financial reporting, and I've been pushing very hard for that."
Roegner did not accept Hayes's reasoning for not including Yost's requests in the bill.
"To me that's a non answer," Roegner said. "I want policy reasons why. To say that others don't want it, that's not a reason not to support it."
She added: "If I was getting good answers, I would stop pushing it."
Roegner's absence led to some political maneuvering both in support of her and Yost's proposals, and against them. Because her amendments would not be included automatically by Hayes, they would have to be proposed by someone at the meeting, then be approved by the full committee to be added to the bill.
Roegner said she could not find any Republican members to put them up for vote today, so she reached out to Democrat John Patterson, of Jefferson. Though he proposed them for her, the committee voted them down.
With Roegner absent, she was replaced on the committee with another Republican, Mike Dovilla of Berea, the bill's other co-sponsor but not normally a member of the education committee. Dovilla was among those voting to table, essentially killing, Roegner's proposals.
Roegner said she was disappointed to be replaced, if even for just this week, because adding in Dovilla provided votes against amendments she wanted.
Fedor, the ranking Democrat on the committee, said she likes some of the changes in the bill, but many are technical and she is till researching details of them. And she thinks the House is skipping making substantive changes.
"It looks good on its face, but the real problems ... I don't think at the end of the day we've gone far enough," Fedor said. "House Bill 2 falls woefully short."
Roegner, despite her disappointment, disagreed and said she would still vote for the bill.
"I still stand by the fact that HB 2 is a very good bill," Roegner said. "It's small steps in the right direction. I'm advocating for big steps in that direction."
Hayes also said he considers the bill a strong one -- and one that the Ohio Senate can adjust when it takes up the issue next. State Sen. Peggy Lehner, chair of the Senate Education Committee, is expected to have her own charter school reform proposal later this week.
"I think we're getting (complaints) from both sides," Hayes said. "We might have reached an equilibrium where we can vote this thing out and let the Senate do their thing with it."
To follow education news from Cleveland and affecting all of Ohio, follow this reporter on Facebook as @PatrickODonnellReporter | {
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\section{Introduction}
Andrew Steane, one of the pioneers of quantum computing, once reportedly quipped,
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{\bf {\em A quantum computer is an error correction machine - computation is just a byproduct.} }
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Steane provides an extremely apt description of how any large-scale active quantum technology will ultimately behave. Quantum information processing suffers from two disadvantages; controllable quantum bits, or qubits, are extremely susceptible to noise from bad control or the external environment, and quantum algorithms are, by nature, exceedingly sensitive to errors. Even a single error during the execution of an algorithm can lead to essentially random output.
Hence, Quantum Error Correction (QEC) was quickly recognized as a necessity for any commercially viable computational or communications protocol, and the theoretical development of error correction techniques is as old as the first architectural models for quantum computers \cite{CZ95,K98,LD98,KLM01}. Developed in the mid 1990's by researchers such as Peter Shor, Andrew Steane, Alexei Kitaev, Daniel Gottesman and Robert Calderbank \cite{CLSZ95,S95,BDSW96,S96,CRSS98}, QEC, when combined with the principle of fault-tolerant quantum computation \cite{DS96,S96+,K97,G98,G00+,K03,G09,DMN13}, leads to arguably the most important theoretical result in quantum computing: the threshold theorem \cite{AB97}.
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{\bf {\em A quantum computation of arbitrary size can be completed successfully with faulty qubits, with a polylogarithmic resource overhead, provided that the physical error rate associated with each qubit and logic gate is below a maximum value, dubbed the fault-tolerant threshold. } }
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What this theorem is basically saying is that provided the error experienced by each qubit is below a certain value (the threshold), error correction will correct more errors than it introduces and a computation, no matter how large, will always be error free by introducing extra qubits.
The value of the fault-tolerant threshold is determined by many factors - the QEC code utilized, the way error correction codes are constructed, and any physical restrictions of the quantum hardware such as if qubits can be coupled together arbitrarily or are interactions restricted to a fixed geometry \cite{KLZ96,AGP08,KBM09,AT06,AP08,PR11,SDT07,DA07,SFH07,S03,SBFRYSGF06}. Initial estimates were very unfavorable, with thresholds for the Steane code and others of the order of 0.01\% or lower \cite{KLZ96,K03,AB97,K97,G97+}. However, this has improved dramatically with the development of topological models of QE. These exhibit fault-tolerant thresholds approaching 1\% for models such as the surface code \cite{DKLP02,FSG08,FMMC12,S14,WFSH09}, and Raussendorf code \cite{RHG06,RH07,RHG07}. These codes are also much more amenable to physical implementation as they are defined on a two-dimensional (surface code) or three-dimensional (Raussendorf code) array of nearest-neighbor interacting qubits.
The high fault-tolerant threshold, the nearest-neighbour nature of these topological codes, and the way in which quantum algorithms are implemented have resulted in them becoming the preferred technique for large-scale quantum computing architectures \cite{DFSG08,DMN08,D09,JMFMKLY10,YJG10,NTDS13,MRRBMD14,LBS10,LHMB15,LWFMDWH15,HPHHFRSH15,ONRMB16}. Essentially all major physical system are now targeting either the surface code or Raussendorf code for their architectures, and physical systems such as Ion Traps and Superconducting qubits are now demonstrating gate and qubit error rates either at, or below, the fault-tolerant threshold \cite{B14,L16}. It is becoming increasingly probable that a functional, commercial quantum computing system will be build using topological QEC as the fundamental computational model.
In Ref.\cite{PFW16}, Paler, Fowler and Wille have compiled a review that details how both computation and error correction is performed in these models. This article continues from this review and will examine both the structure of a topological quantum circuit and how these circuits will ultimately be optimised and implemented on a real world quantum computer.
\section{Topology: Cofffee cups and doughnuts}
Topology, unsurprisingly, plays a crucial role in the function and operation of topological quantum error correction. As with essentially all QEC codes that are considered implementable on large-scale hardware, topological quantum codes are defined in terms of stabiliser operators \cite{G98+}. A quantum state $|\psi\rangle$ is stabilized by an operator $K$, such that $K|\psi\rangle = |\psi\rangle$. A topological quantum code is defined by a set of these operators, which are defined locally. That is, they are defined over a small group of qubits that are nearby to each other\cite{K03}. However, the encoded state defined by these operators have certain global properties. Logical operations, those that define the encoded qubit state, are defined with respect to the entire state - they cannot be defined locally. This is the essential nature of a topological code. Individual stabilizers that are used to perform error correction are defined locally, while logical information is defined globally.
As summarized in Ref. \cite{PFW16}, a two-dimensional lattice of qubits (for the surface code), or a three-dimensional lattice of qubits (for the Raussendorf code) defines a unique quantum state \cite{FMMC12,RHG07}. The eigenvalues of each of the stabilizers associated with the lattices are measured in order to detect and correct for quantum errors that can occur due to imperfect physical qubits and gates \cite{DFTMN10,GP10,GP14,F15+}. Information is encoded into this lattice through the creation of holes, or defects. Defects are regions of the lattice that have been deactivated by having qubits in these regions removed. By removing qubits, or deactivating certain parts of the lattice, degrees of freedom are introduced into the quantum state that can be used to store and manipulate information which is protected from errors due to the properties of the remaining lattice, which is called the bulk.
Interactions (quantum gates) in this model are enacted through an operation called braiding. Braiding is where defects are perturbed such that they {\em move} through the lattice as it evolves in time, and wrap around each other like a tangled ball of string \cite{FMMC12,PF13,DN12}. An example of a large topological circuit that enacts a set of logical gates on encoded qubits, in a fault-tolerant way, is illustrated in Figure \ref{fig:four}. The spatial cross section is illustrated, as well as the temporal axis. The spatial cross section defines the number of qubits used in the surface code while the temporal axis defines how defects are created and manipulated over time \cite{DSMN13}. In the Raussendorf model, all three dimensions of the lattice consist of physical qubits which are sequentially measured along the temporal axis. Measurements are used to define and manipulate the defects through teleportation along this temporal axis of the Raussendorf code.
\begin{figure*}[ht!]
\begin{center}
\resizebox{0.8\linewidth}{!}{0.4\includegraphics{Figure1.jpeg}}
\end{center}
\caption{{\bf How to optimise topological quantum circuits.} Figure a. represents the topological equivalency of a coffee cup and a donut. Figure b. illustrates a {\em plumbing piece}, the basic unit of measure for a topological quantum circuit. Increasing the error correction code required more physical qubits for each pluming piece. Figure c. Topological deformation can be used to reduce the volume, and hence physical resources, without changing its information processing properties.}
\label{fig:one}
\end{figure*}
These structures are topological in nature and, hence, standard definitions of topology apply. The nature of a topological space is that it is preserved through operations known as continuous deformation. Continuous deformation is where a structure is stretched or bent without being cut or glued together at any point. The quintessential example of this is the topological equivalence between a coffee cup and a donut shown in Figure \ref{fig:one}a. Simply by stretching and bending, the coffee cup can be converted to a donut and vice versa. As each structure has only a single hole, they are topologically equivalent.
A qualtum program is therefore, literally, defined and described by a puzzle and this puzzle can be shaped, stretched and molded to change the physical resources needed by a quantum program without changing the program itself \cite{PF13}.
\section{Measuring and benchmarking quantum circuits}
In order to derive relevant metrics when constructing, compiling, and optimizing topologically error-corrected circuits we need to understand how a circuit relates to the number of qubits and the physical computational time when they are implemented \cite{DSMN13}. Regardless of whether we are talking about the Surface code or the Raussendorf lattice, the relationship between a topological circuit and physical resources is identical. The fundamental unit of measure is illustrated in Figure \ref{fig:one}b. A {\em plumbing piece} of a topological quantum circuit is a three-dimensional cubical volume that has an edge length related to the desired strength of the underlying quantum code. For a distance d code (sufficient to correct upto $t = (d-1)/2$ errors), this plumbing piece has an edge length containing $5d/4$ plaquette cells for the surface code),or $5d/4$ cells in the Raussendorf lattice. At the centre of this plumbing piece is the defect, which has a circumference of $d$ plaquettes. Figure \ref{fig:one}b illustrates an example for $d = 4$. The plumbing piece gives a scale independent factor to allow us to measure topological quantum circuits without having to specify the strength of the underlying error correction \cite{FD12,FDJ13}.
Using topological circuit volumes in terms of plumbing pieces allows us to directly calculate the total number of qubits and computational time. For the surface code, the plumbing piece requires a total of $Q = 25d^2/4 + 5d + 1$ qubits and $T = 5d/4$ steps. Each step is defined as a syndrome extraction circuit for both bit-errors and phase-errors. For the Raussendorf lattice, a plumbing piece requires a total of $Q = 6d^3+9d^2+3d$ qubits. For a larger topological circuit, we can use their volume to first calculate the required strength of error correction d to ensure no logical errors occur during implementation, and then calculate the total number of resources needed by converting the volume to physical qubit numbers and computational time.
This method of designing quantum programs is very useful as we do not need to redesign anything about the actual quantum hardware when we change the quantum program. We just need to make sure we have enough qubits to do the job.
\section{Constructing and compiling initial topological circuits}
Before a given computation can be suitably optimized in the topological formalism, quantum circuits need to be compiled and constructed from the original algorithmic specification \cite{G06,GLRSV13,WS14}. Figure \ref{fig:two} illustrates the broad structure of the compilation stack needed for an arbitrary algorithm. The stack is partitioned into several stages;
\begin{figure*}[ht!]
\begin{center}
\resizebox{0.8\linewidth}{!}{0.4\includegraphics{Figure2.jpeg}}
\end{center}
\caption{{\bf Offline compilation and optimisation stack \cite{MD16}}. The conversion of a abstract high level quantum algorithm to an unoptimised topological form. What is addressed in this review is the third level which corresponds to topologically compacting a given circuit specification prior to hardware implementation.}
\label{fig:two}
\end{figure*}
Algorithm to Circuit: A quantum circuit, consisting of single, two and three qubit gate primitives, is derived from the abstract algorithm. This circuit can be optimized for depth, and/or number of qubits \cite{Z98,VI05,WC00,AMMR12,WBCB14}.
Circuit to fault-tolerant primitives: The abstract circuit is further decomposed into gate sets that have well defined fault-tolerant implementations in the topological code. Again, optimizations can occur at this level \cite{GS12,J13+,J13+++,RS14,GKMR14,KMM13}
Fault-tolerant circuit to ICM form: The circuit consisting of fault-tolerant gate primitives is then converted to a form called Initialization, Controlled-Not, Measurement (ICM). This allows us to build in appropriate auxiliary protocols needed before explicitly converting them to a topological implementation \cite{PPND15}.
Canonical Topological Form: Once written in ICM form, the circuit can be converted to an unoptimized canonical form in the topological model, prior to further resource optimization \cite{PDF16}.
The conversion of a higher level circuit to a canonical topological form is a complicated but well defined process, and researchers have designed several software packages to perform this task.
In Figure \ref{fig:four} we illustrate several examples of a canonical topological form and the quantum circuits they are derived from \cite{BH12,J13++}. Each of the circuits shown are known as magic state distillation circuits \cite{BK05+} and are used to enact certain gates that require high-fidelity ancillary states. Each of these circuits have a corresponding volume and, hence, can be used to estimate physical resources.
\section{Topological Optimisation}
The next crucial step in the design stack for error-corrected quantum circuits is topological optimization \cite{PF13}. This also happens to be the most underdeveloped area of the stack. Almost all other elements have been completely understood or are being heavily researched, including efficient methods for circuit construction and optimization, and fault-tolerance. Although it has received little attention, there are strong reasons to believe that topological optimization may result in some of the largest resource savings if implemented well \cite{FD12,FDJ13}.
The basic principal is illustrated in Figure \ref{fig:one}c. The canonical circuit begins with a volume of $V=192$ plumbing pieces, and in the same way as a coffee cup can be deformed topologically into a donut, we can slowly compact the physical three dimensional volume of the circuit without altering its computational function. There are some additional rules not related to continuous deformation, and unique to the computational model, one of which is called bridging, that can be used to reduce the physical volume of a topological circuit significantly \cite{FD12}. After many steps (not illustrated), the final volume of the topological circuit is reduced to $V=18$ - over an order of magnitude smaller than the original canonical form. This amount of optimization is indeed significant. For a surface code quantum computer, the number of qubits required for implementation can be reduced by orders of magnitude by simply compressing these structures. However, two theoretical questions still remain unanswered.
The first is to provide a lower bound, or exact definition for optimality, for a topological circuit. While we can compress, we do not have a condition for optimality given the original circuit specification. The second question concerns the classical complexity of the algorithm required to find this optimal solution. While this problem does appear closely related to the three dimensional bin packing problem \cite{Bin00}, which is known to fall into the complexity class of NP-Hard, there are small differences in the topological QEC model that may imply that these two problems do not directly map on to each other. It is still possible that the optimization of topological quantum circuits may be provably classically efficient to calculate.
Circuits that have been currency compacted have been done so manually. This is obviously not a viable approach for large-scale implementations of error-corrected quantum algorithms. There have been very small steps to try and build automated topological optimization packages \cite{PF13}, but these have, so far, only illustrated the potential difficulties in creating the required software. Being able to optimize even moderately large quantum circuits will not be possible without automated software, and it appears as though techniques in machine learning and artificial Intelligence may be required to provide resource efficient solutions.
\begin{figure*}[ht!]
\begin{center}
\resizebox{0.9\linewidth}{!}{0.4\includegraphics{Figure4.jpeg}}
\end{center}
\caption{{\bf Canonical topological quantum circuits.} In each figure we illustrate a quantum circuit (written in the standard pictorial form) and the corresponding, unoptimised topological quantum circuit. Each circuit can be measured in terms of volume. The temporal axis is defined as the temporal evolution of this structure as the computation proceeds}
\label{fig:four}
\end{figure*}
\section{meQuanics: The Quantum Computer Game}
The approach that we recently took to address this problem was inspired by projects in the biological sciences \cite{foldit,eyewire} that attempt to solve scientifically useful but difficult problems by utilizing the computing capacity of the general public. This technique, sometimes referred to as Citizen Science, was pioneered by projects such as FoldIt (which aimed to find the three-dimensional structure of biological proteins given their constituent sequence of amino acids), and Eyewire (designed to map neural connections in the retina) and has achieved significant success.
Given the relatively simple 3D puzzle structure of topological quantum circuits, and the simple success metric of minimal physical volume, we have tried the same approach. An initial prototype of a platform we have dubbed meQuanics [www.mequanics.com.au], designed to convert the topological optimization problem into a simple 3D puzzle game, has been released online. Designed for touch-based platforms such as smartphones and tablet devices, meQuanics creates an online social media environment where the general public can compete and collaborate to find small volume solutions to various quantum sub-circuits that are critical for large-scale quantum computation.
\begin{figure*}[ht!]
\begin{center}
\resizebox{0.85\linewidth}{!}{0.4\includegraphics{Figure3.jpeg}}
\end{center}
\caption{{\bf Current assets for the prototype client of meQuanics}. Here we show a current screenshot from the meQuanics client [www.mequanics.com.au] and some digital assets that will be used for the final game client.}
\label{fig:three}
\end{figure*}
While it is conceivable that users can derive compact solutions that are significantly smaller than solutions we currently have, the primary goal is not the solutions but rather the process individual players use to generate these solutions. It is well known within the machine learning and AI community that the success of these techniques requires a database of information that the machine system can use to learn. While for many problems there is an existing database of material that can be utilized (such as the AlphaGo platform of the DeepMind project at Google), for this particular problem there are, essentially, no previous examples that we can use to train an appropriate automated program.
While the prototyping stage has demonstrated proof-of-principle client, there is still significant development required. The most important goal is to create a game that is competitive in the larger marketplace of mobile and touch-based games. While the novelty of a game that is very closely related to quantum computing development may induce a large number of users to try it out, long term retention of gamers is required to generate the necessary data sets for the project to be successful.
Gameplay and social interaction in meQuanics is now a major focus of development. The basic narrative is that of an interstellar race, where each ship is powered by a quantum computer. By minimizing the volume of puzzles, players increase their speed and ultimately win over other players working on the same problem. The online interaction environment is being designed under the assumption that each client continuously updates information to central servers informing us how players are tackling problems. Individual players can take a continuously expanding solution tree that begins from a specific canonical circuit structure and either improve on other players solutions, or backtrack and proceed down a different pathway that could lead to better solutions. Elements of the social media and gameplay environment are illustrated in Figure \ref{fig:three}.
\section{The Future}
Future prospects on the software component of large-scale quantum technologies is promising. Not only is there a vast amount of unsolved problems that can be addressed, even by researchers not heavily trained in quantum physics, but the theoretical similarities of essentially all major experimental hardware models implies that software solutions are applicable to all systems. This is now evident from the founding of four private start-up companies, exclusively focused on the software component of quantum technology: QxBranch [www.qxbranch.com.au], 1Qbit [www.1Qbit.com] QCware [www.qcware.com] and Cambridge Quantum Computing [www.cambridgequantum.com]
While each element of the software compilation stack has been addressed at some level, a functional quantum computer will require a completely integrated set of classical software compilation and optimization packages. The expertise of the classical software engineering community will be vital to this. While physicists may be the experts in building quantum hardware, efficient and reliable software control will probably be developed by those already well versed in the advanced techniques of classical software engineering. The fact that these problems are not intrinsically quantum in nature will make it easier for those without explicit training in quantum physics to get involved and make important contributions in this arena.
Quantum information technology is currently experiencing a second renaissance in advancement and investment from both the public and private sectors. As such, there is consensus amongst experts that it is no longer a question of if a large-scale quantum computer can be built, but when. The quantum revolution has the potential to be as significant as the digital revolution of the 20th century, and there is now a worldwide race to be the first to show a commercial advantage in deploying large-scale computers, communication networks, sensors, and other active quantum technology. We stand at the cusp of an exciting new age in computing, with a significant laundry list of problems to interest pioneers.
\section{Acknowledgements}
We would like to thank A. Paler and A.G. Fowler for collaborating on research summarized here. meQuanics was developed in collaboration with K. Nemoto, K. Bruegmann, E. Gray, P. Daouadi, M. Everitt, Y. Quemener and F. Schittig through funds provided by the Hayao Nakayama Foundation for Science and Technology and Culture and the Japanese National Institute of Informatics. S.J. Devitt acknowledges support from the JSPS grant for challenging exploratory research and the JST ImPact project.
\clearpage
\bibliographystyle{unsrt}
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{"url":"https:\/\/webwork.maa.org\/mediawiki_new\/index.php?title=DisableFunctions&direction=prev&oldid=1441","text":"# DisableFunctions\n\n## Disabling Functions in Student Answers: PG Code Snippet\n\nThis code snippet shows the essential PG code to Disabling Functions in Student Answers. Note that these are insertions, not a complete PG file. This code will have to be incorporated into the problem file on which you are working.\n\nIn general, this can be done in a number of ways: there are a number of available limited contexts that limit the types of answers that students may enter, or, alternately, it is possible to specifically indicate within the context what functions or operations are not allowed. In the following, we first consider the available contexts, and, below that, show how to specifically disallow some functions.\n\nPG problem file Explanation\n loadMacros(\"contextLimitedPolynomial.pl\");\n\n\nIn the initialization section of the file, we need to load the Context that we're using. This will be contextContextName.pl, where ContextName is the name of the context used in the problem set-up section of the problem.\n\n Context(\"LimitedPolynomial\");\n# to require that only one monomial of each\n# degree be included in the polynomial, we\n# would also set that flag in the Context:\n# Context()->flags->set(singlePowers=>1);\n\n$ans = Compute(\"x^2 + 3x + 2\"); There are no additions required in the tagging and description or initialization sections of the problem file. In the problem set-up section, we specify the Context that we want. In this case, we have selected the LimitedPolynomial context, which will require all answers to be (expanded) polynomials (i.e., sums of multiples of powers of x). This context allows also requiring that there be only one term with each power of x, as noted in the comment to the right. There are a number of limited contexts like LimitedPolynomial that are available by default: \u2022 LimitedPolynomial-Strict, which disallows mathematical operations even in the coefficients of the polynomial; \u2022 LimitedComplex, which allows complex numbers to be entered, but no complex operations; \u2022 LimitedNumeric, which allows numbers to be entered, but no operations are permitted between them; \u2022 LimitedNumeric-List, which is similar but for lists; \u2022 LimitedPoint, which allows points to be entered, but no operations between them; \u2022 LimitedPowers, which restricts what constants may be raised to powers. There are several available restrictions, only one of which may be selected: \u2022 LimitedPowers::NoBaseE(), which doesn't allow e to be raised to a power, \u2022 LimitedPowers::OnlyIntegers(), which only allows positive and negative integers to be raised to powers, \u2022 LimitedPowers::OnlyPositiveIntegers(), which only allows positive integers, \u2022 LimitedPowers::OnlyNonNegativeIntegers(), which only allows positive integers and zero; \u2022 LimitedVector, which allows vectors to be entered but disallows operations between vectors; there are two variations to this, \u2022 LimitedVector-coordinate, which requires that the vectors be entered in coordinate format (<a,b,c>), and \u2022 LimitedVector-ijk, which requires ijk-format. BEGIN_TEXT Expand the polynomial:$BR\n$$(x+1)(x+2) =$$\n\\{ ans_rule(25) \\}\nEND_TEXT\n\n\nNo changes are required in the text section of the problem file.\n\n ANS( $ans->cmp() ); And in the answer and solution section of the file, we check the answer as usual. Alternately, we may find that we need to disable specific functions or operations in student answers. This can be done manually within an existing Context, as shown in the following snippet. PG problem file Explanation Context(\"Numeric\"); ## to disable specific operations in ## student answers, use the undefine ## method for the operations: Context()->operators->undefine(\"^\",\"**\"); ## we can similarly disable specific ## functions with Context()->functions->undefine(\"sin\",\"cos\", \"tan\",\"sqrt\");$ans = Compute(\"1\/2\");\n\n\nNo changes are necessary in the documentation and tagging section of the problem, or in the initialization section. In the problem set-up section, we declare a context (here, we've chosen the Numeric context), and then change it.\n\nWe can disable specific operations in the Context: in general, predefined operations are * \/ + -\u00a0! >< U ^ ** . ,, for multiplication, division, addition, subtraction, the factorial operation, the cross-product ><, set union, exponentiation (both ^ and ** give exponentiation), the dot product, and list creation (,).\n\nAfter disabling the operation, they can be re-enabled with operators->redefine(), e.g., Context()->operators->redefine(\"^\"). We can also remove operators with operators->remove(), but this is not recommended, as it makes it completely unknown in the Context so that students won't get helpful error messages if they try to use them.\n\nTo disable specific functions in the Context, we similarly undefine them from the predefined functions. The predefined functions are sin, cos, tan, sec, csc, cot, asin, acos, atan, asec, acsc, acot, sinh, cosh, tanh, sech, csch, coth, asinh, acosh, atanh, asech, csch, acoth, ln, log, log10, exp, sqrt, abs, int, sgn, atan2, norm, unit, arg, mod, Re, Im, conj.\n\nIn addition, classes of functions can be disabled with functions->disable():\n\n\u2022 Context()->functions->disable(\"Trig\"); (disables all trig functions in both SimpleTrig and InverseTrig functions)\n\u2022 Context()->functions->disable(\"SimpleTrig\"); (disables sin, cos, tan, sec, csc, cot)\n\u2022 Context()->functions->disable(\"InverseTrig\"); (disables asin, acos, atan, asec, acsc, acot, atan2)\n\u2022 Context()->functions->disable(\"Hyperbolic\"); (disables all hyperbolic functions in both SimpleHyperbolic and InverseHyperbolic functions)\n\u2022 Context()->functions->disable(\"SimpleHyperbolic\"); (disables sinh, cosh, tanh, sech, csch, coth)\n\u2022 Context()->functions->disable(\"InverseHyperbolic\"); (disables asinh, acosh, atanh, asech, acsch, acoth)\n\u2022 Context()->functions->disable(\"Numeric\"); (disables ln, log, log10, exp, sqrt, abs, int, sgn)\n\u2022 Context()->functions->disable(\"Vector\"); (disables norm, unit)\n\u2022 Context()->functions->disable(\"Complex\"); (disables arg, mod, Re, Im, conj)\n\u2022 Context()->functions->disable(\"All\");\n BEGIN_TEXT\nFind the numerical value:\n$$sin^2(\\pi\/4) =$$\n\\{ ans_rule(15) \\}\nEND_TEXT\n\n\nWe don't need to make any changes to the text portion of the problem file.\n\n ANS( \\$ans->cmp() );\n\n\nNor in the answer and solution section.","date":"2022-12-01 12:43:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6598648428916931, \"perplexity\": 6303.5776219894415}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710813.48\/warc\/CC-MAIN-20221201121601-20221201151601-00804.warc.gz\"}"} | null | null |
Kisbér beschreibt:
die ungarische Stadt, siehe Kisbér
die frühere Verwaltungseinheit in Ungarn, siehe Kleingebiet Kisbér
den heutigen Kreis in Ungarn, siehe Kreis Kisbér
das ehemalige Gestüt, siehe Gestüt Kisbér | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,832 |
\section{Background}
\label{sec:background}
This section presents the challenges of effective feature collection and data-driven VNFs in cloud DCs, and, with a comparison of related work, motivates the design of Aquarius.
\subsection{Challenges}
\label{sec:background-challenge}
There is a rising trend of embedding intelligence and applying ML techniques in cloud and distributed systems to dynamically monitor and adaptively configure system parameters and characteristics (\textit{e.g.},\ server configurations, forwarding rules)~\cite{survey-anomaly, tuor2017deep, sivakumar2019mvfst, fu2021use}.
However, this raises a number of challenges and trade-offs:
\textbf{Online and Reliable Feature Collection:}
Few reliable datasets are available and considered as benchmark for ML applications in VNFs and networking systems (\textit{e.g.},\ traffic analysis and anomaly detection)~\cite{kdd-issues, predict, nsl-kdd, caida}.
Though log-based feature collection provides abundant information for various types of applications, it incurs high overhead under heavy traffic, which leads to inaccurate and unreliable measurements and makes it hard to bring ML algorithms ``online'' (making inference in real time)~\cite{survey-anomaly}.
\textbf{Scalability vs. Visibility:}
Active probing is another way of feature collection for VNFs to monitor the system state and make informed decisions~\cite{opnfv2019, openstack, lbas-2020, lalitha2016traffic}.
However, this incurs additional communication overhead and requires modifications on each node to maintain management and communication channels.
\subsection{Requirements}
\label{sec:background-requirements}
Based on the challenges, this paper\ summarizes the following requirements to enable data-driven VNFs in the cloud:
\textbf{Universality:} the feature collection mechanism should cover a wide range of features and be application-agnostic;
\textbf{Reliability:} the collected features should be representative, and have high usability and granularity;
\textbf{Scalability:} the feature collection and exploitation mechanism should incur minimal performance overhead and support large-scale and dynamically changing network topology;
\textbf{Deployability:} the mechanism should be plug-and-play and require no additional installation or configuration.
\begin{figure}[t]
\centering
\includegraphics[width=.9\columnwidth]{figures/intro/overview.pdf}
\caption{Aquarius\ architecture overview.}
\label{fig:design-overview}
\vskip -0.1in
\end{figure}
\subsection{Related Work}
\label{sec:background-related}
Various mechanisms (summarised in Table~\ref{tab:background-related}) dynamically configure and manage VNFs, making data-driven decisions.
ML allows inferring system states from networking features, for, \textit{e.g.},\ intrusion detection systems~\cite{tuor2017deep}, traffic classification~\cite{nguyen2008survey, pacheco2018towards}, and task scheduling~\cite{auto2018sigcomm, decima2018}.
To obtain networking features, these ML applications operate at the Application Layer.
However, they are not application-agnostic and do not generalise to different use cases.
Aquarius\ collects a wide range of features at the Transport Layer and enables generic data-driven network functions with minimal overhead.
Management and Orchestration (MANO) frameworks use centralised controllers to monitor and update VNF configurations~\cite{opnfv2019, openstack}.
Software-Defined Network (SDN) provides programmable APIs to gather per-flow or application-level features in a centralised way, to adaptively update configurations, using network equipments that supports the OpenFlow protocol~\cite{bari2013policycop}.
Smart Network Interface Cards (sNICs)~\cite{le2017uno} and Nitro~\cite{aws-nitro} offload VNFs from host processors to dedicated hardware devices to boost performance and reduce operational cost with centralised management.
Aquarius\ passively extracts networking features from the data plane and let VNFs make decisions in a distributed way.
Distributed VNFs also benefit from periodically polled network states (\textit{e.g.},\ CPU and memory usage), to ensure service availability, improve QoS~\cite{lbas-2020}, or classify networking traffic~\cite{lalitha2016traffic}.
Some network functions gain more visibility via in-network telemetry (INT)~\cite{clove}.
However, these methods require to either deploy agents, or to modify protocol stack on network nodes, which reduce the deployability.
Aquarius\ employs the plug-and-play design and requires no coordinated modification in the network.
Learning algorithms incurs additional inference and processing latencies.
To reduce latency, dedicated hardware, \textit{e.g.},\ CGRA~\cite{taurus2020}, and NetFPGA~\cite{xiong2019switches}, helps accelerate data processing for in-network ML applications.
Yet they lack flexibility when developing ML algorithms for different use cases in elastic networking systems.
MVFST-RL~\cite{sivakumar2019mvfst} proposes--in simulators--to asynchronously update networking configurations from learning algorithms to reduce additional latency in the data plane.
Aquarius\ can incorperate intelligence in a variety of VNFs, requiring no dedicated device, yet it is ready to be deployed in real-world systems.
\section{Conclusion}
\label{sec:conclusion}
Networking features and system state information help VNFs make informed decisions, and intelligently manage and update networking policies in cloud DCs.
Actively collecting features and system state information entails substantial control signalling and management overhead, in particular in large-scale DC networks.
This paper\ has proposed Aquarius, a framework that collects, infers and supplies accurate networking state information with little additional processing latency, in a scalable buffer layout.
The paper\ has illustrated significant performance gains of using of Aquarius\ for various ML-based VNFs and evaluated experimentally the impact of Aquarius\ in the system performance.
\section{Applications}
\label{sec:application}
This section shows $3$ applications of Aquarius\ in cloud DCs in the context of $3$ key VNFs--traffic classification, resource prediction and auto-scaling, and Layer-$4$ load balancing, along with $3$ different ML paradigms.
\subsection{Traffic Classification}
\label{sec:application-classification}
As one of the key VNFs in the cloud, traffic classification allows distinguishing different types of traffic~\cite{tuor2017deep, lalitha2016traffic,xiong2019switches}, to allocate appropriate resources and achieve service level agreements.
It also helps detect anomalies and security threats to prevent potential damages or losses~\cite{tuor2017deep}.
\begin{figure}[t]
\centering
\includegraphics[width=.9\columnwidth]{figures/application/classification/topo.pdf}
\vspace{-.1in}
\caption{Network topology for traffic classification.}
\label{fig:app-class-topo}
\vskip -.1in
\end{figure}
\subsubsection{Task Description and Testbed Configuration}
\label{sec:application-classification-testbed}
This section shows the capability of Aquarius\ to collect reliable features and conduct traffic classification with unsupervised ML algorithms.
A testbed is implemented using Kernel-based Virtual Machine (KVM), where a virtual router embedded with Aquarius\ forwards different types of traffic to $4$ VIPs (Fig.~\ref{fig:app-class-topo}).
In VIP0, a simple PHP \texttt{for}-loop script on each server takes requests for given number of iterations and replies with proportional sizes.
The flow duration and size follow an exponential distribution as in~\cite{facebook-dc-traffic}.
In VIP1, static files of different sizes are served on each server\footnote{The sizes of files are $100$KB, $200$KB, $500$KB, $750$KB, $1$MB, $2$MB, and $5$MB. $50$ files are generated for each size.} as in~\cite{lbas-2020}, to represent IO-bound applications.
In VIP2 and VIP3, each application server is an independent replica of an Apache HTTP server that serves Wikipedia databases.
Two samples of $600$s duration are extracted and replayed from a real-world $24$-hour replay~\cite{wiki_traces}.
In VIP3, an additional $5000$ queries per second SYN flooding traffic is applied to simulate a DoS attack.
\begin{figure}[t]
\centering
\begin{subfigure}{.48\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/classification/pca-full.pdf}
\vskip -0.05in
\caption{PCA clusters (all features).}
\label{fig:app-class-pca-cluster-full}
\end{subfigure}
\begin{subfigure}{.49\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/classification/pca-partial.pdf}
\vskip -0.05in
\caption{PCA clusters ($25$ features).}
\label{fig:app-class-pca-cluster-partial}
\end{subfigure}
\vskip -0.05in
\caption{PCA analysis and $2$D visualisation.}
\label{fig:app-class-pca-cluster}
\vskip -0.2in
\end{figure}
\subsubsection{Principal Component Analysis (PCA) and Clustering}
\label{sec:application-classification-selection}
PCA is conducted to visualise the $4$ clusters for the $4$ different types of network traces in a $2$D projection (Fig.~\ref{fig:app-class-pca-cluster-full}).
Among the $4$ traces, PHP \texttt{for}-loop is pure CPU-bound and PHP file is pure IO-bound.
The \texttt{Wiki} trace consists of both queries for SQL database (CPU-bound) and static files (IO-bound), thus its cluster is located between the former $2$ traces.
The \texttt{Wiki} trace under DoS attack, however, can be clearly noticed as an independent cluster.
More features give multi-dimensional observations, yet at the cost of higher computation and memory overhead.
PCA helps reduce feature dimensionality from $73$ to $25$, while preserving data representation.
As depicted in Fig.~\ref{fig:app-class-pca-cluster-partial}, using $25$ features still gives clear clustering results, yet it reduces data processing time from $30.90$ms to $5.56$ms.
\subsubsection{Unsupervised Learning}
\label{sec:application-classification-unsupervised}
\begin{figure}[t]
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{figures/application/classification/cluster-predict.pdf}}
\vskip -0.1in
\caption{Unsupervised clustering using $25$ features.}
\label{fig:app-class-pca-cluster-predict}
\end{center}
\vskip -0.2in
\end{figure}
As depicted in Fig.~\ref{fig:app-class-pca-cluster-predict}, when applying unsupervised learning algorithms, K-Means and Gaussian Misture are able to generate clusters similar to the ground truth, while they require the number of expected clusters ($4$) as input.
Gaussian Mixture model has the shortest fit time and can be an interesting candidate for online traffic classification.
In case where the number of clusters is not known \textit{a priori}, DBSCAN~\cite{schubert2017dbscan} can distinguish the potential security threat, based only on a predefined distance of $0.1$.
\begin{table}[t]
\resizebox{\columnwidth}{!}{
\begin{tabular}{|ll|r|r|r|r|}
\hline
\multicolumn{2}{|l|}{Configuration} & $0$ Feature & $11$ Features & $73$ Features & PCAP \\ \hline
\multicolumn{1}{|l|}{\multirow{3}{*}{\rotatebox[origin=c]{90}{\begin{tabular}[c]{@{}c@{}}First\\ Packet\end{tabular}}}} & CPU Cycles & $938.232$ & $1635.838$ & $2609.019$ & $1295.284$ \\ \cline{2-6}
\multicolumn{1}{|l|}{} & Delay ($\mu$s) & $0.361$ & $0.629$ & $1.003$ & $0.498$ \\ \cline{2-6}
\multicolumn{1}{|l|}{} & Difference & $1.000\times$ & $1.744\times$ & $2.781\times$ & $1.381\times$ \\ \hline
\multicolumn{1}{|l|}{\multirow{3}{*}{\rotatebox[origin=c]{90}{\begin{tabular}[c]{@{}c@{}}Data\\ Packet\end{tabular}}}} & CPU Cycles & $576.357$ & $1583.798$ & $2602.684$ & $885.041$ \\ \cline{2-6}
\multicolumn{1}{|l|}{} & Delay ($\mu$s) & $0.222$ & $0.609$ & $1.001$ & $0.340$ \\ \cline{2-6}
\multicolumn{1}{|l|}{} & Difference & $1.000\times$ & $2.748\times$ & $4.516\times$ & $1.536\times$ \\ \hline
\multicolumn{2}{|l|}{CPU Usage (\%)} & $26.687$ & $40.858$ & $49.716$ & $31.480$ \\ \hline
\multicolumn{2}{|l|}{CPU Difference} & $1.000\times$ & $1.376\times$ & $1.675\times$ & $1.060\times$ \\ \hline
\multicolumn{2}{|l|}{RAM Usage (GiB)} & $2.652$ & $2.719$ & $2.744$ & $3.305$ \\ \hline
\multicolumn{2}{|l|}{RAM Difference} & $1.000\times$ & $1.025\times$ & $1.034\times$ & $1.246\times$ \\ \hline
\end{tabular}
}
\caption{Per-packet processing overhead (on $2.6$GHz CPU) and system resource consumptions (avg.) comparison.}
\label{tab:app-class-overhead}
\vskip -0.2in
\end{table}
\begin{figure}[t]
\centering
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/classification/cycle.pdf}
\vskip -0.05in
\caption{Per-packet processing latency comparison.}
\label{fig:application-classification-overhead-latency}
\end{subfigure}
\hspace{.35in}
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/classification/system.pdf}
\vskip -0.05in
\caption{System resource consumption.}
\label{fig:application-classification-overhead-resource}
\end{subfigure}
\vspace{.1in}
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/micro/sys-perf.pdf}
\vskip -0.1in
\caption{System performance metric comparison}
\label{fig:micro-flow-table-sys-perf}
\end{subfigure}
\vskip -0.1in
\caption{Aquarius\ feature collection overhead.}
\label{fig:application-classification-overhead}
\vskip -0.1in
\end{figure}
\subsubsection{Overhead Analysis}
\label{sec:application-classification-overhead}
To study the feature collection overhead, Aquarius\ is compared with a vanilla router which collects $0$ feature and a router logging packet information in the memory using \texttt{pcap}.
Under $500$ queries/s PHP \texttt{for}-loop traffic towards a $176$-CPU server cluster, when collecting $11$ features or collecting all $73$ features, Aquarius\ incurs different overhead (Table~\ref{tab:app-class-overhead} and Fig.~\ref{fig:application-classification-overhead-latency}).
On a $2.6$GHz CPU, the additional per-packet processing delays are trivial comparing with the typical round trip time (higher than $200\mu$s) between network equipments.
The mean CPU usage of Aquarius\ is $1.376\times$ and $1.675\times$ higher than the vanilla router when collecting $11$ and $73$ features respectively (Fig.~\ref{fig:application-classification-overhead-resource}).
As expected, log-based feature collection mechanism does not scale in terms of memory consumption\footnote{The results can be machine-dependent. This paper\ aims at showing the order of magnitudes, rather than providing a precise quantification.}.
As depicted in Fig.~\ref{fig:micro-flow-table-sys-perf}, the system performance of Aquarius\ shows advantages over RX interface driven by DPDK on various metrics when using different traffic rates.
\textbf{Take-Away:}
Aquarius\ gathers fine-grained and reliable datasets, which allow feature engineering and conducting in-depth data analysis.
Its fast and configurable design help achieve the right balance between visibility and performance.
\subsection{Resource Prediction and Auto-Scaling}
\label{sec:application-prediction}
To minimize operational cost while guaranteeing QoS, cloud operators~\cite{aws-elastic}) need to elastically provision server capacities.
\begin{figure}[t]
\begin{center}
\centerline{\includegraphics[width=.9\columnwidth]{figures/application/autoscale/topo.pdf}}
\caption{Network topology for autoscaling system.}
\label{fig:app-predict-topo}
\end{center}
\vskip -0.45in
\end{figure}
\subsubsection{Task Description and Testbed Configuration}
\label{sec:application-prediction-testbed}
This section shows the capability of Aquarius\ as a platform to adapt supervised ML algorithms to infer resource utilisation with no actively signaling.
$600$s samples extracted from the real-world $24$-hour Wikipedia trace are replayed on the network topology depicted in Fig.~\ref{fig:app-predict-topo}.
Workloads are randomly distributed among running servers ($2$-CPU each) by way of ECMP.
A learning task can be framed as predicting server load states (CPU usage) on each server with the same set of features as in Sec.~\ref{sec:application-classification}.
The predicted utilisation can be then used to plan and re-scale server cluster to guarantee QoS with reduced operational cost.
This task consists of $2$ steps--offline model training and online prediction.
\subsubsection{Offline Model Training}
\label{sec:application-prediction-offline}
To predict the resource utilisation of server clusters using networking features, $12$ widely used ML algorithms are selected to cover different families of ML algorithms, \textit{e.g.},\ sequential and non-sequential, parametric and non-parametric, linear and non-linear~\cite{fu2021use}.
The first $23$-hour samples are applied on $10$ servers to gather datasets for offline model training.
The distribution of the ground truth CPU usages in the training set covers the other two datasets so that the prediction task is feasible, yet the ML models have not seen the datasets for evaluations.
Based on the offline training performance evaluation, $1$ non-sequential model (linear regression) and $1$ sequential model (WaveNet~\cite{oord2016wavenet}) are selected to be applied for online auto-scaling\footnote{The whole comparison among all $12$ models and additional experimental results will be included supplementary details.}.
\alglanguage{pseudocode}
\begin{algorithm}[t]
\footnotesize
\caption{Auto-scaling Rule}
\label{alg:autoscale}
\begin{algorithmic}[1]
\State $n\_servers\_min,n\_servers\_max \gets 8, 14$ \Comment{Server number range}
\State $\mathbb{S} \gets$ Initial set of running servers
\State $cpu\_lo,cpu\_hi \gets 0.7, 0.8$ \Comment{Desired CPU usage range}
\For {each time step} \Comment{$\Delta t$ = $250$ms}
\State $\delta \gets 0$ \Comment{Initialize server state counter}
\State $y(\mathbb{S}) \gets$ CPU usage prediction of $16$ steps ahead
\State $threshold \gets \lceil\frac{|\mathbb{S}|}{3}\rceil$ \Comment{Threshold that triggers scaling actions}
\For {$s \in \mathbb{S}$}
\If {$y(s) < cpu\_lo$}
\State $\delta++$ \Comment{Increment $\delta$ if $s$ is under-loaded}
\ElsIf {$y(s) > cpu\_hi$}
\State $\delta--$ \Comment{Decrement $\delta$ if $s$ is over-loaded}
\EndIf
\EndFor
\If {$\delta > threshold$ \textbf{and} $|\mathbb{S}| > n\_servers\_min$}
\State $\mathbb{S} \gets downscale(\mathbb{S})$
\State skip $8$ time steps \Comment{Cool-down period}
\ElsIf {$\delta < -threshold$ \textbf{and} $|\mathbb{S}| < n\_servers\_max$}
\State $\mathbb{S} \gets upscale(\mathbb{S})$
\State skip $8$ time steps \Comment{Cool-down period}
\EndIf
\EndFor
\Statex
\end{algorithmic}
\vskip -0.05in
\end{algorithm}
\begin{figure}[t]
\vskip -.2in
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{figures/application/autoscale/timeline.pdf}}
\vskip -0.1in
\caption{Comparison of online auto-scaling performance using different algorithms.
The (discrete) numbers of running servers are plotted for each run in dashed lines, while CPU usage is summarised as avg. $\pm$ stddev across $30$ runs.}
\label{fig:app-predict-timeline}
\end{center}
\vskip -0.35in
\end{figure}
\subsubsection{Online Auto-Scaling}
\label{sec:application-prediction-online}
To test the online performance of offline-trained ML models, a $300$s Wikipedia replay trace sample of the last hour (unseen by the ML models) is synthesized to have scheduled changing traffic rates every $60$s.
Based on the CPU usage predictions of running servers $y(\mathbb{S})$, a simple heuristic (Alg.~\ref{alg:autoscale}), which approximate the threshold-based autoscaling policy as in~\cite{aws-elastic}, is proposed to keep the CPU usage of $\frac{2}{3}$ servers within the desired range ($70\sim80\%$).
Using the same counter $\Delta$ for over- and under-loaded servers reduces the variance induced by imbalanced workload distributions.
As a comparison to~\cite{aws-elastic}, a threshold-based active probing mechanism is implemented, whose predicted CPU usage for running servers $y(\mathbb{S})$ come from periodic polling.
An ``oracle'' benchmark is implemented to over-provision the number of servers proportional to the scheduled traffic rates.
\begin{figure}[t]
\centering
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=.9\columnwidth]{figures/application/autoscale/qos.pdf}
\vskip -0.1in
\caption{QoS.}
\label{fig:app-predict-qos}
\end{subfigure}
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=.9\columnwidth]{figures/application/autoscale/cost.pdf}
\vskip -0.1in
\caption{Operational cost and complexity.}
\label{fig:app-predict-cost}
\end{subfigure}
\vskip -0.05in
\caption{Trade-off between QoS and cost using different autoscaling mechanisms.}
\label{fig:app-predict-trade-off}
\vskip -0.2in
\end{figure}
\begin{figure}[t]
\begin{center}
\centerline{\includegraphics[width=.9\columnwidth]{figures/discuss/probe_lat.pdf}}
\vskip -0.1in
\caption{Feature collection latency comparison between Aquarius\ and active probing techniques.}
\label{fig:online-probe}
\end{center}
\vskip -0.4in
\end{figure}
\subsubsection{Results}
\label{sec:application-prediction-results}
As depicted in Fig.~\ref{fig:app-predict-timeline},
active probing keeps the average CPU usage within the desired range, however, it requires frequent scaling events and leads to oscillating CPU usage with high variance.
Linear regression is simple yet not robust when applied for an online auto-scaling system.
Its under-estimated server load states lead to over-loaded servers.
WaveNet takes sequential features as input and is more robust when applied online.
It keeps the average CPU usage close to the desired range with less oscillations.
As depicted in Fig.~\ref{fig:app-predict-trade-off}, WaveNet is able to provide better QoS than active probing--$78.37$ms less page load time ($26.04\%$) at $90$th percentile and $35.70$ms less ($30.45\%$) on average--with
$3.99\%$ additional server-second cost, and $42.44\%$ less scaling events.
When over-provisioning the server cluster, the page load time is lower than using WaveNet by
$67.13$ms at $90$th percentile and
$28.55$ms average, though it requires $11.86\%$ more server-second operational cost than WaveNet.
Aquarius\ parses features stored in the local shared memory with no control messages, achieving more than $94.18\mu$s less median latency than typical VM- and container-based probing mechanisms (Fig.~\ref{fig:online-probe}).
\textbf{Take-Away:}
Aquarius\ enables agile development and online deployment of learning algorithms to improve network performance.
It makes features quickly accessible while saving management bandwidth for data transmission.
\subsection{Traffic Optimisation and Load Balancing}
\label{sec:application-lb}
As a key component in cloud DCs, Layer-$4$ load balancers (LBs) distribute workloads across servers to provide scalable services.
Yet, in-production LBs requires human intervention which can lead to server weights mis-configurations.
\begin{figure}[t]
\centering
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/lb/timeline.pdf}
\vskip -0.1in
\caption{Mean FCTs (top), ratio between weights assigned to the $2$ groups of servers by RLB (middle), and traffic rates (bottom).}
\label{fig:app-lb-wiki-fct-timeline}
\end{subfigure}
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/lb/rlb-wiki-cdf.pdf}
\vskip -0.1in
\caption{Peak-hour (query rate higher than $500$/s) FCT distribution.}
\label{fig:app-lb-wiki-fct-peak}
\end{subfigure}
\vskip -0.05in
\caption{Wikipedia trace replayed using different LBs.}
\label{fig:app-lb-wiki-fct}
\vskip -0.2in
\end{figure}
\subsubsection{Task Description and Testbed Configuration}
\label{sec:application-lb-description}
This section shows that Aquarius\ can apply RL algorithms to self-configure server weights and optimise load balancing performance with no-prior knowledge of the system.
The configuration of VIP2 (Fig.~\ref{fig:app-class-topo}) is applied--replaying the \texttt{Wiki} trace and load balancing on $2$ groups of servers of different processing capacities.
The task is to extract and infer server processing capacity information from networking features and make informed load balancing decisions.
$3$ benchmark LB algorithms are implemented--(i) ECMP~\cite{silkroad2017}, (ii) statically configured WCMP~\cite{ananta2013}, and (iii) active WCMP~\cite{lbas-2020} based on polled server job queue lengths.
Both ECMP and WCMP are based on the kernel-bypassing implementation of Maglev~\cite{maglev} in VPP, which is the state-of-the-art LB.
\subsubsection{RL Algorithm}
\label{sec:application-lb-algorithm}
This paper\ implements and evaluate RLB~\cite{yao2021reinforced}--implemented and evaluated in simulators--in a realistic testbed using Aquarius.
With Aquarius, RLB (i) counts ongoing flows $\tilde{l}_i$ on servers and (ii) asynchronously updates (every $250$ms) server weights $\tilde{w}_i$ derived from flow durations $\tau_i$ sampled in reservoir buffers.
On receipt of new requests, RLB assigns servers based on $\argmin_{i} \frac{\tilde{l}_{i} + 1}{\tilde{w}_{i}}$, which prioritizes servers with higher processing speed and shorter queue lengths.
Different from~\cite{yao2021reinforced}, which uses actively probed ground truth information, this paper\ derives the reward from locally observed features (FCT) collected by Aquarius, which requires no active signalling between LBs and servers.
RLB is trained using the first hour of \texttt{Wiki} trace sample for $20$ episodes.
The trained RLB model is then tested on unseen traffic and compared with other LB algorithms.
\subsubsection{Results}
\label{sec:application-lb-result}
As depicted in Fig.~\ref{fig:app-lb-wiki-fct-timeline}, during off-peak hours, servers are under-utilised and all algorithms show similar performances in terms of QoS.
As traffic rates grow, RLB achieves lower FCT for both static pages and Wikipedia pages when compared with other LB algorithms (Fig.~\ref{fig:app-lb-wiki-fct-peak}), since RLB is able to dynamically adjust server weights.
As depicted in Fig.~\ref{fig:app-lb-wiki-fct-peak}, for Wikipedia pages the $90$-th percentile FCT of RLB ($0.292$s) is $18.15$x shorter than Maglev ECMP and $8.56$x shorter than Maglev WCMP, and the $95$-th percentile FCT of RLB ($0.552$s) is $17.82$x shorter than Maglev ECMP and $7.82$x shorter than Maglev WCMP.
For static pages, the $90$-th percentile FCT of RLB ($0.013$s) is $351.15$x shorter than ECMP and $124.43$x shorter than WCMP, and the $95$-th percentile FCT of RLB ($0.088$s) is $109.09$x shorter than ECMP and $37.50$x shorter than WCMP.
RLB is trained to learn server processing speed differences and assigns higher weights, thus more queries, to more powerful servers (Fig.~\ref{fig:app-lb-wiki-apache}).
When using RLB, $4$-CPU servers handle respectively $1.258\times$ and $1.523\times$ more tasks than $2$-CPU servers under $676.92$ and $372.01$ queries/s traffic.
\subsubsection{Overhead Analysis}
\label{sec:application-lb-overhead}
\begin{figure}[t]
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{figures/application/lb/apache.pdf}}
\vskip -0.1in
\caption{Query distribution (number of busy Apache threads) on $2$ groups of application servers.}
\label{fig:app-lb-wiki-apache}
\end{center}
\vskip -0.4in
\end{figure}
\begin{figure}[t]
\centering
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/lb/latency.pdf}
\vskip -0.05in
\caption{Per-packet processing latency comparison.}
\label{fig:app-lb-overhead-latency}
\end{subfigure}
\hspace{.35in}
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/lb/overhead.pdf}
\vskip -0.05in
\caption{System resource consumption.}
\label{fig:app-lb-overhead-resource}
\end{subfigure}
\vskip -0.05in
\caption{Overhead comparisons.}
\vskip -0.2in
\label{fig:app-lb-overhead}
\end{figure}
As depicted in Fig.~\ref{fig:app-lb-overhead-latency}, throughout all test runs, RLB consume on average $692.89$ more CPU cycles ($0.26\mu$s on $2.6$GHz CPU) than ECMP, as it computes and compares the server scores when making load balancing decisions.
Fig.~\ref{fig:app-lb-overhead-resource} depicts CPU and memory consumptions of all LBs.
On average, RLB incurs $0.22\times$ additional CPU usage, and $45.99$MiB memory usage, and achieves $87.38\%$ throughput of ECMP.
\textbf{Take-Away:}
Aquarius\ enables closed-loop control (RL) to dynamically adapt to networking systems and optimise performance.
It empowers real-world deployment and evaluation of learning algorithms developed in simulated environments.
\section{Introduction}
\label{sec:intro}
To increase network programmability, and balance the trade-off between capital expenditures and quality of service (QoS), Virtual Network Functions (VNFs) (\textit{e.g.},\ firewalls, load balancers) replace or augment dedicated hardware devices and play a significant role in large-scale data centers (DCs), running on commodity computing platforms.
To dynamically monitor and configure VNFs, the routing and decision-making process (\textit{control plane}) is dissociated from the network packets forwarding process (\textit{data plane}).
Data-driven mechanisms based on machine learning (ML) and reinforcement learning (RL) algorithms are applied in the control plane to adaptively manage networking policies~\cite{decima2018, auto2018sigcomm, yao2021reinforced}.
For instance, auto-scaling systems and load balancers can achieve improved QoS with reduced cost based on periodically polled resource utilisation of distributed nodes~\cite{smartsla}.
However, it is challenging to apply these algorithms in networking systems in real-time, since they require fine-grained observations of network and system states~\cite{sivakumar2019mvfst}.
Reactive polling resource utilisation and system performance incur additional control messages~\cite{lbas-2020, 6lb, smartsla, fu2021use} and reduce system scalability.
Fine-grained networking features are extracted offline or in simulated environments for clustering and RL algorithm development~\cite{yang2018elastic,sivakumar2019mvfst}.
Since the data plane is constrained by low-latency and high-throughput requirements~\cite{vpp}, heuristics--which may not be adaptive to dynamic environments--prevail over advanced learning alrogithms in real-world high performance networks~\cite{aws-elastic, maglev, silkroad2017, 6lb, hpcc, fu2021use}.
This paper\ proposes Aquarius, a fast and scalable data collection and exploitation mechanism that bridges different requirements for data planes (low-latency and high-throughput) and control planes (making informed decisions).
Extensive performance and overhead evaluations of Aquarius\ in a realistic testbed show that Aquarius enable:
\begin{itemize}
\itemsep-0.1em
\item \textbf{unsupervised learning $+$ offline data analysis}:
creating benchmark datasets to gain insight in different networking problems with minimal data collection overhead;
\item \textbf{supervised learning $+$ VNF management}:
embedding ML techniques to achieve self-aware monitoring and self-adaptive orchestration in an elastic compute cloud;
\item \textbf{reinforcement learning $+$ online policy updates}:
enabling closed-loop control to optimise routing policies and improve QoS, with no human intervention.
\end{itemize}
\section{Design}
\label{sec:design}
To meet the $4$ requirements summarised in Section~\ref{sec:background-requirements}, Aquarius\ is designed as a $3$-layer architecture (Fig.~\ref{fig:design-overview}).
Aquarius\ embeds a feature collector at the Transport Layer in the data plane, to efficiently and passively extracts a wide range of features with high granularity and low performance overhead.
It makes the features available via shared memory, for applications of ML algorithms on various use cases.
This paper\ illustrates the design using TCP traffic, which prevails in the cloud of Content Delivery Networks (CDNs)~\cite{facebook-dc-traffic}.
\subsection{Parser Layer}
\label{sec:design-parser}
To balance the tradeoff between scalability and visibility, networking features which indicate system states can be passively collected from the data plane to avoid active probing and additional installations and configurations.
\begin{figure}[t]
\centering
\includegraphics[width=.9\columnwidth]{figures/feature/flow-table}
\caption{Flow table data structure and workflow.}
\label{fig:feature-flow-table}
\vskip -0.1in
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=.9\columnwidth]{figures/feature/connection}
\vskip -0.05in
\caption{A state machine of feature collector for TCP traffic.}
\label{fig:feature-state-machine}
\vskip -0.1in
\end{figure}
\subsubsection{Stateful Feature Collection}
\label{sec:design-parser-collect}
Network traffic consists of flows that traverse different nodes (\textit{e.g.},\ edge routers, load balancers, servers) in the system, whose states can be traced and retrieved from the flows--along with traffic characteristics.
\textbf{Rationale:}
Stateless feature collection mechanisms (\textit{e.g.},\ sketches~\cite{yang2018elastic}) do not track the state of network flows, yet they can gather counters as ordinal features for ML algorithms using hashing functions, with little performance overhead.
However, ordinal features contains less information than quantitative features--time-related features (\textit{e.g.},\ round-trip time, inter-arrival time, flow duration) and throughput information (\textit{e.g.},\ congestion window size, flow size), which are not captured by stateless mechanisms.
\textbf{Design:}
Aquarius\ tracks flow states in bucket entries with a stateful table (Fig.~\ref{fig:feature-flow-table}), which can be configured to collect a wide range of features using a state machine depicted in Fig.~\ref{fig:feature-state-machine}.
In the flow table, Aquarius\ stores the information of each flow into a bucket entry indexed by $hash(fid)\%M$, where $fid$ is the flow ID (\textit{e.g.},\ the $5$-tuple of TCP flows) and $M$ is the flow table size.
An entry in the flow table can be in one of three states--\texttt{SYN}, \texttt{CONN} and \texttt{NULL} (Fig.~\ref{fig:feature-state-machine}).
The flow state transitions trigger the networking feature updates, which are described in the next subsection.
In case where the bucket entry is not available when a new flow arrives, the flow is considered as a ``miss'' and is excluded by the feature collector.
\subsubsection{Network Features}
\label{sec:design-parser-categorize}
Various features can gainfully benefit decision making process for different use cases.
\textbf{Rationale:}
As a generic feature collection mechanism, Aquarius\ should be able to collect as much information as possible with minimal overhead (\textit{e.g.},\ memory space consumption).
Other than ordinal and quantitative features, to capture the system dynamic, it is also important to trace the timestamps of different events, for sequential ML algorithms.
\textbf{Design:}
With the flow table, Aquarius\ allows flexible configuration of attributes, to gather the most significant features and optimise the memory usage overhead for different applications.
Quantitative features are collected as samples, using reservoir sampling (Algorithm~\ref{alg:feature-reservoir}).
Since networking environments are dynamic, it is important to capture not only the features, but also the sequential information of the system.
Reservoir sampling gathers a representative group of samples in fix-sized buffer from a stream.
It captures both the sampling timestamps and exponentially-distributed numbers of samples over a time window, which help analyse patterns sequentially.
\alglanguage{pseudocode}
\begin{algorithm}[t]
\footnotesize
\caption{Reservoir sampling with no rejection}
\label{alg:feature-reservoir}
\begin{algorithmic}[1]
\State $k \gets$ reservoir buffer size
\State $buf \gets [(0, 0), \dots, (0, 0)]$\Comment{Size of $k$}
\For {each observed sample $v$ arriving at $t$}
\State $randomId \gets rand()$
\State $idx \gets randomId\%N$ \Comment{randomly select one index}
\State $buf[idx] \gets (t, v)$ \Comment{register sample in buffer}
\EndFor
\Statex
\end{algorithmic}
\vspace{-0.4cm}%
\end{algorithm}
\begin{figure}[t]
\centering
\includegraphics[width=.9\columnwidth]{figures/architecture/dsr}
\caption{Cloud service topology.}
\label{fig:feature-dsr}
\vskip -0.1in
\end{figure}
\subsection{Partitioner Layer}
\label{sec:design-partitioner}
Cloud services have different characteristics and they are identified by virtual IPs (VIPs) (Fig.~\ref{fig:feature-dsr}), corresponding to clusters of provisioned resources--\textit{e.g.},\ servers, identified by unique direct IPs (DIPs).
In production, cloud DCs are subject to high traffic rates and their environments and topologies change dynamically.
\textbf{Rationale:}
Different cloud services should be separated to (i) avoid multimodal distributions in collected features and to (ii) allow dynamically adding or removing services.
Features should be made available so that both spatial and sequential information can be easily partitioned and accessed.
Even with heavy traffic and dynamically changing network topology, features should be reliable and easy to access with low latency.
\textbf{Design:}
Aquarius\ organises observations of each VIP in independent POSIX shared memory (\texttt{shm}) files, to provide scalable and dynamic service management.
In each \texttt{shm} file, collected features are further partitioned by egress equipments so that spatial information can be distinguished.
Fig.~\ref{fig:architecture-mechanism} exemplifies the \texttt{shm} layout and data flow.
\begin{figure}[t]
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{figures/architecture/shm-layout}}
\caption{Aquarius\ \texttt{shm} layout and data flow pipeline.}
\label{fig:architecture-mechanism}
\end{center}
\vskip -0.2in
\end{figure}
\subsubsection{Bit-Index and Masking}
\label{sec:design-partitioner-mask}
The first byte in the \texttt{shm} file of a VIP defines the max number of egress equipments $N$ ($64$ in Fig.~\ref{fig:architecture-mechanism}).
The $N$-bit \textit{bit-index header} helps quickly identify activated egress and its corresponding memory space--the $i$-th bit is set to $1$ if the $i$-th egress is active and $0$ otherwise.
Adding an activated egress to the system requires only to set the corresponding bit to $1$ after initialising its memory space.
It suffices to simply flip the bit from $1$ to $0$ to deactivate an egress node.
\begin{table}[t]
\footnotesize
\centering
\begin{tabular}{|ll|c|c|}
\hline
\multicolumn{2}{|l|}{Operation / Complexity} & Computation & Memory \\ \hline
\multicolumn{2}{|l|}{Add / Remove VIP} & $\mathcal{O}(1)$ & $\mathcal{O}(kN+mN)$ \\ \hline
\multicolumn{2}{|l|}{Add egress node} & $\mathcal{O}(1)$ & $\mathcal{O}(k+m)$ \\ \hline
\multicolumn{2}{|l|}{Remove egress node} & $\mathcal{O}(1)$ & $\mathcal{O}(1)$ \\ \hline
\multicolumn{2}{|l|}{\begin{tabular}[c]{@{}l@{}}Register reservoir sample\\ Update counter (cache)\end{tabular}} & $\mathcal{O}(1)$ & $\mathcal{O}(1)$ \\ \hline
\multicolumn{2}{|l|}{\begin{tabular}[c]{@{}l@{}}Update counters / actions\\ (multi-buffering)\end{tabular}} & $\mathcal{O}(1)$ & $\mathcal{O}(N)$ \\ \hline
\multicolumn{1}{|l|}{\multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Get the latest \\observation\end{tabular}}} & 1 node & \multirow{2}{*}{$\mathcal{O}(m)$} & $\mathcal{O}(k+m)$ \\ \cline{2-2} \cline{4-4}
\multicolumn{1}{|l|}{} & All nodes & & $\mathcal{O}(kN+mN)$ \\ \hline
\multicolumn{1}{|l|}{\multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Update action in\\ the data plane\end{tabular}}} & 1 node & \multirow{2}{*}{$\mathcal{O}(m)$} & $\mathcal{O}(1)$ \\ \cline{2-2} \cline{4-4}
\multicolumn{1}{|l|}{} & All nodes & & $\mathcal{O}(N)$ \\ \hline
\end{tabular}
\caption{Computation and memory complexity of different operations, where $k$ is the size of reservoir buffer, $N$ is the number of egress nodes, and $m$ is the level of multi-buffering.}
\label{tab:design-complexity}
\vskip -0.1in
\end{table}
\subsubsection{Independent Egress Memory Space}
\label{sec:design-partitioner-egress}
Each egress node has its own independent memory space, storing counters, reservoir samples, and data plane policies (actions).
As depicted in Fig.~\ref{fig:architecture-mechanism}, on receipt of the first \texttt{ACK} from the client to a specific egress node $i$, VNF increments the number of flows in the counters cache of node $i$.
Quatitative features (\textit{e.g.},\ flow duration $t_3 - t_0$ gathered at $t_3$ in Fig.~\ref{fig:architecture-mechanism}) can be stored in the reservoir buffer of node $i$ using Alg.~\ref{alg:feature-reservoir}.
Obtained features for all active egress nodes can then be aggregated and processed to make further inferences or data-driven decisions, which can be written back to the memory space of each egress node.
\subsubsection{Multi-Buffering and Asynchronous I/O}
\label{sec:design-partitioner-io}
Counters and actions are exchanged between cache and buffer using $m$-level multi-buffering with incremental sequence ID.
When copying data, the sequence ID is set to $0$ to avoid I/O conflicts.
ML algorithms can pull the latest observations and push the latest data-driven decisions using multi-buffering with no disruption in the data plane.
This design offers an asynchronous $2$-way communication interface to exchange fine-grained features and data-driven decisions with low latency.
Both computation and memory space complexity is presented in Table~\ref{tab:design-complexity}.
The whole dataflow is asynchronous and avoid stalling in the data exchange process in both the data plane and the control plane.
\subsection{Implementation}
\label{sec:micro-implement}
This paper\ implements Aquarius\ as a plugin to the Vector Packet Processor (VPP)~\cite{vpp}.
The flow table size is configured as $M=65536$.
Each sampled network feature is a $2$-tuple of a $32$-bit float timestamp and a $32$-bit value.
The reservoir buffer size is $k = 128$ for each feature per egress equipment.
The \texttt{shm} file of each VIP consists of $6$KB $3$-level multi-buffering counters and $832$KB reservoir sampling buffers.
In this paper, Aquarius\ is implemented to be able to collect $8$ ordinal features (counters) and $65$ quantitative features in total.
\section{Background}
\label{sec:background}
This section presents the challenges of effective feature collection and data-driven VNFs in cloud DCs, and, with a comparison of related work, motivates the design of Aquarius.
\subsection{Challenges}
\label{sec:background-challenge}
There is a rising trend of embedding intelligence and applying ML techniques in cloud and distributed systems to dynamically monitor and adaptively configure system parameters and characteristics (\textit{e.g.},\ server configurations, forwarding rules)~\cite{survey-anomaly, tuor2017deep, sivakumar2019mvfst, fu2021use}.
However, this raises a number of challenges and trade-offs:
\textbf{Online and Reliable Feature Collection:}
Few reliable datasets are available and considered as benchmark for ML applications in VNFs and networking systems (\textit{e.g.},\ traffic analysis and anomaly detection)~\cite{kdd-issues, predict, nsl-kdd, caida}.
Though log-based feature collection provides abundant information for various types of applications, it incurs high overhead under heavy traffic, which leads to inaccurate and unreliable measurements and makes it hard to bring ML algorithms ``online'' (making inference in real time)~\cite{survey-anomaly}.
\textbf{Scalability vs. Visibility:}
Active probing is another way of feature collection for VNFs to monitor the system state and make informed decisions~\cite{opnfv2019, openstack, lbas-2020, lalitha2016traffic}.
However, this incurs additional communication overhead and requires modifications on each node to maintain management and communication channels.
\subsection{Requirements}
\label{sec:background-requirements}
Based on the challenges, this paper\ summarizes the following requirements to enable data-driven VNFs in the cloud:
\textbf{Universality:} the feature collection mechanism should cover a wide range of features and be application-agnostic;
\textbf{Reliability:} the collected features should be representative, and have high usability and granularity;
\textbf{Scalability:} the feature collection and exploitation mechanism should incur minimal performance overhead and support large-scale and dynamically changing network topology;
\textbf{Deployability:} the mechanism should be plug-and-play and require no additional installation or configuration.
\begin{figure}[t]
\centering
\includegraphics[width=.9\columnwidth]{figures/intro/overview.pdf}
\caption{Aquarius\ architecture overview.}
\label{fig:design-overview}
\vskip -0.1in
\end{figure}
\subsection{Related Work}
\label{sec:background-related}
Various mechanisms (summarised in Table~\ref{tab:background-related}) dynamically configure and manage VNFs, making data-driven decisions.
ML allows inferring system states from networking features, for, \textit{e.g.},\ intrusion detection systems~\cite{tuor2017deep}, traffic classification~\cite{nguyen2008survey, pacheco2018towards}, and task scheduling~\cite{auto2018sigcomm, decima2018}.
To obtain networking features, these ML applications operate at the Application Layer.
However, they are not application-agnostic and do not generalise to different use cases.
Aquarius\ collects a wide range of features at the Transport Layer and enables generic data-driven network functions with minimal overhead.
Management and Orchestration (MANO) frameworks use centralised controllers to monitor and update VNF configurations~\cite{opnfv2019, openstack}.
Software-Defined Network (SDN) provides programmable APIs to gather per-flow or application-level features in a centralised way, to adaptively update configurations, using network equipments that supports the OpenFlow protocol~\cite{bari2013policycop}.
Smart Network Interface Cards (sNICs)~\cite{le2017uno} and Nitro~\cite{aws-nitro} offload VNFs from host processors to dedicated hardware devices to boost performance and reduce operational cost with centralised management.
Aquarius\ passively extracts networking features from the data plane and let VNFs make decisions in a distributed way.
Distributed VNFs also benefit from periodically polled network states (\textit{e.g.},\ CPU and memory usage), to ensure service availability, improve QoS~\cite{lbas-2020}, or classify networking traffic~\cite{lalitha2016traffic}.
Some network functions gain more visibility via in-network telemetry (INT)~\cite{clove}.
However, these methods require to either deploy agents, or to modify protocol stack on network nodes, which reduce the deployability.
Aquarius\ employs the plug-and-play design and requires no coordinated modification in the network.
Learning algorithms incurs additional inference and processing latencies.
To reduce latency, dedicated hardware, \textit{e.g.},\ CGRA~\cite{taurus2020}, and NetFPGA~\cite{xiong2019switches}, helps accelerate data processing for in-network ML applications.
Yet they lack flexibility when developing ML algorithms for different use cases in elastic networking systems.
MVFST-RL~\cite{sivakumar2019mvfst} proposes--in simulators--to asynchronously update networking configurations from learning algorithms to reduce additional latency in the data plane.
Aquarius\ can incorperate intelligence in a variety of VNFs, requiring no dedicated device, yet it is ready to be deployed in real-world systems.
\section{Conclusion}
\label{sec:conclusion}
Networking features and system state information help VNFs make informed decisions, and intelligently manage and update networking policies in cloud DCs.
Actively collecting features and system state information entails substantial control signalling and management overhead, in particular in large-scale DC networks.
This paper\ has proposed Aquarius, a framework that collects, infers and supplies accurate networking state information with little additional processing latency, in a scalable buffer layout.
The paper\ has illustrated significant performance gains of using of Aquarius\ for various ML-based VNFs and evaluated experimentally the impact of Aquarius\ in the system performance.
\section{Applications}
\label{sec:application}
This section shows $3$ applications of Aquarius\ in cloud DCs in the context of $3$ key VNFs--traffic classification, resource prediction and auto-scaling, and Layer-$4$ load balancing, along with $3$ different ML paradigms.
\subsection{Traffic Classification}
\label{sec:application-classification}
As one of the key VNFs in the cloud, traffic classification allows distinguishing different types of traffic~\cite{tuor2017deep, lalitha2016traffic,xiong2019switches}, to allocate appropriate resources and achieve service level agreements.
It also helps detect anomalies and security threats to prevent potential damages or losses~\cite{tuor2017deep}.
\begin{figure}[t]
\centering
\includegraphics[width=.9\columnwidth]{figures/application/classification/topo.pdf}
\vspace{-.1in}
\caption{Network topology for traffic classification.}
\label{fig:app-class-topo}
\vskip -.1in
\end{figure}
\subsubsection{Task Description and Testbed Configuration}
\label{sec:application-classification-testbed}
This section shows the capability of Aquarius\ to collect reliable features and conduct traffic classification with unsupervised ML algorithms.
A testbed is implemented using Kernel-based Virtual Machine (KVM), where a virtual router embedded with Aquarius\ forwards different types of traffic to $4$ VIPs (Fig.~\ref{fig:app-class-topo}).
In VIP0, a simple PHP \texttt{for}-loop script on each server takes requests for given number of iterations and replies with proportional sizes.
The flow duration and size follow an exponential distribution as in~\cite{facebook-dc-traffic}.
In VIP1, static files of different sizes are served on each server\footnote{The sizes of files are $100$KB, $200$KB, $500$KB, $750$KB, $1$MB, $2$MB, and $5$MB. $50$ files are generated for each size.} as in~\cite{lbas-2020}, to represent IO-bound applications.
In VIP2 and VIP3, each application server is an independent replica of an Apache HTTP server that serves Wikipedia databases.
Two samples of $600$s duration are extracted and replayed from a real-world $24$-hour replay~\cite{wiki_traces}.
In VIP3, an additional $5000$ queries per second SYN flooding traffic is applied to simulate a DoS attack.
\begin{figure}[t]
\centering
\begin{subfigure}{.48\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/classification/pca-full.pdf}
\vskip -0.05in
\caption{PCA clusters (all features).}
\label{fig:app-class-pca-cluster-full}
\end{subfigure}
\begin{subfigure}{.49\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/classification/pca-partial.pdf}
\vskip -0.05in
\caption{PCA clusters ($25$ features).}
\label{fig:app-class-pca-cluster-partial}
\end{subfigure}
\vskip -0.05in
\caption{PCA analysis and $2$D visualisation.}
\label{fig:app-class-pca-cluster}
\vskip -0.2in
\end{figure}
\subsubsection{Principal Component Analysis (PCA) and Clustering}
\label{sec:application-classification-selection}
PCA is conducted to visualise the $4$ clusters for the $4$ different types of network traces in a $2$D projection (Fig.~\ref{fig:app-class-pca-cluster-full}).
Among the $4$ traces, PHP \texttt{for}-loop is pure CPU-bound and PHP file is pure IO-bound.
The \texttt{Wiki} trace consists of both queries for SQL database (CPU-bound) and static files (IO-bound), thus its cluster is located between the former $2$ traces.
The \texttt{Wiki} trace under DoS attack, however, can be clearly noticed as an independent cluster.
More features give multi-dimensional observations, yet at the cost of higher computation and memory overhead.
PCA helps reduce feature dimensionality from $73$ to $25$, while preserving data representation.
As depicted in Fig.~\ref{fig:app-class-pca-cluster-partial}, using $25$ features still gives clear clustering results, yet it reduces data processing time from $30.90$ms to $5.56$ms.
\subsubsection{Unsupervised Learning}
\label{sec:application-classification-unsupervised}
\begin{figure}[t]
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{figures/application/classification/cluster-predict.pdf}}
\vskip -0.1in
\caption{Unsupervised clustering using $25$ features.}
\label{fig:app-class-pca-cluster-predict}
\end{center}
\vskip -0.2in
\end{figure}
As depicted in Fig.~\ref{fig:app-class-pca-cluster-predict}, when applying unsupervised learning algorithms, K-Means and Gaussian Misture are able to generate clusters similar to the ground truth, while they require the number of expected clusters ($4$) as input.
Gaussian Mixture model has the shortest fit time and can be an interesting candidate for online traffic classification.
In case where the number of clusters is not known \textit{a priori}, DBSCAN~\cite{schubert2017dbscan} can distinguish the potential security threat, based only on a predefined distance of $0.1$.
\begin{table}[t]
\resizebox{\columnwidth}{!}{
\begin{tabular}{|ll|r|r|r|r|}
\hline
\multicolumn{2}{|l|}{Configuration} & $0$ Feature & $11$ Features & $73$ Features & PCAP \\ \hline
\multicolumn{1}{|l|}{\multirow{3}{*}{\rotatebox[origin=c]{90}{\begin{tabular}[c]{@{}c@{}}First\\ Packet\end{tabular}}}} & CPU Cycles & $938.232$ & $1635.838$ & $2609.019$ & $1295.284$ \\ \cline{2-6}
\multicolumn{1}{|l|}{} & Delay ($\mu$s) & $0.361$ & $0.629$ & $1.003$ & $0.498$ \\ \cline{2-6}
\multicolumn{1}{|l|}{} & Difference & $1.000\times$ & $1.744\times$ & $2.781\times$ & $1.381\times$ \\ \hline
\multicolumn{1}{|l|}{\multirow{3}{*}{\rotatebox[origin=c]{90}{\begin{tabular}[c]{@{}c@{}}Data\\ Packet\end{tabular}}}} & CPU Cycles & $576.357$ & $1583.798$ & $2602.684$ & $885.041$ \\ \cline{2-6}
\multicolumn{1}{|l|}{} & Delay ($\mu$s) & $0.222$ & $0.609$ & $1.001$ & $0.340$ \\ \cline{2-6}
\multicolumn{1}{|l|}{} & Difference & $1.000\times$ & $2.748\times$ & $4.516\times$ & $1.536\times$ \\ \hline
\multicolumn{2}{|l|}{CPU Usage (\%)} & $26.687$ & $40.858$ & $49.716$ & $31.480$ \\ \hline
\multicolumn{2}{|l|}{CPU Difference} & $1.000\times$ & $1.376\times$ & $1.675\times$ & $1.060\times$ \\ \hline
\multicolumn{2}{|l|}{RAM Usage (GiB)} & $2.652$ & $2.719$ & $2.744$ & $3.305$ \\ \hline
\multicolumn{2}{|l|}{RAM Difference} & $1.000\times$ & $1.025\times$ & $1.034\times$ & $1.246\times$ \\ \hline
\end{tabular}
}
\caption{Per-packet processing overhead (on $2.6$GHz CPU) and system resource consumptions (avg.) comparison.}
\label{tab:app-class-overhead}
\vskip -0.2in
\end{table}
\begin{figure}[t]
\centering
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/classification/cycle.pdf}
\vskip -0.05in
\caption{Per-packet processing latency comparison.}
\label{fig:application-classification-overhead-latency}
\end{subfigure}
\hspace{.35in}
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/classification/system.pdf}
\vskip -0.05in
\caption{System resource consumption.}
\label{fig:application-classification-overhead-resource}
\end{subfigure}
\vspace{.1in}
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/micro/sys-perf.pdf}
\vskip -0.1in
\caption{System performance metric comparison}
\label{fig:micro-flow-table-sys-perf}
\end{subfigure}
\vskip -0.1in
\caption{Aquarius\ feature collection overhead.}
\label{fig:application-classification-overhead}
\vskip -0.1in
\end{figure}
\subsubsection{Overhead Analysis}
\label{sec:application-classification-overhead}
To study the feature collection overhead, Aquarius\ is compared with a vanilla router which collects $0$ feature and a router logging packet information in the memory using \texttt{pcap}.
Under $500$ queries/s PHP \texttt{for}-loop traffic towards a $176$-CPU server cluster, when collecting $11$ features or collecting all $73$ features, Aquarius\ incurs different overhead (Table~\ref{tab:app-class-overhead} and Fig.~\ref{fig:application-classification-overhead-latency}).
On a $2.6$GHz CPU, the additional per-packet processing delays are trivial comparing with the typical round trip time (higher than $200\mu$s) between network equipments.
The mean CPU usage of Aquarius\ is $1.376\times$ and $1.675\times$ higher than the vanilla router when collecting $11$ and $73$ features respectively (Fig.~\ref{fig:application-classification-overhead-resource}).
As expected, log-based feature collection mechanism does not scale in terms of memory consumption\footnote{The results can be machine-dependent. This paper\ aims at showing the order of magnitudes, rather than providing a precise quantification.}.
As depicted in Fig.~\ref{fig:micro-flow-table-sys-perf}, the system performance of Aquarius\ shows advantages over RX interface driven by DPDK on various metrics when using different traffic rates.
\textbf{Take-Away:}
Aquarius\ gathers fine-grained and reliable datasets, which allow feature engineering and conducting in-depth data analysis.
Its fast and configurable design help achieve the right balance between visibility and performance.
\subsection{Resource Prediction and Auto-Scaling}
\label{sec:application-prediction}
To minimize operational cost while guaranteeing QoS, cloud operators~\cite{aws-elastic}) need to elastically provision server capacities.
\begin{figure}[t]
\begin{center}
\centerline{\includegraphics[width=.9\columnwidth]{figures/application/autoscale/topo.pdf}}
\caption{Network topology for autoscaling system.}
\label{fig:app-predict-topo}
\end{center}
\vskip -0.45in
\end{figure}
\subsubsection{Task Description and Testbed Configuration}
\label{sec:application-prediction-testbed}
This section shows the capability of Aquarius\ as a platform to adapt supervised ML algorithms to infer resource utilisation with no actively signaling.
$600$s samples extracted from the real-world $24$-hour Wikipedia trace are replayed on the network topology depicted in Fig.~\ref{fig:app-predict-topo}.
Workloads are randomly distributed among running servers ($2$-CPU each) by way of ECMP.
A learning task can be framed as predicting server load states (CPU usage) on each server with the same set of features as in Sec.~\ref{sec:application-classification}.
The predicted utilisation can be then used to plan and re-scale server cluster to guarantee QoS with reduced operational cost.
This task consists of $2$ steps--offline model training and online prediction.
\subsubsection{Offline Model Training}
\label{sec:application-prediction-offline}
To predict the resource utilisation of server clusters using networking features, $12$ widely used ML algorithms are selected to cover different families of ML algorithms, \textit{e.g.},\ sequential and non-sequential, parametric and non-parametric, linear and non-linear~\cite{fu2021use}.
The first $23$-hour samples are applied on $10$ servers to gather datasets for offline model training.
The distribution of the ground truth CPU usages in the training set covers the other two datasets so that the prediction task is feasible, yet the ML models have not seen the datasets for evaluations.
Based on the offline training performance evaluation, $1$ non-sequential model (linear regression) and $1$ sequential model (WaveNet~\cite{oord2016wavenet}) are selected to be applied for online auto-scaling\footnote{The whole comparison among all $12$ models and additional experimental results will be included supplementary details.}.
\alglanguage{pseudocode}
\begin{algorithm}[t]
\footnotesize
\caption{Auto-scaling Rule}
\label{alg:autoscale}
\begin{algorithmic}[1]
\State $n\_servers\_min,n\_servers\_max \gets 8, 14$ \Comment{Server number range}
\State $\mathbb{S} \gets$ Initial set of running servers
\State $cpu\_lo,cpu\_hi \gets 0.7, 0.8$ \Comment{Desired CPU usage range}
\For {each time step} \Comment{$\Delta t$ = $250$ms}
\State $\delta \gets 0$ \Comment{Initialize server state counter}
\State $y(\mathbb{S}) \gets$ CPU usage prediction of $16$ steps ahead
\State $threshold \gets \lceil\frac{|\mathbb{S}|}{3}\rceil$ \Comment{Threshold that triggers scaling actions}
\For {$s \in \mathbb{S}$}
\If {$y(s) < cpu\_lo$}
\State $\delta++$ \Comment{Increment $\delta$ if $s$ is under-loaded}
\ElsIf {$y(s) > cpu\_hi$}
\State $\delta--$ \Comment{Decrement $\delta$ if $s$ is over-loaded}
\EndIf
\EndFor
\If {$\delta > threshold$ \textbf{and} $|\mathbb{S}| > n\_servers\_min$}
\State $\mathbb{S} \gets downscale(\mathbb{S})$
\State skip $8$ time steps \Comment{Cool-down period}
\ElsIf {$\delta < -threshold$ \textbf{and} $|\mathbb{S}| < n\_servers\_max$}
\State $\mathbb{S} \gets upscale(\mathbb{S})$
\State skip $8$ time steps \Comment{Cool-down period}
\EndIf
\EndFor
\Statex
\end{algorithmic}
\vskip -0.05in
\end{algorithm}
\begin{figure}[t]
\vskip -.2in
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{figures/application/autoscale/timeline.pdf}}
\vskip -0.1in
\caption{Comparison of online auto-scaling performance using different algorithms.
The (discrete) numbers of running servers are plotted for each run in dashed lines, while CPU usage is summarised as avg. $\pm$ stddev across $30$ runs.}
\label{fig:app-predict-timeline}
\end{center}
\vskip -0.35in
\end{figure}
\subsubsection{Online Auto-Scaling}
\label{sec:application-prediction-online}
To test the online performance of offline-trained ML models, a $300$s Wikipedia replay trace sample of the last hour (unseen by the ML models) is synthesized to have scheduled changing traffic rates every $60$s.
Based on the CPU usage predictions of running servers $y(\mathbb{S})$, a simple heuristic (Alg.~\ref{alg:autoscale}), which approximate the threshold-based autoscaling policy as in~\cite{aws-elastic}, is proposed to keep the CPU usage of $\frac{2}{3}$ servers within the desired range ($70\sim80\%$).
Using the same counter $\Delta$ for over- and under-loaded servers reduces the variance induced by imbalanced workload distributions.
As a comparison to~\cite{aws-elastic}, a threshold-based active probing mechanism is implemented, whose predicted CPU usage for running servers $y(\mathbb{S})$ come from periodic polling.
An ``oracle'' benchmark is implemented to over-provision the number of servers proportional to the scheduled traffic rates.
\begin{figure}[t]
\centering
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=.9\columnwidth]{figures/application/autoscale/qos.pdf}
\vskip -0.1in
\caption{QoS.}
\label{fig:app-predict-qos}
\end{subfigure}
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=.9\columnwidth]{figures/application/autoscale/cost.pdf}
\vskip -0.1in
\caption{Operational cost and complexity.}
\label{fig:app-predict-cost}
\end{subfigure}
\vskip -0.05in
\caption{Trade-off between QoS and cost using different autoscaling mechanisms.}
\label{fig:app-predict-trade-off}
\vskip -0.2in
\end{figure}
\begin{figure}[t]
\begin{center}
\centerline{\includegraphics[width=.9\columnwidth]{figures/discuss/probe_lat.pdf}}
\vskip -0.1in
\caption{Feature collection latency comparison between Aquarius\ and active probing techniques.}
\label{fig:online-probe}
\end{center}
\vskip -0.4in
\end{figure}
\subsubsection{Results}
\label{sec:application-prediction-results}
As depicted in Fig.~\ref{fig:app-predict-timeline},
active probing keeps the average CPU usage within the desired range, however, it requires frequent scaling events and leads to oscillating CPU usage with high variance.
Linear regression is simple yet not robust when applied for an online auto-scaling system.
Its under-estimated server load states lead to over-loaded servers.
WaveNet takes sequential features as input and is more robust when applied online.
It keeps the average CPU usage close to the desired range with less oscillations.
As depicted in Fig.~\ref{fig:app-predict-trade-off}, WaveNet is able to provide better QoS than active probing--$78.37$ms less page load time ($26.04\%$) at $90$th percentile and $35.70$ms less ($30.45\%$) on average--with
$3.99\%$ additional server-second cost, and $42.44\%$ less scaling events.
When over-provisioning the server cluster, the page load time is lower than using WaveNet by
$67.13$ms at $90$th percentile and
$28.55$ms average, though it requires $11.86\%$ more server-second operational cost than WaveNet.
Aquarius\ parses features stored in the local shared memory with no control messages, achieving more than $94.18\mu$s less median latency than typical VM- and container-based probing mechanisms (Fig.~\ref{fig:online-probe}).
\textbf{Take-Away:}
Aquarius\ enables agile development and online deployment of learning algorithms to improve network performance.
It makes features quickly accessible while saving management bandwidth for data transmission.
\subsection{Traffic Optimisation and Load Balancing}
\label{sec:application-lb}
As a key component in cloud DCs, Layer-$4$ load balancers (LBs) distribute workloads across servers to provide scalable services.
Yet, in-production LBs requires human intervention which can lead to server weights mis-configurations.
\begin{figure}[t]
\centering
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/lb/timeline.pdf}
\vskip -0.1in
\caption{Mean FCTs (top), ratio between weights assigned to the $2$ groups of servers by RLB (middle), and traffic rates (bottom).}
\label{fig:app-lb-wiki-fct-timeline}
\end{subfigure}
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/lb/rlb-wiki-cdf.pdf}
\vskip -0.1in
\caption{Peak-hour (query rate higher than $500$/s) FCT distribution.}
\label{fig:app-lb-wiki-fct-peak}
\end{subfigure}
\vskip -0.05in
\caption{Wikipedia trace replayed using different LBs.}
\label{fig:app-lb-wiki-fct}
\vskip -0.2in
\end{figure}
\subsubsection{Task Description and Testbed Configuration}
\label{sec:application-lb-description}
This section shows that Aquarius\ can apply RL algorithms to self-configure server weights and optimise load balancing performance with no-prior knowledge of the system.
The configuration of VIP2 (Fig.~\ref{fig:app-class-topo}) is applied--replaying the \texttt{Wiki} trace and load balancing on $2$ groups of servers of different processing capacities.
The task is to extract and infer server processing capacity information from networking features and make informed load balancing decisions.
$3$ benchmark LB algorithms are implemented--(i) ECMP~\cite{silkroad2017}, (ii) statically configured WCMP~\cite{ananta2013}, and (iii) active WCMP~\cite{lbas-2020} based on polled server job queue lengths.
Both ECMP and WCMP are based on the kernel-bypassing implementation of Maglev~\cite{maglev} in VPP, which is the state-of-the-art LB.
\subsubsection{RL Algorithm}
\label{sec:application-lb-algorithm}
This paper\ implements and evaluate RLB~\cite{yao2021reinforced}--implemented and evaluated in simulators--in a realistic testbed using Aquarius.
With Aquarius, RLB (i) counts ongoing flows $\tilde{l}_i$ on servers and (ii) asynchronously updates (every $250$ms) server weights $\tilde{w}_i$ derived from flow durations $\tau_i$ sampled in reservoir buffers.
On receipt of new requests, RLB assigns servers based on $\argmin_{i} \frac{\tilde{l}_{i} + 1}{\tilde{w}_{i}}$, which prioritizes servers with higher processing speed and shorter queue lengths.
Different from~\cite{yao2021reinforced}, which uses actively probed ground truth information, this paper\ derives the reward from locally observed features (FCT) collected by Aquarius, which requires no active signalling between LBs and servers.
RLB is trained using the first hour of \texttt{Wiki} trace sample for $20$ episodes.
The trained RLB model is then tested on unseen traffic and compared with other LB algorithms.
\subsubsection{Results}
\label{sec:application-lb-result}
As depicted in Fig.~\ref{fig:app-lb-wiki-fct-timeline}, during off-peak hours, servers are under-utilised and all algorithms show similar performances in terms of QoS.
As traffic rates grow, RLB achieves lower FCT for both static pages and Wikipedia pages when compared with other LB algorithms (Fig.~\ref{fig:app-lb-wiki-fct-peak}), since RLB is able to dynamically adjust server weights.
As depicted in Fig.~\ref{fig:app-lb-wiki-fct-peak}, for Wikipedia pages the $90$-th percentile FCT of RLB ($0.292$s) is $18.15$x shorter than Maglev ECMP and $8.56$x shorter than Maglev WCMP, and the $95$-th percentile FCT of RLB ($0.552$s) is $17.82$x shorter than Maglev ECMP and $7.82$x shorter than Maglev WCMP.
For static pages, the $90$-th percentile FCT of RLB ($0.013$s) is $351.15$x shorter than ECMP and $124.43$x shorter than WCMP, and the $95$-th percentile FCT of RLB ($0.088$s) is $109.09$x shorter than ECMP and $37.50$x shorter than WCMP.
RLB is trained to learn server processing speed differences and assigns higher weights, thus more queries, to more powerful servers (Fig.~\ref{fig:app-lb-wiki-apache}).
When using RLB, $4$-CPU servers handle respectively $1.258\times$ and $1.523\times$ more tasks than $2$-CPU servers under $676.92$ and $372.01$ queries/s traffic.
\subsubsection{Overhead Analysis}
\label{sec:application-lb-overhead}
\begin{figure}[t]
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{figures/application/lb/apache.pdf}}
\vskip -0.1in
\caption{Query distribution (number of busy Apache threads) on $2$ groups of application servers.}
\label{fig:app-lb-wiki-apache}
\end{center}
\vskip -0.4in
\end{figure}
\begin{figure}[t]
\centering
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/lb/latency.pdf}
\vskip -0.05in
\caption{Per-packet processing latency comparison.}
\label{fig:app-lb-overhead-latency}
\end{subfigure}
\hspace{.35in}
\begin{subfigure}{\columnwidth}
\centering
\includegraphics[width=\columnwidth]{figures/application/lb/overhead.pdf}
\vskip -0.05in
\caption{System resource consumption.}
\label{fig:app-lb-overhead-resource}
\end{subfigure}
\vskip -0.05in
\caption{Overhead comparisons.}
\vskip -0.2in
\label{fig:app-lb-overhead}
\end{figure}
As depicted in Fig.~\ref{fig:app-lb-overhead-latency}, throughout all test runs, RLB consume on average $692.89$ more CPU cycles ($0.26\mu$s on $2.6$GHz CPU) than ECMP, as it computes and compares the server scores when making load balancing decisions.
Fig.~\ref{fig:app-lb-overhead-resource} depicts CPU and memory consumptions of all LBs.
On average, RLB incurs $0.22\times$ additional CPU usage, and $45.99$MiB memory usage, and achieves $87.38\%$ throughput of ECMP.
\textbf{Take-Away:}
Aquarius\ enables closed-loop control (RL) to dynamically adapt to networking systems and optimise performance.
It empowers real-world deployment and evaluation of learning algorithms developed in simulated environments.
\section{Introduction}
\label{sec:intro}
To increase network programmability, and balance the trade-off between capital expenditures and quality of service (QoS), Virtual Network Functions (VNFs) (\textit{e.g.},\ firewalls, load balancers) replace or augment dedicated hardware devices and play a significant role in large-scale data centers (DCs), running on commodity computing platforms.
To dynamically monitor and configure VNFs, the routing and decision-making process (\textit{control plane}) is dissociated from the network packets forwarding process (\textit{data plane}).
Data-driven mechanisms based on machine learning (ML) and reinforcement learning (RL) algorithms are applied in the control plane to adaptively manage networking policies~\cite{decima2018, auto2018sigcomm, yao2021reinforced}.
For instance, auto-scaling systems and load balancers can achieve improved QoS with reduced cost based on periodically polled resource utilisation of distributed nodes~\cite{smartsla}.
However, it is challenging to apply these algorithms in networking systems in real-time, since they require fine-grained observations of network and system states~\cite{sivakumar2019mvfst}.
Reactive polling resource utilisation and system performance incur additional control messages~\cite{lbas-2020, 6lb, smartsla, fu2021use} and reduce system scalability.
Fine-grained networking features are extracted offline or in simulated environments for clustering and RL algorithm development~\cite{yang2018elastic,sivakumar2019mvfst}.
Since the data plane is constrained by low-latency and high-throughput requirements~\cite{vpp}, heuristics--which may not be adaptive to dynamic environments--prevail over advanced learning alrogithms in real-world high performance networks~\cite{aws-elastic, maglev, silkroad2017, 6lb, hpcc, fu2021use}.
This paper\ proposes Aquarius, a fast and scalable data collection and exploitation mechanism that bridges different requirements for data planes (low-latency and high-throughput) and control planes (making informed decisions).
Extensive performance and overhead evaluations of Aquarius\ in a realistic testbed show that Aquarius enable:
\begin{itemize}
\itemsep-0.1em
\item \textbf{unsupervised learning $+$ offline data analysis}:
creating benchmark datasets to gain insight in different networking problems with minimal data collection overhead;
\item \textbf{supervised learning $+$ VNF management}:
embedding ML techniques to achieve self-aware monitoring and self-adaptive orchestration in an elastic compute cloud;
\item \textbf{reinforcement learning $+$ online policy updates}:
enabling closed-loop control to optimise routing policies and improve QoS, with no human intervention.
\end{itemize}
\section{Design}
\label{sec:design}
To meet the $4$ requirements summarised in Section~\ref{sec:background-requirements}, Aquarius\ is designed as a $3$-layer architecture (Fig.~\ref{fig:design-overview}).
Aquarius\ embeds a feature collector at the Transport Layer in the data plane, to efficiently and passively extracts a wide range of features with high granularity and low performance overhead.
It makes the features available via shared memory, for applications of ML algorithms on various use cases.
This paper\ illustrates the design using TCP traffic, which prevails in the cloud of Content Delivery Networks (CDNs)~\cite{facebook-dc-traffic}.
\subsection{Parser Layer}
\label{sec:design-parser}
To balance the tradeoff between scalability and visibility, networking features which indicate system states can be passively collected from the data plane to avoid active probing and additional installations and configurations.
\begin{figure}[t]
\centering
\includegraphics[width=.9\columnwidth]{figures/feature/flow-table}
\caption{Flow table data structure and workflow.}
\label{fig:feature-flow-table}
\vskip -0.1in
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=.9\columnwidth]{figures/feature/connection}
\vskip -0.05in
\caption{A state machine of feature collector for TCP traffic.}
\label{fig:feature-state-machine}
\vskip -0.1in
\end{figure}
\subsubsection{Stateful Feature Collection}
\label{sec:design-parser-collect}
Network traffic consists of flows that traverse different nodes (\textit{e.g.},\ edge routers, load balancers, servers) in the system, whose states can be traced and retrieved from the flows--along with traffic characteristics.
\textbf{Rationale:}
Stateless feature collection mechanisms (\textit{e.g.},\ sketches~\cite{yang2018elastic}) do not track the state of network flows, yet they can gather counters as ordinal features for ML algorithms using hashing functions, with little performance overhead.
However, ordinal features contains less information than quantitative features--time-related features (\textit{e.g.},\ round-trip time, inter-arrival time, flow duration) and throughput information (\textit{e.g.},\ congestion window size, flow size), which are not captured by stateless mechanisms.
\textbf{Design:}
Aquarius\ tracks flow states in bucket entries with a stateful table (Fig.~\ref{fig:feature-flow-table}), which can be configured to collect a wide range of features using a state machine depicted in Fig.~\ref{fig:feature-state-machine}.
In the flow table, Aquarius\ stores the information of each flow into a bucket entry indexed by $hash(fid)\%M$, where $fid$ is the flow ID (\textit{e.g.},\ the $5$-tuple of TCP flows) and $M$ is the flow table size.
An entry in the flow table can be in one of three states--\texttt{SYN}, \texttt{CONN} and \texttt{NULL} (Fig.~\ref{fig:feature-state-machine}).
The flow state transitions trigger the networking feature updates, which are described in the next subsection.
In case where the bucket entry is not available when a new flow arrives, the flow is considered as a ``miss'' and is excluded by the feature collector.
\subsubsection{Network Features}
\label{sec:design-parser-categorize}
Various features can gainfully benefit decision making process for different use cases.
\textbf{Rationale:}
As a generic feature collection mechanism, Aquarius\ should be able to collect as much information as possible with minimal overhead (\textit{e.g.},\ memory space consumption).
Other than ordinal and quantitative features, to capture the system dynamic, it is also important to trace the timestamps of different events, for sequential ML algorithms.
\textbf{Design:}
With the flow table, Aquarius\ allows flexible configuration of attributes, to gather the most significant features and optimise the memory usage overhead for different applications.
Quantitative features are collected as samples, using reservoir sampling (Algorithm~\ref{alg:feature-reservoir}).
Since networking environments are dynamic, it is important to capture not only the features, but also the sequential information of the system.
Reservoir sampling gathers a representative group of samples in fix-sized buffer from a stream.
It captures both the sampling timestamps and exponentially-distributed numbers of samples over a time window, which help analyse patterns sequentially.
\alglanguage{pseudocode}
\begin{algorithm}[t]
\footnotesize
\caption{Reservoir sampling with no rejection}
\label{alg:feature-reservoir}
\begin{algorithmic}[1]
\State $k \gets$ reservoir buffer size
\State $buf \gets [(0, 0), \dots, (0, 0)]$\Comment{Size of $k$}
\For {each observed sample $v$ arriving at $t$}
\State $randomId \gets rand()$
\State $idx \gets randomId\%N$ \Comment{randomly select one index}
\State $buf[idx] \gets (t, v)$ \Comment{register sample in buffer}
\EndFor
\Statex
\end{algorithmic}
\vspace{-0.4cm}%
\end{algorithm}
\begin{figure}[t]
\centering
\includegraphics[width=.9\columnwidth]{figures/architecture/dsr}
\caption{Cloud service topology.}
\label{fig:feature-dsr}
\vskip -0.1in
\end{figure}
\subsection{Partitioner Layer}
\label{sec:design-partitioner}
Cloud services have different characteristics and they are identified by virtual IPs (VIPs) (Fig.~\ref{fig:feature-dsr}), corresponding to clusters of provisioned resources--\textit{e.g.},\ servers, identified by unique direct IPs (DIPs).
In production, cloud DCs are subject to high traffic rates and their environments and topologies change dynamically.
\textbf{Rationale:}
Different cloud services should be separated to (i) avoid multimodal distributions in collected features and to (ii) allow dynamically adding or removing services.
Features should be made available so that both spatial and sequential information can be easily partitioned and accessed.
Even with heavy traffic and dynamically changing network topology, features should be reliable and easy to access with low latency.
\textbf{Design:}
Aquarius\ organises observations of each VIP in independent POSIX shared memory (\texttt{shm}) files, to provide scalable and dynamic service management.
In each \texttt{shm} file, collected features are further partitioned by egress equipments so that spatial information can be distinguished.
Fig.~\ref{fig:architecture-mechanism} exemplifies the \texttt{shm} layout and data flow.
\begin{figure}[t]
\begin{center}
\centerline{\includegraphics[width=\columnwidth]{figures/architecture/shm-layout}}
\caption{Aquarius\ \texttt{shm} layout and data flow pipeline.}
\label{fig:architecture-mechanism}
\end{center}
\vskip -0.2in
\end{figure}
\subsubsection{Bit-Index and Masking}
\label{sec:design-partitioner-mask}
The first byte in the \texttt{shm} file of a VIP defines the max number of egress equipments $N$ ($64$ in Fig.~\ref{fig:architecture-mechanism}).
The $N$-bit \textit{bit-index header} helps quickly identify activated egress and its corresponding memory space--the $i$-th bit is set to $1$ if the $i$-th egress is active and $0$ otherwise.
Adding an activated egress to the system requires only to set the corresponding bit to $1$ after initialising its memory space.
It suffices to simply flip the bit from $1$ to $0$ to deactivate an egress node.
\begin{table}[t]
\footnotesize
\centering
\begin{tabular}{|ll|c|c|}
\hline
\multicolumn{2}{|l|}{Operation / Complexity} & Computation & Memory \\ \hline
\multicolumn{2}{|l|}{Add / Remove VIP} & $\mathcal{O}(1)$ & $\mathcal{O}(kN+mN)$ \\ \hline
\multicolumn{2}{|l|}{Add egress node} & $\mathcal{O}(1)$ & $\mathcal{O}(k+m)$ \\ \hline
\multicolumn{2}{|l|}{Remove egress node} & $\mathcal{O}(1)$ & $\mathcal{O}(1)$ \\ \hline
\multicolumn{2}{|l|}{\begin{tabular}[c]{@{}l@{}}Register reservoir sample\\ Update counter (cache)\end{tabular}} & $\mathcal{O}(1)$ & $\mathcal{O}(1)$ \\ \hline
\multicolumn{2}{|l|}{\begin{tabular}[c]{@{}l@{}}Update counters / actions\\ (multi-buffering)\end{tabular}} & $\mathcal{O}(1)$ & $\mathcal{O}(N)$ \\ \hline
\multicolumn{1}{|l|}{\multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Get the latest \\observation\end{tabular}}} & 1 node & \multirow{2}{*}{$\mathcal{O}(m)$} & $\mathcal{O}(k+m)$ \\ \cline{2-2} \cline{4-4}
\multicolumn{1}{|l|}{} & All nodes & & $\mathcal{O}(kN+mN)$ \\ \hline
\multicolumn{1}{|l|}{\multirow{2}{*}{\begin{tabular}[c]{@{}l@{}}Update action in\\ the data plane\end{tabular}}} & 1 node & \multirow{2}{*}{$\mathcal{O}(m)$} & $\mathcal{O}(1)$ \\ \cline{2-2} \cline{4-4}
\multicolumn{1}{|l|}{} & All nodes & & $\mathcal{O}(N)$ \\ \hline
\end{tabular}
\caption{Computation and memory complexity of different operations, where $k$ is the size of reservoir buffer, $N$ is the number of egress nodes, and $m$ is the level of multi-buffering.}
\label{tab:design-complexity}
\vskip -0.1in
\end{table}
\subsubsection{Independent Egress Memory Space}
\label{sec:design-partitioner-egress}
Each egress node has its own independent memory space, storing counters, reservoir samples, and data plane policies (actions).
As depicted in Fig.~\ref{fig:architecture-mechanism}, on receipt of the first \texttt{ACK} from the client to a specific egress node $i$, VNF increments the number of flows in the counters cache of node $i$.
Quatitative features (\textit{e.g.},\ flow duration $t_3 - t_0$ gathered at $t_3$ in Fig.~\ref{fig:architecture-mechanism}) can be stored in the reservoir buffer of node $i$ using Alg.~\ref{alg:feature-reservoir}.
Obtained features for all active egress nodes can then be aggregated and processed to make further inferences or data-driven decisions, which can be written back to the memory space of each egress node.
\subsubsection{Multi-Buffering and Asynchronous I/O}
\label{sec:design-partitioner-io}
Counters and actions are exchanged between cache and buffer using $m$-level multi-buffering with incremental sequence ID.
When copying data, the sequence ID is set to $0$ to avoid I/O conflicts.
ML algorithms can pull the latest observations and push the latest data-driven decisions using multi-buffering with no disruption in the data plane.
This design offers an asynchronous $2$-way communication interface to exchange fine-grained features and data-driven decisions with low latency.
Both computation and memory space complexity is presented in Table~\ref{tab:design-complexity}.
The whole dataflow is asynchronous and avoid stalling in the data exchange process in both the data plane and the control plane.
\subsection{Implementation}
\label{sec:micro-implement}
This paper\ implements Aquarius\ as a plugin to the Vector Packet Processor (VPP)~\cite{vpp}.
The flow table size is configured as $M=65536$.
Each sampled network feature is a $2$-tuple of a $32$-bit float timestamp and a $32$-bit value.
The reservoir buffer size is $k = 128$ for each feature per egress equipment.
The \texttt{shm} file of each VIP consists of $6$KB $3$-level multi-buffering counters and $832$KB reservoir sampling buffers.
In this paper, Aquarius\ is implemented to be able to collect $8$ ordinal features (counters) and $65$ quantitative features in total.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 0 |
The Androscoggin Transportation Resource Center (ATRC), as the designated Metropolitan Planning Organization for the Lewiston/Auburn area in Maine, is seeking submittals for its RFP to conduct a short range transit plan update for Lewiston-Auburn's fixed route transit system, citylink. This webpage will host project materials and updates to the study.
As an addendum to the process, all consultant bids must include with their submission and comply with all applicable components of the Federal Transit Administrations Best Practices Procurement Manual.
#1 What is the projected budget for this study, or a budget range?
The short range transit study is budgeted in ATRC's FY 2014-2015 Unified Planning Work Program as task 10. This study is one of two consultant studies to be completed under this task. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,487 |
function checkNetwork () {
if ( Ti.Network.networkType == Ti.Network.NETWORK_NONE ) {
var alertDialog = Titanium.UI.createAlertDialog({
title: 'WARNING!',
message: 'Your device is not online, please connect to a network',
buttonNames: ['OK']
});
alertDialog.show();
}
}; | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,422 |
<?php
namespace Validation\Assertions;
use Validation\InputValue;
use Validation\ValidationException;
class BooleanTrue extends AbstractAssertion
{
/**
* @param InputValue $input
* @throws ValidationException
*/
public function process(InputValue $input)
{
if ($input->getValue() !== true) {
throw new ValidationException('value is not TRUE');
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,372 |
{"url":"https:\/\/www.varsitytutors.com\/common_core_6th_grade_english_language_arts-help\/reading\/craft-and-structure","text":"# Common Core: 6th Grade English Language Arts : Craft and Structure\n\n## Example Questions\n\n\u2190 Previous 1 3 4 5\n\n### Example Question #1 : Reading To Determine Author's Point Of View\n\nAdapted from\u00a0The Adventures of Tom Sawyer\u00a0by Mark Twain (Samuel Clemens) (1876)\n\nSaturday morning was come, and all the summer world was bright and fresh, and brimming with life. Tom appeared on the sidewalk with a bucket of whitewash and a long-handled brush. He surveyed the fence, and all gladness left him and a deep melancholy settled down upon his spirit.\u00a0Thirty yards of board fence nine feet high.\u00a0Life to him seemed hollow, and existence but a burden. Sighing, he dipped his brush and passed it along the topmost plank; repeated the operation; did it again; compared the insignificant whitewashed streak with the far-reaching continent of unwhitewashed fence, and sat down on a tree-box discouraged.\n\nHe began to think of the fun he had planned for this day, and his sorrows multiplied. Soon the boys would come tripping along on all sorts of delicious expeditions, and they would make a world of fun of him for having to work\u2014the very thought of it burnt him like fire. At this dark and hopeless moment an inspiration burst upon him! Nothing less than a great, magnificent inspiration.\n\nHe took up his brush and went tranquilly to work. Ben Rogers hove in sight presently\u2014the very boy, of all boys, whose ridicule he had been dreading. Ben\u2019s gait was the hop-skip-and-jump\u2014proof enough that his heart was light and his anticipations high. He was eating an apple, and giving a long, melodious whoop, at intervals, followed by a deep-toned ding-dong-dong, ding-dong-dong, for he was personating a steamboat. As he drew near, he slackened speed, took the middle of the street, leaned far over to star-board and rounded to ponderously and with laborious pomp and circumstance\u2014for he was personating the Big Missouri, and considered himself to be drawing nine feet of water. He was boat and captain and engine-bells combined, so he had to imagine himself standing on his own hurricane-deck giving the orders and executing them:\n\n\u201cStop her, sir! Ting-a-ling-ling!\u201d The headway ran almost out, and he drew up slowly toward the sidewalk.\n\nTom went on whitewashing\u2014paid no attention to the steamboat. Ben stared a moment and then said: \u201cHi-yiYou\u2019re\u00a0up a stump, ain\u2019t you!\u201d\n\nNo answer. Tom surveyed his last touch with the eye of an artist, then he gave his brush another gentle sweep and surveyed the result, as before. Ben ranged up alongside of him. Tom\u2019s mouth watered for the apple, but he stuck to his work. Ben said:\n\n\u201cHello, old chap, you got to work, hey?\u201d\n\nTom wheeled suddenly and said:\n\n\u201cWhy, it\u2019s you, Ben! I warn\u2019t noticing.\u201d\n\n\u201cSay\u2014I\u2019m going in a-swimming, I am. Don\u2019t you wish you could? But of course you\u2019d druther\u00a0work\u2014wouldn\u2019t you? Course you would!\u201d\n\nTom contemplated the boy a bit, and said:\n\n\u201cWhat do you call work?\u201d\n\n\u201cWhy, ain\u2019t\u00a0that\u00a0work?\u201d\n\nTom resumed his whitewashing, and answered carelessly:\n\n\u201cWell, maybe it is, and maybe it ain\u2019t. All I know, is, it suits Tom Sawyer.\u201d\n\n\u201cOh come, now, you don\u2019t mean to let on that you\u00a0like\u00a0it?\u201d\n\nThe brush continued to move.\n\n\u201cLike it? Well, I don\u2019t see why I oughtn\u2019t to like it. Does a boy get a chance to whitewash a fence every day?\u201d\n\nThat put the thing in a new light. Ben stopped nibbling his apple. Tom swept his brush daintily back and forth\u2014stepped back to note the effect\u2014added a touch here and there\u2014criticized the effect again\u2014Ben watching every move and getting more and more interested, more and more absorbed. Presently he said:\n\n\u201cSay, Tom, let\u00a0me\u00a0whitewash a little.\u201d\n\nTom considered, was about to consent; but he altered his mind:\n\n\u201cNo\u2014no\u2014I reckon it wouldn\u2019t hardly do, Ben. You see, Aunt Polly\u2019s awful particular about this fence\u2014right here on the street, you know\u2014but if it was the back fence I wouldn\u2019t mind and\u00a0she\u00a0wouldn\u2019t. Yes, she\u2019s awful particular about this fence; it\u2019s got to be done very careful; I reckon there ain\u2019t one boy in a thousand, maybe two thousand, that can do it the way it\u2019s got to be done.\u201d\n\n\u201cNo\u2014is that so? Oh come, now\u2014lemme, just try. Only just a little\u2014I\u2019d let\u00a0you, if you was me, Tom.\u201d\n\n\u201cBen, I\u2019d like to; but Aunt Polly\u2014well, Jim wanted to do it, but she wouldn\u2019t let him; Sid wanted to do it, and she wouldn\u2019t let Sid. Now don\u2019t you see how I\u2019m fixed? If you was to tackle this fence and anything was to happen to it\u2014\u201d\n\n\u201cOh, shucks, I\u2019ll be just as careful. Now lemme try. Say\u2014I\u2019ll give you the core of my apple.\u201d\n\n\u201cWell, here\u2014No, Ben, now don\u2019t. I\u2019m afeard\u2014\u201d\n\n\u201cI\u2019ll give you\u00a0all\u00a0of it!\u201d\n\nTom gave up the brush with reluctance in his face, but alacrity in his heart. And while the late steamer Big Missouri worked and sweated in the sun, the retired artist sat on a barrel in the shade close by, dangled his legs, munched his apple, and planned the slaughter of more innocents. There was no lack of material; boys happened along every little while; they came to jeer, but remained to whitewash. By the time Ben was tired, Tom had traded the next chance to Billy Fisher for a kite, in good repair; and when he played out, Johnny Miller bought in for a dead rat and a string to swing it with\u2014and so on, and so on, hour after hour. And when the middle of the afternoon came, from being a poor poverty-stricken boy in the morning, Tom was literally rolling in wealth. And when the middle of the afternoon came, from being a poor poverty-stricken boy in the morning, Tom was literally rolling in wealth.\u00a0He had besides the things before mentioned, twelve marbles, a piece of blue bottle-glass to look through, a spool cannon, a key that wouldn't unlock anything, a fragment of chalk, a glass stopper of a decanter, a tin soldier, a couple of tadpoles, six fire-crackers, a kitten with only one eye, a brass door-knob, a dog-collar\u2014but no dog\u2014the handle of a knife, four pieces of orange-peel, and a dilapidated old window sash.\n\nHe had had a nice, good, idle time all the while\u2014plenty of company\u2014and the fence had three coats of whitewash on it! If he hadn\u2019t run out of whitewash he would have bankrupted every boy in the village.\n\nWhen Tom asks Ben \u201cDoes a boy get a chance to whitewash a fence every day?\u201d, what can we infer his motives are?\n\nTom is asking about fence-painting so Ben will keep talking to him while he paints the entire fence.\n\nTom is asking the question rhetorically to make fence-painting look like a special, rare event.\n\nTom asks this question to try to convince himself that fence-painting is a fun task.\n\nTom asks this question because he hopes\u00a0Ben will go ask someone else if he or she will help Tom paint the fence.\n\nTom is asking Ben this question because he does not know the answer.\n\nTom is asking the question rhetorically to make fence-painting look like a special, rare event.\n\nExplanation:\n\nTo figure out Tom's motives in asking\u00a0\"Does a boy get a chance to whitewash a fence every day?\", we need to consider the scene that leads up to this statement as well as its effects. The preceding part of the scene demonstrates how Tom has to whitewash a fence on Saturday and doesn't want to work. He gets a great (but ambiguous!) idea about how to get out of work. Then, Ben comes along, pretending to be a steamboat and trying to make fun of Tom for having to work. In talking with Ben, Tom suggests that whitewashing the fence isn't work and that he enjoys it. It's at this point that Tom asks the question in question:\n\n(Ben) \u201cOh come, now, you don\u2019t mean to let on that you\u00a0like\u00a0it?\u201d\n\nThe brush continued to move.\n\n(Tom) \u201cLike it? Well, I don\u2019t see why I oughtn\u2019t to like it. Does a boy get a chance to whitewash a fence every day?\u201d\n\nWhat happens next?\n\nThat put the thing in a new light. Ben stopped nibbling his apple. Tom swept his brush daintily back and forth\u2014stepped back to note the effect\u2014added a touch here and there\u2014criticized the effect again\u2014Ben watching every move and getting more and more interested, more and more absorbed. Presently he said:\n\n\u201cSay, Tom, let\u00a0me\u00a0whitewash a little.\u201d\n\nAha\u2014by portraying his work in such a positive light, Tom switches Ben's perspective on it and makes it seem like a special activity. It's after this switch in perspective that Tom is able to get Ben to give him his apple for the privilege of getting to do his work for him.\u00a0Based on this evidence, we can infer that Tom's great idea is to convince someone else (Ben)\u00a0to whitewash the fence for him, and that he asked the question to make fence-painting look fun and exciting. The story\u00a0supports this idea because after the question, the narrator immediately says, \"That put the thing in a new light.\" The correct answer is \"Tom is asking the question rhetorically to make fence-painting look like a special, rare event.\"\n\n### Example Question #1 : Reading To Determine Author's Point Of View\n\nAdapted from\u00a0The Adventures of Tom Sawyer\u00a0by Mark Twain (Samuel Clemens) (1876)\n\nSaturday morning was come, and all the summer world was bright and fresh, and brimming with life. Tom appeared on the sidewalk with a bucket of whitewash and a long-handled brush. He surveyed the fence, and all gladness left him and a deep melancholy settled down upon his spirit. Thirty yards of board fence nine feet high. Life to him seemed hollow, and existence but a burden. Sighing, he dipped his brush and passed it along the topmost plank; repeated the operation; did it again; compared the insignificant whitewashed streak with the far-reaching continent of unwhitewashed fence, and sat down on a tree-box discouraged.\n\nHe began to think of the fun he had planned for this day, and his sorrows multiplied. Soon the boys would come tripping along on all sorts of delicious expeditions, and they would make a world of fun of him for having to work\u2014the very thought of it burnt him like fire. At this dark and hopeless moment an inspiration burst upon him! Nothing less than a great, magnificent inspiration.\n\nHe took up his brush and went tranquilly to work. Ben Rogers hove in sight presently\u2014the very boy, of all boys, whose ridicule he had been dreading. Ben\u2019s gait was the hop-skip-and-jump\u2014proof enough that his heart was light and his anticipations high. He was eating an apple, and giving a long, melodious whoop, at intervals, followed by a deep-toned ding-dong-dong, ding-dong-dong, for he was personating a steamboat. As he drew near, he slackened speed, took the middle of the street, leaned far over to star-board and rounded to ponderously and with laborious pomp and circumstance\u2014for he was personating the Big Missouri, and considered himself to be drawing nine feet of water. He was boat and captain and engine-bells combined, so he had to imagine himself standing on his own hurricane-deck giving the orders and executing them:\n\n\u201cStop her, sir! Ting-a-ling-ling!\u201d The headway ran almost out, and he drew up slowly toward the sidewalk.\n\nTom went on whitewashing\u2014paid no attention to the steamboat. Ben stared a moment and then said: \u201cHi-yiYou\u2019re\u00a0up a stump, ain\u2019t you!\u201d\n\nNo answer. Tom surveyed his last touch with the eye of an artist, then he gave his brush another gentle sweep and surveyed the result, as before. Ben ranged up alongside of him. Tom\u2019s mouth watered for the apple, but he stuck to his work. Ben said:\n\n\u201cHello, old chap, you got to work, hey?\u201d\n\nTom wheeled suddenly and said:\n\n\u201cWhy, it\u2019s you, Ben! I warn\u2019t noticing.\u201d\n\n\u201cSay\u2014I\u2019m going in a-swimming, I am. Don\u2019t you wish you could? But of course you\u2019d druther\u00a0work\u2014wouldn\u2019t you? Course you would!\u201d\n\nTom contemplated the boy a bit, and said:\n\n\u201cWhat do you call work?\u201d\n\n\u201cWhy, ain\u2019t\u00a0that\u00a0work?\u201d\n\nTom resumed his whitewashing, and answered carelessly:\n\n\u201cWell, maybe it is, and maybe it ain\u2019t. All I know, is, it suits Tom Sawyer.\u201d\n\n\u201cOh come, now, you don\u2019t mean to let on that you\u00a0like\u00a0it?\u201d\n\nThe brush continued to move.\n\n\u201cLike it? Well, I don\u2019t see why I oughtn\u2019t to like it. Does a boy get a chance to whitewash a fence every day?\u201d\n\nThat put the thing in a new light. Ben stopped nibbling his apple. Tom swept his brush daintily back and forth\u2014stepped back to note the effect\u2014added a touch here and there\u2014criticized the effect again\u2014Ben watching every move and getting more and more interested, more and more absorbed. Presently he said:\n\n\u201cSay, Tom, let\u00a0me\u00a0whitewash a little.\u201d\n\nTom considered, was about to consent; but he altered his mind:\n\n\u201cNo\u2014no\u2014I reckon it wouldn\u2019t hardly do, Ben. You see, Aunt Polly\u2019s awful particular about this fence\u2014right here on the street, you know\u2014but if it was the back fence I wouldn\u2019t mind and\u00a0she\u00a0wouldn\u2019t. Yes, she\u2019s awful particular about this fence; it\u2019s got to be done very careful; I reckon there ain\u2019t one boy in a thousand, maybe two thousand, that can do it the way it\u2019s got to be done.\u201d\n\n\u201cNo\u2014is that so? Oh come, now\u2014lemme, just try. Only just a little\u2014I\u2019d let\u00a0you, if you was me, Tom.\u201d\n\n\u201cBen, I\u2019d like to; but Aunt Polly\u2014well, Jim wanted to do it, but she wouldn\u2019t let him; Sid wanted to do it, and she wouldn\u2019t let Sid. Now don\u2019t you see how I\u2019m fixed? If you was to tackle this fence and anything was to happen to it\u2014\u201d\n\n\u201cOh, shucks, I\u2019ll be just as careful. Now lemme try. Say\u2014I\u2019ll give you the core of my apple.\u201d\n\n\u201cWell, here\u2014No, Ben, now don\u2019t. I\u2019m afeard\u2014\u201d\n\n\u201cI\u2019ll give you\u00a0all\u00a0of it!\u201d\n\nTom gave up the brush with reluctance in his face, but alacrity in his heart. And while the late steamer Big Missouri worked and sweated in the sun, the retired artist sat on a barrel in the shade close by, dangled his legs, munched his apple, and planned the slaughter of more innocents. There was no lack of material; boys happened along every little while; they came to jeer, but remained to whitewash. By the time Ben was tired, Tom had traded the next chance to Billy Fisher for a kite, in good repair; and when he played out, Johnny Miller bought in for a dead rat and a string to swing it with\u2014and so on, and so on, hour after hour. And when the middle of the afternoon came, from being a poor poverty-stricken boy in the morning, Tom was literally rolling in wealth. And when the middle of the afternoon came, from being a poor poverty-stricken boy in the morning, Tom was literally rolling in wealth. He had besides the things before mentioned, twelve marbles, a piece of blue bottle-glass to look through, a spool cannon, a key that wouldn't unlock anything, a fragment of chalk, a glass stopper of a decanter, a tin soldier, a couple of tadpoles, six fire-crackers, a kitten with only one eye, a brass door-knob, a dog-collar\u2014but no dog\u2014the handle of a knife, four pieces of orange-peel, and a dilapidated old window sash.\n\nHe had had a nice, good, idle time all the while\u2014plenty of company\u2014and the fence had three coats of whitewash on it! If he hadn\u2019t run out of whitewash he would have bankrupted every boy in the village.\n\nWhich of the following best describes the dialogue in this scene?\n\nSwitching dramatically from formal and fancy to informal and casual\n\nUsing lots of foreign\u00a0words\u00a0borrowed from other languages\n\nSwitching dramatically from informal and casual to\u00a0formal and fancy\n\nFormal and fancy\n\nCasual and informal\n\nCasual and informal\n\nExplanation:\n\nThe dialogue in this scene is the conversation that Tom and Ben have, not the surrounding descriptions:\u00a0the descriptions are the narrator's prose, to which this question is not referring. How do Tom and Ben talk to each other? The language they use is not very formal or fancy: it uses a lot of nonstandard English like \u201cwarn\u2019t,\u201d \u201cdruther,\u201d \u201caint,\u201d and \"afeard.\" This is consistent throughout the whole passage: at no point do they characters shift to speaking in a notably different, more formal style. Their conversation is consistently casual throughout. Thus, the best answer is \"casual and informal.\"\n\n### Example Question #1 : Reading To Determine Author's Point Of View\n\nAdapted from \u201cThe Open Window\u201d in\u00a0Beasts and Super-Beasts\u00a0by H. H. Munro (Saki) (1914)\n\n\"My aunt will be down presently, Mr. Nuttel,\" said a very self-possessed young lady of fifteen. \"In the meantime you must try and put up with me.\"\n\nFramton Nuttel endeavored to say the correct something which should duly flatter the niece of the moment without unduly discounting the aunt that was to come. Privately he doubted more than ever whether these formal visits on a succession of total strangers would do much towards helping the nerve cure which he was supposed to be undergoing.\n\n\"I know how it will be,\" his sister had said when he was preparing to migrate to this rural retreat. \"You will bury yourself down there and not speak to a living soul, and your nerves will be worse than ever from moping. I shall just give you letters of introduction to all the people I know there. Some of them, as far as I can remember, were quite nice.\"\n\nFramton wondered whether Mrs. Sappleton, the lady to whom he was presenting one of the letters of introduction, came into the \"nice\" division.\n\n\"Do you know many of the people round here?\" asked the niece, when she judged that they had had sufficient silent communion.\n\n\"Hardly a soul,\" said Framton. \"My sister was staying here, at the rectory, you know, some four years ago, and she gave me letters of introduction to some of the people here.\"\n\nHe made the last statement in a tone of distinct regret.\n\n\"Then you know practically nothing about my aunt?\" pursued the self-possessed young lady.\n\n\"Only her name and address,\" admitted the caller. He was wondering whether Mrs. Sappleton was in the married or widowed state. An undefinable something about the room seemed to suggest masculine habitation.\n\n\"Her great tragedy happened just three years ago,\" said the child. \"That would be since your sister's time.\"\n\n\"Her tragedy?\" asked Framton; somehow in this restful country spot tragedies seemed out of place.\n\n\"You may wonder why we keep that window wide open on an October afternoon,\" said the niece, indicating a large French window that opened on to a lawn.\n\n\"It is quite warm for the time of the year,\" said Framton, \"but has that window got anything to do with the tragedy?\"\n\n\"Out through that window, three years ago to a day, her husband and her two young brothers went off for their day's shooting. They never came back. In crossing the moor to their favorite snipe-shooting ground they were all three engulfed in a treacherous piece of bog. It had been that dreadful wet summer, you know, and places that were safe in other years gave way suddenly without warning. Their bodies were never recovered. That was the dreadful part of it.\"\u00a0Here the child's voice lost its self-possessed note and became falteringly human.\u00a0\"Poor aunt always thinks that they will come back someday, they and the little brown spaniel that was lost with them, and walk in at that window just as they used to do. That is why the window is kept open every evening till it is quite dusk. Poor dear aunt, she has often told me how they went out, her husband with his white waterproof coat over his arm, and Ronnie, her youngest brother, singing 'Bertie, why do you bound?' as he always did to tease her, because she said it got on her nerves. Do you know, sometimes on still, quiet evenings like this, I almost get a creepy feeling that they will all walk in through that window\u2014\"\n\nShe broke off with a little shudder. It was a relief to Framton when the aunt bustled into the room with a whirl of apologies for being late in making her appearance.\n\n\"I hope Vera has been amusing you?\" she said.\n\n\"She has been very interesting,\" said Framton.\n\n\"I hope you don't mind the open window,\" said Mrs. Sappleton briskly. \"My husband and brothers will be home directly from shooting, and they always come in this way. They've been out for snipe in the marshes today, so they'll make a fine mess over my poor carpets. So like you menfolk, isn't it?\"\n\nShe rattled on cheerfully about the shooting and the scarcity of birds, and the prospects for duck in the winter. To Framton it was all purely horrible. He made a desperate but only partially successful effort to turn the talk on to a less ghastly topic, he was conscious that his hostess was giving him only a fragment of her attention, and her eyes were constantly straying past him to the open window and the lawn beyond. It was certainly an unfortunate coincidence that he should have paid his visit on this tragic anniversary.\n\n\"The doctors agree in ordering me complete rest, an absence of mental excitement, and avoidance of anything in the nature of violent physical exercise,\" announced Framton, who labored under the tolerably widespread delusion that total strangers and chance acquaintances are hungry for the least detail of one's ailments and infirmities, their cause and cure. \"On the matter of diet they are not so much in agreement,\" he continued.\n\n\"No?\" said Mrs. Sappleton, in a voice which only replaced a yawn at the last moment. Then she suddenly brightened into alert attention\u2014but not to what Framton was saying.\n\n\"Here they are at last!\" she cried. \"Just in time for tea, and don't they look as if they were muddy up to the eyes!\"\n\nFramton shivered slightly and turned towards the niece with a look intended to convey sympathetic comprehension. The child was staring out through the open window with a dazed horror in her eyes. In a chill shock of nameless fear Framton swung round in his seat and looked in the same direction.\n\nIn the deepening twilight three figures were walking across the lawn towards the window, they all carried guns under their arms, and one of them was additionally burdened with a white coat hung over his shoulders. A tired brown spaniel kept close at their heels. Noiselessly they neared the house, and then a hoarse young voice chanted out of the dusk: \"I said, Bertie, why do you bound?\"\n\nFramton grabbed wildly at his stick and hat; the hall door, the gravel drive, and the front gate were dimly noted stages in his headlong retreat. A cyclist coming along the road had to run into the hedge to avoid imminent collision.\n\n\"Here we are, my dear,\" said the bearer of the white mackintosh, coming in through the window. \"Fairly muddy, but most of it's dry. Who was that who bolted out as we came up?\"\n\n\"A most extraordinary man, a Mr. Nuttel,\" said Mrs. Sappleton. \"Could only talk about his illnesses, and dashed off without a word of goodbye or apology when you arrived. One would think he had seen a ghost.\"\n\n\"I expect it was the spaniel,\" said the niece calmly. \"He told me he had a horror of dogs. He was once hunted into a cemetery somewhere on the banks of the Ganges by a pack of pariah dogs, and had to spend the night in a newly dug grave with the creatures snarling and grinning and foaming just above him. Enough to make anyone lose their nerve.\"\n\nRomance at short notice was her speciality.\n\nIn the context of the whole story, what is the most likely reason that Mrs. Sappleton\u2019s niece asks, \u201cThen you know practically nothing about my aunt?\u201d in the underlined paragraph?\n\nTo decide whether she wants to keep talking to him\n\nTo figure out if he is interested in going hunting with Mrs. Sappleton\u2019s husband and brothers\n\nTo figure out if she can get him to believe a made-up story about her aunt\n\nTo figure out if he knows about the tragedy that her aunt went through\n\nTo try to figure out what he is afraid of\n\nTo figure out if she can get him to believe a made-up story about her aunt\n\nExplanation:\n\nConsider what happens in the whole story: Mr. Framton is left\u00a0with the niece before Mrs. Sapplton, her aunt, arrives. The two talk\u00a0about why Mr. Nuttel\u00a0is visiting. Then, the niece tells him a horror story about Mrs. Sappleton's family members going hunting and never coming back, and saying that's why her aunt always leaves the window open. (This is essentially a ghost story.) Mr. Nuttel\u00a0believes this. Mrs. Sappleton arrives and talks about the open window as if her family members have just gone hunting. This unnerves Mr. Nuttel.\u00a0The hunting party comes back and the niece looks scared. Mr. Nuttel, thinking he's seeing\u00a0ghosts, runs away. The niece makes up a story about Mr. Nuttel to explain why he ran away.\n\nWhere in that sequence of events does this line appear? At the beginning of the story, when Mr. Nuttel and the niece are talking, but before the niece has told him the scary story about the hunting party. Why might she ask, \u201cThen you know practically nothing about my aunt?\u201d This is before she has made up the scary story. If she can establish that Mr. Nuttel knows nothing about her aunt, she knows that he will be more likely to believe any story she tells him about her aunt and the open window. This is the correct answer: we can infer that the niece asks this question \"to figure out if she can get [Mr. Nuttel]\u00a0to believe a made-up story about her aunt.\"\n\n### Example Question #1 : Reading To Determine Author's Point Of View\n\n\"The Ruby-throated Hummingbird\"\n\nGeographical Range and Migration\n\nThe Ruby-throated Hummingbird is the sole representative of the hummingbird family in eastern North America. It is only a summer visitor in Canada and throughout the greater part of its range in the United States, excepting the southern portions of the Florida peninsula, where it winters to some extent. The majority of these birds migrate south, though, spending the winter in some of the Caribbean islands, while others pass through eastern Mexico into Central America. It usually arrives along our southern border in the latter part of March, rarely reaching the more northern States before the middle of May. It usually goes south again about the latter part of September, the males preceding the females, I believe, in both migrations.\n\nAppearance and Behavior\n\nRuby-throated Hummingbirds have iridescent green feathers on their backs and white feathers on their bellies. The male birds have a patch of red feathers on their throats, from which the species derives its name.\u00a0Both male and female Ruby-throated Hummingbirds\u00a0have relatively short tails and beaks and lack any\u00a0crest of feathers on their heads.\n\nRuby-throated Hummingbirds\u2019 flight is extremely swift, and the rapid motions of its wings in passing back and forth from one cluster of flowers to another causes a humming or buzzing sound, from which the numerous members of this family derive their name of hummingbirds. Notwithstanding the very small size of most of our hummers, they are all extremely\u00a0pugnacious, especially the males, and are constantly quarreling and chasing each other, as well as other birds, some of which are many times larger than themselves. Mr. Manly Hardy writes me that he once saw a male Ruby-throat chase a Robin out of his garden. They are rarely seen entirely at rest for any length of time, and, when not busy preening its feathers, they dart about from one place to another. Although such a small, tiny creature, it is full of energy, and never seems to tire.\n\nThey seem to be especially partial to anything red. Mr. Manly Hardy writes: \"I was once camping on one of the many islands along the coast of Maine during a dense fog, which had held us prisoners for several days, as it was so thick that we could not find our way. We had been living on lobsters, and lots of their red shells lay near the fire in front of our tent, when suddenly a Hummer came out of the fog and darted down at the shells, moving from one to another, seemingly\u00a0loath\u00a0to leave them.\u201d\n\nWhat Do They Eat?\n\nThere appears to be considerable difference of opinion among various observers regarding the nature of the Ruby-throated Hummingbird\u2019s food. Some contend that it consists principally of nectar sipped from flowers, as well as the sweet sap of certain trees. Others, myself included, believe that they subsist mainly on minute insects and small spiders, the latter forming quite an important article of food with them. Mr. Edwin H. Eames, of Bridgeport, Connecticut, mentions finding sixteen young spiders of uniform size in the throat of a young Hummingbird which was about two days old.\n\nMr. W. N. Clute, of Binghamton, New York, writes: \"The swamp thistle, which blooms in August, seems to have great attractions for the Ruby-throated Hummingbirds. I have seen more than a hundred birds about these plants in the course of an hour. Since it has been stated that the bee gets pollen but not honey from the thistle, it would appear that these birds visit these flowers for insects. There is scarcely a flower that contains so many minute insects as a thistle head. Examine one with a lens and it will be found to contain many insects that can hardly be seen with the unaided eye, and if the Ruby-throat eats insects at all, these are the ones it would take; and because the larger ones remained the observer might conclude that none were eaten.\u201d I could quote considerable more testimony showing that the Hummingbirds live to a great extent on minute spiders and insects, but consider it unnecessary.\n\nThat our Hummingbirds live to some extent on the sap of certain trees is undoubtedly true, but that they could exist for any length of time on such food alone is very questionable. They are particularly fond of the sap of the sugar maple, and only slightly less so of that of a few other\u00a0species of trees. They are also fond of\u00a0the nectar secreted in many flowers.\u00a0While stationed at the former cavalry depot at St. Louis, Missouri, in 1873-74, I occupied a set of quarters that were completely overrun with large trumpet vines. When these were in bloom, the place fairly swarmed with Ruby-throats. They were exceedingly inquisitive, and often poised themselves before an open window and looked in my rooms full of curiosity, their bright little eyes sparkling like black beads. I have caught several, while busily engaged sipping nectar in these large, showy flowers, by simply placing my hand over them, and while so imprisoned they never moved, and feigned death, but as soon as I opened my hand they were off like a flash.\n\nPassage adapted from \"Ruby-throated Hummingbird\" from Issue 3 of\u00a0Life Histories of North American Birds, From the Parrots to the Grackles, with Special Reference to Their Breeding Habits and Eggs\u00a0by Charles Bendire (1895)\n\nImage adapted from Giltsch, Adolf, Lithographer, and Ernst Haeckel.\u00a0Trochilidae. - Kolibris. [Leipzig und Wien: Verlag des Bibliographischen Instituts, 1904] Image. Retrieved from the Library of Congress, <https:\/\/www.loc.gov\/item\/2015648985>.\n\nThe author thinks that Ruby-throated Hummingbirds __________.\n\neat nectar and sap as well as large beetles, and that the nectar and sap are more important to their diet\n\neat nectar and sap as well as small bugs and small spiders, and that the spiders are more important to their diet\n\neat nectar and sap as well as large beetles, and that the large beetles are more important to their diet\n\neat nectar and sap only\n\neat nectar and sap as well as small bugs and small spiders, and that the nectar and sap are more important to their diet\n\neat nectar and sap as well as small bugs and small spiders, and that the spiders are more important to their diet\n\nExplanation:\n\nTo figure out what the author thinks Ruby-throated Hummingbirds eat, we should look in the section of the passage titled \"What Do They Eat?\"\u00a0The author begins this section by explaining\u00a0that people\u00a0have come to different conclusions about what these birds eat. The author says:\n\nSome contend that it consists principally of nectar sipped from flowers, as well as the sweet sap of certain trees. Others, myself included, believe that they subsist mainly on minute insects and small spiders, the latter forming quite an important article of food with them.\n\nDid you catch that important phrase, \"myself included\"? By this, the author indicates that this is the group into which he falls. Which group is that? The group that thinks that Ruby-throated Hummingbirds eat mosty small insects and spiders. In addition, the author adds, \"the latter forming quite an important article of food with them.\" What does \"the latter\" refer to? \"The latter\" means the second of two things thing\u00a0just presented. The list that was just presented was \"small insects and spiders.\" So, in this passage, \"the latter\" refers to small\u00a0spiders, and the author thinks\u00a0that these form an important part of the diets of Ruby-throated hummingbirds.\n\nLater in the passage, the author writes,\n\nThat our Hummingbirds live to some extent on the sap of certain trees is undoubtedly true, but that they could exist for any length of time on such food alone is very questionable. They are particularly fond of the sap of the sugar maple, and only slightly less so of that of a few other\u00a0species of trees. They are also fond of\u00a0the nectar secreted in many flowers.\n\nThis tells us that the author thinks that they eat sap and flower nectar, but that this isn't their main form of food.\n\nWe can now answer the question correctly! The author thinks that Ruby-throated Hummingbirds \"eat nectar and sap as well as small bugs and small spiders, and that the spiders are more important to their diet.\"\n\n### Example Question #1 : Reading To Determine Author's Point Of View\n\nAdapted from \u201cTheodore Roosevelt the Rancher.\u201d National Park Service.\u00a0U.S. Department of the Interior, n.d. Web. 1 July 2016. <https:\/\/www.nps.gov\/thro\/learn\/historyculture\/theodore-roosevelt-the-rancher.htm>.\n\nTheodore Roosevelt originally came to Dakota Territory in 1883 to hunt bison. The locals showed little interest in helping this eastern tenderfoot. The promise of quick cash, however, convinced Joe Ferris\u2014a 25-year-old Canadian living in the Badlands\u2014to act as Roosevelt's hunting guide.\n\nThrough terrible weather and awful luck, Roosevelt showed a determination which surprised his exasperated hunting guide. Finding a bison proved difficult; most of the herds had been slaughtered in recent years by commercial hunters. When they were not sleeping outdoors, Roosevelt and Ferris used the small ranch cabin of Gregor Lang as a base camp. Evenings at Lang's ranch saw an exhausted Ferris falling asleep to conversations between Roosevelt and their host. Spirited debates on politics gave way to discussions about ranching, and Roosevelt became interested in raising cattle in the Badlands.\n\nCattle ranching in Dakota was a boom business in the 1880s. With the northern plains recently devoid of bison, cattle were being driven north from Texas to feed on the nutritious grasses. The Northern Pacific Railroad offered a quick route to eastern markets without long drives that reduced the quality of the meat. Entrepreneurs like the Marquis de Mor\u00e8s were bringing money and infrastructure to the region. The opportunity struck Roosevelt as a sound business opportunity.\n\nWith Roosevelt's interest sparked, he entered into business with his guide's brother, Sylvane Ferris, and Bill Merrifield, another Dakota cattleman. Roosevelt put down an initial investment of $14,000\u2014significantly more than his annual salary. Roosevelt returned to New York with instructions for Ferris and Merrifield to build the Maltese Cross Cabin. His investment was not purely for business; Roosevelt saw it as a chance to immerse himself in a western lifestyle he had long romanticized. The purpose of this passage is __________. Possible Answers: to describe how Theodore Roosevelt came to invest in a cattle ranch in the western U.S. to describe the culture of the western U.S. during Roosevelt\u2019s era to talk about the financial investments that Roosevelt made throughout his life to explain Theodore Roosevelt\u2019s motivation for hunting bison in the western U.S. to describe the ranch on which Roosevelt lived during his time in the West Correct answer: to describe how Theodore Roosevelt came to invest in a cattle ranch in the western U.S. Explanation: This biography begins by describing how Theodore Roosevelt's traveled to the Dakota Territories to hunt bison. It then explains how he became interested in owning part of a cattle ranch, and some of his motivations behind this decision. While the passage discusses how hunting bison brought Roosevelt to the Dakota Territories, it does not \"explain [his] motivation for hunting bison in the western U.S.\" because it never talks about why he wanted to hunt bison. The answer choices \"to describe the culture of the western U.S. during Roosevelt\u2019s era\" and \"to talk about the financial investments that Roosevelt made throughout his life\" are both incorrect as they are each too broad. The passage doesn't talk about the culture of the entire western U.S. during Roosevelt's era and it only talks about one financial investment that he made. While it mentions the ranch on which Roosevelt stayed, the purpose of the passage is not \"to describe the ranch on which Roosevelt lived during his time in the West.\" The best answer is that the point of the passage is \"to describe how Theodore Roosevelt came to invest in a cattle ranch in the western U.S.\" This captures the passage's main subject matter: Roosevelt's arrival in the Dakota Territories, his interest in owning a cattle ranch, and his motivations for investing in one. ### Example Question #1 : Reading To Determine Author's Point Of View Young Enterprise Services (YES) is a program created to encourage entrepreneurship in 14- to 18-year-olds who have already shown a clear ability for starting businesses. The program started in 2002, has provided loans, grants, and counseling\u2014in the form of workshops and individual meetings with entrepreneurs\u2014to over 7500 young people. The future of YES, however, is now at risk. One complaint is that the funds that YES distributes have disproportionately gone to young people from low-income families. Though no one has claimed that any of the recipients of YES funds have been undeserving, several families have brought lawsuits claiming that their requests for funding were rejected because of the families\u2019 high levels of income. Another challenge has been the task of making sure that a young person, not his or her family, is receiving the funding. The rules state that the business plan must be created by the youth and that any profits in excess of$1,000 be placed in a bank account. The rules say that the money can only be used for education, investment in the business, and little else. There have been cases of parents or even a neighbor using the money for their business.\n\nOn the other hand, YES has had some real success stories. A 14-year-old girl in Texas used the knowledge and funding she received through the program to connect with a distributor who now carries her line of custom-designed cell phone covers. Two brothers in Alaska have developed an online travel service for young people vacationing with their families. Both of these businesses are doing well and earning money. Unfortunately, these and other successes have received little media coverage. This is a shame, but one that can be fixed.\n\nWhat is the author\u2019s point of view about Young Enterprise Services (YES)?\n\nThe author\u2019s point of view is mainly negative and alludes to the program needing major work. The author believes that there isn\u2019t much hope for it to continue on.\n\nThe author does not provide any evidence or language that would lead readers to a specific point of view.\n\nThe author views the program in a positive manner and does not seem to recognize that there are issues that need to be addressed.\n\nThe author recognizes there are challenges facing a program that has otherwise been successful. The author feels that the program\u2019s tarnished image can be changed.\n\nThe author recognizes there are challenges facing a program that has otherwise been successful. The author feels that the program\u2019s tarnished image can be changed.\n\nExplanation:\n\nThe text is written where both the positive and negative aspects of the program are acknowledged so there is a neutrality to the point of view. The author does lean in the optimistic direction when he or she interjects personal feelings.\n\n### Example Question #1 : Craft And Structure\n\nYoung Enterprise Services (YES) is a program created to encourage entrepreneurship in 14- to 18-year-olds who have already shown a clear ability for starting businesses. The program started in 2002, has provided loans, grants, and counseling\u2014in the form of workshops and individual meetings with entrepreneurs\u2014to over 7500 young people. The future of YES, however, is now at risk.\n\nOne complaint is that the funds that YES distributes have disproportionately gone to young people from low-income families. Though no one has claimed that any of the recipients of YES funds have been undeserving, several families have brought lawsuits claiming that their requests for funding were rejected because of the families\u2019 high levels of income.\n\nAnother challenge has been the task of making sure that a young person, not his or her family, is receiving the funding. The rules state that the business plan must be created by the youth and that any profits in excess of \\$1,000 be placed in a bank account. The rules say that the money can only be used for education, investment in the business, and little else. There have been cases of parents or even a neighbor using the money for their business.\n\nOn the other hand, YES has had some real success stories. A 14-year-old girl in Texas used the knowledge and funding she received through the program to connect with a distributor who now carries her line of custom-designed cell phone covers. Two brothers in Alaska have developed an online travel service for young people vacationing with their families. Both of these businesses are doing well and earning money. Unfortunately, these and other successes have received little media coverage. This is a shame, but one that can be fixed.\n\nWhat personal opinion does the author interject into the story that alludes to his or her point of view?\n\nThere have been cases of parents or even a neighbor using the money for their business.\n\nThe program started in 2002, has provided loans, grants, and counseling\u2014in the form of workshops and individual meetings with entrepreneurs\u2014to over 7500 young people.\n\nA 14-year-old girl in Texas used the knowledge and funding she received through the program to connect with a distributor who now carries her line of custom-designed cell phone covers.\n\nThis is a shame, but one that can be fixed.\n\nThis is a shame, but one that can be fixed.\n\nExplanation:\n\nThe author includes this personal opinion in the last paragraph of the passage. The rest of the answer choices are facts where this alludes to the author\u2019s point of view about the program. He or she believes that the program can fix the image it has and mentions beforehand the need for more positive press coverage.\n\n### Example Question #1 : Reading To Determine Author's Point Of View\n\nWhen I was seven, my father brought home from a business trip a wooden boomerang painted with images of the Australian flag. All summer long I carried that gift with me. I was fascinated by this piece of a continent completely on the other side of the world. Despite promises that if I threw it would immediately return, I had no intention of throwing it, only carrying and admiring it. What if it became stuck in a tree or carried away by a stiff wind? There would go my connection to the magical land of kangaroos, barrier reefs, and untold other pieces of wonder.\n\nAs I walk the shores of Bondi Beach or watch the tourists purchase kangaroo-themed apparel in my adopted hometown of Sydney, I often think back to that boomerang and the world to which it opened my eyes. As an airline pilot, I am fortunate to live out my childhood dream \u2013 inspired by that boomerang \u2013 of exploring faraway lands. Whenever I do, I bring home a trinket for my young daughter such that she might be similarly struck by wanderlust.\n\nHow does the author\u2019s word choice develop the narrator\u2019s point of view?\n\nThe author uses emotional language related to memories and travel to develop the narrator\u2019s point of view.\n\nThe author\u2019s word choice allows readers to understand that the narrator\u2019s point of view towards Australia changes over time.\n\nThe author uses words with a negative connotation to develop the narrator\u2019s point of view regarding his\/her childhood.\n\nThe author\u2019s word choices do not develop the narrator\u2019s point of view.\n\nThe author uses emotional language related to memories and travel to develop the narrator\u2019s point of view.\n\nExplanation:\n\nThe author uses words such as: fascinated, connection, wonder, inspired, and wanderlust to make a connection with emotions and memories to allude to the point of view of the narrator. The language is powerful and personal to the narrator so readers can better understand where the narrator is coming from.\n\n### Example Question #51 : Reading\n\nWhen I was seven, my father brought home from a business trip a wooden boomerang painted with images of the Australian flag. All summer long I carried that gift with me. I was fascinated by this piece of a continent completely on the other side of the world. Despite promises that if I threw it would immediately return, I had no intention of throwing it, only carrying and admiring it. What if it became stuck in a tree or carried away by a stiff wind? There would go my connection to the magical land of kangaroos, barrier reefs, and untold other pieces of wonder.\n\nAs I walk the shores of Bondi Beach or watch the tourists purchase kangaroo-themed apparel in my adopted hometown of Sydney, I often think back to that boomerang and the world to which it opened my eyes. As an airline pilot, I am fortunate to live out my childhood dream \u2013 inspired by that boomerang \u2013 of exploring faraway lands. Whenever I do, I bring home a trinket for my young daughter such that she might be similarly struck by wanderlust.\n\nFrom whose point of view is the text written?\n\nThis text is written from the third-person point of view.\n\nThis text is written from the second-person point of view.\n\nThe text is written from the first-person point of view.\n\nIt is not able to be determined from whose point of view the text is written.\n\nThe text is written from the first-person point of view.\n\nExplanation:\n\nA first-person narrator is a character in the piece. The first-person narrator is only giving us his or her perspective. In this passage, the narrator uses I and my which are first-person pronouns.\n\n### Example Question #1 : Craft And Structure\n\nNearly all the workers of the Lowell textile mills of Massachusetts were unmarried daughters from farm families. Some of the workers were as young as 10. Many people in the 1820s were upset by the idea of working females. The company provided well-kept dormitories for the women to live in. The meals were decent and church attendance was mandatory. Compared to other factories of the time, the Lowell mills were clean and safe. There was even a journal, The Lowell Offering, which contained poems and other material written by the workers, and which became known beyond New England. Ironically, it was at the Lowell mills that dissatisfaction with working conditions brought about the first organization of working women.\n\nThe work was difficult. When wages were cut, the workers organized the Factory Girls Association. 15,000 women decided to \u201cturn out,\u201d or walk off the job. The Offering, meant as a pleasant creative outlet, gave the women a voice that could be heard elsewhere in the country, and even in Europe. However, the ability of women to demand changes was limited. The women could not go for long without wages with which to support themselves and families. This same limitation hampered the effectiveness of the Lowell Female Labor Reform Association (LFLRA), organized in 1844.\n\nNo specific changes can be directly credited to the Lowell workers, but their legacy is unquestionable. The LFLRA\u2019s founder, Sarah Bagley, became a national figure, speaking before the Massachusetts House of Representatives. When the New England Labor Reform League was formed, three of the eight board members were women. Other mill workers took note of the Lowell strikes and were successful in getting better pay, shorter hours, and safer working conditions. Even some existing child labor laws can be traced back to efforts first set in motion by the Lowell mills women.\n\nWhat is the author\u2019s point of view about the Lowell Female Labor Reform Association (LFLRA)?\n\nThe author\u2019s point of view about the Lowell Female Labor Reform Association (LFLRA) is that the women were one of the greatest labor reform groups in history and made a large number of direct changes to the work industry.\n\nThe author\u2019s point of view is mainly negative and alludes to the women\u2019s association not accomplishing very much in labor reform.\n\nThe author recognizes the challenges the women faced and credits their actions for having inspired long-lasting changes.\n\nThe author does not provide any evidence or language that would lead readers to a specific point of view.","date":"2021-01-22 23:06:02","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\": 0.22676102817058563, \"perplexity\": 4301.311971546463}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703531429.49\/warc\/CC-MAIN-20210122210653-20210123000653-00177.warc.gz\"}"} | null | null |
Sprint = setmetatable({}, RaceMode)
Sprint.__index = Sprint
Sprint:register('Sprint')
function Sprint:isApplicable()
return self.checkpointsExist() -- and self.getMapOption('respawn') == 'timelimit'
end
Sprint.modeOptions = {
respawn = 'timelimit',
respawntime = 5,
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 3,914 |
What is economic integration in globalization?
What is the process of economic reforms?
Why is global economic integration important?
What is the impact of economic reform globalization?
What are the reasons for economic reforms?
What is global integration?
What are the reasons of economic reforms?
What are the outcomes of new economic reforms?
When did economic integration begin?
Does economic integration Promote growth in poor countries?
Economic integration, like the name implies, involves the integration of countries' economies. Another term to describe it is globalization, which simply refers to the inter-connectedness of businesses and trading among countries.
(i) Reduction in Restrictions of Export-Import: Restrictions on the exports-imports have almost disappeared leaving only a few items. (ii) Reduction in Export-Import Tax: Export-import tax on some items has been completely abolished and on some other items it has been reduced to the minimum level.
What is the process of economic globalization?
Economic "globalization" is a historical process, the result of human innovation and technological progress. It refers to the increasing integration of economies around the world, particularly through the movement of goods, services, and capital across borders.
What is an example of economic reform?
Economic reform as microeconomic reform is well understood. It dominated government thinking in the 1980s and 90s – a floating dollar, lower tariffs, de-regulation, tax cuts and tax reform, corporatisation and privatisation, labour market reform and the contracting out of government services.
Economic theory and international experience show us that small countries get richer when deeply integrated into the global economy. Economic integration can facilitate access to a larger consumer base, a greater pool of qualified workers, additional sources of financing, and new technologies.
Globalization is basically a process of increasing the economic integration and growing economic interdependence between different countries in the world economy. The processes of economic liberalization and privatization of the public sector enterprises eventually led to the globalization of the Indian economy.
What are the importance of economic reforms?
The reforms were aimed at attaining a high rate of economic growth, reducing the rate of inflation, reducing the current account deficit and overcoming the balance of payments crisis. The important features of the economic reforms were Liberalisation, Privatisation and Globalisation, popularly known as LPG.
What is the main purpose of economic globalization?
Economic globalization improves the efficiency of enterprises and plays a great role in increasing the size of the economy of every country. Economic development improves living standard and level of education of people. It also lays the material basis for the development of education.
The following are the reasons for economic reforms:
(i) Rise in Prices:
(ii) Rise in Fiscal Deficit:
(iii) Increase in Adverse Balance of Payments:
(iv) Iraq War:
(v) Dismal Performance of PSU's (Public Sector Undertakings):
(vi) Fall in Foreign Exchange Reserves:
global integration. noun [ U ] COMMERCE, PRODUCTION. the process by which a company combines different activities around the world so that they operate using the same methods, etc.: Global integration can involve the processes of product standardization and technology development centralization.
What is the meaning of economic integration?
economic integration, process in which two or more states in a broadly defined geographic area reduce a range of trade barriers to advance or protect a set of economic goals.
What do you mean by economic reforms explain?
Economic reforms refer to the fundamental changes that were launched in 1991 with the plan of liberalising the economy and quickening its rate of economic growth. The Narasimha Rao Government, in 1991, started the economic reforms in order to rebuild internal and external faith in the Indian economy.
Reforms led to increased competition in the sectors like banking, leading to more customer choice and increased efficiency. It has also led to increased investment and growth of private players in these sectors.
What is economic integration?
What Is Economic Integration? Economic integration is an arrangement among nations that typically includes the reduction or elimination of trade barriers and the coordination of monetary and fiscal policies.
What is the benefit of global integration?
Existing literature argues that global integration helps MNCs save costs and achieve global efficiencies. For example, global integration minimizes duplication, thus saving costs through standardization (Dunning 1998), and global integration creates efficiencies due to global economies of scale.
While economic integration was increasing throughout the 1970s and 1980s, the extent of integration has come sharply into focus only since the collapse of com- munism in 1989. In 1995 one dominant global economic system is emerg- ing.
In the process we demonstrate the close relationship between economic integration and economic conver- gence, that is, poor countries tend to grow faster than richer countries, as long as the poor and rich countries are linked together by international trade. Poor, closed economies have often performed significantly less
What is the strategic aims of economic reform?
Most programs of economic reform now underway in the developing world and in the post-communist world have as their strategic aim the 1 2 Brookings Paper-s on Economic Activity, 1:1995 integration of the national economy with the world economy. Integra- tion means not only increased market-based trade and financial flows,
Does economic integration promote economic convergence?
In the process we demonstrate the close relationship between economic integration and economic conver- gence, that is, poor countries tend to grow faster than richer countries, as long as the poor and rich countries are linked together by international trade. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,343 |
Never Get Down è un brano musicale interpretato dalla cantautrice italiana Jo Squillo, pubblicato come singolo il 28 giugno 2011.
Il brano
Prodotto da Ben DJ, il brano è stato utilizzato nel 2011 come sigla del programma televisivo TV Moda, condotto dalla stessa Jo Squillo, e in seguito pubblicato come singolo in formato digitale.
Video musicale
Il videoclip ufficiale del brano è stato pubblicato il 2 agosto 2011 sul canale YouTube di TV Moda, e vede Jo Squillo eseguire la canzone a bordo di una cabriolet per le strade di Miami Beach.
Tracce
Download digitale
Never Get Down (Ben DJ Original Edit) - 2:54
Never Get Down (Ben DJ Extended Mix) - 4:56
Never Get Down (Tv Moda Original Soundtrack) - 3:49
Note
Collegamenti esterni | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,045 |
Q: use dll of winform in a website I have a winform application and I'm using an extra dll that I've created. I know that when I open the application and the .dll are not in the same folder, the application won't work.
Now, I want to upload the application to my website so the client will open it without saving, so how can I use the dll and the application on the client computer without any installation?
A: Have a look at Click-once-deployment. It's a technique to deploy your content to a computer where the user has no administrative privileges, by supplying a downloadable package with all components that is easily updatable.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,100 |
Q: Heroku nodejs application error can't find app/index.js I keep on getting the following error log
Heroku Error: Cannot find module '/app/index.js'
Works fine on the local server?
State changed from starting to crashed
rwa@1.0.0 start /app
node index.js
module.js:538
throw err;
Error: Cannot find module '/app/index.js'
at Function.Module._resolveFilename (module.js:536:15)
I can't seem to get it to work on Heroku deploy, any ideas? I've added a correct Procfile (web: node index.js) and even added the npm start in package.json file correctly.
A: By adding start script in the package.json will fix your issues as Heroku looks start script in the package.json if it does not find it run its default start script which causes the error
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,023 |
Lincoln Park assault suspects plead not guilty
KUSI Newsroom,
KUSI Newsroom
One of four people accused of sexually assaulting a
teenage girl after she passed out while binge-drinking with them in a Lincoln
Park canyon pleaded not guilty Monday to a felony charge.
Raatib Quidar Prince, 18, was ordered held on $100,000 bail and faces a
maximum of eight years in state prison if convicted of sexual penetration of an
intoxicated person, according to Deputy District Attorney Karl Husoe.
The other suspects — two 17-year-old boys and a 14-year-old girl — are
being prosecuted in Juvenile Court.
One of the three was arraigned last week and the other teens have a
detention hearing tomorrow, said Deputy District Attorney Cheryl Sueing-Jones.
About 9 p.m. last Monday, a passer-by spotted the blacked-out 16-year-
old victim in a gorge off the 300 block of 49th Street. Medics took her to a
hospital, where she was admitted in "grave condition," SDPD Lt. Anastasia
Smith said. The girl remained unconscious until about 10 a.m. the following
Detectives determined that she and a group of teenage companions had
consumed "large quantities" of straight vodka prior to the alleged assaults.
"Although she was unconscious for several hours, none of the suspects
called for help," the lieutenant said.
Investigators believe the girl had passed out about five hours prior to
being rescued. She has since been released from the hospital and is expected to
fully recover, Smith said.
The alleged victim is acquainted with the suspects and attends the same
high school as several of them, according to police, who declined to identify
the campus. All five of them live near the open area where the unconscious girl
was found.
Prince will be back in court Friday for a bail review. A readiness
conference was set for Dec. 12 and a preliminary hearing for Dec. 14.
Categories: KUSI
California News, Good Evening San Diego, Good Morning San Diego, Politics
Sen. Melissa Melendez: Big Oil isn't the problem, California legislation is | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,771 |
Q: how to copy selected file without moving it to another directory in python i am trying to copy the selected images to another directory. I have tried shutil.move() and shutil.copy and shutil.copy2. these move the selected file. but i want to just make copy that and move that copy to required dir.
code
cwd = os.getcwd()
filepath = os.path.dirname(os.path.abspath(filepath))
shutil.move('{0}/{1}'.format(filepath, filename), '{0}/modeldata/temp/{1}'.format(cwd, filename))
A: You can use shutil.copyfile(src, dst)
import os
import shutil
cwd = os.getcwd()
filepath = os.path.dirname(os.path.abspath(filepath))
shutil.copyfile(os.path.join(filepath, filename), os.path.join(cwd, 'modeldata/temp/' filename))
Read more here:
https://docs.python.org/3/library/shutil.html
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,487 |
{"url":"http:\/\/noahpinionblog.blogspot.com\/2013\/08\/learn-to-stop-worrying-and-love.html","text":"## Tuesday, August 20, 2013\n\n### Learn to stop worrying and love (moderate) inflation\n\nThe Federal Reserve's unprecedented programs of Quantitative Easing have not, as many predicted, resulted in substantially increased inflation. But I view this as a failure of the policy, not a success.\n\nInflation is grossly underappreciated. Economists consistently fail to educate the public about what they mean by the term \"inflation\". People think it just means \"a rise in the price of something\" (though that's not really what it means). And people don't like prices rising, because it seems like it should make stuff more expensive - and who wants that?\n\nWe're told that inflation is a necessary cost of improving the economy. And in fact, that's exactly what monetarist macroeconomists (think of Mike Woodford, Miles Kimball, etc.) tell us that it is. We must accept higher inflation, they tell us, in order to also get better GDP growth. But given our 'druthers, they tell us, we'd rather have very low inflation. No one wants to become like Zimbabwe, or the Weimar Republic, right??\n\nI'm not so sure this is true, and I'll explain why later. But first, let me dispel a couple of popular myths about inflation.\n\nPopular Inflation Myth 1: \"Inflation means I can't buy as much stuff.\"\n\nWrong. Remember, inflation is an increase in the overall price level. But when the price of everything goes up, your wage should rise as well. Why? Because on average, we are all sellers of something. If you work in a tea shop and the price of tea goes up, your wage can be expected to go up as well, and so forth. Remember, every dollar that one person spends becomes the income of another person!\n\nSo when prices go up, wages should go up as well. Read this paper. The authors find that \"higher prices lead to higher wage growth\".\n\nOf course, wages are affected by other things besides inflation - for example, labor's share of total income. So \"price inflation\" and \"wage inflation\" aren't exactly the same. But they tend to be similar:\n\n(source.)\n\nEconomists have a term for how much you can buy with your wages. It's called the \"real wage\". Real wages are wages AFTER accounting for inflation. So to look at how much you can buy, don't look at inflation, look at your real wage. Your real wage tells you your real cost of living; inflation does not.\n\nTo see that inflation doesn't reduce your real wage, just think about Weimar Germany. Prices went up by a factor of one trillion. But people did not starve en masse as a result. Remember that guy with the wheelbarrow full of cash, going to buy bread? HOW DO YOU THINK HE GOT HIS HANDS ON A WHEELBARROW OF CASH IN THE FIRST PLACE?\u00a0The answer: That was not his life's savings. He did not sell the family farm. He had a wheelbarrow full of cash because as prices skyrocketed, wages skyrocketed too!\n\n\"But don't employers take advantage of inflation to screw over workers and make them take wage cuts?\"\n\nMaybe. People don't pay close attention to inflation when it's low, and so a small amount of inflation can allow employers to cut real wages without people noticing. (Actually, some economists who want \"wage flexibility\" like a small amount of inflation for exactly this reason.)\n\nBut for larger amounts of inflation, no. When inflation gets big, people start noticing, and demanding higher wages. See the paper I linked to earlier for proof. Also, check out historical U.S. inflation:\n\nCheck out those huge inflation spikes in the 1910s and 1940s! But workers got much richer in those decades.\n\nAnyway, once more: Inflation does not make your real wage fall.\n\nPopular Inflation Myth 2: \"Inflation punishes savers.\"\n\nThis one is partly right. Surprise inflation punishes past savers, because inflation redistributes wealth from creditors (past savers) to debtors (past borrowers). But in the future, interest rates will adjust to take inflation into account. That's called the Fisher Equation:\n\n$i \\approx r + \\pi$\n\nSo if inflation goes up (and if it's a surprise),\u00a0future savers will be OK, because they will demand - and get - higher interest rates.\n\nOK, so there are the popular myths. What about the benefits of inflation?\n\nInflation Benefit 1: Your debt goes away.\n\nChances are, if you're young and have a mortgage (and maybe some car loans too), you are almost certainly a net borrower. Even if you have some savings, they are probably outweighed by the mortgage. Which means that inflation makes you richer. Remember, surprise inflation helps debtors and hurts creditors. Who are debtors? Mostly the young and the poor. Who are creditors? Mostly the old and the rich.\n\nNow you hopefully see why many conservatives don't like inflation!\n\nInflation Benefit 2: The federal government debt goes away.\n\nAll that scary federal government debt! Slows down economic growth, right? Well realize that when there's inflation, the value of the federal government's debt erodes, just like your mortgage! Debt stays the same in $terms, but nominal GDP goes up, so the debt-to-GDP ratio goes down! That high inflation in the postwar era is exactly how we got rid of our huge World War 2 debt. And remember that today's government debt is tomorrow's taxes. Inflation therefore reduces the size of your future taxes. Inflation is a future tax cut! Remember, inflation means the value of a U.S. dollar goes down. But the dollar value of the debt does not change. So inflation allows you to pay off your share of the government debt with \"funny money\"! Awesome, right? Inflation Benefit 3 (?): \"Balance sheet recession\" might go away! Lots of people believe that the U.S. and other rich countries are experiencing sluggish growth because they are still \"deleveraging\" - in other words, reducing their total stock of gross debt. Now, that's a controversial theory. But if it's true, it means that inflation would help America deleverage and get back on its feet faster. Essentially, inflation is a partial debt jubilee. Now, I have to be fair, so I should mention that of course inflation has its costs as well. One of these is the pure nuisance cost - constantly changing prices is a nuisance, and that nuisance can become extremely economically damaging in a hyperinflation. Second, high inflation leads to variable inflation, increasing uncertainty and depressing investment. And finally there might even be government moral hazard; if the government decides it can simply inflate away its debt, it might engage in more irresponsible spending. These costs are all especially severe for higher levels of inflation. But anyway I hope, after reading this, that you will be a little more wary of all those warnings about the evils of inflation. stop listening to poorly informed politicians, \"Austrian\" forum trolls, and your uncle who thinks he's still in the 70s. Inflation does not rob the poor man of his hard-earned wages; in fact, it is more likely to unburden the poor man from his crippling debt. And inflation helps get rid of all that debt, both public and private, that many people believe is clogging up our economic system. We don't want to let inflation get out of hand. But a higher Fed inflation target for the next decade - say, 4% or 5%, instead of our current 2% - would probably be a good thing for most Americans. Update: Finance blogger Mish Shedlock responds to this post with quite a bit of spluttering. #### 114 comments: 1. Noah, you forgot one more benefit of inflation, highlighted often by Steve Williamson, that inflation is a tax on those who hold currency, and those who hold large amounts of currency are usually illegal enterprises. Raise inflation, and you are taxing them more. At the same time, you ignored all the costs, like: -The cost of updating menus, catalogs, price tags, etc. more frequently -The cost of taking more frequent trips to the ATM as holding currency becomes more costly. -Costs arising from the misallocation of capital across sectors. Of course, at 5% inflation maybe these costs are not significant enough, but I would like to have some quantitative evidence that the benefit exceeds the cost. Moreover, what would a 5% inflation do to creditors, like the holders of mortgage loans with 3% and 4% rates fixed for 30 years? Are you setting us up for another financial crisis? :p 1. At the same time, you ignored all the costs How can you say this when I have a section on costs??? -The cost of updating menus, catalogs, price tags, etc. more frequently -The cost of taking more frequent trips to the ATM as holding currency becomes more costly. I considered these to be included in \"nuisance costs\". -Costs arising from the misallocation of capital across sectors. This I'm not familiar with, but sounds interesting. What's a good paper on the topic? Of course, at 5% inflation maybe these costs are not significant enough Another point I made in the post! ;-) Moreover, what would a 5% inflation do to creditors, like the holders of mortgage loans with 3% and 4% rates fixed for 30 years? Are you setting us up for another financial crisis? Those creditors are also likely to be fairly highly leveraged. Inflation should attenuate their return but not flip it from positive to negative. 2. I have always been a little uncomfortable with that notion of \"taxing.\" You'd be reducing the value of their cash holdings, not \"taxing them\" because you aren't getting the benefit of that loss of value. As for the costs you mention, the first two are specific examples of nuisance costs. The third is just made up (by Mises or Hayek, I can never remember). 3. inflation is a tax. nothing - except for a discounted brand of energy drink- I buy at the Walmart is cheaper than it was just 2 years ago. Now we can re-define inflation to exclude ..umm...inflation. awesome 4. So if inflation is a tax, what is an increase of wages for the same level and amount of work? Did your wages go up from 2 years ago? 5. Noah, sorry, I meant you ignored all the costs BELOW. The word \"below\" never made it. But yes, I suppose the word nuisance may cover them. Anyway, we do agree that moderate inflation is not the monster some people make it to be, but I am not sure how benign it is. On capital and other distortions, see the following: -Feldstein and Summers (1979), \"Inflation and the Taxation of Capital Income in the Corporate Sector.\" published in the National Tax Journal (A 1978 NBER working paper version is also available) and Cooley and Hansen (1991) titled \"The Welfare costs of moderate inflation\" in the Journal of Money, Credit, and Banking. -Frenkel and Mehrez (2000) in Economic Inquiry find that higher inflation induces a misallocation of resources from the production of consumption goods to the financial sector (which manages liquidity). -Fisher (1981) in the Brookings Papers on Economic Activity and Ball and Romer (2003) in Journal of Money, Credit, and Banking show how, with asynchronous price adjustments, inflation results in relative price variability that also distorts the allocation of resources. 6. Thanks!! 7. Don't find any possible reason to cheer up with inflation going up the rates of general commodities is also increasing day by day. Thanks William Martin PPI Claims Made Simple 2. Possible extra benefit - easier to adjust prices? If you're in the widget industry and have a mortgage, and the widget industry has become uncompetitive, better to have two years below inflation wage increases rather than a wage cut. 3. Anonymous1:58 PM \"So inflation allows you to pay off your share of the government debt with \"funny money\"! Awesome, right?\" Yay! Free lunches for everyone! \u201cit might engage in more irresponsible spending\u201d Pfft. What are you? Some kind of backwards liquidationist? 1. Notice that I do mention the possibility of what you're talking about (\"government moral hazard\"). 2. GDP growth 1-2% a year..yea we got a long way to go before that kicks in, although I agree the debt doesn't pose a thread to the economy. 4. Doesn't nominal inflation punish savers through taxes? With 0% inflation, you can maintain purchasing power indefinitely. At 10%, you have lose about 3% per year for example. You might argue that taxes are an orthogonal issue and that taxes could conceivably change to allow an inflation deduction or something like that except for point 2. 1. yes it is a tax. if gas & food is going up 10% a year your wages sure as hell wont be 2. Institute: Well, you might want to consider that you're poor at your job (or in the wrong industry) and are simply being saved from decreasing real wages by downward wage rigidity and corporate inertia. Those of us who can get good wages on the open market aren't afraid of 10% inflation rates. 3. Anonymous1:51 AM Dohsan. Congratulation on currently able to get a good wage. Let us hope for your sake that continues. Look at the overall data and you will find the trend is for wages to lag price increases, significantly so in recent years. Are you saying that the vast majority of the workforce are poor at their jobs or in the wrong industry? Based on your perspective does this adjustment rest solely with the workforce? 4. Doshan, I find your attitude quite irresponsible indeed, as pointed out by Anonymous above. You might find my reply to Noah interesting anyhow. After all, if you got such a great job, putting you in the top 1% and you don't fear 10% inflation, surely you won't fear what I got to propose... http:\/\/theredbanker.blogspot.com\/2013\/08\/noahpinions-latest-on-loving-inflation.html \"So, in conclusion, I still dislike the idea of using inflation as a way to solve our debt issues. Inflation remains the silent thief. Making it the silent Robin Hood does not particularly entice it to me\". 5. Frederic: So you prefer a debt-deflation spiral, debt overhang, and increasing inequality? If inflation is a thief, then what do you call a nominal wage increase for the same amount and level of work? Anonymous above: I looked at the overall data. Did you look at the overall data? Noah posted it above. I do not see a lag there. Certainly not in recent years. Just stating something doesn't make it true. 6. Doshan: Huh? I don't see how what I say can possibly interpreted as Austrian? :) But, to your second point, I would call it ad-hoc wage catch-up. Wages have been trailing productivity gains by an ever wider margin since the mid-late 70s, Labour share of GDP keeps getting smaller. In an economy rightly based on consumption (rather than imperialism, say) then this fact cannot be good. 7. Anonymous8:28 AM Frederic: What you seem to be implying is that there is an element of class warfare going on. Free market purists sometimes can't conceive of the possibility of imperfect markets and so never address the sources for why real wages track productivity and by implication inflation... 8. Hi, Anonymous To be fair (and I acknowledge this in my blogpost), I am not sure how nominal wages track inflation so well... It could be correlation. Inflation went down and nominal wages slowed down for maybe joint reasons (imported deflation, for example, could explain both phenomenons) but, one, I have the sneaky feeling that inflation does not represent correctly the fact that important stuff like education, healthcare and (till recently) housing was getting more expensive way faster than nominal wages and, two, I am still not sure on how to reconcile this data that seems to suggest that things aren't so bad on the wage front with the fact that wages have been stagnating for the great majority of people. As to your point about class warfare and imperfect markets, I am not sure I was consciously referring to this in my reasoning above but it is, imho, a fundamental problem in our economic set-up. So maybe my views seep through? :) 5. Well, I always thought that you could kill several birds with one stone through inflation and I was wondering why so few people agreed with me, so thanks for confirming my attitude. I think the 2% inflation target is too low. Because of this, inequality is too high, growth is too low, debt is too high and we're always close to the zero lower bound. Those menu costs should decrease significantly in the digital age. 6. I object to being called a troll simply because I adhere to the Austrian school of economic thought. Inflation is like heroin. A little bit feels good and then becomes addictive. But after a while, there are only two choices: withdrawal or overdose. Withdrawal is the healthier, albeit initially painful and difficult option. Overdose is the hedonistic, less responsible choice, and results in the death of the user's entire system. Right now, the U.S. is a fairly heavy user. Recognizing this, it's thinking about scaling back on the dosage a bit next month, just to see how it feels. If it's not so bad, maybe they'll eventually get clean and stop using completely. My guess is that it won't feel very good, and it'll choose to up the dosage again after that. Or maybe it'll decide not to scale back at all. We'll have to wait and see. 1. Anonymous2:55 PM Ah yes, in generating 1.2% inflation per annum, the Federal Reserve is no different than that heroin dealer in Pulp Fiction. 2. Hey Brother, A heroin user can appear quite healthy for a while after he starts using, and the symptoms that he exhibits may be minimal. He might even seem like a real cool cat. 3. Think of small inflation as a offering to the Roman Gods, ok? So Uncle Sam gets to write off debt bit by bit. Would we deny him that? On the other hand focus on the hugely productive assets that Uncle Sam produced and produces: Highway systems, B2 bombers, Internet, Tenessee Valley Authority, National Science Foundation. DoD research.....like I said focus on positive NPV assts and keep those assets productive. And don't sweat the small stuff like BIG DEBT and BAD INFLATION. 4. B2 Bombers are hugely productive assets? I always thought they were highly destructive. The NSF is hugely productive? We are obviously on the opposite sides of the economic spectrum when it comes to analyzing productive assets and organizations. The internet is produced by Uncle Sam? Really? 5. Ok B2 bombers we can have a separate discussion (it kind of gets a bit touchy), Highways? ANd yes the Internet was a product of the DoD (it was called TARPA NET. YOu need to wake up and do some actual digging around!) NSF? Man check out the past, present and future discoveries and projects. Literally everything you take for granted is here. And not produced by John Galt! http:\/\/www.nsf.gov\/discoveries\/index.jsp?prio_area=5 6. Sorry meant to type DARPA instead of TARPA (I have too much of the financial crises in my head! :-) 7. If you have a \"school\" of economic thought, why are you offering over-wrought and obviously hyperbolic analogies? 8. Yeah it is the school of economic thought called facts. Where do you see hyperbolas? Much better than getting my economics from reading Atlas Shrugged. 9. This comment has been removed by the author. 10. Som Dasgupta - I think you took my response to Neil as a response to you. 11. Yes I figured with a slight lag. Apologies! 7. foosion2:56 PM Simpler: total spending equals total income. You can't increase one without increasing the other. There are, of course, distributional issues. Also, higher inflation means higher interest rates gives the Fed more room to act. It can't cut nominal rates at the zero lower bound. 8. Anonymous3:06 PM \"Inflation Benefit 1: Your debt goes away. Chances are, if you're young and have a mortgage (and maybe some car loans too), you have a negative net wealth. You are a net borrower. Even if you have some savings, they are probably outweighed by the mortgage. Which means that inflation makes you richer. Remember, surprise inflation helps debtors and hurts creditors. Who are debtors? Mostly the young and the poor. Who are creditors? Mostly the old and the rich. Now you hopefully see why many conservatives don't like inflation!\" Assuming certain definitions that may or may not be correct, does not have to be. Prices go up. Wages do not go up enough. That can make it harder to make interest and principal payments. 1. Alex Bollinger4:08 PM Wages are a price, anon. If they're not going up but prices of goods are going up, then that's not inflation. It also doesn't happen - notice the blue line on Noah's graph is always above zero. 2. Anonymous11:16 PM \"Wrong. Remember, inflation is an increase in the overall price level. But when the price of everything goes up, your wage should rise as well. Why? Because on average, we are all sellers of something. If you work in a tea shop and the price of tea goes up, your wage can be expected to go up as well, and so forth. Remember, every dollar that one person spends becomes the income of another person! So when prices go up, wages should go up as well. Read this paper. The authors find that \"higher prices lead to higher wage growth\". Of course, wages are affected by other things besides inflation - for example, labor's share of total income. So \"price inflation\" and \"wage inflation\" aren't exactly the same. But they tend to be similar:\" And, \"We don't want to let inflation get out of hand. But a higher Fed inflation target for the next decade - say, 4% or 5%, instead of our current 2% - would probably be a good thing for most Americans.\" Sure sounds like Noah and most economists mean goods\/service price inflation and not wage inflation when they say inflation. The best idea is to not use the term inflaton by itself period. I also see places where the blue line is below the dark red line. It's called negative real earnings growth. 9. Noah, Noah, Noah (sad shaking of head). As a relatively old, relatively rich guy I'm in the \"inflation is theft\" camp. First you say expected inflation does not punish savers (because they demand higher nominal returns) and then you say that inflation benefits borrowers. But the only inflation that will benefit borrowers is unexpected inflation. Inflation \"works\" only by: getting around downward \"stickiness\" and wrong footing investors. Inflation can have some other consequences: higher nominal rates can put a cash flow squeeze on households; higher nominal rates on corporate debt is deductible from income for tax purposes reducing the effective real cost of debt; taxation of income from nominal interest rates drives investment to tax exempt structures like pension plans, endowments, 401(k), etc. 1. Are you being forced to hold fixed-income assets...? 2. No, but I am being forced to hold assets outside of 401(k) plans so some of my gains are nominal gains that get taxed. If all investors chose to avoid fixed income assets because of Noah's inflation then it is hard to see how young borrowers would benefit. 3. That's true, someone has to lend, presumably a fixed-income asset, for the borrowers to benefit from inflation. Still, your portfolio should have hedges that make up any loss. Seems to me that inflation is a win-win... 4. Still, your portfolio should have hedges that make up any loss. I am open to suggestions but it seems to me that any sort of hedge or insurance on my portfolio would have costs. 10. Anonymous3:14 PM Also, if you have an ARM on your house and inflation goes way up, you're screwed. 1. Anonymous3:36 PM and\/or a teaser rate also? 2. \"Also, if you have an ARM on your house, you're screwed.\" FTFY 11. On wage inflation: I thought keynesian models had sticky wages causing unemployment. Inflation is supposed to help reduce nominal wages. But if wages go up with inflation then monetary policy causing inflation to go up cannot help with unemployment. Am I missing something? 12. Som Dasgupta, \"However, Noah's basic argument is correct.\" I didn't disagree with it. As I said (way above, my browser doesn't do \"reply\"): \"Either way though, with real assets and nominal debts, you are definitely long surprise inflation.\" I was just making a \"minor\" correction. 1. Yes, K, you're right. I changed the original post. In the interests of not confusing people I deleted your original comment, but yes, you were right! I meant to write that people have net short positions in nominal debt. 2. Anonymous11:35 PM It seems to me you deleted the comment because you got the accounting wrong. Not unusual for economists. 13. I wonder why some countries get super-inflation worse than others. Ruchi Sharma made a point a while back about how Brazil would get rapidly rising inflation even on relatively modest levels of nominal GDP growth, whereas China and the other East Asian countries got pretty solid growth without double-digit inflation. 14. OK so I concur with Keynes: \"By a continuing process of inflation, government can confiscate, secretly and unobserved, an important part of the wealth of their citizens.\" But with such a large debt overhang and with a direct debt jubilee politically very difficult and liquidationism extremely dangerous, there seems to be no safe alternative to a higher degree of inflation to make the nominal private debt load more manageable. 1. So inflation is good theft. Inflation is like Robin Hood. But isn't it also true that low, steady inflation is a safety net protecting an economy from slides into deflation (which most would agree is far worse)? 2. 2% is the Fed's buffer against deflation, as well lubricating the economy a little by devaluing past debts. This seems like the right figure in the long run, but in the next 10 years with the private debt load so high, 4% would be a fairer target. Of course, whether the Fed can actually hit 4% right now is another matter entirely. 3. there seems to be no safe alternative to a higher degree of inflation to make the nominal private debt load more manageable. Inflation rewards all borrowers, regardless of need, and punishes all savers foolish enough to have lent out their money long term. There is no reason to come to the aid of all borrowers. If you want targeted relief then reform the bankruptcy laws - that would benefit the most needy borrowers and lay the cost on the imprudent lenders. That should have been done for mortgages in 2008. One of the consequences of people saying the things that you are saying is that people like me are sitting on our cash and not spending or investing in productive assets. 4. By a continuing process of inflation, government can confiscate, secretly and unobserved, an important part of the wealth of their citizens. Yes, but it automatically transfers that wealth, secretly and unobserved, to a different group of citizens. 5. Yes, but it automatically transfers that wealth, secretly and unobserved, to a different group of citizens. Some of the winners will be richer and less deserving than some of the losers. 6. \"less deserving\" You're opening Pandora's Box with a line like that! :P It seems to me that no matter the proposition in macroeconomics, the debate always boils down to Social Choice. Too many theories have *oughts* in them. 7. Remember that there is no such thing as \"redistribution.\" There are only different distributions. 15. Brett, \"I wonder why some countries get super-inflation worse than others.\" It's just a choice. In developing economies 1) the distorting effects of inflation risk may be lower because capital investments are shorter and 2) other forms of taxation (income, property, etc) may be difficult or impossible to implement. So inflation becomes a relatively more efficient way to raise revenue. I think that's the consensus understanding of why developing countries run higher inflation than developed ones. There's a fairly big literature on the cost of inflation in terms of growth, and the estimates seem to be that for developing economies you start to get an impact somewhere between 7-15% (IIRC). I don't know anything about the specific trade offs in Brazil vs China, but no doubt, any developing country could run whatever inflation rate it wants to. 16. Coincidentally, I just wrote a Myths of inflation posts a few days ago: http:\/\/bessiambre.tumblr.com\/post\/58338607070\/misconceptions-about-money-creation-and-inflation (I am not an economist and my tumbler is more or less just a way of consolidating some thoughts) I make similar points to the ones in this post. I will say that the two first benefits of inflation you mention, government and private debt being partly reduced, are not considered benefits by everyone. It can seem like a free lunch and unfair to lenders. However, in the context of a liquidity trap where market rates are distorted up, it can be justifiable to create a bit of inflation to let the real rates go down to their true free market \"natural\" levels. Otherwise it is lenders and cash hoarders who are getting the free lunch when the rates aren't able to go down enough to reach free market levels. 17. Popular Inflation Myth 1: \"Inflation means I can't buy as much stuff.\" Now we can re-define inflation to exclude ..umm...inflation. Wages goiong up? don't count on it 1. How did the guy get the wheelbarrow full of cash in the first place? 2. It's pretty hard for the overall price level - not prices of specific individual products - to go up without wages also going up. That's because wages are a significant production cost and because you can't sell your stuff for more if people don't have more money to buy it. 3. And by the time he got his wheelbarrow full of cash to the bread shop it was highly likely the price increased by 100%. Also, what about the unemployed and those on fixed incomes? How is inflation good for them? And finally, what is worng with deflation, why would people have a problem with falling prices? 18. Here\u2019s how I\u2019ve thought of it in previous comments: With regard to low inflation targets, like our 2%, I've lately started thinking of the analogy of standing really close to a very dangerous and deep pit, like just 2 feet (or 2 percent) away. It's pretty easy for a tremor or other disturbance to make you fall in, get hurt severely, and have it take a long time before you can climb out. So, as the cost is not that high to stand further away, like 4 or 5 feet (or percent), isn't that the smart, preventative medicine, thing to do? It's widely acknowledged how dangerous the ZLB is, yet your target is to stand super close to it, so it's easy to fall into it? Also, Slipperyslopeaphobia: http:\/\/richardhserlin.blogspot.com\/2008\/07\/slipperyslopeaphobia.html 19. Let me be an annoying pedant: You cannot claim that inflation decreases taxes and future debt directly (NGDP), while also assuming that the fisher equation hold true unless you assume that the government doesn't roll-over the debt. If it does the higher inflation will increase the interest rates through the Fischer equation, right? Having to pay higher interest rates can actually make the debt situation worse. For that to happen inflation must increase RGDP as well, which is I think is pretty important. Another cool thing about inflation - it may increase the taxes you pay by moving you into a higher bracket; but that is conditional if the brackets are setup numerically \"income above$n pays m%\". A lot of jurisdictions just index the taxes using the minimum wage, which is increased regularly. That is assuming progressive taxation, of course. It may happen that the system you live in is creatively regressive though!\n\n1. You cannot claim that inflation decreases taxes and future debt directly (NGDP), while also assuming that the fisher equation hold true unless you assume that the government doesn't roll-over the debt. If it does the higher inflation will increase the interest rates through the Fischer equation, right? Having to pay higher interest rates can actually make the debt situation worse.\n\nCorrect. The government's short-term debt does not decrease, but it's long-term debt decreases. Our government has a lot of long-term debt.\n\n2. Sure, but doesn't the government have to roll over the long-term debt as well?\n\nI can see that its easier to pay interest on a 30-year bond, when starting from the 2 year you have much more inflation, but the next time you try to issue 30 year-old bonds you'll get higher interest rates (assuming things of course).\n\n3. Sure, but doesn't the government have to roll over the long-term debt as well?\n\nEventually. Some of it is coming due immediately, and some isn't coming due for a while.\n\nCheck out that example of WW2 debt.\n\n4. WW2 debt-to-gdp was reduced because of real GDP growth. If anything the biggest reductions in debt were in the low-inflation 50s and 60s.\n\nNow, if the 50s and 60s CB were extreme inflation hawks, I doubt this would have happened. (Just like inflation hawk nations like New Zealand, Canada and Germany have grown less than dovish nations like the United States and Australia )\n\n5. WW2 debt-to-gdp was reduced because of real GDP growth.\n\nThat was important too, of course.\n\n6. Thanks, I will look at the cited results, it is interesting. But I still think that inflation can work both ways on debt servicing.\n\nI have nothing against higher inflation in EU, US and Japan, now of course.\n\n20. A day of very low probability: I have absolutely no disagreement at all with an apparently controversial Noah Smith post.\n\n21. Wow I hate to pull the Scott Sumner card here, but this whole discussion is a classic case of everybody getting confused about inflation because it's such a poorly defined and poorly explained-by-economists concept. Somebody mentions the I-word on the internet and in no time everybody's talking every which way about good vs bad inflation, real interest rates, gold, Wiemar Germany, theft, punishing savers, etc. etc. etc.\n\nAt the end of the day it vastly clarifies the issue to use NGDP as your demand measure and the AS-AD model to determine the split between inflation and RGDP for a given level of NGDP growth.\n\nTo use just one example, unexpected inflation does NOT benefit net borrowers. Unexpected NGDP growth does, as the borrower's debts are denominated in nominal dollars, and thus nominal income is what measures his ability to service debt. If inflation unexpectedly jumps to 4% due to a supply shock, but the Fed keeps NGDP growth on hold at 5%, then RGDP growth winds up being 1%. In that case inflation has done nothing to help or hurt the borrower's ability to repay the debt.\n\nTrying to explain inflation and all its nuances to even relatively economically literate people has proven to be such a hopeless task that NGDP and AS-AD winds up being better in almost all respects.\n\n1. Wow I hate to pull the Scott Sumner card here, but this whole discussion is a classic case of everybody getting confused about inflation because it's such a poorly defined and poorly explained-by-economists concept. Somebody mentions the I-word on the internet and in no time everybody's talking every which way about good vs bad inflation, real interest rates, gold, Wiemar Germany, theft, punishing savers, etc. etc. etc.\n\nYep.\n\nAt the end of the day it vastly clarifies the issue to use NGDP as your demand measure and the AS-AD model to determine the split between inflation and RGDP for a given level of NGDP growth.\n\nIt clarifies the issue but it requires you to commit to a model, thus sacrificing mental and analytical flexibility.\n\n2. \"It clarifies the issue but it requires you to commit to a model, thus sacrificing mental and analytical flexibility.\"\n\nWell, this blog post is presented as a \"debunking popular myths about inflation\" type post, so I would say clarity to your average Joe should be one of the most important factors when deciding how to present your material. Additionally, I'm not so sure you need to commit to a model here. Just about everything you say in your post about inflation applies equally well if not better using NGDP (e.g. my helping net borrowers comment above), mostly independent of model choice. The question of what causes shifts in NGDP and what causes the SRAS curve to be shaped the way it is are largely irrelevant for purposes of your post.\n\n3. Excuse me but I don't see your point.\n\nYou give an example of inflation jumping to 4%, RGDP going down to 1% and stating that this does nothing to help\/hinder borrowers...\n\nWell, hold on. If I got nominal \\$ debt, 4% inflation helps. Period. If that inflation happens to happen during a stagflation, my debt load getting a bit smaller might not be my main concern (losing my income might) but that's besides the point.\n\nBasically, I think inflation is quite clear a concept. We can argue about sectorial inflation inasmuch as I disagree with Noah on the fact that saying 'asset X' or even wage-inflation are fairly clear and understandable narrowing of the term. But I get his counter-arguments. Yours. not so much...\n\n22. Anonymous10:53 PM\n\nWage increases always lag inflation so the common man always loses. Inflation is a game where those at the lower rungs always lag behind inflation so see the smallest potential benefit (if any), and eat we eat the price increase with a lagging wage increase that rarely matches inflation.\n\nIf inflation is 10% do you think we all will get a 10% raise... um sure... Wage appreciation rarely matches real inflation, again many of us lose. With a depressed job market the downward pressure on wage appreciation means that net spending power is diminishing over time. Just like compound interest, compound lag and gap have been very efficient at subtly destroying the net purchasing power of folks.\n\nWith how the job market really is workers are the last in line to see any possible benefit. A weak labour market means wages for most folks are dropping, especially when you take in flat or dropping wage but not flat inflation. Even if inflation was high due to the depressed job market wage appreciation would not match inflation for the bulk of the populace, as it going on right now. Wage increases in a time of oversupply are as rare as Republicans that support socialized medicine. The current situation has been a great opportunity to further erode the share common men get of the pie.\n\nSorry Noah, I am not buying this one, it ignores the lag and gaps that occur that compound over time. I think you looked at the macro level but forgot the over time effect on most people at the person level.\n\n1. Over history, real wage gains have been (slightly) positively correlated with inflation. What do you think about that?\n\n2. agree. I've been reading these 'no inflation' arugments since the bailout and while there is technically no inflation according to the bond market, there is in a lot of other areas such as stuff that people actually need. maybe if you have a plush tenure like krugman with full benefits you aren't going to be impacted by rising living expenses, but many middle\/lower income folks and independent contractors are.\n\n3. rising living expenses\n\nBut inflation does not mean rising living expenses.\n\n4. Anonymous12:34 AM\n\nAre you considering the lag and its effect on disposable income? It is like compound interest in reverse. The period during the lag in wage increase compared to price increases due inflation is a loss of spending power for workers. What workers are paid versus inflation is always a catch up game, where the workers almost always lag. That lag, cumulatively is what hurts workers, possibly more than what they gain in increases.\n\nI understand inflation in and of itself is not a bad thing. Unfortunately the job market is not synchronized to it, so the workers lose out in the catchup period between increasing prices and subsequent (hopefully) wage increases.\n\n5. OK, so define \"inflation\" as \"wage inflation\". Now there's no lag. Voila!\n\n...oh wait, that can't make sense...hmm...how is this more complicated?\n\n6. Anonymous1:20 AM\n\nInflation results in continued adjustment to \"equilibrium\", those adjustments are not simultaneous, the time gap leads to profit and corresponding lose during the adjustment period. Workers usually end up in the losing group especially if you look at the how often wage adjustments lag other adjustments. I have a hard time seeing wages leading except in rare exceptions with limited scale. If they force up the minimum wage we will have some very good data to quantify wages leading the adjustments. I guess I am over focused on the impact to people rather than the economy as a whole.\n\n7. Anonymous1:33 AM\n\nJust some links showing how real wages lag inflation:\nhttp:\/\/www.epi.org\/blog\/real-hourly-wage-growth-last-generation\/\n\nhttp:\/\/www.nytimes.com\/2013\/01\/13\/sunday-review\/americas-productivity-climbs-but-wages-stagnate.html?_r=0&gwh=93ED324DC9346914D01B25C37F93C105\n\nhttp:\/\/www.ilo.org\/wcmsp5\/groups\/public\/---dgreports\/---dcomm\/documents\/publication\/wcms_194844.pdf\n\nInflation is not a bad thing, but badly lagging wages seem to be a cruel way it plays out to working people.\n\n23. There is already a high inflation in prices: fewer and fewer items go on sale. Go to Home Depot and you will notice the area in front of the cashiers is nearly empty. It used to carry lots of stuff on sale. It is the same at every retail store. You can sell less quantity at higher price and still meet revenue targets. You can also let go lots of people because you do not have to stock and display lots of stuff. This also creates a big pressure on jobs and wages; low and mid level jobs are taking pay cuts. And, many of these guys and gals are not able to save or plan for retirement.\n\nFor the rich, it is totally different story. Their pay\/incomes have been going up and their taxes are going down.\n\nWhat I would like to see wage inflation, price deflation, and, significant increase in demand.\n\n1. If there is one thing everyone should agree on it is that it is impossible for you to observe inflation personally. Even if you were capable of keeping track of all that data, we are all severely compromised by confirmation bias and the disproportionate strength of the impression of a higher price over a lower one.\n\nSo, no, don't go to Home Depot to find out what's happening with inflation. Look at the data.\n\n24. Wow! If Noah (who I've long considered too conservative to be worth reading regularly) keeps this up, I'll become a regular reader! I'm shocked, but it seems to me he has presented an even better and broader defense of inflation than Steve Randy Waldmann, one of my favorite bloggers.\n\nSomeday we might actually be thinking about utility generally, and not simply what makes rich people feel more secure. I'm not sure why people think that preserving the value of simple savings is one of the highest social goals. Surely job 1 is getting a good job for everyone, and there is enormous evidence that inflation and job growth go hand in hand, late 1970's for example. IIRC more private jobs created during the Carter administration than any other.\n\nWhat leads to hyperinflation Zimbabwe style is collapse in production, which is exactly the kind of thing that can and will happen if VSP's are too focused on the wrong things...like inflation.\n\n1. Too many Keynesians are focused on the wrong stuff. Throwing money, whether it be \"deficit spending\" from people like Summers, or \"targeted (federal) capital investment\" from people like Stiglitz, show a sore ignorance of political economy. Government spending is already high, and most money just gets transferred, not invested in infrastructure.\n\n2. @Yeah: I hope you understand that transfers are not included in spending, as measured. Transfers are 'negative taxes.'\n\n25. Let me troll the trolls:\n\nHousehold appliances in Sweden: http:\/\/research.stlouisfed.org\/fred2\/series\/CP0530SEM086NEST\n\n-13%!!!!\n\nNow, if I were a fridge-salesman in Sweden I would complain how falling prices are destroying my business, right?\n\nGovernment is obviously lying about the other statistics. Other things like washers also have falling prices! Obviously as a fridge-salesman in Sweden I don't anything else to convince me that extreme deflation is a great threat to the Swedish and World economy!\n\n1. Too bad a middle income family couldn't enjoy a 13% discount for one year on health expenses. But what if the breadwinner is a Business Analyst at a health insurance company? HORROR! JOB LOSS!\n\n26. Ignores the other distributional implications of inflation which is the reason it is not popular:\n\nThe \"inflationist\" argument is a political non-starter.\n\n27. Cameron D9:07 AM\n\nWorth mentioning that there isn't a single measure of inflation across the whole economy. I suspect certain constituencies are more affected by and thus more responsive to price changes in different sectors.\n\n28. I think what many people commenting here don't understand (and correct me if I'm wrong) is the mechanism by which inflation may raise prices.\n\nPeople take raising prices and inflation as one in the same which is false. If lets say money supply increases and prices go up, I would counter this happens because people now have more money to spend on goods and thus increased demand drives up prices. But FIRST people have to be in posetion of this new level of money in order to consider spending it.\n\n1. Inflation stats are really only helpful to two groups of people, and people who watch those two groups; People who do monetary policy, and the Treasury. We have a weird disconnect now, where Economists claim \"low inflation,\" which is true, on average, but people, especially middle income earners, are facing large price increases in areas of health, education, and food.\nThe mainstream economist consensus of deficit spending I think is simply wrong. Most federal government spending is income transfer, not on programs for infrastructure. There is no need to spend more; US citizens are not willing to pay more, the deficit is too high, the debt is too high, and total government spending is about 42% of GDP, enough in my opinion. There is a need to move money around to capital investment, no subsidizing ag. or special parties.\n\n2. Your limit of 42% seems highly arbitrary and I think misses the point of government spending entirely.\n\nYes I would agree that most government spending is a transfer of wealth or reasources but why is this undesirable?\n\nIf global economic forces and technology have been the main culprits of increasing income\/wealth inquality (which is a popular conservative\/libertarian argument) why is it inherantly wrong for a government thru consent of the people to counteract these measures via wealth transfers to maintain a level of equality we as a society find desirable?\n\n3. Generally speaking, my view on government spending, is as local as possible, and were people can't or aren't willing to use market forces, to get goods\/services collectively. Centralization of spending creates disconnects, where the citizens will individually is not enacted collectively.\n\nThe problem with \"increasing wealth inequality\" is the assumptions in the data itself. It is not an accident, in the US, at the time of 91% marginal income brackets, that inequality was squeezed out, but only because of tax evasion, tax shelters, etc.\n\nThe plain fact is that the people do NOT consent to the levels of government spending, and the forms they are taking, that is the problem, but everybody likes everybody elses money. The irony here is that most economists on the left are pushing for \"infrastructure spending,\" basically supply side government spending. My position is we don't need to spend any more money, we just need to reallocate.\n\n29. Anonymous2:43 AM\n\nWage Inflation Fairy, come raise my wage! Some economists saidthat you magically come in the night and transfer newly-created money from Wall Street to wage earners...\n\n30. Anonymous3:04 AM\n\nBy the way the Fisher Equation is bogus given the Fed's control of rates. The QE Equation is\n\ni = r +pi - QE\n\nSo Saver Granny (or as a previous poster accused her of, Drug Dealer Granny) does get screwed by inflation thanks to the Fed.\n\n1. Why is 'saver granny' hiding her money in her mattress? If she were to seek productive investments, she'd be fine. We ought to tax the hell out of 'mattress money.'\n\n31. I have come to the conclusion that the BLS inflation statistics are skewed downwards for low, middle, and high middle income earners. Why? Three major areas that eat a large portion of these groups income due to price increases: College, health insurance AND health care, food and sometimes gasoline. The baskets don't accurately account for these expenses, and the federal subsidies in all cases are probably making the situation worse by perversing incentives (ie getting a college major that is not in demand, corn in places where it shouldn't be).\n\nWould it be possible to have inflation states based on income? Rather than an \"average\" rate?\n\n32. nice post here\ni also love your style of blogging here on noahpinionblog.blogspot.com :)\nyou see, there are 2 blogs that i've found so far to be very helpful and have something interesting for me whenever i visit, this one and http:\/\/danieluyi.com\nKeep it up. I will be visiting again.\n\n33. The author shows what is wrong with economists (esp. Keyensians). They live in a fairy-tale land where equations based on phony assumptions take the place of common sense. Please name the country\/era in which high inflation has not been politically and socially destabilizing? Here's the real world: The German inflation that the author minimizes brought down the Weimar government and led to the rise of Nazi Germany. That's the real world. BTW, you actually have no idea if the money in that wheelbarrow was that person's life savings or not.\n\n34. \"Who are creditors? Mostly the old and the rich.\"\n\nNot true. Although almost by definition a net holder of nominal claims cannot be described as \"poor\", the less well-off (eg the old and financially unsophisticated) do tend to hold a larger proportion of their wealth in nominal claims, especially bank accounts.\n\nBut I would also question whether inflation benefits debtors (and cheats creditors) in general anyway. That is because, not least in view of the fact that the state tends to among the beneficiaries, further inflation is anticipated and factored into interest rates. So the effect of surprise inflation is to benefit today's creditors at the expense of tomorrow's creditors.\n\nAnd let's be honest, that is the real motive of most inflationists; not the lubricating or deflation trap avoiding purposes - 2% is enough for that. The call for higher inflation (including engineered by NGDPLT) is just another way of kicking the can down the road a bit further.\n\n35. Anonymous11:07 AM\n\nPopular Inflation Myth 2: \"Inflation punishes savers.\"\n\nIf this is (partially) a myth, ie, future savers are not harmed, by implication, future borrowers don't benefit either. Thus, when you say\n\nInflation Benefit 1: Your debt goes away.\nor\nInflation Benefit 2: The federal government debt goes away.\nor\nInflation Benefit 3 (?): \"Balance sheet recession\" might go away!\n\nthese are also only partially correct; in that, once the inflation expectation is settled, none of these benefits will materialize. The past govt debt will decrease initially, but without a change of behavior, the interest rates would go up, and the debt servicing cost.\n\n36. Anonymous12:48 PM\n\n\"Who are creditors? Mostly the old and the rich.\n\nNow you hopefully see why many conservatives don't like inflation!\"\n\nOh, so it's ok if the retirement savings of those who are in (or near) retirement are wiped out? Thanks for pointing that out.\n\nOf course, most of these people aren't rich. (They also aren't necessarily conservative). We ARE people who have followed the standard advice of financial advisers and have allocated a large share of their investment portfolios to \"safe\" investments like Treasury securities. Well, \"safe\" only to the extent that idiots like you aren't in control of monetary policy, in which case you will have to stand by helplessly as the value of your retirement portfolios are destroyed by inflation.\n\nMany such people also are on fixed incomes (e.g., pensions that don't adjust to inflation). They, too, will be wiped out.\n\nSo, please, stop pretending that this is an \"old and rich\" versus \"young and poor\" issue.\n\n37. Anonymous4:37 PM\n\nDude, Noah,\nSome of your blog posts are interesting to read. But, this was not \"going out on a high note\" before taking your break.\nIt was basically lots of reasons why inflation is good or not as bad, and then a paragraph giving passing recognition of its costs. And then it ends by simply calls people names if they happen to disagree. This leads to absolutely nothing constructive because it does not advance the debate forward and it immediately goes on the offensive since some people might disagree with you.\nThose last three paragraphs turn it into the format of \"here's what I think and those who disagree are dummies.\" I have yet to see any serious writer make any lasting difference using that formula. It weakens the argument and takes it down to Jerry Springer's level.\n\n38. \"today's government debt is tomorrow's taxes\"\n\nNoah has declared that it is so, so it must be true.\n\nNo evidence or argument is needed. Simply state something, Noah, and it is true, right?\n\nAfter all, everyone *knows* it's true. It's just \"common sense\", right?\n\nWhy not try to substantiate your arguments instead of just stating things as if they were a given?\n\n1. In what weird intellectual funhouse is that statement not true? If the government decided to fully pay off the debt next year, by what mechanism would it do that? I think the real question is 'why does Noah's stated accounting identity bend you so far out of shape?' If you think the identity is wrong, what do you propose instead?\n\n39. There is actually another cost associated with inflation: it increases the real cost of providing interest free credit between firms (e.g. in the form of 30 day payment terms), which is an important tool for businesses seeking to manage their cash flow.\n\n40. Anonymous2:19 PM\n\nI\u2019m not an economist but I can make the observation that Australia\u2019s higher and more flexible inflation target hasn\u2019t seen it\u2019s growth lag behind the US or Europe. Australia hasn\u2019t had a recession for more than two decades and for the bulk of that time, as I understand it, the Reserve Bank has been tasked with keeping core inflation between 2% and 3% on average over the course of the business cycle, a task I think it has achieved.\n\nDuring that time the RBA has been content to let inflation track towards the top of that range for long periods, unlike the Fed and the ECB who have seemed to treat their more fixed 2% targets as upper limits if you believe the liberal economics blogosphere. Inflation in Oz once got away a little, just before the financial crisis, but it was brought back under control without recession. Flexibility may have helped here \u2013 the RBA could look through the inflation that resulted when Australia\u2019s currency plunged and keep slashing rates.","date":"2017-09-21 15:41:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 1, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.440509170293808, \"perplexity\": 2813.9832104812854}, \"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-2017-39\/segments\/1505818687833.62\/warc\/CC-MAIN-20170921153438-20170921173438-00439.warc.gz\"}"} | null | null |
> This page explains how to upgrade to latest version (v3).
Use `streamEvents` instead of `connection.streams`:
```javascript
var stream = connection.streamEvents['streamid'];
// or use this code:
// backward compatibility
connection.streams = connection.streamEvents;
connection.numberOfConnectedUsers = 0;
if (Object.observe) {
Object.observe(connection.streamEvents, function() {
// for backward compatibility
connection.streams = connection.streamEvents;
});
Object.observe(connection.peers, function() {
// for backward compatibility
connection.numberOfConnectedUsers = connection.getAllParticipants().length;
});
}
# you can even use "getStreamById"
var stream = connection.attachStreams.getStreamById('streamid');
# to get remote stream by id
var allRemoteStreams = connection.getRemoteStreams('remote-user-id');
var stream = allRemoteStreams.getStreamByid('streamid');
```
Wanna check `isScreen` or `isVideo` or `isAudio`?
```javascript
connection.onstream = function(event) {
if(event.stream.isScreen) {
// screen stream
}
if(event.stream.isVideo) {
// audio+video or video-only stream
}
if(event.stream.isAudio) {
// audio-only stream
}
};
```
### Mute/UnMute?
```javascript
var stream = connection.streamEvents['streamid'].stream;
stream.mute('audio'); // mute only audio tracks
stream.mute('video'); // mute only video tracks
stream.mute('both'); // mute both video and audio tracks
// or simply
stream.mute(); // mute both video and audio tracks
```
# Other Documents
1. [Getting Started guide for RTCMultiConnection](https://github.com/muaz-khan/RTCMultiConnection/tree/master/docs/getting-started.md)
2. [Installation Guide](https://github.com/muaz-khan/RTCMultiConnection/tree/master/docs/installation-guide.md)
3. [How to Use?](https://github.com/muaz-khan/RTCMultiConnection/tree/master/docs/how-to-use.md)
4. [API Reference](https://github.com/muaz-khan/RTCMultiConnection/tree/master/docs/api.md)
5. [Upgrade from v2 to v3](https://github.com/muaz-khan/RTCMultiConnection/tree/master/docs/upgrade.md)
6. [How to write iOS/Android applications?](https://github.com/muaz-khan/RTCMultiConnection/tree/master/docs/ios-android.md)
7. [Tips & Tricks](https://github.com/muaz-khan/RTCMultiConnection/blob/master/docs/tips-tricks.md)
## Twitter
* https://twitter.com/WebRTCWeb i.e. @WebRTCWeb
## License
[RTCMultiConnection](https://github.com/muaz-khan/RTCMultiConnection) is released under [MIT licence](https://github.com/muaz-khan/RTCMultiConnection/blob/master/LICENSE.md) . Copyright (c) [Muaz Khan](http://www.MuazKhan.com/).
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,125 |
Nicolas Dordevic est un ancien footballeur professionnel français né le à Saint-Germain-en-Laye. Il évoluait au poste de milieu de terrain.
Biographie
Nicolas Dordevic effectue sa formation à l'AS Monaco, puis commence sa carrière professionnelle en 1997 au MUC 72, avec lequel il dispute 61 matchs dans le championnat de France de Division 2.
Il rejoint les rangs amateurs en 2001 et évolue à compter de 2005 au Football club bassin d'Arcachon.
Liens externes
Footballeur français
Joueur du Mans FC
Naissance à Arcachon
Naissance en décembre 1975 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,834 |
Chinese Fountain is the fourth studio album by the Southern California surf rock band The Growlers, released on September 23, 2014 by Everloving Records. The album received generally positive reviews from critics, who called it "one of the best guitar albums of the year".
Track listing
Song#9 "Not the Man" – music by Kyle Mullarky
Charts
References
2014 albums
The Growlers albums | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,748 |
Q: Powershell Copy multiple files with certain numbers Still pretty new to Powershell, I would like to copy mutltiple files with from one server to an other server. The files that i want to copy contain the same two numbers 18 or 19.
Copy-Item -Path \\test\files\07_Export -Filter '*18.csv' -Destination \\test01\files$\2018;
Copy-Item -Path \\test\files\07_Export -Filter '*19.csv' -Destination \\test01\files$\2019;
As example
Test18.csv
question19.csv
searching18.csv
Right now it just copys the empty folder 07_export.
A: This should serve your purpose.
$filename = (Get-ChildItem "<Source Folder>" -Recurse | Where-Object { $_.FullName -imatch "18" -or $_.FullName -imatch "19" }).FullName
foreach ($file in $filename) {
Copy-Item -Path $file -Destination "<Destination Folder>"
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,795 |
\section{Introduction}
It has long been recognized that neutron stars have space velocities much
greater than their progenitors'.
Recent studies of pulsar proper motion give $200-500$~km~s$^{-1}$ as the mean
3D velocity of NSs at birth (e.g., Lyne and Lorimer 1994; Lorimer et al.~1997;
Hansen \& Phinney 1997; Cordes \& Chernoff 1998), with possibly a significant
population having velocities greater than $700$~km~s$^{-1}$. Direct evidence
for pulsar velocities $\simgreat 1000$~km~s$^{-1}$ comes from observations of
the bow shock produced by the Guitar Nebula pulsar (B2224+65)
in the interstellar medium (Cordes et al.~1993)
and studies of pulsar-supernova remnant associations (e.g., Frail et al.~1994;
Kaspi 1999). A natural explanation for such high velocities is that
supernova explosions are asymmetric, and provide kicks to nascent neutron
stars. Support for supernova kicks has come from the detection
of geodetic precession in binary pulsar PSR 1913+16
(Cordes et al.~1990;
Kramer 1998; Wex et al.~1999) and the orbital plane
precession in PSR J0045-7319/B-star binary (Kaspi et al.~1996)
and its fast orbital decay (which indicates retrograde rotation of the B star
with respect to the orbit; Lai 1996); These results demonstrate
that binary break-up (as originally suggested by
Gott et al. 1970; see Iben \& Tutukov 1996) can not be
solely responsible for the observed pulsar velocities, and that
{\it natal kicks} are required.
Evolutionary studies of neutron star binary population also
imply the existence of pulsar kicks (e.g.,
Fryer et al.~1998). Finally, there are many direct observations of
nearby supernovae (e.g.,
Wang et al.~1999) and supernova remnants
which support the notion that supernova explosions are not spherically
symmetric.
Despite decades of theoretical investigations, our understanding
of the physical mechanisms of core-collapse supernovae remains
significantly incomplete. While there is a consensus that neutrino heating
of the stalled shock ($0.1-1$~s after bounce) plays an important role
in driving the explosion, it is unclear whether the heating is sufficient
to produce the observed supernova energetics; It is also unclear whether any
convective motion or hydrodynamical instability is central to the explosion
mechanism (e.g., Herant et al.~1994; Burrows et al.~1995; Janka \& M\"uller
1996; Mezzacappa et al.~1998).
The prevalence of neutron star kicks poses a significant mystery, and indicates
that large-scale, global deviation from spherical symmetry is am important
ingredient in any successful theory of core-collapse supernovae.
In this paper, we concentrate on different classes of
physical mechanisms for generating
neutron star kicks (\S\S 2-4), and then briefly discuss the
astrophysical/observational implications (\S5).
\section{Hydrodynamically Driven Kicks}
The collapsed stellar core and its surrounding mantle are susceptible
to a variety of hydrodynamical (convective) instabilities
(e.g., Herant et al.~1994; Burrows et al.~1995; Janka \& M\"uller 1996;
Keil et al.~1996). It is natural to expect that
the asymmetries in the density, temperature and velocity distributions
associated with the instabilities can lead to asymmetric matter ejection
and/or asymmetric neutrino emission. Numerical simulations, however, indicate
that the local, post-collapse instabilities are not adequate to account
for kick velocities $\simgreat 100$~km~s$^{-1}$ (Janka \& M\"uller 1994;
Burrows \& Hayes 1996; Janka 1998) --- These simulations were done in 2D,
and it is expected that the flow will be smoother on large scale in
3D simulations, and the resulting kick velocity will be even smaller.
Global asymmetric perturbations in presupernova cores are
required to produce the observed kicks hydrodynamically
(Goldreich et al.~1996).
Numerical simulations by Burrows \& Hayes (1996) demonstrate that if the
precollapse core is mildly asymmetric, the newly formed neutron star can
receive a kick velocity comparable to the observed values. (In one simulation,
the density of the collapsing core exterior to $0.9M_\odot$ and within $20^o$
of the pole is artificially reduced by $20\%$, and the resulting kick is about
$500$~km~s$^{-1}$.) Asymmetric motion of the exploding material (since the
shock tends to propagate more ``easily'' through the low-density region)
dominates the kick, although there is also contribution (about $10-20\%$)
from asymmetric neutrino emission. The magnitude of kick velocity is
proportional to the degree of initial asymmetry in the imploding core.
Thus the important question is: What is the origin of the initial asymmetry?
\begin{figure}
\vskip -0.8cm
\epsfig{file=fig_hk.eps, width=10cm, height=8.5cm}
\vskip -3cm
\caption{
Propagation diagram computed for a $15M_\odot$
presupernova model of Weaver and Woosley (1993).
The solid curve shows $N^2$, where $N$ is the Brunt-V\"ais\"al\"a
frequency; the dashed curves show $L_l^2$, where $L_l$ is the
acoustic cutoff frequency, with $l=1,~2,~3$.
The spikes in $N^2$ result from discontinuities in entropy and
composition. The iron core boundary is located at $1.3M_\odot$,
the mass-cut at $1.42M_\odot$.
Convective regions correspond to $N=0$. Gravity modes
(with mode frequency $\omega$) propagate in regions
where $\omega<N$ and $\omega<L_l$, while pressure modes
propagate in regions where $\omega>N$ and $\omega>L_l$.
Note that a g-mode trapped in the core can lose energy
by penetrating the evanescent zones and turning into an
outgoing acoustic wave (see the horizontal line).
Also note that g-modes with higher $n$ (the
radial order) and $l$ (the angular degree) are better trapped in the
core than those with lower $n$ and $l$.
}\label{fig1}\end{figure}
\smallskip
\noindent
{\bf (i) Presupernova Perturbations}
Goldreich et al.~(1996) suggested that overstable g-mode oscillations in the
presupernova core may provide a natural seed for the initial asymmetry.
These overstable g-modes arise as follows. A few hours prior to core collapse,
a massive star ($M\simgreat 8M_\odot$) has gone through a successive stages
of nuclear burning, and attained a configuration with a degenerate iron core
overlaid by an ``onion skin'' mantle of lighter elements. The rapidly growing
iron core is encased in and fed by shells of burning silicon and oxygen, and
the entire assemblage is surrounded by a thick convection zone. The nearly
isothermal core is stably stratified and supports internal gravity
waves. These waves cannot propagate in the unstably stratified convection zone,
hence they are trapped and give rise to core g-modes in which the core
oscillates with respect to the outer parts of the star. The overstability
of the g-mode is due to the ``$\varepsilon$-mechanism'' with driving provided
by temperature sensitive nuclear burning in Si and O shells surrounding
the core before it implodes. It is simplest to see this
by considering a $l=1$ mode: If we perturb the core to the right, the
right-hand-side of the shell will be compressed, resulting in an increase in
temperature; Since the shell nuclear burning rate depends sensitively
on temperature (power-law index $\sim 47$ for Si burning and $\sim 33$ for O
burning), the nuclear burning is greatly enhanced; This generates a large
local pressure, pushing the core back to the left.
The result is an oscillating g-mode with increasing amplitude.
The main damping mechanism comes from the leakage of mode energy.
The local (WKB) dispersion relation for nonradial waves is
\begin{equation}
k_r^2=(\omega^2c_s^2)^{-1}(\omega^2-L_l^2)(\omega^2-N^2),
\end{equation}
where $k_r$ is the radial wavenumber, $L_l=\sqrt{l(l+1)}c_s/r$ ($c_s$
is the sound speed) and $N$ are the acoustic cut-off (Lamb) frequency and the
Brunt-V\"ais\"al\"a frequency, respectively.
Since acoustic waves whose frequencies lie above the acoustic cutoff can
propagate through convective regions, each core g-mode will couple to an
outgoing acoustic wave, which drains energy from the core g-modes (see
Fig.~\ref{fig1}). This leakage of mode energy can be handled with an
outgoing propagation boundary condition in the mode calculation.
Also, neutrino cooling tends to damp the mode.
Since the nuclear energy generation rate depends more sensitively on
temperature than pair neutrino emission (power law index $\sim 9$), cooling is
never comparable to nuclear heating locally. Instead, thermal balance is
mediated by the convective transport of energy from the shells, where the
rate of nuclear energy generation exceeds that of neutrino energy
emission, to the cooler surroundings where the bulk of the neutrino
emission takes place. Calculations (based on the $15M_\odot$ and $25M_\odot$
presupernova models of Weaver \& Woosley 1993) indicate that
a large number of g-modes are overstable, although for low-order modes
(small $l$ and $n$) the results depend sensitively on the detailed
structure and burning rates of the presupernova models
(Lai \& Goldreich 2000b, in preparation).
Our tentative conclusion is that overstable g-modes can potentially grow
to large amplitudes prior to core implosion, although a complete
understanding of the global pre-collapse asymmetries is probably out of reach
at present, given the various uncertainties in the presupernova
models (see Bazan \& Arnett 1998 for complications due to
convective shell burning in presupernova stars).
\smallskip
\noindent
{\bf (ii) Amplification of Perturbation During Core Collapse}
Core collapse proceeds in a self-similar fashion, with
the inner core shrinking subsonically and the outer core falling
supersonically at about half free-fall speed
(Goldreich \& Weber 1980; Yahil 1983). The inner
core is stable to non-radial perturbations because of the
significant role played by pressure in its subsonic collapse.
Pressure is less important in the outer region, making
it more susceptible to large scale instability.
A recent stability analysis of Yahil's self-similar collapse solution
(which is based on Newtonian theory and a polytropic equation of state
$P\propto\rho^\Gamma$, with $\Gamma\sim 1.3$) does not reveal any unstable
global mode before the proto-neutron star forms (Lai 2000).
However, during the subsequent accretion of the outer core
(involving $15\%$ of the core mass) and envelope onto the proto-neutron star,
nonspherical perturbations can grow according to $\delta\rho/\rho\propto
r^{-1/2}$ or even $\delta\rho/\rho \propto r^{-1}$ (Lai \& Goldreich 2000).
The asymmetric density perturbations seeded in the presupernova star,
especially those in the outer region of the iron core, are therefore amplified
(by a factor of 5-10) during collapse. The enhanced asymmetric density
perturbation may lead to asymmetric shock propagation and breakout, which then
give rise to asymmetry in the explosion and a kick velocity to the neutron star
(Goldreich et al.~1996; Burrows \& Hayes 1996).
\section{Neutrino Driven Kicks}
The second class of kick mechanisms rely on asymmetric neutrino emission
induced by strong magnetic fields.
The fractional asymmetry $\alpha$ in the radiated neutrino energy required to
generate a kick velocity $v_{\rm kick}$ is $\alpha=Mv_{\rm kick}c/E_{\rm tot}$
($=0.028$ for $v_{\rm kick}=1000$~km~s$^{-1}$, neutron star mass
$M=1.4\,M_\odot$ and total neutrino energy radiated $E_{\rm tot}
=3\times 10^{53}$~erg).
\smallskip
\noindent
{\bf (i) Effect of Parity Violation}
Because weak interaction is parity violating, the neutrino opacities and
emissivities in a magnetized nuclear medium depend asymmetrically on the
directions of neutrino momenta with respect to the magnetic field, and
this can give rise to asymmetric neutrino emission from the proto-neutron
star. Chugai (1984) (who gave an incorrect expression for the
electron polarization in the relativistic, degenerate regime)
and Vilenkin (1995) considered neutrino-electron scattering, but this is less
important than neutrino-nucleon scattering in determining neutrino transport in
proto-neutron stars. Dorofeev et al.~(1985) considered neutrino emission by
Urca processes, but failed to recognize that in the bulk interior of the star
the asymmetry in neutrino emission is cancelled by that associated with
neutrino absorption (Lai \& Qian 1998a).
Horowitz \& Li (1998) suggested that large asymmetries in the neutrino flux
could result from the cumulative effect of multiple scatterings of neutrinos by
slightly polarized nucleons (see also Janka 1998; Lai \& Qian 1998a). However,
it can be shown that, although the scattering cross-section is asymmetric with
respect to the magnetic field for individual neutrinos, detailed balance
requires that there be no cumulative effect associated with multiple
scatterings in the bulk interior of the star where thermal equilibrium is
maintained to a good approximation (Arras \& Lai 1999a; see also Kusenko et
al.~1998).
For a given neutrino species, there is a drift flux of neutrinos along the
magnetic field in addition to the usual diffusive flux. This
drift flux depends on the deviation of the neutrino distribution function from
thermal equilibrium. Thus asymmetric neutrino flux can be generated
in the outer region of the proto-neutron star (i.e., above the neutrino-matter
decoupling layer, but below the neutrinosphere) where the neutrino
distribution deviates significantly from thermal equilibrium. While the
drift flux associated with $\nu_\mu$'s and $\nu_\tau$'s is exactly canceled by
that associated with $\bar\nu_\mu$'s and $\bar\nu_\tau$'s, there is a
net drift flux due to $\nu_e$'s and $\bar\nu_e$'s. Arras \& Lai (1999b)
found that the asymmetry parameter for the $\nu_e$-$\bar\nu_e$ flux is
dominated for low energy neutrinos ($\simless
15$~MeV) by the effect of ground (Landau) state
electrons in the absorption opacity, $\epsilon_{\rm abs}\simeq
0.6B_{15}(E_\nu/1~{\rm MeV})^{-2}$, where
$B_{15}=B/(10^{15}~{\rm G})$, and for high energy neutrinos by nucleon
polarization ($\sim \mu_mB/T$). Averaging over all neutrino species,
the total asymmetry in neutrino flux is of order
$\alpha\sim 0.2\epsilon_{\rm abs}$, and the resulting
kick velocity $v_{\rm kick}\sim 50\,B_{15}$~km~s$^{-1}$.
There is probably a factor of 3 uncertainty in this estimate. To firm up
this estimate requires solving the neutrino transport equations
in the presence of parity violation for realistic
proto-neutron stars.
\smallskip
\noindent
{\bf (ii) Effect of Asymmetric Field Topology}
A different kick mechanism relies on the asymmetric magnetic field
distribution in proto-neutron stars (see Bisnovatyi-Kogan 1993;
However, he considered neutron decay, which is not directly relevant for
neutrino emission from proto-neutron stars). Since the cross section for
$\nu_e$ ($\bar\nu_e$) absorption on neutrons (protons) depends on the local
magnetic field strength due to the quantization of energy levels for the $e^-$
($e^+$) produced in the final state, the local neutrino fluxes emerged from
different regions of the stellar surface are different. Calculations
indicate that to generate a kick velocity of $\sim 300$~km~s$^{-1}$ using this
mechanism alone would require that the difference
in the field strengths at the two opposite poles of the star
be at least $10^{16}$~G (Lai \& Qian 1998b). Note that unlike
the kick due to parity violation [see (i)], this mechanism does not
require the magnetic field to be ordered, i.e., only the magnitude
of the field matters.
\smallskip
\noindent
{\bf (iii) Exotic Neutrino Physics}
There have also been several interesting ideas on pulsar kicks
which rely on nonstandard neutrino physics. It was
suggested (Kusenko \& Segre 1996) that asymmetric $\nu_\tau$ emission could
result from the Mikheyev-Smirnov-Wolfenstein flavor transformation
between $\nu_\tau$ and $\nu_e$ inside a magnetized proto-neutron star
because a magnetic field changes the resonance condition for the
flavor transformation. This mechanism requires neutrino mass of order
$100$~eV. Another similar idea (Akhmedov et al.~1997)
relies on both the neutrino mass and the neutrino magnetic
moment to facilitate the flavor transformation. More detailed
analysis of neutrino transport (Janka \& Raffelt 1998), however, indicates
that even with favorable neutrino parameters (such as
mass and magnetic moment) for neutrino oscillation, the induced pulsar
kick is much smaller than previously estimated (i.e.,
$B\gg 10^{15}$~G is required
to obtain $100$~km~s$^{-1}$ kick).
\section{EM~ Radiation Driven Kicks}
For completeness, we mention the post-explosion
``rocket'' effect due to electromagnetic (EM) radiation from off-centered
magnetic dipole in the pulsar (Harrison \& Tademaru 1975). In this mechanism,
the neutron star velocity comes at the expense of its spin kinetic energy,
which is radiated away asymmetrically via EM braking;
The kick is attained on the timescale of the initial spindown time of the
pulsar (i.e., this is not a ``natal'' kick). The neutron star velocity
changes according to $M\dot v=\epsilon L/c$, where $L$ is the
EM braking power, and $\epsilon$ is the asymmetry parameter. Typically,
$\epsilon\sim 0.1 (\Omega s/c)(\mu_\phi/\mu_z)$, where $\Omega$ is the spin and
$s$ is the ``off-center'' displacement of the dipole; In fact, there is
theoretical maximum, $\epsilon=0.16$, achieved for $\mu_R=0,~\mu_\phi/\mu_z
=0.63(\Omega s/c)$ (where $\mu_R,\mu_\phi,\mu_z$ are the three cylindrical
components of the dipole). The kick velocity is along the spin axis, and
$v_{\rm kick}\simeq 600\,(\bar\epsilon/0.1)(\nu_0/1~{\rm kHz})^2$~km~s$^{-1}$
(where $\nu_0$ is the initial spin).
Clearly, Even if the neutron star were born with maximum rotation rate
{\it and} ${\epsilon}$ were maintained at near the maximum value, the kick
velocity would still be at most a few hundred km~s$^{-1}$.
Given that most pulsars were born rotating slowly (see Spruit \& Phinney 1998),
we conclude that ``EM rocket'' cannot be the main mechanism for pulsar
kicks.
\section{Discussion}
Statistical studies of pulsar population have revealed no correlation between
$v_{\rm kick}$ and magnetic field strength, or correlation between
the kick direction and the spin axis (e.g., Lorimer et al.~1995;
Cordes \& Chernoff 1998; Deshpande et al.~1999). Given the large
systematic uncertainties, the statistical results, by themselves,
cannot reliably constrain any kick mechanism (see Cordes
\& Chernoff 1998). For example, the magnetic field strengths required for the
neutrino-driven mechanisms are $\simgreat 10^{15}$~G, much larger
than the currently inferred dipolar surface fields of typical radio pulsars;
the internal magnetic fields of neutron stars and their evolution remain
clouded in mystery; and several different mechanisms may contribute to
the observed kick velocities.
It is of interest to note that soft gamma repeaters (``magnetars'':
neutron stars with observed magnetic fields $\simgreat 10^{14}$~G; see
Thompson \& Duncan 1996) have very high velocities, $\simgreat
1000-2000$~km~s$^{-1}$ (e.g., Kaspi 1999; Marsden et al.~1999).
Such a high velocity may well require superstrong magnetic fields
($\simgreat 10^{16}$~G) to be present in the proto-neutron stars,
although hydrodynamical effects remain a viable kick mechanism if
enough presupernova asymmetry can be generated. It has recently been
suggested that MHD jets can play an important role in supernovae (Khokhlov et
al.~1999), but the origin of the jets is unknown, nor it is clear why
the two opposite jets are so different (a necessary condition to
produce a kick).
\smallskip
I thank my collaborators Phil Arras, Peter Goldreich and
Yong-Zhong Qian for their important contribution and insight.
This work is supported by NASA grant NAG 5-8356 and by a fellowship
from the Alfred P. Sloan foundation.
\begin{chapthebibliography}{1}
\bibitem{Akhmedov97}
Akhmedov, E.~K., Lanza, A., \& Sciama, D.~W. 1997, Phys. Rev. D, 56, 6117
\bibitem{arras99a}
Arras, P., \& Lai, D. 1999a, ApJ, 519, 745 (astro-ph/9806285)
\bibitem{arras99b}
Arras, P., \& Lai, D. 1999b, Phys. Rev. D60, 043001 (astro-ph/9811371)
\bibitem{Bisnovatyi-Kogan93}
Bisnovatyi-Kogan, G.~S. 1993, Astron. Astrophys. Trans., 3, 287
\bibitem{Burrows95}
Burrows, A., Hayes, J., \& Fryxell, B.A. 1995, ApJ, 450, 830
\bibitem{}
Burrows, A., \& Hayes, J. 1996, Phys. Rev. Lett., 76, 352
\bibitem{Chugai84}
Chugai, N.N. 1984, Sov. Astron. Lett., 10, 87
\bibitem{Cordes93}
Cordes, J.M., Romani, R.W., \& Lundgren, S.C. 1993, Nature, 362, 133
\bibitem{Cordes90}
Cordes, J.M., Wasserman, I., \& Blaskiewicz, M. 1990, ApJ, 349, 546
\bibitem{}
Cordes, J.M., \& Chernoff, D.F. 1998, ApJ, 505, 315
\bibitem{Dorofeev85}
Dorofeev, O.F., et al. 1985, Sov. Astron. Lett., 11, 123
\bibitem{Frail94}
Frail, D.A., Goss, W.M., \& Whiteoak, J.B.Z. 1994, ApJ, 437, 781
\bibitem{Fryer98}
Fryer, C., Burrows, A., \& Benz, W. 1998, ApJ, 498, 333
\bibitem{Goldreich96}
Goldreich, P., Lai, D., \& Sahrling, M. 1996, in
``Unsolved Problems in Astrophysics", ed. J.N. Bahcall and
J.P. Ostriker (Princeton Univ. Press)
\bibitem{}
Hansen, B.M.S., \& Phinney, E.S. 1997, MNRAS, 291, 569
\bibitem{}
Herant, M., et al. 1994, ApJ, 435, 339
\bibitem{Horowitz97b}
Horowitz, C.J., \& Li, G. 1998, Phys. Rev. Lett., 80, 3694
\bibitem{}
Janka, H.-T., \& M\"uller, E. 1996, A\&A, 306, 167
\bibitem{Janka98}
Janka, H.-T. 1998, in ``Neutrino Astrophysics", ed. M. Altmann et al.
\bibitem{Janka98b}
Janka, H.-T., \& Raffelt, G.G. 1998, Phys. Rev. D59, 023005
\bibitem{Kaspi96}
Kaspi, V.M., et. al. 1996, Nature, 381, 583
\bibitem{kaspi}
Kaspi, V.M. 1999,
astro-ph/9912284
\bibitem{Keil}
Keil, W., Janka, H.-Th., M\"uller, E. 1996, ApJ, 473, L111
\bibitem{}
Khokhlov, A.M., et al.~1999, astro-ph/9904419
\bibitem{}
Kramer, M. 1998, ApJ, 509, 856
\bibitem{Kusenko96}
Kusenko, A., \& Segr\'e, G. 1996, Phys. Rev. Lett., 77, 4872 (also
astro-ph/9811144)
\bibitem{Kusenko98a}
Kusenko, A., Segr\'e, G., \& Vilenkin, A. astro-ph/9806205
\bibitem{}
Lai, D. 1996, ApJ, 466, L35
\bibitem{}
Lai, D. 2000, ApJ, in press.
\bibitem{}
Lai, D., \& Goldreich, P. 2000, ApJ, in press (astro-ph/9906400)
\bibitem{Lai98a}
Lai, D., \& Qian, Y.-Z. 1998a, ApJ, 495, L103 (erratum: 501, L155)
\bibitem{Lai98b}
Lai, D., \& Qian, Y.-Z. 1998b, ApJ, 505, 844
\bibitem{Lorimer97}
Lorimer, D.~R., Bailes, M., \& Harrison, P.~A. 1997, MNRAS, 289, 592
\bibitem{Lyne94}
Lyne, A.G., \& Lorimer, D.R. 1994, Nature, 369, 127
\bibitem{Mazza}
Mazzacappa, A., et al.~1998, ApJ, 495, 911.
\bibitem{Morse95}
Morse, J.A., Winkler, P.F., \& Kirshner, R.P. 1995, AJ, 109, 2104
\bibitem{Spruit}
Spruit, H., \& Phinney, E.S. 1998, Nature, 393, 139
\bibitem{Thompson96}
Thompson, C., \& Duncan, R.~C. 1996, ApJ, 473, 322
\bibitem{Vilenkin95}
Vilenkin, A. 1995, ApJ, 451, 700
\bibitem{wang}
Wang, L., Howell, D.A., H\"oflich, P., \& Wheeler, J.C. 1999,
astro-ph/9912033
\bibitem{wex}
Weaver, T.A., \& Woosley, S.E. 1993, Phys. Rep., 227, 65
\bibitem{wex}
Wex, N., Kalogera, V., \& Kramer, M. 1999, astro-ph/9905331.
\end{chapthebibliography}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,564 |
\chapter{Background}
\section{Scalability in general-purpose processors}
Assuming continued demand for computers where \emph{new functions can
be defined by the end-user by writing and using their own software}
(cf. \cref{sn:gpdemand}), the question remains of how to make
general-purpose processors that are both fast (operations/second) and
efficient (operations/watt). However, conventional approaches on
silicon seem to have reached ``walls'' on both fronts around year
2000~\cite{ronen.01.ieee}.
Since technology progress still delivers increasingly more transistors
per chip (Moore's law), the trend has become to glue individual
processors together on the same chip, i.e. design ``multi-cores.''
The issue with this is that software is mostly written using
sequential algorithms: introducing hardware parallelism (multiple
processors) immediately raises the question of how to introduce
explicit concurrency in software. Software concurrency is hard and
both hardware architects and programming language designers have been
making only baby steps since 2000.
\begin{sidenote}
\caption{On the continued demand for general-purpose
computers.}\label{sn:gpdemand} One needs to accept the premise that
general-purpose computers are highly desirable and that the future of
computing hangs on their continued development to fully value the
remainder of this report. I have explained my own reasons to accept
this premise in \cite[sect.~1.2--1.4]{poss.12}; in short, I propose
that general-purpose computers are, like "stem cells," necessary to
the continuation of computer science. I also propose they are
essential to the democratic freedom of any citizen to create their own
tools (in software) in this numeric age.
Meanwhile, I also acknowledge that it is not the role of mere
academicians to decide "what people really want." If market, fashion
and politics determine that science should research instead all sorts of
maximally efficient special-purpose computing devices, the deconstruction
phase of this report would even be easier: there would be simply
no place at all for D-RISC and Microgrids.
\end{sidenote}
\section{D-RISC and Microgrids: what has been done}
The \emph{Microgrid} many-core architecture is a research project at
the University of Amsterdam, which investigates whether concurrency
management (thread scheduling, synchronization, and inter-thread
communication) traditionally under control of software operating
systems can be accelerated in hardware to obtain higher efficiency and
performance. Microgrids are clusters of a simple RISC core design
called D-RISC~\cite{bolychevsky.96.ieee}; each D-RISC core supports
hardware multi-threading (HMT) using a dataflow scheduler, and is also equipped with a hardware Thread
Management Unit (TMU) which can coordinate with neighbouring TMUs for
automatic thread and data distribution (cf. \cref{sn:tmu}). In short:
\begin{center}
\bfseries
D-RISC = simple RISC + dataflow HMT scheduler + TMU $-$ interrupt management
Microgrid = n$\times$D-RISC + TMU-to-TMU NoC + custom cache/memory protocol
\end{center}
\begin{sidenote}
\caption{Details about D-RISC/Microgrids architecture}\label{sn:tmu}
The rest of the text assumes passing familiarity with the D-RISC
and Microgrids architecture, as presented in chapters 3 and 4 of~\cite{poss.12}.
\end{sidenote}
Prior to 2007, research on D-RISC and the Microgrid was focused on
programmability issues and carried out with high-level simulators:
both using traditional software multithreading and an API to emulate
the TMU services~\cite{tol.09.jsa}, and using a custom functional ISA
emulator~\cite{bousias.06.cj}. As the initial phases of the D-RISC and
Microgrid design were encouraging~\cite{bell.06.jpp,bousias.09.jsa},
the EU-funded project Apple-CORE (2008-2011) was started to study its
implementability in a system, including a full vertical tooling stack
from an FPGA implementation up to benchmarks in higher-level
programming languages.
The outcome of the Apple-CORE project is summarized
in~\cite{poss.12.dsd,poss.12}: the D-RISC core was implemented on FPGA
as UTLEON3~\cite{danek.12}, a model of Microgrids was implemented in
MGSim, software tooling was delivered to program
Microgrids~\cite{saougkos.11,poss.12.sl,grelck.09.cpc}, and D-RISC and
Microgrids were confirmed using both UTLEON3 and MGSim to deliver
higher performance and efficiency for \emph{some of the selected
benchmarks}.
\section{D-RISC and Microgrids: what is going on}
At the time of this writing, research in this area continues on two
fronts. An industry-backed project has funded more effort towards
tailoring D-RISC for real-time embedded systems, by adding
priority-based scheduling and fault tolerance. Next to this, four
doctoral candidates are planning to defend their thesis on extensions
and improvements to D-RISC and Microgrids, and simulations thereof.
The Microgrid-related technology produced so far is also used for
graduate and undergraduate education in computer architecture and
compiler construction.
\section*{Disclaimer}
{\itshape
The arguments presented hereafter are my own, and thus may not be shared
by my colleagues or work partners. To my knowledge, at the time of
this writing there is no acknowledgement or consensus around the
D-RISC/Microgrids enterprise, other than my own experience and
impressions, that give credit to the perspective presented here.
}
\section*{Purpose and rationale}
This report lays flat my personal views on D-RISC and Microgrids as of
March 2013. It reflects the opinions and insights that I have gained
from working on this project during the period 2008-2013.
The origin of this report is a case of cognitive dissonance. On the
one hand, using critical thought against my "achievements" of the past
few years is causing a growing discomfort, discontent and
disappointment at the way the design and implementation of
D-RISC/Microgrids have been carried out so far, both by myself and my
colleagues. On the other hand, my optimism combined with an unusual
combination of curiosity and fascination for theoretical computer
science is sustaining a belief that despite its flaws, the project has
produced an interesting conceptual framework which deserves further
investigation at least by academics and teachers. Only by resolving
this cognitive dissonance can I satisfy myself that my continued work
in this area is compatible with my aspirations as a scientific
researcher. By writing this report, I hope I can resolve this
dissonance by externalizing both sides and construct rationally their
resolution.
\clearpage
\section*{Executive summary}
\paragraph{Shortcomings in the research results so far.}
The D-RISC/Microgrids project was purportedly intended to solve major issues in
micro-architecture research, related to scalability in performance and
efficiency in general-purpose microprocessors.
The strategy to solve these issues was to implement a
combination of dataflow scheduling with hardware support for thread
concurrency management within and across cores on chip.
Implementation
was carried out, but the results are inconclusive. On the one hand, the proposed
hardware does indeed provide higher performance and efficiency in
regular, data-parallel computation kernels. On the other hand, no evidence has yet
been produced that the proposed hardware benefits larger applications
with more irregular workloads.
Power efficiency and intelligent resource
management was regularly advertised but not actively researched.
Effort has been invested into widening the scope of the technology
towards applications and industrial relevance, but these applications
have not yet materialized.
\paragraph{Shortcomings in methodology.}
Research on D-RISC/Microgrids is not following the scientific method.
It is instead currently carried out as an engineering enterprise, but without clear
technology ouputs and without identifying
its potential applications. Its relevance in a university research group is thus questionable.
\paragraph{Obstacles to further progress.}
The D-RISC/Microgrids project has the ambitious aim to produce a
general-purpose processor chip able to disrupt the current
state-of-the-art. However, the limited human resources dedicated to
the project are insufficient to reach this aim in isolation.
The expansion of the research group to a community of users and and research
partners is blocked by a fundamental lack of compatibility with existing operating
systems and application software. This lack of
compatibility is not properly justified, neither by practical nor
theoretical reasons.
Meanwhile, the scientific effort to test the hypotheses that underly
the D-RISC/Microgrids project is poorly directed, and not
enough attention has been given to negative results that
invalidate these hypotheses.
Finally, the multi-core research field is nowadays much
more crowded than it was ten years ago, yet the research on D-RISC/Microgrids
does not acknowledge its competition nor attempts to differentiate
its contributions from the state of the art.
\paragraph{Actual contributions.}
The research has produced interesting discussions
that challenge some tacit assumptions of the research community,
experimental results that can be reused by future work, improvements
to partner technologies and new simulation techniques.
Most of the software designed and implemented during the research
can be reused by third parties, and not only for research
directly related to D-RISC/Microgrids.
The intellectual framework educates practitioners to think about
two general separations of concerns, namely concurrency vs.~ parallelism and
using memory for storage vs.~ synchronization.
\paragraph{Individual architectural features.}
The D-RISC core combines features found in other processors, such as a RISC pipeline
and hardware multithreading, with custom features (e.g.~ its TMU) and optimizations
to the conventional features (e.g.~ switch annotations for the HMT scheduler).
Some architectural optimizations found in D-RISC/Microgrids could be reused
with other processors, for example switch annotations and bulk coherency in the memory network.
The key feature of D-RISC/Microgrids, namely its TMU and inter-TMU control NoC,
does not depend on the other features specific to D-RISC and could be potentially reused with other processors.
\paragraph{Follow-up strategies.}
I can see three follow-up strategies for new investments around
D-RISC/Microgrids: exploitation, i.e.~ apply the technology produced
so far to other uses than research; salvaging and opening the
technology, i.e.~ extracting individual features from the
D-RISC/Microgrids design and evaluating them as extensions of
existing processors; and distillation of the main ideas in the realm
of fundamental computer science.
Ongoing research towards doctoral theses should be careful
to rephrase research questions in the light of our recent shared
understanding of the project's issues.
\chapter{Conclusion}
In traditional academic research projects, the abstract and general
questions receive most attention, and technology and engineering
``happen'' as a by-product. In contrast, the D-RISC/Microgrids project
was primarily a technology and engineering enterprise, with some
occasional and incidental scientific output.
My opinion is that further work in this direction faces two fundamental problems.
Firstly, a continued focus on engineering makes the project increasingly
difficult to host in an academic institution and impedes the growth
of an academic network.
The product of the current and past effort is made of chip
blueprints, simulation software, ancillary programming tools,
education materials and demonstration tools. Unfortunately, the
metrics used in academia to reward scientific effort are
peer-reviewed academic publications, conference attendance, invited
talks and lectures, successfully defended doctoral theses, etc. This
mismatch implies that the work has become extraordinarily difficult to
defend in academic communities. Moreover, any team member expecting
to receive an academic training from this project risks facing a
strong sense of disconnect between expectations and reality that may drive
them away. This is detrimental to the growth of a network of
supporting researchers around the project.
Secondly, the lack of connections with related work, especially
a continued disregard for software compatibility, constitutes a serious
management issue that threatens the project.
This disconnect has not always been an issue. In general, at
the start of a new line of research in computer architecture, compatibility
can be readily sacrificed to simplify the research environment
and quickly obtain preliminary evaluation results using simple, ad-hoc
experiments. Moreover, ten years ago when the research strategy
was being shaped, there did not yet exist any pervasive software culture for
multi-core programming and software interfaces to concurrency management.
In this context, a new, immature approach was simply competing with a host of other equally
new, immature approaches. But this context has thus evolved, and
the research risks facing irrelevance if the circumstancial changes in context and expectations are not addressed soon.
The question then remains: what to do now? For this, I have detailed
in \cref{sec:new} three possible strategies for new investments which
I know are viable from the current status of the research and would
address the two problems identified above.
One is to \emph{exploit}: take the shortest practical route to
maximize visibility of the current results and apply the technology. I
am currently driving exploitation towards academia, using the produced
tools for education in chip architecture and code generation. I am
seeking support from undergraduate students to design a minimal but
working form of exception handling and system-level compatibility with
existing software. I am also keeping ready to partner with industry to
work on exploitation projects that do not require further design.
Another is to \emph{salvage and open}: bring the technology apart and
offer its most salient bits and pieces as reusable components, able
to ground partnerships for follow-up joint research projects using
existing platforms and processor core designs. I may be interested
to support work in this direction, but not to drive the work myself.
The third is \emph{distill and reincarnate}: extract the underlying
fundamental research questions that are still relevant in this day and
age, and create a new research direction to explore. I have started
some preliminary work in this direction myself already.
An incidental, more personal but more fundamental question in the
bigger picture is whether any of these strategies is favorable to the
development of a researcher's career in the current academic
institution where the project is currently hosted. According to my
hierarchy, the answer is currently: ``not likely.'' I may try to
convince them otherwise by sublimating the work somehow, but personal
circumstances may prevent my long-term dedication to D-RISC/Microgrids
in favor of more aligned research topics instead.
\chapter{Actual contributions}
\section{Concrete scientific contributions}
The scientific output of the project is positive on at least four angles.
First, some published discussion-oriented articles have
articulated interesting challenges to the tacit assumptions of the
architecture community about the exploitation of concurrency in processors,
e.g.~~\cite{bell.06.jpp,jesshope.08.samos,tol.11.apc,jesshope.09.parco,bernard.10.ppl,bernard.10,poss.10.amp,poss.12,vantol.13}.
Second, all results-oriented articles accepted by peer review
for publication are based on real experimental results using
``honest and best effort'' implementations of the proposed ideas,
e.g.~~\cite{luo.02,jesshope.09.arcs,hasasneh.07.jsa,vu.08.icamst,bousias.06.cj,bousias.09.jsa,poss.12.dsd}.
Regardless of the conclusions drawn from them, these results
constitute a sound database of prior work to all future
researchers working on related areas.
Third, the research group has indirectly contributed to other projects
via its few partnerships. For example, the close work relationship
with the designers of Single-Assignment C and S-NET have yielded both
joint scientific
outputs~\cite{jesshope.08.apcsac,a-c-d44,poss.12.interact,grelck.09.cpc}
and technology improvements, directly or indirectly inspired by the
work on D-RISC/Microgrids.
Fourth, it the project has enabled ancillary research in novel techniques for system
simulation when cores are hardware multi-threaded~\cite{tol.09.jsa,tol.11.dutc,mirfan.11,poss.12.rapido,mirfan.12,lankamp.13.mgsim}, whose results are
scientific contributions on their own regardless of the specific merits of D-RISC/Microgrids.
\section{Technology products}
The research efforts have produced the following components and tools:
\begin{itemize}
\item svp-ptl and d-utc~\cite{tol.09.jsa,tol.11.dutc}, a library of TMU-like services implemented in software over POSIX threads,
ready to implement concurrent software over multi-cores and distributed memory systems;
\item MGSim~\cite{poss.12.rapido,lankamp.13.mgsim}, a combination of:
\begin{itemize}
\item a general discrete-event, component-based simulation framework in C++, and
\item a library of component models that can be used to simulated D-RISC/Microgrid-based architectures;
\end{itemize}
\item HLSim~\cite{mirfan.11,mirfan.12}, a discrete-event, thread-based simulation of multi-scale systems using the TMU protocol
and the API from svp-ptl;
\item the ``SL core'' package~\cite{poss.12.sl}, a combination of:
\begin{itemize}
\item a code translator from SL to D-RISC code, using any of its 3 possible ISAs,
\item a code translator from SL to the API of HLSim and svp-ptl,
\item a code translator from SL to ``vanilla'', sequential ISO C,
\item an incomplete port of a standard C library suitable for use in the simulated D-RISC/Microgrid environments, and
\item operating system components for resource management and interfacing with I/O services on D-RISC cores;
\end{itemize}
\item a set of micro-benchmarks using the SL language extensions that exercise the architecture and demonstrate the features
of the simulation frameworks;
\item via research partners, the UTLEON3 core design in VHDL~\cite{danek.12} which implements one D-RISC core with a partial TMU
for use on FPGA.
\end{itemize}
Some of these tools are specific to the D-RISC/Microgrid architecture and are not applicable outside of this project, whereas
others could be reused by researchers that have never been exposed to D-RISC/Microgrids. On a different scale,
some of these tools have been explicitly packaged for reuse and tested for portability by 3rd party users, whereas
others are not readily reusable due to dependencies on the local research environment. I summarize how the tools map to
these two scales in \cref{fig:tech}.
\begin{figure}
\centering
\includegraphics[scale=.6]{tech}
\caption{Technology products as of 2012.}\label{fig:tech}
\end{figure}
\section{Conceptual separation of concerns}
The research around D-RISC/Microgrids has strongly promoted two
intellectual exercises and shaped a generation of researchers able to
converse fluently about the following two issue separations.
The first is the separation between concurrency and parallelism, i.e.~
the distinction between \emph{opportunity for parallelism} that can be
encoded in software by relaxing synchronization constraints, and the
\emph{actual simultaneity of execution at run-time} which depends on
resource duplication over space.
This first separation is promoted/enabled by D-RISC/Microgrids by
promoting a TMU protocol which does not guarantee the availability of
parallel resources at run-time. Software that wishes to use the TMU
can only relax synchronization, i.e.~ introduce concurrency, whereas
actual parallelism is introduced later, by the TMU at run-time,
depending on resource availability. Although this separation of
concerns can be promoted by other means, any researcher working on
D-RISC/Microgrids \emph{cannot avoid} acquiring a sharp consciousness
of these issues.
The second separation is between ``memory as storage'' and ``memory as
a synchronization mechanism.'' In commodity architectures, the only
interface between the individual core and its environment in a
multi-core chip is its memory interface. This implies that the same
memory interface is used for both loading and storing values to main
memory within individual threads, and for coordination of work between
cores. The latter, in particular, has historically mandated the
extension of memory systems with transactional mechanisms (bus
locking, compare-and-swap, test-and-set) which would otherwise be
unneeded.
As explained in~\cite[Chap.~7]{poss.12}, the proposed
D-RISC/Microgrids architecture separates\footnotemark{} the memory
network for data storage from a ``control network-on-chip'' in charge
of synchronizing and coordinating work between TMUs. This forces the
researchers writing software for the platform to realize that the data
structures for synchronization traditionally implemented in memory,
such as producer-consumer FIFOs, mutexes or semaphores, are really specific instances of
\emph{abstract synchronization services} whose behavior can be
obtained in other ways, possibly more cheaply and efficiently.
\footnotetext{The aim of this separation was to test whether a custom
NoC can achieve cheaper and more efficient synchronization than a
memory system, and to avoid the research overhead of developing a
complex cache coherency protocol that also supports transactions.
This aim was reached, insofar that a custom NoC is indeed cheaper
than a comprehensive memory protocol, but it requires special
support in software, cf.~~\cite[Chap.~7]{poss.12}.}
These separation of concerns are not yet widely understood and commonly accepted in
the research community. The increased intellectual acuity
of the researchers ``educated'' by working on D-RISC/Microgrids
forms an advantage that can thus be considered a contribution of the enterprise.
\begin{summary}
\begin{itemize}
\item The research has produced interesting discussions
that challenge some tacit assumptions of the research community,
experimental results that can be reused by future work, improvements
to partner technologies and new simulation techniques.
\item Most of the software designed and implemented during the research
can be reused by third parties, and not only for research
directly related to D-RISC/Microgrids.
\item The intellectual framework educates practitioners to think about
two general separations of concerns, namely concurrency vs.~ parallelism and
using memory for storage vs.~ synchronization.
\end{itemize}
\end{summary}
\chapter{Methodology issues}
What the body of published and unpublished materials reveal is a
large, loosely scoped enterprise to define a multi-core processor chip
with the ambitious aim to solve the most significant problems of
architecture research in the period 2010-2020.
\section{Actual methodology}
Although never explicit, a strategy guides the effort:
\begin{enumerate}
\item accumulate technology around the simple ideas of \emph{dataflow
scheduling} and \emph{partial hardware support for concurrency
management}, so as to define an execution platform able to run
parallel benchmark programs;
\item ``try it out'' and measure how it behaves;
\item if the measurements are unsatisfactory, return to step \#1;
otherwise publish results and claim success.
\end{enumerate}
\begin{figure}
\centering
\includegraphics[scale=.4]{process}
\caption{High-level overview of research dynamics around D-RISC/Microgrids.}\label{fig:mgproc}
\end{figure}
An overview of this process is given in \cref{fig:mgproc}. From an
outsider's perspective, this research activity is independent, as its
only input is human effort and financial investment. Its overall
output is scientific articles on measured results, and a regular stream
of educated practitioners.
Over the year, two patterns have emerged. The first is that the
research group often stalls, busy looping from unsatisfactory results
back to implementing more features without questioning the overall
strategy (thick blue arrow in the figure). The consequence is an irregular, unfocused publication
throughput and doctoral candidates abandoning their research from lack
of focus. The second pattern is that only positive, ``competitive''
results are retained as candidates for publication. The consequence is
a lack of visibility on the research process, methodologies and
shortcomings, although these could also be useful and valuable to the
scientific community.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{science-activities}
\caption{Activities related to science.}\label{fig:science}
\end{figure}
\section{Relationship with other scientific activities}
Compare the process above with \cref{fig:science}, explained as follows in \cite[Chap.~1]{poss.12}:
\begin{quote}
\itshape
The traditional purpose of the fundamental sciences is the acquisition
of new knowledge pertaining to observed phenomena, in an
attempt to describe ``what is.'' In parallel to the discovery of
new knowledge through scientific inquiry, philosophers, or
theoreticians, derive ideas of ``what could be.'' Via
formalisms, they construct structures of thought to validate
these ideas and derive iteratively new ideas from them.
We can focus for a moment on the human dynamics around these
activities. On the one hand, the intellectual pleasure that
internally motivates the human scientists is mostly to be found
in the acquisition of knowledge and ideas. For natural scientists, the
focus is on accuracy relative to the observed phenomena, whereas for
philosophers the focus is on consistency. On the other hand, the
external motivation for all fields of science, which materially
sustains their activities, is the need of humans for either
discovery or material benefits to their physical
existence. From this position, the outcome of
scientific inquiry and philosophical thought, namely knowledge and
ideas, is not directly what human audiences are interested in. The
``missing link'' between scientific insight and its practical benefits
is \emph{innovation}, an \emph{engineering} process in two steps.
The first step of innovation is \emph{foundational engineering}: the
creative, nearly artistic process where humans find a new way to
assemble parts into a more complex artifact, following the inspiration
and foreshadowing of their past study of knowledge and ideas, and
guided by human-centered concerns. Foundational engineering, as
an activity, consumes refined matter from the
physical world and produces new more complex things, usually tools and
machines, whose function and behavior are intricate, emergent
composition of their parts. The novelty factor is key: the outcome
must have characteristics yet unseen to qualify as foundational;
merely reproducing the object would just qualify as manufacturing.
The characteristic human factor in this foundational step
is \emph{creativity}, which corresponds to the serendipitously
successful, mostly irrationally motivated selection of ideas,
knowledge and material components in a way that only reveals itself as
useful, and thus can only be justified, \textsl{a posteriori}.
The other step is \emph{applicative engineering}, where humans
assemble artifacts previously engineerd into complex systems that
satisfy the needs of fellow humans. In contrast to foundational
engineering, the characteristic human factor here is meticulousness in
the realization and scrupulousness in recognizing and following an
audience's expectations---if not fabricating them on the spot.
The entire system of activities around science is driven by
a \emph{demand for applications}: the need of mankind to improve its
condition creates demand for man-made systems that solve its problems,
which in turn creates demand for new sorts of devices and artifacts to
construct these systems, which in turn creates demand for basic
materials as input, on the one hand, and intellectual diversity and
background in the form of knowledge and ideas. We illustrate this general view
in \cref{fig:science} [...]. The role of education, in turn, is
to act as a glue, ensuring that the output of the various activities
are duly and faithfully communicated to the interested parties.
\end{quote}
In this context, the activity around D-RISC/Microgrids can be
recognized to actually constitute \emph{foundational engineering}: the process
of invention that produces tools and artefacts that can subsequently
solve ``real-world'' problems.
This immediately highlights two major issues:
\begin{itemize}
\item the output of foundational engineering is measured by the tools
and artefacts it produces, not merely their description in the form
of academic publications. For the effort on D-RISC/Microgrids to be
recognized and valued as innovation, it must be accompanied by the
marketing of its \emph{technology}, including its flaws and
limitations, and ultimately exploitation to real-world applications.
\item it is not the primary purpose of the academic institutions of
science to fund and support foundational engineering. Although it
is not uncommon to see foundational engineering occur in academic
environments, it is usually only accepted as a by-product
exploitation of the other activities of science, namely natural and
fundamental sciences. To justify continued effort on D-RISC/Microgrids
in a university research group, the fundamental principles of the technology
must be extracted, abstracted, studied formally and generalized
in a relevant, technology-independent fashion. This work has not been performed so far
other than via the three isolated publications \cite{tdvu.07.icfem,vu.08.icamst} and \cite[Chap.~7]{poss.12}.
\end{itemize}
\begin{summary}
\begin{itemize}
\item Research on D-RISC/Microgrids is not following the scientific method.
It is instead currently carried out as an engineering enterprise, but without clear
technology ouputs and without identifying
its potential applications.
\item
Its relevance in a university research group is thus questionable.
\end{itemize}
\end{summary}
\chapter{Obstacles to further progress}
In \cref{chap:outcomes} I have presented the status of the research so
far, and identified the areas where it did not deliver on its own
self-set expectations. In this chapter, I follow up by reviewing the
likely causes of these shortcomings.
\section{Large project scope}
The project's aim is to eventually deliver a full general-purpose processor. However, the design
of a new processor architecture requires large research and implementation investment on \emph{all} the following fronts:
\begin{enumerate}
\item micro-processor logic (per core); \label{e:mp}
\item NoC interconnect: flow control, routing, failure management; \label{e:net}
\item cache management and inter-cache coherency; \label{e:caches}
\item core-NoC interfaces; \label{e:if}
\item external memory interfaces;
\item external I/O interfaces;
\item hardware/software interface design: ISA, but also I/O address space layout, MMU control, access to performance counters, etc; \label{e:isa}
\item ISA code generation in low-level compilers; \label{e:cgen}
\item architecture-specific optimizations in compilers; \label{e:opt}
\item architecture-specific support in software operating systems and programming language libraries; \label{e:os}
\item individual component validation, both from formal analysis and unit testing; \label{e:val}
\item component-level, then system-level modeling and simulation; \label{e:sim}
\item design of the integration strategy in a larger system (exploring system parameters and how system-level interconnects will impact intra-chip behavior);
\item circuit synthesis and prototype production (either ASIC or FPGA); \label{e:impl}
\item show casing and marketing via extensive comparisons with competition products; \label{e:cmp}
\item seeking and partnering with industry to define marketable products. \label{e:ind}
\end{enumerate}
Each of these areas would require multiple man-years worth of
investment before evidence of success in reaching the aim ``delivering a
full general-purpose processor'' can be strongly claimed.
To this date, the research effort around D-RISC/Microgrids was focused
on \cref{e:mp,e:caches,e:sim,e:isa}, although for \cref{e:isa} no clear
picture of the interface to virtual memory and I/O has yet emerged. A
lot of effort was also spent towards \cref{e:cgen,e:os} to enable
benchmarking, albeit half-heartedly because the research group lacked
expertise in these areas until recently. Some effort was spent on
\cref{e:impl} with a research partner~\cite{danek.12}, although the resulting FPGA model only
implements 1 core connected to a bus, i.e.~ it only shows limited
benefits of latency tolerance and does not implement the TMU's
multi-core coordination features. The other items have not been
investigated yet. Even if the scope of the research was reduced to
``design a Microgrid component that can be integrated as accelerator
in a larger processor chip,'' as advertised in \cite{poss.12.dsd},
effort should still be invested on
\cref{e:net,e:if,e:opt,e:val,e:ind,e:cmp}. This is out of the reach of
an isolated university research group expected to deliver strong
results in periods of five years with a bandwidth of 1 to 5
contributors per year.
\section{``Here Is My Chip, You Figure It Out''}
Following the original aim thus requires partnerships with other
organizations to carry out the work together. However, in the
scientific community, partnerships only come into existence based on
mutually beneficial arrangements: a peer researcher or instution may
be willing to contribute effort and technology towards the betterment
of D-RISC/Microgrids, only if they get something in return.
However, most of the interactions with potential partners so far were
carried out thus: ``here is our technology, we think it is good for
such-and-such use cases, what about you try to use it and tell us what
you do with it?'' For the reasons discussed extensively
in~\cite[Sect.~1.5, 15.1 \& Chap.~16]{poss.12}, this approach is
subject to the ``Here Is My Chip, You Figure It Out'' (HIMCYFIO)
hazard: faced with alien, unrecognizable technology, a potential
partner or user will be reluctant to invest the effort necessary to
cross the \emph{comprehension} threshold, even before they start to
think about potential shared endeavours. The producer of the
technology must cross this threshold preemptively to avoid the
HIMCYFIO pitfall.
Although my thesis \cite{poss.12} made one step in that direction,
further work is needed by this research group to bring the
D-RISC/Microgrid technology ``to the level'' of its potential
partners. The participants must identify what challenges potential
partners are facing, and preemptively shape the technology into a
palatable solution to the partners' problems.
To this day, the issues faced by the scientific peers in the same
research domain have not been investigated thoroughly by the research
group. The main reason for this is not lack of intent; rather a
crucial technical practical/historical obstacle: validation and peer
recognition in the micro-architecture community heavily relies on the
ability to exchange hardware platforms \emph{without modifying the
software}, so as to enable sound comparisons between
solutions. Although source-level compatibility is sufficient
(recompiling code towards a new platform has become acceptable in the
community), it is also necessary: given the large effort necessary to
design and deliver hardware, little effort can be spent
rewriting/adapting benchmark code, which has often accumulated
man-years of design, towards new platforms. However, for the reasons
described below and as of this writing, D-RISC/Microgrid's
\emph{cannot be made source-compatible} with most existing benchmarks.
This limitation is a high obstacle to publication and thus visibility,
and cuts short most opening discussions with potential partners.
\section{Lack of features needed for compatibility}\label{sec:incompat}
In my opinion, the main obstacle to the usability of D-RISC/Microgrids
by third parties is the lack of the following features:
\begin{itemize}\bfseries
\item \emph{process virtualization}, including per-process virtual memory address spaces and virtual I/O channels;
\item interrupt-like \emph{mechanisms to handle faults and unexpected external events};
\item the \emph{ability to stop a process and inspect it} externally, e.g.~ using a debugger;
\item the ability to \emph{preempt a running program and reclaim its resources}.
\end{itemize}
Programmers, in particular operating system and language implementers,
have been accustomed in the last 50 years to expect these features from any
general-purpose computer; the corollary is that all the existing
\emph{operating software} underlying existing application software
makes pervasive use of these services from the hardware platform.
The standpoint of the research group was that these features
are an ``historical artefact'' that were motivated at a time when
on-chip resources (cores, memory) were limited, and must thus be reconsidered at an
age where there are thousands of cores on chip and 64-bit addresses to
memory. As suggested in~\cite{tol.06,jesshope.08.samos,tol.11.apc,vantol.13}
and \cite[Sect.~3.3.2 \& Chap.~14--15]{poss.12}, this research group's public answer to
queries about these features goes as follows:
\begin{itemize}
\item ``preemptive event handlers should not be needed when any two
system-level tasks waiting on events could be active at the same
time in different hardware threads or cores (of which there are
thousands available);''
\item ``separate virtual address spaces should not be needed when a single virtual 64-bit
address space can be partitioned a thousand-fold while still
providing petabytes addressable to each process;''
\item ``once processes are allocated over space and not over time,
process boundaries are congruent to areas on chip and resource reclamation can be implemented simply by fully resetting the
corresponding hardware resources;''
\item ``issues of debugging and investigation do not need special supports as long
as simulators and emulators are available: debugging can be performed
from the simulator/emulator's host.''
\end{itemize}
As an insider, I can also report the second half of the answer: it is
possible to add support for these features, but the fear is that doing
so would make the research more difficult because a larger set of
issues would need to be considered. An ungrounded assumption is that
introducing support for virtualization would introduce overheads in
the hardware and make D-RISC and Microgrids less competitive against
other processors, including for the workloads where it currently
shines. The assumption is ungrounded because the consequences of
extending D-RISC in that direction have simply not been investigated
yet.
{\bfseries
In short, the position of the research group is ``these are
complicated engineering issues, but we think they are only superficial
usability concerns so they do not deserve our attention yet.''
}
The net effect is that existing operating software cannot be reused
with the proposed platform. Even assuming that new, custom-built
operating software \emph{could potentially show} that these
traditional mechanisms for virtualization can be avoided, the very
lack of software compatibility caused by the current situation may
well form the unsurmountable barrier preventing the group from forming
the partnerships needed for further developments, for the reasons
outlined in the previous section.
(For the sake of clarity, according to my own analysis these features
are both necessary and sufficient to immediately enable porting and
reusing operating software and existing major programming framework on
the proposed platform. To my knowledge, these features constitute
exactly the remaining obstacle to compatibility.)
\section{Weakly grounded and tested hypotheses}
The foundation for the ``D-RISC enterprise'' is the observation
that the static and dynamic costs of OoOE in GP processors is largely caused
by the logic necessary to discover instruction-level parallelism at run-time.
The hypotheses of the D-RISC/Microgrids research are then articulated as follows:
\begin{enumerate}\itshape
\item By shifting the responsibility to
discover concurrency, from the run-time to design-time (or compile-time),
these costs can be avoided. And instead of encoding concurrency the
VLIW way, which is weak when faced with the unpredictable variance of on-chip access
latencies in large chips, the concurrency can be encoded via threads
instead.
\item If thread creation is not more expensive than simple branches,
many sequential patterns including function calls and loops can be
transparently replaced by threads, and this
transformation can be performed casually in code generators for any
language. As a result, just using the platform's compilation tools
can introduce concurrency automatically in any sequential software and
solve the general ``programmability'' challenge of parallel hardware.
\item If concurrency management is encoded in the ISA with lightweight
instructions, the same binary code can be run under any amount of
resources, starting with a single thread on 1 core where it can be
as fast as an equivalent branch-based sequential code.
\item If concurrency management is encoded without explicit reference to parallel
hardware resources (``resource-agnostic''), the execution platform
can adapt the code at run-time to maximize performance and efficiency
to the resources effectively available.
\end{enumerate}
The first hypothesis has been largely confirmed to hold for large
classes of applications, but then not by D-RISC specifically: Intel's
HyperThreaded cores, then Sun/Oracle's Niagara cores, have been endorsing
the benefits of hardware multithreading for general-purpose computing
(especially in the datacenter domain) for a few years already.
The other hypotheses are more problematic.
The 2nd hypothesis heavily relies on the assumption that the
sequential composition of activation frames (for function calls and
loop iterations) can be cheaply and automatically replaced by parallel
composition. As I reveal partially in~\cite[Chap.~9]{poss.12}, this
research group had ``forgotten'' to consider that
\emph{general-purpose computations may use arbitrarily large amounts
of storage} at any step. This implies a local, ``private'' memory
area \emph{of varying size} for each activation frame \emph{while it
is running}.
With sequential execution, this is possible to implement cheaply with
a stack because only the most recently called activation frame is
running and can use the entire remaining memory. With parallel
execution, multiple activation frames are running and compete for the
available memory. Without much care (and thus ``intelligence'' that
must in turn be implemented in the TMU, raising its cost), the management of all these ``memory clients''
that all grow and shrink their local memory dynamically becomes a
bottleneck to parallel scalability.
Note that this problem is avoided in most highly-parallel
``accelerators'' by simply stating they do not support general-purpose
computations~\cite[Sect.~1.4\&12.6]{poss.12}, and in supercomputers by
providing large amounts of local RAM next to each processor. The
``supercomputer approach,'' applied to D-RISC/Microgrids, would imply
embedding large SRAM modules next to each core on chip or DRAM using
3D stacking, a direction not yet envisioned by this research group.
The 3rd hypothesis about automatic sequentialization was partly shown to hold with
D-RISC for threads that mostly perform local computations on
non-shared data and resources. As soon as shared or non-local
resources are used, sequentialization must choose a computation order
that maximizes locality of access and reuse. In sequential code,
typically the specified order of operations carries domain knowledge
from the programmer or compiler about locality; with threaded code,
this knowledge has disappeared from the code and must be reintroduced
at run-time while sequentializing. So far, the results produced show
that the automatic sequentialization is doing a poor
job~\cite[Sect.~10.4]{poss.12} but no research is being performed to
improve this state of affairs.
The 4th hypothesis about performance portability has been only
confirmed with D-RISC for data-parallel code with regular
data access patterns. For other types of concurrent code,
experimentation has shown that performance and efficiency are largely
dependent on the congruence between the data access patterns and
the physical topology of the interconnect between cores and
memory~\cite[Chap.~13]{poss.12}. However, to this date the concurrency
management interface of D-RISC's TMU does not allow to specify data
access patterns, so the TMU cannot make ``intelligent'' decisions
about placement. No further research is being performed in this
direction either.
All in all, these hypotheses seem \emph{attractive}: our peers have
reviewed these hypotheses through our publications and confirmed
implicitly, by accepting publication and funding further research,
that the hypotheses have merit and that our research efforts to test
them are scientifically worthwhile. However, I can also practically
observe that the work is not organized around strategic experiments
that would provide clear answers on these hypotheses. Meanwhile, I
have observed negative results that tend to invalidate the
hypotheses as they now stand, and I have also observed that these negative
results are not publicly exposed; instead, they are casually treated
as ``bugs'' and addressed by ad-hoc workarounds in the architecture
design.
\section{Competition and lack of differentiation}
Research on D-RISC/Microgrids was initiated in the late 1990's: a time
where multi-core chips were not yet widely used, and were there was
still a lot of uncertainty about concurrent programming. Back then,
D-RISC's approach was not only fresh, it was also spearheading
research in this area. There was thus not much to consider in terms of
``competition'' and ``related work.''
This has now changed. Since 2005, we have observed an explosion of
hardware multithreaded cores, multi-core chips and concurrent
programming frameworks. There now even exists architectures where
concurrency management is partly implemented in hardware: NVidia's
Fermi, Kalray's MPPA, Tilera's TILE are examples. Parallel and
multi-core programming is now being studied by students as a basic
course, and a wealth of software frameworks have evolved to manage
large number of ``micro threads'' efficiently on today's multi-core
chips: qthreads, codelets, green threads, Erlang's run-time system,
etc. Moreover, many high-level constructs to expose concurrency in
programming languages have been designed, e.g. in C/C++ (in the new
2011 standards), Scala, Haskell, etc. These have gradually introduced
\emph{expectations} in programmers' sub-cultures about what features
programming environments should and should not provide. Finally, some
architectural features have become widely accepted as fundamental to
the continued relevance of multi-cores, e.g. transactional memory and
heterogeneity, which have yet not been analyzed nor picked up by the
D-RISC/Microgrids enterprise.
In short, the research field has become crowded. To attract attention
and thus gather momentum, it is essential to \emph{acknowledge the
competition}, \emph{stay competitive} by keeping up and integrating
the good ideas from other projects, and simultaneously
\emph{differentiate} the new technology by pitting it against its
competition systematically. To this date, the research effort around
D-RISC/Microgrids has not focused on studying and integrating the
growing state-of-the-art, and differentiation is not expressed in
publications.
\begin{summary}
\begin{itemize}
\item
The D-RISC/Microgrids project has the ambitious aim to produce a
general-purpose processor chip able to disrupt the current
state-of-the-art. However, the limited human resources dedicated to
the project are insufficient to reach this aim in isolation.
\item
The expansion of the research group to a community of users and and research
partners is blocked by a fundamental lack of compatibility with existing operating
systems and application software.
\item This lack of
compatibility is not properly justified, neither by practical nor
theoretical reasons.
\item Meanwhile, the scientific effort to test the hypotheses that underly
the D-RISC/Microgrids project is poorly directed, and not
enough attention has been given to negative results that
invalidate these hypotheses.
\item Finally, the multi-core research field is nowadays much
more crowded than it was ten years ago, yet the research on D-RISC/Microgrids
does not acknowledge its competition nor attempts to differentiate
its contributions from the state of the art.
\end{itemize}
\end{summary}
\chapter{Outcomes vs.~ original intents: a retrospective}
\label{chap:outcomes}
The research on these topics was funded based on multiple research
proposals and statements of intent over time. In this section, I
explore \emph{published} motivations, where the scientific community has
agreed via the publication approval process that the motivations and
justifications were worthwhile.
Although most scientific projects are expected to diverge from their
original intents, published objectives reveal the general motivation
that drives the effort. Note that I do not focus here on
``result-oriented'' papers that report factually on research outcomes;
only on those publications that made statements of intent and
motivation before the corresponding work was carried out.
\section{I. Bell, N. Hasasneh \& C. Jesshope, JPP 2006}
Here we explore the contributions originally advertised in~\cite{bell.06.jpp}.
\subsection{Summary of the article}
Abstract:
\begin{quote}
\itshape Chip multiprocessors (CMPs) hold great promise for achieving
scalability in future systems. Microthreaded CMPs add a means of
exploiting legacy code in such systems. Using this model, compilers
generate parametric concurrency from sequential source code, which
can be used to optimise a range of operational parameters such as
power and performance over many orders of magnitude, given a scalable
implementation. This paper shows scalability in performance, power
and most importantly, in silicon implementation, the main contribution
of this paper. The microthread model requires dynamic register
allocation and a hardware scheduler, which must support hundreds of
microthreads per processor. The scheduler must support thread
creation, context switching and thread rescheduling on every machine
cycle to fully support this model, which is a significant
challenge. Scalable implementations of such support structures are
given and the feasibility of large-scale CMPs is investigated by
giving detailed area estimate of these structures.
\end{quote}
Problem statement:
\begin{quote}\itshape
In general, there are only a few requirements for the design of
efficient and powerful general-purpose CMPs, these are: scalability of
performance, area and power with issue width, and programmability from
legacy sequential code. Issue width is defined here as the number of
instructions issued on chip simultaneously, whether in a single
processor or in multiple processors and no distinction is made
here. To meet these requirements a number of problems must be solved,
including the extraction of ILP from legacy code, managing locality,
minimising global communication, latency tolerance, power-efficient
instruction execution strategies (i.e. avoiding speculation),
effective power management, workload balancing, and finally, the
decoupling of remote and local activity to allow for an asynchronous
composition of synchronous processors. Most CMPs address only some of
these issues as they attempt to reuse elements of existing processor
designs, ignoring the fact that these are suitable only for chips with
relatively few cores.
\end{quote}
Proposed main contribution:
\begin{quote}
\bfseries\itshape
In this paper a CMP is evaluated, that is based on microthreading, which addresses either directly or indirectly, all of the above issues and, theoretically, provides the ability to scale systems to very large number of processors.
\end{quote}
\subsection{Analysis}
The ``CMP'' in the paper's text refers to the D-RISC/Microgrid technology. Does this paper, and all the research
since then (8 years) support the claim that it solves ``all of the above issues''?
We list here how the technology addresses the issues in decreasing order of success.
\paragraph{Extraction of ILP from legacy code:} this was successful. ILP is extracted implicitly by D-RISC's dataflow scheduler, although
the ILP width is limited by the number of active threads and the number of registers per thread, because the reordering
information is stored in registers.
\paragraph{Decoupling of remote and local activity:} this is mostly successful, insofar the D-RISC's TMU control protocol
has different primitives to spawn work locally and remotely.
\paragraph{Scalability of performance, area and power with issue width:} each D-RISC core uses a single-issue pipeline,
so this claim states that performance, area and power scales with the number of cores. The research has indeed
shown this to be true for large, regular, data-parallel computation kernels, but the picture is not so clear for
small or more heterogeneous workloads because of the inter-core latencies in the concurrency management protocol itself
and on-chip network contention due to cache coherency protocols.
\paragraph{Power-efficient instruction execution strategies:} this is only partly successful; with one thread
the execution is not too power efficient compared to traditional
single-issue, in-order designs, and efficiency only increases with the number of threads
active. The problem is that operand availability is only tested at the
read stage of the pipeline. When there are multiple threads,
annotations at the fetch stage prevent ``potentially suspending''
instructions from entering the pipeline, but if only one thread is
active, the instructions with missing operands will still enter the
pipeline, create a bubble after the read stage, and subsequently need
to be rescheduled. This is a form of speculation, thus inherently less
power-efficient than the traditional approach to stall the pipeline during
issue.
\paragraph{Latency tolerance:} this is only partly successful. The latency of intra-core operations is tolerated by intra-thread ILP, but then the same is possible with conventional barrel processors or out-of-order execution\footnote{Albeit possibly at a larger area and power budget. However, the actual area and power requirement of D-RISC are yet to be evaluated.}. Longer latencies can be tolerated as long as there are sufficient threads active on the core to interleave with a waiting thread. If most active threads are busy communicating, then latency tolerance is highly dependent on a full end-to-end support for split-phase transactions, i.e.~ the rest of the system must support
a large number of in-flight transactions. In practice, the caches and
external memory interfaces become a bottleneck, and to this day no clear solution has emerged on this front.
\paragraph{Minimising global communication:} this is only mildly successful. For synchronization and concurrency management, global communication is limited by constraining the TMU's automatic workload distribution to adjacent cores only. The responsibility of choosing an area of the chip where to start the distribution is left to an hypothetical resource manager, not yet researched/implemented. Next to this, one should also consider memory communication. Here all results so far use memory protocols that incur global communication for coherency. No results show yet that global memory communication has been minimized.
\paragraph{Workload balancing:} this is only very mildly successful. D-RISC's TMU can automatically spread a batch of threads to multiple cores using an even N/P distribution, but this is the only form of distribution supported. Due to bulk reuse and synchronization, this simple distribution causes irrecoverable imbalances as soon as the batch is heterogeneous.
\paragraph{Managing locality:} this is not achieved, insofar that the data used by instructions is invisible to D-RISC's ``intelligence'' (its hardware TMU), and the memory and core networks are not topologically congruent. The software has to negociate locality of code and data explicitly with knowledge of the chip's layout.
\paragraph{Programmability from legacy sequential code:} to this date, most existing sequential code cannot
be reused as-is with D-RISC/Microgrids, because of incomplete support
for operating system services, cf.~ also \cref{sec:incompat}.
\paragraph{Effective power management:} to this date, power management has not been explored.
\section{NWO Microgrids: 2006-2010}
\subsection{Research question}
The project NWO Microgrids was funded by the Dutch government based on
the following research question:
\begin{quote} \itshape
Is it possible, through the introduction of simple and explicit concurrency controls, to develop a systematic approach to:
\begin{enumerate}
\item incrementally designing new processor architectures (i.e. based on an existing ISA and infrastructure);
\item dynamically managing and optimising the available resources for a variety of goals such as performance, power and reliability (i.e. resulting in autonomous and self-adaptive microgrids);
\item formally defining the architectures' execution properties;
\item incrementally developing the architectures' infrastructure (i.e. simulators, compilers, binary-to-binary translators and even silicon intellectual property);
\end{enumerate}
all within the context of ten to fifteen years of silicon-technology scaling (i.e. maintaining scalability over a thousand fold increase in chip density)?
\end{quote}
Note that this is a yes/no question.
To answer ``yes,'' it suffices to propose at least one set of ``simple
and explicit concurrency controls'' and a corresponding ``systematic
approach'' that delivers on the four other points. Alternatively,
``yes'' can also be given, perhaps less satisfactorily, if a
theoretical analysis indirectly merely proves the systematic approach
exists (``it is possible to develop it'') without actually developing
it. To answer ``no,'' in contrast, it is necessary to demonstrate
that there cannot exist any set of ``simple and explicit concurrency
controls'' which makes a ``systematic approach'' possible.
Strategically, this question suggests its own answer:
\begin{itemize}
\item a ``no'' answer would be a formidable theoretical endeavour, likely very
difficult to obtain (possibly impossible within the proposed 10-15
years time frame);
\item a ``yes'' answer based on theoretical proof of existence would be equally difficult;
\item therefore, the question suggests a ``yes'' is expected, hinged
on the ability of the researchers to use the features of an existing
architecture as their candidate ``simple and explicit concurrency
controls,'' and consequently show that a corresponding ``systematic
approach'' delivers on the 4 other points.
\end{itemize}
In short, NWO Microgrids's ``declaration of scientific intent'' can be reformulated as follows:
\begin{quote} \itshape
We will show that D-RISC/Microgrids make it possible to develop a systematic approach to:
\begin{enumerate}
\item incrementally designing new processor architectures (i.e. based on an existing ISA and infrastructure);
\item dynamically managing and optimising the available resources for a variety of goals such as performance, power and reliability (i.e. resulting in autonomous and self-adaptive microgrids);
\item formally defining the architectures' execution properties;
\item incrementally developing the architectures' infrastructure (i.e. simulators, compilers, binary-to-binary translators and even silicon intellectual property);
\end{enumerate}
all within the context of ten to fifteen years of silicon-technology scaling.
\end{quote}
\subsection{Outcome vs.~ expectations}
Did NWO Microgrids deliver on its self-set expectations?
\paragraph{Did NWO Microgrids deliver a ``systematic approach'' with the desired properties?} No, the design of D-RISC/Microgrids
was instead carried out in an ad-hoc fashion, with multiple phases of
trial-and-error and backtracking. I highlight an ethical issue here,
because on the one hand the Dutch NWO funded a project on the
assumption that its outcome would be a \emph{systematic approach
(method)} that could be reused with different architectures, and on
the other hand the research team knew well in
advance that systematization would not be studied.
For the sake of deconstruction, let us however stretch the word
``systematic approach'' to encompass ``the process of designing and
building D-RISC/Microgrids.'' Does this extended definition match the
other required properties?
\paragraph{Did the process of designing and building D-RISC/Microgrids incrementally designed a processor architecture, based on an existing ISA and infrastructure?} Here the answer is only partially ``yes'': the design was indeed incremental (starting from a simple, known-to-work RISC pipeline) and used an existing ISA (Alpha), but it did not reuse an existing infrastructure. Instead, all the infrastructure for the project was built from scratch.
\paragraph{Did the process of designing and building D-RISC/Microgrids enabled the dynamic management and optimisation of available resources for a variety of goals such as performance, power and reliability?} The jury is still out on this one; no answer was given yet after many years of research. Even a published doctoral thesis on the topic~\cite{vantol.13}, which merely touched performance-driven resource management, did not yield definite answers. As of this writing, another doctoral candidate is working on the reliability issue, but issues of power are still left untouched.
\paragraph{Did the process of designing and building D-RISC/Microgrids include a formal definition of the architecture's execution properties?} Yes, namely in~\cite{tdvu.07.icfem} and~\cite[Chap.~7]{poss.12}.
\paragraph{Did the process of designing and building D-RISC/Microgrids include an incremental development of the architecture's infrastructure? (i.e. simulators, compilers, binary-to-binary translators and even silicon intellectual property)} Here the answer
is only partially ``yes.'' Simulators and compilers were developed,
and parts of silicon IP (cf.~\cite{danek.12}), however there were no
binary-to-binary translators produced to establish a compatibility
path with existing code, as initially envisioned.
\subsection{Restrospective on the research question}
Since the process of designing D-RISC/Microgrids did not exhibit all
the expected properties, it cannot be used to answer ``yes'' firmly to
that project's research question. However, ``no'' cannot be
confidently given either. In other words, the question is still mostly
left unanswered.
Instead, I can only summarize the situation by proposing that the NWO
simply funded some additional \emph{development} of D-RISC/Microgrids,
and the project's description only merely \emph{guided the development
process} without pressuring it into delivering scientific output.
Moreover, it is clear to me that the original research question
\emph{was so ill-phrased that it cannot be answered scientifically}: I
cannot see any experimental path that would yield a definite answer
within a reasonable time frame.
Consequently, any rephrasing would be equally improductive, for example
determine whether there is any \emph{other} set of concurrency
controls that yield a ``yes'' answer on the same research question, or
whether D-RISC/Microgrids can be fixed/enhanced to this aim.
Therefore, in my opinion, the original phrasing for the project NWO
Microgrids cannot be used to motivate further work in this area. The
corollary is that researchers should not exploit the past attention
given by NWO's to this question as justification to spend more effort
in this area. If justification is needed, it must be found somewhere
else.
\section{C. Jesshope, APC 2008}
This whitepaper/article~\cite{jesshope.08.apc} made a statement of
intent about the applicability and the aims of D-RISC/Microgrids. It
introduces the ``SVP model,'' an intellectual construction used from
2008 to 2011. SVP intended to abstract the specific inner workings of
D-RISC/Microgrids, keeping only the high-level semantics of its TMU
concurrency management protocol visible to programmers.
\begin{table}
\begin{tabular}{p{.48\textwidth}p{.48\textwidth}}
Problem & Proposed answer \\
\hline \hline
How to effectively program distributed multiprocessor systems & Use SVP's simple concurrency control primitives \\
\hline
How to make architectures that are both efficient and can tolerate a large latency in responding to external events &
Use a combination of native support for dataflow scheduling and split-phase transactions throughout the system, such as found
in SVP implementations \\
\hline
How to design programming model that is both deterministic and free from deadlock under concurrent composition & Use SVP's strictly
hierarchical concurrency and forward-only communication patterns \\
\hline
How to ensure binary compatibility across a range of implementations from a single processor to the highest level of concurrency a particular application can support & Use SVP's granularity-independent abstraction of concurrency resources \\
\end{tabular}
\caption{Problems purportedly solved by SVP.}\label{tab:mgprob}
\end{table}
According to this article, D-RISC/Microgrids as abstracted in SVP
should have solved the problems listed in \cref{tab:mgprob}.
Note that this article defined SVP and its benefits \emph{before D-RISC's TMU, and thus Microgrids were fully defined.}
As it happened, the advertised features of SVP
ended up \emph{not being implementable in D-RISC/Microgrids}. Specifically:
\begin{itemize}
\item end-to-end asynchrony stops at the chip boundary, both at
the memory and I/O interfaces, and these latencies cannot be fully
tolerated;
\item the lack of hardware mechanisms to virtualize resources prevented the
proper implementation of deadlock-free composition.
\end{itemize}
Moreover, although binary compatibility is possible across chip
technologies, the execution performance of the code ended up not being
portable between different number of cores and interconnect
topologies.
Since 2011, when D-RISC's TMU was well-enough defined that it was
both obviously \emph{different from and more powerful than SVP's abstractions},
the SVP model has been downplayed and is not a central component of
publications any more.
\section{EU Apple-CORE: 2008-2011}
The project Apple-CORE was funded by the European Union based on a
statement of intent via an \emph{abstract}, and via a list of explicit
\emph{objectives}. There was no ``research question'' per se, as
the goal of the project was to build infrastructure and show
that the objectives were reached as a consequence.
\subsection{Summary of outcomes}
To an outsider,
the EU seemed to have funded research to develop a new general-purpose processor,
that would extend and possibly even replace the technology currently
in use in commodity hardware. Had that objective been reached, the
project would have been disruptive indeed. This \emph{potential} to
both disrupt the state of the art and advance technology in a way
largely beneficial to society was sufficient, to the proposal's
initial reviewers, to justify the investment.
However, there is an ethical issue at hand. First, the Apple-CORE
proposal stated that D-RISC and Microgrids were already
designed and a code generator for $\mu$TC was available prior
to the start of the project, whereas it was known to the authors of the proposal that
this was not true.
Only after the first year, after reviewers had been induced to believe
the issues were minor, did it become clear that the EU was also
funding this prerequisite technology. Also, at that point there was no
evidence that this ``initial'' technology would be sufficient to
research all the project's original objectives in a timely fashion.
Indeed, what happened is that the overhead of producing these
prerequisites prevented the consortium from exploring all the issues.
In short, Apple-CORE \emph{did not actually have the potential to
disrupt the state of the art and advance technology in the way
announced using the budget requested}. Besides, the details of scientific
outcomes (cf.~ next section) do not reveal any strong evidence that the
Apple-CORE technology can replace existing processors (cf.~ also \cref{sec:incompat}).
\paragraph{Disclaimer}
{\itshape The results and circumstances described below have been
brought to the attention of the project's reviewers while
the project was ongoing, and the project was judged successful
by both the reviewers and the project officer \emph{despite} these
issues. As I have learned since then, most
large, publicly-funded projects suffer from more serious issues and
the issues described here pale in comparison.}
\subsection{Outcomes vs.~ project objectives}
I present below how the project's outcome, as can be observed at the
end of 2012, relates to each stated objective in the project's
description of work. I list the objectives in decreasing order of
success, and mark with ``$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$'' those
points that most diverge from the overall initial goal of the project.
\paragraph{Apple-CORE will investigate the support structures in
implementing the SVP model in the LEON 3 processor and develop an SVP soft-core prototype.}
This was achieved, and was quite successful~\cite{danek.10.ddecs,sykora.11.lncs,danek.12}.
\paragraph{Apple-CORE will investigate the integration of instruction-set extensions to support custom accelerators based on both microthreads and families of microthreads.}
This was achieved, and was quite successful~\cite{danek.12}.
\paragraph{Apple-CORE will explore the gains of SVP in the context of data-parallel
programming, investigate the implications of functional concurrency and explore the possible design space.}
This was achieved, and was quite successful~\cite{a-c-d44}.
\paragraph{Apple-CORE will support many-core processors by
capturing concurrency systematically using
instructions in the processors' ISA and by dynamically mapping
and scheduling that concurrency in the processors' implementation (the SVP model).}
This was achieved~\cite{poss.12.dsd}.
\paragraph{Apple-CORE will derive a set of loop transformations to transform iterative computations into a combination of independent and dependent families of threads respecting the communication restrictions in the SVP model.}
This was achieved~\cite{saougkos.09.cpc,saougkos.11}.
\paragraph{Apple-CORE will extract task, loop and, implicitly in the SVP model, instruction level concurrency in the parallelising C compiler.}
This was only partially achieved: only loop and ILP was extracted. Task concurrency wasn't.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will provide binary-code compatibility across generations of multi-cores from few- to many-cores.}
This was achieved, although Apple-CORE also showed that code that performs well on
a small number of cores typically does not scale (performance- and
efficiency-wise) to large number of cores. Conversely, code that runs
with interesting speedups on large number of cores do not run
efficiently on small number of cores. The binary compatibility is thus
merely functional: it is possible to run the same code and obtain the
same results, but the performance is not portable. One can easily
argue that this form of compatibility is not really what was desired.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will implement and evaluate memory models and coherency protocols for many- core systems.}
This was achieved, only to conclude that the proposed models and
protocols were cumbersome to use, inefficient and otherwise
detrimental to performance for any configuration larger than 30-60
cores.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will study the resource management issues that are exposed in exploiting massive concurrency as it arises from data-parallel or functional program specifications.}
This was achieved: the management issues were indeed studied, but only
to conclude that the Apple-CORE strategy did not significantly
simplify the problem, which is otherwise shared by all research projects
in this field.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will provide high-level programming environments that
improve the programming productivity and automate the generation of
concurrency, or at least separate the concerns of concurrent
programming from its implementation, i.e. automate all scheduling and
synchronisation.} This was only partly achieved. By funding extra
development on Single-Assignment C, Apple-CORE did indeed ``improve
the programming productivity and automate the generation of
concurrency.'' However this effort was not directly related to
D-RISC/Microgrids: the improvements on SaC are portable to any
parallel hardware supported by the SaC compiler. Furthermore
Apple-CORE did not ``separate the concerns of concurrent programming
from its implementation,'' and neither did it ``automate all
synchronization.''
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will investigate and implement memory protection and security issues for many- core systems.}
This was investigated but not implemented.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will implement a port of a micro-kernel operating system onto one or more of the processor platforms (emulation and/or soft core).}
This was not investigated and not achieved, cf.~ also \cref{sec:incompat}.
\paragraph{
Apple-CORE will promote the $\mu$TC language as a standard front-end
to the gcc compiler and will use it as a target for all user-level
compiler development.} This did not occur, and $\mu$TC is not being
used any more, for the reasons presented in~\cite[App.~G]{poss.12}. Instead,
the project used another front-end to D-RISC/Microgrids called SL~\cite{poss.12.sl}, which
is riddled with practical limitations and has yet to gain credentials in the scientific community.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will investigate and evaluate programming productivity issues for the tools developed.}
This was not investigated nor evaluated.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Apple-CORE will select a range of benchmark applications of interest to potential users of the SVP model within the European computer industry.}
This was only partly achieved: the industrial participation in the project was low, therefore the relevance of the resulting benchmark selection
cannot be confidently ascertained.
\paragraph{Apple-CORE will build an infrastructure of tools that will enable the SVP model to be evaluated and adopted by the European Computer Industry.}
This objective is untestable. Although an infrastructure of tools was produced, no adoption by the European Computer Industry has yet occurred.
\subsection{Outcomes vs.~ intents in the project's abstract}
The project's abstract also declared research intents not
covered otherwise in the objectives. I review them here:
\paragraph{The benefits are large, [...] as compilers need only capture
concurrency in a virtual way rather than capturing, mapping and
scheduling it.}
These benefits were not observed. Instead, Apple-CORE taught us is not sufficient to capture concurrency;
some semantics that brings an intuition of the machine back to the programmer
was necessary after all.
\paragraph{This separates the
concerns of programming and concurrency engineering and opens the door
for successful parallelising compilers.} There was no breakthrough in
programming and concurrency engineering during Apple-CORE, and no
``successful parallelising compiler'' has been produced as a
result. Technically, there were parallelising compilers produced, but
they are not yet ``successful'' insofar they have not yet gathered any
user base other than their own developers.
\paragraph{Particular benefits can be expected for data-parallel
and functional programming languages as they expose their concurrency
in a way that can be easily captured by a compiler.}
This was indeed shown, although this can be equally shown using
most parallel platforms in this day and age.
\paragraph{Another advantage
of this approach is the binary compatibility the new processor has
with the modified ISA. [...] Once code is compiled with the new
tools, binary-code is executable on an arbitrary numbers of processors
and hence provides future binary-code compatibility.} See above: although
the code is binary-compatible, the performance is not portable.
\paragraph{The concurrency controls also allow for management of
partial failure, which together with the binary-code compatibility
provide the necessary support for reliable systems.} This was
not shown in practice by Apple-CORE, although Apple-CORE's
infrastructure does simplify a research project on this topic,
started later on.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
Finally, this
approach exposes information about the work to be executed on each
processor and how much can be executed at any given time.
This
information can provide powerful mechanisms for the management of
power by load balancing processors based on clock/frequency
scaling. }
This was not researched in Apple-CORE.
\paragraph{
$\filledmedtriangleright\filledmedtriangleright\filledmedtriangleright$
In particular, the binary compatibility provides a unique
opportunity to make an impact on commodity processors in Europe.}
This was not achieved, as there is no binary compatibility with
existing processors. Actually, the argument of ``binary
compatibility'' throughout the project proposal diverges from usual
expectations. ``Binary compatibility'' is usually understood to mean
``backward compatibility with existing binary code,'' meaning that
existing software from other platforms can be reused on the new
platform. However, Apple-CORE instead promotes ``binary compatibility
between multiple instances of the Apple-CORE technology,'' i.e.~ no
binary backward compatibility with other platforms.
\section{C. Jesshope et al., ParCo 2009}
This article~\cite{jesshope.09.parco} rephrased the motivation
behind the D-RISC/Microgrid work, two years in the Apple-CORE project:
\begin{quote}
\itshape In a more general market [than embedded and special-purpose
accelerators], the labour-intensive approach of hand mapping an
application is not feasible, as the effort required is large and
compounded by the many different applications. {\bfseries A more automated
approach from the tool chain is necessary. This investment in the tool
chain, in turn, demands an abstract target to avoid these
compatibility issues. That target or concurrency model then needs to
be implemented on a variety of platforms to give portability, whatever
the granularity of that platform. Our experience suggests that an
abstract target should adopt concurrent rather than sequential
composition, but admit a well-defined sequential schedule. It must
capture locality without specifying explicit communication. Ideally,
it should support asynchrony using data-driven scheduling to allow for
high latency operations. However, above all, it must provide safe
program composition, i.e. guaranteed freedom from deadlock when two
concurrent programs are combined. Our SVP model is designed to meet
all of these requirements.} Whether it is implemented in the ISA of a
conventional core, as described here or encapsulated as a software API
will only effect the parameters described above, which in turn will
determine at what level of granularity one moves from parallel to
sequential execution of the same code.
\end{quote}
A few years afterwards, is the D-RISC/Microgrids management protocol
matching the claims? I review them here in decreasing order of
success.
\paragraph{The model should support asynchrony using data-driven scheduling to
allow for high-latency operations.} This was achieved (primary feature of D-RISC).
\paragraph{The model should adopt concurrent rather than sequential composition.}
This was achieved, although \emph{both} concurrent and sequential composition are equally promoted.
\paragraph{The model must admit a well-defined sequential schedule.}
This was mostly achieved~\cite[Chap.~10]{poss.12}. A sequential schedule is not properly defined
as soon as a program manipulates on-chip resources (in particular cores) explicitly.
\paragraph{The model needs to be implemented on a variety of platforms.} This was not achieved.
A software emulation of D-RISC's \emph{envisioned} TMU was implemented early on~\cite{tol.09.jsa}, but
the actual D-RISC TMU ended up with different semantics which have not yet been implemented elsewhere.
\paragraph{The model must capture locality without specifying explicit communication.}
This was not achieved: explicit communication is required between different threads.
\paragraph{Above all, it must provide safe program composition, i.e. guaranteed freedom from deadlock
when two concurrent programs are combined.}
This was not achieved: the ability of code to manipulate resources explicitly, combined with the lack
of full resource virtualization, may cause compositions to deadlock from resource starvation.
\section{ASCI 5-year research plan, 2010}
This document was submitted to a consortium of Dutch universities
at the start of 2010, to define the overall research plan of the consortium
over the period 2010-2014.
Over D-RISC/Microgrids this report states:
\begin{quote}
\itshape
Our work on fine-grain threaded architectures with data-driven
scheduling using the SVP concurrency model will continue but we will
also explore software implementations of SVP on other emerging
multi-core architectures such as Niagara and Intel's SCC. This will
allow us to explore multi-grain architecture and develop an
infrastructure to support such an approach. One of the major
directions in this work will be the development of a coherent set of
operating system services that support space sharing in these
heterogeneous environments and yet provide a secure operating
environment that can be scaled from chip-level micro-grids to globally
distributed Grids. One of the major challenges, especially in
mainstream computing, will be in making these systems programmable
without specialized concurrency knowledge and we have designed
programming language support to express parallel computations and
systems at a very high level of abstraction and developed compilation
technologies that effectively map the abstract descriptions to
concurrent computing environments. We have international
collaborations developing the functional, data parallel language SAC
(Single Assignment C) and the asynchronous co-ordination language
S-Net. Our long-term vision is in the direction of a compilation
infrastructure that automatically adapts running programs derived from
high-level specifications to a heterogeneous and dynamically varying
execution environment based on continuous reflection of execution
parameters.
\end{quote}
To this date, SVP was not implemented on other architectures. No
operating systems services have yet developed that support space
sharing in heterogeneous environments and provide a secure exeuction
environment. The D-RISC/Microgrids language tools are not yet able to
map the abstraction of concurrent resources to maximize performance
and efficiency. ``Automatic adaption of running programs towards
heterogeneous and dynamically varying execution parameters based
on continuous reflection'' was not achieved either yet.
\begin{summary}
\begin{itemize}
\item
The D-RISC/Microgrids project was purportedly intended to solve major issues in
micro-architecture research, related to scalability in performance and
efficiency in general-purpose microprocessors.
\item
The strategy to solve these issues was to implement a
combination of dataflow scheduling with hardware support for thread
concurrency management within and across cores on chip.
\item
Implementation
was carried out, but the results are inconclusive. On the one hand, the proposed
hardware does indeed provide higher performance and efficiency in
regular, data-parallel workloads, but these are also the ``boring''
applications which benefit equally well from vector units or accelerators
in conventional processors. On the other hand, no evidence has yet
been produced that the proposed hardware benefits larger applications
with more irregular workloads.
\item
Power efficiency and intelligent resource
management was regularly advertised but not actively researched.
\item
Effort has been invested into widening the scope of the technology
towards applications and industrial relevance, but these applications
have not yet materialized.
\end{itemize}
\end{summary}
\section*{Chapter summary}%
}{\end{snugshade}
\end{minipage}
}
\pagestyle{headings}
\begin{document}
\author{Raphael 'kena' Poss\\University of Amsterdam, The Netherlands\\\url{r.poss@uva.nl}}
\title{On whether and how \\ D-RISC and Microgrids can be kept relevant \\
(self-assessment report)}
\maketitle
\input{abstract}
\setcounter{tocdepth}{1}
\tableofcontents
\clearpage
\input{begin}
\input{background}
\part{Deconstruction}\label{part:decons}
\input{outcomes}
\input{methodology}
\input{obstacles}
\part{Reconstruction}
\input{contrib}
\input{parts}
\input{strategy}
\input{conclusion}
\newcommand{\etalchar}[1]{#1}
\addcontentsline{toc}{chapter}{References}
\bibliographystyle{is-alphaurl}
\chapter{Individual architectural features}
Publications, posters and talks usually present the D-RISC core and
Microgrid clusters thereof as a single coherent technology made from
inter-dependent features. In reality, a gradual composition is
possible, as well as adding TMU-like features to other processor cores
than D-RISC.
I shortly present here my understanding of this
composition, which I have started to recognize while
writing~\cite[Chap.~3]{poss.12}.
\section{Overview}
\begin{figure}[b!]
\centering
\includegraphics[width=.6\textwidth]{features2}
\caption{Overview of the characteristic features of D-RISC/Microgrids.}\label{fig:features2}
\end{figure}
To start with, I summarize
the characteristic features in \cref{fig:features2}; in this
diagram I denote with a double edge the features not found
in other processors, and with a striped red edge those
features found in other processors but not on D-RISC/Microgrids. This diagram
exposes the composition of features in the design, as follows:
\begin{itemize}
\item the D-RISC core itself is a composition of the following features:
\begin{itemize}
\item a pretty conventional in-order, single-issue RISC pipeline,
\item a dataflow instruction scheduler that can execution instructions out-of-order while respecting data dependencies,
\item multiple hardware threads (separate program counters and registers),
\item a memory-mapped interface to an I/O subsystem,
\item a custom hardware Thread Management Unit (TMU) that manages logical tasks and maps and schedules them over hardware threads;
\end{itemize}
\item a Microgrid is a cluster of D-RISC cores together with:
\begin{itemize}
\item asynchronous functional units (FUs) between cores for seldom-used instructions;
\item a shared MMU that provides a single virtual address space to all cores;
\item a custom control network-on-chip to coordinate concurrency management between TMUs;
\item optionally, a custom memory interconnect.
\end{itemize}
\end{itemize}
Of these features, we can distinguish those that provide ``added
value'' compared to other processor architectures, namely the TMU,
dataflow scheduler and control NoC, from those that are ``unique'' and
make the design fundamentally incompatible with conventional wisdom, namely
the lack of support for preemption and the lack of per-core (and
per-hardware thread) MMU, which were discussed in \cref{sec:incompat}.
Remarkably, this overview alone reveals that the combination of
features found in D-RISC could be obtained by starting from an existing RISC
core design. The Tera MTA, for example, already features hardware
multithreading, a dataflow scheduler and memory-mapped I/O, and could
be \emph{extended} with a TMU. Similarly, Sun/Oracle's Niagara T4
cores already feature hardware multithreading, memory-mapped I/O and
out-of-order instruction execution (via reservation stations, which is
really a form of dataflow scheduling), and could thus be extended with
a TMU as well.
\section{Details}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{features}
\caption{Overview of the characteristic features of D-RISC/Microgrids (expanded).}\label{fig:features}
\end{figure}
To understand how D-RISC and Microgrids additionally benefit from re-implementing
its own ``version'' of features already found in existing processors,
it is useful to dig one level deeper, as illustrated in
\cref{fig:features}.
(Again, I denote with a double edge the features
not found in other processors, and with a striped red edge those
features found in other processors but not on D-RISC/Microgrids. Features
with dotted borders are experimental, not yet ready for production. The
red triangles highlight the missing features that fundamentally hurt
software reuse.)
Besides highlighting the wealth of features of D-RISC's TMU and the inter-TMU NoC, this diagram draws
attention to the following points.
\subsection{Dataflow scheduling from the register file}
All dataflow schedulers rely on a \emph{matching store}, which retains
information about ``what to do'' when an operation has completed,
while the operation is ongoing. In conventional architectures,
matching stores are implemented:
\begin{itemize}
\item as part of the memory
sub-system, starting at the data cache (e.g. Tera MTA, Niagara T4+), to
determine ``what to do when a memory operation completes,''
\item via reservation stations next to functional units in out-of-order
execution (e.g. PowerPC), to determine ``what to do when a local operation completes.''
\end{itemize}
Like MTA and T4, D-RISC uses the L1 D-cache as matching store; but
contrary to conventional OoOE techniques it uses the main register
file instead of reservation stations for out-of-order execution of
instructions.
This choice \emph{simplifies the circuits} that connects the pipeline,
functional units and the register file together. The trade-off is that
the number of in-flight instructions \emph{per thread} is limited to
the ISA register space, typically 31, whereas it can grow arbitrarily
with reservation stations.
This way, we can recognize that \emph{in the large design domain where
out-of-order instruction execution is desirable, D-RISC is exploring
the sub-domain where per-thread ILP can be traded off with simpler
circuits.}
\subsection{Optimizations to hardware multithreading}
\Cref{fig:features} also highlight two features that optimize
hardware multithreading, regardless of the presence of the TMU
and of a dataflow scheduler.
The first is a tight integration of the schedule queue and the
I-cache. As detailed in~\cite{lankamp.08} and
\cite[Sect.~3.2]{poss.12}, a a thread is not considered by the fetch
stage unless its code is already in the I-cache, so as to prevent
pipeline bubbles due to I-cache misses. This technique decouples code
fetching and thread scheduling and \emph{yields higher utilization of
the pipeline overall}. I do not know whether this feature
is used by other hardware multithreaded core designs, but
I would be surprised if it were not.
The other feature is the ability to \emph{hint} the fetch stage of the
pipeline to switch preemptively to another thread after fetching an
instruction, when that instruction is \emph{likely to cause a pipeline
bubble at a later stage}. This can be used for any instruction that
reads the result of a previously issued long-latency instruction. For
example, in ``ld r1 $\leftarrow$ [x]; add r2 $\leftarrow$ r1+r1'' the ``add''
instruction would be annotated, so that the fetch stage preemptively
switches to another thread: if the ``add'' suspends due to a missing
input (e.g.~ due to a cache miss on the previous ``ld'' instruction), no
other instruction from the same thread needs to be flushed from the
pipeline. This feature also \emph{yields higher utilization of the
pipeline overall}, to my knowledge is not present in other
processors, but could be possibly be added to them.
To this date, switch hints are implemented in D-RISC by
\emph{interleaving} hint bits with the instruction stream, which in
turns requires a custom assembler to generate the binary program
code. However, other implementations are possible, cf.~
\cite[Sect.~4.4]{poss.12}.
\subsection{Bulk coherency in the shared memory network}
As soon as the designer of a chip architecture combines multiple cores
with memory-level parallelism and still wishes
to expose a ``shared memory'' interface to programmers,
a \emph{coherency protocol} must be designed to ensure
that the memory updates performed by one core on one part
of the memory become eventually visible to other cores
connected to other parts of the memory.
(Note that this problem disappears if all cores are physically
connected to the same memory or shared cache with a bus.)
\emph{Necessarily}, a coherency protocol implies communication across
the memory network to propagate the data ``placed into the memory'' by
store requests, to the point where it may be needed by subsequent load
requests. Also, since the memory network cannot ``predict the
future,'' it must make a decision to effect this propagation
preemptively at some granularity: either after each individual store,
or while evicting cache lines, or by allowing the program code to
``grab exclusivity'' for a range of addresses over the entire system (write-invalidate).
In the Microgrid design, knowledge from the TMU about the program's
structure is used to implement an optimization to coherency: when a
bulk of tasks are created together (a feature of the TMU), the TMU
informs the cache network that it can wait to propagate the stores performed by
that bulk until all tasks in it have terminated, or until when a task creates
a sub-task. This ``makes sense,'' i.e.~ it is valid, because D-RISC's programming
model specifies that stores are only visible to other tasks after a
task terminates or to the sub-tasks it subsequently creates. The
benefit of this optimization is a \emph{reduced number of
coherency-related communication events in the memory network} for some workloads.
To my knowledge, this optimization opportunity is also exploited in
SIMD/SPMD accelerators, and in general it could be readily
considered in any design where high-level information about the
clustering of software operations is visible to the hardware.
\section{TMU reusability}
The purpose and consequence of decomposing the D-RISC design is to
isolate its TMU and recognize that the TMU is really a hardware
accelerator for system management functions that would be otherwise
realized in software. It is actually possible to describe the TMU as
an extension to any generic RISC core, even a core that does not
offer the other features of D-RISC:
\begin{itemize}
\item without hardware multithreading, the TMU would be constrained to
schedule logical tasks over a single hardware thread. This would
restrict the amount of instruction-level parallelism (because the
maximum number of in-flight instructions is restricted by ISA
register window), but would still save up the cost of branches and
increments to implement repetition, and thus accelerate loops;
\item without out-of-order execution (either dataflow scheduling or
via other means), instructions that control the TMU would cause the
processor to wait until the TMU operation has completed. This may
imply large waiting times for ``complex'' operations, for example
allocating a group of cores on another part of the chip. It would
also mandate the use of interrupts to signal asynchronous
completion, for example the termination of a task, but would still
save up the overhead of doing the thread management entirely in a
software operating system.
\end{itemize}
\begin{summary}
\begin{itemize}
\item The D-RISC core combines features found in other processors, such as a RISC pipeline
and hardware multithreading, with custom features (e.g.~ its TMU) and optimizations
to the conventional features (e.g.~ switch annotations for the HMT scheduler).
\item Some architectural optimizations found in D-RISC/Microgrids could be reused
with other processors, for example switch annotations and bulk coherency in the memory network.
\item The key feature of D-RISC/Microgrids, namely its TMU and inter-TMU control NoC,
does not depend on the other features specific to D-RISC and could be potentially reused with other processors.
\end{itemize}
\end{summary}
\chapter{Follow-up strategies}
\Cref{part:decons} has highlighted shortcomings in the methodology and
obstacles to further progress. This analysis raised the question of
how to move forward from there \emph{differently}, so as to
avoid these shortcomings and obstacles. This chapter presents my view on this question, articulated in two directions: first,
what would constitute ``sane approaches'' for new projects (\cref{sec:new}); then
how to ``fix'' or ``improve'' ongoing projects/research (\cref{sec:ongoing}).
\section{Possible strategies for new investments}\label{sec:new}
\paragraph{Exploit.}
Apply the technology produced
so far to other uses than research.
A successful application so far has been education: with only
a minor but regular maintenance effort, the simulation tools
can provide support in architecture and compiler courses
for the coming 5-10 years.
However, given the processor is unable to support any C code that requires a ``hosted''
environment (or other
languages whose RTS is written using hosted C), applications in industry will be limited to small
embedded systems.
Possibly, with only a minor effort investment, an ad-hoc form of preemption
and per-core MMU can be added to a simulation model and obtain a
limited compatibility with software frameworks. Without
significant research, this would yield
sub-efficient (non-competitive) performance, but the gained
compatibility might be sufficient to activate further
external interest in the work.
\paragraph{Salvage and open.}
Extract individual features from the D-RISC/Microgrids
design and evaluate them as extensions
of existing processors.
Small ``first steps'' in this direction can be made
by starting with the switch annotations and the coupling
of the fetch stage with the I-cache. These features
seem readily applicable to the Niagara architecture
and the latest ARM multithreaded cores.
A more significant project would be to extract
the TMU and offer it as a reusable accelerator
component where the processor designer
can choose which TMU feature are activated. For example,
the features related to bulk creation/synchronization
or multi-core resource management may not be always relevant,
and a designer should not need to pay the price of
their integration if they end up not being used.
Conversely, the D-RISC core stripped of hardware multithreading and
its TMU could be offered as a SoC building block, marketing its dataflow
scheduler as a lightweight implementation of out-of-order execution. For
this block to be moderately competitive, a branch predictor
may be proposed as an option.
\paragraph{Distill and reincarnate.}
From the perspective of theoretical computer science, the
D-RISC/Microgrids enterprise has raised two questions that may warrant
a wealth of further fundamental research.
The first was opened on purpose: \emph{how does the cost intuition of
programmers evolve when complex operating system services are
available at nearly the cost of basic arithmetic?} This is one of the key
questions that the TMU was designed to answer. The \emph{desired} answer was
originally: ``once programmers are comfortable about the costs of
concurrency, they would use concurrency everywhere and obtain parallel
speedups at every level.'' This particular answer was not obtained by
the research so far, but it may well be that other interesting answers can be
obtained instead.
Further effort in this direction could be bootstrapped as follows.
First, get acquainted with a software community already comfortable
using concurrency without too much assumptions about hardware.
Haskell and Erlang programmers are interesting candidates. Then,
observe and inventory which specific patterns of concurrency they
already use, and those they are striving to implement. Then re-design
a custom TMU that accelerates their favorite language run-time
system. Then, demonstrate the net effect on existing programs, and
document how the programmers modify their software over time to take
advantage of this accelerator.
I discovered the other fundamental question while dismantling
the ``SVP model'' proposed in~\cite{jesshope.08.apc} and used
subsequently in the period 2008-2011. SVP has captured, using its
``places,'' the notion that a group of processors should be considered
as a \emph{single}, \emph{fungible resource} that can be allocated
dynamically from the computing environment and sub-partitioned
dynamically using abstract operators such as those implemented by
D-RISC's TMU. The designers of SVP then claimed that ``places are the
fundamental currency of computing'' and that their abstract operators
were ``general-purpose,'' i.e.~ sufficiently general to carry out any computation.
As I discuss in~\cite[Chap.~9\&12]{poss.12}, I believe this particular claim is invalid,
because a ``general'' model of computing resource should offer and
define memory and means for I/O as well, which SVP
places do not. However, studying SVP raises the complement question:
is it possible to \emph{extend a general model of computation with a
cost model that uses entire virtual parallel computers with multiple
cores and multiple memories as the basic resource unit}? A strategy
for exploring this question would
probably benefit from starting with an inherently concurrent model, which
Turing and queue machines are not. The Actor model~\cite{agha.85} and
Milner's $\pi$-calculus~\cite{milner.92,milner.92.2} may be more suitable candidates, as their intuitive
implementations have well-understood operational semantics already.
Further effort in this direction could be bootstrapped as follows. First,
select a technology which already uses virtualizations of entire
parallel resources as a basic building block. Modern Unix systems
and VM hypervisors are candidates. Then formalize its
basic concurrency operations (e.g.~ fork, wait in Unix) in the conceptual
terminology of a general model. Then, based on this formalization and
expert knowledge of the actual behavior of the technology on parallel hardware,
design a cost algebra that is reasonably predictive. Then implement
a framework that visualizes and predicts cost for existing applications
using that technology. Use the interest gained in this way to attract
funding on the fundamental question.
\section{Possible strategies for ongoing projects}\label{sec:ongoing}
\subsection{Partnership with industry: 150k€ at stake}
An industry partner has recently funded some initial research effort to add
priority scheduling to D-RISC's thread scheduler and to explore fault
detection and recovery. Initial results suggest an opportunity to fund
further development effort in that direction, with the understanding
that the partner can use the benefits of the technology in their embedded
aeronautics controllers, which already use space-hardened custom SPARC
cores, in a 1-core or 2-core configuration.
Here the two strategies ``exploit'' and ``salvage and open'' described
above are applicable.
For the ``exploit'' strategy, the partner would need to fund
simultaneously a rewrite of the D-RISC specification in a language
suitable for both simulation and synthesis, so as to avoid maintaining
two source bases over time, and an extension of the current D-RISC
design to support preemption and resource reclamation, as much as
required by the partner's software.
For the ``salvage and open'' strategy, the partner could simply fund
a rewrite of D-RISC's HMT scheduler and the subset
of D-RISC's TMU that is sufficient for the partner's software as an extension
of the partner's favorite/desired existing processor core.
\subsection{Ongoing PhD theses: 400k€ at stake}
Both my peers who already defended a doctoral thesis founded on D-RISC and
Microgrids~\cite{bernard.10,vantol.13}, and myself, have been assailed during
our defenses with variation of the following:
\begin{itemize}
\item ``why did you choose this platform?''
\item ``what makes this platform especially attractive?''
\item ``why is your evaluation by software applications so poor?''
\end{itemize}
As answers, all three of us formulated variations of ``I was told this
platform was general enough and/or had great potential when I started,
and only later I recognized some of the obstacles, but I did my
part nonetheless. And look, by the way, I found some nice answers to
side research questions of my own, not initially phrased in the
D-RISC/Microgrids enterprise!''
Meanwhile, our unspoken thought was: ``I trusted my supervisor this
was the right place to start my PhD study and obtain the scientific
merit needed to graduate successfully, and as a beginner scientist I
did not have yet the critical acuity to recognize our shared
methodological shortcomings. But everyone can make mistakes, and
should be forgiven for them. After all, my PhD defense committee finds
me worthy of a doctorate, so it couldn't be as bad as it looks.''
In principle, the currently ongoing PhD research projects could
be concluded on the same note, and numerous new projects started
with the understanding they will conclude similarly.
In practice however, as I am sitting next to them and entertain close
social contact, I feel dishonest letting my peers employ this strategy: given
I now understand the shortcomings, is it fair to let my peers struggle
with the large friction to academic publication and peer acceptance
caused by our communal continued use of a flawed approach? The risk
is great also that they recognize this friction but feel the obstacle
is insurmountable, or worse, that this realization engenders distrust
against the potentials of further research in the area.
Here, unfortunately, I do not have the experience sufficient to
guarantee better outcomes with an alternate strategy with any
confidence. The essence of any sane approach, to me, would be to
retrospectively \emph{reverse-engineer properly formulated research
questions} that happen to be suitably answered by the work effectively performed,
independently from the initial initiative. This question should then
be phrased as generally as possible so that it does not hinge on the
specifics of D-RISC/Microgrids. Only in a second phase, subsequently
propose the current implementation of D-RISC/Microgrids as a case
study. To illustrate, I list some possible rephrasings in
\cref{tab:foo}. In nearly all cases, I think it would be useful to
acknowledge early on that the restrictions described in
\cref{sec:incompat} are arbitrary, and seek actively means to overcome
them to gain access to more software benchmarks. This may even imply partial uses of the ``salvage and open''
strategy described earlier.
\begin{table}
\begin{tabular}{p{.35\textwidth}cp{.55\textwidth}}
Initial impulse & & Reverse-engineered research questions \\
\hline
how to build a D-RISC TMU? & $\rightarrow$ & what are the costs/benefits of
accelerating OS functions for thread management with a hardware unit? \\
& $\rightarrow$ & what insights about how the hw/sw interface
influences programming language semantics, are gained while building a TMU? \\
how to build a D-RISC/Microgrids simulator? & $\rightarrow$ & what simulation
framework would be suitable for research in micro-architecture design while
keeping simulation performance high enough to run significant multi-core workloads? \\
& $\rightarrow$ & to which level of accuracy can a model in this framework simulate
the behavior of a hardware implementation? \\
how to improve D-RISC's memory performance? & $\rightarrow$ & what are the costs/benefits
of modifying memory interfaces and protocols to increase the latency tolerance abilities of cores that use HMT and/or dataflow schedulers? \\
& $\rightarrow$ & what are the quantitative benefits of exploiting the concurrency awareness available
in hardware in memory protocols? \\
how to implement priority scheduling in D-RISC? & $\rightarrow$ & what are the cost/benefits
of extending a HMT scheduler with priorities? \\
how to implement fault tolerance in D-RISC/Microgrids? & $\rightarrow$ & what are the costs/benefits
of exploiting the concurrency awareness available in hardware in fault tolerance protocols? \\
& $\rightarrow$ & is it possible to abstract fault tolerance to a general computing model equipped
with a resource/cost model?
\end{tabular}
\caption{Example reverse-engineering of research questions.}\label{tab:foo}
\end{table}
\begin{summary}
\begin{itemize}
\item I can see three follow-up strategies for new investments around
D-RISC/Microgrids: exploitation, i.e.~ apply the technology produced
so far to other uses than research; salvaging and opening the
technology, i.e.~ extracting individual features from the
D-RISC/Microgrids design and evaluating them as extensions of
existing processors; and distillation of the main ideas in the realm
of fundamental computer science.
\item Ongoing research towards doctoral theses should be careful
to rephrase research questions in the light of our recent shared
understanding of the project's issues.
\end{itemize}
\end{summary}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 303 |
Gerhard Bauch from the Hamburg University of Technology was named Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 2015 "for contributions to iterative processing in multiple-input multiple-output systems".
References
Fellow Members of the IEEE
Living people
Year of birth missing (living people)
Academic staff of the Hamburg University of Technology
Place of birth missing (living people) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,205 |
Das Zisterzienserinnenkloster Calatravas ist seit 1219 ein Kloster der Zisterzienserinnen zuerst bei Amaya, ab 1568 in Burgos und seit 1980 in San Cristóbal, einem Vorort von Burgos, in Spanien.
Geschichte
Der zisterziensische Orden von Calatrava gründete 1219 bei Amaya (heute: Barrio de San Felices, Gemeinde Sotresgudo), 45 Kilometer nordwestlich Burgos, für die weiblichen Familienmitglieder das Kloster San Felices de Amaya ("Sankt Felix") oder de Barrios. 1568 wechselte der Konvent in die Stadt Burgos und 1980 in den nordöstlichen Vorort San Cristóbal. Er gehört zur Zisterzienserinnenkongregation San Bernardo (C.C.S.B.). In Barrio de San Felices steht noch ein Nachfolgebau am Ort. In Burgos erinnert der Straßenname Calle Calatravas an das einstige Kloster. In San Cristóbal steht das Kloster Monasterio San Felices de Calatrava (auch: Convento de las Madres Calatravas) in der Straße Barrio San Cristóbal 16.
Literatur
Carlos de Ayala Martínez: San Felices de Amaya, monasterio medieval de la orden de Calatrava. In: Medioevo Hispano. Estudios in memoriam del profesor Derek W. Lomax. Madrid 1995, S. 17–34.
Rafael Sánchez Domingo: Las monjas de la orden militar de Calatrava. Monasterio de San Felices (Burgos) y de la Concepción (Moralzarzal-Madrid). La Olmeda, Burgos 1997.
Ghislain Baury: Les ordres militaires hispaniques et l'économie cistercienne. Le temporel des sœurs de Calatrava (XIIIe-XVe siècles)
Bernard Peugniez: Le Guide Routier de l'Europe Cistercienne. Editions du Signe, Straßburg 2012, S. 812.
Weblinks
Kurzpräsentation, spanisch, mit Bild des heutigen Klosterbaus
Beitrag zur Klostergeschichte, spanisch, bebildert
Ehemaliger Klosterbau in Barrio de San Felices, Foto
Gemeindeseite Sotresgudo zum Nachfolgebau des Klosters in Barrio de San Felices
Zisterzienserinnenkloster in Spanien
Klosterbau in Spanien
Provinz Burgos
Zisterzienserinnenkloster Calatravas
Kloster (13. Jahrhundert) | {
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} | 2,036 |
\section{Introduction}
Black hole accretion discs are believed to be the mechanism of energy release
in active galactic nuclei and in some X-ray binaries.
Their luminosity is constrained by the Eddington limit,
$L_E=4\pi GMm_pc/\sigma_T$ ($M$ is the black hole mass), and
the accretion rate, $\dot{M}$, is usually expressed in the Eddington units,
$\dot{m}=\dot{M} c^2/L_E$. The luminosity becomes equal to the Eddington limit
at $\dot{m}_{cr}=\eta^{-1}$, where $\eta$ is the radiative efficiency of the disc.
Accretion rates of order $\dot{m}_{cr}$ are likely to feed the huge energy
release in quasars, and $\dot{m}\sim \dot{m}_{cr}$ is implied in many models
of the soft X-ray excess observed in quasar spectra
(e.g. Czerny \& Elvis 1987; D\"{o}rrer et al. 1996; Szuszkiewicz, Malkan
\& Abramowicz 1996).
According to the standard model (Shakura 1972; Shakura \& Sunyaev 1973),
the effective temperature of the disc surface scales as $T^s_{eff}\propto
(\dot{m}/M)^{1/4}$. A special feature of discs with $\dot{m}\sim\dot{m}_{cr}$
is that their temperature can be much greater than $T_{eff}^s$ provided
the viscosity parameter $\alpha$ is large enough (Shakura \& Sunyaev 1973).
In the inner region of such a disc, the inflow time-scale is shorter than
the time-scale for relaxation to thermodynamic equilibrium and the accretion
flow is overheated. The overheating might be strong
enough for a transition from the standard radiation
dominated disc to the hot two-temperature ion-pressure supported regime of
accretion proposed by Shapiro, Lightman \& Eardley (1976). This transition
has been discussed in recent works (Liang \& Wandel 1991; Artemova et al.
1996; Bj\"{o}rnsson et al. 1996).
The standard model can be a good approximation only if
$\dot{m} < \dot{m}_{cr}$. When the accretion rate approaches
$\dot{m}_{cr}$ the main assumption of the model, that
the viscously released energy is radiated away locally, becomes unadequate.
Inside some radius $r_t$, the produced radiation
is trapped by the flow and advected eventually to the black hole instead
of being radiated away. This radius can be estimated by taking the internal
energy density, $\varepsilon$, and the density, $\rho$, of the disc from
Shakura \& Sunyaev (1973). Then
one finds that the radial flux of internal energy $=\dot{M}\varepsilon/\rho$
exceeds the total flux of radiation emitted between $r$ and $2r$ at
\begin{equation}
r_t\approx \dot{m} r_g \sqrt{1-\left(\frac{3r_g}{r_t}\right)^{1/2}},
\end{equation}
where $r_g=2GM/c^2$ is the gravitational radius.
For $\dot{m}=\dot{m}_{cr}=12$ equation (1)
yields $r_t\approx 7.1 r_g$. A substantial portion of the
released energy must be swallowed by the black hole even when $\dot{m}<\dot{m}_{cr}$.
On the other hand, the relative height of the Shakura-Sunyaev disc, $H/r$,
equals $\dot{m}/27$ at the maximum.
So, advection becomes essential before the accretion flow
becomes quasi-spherical, and this effect can be
investigated in an extended version of the standard model retaining the
vertically integrated approximation. The corresponding set of equations was
proposed by Paczy{\'n}ski \& Bisnovatyi-Kogan (1981).
Numerical solution of these equations gives adequate description for
the inner transonic edge of the accretion flow and shows that the disc
remains relatively thin at moderately super-Eddington accretion rates
(Abramowicz et al. 1988; Chen \& Taam 1993).
This made natural the extension of the standard model to the super-Eddington
advection dominated regime, called "slim" accretion disc.
All the models of the super-Eddington slim disc employed the
pseudo-Newtonian approximation to the black
hole gravitational field (Paczy\'{n}ski \& Wiita 1980) which is good only in
the case of Schwarzschild black hole.
Recently, the equations of relativistic slim disc has been derived by Lasota
(1994, see also Abramowicz et al. 1996) and applied to another type of
advection dominated accretion flow, optically thin and hot.
In the present paper, we solve the relativistic equations of the
super-Eddington disc. In the case of a rapidly rotating black
hole relativistic effects become especially important. The sonic radius
is typically inside the ergosphere, close to the black hole horizon.
The released power steeply increases when the hole spin
approaches its extreme value, $a_*=Jc/M^2G\rightarrow 1$ ($J$ is the
angular momentum of the black hole). In this paper, we calculate the disc
structure for the case $a_*=0.998$ and compare it with the case of
non-rotating black hole $a_*=0$.
We pay particular attention to the case of a large viscosity parameter,
$\alpha$. In the previously investigated pseudo-Newtonian models, the
super-Eddington disc was assumed to be in thermodynamic equilibrium which
is a good approximation in the case of small $\alpha$. We find that a
significant deviation from the equilibrium occurs when $\alpha\mathrel{\copy\simgreatbox} 0.03$.
Then the flow overheats in the central region. We find that the temperature
of the flow steeply rises at $\dot{m}>\dot{m}_{cr}$ and reaches a maximum in the
advection dominated regime of accretion. The maximum
temperature is especially high in the case of a rapidly rotating black hole.
In next Section, we write down the equations of the relativistic advective
disc. The equations describe a radiation dominated flow and
neglect the thermal pressure of the accreting plasma.
Their solution yields the radiation density, plasma
density, and height of the disc within the equipartition radius of
the standard model, $r_{eq}$, at which the radiation pressure equals the
plasma pressure. The transition from the standard model
to the advective accretion occurs at $r_t\ll r_{eq}$.
We describe the numerical method, present the solution,
and discuss the structure of the disc in Section 3.
Then, in Section 4, we calculate the temperature of the accreting plasma.
Finally, we check that inside $r_{eq}$ our radiation-dominated models never
approach the ion-pressure supported regime of accretion.
The results are summarised in Section 5.
\section{Relativistic disc equations}
In the description of the gas flow around a Kerr black hole
we use the Boyer-Lindquist coordinates $x^i = (t, r, \theta, \varphi)$.
The metric tensor $g_{ij}$ of the Kerr space-time is given,
e.g., by Misner, Thorne \& Wheeler (1973).
The disc is assumed to lie at the equatorial plane of the Kerr geometry.
The four-velocity of the accreting gas has components
$u^i =(u^t, u^r, u^{\theta}, u^{\varphi})$
in the Boyer-Lindquist coordinates, and $u^\theta=0$.
The angular velocity of the gas rotation equals $\Omega = u^\varphi/u^t$
and Lorentz factor measured in the frame of local observers with zero
angular momentum equals $\gamma = u^t(-g^{tt})^{-1/2}$.
To obtain a simple model of the relativistic disc with advection we
employ the vertically integrated (slim) approximation.
The slim disc equations are written for the following
thermodynamic quantities:
surface rest mass density $\Sigma$, surface energy density $U$
(which includes both the rest mass energy $\Sigma c^2$ and the internal energy
$\Pi$), and the vertically integrated pressure $P$.
The dimensionless specific enthalpy is defined as $\mu=(U+P)/\Sigma c^2$.
The equation of state in the radiation dominated disc is $\Pi\approx 3P$.
$F^+$ denotes the surface rate of viscous heating, and
$F^-$ denotes the radiation flux radiated from both faces of the disc.
All the thermodynamic
quantities and both fluxes $F^-, F^+$ are measured in the comoving frame.
The main equations express
the conservation laws for barion number, energy, angular momentum,
and radial momentum. We include in the equations the inertial mass
associated with internal energy accumulated in the flow
(Beloborodov, Abramowicz \& Novikov 1997).
\medskip
\noindent
{\it i) Barion conservation}
\begin{equation}
2\pi r c u^r \Sigma =-\dot{M}.
\end{equation}
\noindent
{\it ii) Conservation of angular momentum}
\begin{equation}
\frac{d}{dr}\left[\mu\left(\frac{\dot{M}u_\varphi}{2\pi}
+2\nu\Sigma r \sigma_{~\varphi}^r\right)\right]=\frac{F^-}{c^2}\;ru_\varphi,
\end{equation}
where $\nu$ is the kinematic viscosity, and $\sigma_{~\varphi}^r$ is the
shear,
\begin{eqnarray}
\sigma_{~\varphi}^r= \frac{1}{2}\;g^{rr}g_{\varphi\varphi}\sqrt{-g^{tt}}
\gamma^3\frac{d\Omega}{dr}. \nonumber
\end{eqnarray}
\noindent
{\it iii) First law of thermodynamics}
\begin{equation}
F^+-F^-=cu^r\left(\frac{d\Pi}{dr}
-\xi \frac{\Pi+P}{\Sigma}\frac{d\Sigma}{dr}\right),
\end{equation}
where $\xi\approx 1$ is a numerical factor accounting for non-homogeneity
of the disc in the vertical direction, hereafter we set $\xi=1$.
The surface heating rate is given by
\begin{eqnarray}
\nonumber
F^+=2\nu\Sigma \mu\; \sigma^2 c^2, \qquad
\sigma^2=\frac{1}{2} g^{rr}g_{\varphi\varphi}\left(-g^{tt}\right)
\gamma^4\left(\frac{d\Omega}{dr}\right)^2.
\end{eqnarray}
\noindent
{\it iv) Radial Euler equation}
\begin{eqnarray}
\nonumber
\frac{1}{2}\;\frac{d}{dr}\left(u_ru^r\right)=
-\frac{1}{2}\;\frac{\partial g_{\varphi\varphi}}{\partial r}
g^{tt}\gamma^2
\left(\Omega-\Omega_K^+\right)\left(\Omega-\Omega_K^-\right)\\
-\frac{1}{c^2\Sigma \mu}\frac{dP}{dr}-\frac{F^+u_r}{c^3\Sigma \mu},
\end{eqnarray}
where $\Omega_K^\pm$ are the Keplerian angular velocities,
\begin{eqnarray}
\nonumber
\Omega_K^\pm=\pm\frac{c}{r(2r/r_g)^{1/2}\pm a}.
\end{eqnarray}
The set of disc structure equations becomes closed when the
viscosity, $\nu$, and radiative cooling, $F^-$, are specified. The standard
$\alpha$-prescription for viscosity is $\nu=\alpha c_sH$, where $\alpha$ is
a constant, $c_s=c(P/U)^{1/2}$ is the isothermal sound speed.
The half-thickness of the disc, $H$, should be
estimated from the vertical balance condition.
\medskip
\noindent
{\it v) Vertical balance}
Near the black hole, relativistic effects become important and the
tidal force contracting the disc in vertical direction depends on $\Omega$.
At $\Omega=\Omega^+_K$, the vertical tidal acceleration in the comoving
tetrade equals (e.g. Riffert \& Herold 1995)
\begin{equation}
a_{(\theta)}=\frac{zr_g(r^2-a_*r_g\sqrt{2r_gr}+0.75a_*^2r_g^2)}
{r^3(2r^2-3r_gr+a_*r_g\sqrt{2r_gr})}=\frac{zr_g}{2r^3}\;J(a_*,r),
\end{equation}
where $z=\sqrt{g_{\theta\theta}}\;(\theta-\pi/2)$ is the height in the disc,
$J$ is a relativistic correction factor becoming unity at $r\gg r_g$.
Then the typical half-thickness of the disc can be estimated as
\begin{equation}
H^2=\frac{P}{U}\;\frac{2r^3}{r_gJ}.
\end{equation}
More accurate expression accounts for a deviation $\Delta\Omega=
\Omega-\Omega_K^+$ of the gas rotation from Keplerian (Abramowicz, Lanza \&
Percival 1997). We will consider only the disc region outside the sonic
radius where the correction to estimation (7) connected with $\Delta\Omega$
does not exceed several percent, and hereafter we use this estimation.
\medskip
\noindent
{\it vi) The radiative losses}
The time-scale for photon diffusion from the
interior of the disc to its surface equals $t_D=H\tau_0/c$ where
$\tau_0=\sigma_T\Sigma/2m_p$.
This is a typical leaking time for the radiation trapped inside the disc,
and the radiative losses can be written as
\begin{equation}
F^-=\chi\; \frac{m_p c\;\Pi}{\sigma_T\Sigma H},
\end{equation}
where $\chi\sim 1$ is a numerical factor. In the standard
model with the vertical structure governed by the radiation diffusion
this factor equals $2/\sqrt{3}$.
In the advection dominated region, $t_D$ exceeds the inflow time-scale $t_a$
and a detailed treatment of the stationary diffusion is not relevant:
$t_a<t_D$ means that the gas accretes faster than a stationary
vertical distribution of the trapped radiation could establish. However,
the estimation (8) gives the wright limit $F^-\ll F^+$ at $r\ll r_t$.
Indeed, when $t_a\ll t_D$
we have $\Pi\approx F^+t_a$, hence $F^-/F^+\approx t_a/t_D\ll 1$.
The radiative losses are "turned off" inside $r_t$, and
the exact value of $F^-$ is not important
for hydrodynamical behaviour of the flow. A detailed
calculation of $F^-$ would require 2D simulation of the radiation
diffusion in the disc with a specified vertical distribution of the viscous
energy release.
For definiteness we hereafter choose $\chi=2/\sqrt{3}$ in equation (8).
\medskip
\noindent
{\it vii) Global energy conservation}
The luminosity of the disc is related to the accretion rate by (Beloborodov
et al. 1997)
\begin{equation}
L^-=-\frac{2\pi}{c}\int\limits_{r_s}^\infty u_t F^- rdr\approx\dot{M}c^2
\left(1+\mu_{s}\frac{u_t^{s}}{c}\right),
\end{equation}
where index "s" refers to the inner transonic edge of the disc.
The advection effect results in that the luminosity is less than the total
power released in the disc,
\begin{eqnarray}
\nonumber
L^+=-\frac{2\pi}{c}\int\limits_{r_s}^\infty u_t F^+ rdr.
\end{eqnarray}
The power "swallowed" by the black hole equals $L_{adv}=L^+-L^-$.
\section{Numerical solution}
To solve numerically the disc structure equations, we choose three
independent variables $\Omega,c_s,$ and
$\zeta=\mu(2\nu\Sigma r\sigma^r_{~\varphi}+\dot{M}u_\varphi/2\pi)$.
From equations (3),(4) we have
\begin{equation}
\frac{d\zeta}{dr}=\frac{F^-}{c^2}\;ru_\varphi,
\end{equation}
\begin{equation}
\frac{P}{\Sigma^2}\frac{d\Sigma}{dr}-
3\frac{d}{dr}\left(\frac{P}{\Sigma}\right)=\frac{2\pi rc}{\dot{M}}
\left(F^+-F^-\right).
\end{equation}
Expressing $u_\varphi,F^\pm,P,\Sigma$ in terms of $\Omega,c_s,\zeta$ we
get two differential equations for the three unknowns. The third differential
equation is the radial equation (5).
We solve the equations (5,10,11) with external boundary conditions
${c_s}^{out},\Omega^{out},\zeta^{out}$ at a radius $r_{out}$ such that
$r_t\ll r_{out}\ll r_{eq}$. The values of $c_s^{out}$ and
$\Omega^{out}=\Omega_K^+$ are taken from the relativistic version of
the standard model (Novikov \& Thorne 1973; Page \& Thorne 1974) with
the corrected vertical balance (Riffert \& Herold 1995).
The value of $\zeta^{out}$ is the eigen value of the problem, see below.
We use the relaxation technique.
As first approximation we assume $\Omega(r)=\Omega^+_K(r)$ and
solve equations (10,11) for $c_s,\zeta$. Substituting the obtained solution to
the radial equation (5), we calculate $\Omega^\prime(x)$ needed to fulfill
this equation. Then we perform the relaxation step changing $\Omega(r)$ to
$\Omega(r)+\epsilon\delta\Omega(r)$ where $\delta\Omega=\Omega^\prime-\Omega$
and $\epsilon<1$ is the relaxation parameter. With new $\Omega(r)$ we solve
again equations (10,11), and so on until $\delta\Omega$ converges
to zero at all $r$. This method is numerically unstable unless some smoothing
procedure is used to suppress numerical oscillations in $\delta\Omega(r)$.
We have found the needed procedure as a combination of two-point and
three-point smoothing, and determined $\Delta\Omega=\Omega-\Omega_K^+$ with
an accuracy of $\sim 0.1$ \%.
This technique allows to obtain a solution of equations (5,10,11) for
any given $\zeta^{out}$. Then we adjust $\zeta^{out}$ to fulfill the
regularity condition at the sonic radius, $r_s$, and find the transonic
solution. The regularity
condition can be written as $N=D=0$ where $N$ and $D$ are the numerator
and denumerator in the radial equation with explicitly expressed derivative
of velocity, $du^r/dr=N/D$. In this way, we find $r_s$ with an accuracy of
$\sim 1$ \%
and calculate the disc structure at $r>r_s$ (inside $r_s$ the gas is almost
in free fall).
\begin{figure*}
\begin{center}
\epsfxsize=17.5cm
\epsfbox{figure1sis.ps}
\end{center}
\label{fig:dc}
\caption{The structure of the disc around a black hole with
$a_*=0$ (a--c) and $a_*=0.998$ (d--f) in the case of viscosity parameter
$\alpha=0.1$. The upper panels show the relative height of the disc,
the panels in the middle show the Thomson optical depth of the flow,
and the bottom panels show the deviation from Keplerian rotation.
All these quantities are independent of the black hole mass.
Curves with different numbers correspond to different accretion rates:
1 -- $\dot{m}=\dot{m}_{cr}$ ($\dot{m}_{cr}=17.5$ in the case of $a_*=0$
and $\dot{m}_{cr}=3.11$ in the case of $a_*=0.998$),
2 -- $\dot{m}=100$, 3 -- $\dot{m}=1000$.
The dashed lines display the relativistic standard model.
}
\end{figure*}
Example solutions outside $r_s$ are shown in Fig.~1 for the cases of $a_*=0$
and $a_*=0.998$ (the represented magnitudes are independent of the black
hole mass). In these models $\alpha=0.1$. To a good accuracy,
solutions with different $\alpha^\prime$ can be
obtained from the solutions with $\alpha=0.1$ using a simple scaling law:
$\Sigma^\prime=\Sigma\alpha/\alpha^\prime$, $P^\prime=P\alpha/\alpha^\prime$,
with the same $H(r)$, $\Delta\Omega(r)$, and $c_s(r)$. Essential
deviation from this law appears only when $\alpha\rightarrow 1$. Then the
sonic radius moves outwards being beyond the radius of the Keplerian
marginally stable orbit, $r_{ms}$. At small $\alpha$, $r_s<r_{ms}$.
The relativistic version of the standard model is shown
by the dashed lines in Fig.~1. The accretion flow deviates from this model
at the trapping radius, $r_t$. Inside $r_t$, the condition $\dot{M}c^2\mu>
2\pi r^2F^-$ takes place which means that the radial flux of internal
energy exceeds the local radiative losses, i.e., the flow is advection
dominated. The trapping radius is practically independent of $\alpha$,
and the dependence on $\dot{m}$ is given in Fig.~3.
From Fig.~1 one can see that: 1) the advection effect reduces the height, $H$,
below the standard value, so that the relative disc height, $H/r$, is kept
modest even at $\dot{m}\mathrel{\copy\simgreatbox} 10\dot{m}_{cr}$, 2) the Thomson optical depth
of the disc, $\tau_0$, has a minimum of $\sim 100\;(\alpha/0.1)^{-1}$ at
$\dot{m}\sim\dot{m}_{cr}$, and 3) the maximum deviation from Keplerian rotation
is $\sim$ 20 \%.
\begin{figure}
\begin{center}
\epsfxsize=8.4cm
\epsfbox{figure2sis.ps}
\end{center}
\caption{The fraction of the total power released in the disc
that is eventually advected into the black hole (see item {\it vii} of
Section 2). Curves 1 and 2 correspond to $a_*=0$ and $a_*=0.998$
respectively.}
\end{figure}
The critical accretion rate $\dot{m}_{cr}=17.5$ in the case of $a_*=0$
and $\dot{m}_{cr}=3.11$ in the case of $a_*=0.998$. At $\dot{m}>\dot{m}_{cr}$
a substantial fraction of the total energy released in the disc is advected
into the black hole, see Fig. 2.
Note that in the standard disc with $\dot{m}<\dot{m}_{cr}$ the density,
$n=\Sigma/2Hm_p$,
scales as $\dot{m}^{-2}$.
In the advection dominated regime ($\dot{m}>\dot{m}_{cr}$, $r\ll r_t$),
the density scales as $\dot{m}$. It means that at $\dot{m}\sim\dot{m}_{cr}$
there must be a minimum of the density. At this minimum a strong overheating
of the flow is possible as discussed in next Section.
\section{Overheating in the central region}
\medskip
An accretion disc can be considered as a two-fluid flow "plasma + radiation".
In the case of a small viscosity, the disc is dense and the plasma and radiation
are close to thermodynamic equilibrium at a common temperature
$T\approx T_{eff}=(w/a_r)^{1/4}$ where $w=3P/2H$ is the radiation density
calculated in Section 3, and $a_r=7.56\cdot 10^{-15}$ is the radiation
constant. [Note that $T_{eff}$ inside the disc differs from the surface
effective temperature $T_{eff}^s=(F^-/2\sigma)^{1/4}$ where $\sigma=a_rc/4$
is the Stefan-Boltzmann constant.]
However, when $\alpha > 0.03$ and $\dot{m}$ approaches $\dot{m}_{cr}$
a strong deviation from the equilibrium occurs in the inner region
of the disc. In this case, the accreting plasma has a low density and therefore,
a low emission capability. As a result, the plasma does not manage to reprocess
the released energy into Planckian radiation, and the balance between heating
and heat-removal through radiative cooling maintains a temperature
$T\gg T_{eff}$. Then the radiation is concentrated in the Wien peak of
temperature $T$ and its density, $w$, is much less than the corresponding
Planckian value, $w_{pl}=a_rT^4$, i.e., the flow is far from the black body
state.
\bigskip
\begin{center}
{\it a) Radius of the overheated region}
\end{center}
\medskip
In the outer "black body" region of the disc, the free-free emission
capability, $\dot{w}_{ff}$, of the plasma with $T=T_{eff}$ exceeds the heating
rate, $\dot{w}^+=F^+/2H$. The boundary $r_*$ between the black body and the
overheated regions can be estimated from the condition
\medskip
\begin{equation}
\dot{w}^+=\dot{w}_{ff}(T_{eff}) \quad {\rm at} \quad r=r_*,
\end{equation}
\begin{eqnarray}
\nonumber
\dot{w}_{ff}=1.6\cdot 10^{-27}n^2\sqrt{T}.
\end{eqnarray}
At exact thermodynamic equilibrium, free-free emission would be
balanced by free-free absorption, and $\dot{w}_{ff}$ can be written as
$\dot{w}_{ff}=cn{\bar\sigma}_{ff}w_{pl}$. $\bar{\sigma}_{ff}$ introduced
in this way has the meaning of an effective cross section for absorption of
the Planckian radiation; it is $\approx 4$ times the Rosseland averaged
$\sigma_{ff}$.
Radius $r_*$ is usually estimated from the condition that the
effective optical depth of the flow $\tau_*=(\tau_0\tau_{ff})^{1/2}=1$
where $\tau_{ff}=Hn{\bar\sigma}_{ff}$ (Rosseland $\sigma_{ff}$ is also
often taken
instead of ${\bar\sigma}_{ff}$). For the standard disc this condition
is equivalent to equation (12). In that case, the radiation
density inside the disc equals $w\approx\dot{w}^+t_D$ where $t_D=\tau_0H/c$
is the diffusion time-scale. Then equality (12) with $w\approx w_{pl}$ gives
$\tau_*\approx 1$, and the decoupling of $w$ below $w_{pl}$ happens if
$\tau_*<1$. The condition $\tau_*<1$ means that a deviation from
thermodynamic equilibrium in the standard disc occurs if the bulk of
radiation diffuses out without absorption and re-emission.
This condition, however, is not relevant in the advection dominated regime.
As the radiation is advected rather than escape, the
time-scale for free-free absorption should be compared
with the inflow time-scale rather than with the diffusion time-scale.
Then the condition $\tau_*<1$ should be replaced by
$\tau_*< (t_D/t_a)^{1/2}\sim r_t/r$ that follows from equation (12).
\begin{figure}
\begin{center}
\epsfxsize=8.4cm
\epsfbox{figure3sis.ps}
\end{center}
\caption{The radius of the overheated region $r_*$ versus $\dot{m}$
in the case of $\alpha=0.3$, $M=10^8M_\odot$, $a_*=0.998$.
For comparison the trapping radius, $r_t$, is also plotted.}
\end{figure}
We have calculated $r_*$ from equation (12) for the case of $\alpha=0.3$,
$M=10^8M_\odot$, $a_*=0.998$. In Fig.~3, $r_*$ is shown versus $\dot{m}$
and compared with the trapping radius, $r_t$. One can see that the entire
advection dominated region is overheated at $\dot{m}< 300$.
\bigskip
\begin{center}
{\it b) Cooling rate inside $r_*$}
\end{center}
The overheated plasma inside $r_*$ is cooled by Comptonized free-free
emission with rate
\begin{equation}
\dot{w}_{cool}=A\dot{w}_{ff}, \qquad r\ll r_*,
\end{equation}
where $A(n,T)$ is an amplification factor due to Compton upscattering of
photons emitted at low energies $h\nu<kT$. This factor is given by
(e.g. Rybicki \& Lightman 1979)
\begin{equation}
A=1+\frac{3}{4}\ln^2 x_{coh},
\end{equation}
where $x_{coh}=h\nu_{coh}/kT$ is the dimensionless photon energy below
which the absorption time-scale is shorter than the time-scale for shifting
in frequency due to Comptonization. $x_{coh}$ is determined by the equation
$\sigma_{ff}(x_{coh})=8\theta\sigma_T$, where $\theta=kT/m_ec^2$, and
$\sigma_{ff}(x)=1.8\cdot 10^{-33}\ln(2.25/x)(1-e^{-x})x^{-3}\theta^{-7/2}
n\sigma_T$. Expression (14) for the Compton enhancement factor assumes that
photons with $x>x_{coh}$ upscatter to the Wien peak before they can
escape the disc or advect to the black hole.
The time-scale for upscattering to the Wien peak is
$t_C=\ln(x_{coh}^{-1})/8\theta n\sigma_T c$. Hence,
$t_C/t_D\sim \ln(x_{coh}^{-1})/y$ where $y=4\theta\tau_0^2$ is the
Kompaneets' parameter. In the considered situation, $y\gg 1$ and photons with
$x>x_{coh}$ do comptonize to the Wien peak before they can escape.
Inside the trapping radius, $t_C$ should be compared with $t_a$.
We check in the calculated models that $t_C$ is shorter than $t_a$ as well.
\begin{center}
{\it c) Heating=cooling balance}
\end{center}
Everywhere in the disc the plasma temperature is determined by the
heating=cooling balance $\dot{w}^+=\dot{w}_{cool}$.
Inside $r_*$, $\dot{w}_{cool}$ is given by equation (13). Outside $r_*$,
the cooling rate is equal to a difference between free-free emission and
absorption, being proportional to a small deviation of $w_{pl}$ from $w$:
$\dot{w}_{cool}=\dot{w}_{ff}(1-w/w_{pl})$. The transition at $r\sim r_*$ can be
smoothly described by the following interpolation for the heating=cooling
balance,
\begin{equation}
\dot{w}^+=\dot{w}_{ff}\left[\frac{w}{w_{pl}}+
A\left(1-\frac{w}{w_{pl}}\right)\right]
\left(1-\frac{w}{w_{pl}}\right).
\end{equation}
In the limit $w\approx w_{pl}$ this equation yields
$\dot{w}^+=\dot{w}_{ff}(1-w/w_{pl})$ while in the limit $w\ll w_{pl}$
it transforms into $\dot{w}^+=A\dot{w}_{ff}$.
\begin{center}
{\it d) Results}
\end{center}
\begin{figure}
\begin{center}
\epsfxsize=8.4cm
\epsfbox{figure4sis.ps}
\end{center}
\caption{ The temperature of the disc with $\alpha=0.3$ in the case of
$a_*=0$:
1 -- $\dot{m}=10$, 2 -- $\dot{m}=100$, 3 -- $\dot{m}=1000$. The dotted
lines show the effective temperature, $T_{eff}$, of the radiation field
in the disc.}
\end{figure}
\begin{figure}
\begin{center}
\epsfxsize=8.4cm
\epsfbox{figure5sis.ps}
\end{center}
\caption{The temperature of the disc with $\alpha=0.3$ in the case of
$a_*=0.998$: 1 -- $\dot{m}=1$, 2 -- $\dot{m}=10$, 3 -- $\dot{m}=1000$.
The dotted lines display the effective temperature, $T_{eff}$, of the
radiation in the disc. The heavy line shows the temperature
for which $\nu_B=\nu_{coh}$ at $B=B_{eq}$ (see the text).}
\end{figure}
\begin{figure}
\begin{center}
\epsfxsize=8.4cm
\epsfbox{figure6sis.ps}
\end{center}
\caption{The maximum temperature in the disc as a function of the
accretion rate for the case of $a_*=0$: 1 -- $\alpha=0.03$, 2 --
$\alpha=0.3$. $T_{eff}$ at the maximum is displayed by the dotted lines.}
\end{figure}
\begin{figure}
\begin{center}
\epsfxsize=8.4cm
\epsfbox{figure7sis.ps}
\end{center}
\caption{The maximum temperature in the disc as a function of the
accretion rate for the case of $a_*=0.998$: 1 -- $\alpha=0.03$,
2 -- $\alpha=0.3$. $T_{eff}$ at the maximum is displayed by the dotted lines.
Above the dashed line, $\nu_B$ exceeds $\nu_{coh}$ in the inner hottest
region of the disc.}
\end{figure}
Using the balance (15), we have calculated the temperature in the disc
around a massive black hole $M=10^8M_\odot$ with spin $a_*=0$ and
$a_*=0.998$.
The results are shown in Fig.~4,5 for the case of $\alpha=0.3$.
The strong overheating, $T\gg T_{eff}$, occurs inside the radius $r_*$.
The maximum temperature is reached in the innermost region.
It is $\sim 3\cdot 10^8$ K in the case of a rapidly rotating black hole
and $\sim 10^7$ K in the case of a Schwarzschild black hole.
In Fig.~6,7 we show the dependence of the maximum temperature on the
accretion rate in the cases of $\alpha=0.03$ and $\alpha=0.3$.
A question of interest is whether the protons can be thermally decoupled from
the electrons at such temperatures. The time-scale for Coulomb energy exchange
between protons of temperature $T_p$ and electrons of temperature $T_e$
is given by (Landau \& Lifshitz 1981)
\begin{eqnarray}
\nonumber
t_{ep}=\sqrt{\frac{\pi}{2}}\frac{m_p}{m_e}
\left(\frac{kT_e}{m_ec^2}\right)^{3/2}
\frac{1}{\ln\Lambda\;n\sigma_Tc}\approx 12.5 \frac{T_e^{3/2}}{n},
\end{eqnarray}
where $\ln\Lambda\approx 20$ is the Coulomb logarithm. Even in the most hot
models with $\alpha=0.3$ and $a_*=0.998$, $t_{ep}\ll t_a$ and the accreting
plasma can be well described in the one-temperature approximation
$T_e\approx T_p\approx T$.
Then we have compared the thermal plasma pressure, $p_{gas}=2nkT$, with
the radiation pressure, $p_{rad}=w/3$. Even in the hottest region of
the disc with $\alpha=0.3$,
the plasma pressure does not exceed $\sim 10^{-3} p_{rad}$.
So, the calculated radiation dominated models are self-consistent.
Note, that accretion flows with $\alpha\rightarrow 1$
are very unlikely (it would imply a turbulence of scale
$H$ with sound speed).
Anyway, such flows could not be much hotter because at higher temperatures
additional cooling mechanisms become important: Comptonization of
cyclotron radiation and collective waves in the plasma,
as noted by Shakura \& Sunyaev (1973).
Cyclotron radiation and collective waves are additional sources of soft
photons which can be upscattered to the Wien peak and cool the plasma efficiently.
Roughly, the cyclotron source starts to be important when the gyrofrequency,
$\nu_B=eB/2\pi m_ec$, exceeds $\nu_{coh}$ (a more detailed treatment including
higher cyclotron harmonics is given in Gnedin \& Sunyaev 1973). In a similar
way, Comptonization of collective waves can be important when $\nu_{pl}
\mathrel{\copy\simgreatbox}\nu_{coh}$, where $\nu_{pl}=(ne^2/\pi m_e)^{1/2}$ is the plasma
frequency. In case the magnetic field is comparable to the equipartition value
$B_{eq}=(8\pi w)^{1/2}$, the gyrofrequency exceeds $\nu_{coh}$ in the disc
sooner than the plasma frequency does.
In Fig.~5, the heavy line shows the temperature at which the
gyrofrequency equals $\nu_{coh}$. The corresponding
maximum temperature in the disc is shown by the dashed line in Fig.~7.
\section{Conclusions}
We have investigated the relativistic accretion disc around a Kerr
black hole with a super-Eddington accretion rate. The disc has a modest
relative height, $H/r < 0.4$, up to $\dot{m}\sim 10\dot{m}_{cr}$, and a
luminosity near the Eddington limit. The bulk of the energy released in the
inner region of the
disc is advected into the black hole. The angular velocity of rotation
deviates from Keplerian up to 20 \%. We paid particular attention to
the case of a large viscosity parameter $\alpha > 0.03$.
In this case, the accretion flow
deviates from thermodynamic equilibrium and overheats in the central region.
The hottest flow has accretion rate $\dot{m}\sim 3\dot{m}_{cr}$.
We have calculated the maximum temperature in the disc around a massive
black hole, $M=10^8M_\odot$, with $\alpha=0.3$.
For a Schwarzschild black hole it equals $\sim 10^7$ K.
For a rapidly rotating black hole, the maximum
temperature is of order $\sim 3\cdot 10^8$ K. It far exceeds
the effective temperature corresponding to thermodynamic equilibrium.
However, it is not large enough for thermal decoupling of the protons and
transition to the ion pressure dominated regime of accretion.
The employed vertically integrated model gives approximate
characteristics of the bulk of the gas flowing in the disc, and does not
describe physical conditions in the upper layers (in the "skin" of the
disc) where the spectrum of emerging radiation is formed.
To evaluate the spectrum
of the super-Eddington disc a more detailed treatment of the vertical energy
transfer is needed which is likely to demand a 2D simulation of the advective
region. Besides, the
disc spectrum may be strongly affected by a corona activity above the surface.
It is clear, however, that the surface of the flow with large $\alpha$ and
a large accretion rate must be much hotter than the effective surface
temperature, $T^s_{eff}=(F^-/2\sigma)^{1/4}$, and an activity of the corona
may make the spectrum only harder.
X-ray emission of the innermost region of the
super-Eddington disc may help to reconcile the observed quasar spectra
with accretion disc models.
\section*{Acknowledgments}
I am grateful to M.A. Abramowicz, I.V. Igumenshchev, and I.D. Novikov
for discussions. I thank the Theoretical Astrophysics Center
for hospitality and acknowledge partial support by RFFI grant 97-02-16975.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,219 |
{"url":"http:\/\/www.mathnet.ru\/php\/archive.phtml?jrnid=aa&wshow=issue&year=1989&volume=1&volume_alt=&issue=3&issue_alt=&option_lang=eng","text":"RUS\u00a0 ENG JOURNALS \u00a0 PEOPLE \u00a0 ORGANISATIONS \u00a0 CONFERENCES \u00a0 SEMINARS \u00a0 VIDEO LIBRARY \u00a0 PACKAGE AMSBIB\n General information Latest issue Archive Impact factor Subscription Search papers Search references RSS Latest issue Current issues Archive issues What is RSS\n\n Algebra i Analiz: Year: Volume: Issue: Page: Find\n\n Expository Surveys The dynamics of analytic transformationsA.\u00a0\u00c8.\u00a0Eremenko, M.\u00a0Yu.\u00a0Lyubich 1 Research Papers Dirichlet series that correspond to representations of zero by indefinite quadratic formsA.\u00a0N.\u00a0Andrianov 71 An element of the disk-algebra that is stationary on a\u00a0set of positive lengthChristopher\u00a0Bishop 83 Closed geodesics on 2-dimensional orbifolds of nonpositive curvatureS.\u00a0V.\u00a0Buyalo 89 Stabilization of the homology of Teichmu\u0308ller modular groupsN.\u00a0V.\u00a0Ivanov 110 Some generalizations of the Strichartz\u2013Brenner inequalityL.\u00a0V.\u00a0Kapitanski 127 Modules of two-component links of codimension twoV.\u00a0L.\u00a0Kobel'skii 160 Stabilization for orthogonal and symplectic algebraic $K$-theoriesI.\u00a0A.\u00a0Panin 172 Automorphic forms on the exceptional group $\\mathrm G_2(\\mathbb C)$N.\u00a0V.\u00a0Proskurin 196 Book Reviews Schro\u0308dinger Operators with Application to Quantum Mechanics and Global Geometry. H.\u2009L.\u00a0Cycon, R.\u2009G.\u00a0Froese, W.\u00a0Kirsch, B.\u00a0Simon. Berlin etc.: Springer, 1987. 319\u00a0p.D.\u00a0R.\u00a0Yafaev 227","date":"2019-09-18 08:42:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.28356078267097473, \"perplexity\": 4136.834909925619}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514573258.74\/warc\/CC-MAIN-20190918065330-20190918091330-00046.warc.gz\"}"} | null | null |
{"url":"http:\/\/pyphi.readthedocs.io\/en\/latest\/api\/compute.distance.html","text":"# compute.distance\u00b6\n\nFunctions for computing distances between various PyPhi objects.\n\npyphi.compute.distance.concept_distance(c1, c2)\n\nReturn the distance between two concepts in concept space.\n\nParameters: c1 (Concept) \u2013 The first concept. c2 (Concept) \u2013 The second concept. The distance between the two concepts in concept space. float\npyphi.compute.distance.constellation_distance(C1, C2)\n\nReturn the distance between two constellations in concept space.\n\nParameters: C1 (Constellation) \u2013 The first constellation. C2 (Constellation) \u2013 The second constellation. The distance between the two constellations in concept space. float\npyphi.compute.distance.small_phi_constellation_distance(C1, C2)\n\nReturn the difference in $$\\varphi$$ between constellations.","date":"2017-08-18 21:51:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7199530601501465, \"perplexity\": 6167.33799474785}, \"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-2017-34\/segments\/1502886105187.53\/warc\/CC-MAIN-20170818213959-20170818233959-00556.warc.gz\"}"} | null | null |
\section{I. Introduction}
Since the initial discovery of superconductivity in fluorine-doped two-dimensional (2D) iron square lattices LaFeAsO, iron based compounds have rapidly developed into one of the most important branches of unconventional superconductors in Condensed Matter Physics and Material Science~\cite{Stewart:Rmp,Johnston:Ap,Dagotto:Rmp,Dai:Rmp}. In the nonsuperconducting parent compounds, their magnetic ground state is known as collinear stripe order, i.e. the so-called C-type antiferromagnetic (AFM) order, with some exceptions such as Block-type and Bicollinear-type~\cite{Dai:Rmp,Dagotto:Rmp,Li:Np,Zhang:RRL}. Since most superconducting phases always emerge next to the suppression of antiferromagnetism by carrier doping or pressure, the AFM order and AFM spin fluctuations are considered to be important for superconductivity~\cite{Mazin:np,Dai:Np}.
Recently, pressure-induced superconductivity was observed in the two-leg quasi-one-dimensional ladder system BaFe$_2$$X_3$ ($X$ = S, Se) with eletronic density $n=6.0$~\cite{Takahashi:Nm,Ying:prb17}, starting a novel field of research for high temperature iron-based superconductors~\cite{Yamauchi:prl15,Arita:prb,Patel:prb16,Wang:prb16,Zhang:prb17,chi:prl,Patel:prb17,Pizarro:prm,Zheng:prb18,Zhang:prb18,Zhang:prb19}. BaFe$_2$S$_3$ displays a stripe-type (CX) AFM order, similar to other 2D iron-based superconductors, below $120$ K~\cite{Takahashi:Nm,chi:prl}.
By applying hydrostatic pressure, this system displayed an insulator-metal transition~\cite{Suzuki:prb,Zhang:prb17} and superconductivity was observed at $P\sim 11$~GPa near a first-order magnetic phase transition where the AFM order was suppressed~\cite{Takahashi:Nm,Materne:prb19}. These recent developments in the context of two-leg iron ladders remind us of the previous results for copper ladders where superconducting tendencies upon doping were theoretically predicated and later confirmed experimentally~\cite{cu-ladder1,cu-ladder2,cu-ladder3}. In fact, in iron ladders similar pairing tendencies were observed upon hole doping by
theoretical research based on density matrix renormalization group (DMRG) at intermediate Hubbard coupling strengths~\cite{Patel:prb16,Patel:prb17}. Another possible explanation of superconductivity is the originally-proposed band narrowing by pressure~\cite{Takahashi:Nm}.
BaFe$_2$Se$_3$ also shows pressure-induced superconductivity~\cite{Ying:prb17,Zhang:prb18}, but in addition it displays an exotic 2$\times$2 Block-type magnetic order below $256$~K under ambient conditions~\cite{Caron:Prb12}. Due to its enhanced degree of electronic correlation~\cite{Caron:Prb,Nambu:Prb}, the physical properties of BaFe$_2$Se$_3$ become more complex. For example, the existence of an ``orbital-selective Mott phase'' (OSMP) was found by neutron experiments at ambient pressure~\cite{mourigal:prl15}. This OSMP of BaFe$_2$Se$_3$ was theoretically discussed based on DMRG methods by using multiorbital Hubbard models as well~\cite{osmp1,osmp2,osmp3}.
However, an additional striking experimental discovery was recently reported for this interesting material: the existence of a polar state with possible polar orbital ordering was confirmed by neutron diffraction methods combined with optical second harmonic generation signals~\cite{Aoyama:prb19}. In fact, this material was first theoretically predicted to be multiferroic because the Block-type magnetic order can produce displacements of Se inducing broken inversion symmetry~\cite{Dong:PRL14}. It should be noted that BaFe$_2$Se$_3$ is the first reported iron-based system to become both superconductor and multiferroic. Moreover, the polar state of BaFe$_2$Se$_3$ is an exotic noncollinear ferrieletric (FiE) phase instead of a plain ferroelectric (FE) one~\cite{Dong:PRL14,Aoyama:prb19}. To our best knowledge, noncollinear FiE order was only proposed in a few compounds, such as $M$O$_2$$X_2$ ($M$= Mo/W, $X$=Br/Cl)~\cite{Lin:PRL} and strained BiFeO$_3$~\cite{Yang:PRL}. It is reasonable to assume that finding this exotic 2$\times$2 Block-type magnetic order with quasi one-dimensional ladders defines an effective feasible path to explore FiE materials. Besides, considering the similar atomic average electronic density in many iron superconductors, it is conceivable to obtain the superconducting phase in other potential $n = 6$ ladders as well.
However, to our best knowledge, there is no other iron ladder reporting to display the 2$\times$2 Block-type magnetic order. According to our previous Hartree-Fock~\cite{luo:prb13} and DFT calculations~\cite{Zhang:prb19}, the Block-type was expected to be stable in a large region of the Hund coupling $J_H$ and Hubbard $U$ phase diagram in $n=6$ iron ladders. Hence, it can be reasonably assumed that the magnetic ground state of $n=6$ Te-based iron ladders, not synthesized yet, may also display the Block-type order if it can be prepared. Actually, the $n=5.5$ iron Te-based ladder was synthesized in experiments~\cite{Klepp:JOAC} but it was recently predicted to display CX-type magnetic order~\cite{Zhang:prb19}. Considering that the ionic radius of Rb$^+$ ($\sim1.47$ \AA) and Ba$^{\rm 2+}$ ($\sim1.434$ \AA) are similar, we believe it should be possible to prepare Te-based $n=6$ iron ladders with chemical formula BaFe$_2$Te$_3$.
In the present publication, we performed first-principles DFT calculations for the BaFe$_2$Te$_3$ system. Our theoretical results indicate that BaFe$_2$Te$_3$ should be stable with a similar crystal structure as BaFe$_2$Se$_3$.
Because in the past DFT has successfully predicted many new compounds before they were truly prepared, such as blue phosphorene~\cite{Guan:prl} and phosphorus carbide~\cite{Guan:nl,Tan:AM}, our structural prediction should be reliable.
Moreover, the 2$\times$2 Block-type spin order is also predicted to be the most likely magnetic ground state in our calculations for this compound. In addition, we found that BaFe$_2$Te$_3$ should display noncollinear ferrielectric order driven by the 2$\times$2 Block-type magnetic order via magnetic exchange striction. The magnetic state and electronic structure similarity with BaFe$_2$Se$_3$ further suggests that BaFe$_2$Te$_3$ could also become superconducting under high pressure.
\section{II. Method}
To understand the physical properties of the BaFe$_2$Te$_3$ system, first-principles DFT calculations were performed based on the projector augmented wave (PAW) pseudopotentials with the Perdew-Burke-Ernzerhof (PBE) exchange functional, as implemented in the Vienna {\it ab initio} simulation package (VASP) code~\cite{Kresse:Prb,Kresse:Prb96,Blochl:Prb,Perdew:Prl}. Since the spin-polarized PBE-GGA function is known to provide an accurate description of the iron based $123$-type two-leg ladder systems~\cite{Suzuki:prb,Arita:prb,Zhang:prb17,Zhang:prb18,Zheng:prb18}, in this publication we do not add an additional Hubbard $U$.
Due to the quasi-one-dimensional ladder structure, the magnetic coupling in-ladder should be the dominant factor affecting the energies and physical properties. Various possible (in-ladder) magnetic configurations were imposed on the iron ladders~\cite{Zhang:prb19} [see Fig.~S1 of the Supplemental Material (SM)~\cite{Supplemental}] to predict the magnetic properties, such as nonmagnetic (NM), ferromagnetic (FM), CX-type with FM rungs and AFM legs, CY-type with AFM rungs and FM legs, G-type with both AFM rungs and legs, and 2$\times$2 magnetic Block-type. Considering previous neutron results for two-leg iron ladders~\cite{Takahashi:Nm,Caron:Prb12}, the ($\pi$, $\pi$, $0$) order was adopted for the CX-AFM and Block-AFM orders. The plane-wave cutoff energy was $500$ eV. Since different magnetic configurations have different minimal unit cells, the mesh was appropriately modified for all the candidates to render the $k$-point densities approximately the same in reciprocal space (as example, $7\times5\times11$ for the FM-type). In addition, we have tested that these $k$-point meshes already lead to converged energies when compared with denser meshes. Both the lattice constants and atomic positions were fully relaxed with different spin configurations until the Hellman-Feynman force on each atom was smaller than $0.01$ eV/{\AA}.
The phonon spectra was calculated using the finite displacement approach and analyzed by the PHONONPY software~\cite{Chaput:prb,Togo:sm}. Furthermore, to estimate the FE polarization, the Berry phase method was adopted~\cite{King-Smith:Prb,Resta:Rmp}. In addition to the standard DFT calculation discussed thus far, the maximally localized Wannier functions (MLWFs) method was employed to fit Fe $3d$'s five bands by using the WANNIER90 packages~\cite{Mostofi:cpc}.
\section{III. Results}
\subsection{A. Crystal structure}
As shown in Fig.~\ref{Fig1}, we constructed two crystal structures [$Cmcm$ (No.63) and $Pbnm$ (No.62) phases] for BaFe$_2$Te$_3$, because those two types of phases were found experimentally in other $123$-type iron ladders~\cite{Caron:Prb12,Svitlyk:JPCM}. In the $Cmcm$ phase [see Fig.~\ref{Fig1}(a)], the FeTe$_4$ tetrahedra are aligned in the $ac$ plane. The $Pbnm$ phase [see Fig.~\ref{Fig1}(b)] can be visualized as adding a tilting of the FeTe$_4$ tetrahedra along the $c$-axis on the $Cmcm$ phase of BaFe$_2$Te$_3$, while the two FeTe$_4$ tetrahedra along the rung direction are rotated counterclockwise/clockwise, respectively. As a consequence, as shown in Figs.~\ref{Fig1}(c-d), the iron ladder would slightly distort with two different iron-iron distances along the leg direction in the $Pbnm$ phase while the iron-iron distances are equal in the ideal $Cmcm$ ladder. By comparing the energies of the two phases with a non-magnetic state, the $Pbnm$ phase was considered to be the most likely crystal structure of BaFe$_2$Te$_3$ ($\sim1.3$ meV/f.u. lower than $Cmcm$ phase) after the lattice constants and atomic positions were fully relaxed. However, note that the energy difference of the two phases for the non-magnetic state are quite small and this issue will be discussed in the next section.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{Fig1.pdf}
\caption{Schematic crystal structure of BaFe$_2$Te$_3$ (electronic density $n = 6$) with the convention: Green = Ba; Blue = Fe; Dark yellow = Te. For better comparison, we used the space group $Pbnm$ instead of the conventional $Pnma$ since the lattice vectors of the $Pbnm$ space group are the same as in the $Cmcm$ space group. Note that the difference between Pbnm and Pnma space groups is only regarding the choice of a unique axis. (a-b) Sketch of the possible crystal structures of BaFe$_2$Te$_3$ for the $Cmcm$ (No. 63) and $Pbnm$ (No. 62) phases, respectively. (c-d) One iron ladder with highlighted FeTe$_4$ tetrahedra for the $Cmcm$ and $Pbnm$ phases, respectively.}
\label{Fig1}
\end{figure}
For completeness, starting from the crystal lattice with $Cmcm$ (No.63) and $Pbnm$ (No.62) symmetry plus various magnetic states, the different spin configurations were fully relaxed. The DFT results indicate that the crystal structure of BaFe$_2$Te$_3$ should be similar to BaFe$_2$Se$_3$ instead of BaFe$_2$S$_3$. In the non-tilting ladder structure, all the energies of different magnetic configurations were higher than the corresponding energies of the tilting ladder structure [see Table~\ref{Table1} and Table~\ref{Table3}]. Hence, we conclude that the $Pbnm$ (No.62) phase should be the favored structure of BaFe$_2$Te$_3$.
\begin{table}
\centering\caption{The optimized lattice constants ({\AA}), local magnetic moments (in $\mu_{\rm B}$/Fe units) within the default PAW sphere, and band gaps (eV) for the various magnetic configurations using the $Cmcm$ structure. Also included the energy differences (meV/Fe) with respect to the Block-B AFM configuration in the $Pbnm$ phase, taken as the reference of energy.}
\begin{tabular*}{0.48\textwidth}{@{\extracolsep{\fill}}llllc}
\hline
\hline
& $a$/$b$/$c$ & $M$ & Gap & Energy \\
\hline
NM & 9.738/11.991/5.672 & 0 & 0 & 385.8 \\
FM & 9.805/13.195/5.634 & 2.92 & 0 & 155.7 \\
CX & 9.805/12.825/5.703 & 2.60 & 0 & 84.3 \\
CY & 9.756/13.214/5.630 & 2.77 & 0.22 & 61.8 \\
G & 9.799/12.812/5.682 & 2.54 & 0 & 160.7 \\
Block-A & 9.837/13.203/5.632 & 2.87 & 0.17 & 44.0 \\
Block-B & 9.823/13.131/5.652 & 2.85 & 0.3& 24.9 \\
\hline
\hline
\end{tabular*}
\label{Table1}
\end{table}
\subsection{B. Stability}
As shown in Figs.~\ref{Fig2}(a-b), there is a small imaginary-frequency branch in the phonon spectrum of the $Cmcm$ structure of BaFe$_2$Te$_3$, which will lead to spontaneous distortions. By removing this unstable phonon modes, the symmetry decreases from $Cmcm$ to $Pbnm$. This small imaginary-frequency issue also corresponds to the small energy difference between the $Cmcm$ and $Pbnm$ phases. According to group theory analysis using the AMPLIMODES software~\cite{Orobengoa:jac,Perez-Mato:aca}, this spontaneous distortion mode is a Y$^{\rm 2+}$ mode. For comparison, the phonon spectrum of the $Pbnm$ structure of BaFe$_2$Te$_3$ is displayed in Figs.~\ref{Fig2}(c-d), which is dynamically stable (no unstable modes). Furthermore, we also investigated the elastic-stability conditions that indicated the $Pbnm$ structure of BaFe$_2$Te$_3$ should be elastically stable.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{Fig2.pdf}
\caption{ Phonon spectrum of BaFe$_2$Te$_3$ (electronic density $n = 6$) for NM state. The coordinates of the high symmetry points in the bulk Brillouin zone (BZ) are given by: $\Gamma$ = (0, 0, 0), X = (0.5, 0, 0), S = (0.5, 0.5, 0), Y = (0, 0.5, 0), Z = (0, 0, 0.5), U = (0, 0, 0.5), R = (0.5, 0.5, 0.5), T (0, 0.5, 0.5). (a) Sketch of the entire BZ for the $Cmcm$ phase of BaFe$_2$Te$_3$. (b) The phonon spectrum of the $Cmcm$ phase along Y-$\Gamma$. (c) Sketch of the entire BZ for the $Pbnm$ phase of BaFe$_2$Te$_3$. (d) The phonon spectrum of the $Pbnm$ phase along Y-$\Gamma$.}
\label{Fig2}
\end{figure}
The elastic matrix of BaFe$_2$Te$_3$ has nine non-zero independent matrix elements ($C_{\rm 11}$, $C_{\rm 12}$, $C_{\rm 13}$, $C_{\rm 22}$, $C_{\rm 23}$, $C_{\rm 33}$, $C_{\rm 44}$, $C_{\rm 55}$, $C_{\rm 66}$) due to the $mmm$ Laue class features of the $Cmcm$ space group (No. 63) and $Pbnm$ space group (No. 62)~\cite{Mouhat:PRB}. These values of the elastic matrix constants $C_{ij}$'s satisfy the Born stability criteria for an orthorhombic system~\cite{Mouhat:PRB}. We calculated the elastic matrix for both $Cmcm$ and $Pbnm$ phase as summarized in Table~\ref{Table2}.
\begin{table} [H]
\centering\caption{The calculated elastic matrix elements ($C_{ij}$, in units of Gpa) corresponding to BaFe$_2$Te$_3$ for the $Cmcm$ and $Pbnm$ phases.}
\begin{tabular*}{0.48\textwidth}{@{\extracolsep{\fill}}lllllllllc}
\hline
\hline
& $C_{\rm 11}$ & $C_{\rm 12}$ &$C_{\rm 13}$ & $C_{\rm 22}$ &$ C_{\rm 23}$ & $ C_{\rm 33}$ &$ C_{\rm 44}$ & $ C_{\rm 55}$ & $C_{\rm 66}$ \\
\hline
No. 63 & 48.4 & 21.1 & 30.8 & 34.3 & 28.2 & 90.8 & 15.3 & 23.5 & 12.9 \\
No. 62 & 48.8 & 23.7 & 29.1 & 38.8 & 27.4 & 88 & 15.7 & 23.1 & 13.9 \\
\hline
\hline
\end{tabular*}
\label{Table2}
\end{table}
The necessary and sufficient Born criteria for an orthorhombic system are the following~\cite{Mouhat:PRB}:
(1) The matrix $C$ is definite positive;
(2) all eigenvalues of $C$ are positive;
(3) all the leading principal minors of $C$ (determinants of its upper-left $k \times k$ submatrix, 1$\leq$$k$$\leq$6) are positive.
(4)\begin{equation}
C_{\rm 11}>0, C_{\rm 11}C_{\rm 22}>C_{\rm 12}^2, \\[-3mm]
\end{equation}
\begin{equation}
C_{\rm 11}C_{\rm 22}C_{\rm 33}+2C_{\rm 12}C_{\rm 13}C_{\rm 23}-C_{\rm 11}C_{\rm 23}^2-C_{\rm 22}C_{\rm 13}^2-C_{\rm 33}C_{\rm 12}^2>0, \\[-1mm]
\end{equation}
\begin{equation}
C_{\rm 44}>0, C_{\rm 55}>0, C_{\rm 66}>0, \\[-4mm]
\end{equation}
\begin{equation}
C_{ii}+C_{jj}-2C_{ij}>0, \\[-1mm]
\end{equation}
As a consequence, BaFe$_2$Te$_3$ should be elastically stable.
\subsection{C. Magnetism}
Previous studies of iron-based superconductors suggest that in the parent compound they all ordered magnetically.
Then, our next task is to understand the magnetic ground state of the here predicted new Te-based ladder. For this purpose,
various possible (in-ladder) magnetic arrangements were tried (see Fig.~S1 in SM~\cite{Supplemental}). Considering previous neutron-based studies for BaFe$_2$Se$_3$~\cite{Caron:Prb12}, the two possible Block AFM orders that were tried here are shown in Fig.~\ref{Fig3}(a). Our main results for BaFe$_2$Te$_3$ are summarized in Table~\ref{Table3}.
Under ambient conditions, our results indicate that the Block-B AFM order is the most stable ground-state magnetic order among all the candidates considered here. For the Block-B AFM state, the calculated local magnetic moment of Fe is $2.85$ $\mu_{\rm B}$/Fe, quite close to the value observed experimentally and theoretically for BaFe$_2$Se$_3$~\cite{Caron:Prb12,Zhang:prb18}. The calculated energy gap of the Block-B AFM order is about $0.32$~eV, which is slightly smaller than the calculated value of BaFe$_2$Se$_3$~\cite{Dong:PRL14,Zhang:prb18}. Moreover, BaFe$_2$Te$_3$ could become metallic under high pressure after considering previous DFT calculations and experiments for related ladders~\cite{Zhang:prb17,Zhang:prb18,Takahashi:Nm,Ying:prb17}. An structural phase transition in the case of BaFe$_2$Se$_3$ was found under pressure, with the tilting ladders becaming non-tilting, corresponding to a transition from the $Pbnm$ symmetry to an ideal $Cmcm$ symmetry~\cite{Svitlyk:JPCM,Zhang:prb18}. Hence, in principle BaFe$_2$Te$_3$ should also display an structural phase transition under pressure although this aspect
requires further detailed calculations beyond the scope of this publication.
\begin{table} [H]
\centering\caption{The optimized lattice constants ({\AA}), local magnetic moments (in $\mu_{\rm B}$/Fe units) within the default PAW sphere,
and band gaps (eV) for the various magnetic configurations considered, as well as the energy differences (meV/Fe) with respect to the Block-B configuration taken as the reference of energy. All the magnetic states discussed here were fully optimized starting from the $Pbnm$ structure.}
\begin{tabular*}{0.48\textwidth}{@{\extracolsep{\fill}}llllc}
\hline
\hline
& $a$/$b$/$c$ & $M$ & Gap & Energy \\
\hline
NM & 9.177/12.200/5.658 & 0 & 0 & 385.2 \\
FM & 9.852/13.397/5.532 & 2.90 & 0 & 125.4 \\
CX & 9.729/13.131/5.652 & 2.63 & 0.05 & 56.6 \\
CY & 9.894/13.178/5.590 & 2.79 & 0.23 & 51.5 \\
G & 9.840/12.852/5.643 & 2.46 & 0.06 & 140.3 \\
Block-A & 9.806/13.424/5.577 & 2.86 & 0.26 & 3.1 \\
Block-B & 9.824/13.182/5.615 & 2.85 & 0.32 & 0 \\
\hline
\hline
\end{tabular*}
\label{Table3}
\end{table}
According to the calculated density of states (DOS) for the Block-B ($\pi$, $\pi$, $0$) AFM order [see Fig.~\ref{Fig3}(b)], the bands near the Fermi level are primarily contributed by Fe-$3d$ orbitals which are highly $hybridized$ with Te-$5p$ orbitals. The total bandwidth of the iron bands corresponding to the magnetic ground state of BaFe$_2$Te$_3$ ($\sim 7$ eV) is slightly smaller than the bandwidths of BaFe$_2$S$_3$ ($\sim 8$ eV)~\cite{Zhang:prb19} and BaFe$_2$Se$_3$ ($\sim 7.6$ eV)~\cite{bandwidthcontex}. This suggests that electronic correlation effects are enhanced in Te-based ladders because
the ratio $U/W$, with $U$ the local atomic Hubbard repulsion and $W$ the bandwidth, is effectively
enhanced. This conclusion agrees with our recent theoretical studies in $n=5.5$ iron Te-based ladders~\cite{Zhang:prb19}. Because many believe that the Block-type magnetic order of BaFe$_2$Se$_3$ is related to an orbital selective Mott state induced by electronic correlations~\cite{Caron:Prb12,osmp1,osmp2,osmp3}, it is reasonable to conclude that BaFe$_2$Te$_3$ could also display this interesting state as well. Of course, more powerful many-body techniques based on multiorbital Hubbard
models are required to confirm this OSMP hypothesis.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{Fig3.pdf}
\caption{(a) Sketch of Block-A and Block-B spin patterns studied here. Spin up and down are represented by blue and brown circles, respectively. The A and B red labels are different ladders located in different layers. The figure was reproduced from our previous publication~\cite{Zhang:prb18}. (b) The density-of-states near the Fermi level based on the Block-B states ($\pi$, $\pi$, $0$) for BaFe$_2$Te$_3$. Blue: Ba; Red: Fe; Green: Te.}
\label{Fig3}
\end{figure}
\subsection{D. Ferrielectricity}
Due to magnetic-exchange striction effect, the Block spin configuration will break parity symmetry but will not break space-inversion symmetry in the iron ladder. However, as shown in Fig.~\ref{Fig4}(a), the displacements of the Te atoms would break space-inversion symmetry. This will induce local dipoles for each iron ladder since irons would move in the same direction perpendicular to the ladder's plane. Due to the phase difference between the A and B ladders in the Block-B AFM state of BaFe$_2$Te$_3$, the induced polarization of each ladder is in principle opposite~\cite{AAcontext}. However, there will be a remaining net polarization along the $c$-axis since the ladders A and B are slightly tilting, as displayed in Fig.~\ref{Fig4}~(b). This conclusion is also supported by group theory analysis~\cite{Groupcontext}.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{Fig4.pdf}
\caption{(a) Sketch of the Fe-Te ladders A and B with Block AFM order. Partial Te atomic displacements induced by the exchange striction effect of the Block order of Fe. (b) Vector of polarizations of the different ladders A and B, as well as the net polarization. (c) Sketch of one ladder with the optimized Block-B AFM order of BaFe$_2$Te$_3$, showing the NN spin up-up (or down-down) and NN spin up-down Fe-Fe distances. (d) Sketch along Ladder B of the optimized Block-B AFM order of BaFe$_2$Te$_3$. The different heights of Te are marked $h_1$ and $h_2$, respectively.}
\label{Fig4}
\end{figure}
In our fully optimized crystal structure of Block-B AFM order of BaFe$_2$Te$_3$, the nearest-neighbor (NN) distances of spin up-up [or down-down] Fe-Fe are $d_1$=$2.583$~\AA and $d_2$=$2.594$~\AA, and the NN distances of spin up-down Fe-Fe are $d_3$=$3.031$~\AA and $d_2$=$3.020$~\AA, as displayed in Fig.~\ref{Fig4}~(c). Meanwhile, our DFT results indicate that the heights of Te(1) and Te(2) are different: $1.987$~\AA and $1.454$~\AA, respectively, which is in agreement with our symmetry and group analysis mentioned in the previous paragraph. Those numerical results show that the ferroelectric dipole of BaFe$_2$Te$_3$ would be larger than the value of BaFe$_2$Se$_3$ by comparing the height difference of different chalcogens (for Te: $\Delta$$h$$\sim0.53$~\AA ~while for Se: $\Delta$$h$$\sim0.22$~\AA ~\cite{Dong:PRL14}). The calculated $P$ of BaFe$_2$Te$_3$ is about $0.31$ $\mu$C/cm$^2$, which is larger than the value for BaFe$_2$Se$_3$ ($0.19$ $\mu$C/cm$^2$)~\cite{Dong:PRL14}. The ferrielectric polarization is directly proportional to the effective ionic charge and relative displacement from paraelectric state, and inversely proportional to volume. Although the difference in distance of the heights of Te is more than twice that of the heights of Se, the value of the polarization of BaFe$_2$Te$_3$ is not twice as large as that of BaFe$_2$Se$_3$. This is because we also have to consider that the effective ionic charge within chalcogenides decreases from Se to Te~\cite{Weakelectronegativitycontext}, while the volume of BaFe$_2$Se$_3$ is smaller than Ba$_2$Fe$_2$Te$_3$.
More importantly, because the ladders for each layer are tilted in different directions, the corresponding induced ferroelectric dipoles would correspond to a non-collinear order as shown in Fig.~\ref{Fig4}(b). Its noncollinearities may be easily modulated by external electric fields. If we only considering the exchange striction without magnetic order, the space group of our fully relaxed FiE structure is $Pnm2_1$ (No.31) which is consistent with recent neutron experiments for BaFe$_2$Se$_3$~\cite{Aoyama:prb19}.
\section{IV. Additional discussion}
In Fig.~\ref{Fig5}~(a), we present the projected band structure of the fully optimized non-magnetic states for the five iron $3d$ orbitals corresponding to BaFe$_2$Te$_3$. For better understanding, we changed the lattice vectors of the $Pbnm$ space group to the conventional $Pnma$ space group where the b-axis is along the ladders direction, the c-axis is perpendicular to the ladders but still in the iron layer, and the a-axis is perpendicular to the iron layer. It is clearly shown that the band structure is more dispersive along the ladder direction ($X-S$ path) than other directions, which indicates the quasi one-dimensional ladder behavior along the $k_y$ axis. Near the Fermi level, the bands are mainly contributed by $d_{xz}$, $d_{yz}$ and $d_{3z^2-r^2}$ (note that the cartesian axes correspond to the lattice axis of $Pnma$ symmetry~\cite{Caron:Prb}; i.e. the $x$ axis is $a$, $y$ axis is $b$ and $z$ axis is $c$ in Fig.~\ref{Fig1}~(b)). Based on the Wannier fitting results, the bandwidth of the five iron orbitals is about $4.1$~eV, as shown in Fig.~\ref{Fig5}~(b). Since the crystal constants of the fully optimized non-magnetic state are usually smaller than the experimental lattice constants in iron ladder systems, the ``real'' non-interacting bandwidth of the five iron orbitals would be smaller than this value. To better understand the electronic correlation of BaFe$_2$Te$_3$ and the electronic structural similarity with BaFe$_2$Se$_3$, we display the electronic structure of BaFe$_2$Se$_3$ for the fully optimized non-magnetic phase in Fig.~\ref{Fig6}. In BaFe$_2$Se$_3$, the band structure is quite similar to BaFe$_2$Te$_3$ because the Fermi level are also mainly contributed by $d_{xz}$, $d_{yz}$ and $d_{3z^2-r^2}$ orbitals. As shown in Fig.~\ref{Fig6}~(b), the bandwidth of the $123$-Se $n = 6$ ladder for the optimized non-magnetic structure is about $4.5$~eV. Because the electronic correlation strength is given by the ratio $U/W$, this analysis indicates that the degree of electronic correlation is enhanced in Te-ladders.
The vast majority of reported 2D iron superconductors~\cite{Stewart:Rmp,Johnston:Ap,Dagotto:Rmp,Dai:Rmp} display a similar structural lattice, magnetic ground state, and electronic structure. Then, it is reasonable that according to our DFT results, the physical and structural properties of our predicted BaFe$_2$Te$_3$ are very similar to those of BaFe$_2$Se$_3$. The BaFe$_2$S$_3$ has a CX-type AFM order at ambient conditions, and the superconducting transition temperature of BaFe$_2$S$_3$ is higher than BaFe$_2$Se$_3$ (the $2\times2$ Block-type AFM order). It suggests that BaFe$_2$Te$_3$ could also become superconducting under pressure although with lower transition temperature than BaFe$_2$S$_3$ because our 123-Te ladder is very similar to BaFe$_2$Se$_3$ which has a lower critical temperature than the 123-S ladder.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{Fig5a.pdf}
\includegraphics[width=0.48\textwidth]{Fig5b.pdf}
\caption{ (a) Projected band structures of BaFe$_2$Te$_3$ (electronic density $n = 6$) for the non-magnetic (NM) state. The Fermi level is shown with dashed lines. The weight of each iron orbital is represented by the size of the circle. (b) The original band dispersion is shown by blue solid, while the Wannier interpolated band dispersion is shown using green dashed curves for BaFe$_2$Te$_3$. The coordinates of the high symmetry points in bulk BZ are given by: $\Gamma$ = (0, 0, 0), Z = (0, 0, 0.5), T = (-0.5, 0, 0.5), X = (-0.5 0 0), S = (-0.5, 0.5, 0), Y = (0, 0,5, 0), U = (0, 0.5, 0.5), R = (-0.5, 0.5, 0.5). }
\label{Fig5}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{Fig6a.pdf}
\includegraphics[width=0.48\textwidth]{Fig6b.pdf}
\caption{ (a) Projected band structures of BaFe$_2$Se$_3$ (electronic density $n = 6$) for the non-magnetic (NM) state. The Fermi level is shown with dashed lines. The weight of each iron orbital is represented by the size of the circle. (b) The original band dispersion is shown by blue solid, while the Wannier interpolated band dispersion is shown using green dashed curves for BaFe$_2$Se$_3$. The coordinates of the high symmetry points in bulk BZ are given by: $\Gamma$ = (0, 0, 0), Z = (0, 0, 0.5), T = (-0.5, 0, 0.5), X = (-0.5 0 0), S = (-0.5, 0.5, 0), Y = (0, 0,5, 0), U = (0, 0.5, 0.5), R = (-0.5, 0.5, 0.5).}
\label{Fig6}
\end{figure}
Since the pressure-induced superconducting phase domes of iron ladders were found near the AFM phase~\cite{Takahashi:Nm,Ying:prb17}, it is reasonable to assume that the driving force of superconductivity are the AFM spin fluctuations~\cite{Arita:prb,Zhang:prb18}. The Block AFM order of BaFe$_2$Se$_3$ was considered to possibly change to the stripe CX-AFM structure due to a structural transition under pressure~\cite{Ying:prb17,Zhang:prb18}. In this scenario, the magnetic fluctuations of BaFe$_2$Se$_3$ were considered to be induced by the competition between Block- and CX-type magnetic orders, while the magnetic fluctuations of BaFe$_2$S$_3$ were attributed entirely to the CX-type AFM order. Hence, it is reasonable to assume that the competition between Block and CX-type magnetic order would be the driving force for superconductivity in high-pressured BaFe$_2$Te$_3$ as well. It should also be noted that, in previous 2D iron superconductors, the $d_{xy}$ orbital was considered to be very important to understand the transition temperature~\cite{Hirschfeld:Rpp,Scalapino:Rmp}, corresponding to our $d_{yz}$ orbital lying in the iron ladder plane. This issue deserves to be studied more in-depth by powerful many-body techniques, beyond the capabilities of the DFT calculations.
\section{V. Conclusion}
In summary, the two-leg iron ladder compound BaFe$_2$Te$_3$, with the iron density $n=6$, was systematically studied by using first-principles calculations. The Block-B type AFM state is here predicted to be the most likely magnetic ground state. The decreased bandwidth of the iron $3d$ bands from S to Te indicates an enhanced degree of electronic correlation. Considering the exotic Block order and strong correlation in this $n=6$ iron Te ladder, the phenomenon of orbital-selective Mott phase is expected. In addition, using the symmetry analysis and DFT calculations, the presence of noncollinear ferrieletricity in BaFe$_2$Te$_3$ is here predicted as induced by the magnetic exchange striction effects of the Block order. Moreover, considering the magnetic state similarity and electronic structure with other iron ladders, BaFe$_2$Te$_3$ may become superconducting under higher pressure. Our overarching conclusion is that the $n=6$ iron Te ladder is worth to be studied by
theoretical and experimental procedures because using Te could lead to interesting results, such as exotic magnetic states, an orbital selective Mott phase, noncollinear ferrieletricity, as well as superconductivity under high pressure.
\section{Acknowledgments}
E.D. and A.M. were supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), Materials Sciences and Engineering Division. S.D., Y.Z., and L.F.L. were supported by the National Natural Science Foundation of China (Grant Nos. 11834002 and 11674055). L.F.L. and Y.Z. were supported by the China Scholarship Council. Y.Z. was also supported by the Scientific Research Foundation of Graduate School of Southeast University. Most calculations were carried out at the Advanced Computing Facility (ACF) of the University of Tennessee Knoxville (UTK).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,388 |
\section{Introduction}
\setcounter{equa}{0}
\subsection{Miscellaneous facts about quadratic differential operators and doubly characteristic pseudodifferential operators}
Since the classical work by J. Sj\"ostrand~\cite{sjostrand}, the study of spectral properties of quadratic diffe\-rential
operators has played a basic r\^ole in the analysis of partial differential operators with double characteristics.
Roughly speaking, if we have, say, a classical pseudodifferential operator $p^w(x,D_x)$ on $\mathbb{R}^n$ with the Weyl
symbol
$p(x,\xi)=p_m(x,\xi)+p_{m-1}(x,\xi)+\ldots$ of order $m$, and if $X_0=(x_0,\xi_0)\in \mathbb{R}^{2n}$ is a point
where
$$p_m(X_0)=dp_m(X_0)=0,$$
then it is natural to consider the quadratic form $q$ which begins the Taylor expansion of
$p_m$ at $X_0$ in order to investigate the properties of the pseudodifferential operator $p^w(x,D_x)$. For example, the study of a priori estimates such as hypoelliptic estimates of the form
$$
\norm{u}_{m-1}\leq C_K \left(\norm{p^w(x,D_x) u}_0+\norm{u}_{m-2}\right),\quad u\in C^{\infty}_0(K), \quad
K\subset \subset \mathbb{R}^n,
$$
then often depends on the spectral analysis of the quadratic operator $q(x,\xi)^w$.
See~\cite{hypoelliptic}, as well as Chapter 22 of~\cite{hormander} together with further references given there.
In the classical work~\cite{sjostrand}, the spectrum of a general quadratic differential operator, that is an operator defined in the Weyl quantization
\begin{equation}\label{3}\stepcounter{equa}
q^w(x,D_x) u(x) =\frac{1}{(2\pi)^n}\int_{\mathbb{R}^{2n}}{e^{i(x-y).\xi}q\Big(\frac{x+y}{2},\xi\Big)u(y)dyd\xi}, \tag{\thesection.\theequa}
\end{equation}
by a symbol $q(x,\xi)$, where $(x,\xi) \in \mathbb{R}^{n} \times \mathbb{R}^n$ and $n \in \mathbb{N}^*$, which
is a complex-valued quadratic form, has been determined under the
basic assumption of global ellipticity of the quadratic symbol
$$(x,\xi) \in \mathbb{R}^{2n}, \ q(x,\xi)=0 \Rightarrow (x,\xi)=0.$$
\medskip
We recently investigated properties of non-elliptic quadratic operators in the works \cite{HiPr} and~\cite{karel}. Considering quadratic operators whose Weyl symbols have real parts with a sign, say here, Weyl symbols with non-negative real parts
\begin{equation}\label{smm1}\stepcounter{equa}
\textrm{Re }q \geq 0, \tag{\thesection.\theequa}
\end{equation}
we pointed out the existence of a particular linear subvector space $S$ in the phase space $\mathbb{R}_x^n \times \mathbb{R}_{\xi}^n$ intrinsically associated to their Weyl symbols $q(x,\xi)$ and
called singular space, which seems to play a basic r\^ole in the understanding of a number of fairly general properties, such as spectral or subelliptic properties, of these non-elliptic quadratic operators.
In particular, we established that when a quadratic symbol fulfilling (\ref{smm1}) satisfies an assumption of partial ellipticity along its singular space $S$, that is,
$$(x,\xi) \in S, \ q(x,\xi)=0 \Rightarrow (x,\xi)=0,$$
then the spectrum of the quadratic operator $q^w(x,D_x)$ is only composed of a countable number of eigenvalues of finite multiplicity, with a structure
similar to the one known in the case of global ellipticity described by J.~Sj\"ostrand in~\cite{sjostrand}.
\medskip
The purpose of the present work is to address the question of how these recent improvements in the understanding of non-elliptic quadratic operators allow to enhance the comprehension of the properties of certain classes of non-selfadjoint semiclassical operators with double characteristics. In this work, which is planned to be the first one in a series on doubly characteristic pseudodifferential operators, we shall study bounds for resolvents and estimates for low lying eigenvalues for non-selfadjoint semiclassical pseudodifferential operators with principal symbols whose quadratic approximations at doubly characteristic points enjoy a partial ellipticity property along their singular spaces. Under these particular assumptions of partial ellipticity for these quadratic approximations, we shall establish a semiclassical hypoelliptic a priori estimate with a loss of the full power of the semiclassical parameter which gives a localization for the low lying spectral values of the operator.
\medskip
Before giving the precise statement of our main result, we shall recall miscellaneous facts and notation
that we will need about quadratic differential operators. Associated to a complex-valued quadratic form
\begin{eqnarray*}
q : \mathbb{R}_x^n \times \mathbb{R}_{\xi}^n &\rightarrow& \mathbb{C}\\
(x,\xi) & \mapsto & q(x,\xi),
\end{eqnarray*}
with $n \in \mathbb{N}^*$, is the Hamilton map $F \in M_{2n}(\mathbb{C})$ uniquely defined by the identity
\begin{equation}\label{10}\stepcounter{equa}
q\big{(}(x,\xi);(y,\eta) \big{)}=\sigma \big{(}(x,\xi),F(y,\eta) \big{)}, \ (x,\xi) \in \mathbb{R}^{2n}, (y,\eta) \in \mathbb{R}^{2n}, \tag{\thesection.\theequa}
\end{equation}
where $q\big{(}\textrm{\textperiodcentered};\textrm{\textperiodcentered} \big{)}$ stands for the polarized form
associated to the quadratic form $q$ and $\sigma$ is the canonical symplectic form on $\mathbb{R}^{2n}$,
\begin{equation}
\label{11}\stepcounter{equa}
\sigma \big{(}(x,\xi),(y,\eta) \big{)}=\xi.y-x.\eta, \ (x,\xi) \in \mathbb{R}^{2n}, (y,\eta) \in \mathbb{R}^{2n}. \tag{\thesection.\theequa}
\end{equation}
It follows directly from the definition of the Hamilton map $F$ that
its real and imaginary parts, denoted respectively by $\textrm{Re } F$ and $\textrm{Im }F$, are the Hamilton maps associated
to the quadratic forms $\textrm{Re } q$ and $\textrm{Im }q$, respectively; and that a
Hamilton map is always skew-symmetric with respect to $\sigma$. This fact is just a consequence of the
properties of skew-symmetry of the symplectic form and symmetry of the polarized form
\begin{equation}\label{12}\stepcounter{equa}
\forall X,Y \in \mathbb{R}^{2n}, \ \sigma(X,FY)=q(X;Y)=q(Y;X)=\sigma(Y,FX)=-\sigma(FX,Y).\tag{\thesection.\theequa}
\end{equation}
We defined in~\cite{HiPr} the singular space $S$ associated to a quadratic symbol $q$ as the following intersection of kernels
\begin{equation}\label{h1}\stepcounter{equa}
S=\Big(\bigcap_{j=0}^{2n-1}\textrm{Ker}\big[\textrm{Re }F(\textrm{Im }F)^j \big]\Big) \cap \mathbb{R}^{2n}, \tag{\thesection.\theequa}
\end{equation}
where $F$ stands for its Hamilton map; and we proved (Theorem~1.2.2 in~\cite{HiPr}) that when a quadratic symbol $q$ with a non-negative real part is elliptic on its singular space $S$,
\begin{equation}\label{sm2}\stepcounter{equa}
(x,\xi) \in S, \ q(x,\xi)=0 \Rightarrow (x,\xi)=0, \tag{\thesection.\theequa}
\end{equation}
then the spectrum of the quadratic operator $q^w(x,D_x)$
is only composed of eigenvalues of finite multiplicity
\begin{equation}\label{sm6}\stepcounter{equa}
\sigma\big{(}q^w(x,D_x)\big{)}=\Big\{ \sum_{\substack{\lambda \in \sigma(F), \\ -i \lambda \in \mathbb{C}_+
\cup (\Sigma(q|_S) \setminus \{0\})
} }
{\big{(}r_{\lambda}+2 k_{\lambda}
\big{)}(-i\lambda) : k_{\lambda} \in \mathbb{N}}
\Big\}, \tag{\thesection.\theequa}
\end{equation}
where $r_{\lambda}$ is the dimension of the space of generalized eigenvectors of $F$ in $\mathbb{C}^{2n}$
belonging to the eigenvalue $\lambda \in \mathbb{C}$,
$$\Sigma(q|_S)=\overline{q(S)} \textrm{ and } \mathbb{C}_+=\{z \in \mathbb{C} : \textrm{Re }z>0\}.$$
Let us finally end these few recollections by mentioning that one can also describe the singular spaces of such quadratic symbols (see Section~1.4 in~\cite{HiPr})
in terms of the eigenspaces associated to the real eigenvalues of their Hamilton maps. Considering such a quadratic symbol $q$, the set of real eigenvalues of its Hamilton map $F$ can then be written as
$$\sigma(F) \cap \mathbb{R} =\{\lambda_1,...,\lambda_r,-\lambda_1,...,-\lambda_r\},$$
with $\lambda_j \neq 0$ and $\lambda_j \neq \pm \lambda_k$ if $j \neq k$; and one can check that its singular space is the direct sum of the symplectically orthogonal spaces
\begin{equation}\label{sm5bis}\stepcounter{equa}
S=S_{\lambda_1} \oplus^{\sigma \perp} S_{\lambda_2} \oplus^{\sigma \perp}... \oplus^{\sigma \perp} S_{\lambda_r}, \tag{\thesection.\theequa}
\end{equation}
where the spaces $S_{\lambda_{j}}$, $1 \leq j \leq r$, are the symplectic spaces
\begin{equation}\label{sm5bis1}\stepcounter{equa}
S_{\lambda_j}=\big(\textrm{Ker}(F -\lambda_j) \oplus \textrm{Ker}(F+\lambda_j) \big) \cap \mathbb{R}^{2n}. \tag{\thesection.\theequa}
\end{equation}
\subsection{Statement of the main result}
Let us now state the main result contained in this paper.
Let $m\geq 1$ be a $C^{\infty}$ order function on $\mathbb{R}^{2n}$ fulfilling
\begin{equation}\label{eq1.1}\stepcounter{equa}
\exists C_0 \geq 1, N_0>0,\ m(X)\leq C_0 \langle{X-Y\rangle}^{N_0} m(Y),\ X,Y\in \mathbb{R}^{2n}, \tag{\thesection.\theequa}
\end{equation}
where $\langle X \rangle=(1+|X|^2)^{\frac{1}{2}}$, and $S(m)$ be the symbol class
$$S(m)=\left\{ a\in C^{\infty}(\mathbb{R}^{2n},\mathbb{C}): \forall \alpha \in \mathbb{N}^{2n}, \exists C_{\alpha}>0, \forall X \in \mathbb{R}^{2n}, \ |\partial_X^{\alpha} a(X)| \leq C_{\alpha} m(X)\right\}.$$
We shall assume in the following, as we may, that $m$ belongs to its own symbol class $m\in S(m)$.
Considering a symbol $P(x,\xi;h)$ with a semiclassical asymptotic expansion in the symbol class $S(m)$,
\begin{equation}\label{xi1}\stepcounter{equa}
P(x,\xi;h) \sim \sum_{j=0}^{+\infty}{p_j(x,\xi)h^j}, \tag{\thesection.\theequa}
\end{equation}
with $p_j \in S(m)$, $j \in \mathbb{N}$; such that its principal symbol $p_0$ has a non-negative real part
\begin{equation}\label{eq1.4}\stepcounter{equa}
\textrm{Re }p_0(X)\geq 0,\ X=(x,\xi) \in \mathbb{R}^{2n}, \tag{\thesection.\theequa}
\end{equation}
we shall study the operator
\begin{equation}\label{eq1.3}\stepcounter{equa}
P = P^w(x,hD_x;h),\ 0<h\leq 1, \tag{\thesection.\theequa}
\end{equation}
defined by the $h$-Weyl quantization of the symbol $P(x,\xi;h)$, that is, the Weyl quantization of the symbol $P(x,h\xi;h)$. When equipped with the domain
$$H(m)=\big(m^w(x,hD_x)\big)^{-1}\big(L^2(\mathbb{R}^n)\big),$$
for $h>0$ sufficiently small, the operator $P$ becomes a closed and densely defined operator on $L^2(\mathbb{R}^n)$ (see Section~3 in~\cite{hager}).
We shall assume that the real part of the principal symbol $p_0$ is elliptic at infinity in the sense that
\begin{equation}\label{eq1.5}\stepcounter{equa}
\exists C>0,\forall \abs{X}\geq C, \ \textrm{Re } p_0(X)\geq \frac{m(X)}{C}. \tag{\thesection.\theequa}
\end{equation}
This assumption ensures (see Section~3 in~\cite{hager}) that for sufficiently small values of the semiclassical parameter $h$, $0 < h \ll 1$, the spectrum of the operator $P$ in a fixed neighborhood{} of $0\in \mathbb{C}$ is discrete and consists of eigenvalues of finite algebraic multiplicity.
We shall also assume that the characteristic set of the real part of the principal symbol $p_0$,
$$(\textrm{Re }p_0)^{-1}(0)\subset \mathbb{R}^{2n},$$
is finite, so that we may write it as
\begin{equation}\label{eq1.6}\stepcounter{equa}
(\textrm{Re } p_0)^{-1}(0) = \{X_1,...,X_N\}.\tag{\thesection.\theequa}
\end{equation}
The sign assumption (\ref{eq1.4}) implies in particular that we have
$$d\textrm{Re }p_0(X_j)=0,$$
for all $1 \leq j \leq N$; and we shall actually assume that these points are all doubly characteristic points for the symbol $p_0$,
\begin{equation}\label{eq1.6.5}\stepcounter{equa}
p_0(X_j)=dp_0(X_j)=0,\ 1\leq j\leq N, \tag{\thesection.\theequa}
\end{equation}
so that we may write
\begin{equation}\label{eq1.6.6}\stepcounter{equa}
p_0(X_j+Y)=q_j(Y)+\mathcal{O}(Y^3),\tag{\thesection.\theequa}
\end{equation}
when $Y\rightarrow 0$; where $q_j$ is the quadratic approximation which begins the Taylor expansion of the principal symbol $p_0$ at $X_j$. Notice that the sign assumption (\ref{eq1.4}) also implies that these complex-valued quadratic forms $q_j$ have non-negative real parts
\begin{equation}\label{kps1}\stepcounter{equa}
{\rm Re\,} q_j\geq 0.\tag{\thesection.\theequa}
\end{equation}
By denoting $S_j$ the singular spaces associated to these quadratic forms $q_j$, the purpose of this work is to establish the following result:
\bigskip
\begin{theo}\label{theo}
Consider a symbol $P(x,\xi;h)$ with a semiclassical expansion in the class $S(m)$ such that its principal symbol $p_0$ fulfills the assumptions {\rm (\ref{eq1.4})}, {\rm (\ref{eq1.5})}, {\rm (\ref{eq1.6})} and {\rm (\ref{eq1.6.5})}. When all the quadratic forms $q_j$, $1 \leq j \leq N$, defined in {\rm (\ref{eq1.6.6})} are elliptic on their associated singular spaces
\begin{equation}\label{kps2}\stepcounter{equa}
X \in S_j, \ q_j(X)=0 \Rightarrow X=0,\tag{\thesection.\theequa}
\end{equation}
then for any constant $C>1$ and any fixed neighborhood{} $\Omega_j \subset \mathbb{C}$ of the spectrum of the quadratic operator associated to the quadratic symbol~$q_j$,
$$\sigma\big(q_j^w(x,D_x)\big) \subset \Omega_j,$$
described in {\rm (\ref{sm6})}, there exist some positive constants $0 < h_0 \leq 1$ and $C_0>0$ such that for all $0<h \leq h_0$, $u \in \mathcal{S}(\mathbb{R}^n)$ and $\abs{z}\leq C$ satisfying
$$z-p_1(X_j) \notin \Omega_j, \ 1 \leq j \leq N,$$
we have
\begin{equation}\label{eq1.7}\stepcounter{equa}
h\|u\|\leq C_0\|(P-hz)u\|,\tag{\thesection.\theequa}
\end{equation}
with $P=P^w(x,hD_x;h)$; where $p_1(X_j)$ stands for the value of the subprincipal symbol of the symbol $P(x,\xi;h)$ evaluated at the doubly characteristic point $X_j$ and $\|\cdot\|$ is $L^2$--norm on $\mathbb{R}^n$.
\end{theo}
\bigskip
Let us begin our few comments about Theorem~\ref{theo} by mentioning that its result was essentially well-known in the case when the quadratic forms $q_j$ are all globally elliptic on $\mathbb{R}^{2n}$,
$$X \in \mathbb{R}^{2n}, \ q_j(X)=0 \Rightarrow X=0,$$
when $1 \leq j \leq N$. We refer the reader to the work \cite{sjostrand} of J. Sj\"ostrand where the case of classical pseudodifferential operators is considered. The novelty of Theorem~\ref{theo} comes therefore from the fact that the semiclassical hypoelliptic a priori estimate with a loss of the full power of the semiclassical parameter (\ref{eq1.7}) remains valid in cases where the global ellipticity of the Hessians of the principal symbol at doubly characteristic points fails. Our result actually shows that this estimate holds only under the weaker assumption of partially ellipticity (\ref{kps2}) for the Hessians of the principal symbol at doubly characteristic points. Let us also stress the fact that Theorem~\ref{theo} actually extends the result of J.~Sj\"ostrand in \cite{sjostrand} since one can check from the definitions (\ref{10}) and (\ref{h1}) that the singular space $S$ of a complex-valued quadratic form $q$ with a non-negative real part is always distinct from the whole phase space $\mathbb{R}^{2n}$ as soon as its real part is a non-zero quadratic form
$$\exists (x_0,\xi_0) \in \mathbb{R}^{2n}, \ \textrm{Re }q(x_0,\xi_0) \neq 0.$$
A noticeable example of non-elliptic quadratic operator fulfilling the assumption of partial ellipticity (\ref{kps2}) is given by the Kramers-Fokker-Planck operator
$$K=-\Delta_v+\frac{v^2}{4}-\frac{1}{2}+v.\partial_x-\big(\partial_xV(x)\big).\partial_v, \ (x,v) \in \mathbb{R}^{2},$$
with the quadratic potential
$$V(x)=\frac{1}{2}ax^2, \ a \in \mathbb{R}^*.$$
One can actually check that this operator can be expressed as
$$K=q^w(x,v,D_x,D_v)-\frac{1}{2},$$
with the Weyl symbol
\begin{equation}\label{kps3}\stepcounter{equa}
q(x,v,\xi,\eta)=\eta^2+\frac{1}{4}v^2+i(v \xi-a x \eta),\tag{\thesection.\theequa}
\end{equation}
which is a non-elliptic complex-valued quadratic form with a non-negative real part and a zero singular space. Starting from this example, we may easily construct models for Hessians with non-negative real parts fulfilling the condition (\ref{kps2}) whose singular spaces $S$ are both
non-trivial and distinct of the whole phase space. Such a model is for instance obtained when adding to the quadratic form $q$ defined in (\ref{kps3}) an elliptic purely imaginary-valued quadratic form $i\tilde{q}$ in other symplectic variables $(x'',\xi'')$,
$$Q(x',x'',\xi',\xi'') = q(x',\xi') + i \tilde{q}(x'',\xi''),$$
since the singular space is in this case given by
$$S=\{(x',x'',\xi',\xi'') \in \mathbb{R}^{2n'+2n''}: x'=\xi'=0\}.$$
\medskip
About the present work, we drew our inspiration quite exclusively from the semiclassical analysis for Kramers-Fokker-Planck equation led by F.~H\'erau, J.~Sj\"ostrand and C.~Stolk in \cite{HeSjSt}. Our proof of Theorem~\ref{theo} relies on a similar construction of a global bounded weight function $G$ with controlled derivatives and the use, on the FBI-Bargmann side, of associated weighted spaces of holomorphic functions on which the quadratic approximations at critical points of the new principal symbol of the operator
$$\tilde{p}_0 \sim p_0+i\delta H_G p_0,$$
become globally elliptic although the quadratic approximations of the original principal symbol may fail global ellipticity since they only fulfill the assumption of partial ellipticity on their singular spaces. The structure of our proof will therefore follow the one of the analysis led in \cite{HeSjSt} for the proof of the first a priori estimate in Theorem~1.2. Parts of our proof will actually be the same and we shall therefore refer directly the reader to some parts of the work by F.~H\'erau, J.~Sj\"ostrand and C.~Stolk when no change of any kind is needed. In the setting considered in \cite{HeSjSt}, the authors make some assumptions of subellipticity for the principal symbol of the operator both locally near critical points, say here $X_0=0$,
\begin{equation}\label{kps4}\stepcounter{equa}
\exists \varepsilon_0>0, \ \textrm{Re }p_0(X)+\varepsilon_0H_{\textrm{Im}p_0}^2\textrm{Re }p_0(X) \sim |X|^2, \tag{\thesection.\theequa}
\end{equation}
and at infinity. In the present work, we shall not consider such a general situation where ellipticity may fail both locally and at infinity. Indeed, the main purpose of the present work being to weaken the assumptions of subellipticity near critical points, we shall simplify parts of the analysis led in \cite{HeSjSt} by requiring a property of ellipticity at infinity for the real part of the principal symbol $p_0$, but we shall consider weaker local assumptions on the doubly characteristic set. Indeed, our assumption of partial ellipticity along the singular spaces for the quadratic approximations of the principal symbol at doubly characteristic points weakens the subelliptic assumption (\ref{kps4}) since, as we shall see in Section~\ref{quadratic}, this subelliptic assumption (\ref{kps4})
induces that the singular space $S$ associated to the Hessian $q$ of the principal symbol $p_0$ at $X_0=0$ is equal to $\{0\}$. More precisely, one can check that the assumption (\ref{kps4}) is actually equivalent to the fact that the singular space $S$ is equal to zero after the intersection of exactly two kernels
\begin{equation}\label{kps5}\stepcounter{equa}
S=\textrm{Ker}(\textrm{Re }F) \cap \textrm{Ker}\big[\textrm{Re }F(\textrm{Im }F)\big] \cap \mathbb{R}^4=\{0\}.\tag{\thesection.\theequa}
\end{equation}
We refer the reader to \cite{karel} for a complete discussion of subelliptic properties of quadratic differential operators where this link between conditions (\ref{kps4}) and (\ref{kps5}) is explained.
Let us finally end this paragraph by mentioning that if one is interested in establishing semiclassical resolvent estimates for the operator $P$ instead of semiclassical hypoelliptic a priori estimates as the ones proved in Theorem~\ref{theo}, one can actually deduce resolvent estimates from that type of a priori estimates in some specific cases. This is discussed by F.~H\'erau, J.~Sj\"ostrand and C.~Stolk in~\cite{HeSjSt}, and we naturally refer the reader to~\cite{HeSjSt} (Section~11.1) for more details about this topic.
\medskip
The plan of this paper is organized as follows. Section~\ref{weight} is devoted to the construction of a global bounded weight function. Following \cite{HeSjSt}, we then recall in Section~\ref{fbi} some basic facts about the FBI-Bargmann transform and weighted spaces of holomorphic functions associated to this bounded weight function. In Section~\ref{quadratic}, we investigate the properties of the differential operators obtained by the Weyl quantization of the quadratic approximations of the principal symbol at doubly characteristic points. This study will allow us to establish in Section~\ref{tiny} some local resolvent estimates in a tiny neighborhood of these doubly characteristic points. After proving other local resolvent estimates in the exterior region (Section~\ref{intermediate}), we finally complete our proof of Theorem~\ref{theo} in Section~\ref{proof}.
\bigskip
\noindent
\textit{Remark}. We are planning to investigate in a future work the precise semiclassical asymptotics of the spectrum (modulo $\mathcal{O}(h^{\infty})$ when $h \rightarrow 0^+$) of the doubly characteristic operator $P^w(x,hD_x;h)$. More specifically, we shall try to establish under the assumptions of Theorem~\ref{theo}, a similar result as the one proved in~\cite{HeSjSt} (Theorem~1.3) for Kramers-Fokker-Planck operators.
\bigskip
\noindent
\textit{Example.} Let $V$ and $W$ be two $C^{\infty}_b(\mathbb{R}^2,\mathbb{R})$ functions such that the non-negative function $V \geq 0$ is elliptic at infinity
$$\exists C >0, \forall \ |x| \geq C, \ V(x)\geq \frac{1}{C},$$
and vanishes only when $x=0$. We assume that
$$V(x)=x_1^2+\mathcal{O}(x^3),$$
while
$$W(x)=\alpha x_1^2 + 2\beta x_1 x_2 +\gamma x_2^2+\mathcal{O}(x^3),$$
when $x\rightarrow 0$, for some constants $\alpha$, $\beta$, $\gamma\in \mathbb{R}$, not all equal to zero. Considering the principal symbol
$$p_0(x,\xi)=\xi^2+V(x)+iW(x),$$
we notice that
$$(\textrm{Re }p_0)^{-1}(0)=\{(0,0,0,0)\},$$
and that this symbol satisfies all the assumptions (\ref{eq1.4}), (\ref{eq1.5}), (\ref{eq1.6}) and (\ref{eq1.6.5}) of Theorem~\ref{theo} with $m(x,\xi)=\langle{\xi}\rangle^2$.
The quadratic approximation of the principal symbol $p_0$ at $(0,0,0,0)$ is then given by the following quadratic form
\begin{equation}\label{eq1.7.1}\stepcounter{equa}
q(x_1,x_2,\xi_1,\xi_2)=\xi_1^2 + \xi_2^2+ x_1^2 + i(\alpha x_1^2 + 2\beta x_1 x_2 +\gamma x_2^2),\tag{\thesection.\theequa}
\end{equation}
which is globally elliptic precisely when $\gamma \neq 0$. In general, a direct computation using (\ref{10}) and (\ref{h1}) shows that the singular space $S$ associated to $q$ is reduced to zero
precisely when $\beta^2+\gamma^2\neq 0$. In particular, when $\gamma=0$ and $\beta\neq 0$, the quadratic approximation $q$ is not globally elliptic but it obviously fulfills the assumption of partial ellipticity along its singular space $S=\{0\}$. Theorem~\ref{theo} can therefore be applied to any operator $P^w(x,hD_x;h)$ whose symbol $P(x,\xi;h)$ satisfies the following semiclassical asymptotic expansion
$$P(x,\xi;h) \sim \sum_{j=0}^{+\infty}{p_j(x,\xi)h^j},$$
with $p_j \in S(m)$ for $j \geq 1$,
despite the lack of global ellipticity of the quadratic form $q$. Finally, in the case when $\beta=\gamma=0$, the singular space $S$ is then a one-dimensional subspace and the quadratic form $q$ fails ellipticity on $S$. One can actually check in this case that the quadratic form $q$ vanishes identically on its singular space and notice that the spectrum of the associated operator
$$q^w(x,D_x)=D_{x_1}^2 + D_{x_2}^2+ (1+i\alpha)x_1^2,$$
is no longer discrete.
\bigskip
We shall finish this introduction by explaining that it is actually sufficient to establish Theorem~\ref{theo} in the special case when $m=1$. Indeed, when assuming that Theorem~\ref{theo} has already been proved when $m=1$, we may consider an order function $m \geq 1$ as in (\ref{eq1.1}) such that $m \in S(m)$; and a symbol $P(x,\xi;h)$ satisfying the associated assumptions of Theorem~\ref{theo}. Then, one can choose a symbol $\widetilde{p}_0\in S(1)$ with a non-negative real part $\textrm{Re } \widetilde{p}_0\geq 0$ which is elliptic near infinity in the symbol class $S(1)$; and such that $\widetilde{p}_0=p_0$ on a large compact set containing $p_0^{-1}(0)$ where $p_0$ stands for the principal symbol of $P(x,\xi;h)$. This is for instance the case when taking $\chi_0\in C^{\infty}_0(\mathbb{R}^{2n};[0,1])$ such that $\chi_0=1$ near $p_0^{-1}(0)$ and setting
$$\widetilde{p}_0=\chi_0 p_0 + (1-\chi_0).$$
Defining also the symbols
$$\widetilde{p}_j=\chi_0 p_j + (1-\chi_0) \in S(1),$$
when $j \geq 1$, we may choose $\chi\in C^{\infty}_0(\mathbb{R}^{2n},[0,1])$ such that $\chi=1$ near $p_0^{-1}(0)$ and $\chi_0=1$ near $\rm{supp}\, \chi$. By setting $P=P^w(x,hD_x;h)$ and $\tilde{P}=\tilde{P}^w(x,hD_x;h)$, where
$$\tilde{P}(x,\xi;h) \sim \sum_{j=0}^{+\infty}{\tilde{p}_j(x,\xi)h^j},$$
in the symbol class $S(1)$;
and using $L^2$--norms throughout, we deduce from the semiclassical elliptic regularity that
\stepcounter{equa}\begin{align*}\label{eq1.8}
h\|u\| \leq & \ h\|\chi^w(x,hD_x) u\|+h\|(1-\chi)^w(x,hD_x)u\| \tag{\thesection.\theequa} \\
\leq & \ h\|\chi^w(x,hD_x) u\|+\mathcal{O}(h) \|(P-hz)u\|+\mathcal{O}(h^{\infty})\|u\|,
\end{align*}
since the principal symbol $p_0$ of the operator $P$ is elliptic near the support of the function $1-\chi$. By using that Theorem~\ref{theo} is valid when $m=1$, we may apply it to the operator $\widetilde{P}$ to get that if $z$ is as in Theorem~\ref{theo},
\stepcounter{equa}\begin{align*}\label{eq1.9}
h \|\chi^w(x,hD_x) u\| \leq & \ \mathcal{O}(1)\|(\widetilde{P}-hz)\chi^w(x,hD_x) u\| \tag{\thesection.\theequa} \\
\leq & \ \mathcal{O}(1)\|(P-hz)\chi^w(x,hD_x) u\|+\mathcal{O}(h^{\infty})\|u\|,
\end{align*}
since $(\widetilde{P}-P)\chi^w(x,hD_x) = \mathcal{O}(h^{\infty})$ in $\mathcal{L}(L^2)$ when $h \rightarrow 0^+$.
We get that
\begin{equation}\label{eq1.10}\stepcounter{equa}
h \|\chi^w(x,hD_x) u\| \leq \mathcal{O}(1)\|(P-hz)u\|+\mathcal{O}(1)\|[P,\chi^w(x,hD_x)]u\|+\mathcal{O}(h^{\infty})\|u\|.\tag{\thesection.\theequa}
\end{equation}
When estimating the commutator term in the right hand side of (\ref{eq1.10}), we take $\widetilde{\chi}\in C^{\infty}_0(\mathbb{R}^{2n},[0,1])$ such that $\widetilde{\chi}=1$ near $p_0^{-1}(0)$ and $\chi=1$ near $\rm{supp}\, \widetilde{\chi}$. Then, by using that
$$[P,\chi^w(x,hD_x)]\widetilde{\chi}^w(x,hD_x)=\mathcal{O}(h^{\infty}),$$
in $\mathcal{L}(L^2)$, together with the fact that $p_0$ is elliptic near the support of $1-\widetilde{\chi}$, we get that
\begin{equation}\label{eq1.11}\stepcounter{equa}
h \|\chi^w(x,hD_x) u\| \leq \mathcal{O}(1)\|(P-hz)u\| + \mathcal{O}(h^{\infty})\|u\|,\tag{\thesection.\theequa}
\end{equation}
which in view of (\ref{eq1.8}) completes the proof of the reduction to the case when $m=1$. In what follows, we shall therefore be concerned exclusively with the case when $m=1$.
\bigskip
\noindent
\textbf{Acknowledgments:} The first author is grateful to the partial support of the National Science Foundation under grant DMS-0653275 and the Alfred P. Sloan Research Fellowship.
\section{Construction of a bounded weight function}\label{weight}
\setcounter{equa}{0}
The purpose of this section is to achieve the construction of a bounded weight function whose properties will be summarized below in Proposition~\ref{prop1}. When assuming that the assumptions of Theorem~\ref{theo} are all fulfilled and beginning our construction of this weight function, we shall first work in a small neighborhood{} of a fixed doubly characteristic point of the principal symbol, say for example $X_1 \in p_0^{-1}(0)$. We shall assume, for notational simplicity only, that $X_1=(0,0) \in \mathbb{R}^{2n}$; and drop the index 1 by denoting simply $q$ the quadratic approximation of the principal symbol $p_0$ at $(0,0)$ appearing in (\ref{eq1.6.6}) and $S$ its associated singular space. We may therefore write that
\begin{equation}\label{eq2.0}\stepcounter{equa}
p_0(X)=q(X)+\mathcal{O}(X^3),\tag{\thesection.\theequa}
\end{equation}
when $X \rightarrow 0$; and recall that under the assumptions of Theorem~\ref{theo}, the quadratic form $q$ is assumed to have a non-negative real part, $\textrm{Re }q \geq 0$, and to be elliptic along its singular space $S$.
Under these two assumptions, we established in \cite{HiPr} (see Section~1.4.1 and Proposition~2.0.1) that the singular space $S$ of the quadratic form $q$ has necessarily a symplectic structure and that new symplectic linear coordinates
$$X=(x,\xi)=(x',x'';\xi',\xi'') \in \mathbb{R}^{2n}=\mathbb{R}^{2n'+2n''},$$
can be chosen such that $(x'',\xi'')$ and $(x',\xi')$ are, respectively, some linear symplectic coordinates in $S$ and its symplectic orthogonal space $S^{\sigma \perp}$, so that in these coordinates, the symbol $q$ can be decomposed as the sum of two quadratic forms
\begin{equation}\label{k20b}\stepcounter{equa}
q(x,\xi)=q|_{S^{\sigma \perp}}(x',\xi')+q|_{S}(x'',\xi''), \tag{\thesection.\theequa}
\end{equation}
where the average of the real part of first one by the flow defined by the Hamilton vector field of its imaginary part
\begin{equation}\label{k21b}\stepcounter{equa}
\langle \textrm{Re }q|_{S^{\sigma \perp}}\rangle_{T,\textrm{Im}q|_{S^{\sigma \perp}}}(X')=\frac{1}{T}\int_{0}^{T}{\textrm{Re }q|_{S^{\sigma \perp}}(e^{tH_{\textrm{Im}q|_{S^{\sigma \perp}}}}X')dt}, \tag{\thesection.\theequa}
\end{equation}
with $X'=(x',\xi') \in \mathbb{R}^{2n'}$, is a positive definite quadratic form for all $T>0$; and
\begin{equation}\label{k22b}\stepcounter{equa}
q|_S(x'',\xi'')=i \tilde{\varepsilon}_0 \sum_{j=1}^{n''}{\lambda_j(\xi_j''^2+x_j''^2)}, \tag{\thesection.\theequa}
\end{equation}
with $\tilde{\varepsilon}_0 \in \{\pm 1\}$, $0 \leq n'' \leq n$ and $\lambda_j>0$ for all $j=1,...,n''$.
Here the notation
$$H_f=\frac{\partial f}{\partial \xi}\cdot \frac{\partial}{\partial x} - \frac{\partial f}{\partial x} \cdot \frac{\partial}{\partial \xi},$$
stands for the Hamilton vector field of a $C^1(\mathbb{R}^{2d},\mathbb{C})$ function $f$. More specifically, we checked in the proof of Proposition~2.0.1 in \cite{HiPr} that the two subspaces $S$ and $S^{\sigma \perp}$ are stable by the real and imaginary parts $\textrm{Re }F$ and $\textrm{Im }F$ of the Hamilton map of the symbol $q$. By using that the flow defined by the Hamilton vector field of $\textrm{Im }q$ is the linear transformation, $e^{tH_{\textrm{Im}q}}X=e^{2t\textrm{Im}F}X$, since a direct computation using (\ref{10}) shows that $H_{\textrm{Im}q}=2\textrm{Im }F$, we deduce from (\ref{k20b}), (\ref{k21b}) and (\ref{k22b}) that
\begin{equation}\label{chel3}\stepcounter{equa}
\textrm{Re }q(e^{tH_{\textrm{Im}q}}X)= \textrm{Re }q|_{S^{\sigma \perp}}(e^{tH_{\textrm{Im}q|_{S^{\sigma \perp}}}}X') \tag{\thesection.\theequa}
\end{equation}
and
\begin{equation}\label{chel2}\stepcounter{equa}
\langle \textrm{Re }q\rangle_{T,\textrm{Im}q}(X)\stackrel{\textrm{def}}{=}\frac{1}{T}\int_{0}^{T}{\textrm{Re }q(e^{tH_{\textrm{Im}q}}X)dt}=
\langle \textrm{Re }q|_{S^{\sigma \perp}}\rangle_{T,\textrm{Im}q|_{S^{\sigma \perp}}}(X'), \tag{\thesection.\theequa}
\end{equation}
where $X=(X',X'')$, $X' \in S^{\sigma \perp}$, $X'' \in S$.
We shall now use the following general observation:
\medskip
\begin{prop}\label{prop0}
For each fixed $T>0$, we have
$$\langle{\emph{\textrm{Re }}p_0\rangle}_{T,\emph{\textrm{Im}} p_0}(X)\stackrel{\emph{\textrm{def}}}{=}\frac{1}{T} \int_{0}^{T} \emph{\textrm{Re }}p_0(e^{tH_{\emph{\textrm{Im}}p_0}}X)dt =
\langle{\emph{\textrm{Re }}q\rangle}_{T,\emph{\textrm{Im}}q}(X)+\mathcal{O}(X^3),$$
when $X\rightarrow 0$; where
$$\langle{\emph{\textrm{Re }}q\rangle}_{T,\emph{\textrm{Im}} q}(X)=\frac{1}{T}\int_{0}^{T}\emph{\textrm{Re }} q(e^{tH_{\emph{\textrm{Im}}q}}X)\,dt.$$
\end{prop}
\medskip
\begin{proof}
Let $T>0$ be a fixed constant. We begin by noticing that there exists $c>0$ such that for all $0 \leq t \leq T$ and $X \in \mathbb{R}^{2n}$,
\begin{equation}\label{eq2.6}\stepcounter{equa}
|e^{tH_{\textrm{Im}p_0}}X-X| \leq c t|X| \textrm{ and } |e^{tH_{\textrm{Im}q}}X-X| \leq ct|X|. \tag{\thesection.\theequa}
\end{equation}
Indeed, there exists $C>0$ such that for all $X \in \mathbb{R}^{2n}$,
\begin{equation}\label{serena1}\stepcounter{equa}
|H_{\textrm{Im}p_0}(X)| \leq C|X| \textrm{ and } |H_{\textrm{Im}q}(X)| \leq C|X|,\tag{\thesection.\theequa}
\end{equation}
since $H_{\textrm{Im}q}=2\textrm{Im }F$, $p_0 \in S(1)$ and that $X=0$ is a doubly characteristic point of $p_0$. By writing that
\begin{equation}\label{ju10}\stepcounter{equa}
e^{tH_{\textrm{Im}p_0}}X=X+\int_0^tH_{\textrm{Im}p_0}(e^{sH_{\textrm{Im}p_0}}X)ds, \tag{\thesection.\theequa}
\end{equation}
we notice that
$$|e^{tH_{\textrm{Im}p_0}}X| \leq |e^{tH_{\textrm{Im}p_0}}X-X|+|X| \leq |X| +C\int_0^t{|e^{sH_{\textrm{Im}p_0}}X|ds},$$
induces that
\begin{equation}\label{ju11}\stepcounter{equa}
|e^{tH_{\textrm{Im}p_0}}X| \leq e^{Ct}|X|, \ t \geq 0, \ X \in \mathbb{R}^{2n},\tag{\thesection.\theequa}
\end{equation}
by Gronwall's Lemma. By coming back to (\ref{ju10}), we easily obtain from (\ref{serena1}) the first estimate in (\ref{eq2.6}), the second one being obtained using exactly the same arguments. Then, it directly follows from (\ref{eq2.0}) and (\ref{ju11}) that
$$\langle{\textrm{Re }p_0\rangle}_{T,\textrm{Im}p_0}(X)=\langle{\textrm{Re }q\rangle}_{T,\textrm{Im}p_0}(X)+\mathcal{O}(X^3),$$
so we only need to compare the two flow averages $\langle{\textrm{Re }q\rangle}_{T,\textrm{Im}p_0}$ and $\langle{\textrm{Re }q\rangle}_{T,\textrm{Im}q}$. When doing so, it is sufficient to argue at the level of formal Taylor
expansions. Starting from (\ref{eq2.0}) and writing
\begin{equation}\label{eq2.4}\stepcounter{equa}
\textrm{Im } p_0\sim \sum_{j=1}^{+\infty} \textrm{Im }p_{0,j}, \tag{\thesection.\theequa}
\end{equation}
where for any $j \in \mathbb{N}^*$, the functions $\textrm{Im }p_{0,j}$ are homogeneous of degree $j+1$ in the variables $X=(x,\xi)$, so that in particular we have $\textrm{Im }p_{0,1}=\textrm{Im }q$; we get that
\begin{align*}
\textrm{Re }q(e^{tH_{\textrm{Im}p_0}}X) = & \ \textrm{Re }q(X) + \sum_{k=1}^{+\infty} \frac{t^k}{k!}H_{\textrm{Im}p_0}^k \textrm{Re }q(X) \\
= & \ \textrm{Re }q(X) + \sum_{k=1}^{+\infty}\sum_{j_1=1}^{+\infty}\ldots \sum_{j_k=1}^{+\infty} \frac{t^k}{k!} H_{\textrm{Im}p_{0,j_1}}\ldots H_{\textrm{Im}p_{0,j_k}} \textrm{Re }q(X).
\end{align*}
Here, we are only interested in terms that are homogeneous of degree 2. By noticing that when $f$ is a homogeneous function of degree $j$ then $H_{\textrm{Im}p_{0,\ell}}f$ is a homogeneous function of degree $\ell+j-1$, it follows that the term $H_{\textrm{Im}p_{0,j_1}}\ldots H_{\textrm{Im}p_{0,j_k}}\textrm{Re }q$ is a homogeneous function of degree $j_1+\ldots\,+j_k+2-k$, which has a degree equal to 2 precisely when $j_1+\ldots \, +j_k=k$. We conclude that the quadratic contributions in the term $\textrm{Re }q(e^{tH_{\textrm{Im}p_0}}X)$ only come from the terms when $j_1=\ldots \,=j_k=1$,
$$\frac{t^k}{k!} H_{\textrm{Im}p_{0,1}}^k \textrm{Re }q,$$
for $k\geq 0$. This proves that
$$\textrm{Re }q(e^{tH_{\textrm{Im}p_0}}X)=\textrm{Re }q(e^{tH_{\textrm{Im}q}}X)+\mathcal{O}(X^3),$$
and ends the proof of Proposition~\ref{prop0}.
\end{proof}
\bigskip
\noindent
\textit{Remark.} Alternatively, the statement in Proposition~\ref{prop0} is an easy consequence of (\ref{eq2.6}) and the observation that there exists $\tilde{c}>0$ such that for all $0 \leq t \leq T$ and $X \in \mathbb{R}^{2n}$,
\begin{equation}\label{eq2.5}\stepcounter{equa}
|e^{tH_{\textrm{Im}p_0}}X-e^{tH_{\textrm{Im}q}}X| \leq \tilde{c}t |X|^2, \tag{\thesection.\theequa}
\end{equation}
that we shall have the occasion to use directly later on. Setting $r=\textrm{Im }p_0-\textrm{Im }q$ and writing that
\begin{align*}
e^{tH_{\textrm{Im}p_0}}X-e^{tH_{\textrm{Im}q}}X= & \ \int_0^t\big[H_{\textrm{Im}p_0}(e^{sH_{\textrm{Im}p_0}}X)-H_{\textrm{Im}q}(e^{sH_{\textrm{Im}q}}X)\big]ds \\
= & \ \int_0^tH_{r}(e^{sH_{\textrm{Im}p_0}}X)ds+\int_0^t H_{\textrm{Im}q}(e^{sH_{\textrm{Im}p_0}}X-e^{sH_{\textrm{Im}q}}X)ds,
\end{align*}
since $H_{\textrm{Im}q}$ is a linear map, the estimate (\ref{eq2.5}) directly follows from another use of Gronwall's Lemma together with (\ref{serena1}), (\ref{ju11}) and the fact that $H_r(X)=\mathcal{O}(X^2)$ uniformly on $\mathbb{R}^{2n}$.
\bigskip
We can therefore deduce from (\ref{chel2}) and Proposition~\ref{prop0} that for each fixed $T>0$,
\begin{equation}\label{eq2.7}\stepcounter{equa}
\langle{\textrm{Re }p_0\rangle}_{T,\textrm{Im}p_0}(X) = \widetilde{q}(X')+\mathcal{O}(X^3), \tag{\thesection.\theequa}
\end{equation}
where $\widetilde{q}(X')\stackrel{\textrm{def}}{=}\langle{\textrm{Re }q|_{S^{\sigma \perp}}\rangle}_{T,\textrm{Im}q|_{S^{\sigma \perp}}}(X')\sim |X'|^2$ is the positive definite quadratic form defined in (\ref{k21b}).
Let us now begin our construction of the weight function in a neighborhood{} of the point $(0,0)$. When doing so, we shall follow an idea of~\cite{HeHiSj} (Section~4), and consider $g$ a decreasing $C^{\infty}(\mathbb{R}_+, [0,1])$ function satisfying
\begin{equation}\label{eq2.75}\stepcounter{equa}
g(t)=1, \ t\in [0,1]; \textrm{ and } g(t)=t^{-1}, \ t\geq 2. \tag{\thesection.\theequa}
\end{equation}
Notice that this choice induces that for each $k\in \mathbb{N}$,
\begin{equation}\label{eq2.76}\stepcounter{equa}
g^{(k)}(t)=\mathcal{O}(\langle{t\rangle}^{-1-k}),\tag{\thesection.\theequa}
\end{equation}
when $t \rightarrow +\infty$, where $\langle t \rangle=(1+t^2)^{1/2}$.
Setting
\begin{equation}\label{eq2.8}\stepcounter{equa}
(\textrm{Re }p_0)_{\varepsilon}(X)=g\Big(\frac{|X|^2}{\varepsilon}\Big) \textrm{Re }p_0(X),\tag{\thesection.\theequa}
\end{equation}
for any $\varepsilon >0$, and recalling that $p_0 \in S(1)$; we easily see from (\ref{eq2.0}) that we have the following uniform bound on $\mathbb{R}^{2n}$,
\begin{equation}\label{eq2.80}\stepcounter{equa}
(\textrm{Re }p_0)_{\varepsilon}(X)=\mathcal{O}(\varepsilon),\tag{\thesection.\theequa}
\end{equation}
when $\varepsilon \rightarrow 0$.
Recalling now the well-known inequality
\begin{equation}\label{eq2.81}\stepcounter{equa}
|f'(x)|^2 \leq 2f(x)\|f''\|_{L^{\infty}(\mathbb{R})},\tag{\thesection.\theequa}
\end{equation}
fulfilled by any non-negative smooth function with bounded second derivative, we notice that the estimate $|\nabla \textrm{Re }p_0| =\mathcal{O}\big((\textrm{Re }p_0)^{1/2}\big)$ induces that
\begin{equation}\label{eq2.9}\stepcounter{equa}
\partial_{X}^{\alpha}(\textrm{Re }p_0)_{\varepsilon}(X)=\mathcal{O}(\varepsilon^{1/2}),\tag{\thesection.\theequa}
\end{equation}
for any $\alpha\in \mathbb{N}^{2n}$ with $|\alpha|=1$. By noticing from (\ref{eq2.75}) that
\begin{equation}\label{eq2.9b}\stepcounter{equa}
(\textrm{Re }p_0)_{\varepsilon}=\textrm{Re } p_0, \tag{\thesection.\theequa}
\end{equation}
in the region where $|X|\leq \varepsilon^{1/2}$,
we find from (\ref{eq2.0}) that the bound (\ref{eq2.9}) improves to
\begin{equation}\label{eq2.10}\stepcounter{equa}
\forall \alpha \in \mathbb{N}^{2n}, \ |\alpha|=1, \ \partial_{X}^{\alpha} (\textrm{Re }p_0)_{\varepsilon}(X)=\mathcal{O}(X),\tag{\thesection.\theequa}
\end{equation}
there. One can then check furthermore that for all $\alpha\in \mathbb{N}^{2n}$, with $\abs{\alpha}=2$, we have
\begin{equation}\label{eq2.11}\stepcounter{equa}
\partial_X^{\alpha} (\textrm{Re }p_0)_{\varepsilon}(X)=\mathcal{O}(1), \tag{\thesection.\theequa}
\end{equation}
uniformly on $\mathbb{R}^{2n}$.
\bigskip
\noindent
\textit{Remark}. Bounds on higher derivatives can easily be derived. In the region where $|X|\leq \varepsilon^{1/2}$, we have
\begin{equation}\label{eq2.12}\stepcounter{equa}
\partial_X^{\alpha} (\textrm{Re }p_0)_{\varepsilon}(X)=\mathcal{O}\big(X^{(2-|\alpha|)_+}\big), \tag{\thesection.\theequa}
\end{equation}
for any $\alpha\in \mathbb{N}^{2n}$; while in the region where $|X| \geq \varepsilon^{1/2}$, we check that
\begin{equation}\label{eq2.13}\stepcounter{equa}
\partial_X^{\alpha} (\textrm{Re }p_0)_{\varepsilon}(X)=\mathcal{O}(\varepsilon^{1-|\alpha|/2}).\tag{\thesection.\theequa}
\end{equation}
\bigskip
\noindent
For $T>0$, we define
\begin{equation}\label{eq2.14}\stepcounter{equa}
G_{\varepsilon}(X) = -\int_{-\infty}^{+\infty} J\Big(-\frac{t}{T}\Big) (\textrm{Re }p_0)_{\varepsilon}(e^{tH_{\textrm{Im}p_0}}X)dt, \tag{\thesection.\theequa}
\end{equation}
where $J$ stands for a compactly supported piecewise affine function solving the equation
$$J'(t)=\delta(t)-{\mathrm{1~\hspace{-1.4ex}l}}_{[-1,0]}(t)$$
and ${\mathrm{1~\hspace{-1.4ex}l}}_{[-1,0]}$ the characteristic function of the set $[-1,0]$.
A direct computation as in~\cite{HeHiSj} (Section~4) using an integration by parts gives that
\begin{equation}\label{eq2.15}\stepcounter{equa}
H_{\textrm{Im}p_0} G_{\varepsilon} =\langle{ (\textrm{Re }p_0)_{\varepsilon} \rangle}_{T,\textrm{Im}p_0}-(\textrm{Re }p_0)_{\varepsilon},\tag{\thesection.\theequa}
\end{equation}
where
$$\langle (\textrm{Re }p_0)_{\varepsilon} \rangle_{T,\textrm{Im}p_0}(X)=\frac{1}{T}\int_{0}^{T}{(\textrm{Re }p_0)_{\varepsilon}(e^{tH_{\textrm{Im}p_0}}} X)dt.$$
We notice from (\ref{eq2.80}) and (\ref{eq2.14}) that $G_{\varepsilon}=\mathcal{O}(\varepsilon)$; and that the estimates (\ref{eq2.9}) and (\ref{eq2.11}) hold for the derivatives of the function $G_{\varepsilon}$ as well. Let us also notice from (\ref{eq2.6}), (\ref{eq2.75}), (\ref{eq2.8}) and (\ref{eq2.14}) that we have in the region, where $|X|^2 \leq \varepsilon/2$, that
\begin{equation}\label{eq2.15.1}\stepcounter{equa}
G_{\varepsilon}(X) = -\int_{-\infty}^{+\infty}J\Big(-\frac{t}{T}\Big) \textrm{Re }p_0(e^{tH_{\textrm{Im}p_0}}X)dt, \tag{\thesection.\theequa}
\end{equation}
fulfills
\begin{equation}\label{chel5}\stepcounter{equa}
H_{\textrm{Im}p_0}G_{\varepsilon}(X)=\langle{\textrm{Re }p_0 \rangle}_{T,\textrm{Im}p_0}(X)-\textrm{Re }p_0(X);\tag{\thesection.\theequa}
\end{equation}
while we have, where $|X|^2 \geq 8\varepsilon$, that
\begin{equation}\label{eq2.15.2}\stepcounter{equa}
G_{\varepsilon}(X) = - \int_{-\infty}^{+\infty}J\Big(-\frac{t}{T}\Big) \frac{\varepsilon \textrm{Re }p_0(e^{tH_{\textrm{Im}p_0}}X)}{|e^{tH_{\textrm{Im}p_0}}X|^2}dt, \tag{\thesection.\theequa}
\end{equation}
provided that the constant $T>0$ is chosen fixed sufficiently small.
\bigskip
\noindent
\textit{Remark.} We notice from (\ref{chel3}), (\ref{eq2.15.1}) and the proof of Proposition~\ref{prop0} that in the region, where $|X|^2\leq \varepsilon/2$, we have
\begin{equation}\label{keq}\stepcounter{equa}
G_{\varepsilon}(X)=G^o(X')+\mathcal{O}(X^3),\tag{\thesection.\theequa}
\end{equation}
where
\begin{equation}\label{eq2.15.2.1}\stepcounter{equa}
G^o(X')=-\int_{-\infty}^{+\infty}J\Big(-\frac{t}{T}\Big) \textrm{Re }q|_{S^{\sigma \perp}}(e^{tH_{\textrm{Im}q|_{S^{\sigma \perp}}}}X')dt, \tag{\thesection.\theequa}
\end{equation}
is a quadratic form in the variables $X'=(x',\xi')$.
\bigskip
When considering a constant $0<\delta \ll 1$ whose value will be chosen later on, we get that
\begin{equation}\label{eq2.15.3}\stepcounter{equa}
\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)=p_0(X)+i \delta H_{G_{\varepsilon}} p_0 + \mathcal{O}\left(\delta^2 |\nabla G_{\varepsilon}|^2\right), \tag{\thesection.\theequa}
\end{equation}
when $X\in \mathbb{R}^{2n}$ varies in a small neighborhood{} of $0$; if we denote by $\widetilde{p}_0$ an almost analytic extension of the symbol $p_0$, which is bounded together with all its derivatives in a fixed tubular neighborhood{} of $\mathbb{R}^{2n}$. Writing
\begin{equation}\label{eq2.16}\stepcounter{equa}
\textrm{Re}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right)=\textrm{Re }p_0+\delta H_{\textrm{Im}p_0}G_{\varepsilon}+\mathcal{O}\left(\delta^2 |\nabla G_{\varepsilon}|^2\right) \tag{\thesection.\theequa}
\end{equation}
and
\begin{equation}\label{eq2.17}\stepcounter{equa}
\textrm{Im}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right) = \textrm{Im }p_0+\delta H_{G_{\varepsilon}} \textrm{Re }p_0+\mathcal{O}\left(\delta^2 |\nabla G_{\varepsilon}|^2\right), \tag{\thesection.\theequa}
\end{equation}
we shall first consider the region, where $|X|^2\leq \varepsilon/2$. Then, by using (\ref{eq2.7}), (\ref{chel5}), as well as the fact that from (\ref{keq}), the estimate
\begin{equation}\label{chel6}\stepcounter{equa}
\nabla G_{\varepsilon}(X)=\mathcal{O}(X), \tag{\thesection.\theequa}
\end{equation}
is fulfilled in this region;
we get from (\ref{eq2.16}) that
\stepcounter{equa}\begin{align*}\label{eq2.18}
\textrm{Re}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right) = & \ \textrm{Re }p_0+\delta\big(\langle{\textrm{Re }p_0}\rangle_{T,\textrm{Im}p_0}-\textrm{Re }p_0\big)+\mathcal{O}\left(\delta^2 X^2\right) \tag{\thesection.\theequa} \\
= & \ (1-\delta)\textrm{Re }p_0+\delta \widetilde{q}(X')+\mathcal{O}\left(\delta |X|^3+\delta^2 |X|^2\right).
\end{align*}
By noticing that (\ref{chel6}) implies that $H_{G_{\varepsilon}} \textrm{Re }p_0(X)=\mathcal{O}(X^2)$, since $0$ is a doubly characteristic point for the symbol $p_0$; we get from (\ref{eq2.0}), (\ref{k20b}), (\ref{eq2.17}) and (\ref{chel6}) that
$$\delta \textrm{Im}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right)=\delta \textrm{Im }p_0 + \mathcal{O}\left(\delta^2 X^2\right)=\delta \textrm{Im }q + \mathcal{O}\left(\delta^2 |X|^2+\delta |X|^3\right),$$
then
\stepcounter{equa}\begin{multline*}\label{eq2.19}
i\delta \textrm{Im}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right)=i\delta \textrm{Im }q|_{S^{\sigma \perp}}(X')+i\delta \textrm{Im }q|_S(X'')\\ +\mathcal{O}\left(\delta^2 |X|^2+\delta |X|^3\right),\tag{\thesection.\theequa}
\end{multline*}
when $|X|^2\leq \varepsilon/2$. By combining (\ref{eq2.18}) and (\ref{eq2.19}), we get that
\stepcounter{equa}\begin{multline*}\label{eq2.20}
\textrm{Re}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right)+i\delta \textrm{Im}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right) \tag{\thesection.\theequa}
= (1-\delta)\textrm{Re }p_0\\ +\delta \big(\widetilde{q}(X')+i \textrm{Im }q|_{S^{\sigma \perp}}(X')+i \textrm{Im }q|_S(X'')\big) +\mathcal{O}\left(\delta^2 |X|^2+\delta |X|^3\right).
\end{multline*}
Let us notice that the quadratic form
$$X=(X',X'')\mapsto \widetilde{q}(X')+i \textrm{Im }q|_{S^{\sigma \perp}}(X')+i \textrm{Im }q|_S(X''),$$
is elliptic on $\mathbb{R}^{2n}$. This comes from the facts that on one hand, the quadratic form $\widetilde{q}$ defined in (\ref{eq2.7}) is positive definite in the variables $X'$; and that on the other hand the quadratic form $\textrm{Im }q|_{S}$ defined in (\ref{k22b}) is also obviously elliptic in the variables $X''$. Combining this with the fact that, $(1-\delta)\textrm{Re }p_0\geq 0$, if $0 <\delta \leq 1$; we obtain from (\ref{eq2.20}) that there exists a positive constant $\widehat{C}>0$ such that we have in the region
where $\abs{X}^2\leq \varepsilon/2$,
\begin{equation}\label{eq2.21}\stepcounter{equa}
\big|\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\big|\geq \frac{\delta |X|^2}{\widehat{C}}-\mathcal{O}(\delta^2 X^2)-\mathcal{O}(\delta X^3)\geq \frac{\delta \abs{X}^2}{2\widehat{C}},\tag{\thesection.\theequa}
\end{equation}
when $0<\delta \leq \delta_0$ and $0<\varepsilon \leq \varepsilon_0$, if the positive constants $\delta_0$ and $\varepsilon_0$ are chosen sufficiently small. This comes from the fact that one can estimate from below the quantity
\begin{align*}
& \ \big|(1-\delta)\textrm{Re }p_0 +\delta \big(\widetilde{q}(X')+i \textrm{Im }q|_{S^{\sigma \perp}}(X')+i \textrm{Im }q|_S(X'')\big)\big|\\
=& \ \Big[\big((1-\delta)\textrm{Re }p_0 +\delta \widetilde{q}(X')\big)^2+ \delta^2 \big(\textrm{Im }q|_{S^{\sigma \perp}}(X')+\textrm{Im }q|_S(X'')\big)\big)^2\Big]^{\frac{1}{2}}
\end{align*}
by the quantity
\begin{align*}
& \ \big[\delta^2 \widetilde{q}(X')^2+ \delta^2 \big(\textrm{Im }q|_{S^{\sigma \perp}}(X')+\textrm{Im }q|_S(X'')\big)^2\big]^{\frac{1}{2}}\\
=& \ \delta |\widetilde{q}(X')+i \textrm{Im }q|_{S^{\sigma \perp}}(X')+i \textrm{Im }q|_S(X'')|,
\end{align*}
since $\widetilde{q} \geq 0$.
We now consider a region of the phase space where $X$ belongs to a fixed neighborhood of $0$ and where $\abs{X}^2 \geq 8\varepsilon$. It follows from (\ref{eq2.0}), (\ref{eq2.15}), (\ref{eq2.16}) and (\ref{eq2.17}) that
\begin{equation}\label{eq2.22}\stepcounter{equa}
\textrm{Re}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right)=\textrm{Re }p_0+\delta \big(\langle{(\textrm{Re }p_0)_{\varepsilon}\rangle}_{T,\textrm{Im}p_0} -(\textrm{Re }p_0)_{\varepsilon}\big)+\mathcal{O}(\delta^2 \varepsilon)\tag{\thesection.\theequa}
\end{equation}
and
\stepcounter{equa}\begin{align*}\label{eq2.23}
\textrm{Im}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right)= & \ \textrm{Im }p_0+\delta H_{G_{\varepsilon}}\textrm{Re }p_0+\mathcal{O}(\delta^2 \varepsilon) \tag{\thesection.\theequa} \\
= & \ \textrm{Im }p_0+\mathcal{O}(\delta \varepsilon^{1/2} |X| +\delta^2 \varepsilon)\\
= & \ \textrm{Im }q+\mathcal{O}(\delta \varepsilon^{1/2}|X|+|X|^3),
\end{align*}
since, according to our construction of the weight function, $\nabla G_{\varepsilon} =\mathcal{O}(\varepsilon^{1/2})$; while we can use that $\nabla \textrm{Re }p_0(X)=\mathcal{O}(X)$, because $0$ is a doubly characteristic point of the symbol $p_0$ and that $p_0 \in S(1)$. To understand the right hand side of (\ref{eq2.22}), we first notice from (\ref{eq2.6}), (\ref{eq2.75}), (\ref{eq2.8}) and the proof of Proposition~\ref{prop0} that
\begin{align*}
\langle{(\textrm{Re }p_0)_{\varepsilon}\rangle}_{T,\textrm{Im}p_0}(X)=& \ \varepsilon\frac{1}{T}\int_0^T \frac{\textrm{Re }p_0(e^{tH_{\textrm{Im}p_0}}X)}{|e^{tH_{\textrm{Im}p_0}}X|^2}dt \\
= & \ \varepsilon \frac{1}{T}\int_0^T \frac{\textrm{Re }q(e^{tH_{\textrm{Im}q}}X)} {|e^{tH_{\textrm{Im}p_0}}X|^2}dt+\mathcal{O}(\varepsilon X),
\end{align*}
provided that $T>0$ is chosen sufficiently small. When simplifying the denominator in the right hand side of the previous equation, we use (\ref{eq2.6}) and (\ref{eq2.5}) to get that
$$\frac{1}{|e^{tH_{\textrm{Im}p_0}}X|^2}-\frac{1}{|e^{tH_{\textrm{Im}q}}X|^2}=\mathcal{O}\Big(\frac{t\abs{X}^3}{\abs{X}^4}\Big),$$
and then obtain that
\begin{equation}\label{eq2.23.1}\stepcounter{equa}
\langle{(\textrm{Re }p_0)_{\varepsilon}\rangle}_{T,\textrm{Im}p_0}(X)=\varepsilon \frac{1}{T}\int_0^T \frac{\textrm{Re }q(e^{tH_{\textrm{Im}q}}X)} {|e^{tH_{\textrm{Im}q}}X|^2}dt+\mathcal{O}(\varepsilon X), \tag{\thesection.\theequa}
\end{equation}
since $\textrm{Re }q(e^{tH_{\textrm{Im}q}}X)$ is a quadratic form.
Considering the following non-negative homogeneous function of degree $0$,
$$f(X)=\frac{1}{T}\int_0^T \frac{\textrm{Re }q(e^{tH_{\textrm{Im}q}}X)}{|e^{tH_{\textrm{Im}q}}X|^2}dt,$$
we notice from (\ref{chel3}) and (\ref{eq2.7}) that, if $f(X)=0$ then we necessarily have $\widetilde{q}(X')=0$, and therefore $X'=0$, since $\widetilde{q}$ is a positive definite quadratic form.
We may even be more precise and observe that it follows from (\ref{eq2.6}) that
\begin{equation}\label{eq2.23.2}\stepcounter{equa}
f(X)\geq \frac{1}{\mathcal{O}(1)}\frac{|X'|^2}{|X|^2},\ X'=(x',\xi')\in S^{\sigma \perp}. \tag{\thesection.\theequa}
\end{equation}
It follows from (\ref{eq2.22}) and (\ref{eq2.23.1}) that
\begin{equation}\label{eq2.24}\stepcounter{equa}
\textrm{Re}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right)=\textrm{Re }p_0-\delta (\textrm{Re }p_0)_{\varepsilon}+\delta \varepsilon f(X)+\mathcal{O}(\delta^2 \varepsilon+\delta \varepsilon |X|). \tag{\thesection.\theequa}
\end{equation}
On the other hand, we deduce from (\ref{k20b}) and (\ref{eq2.23}) that
\begin{align*}
\textrm{Im}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right)= & \ \textrm{Im }q(X)+\mathcal{O}\big(\delta \varepsilon^{1/2} |X| +|X|^3\big) \\
= & \ \textrm{Im }q|_{S^{\sigma \perp}}(X')+\textrm{Im }q|_S(X'')+\mathcal{O}\big(\delta \varepsilon^{1/2} |X| +|X|^3\big),
\end{align*}
which implies, when $\abs{X}^2 \geq 8\varepsilon$, that
\stepcounter{equa}\begin{multline*}\label{eq2.25}
\frac{\delta \varepsilon}{|X|^2}\textrm{Im}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right)= \frac{\delta \varepsilon}{|X|^2}
\big(\textrm{Im }q|_{S^{\sigma \perp}}(X')+\textrm{Im }q|_S(X'')\big) \\ +\mathcal{O}(\delta^2 \varepsilon+\delta\varepsilon |X|). \tag{\thesection.\theequa}
\end{multline*}
It follows from (\ref{eq2.24}) and (\ref{eq2.25}) that
\stepcounter{equa}\begin{multline*}\label{eq2.25.1}
\textrm{Re}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right)+i\frac{\delta \varepsilon}{|X|^2}\textrm{Im}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right) \tag{\thesection.\theequa}
= \textrm{Re }p_0\\ -\delta (\textrm{Re }p_0)_{\varepsilon} +\frac{\delta \varepsilon}{|X|^2}\left(|X|^2 f(X)+i \textrm{Im }q|_{S^{\sigma\perp}}(X')+i \textrm{Im }q|_S(X'')\right)+\mathcal{O}\left(\delta^2 \varepsilon+\delta \varepsilon|X|\right),
\end{multline*}
where the continuous function of $X\neq 0$,
$$X \mapsto |X|^2 f(X)+i \textrm{Im }q|_{S^{\sigma\perp}}(X')+i \textrm{Im }q |_S(X''),$$
is homogeneous of degree 2 and does not vanish when $|X|=1$. This comes from (\ref{k22b}) and the fact that $f(X)=0$ implies that $X'=0$. By using from (\ref{eq2.75}) and (\ref{eq2.8}) that
$$\textrm{Re }p_0-\delta (\textrm{Re }p_0)_{\varepsilon}\geq 0,$$
when $0<\delta \leq 1$; and by possibly considering a smaller constant $0<\delta_0 \leq 1$,
we deduce from (\ref{eq2.25.1}) that there exist some positive constants $C>1$ and $\tilde{C}>1$ such that for all $0<\delta \leq \delta_0$, $0<\varepsilon \leq \varepsilon_0$ and $8 \varepsilon \leq |X|^2\leq 1/C$; we have
\begin{equation}\label{eq2.26}\stepcounter{equa}
\big|\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\big| \geq \frac{\delta \varepsilon}{\tilde{C}}.\tag{\thesection.\theequa}
\end{equation}
This follows from the fact that one can estimate from below the quantity
\begin{align*}
& \ \Big|\textrm{Re }p_0-\delta (\textrm{Re }p_0)_{\varepsilon}+\frac{\delta \varepsilon}{|X|^2}\big(|X|^2 f(X)+i \textrm{Im }q|_{S^{\sigma\perp}}(X')+i \textrm{Im }q|_S(X'')\big)\Big|\\
= & \ \Big[\big(\textrm{Re }p_0-\delta (\textrm{Re }p_0)_{\varepsilon}+\delta \varepsilon f(X)\big)^2+ \frac{\delta^2 \varepsilon^2}{|X|^4}\big(\textrm{Im }q|_{S^{\sigma\perp}}(X')+ \textrm{Im }q|_S(X'')\big)^2\Big]^{\frac{1}{2}},
\end{align*}
by
\begin{align*}
& \ \Big[\delta^2 \varepsilon^2 f(X)^2+ \frac{\delta^2 \varepsilon^2}{|X|^4}\big(\textrm{Im }q|_{S^{\sigma\perp}}(X')+ \textrm{Im }q|_S(X'')\big)^2\Big]^{\frac{1}{2}}\\
=& \ \Big|\frac{\delta \varepsilon}{|X|^2}\big(|X|^2 f(X)+i \textrm{Im }q|_{S^{\sigma\perp}}(X')+i \textrm{Im }q|_S(X'')\big)\Big|,
\end{align*}
since $f(X) \geq 0$.
Recalling (\ref{k20b}) and (\ref{k22b}), we notice that $\tilde{\varepsilon}_0\textrm{Im }q|_S$ is a positive definite quadratic form
\begin{equation}\label{fa1}\stepcounter{equa}
\tilde{\varepsilon}_0 \textrm{Im }q|_S(X'')\gtrsim |X''|^2,\tag{\thesection.\theequa}
\end{equation}
while
\begin{equation}\label{fa2}\stepcounter{equa}
\tilde{\varepsilon}_0\textrm{Im }q|_{S^{\sigma \perp}}(X')=\mathcal{O}(X'^2).\tag{\thesection.\theequa}
\end{equation}
Continuing to work in the region where $8 \varepsilon \leq |X|^2 \leq 1/C$ and writing
\stepcounter{equa}\begin{multline*}\label{eq2.24.2}
\textrm{Re}\Big(\Big(1-i\tilde{\varepsilon}_0\frac{\delta \eta \varepsilon}{|X|^2}\Big)\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\Big) = \textrm{Re}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right)
\\ +\tilde{\varepsilon}_0\frac{\delta \eta \varepsilon}{|X|^2} \textrm{Im}\left(\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\right), \tag{\thesection.\theequa}
\end{multline*}
where $0< \eta \leq 1$ stands for a constant whose value will be chosen later on, it follows from (\ref{eq2.24}) and (\ref{eq2.25}) that
\stepcounter{equa}\begin{multline*}\label{eq2.25.1bis}
\textrm{Re}\Big(\Big(1-i\tilde{\varepsilon}_0\frac{\delta \eta \varepsilon}{|X|^2}\Big)\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\Big) =\textrm{Re }p_0(X)-\delta (\textrm{Re }p)_{\varepsilon}(X)\\ +\frac{\delta \varepsilon}{|X|^2}\left(|X|^2 f(X)+ \eta \tilde{\varepsilon}_0\textrm{Im }q|_S(X'') + \eta \tilde{\varepsilon}_0\textrm{Im }q|_{S^{\sigma \perp}}(X')\right)
+\mathcal{O}(\delta^2 \varepsilon+\delta \varepsilon |X|). \tag{\thesection.\theequa}
\end{multline*}
By noticing from (\ref{eq2.23.2}), (\ref{fa1}) and (\ref{fa2}) that the degree 2 homogeneous continuous function of the variable $X=(X',X'')\neq 0$,
$$X \mapsto |X|^2 f(X) + \eta \tilde{\varepsilon}_0\textrm{Im }q |_S(X'')+\eta \tilde{\varepsilon}_0\textrm{Im }q|_{S^{\sigma \perp}}(X'),$$
can be estimated from below by
$$|X|^2 f(X) + \eta \tilde{\varepsilon}_0\textrm{Im }q |_S(X'')+\eta \tilde{\varepsilon}_0\textrm{Im }q|_{S^{\sigma \perp}}(X') \geq \frac{|X|^2}{\mathcal{O}(1)},$$
provided that the constant $0 <\eta \leq 1$ is chosen sufficiently small. By using again that
$$\textrm{Re }p_0-\delta (\textrm{Re }p_0)_{\varepsilon}\geq 0,$$
we get from (\ref{eq2.25.1bis}) after possibly decreasing the value of the constant $0<\delta_0 \leq 1$ and increasing the value of the constant $C>1$ that for all $0<\delta \leq \delta_0$ and $0<\varepsilon\leq \varepsilon_0$,
\begin{equation}\label{eq2.26bis}\stepcounter{equa}
\textrm{Re}\Big(\Big(1-i\tilde{\varepsilon}_0\frac{\delta \eta \varepsilon}{|X|^2}\Big)\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}} (X)\big)\Big) \geq \frac{\delta \varepsilon}{\mathcal{O}(1)}, \tag{\thesection.\theequa}
\end{equation}
when $8 \varepsilon \leq |X|^2\leq 1/C$.
Having established the estimates (\ref{eq2.21}), (\ref{eq2.26}) and (\ref{eq2.26bis}), it finally remains to consider the intermediate region where $\varepsilon/2 \leq |X|^2 \leq 8\varepsilon$. We notice that in this region, the estimate (\ref{eq2.22}) also holds true, while we may write by using (\ref{chel3}) and Proposition~\ref{prop0} that
\begin{align*}
\langle{(\textrm{Re }p_0)_{\varepsilon}\rangle}_{T,\textrm{Im}p_0}(X) = & \ \frac{1}{T}\int_0^T g\Big(\frac{|e^{tH_{\textrm{Im}p_0}}X|^2}{\varepsilon}\Big) \textrm{Re }q\left(e^{tH_{\textrm{Im}q}}X\right)dt+\mathcal{O}(X^3)\\
= & \ \frac{1}{T}\int_0^T g\Big(\frac{|e^{tH_{\textrm{Im}p_0}}X|^2}{\varepsilon}\Big) \textrm{Re }q|_{S^{\sigma \perp}}(e^{tH_{\textrm{Im}q|_{S^{\sigma \perp}}}}X')dt+\mathcal{O}(X^3),
\end{align*}
where we have here
$$g\Big(\frac{|e^{tH_{\textrm{Im}p_0}}X|^2}{\varepsilon}\Big) \sim 1,$$
uniformly with respect to the parameters $0<\varepsilon \leq \varepsilon_0$ and $t\in [0,T]$; if the positive constant $T$ is chosen sufficiently small. While using the same arguments as before, it is then easy to check that the estimates (\ref{eq2.26}) and (\ref{eq2.26bis}) also hold in this region.
To summarize our discussion so far, we have shown that there exist some positive constants $C>1$, $\tilde{C}>1$, $\tilde{c} \neq 0$, $0<\varepsilon_0 \leq 1$ and $0<\delta_0 \leq 1$ such that the local $C^{\infty}$ weight function
$$G_{1,\varepsilon}\stackrel{\textrm{def}}{=}G_{\varepsilon},$$
defined in a neighborhood of $0$
satisfies for all $0 < \varepsilon \leq \varepsilon_0$ and $0<\delta \leq \delta_0$ that
\begin{equation}\label{eq2.27}\stepcounter{equa}
\abs{\widetilde{p}_0\big(X+i\delta H_{G_{1,\varepsilon}}(X)\big)}\geq \frac{\delta}{\tilde{C}} {\rm min}(|X|^2,\varepsilon), \tag{\thesection.\theequa}
\end{equation}
when $|X| \leq 1/C$; and
\begin{equation}\label{fan10}\stepcounter{equa}
\textrm{Re}\Big(\Big(1-i\tilde{c}\frac{\delta \varepsilon}{|X|^2}\Big)\widetilde{p}_0\big(X+i\delta H_{G_{1,\varepsilon}} (X)\big)\Big) \geq \frac{\delta \varepsilon}{\tilde{C}}, \tag{\thesection.\theequa}
\end{equation}
when $\varepsilon^{1/2} \leq |X| \leq 1/C$.
Proceeding similarly and working near each of the other doubly characteristic points $X_j\in p_0^{-1}(0)$, we obtain locally defined weight functions
$$G_{j,\varepsilon}\in C^{\infty}({\rm neigh}(X_j,\mathbb{R}^{2n})),$$
for $2\leq j\leq N$, so that the natural analogues of (\ref{eq2.27}) and (\ref{fan10}) hold for $G_{j,\varepsilon}$ in the regions where respectively $|X-X_j|\leq 1/C$ and $\varepsilon^{1/2} \leq |X-X_j| \leq 1/C$.
To obtain a definition of the global weight function that we shall denote again $G_{\varepsilon}$, we consider small open sets $\Omega_j \subset \mathbb{R}^{2n}$, $1\leq j\leq N$, with $X_j\in \Omega_j$ and
$\overline{\Omega_j}\cap \overline{\Omega_k}=\emptyset$, when $j\neq k$.
By taking some $C^{\infty}_0(\Omega_j,[0,1])$ functions $\chi_j$ such that $\chi_j=1$ in a neighborhood{} of the doubly characteristic point $X_j$, we define the global $C^{\infty}_0(\mathbb{R}^{2n})$ weight function
$$G_{\varepsilon} = \sum_{j=1}^N \chi_j G_{j,\varepsilon},$$
and by restricting our attention to a fixed open set $\Omega_j$, we notice from (\ref{eq1.5}) and (\ref{eq1.6}) that the support of the functions $\nabla \chi_j$ is contained in a region where the real part of the principal symbol $p_0$ is elliptic
$$\textrm{Re }p_0(X)\geq 1/\mathcal{O}(1).$$
We have therefore proved the following result which sums up the whole discussion led in this section.
\bigskip
\begin{prop}\label{prop1}
Let $\widetilde{p}_0$ be an almost analytic extension of the principal symbol $p_0$ of the symbol $P(x,\xi;h)$ considered in Theorem~\emph{\ref{theo}} to a tubular neighborhood{} of $\mathbb{R}^{2n}$; bounded together with all its derivatives in this neighborhood. Under the assumptions of Theorem~\emph{\ref{theo}}, one can find some constants
$$C>1, \ \tilde{C}>1, \ 0<\delta_0 \leq 1, \ 0<\varepsilon_0 \leq 1,\ \tilde{c} \neq 0;$$
and a $C^{\infty}_0(\mathbb{R}^{2n},\mathbb{R})$ weight function $G_{\varepsilon}$ depending on the parameter $0<\varepsilon \leq \varepsilon_0$ and supported in a neighborhood of the doubly characteristic set
$${\rm supp} \ G_{\varepsilon} \subset \big\{X \in \mathbb{R}^{2n} : {\rm dist}\big(X,p_0^{-1}(0)\big)\leq 2/C\big\},$$
such that we have uniformly for all $0< \varepsilon \leq \varepsilon_0$ and $0<\delta \leq \delta_0$ that
\begin{enumerate}
\item[$(i)$] $G_{\varepsilon}=\mathcal{O}(\varepsilon)$, $\partial^{2}G_{\varepsilon}=\mathcal{O}(1)$ on $\mathbb{R}^{2n}$
\item[$(ii)$] $\nabla G_{\varepsilon}=\mathcal{O}\big({\rm dist}\big(X,p_0^{-1}(0)\big)\big)$ in the region where ${\rm dist}\big(X,p_0^{-1}(0)\big) \leq \varepsilon^{1/2}$
\item[$(iii)$] $\nabla G_{\varepsilon}=\mathcal{O}(\varepsilon^{1/2})$ in the region where ${\rm dist}\big(X,p_0^{-1}(0)\big)\geq \varepsilon^{1/2}$
\item[$(iv)$] We have
$$\big|\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)\big| \geq \frac{\delta}{\tilde{C}}\emph{\textrm{min}}\big[{\rm dist}\big(X,p_0^{-1}(0)\big)^2,\varepsilon\big],$$
in the region where ${\rm dist}\big(X,p_0^{-1}(0)\big)\leq 1/C$
\item[$(v)$] We have
$$\emph{\textrm{Re}}\Big(\Big(1-i \tilde{c}\frac{\delta \varepsilon}{{\rm dist}\big(X,p_0^{-1}(0)\big)^2}\Big){\widetilde{p}_0\big(X+i\delta H_{G_{\varepsilon}}(X)\big)}\Big) \geq \frac{\delta \varepsilon}{\widetilde{C}}$$
in the region where ${\rm dist}\big(X,p_0^{-1}(0)\big) \geq \varepsilon^{1/2}$
\end{enumerate}
\end{prop}
\section{Review of FBI tools}\label{fbi}
\setcounter{equa}{0}
The purpose of this section is to recall the definition of the weighted spaces of holomorphic functions associated to the weight function $G_{\varepsilon}$ constructed in the previous section; and the action of the operator $P$ in these spaces. The following discussion will very much follow the corresponding discussion in~\cite{HeSjSt} (Section 3) and will therefore be somewhat brief.
Throughout this paper, we shall work with the usual semiclassical FBI-Bargmann transform
\begin{equation}\label{eq3.1}\stepcounter{equa}
Tu (x) = \widetilde{C} h^{-3n/4} \int_{\mathbb{R}^{2n}} e^{\frac{i}{h}\varphi(x,y)} u(y)dy,\ x\in \mathbb{C}^n,\ \widetilde{C}>0,\tag{\thesection.\theequa}
\end{equation}
with the phase $\varphi(x,y)=\frac{i}{2}(x-y)^2$. Associated to the FBI-Bargmann transform $T$, there is the linear canonical transformation
\begin{equation}\label{eq3.2}\stepcounter{equa}
\kappa_T: T^*\mathbb{C}^n \ni (y,\eta)\mapsto (x,\xi)=(y-i\eta,\eta)\in T^*\mathbb{C}^n, \tag{\thesection.\theequa}
\end{equation}
mapping the real phase space $T^*\mathbb{R}^n$ onto the IR-manifold
\begin{equation}\label{eq3.2.1}\stepcounter{equa}
\Lambda_{\Phi_0}=\Big\{ \Big(x,\frac{2}{i} \frac{\partial \Phi_0}{\partial x}(x)\Big) : x \in \mathbb{C}^n \Big\},\tag{\thesection.\theequa}
\end{equation}
where
\begin{equation}\label{eq3.2.1b}\stepcounter{equa}
\Phi_0(x) =\frac{1}{2} \left({\rm Im\,} x\right)^2. \tag{\thesection.\theequa}
\end{equation}
We recall that for a suitable choice of the constant $\widetilde{C}>0$ in (\ref{eq3.1}), the transformation
\begin{equation}\label{eq3.3}\stepcounter{equa}
T: L^2(\mathbb{R}^n) \rightarrow H_{\Phi_0}(\mathbb{C}^n) \tag{\thesection.\theequa}
\end{equation}
is unitary; where here and in what follows, when $\Phi \in C^{\infty}(\mathbb{C}^n)$ is a suitable strictly plurisubharmonic weight function close to $\Phi_0$, we shall let $H_{\Phi}(\mathbb{C}^n)$ stand for the closed subspace of $L^2(\mathbb{C}^n,e^{-\frac{2\Phi}{h}}L(dx))$, consisting of functions that are entire holomorphic. The integration element $L(dx)$ stands here for the Lebesgue measure on $\mathbb{C}^n$.
We have an exact version of the Egorov theorem which says that
\begin{equation}\label{eq3.4}\stepcounter{equa}
T a^w T^{-1} = \mathfrak{a}^w, \ a\in S(1), \tag{\thesection.\theequa}
\end{equation}
where the symbol $\mathfrak{a}\in S(\Lambda_{\Phi_0},1)$ is given by the formula
$$\mathfrak{a} = a\circ \kappa_T^{-1},$$
where $\kappa_T$ is the linear canonical transformation defined in (\ref{eq3.2}).
Here the Weyl quantization of the symbol $\mathfrak{a}\in S(\Lambda_{\Phi_0},1)$, given by
$$\mathfrak{a}^w (x,hD_x) u(x) = \frac{1}{(2\pi h)^n} \int\!\!\!\int_{\Gamma(x)} e^{\frac{i}{h}(x-y)\cdot \theta}\mathfrak{a}\Big(\frac{x+y}{2},\theta\Big) u(y)dy d\theta,$$
where $\Gamma(x)$ is the contour $\theta = \frac{2}{i}\frac{\partial \Phi_0}{\partial x}\left(\frac{x+y}{2}\right)$, gives rise to a uniformly bounded operator on $H_{\Phi_0}(\mathbb{C}^n)$. We refer to~\cite{HeSjSt} (Section~3), as well as to~\cite{Sj95}
for a proof of this fact, based on a contour deformation argument for an almost holomorphic extension of the symbol $\mathfrak{a}$ to a tubular neighborhood{} of $\Lambda_{\Phi_0}$ in $\mathbb{C}^{2n}$.
Applying the remarks above to the operator $P=P^w(x,hD_x;h)$ appearing in Theorem~\ref{theo} and setting ${P}_0=T P T^{-1}$, we get that
$${P}_0 =\mathcal{O}(1): H_{\Phi_0}(\mathbb{C}^n) \rightarrow H_{\Phi_0}(\mathbb{C}^n),$$
if $P_0$ stands for the operator $P_0^w(x,hD_x;h)$ whose symbol has the following semiclassical asymptotic expansion
\begin{equation}\label{xi13}\stepcounter{equa}
P_0(x,\xi;h) \sim \sum_{j=0}^{+\infty}{\mathfrak{p}_j(x,\xi) h^j},\tag{\thesection.\theequa}
\end{equation}
where $\mathfrak{p}_j=p_j\circ \kappa_T^{-1}$, for any $j \geq 0$.
Then, associated with the weight function $G_{\varepsilon}$ constructed in Section~\ref{weight}, there is the IR-manifold
\begin{equation}\label{eq3.4.5}\stepcounter{equa}
\Lambda_{\delta,\varepsilon} = \left\{X+i\delta H_{G_{\varepsilon}}(X) : X\in \mathbb{R}^{2n} \right \}, \tag{\thesection.\theequa}
\end{equation}
for $0 \leq \delta \leq 1$ and $0<\varepsilon \leq \varepsilon_0$.
Arguing as in~\cite{HeSjSt}, we see that we have
\begin{equation}\label{eq3.5}\stepcounter{equa}
\kappa_T\left(\Lambda_{\delta,\varepsilon}\right)=\Lambda_{\Phi_{\delta,\varepsilon}}\stackrel{\textrm{def}}{=}\Big\{\Big(x,\frac{2}{i} \frac{\partial \Phi_{\delta,\varepsilon}}{\partial x}(x)\Big) : x \in \mathbb{C}^{2n} \Big\},\tag{\thesection.\theequa}
\end{equation}
when $0<\delta \ll 1$; where the function $\Phi_{\delta,\varepsilon}(x)$ is the critical value with respect to the variables $(y,\eta)$ of the following functional
\begin{equation}\label{ju2}\stepcounter{equa}
\Phi_{\delta,\varepsilon}(x)=\textrm{v.c.}_{(y,\eta) \in \mathbb{C}^n \times \mathbb{R}^n}\big(-\textrm{Im }\varphi(x,y)-(\textrm{Im }y).\eta+\delta G_{\varepsilon}(\textrm{Re }y,\eta)\big).\tag{\thesection.\theequa}
\end{equation}
Following the discussion led in \cite{HeSjSt} (Section~3.2), one can check that
$\Phi_{\delta,\varepsilon} \in C^{\infty}(\mathbb{C}^n)$ is a strictly plurisubharmonic function satisfying
\begin{equation}\label{eq3.6}\stepcounter{equa}
\Phi_{\delta,\varepsilon}(x)=\Phi_0(x)+\delta G_{\varepsilon}({\rm Re\,} x, -{\rm Im\,} x)+\mathcal{O}(\delta^2 \varepsilon), \tag{\thesection.\theequa}
\end{equation}
uniformly on $\mathbb{C}^n$, when $0< \delta \ll 1$. Since from Proposition~\ref{prop1},
$${\rm supp} \ G_{\varepsilon} \subset \big\{X \in \mathbb{R}^{2n} : {\rm dist}\big(X,p_0^{-1}(0)\big)\leq 2/C\big\},$$
we furthermore know that this strictly plurisubharmonic function $\Phi_{\delta,\varepsilon}$ agrees with $\Phi_0$ outside a bounded set and that
\begin{equation}\label{eq3.7}\stepcounter{equa}
\nabla\left(\Phi_{\delta,\varepsilon}-\Phi_0\right)=\mathcal{O}(\delta \varepsilon^{1/2}), \tag{\thesection.\theequa}
\end{equation}
uniformly on $\mathbb{C}^n$;
with $\nabla^2 \Phi_{\delta,\varepsilon}\in L^{\infty}(\mathbb{C}^n)$, uniformly with respect to the parameters $\delta$ and $\varepsilon$. Since from another use of Proposition~\ref{prop1} and (\ref{eq3.6}), we globally have
on $\mathbb{C}^n$,
\begin{equation}\label{ju6}\stepcounter{equa}
\Phi_{\delta,\varepsilon}-\Phi_0=\mathcal{O}(\varepsilon), \tag{\thesection.\theequa}
\end{equation}
we notice, by choosing the small parameter $\varepsilon$ appearing in the construction of the weight $G_{\varepsilon}$ to be equal to $\varepsilon= Ah$,
where $h$ is the semiclassical parameter and $A \gg 1$ a large constant to be chosen in the
following; that for each fixed $A>0$, the norms in the weighted spaces $H_{\Phi_0}(\mathbb{C}^n)$ and $H_{\Phi_{\delta,\varepsilon}}(\mathbb{C}^n)$ are uniformly equivalent in the semiclassical limit $h\rightarrow 0^+$. Carrying out an additional contour deformation, as in~\cite{HeSjSt}, and using (\ref{eq3.7}), we get a bounded operator
$${P}_0 = \mathcal{O}(1): H_{\Phi_{\delta,\varepsilon}}(\mathbb{C}^n) \rightarrow H_{\Phi_{\delta,\varepsilon}}(\mathbb{C}^n),$$
given, for $u$ in $H_{\Phi_{\delta,\varepsilon}}(\mathbb{C}^n)$, by
\begin{equation}\label{eq3.8}\stepcounter{equa}
{P}_0 u(x) = \frac{1}{(2\pi h)^n} \int\!\!\!\int_{\Gamma_{\delta,\varepsilon}(x)} e^{\frac{i}{h}(x-y)\cdot \theta} \psi_0(x-y) {P}_0\Big(\frac{x+y}{2},\theta;h\Big) u(y)dyd\theta + Ru, \tag{\thesection.\theequa}
\end{equation}
where $\psi_0$ stands for a $C^{\infty}_0(\mathbb{C}^n)$ function such that $\psi_0=1$ near $0$; and $\Gamma_{\delta,\varepsilon}(x)$ is the contour given by
$$\theta = \frac{2}{i} \frac{\partial \Phi_{\delta,\varepsilon}}{\partial x}\Big(\frac{x+y}{2}\Big)+it_0\overline{(x-y)},\ t_0>0.$$
We have that the operator $R$ appearing in (\ref{eq3.8}) satisfies
$$R=\mathcal{O}_A(h^{\infty}): L^2\big(\mathbb{C}^n,e^{-2\Phi_{\delta,\varepsilon}/h}L(dx)\big)\rightarrow L^2\big(\mathbb{C}^n,e^{-2\Phi_{\delta,\varepsilon}/h}L(dx)\big).$$
Let us mention that in (\ref{eq3.8}), we continue to write ${P}_0$ for an almost holomorphic extension of the symbol ${P}_0 \in S(\Lambda_{\Phi_0},1)$ in a tubular neighborhood{} of $\Lambda_{\Phi_0}$, bounded together with all of its derivatives. Similarly, we shall also write $\mathfrak{p}_j$ for some almost holomorphic extensions of the symbols $\mathfrak{p}_j \in S(\Lambda_{\Phi_0},1)$ in a tubular neighborhood{} of $\Lambda_{\Phi_0}$, bounded together with all of their derivatives when $j \geq 0$.
\section{The quadratic case}\label{quadratic}
\setcounter{equa}{0}
The main purpose of this section is to get localized resolvent estimates for quadratic operators whose Weyl symbols fulfill the same properties as the quadratic approximations~$q_j$ of the principal symbol $p_0$ of the operator considered in Theorem~\ref{theo}. We shall therefore be concerned in this section with complex-valued quadratic forms with non-negative real part
\begin{equation}\label{k1}\stepcounter{equa}
\textrm{Re } q(X) \geq 0, \ X \in \mathbb{R}^{2n}, \ n \in \mathbb{N}^*,\tag{\thesection.\theequa}
\end{equation}
which enjoy a property of ellipticity on their singular spaces
\begin{equation}\label{k2}\stepcounter{equa}
X \in S, \ q(X)=0 \Rightarrow X=0. \tag{\thesection.\theequa}
\end{equation}
The main reference about quadratic operators is the work of J.~Sj\"ostrand \cite{sjostrand} and we shall actually follow very closely the analysis relying on this basic work which is led in \cite{HeSjSt} (Section~5), to explain how in view of recent improvements in the understanding of non-elliptic quadratic operators obtained in~\cite{HiPr}, the localized resolvent estimates established in~\cite{HeSjSt} (Proposition~5.2) for quadratic operators whose symbols fulfill (\ref{k1}) and the subelliptic assumption
\begin{equation}\label{k3}\stepcounter{equa}
\exists \varepsilon_0,C>0, \ {\rm Re\,} q(X) +\varepsilon_0 H_{\textrm{Im}q}^2\textrm{Re }q(X) \geq \frac{\varepsilon_0}{C}|X|^2, \ X \in \mathbb{R}^{2n}, \tag{\thesection.\theequa}
\end{equation}
can be extended to the class of quadratic operators with symbols satisfying only (\ref{k1}) and (\ref{k2}).
Let us start by verifying that the assumption (\ref{k2}) really weakens the subelliptic assumption (\ref{k3}). This is the case because condition (\ref{k3}) actually implies that the singular space is necessarily zero.
To check this, we first notice from Lemma~2 in \cite{mz} that the Hamilton map of the quadratic form $H_{\textrm{Im}q}^2\textrm{Re }q$ is given by the double commutator
$$4[\textrm{Im }F,[\textrm{Im }F, \textrm{Re }F]].$$
One can then deduce from the definition of the singular space (\ref{h1}) and the properties of skew-symmetry of the symplectic form and Hamilton maps (\ref{12}) that the two quadratic forms $\textrm{Re }q$ and $H_{\textrm{Im}q}^2\textrm{Re }q$ identically vanish on $S$,
$$\textrm{Re }q(X)=\sigma(X, \textrm{Re }F X),$$
\begin{multline*}
H_{\textrm{Im}q}^2\textrm{Re }q(X)=\sigma\big(X,4[\textrm{Im }F,[\textrm{Im }F, \textrm{Re }F]]X\big)=-8\sigma\big((\textrm{Im }F) X,[\textrm{Im }F, \textrm{Re }F]X\big) \\
= -8 \sigma\big((\textrm{Im }F) X,(\textrm{Im }F)(\textrm{Re }F)X\big)+8\sigma\big((\textrm{Im }F) X,(\textrm{Re }F)(\textrm{Im }F)X\big),
\end{multline*}
implying that when (\ref{k3}) is fulfilled, then $S=\{0\}$.
By considering from now a complex-valued quadratic form $q$ satisfying (\ref{k1}) and (\ref{k2}), we shall study the associated quadratic operator $Q=q^w(x,hD_x)$ on the FBI-Bargmann side. By using the FBI-Bargmann transform $T$ introduced in (\ref{eq3.1}),
$$T: L^2(\mathbb{R}^n)\rightarrow H_{\Phi_0}(\mathbb{C}^n),$$
and the Egorov property recalled in (\ref{eq3.4}), we may write
\begin{equation}\label{k7}\stepcounter{equa}
TQu=Q_0Tu,\ u\in \mathcal{S}(\mathbb{R}^n), \tag{\thesection.\theequa}
\end{equation}
where $Q_0$ is the quadratic differential operator on $\mathbb{C}^n$ whose Weyl symbol $\mathfrak{q}_0$ is defined by the identity
\begin{equation}\label{k8}\stepcounter{equa}
\mathfrak{q}_0\circ \kappa_T = q,\tag{\thesection.\theequa}
\end{equation}
with $\kappa_T$ being the linear canonical transformation given in (\ref{eq3.2}).
Following~\cite{Sj95}, we recall that when realizing $Q_0$ as an unbounded operator on $H_{\Phi_0}(\mathbb{C}^n)$, we may first use the contour integral representation
$$Q_0 u(x) = \frac{1}{(2\pi h)^n}\int\!\!\!\int_{\theta=\frac{2}{i}\frac{\partial \Phi_0}{\partial x}\left(\frac{x+y}{2}\right)} e^{\frac{i}{h}(x-y)\cdot \theta} \mathfrak{q}_0\Big(\frac{x+y}{2},\theta\Big)u(y)\,dy\,d\theta,$$
and then, by using that the symbol $\mathfrak{q}_0$ is holomorphic, we obtain from a contour deformation the following formula for $Q_0$ as an unbounded operator on $H_{\Phi_0}(\mathbb{C}^n)$,
\begin{equation}\label{k12}\stepcounter{equa}
Q_0 u(x) = \frac{1}{(2\pi h)^n}\int\!\!\!\int_{\theta=\frac{2}{i}\frac{\partial \Phi_0}{\partial x}\left(\frac{x+y}{2}\right)+it\overline{(x-y)}} e^{\frac{i}{h}(x-y)\cdot \theta} \mathfrak{q}_0\Big(\frac{x+y}{2},\theta\Big)u(y)\,dy\,d\theta, \tag{\thesection.\theequa}
\end{equation}
for any $t>0$. We now consider as in (\ref{eq2.15.2.1}) the real-valued quadratic weight
\begin{equation}\label{k13}\stepcounter{equa}
G^o(X) = - \int_{-\infty}^{+\infty} J\Big(-\frac{t}{T}\Big) \textrm{Re } q(e^{t H_{\textrm{Im} q}}X)dt, \tag{\thesection.\theequa}
\end{equation}
which fulfills as in (\ref{eq2.15}) the identity
\begin{equation}\label{k14}\stepcounter{equa}
H_{\textrm{Im}q} G^o =\langle{\textrm{Re }q}\rangle_{T,\textrm{Im}q} -\textrm{Re }q, \tag{\thesection.\theequa}
\end{equation}
where
$$\langle \textrm{Re }q \rangle_{T,\textrm{Im}q}(X)=\frac{1}{T}\int_{0}^{T} \textrm{Re } q (e^{tH_{\textrm{Im}q}}X)dt.$$
As in~\cite{HeSjSt} and~\cite{HiPr}, we shall consider an IR-deformation of the real phase space $\mathbb{R}^{2n}$ associated to this quadratic weight $G^o$.
Setting
\begin{equation}\label{k15}\stepcounter{equa}
\Lambda_{\delta}=\big\{X+i\delta H_{G^o}(X): X \in \mathbb{R}^{2n} \big\} \subset \mathbb{C}^{2n}, \tag{\thesection.\theequa}
\end{equation}
for $0 \leq \delta \leq 1$, where $H_{G^o}$ stands for the Hamilton vector field of $G^o$, we find as in~\cite{HiPr} and in the previous section that we have
\begin{equation}\label{k16}\stepcounter{equa}
\kappa_T(\Lambda_{\delta})=\Lambda_{\Phi_{\delta}^o}\stackrel{\textrm{def}}{=}\Big\{\Big(x,\frac{2}{i}\frac{\partial \Phi_{\delta}^o}{\partial x}(x)\Big) : x\in \mathbb{C}^n\Big\}, \tag{\thesection.\theequa}
\end{equation}
for all $0\leq \delta \leq \delta_0$, with $\delta_0>0$ small enough, where $\Phi_{\delta}^o$ is a strictly plurisubharmonic quadratic form on $\mathbb{C}^n$ verifying
\begin{equation}\label{k17}\stepcounter{equa}
\Phi_{\delta}^o(x)=\Phi_0(x)+\delta G^o({\rm Re\,} x, -{\rm Im\,} x)+\mathcal{O}(\delta^2 \abs{x}^2). \tag{\thesection.\theequa}
\end{equation}
Associated to the quadratic form $\Phi_{\delta}^o$ is the weighted space of holomorphic functions $H_{\Phi_{\delta}^o}(\mathbb{C}^n)$ defined as in Section~\ref{fbi}. We can now view the operator $Q_0$ as an unbounded operator
$$Q_0: H_{\Phi_{\delta}^o}(\mathbb{C}^n) \rightarrow H_{\Phi_{\delta}^o}(\mathbb{C}^n),$$
if we make a new contour deformation as in (\ref{k12}) and set
\begin{equation}\label{k18}\stepcounter{equa}
Q_0 u(x) = \frac{1}{(2\pi h)^n}\int\!\!\!\int_{\theta=\frac{2}{i}\frac{\partial \Phi_{\delta}^o}{\partial x}\left(\frac{x+y}{2}\right)+it_0\overline{(x-y)}} e^{\frac{i}{h}(x-y)\cdot \theta}\mathfrak{q}_0\Big(\frac{x+y}{2},\theta\Big)u(y)\,dy\,d\theta, \tag{\thesection.\theequa}
\end{equation}
for $t_0>0$, when $\delta$ is sufficiently small, $0 < \delta \ll 1$. By coming back to the real side via the FBI-Bargmann transform, the operator $Q_0$ can then be viewed as an unbounded operator on $L^2(\mathbb{R}^n)$ whose Weyl symbol is given by the following quadratic form
\begin{equation}\label{k19}\stepcounter{equa}
\widetilde{q}(X)=q\big(X+i \delta H_{G^o}(X)\big).\tag{\thesection.\theequa}
\end{equation}
As in Section~\ref{weight}, we recall that the singular space $S$ of a quadratic symbol $q$ satisfying the assumptions (\ref{k1}) and (\ref{k2}) always has a symplectic structure (see Section~1.4.1 in~\cite{HiPr}) and that according to~\cite{HiPr} (Proposition 2.0.1), new symplectic linear coordinates
$$X=(x,\xi)=(x',x'';\xi',\xi'') \in \mathbb{R}^{2n}=\mathbb{R}^{2n'+2n''},$$
can be chosen such that $(x'',\xi'')$ and $(x',\xi')$ are, respectively, some linear symplectic coordinates in $S$ and its symplectic orthogonal space $S^{\sigma \perp}$, so that in these coordinates, the symbol $q$ can be decomposed as the sum of two quadratic forms
\begin{equation}\label{k20}\stepcounter{equa}
q(x,\xi)=q|_{S^{\sigma \perp}}(x',\xi')+q|_{S}(x'',\xi''), \tag{\thesection.\theequa}
\end{equation}
such that the average of the real part of first one by the flow defined by the Hamilton vector field of its imaginary part
\begin{equation}\label{k21}\stepcounter{equa}
\langle \textrm{Re }q|_{S^{\sigma \perp}}\rangle_{T,\textrm{Im}q|_{S^{\sigma \perp}}}(X')=\frac{1}{T}\int_{0}^{T}{\textrm{Re }q|_{S^{\sigma \perp}}(e^{tH_{\textrm{Im}q|_{S^{\sigma \perp}}}}X')dt}, \tag{\thesection.\theequa}
\end{equation}
where $X'=(x',\xi') \in \mathbb{R}^{2n'}$;
is a positive definite quadratic form for all $T>0$; and
\begin{equation}\label{k22}\stepcounter{equa}
q|_S(x'',\xi'')=i \tilde{\varepsilon}_0 \sum_{j=1}^{n''}{\lambda_j(\xi_j''^2+x_j''^2)}, \tag{\thesection.\theequa}
\end{equation}
with $\tilde{\varepsilon}_0 \in \{\pm 1\}$, $0 \leq n'' \leq n$ and $\lambda_j>0$ for all $j=1,...,n''$.
Then, by noticing that a direct computation gives that
$$H_{\textrm{Im}q}=2 \textrm{Im }F,$$
we deduce from (\ref{k13}), (\ref{k14}), (\ref{k20}), (\ref{k22}); and the stability property of the spaces $S$ and $S^{\sigma \perp}$ by the map $\textrm{Im }F$ \big(see (2.0.4) in \cite{HiPr}\big); that the quadratic weight $G^o$ only depends on the variables $X'=(x',\xi') \in \mathbb{R}^{2n'}$,
\begin{equation}\label{k23}\stepcounter{equa}
G^o(X')=- \int_{-\infty}^{+\infty} J\Big(-\frac{t}{T}\Big) {\rm Re\,} q|_{S^{\sigma \perp}}(e^{tH_{\textrm{Im} q|_{S^{\sigma \perp}}}}X')\,dt \tag{\thesection.\theequa}
\end{equation}
and satisfies
\begin{equation}\label{k24}\stepcounter{equa}
H_{\textrm{Im} q|_{S^{\sigma \perp}}} G^o =\langle{{\rm Re\,} q|_{S^{\sigma \perp}}}\rangle_{T,\textrm{Im}q|_{S^{\sigma \perp}}} -{\rm Re\,} q|_{S^{\sigma \perp}}. \tag{\thesection.\theequa}
\end{equation}
Since the symbol $\widetilde{q}$ in (\ref{k19}) is easily seen to be equal to
\begin{equation}\label{k25}\stepcounter{equa}
\widetilde{q}=q-i \delta H_q G^o +\mathcal{O}(\delta^2\abs{\nabla G^o}^2), \tag{\thesection.\theequa}
\end{equation}
we deduce from the previous tensorization of the variables (\ref{k20}), (\ref{k22}) and (\ref{k23}) that this symbol can be written as
\begin{equation}\label{k26}\stepcounter{equa}
\widetilde{q}(X)=r(X')+i \tilde{\varepsilon}_0 \sum_{j=1}^{n''}{\lambda_j(\xi_j''^2+x_j''^2)}, \tag{\thesection.\theequa}
\end{equation}
with
$$r(X')=q|_{S^{\sigma \perp}}(X')-i \delta H_{q|_{S^{\sigma \perp}}}G^o(X')+\mathcal{O}(\delta^2\abs{X'}^2).$$
Using now (\ref{k1}), (\ref{k20}), (\ref{k21}) and (\ref{k24}), we notice that the real part of the quadratic symbol $r$,
\begin{multline*}
\textrm{Re }q|_{S^{\sigma \perp}}(X')+\delta H_{\textrm{Im}q|_{S^{\sigma \perp}}}G^o(X')+\mathcal{O}(\delta^2\abs{X'}^2) \\
=(1-\delta)\textrm{Re }q|_{S^{\sigma \perp}}(X')+\delta \langle \textrm{Re }q|_{S^{\sigma \perp}}\rangle_{T,\textrm{Im}q|_{S^{\sigma \perp}}}(X') +\mathcal{O}(\delta^2\abs{X'}^2)
\geq \frac{\delta}{C}|X'|^2,
\end{multline*}
for $C>1$; is a positive definite quadratic form for all $0<\delta \ll 1$ sufficiently small. In view of (\ref{k26}), this particular property implies the ellipticity of the quadratic symbol $\widetilde{q}$ on $\mathbb{R}^{2n}$. We can then apply the classical result of J.~Sj\"ostrand (Theorem 3.5 in \cite{sjostrand}) to the operator $Q_0$ viewed as an unbounded operator on $H_{\Phi_{\delta}^o}(\mathbb{C}^n)$, and using similar arguments as the ones used by F.~H\'erau, J.~Sj\"ostrand and C.~Stolk in their proof of Proposition 5.1 in \cite{HeSjSt}, we get that this operator $Q_0$ fulfills all the properties stated in~\cite{HeSjSt} (Proposition~5.1), namely that its spectrum, as an unbounded operator on $H_{\Phi_{\delta}^o}(\mathbb{C}^n)$ for $0<\delta \ll 1$, is only composed of eigenvalues of finite multiplicity with the following structure
\begin{equation}\label{k27}\stepcounter{equa}
\sigma\big{(}Q_0\big{)}=\Big\{ \sum_{\substack{\lambda \in \sigma(F), \\ -i \lambda \in \mathbb{C}_+ \cup (\Sigma(q|_S) \setminus \{0\})}}{h\big{(}r_{\lambda}+2 k_{\lambda}\big{)}(-i\lambda) : k_{\lambda} \in \mathbb{N}}\Big\}, \tag{\thesection.\theequa}
\end{equation}
where $F$ is the Hamilton map associated to the quadratic form $q$, $r_{\lambda}$ the dimension of the space of generalized eigenvectors of $F$ in $\mathbb{C}^{2n}$ belonging to the eigenvalue $\lambda \in \mathbb{C}$,
$$\Sigma(q|_S)=\overline{q(S)} \textrm{ and } \mathbb{C}_+=\{z \in \mathbb{C}: \textrm{Re }z>0\};$$
and that if $z$ remains in a compact set of empty intersection with $\sigma(Q_0)$, then we have with $d(x)=|x|$,
\begin{equation}\label{k28}\stepcounter{equa}
\|(h+d^2)u\| \leq \mathcal{O}(1) \|(Q_0-hz)u\|, \tag{\thesection.\theequa}
\end{equation}
and
\begin{equation}\label{k29}\stepcounter{equa}
\|(h+d^2)^{\frac{1}{2}}u\| \leq \mathcal{O}(1) \|(h+d^2)^{-\frac{1}{2}}(Q_0-hz)u\|, \tag{\thesection.\theequa}
\end{equation}
for all holomorphic function $u$ satisfying respectively
$$(h+d^2)u \in L^2\big(\mathbb{C}^n,e^{-2\Phi_{\delta}^o(x)/h}L(dx)\big) \textrm{ and }(h+d^2)^{\frac{1}{2}}u \in L^2\big(\mathbb{C}^n,e^{-2\Phi_{\delta}^o(x)/h}L(dx)\big).$$
The only difference with Proposition~5.1 in \cite{HeSjSt} is that we describe here the spectrum of $Q_0$ by the mean of the singular space as we did in \cite{HiPr} (Theorem 1.2.2).
We have therefore checked that any quadratic operator whose symbol $q$ fulfills the assumptions (\ref{k1}) and (\ref{k2}), verifies all the properties stated in~\cite{HeSjSt} (Proposition~5.1). We can then deduce from Proposition~5.2 in \cite{HeSjSt} that the operator $Q_0$ also verifies the estimates stated in~\cite{HeSjSt} (Proposition~5.2), since the proof of Proposition~5.2 only relies on estimates obtained in~\cite{HeSjSt} (Proposition~5.1). This proves that any quadratic operator whose Weyl symbol fulfills the assumptions (\ref{k1}) and (\ref{k2}), defines on the FBI transform side, an unbounded operator on the spaces $H_{\Phi_{\delta}^o}(\mathbb{C}^n)$, with $0 < \delta \ll 1$, which fulfills the following localized resolvent estimates:
\bigskip
\begin{lemma}\label{klem1}
Let $\chi_0 \in C_0^{\infty}(\mathbb{C}^n)$ be fixed and equal to {\rm 1} near {\rm 0}, and fix $k \in \mathbb{R}$. Then for $z$ varying in a compact set that does not contain any eigenvalues of the operator $Q_0|_{h=1}$ described in \emph{(\ref{k27})}, we have with $d(x)=|x|$,
\begin{equation}\label{k30}\stepcounter{equa}
\|(h+d^2)^{1-k}\chi_0 u\| \leq \mathcal{O}(1)\|(h+d^2)^{-k}\chi_0(Q_0-hz)u\| + \mathcal{O} (h^{\frac{1}{2}})\|{\mathrm{1~\hspace{-1.4ex}l}}_K u\|, \tag{\thesection.\theequa}
\end{equation}
where $K$ is any fixed neighborhood of $\emph{\textrm{supp}}(\nabla\chi_0)$ and ${\mathrm{1~\hspace{-1.4ex}l}}_K$ stands for its characteristic function. Here the norm is taken in the space $L^2(\mathbb{C}^n, e^{-2\Phi_{\delta}^o/h}L(dx))$.
\end{lemma}
\bigskip
By using these localized resolvent estimates satisfied by quadratic operators defined by the quadratic approximations of the principal symbol $p_0$ at double characteristic points, we shall establish in the next section local resolvent estimates for these operators and the operator $P=P^w(x,hD_x;h)$ in a tiny neighborhood of the doubly characteristic set.
\section{Local resolvent estimates in a tiny neighborhood{} of the doubly characteristic set}\label{tiny}
\setcounter{equa}{0}
In all of this section, $G_{\varepsilon}$ stands for the weight function constructed in Proposition~\ref{prop1} and we choose the small parameter $\varepsilon$ to be equal to
\begin{equation}\label{k31}\stepcounter{equa}
\varepsilon=Ah, \ A \gg 1, \tag{\thesection.\theequa}
\end{equation}
where $h$ is the semiclassical parameter and $A$ a large constant to be chosen in the following.
We recall from Section~\ref{fbi} that the IR-manifold $\Lambda_{\delta,\varepsilon}$ defined in (\ref{eq3.4.5}) is represented on the FBI-transform side by
\begin{equation}\label{k32}\stepcounter{equa}
\kappa_{T}(\Lambda_{\delta,\varepsilon})=\Lambda_{ \Phi_{\delta,\varepsilon}}=\Big\{\Big(x,\frac{2}{i}\frac{\partial \Phi_{\delta,\varepsilon}}{\partial x}(x)\Big): x \in \mathbb{C}^n\Big\}, \tag{\thesection.\theequa}
\end{equation}
where $ \Phi_{\delta,\varepsilon} \in C^{\infty}(\mathbb{C}^n)$ is the strictly plurisubharmonic function introduced in (\ref{ju2}) for $0<\varepsilon \leq \varepsilon_0$ and sufficiently small values of the parameter $0<\delta \ll 1$.
Let us notice directly from (\ref{eq2.15.1}) and (\ref{ju2}) that this weight function $\Phi_{\delta,\varepsilon}$ is independent of the parameter $\varepsilon$ in a region where $|x| \leq \sqrt{\varepsilon}/C$, after a suitable choice of the constant $C>0$. By making a rescaling in $\varepsilon$, we may and will assume in the following that we have $C=1$.
By recalling from Section~\ref{fbi} that we have a uniformly bounded operator
\begin{equation}\label{k33.1}\stepcounter{equa}
{P}_0: H_{\Phi_{\delta,\varepsilon}}(\mathbb{C}^n) \rightarrow H_{\Phi_{\delta,\varepsilon}}(\mathbb{C}^n), \tag{\thesection.\theequa}
\end{equation}
we shall be concerned with the almost holomorphic extension of the principal symbol $\mathfrak{p}_0$ of the operator ${P}_0$, restricted to $\Lambda_{\Phi_{\delta,\varepsilon}}$, and to ease the notation in this section, we shall simply write
$$\mathfrak{p}_0\stackrel{\textrm{def}}{=} \mathfrak{p}_0|_{\Lambda_{\Phi_{\delta,\varepsilon}}}.$$
We shall also assume for simplicity that the characteristic set $p_0^{-1}(0)$ is composed of an unique point, say here $(0,0)$, which corresponds to the point $\mathbb{C}^n\ni x=0$ on the FBI-transform side.
Considering the quadratic approximation of the principal symbol $p_0$,
\begin{equation}\label{k34}\stepcounter{equa}
q_0(x,\xi)=\sum_{|\alpha+\beta|=2}\frac{\partial_x^{\alpha}\partial_{\xi}^{\beta}p_0(0,0)}{\alpha! \beta!}x^{\alpha}\xi^{\beta}, \tag{\thesection.\theequa}
\end{equation}
at the critical point $(0,0)$, we may write
\begin{equation}\label{k35}\stepcounter{equa}
p_0(x,\xi)+hp_1(x,\xi)-q_0(x,\xi)-hp_1(0,0)=\mathcal{O}\big((h+|(x,\xi)|^2)^{\frac{3}{2}}\big), \ \ 0<h \leq 1, \tag{\thesection.\theequa}
\end{equation}
where $p_1$ stands for the subprincipal symbol of the symbol $P(x,\xi;h)$ appearing in its asymptotic expansion (\ref{xi1}).
Using this quadratic approximation, we aim in this section at getting local resolvent estimates for the operator $P_0$ in the following tiny neighborhood of the doubly characteristic point
$$|x| \leq \sqrt{\varepsilon},$$
on the FBI-transform side. Working in this region, we can first notice that we may replace $\Phi_{\delta,\varepsilon}$ by the strictly plurisubharmonic quadratic form $\Phi_{\delta}^o$, introduced in (\ref{k16}), since the identities (\ref{keq}), (\ref{eq3.6}) and (\ref{k17}) induce that the two $L^2$-norms associated to these weight functions are equivalent in the region where $|x| \leq \sqrt{\varepsilon}$,
\begin{equation}\label{ju1}\stepcounter{equa}
\exp\big(-\mathcal{O}(1)(A+ A^{3/2} h^{1/2})\big) \leq e^{-\frac{2\Phi_{\delta,\varepsilon}(x)}{h}} e^{\frac{2\Phi_{\delta}^o(x)}{h}} \leq \exp\big(\mathcal{O}(1)(A+ A^{3/2}h^{1/2})\big), \tag{\thesection.\theequa}
\end{equation}
for each fixed constant $A\gg 1$.
By assuming therefore that the weight function $\Phi_{\delta,\varepsilon}$ is equal to the quadratic form $\Phi_{\delta}^o$ in the region where $|x| \leq \sqrt{\varepsilon}$, we realize the operators $\mathfrak{p}_0^w(x,hD_x)$, $\mathfrak{p}_1^w(x,D_x)$, $P_0$ and $Q_0$ with a contour as in (\ref{eq3.8}). We deduce from this realization and (\ref{k35}) that the difference between the corresponding effective kernels of $\mathfrak{p}_0^w(x,D_x)+h\mathfrak{p}_1^w(x,hD_x)$ and $Q_0+hp_1(0,0)$ is, for some $D>0$,
$$\mathcal{O}_A(1)h^{-n}e^{-\frac{D}{h}|x-y|^2}(h+|x|^2+|y|^2)^{\frac{3}{2}}=\mathcal{O}_A(1)h^{-n}e^{-\frac{D}{h}|x-y|^2}(h^{\frac{3}{2}}+|x|^3+|x-y|^3),$$
which implies as in~\cite{HeSjSt} (see (6.9)) that
\stepcounter{equa}\begin{multline*}\label{k36}
\|\mathfrak{p}_0^w(x,hD_x)u+h\mathfrak{p}_1^w(x,hD_x)u-Q_0u-hp_1(0,0)u\|_{H_{\Phi_{\delta,\varepsilon}}(|x| \leq \sqrt{Ah})}\\
=\mathcal{O}_A(h^{\frac{3}{2}})\|u\|_{H_{\Phi_{\delta,\varepsilon}}}, \tag{\thesection.\theequa}
\end{multline*}
where the notation $\mathcal{O}_A$ is here to emphasize that the constant may depend on the large parameter $A$ which will be chosen later on.
Keeping in mind that the operator $Q_0$ is realized with a contour as in (\ref{eq3.8}) and using the fact that its symbol is a quadratic polynomial, we check that if the quantity $Q_0u$ is replaced by the corresponding differential expression
$$\sum_{|\alpha+\beta|=2}\frac{\partial_x^{\alpha}\partial_{\xi}^{\beta}p_0(0,0)}{\alpha! \beta!}\big(x^{\alpha}(hD_x)^{\beta}\big)^wu,$$
on the FBI-transform side, we then commit an error $w$ satisfying
\begin{equation}\label{k37}\stepcounter{equa}
\|w\|_{H_{\Phi_{\delta,\varepsilon}}(|x| \leq \sqrt{Ah})} \leq e^{-\frac{1}{Ch}} \|u\|_{H_{\Phi_{\delta,\varepsilon}}}, \tag{\thesection.\theequa}
\end{equation}
for some $C>0$.
Continuing to follow \cite{HeSjSt} (Section~6), we can now consider the change of variables on the FBI-transform side
$$x=\sqrt{\varepsilon}\tilde{x}, \ hD_{x}=\sqrt{\varepsilon}\tilde{h}D_{\tilde{x}},$$
with $\tilde{h}=A^{-1}$. Then, setting $\tilde{Q}_0=\mathfrak{q}_0(\tilde{x},\tilde{h}D_{\tilde{x}})^w$, homogeneity properties imply that
\begin{equation}\label{k38}\stepcounter{equa}
Q_0=\mathfrak{q}_0^w(x,hD_x)=\frac{h}{\tilde{h}}\mathfrak{q}_0^w(\tilde{x},\tilde{h}D_{\tilde{x}})=\frac{h}{\tilde{h}}\tilde{Q}_0, \tag{\thesection.\theequa}
\end{equation}
and, with $d=d(x)=|x|$, $\tilde{d}=d(\tilde{x})=|\tilde{x}|$; we have the following identities, since $\Phi_{\delta}^o(x)$ is a quadratic form
\begin{equation}\label{k39}\stepcounter{equa}
h+d^2=\frac{h}{\tilde{h}}(\tilde{h}+\tilde{d}^2), \ e^{-\frac{2\Phi_{\delta}^o(x)}{h}}=e^{-\frac{2\Phi_{\delta}^o(\tilde{x})}{\tilde{h}}}. \tag{\thesection.\theequa}
\end{equation}
Let $\chi_0 \in C_0^{\infty}(\mathbb{C}^n)$ be fixed and equal to 1 near the set where $\abs{x}\leq 1$. For $k \in \mathbb{R}$, we can then apply Lemma~\ref{klem1} to the operator $\tilde{Q}_0$ to get that for any $z$ belonging to a fixed compact set avoiding the eigenvalues of the operator $\tilde{Q}_0|_{\tilde{h}=1}$, the following estimate is fulfilled
\begin{equation}\label{k40}\stepcounter{equa}
\|(\tilde{h}+\tilde{d}^2)^{1-k}\chi_0(\tilde{x}) \tilde{u}\| \leq \mathcal{O}(1) \|(\tilde{h}+\tilde{d}^2)^{-k}\chi_0(\tilde{x})(\tilde{Q}_0-\tilde{h}z)\tilde{u}\| + \mathcal{O}(\tilde{h}^{\frac{1}{2}})\|{\mathrm{1~\hspace{-1.4ex}l}}_{K}\tilde{u}\|, \tag{\thesection.\theequa}
\end{equation}
where $K$ is a fixed neighborhood of $\textrm{supp}(\nabla \chi_0)$. Here the norm is taken with respect to the weight $e^{-2\Phi_{\delta}^o(\tilde{x})/\tilde{h}}$. By noticing that we can replace the term $\mathcal{O}(\tilde{h}^{\frac{1}{2}})\|{\mathrm{1~\hspace{-1.4ex}l}}_{K}\tilde{u}\|$ by
$$\mathcal{O}(\tilde{h}^{\frac{1}{2}})\|(\tilde{h}+\tilde{d}^2)^{1-k}{\mathrm{1~\hspace{-1.4ex}l}}_{K}\tilde{u}\|,$$
and that in view of (\ref{k38}) the two operators $Q_0|_{h=1}$ and $\tilde{Q}_0|_{\tilde{h}=1}$ have the same spectra, we get from (\ref{k38}), (\ref{k39}) and (\ref{k40}) by coming back to initial variables $x$, $h$, that for any $z$ belonging to a fixed compact set avoiding the eigenvalues of the operator $Q_0|_{h=1}$, we have
\begin{multline*}
\Big\| \Big(\frac{\tilde{h}}{h}\Big)^{1-k} (h+d^2)^{1-k}\chi_0\Big(\frac{x}{\sqrt{Ah}}\Big) u\Big\| \leq \mathcal{O}(1) \Big\| \Big(\frac{\tilde{h}}{h}\Big)^{1-k} (h+d^2)^{-k}\chi_0\Big(\frac{x}{\sqrt{Ah}}\Big) (Q_0-hz)u\Big\|\\ +\tilde{C}\tilde{h}^{\frac{1}{2}} \Big\|\Big(\frac{\tilde{h}}{h}\Big)^{1-k}(h+d^2)^{1-k}{\mathrm{1~\hspace{-1.4ex}l}}_{K}\Big(\frac{x}{\sqrt{Ah}}\Big)u\Big\|,
\end{multline*}
which finally induces the following result:
\bigskip
\begin{lemma}\label{klem1cel}
Let $\chi_0 \in C_0^{\infty}(\mathbb{C}^n)$ be fixed and equal to {\rm 1} near the set where $\abs{x}\leq 1$, and fix $k \in \mathbb{R}$. Then for $z$ varying in a compact set that does not contain any eigenvalues of the operator $Q_0|_{h=1}$ described in \emph{(\ref{k27})}, we have with $d(x)=|x|$,
\stepcounter{equa}\begin{multline}\label{k41}
\Big\| (h+d^2)^{1-k}\chi_0\Big(\frac{x}{\sqrt{Ah}}\Big) u\Big\| \leq \mathcal{O}(1) \Big\| (h+d^2)^{-k}\chi_0\Big(\frac{x}{\sqrt{Ah}}\Big) (Q_0-hz)u\Big\|\\ +\mathcal{O}\Big(\frac{1}{\sqrt{A}}\Big) \Big\|(h+d^2)^{1-k}{\mathrm{1~\hspace{-1.4ex}l}}_{K}\Big(\frac{x}{\sqrt{Ah}}\Big)u\Big\|,\tag{\thesection.\theequa}
\end{multline}
when $0<h \ll 1$ and $A \gg 1$,
where $K$ is any fixed neighborhood of $\emph{\textrm{supp}}(\nabla\chi_0)$ and ${\mathrm{1~\hspace{-1.4ex}l}}_K$ stands for its characteristic function. Here the norm is taken with respect to the weight $e^{-2\Phi_{\delta}^o(x)/h}$. According to \emph{(\ref{ju1})}, all these norms with respect to the weight $e^{-2\Phi_{\delta}^o/h}$ can then be replaced by norms with respect to the weight $e^{-2\Phi_{\delta,\varepsilon}/h}$ in the previous estimate.
\end{lemma}
\section{Local resolvent estimates in the exterior region}\label{intermediate}
\setcounter{equa}{0}
We shall now establish local resolvent estimates in the region outside the tiny neighborhood of the doubly characteristic set considered in the previous section. As in Section~\ref{tiny}, we shall continue to assume for simplicity only that the doubly characteristic set $p_0^{-1}(0)$ is composed of a unique point, say here $(0,0) \in \mathbb{R}^{2n}$, which corresponds to the point $\mathbb{C}^n\ni x=0$ on the FBI-transform side.
Considering always the same IR-manifold
$$\Lambda_{\Phi_{\delta,\varepsilon}}=\kappa_{T}(\Lambda_{\delta,\varepsilon})=\Big\{\Big(x,\frac{2}{i}\frac{\partial \Phi_{\delta,\varepsilon}}{\partial x}(x)\Big) : x \in \mathbb{C}^n\Big\},$$
associated to the weight $G_{\varepsilon}$ as in the previous section, we recall that the small parameter $\varepsilon$ is taken equal to
$$\varepsilon=Ah, \ A \gg 1,$$
where $h$ stands for the semiclassical parameter and $A$ a large parameter still remaining to be chosen. The purpose of this section is to get a local resolvent estimate in the region outside the tiny $\sqrt{\varepsilon}$--neighborhood of the doubly characteristic point $0$ studied in the previous section. We shall therefore be concerned with studying the following region on the FBI-transform side of the IR-manifold~$\Lambda_{\Phi_{\delta,\varepsilon}}$,
\begin{equation}\label{k42}\stepcounter{equa}
|x| \geq \sqrt{\varepsilon}. \tag{\thesection.\theequa}
\end{equation}
We begin by noticing from (\ref{eq3.2.1b}) and (\ref{eq3.6}) that we have
\begin{equation}\label{eq6.2xi}\stepcounter{equa}
\xi = -\textrm{Im }x +\mathcal{O}(\delta \sqrt{\varepsilon}),\tag{\thesection.\theequa}
\end{equation}
when $(x,\xi)\in \Lambda_{\Phi_{\delta,\varepsilon}}$.
When working in the unbounded region (\ref{k42}), we recall from Proposition~\ref{prop1} that we have
\begin{equation}\label{eq6.2}\stepcounter{equa}
\textrm{Re}\Big(\Big(1-i \tilde{c} \frac{\delta \varepsilon}{|x|^2}\Big) \mathfrak{p}_0\Big(x,\frac{2}{i}\frac{\partial \Phi_{\delta,\varepsilon}(x)}{\partial x}\Big)\Big) \geq \frac{\delta \varepsilon}{\tilde{C}},\tag{\thesection.\theequa}
\end{equation}
when $|x| \geq \sqrt{\varepsilon}$.
It is therefore convenient to consider again the new variables
\begin{equation}\label{eq6.3}\stepcounter{equa}
x = \sqrt{\varepsilon} \widetilde{x}.\tag{\thesection.\theequa}
\end{equation}
In these new coordinates, the IR-manifold $\Lambda_{\Phi_{\delta,\varepsilon}}$ then becomes replaced by
\begin{equation}\label{eq6.4}\stepcounter{equa}
\Lambda_{\widetilde{\Phi}_{\delta,\varepsilon}}= \Big\{\Big(\widetilde{x},\frac{2}{i} \frac{\partial \widetilde{\Phi}_{\delta,\varepsilon}(\widetilde{x})}{\partial
\widetilde{x}},\Big) : \widetilde{x} \in \mathbb{C}^n\Big\},\tag{\thesection.\theequa}
\end{equation}
with
\begin{equation}\label{eq6.4xi}\stepcounter{equa}
\widetilde{\Phi}_{\delta,\varepsilon}(\widetilde{x}) = \frac{1}{\varepsilon} \Phi_{\delta,\varepsilon}(\sqrt{\varepsilon}\widetilde{x}).\tag{\thesection.\theequa}
\end{equation}
We notice from (\ref{eq6.2xi}) and (\ref{eq6.4xi}) that the function $\nabla^2 \widetilde{\Phi}_{\delta,\varepsilon}$ belongs to the space $L^{\infty}(\mathbb{C}^{n})$ uniformly with respect to the parameters $0<\delta \leq \delta_0$ and $0<\varepsilon \leq \varepsilon_0$, since it is the case for the function $\nabla^2 \Phi_{\delta,\varepsilon}$; and that
$$\widetilde{\xi} = -{\rm Im\,} \widetilde{x} +\mathcal{O}(\delta),$$
along the IR-manifold $\Lambda_{\widetilde{\Phi}_{\delta,\varepsilon}}$.
Writing
\begin{equation}\label{eq6.5}\stepcounter{equa}
\frac{1}{\varepsilon} \mathfrak{p}_0^w(x, hD_x) = \frac{1}{\varepsilon}\mathfrak{p}_0^w\big(\sqrt{\varepsilon}(\widetilde{x},\widetilde{h}D_{\widetilde{x}})\big),\tag{\thesection.\theequa}
\end{equation}
with
\begin{equation}\label{eq6.5xi}\stepcounter{equa}
\widetilde{h}=\frac{h}{\varepsilon}=\frac{1}{A},\tag{\thesection.\theequa}
\end{equation}
we shall work with the $\widetilde{h}$--pseudodifferential operator
$$\widetilde{P}_{\varepsilon} = \frac{1}{\varepsilon} \mathfrak{p}_0^w(x, hD_x),$$
whose Weyl symbol is given by
\begin{equation}\label{eq6.6}\stepcounter{equa}
\widetilde{\mathfrak{p}}_{\varepsilon}(\widetilde{x},\widetilde{\xi}) = \frac{1}{\varepsilon} \mathfrak{p}_0\big(\sqrt{\varepsilon}(\widetilde{x},\widetilde{\xi})\big), \tag{\thesection.\theequa}
\end{equation}
fulfilling for any $k\geq 0$,
$$\nabla^{k}\widetilde{\mathfrak{p}}_{\varepsilon} = \mathcal{O}(\varepsilon^{k/2-1}),$$
so that in particular, we have
\begin{equation}\label{eq6.6.1}\stepcounter{equa}
\nabla^k \widetilde{\mathfrak{p}}_{\varepsilon} = \mathcal{O}(1),\tag{\thesection.\theequa}
\end{equation}
uniformly with respect to the parameter $0 < \varepsilon \leq \varepsilon_0$ on $\Lambda_{\widetilde{\Phi}_{\delta,\varepsilon}}$, when $k\geq 2$. Recalling that
$$\mathfrak{p}_0=p_0 \circ \kappa_T^{-1},$$
where $\kappa_T$ is the linear canonical transformation defined in (\ref{eq3.2}), this remark together with the fact that we have the estimate
$$\mathfrak{p}_0(x,\xi)=\mathcal{O}(|(x,\xi)|^2),$$
near the origin since $(0,0)$ is a doubly characteristic point for the symbol $p_0$; imply that in any region along $\Lambda_{\widetilde{\Phi}_{\delta,\varepsilon}}$ where the variables $\widetilde{x}$ remains bounded, we have
$$\widetilde{\mathfrak{p}}_{\varepsilon}=\mathcal{O}(1),$$
uniformly with respect to the parameter $0<\varepsilon \leq \varepsilon_0$; while
$$\widetilde{\mathfrak{p}}_{\varepsilon}=\mathcal{O}(|\tilde{x}|^2),$$
when $\abs{\widetilde{x}}\rightarrow +\infty$. It follows from (\ref{eq6.2}) that along the IR-manifold $\Lambda_{\widetilde{\Phi}_{\delta,\varepsilon}}$ the symbol (\ref{eq6.6}) satisfies the following estimate
\begin{equation}\label{eq6.7}\stepcounter{equa}
\textrm{Re}\Big(\Big(1-i\tilde{c}\frac{\delta}{|\widetilde{x}^2|}\Big) \widetilde{\mathfrak{p}}_{\varepsilon}\Big(\widetilde{x},\frac{2}{i}\frac{\partial \widetilde{\Phi}_{\delta,\varepsilon}(\widetilde{x})}{\partial \widetilde{x}}\Big)\Big)\geq \frac{\delta}{\tilde{C}}.\tag{\thesection.\theequa}
\end{equation}
in the region where $\abs{\widetilde{x}}\geq 1$; and we recall that associated with the IR-manifold $\Lambda_{\widetilde{\Phi}_{\delta,\varepsilon}}$ is the weighted space
$$H_{\widetilde{\Phi}_{\delta,\varepsilon},\widetilde{h}}(\mathbb{C}^n),$$
where the index $\tilde{h}$ is here to remind us that the new semiclassical parameter used in the definition of this space is now $\tilde{h}$. Since from (\ref{eq6.4xi}), we have
$$\frac{\widetilde{\Phi}_{\delta,\varepsilon}(\widetilde{x})}{\widetilde{h}} = \frac{\Phi_{\delta,\varepsilon}(x)}{h},$$
we notice that the map
$$u(x)\mapsto \widetilde{u}(\widetilde{x})=\varepsilon^{n/2} u(\sqrt{\varepsilon}\widetilde{x}),$$
then maps unitarily the spaces $H_{\Phi_{\delta,\varepsilon},h}(\mathbb{C}^n)=H_{\Phi_{\delta,\varepsilon}}(\mathbb{C}^n)$ to $H_{\widetilde{\Phi}_{\delta,\varepsilon},\widetilde{h}}(\mathbb{C}^n)$.
The idea is then to make use of the fundamental quantization \textit{vs.} multiplication formula in the space $H_{\widetilde{\Phi}_{\delta,\varepsilon},\widetilde{h}}(\mathbb{C}^n)$ applied to the operator $\widetilde{P}_{\varepsilon}$. This fundamental formula was established in the analytic case in~\cite{SjDuke} and in the $C^{\infty}$ case in~\cite{HeSjSt} (Section~3.4). Since, as we have already observed, the symbol $\widetilde{\mathfrak{p}}_{\varepsilon}$ may exhibit some quadratic growth, its use requires a bit some care. We verify in the following Proposition that this formula can still be applied in our case.
\bigskip
\begin{prop}\label{prop89}
Let $\psi(\widetilde{x}) \in C^{\infty}_b(\mathbb{C}^n)$ be such that $\nabla \psi = \mathcal{O}(\abs{\widetilde{x}}^{-1})$ when $|\widetilde{x}| \rightarrow +\infty$. Then the quantization \textit{vs.} multiplication formula holds
\stepcounter{equa}\begin{multline*}\label{eq6.8}
(\psi \widetilde{P}_{\varepsilon}\widetilde{u},\widetilde{u})_{\widetilde{\Phi}_{\delta,\varepsilon},\tilde{h}}= \int_{\mathbb{C}^n} \psi(\widetilde{x}) \widetilde{\mathfrak{p}}_{\varepsilon}\Big(\widetilde{x},\frac{2}{i} \frac{\partial \widetilde{\Phi}_{\delta,\varepsilon}(\widetilde{x})}{\partial \widetilde{x}}\Big) |\widetilde{u}(\widetilde{x})|^2 e^{-\frac{2\widetilde{\Phi}_{\delta,\varepsilon}(\widetilde{x})}{\widetilde{h}}}L(d\widetilde{x}) \\+\mathcal{O}(\widetilde{h})\|\widetilde{u}\|^2_{\widetilde{\Phi}_{\delta,\varepsilon},\tilde{h}}.\tag{\thesection.\theequa}
\end{multline*}
\end{prop}
\bigskip
\begin{proof}
The proof of this proposition follows by inspection of the arguments given in~\cite{HeSjSt} (Section~3.4). Writing a contour integral representation for the quantity $\widetilde{P}_{\varepsilon} \widetilde{u}$ obtained by making a change of scales in the contour integral (\ref{eq3.8}), we shall consider the Taylor expansion of the expression
$$\widetilde{\mathfrak{p}}_{\varepsilon}\Big(\frac{\widetilde{x}+\widetilde{y}}{2},\widetilde{\theta}\Big),$$
along the corresponding contour to get that
\stepcounter{equa}\begin{multline*}\label{eq6.9}
\widetilde{\mathfrak{p}}_{\varepsilon}\Big(\frac{\widetilde{x}+\widetilde{y}}{2},\widetilde{\theta}\Big) = \widetilde{\mathfrak{p}}_{\varepsilon}\big(\widetilde{x},\xi(\widetilde{x})\big) + \sum_{j=1}^n f_{j,\varepsilon}(\widetilde{x}) \big(\widetilde{\theta}_j - \xi_j(\widetilde{x})\big)+ \sum_{j=1}^n g_{j,\varepsilon}(\widetilde{x})(\widetilde{y}_j-\widetilde{x}_j) \\ + \widetilde{r}_{\varepsilon}(\widetilde{x},\widetilde{y};\tilde{\theta}), \tag{\thesection.\theequa}
\end{multline*}
where
$$f_{j,\varepsilon}(\widetilde{x}) = \partial_{\theta_j} \widetilde{\mathfrak{p}}_{\varepsilon}\big(\widetilde{x},\xi(\widetilde{x})\big),\ g_{j,\varepsilon}(\widetilde{x}) = \frac{1}{2} \partial_{x_j} \widetilde{\mathfrak{p}}_{\varepsilon}\big(\widetilde{x},\xi(\widetilde{x})\big) \textrm{ and } \xi(\widetilde{x})=\frac{2}{i} \frac{\partial \widetilde{\Phi}_{\delta,\varepsilon}(\widetilde{x})}{\partial \widetilde{x}}.$$
We notice in particular that
\begin{equation}\label{eq6.9.1}\stepcounter{equa}
f_{j,\varepsilon}(\widetilde{x})=\mathcal{O}(|\widetilde{x}|), \ \nabla f_{j,\varepsilon}(\widetilde{x})=\mathcal{O}(1),\tag{\thesection.\theequa}
\end{equation}
uniformly with respect to $0<\varepsilon \leq \varepsilon_0$ and $\tilde{x} \in \mathbb{C}^n$; and that the remainder $\widetilde{r}_{\varepsilon}$ appearing in right hand side of (\ref{eq6.9}) satisfies
$$\widetilde{r}_{\varepsilon}(\widetilde{x},\widetilde{y};\tilde{\theta}) = \mathcal{O}(|\widetilde{x}-\widetilde{y}|^2+h^{\infty}),$$
uniformly with respect to the parameter $0<\varepsilon \leq \varepsilon_0$; as we know that
$$\nabla^2 \widetilde{\mathfrak{p}}_{\varepsilon} \in L^{\infty} \textrm{ and } \widetilde{\theta}-\xi(\widetilde{x})=\mathcal{O}(|\widetilde{x}-\widetilde{y}|),$$
uniformly with respect to the parameter $0<\varepsilon \leq 1$. Arguing as in~\cite{HeSjSt} (Section~3.4), we see that the contribution to the scalar product $(\psi \widetilde{P}_{\varepsilon}\widetilde{u},\widetilde{u})_{\widetilde{\Phi}_{\delta,\varepsilon},\tilde{h}}$ coming from this remainder term $\widetilde{r}_{\varepsilon}$ can be estimated from above by
$$\mathcal{O}(\widetilde{h}) \|\widetilde{u}\|^2_{\widetilde{\Phi}_{\delta,\varepsilon},\tilde{h}}.$$
The contribution from the third term in the right hand side of (\ref{eq6.9}) vanishes as noticed in~\cite{HeSjSt} (Section~3.4). It therefore remains to study the contribution coming from the second term in (\ref{eq6.9}). Here, arguing as in~\cite{HeSjSt} and~\cite{SjDuke}, we see that this term contributes the following sum of integrals
$$\sum_{j=1}^n \int_{\mathbb{C}^n} \psi(\widetilde{x}) f_{j,\varepsilon}(\widetilde{x}) \big(\widetilde{h}D_{\widetilde{x}_j}-\xi_j(\widetilde{x})\big)\widetilde{u}(\widetilde{x}) \overline{\widetilde{u}(\widetilde{x})} e^{-\frac{2\widetilde{\Phi}_{\delta,\varepsilon}(\widetilde{x})}{\widetilde{h}}}L(d\widetilde{x}).$$
Integrating by parts, as in~\cite{SjDuke} (p.9), and using that $\widetilde{u}$ is a holomorphic function, we find that each of the terms in this sum is equal to
$$-\int_{\mathbb{C}^n} \widetilde{h} D_{\widetilde{x}_j} \big( \psi(\widetilde{x}) f_{j,\varepsilon} (\widetilde{x})\big)|\widetilde{u}(\widetilde{x})|^2 e^{-\frac{2\widetilde{\Phi}_{\delta,\varepsilon}(\widetilde{x})}{\widetilde{h}}}L(d\widetilde{x}).$$
Considering
$$D_{\widetilde{x}_j}\big(\psi(\widetilde{x}) f_{j,\varepsilon}(\widetilde{x})\big) = D_{\widetilde{x}_j}\psi(\widetilde{x})f_{j,\varepsilon}(\widetilde{x}) + \psi(\widetilde{x})\, D_{\widetilde{x}_j} f_{j,\varepsilon}(\widetilde{x}),$$
we notice that the second term appearing in the right hand side of the last equality is uniformly bounded since from (\ref{eq6.9.1}), we know that $\nabla f_{j,\varepsilon} =\mathcal{O}(1)$. As for the first one, we see using (\ref{eq6.9.1}) and the bound $\nabla \psi = \mathcal{O}(|\widetilde{x}|^{-1})$, that it is bounded as well. This completes the proof of Proposition~\ref{prop89}.
\end{proof}
\bigskip
Considering $\chi$ a $C^{\infty}_b(\mathbb{C}^n;[0,1])$ function such that $\chi=1$ when $|\widetilde{x}|$ is large enough with
$$\textrm{supp } \chi \subset \{\widetilde{x} \in \mathbb{C}^n : |\widetilde{x}|\geq 1\};$$
and
$$m(\widetilde{x})=1-i \tilde{c}\frac{\delta}{|\widetilde{x}|^2},$$
we shall apply Proposition~\ref{prop89} with the function $\psi(\widetilde{x})=\chi(\widetilde{x}) m(\widetilde{x})$ and the operator $\widetilde{P}_{\varepsilon}$ replaced by $\widetilde{P}_{\varepsilon}-\widetilde{h}z$; where the spectral parameter $z\in \mathbb{C}$ satisfies $\abs{z}\leq C$ for some fixed $C>0$.
From now on, the parameter $\delta>0$ is going to be kept sufficiently small but fixed.
We get from Proposition~\ref{prop89} that the scalar product
\begin{equation}\label{eq6.10}\stepcounter{equa}
\big(\chi m (\widetilde{P}_{\varepsilon}-\widetilde{h}z)\widetilde{u},\widetilde{u}\big)_{\widetilde{\Phi}_{\delta,\varepsilon},\tilde{h}}, \tag{\thesection.\theequa}
\end{equation}
is equal to
$$\int_{\mathbb{C}^n} \chi(\widetilde{x})m(\widetilde{x})\widetilde{\mathfrak{p}}_{\varepsilon}\Big(\widetilde{x},\frac{2}{i} \frac{\partial \widetilde{\Phi}_{\delta,\varepsilon}(\widetilde{x})}{\partial \widetilde{x}}\Big) \abs{\widetilde{u}(\widetilde{x})}^2e^{-\frac{2\widetilde{\Phi}_{\delta,\varepsilon}(\widetilde{x})}{\widetilde{h}}}L(d\widetilde{x}) +\mathcal{O}(\widetilde{h})\|\widetilde{u}\|^2_{\widetilde{\Phi}_{\delta,\varepsilon},\tilde{h}}.$$
Since
\begin{multline*}
\textrm{Re}\big(\chi m (\widetilde{P}_{\varepsilon}-\widetilde{h}z)\widetilde{u},\widetilde{u}\big)_{\widetilde{\Phi}_{\delta,\varepsilon},\tilde{h}} \\
= \int_{\mathbb{C}^n} \chi(\widetilde{x}) \textrm{Re}\Big[m(\widetilde{x})\widetilde{\mathfrak{p}}_{\varepsilon}\Big(\widetilde{x},\frac{2}{i} \frac{\partial \widetilde{\Phi}_{\delta,\varepsilon}(\widetilde{x})}{\partial \widetilde{x}}\Big)\Big]|\widetilde{u}(\widetilde{x})|^2 e^{-\frac{2\widetilde{\Phi}_{\delta,\varepsilon}(\widetilde{x})}{\widetilde{h}}}L(d\widetilde{x}) +\mathcal{O}(\widetilde{h})\|\widetilde{u}\|^2_{\widetilde{\Phi}_{\delta,\varepsilon},\tilde{h}},
\end{multline*}
we deduce from the Cauchy-Schwarz inequality and the estimate (\ref{eq6.7}) holding on the support of the function $\chi$ that
$$\int_{\mathbb{C}^n} \chi(\widetilde{x}) |\widetilde{u}(\widetilde{x})|^2 e^{-\frac{2\widetilde{\Phi}_{\delta,\varepsilon}(\widetilde{x})}{\widetilde{h}}}L(d\widetilde{x}) \leq \mathcal{O}(1) \|(\widetilde{P}_{\varepsilon}-\widetilde{h}z)\widetilde{u}\|_{\widetilde{\Phi}_{\delta,\varepsilon},\tilde{h}} \|\widetilde{u}\|_{\widetilde{\Phi}_{\delta,\varepsilon},\tilde{h}} + \mathcal{O}(\widetilde{h})\|\widetilde{u}\|^2_{\widetilde{\Phi}_{\delta,\varepsilon},\tilde{h}}.$$
We can now come back to the original variables $x=\sqrt{\varepsilon}\widetilde{x}$ and obtain that
\begin{multline*}
\varepsilon \int_{\mathbb{C}^n} \chi\Big(\frac{x}{\sqrt{\varepsilon}}\Big) |u(x)|^2 e^{-\frac{2\Phi_{\delta,\varepsilon}(x)}{h}}L(dx) \leq \mathcal{O}(1) \|(\mathfrak{p}_0^w(x,hD_x)-hz)u\|_{\Phi_{\delta,\varepsilon}}\|u\|_{\Phi_{\delta,\varepsilon}}\\ +\mathcal{O}(h)\|u\|^2_{\Phi_{\delta,\varepsilon}}.
\end{multline*}
It follows from (\ref{xi13}) that the operator $P_0=P_0^w(x,hD_x;h)$ also fulfills
$$\varepsilon \int_{\mathbb{C}^n} \chi\Big(\frac{x}{\sqrt{\varepsilon}}\Big) |u(x)|^2 e^{-\frac{2\Phi_{\delta,\varepsilon}(x)}{h}}L(dx) \leq \mathcal{O}(1) \|(P_0-hz)u\|_{\Phi_{\delta,\varepsilon}}\|u\|_{\Phi_{\delta,\varepsilon}} +\mathcal{O}(h)\|u\|^2_{\Phi_{\delta,\varepsilon}}.$$
Finally, by recalling that $\varepsilon=Ah$, we can rewrite this estimate as
\stepcounter{equa}\begin{multline}\label{eq6.11}
h \int_{\mathbb{C}^n} \chi\Big(\frac{x}{\sqrt{A h}}\Big)|u(x)|^2 e^{-\frac{2\Phi_{\delta,\varepsilon}(x)}{h}}L(dx) \leq \mathcal{O}(1) \|(P_0-hz)u\|_{\Phi_{\delta,\varepsilon}}\|u\|_{\Phi_{\delta,\varepsilon}}
\\ + \mathcal{O}\Big(\frac{h}{A}\Big)\|u\|_{\Phi_{\delta,\varepsilon}}^2, \tag{\thesection.\theequa}
\end{multline}
where $A \gg 1$ is a large parameter remaining to be chosen.
\section{Proof of Theorem~\ref{theo}}\label{proof}
\setcounter{equa}{0}
In this section, we shall glue together the local resolvent estimates that we established first in a tiny neighborhood of the doubly characteristic set (Section~\ref{tiny}) and then in the exterior region considered in the previous section in order to complete the proof of Theorem~\ref{theo}.
By considering the same cutoff function $\chi_0 \in C_0^{\infty}(\mathbb{C}^n,[0,1])$ as in Section~\ref{tiny} and going back to (\ref{k41}) with $k=1/2$, we deduce from this estimate and the triangle inequality that there exists $C>0$ such that
\stepcounter{equa}\begin{align*}\label{chel1}
& \ \Big\| (h+d^2)^{\frac{1}{2}}\chi_0\Big(\frac{x}{\sqrt{Ah}}\Big) u\Big\|_{\Phi_{\delta,\varepsilon}} \tag{\thesection.\theequa}\\
\leq & \ C \Big\| (h+d^2)^{-\frac{1}{2}}\chi_0\Big(\frac{x}{\sqrt{Ah}}\Big) \big(\mathfrak{p}_0^w(x,hD_x)+h\mathfrak{p}_1^w(x,hD_x)-hz\big)u\Big\|_{\Phi_{\delta,\varepsilon}}\\
+ & \ C \Big\| (h+d^2)^{-\frac{1}{2}}\chi_0\Big(\frac{x}{\sqrt{Ah}}\Big) \big(\mathfrak{p}_0^w(x,hD_x)+h\mathfrak{p}_1^w(x,hD_x)-Q_0-hp_1(0,0)\big)u\Big\|_{\Phi_{\delta,\varepsilon}}\\
+ & \ \frac{\tilde{C}}{\sqrt{A}} \Big\|(h+d^2)^{\frac{1}{2}}{\mathrm{1~\hspace{-1.4ex}l}}_{K}\Big(\frac{x}{\sqrt{Ah}}\Big)u\Big\|_{\Phi_{\delta,\varepsilon}},
\end{align*}
for all $z$ varying in a compact set such that the set $z-p_1(0,0)$ does not contain any eigenvalues of the operator $Q_0|_{h=1}$ described in (\ref{k27}). We recall that $Q_0$ stands here for the operator appearing in (\ref{k36}). Since on the supports of the functions $\chi_0(x/\sqrt{Ah})$ and ${\mathrm{1~\hspace{-1.4ex}l}}_{K}(x/\sqrt{Ah})$, one can estimate the quantity $h+d^2$ by
$$h \leq h+d^2 \leq c_1 Ah,$$
with $c_1>0$ a positive constant independent of the parameters $A$ and $h$; we deduce from (\ref{k36}) and (\ref{chel1}) that there exist some positive constants $c_{2}$ and $c_{3,A}$, where the constant $c_2$ is independent of the parameters $A$ and $h$, whereas $c_{3,A}$ may actually depend on $A$ but not on $h$; such that
\stepcounter{equa}\begin{multline*}\label{k51xi}
h^{\frac{1}{2}}\Big\| \chi_0\Big(\frac{x}{\sqrt{Ah}}\Big) u\Big\|_{\Phi_{\delta,\varepsilon}} \leq c_2 h^{-\frac{1}{2}} \big\|\big(\mathfrak{p}_0^w(x,hD_x)+h\mathfrak{p}_1^w(x,hD_x)-hz\big)u\big\|_{\Phi_{\delta,\varepsilon}} \\ +c_{3,A} h \|u\|_{\Phi_{\delta,\varepsilon}} +c_2 h^{\frac{1}{2}} \Big\|{\mathrm{1~\hspace{-1.4ex}l}}_{K}\Big(\frac{x}{\sqrt{Ah}}\Big)u\Big\|_{\Phi_{\delta,\varepsilon}}. \tag{\thesection.\theequa}
\end{multline*}
It follows from (\ref{xi13}) that the operator $P_0=P_0^w(x,hD_x;h)$ also fulfills
\stepcounter{equa}\begin{multline*}\label{k51}
h^{\frac{1}{2}}\Big\| \chi_0\Big(\frac{x}{\sqrt{Ah}}\Big) u\Big\|_{\Phi_{\delta,\varepsilon}} \leq c_2 h^{-\frac{1}{2}} \big\|\big(P_0-hz\big)u\big\|_{\Phi_{\delta,\varepsilon}} +c_{3,A} h \|u\|_{\Phi_{\delta,\varepsilon}} \\ +c_2 h^{\frac{1}{2}} \Big\|{\mathrm{1~\hspace{-1.4ex}l}}_{K}\Big(\frac{x}{\sqrt{Ah}}\Big)u\Big\|_{\Phi_{\delta,\varepsilon}}, \tag{\thesection.\theequa}
\end{multline*}
where $c_{3,A}$ stands for a new constant depending on the parameter $A$ but not on $h$.
Recalling that here $K$ stands for a fixed neighborhood of the support of the function $\nabla \chi_0$, we get from (\ref{k51}) upon squaring that
\stepcounter{equa}\begin{multline}\label{eq51.1}
h \Big\| \chi_0\Big(\frac{x}{\sqrt{Ah}}\Big) u\Big\|_{\Phi_{\delta,\varepsilon}}^2
\leq \frac{\mathcal{O}(1)}{h} \|(P_0-hz)u\|_{\Phi_{\delta,\varepsilon}}^2 + \mathcal{O}_A(h^2)\|u\|_{\Phi_{\delta,\varepsilon}}^2 \\ + \mathcal{O}(h) \Big\|{\mathrm{1~\hspace{-1.4ex}l}}_{K}\Big(\frac{x}{\sqrt{Ah}}\Big)u\Big\|_{\Phi_{\delta,\varepsilon}}^2. \tag{\thesection.\theequa}
\end{multline}
On the other hand, we get from the estimate (\ref{eq6.11}) that
\stepcounter{equa}\begin{multline}\label{eq51.2}
h \int_{\mathbb{C}^n} \chi\Big(\frac{x}{\sqrt{Ah}}\Big) |u(x)|^2 e^{-\frac{2\Phi_{\delta,\varepsilon}(x)}{h}}L(dx)+ h \Big\|{\mathrm{1~\hspace{-1.4ex}l}}_{K}\Big(\frac{x}{\sqrt{Ah}}\Big)u\Big\|_{\Phi_{\delta,\varepsilon}}^2 \\
\leq \mathcal{O}(1) \|(P_0-hz)u\|_{\Phi_{\delta,\varepsilon}} \|u\|_{\Phi_{\delta,\varepsilon}}+\mathcal{O}\Big(\frac{h}{A}\Big)\|u\|_{\Phi_{\delta,\varepsilon}}^2, \tag{\thesection.\theequa}
\end{multline}
and we then obtain by collecting (\ref{eq51.1}) and (\ref{eq51.2}) that
\begin{multline*}
h \int_{\mathbb{C}^n}|u(x)|^2 e^{-\frac{2\Phi_{\delta,\varepsilon}(x)}{h}}L(dx) \leq \mathcal{O}(1) \|(P_0-hz)u\|_{\Phi_{\delta,\varepsilon}}\|u\|_{\Phi_{\delta,\varepsilon}} \\
+\frac{\mathcal{O}(1)}{h} \|(P_0-hz)u\|_{\Phi_{\delta,\varepsilon}}^2 + \Big(\mathcal{O}\Big(\frac{h}{A}\Big)+\mathcal{O}_A(h^2)\Big)\|u\|_{\Phi_{\delta,\varepsilon}}^2.
\end{multline*}
Here we have also used that we arrange, as we may, that $\chi+\chi_0^2\geq 1$ on $\mathbb{C}^n$.
By multiplying by the parameter $h$ and using that
$$\mathcal{O}(1) h \|(P_0-hz)u\|_{\Phi_{\delta,\varepsilon}} \|u\|_{\Phi_{\delta,\varepsilon}} \leq \mathcal{O}(1) \|(P_0-hz)u\|_{\Phi_{\delta,\varepsilon}}^2 + \frac{h^2}{2}\|u\|_{\Phi_{\delta,\varepsilon}}^2,$$
we get that
\begin{multline*}
h^2 \int_{\mathbb{C}^n} |u(x)|^2 e^{-\frac{2\Phi_{\delta,\varepsilon}(x)}{h}}L(dx) \leq \mathcal{O}(1) \|(P_0-hz)u\|_{\Phi_{\delta,\varepsilon}}^2 + \frac{h^2}{2}\|u\|_{\Phi_{\delta,\varepsilon}}^2 \\
+ \Big(\mathcal{O}\Big(\frac{h^2}{A}\Big) + \mathcal{O}_A(h^3)\Big)\|u\|_{\Phi_{\delta,\varepsilon}}^2.
\end{multline*}
By now choosing the parameter $A$ sufficiently large, but fixed; and then considering a positive constant $h_0>0$ sufficiently small, $0<h_0 \ll 1$, depending on the choice done for the value of the constant $A$, we obtain that for all $0 < h \leq h_0$,
\stepcounter{equa}\begin{equation}\label{eq51.3}
h \|u\|_{\Phi_{\delta,\varepsilon}} \leq \mathcal{O}(1)\|(P_0-hz)u\|_{\Phi_{\delta,\varepsilon}}. \tag{\thesection.\theequa}
\end{equation}
Let us underline that one can then replace in this estimate the weight function $\Phi_{\delta,\varepsilon}$ by the standard quadratic weight $\Phi_0$ defined in (\ref{eq3.2.1b}) at the expense of an $\mathcal{O}(1)$--loss. Indeed, this follows from the fact that according to (\ref{ju6}), the two associated $L^2$-norms are equivalent since we have
$$\textrm{exp}\big(-\mathcal{O}(A)\big) \leq e^{-\frac{2\Phi_{\delta,\varepsilon}}{h}} e^{\frac{2\Phi_0}{h}}\leq \textrm{exp}\big(\mathcal{O}(A)\big).$$
It is now easy to complete the proof of Theorem~\ref{theo} . In doing so, it is sufficient to go back to the $L^2$--side by undoing the FBI transform $T$ in (\ref{eq51.3}) to get that for all $u\in L^2(\mathbb{R}^n)$ and $0<h \leq h_0$,
\begin{equation}\label{eq51.5xi}\stepcounter{equa}
h\|u\|_{L^2} \leq \mathcal{O}(1) \|(P-hz)u\|_{L^2}, \tag{\thesection.\theequa}
\end{equation}
for all $z$ varying in a compact set such that the set $z-p_1(0,0)$ does not contain any eigenvalues of the operator $q_1^w(x,D_x)$ described in (\ref{k27}). This ends the proof of Theorem~\ref{theo} since we recall that we were considering here, for simplicity only, the case when the doubly characteristic set is reduced to an unique point $X_1=(0,0)$.
In the general case when the doubly characteristic set is composed of a finite number of points
$$p_0^{-1}(0)=\{X_1,...,X_N\},$$
a simple adaptation of the previous arguments allows to establish similar estimates as (\ref{eq51.1}) near each doubly characteristic point $X_j$ when the spectral parameter $z$ stays in a compact set as described in the statement of Theorem~\ref{theo}. We then conclude the proof of Theorem~\ref{theo} by using a similar a priori estimate as (\ref{eq51.2}) in the exterior region.~$\Box$
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,635 |
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